Use of Modulated Excitation Signals in Medical Ultrasound. Part I

ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 2, february 2005
177
Use of Modulated Excitation Signals in
Medical Ultrasound.
Part I: Basic Concepts and Expected Benefits
Thanassis Misaridis and Jørgen Arendt Jensen, Senior Member, IEEE
Abstract—This paper, the first from a series of three
papers on the application of coded excitation signals in
medical ultrasound, discusses the basic principles and
ultrasound-related problems of pulse compression. The con­
cepts of signal modulation and matched filtering are given,
and a simple model of attenuation relates the matched filter
response with the ambiguity function, known from radar.
Based on this analysis and the properties of the ambigu­
ity function, the selection of coded waveforms suitable for
ultrasound imaging is discussed. It is shown that linear fre­
quency modulation (FM) signals have the best and most
robust features for ultrasound imaging. Other coded sig­
nals such as nonlinear FM and binary complementary Golay
codes also have been considered and characterized in terms
of signal-to-noise ratio (SNR) and sensitivity to frequency
shifts. Using the simulation program Field II, it is found
that in the case of linear FM signals, a SNR improvement
of 12 to 18 dB can be expected for large imaging depths
in attenuating media, without any depth-dependent filter
compensation. In contrast, nonlinear FM modulation and
binary codes are shown to give a SNR improvement of only
4 to 9 dB when processed with a matched filter. Other is­
sues, such as depth-dependent matched filtering and use of
filters other than the matched filter (inverse and Wiener
filters) also are addressed.
I. Introduction
he impetus for this series of papers has been the in­
creasing interest over the last decade in the medical
ultrasound community in the use of more sophisticated
excitation signals than the single-carrier, short pulses cur­
rently used in ultrasound scanners. The potential advan­
tages of such coded signals are an increase in penetration
depth and/or an increase in signal-to-noise ratio (SNR),
and an increase in frame rate. Both SNR and frame rate
are very valuable resources in medical ultrasound imaging.
Higher SNR will allow imaging of structures that are lo­
cated deep inside the human body. Higher SNR also can
allow migration to higher frequencies, which in turn will
result in images with better resolution. High frame rates
will make real-time, three-dimensional ultrasound imaging
T
Manuscript received July 31, 2002; accepted September 7, 2004.
This work was supported by grant 9700883 and 9700563 from the
Danish Science Foundation and by B-K Medical A/S.
T. Misaridis is currently with the National Technical University of
Athens, 10024 Athens, Greece (email: [email protected]).
J. A. Jensen is with the Center for Fast Ultrasound Imaging,
Ørsted•DTU, Technical University of Denmark, DK-2800 Lyngby,
Denmark (email: [email protected]).
possible and will allow imaging of fast moving objects such
as the heart [1].
Coded signals have been used successfully in other en­
gineering disciplines such as radars and mobile commu­
nication systems. It is, therefore, natural for one to ask
for the reasons why coded excitation has not been ex­
plored and used in medical ultrasound imaging as much
as in the other areas. The answer to this question (apart
from the required complexity in electronics and implemen­
tation issues) is that ultrasound imaging with codes is a
far more challenging and difficult task. In radar systems,
the problem is the detection of isolated targets. In imag­
ing, the problem is mapping of distributed scatterers in
which no decision-making is possible. The high require­
ments in the displayed dynamic range of the ultrasound
images is translated to increased requirements for the cor­
relation properties of the coded signals. The problem is
further complicated by the frequency-dependent attenua­
tion in the tissues and by the presence of speckle. In com­
munication systems, codes are used as modulated carriers
of binary data and separation of users is based on thresh­
old detectors. For fast ultrasound imaging, any cross-talk
between simultaneously transmitted coded beams will ap­
pear as ghost echoes in the image. Apart from having a
more difficult task to accomplish, the ultrasound engineer
has to work with far more limited system bandwidth and
code length. Unfortunately, the performance of coded ex­
citation is based exactly on these two parameters.
A. Literature Review
The first investigator that considered the application
of coded excitation in medical ultrasound systems was
Takeuchi [2] in a paper dating back to 1979. Takeuchi
pointed out the time-bandwidth limitations in the applica­
tion of coded signals in ultrasound imaging. Possibly due
to this limitation, as well as the anticipated limitation im­
posed by the frequency-dependent attenuation in tissues,
there is no such contribution during the following years in
the literature on this topic to our knowledge. It is only in
the last decade that there is a renewed interest in coded
excitation within the medical ultrasound community, re­
sulting in a rather vast amount of published papers.
O’Donnell [3] discussed the expected improvement in
SNR, concluding that coded excitation can potentially
yield an improvement of 15 to 20 dB. His system was using
a single correlator on the output of a digital beamformer,
c 2005 IEEE
0885–3010/$20.00 ©
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 2, february 2005
i.e., beamforming was done prior to compression. Subse­
quently, several contributions have been made, primarily
on pulse compression mechanism and range sidelobe re­
duction problems. Most of the authors have used chirp
[linear frequency modulation (FM)] or pseudo-chirp exci­
tation [3]–[7], others have considered binary codes, such
as m-sequences [8], [9] and orthogonal Golay sequences
[10], [11], and others have considered both [12], [13]. Rao
[6] pointed out that ultrasonic attenuation will result in
a reduction of the effective time-bandwidth product and,
thus, in SNR degradation. Eck et al. [14] have designed
depth-dependent mismatched filters for linear FM excita­
tion, based on a simple model of ultrasound attenuation.
Pollakowski and Ermert [5] discussed the design of nonlin­
ear FM signals. Others [15] also have considered prefilter­
ing of the excitation nonlinear FM signals, transmitting
higher frequency spectral components for a longer time,
in order to compensate for the attenuation. Some authors
[3], [8], [12] have considered the application of inverse fil­
tering instead of matched filtering for more efficient range
sidelobe reduction.
Considerably fewer authors [8]–[10], [16], [17] have con­
sidered fast ultrasound imaging using coded signals. High
frame rate coded imaging will be discussed in the third
paper of this series [18].
B. Paper Structure
[18] investigates the use of codes for increasing the frame
rate in ultrasound imaging.
II. Signal Modulation
Let s(t) = α(t) · cos[2πf0 t + φ(t)] be a real modulated
signal transmitted by the ultrasound transducer, where
f0 is the center frequency of the transducer and α(t), φ(t)
are the amplitude and phase modulation functions, respec­
tively. The complex representation of s(t) is:
ψ(t) = µ(t) · ej2πf0 t ,
(1)
where µ(t), a complex function often called the complex
envelope of the signal, combines amplitude and phase mod­
ulation:
µ(t) = |µ(t)| · ejφ(t) .
(2)
If s(t) is a narrowband signal, ψ(t) is analytic, i.e.:
ψ(t) = s(t) + jH{s(t)},
(3)
where H denotes the Hilbert transform; ψ(t) has no neg­
ative frequencies and double the amplitude of the positive
frequencies. If Ψ(f ) and M (f ) are the Fourier transforms
of the analytic signal ψ(t) and the complex envelope µ(t),
respectively, the Fourier transform of (1) yields:
M (f ) = Ψ(f + f0 ).
The aim of this paper, the first in a series of three pa­
pers, is to investigate systematically the applicability of
modulated signals in medical ultrasound imaging, and to
provide an overview of the problems and expected benefits.
In other words, the choice of the appropriate coded wave­
forms and compression filters, as well as the anticipated
SNR improvements are dictated by the ultrasound-specific
properties, such as the available bandwidth, the transducer
weighting effect, the frequency-dependent tissue attenuation, and the presence of speckle. These factors are taken
into account and discussed throughout the paper, which is
organized as follows.
Section II gives representations of modulated signals.
Section III describes the matched filter, the center element
of pulse compression. Subsequently, a simple model for the
gross effect of attenuation on pulse compression leads to
the definition of the ambiguity function of a coded waveform. Using the ambiguity function (Section V) and its
properties (Section VI) as a tool, a discussion is given in
Section VI for the choice of the appropriate waveforms. It
is argued that the FM-modulated family of signals have a
strong advantage over binary codes for ultrasound systems.
This is further enforced by the results on the expected
SNR improvement that are presented in Section VII. Mis­
matched filtering is discussed in Section VIII. Alternatives
to matched filtering, such as the inverse or the Wiener
filters, are addressed in Section IX.
The second paper of the series [19] suggests signal and
filter design methods for coded imaging and gives experi­
mental evaluation of the proposed signals. The third paper
(4)
Thus, the frequency spectrum of the complex envelope
is the shifted spectrum of the signal with the carrier fre­
quency removed. The real signal becomes:
s(t) = Re {ψ(t)} = |µ(t)| cos[2πf0 t + φ(t)].
(5)
The autocorrelation of the complex signal ψ(t) is:
∞
∞
∗
Rψψ (τ ) =
ψ(t)ψ (t + τ )dt =
−∞
−∞
|Ψ(f )|2 ej2πf τ df,
(6)
and can be expressed as a function of the modulation function:
∞
Rψψ (τ ) = e
µ(t)µ∗ (t + τ )dt
j2πf0 τ
−∞
∞
=e
(7)
|M (f )| e
j2πf0 τ
2 j2πf τ
df.
−∞
The complex notation offers many advantages, particu­
larly in expressing correlation integrals. Using (1) and the
symmetry properties of the Hilbert transform (or taking
the real part of (6) and use the one-sided spectrum prop­
erty of ψ(t)), the autocorrelation function Rψψ of ψ(t) is
related to the envelope of the autocorrelation function of
the real signal Rss [20]:
Rψψ (τ ) = 2[Rss (τ ) + j · H{Rss (τ )}] = 2 · Env{Rss (τ )}.
(8)
misaridis and jensen: use of modulated excitation signals in medical ultrasound. part i
III. Matched Filtering and Pulse Compression
The matched-filter concept is the solution to the prob-
lem of finding a linear time-invariant filter, which maximizes the SNR (the peak voltage to noise power) of the
receiver output in the presence of white Gaussian noise.
However, the matched filter is also the ideal receiver from
a statistical decision theory point of view (using the like­
lihood criterion), as well as from an information theory
point of view (using the inverse probability criterion). The
Neyman-Pearson and Bayes criteria for optimizing the
probability of detection maximize the SNR at the output
of the receiver [21]. Thus, the matched filter results in a
receiver that maximizes the SNR and at the same time
minimizes the probability of error in decision and param­
eter estimation. Whether such criteria yield the optimal
receiver for imaging will be discussed in a following sec­
tion.
The matched filter has an impulse response h(t) equal
to the input waveform with reversed time axis, except for
a gain factor k and a translation in time τd for physical
realization. For a complex signal ψ(t), the corresponding
matched filter’s impulse response is denoted as η(t) and is
an analytic signal. The impulse responses of the real and
complex matched filters h(t) and η(t) then are given by:
h(t) = k · s(τd − t), η(t) = k · ψ ∗ (τd − t).
(9)
The transfer function of the matched filter is the com­
plex conjugate of the signal spectrum:
H(f ) = ke−j2πf τd · Ψ∗ (f ).
(10)
For the derivation of the impulse response of the
matched filter using the aforementioned optimization cri­
teria, the reader is referred to [22]. The matched filter min­
imizes the effect of noise by suppressing the noise outside
the frequency band of the input signal, essentially making
the noise colored. A complex filter’s output is given by the
fundamental convolution theorem of signal analysis:
∞
∞
ψ(t)η(τ − t)dt =
γ(τ ) =
0
Ψ(f )H(f )ej2πf τ df.
0
(11)
The matched filter response for the signal of (1) easily
can be obtained [23] by substituting (9) or (10) into (11):
∞
x(τ ) = k
0
|Ψ(f )|2 ej2πf (τ −τd ) df = k · Rψψ (τ − τd ),
(12)
where (6) has been used in the last part of the equa­
tion. Thus, if the input signal to the filter was the same
as the excitation signal (i.e., if the input noise is negligi­
ble), the matched-filter response would be mathematically
equivalent to the autocorrelation of the transmitted signal,
shifted by τd .
In a coded imaging system, the displayed quantity is
the envelope of the output of the matched filter. Using (6)
and (7), we obtain:
179
1
Env{Rss (τ )} =
|Rψψ (τ )|
2
1
=
2
∞
−∞
|M (f )|2 ej2πf τ df .
(13)
The displayed echo from a point scatterer is, thus,
the inverse Fourier transform of the modulation’s energy
density spectrum |M (f )|2 . If the transmitted pulse is a
rectangular-enveloped sinusoid, the output of the matched
filter will have a triangular envelope. Mathematically, this
can be drawn out from (13), as µ(t) is a real-valued rectan­
gular window, the modulus of its Fourier transform M (f )
is a sinc function, and the inverse Fourier transform of a
sinc2 function is the triangle function. For a linear FM sig­
nal, it will be shown in the second paper [19] that M (f )
is rectangular, which inverse Fourier transform is approx­
imately a sinc function.
A. The Time-Bandwidth Product
The concepts of signal modulation and pulse compres­
sion can be described by a single measure of a signal: the
time-bandwidth product (TB). It can be shown [23] that
there is a lower limit for the time-bandwidth product of a
waveform, which is on the order of one. In fact, the signal
with the lowest time-bandwidth product is a single-carrier
pulse with a Gaussian envelope. The important point to
be mentioned here is that any modulation will increase
the time-bandwidth product. Any waveform with a time­
bandwidth product larger than one is referred to as a pulse
compression waveform. Therefore, the difference between
a pulse compression waveform and a single-carrier pulse is
the time-bandwidth product.
The increase in the time-bandwidth product of a sig­
nal can serve the need for increasing either its duration
or bandwidth, or both. Modulation can increase the sig­
nal bandwidth for the same signal duration. In an imaging
system, a practical requirement is the use of the avail­
able system bandwidth. In ultrasound imaging, the system
bandwidth is determined by the ultrasound transducer. In
a conventional system, this requirement is met by simply
transmitting a short pulse. If more bandwidth is available,
the single-carrier pulse should be even shorter. This is ob­
viously an inefficient way of using the available bandwidth.
A pulse-compression waveform can provide the necessary
bandwidth without reducing the pulse duration. A short
pulse of duration T at a carrier frequency f0 is a broadband
signal containing all frequencies in a bandwidth B = 1/T
around f0 . Therefore, the time-bandwidth product of such
an unmodulated pulse is in the order of unity, which is
the smallest possible value for all signals with the same
envelope, and is due to the precise phase relationship of
the individual frequency components. If this precise phase
relationship is altered by using any nonlinear phase func­
tion, a longer signal will result with the same bandwidth as
the pulse. With a proper arrangement of the phases, the
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 2, february 2005
time-bandwidth product of the new signal can be much
larger than one.
B. Range Resolution
signal. Therefore, pulse-compression waveforms processed
with a correlation-based filter combine the advantages of
an optimal system in terms of high axial resolution and
signal detection in noise.
Rihaczek [24] has reported an approximate equation
for the half-power width τb of the main matched-filter re­
sponse peak:
IV. Matched Filtering in Attenuating Media
π/2
τb =
,
β
When the medium has no attenuation, the returned sig­
nal r(t) from a single scatterer is simply a time-shifted
version of the transmitted signal:
(14)
where, β is the root mean square bandwidth. The latter
equation can be verified heuristically easily by taking the
inverse Fourier transform of |M (f )|2 and then finding its
half-power width. The compression ratio, i.e., the ratio of
the uncompressed pulse and compressed pulse widths is
T /(1/B) = T B. The matched-filter response has a reso­
lution on the order of the inverse bandwidth. The impor­
tance of (14) is that, in a pulse compression system with
modulated signals, range resolution can be set indepen­
dently of signal duration. Pulse duration can be adjusted
first according to power requirements, and achievable res­
olution is limited only by the available system bandwidth.
C. Signal-to-Noise Ratio
2 × (received signal energy)
2E
=
=
.
noise spectral density (W/Hz)
N0
(15)
The maximum probability of detection, therefore, is in­
dependent of pulse bandwidth and modulation, and de­
pends only on the transmitted energy E and the noise
power density N0 .
Let the received signal (the input to the matched fil­
ter) have an average input power S over the pulse dura­
tion T . The average noise power within the input signal
bandwidth B is BN0 . For narrowband signals, the instan­
taneous peak power at any time instance is approximately
twice the average power at the same time instance. Then
the matched filter processing gain or gain in SNR (GSNR)
is given by [25]:
SN Rout
SN Rmax,out /2
=
SN Rin
SN Rin
(2E/N0 ) /2
ST /N0
=
=
= T B.
S/ (BN0 )
S/ (BN0 )
(16)
The improvement in SNR will be equal to the time­
bandwidth product of the transmitted waveform.
A correlation filter applied on a pulse-compression
waveform readjusts the phases, restoring the short pulse
through the removal of any nonlinear phase modulation; or
more precisely, reduces the time-bandwidth product back
to an order of one. At the same time, the SNR is maximized
through the weighting of the received-signal spectrum ac­
cording to the spectral components of the transmitted
(18)
where f0 and Br are the center frequency and the rela­
tive bandwidth of the transmitted pulse, respectively, β is
the frequency-dependent attenuation in dB/[MHz × cm],
and z is the depth in tissue. For the typical transducer
carrier frequencies in the range of 2 to 10 MHz used in
ultrasound, a significant downshift of several hundreds of
kilohertz can be seen even for moderate tissue depths. For
moving targets, there is an additional Doppler shift, which
is, however, two orders of magnitude less than the fre­
quency downshift due to attenuation.
Assuming that attenuation does not distort the complex
envelope of the modulated signal, and it only causes a
downshift fd of the central frequency, a simplified model
for the returned signal in an attenuating medium will yield:
r(t) c µ(t − τ0 ) · ej2π[(f0 −fd )(t−τ0 )]
c µ(t − τ0 ) · ej2π[f0 (t−τ0 )−fd (t−τ0 )] .
GSN R =
(17)
where τ0 is the time instant after the start of transmission
(t = 0), at which the signal is being received. The effect of
the frequency dependence of the attenuation is a larger at­
tenuation of the high frequencies of the transmitted signal
compared to the lower when this signal propagates through
the tissue. This will decrease the upper part of the band­
width, effectively causing a decrease in the mean frequency.
For a simple model of attenuation and a Gaussian-shaped
transmitted pulse, Jensen [26] showed that the mean fre­
quency of the propagating pulse decreases linearly with
depth:
fmean = f0 − βBr2 f02 z,
The maximum SNR occurs at τ = τd and is equal to:
SN Rmax
r(t) = ψ(t − τ0 ) = µ(t − τ0 ) · ej2πf0 (t−τ0 ) ,
(19)
The latter equation is based on the narrowband approxi­
mation, i.e., the spectrum shape is not distorted.
Fig. 1 shows simulation results of coded phased-array
imaging using the simulation program Field II [27], [28]. A
fixed focus at 10 cm is used both in transmit and receive.
The graph shows the frequency contents of the transmitted
linear FM signal after being convolved with the impulse
response of the 4 MHz transducer with a 65% fractional
bandwidth. It also shows the frequency contents of the re­
turned and beamformed uncompressed signal from a point
scatterer positioned at a depth of 20 cm in a medium with
frequency-dependent attenuation of 0.7 dB/[MHz × cm].
The mean frequency is shifted down by about 400 kHz.
misaridis and jensen: use of modulated excitation signals in medical ultrasound. part i
181
where χ(τ, fd ) is defined as the ambiguity function (AF),
and is given by:
∞
µ(t) · µ∗ (t − τ ) · ej2πfd t dt
χ(τ, fd ) =
−∞
∞
(22)
ψ(t) · ψ ∗ (t − τ ) · ej2πfd t dt.
=
−∞
Fig. 1. Frequency contents of the transmitted linear FM signal
and the response from a point scatterer positioned at a depth
of 20 cm in a medium with frequency-dependent attenuation of
0.7 dB/[MHz × cm]. The graph shows the mean frequency downshift
and the reduction in bandwidth caused by the ultrasound attenua­
tion in tissues.
These simulation results also show that the shape of the
spectrum is not distorted; and, thus, the approximation
in (19) is valid. However, the bandwidth of the received
echoes also is reduced by 6% due to attenuation, a fact
that (19) does not take into account.
Matched filtering of the received signal given in (19)
requires a priori knowledge of the unknown parameters τ0
and fd . Practically, there is no need to match to the target
delay τ0 because a change in delay merely changes the time
at which the output occurs.
In the general case, the returned signal is matched to the
signal rF (t) = µ(t−τm )·ej2π(f0 −fm )(t−τm ) with parameters
τm and fm . This corresponds to a filter with an impulse
response:
hτm ,fm (t) = rF∗ (T − t + τm ).
Using the fact that hτm ,fm (τ − t) =
the time constant T we get:
rF∗ (t − τ ),
V. Properties of the Ambiguity Function
From a mathematical point of view, there are some
strong restrictions for a two-dimensional function to be an
ambiguity function. In the absence of attenuation (or gen­
erally when fm − fd = 0), the matched filter output is the
autocorrelation function of the waveform. The maximum
of the ambiguity function occurs at the origin:
∞
and ignoring
hτm ,fm (τ − t) = µ (t − τ − τm )
· exp{j2π[f0 (−t + τ + τm ) + fm (t − τ − τm )]}. (21)
Substituting the returned signal given in (19) and the fil­
ter impulse response from (21) into (11), one obtains the
receiver output:
∞
µ(t) · µ∗ (t − τ + τ0 − τm )
−∞
· ej2π[(f0 −fm )(τ −τ0 +τm )+(fm −fd )t] dt
= ej2π(f0 −fm )(τ −τ0 +τm )
· χ (τ − τ0 + τm , fm − fd ) ,
∞
|ψ(t)| dt =
|µ(t)|2 dt = 2E.
2
χ(0, 0) =
−∞
(23)
−∞
The squared modulus of the ambiguity function is re­
ferred to as the ambiguity surface (AS) [24]. The am­
biguity surface is symmetrical around the origin, i.e.,
|χ(−τ, −fd )| = |χ(t, fd )|. The volume under the ambi­
guity surface is given by [22], [24]:
(20)
∗
x(τ ) =
If the delay is translated by τ0 − τm , so that the maxi­
mum occurs at τ = 0 rather than at τ = τ0 − τm , and the
frequency is translated by fm , so that matching occurs at
fd = 0 rather than at fd = fm , the ambiguity function
is the matched-filter response. The value of the ambiguity
function at point (τ, fd ) away from the origin τ = fd = 0
shows the response of a filter that is mismatched by τ and
fd relatively to the input signal.
Thus, the matched filter output at a given depth in
a coded ultrasound system is a cross section of the two­
dimensional ambiguity function parallel to the frequency­
shift axis.
∞
∞
|χ(τ, fd )| dτ dfd =
Vamb =
2
∞
|µ(t)| dt
2
−∞ −∞
= |χ(0, 0)|2 = 4E 2 .
2
−∞
(24)
This equation states that the total volume under the
AS is only a function of the signal’s energy and, therefore,
independent of modulation. It also states that the volume
under the surface is equal to the maximum squared. If a
waveform is chosen whose AF has a narrow central spike,
the bulk of the fixed volume under the AS must appear
as a pedestal on the τ − fd plane around the spike. This
AF is referred to as the “thumbtack” function in the radar
literature. This sets a fundamental limit on achievable res­
olution.
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 2, february 2005
In addition to the property of the total volume invari­
ance, the AF is constrained on the distribution of this
volume in τ and fd . The AS is its own two-dimensional
Fourier transform. The integrated volume distribution in
frequency shift is completely determined by the value of
the ambiguity function on the delay axis. This is based on
the following property of the ambiguity function [24]:
∞
∞
|χ(τ, 0)|2 e−j2πfd τ dτ.
|χ(τ, fd )| dτ =
2
−∞
−∞
(25)
More precisely, the distribution of the ambiguity vol­
ume along the frequency shift axis is the Fourier transform
of the squared envelope of the ambiguity function on the
delay axis, i.e., the squared envelope of the autocorrela­
tion function of the transmitted signal. The dual integral
of (25) states that the integrated volume distribution in
delay is given as the Fourier transform of the ambiguity
surface on the frequency shift axis, which depends only on
the autocorrelation function on this axis. Therefore, this
distribution is independent of any phase modulation for a
given envelope µ(t). What modulation does, is simply to
redistribute the fixed volume in each delay cross section,
but no volume can be moved from one cross section to
another.
These transform relations imply that, if the central peak
is narrowed along one axis, the volume must spread out
in the other axis. The spike cannot be made arbitrarily
narrow. The narrower it is on one axis, the broader it will
be on the other.
Fig. 2. The ambiguity functions of a short (left) and a long (right)
unmodulated pulse. They have a triangular shape on the time axis
and a sinc shape on the frequency axis.
of a continuous wave (CW) signal. In spread spectrum ap­
plication, this coding is called phase-shift coding or key­
ing (PSK). Frequency shift keying (FSK) yields FM dis­
crete waveforms in which the frequency changes in discrete
steps. Shifting of the phase in the frequency domain rather
than in the time domain, although possible, yields wave­
forms with nonconstant amplitude which are not used in
practice. Binary or phase-reversal codes are referred to as
the phase-modulated codes in which the phase takes only
two values: 0◦ and 180◦. Polyphase codes can have more
than two-value phase encoding.
However, the relevant quantity for imaging is the corre­
sponding ambiguity function of the coded waveform. Sig­
nals can be categorized according to their ambiguity func­
tion [29] into basically four groups:
single-carrier pulses,
waveforms with a ridge ambiguity function,
• waveforms with thumbtack ambiguity function,
• waveforms with AF of discrete ambiguities and clear
areas (pulse trains).
•
•
VI. Waveform Selection
A. Pulse Compression Waveforms
The time-bandwidth product of a waveform can be in­
creased by modulating either the amplitude or the phase.
Amplitude modulation is suboptimal in terms of signal
energy, and it will not be considered further. Phase­
modulated signals can be categorized in two main groups,
based on the kind of phase modulation that is applied;
phase modulation can be either continuous or discrete.
This gives the main classification of pulse compression
waveforms in analog (or FM) signals and discrete (usu­
ally phase-modulated) codes. A third way to increase the
time-bandwidth product is the pulse trains, which will not
be discussed further due to their long duration.
The FM waveforms have a continuous, nonlinear phase­
modulation function of time. The simplest and most im­
portant of all FM waveforms is the linear FM with a
quadratic-phase modulation (and, therefore, linear instan­
taneous frequency). Other commonly used nonlinear FM
signals are the cubic FM and the sinusoidal frequency mod­
ulated signal (SFM).
Discrete coded waveforms are signals in which the phase
changes in discrete steps, usually every one or more cycles
1. Unmodulated Pulses: Fig. 2 shows the ambiguity
function of a short and a long pulse. The short pulse is
4 cycles of a 4 MHz sinusoidal signal and the long pulse is
80 cycles. The axial resolution (on the time axis) is on the
order of the inverse bandwidth, and the frequency resolu­
tion is on the order of the inverse duration of the pulse.
2. Ridge Ambiguity Function: Linear frequency mod­
ulation shears the ridge of Fig. 2(right) away from the
delay axis. These waveforms have a group delay, which
is proportional to frequency, i.e., a quadratic phase func­
tion. The ambiguity function consists of a long ridge with
a slope B/T (Fig. 3). Waveforms with this property are
the linear FM, discrete phase-coded waveforms such as the
Frank polyphase code [30], in which the carrier frequency
is stepped in a linear manner, as well as coherent pulse
trains with linear frequency shifting from pulse to pulse.
3. Thumbtack Ambiguity Function: These waveforms
have a narrow spike at the center and sidelobes, usually
uniform, covering the entire τ −fd plane. Fig. 4(left) shows
the ambiguity function of a Barker code of length N = 13.
misaridis and jensen: use of modulated excitation signals in medical ultrasound. part i
Fig. 3. Ambiguity function of a weighted linear FM signal with a
time-bandwidth product of 140.
Fig. 4. The ambiguity functions of the Barker sequence of length 13
(left) and of an m-sequence of length 64 (right).
Barker codes are optimal in the sense that they have the
lowest uniform sidelobe level of height 1/N along the de­
lay axis (Fig. 5). Unfortunately, there are no codes with
a length higher than 13 with this property. For higher
lengths, phase-coded waveforms have peak sidelobes of
2/N or higher along the delay axis. Waveforms whose
ambiguity function resembles the thumbtack function are
generally those in which some kind of random coding is in­
volved, such as the PN (pseudonoise) sequences. The PN
sequences include maximal-length linear feedback shift­
register sequences (m-sequences), Gold and Kasami se­
quences [31]. Fig. 4(right) shows the ambiguity function of
an m-sequence (maximum-length sequence) of length 64.
A special family of binary codes are the complemen­
tary codes. Because a delta ambiguity function is utopian,
one solution to the sidelobe problem can be the use of a
183
set of waveforms with complex ambiguity functions, which
are in some sense as different as possible. By coherently
combining the matched-filter responses (the complex am­
biguity functions of the set), the restrictions imposed on
a single ambiguity function can be bypassed. Guey and
Bell [32] proved that, for a given energy of the waveforms
in a set, the volume under a combined (or composite) AS
always will be less than the volume under the AS of any
single waveform in the set. Furthermore, the minimum oc­
curs when the waveform is a set of equal-energy orthogonal
signals.
The simplest application of this method is the use of
complementary codes consisting of two or more sequences
of equal length, whose ambiguity functions on the delay
axis (i.e., their autocorrelation functions) have sidelobes
equal in magnitude but with inverse signs. The sidelobes
on the delay axis theoretically can be cancelled entirely
by addition, and the mainlobe is doubled. The most well­
known complementary codes are the Golay codes [33].
Properties and construction methods of Golay codes for
a given code length N are discussed in [34] and [35]. Com­
plementary sets, in which the autocorrelation functions of
more than two sequences have to be added together, also
have been derived [36], [37].
In practice, the two sequences of a Golay pair have to
be transmitted in different transmit events. Thus, apart
from increasing the acquisition rate by a factor of two,
some decorrelation on the received radio frequency (rf)­
data will prevent complete cancellation of the sidelobes.
Additionally, the complementary property of Golay pairs is
degraded in the presence of ultrasound attenuation, as will
be discussed later. Consideration of extended complemen­
tary waveforms that can cancel the ambiguity everywhere
on the τ − fd plane has been discussed by Sivaswamy [38].
He showed that, from any coded waveform with a dura­
tion τ0 , N waveforms of duration N τ can be constructed,
that their autocorrelation sum is zero anywhere except the
region |τ | < τ0 . He named these sequences subcomplemen­
tary, and he showed that cancellation is full only when the
basic waveform is repeated using Hadamard encoding.
A low-cost and straightforward implementation of bi­
nary codes is the direct transmission of the code, i.e., the
+1 and −1 of the code are multiplied on a sampled ver­
sion of a rectangular pulse. This approach does not re­
quire linear amplifiers on the transmitter, but it has low
efficiency because the spectrum of the transmitted code
peaks at direct current (DC) and has most of its energy
outside the transducer bandwidth. For better spectral con­
trol, more complex bipolar signals can replace the rectan­
gular pulse to modulate the codes. Four-digit carrier base
sequences such as [1, −1, 1, −1] [39] and [1, −1, −1, 1] [13]
have been suggested in the literature.
B. Discussion on Waveform Selection
Fig. 5. Autocorrelation function (ACF) for a Barker code of
length 13.
An image of a number of scatterers will be the superpo­
sition of weighted and time-shifted cross sections of ambi­
guity functions. The objective of a coded waveform design
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 2, february 2005
is the control of the shape of the ambiguity function for
a given imaging medium. In radar systems, the objective
of pulse compression is the detection of isolated targets in
noise and the estimation of range and velocity of each of
the targets with minimum root mean square error. The
task for the radar designer is to design signals and cor­
responding ambiguity functions that have good resolution
properties in both range and Doppler. If the medium con­
sists of only a few scatterers, the thumbtack AF would
be the appropriate choice. The range resolution of such a
spike will be proportional to the inverse bandwidth, and
the Doppler resolution will be proportional to the inverse
of the duration of the waveform. The volume constraint
of the AS, discussed in the last section, will result in a
high pedestal in which the bulk of the AS volume will
be concentrated. The thumbtack ambiguity function of­
fers good resolution both in range and frequency shift,
with two tradeoffs: a high pedestal and that, when the
frequency shift mismatch exceeds 1/T , effectively no com­
pression occurs. For a coded waveform with a duration of
20 µs, the corresponding frequency shift is 50 kHz.
In ultrasound, although attenuation causes a frequency
shift to the returned echoes similar to the Doppler effect,
there is no need for retaining good resolution in the fre­
quency shift axis. The reason for this is that blood ve­
locity cannot be measured anyway in ultrasound from a
single measurement because the Doppler shift is obscured
by the frequency shift from the attenuation. Blood veloc­
ity is measured from the phase shift between successive
pulses [26]. Therefore, the ideal ambiguity function for ul­
trasound should be as insensitive to frequency shifts as
possible because the frequency shift of the returned signal
due to attenuation is not known. This precludes the so­
called in radars “Doppler-sensitive waveforms”, that their
ambiguity function is an approximation of the thumbtack
function, such as PN sequences, Barker codes etc.1 In ul­
trasound, a coded waveform should have only good range
resolution for a large range of frequency shifts.
A good choice for ultrasound imaging would then be a
very short pulse. It has good range resolution because of its
large bandwidth and is insensitive to frequency shifts due
to its short duration. Note that if both range and frequency
shift are to be measured, as in radar systems, such a wave­
form is inappropriate. As was mentioned before, the prob­
lem with the short-pulse excitation is its low transmitted
energy and low time-bandwidth product. High TB results
in high SNR, which in turn provides the optimal detec­
tion and measurement accuracy. As was mentioned earlier,
a Gaussian short pulse has the smallest time-bandwidth
product, and, thus, would be the worst possible choice in
terms of SNR.
The linear FM signal is a good candidate as long as
appropriate weighting assures good range resolution and
1 A use of a frequency-shift-sensitive waveform in ultrasound can
be a method for estimating ultrasound attenuation. Because such a
waveform has good frequency resolution, a bank of matched filters,
each tuned at a different frequency, can yield the frequency shift due
to attenuation for a given depth.
sidelobe suppression. Compression is satisfactory for a
large range of frequencies due to the ambiguity ridge and
is ideal for uncompensated frequency shifts [40]. The rect­
angular signal spectrum allows full use of the system band­
width and good SNR improvement. Waveforms with a
ridge ambiguity function have the property that pulse com­
pression still occurs in the presence of a mismatch. The
effect of a frequency shift is only a time translation of the
compressed pulse proportional to the frequency shift mis­
match.
VII. Expected Signal-to-Noise Ratio
Improvement
It was shown in (16) that the theoretical improvement
in SNR by using modulated signals will be equal to the
time-bandwidth product. The echoes, however, that will
be compressed will be frequency shifted due to attenua­
tion. From the ambiguity function analysis, it is expected
to have a loss in GSNR for certain waveforms. Mismatched
filtering for range sidelobe reduction will reduce the GSNR
further. Therefore, it is necessary to evaluate the appar­
ent improvement in SNR of coded excitation in ultrasound
imaging. This is done in this section with the simulating
program Field II [27].
From hydrophone measurements in a water tank, peak
intensities were calculated for pulsed and coded excitation.
It was found that pulse and coded excitation of equal peak
voltage amplitude resulted in a measured Isptp (spatial­
peak-temporal-peak) about 1.6 times greater for the coded
signal at the acoustical focus. This is due to different focus­
ing properties of the transducer for a much longer signal.
The relationship is nonlinear and is between 1.1 and 1.8,
depending on the applied voltage amplitude. For instance,
a conventional pulse of 80 V and a chirp signal of 50 V give
the same Isptp at the focal point. The simulations showed
a similar dependence, and, therefore, for all SNR gain sim­
ulation results, the excitation voltages were normalized to
yield the same Isptp at the focal depth for all excitation
signals. The calculation of the SNR has been based on the
following equation [41]:
TP
SN Ri =
p2i (t, fri )dt
0
Pnoise
,
(26)
where p is the pressure field at the axial position fri , TP
is the total duration of the received echoes, and Pnoise is
the power of the simulated band-pass thermal (Johnson)
noise of the system. The energy of the received echoes has
been calculated as an integration over time of the squared
pressure field. The results have been normalized so that
the peak SNR in the case of pulsed excitation is always
60 dB.
This section presents simulation results on the expected
SNR gain from coded excitation for four different coded
signals. The four coded signals (which are shown in the
misaridis and jensen: use of modulated excitation signals in medical ultrasound. part i
185
Fig. 7. The frequency response of the last three coded excitation
signals shown on the left column of Fig. 6.
Fig. 6. The four coded excitation signals used in the SNR simulations
are shown on the left. On the right plots are the actual propagating
signals after convolution with the transducer impulse response. The
presence of the transducer affects the transmitted energy of the linear
FM signals the most.
left column of Fig. 6), are a standard linear FM signal,
an amplitude-tapered FM signal that will be presented in
detail in the second of this paper series [19], a nonlinear
FM signal, and a Golay sequence of length 40. The Golay
sequence is not a direct bipolar Golay code of +1 and −1
values, but a sinusoid at the center frequency of the trans­
ducer, with each of its cycle phase-shifted using a Golay
code (where +1 corresponds to no phase shift and −1 cor­
responds to a 180◦ phase shift). The frequency response of
the last three signals and the transducer transfer function
are shown in Fig. 7.
The right plots of Fig. 6 show the actual transmit­
ted coded signals after convolution with the transducer
impulse response. A transducer with center frequency of
4 MHz, 65% −6 dB bandwidth, and a fixed acoustic focus
at 6 cm was used in the simulations. These simulation pa­
rameters correspond to the specifications of the transducer
used in the experiments presented in the second paper of
the series [19]. The energy of the transmitted signals is re­
duced due to the transducer filtering. The energy loss for
the various signals can be explained by looking at the fre­
quency characteristics of the signals in Fig. 7. The Golay
sequence is a narrow-band signal and suffers the smallest
energy loss. A direct bipolar Golay code, which contains
a big portion of its energy in higher harmonics, would un­
dergo a much greater energy loss. The same is true for a
Fig. 8. Simulation results on the expected SNR from pulsed and
coded excitation for four different coded signals using Field II. There
is no ultrasonic attenuation in the simulated medium. The higher
transmitted energy of Golay codes results in higher SNR gain com­
pared to the linear FM signals. The expected SNR gain relative to
pulse excitation is 11 dB for the linear FM, 14.7 dB for the nonlinear
FM and 16 dB for the Golay coded for all depths.
bipolar pseudo-chirp. The energy reduction of the trans­
mitted coded signals is 74% for the linear FM, 72% for the
tapered linear FM, 41% for the nonlinear FM, and 23%
for the Golay code.
Fig. 8 shows simulation results on the expected SNR
gain for the four coded signals when there is no attenu­
ation in the medium. The reduced transmitted energy of
the linear FM signals explains the results of Fig. 8. The
expected SNR improvement is 11.1 dB for the linear FM
signal, 10.4 dB for the tapered linear FM signal, 14.7 dB
for the nonlinear FM signal, and 16 dB for the Golay code.
The SNR gain is independent of depth, and the overall
SNR reduction is due to the 1/r dependence of acoustic
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 2, february 2005
Fig. 9. Same as in Fig. 8 in tissues with attenuation of
0.5 dB/[MHz × cm]. In this case, the linear FM signals exhibit higher
SNR gain relative to the pulsed excitation than the nonlinear FM and
Golay-coded signals.
diffraction. Note that the driving voltages have been cal­
ibrated to yield the same Isptp at the focal point for all
signals. The higher transmitted energy and SNR gain for
the Golay code also implies that Golay code will reach the
Ispta allowed upper limit faster than the FM signal as the
code length increases.
Fig. 9 shows simulation results on the expected SNR
gain of the same coded signals when attenuation of
0.5 dB/[MHz × cm] has been simulated for the medium.
The expected SNR improvement is depth dependent, and
it improves significantly for the linear FM signals. At the
acoustic focus, the expected SNR improvement is 10.9 dB
for the linear FM signal, 10.2 dB for the tapered linear FM
signal, 12.8 dB for the nonlinear FM signal, and 14.5 dB
for the Golay code. At higher depths in which we expect to
benefit from coded excitation the most, the expected SNR
gain of the linear chirps relative to the pulsed excitation
increases. At a depth of 20 cm, the gain in SNR relative
to the regular pulse is 15.9 dB for the tapered linear FM,
12.2 dB for the nonlinear FM, and 13.2 dB for the Golay
sequence.
In the next set of simulation results, shown in Fig. 10,
matched filtering has been applied on the received rf-data,
and the SNR has been calculated from the energy of the
compressed pulses. Matched filters have been applied on
all signals. The graphs in Fig. 10 show the effect of dis­
tortions from frequency shifting due to attenuation on the
SNR loss. The gain in SNR relative to the short pulse im­
proves with depth only for the linear FM signals, and it
is reduced for the nonlinear FM and Golay signals. These
graphs in Fig. 10 show the robustness of linear FM signals
Fig. 10. Expected SNR for pulsed and coded excitation in tissues
with attenuation of 0.5 dB[MHz × cm] after matched filtering.
in attenuation in respect to gain in SNR. In the second
paper [19], the robustness of different coded signals in at­
tenuation will be investigated in respect to range sidelobes.
VIII. Mismatched Filtering for Range Sidelobe
Reduction
Range sidelobes represent an inherent part of the pulse
compression mechanism. For FM signals, it is the sharp
edges of the rectangular spectrum amplitude that generate
the high, slowly decreasing sidelobes of the sinc function.
As will be shown in the second paper of the series [19], if a
window function is applied on the impulse response of the
matched filter either in the time or in the frequency do­
main, these sidelobes can be reduced well below the level
required for diagnostic imaging. The weighted matched
filter usually is referred to as the mismatched filter. For
small frequency shifts, mismatched filtering is equivalent
to the transmission of a weighted spectrum. In the pres­
ence of frequency shifts, an amplitude-weighted receiver
filter passes an unsymmetrical spectrum, distorting the
weighting effect and can introduce high-range sidelobes.
In this case, it is necessary to weight both the transmitted
spectrum and the filter spectrum, a design often called bi­
lateral weighting. Mismatched filter design for FM signals
will be discussed in detail in [19].
There mainly are two drawbacks of mismatched filter­
ing. The first is a widening of the axial mainlobe. For ultra­
sound, even when heavy weighting is applied, the widen­
ing will not exceed two times the width of the unweighted
mainlobe; a resolution that is acceptable in most cases.
Design guidelines for the trade-off between axial sidelobe
misaridis and jensen: use of modulated excitation signals in medical ultrasound. part i
187
A. Sidelobe Reduction for Phase-Encoded Sequences
Fig. 11. Using the data from Figs. 9 and 10, this plot shows the
expected SNR improvement relative to pulsed excitation in tissues
with attenuation of 0.5 dB/[MHz × cm] before (left) and after (right)
matched filtering.
level and mainlobe width will be given in [19]. The second
undesired effect of mismatched filtering is a small degrada­
tion in SNR improvement, which is—for useful ultrasound
signals—about 1 dB compared to the matched case. The
reason for such a small loss is that mismatching does not
change the phase characteristics of the compression filter,
but it only shapes the amplitude. It can be seen from
Figs. 10 and 11 that the GSNR loss attributed to mis­
matching for the linear FM signal is in fact less than 1 dB.
The range sidelobes effects, however, are more deleterious,
and a sacrifice in that range in SNR is often a reasonable
price to pay.
Although mismatch in the phase function rather than in
the amplitude is possible, phase mismatching is a sensitive
operation, and there is an increased difficulty in control­
ling the resulting axial sidelobes. Even small mismatches
in the phase can have an effect on the compressed pulse
comparable to that from large amplitude weighting [24].
If µ2 (t) is the modulation function of a mismatched fil­
ter to a transmitted signal with modulation function µ1 (t),
the cross-ambiguity function can be defined in accordance
to the ambiguity function:
∞
µ1 (t) · µ∗2 (t − τ ) · ej2πfd t dt.
χ12 (τ, fd ) =
−∞
(27)
Rihaczek [24] showed that the volume of the cross­
ambiguity surface is constant for a given filter, if the input
signal is varied so that its energy remains unchanged. That
implies that mismatched filtering does not release the vol­
ume constraints of the cross-ambiguity function. He also
showed that the average volume below a cross-ambiguity
surface cannot be lower than that of an ambiguity sur­
face, i.e., the effective decrease in the filter passband does
not solve the sidelobe problem on the entire (τ, fd ) plane.
However, mismatching can be very useful in suppressing
sidelobes on a plane along the delay axis or shifting part
of the volume into regions with less interference, at the
expense of a slight increase in the mainlobe width.
As opposed to the sidelobes of the linear FM signal,
which generally decrease as a function of time displace­
ment away from the peak, the sidelobes of most phase­
coded signals tend to have an almost constant level ex­
tending out to ±T seconds on either side of the compressed
pulse (see Fig. 5). The long-time duration of the range
sidelobes of the phase-coded signals will degrade image
contrast, particularly in a dense medium [22].
Mismatched filters for range sidelobe reduction in FM
type of signals are designed easily by weighting the spec­
trum with appropriate window functions. In phase-coded
signals, the lack of symmetry in the signal spectrum re­
quires optimization methods. The filter coefficients are
calculated by minimization of quantities such as the inte­
grated sidelobe level (ISL filters) or the peak sidelobe level
(PSL filters). Mismatched filter design techniques can be
found in [42] and [43]. For direct binary codes, mismatch­
ing can be done by the so-called spiking filter. Spiking
filters can reduce sidelobes more than a factor of N com­
pared to the matched filter case, but their performance
is a function of filter length [13], [39]. Because such mis­
matched filters are not optimized for SNR, there is a loss
in SNR gain.
B. Depth-Dependent Mismatched Filtering
In order to bypass the constraints of the ambiguity func­
tion of frequency-sensitive coded waveforms in an atten­
uating medium, a range-variant matched filter has to be
applied. That is, a filter bank of i depth-dependent filters
with impulse responses hτ,fmi , so that fmi = fdi , where
fdi is the estimated frequency shift at a certain depth. The
design of such filters for binary codes has to be based on
optimization techniques for each depth. In contrast, depth­
dependent matched filters for FM signals are constructed
easily from weighted FM signals with variable center fre­
quency. This approach can be very useful to nonlinear FM
signals that are also frequency-sensitive waveforms. It has
been applied by Eck et al. [14]. Nonlinear FM signals have
some advantages over the linear FM signals, such as better
axial resolution (narrower mainlobe) and low-range side­
lobes without any weighting.
Depth-dependent matched filtering requires an assump­
tion on the value of attenuation. Depending on the in­
terrogating medium, there always will be deviations from
the assumed values, and the requirement for robustness
of compression in frequency shifts remains. If a linear FM
signal is used, a single matched filter can be used for all
depths, with no need to compensate for the frequency shift­
ing caused by attenuation. This reduces significantly the
required signal processing. Application of depth-dependent
filtering for linear FM signals has limited use because it
will not improve resolution or range sidelobe levels and
will increase only slightly the GSNR.
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 2, february 2005
cases in which SNR is not a problem, the ideal filter should
have a transfer function equal to the inverse of the input
spectrum:
IX. Alternative Compression Filtering
A. Generalized Matched Filter
The matched filtering concept can be extended to non­
white or colored noise, i.e., noise that its power density
N (f ) is a function of frequency. In this case, the filter
that maximizes the SNR at its output has a transfer func­
tion [22]:
H(f ) = ke−j2πf τd ·
Ψ∗ (f )
.
N (f )
(28)
Assuming that the noise power density N (f ) can be
∗
factored such as N (f ) = Nm (f ) · Nm
(f ) = |Nm (f )|2 , the
generalized matched filter is equivalent to a cascade of two
Ψ∗ (f )
filters, with transfer functions Nm1(f ) and N
. When the
∗
m (f )
noise passes through the first filter (which is sometimes
referred to as the whitening filter), its power density be­
comes:
1
N (f ) ·
Nm (f )
2
= 1,
i.e., the noise at the first filter output is white. At the
same time, the signal spectrum is altered by the whitening
filter. The second filter is the conventional matched filter,
which is matched to the distorted signal in the presence
of white noise. Thus, optimal detection in nonwhite noise
is achieved by first whitening the noise then applying a
conventional matched filter on the distorted signal.
B. Inverse Filter—Noise-Free Speckle
1. The inverse filter: The returned signal r(t) is given
by the temporal convolution of the transmitted signal ψ(t)
with a scattering function γ [44]:
r(fr), t) = ψ(t) ∗ fm (fr) ∗ hpe (fr, t) = ψ(t) ∗ γ(fr, t),
t
r
t
,
'
(29)
γ((
r,t)
where fm is the scattering (or reflectivity) function of the
medium due to the perturbations in density and speed of
sound, and hpe is the pulse-echo spatial impulse response
that relates the transducer geometry to the spatial extent
of the scattered field [44].
In a matched filter receiver, r(t) is correlated with the
transmitted signal:
∗
∗
x(fr, t) = ψ (t) ∗ r(t) = ψ (t) ∗ ψ(t) ∗ γ(fr, t),
t
t
t
(30)
where x(fr, t) is the matched filter output. The objective of
imaging is the extraction of the scattering function γ(fr, t).
However, the output of the matched filter is the convo­
lution of γ(fr, t) with the autocorrelation function of the
transmitted signal. The autocorrelation function is, thus,
the point spread function or the imaging kernel of the
matched-filter imaging system in the axial direction. In
H(f ) =
1
.
Ψ(f )
(31)
In this way, the convolution of the signal with the reflec­
tivity function of the medium is cancelled. In practice, such
inverse (or equalization) filter cannot be realized because
it has infinite response outside the signal bandpass, and
implementation is based on approximation of the inverse
filter only over a limited frequency range (pseudo-inverse
filter). By suppressing the passband response, the inverse
filter effectively widens the bandwidth and thereby im­
proves the axial resolution, with the expense of amplifying
the relative noise power and degrading the SNR. Inverse
filtering has been used in coded ultrasound imaging by
some investigators [3], [12].
2. Speckle: The human tissue can be considered as
composed by an ensemble of scatterers, which are spaced
much closer than the effective waveform resolution at the
matched-filter output and thus cannot be resolved. The
resulting speckle pattern in an ultrasound image is only
a constructive and destructive interference (depending on
the phases) of the scattering strengths. The speckle sig­
nal is deterministic in the sense that observations are not
independent, i.e., the same signal will result for repeated
measurements. This implies that averaging cannot reduce
the speckle signal as it does with noise. Additionally, the
signal-to-thermal-noise-ratio can be increased by increas­
ing the transmitted power, but this does not change the
signal-to-clutter-ratio because speckle is a result of a con­
volution process.
However, the phase of the speckle is random, but the
amplitude is dependent on the signal spectrum amplitude
[25]. For a large number of randomly distributed scatter­
ers, the speckle signal has Gaussian character; and, if it
is uniformly distributed in range, it can be considered as
stationary Gaussian random signal [24]. The difference be­
tween thermal noise and clutter is that the power spec­
tral density of speckle is frequency dependent, so that the
speckle signal is nonwhite noise. Assuming independent
scatterers, the echoes will have the same frequency spec­
trum as that of the transmitted signal s(t), and the power
density spectrum of speckle Nc will be [22]:
Nc (f ) = kc |S(f )|2 ,
(32)
where kc is the average scattering strength.
When the medium consists of speckle with power den­
sity spectrum given by (32), the generalized matched fil­
ter of (28) gives the inverse filter. The inverse filter will
be the optimal filter when no noise is present. This is,
of course, not a realistic condition, especially when the
spectrum amplitude is a weighted function, as it is when
it passes through the transducer. Even when the noise is
low, it will be significant at the edges of the spectrum.
misaridis and jensen: use of modulated excitation signals in medical ultrasound. part i
C. Wiener Filter—Noise + Speckle
The Wiener filter is derived as the solution to the opti­
mization criterion of the best mean-square fit to a desired
signal over a given time or frequency range, usually with
the best fit over the passband of the input signal. For an
input signal with spectrum Ψ(f ) in the presence of noise
with a spectrum Nf (f ), the Wiener filter has a transfer
function [45], [46]:
H=
ΨΨ∗
Ψ∗
.
+ Nf Nf∗
(33)
If the generalized matched filter is applied to the prob­
lem of detection in the presence of speckle, when Gaussian
receiver noise is added to the filter input, the optimum
receiver should have a transfer function:
H(f ) =
Ψ∗ (f )
.
N0 /2 + kc |Ψ(f )|2
189
The optimality of the transfer function of the Wiener
filter lies on a relative amplification of the spectrum out­
skirts. This effect can be achieved by widening the band­
width of the transmitted signal slightly over the bandwidth
of the transducer.
In [22] it was discussed that the optimum SNR of the
Wiener filter for a fixed signal bandwidth yields a sig­
nal with a rectangular spectrum. It was also shown that
weighting causes only small degradation, which for the
matched filter is independent of the speckle-to-noise ratio.
From all compression waveforms, the one with a rectan­
gular amplitude spectrum is the linear FM signal. These
notes indicate that, in the presence of speckle and receiver
noise, the use of the optimal but speckle-dependent Wiener
filter might not be necessary. It can be sufficient to use a
coded signal with appropriate ambiguity function, such as
the linear FM signal, in conjunction with a mismatched
filter.
(34)
A comparison of (34) with (33) shows that, in this imag­
ing situation, the optimum filter is the Wiener filter. In the
absence of clutter, the Wiener filter is simply the matched
filter, and in the absence of noise, it is the inverse filter.
The transfer function at the outskirts of the spectrum re­
sembles that of the matched filter and at the passband
resembles that of the inverse filter. Therefore, what the fil­
ter does is suppress the bandpass region and effectively en­
hance the outskirts of the spectrum. In this way, the clutter
power is spread and the peak signal-to-average-clutter ra­
tio is improved. Close to the edges, at which the clutter is
not significant but the noise is, the filter is matched to the
signal.
The problem in constructing such a filter is that the
mean scattering strength kc relative to the noise level is
not known. The performance of the filter depends on the
clutter-to-noise ratio, which is unknown and may be vari­
able, depending on the interrogated tissue. However, the
ratio can be estimated based on a measurement of the
noise and the pulse used [47].
D. Discussion on Filter Selection
The optimal filter in ultrasound imaging is the Wiener
filter described in Section IX-C. Rihaczek [24] has studied
the performance loss for the matched and inverse filters
over the optimal Wiener filter as a function of the clutter­
to-noise ratio. By choosing a realistic, weighted spectrum,
he calculated the SNR loss for the three filters under the
constraint of constant power, and reported a performance
loss of 1.4 dB for the inverse filter and only 0.48 dB for
the matched filter. This should not be a surprising result
because the phase characteristics of all three filters are
similar. Considering the practical difficulty of selecting the
clutter-to-noise ratio in the Wiener filter, Rihaczek [24]
questioned the usefulness of using an optimal filter instead
of the matched filter.
X. Conclusions
In this paper, the properties of modulated signals are
stated mathematically, and it is shown how the coded sig­
nals, when processed with a matched filter, can be evalu­
ated in the presence of ultrasonic attenuation using ambi­
guity functions. It is shown that if matched-filter receiver
processing is used, the compressed output is not the au­
tocorrelation function of the code, but a cross section of
the ambiguity function for a certain frequency downshift.
Therefore, the AF of the transmitted waveform ought to
have desired properties in the entire delay-frequency shift
plane. The criteria of selecting the appropriate coded wave­
forms and receiver processing filters have been discussed in
detail. One of the main results is the conclusion that lin­
ear FM signals have the best and most robust features for
ultrasound imaging. Other coded signals such as nonlin­
ear FM and binary complementary Golay codes also have
been considered and characterized in terms of SNR and
sensitivity to frequency shifts. These results are demon­
strated in this paper through computer simulations using
Field II [27].
It is found that, in the case of linear FM signals, a SNR
improvement of 12 to 18 dB can be expected for large
imaging depths of attenuating media, without any depth­
dependent filter compensation. In contrast, nonlinear FM
modulation and binary codes are shown to give a SNR
improvement of only 4 to 9 dB when processed with a
matched filter. In the second paper of this series [19], the
design of appropriate FM coded signals will be described
and evaluated experimentally in phantom and in-vivo mea­
surements. It will be shown then how the higher demands
on the codes in medical ultrasound can be met by ampli­
tude tapering of the emitted signal and by using a mis­
matched filter during receive processing to keep temporal
sidelobes below 60 to 100 dB.
190
ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 2, february 2005
Acknowledgments
The authors would like to thank the reviewers for their
excellent comments, which have helped us to improve this
work.
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misaridis and jensen: use of modulated excitation signals in medical ultrasound. part i
Thanassis Misaridis received the B.E. and
M.S. degrees in 1992 from the National Tech­
nical University of Athens in electrical engi­
neering. He received the M.S. degree in 1997
from the Pennsylvania State University, in
bioengineering, and the Ph.D. degree in 2001
from the Technical University of Denmark,
Lyngby, Denmark.
He was a research scientist at the Labo­
ratoire Ondes et Acoustique in Paris, France,
until 2003. He is currently a collaborator of
Kretz GE Medical Systems in conjunction
with the National Technical University of Athens (NTUA), Greece.
He teaches ultrasound imaging at the University of Patras, Patras,
Greece, and at NTUA.
Dr. Misaridis’ current research interests include coded excitation
in medical ultrasound, array processing, synthetic aperture, and non­
linear imaging.
191
Jørgen Arendt Jensen (M’93–S’02) earned
his Master of Science degree in electrical en­
gineering in 1985 and the Ph.D. degree in
1989, both from the Technical University of
Denmark, Lyngby, Denmark. He received the
Dr.Techn. degree from the Technical Univer­
sity of Denmark in 1996.
He has published a number of papers on
signal processing and medical ultrasound, and
the book Estimation of Blood Velocities Us­
ing Ultrasound, Cambridge University Press
in 1996. He is also developer of the Field II
simulation program. He has been a visiting scientist at Duke Univer­
sity, Durham, North Carolina, Stanford University, Stanford, Cali­
fornia, and the University of Illinois at Urbana-Champaign. He is
currently a full professor of biomedical signal processing at the Tech­
nical University of Denmark at Ørsted•DTU and head of the Center
for Fast Ultrasound Imaging, Technical University of Denmark. He
has given courses on blood velocity estimation at both Duke Uni­
versity and the University of Illinois and teaches biomedical signal
processing and medical imaging at the Technical University of Den­
mark. He has given several short courses on simulation, synthetic
aperture imaging, and flow estimation at international scientific con­
ferences. He is also the co-organizer of a new biomedical engineering
education offered by the Technical University of Denmark and the
University of Copenhagen.
His research is centered around simulation of ultrasound imag­
ing, synthetic aperture imaging, and blood flow estimation, and con­
structing systems for such imaging.