Multilevel optical modulation formats with direct detection

Transcrição

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
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
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
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
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
maximum tolerable positive accumulated dispersion for x-dB optical
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)
0
−0.5
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
(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
×.
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
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
2.6
18.5
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
(a)
18.5
19
0.6 0.8 1.0 1.2 1.4 1.6
(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
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
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
(a)
(b)
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 ×.
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
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
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
33
30
27
24
NRZ−2−ASK
RZ−2−ASK
RZ−2−DPSK−MZM
NRZ−2−DPSK−PM
RZ−2−DPSK−PM
21
18
−300
−200
−100
0
100
200
300
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)
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
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
2.4
21.5
5
2.6
21
.5
20.5
21
0.6 0.8 1.0 1.2 1.4 1.6
1.0
0.4
(a)
0.6 0.8 1.0 1.2 1.4 1.6
(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.
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
5
3.2
23
3.0
20.5
23.
22.5
21.5
2322.5
3.2
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
(a)
21
0.6 0.8 1.0 1.2 1.4 1.6
(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.
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
21
1.0
0.4
(a)
21.5
22
22.5
0.6 0.8 1.0 1.2 1.4 1.6
(b)
receiver bandwidths for (a) NRZ-4-DPSK-Single and (b) RZ-4-DPSK-Single.
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
.2
22
2.0
22.
24
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)
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 5.12: 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
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
NRZ−2−ASK
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
(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
33
30
27
24
NRZ−2−ASK
NRZ−4−DPSK−Serial
NRZ−4−DPSK−Parallel
NRZ−4−DPSK−Single
21
18
−300
−200
−100
0
100
200
300
(a)
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
200
300
(b)
(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)
LPF
decision
- Photodiode
Photodiode
τ
ψ
Photodiode
2-DPSK path
ĩ2 (t)
R
Sampl.
3rd ord.
i2 (t)
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
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
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
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
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
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
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
(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
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
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
1
1
0
0
−1
−1
−0.5
0
Time t/Ts
−0.5
0.5
0.5
(b)
(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
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
(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
33
30
27
24
NRZ−4−ASK−DPSK: b/a=2.0
NRZ−4−DPSK
21
18
−300
−200
−100
0
100
200
300
(a)
33
30
27
24
RZ−4−ASK−DPSK: b/a=2.0
RZ−4−DPSK
21
18
−300
−200
−100
0
100
200
300
(b)
(a) NRZ-4-ASK-DPSK and (b) RZ-4-ASK-DPSK with selected amplitude ratios
b/a at Rb = 40 Gbit/s
79
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
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
33
30
27
24
NRZ−2−ASK
NRZ−2−DPSK
NRZ−4−DPSK
21
18
−300
−200
−100
0
100
200
300
(a)
33
30
27
24
RZ−2−ASK
RZ−2−DPSK
RZ−4−DPSK
RZ−4−ASK−DPSK
21
18
−300
−200
−100
0
100
200
300
(b)
(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.
LPF
decision
4-DPSK path
-
Photodiode
τ
i3 (t)
R
DAF 2
ψ2
Photodiode
ĩ3 (t)
Sampl.
3rd ord.
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
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
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
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
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
25
23
2.4
24
25 24.5
.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
27
1.0
0.4
(a)
25
24
24
0.6 0.8 1.0 1.2 1.4 1.6
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
1
1
0
0
−1
−1
−0.5
0
Time t/Ts
−0.5
0.5
0.5
(b)
(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
NRZ−2−ASK
NRZ−2−DPSK
NRZ−4−DPSK
10−3
10−6
10−9
10−12
12
15
18
21
OSNR [dB]
24
27
30
(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
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
33
30
27
24
NRZ−8−ASK−DPSK: d/c=1.8
NRZ−4−DPSK
21
18
−300
−200
−100
0
100
200
300
(a)
33
30
27
24
RZ−8−ASK−DPSK: d/c=1.8
RZ−4−DPSK
21
18
−300
−200
−100
0
100
200
300
(b)
(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
200
300
Optimum amplitude ratio d/c
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
33
30
27
24
NRZ−2−ASK
NRZ−2−DPSK
NRZ−4−DPSK
21
18
−300
−200
−100
0
100
200
300
(a)
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
200
300
(b)
(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
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
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
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
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
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.
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.
LPF
decision
i4 (t)
R
ĩ4 (t)
Sampl.
3rd ord.
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
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
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
34
32
30
0.6 0.8 1.0 1.2 1.4 1.6
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
39
2.8
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)
receiver bandwidths for (a) NRZ-8-DPSK-4L and (b) RZ-8-DPSK-4L. Dashed
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
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
2.8
28
2.8
2.6
25
3.0
28
27
0.6 0.8 1.0 1.2 1.4 1.6
(b)
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
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
26
2.4
28
29
2.6
1.0
0.4
(a)
0.6 0.8 1.0 1.2 1.4 1.6
(b)
marked by ×.
112
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)
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 8.16: 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)
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
1
1
0
0
−1
−1
−0.5
0
Time t/Ts
−0.5
0.5
(a)
0
Time t/Ts
0.5
(b)
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
NRZ−2−ASK
NRZ−2−DPSK
NRZ−4−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
(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
33
30
27
24
NRZ−8−DPSK−2L
NRZ−8−DPSK−3L
NRZ−8−DPSK−4L
21
18
−300
−200
−100
0
100
200
300
(a)
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
200
300
(b)
(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
33
30
27
NRZ−2−ASK
NRZ−2−DPSK
NRZ−4−DPSK
NRZ−8−DPSK−2L
NRZ−8−DPSK−3L
24
21
18
−300
−200
−100
0
100
200
300
(a)
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
200
300
(b)
(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.
This means that Aµ and αµ are also of the unit W, and therefore the power spectrum |αn |2
is of the unit W.
All power spectra in this thesis are calculated by DFT. However, as you may have noticed,
the ordinate labels use A( f ) instead of the actual A(µ 2π f p ). This mathematical inaccuracy
was introduced intentionally, because the period Tp = 1/ f p was chosen long enough such
that individual points in the diagrams appear like continuous curves. Further, the abscissae
can be labeled with continuous frequencies f instead of frequency indices µ , making the
diagrams more intuitively understandable.
131
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