Multilevel optical modulation formats with direct detection
Transcrição
Multilevel optical modulation formats with direct detection
Multilevel Optical Modulation Formats with Direct Detection Von der Fakultät Informatik, Elektrotechnik und Informationstechnik der Universität Stuttgart zur Erlangung der Würde eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung Vorgelegt von Michael Ohm aus Braunschweig Hauptberichter: Mitberichter: Tag der mündlichen Prüfung: Prof. Dr.-Ing. Joachim Speidel Prof. Dr.-Ing. Manfred Berroth 14. Juni 2006 Institut für Nachrichtenübertragung der Universität Stuttgart 2006 This thesis presents results from my research activities at the Institute of Telecommunications (INÜ) at the University of Stuttgart. My very special thanks go to Prof. Dr.-Ing. Joachim Speidel for giving me the oppurtunity to work under his supervision on such an exciting topic. Numerous fruitful discussions and valuable suggestions have substantially contributed to the success of this work. Moreover, I would like to thank him for his permanent encouragement and support to publish my research results. Also, I cordially thank Prof. Dr.-Ing. Manfred Berroth for the assessment of this thesis. Of course, I am grateful for the help and assistance that I got from all my colleagues at the Institute of Telecommunications. I want to point out Dr. Frieder Sanzi and Dr. Romed Schur, who encouraged me to become a research assistant at this institute. Robert Fritsch was always willing to discuss even the most advanced topics. Dr. Alexander Boronka was an invaluable source of help on LATEX. In the final phase of writing, Stephan Saur had to endure my many complaints about various software packages. Our technical support team provided a most reliable infrastructure. I would further like to thank all students who have contributed to this work by their study, diploma or master theses. Especially, I want to mention Torsten Freckmann, who has implemented the semi-analytical error calculation method in Matlab, and Timo Pfau, who had great ideas for simple 8-DPSK receivers. My thanks also go to Dr. Henning Bülow and his colleagues at Alcatel in Stuttgart for interesting discussions. Dr. Jörg-Peter Elbers of Ericsson in Backnang has to be mentioned for pointing out most useful implementation aspects for the semi-analytical error calculation method. I will never forget the great time I had at Bell Laboratories on Crawford Hill, which was made possible by Dr. habil. Peter J. Winzer and Dr. René-Jean Essiambre. Parts of the work presented in this thesis were funded by the German Ministry of Education and Research (BMBF) within the MultiTeraNet project. Frank Mösle spent his valuable time on proof-reading the manuscript, for which I am very grateful. Finally, I want to thank Christina Pusch for all the love she has for me. Contents Acronyms and Abbreviations ix Symbols xi Abstract xix Kurzfassung xix 1 Introduction 1 2 Fundamentals 3 2.1 Multilevel digital modulation . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The optical carrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Digital modulation of the optical carrier . . . . . . . . . . . . . . . . . . . 6 2.4 Optical fiber transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5 Noise from optical amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Optical receivers with direct detection . . . . . . . . . . . . . . . . . . . . 17 2.7 Simulation of optical transmission systems and semi-analytical bit error probability calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Binary Amplitude-Shift Keying (2-ASK) 28 3.1 2-ASK receiver and transmitter . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 2-ASK performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 vii 4 Binary Differential Phase-Shift Keying (2-DPSK) 35 4.1 2-DPSK transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2 2-DPSK receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Extension of the bit error probability calculation method to 2-DPSK with balanced detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2-DPSK performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 5 6 7 8 9 4-level Differential Phase-Shift Keying (4-DPSK) 50 5.1 4-DPSK transmitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.2 4-DPSK receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.3 4-DPSK performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4-level Amplitude-/Differential Phase-Shift Keying (4-ASK-DPSK) 67 6.1 4-ASK-DPSK transmitter and receiver . . . . . . . . . . . . . . . . . . . . 67 6.2 4-ASK-DPSK optimum signal point amplitude ratio and performance . . . 70 8-level Amplitude-/Differential Phase-Shift Keying (8-ASK-DPSK) 83 7.1 8-ASK-DPSK transmitter and receiver . . . . . . . . . . . . . . . . . . . . 83 7.2 8-ASK-DPSK optimum signal point amplitude ratio and performance . . . 86 8-level Differential Phase-Shift Keying (8-DPSK) 100 8.1 8-DPSK transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.2 8-DPSK receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.3 8-DPSK performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Conclusion 120 A Optical phase shifters 122 B Region of convergence of Φik (−s) 124 C Common System Parameters 125 D Optical and Electrical Filters 126 E Power Spectra 129 viii Acronyms and Abbreviations AM ASE ASK AWGN BEP BER BPF CW DAF DBBS DCF DFT DPSK DQPSK EDFA FFT FWM IDFT IFWM IM ISI IXPM L LPF MGF MZM NLS NRZ OOK OSNR PD PDF PM amplitude modulator amplified spontaneous emission amplitude-shift keying additive white Gaussian noise bit error probability bit error ratio bandpass filter continuous-wave delay & add filter DeBruijn binary sequence dispersion-compensating fiber discrete Fourier transform differential phase-shift keying differential quadrature phase-shift keying Erbium-doped fiber amplifier fast Fourier transform four-wave mixing inverse discrete Fourier transform intrachannel four-wave mixing intensity modulation intersymbol interference intrachannel cross-phase modulation level low-pass filter moment-generating function Mach-Zehnder modulator nonlinear Schrödinger non-return-to-zero on-off keying optical signal-to-noise ratio photodiode probability density function phase modulator ix PMD PRBS PSK RC ROC RZ SEP SMF SPM WDM w/o XPM polarization-mode dispersion pseudo-random binary sequence phase-shift keying raised-cosine region of convergence return-to-zero symbol error probability single-mode fiber self-phase modulation wavelength-division multiplexing without cross-phase modulation x Symbols ◦−• •−◦ ⊕ (·)∗ (·)H 0 1t a a A a a a a A a1 , a2 A1 , A2 ãrx ãrx,1,k , ãrx,2,k Aeff ak Ak ãrx,k ărx,k al ãl Fourier transform inverse Fourier transform logical X OR logical N OT complex conjugate complex-conjugate transpose of vector or matrix vector containing zeros symbol carrying the unit of time complex envelope of electric field column vector containing samples of complex envelope of electric field spectrum of complex envelope of electric field value of arbitrary signal point magnitude of two inner signal points of 4-level combined amplitude-shift and differential phase-shift keying arbitrary signal vector containing samples of arbitrary signal vector containing discrete Fourier transform of samples of arbitrary signal destructive and constructive output signals of delay & add filter spectra of destructive and constructive output signals of delay & add filter column vector containing samples of complex envelope of electric field after filtering without noise reordered versions of column vectors containing samples of destructive and constructive output signals of delay & add filter without noise effective fiber area symbol sequence at receiver samples of spectrum of complex envelope reordered version of column vector containing samples of complex envelope of electric field after filtering without noise concatenation of reordered versions of column vectors containing samples of destructive and constructive output signals of delay & add filter without noise samples of complex envelope of electric field samples of complex envelope of electric field after filtering xi alow,in , alow,out an An aNRZ ANRZ aNRZ,n arx arx,l aRZ ARZ ap Ap a p,a A p,a a p,smp asmp aup,in , aup,out Aµ aν b b B b̂k bm b0n BN b0i,n bi,n c c Cx Cv Cv̆ d D signals at lower input and output ports of cross coupler symbol sequence at transmitter magnitude of symbol an at transmitter complex envelope of electric field with non-return-to-zero pulse shaping spectrum of complex envelope of electric field with non-return-to-zero pulse shaping samples of complex envelope of electric field with non-return-to-zero pulse shaping complex envelope of received electric field samples of complex envelope of received electric field complex envelope of electric field with return-to-zero pulse shaping spectrum of complex envelope of electric field with return-to-zero pulse shaping primitive period of arbitrary signal spectrum of primitive period of a signal sampled primitive period of arbitrary signal spectrum of sampled primitive period of arbitrary signal primitive period of sampled complex envelope of electric field sampled complex envelope of electric field signals at upper input and output ports of cross coupler discrete Fourier transform of samples of arbitrary signal samples of arbitrary signal magnitude of complex envelope a of electric field magnitude of two outer signal points of 4-level combined amplitude-shift and differential phase-shift keying reference bandwidth estimated bit sequence at receiver bit sequence before mapper bit sequence after differential encoder N-th order Bessel polynomial bit sequence i after differential encoder bit sequence i after mapper free-space velocity of light magnitude of inner signal points of 8-level combined amplitude-shift and differential phase-shift keying covariance matrix of received optical signal covariance matrix of received optical signal after filtering covariance matrix of destructive and constructive output signals of delay & add filter magnitude of outer signal points of 8-level combined amplitude-shift and differential phase-shift keying dispersion coefficient xii det(·) Di diag{·} dk dRZ e e e E e0 ein eout erx ẽrx ex , ey , ez ex0 , ey0 e0x0 , e0y0 f f0 f3 dB f3 dB,el ∆ f3 dB ∆ f3 dB,opt f ik fp fx fv fvk fv̆k G h H H1 , H2 HBessel hel matrix determinant dispersion coefficient of i-th fiber diagonal matrix formed from the arguments coefficients of Bessel polynomials duty cycle of return-to-zero pulse Eulers’s constant electric field vector short notation for electric field vector component ex , optical signal spectrum of electric field short notation for amplitude ex0 of electric field vector component ex optical input signal of Mach-Zehnder modulator optical output signal of Mach-Zehnder modulator received electric field vector received electric field vector after filtering components of the electric field vector e in x-, y-, and z-direction amplitudes of electric field vector components ex and ey unnormalized amplitudes of electric field vector components ex and ey frequency carrier frequency 3-dB cut-off frequency electrical 3-dB cut-off frequency 3-dB bandwidth optical 3-dB bandwidth probability density function of electrical signal after sampling device base frequency for Fourier series expansion multivariate Gaussian probability density function of samples of received optical signal multivariate Gaussian probability density function of samples of received optical signal after filtering multivariate Gaussian probability density function of reordered samples of received optical signal after filtering multivariate Gaussian probability density function of concatenation of reordered versions of samples destructive and constructive output signals of delay & add filter amplifier gain impulse response of pulse shaper transfer function of pulse shaper transfer functions of destructive and constructive ports of delay & add filter transfer function of Bessel filter diagonal matrix containing samples of impulse response of electrical filter xiii h̆el Hel Hel,k Hf H f ,k HGauss Hk hl Hopt Hopt,k hRZ,Gauss i ĩ I i1 , i2 ij ĩ j ∆i j Ij j Ii ik i0l ĩ0l Im{·} ith j k L li m, n M n̄ N N0 N0 n2 ncore nclad NISI diagonal matrix containing regular and inverted samples of impulse response of electrical filter electrical filter transfer function samples of electrical filter transfer function fiber transfer function samples of fiber transfer function transfer function of Gaussian filter samples of arbitrary transfer function samples of impulse response of electrical filter optical filter transfer function samples of optical filter transfer function shape of Gaussian return-to-zero pulse photocurrent, electrical signal at receiver filtered electrical signal at receiver identity matrix electrical signals after photodetection of destructive and constructive output signals of delay & add filter electrical signals in j-th receiver branch electrical signals after balanced detection and filtering in j-th receiver branch eye openings in j-th receiver branch j-th set of values i-th element in j-th set of values electrical signal after sampling device samples of electrical signal samples of electrical signal after filtering imaginary part of a complex number decision threshold imaginary unit time index at receiver size of symbol alphabet length of i-th fiber time indices at transmitter number of bits per symbol effective refractive index of fiber arbitrary number noise power spectral density sampling instant offset in number of samples nonlinear index coefficient refractive index of fiber core refractive index of fiber cladding number of symbols affected by intersymbol interference xiv Nj Np N̆ p Ns p p p̄ P[·] Pe Pe,ik Pe,ik ,0 Pe,ik ,1 Pnoise Psignal r q ql R R+ ,R− Re{·} Ra Rb rcore rD ∆rD ∆rD,x dB neg rD,x dB pos rD,x dB rl Rs s S Sk t number of elements in j-th set of values number of samples within primitive period or block twice the number of samples within primitive period or block number of symbols within primitive period or block complex variable instantaneous power average power probability of specified event total error probability probability of wrong decision probability of wrong decision of zero-bit probability of wrong decision of one-bit noise power within reference bandwidth total signal power vector containing the samples of the autocorrelation function transformed version of reordered column vector containing samples of complex envelope of electric field after filtering elements of transformed version of reordered column vector containing samples of complex envelope of electric field after filtering photodiode responsivity sets of positive or negative real numbers real part of a complex number sampling rate bit rate fiber core radius accumulated dispersion difference between maximum tolerable positive accumulated dispersion and minimum tolerable negative accumulated dispersion for arbitrary optical signal-to-noise ratio penalty difference between maximum tolerable positive accumulated dispersion and minimum tolerable negative accumulated dispersion for x-dB optical signal-to-noise ratio penalty minimum tolerable negative accumulated dispersion for x-dB optical signal-to-noise ratio penalty maximum tolerable positive accumulated dispersion for x-dB optical signal-to-noise ratio penalty samples of autocorrelation function symbol rate complex variable dispersion slope samples of the noise power spectral density after filtering time xv t0 T Ta Tb TFDHM Tp Ts u U1 ubias,i ui ûi Ui ui,MZM uRZ,Gauss us Uπ v v1 , v2 vk v̆k v1,k , v2,k vl w w̃ w̃k wl w̃l wWGN wx , wy wx,l , wy,l x, y, z sampling instant offset arbitrary duration sampling period bit interval full duration at half maximum of return-to-zero pulse duration of primitive period symbol interval electrical drive signal modal matrix of covariance matrix of received optical signal after filtering bias of electrical drive signal at i-th Mach-Zehnder modulator electrical drive signal i amplitude of electrical drive signal at i-th Mach-Zehnder modulator spectrum of the electrical drive signal ui electrical drive signal at i-th Mach-Zehnder modulator electrical drive signal for Gaussian return-to-zero pulse shaping unit step function reference voltage of Mach-Zehnder modulator column vector containing samples of filtered optical signal including noise destructive and constructive output signals of delay & add filter after filtering including noise reordered version of column vector containing samples of filtered optical signal including noise concatenation of reordered versions of column vectors containing samples of destructive and constructive output signals of delay & add filter including noise reordered versions of column vectors containing samples of destructive and constructive output signals of delay & add filter including noise samples of filtered optical signal including noise column vector containing samples of complex baseband noise in xpolarization column vector containing samples of complex baseband noise in xpolarization after filtering reordered version of column vector containing samples of complex baseband noise in x-polarization after filtering short notation for samples of complex baseband noise in x-polarization samples of complex baseband noise in x-polarization after filtering white Gaussian noise vector complex baseband noise processes in x- and y-polarization samples of complex baseband noise in x- and y-polarization cartesian coordinates xvi x xl Z ∆z α αf α f ,dB αn β βi γ γi,k γi j,k Γi Γi j δ ∂ ∂t ε εdB κN λ λ0 λl + λmax − λmin ν, µ π ρ σ τ τG ϒ ϕ ∆ϕ ϕi,MZM ϕMZM ϕn ϕk Φ ik column vector containing samples of the received optical signal including noise samples of received optical signal including noise set of integer numbers split-step Fourier algorithm step size roll-off factor of raised-cosine pulse shaper fiber attenuation fiber attenuation in dB Fourier series expansion coefficients fiber propagation coefficient coefficients of Taylor series expansion of propagation coefficient nonlinear fiber parameter binary sequence after decision device for i-th electrical signal binary sequence after j-th decision device for i-th electrical signal decision threshold for i-th electrical signal j-th decision threshold for i-th electrical signal Dirac’s delta function time derivative extrinsic extinction ratio of Mach-Zehnder modulator extrinsic extinction ratio of Mach-Zehnder modulator in dB frequency correction factor for Bessel filter vector containing eigenvalues of hel Cv carrier wavelength eigenvalues of hel Cv largest eigenvalue smallest eigenvalue index variables ratio of circle’s circumference to its diameter parameter related to presence of noise in x- and y-polarization real part of complex variable s time delay in delay & add filter group delay factor related to power split ratio of Mach-Zehnder modulator phase of complex envelope a of electric field difference between phase angles of complex envelope of electric field at time instants t and t − Ts phase shift in one arm of i-th Mach-Zehnder modulator phase shift in one arm of Mach-Zehnder modulator phase of symbol an at transmitter phase of symbol ak at receiver moment-generating function of sample ik of electrical signal after sampling device xvii φNL χ ψ ψi ω ω ω0 ∆ω3 dB ωa ωp ζ nonlinear phase shift normalization factor for electric field vector phase shift in delay & add filter phase shift in i-th delay & add filter angular frequency imaginary part of complex variable s angular carrier frequency angular 3-dB bandwidth angular frequency related to the sampling rate angular base frequency for Fourier series expansion real variable xviii Abstract Modern fiber optical communication networks require highly spectral efficient transmission, and should at the same time be tolerant against transmission impairments. Nowadays, binary modulation formats with inherently low spectral efficiency are dominantly used in optical transmission systems because of their simple receivers and transmitters. Multilevel modulation formats, which are widely used in other areas of communications, may be a solution to overcome some problems associated with binary formats in optical transmission systems, although they require more elaborate transmitters and receivers. This thesis proposes several multilevel optical modulation formats with direct detection receivers. Their transmitter and receiver structures are analyzed, and the performance of the multilevel formats is assessed and compared to the binary formats. The goal is the systematic comparison of different modulation formats regarding fundamental performance characteristics. For this task, a common parameter set is chosen. Modulation format specific parameters as well as receiver filter bandwidths are optimized. The investigations are based on a semi-analytical method for calculating error probabilities in direct detection optical systems. Kurzfassung Moderne optische Glasfaserübertragungssysteme verlangen nach einer Übertragung mit hoher spektraler Effizienz und sollen gleichzeitig tolerant gegen Störungen sein. Heutzutage werden hauptsächlich binäre Modulationsverfahren mit inhärent niedriger spektraler Effizienz in optischen Übertragungssystemen eingesetzt, da sie sehr einfache Sender und Empfänger benötigen. Mehrstufige Modulationsverfahren, die in anderen Gebieten der Übertragungstechnik weit verbreitet sind, könnten eine Möglichkeit zur Lösung der Probleme sein, die mit binären Modulationsverfahren verbunden sind. Allerdings benötigen sie aufwendigere Sender und Empfänger. Die vorliegende Arbeit untersucht verschiedene mehrstufige Modulationsverfahren mit Direktempfang. Die Sender- und Empfängerstrukturen werden analysiert, und die Leistungsfähigkeit der mehrstufigen Verfahren wird sowohl untereinander als auch mit der von binären Verfahren verglichen. Das Ziel ist der systematische Vergleich unterschiedlicher Modulationsverfahren bezüglich grundlegender Leistungskriterien. Für diese Aufgabe werden bei allen Verfahren die gleichen Parameter verwendet. Parameter, die für ein Verfahren spezifisch sind, sowie Empfängerfilterbandbreiten werden optimiert. Die Untersuchungen basieren auf einem halb-analytischen Verfahren zur Berechnung von Fehlerwahrscheinlichkeiten in optischen Übertragungssystemen mit Direktempfang. xix Chapter 1 Introduction Fiber optics have become the core of the global telecommunications infrastructure. Telephony and the ever increasing demand for information retrieval over world-wide data networks, especially the Internet, but also future mobile broadband applications, are pushing the needs for higher and higher transmission rates in the core and also regional optical networks. The use of wavelength-division multiplexing, where several wavelength channels are used simultaneously for transmission over a single optical fiber, together with high channel bit rates, enables transmission beyond 1 Tbit/s. At the same time, it becomes desirable to increase the network’s flexibility by moving from static point-to-point connections with opto-electronic and electro-optic conversion at each node to switched all-optical networks with optical add-drop multiplexers or even optical cross-connects for the optical wavelength channels. The actual paths for a connection between a node A and a node B within such a switched network may change over time with respect to the overall network conditions. This flexibility together with increasing transmission rates requires techniques that are tolerant against typical impairments in fiber optical communication systems, such as pulse broadening from chromatic dispersion. Since the advent of fiber optical communications in the 1970s, enabled by the dramatic reduction of the fiber loss and the availability of compact light sources, binary amplitudeshift keying has been the dominant modulation format in digital optical transmission systems. Basically, the light is switched on for the transmission of the data bit ’1’ and switched off for the data bit ’0’. Therefore, this modulation format is also known as on-off keying. The reason for its dominance lies in the simplicity, especially of the receiver. In principal, only a photodiode is needed for detection. It generates an electrical current, if the light is switched on for bit ’1’, or produces no current, if the light is switched off for bit ’0’. Another binary modulation format is binary differential phase-shift keying, in which the phase of an optical carrier is modulated with respect to the data signal. The corresponding receiver is more complex, but also relies on direct detection with photodiodes. Further, it has a better receiver sensitivity than binary amplitude-shift keying. The drawback of binary modulation formats, however, is their low spectral efficiency, because they can transmit only 1 bit/symbol. An increase of the channel bit rate requires shorter pulse durations with larger signal bandwidth, 1 and thus transmitter and receiver components with larger bandwidth are needed. Further, high signal bandwidth prevents close channel spacing in wavelength-division multiplexing and decreases the tolerance against transmission impairments. In the search for high spectral efficiency, multilevel modulation formats may be an answer. Here, two or more bits are carried by each transmitted pulse or symbol. So in order to increase the channel bit rate, the symbol’s duration stays the same, but certain characteristics can take on more than two values, e.g. its phase may have eight different values instead of the simple 0 and π for binary differential phase-shift keying. Thus, lower signal bandwidth is achieved at high bit rates, solving the drawbacks of binary formats. Multilevel modulation formats are widely used in other areas of telecommunications such as mobile wireless communications or digital subscriber lines over copper wire. There, the receivers use coherent detection based on the multiplication of the received signal with the signal from a local oscillator. However, in an optical receiver, a costly laser diode would be needed as a local oscillator, so that it appears desirable to find receiver structures that are based on direct detection using photodiodes and some optical and electrical processing. It is the aim of this thesis to investigate fiber optical communication systems using multilevel modulation formats and direct detection receivers by means of analytical and numerical methods. The transmitters and especially the corresponding receivers are discussed. The performance is studied with respect to optimum optical and electrical receiver filter combinations, required optical signal-to-noise ratios for achieving a given bit error probability, and tolerance to the fiber’s chromatic dispersion. The goal is the systematic comparison of a number of basic multilevel modulation formats regarding the above mentioned fundamental characteristics, and not the optimization of a large parameter set for reaching the global optimum for a single modulation format. The thesis is structured as follows. Chapter 2 discusses fundamentals of digital communications, optical transmission systems, and the calculation of bit error probabilities. Chapter 3 and Chapter 4 deal with binary amplitude-shift keying and binary differential phase-shift keying, for which two different transmitters are used. The results from these two chapters are needed for comparisons with the multilevel modulation formats. Next, two 4-level formats are considered. Chapter 5 is about 4-level differential phase-shift keying. It presents three transmitters and the performance of the corresponding systems. The 4-level format in Chapter 6 is the combination of binary amplitude-shift keying and binary differential phaseshift keying. Here, the impact of the amplitude values in the amplitude-shift keying part must be taken into account. Then, two 8-level formats are evaluated. Following the idea of Chapter 6, the modulation format in Chapter 7 now is the combination of binary amplitude-shift keying and 4-level differential phase-shift keying. In contrast, Chapter 8 is about 8-level differential phase-shift keying, which may be seen as the extension of 4-level differential phase-shift keying from Chapter 5. However, the concepts of the three included receivers are much more elaborated. Finally, Chapter 9 gives a conclusion of the previous investigations. 2 Chapter 2 Fundamentals 2.1 Multilevel digital modulation In a digital communication system, information from a source is sent to a remote sink. The source may produce either analog or a digital signals, which are converted into a sequence of binary digits or bit sequence at the transmitter. (’Bit’ is short for binary digit). Digital signals are often obtained by analog-to-digital conversion of analog signals, which comprises the steps of sampling and quantization. The elements of the bit sequence take on either the discrete value ’0’ or the discrete value ’1’, called ’bit 0’ or ’bit 1’. The time between two bits produced in the above process is called bit interval Tb . The bit rate is the inverse of the bit interval: Rb = 1/Tb . It is further the task of the transmitter to assign a signal to the bit sequence, which can be sent over the transmission channel to the receiver. The receiver then recovers the bit sequence from the received signal. For the transmission of binary data and the corresponding signals over a bandpass channel, a digital modulator is required at the transmitter. A bandpass channel is characterized by the property that it does not pass spectral signal components with frequencies around zero. In this thesis, the transmission over optical fibers is considered, which have the property of a bandpass channel. So first, the concepts of digital modulation will be shortly reviewed in this section, and then, the application to optical transmission will be presented in Section 2.2. The basic block diagram of the digital modulator is shown in Figure 2.1. First, a 1 : M serial-to-parallel conversion of the bit sequence bm ∈ {0, 1} is performed, yielding M bit sequences bi,n (i = 1 . . . M). Then, the mapper forms a symbol an from the bit sequences bi,n . The symbol alphabet contains L = 2M symbols. In general, the symbol an is complex valued and can thus be expressed by its magnitude |an | and and its phase ϕn = arg{an } as an = |an | · ejϕn . A constellation diagram of the symbols for a simple example is given in Figure 2.2. There, two consecutive bits are taken to form one out of four complex valued symbols with identical magnitudes A but different phase angles. The symbols an are also called signal points. If a constellation diagram consists of more than two signal points as in Figure 2.2, the modulation format is called a multilevel format, whereas it is called a binary format, if there are only two signal points. 3 ejω0 t bm - S P b1,n .. . Mapper bM,n - an ? Pulse - shaper h(t) u1 (t) u2 (t) - - Re{·} u3 (t) - Figure 2.1: Block diagram of the digital modulator Im{an } 6 0 1 7→ A · ej3π /4 0 0 7→ A · ejπ /4 - Re{an } 1 1 7→ A · ej5π /4 1 0 7→ A · ej7π /4 Figure 2.2: Example of a constellation diagram with four signal points The output signal of the pulse shaper in the digital modulator is u1 (t) = ∞ X n=−∞ an h (t − nTs ) = u1 (t) · ej arg{u1 (t)} , (2.1) where h(t) is the real valued impulse response of the pulse shaper and Ts = MTb is the symbol interval. The symbol rate is defined as Rs = 1/Ts . It is often desirable that h(t) has the property h(0) if n = 0, h(nTs ) = 0 if n 6= 0, (2.2) in order to avoid inter-symbol interference (ISI) at the transmitter. The next step is the modulation of the carrier ejω0t with angular frequency ω0 , and u2 (t) can be written as u2 (t) = u1 (t) · e jω0 t j arg{u1 (t)} = u1 (t) · e jω0 t ·e = ∞ X n=−∞ an h (t − nTs ) · ejω0t . (2.3) This multiplication shifts the spectrum of U1 (ω ) •−◦ u1 (t) into the bandpass region, according to the Fourier transform relation u2 (t) = u1 (t) · ejω0t ◦−•U1 (ω − ω0 ) = U2 (ω ). The digital modulation of the amplitude is called amplitude-shift keying (ASK), and the digital modulation of thenphaseois called n phase-shiftokeying (PSK). Finally for transmission, the real part u3 (t) = Re u2 (t) = Re u1 (t) · ejω0t is taken, as physical signals for transmission over 4 physical channels are always real valued signals. One way of demodulating the band-pass signal at the receiver is the multiplication of the received signal with the conjugate complex of the carrier e−jω0t . This is the standard technique in electrical wireline and wireless systems, but it requires a local oscillator (i.e. a laser) at the receiver. In optical telecommunications it is preferred to omit the costly local oscillator and use direct detection explained later in Section 2.6. However, optical receivers with local oscillators offer better performance and may be an interesting alternative if the cost for lasers can be further reduced by technological progress. As the impulse response of the pulse shaper is usually scaled by the symbol interval Ts , i.e. h(t) = h̃(t/Ts ) = h̃(t/(MTb )), an increase of the number M of bits per symbol leads to a broader pulse shape and thus a reduced spectral width according to the Fourier transform relation h̃(t/(MTb )) ◦−• MTb · H̃(MTb ω ). 2.2 The optical carrier In optical telecommunications, light is the physical signal that carries the data. This optical signal is an electro-magnetic wave and its propagation can be described by Maxwell’s equations [1]. The electro-magnetic wave consists of both an electric field and a magnetic field. As the magnetic field is proportional to the electric field in an optical fiber, it is sufficient to use the electric field vector e in the study of optical signals. In cartesian coordinates x, y and z the electric field vector can be written as ex (x, y, z,t) e(x, y, z,t) = ey (x, y, z,t) ez (x, y, z,t) (2.4) The planar electro-magnetic wave is a simple solution of Maxwell’s equations, for which ez = 0 if the signal propagates along the z-axis. For the optical signals and transmission media considered in this thesis the planar wave is an appropriate solution. ex is the so-called field component in x-polarization, and ey is the field component in y-polarization. In order to transmit data, we have to modulate the amplitude or the phase of the optical carrier at the transmitter, i.e. at location z = 0. If we assume that the spatial distribution of the field (the dependence on x and y) can be neglected, the unmodulated carrier in the x- and y-polarization is e(x, y, 0,t) = e(0,t) = e(t) = ! ex (t) = ey (t) ! ex0 · ejω0t . ey0 (2.5) Note, that in this thesis a complex notation for optical signals will be used because of its simplicity. However, in reality the physical signals are always the real parts of their complex descriptions [1, 2]. 5 Throughout this thesis it will be assumed, that the fiber input signal is x-polarized, i.e. ey0 = 0. Thus, the optical carrier from a single-mode laser, e.g. a distributed-feedback laser [3], before modulation in complex notation takes the form ec (0,t) = ec (t) = ex0 · ejω0t = e0 · ejω0t . (2.6) A laser that generates such an unmodulated carrier is called continuous-wave (CW) laser. Typically, the wavelength of the laser is around λ0 = 1550 µ m for optical fiber communications, as the fiber has low attenuation in that region. This amounts to a carrier frequency of f0 = ω0 /(2π ) = c/λ0 ≈ 193.5 THz. c is the free-space velocity of light. The optical signal is assumed to be normalized [2] such that the instantaneous power of the signal from (2.5) is 2 2 2 p(t) = e(t) = ex (t) + ey (t) . (2.7) This normalization means that ex0 = χ e0x0 and ey0 = χ e0y0 with the normalization factor χ in p units of m2 /Ω related to the wave impedance and other physical parameters. e0x0 and e0y0 are expressed in units of V/m, i.e. in units of the electric field strength, and ex0 and ey0 in √ units of W. With (2.7) the power of the unmodulated carrier in (2.6) is p0 = e20 . (2.8) in units of W. 2.3 Digital modulation of the optical carrier This section describes the digital modulation of the optical carrier from (2.6). One way is the direct modulation of the laser with an electrical drive signal, so that the output field of the laser is already modulated. Another way is to modulate the light from a CW laser in an external modulator. All modulation formats discussed in this thesis will be generated with external modulators. There are different external modulator types based on different physical principles [3]. In the following the so-called Mach-Zehnder modulator (MZM) type will be used, because it allows better control over both the amplitude and phase of the optical signal than other modulator types or direct modulation. Figure 2.3 illustrates the schematic of an MZM [4, 5]. The incoming optical signal ein (t) is split by an Y-coupler. The two waves propagate through two different arms and are recombined in another coupler forming the optical output signal eout (t). The wave guide in an MZM is made of an electro-optic material, typically lithium niobate (LiNbO3 ). By applying a voltage across two electrodes enclosing such a wave guide, a phase shift can be induced on 6 u1,MZM (t) ? ϕ1,MZM (t) ein (t) - eout (t) - ϕ2,MZM (t) 6 u2,MZM (t) Figure 2.3: Schematic of a Mach-Zehnder modulator the optical signal propagating in the the wave guide (cf. Appendix A for a short discussion of optical phase shifters). Figure 2.3 shows a so-called differential MZM where the drive voltages u1,MZM (t) and u2,MZM (t) can individually control the phase shifts in the upper and lower arm, respectively. Mathematically, the relation between the optical input and output of an MZM can be described as p eout (t) 1 jϕ1,MZM ±jϕ2,MZM 2 + 1−ϒ ·e . = √ ϒ·e ein (t) 2 (2.9) The power split ratio ϒ2 /(1 − ϒ2 ) between the p upper and lower arm of the MZM is related 2 to the intrinsic extinction ratio ε by ϒ = 0.5 + 1/ε . Generally, the extinction ratio stands for the ratio between the maximum and minimum power of a signal. The intrinsic extinction ratio represents the highest extinction ratio that can be achieved with the respective MZM and is therefore considered an MZM parameter. The extinction ratio is usually given in dB as εdB = 10 · log10 (ε ). The linear relations between the phase shifts in the upper and lower arms of the modulator and the drive voltages are u1,MZM (t) Uπ u2,MZM (t) ϕ2,MZM (t) = π . Uπ ϕ1,MZM (t) = π (2.10) Uπ is a reference voltage, which causes a π phase shift. For LiNbO3 it is typically around 5 V [4, ch. 15]. The sign of the second exponent in (2.9) can be either ‘+’ or ‘−’ depending on the actual physical realization of the MZM. The MZM can be used for amplitude as well as for phase modulation. Rewriting (2.9) as 7 i √1 − ϒ2 − ϒ eout (t) ϒ h jϕ1,MZM (t) ±jϕ2,MZM (t) √ +e + e±jϕ2,MZM (t) =√ e ein (t) 2 2 √ h i ϕ ϕ1,MZM (t) ∓ ϕ2,MZM (t) j 1,MZM (t)±ϕ2,MZM (t) 2ϒ 1 − ϒ2 − ϒ ±jϕ2,MZM (t) 2 √ = √ cos e e + 2 2 2 (2.11) and assuming identical drive voltages u(t) = u1,MZM (t) = u2,MZM (t) and therefore identical phase shifts ϕMZM (t) = ϕ1,MZM (t) = ϕ2,MZM (t), we arrive at √ eout (t) ϒ + 1 − ϒ2 jϕMZM (t) √ e = ein (t) 2 (2.12) for ‘+’ in (2.9) and at eout (t) 2ϒ = √ cos ϕMZM (t) + ein (t) 2 √ 1 − ϒ2 − ϒ −jϕMZM (t) √ e 2 (2.13) for ‘−’ in (2.9). A MZM with the input-output relation according to (2.12) is a phase modulator (PM). A MZM with the input-output relation according to (2.13) can be used either as an amplitude modulator or a binary phase modulator depending on the range of the phase shift ϕMZM (t) and the drive voltage u(t), respectively. There is a term responsible for spurious phase modulation, which is small in magnitude compared p to the desired term. This term vanishes for infinite intrinsic extinction ratio ε , i.e. ϒ = 1/2. Figure 2.4 illustrates that this MZM operates as an amplitude modulator (AM), if it is driven by a voltage in the range ∆u(t) = Uπ /2, whereas it operates as a PM, if it is driven by a voltage in the range ∆u(t) = Uπ . In the rest of this thesis, the acronym MZM will denote a Mach-Zehnder modulator with an input-output relation according to (2.13), regardless of whether the modulator is used for phase or amplitude modulation but assuming the proper range of the drive voltage. The acronym PM will denote an Mach-Zehnder modulator with an input-output relation according to (2.12), which is solely used for phase modulation. In principle, a single Mach-Zehnder modulator could be used for arbitrary optical multilevel amplitude and phase modulation formats [6, 7], if the electrical drive signals u1,MZM (t) and u2,MZM (t) had multiple levels. However, it is more common to have optical transmitters that consist of a combination of M Mach-Zehnder modulators driven by binary electrical signals with u(t) = u1,MZM (t) = u2,MZM (t) as above to create the L = 2M signal points. It is easier to generate clear binary drive signals than clear multilevel drive signals at the desired bit rates of 10 Gbit/s or 40 Gbit/s, especially because of relaxed amplifier linearity requirements for binary signals in real systems. The electrical drive signals to the M MachZehnder modulators are thus 8 Phase modulation: ∆ϕMZM (t) = π eout (t) ∆u(t) = Uπ ein (t) 1 π π /2 0 2π 3π /2 ϕMZM (t) = π u(t) Uπ −1 Amplitude modulation: ∆ϕMZM (t) = π /2 ∆u(t) = Uπ /2 Figure 2.4: Mach-Zehnder modulator input-output relation according to (2.13) with ϒ = ui (t) = ûi · ∞ X n=−∞ p 1/2 and fixed time t bi,n · h(t − nTs ) + ubias,i (i = 1 . . . M). (2.14) bi,n is the bit sequence, ûi controls the amplitude of the drive signal and ubias,i ensures the correct bias. For example, we have to choose ûi = −Uπ /2 and ubias,i = Uπ /2 for binary amplitude modulation, or ûi = Uπ and ubias,i = 0 for binary phase modulation as shown in Figure 2.4. The pulse shaper used for generating the electrical drive signals in this thesis has a raised-cosine pulse shape [8] h(t) = cos2 h i , |t| ≤ 2 (1 − α ) π 2|t|−Ts (1−α ) , T2s (1 − α ) < |t| < 4 α Ts Ts 1 0 , |t| ≥ Ts 2 (1 + α ) . (2.15) Ts 2 (1 + α ) 0 ≤ α ≤ 1 is the roll-off factor, which controls the edges of the impulse response. α = 0 leads to a rectangular impulse response with a total duration of Ts , whereas α = 1 leads to an impulse response with cosine-shaped edges and a total duration of 2Ts as illustrated in Figure 2.5(a). Figure 2.5(b) gives the corresponding spectra H(ω ) •−◦ h(t). This impulse response satisfies (2.2) and is a good approximation for pulse shapers that can be technologically realized at the desired bit rates of 10 Gbit/s or 40 Gbit/s. Obviously, it is closely related to the well-known Nyquist pulse, which has a raised-cosine shape in the frequency domain [9]. After digital modulation in one or more Mach-Zehnder modulators, an additional returnto-zero (RZ) pulse shaping can be applied to the optical signal. RZ pulse shaping may for example enhance receiver sensitivity [10, 11] or improve nonlinear signal transmission 9 0 20 · log10 H(ω )/H(0) 1 h(t)/h(0) α =0 α = 0.5 0.5 0 −1 α =1 −0.5 0 t/Ts 0.5 −20 α =0 −40 α = 0.5 −60 α =1 −80 −100 −4 −3 −2 −1 0 1 ω · Ts /(2π ) 1 (a) 2 3 4 (b) Figure 2.5: (a) Impulse responses and (b) spectra of time-domain raised-cosine impulse responses for different roll-off factors α [12, 13, 14, 15]. RZ pulse shaping means that the optical signal is modulated in such a way that the instantaneous power between consecutive symbols becomes zero. Figure 2.6 illustrates RZ pulse shaping of an unmodulated carrier. The RZ duty cycle is defined as dRZ = TFDHM /Ts , where TFDHM is the full duration at half maximum of the power of the optical pulses. RZ pulse shaping is achieved by driving a Mach-Zehnder modulator with a periodic electrical drive signal, for example with a cosine-shaped clock signal at the symbol rate Rs leading to the duty cycle dRZ = 0.5 as in Figure 2.6. A periodic sequence of electrical Gaussian pulses is used in this thesis for RZ pulse shaping, as it allows for arbitrary duty cycles dRZ . The individual Gaussian pulse and the periodic signal are hRZ,Gauss (t) = and e(− ln 2)·(2t/TFDHM )2 0 uRZ,Gauss (t) = ûRZ · ∞ X n=−∞ for |t| < Ts /2 else hRZ,Gauss (t − nTs ). (2.16) (2.17) Note, that TFDHM is the same for the electrical pulses and the power of the optical pulses if the Mach-Zehnder modulator has infinite extinction ratio. The case where no explicit RZ pulse shaping is applied is called non-return-to-zero (NRZ) pulse shaping. With electrical timedomain raised cosine pulse shaping according to (2.15), the power between two identical symbols does not become zero for NRZ pulse shaping. After digital modulation or after RZ pulse shaping , the optical signal can be expressed in general form as e(t) = a(t) · ejω0t = a(t) · ej arg {a(t)} · ejω0t = e0 · b(t) · ejϕ (t) · ejω0t = b(t) · ejϕ (t) · ec (t) (2.18) 10 TFDHM Optical power p(t)/p0 1 - 0.5 00 1 2 t/Ts 3 4 Figure 2.6: Return-to-zero pulse shaping of the unmodulated carrier with the complex envelope a(t) or its corresponding magnitude 0 ≤ b(t) = | a(t) e0 |≤ 1 and phase ϕ (t) = arg a(t) . The chirp of an optical signal is the time-derivative of the phase ∂ ϕ (t)/∂ t. At the time instants t = nTs the complex envelope a(nTs ) = an represents the complex signal points of the digital (multilevel) modulation from Section 2.1. According to (2.7) the instantaneous power of the modulated carrier is p(t) = |e(t)|2 = |a(t)|2 = p0 · |b(t)|2 , with p0 = e20 as in (2.8). The average power is 1 p̄ = lim T →∞ 2T 2.4 Z T −T 1 p(t)dt = lim T →∞ 2T Z T 1 |a(t)| dt = p0 · lim T →∞ 2T −T 2 Z T −T |b(t)|2 dt. (2.19) Optical fiber transmission This section gives a brief overview of the effects that govern the transmission of optical signals over an optical fiber. Up to now we have considered optical signals at the location z = 0, i.e. at the transmitter, depending only on the time variable t. It is the purpose of this section to study the dependence of the optical signal on the coordinate z, where we keep the assumption of a planar optical wave from Section 2.2, which propagates in z direction. The optical fiber is a homogeneous cylindrical silica glass fiber with a core of radius rcore and a surrounding cladding. The optical refractive index ncore of the core is slightly higher than the refractive index nclad of the cladding. Solving Maxwell’s equations for this system shows [4, ch. 3], that there is only one possible electric field distribution or mode for a propagating harmonic wave if the wavelength satisfies 2π c ≥ 2π rcore λ0 = ω0 q n2core − n2clad 2.4 . (2.20) Throughout this thesis the case of such a single-mode fiber (SMF) will be assumed. For the unmodulated optical carrier ec (t) from (2.6) with the frequency ω0 and constant power, the 11 propagation in z-direction through the fiber can be described by the equation of a harmonic wave ec (z,t) = ec (0,t) · e− αf 2 z · e−jβ z = e0 · e− αf 2 z · ej(ω0t−β z) (2.21) with the fiber attenuation constant α f and the propagation constant β both in units of 1/km. The fiber attenuation coefficient in units of dB/km is α f ,dB = α f · 10 · log10 e. For SMF the minimum of α f ,dB lies at wavelengths around 1550 nm and amounts approximately to 0.2 dB/km. Therefore, most optical fiber communication systems operate around this wavelength. The propagation constant β is related to the effective refractive index of the fiber n̄ with ncore ≥ n̄ ≥ nclad by β = n̄ω /c [16]. The effective index n̄ and thus the propagation coefficient β are both frequency and power dependent in an SMF. Note, that the spatial field distribution is again neglected in (2.21), as it is only needed for a parameter in nonlinear fiber transmission (cf. Subsection 2.4.2). 2.4.1 Linear fiber transmission Let us first consider the frequency dependence of the propagation constant β = β (ω ) only and study the propagation of the modulated optical signal e(t) from (2.18) for this case. The Fourier spectrum of the optical input signal into the fiber at z = 0 E(0, ω ) = A(0, ω − ω0 ) •−◦ e(0,t) = e(t) = a(0,t) · ejω0t = a(t) · ejω0t (2.22) and the Fourier spectrum of the optical signal at a position z along the fiber E(z, ω ) = A(z, ω − ω0 ) •−◦ e(z,t) = a(z,t) · ejω0t (2.23) are related by E(z, ω ) = E(0, ω )e− αf 2 z −jβ (ω )z (2.24) e or for the complex envelopes A(z, ω − ω0 ) = A(0, ω − ω0 )e− αf 2 z −jβ (ω )z e . (2.25) The frequency dependence of the propagation constant leads to a frequency dependence of the group delay τG = z · ∂ β (ω )/∂ ω or the group velocity vG = z/τG , respectively. This property results in chromatic dispersion or simply dispersion. It causes different spectral components of the optical signal to travel at different velocities through the optical fiber, which then arrive at the fiber output with a time difference. Dispersion leads to the broadening of transmitted pulses and is a major source of signal distortion in optical fiber transmission. 12 It is common to expand β (ω ) into a Taylor series around the carrier frequency ω0 1 1 β (ω ) = β0 + (ω − ω0 )β1 + (ω − ω0 )2 β2 + (ω − ω0 )3 β3 + . . . 2 6 (2.26) with βi = ∂ i β (ω ) ∂ ωi , (i = 0, 1, 2, . . .) (2.27) ω =ω0 and therefore we arrive at A(z, ω − ω0 ) = A(0, ω − ω0 )e− αf 2 z −j[β0 +(ω −ω0 )β1 + 21 (ω −ω0 )2 β2 + 61 (ω −ω0 )3 β3 +...]z e . (2.28) The coefficient β0 imposes a constant phase shift on the complex envelope, whereas β1 is responsible for a constant time delay. It is therefore common to set them to zero as they do not cause signal distortion. Out of the higher order coefficients it is sufficient to retain β2 and β3 and set βi = 0 (i > 3) in order to capture the dominant distortion effects at the considered bit rates. With these assumptions we can define a fiber transfer function for the optical signal H 0f (z, ω ) = αf 1 1 2 3 A(z, ω − ω0 ) E(z, ω ) = e− 2 z e−j[ 2 (ω −ω0 ) β2 + 6 (ω −ω0 ) β3 ]z , = E(0, ω ) A(0, ω − ω0 ) (2.29) or with a shift of variables for the complex envelope H f (z, ω ) = αf 1 2 1 3 A(z, ω ) = e− 2 z e−j[ 2 ω β2 + 6 ω β3 ]z . A(0, ω ) (2.30) It is common to characterize an optical fiber by specifying the dispersion coefficient D and the dispersion slope S as the wavelength dependence of the group delay evaluated at the carrier wavelength λ0 = 2π c/ω0 : D= 1 ∂ τG z ∂λ = λ =λ0 1 ∂ τG ∂ ω z ∂ω ∂λ = λ =λ0 13 ∂ 2β ∂ ω ∂ ω2 ∂ λ λ =λ0 =− 2π c β2 , λ02 (2.31) and S= 1 ∂ 2 τG z ∂λ2 = λ =λ0 ∂ω 1 = z ∂λ !2 = β3 + 2π c λ02 !2 1 ∂ z ∂λ ∂ τG ∂ ω ∂ω ∂λ ! = λ =λ0 ∂ 2 τG ∂ τG ∂ 2 ω + ∂ ω2 ∂ω ∂λ2 1 ∂ω ∂ z ∂λ ∂λ λ =λ0 4π c β2 . λ03 ∂ω = ∂λ !2 ∂ τG ∂ω ! + ∂ 2ω ∂ τG ∂ω ∂λ2 λ =λ0 ∂ 3β ∂ 2β ∂ 2ω + ∂ ω3 ∂ ω2 ∂ λ 2 λ =λ0 (2.32) The dispersion coefficient D and the dispersion slope S depend on the fiber type and are functions of the wavelength λ0 = 2π c/ω0 , at which the wavelength dependence of the group delay is evaluated. For a standard SMF typical values according to [17] are DSMF = 17 ps/(nm · km) and SSMF = 0.056 ps/(nm2 · km) at λ0 = 1550 nm. It is possible to compensate for the dispersion-induced signal distortions in an SMF with length l1 by a dispersion-compensating fiber (DCF) with length l2 , dispersion coefficient DDCF and dispersion slope SDCF . Full dispersion compensation is achieved if H f ,SMF (l1 , ω ) · H f ,DCF (l2 , ω ) = exp [−(α f ,1 /2)l1 ] exp [−(α f ,2 /2)l2 ] (2.33) with the corresponding attenuation constants α f ,1 and α f ,2 . In terms of the dispersion coefficients and dispersion slopes, this is the case for DSMF · l1 = −DDCF · l2 and SSMF /DSMF = SDCF /DDCF with DDCF < 0 and SDCF < 0. In an optical fiber transmission system, in which the optical signal is transmitted over N possibly different fiber links the residual or accumulated dispersion is defined as rD = N X i=1 Di · li . (2.34) In later chapters of this thesis, dispersion induced signal distortions will be investigated as a function of rD . So-called polarization-mode dispersion (PMD) is another source of signal distortions in optical fiber transmission. PMD is caused by the polarization dependence of the propagation coefficient β (ω ) and may result in a differential group delay of the signal components in the x- and y-polarization or other effects [18, ch. 15]. However, PMD will not be considered in this thesis as degradations from chromatic dispersion and PMD are qualitatively similar in the comparison of multilevel modulation formats [19]. 14 2.4.2 Nonlinear fiber transmission If we also take into account the nonlinear power dependence of the effective index n̄ (Kerr effect), the propagation of the complex envelope a(z,t) is described by a differential equation, the nonlinear Schrödinger (NLS) equation [2, 4, ch. 4] jβ2 ∂ 2 a(z,t) β3 ∂ 3 a(z,t) ∂ a(z,t) α f + a(z,t) − − = −jγ |a(z,t)|2 a(z,t) 2 3 ∂z 2 2 ∂t 6 ∂t (2.35) with a nonlinear term on the right-hand side. The nonlinear parameter γ is γ= n2 ω0 cAeff (2.36) with the nonlinear index coefficient n2 in units of m2 /W and the effective core area Aeff , which is related to the spatial field distribution in the optical fiber [2]. Typical values for SMF are n2 = 2.6 · 10− 20 m2 /W [2] and Aeff = 87 µ m2 [18, ch. 6]. The coefficients β0 and β1 are again omitted in the NLS equation for reasons explained in Subsection 2.4.1. The nonlinear term in (2.35) leads to an effect called self-phase modulation (SPM) [2]. In the absence of chromatic dispersion (β2 = 0, β3 = 0) the transmitted signal acquires a nonlinear phase shift depending on its own instantaneous power according to a(z,t) = a(0,t) exp jφNL (z,t) . The nonlinear phase shift is φNL = −γ |a(0,t)|2 [1 − exp (−α f z)]/α f . Together with chromatic dispersion, SPM in general leads to signal distortions during fiber transmission. Only for special optical pulses called solitons the interplay between chromatic dispersion and SPM can lead to undistorted signal transmission. Note, that in principal the complex envelope a(z,t) in (2.35) may also represent the total field of a wavelength-division multiplexed (WDM) transmission system [18, ch. 6]. The optical field for 2N + 1 WDM signals with carrier frequencies ωi at the fiber input is e(0,t) = N X i=−N ai (0,t) · ejωit = N X i=−N ai (0,t) · ej(ωi −ω0 )t · ejω0t = a(0,t) · ejω0t . (2.37) Then, the solutions of the nonlinear Schrödinger equation in (2.35) also includes cross-phase modulation (XPM) and four-wave mixing (FWM). XPM means the nonlinear phase shift of a signal imposed by a signal in another wavelength channel transmitted over the same fiber. FWM causes cross-talk between copropagating signals. Further, for symbol rates Rs ≥ 10 Gsymbols/s nonlinear inter-symbol interference from intrachannel cross-phase modulation (IXPM) and intra-channel four-wave mixing (IFWM) is also considered in (2.35) [18, ch. 6]. 15 Opposed to linear fiber transmission, the order of SMF and DCF (precompensation, postcompensation, or both) as well as the amount of DCF (under- or overcompensation) has an impact on the actual signal distortions. For long-haul systems, proper dispersion mapping is required in order to keep distortions as low as possible [18, ch. 6]. The nonlinear Schrödinger equation is usually solved by the split-step Fourier algorithm [2]. In this algorithm the fiber is divided in short segments of length ∆z and it is assumed that in the segments dispersion and nonlinearity act independently. In a first step, nonlinearity alone is considered by imposing the nonlinear phase shift φNL (∆z,t) in the time-domain. In a second step, dispersion alone is accounted for by the fiber transfer function H f (∆z, ω ) in the frequency domain. This is repeated for all segments of the fiber. 2.5 Noise from optical amplifiers Optical amplifiers can be used for the compensation of attenuation from fibers or other sources. One realization of an optical amplifier is the Erbium-doped fiber amplifier (EDFA) [20, 4]. Within a limited wavelength window around 1550 nm, the EDFA is transparent with respect to wavelength, bit rate and modulation format. Power from a pump laser excites the Erbium ions in the fiber so that the input optical signal can stimulate emission and is thus being amplified as it travels through the EDFA. Besides the desired stimulated emission, however, there is also some spontaneous emission, which generates noise. The noise from spontaneous emission is in turn amplified by stimulated emission, so that the noise from the EDFA is called amplified spontaneous emission (ASE) noise. There is independent ASE noise in both polarizations, but if the signal is fully x-polarized, a polarization filter can be used to remove the ASE noise in the y-polarization. We assume that ASE noise from optical amplifiers is the dominant noise source in the considered systems. The linear amplifier model used in this thesis assumes constant amplification G of the optical signal e(t) and additive white Gaussian noise (AWGN) wWGN (t) with zero mean in both polarizations over the bandwidth of interest around the carrier frequency ω0 [21, 22, 4] as depicted in Figure 2.7. Consistent with the complex signal notation from the above sections, the baseband noise processes wx (t) and wy (t) are also complex valued. The real and imaginary parts of wx (t) and wy (t) are independent. We assume a constant noise power spectral density N0 per polarization, which is related to the EDFA parameters, and define the optical signal-to-noise ratio OSNR = 10 · log10 Psignal Psignal = 10 · log10 . Pnoise ρ · N0 · B (2.38) Psignal is the average signal power and Pnoise is the average noise power within the reference bandwidth B. It is common to set B = 12.5 GHz, which amounts approximately to 0.1 nm at a carrier frequency of λ0 = 1550 nm. For noise in both polarizations ρ = 2, and for noise only in the signal polarization ρ = 1. The noise power spectral density N0 is the same for the 16 AGWN: wWGN (t) = e(t) G G · e(t) ? - wx (t) · ejω0 t wy (t) G · e(t) + wWGN (t) - Figure 2.7: Linear amplifier model: Constant amplification G and additive white Gaussian noise wWGN (t) baseband and bandpass processes [23]. Although in reality the amplifiers and noise sources are distributed over the whole transmission system, we assume for linear fiber transmission that all amplification and noise is located just before the optical receiver. For nonlinear fiber transmission, however, the actual locations of the noise sources must be taken into account as they may have a strong impact on the system performance [24, 25]. Another way of compensating for attenuation is the use of coherent detection. It can be shown that coherent detection and direct detection with optical amplification have similar performance [26] under certain conditions. However, this thesis focuses on direct detection and coherent detection will not be discussed. 2.6 Optical receivers with direct detection This section gives a short overview over basic elements of optical receivers with direct detection using a binary ASK receiver shown in Figure 2.8 as an example. The direct detection receivers for multilevel modulation formats discussed in the following chapters rely on the same principals, but obviously have different structures. Direct detection means that the optical bandpass signal is directly converted into an electrical baseband signal by a photodiode without the use of a local oscillator laser. Under the assumption of a fully x-polarized signal part, the received optical field including AWGN is arx (t) + wx (t) ex,rx (t) · ejω0t . = erx (t) = wy (t) ey,rx (t) (2.39) The complex envelope of the signal arx (t) may be distorted from fiber dispersion and nonlinearities. The received optical field is first passed through an optical bandpass filter (BPF) with the polarization independent transfer function Hopt (ω ). Optical filters are based on different technologies [3], however, we just classify and describe them by their transfer functions, e.g. with Gaussian magnitude. The optical filter usually represents the optical demultiplexer of a WDM system. The optical noise after filtering is colored but still has Gaussian probability density functions (PDF). After filtering, the optical signal ẽrx (t) is converted to an 17 electrical current i(t) by a photodiode (PD). The photodiode is made of a semiconductor with a pn-junction or, in order to increase conversion efficiency, an intrinsic semiconductor between p-type and n-type semiconductors (pin photodiode). The photodiode is usually reverse-biased. The incident optical signal creates electron-hole pairs, which give rise to an electrical photocurrent. The photocurrent i(t) is proportional to the received instantaneous optical power and therefore, according to (2.7), proportional to the squared magnitude of the optical field. Thus, 2 2 2 i(t) = R · p̃rx (t) = R · ẽrx (t) = R · ẽx,rx (t) + ẽy,rx (t) n o 2 2 2 ∗ = R · ãrx (t) + 2R · Re ãrx (t)w̃x (t) + R · w̃x (t) + R · w̃y (t) . | {z } | {z } | {z } baseband signal signal-spontaneous noise (2.40) spontaneous-spontaneous noise The proportionality factor R is the photodiode’s responsivity in units of A/W. For convenience we assume R = 1 A/W throughout this work. The star (∗) denotes the conjugate complex. Besides the desired baseband signal, there are two noise terms in (2.40): signaldependent signal-spontaneous noise (mixing of signal and ASE) and signal-independent spontaneous-spontaneous noise (mixing of noise with itself). The photodiode’s thermal and shot noise can be safely neglected for an average received optical power around 1 mW typical for optically amplified transmission systems [4, ch. 25]. The electrical noise is now no longer Gaussian distributed but rather chi-square distributed [9]. In the receiver model, the photocurrent passes an electrical low-pass filter (LPF) with transfer function Hel (ω ), e.g. an electrical Bessel filter. This electrical filter represents the bandlimitation of the photodiode combined with an actual electrical filter. The filtered current ĩ(t) is then sampled at the time instants t = t0 + kTs , k ∈ Z, and the samples ik = ĩ(t0 + kTs ) are passed to a binary decision device with threshold ith . For ik > ith the estimated bit b̂k becomes ’1’, whereas for ik ≤ ith the estimated bit b̂k becomes ’0’. There are no closed-form probability density functions for the noise in ĩ(t) and ik . Both the sampling instant offset t0 and the decision threshold ith need to be optimized in order to reach the lowest possible bit error ratio (BER) or bit error probability (BEP). The term BER is used if the ratio of erroneous bits to the total number of transmitted bits is obtained from error counting in a measurement or simulation, whereas the term BEP states that the error probability is calculated from probability density functions or their moment-generating functions (MGF). The calculation of bit error probabilities normally requires the knowledge of the exact PDF, which does not readily exist for the ik . A common assumption for binary optical ASK is that the signal-spontaneous and spontaneous-spontaneous noise are Gaussian distributed [27]. Then, the noise on the samples ik is also Gaussian distributed. This assumption leads to very precise results in the calculation of the BEP, which happens by pure mathematical coincidence [28], but totally fails in the calculation of the optimum decision threshold. 18 White Gaussian noise Colored Gaussian noise ? erx (t) Optical - BPF Hopt (ω ) Signal-dependent chi-square noise ? ẽrx (t) Photo- diode |·|2 No closed-form PDF ? i(t) Electrical - LPF Hel (ω ) No closed-form PDF ? Sampling ĩ(t) t= t0 + kTs ? Binary ik - decision ik ≷ ith b̂k - Figure 2.8: Optical receiver for binary ASK: Received optical fields and electrical currents and the corresponding noise types. For optical differential PSK systems, which will be the subject of this thesis in later sections, the Gaussian assumption even leads to inaccurate and unreliable BEP results [29, 30]. For some special cases regarding optical and electrical filters, BEP was calculated with the help of moment-generating functions accounting for correct statistical properties of the noise [31, 22]. However, this thesis uses a a more elaborate method shortly outlined in Section 2.7, which can handle arbitrary optical and electrical filters as well as arbitrary signal distortions, for the case of optical AWGN in front of the receiver. 2.7 Simulation of optical transmission systems and semianalytical bit error probability calculation This section gives information about computer simulation of optical transmission systems and reviews a method for the calculation of bit error probabilities with arbitrary receiver filters, which is used for the performance evaluation of multilevel modulation formats in the following chapters. 2.7.1 Baseband representation, periodicity, and sampled signals First of all, it can be noted that modulation, propagation, and detection of an optical signal e(t) are well described by operations on its complex envelope a(t). Therefore, the factor exp (jω0t) representing the carrier can be omitted for a baseband representation geared towards computer simulations. The MZM input-output relation in (2.9) describes either the relation of optical signals or their complex envelopes. The fiber transfer function from (2.30) or the nonlinear Schrödinger equation from (2.35) are already tailored for complex envelopes. The transfer functions of optical bandpass filters within the transmission system need to be replaced by their respective baseband transfer functions (cf. Appendix D). Although the factor exp (jω0t) does not show up explicitly in the baseband representation, the carrier frequency ω0 may well be an important parameter, e.g. for the series expansion of the propagation constant β (ω ) in (2.26) or the nonlinear parameter γ in (2.36). All electrical signals at the transmitter and receiver are of course real valued baseband signals per se. 19 The next step is to find a way of representing real-world continuous-time signals for discretetime computer simulation or processing. Let us shortly review the most important features required in this thesis loosely following [32], which is an extensive work of reference on this topic. For computer simulations, the continuous-time signal is replaced by a sampled version of P itself, e.g. the complex envelope a(t) is replaced by asmp (t) = ∞ l=−∞ a(lTa ) · δ (t − lTa ) or simply by the sequence al = a(lTa ), l ∈ Z. Ta is the sampling period, and Ra = 1/Ta is the corresponding sampling rate. The sampling rate should be chosen according to the wellknown sampling theorem in order to avoid aliasing. In our case there is a small amount of aliasing, because we consider non-bandlimited pulse shapers (cf. Section 2.3). However, if the sampling rate is selected high enough, aliasing is reasonably low and can be neglected. We further assume that all baseband signals are periodic with period Tp , such that for example the complex envelope a(t) = a(t + Tp ) for all t, or al = al+Np , respectively. N p = Tp /Ta is the number of samples in the primitive period of the sampled signal. The sampled primitive PNp −1 period is described by a p,smp (t) = l=0 al · δ (t − lTa ). With the periodicity assumption, all filter operations can be described by circular convolutions. There are Na = Ts /Ta samples per symbol interval. This periodicity assumption is the key to the most frequently used simulation technique for optical transmission systems. It allows for the computation of filter operations with a discrete Fourier transform (DFT) based method illustrated in Figure 2.9. The DFT with its computationally efficient fast Fourier transform (FFT) algorithm is equivalent to the Fourier transform of a periodic and sampled signal. The block of N p samples al , l = 0 . . . N p − 1, in the time domain is transformed into a block of N p samples Ak , k = 0 . . . N p − 1 in the frequency domain. The filter operations are now easily realized by the multiplication of the Ak with the samples of the filter transfer functions, e.g. the samples of the linear fiber transfer function H f ,k (z) = H f (z, 2π k/(N p Ta )) from (2.30). The filtered signal in the time domain is then obtained by the inverse DFT (IDFT) or inverse FFT (IFFT), respectively. Filter operations for periodic sampled signals are equivalently described by the DFT based method or by circular convolution. The reason for the wide use of this simulation technique is the split-step Fourier algorithm from Subsection 2.4.2, which divides the fiber into many individual parts and requires the computation of the signal transmission over those parts. The periodicity of the signals of course requires periodic symbol sequences. In order to capture all possible inter-symbol interference effects in the received signal, all possible symbol transitions of the length reflecting the area of influence of the inter-symbol interference need to be included in the symbol sequence. This choice defines the length of the symbol sequence and thus the required period Tp . Figure 2.10 shows eye diagrams for a binary ASK signal with inter-symbol interference affecting one previous and one following bit interval Tb . For binary ASK, the bit interval Tb equals the symbol interval Ts . The bit sequence is 0 1 0 1 1 1 0 0 and contains all eight 3-bit transitions 0 0 0, 0 0 1, . . ., 1 1 1, if it is periodically repeated. The frequencies of occurrence of ones and zeros are equal. Such a sequence is called a DeBruijn binary sequence (DBBS). The DBBS is generated in a shift-register pseudo-random bit se- 20 Sampled signal al = al+Np a0 a1 a2 Sampled spectrum Ak aNp −2 aNp −1 DFT- A0 A1 A2 · · · H0 H1 H2 ANp −2 ANp −1 · · HNp −2 HNp −1 Transfer function Hk ? Filtered signal a0l = a0l+Np a00 a01 a02 Filtered spectrum Ak · Hk = A0k a0Np −2 a0Np −1 IDFT A00 A01 A0Np −2 A0Np −1 A02 Figure 2.9: Filtering of sampled signals by using DFT and IDFT 0.02 ? Subfigure (c) Subfigure (b) 6 0 t/Tb (a) 0.5 1 101 ? 001 100 0 −0.25 000 0 t/Tb (b) Optical power p(t)/p0 0 −0.5 Optical power p(t)/p0 Optical power p(t)/p0 1 0.25 111 011 - 0.8 −0.25 6 010 110 0 t/Tb 0.25 (c) Figure 2.10: Eye diagrams for binary ASK with inter-symbol interference affecting one previous and one following bit interval Tb : (a) all traces, (b) traces for bit 0, and (c) traces for bit 1. quence (PRBS) generator by introducing an additional zero in the longest run of zeros. This concept can be easily extended to multilevel symbol sequences. In general, if the intersymbol interference affects (NISI − 1)/2 previous symbols and (NISI − 1)/2 following symbols, the symbol sequence must be of the length Ns = LNISI , e.g. for binary symbols with L = 2 and NISI = 3 the length of the sequence must be Ns = 23 = 8 as above, or for quaternary symbols with L = 4 and NISI = 5 the length of the sequence must be Ns = 45 = 1024. 2.7.2 Moment-generating function Using the above introduced baseband model, periodicity, and sampled signals, we shall now review a method for calculating bit-error probabilities for optical transmission systems using binary ASK. The method is originally based on [33] and is described in great detail in [34], which also includes the straightforward extension to other modulation formats and some implementation details. The block diagram from Figure 2.8 is slightly modified in Figure 2.11 21 i0l xl = arx,l + wx,l Optical vl = ãrx,l + w̃x,l Photo- BPF - diode Hopt,k |·|2 Electrical ĩ0l - LPF Hel,k Sampling l= N0 + kNa Binary ik - decision ik ≷ ith b̂k - Figure 2.11: Optical receiver for binary ASK: Periodic baseband model with noise in the signal polarization only. to account for sampled baseband signals and noise in the signal polarization (x-polarization). Noise in the orthogonal polarization can be easily included as shown later. We consider sampled baseband signal blocks of length N p = Na · Ns (Na , Ns ∈ N). The frequency-continuous filter transfer functions are replaced by their frequency-discrete baseband equivalents Hopt,k and Hel,k of length N p . The block of samples of the received signal xl = arx,l + wx,l , l = 0, . . . , N p − 1, or for a shorter notation xl = arx,l + wl , can be written as a complex random column vector with a deterministic signal part (mean values) and a random noise part as x0 x1 x= . = .. xNp −1 w0 arx,0 arx,1 w1 . + . = arx + w .. .. wNp −1 arx,Np −1 {z } | {z } | Deterministic part (2.41) Random part with the multivariate Gaussian probability density function fx (x) = 1 π Np det Cx H C−1 (x−a ) rx x e−(x−arx ) . (2.42) The operator (·)H denotes the complex-conjugate transpose of a vector or a matrix, and det(·) means the determinant of a matrix. As the noise in front of the optical filter is assumed to be AWGN with constant power spectral density N0 in Section 2.5, the covariance matrix Cx is simply given by Cx = N0 · Ra · INp ×Np with the N p × N p identity matrix INp ×Np . The multiplication with the sampling rate Ra accounts for the fact that we have a sampled version of the noise [32]. The samples vl = ãrx,l + w̃l are the filtered versions of the xl and can be obtained by circular convolution or the DFT method described above. The complex random column vector vl of course also has a deterministic and a random part v0 v1 .. . = v= vNp −1 ãrx,0 ãrx,1 .. . w̃0 w̃1 .. . + = ãrx + w̃. ãrx,Np −1 w̃Np −1 | {z } | {z } Deterministic part 22 Random part (2.43) Similar to (2.42), its multivariate probability density function is fv (v) = 1 −(v−ãrx )H C−1 v (v−ãrx ) . e N p π det Cv (2.44) The covariance matrix Cv for the filtered noise can be obtained as follows: Let the vector r contain the autocorrelation function rl of the filtered noise obtained numerically from ij the IDFT of the noise power spectral density Sk = N0 Ra |Hopt,k |2 . The element cv (i, j = ij 0, . . . , N p − 1) in the ith row and jth column of Cv is then given by cv = ri− j using the symmetry of the autocorrelation function r−l = rl∗ . The samples of the current directly after the photodiode are i0l = |vl |2 = |ãrx,l + w̃l |2 . With electrical filtering they become ĩ0l = N p −1 X n=0 i0n · hl−n = N p −1 X n=0 2 |vn | ·hl−n = N p −1 X n=0 |ãrx,n + w̃n |2 ·hl−n , l = 0 . . . Np − 1 (2.45) using the circular convolution sum. The real-valued filter coefficients hl = hl+Np are obtained from the block of N p samples Hel,k by IDFT. The sampling device in Fig. 2.11 takes every Na th sample out of ĩ0l with an offset of N0 samples, accounting for the time offset t0 in Figure 2.8. We assume that N0 = t0 /Ta ∈ N. This leads to Ns = N p /Na samples ik according to the Ns symbols represented by the original block of N p samples. For the samples we get ik = ĩ0kNa +N0 = = N p −1 N p −1 X in · hkNa +N0 −n = X v∗kNa +N0 −n · hn · vkNa +N0 −n n=0 N p −1 n=0 X n=0 v∗n · hkNa +N0 −n · vn (2.46) , k = 0, . . . , Ns − 1, which can be rewritten in matrix notation as a quadratic form for each of the Ns samples ik = vH k hel vk , k = 0, . . . , Ns − 1. (2.47) The N p × N p diagonal matrix hel = diag{h0 , h1 , . . . , hNp −1 } contains the electrical filter coefficients. The complex random column vector vkNa +N0 ãrx,kNa +N0 ãrx,kNa +N0 −1 .. . w̃kNa +N0 w̃kNa +N0 −1 .. . vkNa +N0 −1 = + = ãrx,k + w̃k . (2.48) vk = . . . vkNa +N0 −Np +1 ãrx,kNa +N0 −Np +1 w̃kNa +N0 −Np +1 23 is just a reordered version of (2.43). Its multivariate probability density function fvk (vk ) is therefore the same as in (2.44) with v replaced by vk and ãrx replaced by ãrx,k . Although it is not possible to give a closed form version of the probability density function of a sample ik as noted in Section 2.6, we shall now investigate its MGF and how this function can be used to calculate the error probability for binary threshold decision. For the definition and properties of the moment-generating function and other statistical terms in the following, refer to [35]. In principal, the MGF is closely related to the two-sided Laplace transform of the probability density function of a random variable. Here, the MGF of a particular ik is given by Φik (s) = Z vk svH k hel vk e 1 fvk (vk )dvk = Np π det Cv Z H H C−1 (v −ã k rx,k ) v esvk hel vk −(vk −ãrx,k ) dvk , (2.49) vk with the complex variable s. After diagonalization of the exponent and evaluation of the integrals in (2.49) the MGF can be rewritten as N p −1 sλl |ql |2 Y e 1−sλl . Φik (s) = 1 − sλl (2.50) l=0 The elements in this equation are as follows [33, 34]: The vector q with its elements ql , l = 0, . . . , N p − 1, is given by the coordinate transformation ãrx,k = U1 M−1 U2 q. The matrix U1 is the normalized modal matrix of the covariance matrix Cv (and also of C−1 v ), and M 2 H −1 is obtained from the corresponding diagonalization M = U1 Cv U1 . The matrix U2 is the −1 normalized modal matrix of M−1 UH 1 hel U1 M . The vector λ with its elements λl , l = 0, . . . , N p − 1 contains the eigenvalues of hel Cv . The eigenvalues are real-valued since hel is a diagonal matrix and Cv = CH v . Despite the seeming complexity, all these elements can be easily calculated numerically. Until now we have considered noise in the signal polarization only. Noise in the orthogonal polarization (y-polarization) leads to an additional noise term independent from the signal and the noise in the x-polarization given by the last term in (2.40). The MGF of this noise term after electrical filtering and sampling is obtained as explained above, by setting the vectors ãrx = ãrx,k = 0 and consequently q = 0. The MGF for the sample ik is then given by the multiplication of both MGF resulting in Φik (s) = N p −1 Y l=0 sλl |ql |2 1−sλl e . (1 − sλl )2 (2.51) The method for obtaining the MGF is sometimes called a semi-analytical approach [36]. This term shall point out that in (2.49) the noise is treated analytically with its statistical 24 properties in the covariance matrix, whereas the values of the signal samples (e.g. after fiber transmission and optical filtering) actually have to be computed with the DFT method or circular convolution sums. 2.7.3 Evaluation of the moment-generating function for bit error probability calculation For binary ASK, an error in the estimated bit b̂k occurs, if the sample ik is greater than a threshold ith and if the originally transmitted bit bk was a zero-bit, or if the sample ik is smaller than ith and if the originally transmitted bit bk was a one-bit. Using the probability density function fik (ik ) for a sample ik and the unit step function 0 if ζ ≤ 0 us (ζ ) = 1 if ζ > 0 (2.52) the probability for a wrong decision becomes Pe,ik R∞ Pe,ik ,0 = P[ik > ith ] = −∞ fik (ik )us (ik − ith )dik for zero-bit bk = 0, = R Pe,i ,1 = P[ik ≤ ith ] = ∞ fik (ik )us (ith − ik )dik for one-bit bk = 1, k −∞ (2.53) which leads to the mean bit error probability of the block of Ns bits Ns −1 1 X Pe = Pe,ik . Ns (2.54) k=0 The DBBS length was selected according to the area of influence of the intersymbol interference NISI (cf. Subsection 2.7.1), and thus, all possible intersymbol interference effects have been captured. Therefore, Pe is also the bit error probability for an infinitely long and uncorrelated bit sequence, in which the probabilities of occurrence for the zero-bit and one-bit are equal. The PDF in (2.53) is given by the inverse Laplace transform of Φik (−s) from (2.50) or (2.51), respectively, as follows 1 fik (ik ) = lim ω →∞ 2π j Z σ 0 +jω σ 0 −jω Φik (−s )e 0 s 0 ik 1 ds = lim ω →∞ 2π j 0 Z σ +jω σ −jω Φik (s)e−sik ds. (2.55) The inverse Laplace integral only converges in the region of convergence (ROC), which is determined here by the eigenvalues λl , as they define the singularities of Φik (−s). With the 25 smallest eigenvalue λmin = minl {λl } and the largest eigenvalue λmax = maxl {λl }, the region of convergence of Φik (−s) is −1/λmax < σ < −1/λmin (cf. Appendix B). Inserting (2.55) into (2.53) yields Pe,ik = R∞ 1 σ R+j∞ Φ (s) P = us (ik − ith )e−sik dik ds for zero-bit bk = 0, i e,i ,0 k 2π j σ −j∞ k −∞ R∞ 1 σ R+j∞ us (ith − ik )e−sik dik ds for one-bit bk = 1. Φ (s) P = i e,ik ,1 2π j σ −j∞ k −∞ (2.56) For convenience of notation, we replaced the limit operator by writing σ ±j∞ into the integral limits. We use the the Laplace transforms of the shifted step functions Z us (ik − ith )e−sik dik = e−sith /s ; ROC: Re{s} > 0 us (ith − ik )e−sik dik = −e−sith /s ; ROC: Re{s} < 0 ∞ −∞ and Z ∞ −∞ and we finally arrive at Pe,ik = , if ith ≥ 0, ∞ > Re{s} > 0 , if ith < 0, , if ith < 0, −∞ < Re{s} < 0 , if ith ≥ 0, (2.57) (2.58) 1 σ R+j∞ −si P = Φik (s) e s th ds for zero-bit bk = 0, ROC: 0 < σ < −1/λmin , e,ik ,0 2π j σ −j∞ −1 σ R+j∞ −si Φik (s) e s th ds for one-bit bk = 1, Pe,ik ,1 = 2π j σ −j∞ ROC: −1/λmax < σ < 0. (2.59) The integrals in (2.59) must be solved numerically in order to calculate the error probabilities for the individual bits. This can be done by a method called steepest descent integration described in [37] together with an approximation formula. The basic idea of this method is to find an integration path in the complex plane, for which the integrand in (2.59) decreases most rapidly to zero. Such a path goes through the saddlepoint of the integrand. Therefore, the approximation formula based on a series expansion of the integrand in the saddle point is also known as saddlepoint approximation. As it turns out that this approximation produces extremely accurate results [38, 34], it will be used for error probability calculations throughout this thesis. 26 It is easily possible to extend the error probability computation to other optical modulation formats with direct detection such as binary or multilevel differential PSK or combinations of binary ASK with the latter. Basically, all operations on the received optical signal such as time delays or phase shifts must be represented as filter transfer functions in order to properly calculate the covariance matrix Cv of the noise before the photodiode, which may be different for different receiver parts. Further processing in the electrical domain needs to be accounted for. In the receiver descriptions for the other modulation formats in the following sections, modifications in the calculation method will be explained. 27 Chapter 3 Binary Amplitude-Shift Keying (2-ASK) Binary amplitude-shift keying (2-ASK) with direct detection is the most commonly used modulation format for optical communication systems. The terms on-off keying (OOK) or intensity modulation (IM) are often used synonymously for 2-ASK. Although this thesis focuses on multilevel modulation formats, some properties of 2-ASK are reviewed in order to use it as a reference. Basic parameters, which will be the same for each modulation format, are also introduced here and summarized in Appendix C. 3.1 2-ASK receiver and transmitter The transmitter and receiver block diagrams are given in Figures 3.1(a) and (b) with the signals according to the baseband model. Light from a CW laser is modulated in an MZM by the binary electrical drive signal u(t). (In the baseband notation, a CW signal is represented by a constant complex envelope a(t) = const. The acronym CW is kept as a reference to the actual physical signal.) The drive signal u(t) is generated in the raised-cosine (RC) pulse shaper from the bit sequence bn according to (2.14) and (2.15). The optical RZ signal aRZ (t) with duty cycle dRZ = 0.5 is obtained by modulation of the NRZ signal aNRZ (t) in a subsequent MZM with a periodic sequence of Gaussian pulses according to (2.16) and (2.17). The intrinsic extinction ratio of the MZM is assumed as εdB = 35 dB. The roll-off factor of the RC pulse shaper is set to α = 0.5 as an example for nonrectangular pulse shaping without ISI at the transmitter. All ISI observed at the receiver must thus be introduced by the channel or the receiver filters. The same extinction ratio and roll-off factor will also be used at the transmitters for the other modulation formats in later chapters. The complex envelopes of the optical NRZ and RZ signals normalized to the square root of the respective average power p̄ are plotted into the constellation diagrams of Figures 3.2(a) and (b). The two signal points lie on the real axis of the constellation diagrams. The transitions between the two signal points also follow the real axis. Thus, the chirp of the ASK signal is zero. Although chirped ASK transmission may be advantageous for transmission systems in the nonlinear regime [39], we consider only chirp-free ASK throughout this thesis. 28 Pulse bn - shaper u(t) ? Laser CW e0 MZM uRZ,Gauss (t) ? MZM aNRZ (t) aRZ (t)- (a) arx (t) + w(t) 2nd ord. - Gauss BPF - Photodiode ĩ(t) Sampling - & binary i(t) 3rd ord. - Bessel LPF b̂k - decision (b) 1 √ Im{aRZ (t)}/ p̄ √ Im{aNRZ (t)}/ p̄ Figure 3.1: (a) 2-ASK transmitter and (b) 2-ASK receiver 0 −1 1 0 −1 0 1 2√ Re{aNRZ (t)}/ p̄ 3 0 (a) 1 2√ Re{aRZ (t)}/ p̄ 3 (b) Figure 3.2: Constellation diagram with phase transitions for (a) NRZ-2-ASK and (b) RZ-2-ASK The signal points of RZ-2-ASK have a greater distance than the signal points of NRZ-2-ASK, if the NRZ and RZ signals have the same average power p̄. The power spectra for Rs = Rb = 40 Gbit/s normalized to the average power p̄ are shown in Figures 3.3(a) and (b). Please refer to Appendix E for details on the calculation of the power spectra. Obviously, the RZ signal bandwidth is greater than the NRZ signal bandwidth, because of the shorter pulse duration. This is a general feature of RZ modulated signals, as will also be seen in later chapters for other modulation formats. The higher RZ signal bandwidth has especially to be considered in WDM transmission systems, where the frequency spacing between adjacent channels must be adjusted to the signal bandwidth in order to avoid interchannel interference. As the 2-ASK signals do not have zero mean, both spectra show strong contributions at frequency f = 0 marking the carrier frequency in a bandpass description, and integer multiples of 40 GHz corresponding to the symbol rate Rs . They would lie at the carrier frequency in the passband description. 29 20 n o 10 · log10 |ARZ ( f )|2 / p̄ n o 10 · log10 |ANRZ ( f )|2 / p̄ 20 0 0 −20 −20 −40 −40 −120 −80 −40 0 40 Frequency f [GHz] 80 120 (a) −120 −80 −40 0 40 Frequency f [GHz] 80 120 (b) Figure 3.3: Power spectra for (a) NRZ-2-ASK and (b) RZ-2-ASK with Rb = 40 Gbit/s The general 2-ASK receiver has already been presented in Section 2.6 with unspecified optical and electrical filters. The 2-ASK receiver in Figure 3.1(b) has an optical 2nd order Gaussian bandpass filter and an electrical 3rd order Bessel low-pass filter (cf. Appendix D). Note, that the bandpass filter is of course represented by its corresponding baseband description. The receiver filter types will be the same in the investigations of all modulation formats throughout this thesis. In Figure 3.1(b) the sampling device and the binary decision are merged into a single block for simplicity. Their functionality is of course still the same as in Section 2.6, where they were represented by individual blocks in the block diagram. 3.2 2-ASK performance For the following investigations of 2-ASK and other modulation formats, the optical noise according to Section 2.5 has only a component in the signal polarization (x-polarization). The component in the orthogonal polarization shall be removed by appropriate means. The parameter ρ in the definition of the OSNR in (2.38) is therefore set to ρ = 1. All presented bit error probabilities and required OSNR values for achieving certain bit error probabilities are calculated based on the methods from Subsections 2.7.2 and 2.7.3. In order to achieve the best performance, i.e. the lowest bit error probability, of an optical transmission system, the optical and electrical receiver filter bandwidths need to optimized with respect to the transmitted signal waveform [8, 34]. This optimization is a trade-off between the filter induced ISI, which leads to a closure of the eye diagram at the decision device, and the amount of noise that passes through the filters together with the signal. On the one hand, the lower the filter bandwidths, the more noise is filtered out but the higher is 30 Table 3.1: Optimized optical and electrical receiver filter bandwidths, required OSNR for BEP = 10−9 , and chromatic dispersion tolerances ∆rD for 1-dB and 2-dB OSNR penalties 2-ASK Format NRZ RZ ∆ f3 dB,opt /Rs 1.20 2.20 f3 dB,el /Rs 1.05 0.55 Req. OSNR [dB] 21.76 21.11 ps ∆rD,1 dB [ nm ] ps ∆rD,2 dB [ nm ] 63 62 98 82 the ISI. On the other hand, the greater the filter bandwidths, the lower is the impact of ISI but the signal at the decision device is more strongly corrupted by noise. In Figures 3.4(a) and (b) the required OSNR for achieving the bit error probability BEP = 10−9 are shown in contour plots vs. the electrical 3-dB cut-off frequency f3 dB,el and the optical 3-dB bandwidth ∆ f3 dB,opt for NRZ and RZ, respectively. BEP = 10−9 is often chosen in laboratory measurements for shorter measurement times, whereas the operators of actual optical transmission systems often demand much lower values, e.g. BEP = 10−18 . The optimal bandwidth pairs for NRZ and RZ are marked by × in the contour plots and given in Table 3.1 together with the required OSNR value. The dashed lines delimit the regions with an OSNR penalty below 0.5 dB with respect to the minimum required OSNR. These values represent an optically amplified transmission system with linear fiber transmission and full dispersion and dispersion slope compensation, or simply the cascade of the transmitter, an attenuator followed by an optical amplifier, and the receiver. Therefore, this is also commonly called the back-to-back case. The eye diagrams of the electrical signals without noise after the Bessel low-pass filter are depicted in Figures 3.5(a) and (b) for NRZ and RZ, respectively. Figure 3.4 shows that the optical filter bandwidth for the RZ signal needs to be significantly greater than for NRZ in order to achieve the lowest required OSNR. For the electrical filter bandwidths this is just the other way round. Because of the narrow electrical filter, the RZ signal also becomes NRZ-like, i.e. the electrical signal does not go to zero between bit periods, as can be seen in Figure 3.5. The required OSNR of 21.11 dB for RZ is 0.65 dB lower than for NRZ with 21.76 dB. If the electrical filter bandwidths are chosen properly, NRZ allows optical filter bandwidths below 1.0 · Rs and RZ allows an optical filter bandwidth as low as 1.2 · Rs for an OSNR penalty of 0.5 dB with respect to the minimum. This tolerance to narrow optical filtering is important in WDM systems, in which the optical filters are used as demultiplexers, because it allows for close channel spacing. For the optimized receiver filter bandwidths, BEP vs. OSNR is plotted in Figure 3.6. This figure can be used to determine the required OSNR for other BEP values than 10−9 . Both 31 24 23 3.2 24 23 2.2 23 .5 2.0 1.8 22.5 1.6 22.5 23.5 1.4 2.2 2.0 1.8 21.25 21.5 1.6 1.4 22 1.2 23 24 22 22 22.5 23 1.2 1.0 0.4 21.5 21.5 22 22 2.4 .2 5 23 2.4 2.6 21 Optical bandwidth ∆ f3 dB,opt /Rs 22.5 2.6 22.5 1.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs 22 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (a) (b) 3 3 Electrical signal ĩ(t)/(R · p̄) Figure 3.4: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-2-ASK and (b) RZ-2-ASK. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. Electrical signal ĩ(t)/(R · p̄) Optical bandwidth ∆ f3 dB,opt /Rs 2.8 21.5 21.25 .5 2.8 22 21.25 3.0 24 23.5 3.0 21.5 3.2 3.4 23.5 23 3.4 2 2 1 1 0 −0.5 0 Time t/Ts 0 −0.5 0.5 (a) 0 Time t/Ts 0.5 (b) Figure 3.5: Eye diagrams after optical and electrical filtering for (a) NRZ-2-ASK and (b) RZ-2-ASK 32 Bit error probability NRZ−2−ASK RZ−2−ASK 10−3 10−6 10−9 10−12 12 15 18 21 OSNR [dB] 24 27 30 Figure 3.6: Bit error probability vs. OSNR for NRZ-2-ASK and RZ-2-ASK Required OSNR [dB] for BEP = 10−9 33 30 27 24 21 NRZ−2−ASK RZ−2−ASK 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 Figure 3.7: Required OSNR for BEP = 10−9 vs. accumulated dispersion rD for NRZ-2-ASK and RZ-2-ASK the RZ and the NRZ curves are almost straight lines with the double logarithmic scale for the ordinate (BEP axis) over the considered OSNR range. The curves for all bandwidth pairs other than the optimum ones would lie above those two depicted curves. Therefore, the optimum bandwidth pairs for the lowest required OSNR for BEP = 10−9 are also the optimum bandwidth pairs for the lowest required OSNR for any other BEP greater or lower than 10−9 . Actually, the optimum bandwidth pairs could also be determined by finding the pair with the lowest BEP for a fixed OSNR as it was done in [34]. Finally, let us look at the dispersion tolerance of 2-ASK. In Figure 3.7, the required OSNR for BEP = 10−9 for both NRZ and RZ is plotted vs. accumulated dispersion rD according to (2.34) for linear fiber transmission with zero dispersion slope S. Dispersion tolerance can 33 be quantified by the amount of accumulated dispersion that can be tolerated for an OSNR penalty below a fixed threshold. As the accumulated dispersion can be either negative or pos neg positive, we choose the difference ∆rD,x dB = rD,x dB − rD,x dB between the maximum tolerable pos positive accumulated dispersion rD,x dB > 0 and the minimum tolerable negative accumulated neg dispersion rD,x dB < 0 for an x-dB OSNR penalty as a measure for chromatic dispersion tolerance. As can be seen from Figure 3.7, the RZ signal has a slightly smaller chromatic dispersion tolerance than the NRZ signal because of the greater RZ signal bandwidth. The values of ∆rD for OSNR penalties of 1 dB and 2 dB are also listed in Table 3.1. For 1-dB OSNR penalty, the difference between RZ and NRZ is just 1 ps/nm in favor of NRZ, as the RZ curve is flater around the minimum at rD = 0. For 2-dB OSNR penalty, the difference is already 16 ps/nm in favor of NRZ. Note, that for 2-ASK the OSNR vs. rD curves are fully symmetric, as the the signals are chirp-free. 34 Chapter 4 Binary Differential Phase-Shift Keying (2-DPSK) Optical binary differential phase-shift keying (2-DPSK) with direct detection [40] has recently been attracting attention because of an improved receiver sensitivity [41, 42, 43] and better tolerance to some nonlinear transmission effects [44, 45] compared to 2-ASK. It is not the aim of this thesis to study 2-DPSK in all details, however, 2-DPSK is used as a reference for the multilevel modulation formats in later chapters and thus needs to be shortly reviewed. Further, basic receiver concepts and especially the extension of the bit error probability calculation method from Section 2.7 are also the underlying methods for the multilevel modulation formats and therefore presented here. In 2-DPSK transmission, the information is contained in the phase differences of consecutively transmitted symbols, not in the phase angles or the amplitudes. The phase difference can be expressed as ∆ϕ (t) = ϕ (t) − ϕ (t − Ts ), according to the definition of the complex envelope in (2.18). We will first look at the 2-DPSK transmitter and receiver with direct detection and then investigate the performance of 2-DPSK. 4.1 2-DPSK transmitter At first sight, the 2-DPSK transmitter in Figure 4.1 looks very similar to the the 2-ASK transmitter in Figure 3.1(a). Light from a CW laser is modulated by a binary electrical drive signal u(t). The modulator can be either an MZM with the characteristic of (2.12) or a PM with the characteristic of (2.13), as discussed in Section 2.3. If an MZM is used, twice the amplitude of the drive voltage u(t) is required compared to 2-ASK. In both cases, the electrical drive signal u(t) is generated by an RC pulse shaper with roll-off factor α = 0.5. The input bit sequence b0n to the RC pulse shaper is a differentially encoded version of the original bit sequence bn . If 2-DPSK modulation is accomplished with an MZM, it will be labeled 2-DPSK-MZM, whereas it will be labeled 2-DPSK-PM, if the PM is used in the transmitter. Differential encoding ensures that the modulated signal aNRZ (t) has the phase 35 bn - Diff. encoder Pulse b0n - shaper u(t) ? Laser CW e0 - MZM or PM uRZ,Gauss (t) ? MZM aNRZ (t) aRZ (t)- Figure 4.1: 2-DPSK transmitter difference ∆ϕ (nTs ) = 0 for a transmitted original one-bit bn = 1 and the the phase difference ∆ϕ (nTs ) = π for a transmitted original zero-bit bn = 0. The differential encoding function can easily be derived with knowledge of the receiver and will thus be given later. RZ pulse shaping with duty cycle dRZ = 0.5 is again achieved in a second MZM by modulation of the NRZ signal aNRZ (t) with a periodic sequence of Gaussian pulses. Let us now look at the constellation diagrams of the normalized complex envelopes in Figures 4.2 and 4.3 for 2-DPSK-MZM and 2-DPSK-PM. They show the signal points with their magnitudes and phase angles. The information is contained in the phase differences as explained above. The difference between 2-DPSK-MZM and 2-DPSK-PM becomes especially obvious for the NRZ signals in Figures 4.2(a) and 4.3(a). For 2-DPSK-MZM the transitions go through zero and the amplitude of the signal is therefore not constant over time. However, the signal is chirp-free except for a small residual chirp caused by the finite extinction ratio of the modulator (cf. Eq. (2.13)). In contrast to that, the amplitude of the 2-DPSK-PM signal is approximately constant over time, as the transitions between the two signal points follow a circle in the complex plane. The signal has significant chirp as ∂ ϕ /∂ t 6= 0 during a full transition. The constellation diagrams after RZ pulse shaping are depicted in Figures 4.2(b) and 4.3(b). The RZ constellation diagrams look more similar than the two NRZ constellation diagrams. Naturally, the amplitude of the 2-DPSK-PM signal is no longer constant over time, but the more important effect is that the chirp is significantly reduced compared to the NRZ signal. RZ pulse shaping cuts out the middle portions of a symbol period Ts , in which the major part of the phase transition has already taken place. Note, that the transitions for RZ2-DPSK-PM do not exactly go through zero as the used Gaussian pulses with TFDHM = Ts /2 still have a small but not completely negligible amplitude at the symbol period boundaries. In comparison to the ASK constellation diagrams from Figure 3.2, we observe a larger signal point distance for 2-DPSK for signals with the same average power as is well-known from communication theory . This can be explained as follows: With rectangular pulse shaping, the average power of a 2-ASK constellation with equally probable signal points at zero and a is √ p̄ = 21 · 02 +√21 · a2 = 12 · a2 . A DPSK constellation with equally probable signal points 1 1 2 2 2 at √ − 2a/2 and 2a/2 has the same average power p̄ = 2 · 2 · 4 · a = 2 · a but a by factor 2 larger signal point distance. In our case, in which the transitions follow the√RC pulse over time and we do not have phase jumps, the factor may slightly differ from 2, as the power in the transitions also has to be considered. A closer look at the NRZ constellation 36 √ Im{aRZ (t)}/ p̄ √ Im{aNRZ (t)}/ p̄ 1 0 −1 1 0 −1 −1 0 √1 Re{aNRZ (t)}/ p̄ −1 0 √ 1 Re{aRZ (t)}/ p̄ (a) (b) 1 √ Im{aRZ (t)}/ p̄ √ Im{aNRZ (t)}/ p̄ Figure 4.2: Constellation diagram with phase transitions for (a) NRZ-2-DPSKMZM and (b) RZ-2-DPSK-MZM 0 −1 1 0 −1 −1 0 √1 Re{aNRZ (t)}/ p̄ −1 0 √ 1 Re{aRZ (t)}/ p̄ (a) (b) Figure 4.3: Constellation diagram with phase transitions for (a) NRZ-2-DPSKPM and (b) RZ-2-DPSK-PM diagrams in Figures 4.2(a) and 4.3(a) further reveals that the distance is somewhat larger for 2-DPSK-MZM than for 2-DPSK-PM. For 2-DPSK-PM the instantaneous power is constant over time and thus the same for a signal point or at some point during the transition. For 2-DPSK-MZM, however, the power during the transitions is lower than at the signal points. Thus, for the same average power, the signal point distance for 2-DPSK-MZM must be larger than for 2-DPSK-PM. The power spectra of the 2-DPSK signals with Rb = Rs = 40 Gbit/s are shown in Figures 4.4(a) and (b), and 4.5(a) and (b). Because the 2-DPSK-MZM signals have zero mean, there are no observable spikes at the center frequency f = 0 or multiples of 40 GHz, as they could be found for 2-ASK in Figure 3.3. However, these spikes can be clearly observed for NRZ-2-DPSK-PM and also to some extent for RZ-2-DPSK-PM. This can be attributed to the significant mean value of the imaginary part of the NRZ-2-DPSK-PM signals. As this mean value is reduced by RZ pulse shaping, the spikes are weaker for RZ-2-DPSK-PM. Of course, 37 20 n o 10 · log10 |ARZ ( f )|2 / p̄ n o 10 · log10 |ANRZ ( f )|2 / p̄ 20 0 0 −20 −20 −40 −40 −120 −80 −40 0 40 Frequency f [GHz] 80 −120 −80 120 (a) −40 0 40 Frequency f [GHz] 80 120 80 120 (b) Figure 4.4: Power spectra for (a) NRZ-2-DPSK-MZM and (b) RZ-2-DPSKMZM with Rb = 40 Gbit/s 20 n o 10 · log10 |ARZ ( f )|2 / p̄ n o 10 · log10 |ANRZ ( f )|2 / p̄ 20 0 0 −20 −20 −40 −40 −120 −80 −40 0 40 Frequency f [GHz] 80 120 (a) −120 −80 −40 0 40 Frequency f [GHz] (b) Figure 4.5: Power spectra for (a) NRZ-2-DPSK-PM and (b) RZ-2-DPSK-PM with Rb = 40 Gbit/s the bandwidths for the RZ signals are larger than for the NRZ signals, as it could already be observed for ASK in Figure 3.3. This effect will not be pointed out again explicitly for the multilevel modulation formats. At this point it should be noted that the mean values of the real parts of the 2-DPSK complex envelopes do not completely vanish, if the bit sequence bn is a DBBS as introduced in Subsection 2.7.1. The differentially encoded bit sequence b0n and therefore the sequence of phase angles ϕn = ϕ (nTs ) are of DBBS nature in the differences of consecutive values and not in the values themselves. For such an encoded sequence, the number of one-bits and zero-bits in b0n or the number of zero-phase and π -phase angles in ϕn can be slightly different. 38 Balanced detector a1 (t) a(t) - τ DAF ψ a2 (t) Photodiode Photodiode i1 (t) R aup,in - i(t) - X coupler alow,in - aup,out alow,out - i2 (t) (a) (b) Electrical signal i(t)/(Re20 ) 1 i(t) ∼ cos ∆ϕ (t) + 0 0 i(t) ∼ cos ∆ϕ (t) + π −1 0 1 Differential phase ∆ϕ (t)/π 2 (c) Figure 4.6: (a) Delay & add filter with balanced detector, (b) a single cross coupler, and (c) the output signal i of the delay & add filter with balanced detector with respect to the differential phase ∆ϕ for ψ = 0 and ψ = π 4.2 2-DPSK receiver 4.2.1 Delay & add filter with balanced detector Next comes the analysis of the 2-DPSK receiver with direct detection. The key element in this receiver is the so-called delay & add filter (DAF) with the balanced detector comprised of two photodiodes shown in Figure 4.6(a). The DAF is also called Mach-Zehnder delay interferometer filter or simply Mach-Zehnder interferometer in the literature. Because of its importance for all the following multilevel modulation formats, it will first be discussed in more detail before looking at the full 2-DPSK receiver. Figure 4.6(a) shows that the optical input signal a(t) = e0 · b(t) · ejϕ (t) into the DAF is split into two paths by a cross coupler. The input-output relation for the upper and lower input and output ports of a single cross coupler depicted in Figure 4.6(b) is aup,out alow,out ! 1 =√ 2 ! ! 1 j aup,in j 1 alow,in (4.1) The signal in the upper path of the DAF is delayed by τ , whereas a phase shift ψ is imposed 39 onto the signal in the lower path. As the next step, the two signals are recombined in a second cross coupler. The two optical output signals of the DAF are i i e h 1h 0 jψ jϕ (t) jψ jϕ (t−τ ) − b(t)e e a1 (t) = a(t − τ ) − e a(t) = b(t − τ )e 2 2 (4.2) i i je h jh 0 jψ jϕ (t) jψ jϕ (t−τ ) a2 (t) = + b(t)e e . a(t − τ ) + e a(t) = b(t − τ )e 2 2 (4.3) and The upper output port of the DAF is called destructive port, as a1 (t) = 0 for an optical CW input signal into the DAF with ψ = 0. Naturally, the lower output port of the DAF is called constructive port, as a2 (t) = ja(t) = const in this case. The transfer functions of the DAF are obtained by Fourier transformation from (4.2) and (4.3) as i h A1 (ω ) 1 −jωτ jψ j 21 (ωτ +ψ ) 1 H1 (ω ) = e −e = j·e · sin 2 (ωτ − ψ ) = A(ω ) 2 (4.4) i h A2 (ω ) j −jωτ jψ j 21 (ωτ +ψ ) 1 e H2 (ω ) = +e = j·e · cos 2 (ωτ − ψ ) . = A(ω ) 2 (4.5) and The electrical signals after detection become 2 i1 (t) = R a1 (t) = and 2 i2 (t) = R a2 (t) = o Re20 n 2 b (t) + b2 (t − Ts ) − 2b(t)b(t − Ts ) cos ∆ϕ (t) + ψ 4 o Re20 n 2 b (t) + b2 (t − Ts ) + 2b(t)b(t − Ts ) cos ∆ϕ (t) + ψ . 4 (4.6) (4.7) where we already set τ = Ts , as we want to evaluate the phase differences between two consecutive symbols, and used the above introduced notation ∆ϕ (t) = ϕ (t) − ϕ (t − Ts ). For modulation formats that evaluate the phase differences between symbols, which are further apart, the delay τ can of course also be set to multiples of Ts [46]. Finally, after subtraction we arrive at i(t) = i2 (t) − i1 (t) = Re20 b(t)b(t − Ts ) cos ∆ϕ (t) + ψ . 40 (4.8) Balanced detector arx (t) + w(t) 2nd ord. - Gauss BPF v1 (t) - τ DAF ψ v2 (t) Photodiode Photodiode R i(t) 3rd ord. - Bessel LPF ĩ(t) Sampling b̂k - & binary decision Figure 4.7: 2-DPSK receiver If the receiver uses only one optical signal a1 (t) or a2 (t) from the DAF and thus only one electrical signal i1 (t) or i2 (t), it is called a receiver with single-ended detection of the destructive or constructive port, respectively. Otherwise and as shown in Figure 4.6(a), it is called a receiver with balanced detection. Figure 4.6(c) shows the electrical signal i(t) vs. the differential phase ∆ϕ (t) under the assumption of constant magnitude b(t) = 1. The solid line results if the phase shift in the DAF is set to ψ = 0, whereas the dashed line represents a phase shift of ψ = π . For 2-DPSK the differential phase ∆ϕ (t) nominally takes on two different values at the sampling instants, 0 or π . Both choices of ψ thus lead to electrical signals with the maximum possible amplitude: For ψ = 0 we get i(t0 + kTs ) = Re20 if ∆ϕ (t0 + kTs ) = 0, and i(t) = −Re20 if ∆ϕ (t0 + kTs ) = π . For ψ = π we get i(t0 + kTs ) = −Re20 if ∆ϕ (t0 + kTs ) = 0, and i(t0 + kTs ) = Re20 if ∆ϕ (t0 + kTs ) = π , which is just the inverted signal. 4.2.2 2-DPSK receiver with delay & add filter and balanced detector With a better understanding of the DAF with balanced detection, let us now look the 2-DPSK receiver considered in this thesis and illustrated in Figure 4.7. As in the 2-ASK receiver, the received signal corrupted by AWGN is first filtered in an optical 2nd order Gaussian bandpass filter. Then it is fed into the DAF. For the 2-DPSK receivers in this thesis, the phase shift is always set to ψ = 0. The two DAF output signals are labeled v1 (t) and v2 (t).The electrical signal after balanced detection is again filtered in an electrical 3rd order Bessel low-pass filter, before we get the estimates of the bit sequence b̂k after sampling and binary threshold decision. 4.2.3 2-DPSK differential encoder With knowledge of both the transmitter and the receiver, it is now possible to find a proper differential encoding function, as promised at the beginning of this section. This task is illustrated in Figure 4.8. The cascade of the transmitter without (w/o) the differential encoder and the receiver (second and third block in the figure) converts the bit sequence b0n into the bit sequence bk in the noise-free case. The differential encoder (first block in the figure) must thus 41 bn - Diff. encoder b0n 2-DPSK -transmitter w/o diff. encoder ! b̂ = bn - 2-DPSK k receiver Figure 4.8: Differential encoding for 2-DPSK Table 4.1: Truth table for deriving the 2-DPSK differential encoding function ! b0n b0n−1 ϕk ϕk−1 ∆ ϕk ik b̂k = bn 0 0 0 0 0 pos 1 0 1 0 π neg 0 1 0 π 0 −π π neg 0 1 1 π π 0 pos 1 transfer the original bit sequence bn into the bit sequence b0n in such a way that the originally transmitted and the received bit sequences are the same. Note, that for simplicity the indices n and k are used interchangeably, which assumes zero delay from transmission and processing, but gives an adequate model for deriving the differential encoding function. Table 4.1 shows the phase angles ϕk , the differential phase ∆ϕk , the electrical signal after balanced detection ik at the sampling instants, and the estimated bit b̂k with respect of the encoded bits b0n . It is a truth table that can be used to straightforwardly determine the differential encoding function as b0n = bn · b0n−1 + bn · b0n−1 = bn ⊕ b0n−1 . (4.9) In (4.9) the operators have the following meaning: ’·’ denotes logical A ND, ’+’ stands for logical O R, ’⊕’ means logical X OR, and ’ ’ represents logical N OT. Note, that the truth table would have had different entries in the last two columns and thus the differential encoding function would have been different, if we had selected the phase shift ψ = π in the DAF. Differential encoding functions for the multilevel modulation formats in later chapters can be derived with the same methodology and will not be explained in such detail as for 2-DPSK. 4.3 Extension of the bit error probability calculation method to 2-DPSK with balanced detection The bit error calculation method for 2-ASK described in Section 2.7 can be extended to 2-DPSK in the following way [34]: The electrical signal at the decision device ik is now 42 the sum of two quadratic forms of the sampled versions of v1 (t) and v2 (t). One summand represents the contribution from the upper port of the DAF and the upper photodiode, the other one is the contribution from the lower port of the DAF and the lower photodiode in Figure 4.7. Based on the MGF for 2-ASK in (2.49) and using the same notation introduced in Section 2.7, we write the MGF for the 2-DPSK receiver as Φik (s) = Z Z H H es(−v1,k hel v1,k +v2,k hel v2,k ) fv1,k ,v2,k (v1,k , v2,k )dv1,k dv2,k . (4.10) v1,k v2,k The column vectors v1,k and v2,k are the reordered versions of the column vectors containing the samples of the destructive and constructive output signals of the DAF including noise. With ãrx,1,k and ãrx,2,k as the noise-free versions of v1,k and v2,k , we then define new vectors ã v ) as the stacking of the two individual vectors. The multivariate v̆k = ( v1,k ) and ărx,k = ( ãrx,1,k 2,k rx,2,k PDF is then of the same form as in (2.44) fv̆k (v̆k ) = 1 −(v̆k −ărx,k )H C−1 v̆ (v̆k −ărx,k ) , e 2N p π det Cv̆ (4.11) with the difference that the vectors are now of length 2N p , the covariance matrix Cv̆ is of size 2N p × 2N p , and the exponent of π is therefore 2N p = N̆ p . The covariance matrix Cv̆ is obtained numerically as described in Section 2.7, but with taking the optical bandpass filter function and additionally the DAF transfer functions from (4.4) and (4.5) into account. After introducing a new 2N p × 2N p diagonal electrical filter matrix h̆el simply constructed from the original matrix as h̆el = ( −h0 el h0el ), the MGF is rewritten as Φik (s) = Z v̆k sv̆H k h̆el v̆k e fv̆k (v̆k )d v̆k = 1 π N̆p det Cv̆ Z H H C−1 (v̆ −ă k rx,k ) v̆ esv̆k h̆el v̆k −(v̆k −ărx,k ) d v̆k . (4.12) v̆k Now, we have exactly the same form as in (2.49) and the process of diagonalization, inverse Laplace transformation and error probability calculation is exactly the same as detailed in Section 2.7 for 2-ASK. 4.4 2-DPSK performance The results of the receiver filter bandwidth optimization are shown in the contour plots of Figures 4.9(a) and (b) for 2-DPSK-MZM and of Figures 4.10(a) and (b) for 2-DPSK-PM. The optimal bandwidth pairs that lead to the lowest required OSNR for BEP = 10−9 are marked by × and listed in Table 4.2 together with the required OSNR values. Eye diagrams for the electrical signal ĩ(t) at the sampling and decision device with these optimal bandwidth pairs are given in Figures 4.11(a) and (b) for 2-DPSK-MZM and Figures 4.11(a) and (b) 43 Table 4.2: Optimized optical and electrical receiver filter bandwidths, required OSNR for BEP = 10−9 , and chromatic dispersion tolerances ∆rD for 1-dB and 2-dB OSNR penalties 2-DPSK-MZM 2-DPSK-PM NRZ RZ NRZ RZ ∆ f3 dB,opt /Rs 1.20 1.95 1.45 1.95 f3 dB,el /Rs 1.35 0.70 1.05 0.70 Req. OSNR [dB] 18.37 18.14 19.30 18.19 ps ∆rD,1 dB [ nm ] ps ∆rD,2 dB [ nm ] 101 71 79 70 149 96 117 95 Format for 2-DPSK-PM. The eye diagrams show that the electrical signals are binary. A negative value represents a zero-bit, a positive value a one-bit. The optimized decision thresholds are approximately zero. Principally, the optical filter bandwidths need to be greater for RZ than for NRZ, and the other way round for the electrical filter cut-off frequencies, similar to 2-ASK. The electrical RZ signal therefore becomes again NRZ-like as can be seen in the eye diagrams. The lowest required OSNR of 18.14 dB is achieved with RZ-2-DPSK-MZM, which is only 0.23 dB lower than for the NRZ variant. The difference is larger for 2-DPSK-PM. Here, the OSNR of 18.19 dB for RZ is by 1.11 dB lower than for NRZ. The similarity between both 2-DPSK variants for RZ pulse shaping is obvious. In the previous subsection we have already seen that both signals at the transmitter and consequently at the receiver are very similar. Here, we find that the same receiver bandwidths lead to a required OSNR, which differs by only 0.05 dB in favor of 2-DPSK-MZM. If a 0.5-dB OSNR penalty with respect to the minimum required OSNR is tolerable and the electrical filter cut-off frequencies are chosen properly, optical filter bandwidths below 1.0 · Rs and as low as 1.2 · Rs are possible for 2-DPSK-MZM with NRZ and RZ pulse shaping, respectively. For 2-DPSK-PM these lower filter bandwidth bounds are slightly larger: 1.1 · Rs for NRZ and 1.25 · Rs for RZ. Figure 4.13 shows a plot of the bit error probability vs. OSNR for 40-Gbit/s 2-DPSK with optimized receiver filter bandwidths. The ASK curves are included for comparison. The two RZ-2-DPSK curves lie almost on top of each other. Only the inset reveals the slightly lower bit error probabilities for RZ-2-DPSK-MZM. The poor performance of NRZ-2-DPSK-PM compared to the other 2-DPSK variants becomes clearly visible. All 2-DPSK curves are lower than the 2-ASK curves. An approximately 3-dB advantage of RZ-2-DPSK over RZ2-ASK is maintained over a wide range of OSNR. Note again, that we only consider balanced 2-DPSK detection in this thesis because it has been found [43, 47], that balanced detection has an approximately 3-dB advantage over single-ended detection. The 3-dB advantage of 2-DPSK over 2-ASK can thus only be 44 19 3.0 18.5 20 2.0 19.5 1.6 18.5 1.4 .25 18 1.4 19 192.50 202.15 21.5 1.2 1.2 18.5 5 20 20.21 18.5 1.0 0.4 19 1.8 25 19 1.6 2.0 18. 1.8 18.5 2.2 25 20 2.4 18.25 20.5 20 19.5 19 2.2 5 .25 18 20. 2.4 2.6 19 19.5 Optical bandwidth ∆ f3 dB,opt /Rs 2.6 18.5 Optical bandwidth ∆ f3 dB,opt /Rs 2.8 19.5 2.8 18. 3.0 18.5 3.2 .5 21 21 20.5 3.2 3.4 20.5 20 19 .5 3.4 1.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (a) 18.5 19 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (b) Figure 4.9: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-2-DPSK-MZM and (b) RZ-2-DPSK-MZM. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. 45 20.5 21 21.5 21 2.0 20 19 20 1.8 .5 1.6 19.5 19 18.5 19 19.5 .5 2.4 2.2 2.0 18.25 21 20 18.25 2.4 2.6 18.5 20 Optical bandwidth ∆ f3 dB,opt /Rs 20 2.8 20.5 1.8 1.6 21 20 20.5 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs 5 18. 1.0 0.4 1.2 18.5 20 21 22 1.5 5 222 2223. 19.5 1.2 1.4 19 5 19. .5 20 19.5 1.4 1.0 0.4 0.6 0.8 1.0 1.2 1.4 Electrical bandwidth f3 dB,el /Rs (a) (b) Electrical signal ĩ(t)/(R · p̄) Figure 4.10: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-2-DPSK-PM and (b) RZ-2-DPSK-PM. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. Electrical signal ĩ(t)/(R · p̄) Optical bandwidth ∆ f3 dB,opt /Rs 3.0 2.6 2.2 19 2.8 3.2 18.5 3.0 22 21.5 21 2 0 .5 3.2 3.4 21.5 3.4 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 (a) 0 Time t/Ts 0.5 (b) Figure 4.11: Eye diagrams after optical and electrical filtering for (a) NRZ-2DPSK-MZM and (b) RZ-2-DPSK-MZM 46 19 1.6 Electrical signal ĩ(t)/(R · p̄) Electrical signal ĩ(t)/(R · p̄) 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 0 Time t/Ts (a) 0.5 (b) Figure 4.12: Eye diagrams after optical and electrical filtering for (a) NRZ-2DPSK-PM and (b) RZ-2-DPSK-PM achieved with balanced detection. It can be attributed to the somewhat different decision process for balanced and single-ended detection: In single-ended detection, a noisy sample is compared against a threshold (ik ≷ith ), whereas the decision process in balanced detection can be seen as the comparison of two noisy samples against each other (i2,k − i1,k = ik ≷ ith seen as i2,k ≷ i1,k + ith ; the optimized ’offset’ ith is always close to zero and is not the same as for single-ended detection). The decision is based on whether the noisy sample i2,k is greater or smaller than the noisy sample i1,k . The following example shall illustrate this process. For a transmitted one-bit bk = 1, we expect i2,k > i1,k + ith at the receiver for a correct decision. Let fi1,k (i1,k ) and fi2,k (i2,k ) be the probability density functions for the noisy samples i1,k and Rx i2,k and Fi2,k (x) = −∞ fi2,k (x)dx the cumulative distribution function of i2,k . The probability for a decision error for balanced detection is then given by [48] h i Pe = P i2,k < i1,k + ith = Z ∞ −∞ Fi2,k (i1,k + ith ) fi1,k (i1,k )di1,k . (4.13) The error probability for single-ended detection of the constructive port for the same case is h i Pe = P i2,k < ith = Z ith −∞ fi2,k (i2,k )di2,k . (4.14) The special shapes of the probability density and cumulative distribution functions are the reason [28], why the error probability for balanced detection according to (4.13) is lower by orders of magnitude than the error probability for single-ended detection according to (4.14). Finally, we look at the dispersion tolerance for linear fiber transmission of 40-Gbit/s 2-DPSK. In Figure 4.14 the required OSNR for BEP = 10−9 is plotted vs. accumulated dispersion rD for 2-DPSK and 2-ASK. The maximum tolerable accumulated dispersion values ∆rD,x dB for 1-dB and 2-dB OSNR penalties are read from the plot and put into Table 4.2. The most obvious observation is again that the curves for RZ-2-DPSK-MZM and RZ-2-DPSK-PM are 47 Bit error probability NRZ−2−ASK RZ−2−ASK NRZ−2−DPSK−MZM RZ−2−DPSK−MZM NRZ−2−DPSK−PM RZ−2−DPSK−PM 10−3 - 10−8 10−6 test 10−9 10−9 10−12 12 18 17.5 15 18 21 OSNR [dB] 24 18.5 27 30 Figure 4.13: Bit error probability vs. OSNR for NRZ-2-DPSK and RZ-2-DPSK Required OSNR [dB] for BEP = 10−9 33 30 27 24 NRZ−2−ASK RZ−2−ASK NRZ−2−DPSK−MZM RZ−2−DPSK−MZM NRZ−2−DPSK−PM RZ−2−DPSK−PM 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 Figure 4.14: Required OSNR for BEP = 10−9 vs. accumulated dispersion rD for 2-DPSK almost identical. As a consequence, their ∆rD values differ by only 1 ps/nm. The plot further reveals a poor dispersion tolerance of 2-DPSK-PM compared to 2-DPSK-MZM, which by far has the greatest tolerance. Both 2-DPSK variants have a greater dispersion tolerance than 2-ASK with the respective pulse shaping, NRZ or RZ. Further, Figure 4.14 shows that the curve for NRZ-2-DPSK-PM is not longer fully symmetric to the origin. For the 2-dB pos OSNR penalty the magnitude of the maximum positive accumulated dispersion |rD,x dB | is 2 ps/nm larger than the magnitude of the minimum tolerable negative accumulated dispersion neg |rD,x dB |. This effect can be attributed to the significant chirp of the signal. Consequently, in the three other cases the difference is well below 1 ps/nm as the chirp is negligible. 48 The performance investigations in this subsection have shown, that 2-DPSK outperforms 2-ASK with respect to both chromatic dispersion tolerance and especially receiver sensitivity. 2-DPSK using an MZM at the transmitter leads to better values than 2-DPSK using a PM. This statement holds particularly for NRZ; for RZ the differences are negligibly small. Nevertheless, we will only use 2-DPSK-MZM as a reference in the following chapters to keep the plots clear and concise. 2-DPSK will therefore always mean 2-DPSK-MZM unless otherwise specified. 49 Chapter 5 4-level Differential Phase-Shift Keying (4-DPSK) The first optical multilevel modulation format with direct detection discussed in this thesis is 4-level (or quaternary) differential phase-shift keying (4-DPSK). As for 2-DPSK, the information is contained in the phase differences ∆ϕ (t) of consecutive symbols, but there are now four possible phase differences instead of only two. 4-DPSK is most often called differential quadrature phase-shift keying (DQPSK) in the literature. However, this thesis uses the term 4-DPSK, as it explicitly points out the 4-level nature. This chapter shows three different 4-DPSK transmitters and the 4-DPSK receiver with direct detection based on delay & add filters with balanced detectors. Then, the performance for 4DPSK systems with the different transmitters is investigated and compared to 2-DPSK and 2-ASK. 5.1 4-DPSK transmitters The task of a 4-DPSK transmitter is to generate an optical signal that has one out of four different phase angles ϕn at the time instants t = nT and therefore four different phase differences ∆ϕn ∈ {0, π2 , π , 32π }. As there are now L = 4 symbols available, 4-DPSK transmits 2 bit/symbol instead of only 1 bit/symbol as 2-ASK or 2-DPSK do. For a given bit rate Rb , the 4-DPSK symbol rate is Rs = Rb /2. This means that all 4-DPSK transmitter elements as well as the receiver elements can be designed for Rb /2 instead of Rb . Three different 4-DPSK transmitters have been proposed or demonstrated so far. Their block diagrams are given in Figures 5.1(a)-(c). The common features of the three transmitters are that two input bit sequences b1,n and b2,n are differentially encoded, and that the encoded bit sequences b01,n and b02,n are used to generate the electrical drive signals for the optical modulators with two pulse shapers. Note, however, that the encoding functions in the differential encoders are different for the three transmitters. The two input bit sequences b1,n and b2,n 50 b1,n - b2,n - Differential encoder u1 (t) b01,n Pulse - shaper b02,n - Pulse shaper ? - MZM X coupler Laser CW e0 Phase shift - 3π 2 a1 (t) X coupler aNRZ (t) MZM aRZ (t) - MZM u2 (t) 6 a2 (t) uRZ,Gauss (t) 6 b1,n - b2,n - Differential encoder (a) u1 (t) b01,n Pulse - shaper b02,n Pulse - shaper ? 0 Laser eMZM CW a1 (t) aNRZ (t) - - PM MZM aRZ (t) - 6uRZ,Gauss (t) 6 u2 (t) (b) b1,n - b2,n - Differential encoder Laser CW b01,n Pulse - shaper b02,n Pulse - shaper u1 (t) ? 2 - 6 2u2 (t) e0 ? PM aNRZ (t) - MZM aRZ (t) - uRZ,Gauss (t) 6 (c) Figure 5.1: 4-DPSK transmitters with (a) parallel MZM, (b) an MZM and a PM in series, and (c) a single PM can for example be obtained from 1 : 2 serial-to-parallel conversion of a bit sequence bm with bit rate Rb , as mentioned in Section 2.1. The bit rate of b1,n and b2,n is thus half the original (total) bit rate Rb and therefore the same as the resulting symbol rate Rs . The first 4-DPSK transmitter in Figure 5.1(a) has two parallel MZM [49,50]. This transmitter will therefore be labeled 4-DPSK-Parallel transmitter. Light from a CW laser with constant envelope e0 is split into two paths by a cross coupler. The amplitudes and biases of the two electrical drive signals u1 (t) and u2 (t) are chosen such that a binary phase modulation is performed in the upper as well as in the lower MZM. The two modulation processes are the same as in a 2-DPSK-MZM transmitter. The two binary optical √ √ signals in baseband notation are a1 (t) = (e0 / 2) · b1 (t) · exp jϕ1 (t) and a2 (t) = (e0 / 2) · b2 (t) · exp jϕ2 (t) with phase angles ϕn,i ∈ {0, π }, i = 1, 2 at t = nT . For a2 (t), the π /2 phase shift from the first cross coupler and the additional 3π /2 phase shift are already taken p into account. The second cross coupler combines a1 (t) and a2 (t) according to aNRZ = 1/2 a1 (t) + ja2 (t) such that the phase angles of aNRZ (nT ) are ϕn ∈ { π4 , 34π , 54π , 74π }. RZ pulse shaping is done with a subsequent MZM. The 4-DPSK-Parallel constellation diagrams with the transitions are shown in Figures 5.2(a) and (b) for NRZ and RZ, respectively. 51 The second 4-DPSK transmitter in Figure 5.1(b) has an MZM and a PM in series [51]. This transmitter will thus be labeled 4-DPSK-Serial transmitter. Light from a CW laser with constant envelope e0 is first modulated in an MZM in such a way that a1 (t) is a binary phase-modulated signal a1 (t) = e0 b1 (t) exp jϕ1 (t) with phase angles ϕn ∈ {0, π } at t = nT . This modulation step is the same as in the 2-DPSK-MZM transmitter. In the following PM an additional π /2 phase shift is applied for a one-bit in b02,n or the the signal is left unaltered for a zero-bit in b02,n . This second modulation step is similar to the 2-DPSK-PM transmitter, with the only difference being the reduced phase shift of π /2 instead of π . The resulting signal aNRZ (t) has the phase angles ϕn ∈ {0, π2 , π , 32π } at t = nT . RZ pulse shaping is again performed by a subsequent MZM. The 4-DPSK-Serial constellation diagrams with the transitions are shown in Figures 5.3(a) and (b) for NRZ and RZ, respectively. Compared to the 4-DPSK-Parallel constellation diagram we observe that the signal points are rotated by π /4 (or some odd integer multiple of π /4) and thus now lie on the real and imaginary axes. This indicates again that not the phase angles but only the phase differences are important in a differential scheme. The third 4-DPSK transmitter in Figure 5.1(c) with reduced optical hardware effort uses only a single PM [52,53,15] and thus requires a multilevel electrical drive signal. This transmitter will be labeled 4-DPSK-Single transmitter. It is the only transmitter in this thesis, which does not follow the rule from Section 2.3 that the optical modulators are used for binary modulation only. However, the multilevel electrical drive signal for the PM is generated by addition of two binary drive signals u(t) = u1 (t) + 2u2 (t). Both u1 (t) and u2 (t) have the same amplitudes and biases, which are chosen such that the resulting 4-level drive signal u(t) produces an optical 4-level phase modulated signal aNRZ (t) from the light of the CW laser e0 . The phase angles of aNRZ are ϕn ∈ {0, π2 , π , 32π } at t = nT as after the 4-DPSK-Serial transmitter. The subsequent MZM can produce the RZ version, if desired. The NRZ and RZ constellation diagrams in Figures 5.3(a) and (b) show however, that the transitions between the signal points are different: For 4-DPSK-Single they follow a circle around the origin in the complex plane. (It is approximately a circle, as the PM intrinsic extinction ratio is not infinite but 35 dB as stated in Chapter 3.) For the roll-off factor α = 0 in the pulse shapers, all three 4-DPSK transmitters would in principal produce the same optical signals (except for the different differential encoding and a rotation of the signal points), and systems using different transmitters would show the same performance [15]. However, in Chapter 3 we have selected α = 0.5 as an example of real-world nonrectangular pulse shaping and can thus expect performance differences. Figures 5.5(a) and (b), 5.6(a) and (b), and 5.7(a) and (b) present the power spectra of the NRZ and RZ signals from the 4-DPSK-Parallel, 4-DPSK-Serial, and 4-DPSK-Single transmitters at the bit rate Rb = 2Rs = 40 Gbit/s. The power spectra of the NRZ signals in Figures 5.5(a), 5.6(a), and 5.7(a) show a reduced bandwidth compared to 2-DPSK. The 4-DPSK main lobes have only half the width of the 2-DPSK main lobes. Further, the magnitudes of side lobes decrease faster than for 2-DPSK. The NRZ-4-DPSK-Single spectrum shows spikes at multiples of 20 GHz, a small spike at f = 0 can also be observed. The reason is the mean value of the 52 √ Im{aRZ (t)}/ p̄ √ Im{aNRZ (t)}/ p̄ 1 0 −1 1 0 −1 −1 0 √1 Re{aNRZ (t)}/ p̄ −1 0 √ 1 Re{aRZ (t)}/ p̄ (a) (b) 1 √ Im{aRZ (t)}/ p̄ √ Im{aNRZ (t)}/ p̄ Figure 5.2: Constellation diagram with phase transitions for (a) NRZ-4-DPSKParallel and (b) RZ-4-DPSK-Parallel 0 −1 1 0 −1 −1 0 √1 Re{aNRZ (t)}/ p̄ −1 0 √ 1 Re{aRZ (t)}/ p̄ (a) (b) 1 √ Im{aRZ (t)}/ p̄ √ Im{aNRZ (t)}/ p̄ Figure 5.3: Constellation diagram with phase transitions for (a) NRZ-4-DPSKSerial and (b) RZ-4-DPSK-Serial 0 −1 1 0 −1 −1 0 √1 Re{aNRZ (t)}/ p̄ −1 0 √ 1 Re{aRZ (t)}/ p̄ (a) (b) Figure 5.4: Constellation diagram with phase transitions for (a) NRZ-4-DPSKSingle and (b) RZ-4-DPSK-Single 53 20 n o 10 · log10 |ARZ ( f )|2 / p̄ n o 10 · log10 |ANRZ ( f )|2 / p̄ 20 0 0 −20 −20 −40 −40 −120 −80 −40 0 40 Frequency f [GHz] 80 −120 −80 120 (a) −40 0 40 Frequency f [GHz] 80 120 80 120 (b) Figure 5.5: Power spectra for (a) NRZ-4-DPSK-Parallel and (b) RZ-4-DPSKParallel with Rb = 40 Gbit/s 20 n o 10 · log10 |ARZ ( f )|2 / p̄ n o 10 · log10 |ANRZ ( f )|2 / p̄ 20 0 0 −20 −20 −40 −40 −120 −80 −40 0 40 Frequency f [GHz] 80 120 (a) −120 −80 −40 0 40 Frequency f [GHz] (b) Figure 5.6: Power spectra for (a) NRZ-4-DPSK-Serial and (b) RZ-4-DPSKSerial with Rb = 40 Gbit/s 54 20 n o 10 · log10 |ARZ ( f )|2 / p̄ n o 10 · log10 |ANRZ ( f )|2 / p̄ 20 0 0 −20 −20 −40 −40 −120 −80 −40 0 40 Frequency f [GHz] 80 120 (a) −120 −80 −40 0 40 Frequency f [GHz] 80 120 (b) Figure 5.7: Power spectra for (a) NRZ-4-DPSK-Single and (b) RZ-4-DPSKSingle with Rb = 40 Gbit/s complex envelope, as it was already the case for 2-DPSK-PM. The other two NRZ spectra do not show such spikes. The RZ power spectra in Figures 5.5(b), 5.6(b), and 5.7(b) unsurprisingly have wider main lobes than the NRZ spectra because of the short RZ pulse widths as we have also seen for 2-ASK and 2-DPSK. It can be observed that RZ pulse shaping somewhat eliminates the differences between the 4-DPSK signals. The main lobes and the first two side lobes are almost identical for the three signals, except for the 4-DPSK-Single spikes, which are reduced in power though compared to NRZ. The symbol rate reduction obviously not only brings relaxed transmitter component requirements but also an optical signal bandwidth reduction by factor 2 compared to 2-DPSK. 5.2 4-DPSK receiver The 4-DPSK receiver depicted in Figure 5.8(a) is based on the 2-DPSK receiver from Subsection 4.2.2. The received optical signal arx (t) corrupted by AWGN w(t) is first filtered in a 2nd order Gaussian filter, and then split by a cross coupler into an upper and a lower path. Then the two signals are fed into DAF with subsequent balanced detectors. The delay in both DAF is τ = Ts , but the phase shifts in the lower arms are ψ1 = π /4 for DAF 1 and ψ2 = −π /4 for DAF 2. Following the DAF analysis in Subsection 4.2.1, the normalized electrical output signals are plotted vs. the differential phase for these settings in Figure 5.8(b) under the assumption of constant magnitude of the optical input signal. The grid lines in the diagram are chosen such that they mark the set of phase differences ∆ϕ (t) ∈ {0, π2 , π , 32π } and the resulting electrical signal values i1 (t) and i2 (t) at the sampling instants t = t0 + kT at the receiver. Note, that the factor 2 is included in the normalization factor in order to account for the power splitting of cross coupler. 55 - Photodiode τ ĩ1 (t) R Sampl. 3rd ord. i1 (t) - Bessel - & binary b̂1,k LPF decision DAF 1 ψ1 arx (t) + w(t) 2nd ord. - Gauss BPF Photodiode X coupler - - Photodiode τ ĩ2 (t) R Sampl. 3rd ord. i2 (t) - Bessel - & binary b̂2,k LPF decision DAF 2 ψ2 Photodiode Electrical signal 2i j (t)/(R · e20 ), j = 1, 2 (a) 1 h i i1 (t) ∼ cos ∆ϕ (t) + π4 0 −1 0 h i i2 (t) ∼ cos ∆ϕ (t) − π4 1 2 3 Differential phase ∆ϕ (t)/(π /2) (b) 4 Figure 5.8: (a) 4-DPSK receiver and (b) the DAF output signals i1 and i2 with respect to the differential phase ∆ϕ First of all, it can be observed in Figure 5.8(b) that for |ψ1 − ψ2 | = π2 the combinations of i1 and i2 are unique for any ∆ϕ . From this fact it can be concluded that the receiver structure in Figure 5.8(a) represents the general receiver for DPSK with any 2M , M ≥ 2, number of differential phase levels. The actual choice of the phase shifts ψ1 and ψ2 should however be matched to M, what we will now look at for 4-DPSK. i1 and The above choice of ψ1 and ψ2 leads to a situation, in which all combinations of p i2 are not only unique but can only take on values proportional to ± cos (π /4) = ± 1/2, as marked by the grid lines. Together with proper differential encoding at the transmitter following the ideas presented in Subsection 4.2.3, a simple binary threshold decision can be used for obtaining the bit sequence estimates b̂1,k and b̂2,k according to the decision rule b̂ j,k = 0, if ĩ j (t0 + kTs ) ≤ 0, and b̂ j,k = 1, if ĩ j (t0 + kTs ) > 0, ( j = 1, 2). If the phase shifts had been set to ψ1 = π /2 and ψ2 = 0, the condition |ψ1 − ψ2 | = π2 would also have been satisfied. However, the electrical signals i1 and i2 could have taken values proportional to {−1, 0, 1} and more effort would have been needed to estimate the bit sequences. 6 0, Note further, that all combinations of i1 and i2 are unique for any ∆ϕ even if |ψ1 − ψ2 | = 56 but then the number of electrical signal values is not the same for i1 and i2 and may further be not as low as possible. In general, the number of elements NI in a set of values I I = {I0I , I1I , . . .} and NII in a set of values I II = {I0II , I1II , . . .}, which are used to distinguish 2M combinations {i1 , i2 } with i1 ∈ I I and i2 ∈ I II must satisfy NI · NII ≥ 2M . (5.1) The actual 4-DPSK receiver with the phase shifts ψ1 = π /4 and ψ2 = −π /4 leads to I I = p p I II = {− 1/2, 1/2} and thus NI = NII = 2 and NI · NII = 4 = 2M . This is the minimum NI · NII , which satisfies the condition (5.1). The other 4-DPSK receiver example with ψ1 = π /2 and ψ2 = 0 leads to I I = {−1, 0, 1} and I II = {I0II , 1} with the metasymbol I0II that combines {−1, 0}. Therefore, NI = 3 and NII = 2 and NI · NII = 6 ≥ 2M . The closer NI · NII comes to 2M , the simpler the bit sequence estimation rules at the receiver. For the actual 4-DPSK, bit sequence estimation reduces to two binary threshold decision (one for each bit sequence), as explained above. It is the most simple 4-DPSK receiver with direct detection possible. The introduced methodology will be used for finding simple 8DPSK receivers in Chapter 8, but could be easily extended to even higher order multilevel modulation formats. The bit error probability calculation for 4-DPSK relies on the 2-DPSK extension of the method explained in previous chapters. With the assumption that decision errors in the upper and lower paths of the receiver are independent, decision errors in both paths are calculated as for 2-DPSK. The total bit error probability then is the mean of both individual error probabilities. 5.3 4-DPSK performance The first step in the performance evaluation of the three 4-DPSK variants is the receiver bandwidth optimization. The contour diagrams in Figures 5.9(a) and (b), 5.10(a) and (b), and 5.11(a) and (b) show the results in the now well-known form of required OSNR for BEP = 10−9 vs. the electrical 3-dB cut-off frequency f3 dB,el and the optical 3-dB bandwidth ∆ f3 dB,opt . The optimum bandwidth pairs are marked by × and listed in Table 5.1 together with the required OSNR values. Electrical eye diagrams in front of the sampling & decision device for the optimal bandwidths pairs are shown in Figures 5.12(a) and (b), 5.13(a) and (b), and 5.14(a) and (b). Let us now look at NRZ pulse shaping in Figures 5.9(a), 5.10(a), and 5.11(a). With 20.19 dB, 4-DPSK-Parallel requires the lowest OSNR of the three variants: 1.07 dB less than 4-DPSKSerial and 1.84 dB less than 4-DPSK-Single. Furthermore, the minimum required OSNR for 4-DPSK-Parallel lies at an optical receiver bandwidth of only ∆ f3 dB,opt = 1.2 · Rs compared 57 Table 5.1: Optimized optical and electrical receiver filter bandwidths, required OSNR for BEP = 10−9 , and chromatic dispersion tolerances ∆rD for 1-dB and 2-dB OSNR penalties for 4-DPSK 4-DPSK-Parallel 4-DPSK-Serial 4-DPSK-Single NRZ RZ NRZ RZ NRZ RZ ∆ f3 dB,opt /Rs 1.20 2.10 2.15 2.20 2.10 2.15 f3 dB,el /Rs 1.05 0.60 0.65 0.60 1.05 0.85 Req. OSNR [dB] 20.19 20.02 21.26 20.07 22.03 20.17 ps ∆rD,1 dB [ nm ] ps ∆rD,2 dB [ nm ] 206 249 160 237 113 203 310 321 234 305 155 262 Format to 2.15 · Rs for 4-DPSK-Serial and 2.1 · Rs for 4-DPSK-Single. If the electrical filter cutoff frequency is selected properly, the 0.5-dB tolerance region with respect to the minimum required OSNR reaches down to approx. ∆ f3 dB,opt = 1.05 · Rs for 4-DPSK-Parallel, but only down to 1.3·Rs for 4-DPSK-Serial and 1.8·Rs and 4-DPSK-Single. Thus, in a WDM system, the closest channel spacing could be achieved with the 4-DPSK-Parallel system. Next, we study RZ pulse shaping in Figures 5.9(b), 5.10(b), and 5.11(b). The results are qualitatively the same as for NRZ. 4-DPSK-Parallel requires 20.02 dB OSNR for BEP = 10−9 , but now the two other 4-DPSK variants come very close to this value: 4-DPSK-Serial needs only 0.05 dB and 4-DPSK-Single needs only 0.15 dB more OSNR. RZ pulse shaping significantly enhanced the receiver sensitivity for the two 4-DPSK variants, which performed rather poorly for NRZ. As for 2-DPSK, this can be attributed to the fact that the RZ pulse shaper passes only the symbol slot centers, where the phase has almost reached its nominal value. The rest of the symbol interval, where all the phase transitions take place, is taken out. The optimal optical and electrical receiver bandwidths are similar for all three variants, only 4-DPSK-Single requires a somewhat higher electrical filter cut-off frequency. If we look at the 0.5-dB tolerance region around the optimum, we see that, if the electrical cut-off frequency is chosen properly, the optical filter bandwidth can be as low as 1.7 · Rs for 4DPSK-Single, as low as 1.5 · Rs for 4-DPSK-Serial, and even as low as 1.2 · Rs for 4-DPSKParallel. Thus again, in a WDM system with RZ-4-DPSK, the closest channel spacing could be achieved with the 4-DPSK-Parallel system. Note again, that the symbol rate of 4-DPSK is only half the symbol rate of 2-DPSK or 2ASK at the same bit rate, so that 4-DPSK in general allows lower absolute receiver filter bandwidths than the two binary formats. The 4-DPSK eye diagrams in Figures 5.12(a) and (b), 5.13(a) and (b), and 5.14(a) and (b) are all binary with negative values for zero-bits and positive values for one-bits as explained in the previous Section 5.2. The optimum decision thresholds are close to zero. 58 3.4 21 20.5 .5 23 3.2 21.5 3.0 23 2.8 20.2 2.8 22 21.5 21 3.0 22.5 21 3.2 20.5 20.25 3.4 22 21 .5 21 2.0 21 2.2 1.8 1.6 20.5 2.4 2.2 2.0 1.8 20.25 1.4 20.5 1.2 20.5 .5 2212 1.0 0.4 25 21 20.5 5 222223. 24 1.2 20. 1.6 21 1.4 2.6 20.25 20.5 Optical bandwidth ∆ f3 dB,opt /Rs 2.4 21.5 Optical bandwidth ∆ f3 dB,opt /Rs 5 2.6 21 .5 20.5 21 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs 1.0 0.4 (a) 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (b) Figure 5.9: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-4-DPSK-Parallel and (b) RZ-4-DPSK-Parallel. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. 59 3.4 22 22 21.5 2322.5 2.2 22 2.0 20.25 2.2 20.25 2.0 1.8 1.6 1.4 20.5 2.4 20 20.25 .25 21 .5 21 1.6 2.6 20.5 1.8 25 20. 2.4 20.5 21.5 2.6 2.8 21 3.0 22 2.8 Optical bandwidth ∆ f3 dB,opt /Rs 5 3.2 23 3.0 20.5 23. 22.5 21.5 2322.5 3.2 Optical bandwidth ∆ f3 dB,opt /Rs 3.4 20.5 1.4 1.2 22 23.5 23 25 1.0 0.4 22.5 21.5 22 22 24 1.2 1.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (a) 21 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (b) Figure 5.10: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-4-DPSK-Serial and (b) RZ-4-DPSK-Serial. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. 60 3.4 5 5 .2 22 .25 22 5 22. 1.8 1.6 29 1.2 31 1.0 0.4 33 5 20.5 5 2.2 2.0 20. 25 20.2 5 1.6 1.2 33 33 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs 21 1.0 0.4 (a) 21.5 22 22.5 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (b) Figure 5.11: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-4-DPSK-Single and (b) RZ-4-DPSK-Single. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. 61 25 20.5 1.4 27 29 31 20. 1.8 22.5 23 1.4 20. 20.2 22 27 2.4 21.5 5 23 23.5 24 24.5 25 2.6 20.5 21 Optical bandwidth ∆ f3 dB,opt /Rs .2 22 2.0 22. 24 Optical bandwidth ∆ f3 dB,opt /Rs 22.25 23 22.5 23 23.5 .5 2425 2.2 2 0 .5 24 2.4 21 20.25 3.0 2.8 2.6 5 20.2 24 23.5 22.5 3.0 2.8 3.2 23 22.5 23 23.5 24.5 25 3.2 3.4 Electrical signal 2ĩ1 (t)/(R · p̄) Electrical signal 2ĩ1 (t)/(R · p̄) 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 (a) 0 Time t/Ts 0.5 (b) Electrical signal 2ĩ1 (t)/(R · p̄) Electrical signal 2ĩ1 (t)/(R · p̄) Figure 5.12: Eye diagrams after optical and electrical filtering for (a) NRZ-4DPSK-Parallel and (b) RZ-4-DPSK-Parallel 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 (a) 0 Time t/Ts 0.5 (b) Figure 5.13: Eye diagrams after optical and electrical filtering for (a) NRZ-4DPSK-Serial and (b) RZ-4-DPSK-Serial The electrical signals are now normalized to R · p̄/2, where the factor 1/2 accounts for the cross coupler at the receiver, which directs half the optical power to the upper and half the optical power to the lower receiver path. Only the eye diagrams in the upper receiver paths are shown, as they look the same as in the lower paths. The receiver eye diagrams of 4DPSK-Parallel and 4-DPSK-Serial in Figures 5.12 and 5.13 are quite similar for both NRZ and RZ. The NRZ-4-DPSK-Single eye diagram in Figure 5.14(a) shows a reduced horizontal opening compared to the two other NRZ eye diagrams. In an actual receiver, this will lead to more stringent requirements for the sampling jitter in the sampling & decision device. The RZ-4-DPSK-Single eye diagram in Figure 5.14(b) looks more RZ-like than the two other RZ eye diagrams, for which the lower electrical filter cut-off frequencies have led to a more NRZ-like eye diagram. 4-DPSK is compared to 2-DPSK and 2-ASK in Figures 5.15(a) and (b) in terms of bit error probability vs. OSNR at Rb = 40 Gbit/s and the optimal receiver filter bandwidths. For 62 Electrical signal 2ĩ1 (t)/(R · p̄) Electrical signal 2ĩ1 (t)/(R · p̄) 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 (a) 0 Time t/Ts 0.5 (b) Figure 5.14: Eye diagrams after optical and electrical filtering for (a) NRZ-4DPSK-Single and (b) RZ-4-DPSK-Single NRZ pulse shaping in Figure 5.15(a), the 4-DPSK-Parallel and 4-DPSK-Serial curves lie above the 2-DPSK curve, but below the 2-ASK curve. 4-DPSK-Single has a slightly worse performance than 2-ASK. For a bit error probability of 10−9 , the best 4-DPSK variant, 4DPSK-Parallel, requires 1.82 dB more OSNR than 2-DPSK, but 1.57 dB less OSNR than 2-ASK. For RZ pulse shaping in Figure 5.15(b), the three 4-DPSK curves almost lie on top of each other. This had of course to be expected from the previous results regarding the filter bandwidth optimization. The inset in Figure 5.15(b) then shows that the 4-DPSK-Parallel curve is the lowest and the 4-DPSK-Single curve is the highest curve. All three 4-DPSK curves lie well in between the 2-DPSK and the 2-ASK curves. If we again take 4-DPSKParallel, it requires 1.88 dB more OSNR than 2-DPSK but 1.09 dB less OSNR than 2-ASK for a bit error probability of 10−9 . The last two diagrams in Figures 5.16(a) and (b) compare the dispersion tolerance of 4-DPSK with 2-DPSK and 2-ASK. For NRZ pulse shaping depicted in Figure 5.16(a), we have qualitatively the same situation as for the other performance criteria, if we compare the three 4-DPSK variants. 4-DPSK-Parallel has the greatest dispersion tolerance manifesting itself in the widest curve in the diagram, and the dispersion tolerance is reduced for 4-DPSK-Serial and even more for 4-DPSK-Single. The curve widths at 1-dB and 2-dB OSNR penalty are listed in Table 5.1 as ∆rD,x dB (x = 1 or 2). At 2-dB OSNR penalty, for example, 4-DPSKParallel tolerates by factor 1.3 more accumulated dispersion than 4-DPSK-Serial and by factor 2.0 more than 4-DPSK-Single. Compared to 2-DPSK and 2-ASK, 4-DPSK-Parallel tolerates by factor 2.1 and 3.2 more accumulated dispersion, respectively. The asymmetry of the 4-DPSK curves at 2-dB OSNR penalty is below 1 ps/nm. The results for RZ pulse shaping are given in Figure 5.16(b). All three 4-DPSK curves start out at approximately the same OSNR value at rD = 0 ps/nm. There they have a rather flat minimum so that the tolerable accumulated dispersion values at the 1-dB and 2-dB OSNR penalties are significantly higher than for NRZ. This holds especially for the 4-DPSK-Single case, where RZ tolerates by factor 1.7 more accumulated dispersion than NRZ for a 2-dB OSNR penalty. 63 Bit error probability NRZ−2−ASK NRZ−2−DPSK−MZM NRZ−4−DPSK−Serial NRZ−4−DPSK−Parallel NRZ−4−DPSK−Single 10−3 10−6 10−9 10−12 12 15 18 21 OSNR [dB] 24 27 30 Bit error probability (a) RZ−2−ASK RZ−2−DPSK−MZM RZ−4−DPSK−Serial RZ−4−DPSK−Parallel RZ−4−DPSK−Single 10−3 10−6 - 10−8 test 10−9 10−12 12 10−9 19.5 15 18 21 OSNR [dB] 24 20 27 20.5 30 (b) Figure 5.15: Bit error probability vs. OSNR for (a) NRZ-4-DPSK (b) RZ-4DPSK at Rb = 40 Gbit/s 64 Required OSNR [dB] for BEP = 10−9 33 30 27 24 NRZ−2−ASK NRZ−2−DPSK−MZM NRZ−4−DPSK−Serial NRZ−4−DPSK−Parallel NRZ−4−DPSK−Single 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (a) Required OSNR [dB] for BEP = 10−9 33 30 27 24 RZ−2−ASK RZ−2−DPSK−MZM RZ−4−DPSK−Serial RZ−4−DPSK−Parallel RZ−4−DPSK−Single 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (b) Figure 5.16: Required OSNR for BEP = 10−9 vs. accumulated dispersion rD for (a) NRZ-4-DPSK and (b) RZ-4-DPSK at Rb = 40 Gbit/s 65 Only for very high or low accumulated dispersion values, NRZ requires less OSNR than RZ. However, the OSNR penalties are already extremely high in this regions. Further, it can be observed that the three 4-DPSK curves keep closer together as the NRZ curves with increasing or decreasing accumulated dispersion. However, a clear advantage of 4-DPSK-Parallel over 4-DPSK-Serial and 4-DPSK-Single remains. At 2-dB OSNR penalty, for example, 4DPSK-Parallel still tolerates by factor 1.1 more accumulated dispersion than 4-DPSK-Serial and by factor 1.2 more than 4-DPSK-Single. The advantage in tolerable accumulated dispersion of 4-DPSK-Parallel over 2-DPSK and 2-ASK is by a factor of 3.3 and 3.9, respectively, in the RZ case. In addition to the above presented performance results, it has been shown [15] that for nonlinear fiber transmission described in Section 2.4.2, 4-DPSK-Single suffers the most from nonlinear signal distortions, whereas 4-DPSK-Parallel shows the greatest tolerance to these nonlinear effects. Also, the differences between the three 4-DPSK variants are less pronounced for RZ than for NRZ pulse shaping, which fits in with the previous results. In conclusion of the 4-DPSK performance evaluation, it was found that the 4-DPSK system with a single PM, which is attractive because of a reduced optical hardware effort, exhibits the worst performance regarding receiver sensitivity, tolerance to optical filtering, and dispersion and nonlinear tolerance, whereas the 4-DPSK system with two parallel MZM shows the best performance. The system with an MZM and PM in series always lies in between the other two systems with its results. The rather large differences for NRZ pulse shaping can be somewhat reduced by RZ pulse shaping. In comparison to the previously studied binary modulation formats 2-ASK and 2-DPSK, 4-DPSK lies between 2-DPSK and 2-ASK in terms of receiver sensitivity. It has lower absolute filter bandwidths than the binary formats because of the reduced signal bandwidth. This reduced signal bandwidth leads to a significantly improved dispersion tolerance of 4-DPSK. In the following chapters, 4-DPSK-Parallel will be used for comparisons to the other modulation formats, unless otherwise stated, because it is the best of the three variants. The term 4-DPSK will then be synonymously used for 4-DPSK-Parallel for shortness of notation. 66 Chapter 6 4-level Combined Amplitude- and Differential Phase-Shift Keying (4-ASK-DPSK) Another possibility for a 4-level optical modulation format with direct detection is the combination of 2-ASK and 2-DPSK. This combination is labeled 4-ASK-DPSK in this thesis. In 4-ASK-DPSK one bit of information is contained in the amplitude of the transmitted symbols, and one bit of information is contained in the phase differences of two consecutive symbols. 4-ASK-DPSK has first been proposed as a modulation format in [54] and then experimentally investigated in [55]. Nonlinear performance has been studied in [56] and the use of forward-error correction in 4-ASK-DPSK compared to 4-DPSK in [57]. Just recently, 4-ASK-DPSK has been demonstrated for high-speed single-channel transmission [58]. Besides the use as a 4-level modulation format, 4-ASK-DPSK is also of interest for optical packet labeling in packet-switched optical networks [59, 60]. 4-ASK-DPSK has especially to be compared to 4-DPSK as those two modulation formats are natural competitors. Thus, this chapter discusses the 4-ASK-DPSK transmitter and receiver and the 4-ASK-DPSK performance with a special focus on the comparison to 4-DPSK. Binary modulation formats are also included for completeness. 6.1 4-ASK-DPSK transmitter and receiver The task of the 4-ASK-DPSK transmitter in Figure 6.1 is to generate an optical signal, which is the combination of a 2-ASK signal and a 2-DPSK signal. In contrast to conventional 2ASK in Chapter 3, the ASK part requires two nonzero amplitudes at the sampling instants, because it would not make sense in the DPSK part to associate an optical phase with a zero a amplitude. In terms of the transmitted signal, the symbol alphabet is made up of NRZ,n e0 = aNRZ (nTs ) e0 ∈ {bejπ , aejπ , aej0 , bej0 } with the two positive amplitude values b > a and phase 67 Pulse b1,n - shaper b02,n Pulse b2,n - Diff. - shaper encoder u1 (t) ? 0 Laser eMZM CW u2 (t) a1 (t) aNRZ (t) - - PM MZM aRZ (t) - 6uRZ,Gauss (t) 6 Figure 6.1: 4-ASK-DPSK transmitter with an MZM and a PM in series angles 0 and π . With L = 4 available symbols, 4-ASK-DPSK transmits 2 bit/symbol and thus the symbol rate Rs equals half the bit rate Rb , which is the same as for 4-DPSK. A first comparison of the 4-ASK-DPSK transmitter in Figure 6.1 reveals a great similarity to the 4-DPSK-Serial transmitter in Figure 5.1(b) of Section 5.1. Both transmitters consist of an MZM and a PM modulator in series. Here, the electrical drive signal u1 (t) for the MZM is generated in a pulse shaper from the bit sequence b1,n . Its amplitude and bias are adjusted such that the MZM performs a binary amplitude modulation of the optical CW signal e0 resulting in the optical amplitude e0 · b for a one-bit b1,n or in e0 · a for a zero-bit b1,n . Next comes the phase modulation in the PM, which is the same as for 2-DPSK-PM. The bit sequence b2,n is differentially encoded in a standard 2-DPSK encoder, and the encoded version b02,n is used to generate the electrical drive signal u2 (t). The amplitude and bias of u2 (t) are chosen such that the PM induces a π -phase shift for a one-bit b02,n or leaves the signal unaltered for a zero bit b02,n . RZ pulse shaping is achieved in a subsequent MZM. In contrast to the 4-DPSK-Serial transmitter, the optical signal a1 (t) after the first MZM is a 2-ASK signal, whereas it has been a 2-DPSK signal there. If the same MZM is used, the amplitude of u1 (t) can thus be by factor 2 or more lower in the 4-ASK-DPSK transmitter than in the 4-DPSK-Serial transmitter. However, the PM in the 4-DPSK-Serial transmitter induces only a π /2-phase shift for a one-bit opposed to a π -phase shift here. Therefore, the amplitude of u2 (t) needs to be by factor 2 larger in the 4-ASK-DPSK transmitter, if the same PM is used. But in principal, the structure of both transmitters is the same. Figures 6.2(a) and (b) show the 4-ASK-DPSK constellation diagrams with the four signal points and transitions for NRZ and RZ pulse shaping for one specific choice of the amplitude ratio b/a = 3. The NRZ signal exhibits significant chirp because of the PM, which is strongly reduced by RZ pulse shaping. If we now take a look at the 40-Gbit/s NRZ and RZ power spectra in Figures 6.3(a) and (b) we further see the typical spikes in the NRZ spectrum at multiples of 20 GHz and a small one at the center because the NRZ signal has a nonzero mean, not because of the 2-ASK part but because of the 2-DPSK phase transitions in the complex half-plane with Im aNRZ (t) > 0. These spikes are therefore reduced in the RZ spectrum. The widths of the NRZ and RZ main and side lobes are comparable to those of the 4-DPSK-Serial power spectra in Figures 5.6(a) and (b). So in terms of optical transmit signal bandwidth there is no clear advantage for one of the 4-level formats. Figure 6.4 presents the block diagram of the 4-ASK-DPSK receiver. The received optical signal arx (t) corrupted by optical AWGN is first filtered in an optical 2nd order Gaussian 68 0 −b −a a 1 √ Im{aRZ (t)}/ p̄ √ Im{aNRZ (t)}/ p̄ 1 b −1 0 −b −a a b −1 −1 0 √ 1 Re{aRZ (t)}/ p̄ −1 0 √1 Re{aNRZ (t)}/ p̄ (a) (b) Figure 6.2: Constellation diagram with phase transitions for (a) NRZ-4-ASKDPSK and (b) RZ-4-ASK-DPSK with amplitude ratio b/a = 3 20 n o 10 · log10 |ANRZ ( f )|2 / p̄ n o 10 · log10 |ANRZ ( f )|2 / p̄ 20 0 0 −20 −20 −40 −40 −120 −80 −40 0 40 Frequency f [GHz] 80 120 (a) −120 −80 −40 0 40 Frequency f [GHz] (b) Figure 6.3: Power spectra for (a) NRZ-4-ASK-DPSK and (b) RZ-4-ASK-DPSK with Rb = 40 Gbit/s and amplitude ratio b/a = 3 69 80 120 ĩ1 (t) 2-ASK path X coupler arx (t) + w(t) 2nd ord. - Gauss BPF Sampl. 3rd ord. i1 (t) - Bessel - & binary b̂1,k LPF decision - Photodiode Photodiode τ ψ Photodiode 2-DPSK path ĩ2 (t) R Sampl. 3rd ord. i2 (t) - Bessel - & binary b̂2,k LPF decision Figure 6.4: 4-ASK-DPSK receiver with 2-ASK and 2-DPSK path bandpass filter and then split into an 2-ASK and a 2-DPSK path by a cross coupler. The 2-ASK path consists of a standard 2-ASK receiver as in Section 3.1, whereas the 2-DPSK path directly corresponds to the 2-DPSK receiver with the same phase shift ψ as in Subsection 4.2.2. The 4-ASK-DPSK receiver is simpler compared to the 4-DPSK receiver, as it requires only one optical delay & add filter instead of two, and three photodiodes instead of four. 6.2 4-ASK-DPSK optimum signal point amplitude ratio and performance A parameter with major influence on the 4-ASK-DPSK performance is the amplitude ratio b/a of the signal points. This statement becomes clear, if the impact on the eye openings in the 2-ASK and 2-DPSK receiver paths is examined in more detail. If we make the assumption of a noise-free received signal that is the same as the transmitted signal, the received s) symbols are ae0k = arx (t0e+kT ∈ {bejπ , aejπ , aej0 , bej0 }. Let us further assume that there are 0 no receiver filters. Then, the sampled electrical signal i1,k = ĩ1 (t0 + kTs ) in the 2-ASK path can take on 2 values. According to (4.8), the sampled electrical signal i2,k = ĩ2 (t0 + kTs ) in the 2-DPSK path can take on 6 values depending on |ak |, |ak−1 |, and ∆ϕk . These values are listed in Table 6.1 together with the electrical signal values in the 2-ASK path and the corresponding estimated bit sequences. Because of the cross coupler, an additional factor of 1/2 is considered in the eye openings of both paths. First of all, Table 6.1 reconfirms that standard 2-ASK and 2-DPSK receivers can be used: In the noise-free case, b̂1,k = 0 if i1,k = Re20 a2 /2 and b̂1,k = 1 if i1,k = Re20 b2 /2, and b̂2,k = 0 if i1,k < 0 and b̂2,k = 1 if i2,k > 0. We can further read the electrical eye openings as ∆i1 = Re20 (b2 − a2 )/2 in the 2-ASK path and ∆i2 = Re20 a2 in the 2-DPSK path. If k1 = a2 + b2 ∼ p̄ remains constant, the eye openings can be rewritten as 70 Table 6.1: Sampled electrical signals i1,k and i2,k and estimated bits b̂1,k and b̂2,k in the 2-ASK and 2-DPSK paths with respect to the received symbols ak ak−1 e0 | ∆ ϕk 2i1,k Re20 2i2,k Re20 b̂1,k b̂2,k a a 0 a2 a2 0 1 a b 0 a2 ab 0 1 0 b2 ab 1 1 b2 1 1 −a2 0 0 −ab 0 0 −ab 1 0 −b2 1 0 | ae0k | b | a b b 0 b2 a a π a2 a b π a2 b a π b2 b π b2 b ∆i1 = Re20 (b/a)2 − 1 · k1 · 2 (b/a)2 + 1 (6.1) and ∆i2 = Re20 · k1 . (b/a)2 + 1 (6.2) The amplitude ratio b/a can take on values between 1 and ∞. For b/a = 1, the 2-ASK eye opening becomes ∆i1 = 0 and the 2-DPSK eye opening ∆i2 = Re20 k1 /2, thus this case represents pure 2-DPSK. For b/a → ∞, the 2-ASK eye opening is ∆i1 → Re20 k1 /2 and the 2-DPSK eye opening ∆i2 → 0, thus this case represents pure 2-ASK. Obviously, b/a must lie somewhere in between in order to obtain the optimum 4-ASK-DPSK performance. Figures 6.5(a) and (b) show the bit error probabilities vs. the amplitude ratio b/a for 40Gbit/s NRZ- and RZ-4-ASK-DPSK at OSNR = 23 dB. For each value of b/a the receiver filter bandwidths have been optimized for lowest bit error probability. In both cases, b/a = 3 leads to the lowest total bit error probability. As already depicted in Figures 6.2(a) and (b), all neighboring signal points have equal distances for this optimum amplitude ratio. If b/a < 3, errors in the 2-ASK path dominate, whereas there are more errors in the 2DPSK path for b/a > 3, which complies with the decrease or increase of the respective eye openings according to (6.1) and (6.2). For completeness and as a reference, the optimum filter bandwidths for the considered amplitude ratios are given in Table 6.2. Although the electrical and optical receiver filter bandwidths have been optimized with respect to minimum bit error probability at a fixed OSNR in the investigation of the optimum amplitude ratio, contour diagrams showing the required OSNR for a bit error probability of 71 Bit error probability 10−6 10−9 10−12 Total 2−ASK path 2−DPSK path 10−15 10−18 2.0 2.5 3.0 4.0 3.5 Amplitude ratio b/a (a) 4.5 5.0 3.0 4.0 3.5 Amplitude ratio b/a (b) 4.5 5.0 Total 2−ASK path 2−DPSK path Bit error probability 10−6 10−9 10−12 10−15 10−18 2.0 2.5 Figure 6.5: Bit error probability vs. amplitude ratio b/a for (a) NRZ-4-ASKDPSK and (b) RZ-4-ASK-DPSK at Rb = 40 Gbit/s at OSNR = 23 dB 72 Table 6.2: Optimized optical and electrical receiver filter bandwidths for various 4-ASK-DPSK amplitude ratios b/a Amplitude ratio b/a 2.0 2.5 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.2 4.4 4.6 4.8 5.0 NRZ-4-ASK-DPSK ∆ f3 dB,opt /Rs f3 dB,el /Rs 1.60 0.65 1.65 0.65 1.65 0.65 1.70 0.65 1.70 0.65 1.60 0.70 1.45 0.85 1.45 0.85 1.45 0.85 1.45 0.85 1.45 0.85 1.45 0.85 1.45 0.85 1.40 0.85 1.40 0.85 1.40 0.85 1.40 0.85 1.40 0.85 1.40 0.85 1.35 0.85 1.35 0.85 73 RZ-4-ASK-DPSK ∆ f3 dB,opt /Rs f3 dB,el /Rs 2.60 0.55 2.60 0.55 2.60 0.55 2.60 0.55 2.55 0.55 2.30 0.60 2.20 0.60 2.20 0.60 2.20 0.60 2.20 0.60 2.20 0.60 2.20 0.60 2.20 0.60 2.20 0.60 2.15 0.60 2.15 0.60 2.15 0.60 2.15 0.60 2.15 0.60 2.15 0.65 2.15 0.65 Optical bandwidth ∆ f3 dB,opt /Rs 23.5 2524 22 23 22.5 2.0 22 1.8 23 22.5 1.6 .5 23 23.5 1.4 26 0.6 0.8 1.0 1.2 1.4 Electrical bandwidth f3 dB,el /Rs 24 1.6 1.0 0.4 (a) 24 23 1.2 23.5 3.5 2426 8 2 2 24 5 2 1.0 0.4 2.2 24 23 23.5 1.2 2830 1.4 2.4 2423.5 23.25 1.6 23.25 1.8 22.5 2.6 22 25 24 2 3 .5 2.4 26 Optical bandwidth ∆ f3 dB,opt /Rs 2.8 2.6 2.0 23.524 3.0 2.8 2.2 22.5 25 24 26 3.2 23 3.0 25 3.2 3.4 24 3.4 24 23.5 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (b) Figure 6.6: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-4-ASK-DPSK and (b) RZ-4-ASK-DPSK. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. 10−9 have also been created for better comparability to the other modulation formats. These contour diagrams represent 4-ASK-DPSK with amplitude ratio b/a = 3 and are given in Figures 6.6(a) and (b) for NRZ and RZ pulse shaping. As usual the optimum bandwidth pairs are marked by × and the 0.5-dB tolerance regions with respect to the optimum are indicated by dashed lines. The optimum bandwidth pairs are listed in Table 6.3 and the electrical eye diagrams in the 2-ASK and 2-DPSK paths for NRZ and RZ pulse shaping are shown in Figures 6.7(a) through (d). In the NRZ case depicted in Figure 6.6(a), we can achieve a required OSNR of 23.11 dB at the optical bandwidth ∆ f3 dB,opt = 1.6 · Rs . For a properly selected electrical filter 3-dB cutoff frequency f3 dB,el , the 0.5-dB tolerance region reaches down to ∆ f3 dB,opt = 1.15 · Rs . This is not as low as for 4-DPSK-Parallel (1.05 · Rs ) but lower than for 4-DPSK-Serial (1.3 · Rs ). However, the absolute required OSNR value for 4-ASK-DPSK is significantly higher than 74 Table 6.3: Optimized optical and electrical receiver filter bandwidths, required OSNR for BEP = 10−9 , and chromatic dispersion tolerances ∆rD for 1-dB and 2-dB OSNR penalties for 4-ASK-DPSK Format 4-ASK-DPSK (b/a = 3) 4-ASK-DPSK (opt. b/a) NRZ NRZ RZ — — RZ 2.30 ∆ f3 dB,opt /Rs 1.60 f3 dB,el /Rs 0.70 0.60 — — Req. OSNR [dB] 23.11 21.89 — — ps ] ∆rD,1 dB [ nm 84 178 111 180 ps ] ∆rD,2 dB [ nm 132 237 180 243 (2.25) those of all considered 4-DPSK variants. Nevertheless, the possibility of narrow optical filtering of 4-ASK-DPSK enables close channel spacing in WDM systems. Next, in Figure 6.6(b) for RZ pulse shaping, we see that the lowest required OSNR of 21.89 dB is achieved for the optical bandwidth ∆ f3 dB,opt = 2.25 · Rs . If the electrical cut-off frequency is chosen properly, the optical bandwidth can be as low as 1.65 · Rs for a tolerance of 0.5 dB. This is significantly more than for 4-DPSK-Parallel (1.2 · Rs ) but only moderately more than for 4-DPSK-Serial (1.5 · Rs ). WDM channels could not be spaced as closely as for 4-DPSK. It should further be noted that the required OSNR is by more than 1 dB higher than for 4-DPSK. For RZ pulse shaping, the bandwidth optimization with respect to the bit error probability, used for finding the optimum amplitude ratio, led to the same electrical cut-off frequency f3 dB,el = 0.6 · Rs as for the optimization using required OSNR values, but to a slightly different optical bandwidth ∆ f3 dB,opt = 2.3 · Rs . In the selected resolution with two places beyond the decimal point, this bandwidth pair has the same required OSNR. Only after increasing the resolution to 4 places beyond the decimal point, it turns out that the required OSNR is 0.0004 dB higher. The difference between both bandwidth optimization runs must be attributed to numerical effects in the evaluation of the moment-generating function in conjunction with the rather flat minimum. In the following, the value pair ∆ f3 dB,opt = 2.3 · Rs and f3 dB,el = 0.6 · Rs will be used. After amplitude ratio and receiver bandwidth optimization, let us now take a look at the eye diagrams in Figure 6.7. Figures 6.7(a) and (c) show the 2-ASK eye diagrams, and Figures 6.7(b) and (d) the 2-DPSK eye diagrams for NRZ and RZ pulse shaping. As the receiver filters are now included, the levels do only approximately match those according to (6.1) and (6.2). The two-level nature of the 2-ASK eye diagrams and the six-level nature of the 2DPSK eye diagrams, however, can be clearly observed. Of course, the six electrical levels 75 Electrical signal 2ĩ2 (t)/(R · p̄) Electrical signal 2ĩ1 (t)/(R · p̄) 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 0.5 (b) Electrical signal 2ĩ1 (t)/(R · p̄) Electrical signal 2ĩ2 (t)/(R · p̄) (a) 0 Time t/Ts 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 (c) 0 Time t/Ts 0.5 (d) Figure 6.7: Eye diagrams after optical and electrical filtering for (a) the 2-ASK path of NRZ-4-ASK-DPSK and (b) the 2-DPSK path of NRZ-4-ASK-DPSK, and (c) the 2-ASK path of RZ-4-ASK-DPSK and (d) the 2-DPSK path of RZ-4-ASKDPSK 76 represent only two logical levels distinguished by the decision thresholds at approximately zero. Although the innermost eye openings in the 2-DPSK eye diagrams are much smaller than the 2-ASK eye openings, the fact that the error probabilities in both paths are still the same, can be attributed to the general sensitivity advantage of 2-DPSK over 2-ASK found in Chapter 4. Figure 6.8 gives 4-ASK-DPSK bit error probabilities vs. OSNR compared to 2-ASK, 2DPSK, and 4-DPSK at Rb = 40 Gbit/s. For NRZ pulse shaping in Figure 6.8(a) as well as for RZ pulse shaping in Figure 6.8(b), 4-ASK-DPSK has the highest bit error probabilities for a given OSNR values of all considered modulation formats. Compared to 4-DPSK-Parallel, 4-ASK-DPSK needs 2.92 dB more OSNR for NRZ pulse shaping and 1.87 dB more OSNR for RZ pulse shaping. The disadvantage of 4-ASK-DPSK is not as large but still significant, if 4-DPSK-Serial is taken for comparison. Although the actual OSNR difference depends on many system parameters, the advantage of 4-DPSK can be attributed to a larger signal point distance for a given average optical power, as can be seen in the comparison of the 4-DPSK constellation diagrams in Figures 5.2 through 5.4 on the one hand, and the 4-ASK-DPSK constellation diagrams in Figure 6.2 on the other hand. Now, the dispersion tolerance of 4-ASK-DPSK is studied. Figures 6.9(a) and (b) show required OSNR for BEP = 10−9 vs. accumulated dispersion rD for NRZ and RZ pulse shaping. In each case, the curves for three amplitude ratios are given: The optimal value b/a = 3, a value below the optimum b/a = 2, and a value above the optimum b/a = 4. 4-DPSK curves are included for comparison. In the NRZ plot in Figure 6.9(a) it can first be observed, that b/a = 3 requires the lowest OSNR for rD = 0 ps/nm. This is of course to be expected, as it was the objective of the amplitude ratio optimization. Then, the curve for b/a = 2 has its minimum at a higher OSNR and the curve exhibits an increased slope. Finally, the curve for b/a = 4 has also a higher minimum, but a decreased slope, which leads to two points of intersection with the curve for b/a = 3. This decreased slope is because the 2-ASK path benefits from the increased amplitude ratio, whereas the dispersion tolerance of 2-DPSK is generally higher. The points of intersection suggest that for fiber transmission without exact dispersion compensation the lowest required OSNR may be achieved with an amplitude ratio b/a ≥ 3 selected according to the amount of accumulated dispersion rd . For RZ pulse shaping in Figure 6.9(b), such points of intersection cannot be observed for the considered amplitude ratios, and the curve for b/a = 3 exhibits the lowest OSNR values. Figure 6.10 shows a diagram, where the above dispersion behavior of 4-ASK-DPSK is exploited. On the abscissa, we have again the accumulated dispersion rD . For each value of rD , the amplitude ratio b/a is chosen such the lowest required OSNR is achieved. This optimum amplitude ratio is found on the right ordinate of the diagram, whereas the corresponding minimum required OSNR is given by the left ordinate. Note that the scaling of the OSNR axis is the same as in Figure 6.9, but the range is adjusted. The diagram shows, that in the NRZ case the amplitude ratio needs to be approximately linearly increased with respect to the magnitude of the accumulated dispersion in order to obtain the lowest OSNR. If we compare the resulting dispersion curve with the one for the fixed amplitude ratio b/a = 3 77 Bit error probability NRZ−2−ASK NRZ−2−DPSK NRZ−4−DPSK NRZ−4−ASK−DPSK 10−3 10−6 10−9 10−12 12 15 18 21 OSNR [dB] 24 27 30 Bit error probability (a) RZ−2−ASK RZ−2−DPSK RZ−4−DPSK RZ−4−ASK−DPSK 10−3 10−6 10−9 10−12 12 15 18 21 OSNR [dB] 24 27 30 (b) Figure 6.8: Bit error probability vs. OSNR for (a) NRZ-4-ASK-DPSK (b) RZ-4ASK-DPSK with b/a = 3 at Rb = 40 Gbit/s 78 Required OSNR [dB] for BEP = 10−9 33 30 27 24 NRZ−4−ASK−DPSK: b/a=2.0 NRZ−4−ASK−DPSK: b/a=3.0 NRZ−4−ASK−DPSK: b/a=4.0 NRZ−4−DPSK 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (a) Required OSNR [dB] for BEP = 10−9 33 30 27 24 RZ−4−ASK−DPSK: b/a=2.0 RZ−4−ASK−DPSK: b/a=3.0 RZ−4−ASK−DPSK: b/a=4.0 RZ−4−DPSK 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (b) Figure 6.9: Required OSNR for BEP = 10−9 vs. accumulated dispersion rD for (a) NRZ-4-ASK-DPSK and (b) RZ-4-ASK-DPSK with selected amplitude ratios b/a at Rb = 40 Gbit/s 79 Required OSNR [dB] for BEP = 10−9 4.2 32 3.9 29 3.6 26 NRZ: OSNR RZ: OSNR NRZ: b/a RZ: b/a 3.0 23 20 3.3 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 Optimum amplitude ratio b/a 35 2.7 Figure 6.10: Optimum amplitude ratio b/a (right ordinate) and corresponding required OSNR for BEP = 10−9 (left ordinate) vs. accumulated dispersion rD for 4-ASK-DPSK at Rb = 40 Gbit/s in Figure 6.9(a), we find an improved dispersion tolerance for the curve with adjusted amplitude ratios. At 1-dB OSNR penalty, for example, the tolerable accumulated dispersion is increased from ∆rD,1 dB = 84 ps/nm for fixed b/a = 3 to ∆rD,1 dB = 111 ps/nm for adjusted b/a. The values for 2-dB OSNR penalty are listed in Table 6.3. In the RZ case, the best amplitude ratios remain around b/a = 3, the optimum for rD = 0 ps/nm. Consequently, the OSNR curves from Figure 6.9(b) for b/a = 3 and the one from Figure 6.10 are almost identical. Here, the tolerable accumulated dispersion at 1-dB OSNR penalty is only increased from ∆rD,1 dB = 178 ps/nm for fixed b/a = 3 to ∆rD,1 dB = 180 ps/nm for adjusted b/a. The 2-dB values are listed in Table 6.3 Figures 6.11(a) and (b) put the dispersion tolerances of NRZ- and RZ-4-ASK-DPSK in context with the other modulation formats. Figure 6.11(a) for NRZ pulse shaping includes both the dispersion curves for fixed b/a = 3 and adjusted b/a and again visualizes the benefit of the adjustment. We observe that with adjusted b/a, 4-ASK-DPSK has a better dispersion tolerance than the binary formats 2-ASK and 2-DPSK. However, for fixed b/a = 3, 4-ASK-DPSK tolerates less accumulated dispersion than 2-DPSK but still more than 2-ASK at 1-dB and 2-dB OSNR penalty. 4-DPSK remains the most tolerant format. Its tolerable accumulated dispersion at 1-dB OSNR penalty is by factor 1.9 higher than for 4-ASK-DPSK with adjusted b/a for 4-DPSK-Parallel. For 4-DPSK-Serial it is still by factor 1.4 higher. Figure 6.11(b) for RZ pulse shaping shows only the curve for adjusted b/a, as it is approximately the same as for fixed b/a = 3. Here, 4-ASK-DPSK clearly exhibits greater dispersion tolerance than the binary formats, but again less than 4-DPSK. At 1-dB OSNR penalty, 4DPSK-Parallel tolerates by factor 1.4 and 4-DPSK-Serial by factor 1.3 more accumulated dispersion. 80 Required OSNR [dB] for BEP = 10−9 33 30 27 24 NRZ−2−ASK NRZ−2−DPSK NRZ−4−DPSK NRZ−4−ASK−DPSK: b/a=3.0 NRZ−4−ASK−DPSK 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (a) Required OSNR [dB] for BEP = 10−9 33 30 27 24 RZ−2−ASK RZ−2−DPSK RZ−4−DPSK RZ−4−ASK−DPSK 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (b) Figure 6.11: Required OSNR for BEP = 10−9 vs. accumulated dispersion rD for (a) NRZ-4-ASK-DPSK and (b) RZ-4-ASK-DPSK with adjusted amplitude ratios b/a at Rb = 40 Gbit/s 81 In conclusion of the 4-ASK-DPSK performance evaluation, we found that the signal point amplitude ratio b/a needs to be optimized for lowest bit error probabilities. An amplitude ratio adjustment especially improves dispersion tolerance for NRZ pulse shaping. Then, 4-ASK-DPSK has a better tolerance than 2-ASK and 2-DPSK but performs worse than 4DPSK. In order to achieve a given bit error probability, 4-ASK-DPSK requires the highest OSNR of all modulation formats considered so far. These features suggest that 4-ASKDPSK may be used in a case, where better dispersion tolerance and lower spectral width than for binary formats are needed, optical noise is not a major impairment, and further the receiver should be simpler than for 4-DPSK. If NRZ pulse shaping is used, the amplitude ratio can be either preadjusted for a known fiber length or adjusted using some feedback information from the receiver to the transmitter. RZ pulse shaping does not require amplitude ratio adjustment, as the improvement is only marginal. In the following chapters, the term 4-ASK-DPSK will implicitly assume optimized amplitude ratios. 82 Chapter 7 8-level Combined Amplitude- and Differential Phase-Shift Keying (8-ASK-DPSK) The first 8-level optical modulation format with direct detection in this thesis follows the idea of 4-ASK-DPSK. Now, 2-ASK is combined with 4-DPSK, which will be labeled 8-ASKDPSK. In this modulation format, one bit of information is contained in the amplitude of the transmitted symbols, whereas two bits of information are contained in the phase differences of two consecutive symbols. It has first been proposed and investigated in [61] and further studied especially with respect to nonlinear single-channel and WDM transmission in [62, 63, 64] Just recently, 8-ASK-DPSK has been demonstrated for high-speed single-channel transmission [65]. This chapter uses the same methodology as the previous chapter on 4-ASK-DPSK and further puts 8-ASK-DPSK into context with the other modulation formats. Thus, first the transmitter and receiver are presented, and then the performance is discussed with a special focus on the amplitude ratio of the signal points as in [66]. 7.1 8-ASK-DPSK transmitter and receiver The 8-ASK-DPSK transmitter has to generate an optical signal with two amplitude levels and four phase levels. As for 4-ASK-DPSK in Chapter 6, two nonzero amplitude levels are needed in order to properly associate phase angles to them. In the terms of the transmitted a aNRZ (nTs ) signal aNRZ (t), the set of eight symbols is constructed as | NRZ,n | ∈ {d, c} and e0 | = | e0 3π π arg{aNRZ,n } ∈ {0, 2 , π , 2 } with the two positive amplitude values d > c and four phase angles 0, π /2, π , and 3π /2. Because of L = 8 available symbols, 8-ASK-DPSK transmits 3 bit/symbol and the symbol rate Rs is as low as one third of the bit rate Rb . Figure 7.1 depicts the block diagram of the 8-ASK-DPSK transmitter. The first MZM makes up the 2-ASK part of the transmitter and modulates the light from a CW laser in such a 83 b2,n - b3,n - Differential encoder Pulse b1,n - shaper b02,n Pulse - shaper b03,n Pulse - shaper u1 (t) ? 0 Laser eMZM CW a1 (t) aNRZ (t) - MZM 6 u2 (t) - PM - MZM aRZ (t) - 6uRZ,Gauss (t) 6 u3 (t) Figure 7.1: 8-ASK-DPSK transmitters with two MZM and a PM in series way that the optical signal a1 (t) is a 2-ASK signal with amplitude levels e0 · d for a one-bit b1,n and e0 · c for a zero-bit b1,n . The following MZM and PM make up the 4-DPSK part of the transmitter and represent the basis of the 4-DPSK-Serial transmitter from Chapter 5. Two bit sequences b2,n and b3,n are differentially encoded. The MZM modulates the optical signal a1 (t) such that the phase becomes π for a one-bit b02,n or remains 0 for a zero-bit b02,n . Typically for an MZM, the transitions have to go through the origin in the complex plane. The PM finally applies an additional phase shift of π /2 for a one-bit b03,n or leaves the signal unaltered for a zero-bit b03,n , so that the desired 8-level signal aNRZ (t) is obtained. RZ pulse shaping is achieved in a subsequent MZM. The three bit sequences may be obtained from 1 : 3 serial-to-parallel conversion of a bit sequence at the bit rate Rb . In principal, the 4-level phase modulation could also be obtained by the parallel MZM or the single PM as discussed in Chapter 5. We use the MZM and PM in series, as this is the setup, which has been proposed first. The results from Chapter 5 suggest, that there may be performance differences for NRZ pulse shaping, whereas for RZ pulse shaping the actual implementation of the 4-DPSK part of the 8-ASK-DPSK transmitter should not matter. Figures 7.2(a) and (b) show the resulting 8-ASK-DPSK constellation diagrams for NRZ and RZ pulse shaping with the signal points and the transitions between them. The NRZ signal in Figure 7.2(a) is chirped because of the PM. The signal point amplitude ratio for the figure is d/c = 2.15. For RZ pulse shaping in Figure 7.2(b) the chirp is strongly reduced. Here, the amplitude ratio d/c = 2.1 is selected. These amplitude ratios are the optimum ones, as will be shown in the next section. The corresponding 40-Gbit/s power spectra are given in Figures 7.3(a) and (b). Compared to the 2-level and 4-level modulation formats, the spectral width is further reduced. In contrast to 4-ASK-DPSK, the 8-ASK-DPSK spectra do not show spikes at multiples of the frequency corresponding to the symbol rate, i.e. 13.3 GHz, because the signal has zero mean. The 8-ASK-DPSK receiver in Figure 7.4 consists of a 2-ASK and a 4-DPSK path. The received optical signal arx (t) corrupted by AWGN w(t) is filtered in an optical 2nd order Gaussian bandpass filter and then split by a first cross coupler. The signal from the upper output port of that cross coupler is used to detect the amplitude of the signal in a standard 2-ASK receiver according to Section 3.1. The signal from the lower port of that first cross 84 d √ Im{aRZ (t)}/ p̄ √ Im{aNRZ (t)}/ p̄ d 1 c 0 −d −c c d −c −1 1 0 c −d −c c −c −1 −d −1 −1 0 √1 Re{aNRZ (t)}/ p̄ d −d 0 √1 Re{aRZ (t)}/ p̄ (b) (a) Figure 7.2: Constellation diagram with phase transitions for (a) NRZ-8-ASKDPSK with amplitude ratio d/c = 2.15 and (b) RZ-8-ASK-DPSK with amplitude ratio d/c = 2.1 20 n o 10 · log10 |ARZ ( f )|2 / p̄ n o 10 · log10 |ANRZ ( f )|2 / p̄ 20 0 0 −20 −20 −40 −40 −120 −80 −40 0 40 Frequency f [GHz] 80 120 (a) −120 −80 −40 0 40 Frequency f [GHz] (b) Figure 7.3: Power spectra for (a) NRZ-8-ASK-DPSK with amplitude ratio d/c = 2.15 and (b) RZ-8-ASK-DPSK with amplitude ratio d/c = 2.1, both with with Rb = 40 Gbit/s 85 80 120 ĩ1 (t) i1 (t)- 3rd ord. - Sampl. b̂1,k & binary Bessel LPF decision 2-ASK path - Photodiode arx (t) + w(t) 2nd ord. - Gauss BPF X coupler - Photodiode τ i2 (t) R DAF 1 ψ1 Photodiode X coupler - ĩ2 (t) Sampl. 3rd ord. - Bessel - & binary b̂2,k LPF decision 4-DPSK path - Photodiode τ i3 (t) R DAF 2 ψ2 Photodiode ĩ3 (t) Sampl. 3rd ord. - Bessel - & binary b̂3,k LPF decision Figure 7.4: 8-ASK-DPSK receiver with 2-ASK and 4-DPSK path coupler is used to detect the phase differences in the signal in the standard 4-DPSK receiver from Section 5.2 with the same phase shifts ψ1 = π /4 and ψ2 = −π /4. This 4-DPSK receiver includes a second cross coupler and two delay & add filters with balanced detectors. Thus, a total of five photodiodes and two delay & add filters are needed for the 8-ASK-DPSK receiver. 7.2 8-ASK-DPSK optimum signal point amplitude ratio and performance Obviously, the performance of 8-ASK-DPSK will be strongly impacted by the amplitude ratio d/c of its signal points. So first, the optimum amplitude ratios for NRZ and RZ pulse need to be found in order to assess and compare the performance with respect to required OSNR and chromatic dispersion tolerance in a fair way. Let us start in the same way as in Section 6.2 for 4-ASK-DPSK by considering the eye openings of the electrical signals ĩ1 (t) through ĩ3 (t) under the assumption of a noise-free received signal, equal to the transmitted signal, and neither optical nor electrical filters. Then, the sampled electrical signal i1,k = ĩ1 (kTs +t0 ) in the 2-ASK path takes on two values as for 4-ASK-DPSK. We can use the same values as there in Table 6.1 with simply replacing b by d, and a by c. According to the DAF characteristic in (4.8), both electrical signals i2,k and i3,k in the 4-DPSK path can take on six different values depending on the magnitude and differential phase of the received optical signal ak = arx (kTs + t0 ), i.e. |ak |, |ak−1 |, and ∆ϕk . They are listed in Table 7.1 together with the electrical signals i1,k through i3,k and the corresponding estimated bit sequences b̂1,k through b̂3,k . Note, that the normalization factor for i2,k and i3,k contains 1/4 because of the 86 Table 7.1: Sampled electrical signals i1,k through i3,k and estimated bits b̂1,k through b̂3,k in the 2-ASK and 4-DPSK paths with respect to the received symbols ak ∆ ϕk 2i1,k Re20 √ 4 2i2,k Re20 √ 4 2i3,k Re20 b̂1,k b̂2,k b̂2,k c 0 c2 c2 c2 0 1 1 d 0 c2 cd cd 0 1 1 0 d2 cd cd 1 1 1 d2 d2 1 1 1 −c2 c2 0 0 1 −cd cd 0 0 1 −cd cd 1 0 1 −d 2 d2 1 0 1 −c2 −c2 0 0 0 | ae0k | a | k−1 e0 | c c d c d d 0 d2 c c π /2 c2 c d π /2 c2 d c π /2 d2 d d π /2 d2 c c π c2 c d π c2 d c π d2 d d π d2 c c 3π /2 c2 c d 3π /2 c2 cd d c 3π /2 d2 cd d d 3π /2 d2 d2 −cd −cd 0 0 0 1 0 0 −d 2 −cd −d 2 1 0 0 −c2 0 1 0 −cd 0 1 0 −cd 1 1 0 −d 2 1 1 0 −cd c2 p two cross couplers and 1/2 because of the choice of ψ1 and ψ2 . The successive power splitting in the cross couplers does not impact the bit error probability, as both signal and noise power are reduced and the generally small thermal noise is neglected. Table 7.1 reconfirms that standard 2-ASK and 4-DPSK receiver parts can be used. In the noise-free case, the 2-ASK path leads to b̂1,k = 0 if i1,k = Re20 c2 /2, and b̂1,k = 1 if i1,k = Re20 d 2 /2. For both outputs of the 4-DPSK path, we find b̂ j,k = 0 if i j,k < 0, and b̂ j,k = 1 if i j,k > 0 ( j = 2, 3). The eye openings can be found from Table 7.1 as ∆i1 = Re20 (d 2 − c2 )/2 √ in the 2-ASK path, and ∆i2 = ∆i3 = Re20 c2 /(2 2) in the 4-DPSK path. Choosing constant optical signal power according to k2 = c2 + d 2 ∼ p̄, the eye openings become ∆i1 = Re20 (d/c)2 − 1 · k2 · 2 (d/c)2 + 1 87 (7.1) and Re2 k2 . ∆i2 = ∆i3 = √0 · 2 2 (d/c)2 + 1 (7.2) Basically, we have the same situation as for 4-ASK-DPSK in Section 6.2. The amplitude ratio d/c can take on values between 1 and ∞. For d/c = 1, the √ 2-ASK eye opening becomes 2 ∆i1 = 0 and the 4-DPSK eye openings are ∆i2 = ∆i3 = Re0 k2 /(4 2), thus this case represents pure 4-DPSK. For d/c → ∞, the 2-ASK eye opening amounts to ∆i1 → Re20 k2 /2 and the 4-DPSK eye openings are ∆i2 = ∆i2 → 0, thus this case represents pure 2-ASK. Therefore, 8-ASK-DPSK with optimum performance has an amplitude ratio d/c, which lies somewhere in between 1 and ∞. Figures 7.5(a) and (b) show the bit error probabilities of NRZ- and RZ-8-ASK-DPSK with Rb = 40 Gbit/s and an OSNR of 23 dB vs. the amplitude ratio d/c. For each amplitude ratio the receiver filter bandwidths are optimized in order to achieve the lowest bit error probability. Note, that in the figure the 4-DPSK error probability needs to be weighted with the factor 2/3 whereas the 2-ASK error probability needs to be weighted with the factor 1/3 to get the total error probability. Figure 7.5 shows that the optimum amplitude ratio for NRZ pulse shaping is d/c = 2.15, whereas the slightly smaller value d/c = 2.1 leads to the lowest bit error probability for RZ pulse shaping. For amplitude ratios above the optimum values, errors from the 4-DPSK path dominate the total error probability, whereas the 2-ASK is the dominant source of errors for amplitude ratios below the optimum values. This behavior corresponds to the increase or decrease in the eye openings according to (7.1) and (7.2). The receiver filter bandwidths that have been used in the amplitude ratio optimization are given for completeness and as a reference in Table 7.2 For better comparison to the other modulation formats, Figures 7.6(a) and (b) also show contour plots of the required OSNR for a bit error probability of 10−9 vs. the optical and electrical receiver filter bandwidths for NRZ and RZ pulse shaping with the optimum amplitude ratios. The bandwidth pairs that lead to the lowest required OSNR are marked by ×, and the 0.5-dB tolerance regions are delimited by dashed lines as usual. The comparison of Table 7.2 and Figure 7.6 reveals, that the two bandwidths optimization methods come to slightly different optimum filter values, which must be attributed to the rather flat minimum and numerical effects in the evaluation of the moment-generating function. However, the differences are very small as discussed below. For NRZ pulse shaping in Figure 7.6(a) we read the optimum bandwidth pair as ∆ f3 dB,opt = 2.3 · Rs and f3 dB,el = 0.65 · Rs leading to a required OSNR = 23.63 dB. With two places beyond the decimal point, the optical bandwidth ∆ f3 dB,opt = 2.2 · Rs from Table 7.2 leads to the required OSNR = 23.64 dB . If four places beyond the decimal point are considered, we get 23.6345 dB for ∆ f3 dB,opt = 2.3·Rs compared to 23.6371 dB for ∆ f3 dB,opt = 2.2·Rs , which is only a difference of 0.0026 dB. Same as for 4-ASK-DPSK we continue to use the values from Table 7.2 from the amplitude ratio optimization because the difference is so small. Further, Figure 7.6(a) shows, that the optical filter bandwidth can be ∆ f3 dB,opt = 1.45 · Rs 88 Bit error probability 10−6 10−9 10−12 10−15 10−18 Total 2−ASK path 4−DPSK path 2.0 2.5 3.0 Amplitude ratio d/c (a) 3.5 4.0 Bit error probability 10−6 10−9 10−12 Total 2−ASK path 4−DPSK path 10−15 10−18 2.0 2.5 3.0 Amplitude ratio d/c (b) 3.5 4.0 Figure 7.5: Bit error probability vs. amplitude ratio d/c for (a) NRZ-8-ASKDPSK and (b) RZ-8-ASK-DPSK at Rb = 40 Gbit/s at OSNR = 23 dB 89 Table 7.2: Optimum optical and electrical receiver filter bandwidths for various 8-ASK-DPSK amplitude ratios d/c used in the amplitude ratio optimization Amplitude ratio b/a 1.8 1.9 2.0 2.05 2.1 2.15 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.0 3.2 3.4 3.6 3.8 4.0 NRZ-8-ASK-DPSK ∆ f3 dB,opt /Rs f3 dB,el /Rs 1.85 0.70 1.90 0.70 1.90 0.70 — — 2.10 0.65 2.20 0.65 2.25 0.65 2.25 0.65 2.25 0.65 2.20 0.65 2.15 0.65 2.15 0.65 2.10 0.65 2.05 0.65 2.05 0.65 1.95 0.65 1.90 0.65 1.80 0.65 1.75 0.65 90 RZ-8-ASK-DPSK ∆ f3 dB,opt /Rs f3 dB,el /Rs 2.55 0.55 2.55 0.55 2.55 0.55 2.55 0.55 2.50 0.55 2.50 0.55 2.45 0.55 2.45 0.60 2.30 0.60 2.30 0.60 2.30 0.60 2.30 0.60 2.30 0.60 2.30 0.60 2.30 0.60 2.25 0.60 2.25 0.60 2.25 0.60 2.25 0.60 with a properly chosen electrical filter cut-off frequency, if a 0.5-dB penalty with respect to the minimum can be tolerated. The lower edge of the 0.5-dB tolerance region at ∆ f3 dB,opt = 1.45 · Rs is somewhat higher compared to other modulation formats in terms of bandwidth values normalized to the symbol rate Rs , e.g. the ∆ f3 dB,opt = 1.05 · Rs for 4-DPSK-Parallel or ∆ f3 dB,opt = 1.3 · Rs for 4-DPSK-Serial. However, in terms of absolute bandwidth values 8-ASK-DPSK still has lower optical filter bandwidths because of the lower symbol rate Rs . The contour diagram for RZ pulse shaping in Figure 7.6(b) gives us the optimum bandwidth pair as ∆ f3 dB,opt = 2.45 · Rs and f3 dB,el = 0.55 · Rs . The corresponding required OSNR is 22.38 dB. The pair from Table 7.2 is ∆ f3 dB,opt = 2.5 · Rs and f3 dB,el = 0.55 · Rs , also with required OSNR = 22.38 dB, if we take two digits after the decimal point. But taking into account four digits, we get 22.3806 dB vs. 22.3807 dB in favor of the bandwidth pair ∆ f3 dB,opt = 2.45 · Rs and f3 dB,el = 0.55 · Rs . Again, we can safely keep the original value from the amplitude ratio optimization in Table 7.2 because of the negligible OSNR difference. Further, the 0.5-dB tolerance region reaches down to ∆ f3 dB,opt = 1.70 · Rs for a properly selected electrical filter cut-off frequency. In terms of normalized bandwidth this is higher than for 4-DPSK (1.2 · Rs for 4-DPSK-Parallel) but comparable to 4-ASK-DPSK (1.65 · Rs ). In terms of absolute bandwidths, 8-ASK-DPSK still has an advantage over 4DPSK and 4-ASK-DPSK. Table 7.3 lists the optimum receiver filter bandwidths with the corresponding required OSNR values for optimum amplitude ratios d/c. The values in parentheses represent the bandwidth values from Figure 7.6. In the following and as already stated, only the bandwidth values according to Table 7.2 will be used. The electrical eye diagrams at the receiver are depicted in Figures 7.7(a) through (d). Figures 7.7(a) and (c) show the eye diagrams in the 2-ASK path for NRZ and RZ pulse shaping. The eye diagrams in the 4-DPSK path are given in Figures 7.7(b) and (d). The eye diagrams in the upper and lower parts of the 4-DPSK path look the same, so only one is shown. The levels match those from (7.1) and (7.2) only approximately, as now intersymbol interference from the receiver filters is included. However, both the 2-level nature in the 2-ASK path and the 6-level nature in the 4-DPSK paths are clearly observably. Same as for 4-ASK-DPSK, the upper three and the lower three electrical levels form one logical level each, and simple binary decision devices with thresholds at approximately zero can be used in the 4-DPSK path. In order to compare the back-to-back performance of 8-ASK-DPSK to other modulation formats, Figures 7.8(a) and (b) present bit error probabilities vs. OSNR. For bit error probabilities below 10−3 , 8-ASK-DPSK has higher bit error probabilities for given OSNR values than the other considered formats. However, at a bit error probability of 10−9 , the 4-level format 4-ASK-DPSK is only 0.53 dB better for NRZ pulse shaping and 0.49 dB better for RZ pulse shaping. As the slopes of the 8-ASK-DPSK curves are less steep, however, the disadvantage of 8-ASK-DPSK will become more severe for lower bit error probabilities. Next, we take a look at the dispersion tolerance of 8-ASK-DPSK. First, Figures 7.9(a) and (b) show the required OSNR for BEP = 10−9 vs. accumulated dispersion rD for NRZ and RZ 91 3.4 3.4 23 .5 26 25 24 25 24.5 3.2 3.0 3.0 23 24 22.5 .5 23 23.5 24 25 24.5 1.2 25 26 1.0 0.4 22.5 1.4 25 30 1.2 1.8 1.6 2256.5 1.4 2.0 24 1.6 2.2 23 27 1.8 24.5 2.0 2.4 22 2.2 2.6 23.5 Optical bandwidth ∆ f3 dB,opt /Rs 25 23 2.4 24 25 24.5 Optical bandwidth ∆ f3 dB,opt /Rs .5 2.8 25.5 26 2.6 25 24 2.8 22.5 2 4 .5 23.5 3.2 25 25.5 26 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs 27 1.0 0.4 (a) 25 24 24 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs 24 (b) Figure 7.6: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-8-ASK-DPSK with d/c = 2.15, and (b) RZ-8ASK-DPSK with d/c = 2.1. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. 92 Electrical signal 4ĩ2 (t)/(R · p̄) Electrical signal 2ĩ1 (t)/(R · p̄) 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 0.5 (b) Electrical signal 2ĩ1 (t)/(R · p̄) Electrical signal 4ĩ2 (t)/(R · p̄) (a) 0 Time t/Ts 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 (c) 0 Time t/Ts 0.5 (d) Figure 7.7: Eye diagrams after optical and electrical filtering for (a) the 2-ASK path of NRZ-8-ASK-DPSK and (b) the upper 4-DPSK path of NRZ-8-ASKDPSK, and (c) the 2-ASK path of RZ-8-ASK-DPSK and (d) the upper 4-DPSK path of RZ-8-ASK-DPSK 93 Bit error probability NRZ−2−ASK NRZ−2−DPSK NRZ−4−DPSK NRZ−4−ASK−DPSK NRZ−8−ASK−DPSK 10−3 10−6 10−9 10−12 12 15 18 21 OSNR [dB] 24 27 30 Bit error probability (a) RZ−2−ASK RZ−2−DPSK RZ−4−DPSK RZ−4−ASK−DPSK RZ−8−ASK−DPSK 10−3 10−6 10−9 10−12 12 15 18 21 OSNR [dB] 24 27 30 (b) Figure 7.8: Bit error probability vs. OSNR for (a) NRZ-8-ASK-DPSK with d/c = 2.15 (b) RZ-8-ASK-DPSK with d/c = 2.1, both at Rb = 40 Gbit/s 94 Table 7.3: Optimized optical and electrical receiver filter bandwidths, required OSNR for BEP = 10−9 , and chromatic dispersion tolerances ∆rD for 1-dB and 2-dB OSNR penalties for 8-ASK-DPSK 8-ASK-DPSK d/c = 2.15 d/c = 2.15 NRZ RZ 2.20 2.50 (2.30) (2.45) 0.65 0.55 23.64 22.38 (23.63) (22.38) ps ∆rD,1 dB [ nm ] 223 ps ∆rD,2 dB [ nm ] 302 Format ∆ f3 dB,opt /Rs f3 dB,el /Rs Req. OSNR [dB] 8-ASK-DPSK opt. d/c NRZ RZ — — — — — — 456 227 456 581 317 581 pulse shaping. In each case, three amplitude ratios are considered. In addition to the backto-back optimum values d/c = 2.15 and d/c = 2.1, respectively, a value below the optimum (d/c = 1.8) and a value above the optimum (d/c = 2.4) is included. 4-DPSK shall serve as a reference in the diagrams. Qualitatively, we see a similar behavior as for 4-ASK-DPSK in Figure 6.9. For NRZ-8-ASK-DPSK in Figure 7.9(a), the curve for d/c = 2.15 leads to the required OSNR minimum at rD = 0 ps/nm. Both other amplitude ratios have greater minima. The curve with d/c = 1.8 has the greatest slope and this amplitude ratio exhibits the lowest dispersion tolerance. Despite the higher minimum, the curve for d/c = 2.4 has a lower slope, and thus, there are points of intersection with the curve for d/c = 2.15 at moderate values of rD . This suggests that dispersion tolerance of 8-ASK-DPSK will benefit from an adjustment of the amplitude ratio, just as the dispersion tolerance of 4-ASK-DPSK did. The RZ curves in Figure 7.9(b) do not intersect with the one for d/c = 2.1, which exhibits the lowest values for required OSNR over the whole considered range of accumulated dispersion. Then, if the amplitude ratios are adjusted with accumulated dispersion, we arrive at the results presented in Figure 7.10. This diagram is constructed in the same way as the one in Figure 6.10 for 4-ASK-DPSK. The abscissa shows the accumulated dispersion rD . For each value of rD the amplitude ratio d/c is chosen such that the lowest required OSNR is achieved. This optimum d/c is represented by the right ordinate. The corresponding required OSNR values are given by the left ordinate of the diagram. Note, that the OSNR range is different from Figure 7.9 but the scaling is the same. The diagram shows that for NRZ pulse shaping, the amplitude ratio d/c needs to be steadily increased with increasing magnitude of the accumulated dispersion rD in order to get the lowest OSNR values. The range, over which the amplitude ratio is increased, is not as large as for 4-ASK-DPSK. Further, the resulting OSNR curve shows only a slightly improved dispersion tolerance compared to the the one for fixed 95 Required OSNR [dB] for BEP = 10−9 33 30 27 24 NRZ−8−ASK−DPSK: d/c=1.8 NRZ−8−ASK−DPSK: d/c=2.15 NRZ−8−ASK−DPSK: d/c=2.4 NRZ−4−DPSK 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (a) Required OSNR [dB] for BEP = 10−9 33 30 27 24 RZ−8−ASK−DPSK: d/c=1.8 RZ−8−ASK−DPSK: d/c=2.1 RZ−8−ASK−DPSK: d/c=2.4 RZ−4−DPSK 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (b) Figure 7.9: Required OSNR for BEP = 10−9 vs. accumulated dispersion rD for (a) NRZ-8-ASK-DPSK and (b) RZ-8-ASK-DPSK with selected amplitude ratios d/c at Rb = 40 Gbit/s 96 2.5 NRZ: OSNR RZ: OSNR NRZ: d/c RZ: d/c 32 2.4 29 2.3 26 2.2 23 2.1 20 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 Optimum amplitude ratio d/c Required OSNR [dB] for BEP = 10−9 35 2.0 Figure 7.10: Optimum amplitude ratio d/c (right ordinate) and corresponding required OSNR for BEP = 10−9 (left ordinate) vs. accumulated dispersion rD for 8-ASK-DPSK at Rb = 40 Gbit/s d/c = 2.15 from Figure 7.9. At 1-dB OSNR penalty, for example, the tolerable accumulated dispersion is increased from ∆rD,1 dB = 223 ps/nm for fixed amplitude ratio d/c = 2.15 to ∆rD,1 dB = 227 ps/nm for the adjusted amplitude ratios. Obviously, the benefit from the amplitude adjustment is not as large as for 4-ASK-DPSK. For RZ pulse shaping, Figure 7.10 shows a constant amplitude ratio d/c = 2.1 over most of the accumulated dispersion. Consequently, there is no increase in tolerable dispersion ∆rD,1 dB or ∆rD,2 dB . As for the other multilevel formats, the tolerable accumulated dispersion for 1-dB and 2-dB OSNR penalties is larger for RZ pulse shaping than for NRZ pulse shaping. There, the RZ curves had very flat minima but finally intersected with the NRZ curves. Here, however, the points of intersection probably lie far outside the range of the diagram. The results on the dispersion tolerance from this paragraph are summarized in Table 7.3. Figures 7.11(a) and (b) compare the dispersion tolerance of 8-ASK-DPSK with the previously considered formats. Figure 7.11(a) for NRZ pulse shaping contains both the 8-ASKDPSK curves with fixed d/c = 2.15 and with adjusted d/c, which visualizes again that the benefit of the adjustment is low for reasonable OSNR penalties. Despite the highest required OSNR for zero dispersion rD = 0 ps/nm, 8-ASK-DPSK outperforms all previously considered modulation formats with respect to the dispersion tolerance. It is slightly better than 4-DPSK (factor 1.1 at 1-dB OSNR penalty), but much better than the other ones. 8-ASKDPSK unfolds its full superiority for RZ pulse shaping as can be seen in Figure 7.11(b). Compared to 4-DPSK, the second best performer, it tolerates by factor 1.8 more accumulated dispersion at both 1-dB and 2-dB OSNR penalty. 97 Required OSNR [dB] for BEP = 10−9 33 30 27 24 NRZ−2−ASK NRZ−2−DPSK NRZ−4−DPSK NRZ−4−ASK−DPSK NRZ−8−ASK−DPSK: d/c=2.15 NRZ−8−ASK−DPSK 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (a) Required OSNR [dB] for BEP = 10−9 33 30 27 24 RZ−2−ASK RZ−2−DPSK RZ−4−DPSK RZ−4−ASK−DPSK RZ−8−ASK−DPSK 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (b) Figure 7.11: Required OSNR for BEP = 10−9 vs. accumulated dispersion rD for (a) NRZ-8-ASK-DPSK and (b) RZ-8-ASK-DPSK with adjusted amplitude ratios d/c compared to other modulation formats at Rb = 40 Gbit/s 98 In conclusion of the 8-ASK-DPSK performance evaluation, we found that the amplitude ratio of the signal points needs to be optimized in order to achieve the lowest possible bit error probability. The amplitude ratio adjustment increases the NRZ dispersion tolerance, but the effect is not as pronounced as found for 4-ASK-DPSK in Section 6.2. 8-ASK-DPSK exhibits the greatest dispersion tolerance of all considered formats so far. However, it also requires the highest OSNR for a given bit error probability. Further, the transmitter and receiver require more elements, but their bandwidth requirements are not as high because of the low symbol rate. This suggests that 8-ASK-DPSK may be used in a scenario, which mainly requires a high dispersion tolerance, but the hardware complexity and receiver sensitivity are not as important. 99 Chapter 8 8-level Differential Phase-Shift Keying (8-DPSK) Another possibility for an 8-level optical modulation format with direct detection is 8-level differential phase-shift keying (8-DPSK). Three bits of information are contained in the phase differences of two consecutive symbols. Optical 8-DPSK with direct detection has first been proposed in [67] and in more detail in [68]. There, the receiver was rather complicated and required three delay & add filters and analog electrical signal processing. It provided electrical output signals with only two electrical levels, from which the bit sequences could be estimated with binary decision devices. In [69] and further in [14, 70] a much simpler receiver with only two delay & add filters as in the 4-DPSK receiver was proposed and investigated. There, the electrical signals can take on more than two levels and binary decision devices in combination with a binary logic are required. An 8-DPSK receiver with four delay & add filters and a binary logic has been proposed in [71] and [72]. Finally, a slightly modified version of the receiver with two delay & add filters from [69] has been investigated in [73] and [74]. First 8-DPSK implementations [75] and transmission experiments [76] have already been reported. So obviously, 8-DPSK has recently attracted quite some attention. This chapter first presents the 8-DPSK transmitter and then discusses three 8-DPSK receivers based on delay & add filters and direct detection. The performance of 8-DPSK using the different receivers is compared to the other modulation formats. 8.1 8-DPSK transmitter The task of the 8-DPSK transmitter is to generate an optical signal, which has eight different phase angles ϕn at the time instants t = nT and consequently eight different phase differences ∆ϕn ∈ {l π /4; l = 0, 1, . . . , 7}. There are L = 8 symbols available, and therefore 3 bit/symbol are transmitted. The symbol rate Rs is thus only one third of the bit rate Rb . The 8-DPSK transmitter in Figure 8.1 is the combination of the 4-DPSK-Parallel transmitter from Section 5.1 with two parallel MZM and an additional PM. The two electrical drive sig- 100 u1 (t) b01,n Pulse shaper Differential encoder b1,n - b2,n - b02,n - Pulse shaper - MZM X coupler Laser CW b03,n Pulse shaper b3,n - ? Phase shift - e0 3π 2 X coupler a1 (t) - PM - MZM u2 (t) 6 aNRZ (t) - MZM aRZ (t) - 6 6 uRZ,Gauss (t) u3 (t) 1 √ Im{aRZ (t)}/ p̄ √ Im{aNRZ (t)}/ p̄ Figure 8.1: 8-DPSK transmitter with two parallel MZM and a PM in series 0 −1 1 0 −1 −1 0 √1 Re{aNRZ (t)}/ p̄ −1 0 √ 1 Re{aRZ (t)}/ p̄ (a) (b) Figure 8.2: Constellation diagram with phase transitions for (a) NRZ-8-DPSK and (b) RZ-8-DPSK nals u1 (t) and u2 (t), which are generated from the differentially encoded bit sequences b01,n and b02,n , modulate the light from a CW laser in the two MZM in such a way, that the optical signal a1 (t) is a 4-level phase-modulated signal. The signal a1 (t) directly corresponds to the NRZ output signal in the 4-DPSK-Parallel transmitter in Figure 5.1(a) with the constellation diagram in Figure 5.2(a). The phase of a1 (t) is modulated in the subsequent PM by the electrical drive signal u3 (t), which is generated from the differentially encoded bit sequence b03,n in such a way that an additional phase shift of π /4 is applied for a one-bit in b03,n or the phase is left unaltered for a zero-bit in b03,n . aNRZ (t) is the desired 8-level signal. RZ pulse shaping is achieved in a subsequent MZM. The three bit sequences may be obtained from 1 : 3 serial-to-parallel conversion of a bit sequence at the bit rate Rb . In principal, the 4-DPSK-Parallel transmitter part could be replaced by any of the two other 4-DPSK transmitters from Section 5.1. The complete 8-DPSK transmitter could even be implemented with a single PM, following the idea of the 4-DPSK-Single transmitter. However, as the investigation in this thesis focuses on the 8-DPSK receivers anyway, only the 8-DPSK transmitter in Figure 8.1 is considered, as it was the first to be proposed [68, 67]. Further, the results from Chapter 5 suggest that this may be also the best transmitter. Figures 8.2(a) and (b) show the 8-DPSK constellation diagrams for NRZ and RZ pulse shaping. All eight signal points lie on a circle around the origin. The minimum distance between signal points is quite similar to 8-ASK-DPSK in Figure 7.2 of Section 7.1. E.g. with RZ 101 20 n o 10 · log10 |ARZ ( f )|2 / p̄ n o 10 · log10 |ANRZ ( f )|2 / p̄ 20 0 0 −20 −20 −40 −40 −120 −80 −40 0 40 Frequency f [GHz] 80 120 (a) −120 −80 −40 0 40 Frequency f [GHz] 80 120 (b) Figure 8.3: Power spectra for (a) NRZ-8-DPSK and (b) RZ-8-DPSK with Rb = 40 Gbit/s √ pulse shaping, we have a minimum signal point distance of approx. 0.94 · p̄ for 8-DPSK √ and 0.93 · p̄ for 8-ASK-DPSK. For 8-DPSK, every signal point has two nearest neighbors with the minimum distance, whereas for 8-ASK-DPSK every signal point has only one near√ est neighbor at the minimum distance and the second nearest neighbor is at least 1.21 · p̄ away. So 8-ASK-DPSK should have some advantage in terms of receiver sensitivity. As for other modulation formats, the NRZ-8-DPSK signal contains chirp, which is greatly reduced by RZ pulse shaping. Figures 8.3(a) and (b) depict the 8-DPSK power spectra for NRZ and RZ pulse shaping. In both cases the widths of the main lobes are the same as for 8-ASK-DPSK, for which the spectra were shown in Figures 7.3(a) and (b). Even the first few side lobes are almost congruent for both NRZ and RZ pulse shaping. Then, however, the power in the side lobes decreases somewhat stronger for NRZ-8-DPSK, but this happens for power levels already 60 dB below the maximum. The signal bandwidth for both 8-level modulation formats can therefore be considered identical. Because the 8-DPSK signals have zero mean, the power spectra do not show spikes at multiples of the symbol rate. 8.2 8-DPSK receivers In Section 5.2 we found that the 4-DPSK receiver in Figure 5.8 can be used as a receiver for optical DPSK with any 2M , M ≥ 2, number of differential phase levels ∆ϕ , as the two electrical output signals of the balanced detectors i1 and i2 lead to unique combinations for any value ∆ϕ . We further recall from that section that the number of elements NI in a set of values I I = {I0I , I1I , . . .} and NII in a set of values I II = {I0II , I1II , . . .}, which are used to 102 distinguish 2M combinations {i1 , i2 } with i1 ∈ I I and i2 ∈ I II , must satisfy the condition from (5.1): NI · NII ≥ 2M . Two simple choices satisfying this condition for 8-DPSK are NI = 4 and NII = 2 as well as NI = NII = 3. These choices form the bases of the first two considered 8-DPSK receivers with two optical delay & add filters with balanced detectors. Now assume that we use four optical delay & add filters with balanced detectors and therefore have four electrical signals i1 through i4 with i1 ∈ I I through i4 ∈ I IV . The number of values in each set is N j , j = I, . . . , IV. For distinguishing 2M combinations {i1 , i2 , i3 , i4 }, the condition NI · NII · NIII · NIV ≥ 2M (8.1) must be fulfilled, which has one very simple solution NI = NII = NIII = NIV = 2. This forms the basis of the third considered 8-DPSK receiver. Obviously, the condition (8.1) suggests NI = NII = NIII = 2 and NIV = 1 as the simplest solution, which in turn leads to an 8-DPSK receiver with three delay & add filters. For this receiver, however, the symbol sets I j , j = I, II, III for the three electrical signals would contain different elements, whereas the symbol sets I j , j = I, II or j = I, . . . , IV, respectively, in the previous cases contain identical symbols or groups thereof, simplifying the detection process. Let us now take a closer look at the first of the three considered 8-DPSK receivers in Figure 8.4. The received 8-DPSK signal arx (t), which is corrupted by AWGN w(t), is filtered in a 2nd order Gaussian bandpass filter, split by a cross coupler and fed to two delay & add filters. The phase shift in the upper delay & add filter is set to ψ1 = −π /8, and in the lower delay & add filter to ψ2 = 3π /8. Figure 8.5 is a plot of the electrical signals i1 and i2 after the balanced detectors with respect to the differential phase ∆ϕ , according to the study of the delay & add filter in Subsection 4.2.1 for an optical signal with constant magnitude e0 . The eight nominal differential phase values ∆ϕn ∈ {l π /4; l = 0, 1, . . . , 7} lead to four electrical signal levels or symbols I14L through I44L . Therefore, the receiver in Figure 8.4 is labeled 8-DPSK-4L receiver. We now form the symbol sets I I = {I14L , I24L , I34L , I44L }, i1 ∈ I I , and I II = {IA4L , IB4L }, i2 ∈ I II , where IA4L is obtained from grouping I14L and I24L , and IB4L from grouping I34L and I44L . As illustrated in Figure 8.6, condition (5.1) is satisfied with NI = 4 and NII = 2. Three thresholds Γ11 , Γ12 , and Γ13 are needed to distinguish the four symbols of I I , whereas only one threshold Γ21 is needed to distinguish the two symbols of I II . Thus, as can be seen in Figure 8.4, three sampling & binary decision devices are placed after the electrical 3rd order Bessel low-pass filter in the upper path, whereas only one sampling & binary decision device is needed in the lower path. The sampling & binary decision device with the threshold Γi j produces the binary signal γi j,k . It is straightforward to find a logic, which combines γ11,k , γ12,k , γ13,k , and γ21,k in such a way that the estimated bit sequences b̂1,k , b̂2,k , and b̂3,k are obtained. Each combination {γ11,k , γ12,k , γ13,k , γ21,k } corresponds to one specific value of the differential phase ∆ϕ , to which in turn is associated one specific 3-bit combination {b̂1,k , b̂2,k , b̂3,k } using a Gray mapping [77]. These values form the Truth Table 8.1. The logic functions can be directly read 103 arx (t) + w(t) τ i1 (t) R DAF 1 ψ1 2nd ord. - Gauss BPF X coupler - - Photodiode Photodiode τ Photodiode 3rd ord. - Bessel LPF Sampl. - & binary γ11,k decision b̂1,k Sampl. - & binary γ12,k decision Sampl. - & binary γ13,k - i2 (t) R DAF 2 ψ2 ĩ1 (t) 3rd ord. - Bessel LPF ĩ2 (t) Logic - Photodiode decision b̂3,k Sampl. - & binary γ21,k decision Electrical signal 2i j (t)/(R · e20 ), j = 1, 2 Figure 8.4: 8-DPSK-4L receiver: Electrical signals with four logical levels I14L I24L i h i2 (t) ∼ cos ∆ϕ (t) + 38π i h i1 (t) ∼ cos ∆ϕ (t) − π8 I34L I44L 0 1 2 3 4 7 5 6 Differential phase ∆ϕ (t)/(π /4) 8 Figure 8.5: 8-DPSK-4L receiver: DAF output signals i1 and i2 with respect to the differential phase ∆ϕ I44L Γ13 I34L I34L I44L Γ12 Γ21 IB4L I24L Γ11 I24L I14L I14L - i1 - i2 IA4L Figure 8.6: 8-DPSK-4L receiver: Electrical and logical levels at the receiver together with the decision thresholds 104 b̂2,k Table 8.1: Truth table for 8-DPSK-4L receiver logic (X denotes “don’t care”) ∆ϕ /(π /4) b̂1,k b̂2,k b̂3,k i1 i2 γ11,k γ12,k γ13,k γ21,k 0 1 2 3 4 5 6 7 — — — — — — — — 1 1 1 0 0 0 0 1 X X X X X X X X 1 1 0 0 1 1 0 0 X X X X X X X X 1 0 0 0 0 1 1 1 X X X X X X X X I14L I14L I24L I34L I44L I44L I34L I24L — — — — — — — — IA4L IB4L IB4L IB4L IB4L IA4L IA4L IA4L — — — — — — — — 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 from the truth table, in this case b̂1,k = γ12,k , b̂2,k = γ11,k + γ13,k , and b̂3,k = γ21,k . As the logic depends on the mapping, the mapping may be tailored in order to arrive at other logic functions. For Gray mapping the bit error probability can be well approximated using the symbol error probability (SEP) according to BEP ≈ SEP/3. The semi-analytical method for calculating 4-DPSK bit error probabilities can be easily extended to calculating 8-DPSK symbol error probabilities. For 8-DPSK, the moment-generating functions are not just evaluated at single thresholds but a several thresholds. Naturally, the differential encoder at the transmitter has to be designed in order to achieve the desired mapping of bits to the differential phase values. Figure 8.7 shows the second 8-DPSK receiver. It is obviously very similar to the 8-DPSK4L receiver from Figure 8.4. From a first comparison we see that the hardware amount is the same for both receivers. However, the phase shifts in the delay & add filters are now ψ1 = −π /4 and ψ2 = π /4. (The phase shifts are swapped compared to the 4-DPSK receiver.) As the plot of the electrical signals i1 and i2 after the balanced detectors vs. the differential phase ∆ϕ in Figure 8.8 shows, this choice of ψ1 and ψ2 leads to five electrical signal levels I13L through I53L at the nominal values of ∆ϕ . The five electrical signal levels or symbols are again shown in Figure 8.9, which further introduces the two symbols IA3L obtained from grouping I13L and I23L , and IB3L obtained from grouping I43L and I53L . The symbol sets I I = I II = {IA3L , I33L , IB3L }, i1 ∈ I I , i2 ∈ I II , contain three symbols, which need to be distinguished. 105 τ i1 (t) R DAF 1 ψ1 2nd ord. - Gauss BPF X coupler - - Photodiode Photodiode τ Photodiode 3rd ord. - Bessel LPF Sampl. - & binary γ11,k decision Sampl. - & binary γ21,k - i2 (t) R 3rd ord. - Bessel LPF ĩ2 (t) b̂1,k Sampl. - & binary γ12,k decision DAF 2 ψ2 ĩ1 (t) Logic arx (t) + w(t) Photodiode decision b̂2,k b̂3,k Sampl. - & binary γ22,k decision Electrical signal 2i j (t)/(R · e20 ), j = 1, 2 Figure 8.7: 8-DPSK-3L receiver: Electrical signals with three logical levels I13L I23L i h i2 (t) ∼ cos ∆ϕ (t) + π4 I33L i h i1 (t) ∼ cos ∆ϕ (t) − π4 I43L I53L 0 1 2 3 4 7 5 6 Differential phase ∆ϕ (t)/(π /4) 8 Figure 8.8: 8-DPSK-3L receiver: DAF output signals i1 and i2 with respect to the differential phase ∆ϕ Therefore, this receiver is labeled 8-DPSK-3L. Condition (5.1) is satisfied with NI = NII = 3. Two thresholds Γ11 and Γ12 are needed for i1 and again two thresholds Γ21 and Γ22 are needed for i2 in order to differentiate between the levels. In the 8-DPSK-3L receiver there are now two sampling & binary decision devices both in the upper and lower parts, generating the binary sequences γ11,k , γ12,k , γ21,k , and γ22,k . The derivation of the logic functions relating the estimated bit sequences b̂ j,k , j ∈ {1, 2, 3}, to the sequences γi j,k , i, j ∈ {1, 2}, follows the same methodology as described in detail for the 8-DPSK-4L receiver. Therefore, it is not included here but left as an exercise to the reader. One solution can be found in [78]. If the mapping of bits to the differential phase is the same as above for 8-DPSK-4L, the differential encoder at the transmitter is also the same as above. If the mapping is changed in order to arrive at other logic functions, the differential encoder has also to be adjusted. The third and last receiver for 8-DPSK is the one shown in Figure 8.10. Compared to the previous receivers, the hardware amount is approximately doubled. The received signal arx (t) and AWGN w(t) are now split into four paths after optical filtering. The phase shifts in the four delay & add filters are set to ψ1 = π /8, ψ2 = 3π /8, ψ3 = 5π /8, and ψ4 = 7π /8. 106 I53L I43L Γ12 Γ11 I33L I23L IB3L I53L I13L - i1 IA3L I43L Γ22 Γ21 I33L I23L IB3L I13L - i2 IA3L Figure 8.9: 8-DPSK-3L receiver: Electrical and logical levels at the receiver together with the decision thresholds τ i1 (t) R DAF 1 ψ1 X coupler - X coupler - τ Photodiode Photodiode Photodiode Sampl. 3rd ord. - Bessel - & binary γ1,k LPF decision b̂1,k i2 (t) R DAF 2 ψ2 ĩ1 (t) ĩ2 (t) Sampl. 3rd ord. - Bessel - & binary γ2,k LPF decision 6 2nd ord. Gauss BPF arx (t) 6 + w(t) - τ Photodiode i3 (t) R DAF 3 ψ3 X coupler - - τ Photodiode Photodiode Photodiode Sampl. 3rd ord. - Bessel - & binary γ3,k LPF decision i4 (t) R ĩ4 (t) Sampl. 3rd ord. - Bessel - & binary γ4,k LPF decision Figure 8.10: 8-DPSK-2L receiver: Electrical signals with two logical levels 107 b̂2,k b̂3,k DAF 4 ψ4 ĩ3 (t) Logic - Photodiode Electrical signal 4i j (t)/(R · e20 ), j = 1, . . . , 4 h i i1 (t) ∼ cos ∆ϕ (t) + π8 I42L h i i2 (t) ∼ cos ∆ϕ (t) + 38π I32L h i i3 (t) ∼ cos ∆ϕ (t) + 58π I22L I12L 0 1 2 3 4 7 5 6 Differential phase ∆ϕ (t)/(π /4) 8 h i i4 (t) ∼ cos ∆ϕ (t) + 78π Figure 8.11: 8-DPSK-2L receiver: DAF output signals i1 through i4 with respect to the differential phase ∆ϕ I32L I42L Γ1 I22L IB2L Γ2 I22L IB2L Γ3 I22L IB2L - i2 I12L - i3 IA2L I32L I42L I12L IA2L I32L I42L - i1 IA2L I32L I42L I12L Γ4 IB2L I22L I12L - i4 IA2L Figure 8.12: 8-DPSK-2L receiver: Electrical and logical levels at the receiver together with the decision thresholds 108 In Figure 8.11 the electrical signals i1 through i4 after the balanced detectors are plotted vs. the differential phase ∆ϕ . This choice of the phase shifts leads to four electrical signal levels I12L through I42L for the nominal values of the differential phase. These signal levels are the same as for the 8-DPSK-4L receiver. Here, however, the levels are grouped as illustrated by Figure 8.12. For each of the four electrical signals, the symbol IA2L is obtained by grouping I12L and I22L , and the symbol IB2L is obtained by grouping I32L and I42L . The symbol sets I I = I II = I III = I IV = {IA2L , IB2L }, i1 ∈ I I , i2 ∈ I II , i3 ∈ I III , i4 ∈ I IV , contain two symbols each. Therefore, the receiver is labeled 8-DPSK-2L. Condition (8.1) is satisfied with NI = NII = NIII = NIV = 2. Each receiver path needs only one sampling & binary decision device with threshold Γ j , j ∈ {1, 2, 3, 4}, to distinguish between the two symbols. The logic evaluates the binary sequences γ1,k through γ4,k in order to arrive at the estimated bit sequences b̂1,k through b̂2,k . The derivation of the logic functions is again straightforward and follows the same methodology as described for the 8-DPSK-4L receiver. The same can be said for the mapping and differential encoding. 8.3 8-DPSK performance The first step in the performance evaluation is the receiver bandwidth optimization. The contour diagrams in Figures 8.13(a) and (b), 8.14(a) and (b), and 8.15(a) and (b) show the required OSNR for BEP = 10−9 vs. the 3-dB cut-off frequencies f3 dB,el of the electrical 3rd order Bessel low-pass filters and the 3-dB bandwidths ∆ f3 dB,opt of the optical 2nd order Gaussian bandpass filters. The optimum bandwidth pairs are marked by × and listed in Table 8.2 together with the required OSNR values. For some bandwidths pairs, the computation of required OSNR failed, because the decision threshold optimization did not converge. Although this problem might have been solved by adjusting some numerical parameters, these bandwidth pairs have just been left out in the contour diagrams, because they are well away from the optimum bandwidth combination. The respective areas are marked with gray boxes in the diagrams. Electrical eye diagrams before the sampling & binary decision devices for the optimum bandwidth pairs are given in Figures 8.16(a) and (b), 8.17(a) and (b), and 8.18(a) and (b). Let us first compare the contour diagrams for NRZ pulse shaping in Figures 8.13(a), 8.14(a), and 8.15(a). It is common to all three receivers, that the 0.5-dB tolerance region with respect to the optimum bandwidth pair is rather narrow around f3 dB,el = 0.65 · Rs for the electrical 3-dB cut-off frequency, but covers a large range of optical bandwidths. The 8-DPSK-3L receiver has the lowest optimum optical 3-dB bandwidth ∆ f3 dB,opt = 1.80·Rs and also its 0.5dB tolerance region reaches down to the lowest optical bandwidth of 1.3 · Rs . This is slightly lower than the 1.45 · Rs for the competitor 8-ASK-DPSK in Figure 7.6(a). The 8-DPSK2L receiver has its optimum optical bandwidth at ∆ f3 dB,opt = 2.20 · Rs with the tolerance region going down to 1.45 · Rs . For the 8-DPSK-4L receiver, the optimum lies as high as ∆ f3 dB,opt = 2.55 · Rs and comes down to only 1.5 · Rs . The lowest required OSNR for BEP = 10−9 is achieved with the 8-DPSK-2L receiver, however. It needs 25.34 dB OSNR. The 109 3.4 36 31 28 29 2.8 2.6 31 2.4 29 28.5 36 31 1.8 2.4 2.2 2.0 28 1.8 29 31 29 1.4 1.4 34 36 30 30 1.2 31 1.0 0.4 32 0.6 0.8 1.0 1.2 1.4 Electrical bandwidth f3 dB,el /Rs 34 32 30 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs 30 33 3936 1.6 32 1.2 30 30 1.6 31 1.6 30 27.5 28 29 33 39 2.0 30 30 2.2 2.6 27.5 Optical bandwidth ∆ f3 dB,opt /Rs 39 2.8 Optical bandwidth ∆ f3 dB,opt /Rs 3.0 27 28.5 3.0 28.5 33 29 3.2 27.5 30 3.2 27.5 28 30 3.4 1.0 0.4 (a) (b) Figure 8.13: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-8-DPSK-4L and (b) RZ-8-DPSK-4L. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. 110 3.4 3.4 29 30 3.2 26.5 28 27 3.2 3.0 .5 27 26.5 29 30 2.2 26 26 27 28 26 1.6 28 .5 27 0 28293 1.2 27 28 33 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs 1.0 0.4 (a) 29 30 27 26 27 30 1.0 0.4 25.5 1.8 27 26.5 29 1.2 2.0 1.4 33 1.4 28 1.6 2.2 5 1.8 2.4 25. 26 27 26.5 2.0 25 2.4 2.6 25 Optical bandwidth ∆ f3 dB,opt /Rs 2.8 28 Optical bandwidth ∆ f3 dB,opt /Rs 2.8 2.6 25 3.0 28 27 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (b) Figure 8.14: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-8-DPSK-3L and (b) RZ-8-DPSK-3L. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. 111 29 3.4 3.4 3.0 27 2.8 25 26 3.0 24.5 3.2 24.5 25.5 27 25.5 28 3.2 2.8 2.2 2.0 1.8 26 28 29 25 26 25.5 27 28 26 28 24.5 26 27 1.2 27 28 7 2282390 30 36 33 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs 29 2930 1.4 1.0 0.4 5 1.6 1.4 1.2 24. 2.4 25 27 1.8 1.6 27 26 25.5 2.0 2.6 24.5 25.5 2.2 Optical bandwidth ∆ f3 dB,opt /Rs 26 2.4 28 Optical bandwidth ∆ f3 dB,opt /Rs 29 2.6 1.0 0.4 (a) 0.6 0.8 1.0 1.2 1.4 1.6 Electrical bandwidth f3 dB,el /Rs (b) Figure 8.15: Required OSNR [dB] for BEP = 10−9 vs. electrical and optical receiver bandwidths for (a) NRZ-8-DPSK-2L and (b) RZ-8-DPSK-2L. Dashed lines delimit a 0.5-dB penalty region with respect to the minimum required OSNR marked by ×. 112 Table 8.2: Optimized optical and electrical receiver filter bandwidths, required OSNR for BEP = 10−9 , and chromatic dispersion tolerances ∆rD for 1-dB and 2-dB OSNR penalties for 8-DPSK 8-DPSK-4L 8-DPSK-3L 8-DPSK-2L NRZ RZ NRZ RZ NRZ RZ ∆ f3 dB,opt /Rs 2.55 2.85 1.80 2.60 2.20 2.55 f3 dB,el /Rs 0.65 0.60 0.65 0.60 0.65 0.60 Req. OSNR [dB] 28.35 26.99 25.96 24.77 25.34 24.07 ps ∆rD,1 dB [ nm ] ps ∆rD,2 dB [ nm ] 187 442 217 467 235 478 275 542 325 575 343 595 Format second best receiver is the 8-DPSK-3L receiver with only 0.62 dB more required OSNR. The 8-DPSK-4L receiver is clearly the worst performer with an OSNR difference of 3.01 dB to the best receiver. Still, even the 25.34 dB for 8-DPSK-2L are by 1.70 dB higher than for 8-ASK-DPSK. Next, RZ pulse shaping in Figures 8.13(b), 8.14(b), and 8.15(b) is investigated. Here, the contour diagrams are also qualitatively similar. All three receivers have the same optimum electrical 3-dB cut-off frequency f3 dB,el = 0.60 · Rs . The differences in the optimum optical 3-dB bandwidth ∆ f3 dB,opt are not as large as for NRZ. Here, the 8-DPSK-2L and 8-DPSK-2L receivers have approx. the same optimum value of 2.55 · Rs and 2.60 · Rs , respectively. This is quite close to 8-ASK-DPSK in Figure 7.6(b), where we obtained 2.50 · Rs as the optimum optical bandwidth. For the 8-DPSK-4L receiver it lies somewhat higher at 2.85 · Rs . The lowest possible optical bandwidth within the 0.5-dB tolerance region is quite similar for the three receivers. If the electrical bandwidths are chosen properly, the optical bandwidth can be as low as 2.05 · Rs , 1.90 · Rs , and 1.85 · Rs with 0.5 dB tolerance to the minimum for the 8-DPSK-4L, 8-DPSK-3L and 8-DPSK-2L receiver, respectively. This is slightly higher than the 8-ASK-DPSK value of 1.70 · Rs . Again, the 8-DPSK-2L receiver requires the lowest OSNR for BEP = 10−9 . It achieves 24.07 dB. The 8-DPSK-3L receiver requires just 0.7 dB more OSNR, but the 8-DPSK-4L receiver is far behind with its 26.99 dB. The best 8-DPSK receiver requires 1.69 dB more OSNR than 8-ASK-DPSK. The order of the three 8-DPSK receivers with respect to the required OSNR can be understood from the eye diagrams in Figures 8.16, 8.17, and 8.18 in conjunction with Figures 8.6, 8.9, and 8.12, which put the signal levels in context with the decision thresholds. In the 8-DPSK-4L receiver, decisions involving the thresholds Γ11 and Γ13 , i.e. the thresholds lying in the upper and lower eye openings of Figures 8.16(a) and (b), contribute dominantly to the error probability. Without receiver filters, these two eye openings are (Re20 /2) · cos(π /8) − cos(3π /8) ≈ 0.541 · (Re20 /2), whereas the middle eye opening is (Re20 /2) · 2 cos(3π /8) ≈ 0.765 · (Re20 /2). Decisions involving the thresholds Γ12 and Γ21 113 Electrical signal 2ĩ(t)/(R · p̄) Electrical signal 2ĩ(t)/(R · p̄) 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 (a) 0 Time t/Ts 0.5 (b) Electrical signal 2ĩ(t)/(R · p̄) Electrical signal 2ĩ(t)/(R · p̄) Figure 8.16: Eye diagrams after optical and electrical filtering for (a) NRZ-8DPSK-4L and (b) RZ-8-DPSK-4L 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 (a) 0 Time t/Ts 0.5 (b) Figure 8.17: Eye diagrams after optical and electrical filtering for (a) NRZ-8DPSK-3L and (b) RZ-8-DPSK-3L therefore have a much smaller contribution to the error probability. In the 8-DPSK-3L receiver, there is also a small upper and a small lower eye opening, as can be seen in Figures 8.17(a) and (b). However, the decision thresholds lie inside the two large middle eye openings, which are (Re20 /2) · cos(π /4) ≈ 0.707 · (Re20 /2) in height, because the upper two and the lower two levels have been grouped to form one logical level each. Finally, in the 8-DPSK-2L receiver, there are also four electrical signal levels as depicted in Figures 8.18(a) and (b), but only the large middle eye opening is used for decisions. It has the same height as the middle eye opening in the 8-DPSK-4L receiver, apart from an additional factor 1/2 because of the additional cross couplers. Hence, its height is (Re20 /4)·2·cos(3π /8) ≈ 0.765·(Re20 /4), which is the greatest of the three receivers. This order of the values of the relevant vertical eye openings in the three 8-DPSK receivers explains their order with respect to the required OSNR. The back-to-back performance of 40-Gbit/s 8-DPSK is put into context with the other mod- 114 Electrical signal 4ĩ(t)/(R · p̄) Electrical signal 4ĩ(t)/(R · p̄) 1 1 0 0 −1 −1 −0.5 0 Time t/Ts −0.5 0.5 (a) 0 Time t/Ts 0.5 (b) Figure 8.18: Eye diagrams after optical and electrical filtering for (a) NRZ-8DPSK-2L and (b) RZ-8-DPSK-2L ulation formats in Figure 8.19. For both NRZ pulse shaping in Figure 8.19(a) and RZ pulse shaping in Figure 8.19(b), the three BEP curves for the 8-DPSK receivers are the rightmost ones. The OSNR differences between the three curves at a given BEP are approximately the same for NRZ and RZ pulse shaping over a wide range of OSNR. As the 8-DPSK-4L curve has a slightly lower slope as the two other 8-DPSK curves, its difference to them increases with increasing OSNR, making the 8-DPSK-4L receiver even more disadvantageous at low BEP. The OSNR difference of 8-DPSK-2L to 8-ASK-DPSK remains approximately constant over the considered OSNR range for RZ pulse shaping, whereas it slightly decreases but still stays substantial for NRZ pulse shaping. Let us now study the dispersion tolerance of 8-DPSK. Figures 8.20(a) and (b) give the required OSNR for a bit error probability of 10−9 vs. the accumulated dispersion rD for NRZ and RZ pulse shaping. 8-ASK-DPSK is included for orientation. Apart from the back-toback OSNR differences, we again observe for NRZ pulse shaping in Figure 8.20(a), that the 8-DPSK-3L and 8-DPSK-2L receivers exhibit similar performance, whereas the 8-DPSK4L receiver performs clearly worse. The 1-dB and 2-dB OSNR penalties are listed in Table 8.2. If we take the 1-dB OSNR penalty, for example, the 8-DPSK-2L receiver tolerates the maximum of 235 ps/nm accumulated dispersion. This is only by factor 1.08 larger than the 217 ps/nm for the 8-DPSK-3L receiver, but by factor 1.26 larger than the 187 ps/nm for the 8-DPSK-4L receiver. In comparison to its direct competitor 8-ASK-DPSK, the 8-DPSK2L receiver achieves a slightly better (by factor 1.04) and the 8-DPSK-3L receiver a slightly worse (by factor 0.96) dispersion tolerance, if the 1-dB OSNR penalty is taken into account. Both 8-DPSK receivers can be attributed with a slightly better dispersion tolerance, if the 2-dB OSNR penalty is used, as the 8-ASK-DPSK curve has a greater slope than the two 8-DPSK curves. The 8-DPSK-4L receiver performs worse than 8-ASK-DPSK. Most notably, RZ pulse shaping considered in Figure 8.20(b) increases the dispersion tolerance for all three 8-DPSK receivers significantly, as it was also observed for 8-ASK-DPSK. Within an OSNR penalty of 1 dB, the 8-DPSK-2L now reaches a tolerable accumulated dispersion 115 Bit error probability NRZ−2−ASK NRZ−2−DPSK NRZ−4−DPSK NRZ−4−ASK−DPSK NRZ−8−ASK−DPSK NRZ−8−DPSK−2L NRZ−8−DPSK−3L NRZ−8−DPSK−4L 10−3 10−6 10−9 10−12 12 15 18 21 OSNR [dB] 24 27 30 Bit error probability (a) RZ−2−ASK RZ−2−DPSK RZ−4−DPSK RZ−4−ASK−DPSK RZ−8−ASK−DPSK RZ−8−DPSK−2L RZ−8−DPSK−3L RZ−8−DPSK−4L 10−3 10−6 10−9 10−12 12 15 18 21 OSNR [dB] 24 27 30 (b) Figure 8.19: Bit error probability vs. OSNR for (a) NRZ-8-DPSK and (b) RZ-8DPSK at Rb = 40 Gbit/s with the different receivers 116 Required OSNR [dB] for BEP = 10−9 33 30 27 24 NRZ−8−ASK−DPSK NRZ−8−DPSK−2L NRZ−8−DPSK−3L NRZ−8−DPSK−4L 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (a) Required OSNR [dB] for BEP = 10−9 33 30 27 24 RZ−8−ASK−DPSK RZ−8−DPSK−2L RZ−8−DPSK−3L RZ−8−DPSK−4L 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (b) Figure 8.20: Required OSNR for BEP = 10−9 vs. accumulated dispersion rD for (a) NRZ-8-DPSK and (b) RZ-8-DPSK at Rb = 40 Gbit/s 117 of 478 ps/nm. This value is only by factor 1.02 larger than the 467 ps/nm for the 8-DPSK-3L receiver, and by 1.08 larger than the 442 ps/nm for the 8-DPSK-4L. Here, the performance of the receiver comes closer together than for NRZ pulse shaping. Compared to 8-ASK-DPSK at the 1-dB OSNR penalty, the 8-DPSK-2L receiver can tolerate by factor 1.05 and the 8DPSK-3L receiver by factor 1.02 more accumulated dispersion. So the dispersion tolerances for RZ pulse shaping are almost identical. Only the 8-DPSK-4L receiver performs slightly worse than 8-ASK-DPSK. Finally, Figures 8.21(a) and (b) are used to compare the 8-DPSK dispersion tolerance to the previously discussed modulation formats with NRZ and RZ pulse shaping. Here, the 8-DPSK-4L receiver is left out for more clarity, as it is worse than the other two 8-DPSK receivers anyway. Obviously, 8-DPSK outperforms the binary formats. For example with RZ pulse shaping, the 8-DPSK-2L receiver tolerates by factor 6.73 more accumulated dispersion than 2-DPSK at the 1-dB OSNR penalty. Similar as for 8-ASK-DPSK, there is only a small advantage compared to 4-DPSK for NRZ pulse shaping, but for RZ pulse shaping, 8-DPSK is clearly better. For example, for the 1-dB OSNR penalty, the 8-DPSK-2L receiver tolerates by factor 1.92 more accumulated dispersion than 4-DPSK. In conclusion of the 8-DPSK performance evaluation, it must be said that the choice of a particular receiver for 8-DPSK has a significant impact on the results for the required OSNR and the dispersion tolerance. The 8-DPSK-2L receiver is the best solution with respect to both performance measures, whereas the 8-DPSK-4L receiver is the worst. The 8-DPSK-3L receiver comes very close to the 8-DPSK-2L receiver, but has only half the hardware amount. Thus, it may be the favorable candidate for an actual 8-DPSK system implementation. The great advantage of 8-DPSK is its large dispersion tolerance, which unfortunately comes together with high required OSNR, even compared to the direct competitor 8-ASK-DPSK. However, it must be noted, that the 8-DPSK-3L receiver requires only four photodiodes, whereas the 8-ASK-DPSK receiver uses five. Their transmitter hardware amount is comparable. 8-DPSK may be used in similar scenarios as 8-ASK-DPSK, i.e. with the requirement for low signal bandwidth and high dispersion tolerance, but with a stronger emphasis on low receiver component count. 118 Required OSNR [dB] for BEP = 10−9 33 30 27 NRZ−2−ASK NRZ−2−DPSK NRZ−4−DPSK NRZ−4−ASK−DPSK NRZ−8−ASK−DPSK NRZ−8−DPSK−2L NRZ−8−DPSK−3L 24 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (a) Required OSNR [dB] for BEP = 10−9 33 30 27 RZ−2−ASK RZ−2−DPSK RZ−4−DPSK RZ−4−ASK−DPSK RZ−8−ASK−DPSK RZ−8−DPSK−2L RZ−8−DPSK−3L 24 21 18 −300 −200 −100 0 100 Accumulated dispersion rD [ps/nm] 200 300 (b) Figure 8.21: Required OSNR for BEP = 10−9 vs. accumulated dispersion rD for (a) NRZ-8-DPSK and (b) RZ-8-DPSK compared to other modulation formats at Rb = 40 Gbit/s 119 Chapter 9 Conclusion This thesis has compared multilevel optical modulation formats with direct detection to binary formats. The comparison is motivated by their higher spectral efficiencies and anticipated better tolerance to chromatic dispersion, a major impairment in high-speed fiber optical transmission systems. Two 4-level and two 8-level optical modulation formats as well as two binary modulation formats are taken into account, and both NRZ and RZ pulse shaping are included. The results are obtained from numerical studies based on a semi-analytical bit error probability calculation method for direct detection systems. A common parameter set is selected for all formats in order to make the individual results directly comparable. 2-ASK, which is the most common modulation format in today’s optical communication systems, and 2-DPSK serve as references. They both carry 1 bit/symbol, i.e. the symbol rate equals the bit rate. Two different 2-DPSK transmitters are included. For both formats, the required OSNR for a fixed bit error probability of 10−9 and the tolerable accumulated chromatic dispersion for linear fiber transmission are determined after the optimization of the optical and electrical receiver filter bandwidths. For all modulation formats, the receiver bandwidths must be optimized in order to get the lowest possible required OSNR. The 2-ASK receiver with a single photodiode is the simplest receiver. The 2-DPSK receiver requires an optical delay & add filter and two photodiodes, but has a significantly lower required OSNR. NRZ pulse shaping leads to a better dispersion tolerance than RZ pulse shaping. 4-DPSK carries 2 bit/symbol and thus has a higher spectral efficiency than the binary formats. Here, the symbol rate is only half the bit rate. Three different transmitters are taken into account. They are more complex than the transmitters for the binary formats. Two of them need an additional optical modulator and one relies on a multilevel electrical drive signal. The 4-DPSK receiver requires twice the hardware amount of the 2-DPSK receiver, i.e. two optical delay & add filters and four photodiodes, but their bandwidth requirements are only half that of 2-DPSK because of the lower symbol rate. Out of the three transmitters, the one with two parallel Mach-Zehnder modulators leads to the lowest required OSNR. For NRZ pulse shaping, there are significant differences in required OSNR, which are almost removed by RZ pulse shaping. 4-DPSK has a higher required OSNR than 2-DPSK, but a lower one than 2-ASK. Its dispersion tolerance is greater than for the binary formats. For 120 OSNR penalties of 1 or 2 dB, RZ pulse shaping leads to better dispersion tolerances than NRZ pulse shaping. 4-ASK-DPSK also transmits 2 bit/symbol. The transmitter comprises again two optical modulators, but the receiver just needs one delay & add filter and three photodiodes, with the same bandwidth requirements as for 4-DPSK. The optimization of the amplitude ratio of the format’s signal points turns out to be very important. 4-ASK-DPSK has a higher required OSNR than 4-DPSK and the binary formats. The dispersion tolerance is much better than for the binary formats, but slightly worse than for 4-DPSK. RZ pulse shaping leads to better results than NRZ pulse shaping. 8-ASK-DPSK carries 3 bit/symbol, leading to an even higher spectral efficiency. The symbol rate is now just a third of the bit rate. The transmitter consists of three optical modulators, and the receiver uses two optical delay & add filters and five photodiodes. All components have even further relaxed bandwidth requirements. Again, the amplitude ratio of the signal points has to be optimized. The required OSNR is greater than for the binary and 4-level formats. However, the dispersion tolerance is significantly improved, especially for RZ pulse shaping. 8-DPSK also transmits 3 bit/symbol and uses three modulators at the transmitter. Three receivers are considered. Two of them require two delay & add filters and four photodiodes, the other one uses four delay & add filters and eight photodiodes. Although the latter one has the lowest required OSNR, one of the low complexity receivers performs only slightly worse, so that it is the favorable solution. The required OSNR is greater than for 8-ASKDPSK, whereas the dispersion tolerance is a little bit better. As for all multilevel formats, the dispersion tolerance for common OSNR penalties is better for RZ pulse shaping than for NRZ pulse shaping. In summary, multilevel modulation formats offer better spectral efficiency and therefore better chromatic dispersion tolerance than binary formats. However, the transmitters and receivers are more complex and the receiver sensitivity is lower. 121 Appendix A Optical phase shifters This appendix shortly discusses the operation of optical phase shifters, which are needed for example in Mach-Zehnder modulators or delay & add filters. Physically, a phase shifter is realized as a wave guide made of an electro-optic material, typically lithium niobate (LiNbO3 ). An optical signal propagating through this wave guide is delayed in time by τϑ , if a voltage is applied across two electrodes enclosing such a wave guide. For an unmodulated optical signal ec (t) = e0 · exp (jω0t) the time delay τϑ directly translates into a phase shift according to ec (t − τϑ ) = e0 · ejω0 (t−τϑ ) = e0 · ejω0t · e−jω0 τϑ = e0 · ejω0t · e−jϑ = ec (t) · e−jϑ (A.1) with the phase shift ϑ = ω0 τϑ . At a carrier frequency of ω0 /2π = 193.5 THz, phase shifts 0 ≤ ϑ ≤ 2π are obtained with time delays 0 ≤ τϑ ≤ 5.2 · 10−15 s. For a modulated optical signal e(t) = a(t) · exp jω0t with complex envelope a(t) we similarly obtain e(t − τϑ ) = a(t − τϑ ) · ejω0 (t−τϑ ) . (A.2) If a(t) ◦−• A(ω ), then a(t − τϑ ) ◦−• A(ω ) · exp (−jωτϑ ). However, within typical spectral widths of A(ω ), e.g. 240 GHz for RZ-2-ASK in Chapter 3, and with τϑ in the range given above, A(ω ) · exp (−jωτϑ ) ≈ A(ω ) and consequently a(t − τϑ ) ≈ a(t). Using this result in (A.2), we get e(t − τϑ ) = a(t − τϑ ) · ejω0 (t−τϑ ) ≈ a(t) · ejω0t · e−jω0 τϑ = a(t) · ejω0t · e−jϑ = e(t) · e−jϑ . (A.3) For this thesis, it is assumed that the approximately equal sign (≈) in (A.3) can always be replaced by the equal sign (=). Note that in (A.2) and (A.3), the actual phase shift is negative 122 because of the minus in the exponential factor. Positive phase shifts can be realized either by using the periodicity of the exponential factor or by controlling the voltage around some bias point so that positive or ’negative’ time delays are achieved. Such a bias point will not be explicitely considered in this thesis, but it is simply assumed that both positive and negative phase shifts can be applied. 123 Appendix B Region of convergence of Φik (−s) In this appendix, the ROC of Φik (−s) required in (2.55) is derived. Generally, a two-sided Laplace transform can have the same algebraic expression Φik (−s) for two different original functions. However, the ROC is unique for each original function. In our case, Φik (−s) represents the two-sided Laplace transform of the PDF fik (ik ). The task is to find the correct ROC such that Φik (−s) from (2.51) is the Laplace transform of a PDF. To solve this, we use some general properties for the ROC of two-sided Laplace transforms [79] and the PDF: The ROC consists of strips parallel to the line Re{s} = 0, i.e. the imaginary axis. The strips, which can also become right-sided or left-sided half planes, are bounded by poles. As a PDF, R∞ fik (ik ) satisfies −∞ fik (ik )dik = 1 < ∞. Therefore, it is required that the line Re{s} = 0 lies within the ROC. In our case, the real eigenvalues λl , l = 0, . . . , N p , determine the poles 1/λl of Φik (−s). The smallest and the largest eigenvalue are denoted by λmin = minl {λl } and λmax = maxl {λl }, respectively. Depending on the λl , we can have three different scenarios: 1. All poles lie on the positive real axis. Then, the ROC is a left-sided half plane with Re{s} < −1/λmin . 2. All poles lie on the negative real axis. Here, the ROC is a right-sided half plane with Re{s} > −1/λmax . 3. There are poles on both the negative and the positive real axis. Now, the ROC is a strip −1/λmax < Re{s} < −1/λmin . For the receivers encountered in this thesis, we always get the third scenario, which is therefore used in the evaluation of the MGF in Subsection 2.7.3. Some special cases that may lead to the first or second scenario would not change the error probability calculation, as in subsequent steps the ROC is further resticted anyway. 124 Appendix C Common System Parameters Table C.1 summarizes common system parameters used in the investigation of all modulation formats. Table C.1: Common system parameters NRZ pulse shaping Roll-off factor α = 0.5 RZ pulse shaping Gaussian pulses dRZ = 0.5 Mach-Zehnder modulator Intrinsic extinction ratio εdB = 35 dB Bit rate Rb = 40 Gbit/s Receiver filters 2nd order Gaussian bandpass filters 3rd order Bessel low-pass filters Optical noise Signal polarization only Linear fiber transmission Dispersion slope S = 0 125 Appendix D Optical and Electrical Filters This appendix shortly presents the transfer functions of the optical and electrical filters and discusses some related notation used in this thesis. Here, only filters with basic Gaussian and Bessel shapes are considered. The actual implementation, especially of the optical bandpass filters, may however be based on different technologies such as gratings, Fabry-Perot filters, thin-film filters, or Mach-Zehnder interferometers [3]. Gaussian filter In this thesis, filters with Gaussian shape of the transfer function are used as optical bandpass filters at the receivers. Their transfer functions in the baseband representation are [5] HGauss ( f ) = e 2N √ 2f − loge 2 · ∆ f 3 dB . (D.1) N denotes the order of the filter, and ∆ f3 dB,opt stands for the 3-dB bandwidth of the bandpass filter as illustrated in Fig. D.1. The transfer function with respect to the angular frequency can be obtained by the common relations ω = 2π f and ∆ω3 dB = 2π ∆ f3 dB . Note that the Gaussian filter has constant vanishing phase arg HGauss ( f ) = 0. Bessel filter Bessel filters are used for electrical low-pass filters in the various receivers. Their transfer functions in the baseband representation are of the form [32] HBessel (p) = 126 d0 BN (p) (D.2) 0 −3 20 · log10 HGauss ( f )/HGauss (0) ∆ f3 dB −10 −20 −30 −40 −3 −2 −1 0 f /∆ f3 dB 1 2 3 Figure D.1: Magnitude of the transfer function of a second order Gaussian filter with d0 = (2N)!/(2N N)!. The Bessel polynomial of order N BN (p) = N X dk pk (D.3) k=0 h i has the coefficients dk = (2N − k)!/ 2N−k k!(N − k)! . The frequency f = ω /(2π ) is related to the complex variable p by p = jκN 2f ∆ f3 dB (D.4) with the correction factor κN . Without the correction factor κN , the 3-dB bandwidth of the Bessel filter would √ vary with its order N. The correction factor can be obtained from solving |HBessel (jκN )| = 1/ 2. Figs. D.2(a) and (b) depict the magnitude and the phase of the transfer function. The magnitude roll-off is by far not as steep as for the Gaussian filter. Within the 3-dB bandwidth ∆ f3 dB the phase is almost linear, leading to a nonzero but constant group delay. Note again that Bessel filters are used as electrical low-pass filters only throughout this thesis. Therefore, the 3-dB cut-off frequency f3 dB = ∆ f3 dB /2 is used for their characterization instead of the 3-dB bandwidth ∆ f3 dB , which is more commonly used for bandpass filters. 127 0 −3 20 · log10 HBessel ( f )/HBessel (0) ∆ f3 dB arg HBessel ( f ) −10 −20 0 −30 −40 −3 ∆ f3 dB π −π −2 −1 0 f /∆ f3 dB 1 2 3 (a) −3 −2 −1 0 f /∆ f3 dB 1 (b) Figure D.2: (a) Magnitude and (b) phase of the transfer function of a third order Bessel filter 128 2 3 Appendix E Power Spectra The purpose of this appendix is to shortly explain how the various power spectra in this thesis have been computed and why they are called power spectra. This appendix loosely follows [32] and [80]. We recall from Section 2.7 that all time-continuous signals are replaced by sampled versions of themselves in order to capture them for computer processing. All signals are further chosen to be periodic with period Tp , which enables the use of the DFT and IDFT for filter operations. The period Tp captures a signal representing a DBBS of a certain order, and thus contains all relevant symbol transitions. The number of samples within one period is determined by the sample rate Ra = 1/Ta as N p = Tp /Ta . Let us start with the primitive period a (t) for 0 ≤ t ≤ T , p p a p (t) = 0 else, of a periodic signal a(t) = ∞ X µ =−∞ a p (t − µ Tp ). (E.1) (E.2) We can expand a(t) into a Fourier series according to a(t) = ∞ X n=−∞ αn · ejnω pt (E.3) with ω p = 2π /Tp = 2π f p and the Fourier coefficients 1 αn = Tp Z 0 Tp a(t) · e −jnω p t 1 dt = Tp 129 Z ∞ −∞ a p (t) · e−jnω pt dt. (E.4) We note that (E.4) can be rewritten using the spectrum A p (ω ) = αn = 1 A p (nω p ). Tp R∞ −∞ a p (t) exp (−jω t)dt as (E.5) Parseval’s theorem for Fourier series states that the average signal power is the sum of the powers of the absolute values of the Fourier coefficients, i.e. 1 Tp Z Tp 0 ∞ X 2 |a(t)| dt = |αn |2 . (E.6) n=−∞ As the |αn |2 denote the average power of the signal at discrete frequency components nω p , the |αn |2 are called power spectrum. Now, sampling with rate Ra = 1/Ta = ωa /(2π ) is considered. The sampled signal in the primitive period is a p,a (t) = a p (t) · ∞ X m=−∞ δ (t − mTa ) = N p −1 X m=0 a p (mTa ) · δ (t − mTa ) (E.7) with Dirac’s delta function δ (t). The corresponding spectrum can be expressed in two forms using either the spectrum A p (ω ) or the signal samples a p (mTa ) resulting in ∞ ∞ X 1 X a p (ν Ta ) · e−jνω Ta . A p,a (ω ) = 1t · A p (ω − kωa ) = 1t · Ta ν =−∞ (E.8) k=−∞ The symbol 1t stands for the number 1 from a system theoretical view. However, from a physical view, the unit of time is associated with it, as a spectrum has the unit of time multiplied by the unit of the original function. A p,a (ω ) is the periodic repetition of A p (ω ). The period is ωa = N p ω p . Note, that the individual frequency shifted spectra A p (ω − kωa ) in the periodic repetition do not overlap, if the sampling rate Ra is chosen properly. Next, the spectrum in (E.8) is sampled at discrete frequencies µω p , which leads to A p,a (µω p ) = 1t · ∞ X ν =−∞ a p (ν Ta ) · e −jν µω p Ta = 1t · N p −1 X ν =0 a p (ν Ta ) · e µ −j 2πν Np . (E.9) Because of the periodicity with ωa , it suffices in principal to consider µ = 0, 1, . . . , N p − 1. Without 1t , (E.9) is just the definition of the DFT. If a denotes the vector of signal samples aν = a p (ν Ta ), ν = 0, 1, . . . , N p − 1, the DFT transforms a into a vector A with the elements Aµ , µ = 0, 1, . . . , N p − 1, according to N p −1 X 1 −j 2πν µ Aµ = · A p,a (µω p ) = aν · e Np . 1t ν =0 130 (E.10) The elements satisfy A−µ = ANp −µ , which may be used if spectra centered around zero frequency are considered. If we assume non-overlapping spectra in (E.8), we obtain A p,a (ω ) = 1t · 1 · A p (ω ) for −ωa /2 ≤ ω ≤ ωa /2 Ta (E.11) and consequently A p,a (µω p ) = 1t · 1 · A p (µω p ) for −N p /2 ≤ µ ≤ N p /2 − 1. Ta (E.12) Using (E.5) and (E.10) with (E.12) we finally arrive at 2 Aµ = T 2 p Ta 2 · αµ . (E.13) This means that the power spectrum |αn |2 of the periodic signal a(t) can be easily calculated by the DFT of the signal samples a p (ν Ta ) of the primitive period. In this thesis, a(t) denotes √ the complex envelopes of the optical signals and√is normalized such that it is of the unit W. 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