signal reconstruction from discrete-time wigner … · 2020-01-21 · far, few synthesis methods...
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SIGNAL RECONSTRUCTION FROM DISCRETE-TIME WIGNER DISTRIBUTION
bySiuling Cheng
Thesis submitted to the Faculty cf the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
. Master of ScienceL
in
Electrical Engineering
APPROVED:
zu-Dr. Kai-Bor Yu
{5/J/W
5 "'N . "”.R4 /.
Dr. David dewolf Dr. Tri Thuc Ha
March 18, 1985
Blacksburg, Virginia
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pg SIGNAL RECONSTRUCTION FROM DISCRETE-TIME WIGNER DISTRIBUTION
byk\ Siuling Cheng
EElectrical Engineering
(ABSTRACT)
Wigner distribution is considered to be one of the most pow-
erful tools for time-frequency analysis of rumvstationary
signals. Wigner distribution is a bilinear signal transfor-
mation which provides two dimensional time-frequency charac-
terization of one dimensional signals. Although much work has
been done recently in signal analysis and applications using
Wigner distribution, not many synthesis methods for Wigner
distribution have been reported in the literature.
This thesis is concerned with signal synthesis from
discrete-time Wigner distribution and from discrete-time
pseudo-Wigner distribution and their applications in noise
filtering and signal separation. Various algorithms are de-
veloped to reconstruct signals from the modified or specified
Wigner distribution and pseudo-Wigner distribution which
generally do not have a valid Wigner distributions or valid
pseudo-Wigner distribution structures. These algorithms are
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successfully applied to the noise filtering and signal sepa-
ration problems.
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TABLE OF CONTENTS
1.0 INTRODUCTION .................. 1
2.0 DISCRETE-TIME WIGNER DISTRIBUTION AND ITS PROPERTIES 8
2.1 Definition of Wigner Distribution ........ 8
2.2 Properties of Wigner Distribution ........ 9
3.0 SAMPLING TECHNIQUES AND ALIASING EFFECTS .... 14
4.0 SIGNAL SYNTHESIS FROM WIGNER DISTRIBUTION .... 23
4.1 Least-Square Synthesis Using Basis Functions Expan—l
sion ........................ 23
4.1.1 Basis functions and their properties .... 24
4.1.2 Signal estimation from modified auto-Wigner
distribution ................... 29
4.1.3 Signal estimation from modified cross-Wigner
distribution ................... 31
4.2 The Second Method of Least Squares Error Estimation
from Modified Wigner Distribution .......... 33
4.3 Signal Reconstruction Using Outer Product Approxi-
mation ....................... 38
5.0 DISCRETE-TIME PSEUDO-WIGNER DISTRIBUTION AND ITS
PROPERTIES ..................... 48
Table of Contents V
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U6.0 SIGNAL SYNTHESIS FROM PSEUDO-WIGNER DISTRIBUTION 55
6.1 Outer Product Approximation for Signal Synthesis
from Pseudo-Wigner Distribution ........... 55
6.2 Overlapping Method for Signal Synthesis from
Pseudo—Wigner Distribution ............. 60
7.0 APPLICATION ................... 64
7.1 Noise Filtering ................. 64
7.2 Signal Separation ................ 68
8.0 CONCLUSION .................. 103
BIBLIOGRAPHY ..............._ ..... 105
VITA ........................ 109
Table of ContentsN
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LIST 0F ILLUSTRATIONS
Figure 1. The WD of Eq.(3.4) with aliasing. ..... 19
Figure 2. The WD of Eq.(3.5) with aliasing. ..... 20
Figure 3. The WD of Eq.(3.4) with no aliasing. . . . 21
Figure 4. The WD of Eq.(3.5) with no aliasing. . . . 22
Figure 5. The FM chirp signal f(t). ......... 43
Figure 6. The WD of Fig. 5. ............. 44
Figure 7. f(t) recovered by using the method in Sec.4.1. 45
Figure 8. f(t) recovered by using the method in Sec.4.2. 46
Figure 9. f(t) recovered by using the method in Sec.4.3. 47
Figure 10. The PWD of Eq.(5.9) with L=24. ...... 52
Figure 11. The PWD of Eq.(5.9) with L=60. ...... 53
Figure 12. The PWD of Eq.(5.10). ........... 54
Figure 13. The PWD of Eq.(7.2). ........... 70
Figure 14. The signal x(t) in time domain. ...... 71
Figure 15. The signal f(t1=cos(nt/8). ........ 72
Figure 16. The PWD of Fig. 15. ............ 73
Figure 17. The smoothed PWD with M=6. ........ 74
Figure 18. The smoothed PWD with M=24. ........ 75
Figure 19. The smoothed PWD with M=36. ........ 76
Figure 20. The smoothed PWD with M=50. ........ 77
Figure 21. The f(t) recovered from Fig.17. ...... 78
Figure 22. The f(t) recovered from Fig.20. ...... 79
Figure 23. The f(t) recovered from Fig.17. ...... 80
Figure 24. The f(t) recovered from Fig.20. ...... 81
List of Illustrations vii
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Figure 25. The signal f(t) of Eq.(7.6). ....... 82
Figure 26. The signal x(t) of Eq.(7.7). ....... 83
Figure 27. The PWD of Fig.25. ............ 84
Figure 28. The PWD of Fig.26 ............. 85
Figure 29. The smoothed PWD of Fig.28 with M=6. . . . 86
Figure 30. The smoothed PWD of Fig.28 with M=12. . . . 87
Figure 31. The f(t) recovered from Fig.30. ...... 88
Figure 32. The f(t) recovered from Fig.30. ...... 89
Figure 33. The signal f(t). ............. 90
Figure 34. The signal g(t). ............. 91
Figure 35. The signal x(t)=f(t)+g(t). ........ 92
Figure 36. The PWD of x(t). ............. 93' Figure 37. The modified PWD of Fig.36. ........ 94
Figure 38. The contour plot of Fig.36. ........ 95
Figure 39. The contour plot of Fig.37. ........ 96
Figure 40. The smoothed PWD of Fig.36. ........ 97
Figure 41. The modified PWD of Fig.40. ........ 98
Figure 42. The contour plot of Fig.40. ........ 99
Figure 43. The contour plot of Fig.41. ....... 100
Figure 44. The f(t) recovered from Fig.37. ..... 101
Figure 45. The f(t) recovered from Fig.4l. ..... 102
List of Illustrationsl
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l.O INTRODUCTIONIn signal analysis, a stationary signal can be fully de-
scribed either in the time domain f(t) or in the frequency
domain F(w). Because the stationary signal spectral contents
do not change with time, f(t) and F(w) each give complete,
equivalent information of a signal.
F(w) = T f(t) e-jwtdt . (l.l)
f(t) =
EäHowever,these two monodimensional representations (in time
t or in frequency w) do not always display sufficient infor-
mation for signal processing and analysis, especially when
the signal spectral contents vary with time (non-stationary
signal). This problem has received considerable attention in
recent years. Different approaches have been used 1x> deal
with this situation. One of them treats the non-stationary
situation as a concatenation of quasi-stationary ones, such
as short-time Fourier transform approach [l]. In other words,windowing is applied such that the signal is assumed to be
stationary· over· the duration of the window. The Fourier
transform of this windowed signal can be used to characterize
Introduction Q l
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the energy distribution of the signal at a time that is given
by the center of the window. Sliding the window over the
signal allows us to display the variations of this distrib-
ution with time. This yields the spectrogram of the signal.
However, short-time analyses are known to suffer drawbacks,
especially according to the quasi-stationary assumptions. The
short-time Fourier transform then faces the well-known com-
promise of either widening the window for a better resol-
ution, or shortening this duration to assume a better
quasi-stationary but suffering then a poorer resolution. An-
other approach is to look for a true time—frequency repre-
sentation. Various definitions of time—frequency
representation have been proposed [2-5], each with its own
merits and drawbacks. A recent series of papers by Claasen
and Mecklenbrauker [6,7] has shown that the Wigner distrib-
ution auui its evolutional version (the pseudo-Wigner dis-
tribution) were good candidates. The Wigner distribution is
defined as [6]:
WfIg(t,w) =-E e-jwTf(t+1/2)g*(t-1/2)d1 . (1.3)
Wigner distribution has many important properties which make
it very attractive for time-frequency analysis of
non—stationary signals and systems. Many other time-frequency
representations, such as the ambiguity function used in radar
and sonar, and the spectrogram used in speech enhancement and
Introduction I 2
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compression, can. be derived from it via linear transf-
ormations [8]. There are various methods to evaluate theWigner distribution. One efficient method is based cui dig-
ital signal processing which is to convert an analog signal
into a discrete-time signal and to determine a discrete-time
version of the Wigner distribution. The discrete-time Wigner
distribution is defined as [7]:
(1.4)
The determination of this distribution function involves in-
finite summation which requires the signals to be known forall time. In practical implementations it is necessary to Vwindow the signals before the discrete-time Wigner distrib-ution can be computed. This leads to the definition of thepseudo-Wigner distribution. Pseudo-Wigner distribution is a
filtered version of the Wigner distribution of the signals,
where filtering takes place in the frequency variable. The
discrete-time[pseudo-Wigner distribution is defined as [7]:
pwof (¤,6) =2'gk=—L(1.5)
Utilizing Wigner distribution for signal analysis and appli-
cations has rapidly increased. This causes the demand of de-
veloping the synthesis technique for Wigner distribution. So
Introduction 3
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far, few synthesis methods have been reported in the litera-
ture. In this thesis, the synthesis algorithms for
discrete-time Wigner distribution and discrete-time
pseudo-Wigner distribution are investigated. These synthesis
procedures can be used for noise filtering and signal sepa-
ration in applications.
Signal synthesis from Wigner distribution is a least squares
approximation problem [23]. Three algorithms of signal syn-[
thesis from discrete-time Wigner distribution are investi-
gated in this thesis. The first algorithm is a least squares
procedure using basis functions expansion. Two sets of basis
functions are investigated for signal synthesis applications.
The second algorithm is first developed by Boudreaux-Bartels _and Parks [9] using least squares procedures. Since the
even-indexed and odd-indexed samples are not coupled, this
synthesis procedure leads us to find the even-indexed and
odd-indexed sequences separately. Wigner distribution gives
a description of an one-dimensional signal f(n) by a
two-dimensional pattern. Inverse Fourier transform of Wigner
distribution gives a two-dimensional discrete-time function.
This two-dimensional discrete-time sequence is recognized to
be the outer product of two one-dimensional sequences. The
third algorithm then involves the factorization procedure
where the desired one-dimensional signal sequence can be re-
covered as the solution of this factorization problem.
Introduction :4
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Due to the fact that even-indexed and odd-indexed samples are
not coupled, signal recovery from discrete-time Wigner dis-
tribution is possible up to a constant factor, but it re-
quires different procedures for even-indexed and odd-indexed
samples. In general, it is not possible to determine both
even-indexed and odd-indexed samples of a signal up to the
same constant factor. Combining the even-indexed and
odd-indexed sequences is then not unique unless we assume
that L samples of signal f(n) are known with L 2 2. If the
discrete signal f(n) is obtained by sampling za continuous
signal with a sampling rate of at least twice the Nyquist
rate, the even-indexed or the odd-indexed samples fully de-
scribe the signal. Two approaches to combine the even-indexed
and odd-indexed samples are discussed in this thesis.
In a number of practical applications [10,11], it is desira—
ble to modify the pseudo-Wigner distribution and then esti-
mate the processed signal from the uwdified pseudo-Wigner
distribution. In speech, for example, the original signals
are often corrupted with noise. We use a smoothed
pseudo-Wigner estimator to smooth the noise in a
time—frequency plane and then the signal is reconstructed
from the modified version of the pseudo-Wigner distribution.
As another example, in the multicomponent signal case, the
bilinear structure of the Wigner distribution creates
cross-terms without any real physical significance. In order
Introduction 5
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to separate the signal from the cross-terms, we use a
smoothing process to reduce the cross-terms in time-frequency
plane; then signals are estimated from this modified version
of the pseudo-Wigner distribution. For application purposes,
two synthesis algorithms for pseudo-Wigner distribution are
developed in this thesis. Both algorithms are formulated in
the similar way as the third synthesis methmd for Wigner
distribution which involves the outer product approximation.
The first method does not involve overlapped segments. The“
second synthesis method is an overlapping method where a
section of unknown samples can be determined by using a por-
tion of known samples determined in a previous section. This
process continues as the reconstruction of a new section
makes the reconstruction of the next overlapping section
possible. This procedure is motivated by an overlapped method
used in signal synthesis from Short Time Fourier Transform
[12].
This thesis consists of eight chapters. Chapter 2 gives a
brief review of vügner distribution. Since most practical
applications of the Wigner distribution will involve digital
_ processing of sampled data, a numerical evaluation of the
time-frequency distribution is required. The sampling tech-
niques and aliasing effects are discussed in Chapter 3
[14,15]. Chapter 4 is devoted to signal synthesis from Wigner
distribution. Three synthesis algorithms are presented there.
IntroductionN
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The concept of pseudo—Wigner distribution and its properties
are discussed in Chapter 5. Two synthesis methods for
pseudo-Wigner distribution are introduced in Chapter 6. Ap-
plications are discussed in Chapter 7. Chapter 8 concludes
this thesis.
Introduction 7I
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2.0 DISCRETE-TIME WIGNER DISTRIBUTION AND ITS PROPERTIES
Discrete-time Wigner distribution is a bilinear signal
transformation which provides a time-frequency characteriza-
tion of a signal. It has some important properties which make
it attractive for analysis of non-stationary signals and
systems. The Wigner distribution was proposed by Wigner for
application in quantum mechanics by means of joint distrib-
utions of quasi-probabilities in 1932 [2]. The theory of
Wigner distribution has been recently reviewed from a signal
processing view point by ·Claasen and Mecklenbrauker
[6,7,8,13].
2.1 DEFINITION OF WIGNER DISTRIBUTION
The Wigner distribution can be evaluated both from the dis-
crete time signals and from the Eburier transform of the
signals. The cross-Wigner distribution of two signals f(n)
and g(n) is defined by [7]:
wf (¤,6) = 2 2 e—j2kBf(n+k) g*(n-k) . (2.1)lg k=_„
A similar expression of Wigner distribution for spectra F(8)
and G(6) can be obtained by:
Discrete-Time Wigner Distribution and its PropertiesN
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1 “ jzm) +=WF G(8,n) = — f e F(8+w) G (8-w) dw. (2.2)
' n —n
By letting g(n) = f(n) or G(6) = F(8), we obtain the
auto—Wigner distribution Wf(n,8). The two distributions (2.1)
and (2.2) have the relation:
WF’G(8,n) = Wf’g(n,B). (2.3)
2.2 PRQPERTIES OF WIGNER DISTRIBUTION
The properties of Wigner distribution reviewed here are based
on the articles by Claasen and Mecklenbrauker [6,7] and a
very detail description of the signal theoretical background
of the Wigner distribution can be found in those references.
Pl: The Wigner distribution is real for both real or complex
discrete—time signal f(n).
Wf(¤„9) — Wf(¤, ) (2-4)
P2: Wigner distribution is discrete in time domain n and
continuous in frequency domain 8. With respect to 9, thel
function is periodic with period n.
Discrete—Time Wigner Distribution and its Properties 9
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Wflg(n,6) = WfIg(n,8+w) (2.5)
P3: Wigner distribution has the same time support as
signals, i.e.,4
if f(n) = g(n) = O, n < Ha or n > nb
then WfIg(n,6) = O, n < Ha or n > nb. (2.6)
P4: The band limited characteristics of discrete-time signal
on the Wigner distribution is different from the
continuous—time case due to the fact that the Wigner dis-
tribution for continuous-time case is not periodic, while the
Wigner distribution for the discrete-time signal is periodic
in 8 with a period n, i.e.,
if F(6) = G(G) = O, Ba < 6 < Bb and Bb — Ba > n
then Wflg(n,9) = O, Ga < 6 < Gb- n. (2.7)
P5: The time shift of the signal f(n) yields the same time
shift for the Wigner distribution. Similarly, modulation of
a signal f(n), which corresponds to za frequency shift of
Discrete-Time Wigner Distribution and its Propertiesq
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F(8), yields the same frequency shift for the Wigner dis-
tribution, i.e.,
if f(¤) = g(¤-k), ·then Wf(n,8) = Wg(n-k,8). (2.8)
Similarly,
if f(¤) = q(¤) ejnw,then Wf(n,6) = Wg(n,G—w). (2.9)
P6: The Wigner distribution of a modulated signal is a con-
volution in frequency domain, i.e.,
if f(¤> = q(¤> m(¤),n/2then wf(¤,6) = gg 1 wg(¤,w) wm(¤,6—w) dw. (2·lO)
-n/2
P7: An inverse Fourier transform yields a two-dimensional
time function y(n,m),
TI' .y(¤,m) = gg 1 ejgm Wf(n,8/2)d8, (2·ll)"TT
or this can be written in the form
Discrete-Time Wiqner Distribution and its Properties ll
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w/2 . 6) dg. (2.12)-n/2
If Wf(n,6) is a valid Wigner distribution, then y(n1,nz) =
f(n1)f*(nz). That is, the necessary and sufficient condition
of Wf(n,6) that describes signal f(n) is that the matrix
[y(n1,¤z)] is a rank 1 matrix. Then we can rewrite (2.12) as:
1 "/2 j(¤ -¤ )6 ¤ +¤ * (2-13)E; J e 1 2 Wf(—1E—2,8)d8 = f(n1)f (nz). °
-n/2
iBy letting nl = 2n, Hz = O, we obtain:
n/2 .1 eJ2“° w (¤,6) da. (2·14)_ 2n f
-n/2
Similarly if nl = 2n—l, nz = l gives
1 n/2 . _£(2¤-1) f*(l) = Eé 1 e12(“ 116 Wf(n,9) aa. (2·15)
” -w/2
From the above two equations, we find the synthesis proce-
dures of reconstructing the even and odd numbered samples of
signal f(n) are different. This is due to the fact that in
the Eq. (2.1) only the even or the odd samples occur simul-
taneously.
Discrete-Time Wigner Distribution and its Properties 12
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P8: Integral Wigner distribution over one period in fre-
quency domain is equal to the instantaneous signal power,
/2 .A“ _ 2 (2.16)
Zw J Wf(n,8) d6 — |f(n)| .-n/2
P9: The summation of the Wigner distribution defined in Eq.
(2.1) over the time variable at certain frequency 9 does not
yield the energy density spectrum of f(n) at this frequency.
Instead we have,
E Wf(n,6) = |F(9)|’+ |F(9+¤)|’. (2.17)¤=·¤
The aliasing term in (2.17) will cause some problems later
in various equations. In next section, we will discuss the
sampling methods to avoid aliasing if the signal f(n) is the
sampled data.
Discrete-Time Wigner Distribution and its PropertiesM
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3.0 SAMPLING TECHNIQUES AND ALIASING EFFECTS
Although the Wigner distribution for continuous-time signall
fc(t) (we use fc(t) to denote a continuous time signal here-
after) has a number of attractive properties, the evaluation
for continuous-time Wigner distribution is very difficult and
may even be impossible. One solution is to sample the
continuous-time signal. The discrete-time Wigner distrib-
ution can be determined efficiently by digital computer or
by other digital hardwares. The determination of the
discrete-time Wigner distribution involves infinite summa-
tion which requires the signal to be known for all time. In
practical implementations it is necessary to window the sig-
nal before the discrete-time Wigner distribution can be com-
puted. This leads to the pseudo-Wigner distribution, which
will be discussed in Chapter 5.
If the continuous-time signal fc(t) is band-limited,
' F(w) = 0, |w| > wc,
the Wigner distribution can be completely determined by the
samples {f(nT)}, where {f(nT)} is a sequence of samples of _
the continuous-time signal fc(t) taken periodically with
sampling period T.e
Sampling Techniques and Aliasing Effects 14
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Wf(nT,w) = 2T Z e-j2wkT[f(n+k)T][f*(n-k)T],k=·~
for T S n/Zwc. (3.1)
The continuous-time Wigner distribution can be recovered by
using the interpolation:
=N sin w(tgT - n)Wf(t,w) Z Wf(nT,w) _ H) ,
nz-” for T S n/2wc. (3'Z)
When T = 1 of (3.1), we can obtain the Wigner distribution
for a discrete-time signal f(n) with a unit sample period:
Wf(n,6) = 2 E e-jzka f(n+k) f*(n-k). (3.3)k=·~ 1
The problem with discrete-time Wigner distribution is that
it suffers from the effect of aliasing. The aliasing problem
occurs because of the fact that Wf(n,8) has a period 11
whereas F(6) has a period of Zn.
In order to illustrate the effect of aliasing on Wigner dis-
tribution, we present two examples below. For convenience,
we restrict ourselves to real signals only.
As a first example, Fig. 1 presents discrete-time Wigner
distribution for sampled continuous-time FM signal fc(t):
Sampling Techniques and Aliasing Effects5
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fc(t) = cos [w„t f ßsinwmtl 100 S t S 200, (3.4)= O otherwise.
where «»„ = w/30, wm = w/150, B = lO7. We sample the signal
fC(t) at a rate higher than the Nyquist rate, but lower than
twice of the Nyquist rate. As a second example, Fig. 2 is the
discrete-time Wigner distribution of a linear chirp signal
which is given by
- 2gc(t) cos (at ) O S t S 240, (3.5)= 0 otherwise.
where a == n/500. We again sample the chirp signal at a rate
higher than the Nyquist rate but lower than twice of the
Nyquist rate. In both examples, frequency and time increments
are respectively Aw = w/121 and An = 5. It can be seen that _
severe aliasing contributions occur because Wigner distrib-
ution repeats itself around 6 = w. (Since Wigner distribution
involves infinite duration, the Wigner distribution we com-
puted above is actually the pseudo-Wigner distribution.)
There are several methods to avoid the aliasing:
OVERSAMPLING: The signal is sampled with at least twice the
Nyquist rate; that is, the discrete signal has a Fourier
transform F(8) that is zero over an interval covering at
least half of its period, i.e.,
Sampling Techniques and Aliasing Effects 16
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F(8) = O n/2 < |6| < n. (3.6)
Now we want to test the two signals in example 1 and 2 with
a sampling rate higher than twice the Nyquist rate. Fig 3 and
4 shows that the Wigner distribution has no aliasing con-
tributions for any signal that samples at least twice of the
Nyquist rate, where Fig. 3 corresponds to Fig. 1, and Fig. 4
corresponds to Fig. 2.
INTERPOLATION: The second method involves interpolation.
An analog waveform f(n) = fc(nT) is sampled at the Nyquist
rate and then interpolated by a factor 2, i.e.,
g(n) = fc(nT') = fc(nT/2). (3.7)
Clearly, g(n) = f(n/2) for n = O, :2, :4, ... but we must filll
in the unknown samples for all other values of n by an in-
terpolation process. This can be done using a digital filter
[1].
ANALYTIC SIGNAL: For an analytic signal, we can avoid the
aliasing by using data sampled at the Nyquist rate. This is
because the frequency spectrum of the analytic signal van-
ishes for negative frequencies. .A discrete—time analytic
signal is defined in [16] by
Sampling Techniques and Aliasing EffectsN
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Afa(¤) = f<¤> + i f<¤) (3-8)
where f(n) is real and is the discrete Hilbert transform
of f, defined by
A ‘ 2= sin (m—n)wg2 (3.9)f(n) 2 f(m) (m_n) U/2
m¢n
The relation between the spectrum F(B) of the original signal
f(n) and the spectrum Fa(8) is given by:
Fa(8) = 2F(6) 0 < 6 < n,= E(9) 8 = O,= 0 -11 < 8 < 0. (3.10)
By evaluating the Wigner distribution from the analytic sig-
nal rather than the signal itself, we avoid the interference
between positive and negative frequency components.
Sampling Techniques and Aliasing EffectsH
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l ffll ly l
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I
III2'·
I‘I„IIIe·»—Iz I , II I I II,„é„„2ae2a=e·IIIIIIIIII I II I I I ‘’II
I II II‘I·'·I ’ I I ' IIII· · ,:222%,} I I, J II. I I _ II I; „ I,
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. .. ' •
g.Figure 2. The wm of Eq (3·5) with allasln
· · ecff ts JJ 20 JSampling Techniques and Aliasing E
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I./I? IIIII II\I·III M III·I -. I < I 4 · E * 2z;2asee2eezz='Ä II* I I II I I~« Al I
. IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII||I||||||||I||||||||l|||||||||||I||||l
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· h. liasihg.Figure 3. The WD of Eq.(3.4) Wlt no a
~ md Aliasing Effects 21Samplimg·Tech¤1ques a
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ling Techniques and Aliasing
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4.0 SIGNAL SYNTHESIS FROM WIGNER DISTRIBUTION
This chapter is concerned with signal synthesis from Wigner
distribution. Three algorithms are investigated for deter-
mining the discrete-time sequence of which the Wigner dis-
tribution best approximates a modified or specified Wigner
distribution. In section 4.1 we derive a least-square syn-
thesis procedure using basis functions expansion. In section
4.2 we discuss another least—square synthesis method in which
the error minimization procedure can be broken up into two Iseparable time-domain minimization procedures that depend
upon only the even indexed samples or the odd indexed samples
for the optimal solution. A third algorithm is given in sec-
tion 4.3. This is an ad—hoc synthesis procedure which in-
volves factorization by using outer product.
4.1 LEAST—SQUARE SYNTHESIS USING BASIS FUNCTIONS EXPANSION
In this section we present a least square fitting algorithm
for signal reconstruction for modified Wigner distribution.
The synthesis algorithm developed here follows the synthesisW procedure of radar ambiguity function described in [17].
Signal Synthesis from Wigner Distribution 23
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4.1.1 BASIS FUNCTIONS AND THEIR PROPERTIES
Wigner distribution can be expanded according to cross-Wigner
distribution associated with aa set of special orthonormal
basis. Two sets of basis functions are investigated in this
section. Related work can be found in [22]. We suppose that
for a discrete—time signal f(n) there exists a set of basis
functions {¢i(n)} which satisfy:
Lf(n) = 2 ai¢i(n). (4.1)
i=—L
The basis functions are assumed to vanish outside an interval
-L s n s L and to be orthonormal:
¢i(¤)'= O I¤I > L,2
L ·k .ni-L¢i(¤) ¢>j(¤) - öilj- (4-2)
Because the even—indexed and the odd-indexed values of a
cross-Wigner distribution of the basis functions are not
coupled, the sequence {¢i(n)} is required to satisfy an ad-
ditional constraint below in order to obtain an orthogonal
set of the cross-Wigner distribution of the basis functions,
+ - - = . .L * * 6 6 43Z_ m) llk ßlp ( )
n,m——L
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Let's.illustrate the sequence of basis functions by two ex-
amples.
Example 1:
6i n -L S n S L¢1(“) = { 0 atherwise. . (4·4)
L * L=ni-Löilnöj n = 6i j, (4.5)
which satisfies (4.2), the basis functions are orthonormal.
Änd
L L ·k 1%E Z ¢-(¤+m) ¢ (¤+m) ¢ (¤-m) ¢ (¤-m)n=—L m=-L 1 k 2 P
L L‘ U
=m=§L mé_Löi,n+m 6k,n+m 62,n-m öp,n—m
L= 2 6 . 6 . 6 .
n=_L k,1 £,2n 1 p,2n 1
= öilk ößlp , (4.6)
satisfies the condition (4.3).
Example 2:
. 2n . ———— (4.7)———— /2L+l -L S S L¢ (n) Z { exp(3 2L+l in) / ni O otherwise.
Signal Synthesis from Wigner DistributionN
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L * 1 L 2n ZnEEn=-Ln=-L
L_ 1 . 2n ._— 21+ 1 E exp(J2L+l¤(l k>>n=-L
= öi k. (4.8)
And
. n+m n+m n-m n-mg g ¢ ( ) ¢*( ) ¢ ( ) ¢*( )k ß pn=-L m=·L 1LZL
_ 1 . 2u ._ _k+p “>]n=-L
L . 2n .•Em=·L
= 61-k+p-1,0 Ö1-k—p+z,o (4*9)
The last expression in the above equation equals 1 when
i-k+p-2 = 0{ (4.10)i-k—p+£ = 0
From (4.10) we have
Signal Synthesis from Wigner Distribution 26
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i = k{ (4.11)P = Z
Rewriting (4.9)
L 4 4Z ¢i(n+m)¢k(n+m)¢ß(n-m)¢p(n—m) = öi kön p. (4.12)
n,m=—L ’ '
The above two examples demonstrate that both sets of basis
functions satisfy the requirements (4.2) and (4.3).
Substituting (4.1) into the auto—Wigner distribution (3.3)
and interchanging the summation gives
L *” -j2k6 4
Wf(n,6) = 2 E a.a. Z e ¢.(n+k)¢.(n—k) (4.13). ._ 1 3 _ 1 31,3--L k--»
Ki j(n,G) which is denoted as aa cross—Wigner distribution
between a pair of basis functions:
K. .(¤,6) = 221,3k=__ 1 3K -j2k8 2
= 2 Z e ¢i(n+k) ¢j(n-k) (4.14)k=·K
lwhere |n| S L, and K 2 L — |n|.
Substituting (4.14) in (4.13) gives
L L *W (n,9) = E 2 a.a. K. .(n,8) (4.15)f i=-L j=—L 1 J l']
Signal Synthesis from Wigner Distribution 27
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FIt is easy to show that the set {Ki j(n,B)} is orthogonal,
11 n/2
*K. .,K = — J K. . ,9 K ,9 d9 4.16
Inserting (4.14) in (4.16) and interchanging the order of
integration and summation give
K. .,K =( 113 Plq)
. n . n- n n-4 ä ä ¢ +4>¢ ( 4)n=-L k,£=-K 1 J p q
1 n/2 . _• — 1 eJ2°(“ k)d6 (4.17)n —n/2
The integration of the above expression gives
1 n/2 . _ 1, Z = k,— 1 6J2B(“ k)a6 =.( (4.18)n -w/2 0, 2 # k.
Return to (4.17) we have
L K 1l· ·kK. .,K = 4 2 2 . +k +k -k . —k 4.19
By setting the summation limit K = L and employing (4.3) to
above equation, we finally obtain
L 1 n/2*Z — J K. .(n,9)K (n,9)d9 = 46. 6. (4.20)ll] p/q 1/p Jrq
Signal Synthesis from Wigner DistributionN
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* 4.1.2 SIGNAL ESTIMATION FROM MODIFIED AUTO-WIGNER „.
DISTRIBUTION
Y(n,6) is a given modified auto-Wigner distribution. In
general, Y(n,8) is not a valid auto—Wigner distribution in
the sense that there is not a sequence of which the
auto—Wigner distribution is given by Y(n,6). We develop the
algorithm to estimate a sequence f(n) whose auto—Wigner dis-
tribution Wf(n,9) is closest to Y(n,6) in the squared error -
sense. In other words, we use the least-square algorithm that
Jminimizes _J
L 1 n/26 = Z — f |Y(n,8) - Wf(n,8)|1d6 (4.21)
n=-L n -n/2
Inserting (4.15) into (4.21) gives
L 1 n/2 L*s = 2 — J |Y(n,8) - 2 aia.Ki .(n,8)|1d8. (4.22)
n=-L n —n/2 i,j=—L J ’J
This expression is to be minimized by choosing the coeffi-
cients {ak}. Expanding the equation and utilizing the
orthogonality of Ki j(n,6) gives
s = + a.a. — a.a. ..- a.a. ..||Yu= 4 IS g *|= IE *6* I5 * 61,j=—L 1 J 1,j=-L 1 J 1J 1,j=-L 1 J 1J
(4.23)
where
Signal Synthesis from Wigner DistributionM
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1 n/2HYH2 = E — f |Y(n,9)|2d9, (4.24)
n n —n/2
1 w/2*Bi . = 2 — J Y(n,B) Ki .(n,8) dB. (4.25)'J ¤ « -«/2 ·J
The minimization of s with respect to the coefficients ai
will provide a least-square approximation to the desired
function. We treat ai and ai* as independent variables whenminimizing the mean square error s.
86 L L *L——*= 0 = 8 a Z lailz — Z aiBi - 2 aiB i, (4.26)
aa p i=—L i=—L p i=—L pP
Lwhere E lailz is the signal energy E. The equation (4.26)
i=—Lcan be formulated in matrix notation as
(B + B) a = 8E a = X B, (4.27)
where B == (Bi j) is a (2L+l) by (2L+l) matrix and B is the
conjugate transpose of B. _a is the normalized eigenvector
of signal coefficients that needed to be determined. By
carrying out the eigenvalues and eigenvectors decomposition
of the Hermitian matrix (B + B), we observe that there are
two significant positive eigenvalues X1 and X2, i.e.,
(E + E) B X1 ä<1)ä<1) + X2 ä<2>ä<2> (4·28)
Signal Synthesis from Wigner Distribution 30
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Lf,(n) = 2 /X1/8 ai"’¢i(n), (4.29)
i=-L
and
Lf,(n) = 2 /X,/8 ai‘2’¢i(n), (4.30)
i=-L
where ai"’ is the ith element of eigenvector a"’, and ai‘“’
is the ith element of eigenvectorg‘2’.
We observe that f1(n)
is the even indexed sequence of f(n) since all the
odd—indexed values of f1(n) are zero. Similarly, f2(n) is the
odd—indexed sequence of f(n).
L L/2 L/2-1{f(n)} = ¤{f1(2n)} + ß{f2(2n+1)} , (4.31)
n=-L n=-L/2 n=-L/2
where L is an even integer. From the observation, we find the
reconstructed signal f(n) is a separable sequence which di-
vides into an even—indexed sequence and an odd—indexed se-
quence. This is due to the fact that in the sum (3.3) only
even-indexed or odd-indexed samples occur simultaneously.
4.1.3 SIGNAL ESTIMATION FROM MODIFIED CROSS—WIGNER
DISTRIBUTION
For signal synthesis from a given modified cross-Wigner dis-
tribution Y(n,9), we write the total square error of the ap-
proximation as in section 4.1.2.
Signal Synthesis from Wigner DistributionJ
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L 1 w/2/
6 = E — f |Y(n,6) — Wf (n,8)|2d8 (4.32)n=-L n -n/2 ’g
We suppose the signals f(n) and g(n) satisfy:
Lf(n) = 2 ai¢i(n) (4.33)
i=-LL
g(n) = E bi¢i(n) (4.34)i=-L
Substituting (4.33) and (4.34) into (2.1), the cross—Wigner
distribution can be written as
L L *wflg(n,8)=i=EL
ji-Laibj Ki j(n,6), (4.35)
and the mean-square error is given by '
L 1 n/2 L L *6 = Z — f |Y(n,6) — E E aib.Ki .(n,9)|2d9 (4.36)n=-L n -n/2 i=-L j=-L J ’J
Following the same procedure as in section 4.1.2, equation
(4.36) becomes
L L * *6 = HYH2 - 2 2 aib.Bi.i=-L j=-L J J
L L*
L L- E 2 a.b.B..+ 4 2 2 |a.l2|b.|‘ (4.37)1 3 13 1 3i=-L j=-L i=-L j=-L
Now we treat ai, aä, bj and b; as four independent vari-
ables when minimizing the mean-square error 6.
Signal Synthesis from Wigner Distribution 32
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8: L L—— = 0 = — 2 b. B .+ 4 a 2 lb.|2 (4.38)3a* j=_L 1 pi pj=_L 1P
3: L*
L—— = 0 = - E a. B. + 4 b Z |a.|2 (4.39)8a* i=-L 1 lp pi=-L 1
P
Formulating (4.38) and (4.39) in matrix notation, we have
B B = 4Eb B (4.40)
B B = 4Ea B (4.41)
Substituting (4.40) into (4.41) and vice versa gives
B B B = 16 EaEb B (4.42)
B B B = 16 EaEb B (4.43)
By carrying out the eigenvalues and eigenvectors decompos-
ition of matrices BB and BB, we obtain the reconstructed
signals f(n) and g(n) respectively up to a constant factor.
4.2 THE SECOND METHOD OF LEAST SQUARES ERROR ESTIMATION FROM
MODIFIED WIGNER DISTRIBUTION
Let f(n) and v%(n,9) denote a sequence of signal and its
auto—Wigner distribution. From the definition, the
auto—Wigner distribution is
Signal Synthesis from Wigner Distribution 33
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Wf(n,8) = 2 x e_j2k6f(n+k) f*(n-k) (4.44)k=·¤
Given a modified auto—Wigner distribution Y(n,6), the inverse
transform of this distribution is a two dimensional time
function
1 n/2 .m6 .y(n,m) = — J Y(n,6) el dB (4.45)
n —n/2
In this section we discuss another algorithm to estimate asequence f(n) of which the modified. Wigner distribution
Wf(n,9) is closest to Y(n,9) in the total square error sense.
This algorithm is first developed by Boudreaux—Bartels and
Parks [9].
1 w/26 = Z — f |Y(n,9) — Wf(n,6)|2d9, (4.46)
n n -n/2
where Wf(n,9) is a valid Wigner distribution while Y(n,6) is
not necessarily a valid Wigner distribution. By Parseval's
theorem, equation (4.46) can be written as
6 = 2 2 |y(n,2m) - 2 f(n+m)f (n-m)|’ (4.47)n m
Recalling the Wigner distribution property (7) in section 2,
or Eq. (2.14) and Eq. (2.15), we see that even—indexed or
odd-indexed sample sequences can be reconstructed up to a
constant factor with different procedure. To ndnimize the
Signal Synthesis from Wigner Distribution 34
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mean square error, Equation (4.47) can be broken up into two
separate parts. That will minimize the mean square error of
even indexed sequence and odd indexed sequence separately.
Equation (4.47) can be rewritten asl
s = s + se o= E Z |y(n,4k—2n) - 2f(2k)f*(2n—2k)|2
n k+ E E |y(n,4k-2n—2) — 2f(2k-1)f*(2n—2k+l)|2 (4.48)
n k
To minimize the mean square error se, viz.,
se = 2 Z |y(n,4k-2n) — 2f(2k)f*(2n-2k)|’, (4.49)n k
can be obtained by treating f(n) and f*(n) as independent
variables.
(See/8f*(2p)) = 0 = 4f(2p) E |f(2m)|2m
— Z [y(m+p,2p—2m) + y*(m+p,2m—2p)] x(2m) (4.50)m
The above expression can be formulated in matrix notation as
X f = 4 Hf Hzf (4.51)e —e —e —e
where fe is the normalized eigenvector and Xe = (ym p) with
al-_ ym P = y(m+p,2p-2m) + y (m+p,2m—2p) · (4-52)
Signal Synthesis from Wigner Distribution 35
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Similarly, we have
so = Z 2 |y(n,4k—2n-2) — 2f(2k-l)f*(2n—2k+l)|’ (4.53)n k
(8:0/3f*(2p-1)) = O = 4f(2p—l) E |f(2m-l)|2m
*- E [y(m+p-l,2p-2m) + y (m+p—l,2m-2p)] x(2m-1) (4.54)m
XC fo = 4 l|_-iollz io (4-55)
where
ym p = y(m+p-l,2p—2m) + y*(m+p—l,2m—2p). (4.56)
It can be shown [9] that optimal solutions of even-indexed
sequence fe and odd-indexed sequence fo are normalized
eigenvectors of X6 and XC associated with the largest
eigenvalues ke and ko, respectively.
There are two approaches of combining the even—indexed se- '
quence and odd-indexed sequence. One of them, which was in-
troduced by Boudreaux—Bartels and Parks, is to find the real
parameters a and B such that f(n) is closest to the original
sequence s(n), i.e. minimize
se(¤) = E |s(2n) - fe(2n)ej°|“ . (4.57)I‘1
and
Signal Synthesis from Wigner Distribution 36
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sO(ß) = ! |s(2m—1) - fO(2m-l)ejß|’ . (4.58)m
Clearly, the mean square error se is a function of a and is
minimum if
- - 'ja * 2ase(¤)/aa — 0 - -2Imle Z s(2n)f (2n)—H§€H ] .n
(4.59)
The above equation yields
-1 4 4a = tan {Rel! s(2n)fe(2n)]/Iml! s(2n)fe(2n)]} .
n n(4.60)
Similarly,
-1 + 4ß=tan Rel!s(2n-l)fO(2n—l)]/Iml!s(2n—1)fO(2n-1)] .
n n(4.61)
The second approach is based on a smoothness criterion of
which the phase is chosen in such a way that the resulting
sequence will be as smooth as possible.
E = 2 |fe(2n)ej¤- fO(2n-l)ejß|2 (4.62)n
Error s will be minimum if
Signal Synthesis from Wigner Distributionl
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8s/Ba = O,
86/BB = O (4.63)
‘Then the reconstructed signal is given by
f = f eja + f ejß (4.64)— —e —o
4.3 SIGNAL RECONSTRUCTION USING OUTER PRODUCT APPROXIMATION
The third algorithm is an ad-hoc procedure. Given a desired
time—frequency function. Y(n,9), the inverse transform of
Y(n,6) gives
l n/2 .y(n,2m) = — f Y(n,9) eJ2m8d6. (4.65)
w —w/2
Measuring the error distance between y(n,2m) and a valid
auto-Wigner distribution Wf(n,6) gives
l n/2 .I@(¤,m)I = ly(¤,2m) - — f Wf(¤,9) elzmödßl
n —n/2
= |y(n,2m) - 2 f(n+m) f*(n—m)|. (4.66)
The total square error is
s = E Z e2(n,m) = E Z |y(n,2m) - 2f(n+m)f*(n—m)|2, (4.67)n m n m
Signal Synthesis from Wigner Distribution 38
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which is the same expression we obtained in previous section.
If the given distribution Y(n,S) is a valid Wigner distrib-
ution and if there is no noise in the measurement, we should
have
y(n,2m) = 2 f(n+m) f*(n-m) (4.68)
¤rSince (n+m)/2 must be an integer, n and m can only occur with
both even or odd integer. This tells us that the even-indexed
samples and the odd—indexed samples are not coupled, that is,
the even-indexed samples and the odd-indexed samples cannot
occur simultaneously. If f(n) is aa time-limited signal,
i.e.,
f(n) = 0 n < Ha or n > nb (4.70)
We can form an outer product expression for even-indexed
samples or odd-indexed samples separately. For convenience
we let both Ha and nb be even integer numbers, the even part
of the outer product expression can be written as
+2)lfa I a a b(¤ +2)I a II - IIf‘"" IL.f(¤b) .I
Signal Synthesis from Wigner Distribution 39
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1 gyll ylz ... ... ... ylN}= — Iyzl y22 ... ... ... y2N|
2 I' ' ILyN1 yN2 ... . ... yNNJ (4.71)
where
yi’j<m,¤> = y(@§§. m—¤)1 f¤r 1 2 j,4
yi’j(m,n) = yjli(m,n) for i < j. (4.72)
where i,j are the indices of the matrix, m,n are the corre-
sponding coefficients of the signal, and both I1 and ux are
even integers. Equation (4.71) can be written in matrix no-
tation as
+X9 = 2 je X6 (4.73)
where X; denotes the conjugate transpose of X6. The
even—indexed samples of f(n) can be reconstructed by solving
the eigenvalues and eigenvectors problem of matrix X6,
X f = 2Hf H2 f = X f . (4.74)e —e —e —e e—e
Xe is a rank 1 Hermitian matrix if Y(n,9) corresponds to a
valid Wigner distribution. This can be used as a test for
validity. A similar procedure can be carried out for the
odd—indexed samples, viz.,
Signal Synthesis from Wigner Distribution 40
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Z 2 ZXOQO 2 ngen fo kOfO (4.75)
The expression of (4.72) can also be used to determine the
matrix XC, but now both n and m are odd integers.
If the given distribution Y(n,9) is a modified Wigner dis-
tribution, the matrices X6 and XC may have rank greater than
1. Then we have mean square errors as:
- _ + 2Se - Nie/2 ieéell (4-76)
- _ + 2SO — IIXO/2 ioicll (4-77)
The eigenvectors that minimize the mean square error func-
tions are the normalized eigenvectors of XG and XC associated
with the largest eigenvalues ke and ko, respectively.
Three algorithms for signal synthesis from the discrete-time
Wigner distribution have been discussed. Let us test those
algorithms by an example. Fig. 5 and Fig. 6 show an FM chirp
signal and its Wigner distribution.
f(t) = 50 cos(nt2/1000) -40 S t S 40
= 0 otherwise (4.78)
Signal Synthesis from Wigner Distribution 41
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Fig. 7, Fig. 8 and Fig. 9 show the reconstructed signal from
Fig. 6 using the synthesis methods discussed in Section 4.1,
4.2, and 4.3, respectively.
Signal Synthesis from Wigner DistributionU 4
42
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Y1.0
0.8
0.6
0.H
0.2 ·
0.0
I-0.2
*~0.4g· -0.6
I
-0.8·
-50 g -35 -20 -5 10 25 Q0URIGINAL SIGNAL
Figure 5. The FM chirp signal f(t). _
Signal Synthesis from Wigner Distribution 43
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L
. r' ?„~; ‘ 1l ; 4·¤un•· ‘¤"'l"’ ” [[[[W"F"'' . . °” l.• •' A . ."'!!|I||"!Y
H[}i"*4·•~ ._ ‘‘·4
din!u?muIII|II!1in!.§!sä1III"ml!}niuzzhnul|||IInn•%€é7mä'Ä.; VIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIYIIIIIIIIIIIIIII|||IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII»
Figure 6. The WD of Fig. 5-
· ‘b tion 44Signal Synthesis from Wig¤€r Dlstrl u
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Q T1.0
0.6
0.6
0.*1
0.2 }0.0
-0.2-0.*1x -0.6
-0.6 ·
*1 .0I
1 _'-50 -35
-20Figure7. f(t) recovered by using the method in Sec.4.l.
Signal Synthesis from Wigner Distribution 45
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1.00.80.6 .0.li0.20.0 '
-0.2 §· -0.u i-0.8
. r 1-1.0 l 'V -50 -55 - -20 -5 10 25 ED
Figure 8. f(t) recovered by using the method in Sec.4.2.
Signal Synthesis from Wigner Distribution 46
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71.0 U
0.8
0.6
0.¤i
0. 0{ I-0.2
” ·0.¤l
•SO -85 — -20 -5 10 25 40
Figure 9. f(t) recovered by using the method in Sec.4.3.
Signal Synthesis from Wigner Distribution 47
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5.0 DISCRETE-TIME PSEUDO-WIGNER DISTRIBUTION AND ITS
PROPERTIES
For practical implementations it is necessary to weight the
signals f and g by functions hf and hg, respectively, before
the Wigner distribution can be computed. These weighting
functions are called windows and will slide along the time
axis with the instant k when the Wigner distribution has to
be evaluated. Denoting the windowed signals be fk(n) and
gk(n). HenceU
I
fk(n) = f(n) hf(n—k)
qk(¤) = q(¤) hg(¤-k) (5 1)
With Eq.(2.lO) we can evaluate the Wigner distribution of the
windowed signals with each fixed k:
l n/2 —W (n,6) = —— f W (n,w)W (n-k,9-w)dw (5.2)fk’gk 2n -«/2 f'g h£'hg I
For each window position one gets another Wigner distrib-
ution. We consider only the case when n=k. In this case, the
window is symmetrically located around n; and we obtain
l V/2W (n,6)| = —— f W (n,w)W (O,6—w)dw. (5.3)fklgk ¤=k 2n -«/2 f'g h£'hg
Discrete-Time Pseudo-Wigner Distribution5
48l
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The new function of I1 and 6 is the so called pseudo—Wigner
distribution proposed by Claasen and Mecklenbrauker [7].
PWDf’g(n,6) = Wf lg (n,B) | ¤=k (5.4)k k
Using the Wigner distribution definition (2.1), the above
equation can be rewritten as
_ ” —j2k6 * *PWDf g(n,6) — 2 2 e f(n+k)h(k)g (n—k)h (-k) (5.5)I k=_”
If the windows have a duration 2L-1, i.e., if
h(k) = O, |k| 2 L (5.6)
then
L-1 ._ -32kB * *PWDf (n,6) — 2 Z e f(n+k)h(k)g (n—k)h (-k) (5.7)I g 1k--L+1
The explicit derivation of (5.3) shows that the pseudo-Wigner
distribution is nothing but a frequency smoothed Wigner dis-
tribution [13,18]. This smoothing is equivalent to a convo-
lution in the frequency direction of the Wigner distribution
of the non-windowed signals with the Wigner distribution of
the window functions. An important point is that the
pseudo-Wigner distribution does not give a spread in the time
direction. Consequently, time properties which are held in
Discrete-Time Pseudo-Wigner Distribution 49
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the Wigner distribution are still held in the pseudo-Wigner
distribution. In particular, properties P1, P3, P5, and P8
of Wigner distribution in Section 2 hold for the
- pseudo—Wigner distribution.
We shall illustrate the concept of the pseudo-Wigner dis-
tribution by some examples. We use a normed Hamming window
in all the examples below, i.e.,
h(n) = [0.54+0.46cos(2wn/2L)]/(2L+1), -L S n S L
= 0 otherwise. (5.8)
As a first example, we consider a cosine wave form:
f(n) = cos (nn/8)4
· (5.9)
with L = 24, Aw = 11/49, and An = 2. Fig. 10 shows the
pseudo-Wigner distribution result. Fig. 11 is the
pseudo-Wigner distribution of the same signal with L = 60,
Aw = n/121, and An = 5. From these two plots, we observe that
the wider the window duration, the smaller the spread version
occurs in frequency direction. Furthermore it can be seen
that the spectral contents do not change with time.
As a second example, we consider a chirp signal:
Discrete-Time Pseudo—Wigner DistributionN
50
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f(n) = cos (nnz/1000) (5.10)
Fig. 12 shows the pseudo—Wigner distribution of the signal
f(n) with window duration L = 60, Aw = n/121, and An = 5.
This pseudo-Wigner distribution shows that the frequency
linearly increases with time.
Discrete—Time Pseudo—Wigner Distribution 51
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IFigure10. The PWD of Eq.(5.9) with L=24.
Discrete#Time Pseudo-Wigner Distribution 52
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6.0 SIGNAL SYNTHESIS FROM PSEUDO-WIGNER DISTRIBUTION
In a number of applications, it is desirable to modify the
pseudo-Wigner distribution and then estimate the processedl
signal from it. In this section two synthesis algorithms are
developed for estimating a signal from the time—frequency
function.
6.1 OUTER PRODUCT APPROXIMATION FOR SIGNAL SYNTHESIS FROM
PSEUDO-WIGNER DISTRIBUTION
The synthesis algorithm developed here is an ad—hoc procedure
which follows the same algorithm we introduce in section 4.3.
For a given pseudo-Wigner.distribution
_ L°l * * -j2k6Y(n,9) — 2 E f(n+k)f (n—k)h(k)h (—k)e , (6.1)k=-L+1
with
h(k) = 0, |k| 2 L. (6.2)
The inverse transform of Y(n,B) gives
1 n/2 .y(n,2k) = — f Y(n,9) eJ2kGd8
n —n/2= 2 f(n+k)f*(n—k)h(k)h*(-k) ‘ (6.3)
Signal Synthesis from Pseudo—Wigner Distribution 55
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Hence ..
* _ n 2k (6.4)(H ) (H ) 21:(k)h(-1:)
or
ul-f(n)f (m) = c(n,m)
— * - 6.5= y(H§H„¤—m) / 2h<H;H>h <H;H> ( )
This is in the outer product form again and thus we can syn-
thesize the desired even—indexed and odd-indexed sequences
using an outer product approximation procedure discussed in
section 4.3. We assume the window h(k) is known within the
interval -L < K < L and no zero within this interval, i.e.,
h(k) # O -L < k < L
h(k) = O |k| 2 L (6.6)
The desired even indexed sequence can again be recovered by
solving the eigenvalues and eigenvectors problem:
C f = Hf H2 f = X f , (6.7)—e —e —e —e e —ewhere
-
1%SignalSynthesis from Pseudo-Wigner DistributionH
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with both I1 and H1 are even integers. Similarly, for the
odd-indexed sequence we have,
=2
=EO io |l£O|| io ko io- (6-9)
The expression for CO(n,m) is exactly the same as Eq.(6.8)
except that both n and m are odd integers now. Both ge and
go are rank 1 Hermitian matrix if Y(n,9) corresponds to a
valid pseudo-Wigner distribution. However, for aa modified
pseudo—Wigner distribution, the matrix ge or go may have rank
greater than l. Then we have mean square error functions as:
6 =uc -6 6‘°u=e —e —e —e
6 = nc — f f+H* (6.10)o —o —o —o
The eigenvectors that minimize the mean square error func-
tions are vectors associated with the largest positive
eigenvalues. Thus, the synthesized signals are given by,
f =/x—
u , l—e e —e
fo = Jxo vc, 4 (6.11)and f = a f + B f— —e -o
where xa and xo are the largest eigenvalues corresponding to
ge and go, and ue and vo are the normalized eigenvectors as-
Signal Synthesis from Pseudo-Wigner Distribution 57
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sociated with Xe and XO, respectively. a and ß are constant
factors. The approximation errors are given by
s = nc - X u u+H’= 2 X.e —e e —e—e i¢e 1 _
.. _ "' z ..6O — HQO XO yoyoü — -2 Xi (6.12)l¢O
The outer product procedure can also be used for signal syn-
thesis·from cross-pseudo-Wigner distribution. Inverse trans-
form of cross—pseudo-Wigner distribution of Y(n,6) gives
·k sl-y(n,2k) = 2 f(n+k)g (n—k)h(k)h (-k), (6.13)
orab 9:¢(¤,k) = f(¤><z (k)
k -k * k- 6.14
In general, matrix Q is a rectangular matrix and the desired
signal f(n) and g(n) can be recovered by carrying out the
singular value decomposition procedure [21]. Thus
- rw V7%V7+ge " LE.1 LM (6-15)
where [A] is a diagonal matrix, and matrices [u] and [v] are
unitary matrices, i.e.,
. +M Eu] = [I] — <6.16>Ev] [1/+] = [I] ‘ (6-17)
Signal Synthesis from Pseudo-Wigner Distribution 58
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and
[u] = [ul Q2 ... up] (6.18)[V] = [gl g, gp] (6.19)
We can reconstruct even-indexed sequence of f(n) by finding
the largest eigenvalue and eigenvector of matrix QeQ*e; viz.,
Qe Q; = Xlulut + Xzuzui + ... + Xpupgg (6.20)
fe = ul , (6.21)
where we assume X1 is the largest eigenvalue which is the
product of the energy of signals f and g, and ul is the cor-
responding eigenvector. Similarly, we can reconstruct
even—indexed sequence of g(n) by finding the largest
eigenvalue and eigenvector of matrix Q;Qe;
+ + + +
ge = gl , (6.23)
A similar procedure can be carried out for reconstructing
odd-indexed samples of f(n) and g(n).
Signal Synthesis from Pseudo—Wigner Distribution 59
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.. 6.2 OVERLAPPING METHOD FOR SIGNAL SYNTHESIS FROM
PSEUDO—WIGNER DISTRIBUTION
We assume the window function h(n) is a known sequence with
no zero samples over its finite length and ZP samples of f(n)
for 1 S n S ZP are known. Given a desired pseudo—Wigner dis-
tribution Y(n,8), we can write the outer product expansion
as in section 6.1.
f 6+ = c (6.24)-6 -6 -6
where
rf(2) 1|f(4) | _ (6.25)
f = I - I—e }f(Zp)}
_ n+m _ n—m * m—n (6.26)Ce(¤,mI — y(——2 I Y1 m) / 2h(—2 Ib (TI
with
h(k) = O |k| 2 L. (6.27)
Now let
Signal Synthesis from Pseudo-Wigner Distribution 60
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f IXe
I_. (6.28)
°€ L ge J
with vector X6 contains all the known even samples, and vec-
tor Aa is the unknown samples needed to be determined, i.e.,
V X1 1 V f(Z) 1I x2 I I f(4) I (6.29)
X = . = .—¤ I- I I- II.Xp.I Lf(2p)..I
_ V al 1 Vf(2p+2)7I
azI
If(2p+4)I (6.30)A = . = .‘€ I·I I,-·2 IaL_p ( L)
Let
Cl PI ClIp+l ... ... CIIQI
IC ,1... C I I_C I +1 ... ...C-
I. . | . . I‘
plV
U I V+ 1Z I — I — I (6-3l)
I V I W IL - I - J
Signal Synthesis from Pseudo-Wigner Distribution 6l
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Inserting (6.28), and (6.31) in (6.24), we have
V X 1 V X+ A+ 1I—€
I L __<== _6 JIA IL..e.I
V U I V+ 1=I — I — I (6-32)
I V I W IL - I - .1
where
+X X = U (6.33)
+A X = V (6.34)
+A A = W (6.35)—·€ —€. —
Multiply vector Xe to both sides of (6.34) we have
2 -ge IIXeII —VXel (6-36)
Signal Synthesis from Pseudo-Wigner DistributionJ
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The unknown vector A6 can be determined by
= 2ge Y E8 / lläell (6-37)
Once Aa is determined, we can use A6 as known samples to de-
termine the next unknown section. This process continues un-
til all the unknown samples are determined. It turns out that
this synthesis algorithm is the most straight forward one.
It does not involve eigenvalue and eigenvector decomposition.
It can compute much faster than any of the algorithms that
were introduced before. The approximation error is given by
- "' 2s — HW — A A H (6.38)e — —e—e
A similar procedure can also be carried out to determine the
odd—indexed sequence.
Signal Synthesis from Pseudo—Wigner Distribution 63
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7.0 APPLICATION
This chapter presents two experiments 1x> test the recon-
struction algorithms discussed in Chapter 6. The first ex-
periment is an application for noise filtering. It is, in
essence, to reconstruct a signal from the modified version
of the pseudo—Wigner distribution after a smoothing process
in the time-frequency plane. In the second experiment, sep-
aration of multicomponent signals is considered. In the case
of multicomponent signals, the bilinear structure of Wigner
distribution is known to create cross-terms without any
physical significance. Ihi order to reduce these unwanted
l cross-terms, a smoothing process is applied to the
time-frequency distribution. In our example, two components
of the signal overlap in both time and frequency domains.
However, they are discernible and separable in the
time—frequency' plane. The smoothed pseudo-Wigner distrib-
ution is modified, and a component of the signal is separated
and reconstructed from modified smoothed Wigner distribution.
7.1 NOISE FILTERING
In this experiment, we reconstruct a signal from the smoothed
version of the pseudo—Wigner distribution. A smoothed
pseudo-Wigner estimator is defined as [10],
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N-1SPW(n,8) = E |h(k)|2
k=-N+l
M-1 .• 2 g(£)x(n+£+k)x*(n+£—k)e_22k8 (7.1)ß=-M+1
where g(ß) is a symmetric, normed data window, this smoothing
process only smooths the distribution in the time direction
independently from the frequency smoothing performed by h(k).
It is then much more versatile than the double smoothing used
in spectrogram.
As a first example, Fig. 13 shows the pseudo—Wigner distrib-
ution for a signal+noise case, i.e.,
x(t) = 50 cos (nt/8) + n (t), t=0,l,...,300. (7.2)
where n(t) is a zero-mean Gaussian white noise with variancec2 = 252, h(k) is a normed Hamming window, i.e.,
_ 0.54 0.46 (2k (2L)h(k) (k) < L
= 0 · |k| 2 L (7.3)
We set L=25, Aw=w/49 and sampling period T=1. Time origin
t„=48. Fig. 14 shows x(t) in time domain. Fig. 15 and Fig.
16 show the uncorrupted signal and its pseudo-Wigner dis-
tribution. Smoothed versions of the pseudo-Wigner distrib-
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ution are shown in Figs. 17-20 by using the smoothed
pseudo-Wigner estimator (7.1). The data window g(9.) is a
rectangular window and the degree of smoothing is specified
on each figure as M. Increasing M from 6 to 50 clearly shows
the noise reduction. In order to compare the synthesis re-
sults with the input, we define the input signal-to-noise
ratio as:
SNRi¤= 2 f2(k) / Z n’(k) (7.4)k k
And the output signal-to-noise ratio is defined as:
SNROut= E f2(k) / Z (f(k) - cf(k))“ (7.5)k k
where
c
=f(k)is the reconstructed signal. The reconstructions from
Fig. 17 and Fig. 20 are given in Fig. 21 and Fig. 22, re-
spectively, using the synthesis method developed in Section
6.1. Input signal-to-noise ratio is 2.5 dB. Signal-to—noise
ratios corresponding to Fig. 21 and Fig. 22 are -3 dB and 24.3
dB, respectively. Fig. 21 and Fig. 22 show that the phase of
the reconstructed signal is different from the original sig-
nal (Fig. 15). Consequently, the signal can be recovered up
to a phase factor only. Fig. 23 and Fig. 24 show the recon-
structed signal corresponding to Fig. 17, and Fig. 20 using
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the overlapping method developed in Section 6.2 with l0 un-
known samples being reconstructed in each window section and
the length of each section is 49. Input signal-to—noise ratio
is 3.2 dB. Signal-to—noise ratios corresponding to Fig. 23
and Fig. 24 are 3.9 dB and l7J7 dB. Fig. 23 and Fig. 24 show
that the phase of the reconstructed signal is same as the
original signal. Indeed, the overlapping method reserves the
phase information.
In the second example, we consider the cases:
f(t) = 50 cos (ntz/1000), t=0,l,...,300 (7.6)
x(t) = f(t) + n(t) t=0,l,...,300 (7.7)
Fig. 25 and Fig. 26 show the signals f(t) and x(t) in time
domain, and Fig. 27 and Fig. 28 are the pseudo-Wigner dis-
tribution of the signals. We use the same white noise func-
tion n(t) and same window function h(k) as in the first
example with L=61, Aw=w/121, and sampling period T=l. Time
origin t„=60. The smoothed versions of pseudo-Wigner dis-
tribution of signal x(t) are given in Fig. 29 and Fig. 30.
The reconstructed signals from Fig. 30 are given in Fig. 31
and Fig. 32 using the synthesis methods developed in Section
6.1 and Section 6.2, respectively.
Application 67
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The above two examples show the ability of the smoothed
pseudo—Wigner estimator to discriminate the signal from the
noise, and the signals are successfully reconstructed from
these modified versions of pseudo-Wigner distribution using
the synthesis methods developed in Chapter 6. '
7.2 SIGNAL SEPARATION
In the second experiment, we reconstruct monocomponent
signals from the smoothing version of multicomponent
pseudo—Wigner distribution. If a signal x(n) happens to be
multicomponent, it could be written as combination of
monocomponent ones:
N .x(n) = 2 xi(n) (7.8)
i=1
The Wigner distribution of this signal x(n) is given by
N N iWx(n,6) =·E Wx(n,8) + 2 Re_E li (7.9)
1=1 1 1-1 3-1 1 3
and a similar relation holds for pseudo—Wigner distribution.
Hence, the bilinear structure of Wigner distribution adds
cross—terms without any real physical significance to the sum
of the Wigner distribution of each component. Fig. 33, Fig.
34 and Fig. 35 show two windowed FM chirp signals and the sum
of them, i.e.,
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f(t) = so cos («t=/1450) e“(t'l4°)2/looo (7.10)
g(t) = 50 cos (nt:/900) e-(t-14O)2/1000 (7.11)
x(t) = f(t) + g(t) (7-12)
f(t) and g(t) overlap in both time and frequency domains. The
pseudo-Wigner distribution of x(t) is given in Fig. 36 using
the same Hamming window Eq.(7.3), where L=61,Aw=w/121 and
sampling period T=l. Fig. 36 shows that f(t) and g(t) are
clearly discernible and separable in the time-frequency
plane, and the pseudo-Wigner distribution in Fig. 36 is mod-
ified to produce the modified pseudo-Wigner distribution of
f(t) in Fig. 37. Fig. 38 and Fig. 39 show the contour plots
of Fig. 36 and Fig. 37. Fig. 40 shows the smoothed
pseudo—Wigner distribution of Fig. 36 with the degree of
smoothing‘ M=7, and Fig. 40 is modified to produce the
smoothed time-frequency function of f(t) in Fig. 41. Fig. 42
and Fig. 43 show the contour plots of Fig. 40 and Fig. 41.
The reconstructed signal from Fig. 37 and Fig. 41 are shown
in Fig. 44 and Fig. 45, respectively, using the outer product
synthesis method. This example shows that the pseudo-Wigner
distribution synthesis algorithm can be used for signal sep-
aration.
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·II'III III.I I. I. . -. I.I II I· II III I_I„ In .,I II II II $+2. ·I}I;r..; II IL I' WIR ‘ II ._ I I III I
II'
I-—fI __ IIÄITI.gg; _ _!
II__
LI*IÄITII·-ISI"
I III I IIIIIIIIII.,_r ._ ':I _x , __ I n' III __ __ I I I I I I II I I _ TI-I II \•_·
IIII ,· I' IIII I II‘ III
V ·
Figure 13. The PWD of Eq.(7.2).
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Ai MF, F W W W
Figure 14. -'I‘he signal x(t) in time domain.l
i
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Figure 15. The signal f(t)=cos(nt/8).
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{1I·’ 'I· ‘·II
I·- 5. _—
I __ _‘*1 ·~II I g _eII§III
I_III
III _IIgII __ IIII II IIIIIIIIII.=
IFigure16. The PWD of Fig. 15.
Applicatien 73
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IVIQI .‘ ‘·4•;¤·?‘: •.·'• 1} &· *·e· '} 1,4 '¤. 1 ,751 2%.· ,1IX 'I} II} QII}
Ä II}IIIXII
_'Iggga, "¢1 Ö: V.I} ,. 2}}.; [5, -_'·f•'.”','_,¤ _ XI Ih
"-I IH'! XII I?} Ü} ?-I___ I,
I1; il~ -„ -° rl ‘I 2 I, ‘; I_ In F ’• *7 lo- j[ ’.ä
·I‘I
I I‘_ g' '.F<' ' ', 1 Ä, „·„ «lu:.Figure
17. The smoothed PWD with M=6.
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III
’II|ÜII III I II
I IIII II Igöe I 'III II IIIIIII IIIIIIIIIIIIII I_ IIIIII
III III; IIIIII I
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Application
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Figure 20. The smeothed PWD with M=50.
Applicatien 77
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Figure 21. The f(t) recovered from Fig.l7.
Application 78
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Figure 22. The f(t) recovered from Fig.20.
. Application A 79
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Figure 23. The f(t) recovered from Fig.17.
Application 780
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Figure 24. The f(t) recovered from Fig.20.
Application 81
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SBE
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Figure 25. The signal f(t) of Eq.(7.6). I
Applicati·on 82
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1 If 1. 1 'I _ ,*1 ri :1 ··.;.3IlI
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II I1 •I „„ Z., I
Z 1; 1 I .
Figure 26. The signal x(t) of Eq.(7.7).
Application 183
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: II ·‘
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IJFigure27. The PWD of Fig.25.
Application 84
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I."I. II
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In { III! ~· II». · ·I1 I KLV ' VW III; ly· KV -. II 1 = I ·, I
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Figure 28. The PWD of Fig.26I
Application 85
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III III II II {IIII IIIIIIIIÄII I I{I{III III I I IMIL {III {II ;I« II {· .•·I.·a II{
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Figure 29. The smoothed PWD of Fig.28 with M=6.
Application · 86
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/~.I.•IT! ,AIIIIEIIII II::I~I 'IIII III!
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Figure 30. The smocthed PWD of Fig.28 with M=l2.
Applicatien 87
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0.3-J Ä Igß „ A,‘
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Figure 31. The f(t) recovered from Fig.30.
Application 88
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sa IQ{X
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' *12 1·6 =* MÄ 6 Y6 6 6 6 6% 61fY Y¤' Y6,Y66!Y6Y '“? ' Y Y1, Y6Y‘fYY‘6 >Y+166=2*+‘¤J1 ,111 662 'J ? *6,* 1:°Y.$äf YY;:YYJYY1‘YYY‘@1 J '6 Y EJ 1.1 1,2 f 66 LJ Yi; Z? 1* *61 E J ’.» ~: ‘ ' ‘ 66 2.+ · 22 ~ eJ " ' ‘ ' g
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Figure 32. The f(t) recovered from Fig.30.
Application 89
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J
II
Figure 33. The signal f(t).
Application 90
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I
Figure 34. The signal g(t).
Application · 91
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•‘.
Figure 35. The signal x(t)=f(t)+g(t).
Applicatibh 92
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;: h
WäägggF1gure«36.The PWD ¤f X(‘¤)·
‘'
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QQQQ QQ;QQQQQQQQQQ
Figure 37. 'The medified PWD of Fig·3é·V
Applicatiorx 'l 94
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·vsse _
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‘·°I 110 I 1N I 0
I1 1 1 0 160 190 2
x
Figure 38. The contour plot of Fig.36.
Application 95
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—v
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Figure 39. The contour plot of Fig.37.
Application 96
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' I
I I·I
I:II .=. 1 I
II 4 IV1 Q .„
V.:
V'LI I ^ e I¢- : I I ~V , I
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I .6I
I I·-'llllll’ ·» Iyyyppyi II'
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Applicatiam
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I =. IP;„mjUFIT Z I. QT I E
IIF Ü.I
Ü L? T F‘— •‘: '~rl' EFFIUTT" 1ääszä
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T?Figure41. The modified PWD of Fig.40.
Application 98
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Figure 42. The contour plot of Fig.40.
Application 99
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. ' „ ¤ '
Figure 43. The contour plot of Fig.4l.
Application · _U
100
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9Q
In I I“ "? IIII { _= ' I 2FI II II .61 I 66,
I' II {I I I I4 I I ' I I-, -- f I ' 6I ;_ ·, Ö VII _·JI II I I I I I I I II2 I II ·.I I- ·. I r“r' I I I . Q { ,' I I ,: I . {1
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Figure 44. The f(t) recovered from Fig.37.
A licatiom lOlQ
P
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··=* ‘ H 5 Ä 5 li 5*I 5 5 l } E 5 5 5*.l ¤-vg _ g g g g g' ig f•·•\
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Figure 45. The f(t) recovered from Fig.4l.
Application · 102
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8.0 CONCLUSION
Wigner distribution is a mapping of a one dimensional signal
into a two dimensional signal on the time—frequency plane.
It possesses many important properties which make it very
attractive for non-stationary signal analysis. Some important
theoretical properties of discrete-time Wigner distribution
have been reviewed in the first part of the thesis. Aliasing
behavior of discrete-time Wigner distribution has been dis-
cussed with three methods being given to avoid the aliasing
effect. The particular attenthmi of this thesis has been
given to develop signal synthesis algorithms. It has been
shown that the inverse Fourier Transform of discrete-time
Wigner distribution was a two dimensional time function which
can be viewed as an outer product of 1n«> one dimensional
functions.
The contribution of this work is developing the synthesis
algorithms to reconstruct the signal from discrete-time
Wigner distribution or from discrete-time pseudo-Wigner dis-
tribution. If the given distribution is a valid Wigner dis-
tribution or valid pseudo-Wigner distribution, signal
recovery is possible up to a constant factor, and it requires
a different procedure for even-indexed samples and
odd-indexed samples because even-indexed and odd-indexed
ConclusionM
103n
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samples are not coupled. If the given distribution is not a
valhd Wigner distribution cu: valid pseudo-Wigner distrib-
ution, a least-square approximation procedure is formulated
to recover the signal which minimizes the total square error.
.A previous work in synthesis of discrete-time Wigner dis-
tribution has been done by Boudreaux-Bartels and Parks [9]
using least-square procedures. A new synthesis algorithm has
been developed in the thesis which involves a factorization
procedure where the desired one—dimensional signal sequence
can be recovered as the solution of this factorization prob-
lem. In addition, two synthesis methods for discrete-time
pseudo-Wigner distribution have been developed in the thesis
and successfully applied to noise filtering and signal sepa-
ration problems. Furthermore, these synthesis methods can be
applied to test whether a given time-frequency function is
a valid Wigner distribution.
An open question arises from these results and serves as a
point of departure for further research. Since the synthesis
methods developed in the thesis can only' be applied to
deterministic signals, to develop a synthesis algorithm for
stochastic signal processing is one of the interesting topic
for further research. In addition, the comparison of smooth-
ing in time—frequency plane with smoothing in time domain,
the window sizes and types, and the smoothing effects on
different signals should receive further investigations.
Conclusion” 104
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BIBLIOGRAPHY
1. Rabiner, L.R. and R.W. Schafer, Digital Processing of
Speech Signals, Englewood Cliffs, NJ: Prentice-Hall,
1978.
2. Wigner, E., "On the Quantum Correction for Thermodynamic —
Equilibrium," Phys Rev. Vol. 40, June, 1932, pp. 747-759.
3. Page, C.H., "Instantaneous Power Spectrum", Journal ofl
Appl. Physics, Vol. 23, January 1952, pp. 103-106.
4. Woodward, P.W., Probability and Information Theory with
Applications to Radar, Pergamon Press, Inc., New York,
N.Y., Ch.7, 1953.
5. Rihaczek, A.W., "Signal Energy Distribution in Time and
Erequency", IEEE Trans. of Inf. Th., IT—14, 1968, pp. _
369-374.
6. Claasen, T.A.C.M. and W.F.G. Mecklenbrauker, "The Wigner
Distribution - A Tool for Time-Frequency Signal Analysis;
Part I: Continuous-Time Signals", Philips Journal of Re-
search, Vol. 35, 1980, pp. 217-250.
BibliographyM
1054 I
![Page 114: SIGNAL RECONSTRUCTION FROM DISCRETE-TIME WIGNER … · 2020-01-21 · far, few synthesis methods have been reported in the litera-ture. In this thesis, the synthesis algorithms for](https://reader034.vdocuments.us/reader034/viewer/2022042417/5f32a7e68b2e1573097776d4/html5/thumbnails/114.jpg)
7. Claasen, T.A.C.M. and W.F.G. Mecklenbrauker, "The Wigner
Distribution - A Tool for Time-Frequency Signal Analysis;
Part II: Discrete—Time Signals", Philips Journal of Re-
search, Vol. 35, 1980, pp. 276-300.
8. Claasen, T.A.C.M. and W.F.G. Mecklenbrauker, "The Wigner
Distribution - A Tool for Time-Frequency Signal Analysis;
Part III: Relations with Other Time-Frequency Signal
Transformation", Philips Journal of Research, Vol. 35,
1980, pp. 372-389.
9. Boudreaux-Bartels, G.F. and T.W. Parks, "Signal Esti-
mation Using Modified Wigner Distributions", IEEE Inter-
national Conference on Acoustics, Speech, and Signal
Processing, San Diego, March 1984.
10. Flandrin, P. and W. Martin, "Pseudo—Wigner Estimators for
the Analysis of Non-Stationary Processes", IEEE Interna-
tional Conference on Acoustics, Speech, and Signal Proc-
essing, San Diego, March 1984.V
11. Flandrin, P., "Some Features of Time-Frequency Repres-
entations of Multicomponent Signals", IEEE International
Conference on Acoustics, Speech, and Signal Processing,
San Diego, March 1984.
Bibliography 106
![Page 115: SIGNAL RECONSTRUCTION FROM DISCRETE-TIME WIGNER … · 2020-01-21 · far, few synthesis methods have been reported in the litera-ture. In this thesis, the synthesis algorithms for](https://reader034.vdocuments.us/reader034/viewer/2022042417/5f32a7e68b2e1573097776d4/html5/thumbnails/115.jpg)
12. Nawb, S.H., T.F. Quatieri, and J.S. Lim, "Algorithms for
Signal Reconstruction from Short—Time Fourier Transform
Magnitude", IEEE International Conference on Acoustics,
Speech, and Signal Processing, April 1983, pp. 800-803.
13. Claasen, T.A.C.M. and W.F.G. Mecklenbrauker,
!"Time-Frequency Signal Analysis by Means of the Wigner
Distribution", Proc. IEEE International Conference on
Acoustics, Speech, and Signal Processing, 1981, pp.69-72.
14. Chan, D.S.K., "A Non-Aliased Discrete-Time Wigner Dis-
tribution for Time-Frequency Signal Analysis", Proc
ICASSP-82, Paris, France, May 1982, pp. 1333-1336.
15. Claasen, T.A.C.M. and W.F.G. Mecklenbrauker, "The Alias-
ing Problem in Discrete-Time Wigner Distribution", IEEE
Transactions on Acoustics, Speech, and Signal Processing,
Vol. 31, No. 5, Oct. 1983, pp. 1067-1072.
16. Oppenheim, A.V. and R.W. Schafer, Digital Signal Proc-
essing, Prentice Hall, Englewood Cliffs, 1975.
17. Sussman, S.M., "Least—Square Synthesis of Radar Ambiguity
Functions", IRE Transactions on Information Theory, IT-8,l
April 1962, PP. 246-254.
BibliographyÜ
107A
![Page 116: SIGNAL RECONSTRUCTION FROM DISCRETE-TIME WIGNER … · 2020-01-21 · far, few synthesis methods have been reported in the litera-ture. In this thesis, the synthesis algorithms for](https://reader034.vdocuments.us/reader034/viewer/2022042417/5f32a7e68b2e1573097776d4/html5/thumbnails/116.jpg)
18. Janse, C.P. and A.J.M. Kaizer, "Time-Frequency Distrib-
utions of Loudspeakers: The Applicatmma of the Wigner
Distribution", Journal of the Audio Engineering Soc.,
Vol. 31, No. 4, April 1983, pp. 198-222
19. Tatarski, V.I., "The Wigner Representation of Quantum
Mechanics", Sov. Phys. Usp., 26(4), April 1983, pp.
11-327.
20. Griffin, D.W. and J.S. Lim, "Signal Estimation from Mod-
ified Short-Time Fourier Transform", IEEE International
Conference on Acoustics, Speech, and Signal Processing,
April 1983, pp. 804-807.
21. Andrews, H.C. and B.R. Hunt, Digital Image Restoration,
Englewood Cliffs, NJ: Prentice-Hall, 1977.
22. Auslander, Louis and Richard Tolimieri, "Characterizing
the Radar Ambiguity Functions", IEEE Transactions on In-
formation Theory, IT-30, Nov. 1984, pp 832-836.
l23. Huang, N.C. and J.K. Aggarwal, "Synthesis and Implemen-
tation of Recursive Linear Shift-Variant. Digital
Filters", IEEE Transactions on Circuits and Systems,
CAS-30, Jan. 1983, PP 29-36.
Bibliography 108
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