SIGNAL RECONSTRUCTION FROM DISCRETE-TIME WIGNER … · 2020-01-21 · far, few synthesis methods...

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

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

successfully applied to the noise filtering and signal sepa-

ration problems.

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

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

vi

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 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 21. The f(t) recovered from Fig.17. ...... 78




List of Illustrations vii

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 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 40. The smoothed PWD of Fig.36. ........ 97

Figure 41. The modified PWD of Fig.40. ........ 98


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

Viii

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

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

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

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

V

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

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

6

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

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

8l

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

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

lO

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

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

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

13A

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

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

15

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

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

17

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

18

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U 7 II — .U

· .I„ a II

1 I}IE

3 5 with no a iasin .Figure 4. The WD 0 q· ·

ling Techniques and Aliasing

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

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


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

25

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


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']


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

28

* 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

29J

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)


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

31i

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.


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


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


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)


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


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

37

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


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


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.,


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)


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

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). _


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

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.


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.


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.


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

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

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

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

I Iä ä

IIII

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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

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

56(

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

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)


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).


.. 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


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

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

62

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.


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],

ApplicationQ

64u

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-

Application2

652

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

ApplicationN

66l

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

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.,

Application 68

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.

ApplicationA

69l

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V ·

Figure 13. The PWD of Eq.(7.2).

Application 70

Ai MF, F W W W

Figure 14. -'I‘he signal x(t) in time domain.l

i

Application 71

Figure 15. The signal f(t)=cos(nt/8).

Application · q 72

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IFigure16. The PWD of Fig. 15.

Applicatien 73

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17. The smoothed PWD with M=6.

Application 74

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Figure 18 ThI - e smoothed PWD ’with M=24

Application

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EI'ac-

m<>O1

<¤ ¤

th

4; __ _IIIII,...•»7

Gd P

"

WD

w•1th M

.

_

76

,69;;QL?;>,;~§·,¤,·3g r .·:;9.;«.·2;:Qr#;!;’*T¤·. J I$«.·.ä?;c.¤I'Q‘;-gc Q {49;;/::¤p:;4@l«· Q Q .._.„.I IQi**‘——‘€***3,6 ""{ä9äy'·'·;‘?'*'=£;?*I·——·—-=-{' « ¢%I§"*‘€¢* 6*"7“*—=*$==‘°?*,·.,,,..’*·-——·.,,,..;—=-—-«··¤ <·‘-.>»‘=-·—· ‘-

Q.-„-...... ... ~„ ,«.• ..-- - . 4- .. .¥-

shdH! A„Q ‘·i:7°.l;-;~—I;q-‘·{'Q Q :«4Ö"r._ ·{-j?‘é;‘==-:;;=5§·£*;-4?.E,‘·£4·*i5;_';·‘··—;=‘v‘==:$Q Q

II Ü I ‘· ·•1 Q .„Z 6 ~-

I I I IQ’

· ' Q I I ^="";.=>:‘i'·~—-—-"""7‘=:·‘~3?¤IQQI'IQ¢€QQQQ{„QQQQQ 'QQe?Q‘QQ„QQQQI_II@Q•¤¥rg§I Q Q Q Q : · · · ' 2 I 2 - 3 I

_ «QQQQQQQQQQ QQ,QQ»¥QQQ¤§''Q ‘I;‘I}!;;¤.Q·Q{I ;Q·Z·I;=·

”··:···"—'#I;IQQ«IQ.¤:{{¤¤-{QQQ ·aQQQIe;; I·I QI3IIQI;¢

Figure 20. The smeothed PWD with M=50.

Applicatien 77

I

Ä nLW Ü I G I I Ä I I

I

II IIII I II-II3

I IIIIIIIIII IMIIII IIIILW I I I II {I II II • II FI II =I IZ III ' f I. I · · . I I '· „ I I · II I E I I I I · I I II I I I I I I I ° I IILIII ° I I II ‘ I II I I 5 I I I I I I-·] III III¥II*?‘£II*’I‘II=I‘°‘II IIIIIIIIIIIIIIIIIIII - I I I · I II I I I I I I ‘ I_ . „ I I I . I I , I , I , _· I I ·_ ,

,•I II I I

- II · - · II}__-•L I I II L} I . I Ox Öl I

1 '_L' _: _I I IL I« · In I II II< n I II II I I I |I I} I_ IL_„ LI

— j Ii ‘IJ IL ' I.? II

IILIIIII ' gi: · * I

__ _ ‘

Figure 21. The f(t) recovered from Fig.l7.

Application 78

' ä1 I W 5 5 Ä L 5 {Iß Iß ,= Il aß a· ,A Ilß I2 ea ß I29.27; I} 7L gg ai _:• 'I nk 9, gaÄ 6'I II II ZT II II II -·IÄ ?'I‘ ***2:I I I'_I I I II ‘ I I I II

' II I I , 1 I IIÄ

¢I_ [I II I I I- I :I } I I :I II III‘· ‘ 5 I T I I 'a I I·:·°‘°* IIIIIIIIIIIIIIIIIIIIIIII

Ä I I I I ß I II | I I I: I I 6 I ßII I I f 2

ßIßß2_*III; II II II :2 :I I; II aa'6I Ia II Ia II .* I- II I; II TI[I II II III E-‘ I.' III II6 I “‘ *a' 'J ' Ia 4

-0 ° J_•—-·—————-——————·-·—·—--·—--··—-$040 e ·T¤ E J l¤I·¤I¤ L .L¤l• L Ian lt ·• L Z·§¤ L •I••I¤ L - ia L4-:

rL.1E

Figure 22. The f(t) recovered from Fig.20.

. Application A 79

¤ü·jÄÄ I It fe ..eÄ‘| ‘I

v Ä ' ·wi I •, Q2 „ I'! II 1*1IIII II II JI In ·» M II II II • I2 II II 'I FI :4 II II II I EI II¢° I II :1 · I :I I I II II I• ·‘ IIÄ ‘

I I I JI I I I ; 4 I II I I‘

I 7 I‘I I I I II I I I I I I I I I I I I II07I

IIIf‘IIII-‘I=4I‘~."I.r;‘.=1 II>=I.:I„I1?-I? ;:z?I+::2;2I-;¤-I I ·„ I I I., I. II II ‘ II I; ;s

edßig I O: IJ III3 I.! EI! ° .1E—66J_________„„__,„„__,„.„„.„.„.„„„„,.„.„„.,„...„„..„.„„.„—.-„—.


Application 780

l

il;

I3 I I*63 I ;, III _,_ ,·. fg fg „F‘I 6;I I 1 I ;· "I ·§ II "I V-,3 I I I II I‘ I3 II II rt I1 -3 ,*3 II·°ä ·I :1 ·II'* ·II¤3II¥I¤a .I!I.'1I§:=3·III?IIII*I%:1%3 1 13 IIIIIMSIIIIIIIIIIIIII‘ '. E; E 5 I TF I1 2 'I- I IS ä3 IL ll äf II L} if II I} I; II .‘ LQM II Ii ·· sg.!} 1J 3: II I= I: 3 21 I I I . '.. ' I 1 ' V ‘ 3 I 'I L . I I ·3 g I I 3 ( v I _I . T 1.3 II I I I; L;

dgl J I • ~‘· V • • Ä:5-‘··I; I; „ * 'ä

•;• \‘• •

.e_,}.1i_~·—_—___T____r____?_______—i__—-—_—~—_—_7-______i____’i_____________i_____i______________________

¤T• LI) -¤~Z¤ ·=·'= *·T· IIIIIJ 1L·.• .<·I• Le-· ;E·· L-;·.• LL-'I


Application 81

SBE

I ,*·I {I G I 1 II I I II I I I II I IIII Q! ··I VI I I •‘°III II II II II II I II I I· ·; I• II I I

II I ‘

Il „IIII

I1 I•II I? „I II IIIIIFI:I'5II§°”31 II II IIIIJIIIIIIIII+III·I,1 I I I

IIIIII IIIII "I.I&ITI§.II,I„„1 I I I II,·é„IIIII‘IIIII‘I•=II‘I-,_I I I

I I v I I I: I I II II

I L I : I I II I I I · I'A ' I I I I. I I I 7 I It _ 7 · • I, I E · I E I I II 1Z { I ' ' I · . I I 6 I I I I „ I I ¤ II I I I. III I I V I 1 I . I l_.

III I I I I I I III II III II III I.I I ’·§ I? II ·» II II ‘* ¤III=I'IiI§iII II I I L.I II ,I III

I IJ 'II 2 .‘ 2* I' II ·’ Ii E ' °· ·I·=·>—I.....,..„_.,..,..........„..--...-.._--.._,.........-............_.,,,.,_„„.,_____,___,___,,,„..,„...,.,.,__,.,..___,,__,

Figure 25. The signal f(t) of Eq.(7.6). I

Applicati·on 82

1 If 1. 1 'I _ ,*1 ri :1 ··.;.3IlI

IZ ·! ‘1l{ „= • A'. 1 1-, 1 1*1*. **1* :1 l“1'1 I 1 1 ll 11E °· 1 11.111111 1::111-. ala; I ZI I 1 1 I1‘.1 . - · 1 il I || li 11 I 111 Y1, 1

E 1I' 1 .* :1* * 11 1 111*1;; 1 ¥Ü?Ä1-I-.·. ~l 1 °' Z 1 ‘

1 * 1* § 1 A · 1 1 * ‘· • 1* 1I *»¥ *...1' I ~ \ I . I I I ~ I, II,1-*1* *,1 1* 1 1E 11I 11 1* *‘1*'‘; Ö. .' T1 1 1} fl 1* ZI I *1 1 l—‘

II I1 •I „„ Z., I

Z 1; 1 I .

Figure 26. The signal x(t) of Eq.(7.7).

Application 183

: II ·‘

*

il

II, I ~ ·:,III “··'I‘I'I I I I': III I I .III., ..:.¤‘

I II :::1: ::|° ?"{I'I:"=II;'§‘”II:I I [II I I .I:I':I':IIIII: II I: II' IIIII I I:II°I I,„IIIIII I IIIII I I.„„„„.„·—————-==···——·—”””‘°%°‘F_——*————— III I:II:,+Iä:Ü%*:IIII III I :,,,.„..,,..,_.I'·:—I——«=—=—»I IIIIIII

%.—..:»»:.—""'°%‘·~° I I " °

!.I'i¥,‘j:§'III‘;QI III

IJFigure27. The PWD of Fig.25.

Application 84

I."I. II

IIIII I I II II I _lIII IIIIII II IIIIII IIII IIIIII I „IIII· _ II_,«I ju {ITIIIIKÄ I III ,_ 'I III. IIIII I ·I?I IIIIII iwi ‘I IIII I II II III III [ I;·I IIII,IIII.„ II ·

In { III! ~· II». · ·I1 I KLV ' VW III; ly· KV -. II 1 = I ·, I

I liygj]

Figure 28. The PWD of Fig.26I

Application 85

III III II II {IIII IIIIIIIIÄII I I{I{III III I I IMIL {III {II ;I« II {· .•·I.·a II{

I1=¤¤.a'%"I ‘ '· ‘ 'lIII{ { ( II

_ ·{III; I{ 'IIII II I {III { {III {{II IIIi' L Ü}; L

?'I‘~_I I II I„I II In

". „·

L; II I I·*¥I,' CI. II I}? AI. T: II III;¢?iIIIIIwIII—I;'2I;III+äIÜIII;I,?II II II_II.I I II II {·{W I¢»?.;~*·»*I‘·%=>+;’ +IIEI‘:¤·‘·,‘ I.I§III IIIIIII„I,I.I‘II|I?I"·'III"II II·.I&léII IIII I I II I°IIä{I{IIIIIII"II.ÄI"III·ÜIIIt*;i'IIIIIIIIIIII I III«Ä.·m ,I~T~{-+«-—-—„——I.fI'I.?fI;°f"IIf,.-+{„-„‘iI‘*‘.Li‘r;.L;1i;Ü{ ·‘ ·' II If

{{I ° *‘·;j{£Ü{I{{{II>*

Figure 29. The smoothed PWD of Fig.28 with M=6.

Application · 86

/~.I.•IT! ,AIIIIEIIII II::I~I 'IIII III!

I II III I.} I M I III I.. I II*g§EII.’$I IIIIIII IIIIIIIIII III II III I IIIIIIIIIIII III IIIIIII IIIII lu 1II.IIII.IIIIIIIIsI IIII I II ‘I.I_II’._.III’IIIIIIIIII II IIIIII IIII2 IIIIIIIII IIIIII IIIII II'·II,III1 ·I ' I ‘. II.I II I= II I·II,*'I IIIIIIIIII.I.II‘IIItg":}-°l’AI·!°_Ä„__„~$4L(1_§,!•*1é‘I*,I‘,¤§‘.;'I·‘IIiI IQIA ,I‘*;p·\II I II I I {III I'g,I‘If·.!iI._.III f!°{ III ggf .I| fi 4I=«Ig„.‘„·'·'—U*°I' I Ii; I I II I I 'III I I.—I· I I, III. II I I I IIIß I**}I.·°¥;=I§i<r·i§*?II*° Im; II%“"II'I 'I. I I··"‘1·.·II-~“I. ·¢;c*°:‘Ti‘ I II I II ILQII II.‘I*‘*I*II*I1II§$;'IÜI?v*IP+Ä‘I=-{I*‘I=‘IIiÜ‘ä(*·IIIIIIÄIIIII .*2* II II I I II 'I III , · I III I-II II I I·'I"I II I I II II I ·‘ I Ii I I I I. I I I ä II II I ' I IIIY. '§ I ffI.‘

Figure 30. The smocthed PWD of Fig.28 with M=l2.

Applicatien 87

0.3-J Ä Igß „ A,‘

II II II' I 'I ,! IIJ I I , A gl · J, ,‘ _·, IJ 7/ •·,

I JI • .I III II„I• •— IL6.2-I ·’ I II I I II I' II- II II II II

3 I I IJ II I II II II InJ I 7 I , I I III II II ~’I II I I II .~I I III IIIIIIIIIIII3 I II I I II II I III IIIIII.~III·I

I I ‘II=¢II III ~'“-z^IIéII'III I IIII IIII IIIIIIIIIIIIQ ß I II I II I I I I I -_

I I I I I II II-II I I “· II I I II II II I IIIIII I I I · U II H Ii IÜ;é‘*q U'”°°°I I, I III ° ”‘

IE EI EI '5 I 'Ij II III I II IIIn III I

3·•I•.4j


Application 88

sa IQ{X

lß. ['=” Y JY Y 6 Y6 6 61. 62 62936 Yxaßdä6 al 6- 6Y

' *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

6 -1·¤L....„„...„.„„...Ä:°.„.„„-.......„..„.......„....„....„.„„.0

1-1-—·3 e··a Z-) LZ--T- 6;-) :4-- 1-;-;- 6;-J ;a;6•j6 ;;„;6


Application 89

J

II

Figure 33. The signal f(t).

Application 90

I

Figure 34. The signal g(t).

Application · 91

•‘.

Figure 35. The signal x(t)=f(t)+g(t).

Applicatibh 92

;: h

WäägggF1gure«36.The PWD ¤f X(‘¤)·

‘'

QQQQ QQ;QQQQQQQQQQ

Figure 37. 'The medified PWD of Fig·3é·V

Applicatiorx 'l 94

·vsse _

,4;%:/</:5/1/’},·,·’=‘~.%;.

· · ,·"\ 1 ‘„Ü‘%€‘If·’ ”*",.-··--·”;·‘”

° _/

Ql§_,..«¤""·

·•:.•:• °° :V^x , A' -/7 --<‘7ÄI*l‘>- '?§= 5 * ° - °/’( ·//§,‘«.-’,.·f!\-:£·x ( ; Q g , ·f3 q} \_i ‘„ E

‘·°I 110 I 1N I 0

I1 1 1 0 160 190 2

x

Figure 38. The contour plot of Fig.36.

Application 95

—v

Nl

· . l

/«·‘i__'_<;;:1/{;·;//'

.61. " 1 1_/’ 1 fi ! E/ (1/213 E ä z 12 ; :1j ff'; "/‘\ ¤

·’}-\‘·/"·Ä"‘°‘$'¥ä’*x ‘

¤ • r1': E 1'S:F:':x ‘ Y; ‘~ f F n° 1 o So 120 1:16: 1 ¤ 1 ¤ l60 1 o 1a¤ 1so zu

* „


Application 96

' I

I I·I

I:II .=. 1 I

II 4 IV1 Q .„

V.:

V'LI I ^ e I¢- : I I ~V , I

.*—¢ I ‘·

I*‘

l

V .V

I .6I

I I·-'llllll’ ·» Iyyyppyi II'

'llllllfllalfjfx I'

u'3· ?‘I “‘II I'•I IÖ" 'Ir I

Iilllßllll_·I •I 1 1II IIII•I II I II '

· v Y 0•,

· •I

1u~Iur1n¢1 " I ‘ I V :1;;; _ —I ·•

lvruuutuiv.

,1$$1521522225 L I}WI [•,I · I )' •

Iuuvnunvrrur· .

flllllllllfllllll0.1I1l~Iv¢•'•lHIl” . . « \ I _ /I/ '[Iunununnnnn

1 illlllllllllltlllllll ' 1·|11 llrllrllllllllilllllllll ·,

Q Q" ll/1/I I IQ']IIIIIIIIIIIldlllllllllllllll II 1ll" ’l1a¤/ll '— • Jlllllre llIII,l,I' ’lI11;411111a11111r•n1uunv

111pn11I¢aua1¢:au11;11uv11¢lllin!IIIIIIIIIIIIIIIIIIIINI

IraIII!1i1:lII1111111:1Ir1I1-al

1 1 1nurunununun IIIIII I{III. _ I ‘

I I

· 40 The smcothed P Of F gF1 gl] I° G ·

Applicatiam

'„

I =. IP;„mjUFIT Z I. QT I E

IIF Ü.I

Ü L? T F‘— •‘: '~rl' EFFIUTT" 1ääszä

I IEFTT TIT T 1 M6 3 3 T Ü;TT? TT Ü

T?Figure41. The modified PWD of Fig.40.

Application 98

' Y

y//‘{ ·

/¢'f,7Q‘;/-.,

~ .·

Ä'·'_,«-"‘~—, s\~—\&/E)g . sw \-

(I#:2 ~“"~··{,-•·«-——··~~,3 "*-•:•·"‘°,··‘°,, _/. Ü

/’,,•·;;;:•"·1""::’;"Z,/J __,„••""

(I0

00 1 1l0 120 1 I 0 I 1 I 0 I 100 2I X


Application 99

'TBIO

s¤ „./" 0 ‘

. ‘-—-——····"’

fk .VVK°

. "· F P1. =• ‘- •· 110 120 1 • 1 • 1 • I60 10 180 190 20

. ' „ ¤ '

Figure 43. The contour plot of Fig.4l.

Application · _U

100

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

III II I II 6 g I 6.

•Q _iTI

Ii .

I III II I·

I-6FE2 6-) '·Z• i ·Z• ··i• Z -Z·~I· 1 2 . ·· . L : -::6 3 = :» ,

-.;• , 7 .;. I ,.3 _ j- .;:.3


A licatiom lOlQ

P

··=* ‘ H 5 Ä 5 li 5*I 5 5 l } E 5 5 5*.l ¤-vg _ g g g g g' ig f•·•\

' - -1 I “’-,,_,.§ \.5’ gl g g g g tg! '„“

*.III vin;

E I-6.3-: ' I_•

·xJ45

‘Q.) ,,3 1.1-0 1* -i· 1,..3 17.‘¤ 1.-·i· 1913 L·.··.·

Figure 45. The f(t) recovered from Fig.4l.

Application · 102

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

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

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



Part II: Discrete—Time Signals", Philips Journal of Re-

search, Vol. 35, 1980, pp. 276-300.



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

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

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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