Legal Wigner space filtering and its interpretation

SIGNAL

PROCESSING ELSEVIER

Signal Processing 39 (1994) 179-191

Legal Wigner space filtering and its interpretation D. Adler, S. Raz* Department of Electrical Engineering, Technion - Israel Institute of Technology, Haifa 32000, Israel

Received 23 April 1992; revised 28 January 1994

Abstract Filtering in Wigner space generates a time-frequency function that does not generally constitute a legal Wigner distribution (WD). The notion 'legal WD' implies a time-frequency function, specified in Wigner space, for which a time-domain signal exists. This illegality prevents an exact synthesis of the corresponding time signal, and resorting to an approximate synthesis procedure is unavoidable. Presented herein are necessary and sufficient conditions, on a filter specified in Wigner space, that guarantee the legality of the output signal. We prove that legal Wigner space filters belong either to the class of linear time-variant filters or to the class of nonlinear (quadratic) filters. These two classes possess vastly different characteristics.

Zusammenfassung Bei der Filterung im Wigner-Bereich wird eine Zeit-Frequenzfunktion erzeugt, die im allgemeinen keine giiltige WignerVerteilung (WV) darstellt. Die Bezeichnung 'giiltige WV' impliziert eine Funktion im Wigner-Bereich, zu der ein Zeithereichssignal existiert. Diese Tatsache verhindert eine exakte Synthese des entsprechenden Zeitsignals und erfordert das Ausweichen auf nfiherungsweise Syntheseverfahren. Hier werden notwendige und hinreichende Bedingungen f'tir ein Filter im WignerBereich derart angegeben, dab ein giJltiges Ausgangssignal garantiert ist. Wir beweisen, dab derartige Filter entweder linear und zeitvariant oder nichtlinear sein mfissen. Beide Klassen weisen stark unterschiedliche Charakteristiken auf.

R~sume Le filtrage darts l'espace de Wigner g6n~re une fonction temps-fr6quence qui ne constitue g6n~rallement pas une distribution de Wigner 16gale (WD). La notion de '16gale WD' implique une fonction temps-fr6quence, sp6cifi6e dans l'espaee de Wigner, pour laquelle le signal temps-domaine existe. Cette ill6galit6 emp~che une synth6se exacte du signal temporelle correspondant, ce qui m6ne inexorablement fi une prockdure de synth6se approximative. Des conditions n6cessaires et suffisantes pour qu'un filtre sp6cifique dans l'espace de Wigner guarantisse la 16galit6 du signal de sortie sont pr6sent6es. Nous prouvons que l'espace des filtres de Wigner 16gaux appartient 6galement ~i la classe des filtres variant lin6airement dans le temps ou ~ la classe des filtres non lin6raires (quadratique). Ces deux classes poss6dent des caract6ristiques extr6mement diff6rentes. Key words: Wigner distribution; Filtering; Signal processing; Time-varying; Matched filter

* Corresponding author. 0165-1684/94/$7.00 © 1994 Elsevier Science B.V. All rights reserved SSD1 0 1 6 5 - 1 6 8 4 ( 9 4 ) 0 0 0 4 8 - 5

180

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

1. Introduction The potential advantages of analyzing and processing signals in a combined time-frequency space have been long recognized and well documented [3, 6]. Among all time-frequency distributions the Wigner distribution (WD) possesses well-known attractive features (e.g., [5, 8, 9, 11]), and is therefore a good candidate for a time-frequency analysis. Alternative definitions of the discrete WD were suggested [4, 13], of which the latter appears to be the more attractive, for our purposes. A major difficulty associated with processing signals in Wigner space stems from the fact that the Wigner space output does not generally constitute a legal WD. Consequently, only an approximate reconstruction of a finite-energy signal is possible and it may entail a massive computational effort [2, 15, 17, 19-21]. Furthermore, the procedure possesses a nonlinear and discontinuous operation. These draw-backs strongly motivate the introduction of constraints on a desired Wigner space filter that will ensure the legality of the output signal. Some work concerning this problem has already been done, but direct constraints on the Wigner space filter have not been established. The problem of describing a legal Wigner filter was first raised (without solution) by Saleh and Subotic [16] in the context of a masking filter. Yu [22] addressed this problem by introducing a generalized linear timevariant operator in Wigner space but the legality question remained unresolved. A sufficiency constraint ensuring the legality of a Wigner space filter was introduced by Pei and Wang [14]. It led to a linear time-variant filter. Cohen [7] suggested a necessary condition rendering Wigner space masking legal. Partial constraints were also introduced by Topkar et al. [18]. A new approach to legal filtering was proposed lately by Hlawatsch and Kozek [10], who deal with orthogonal projections which constitutes a special case of linear time-variant filtering. The essence of the present work is the presentation of global necessary and sufficient constraints that guarantee the legality of Wigner space filters. The requirement of legality of the output Wigner space signal [1] leads to combined constraints involving the filter expansion coefficients as well as

the coefficients representing the input signal. These constraints generate legal filters that fall into two distinct categories. One that is recognizable as a linear time-variant filter and a second which is intrinsically nonlinear (quadratic).

2. Background Among all bilinear time-frequency distributions [6], the WD [3] appears to be a good candidate for time-frequency analysis tool, owing to several desirable properties summarized in [8]. The WD of a continuous-time signal x(t) defined as

W~(t, co) a_ fR x(t + z/2)x* (t -- ~/2)exp( -- jcoz) d~ (2.1) is a real valued function of time (t) and frequency (co) which, in a sense, describes the signal energy distribution in a time-frequency plane [3]. For computational purposes, a discrete version of the WD is essential. Alternative definitions of the discrete WD have been proposed [4, 13]. Herein, we adapt the definition 1 N-1

Wx(n,p) = ~-~ k~o x(k)x*(n - k) x exp [ - jn(2k -

n)p/N]

(2.2)

proposed by Peyrin and Prost [13]. Wexler and Raz [20, 21] (for discrete time) and Adler and Raz (for continuous time) have shown that both definitions transform a set of complete orthonormal time-domain basis functions into a complete orthonormal Wigner space basis. This crucial property enables us to formulate the results in terms of readily computable expansion coefficients. Theorems 3 and 4 in [1] may be stated as follows. Let {q~k} be an orthonormal basis of .La2(R). Then the set of functions {@kZ};k, l e Z defined by

~Ok,(t,co) zx=W~,,.o,(t,co) zx=f ckk(t + z/2)~b~'(t - z/2) × exp( - jcoz)dz

(2.3)

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

is an orthonormal basis of ~'2(R2), satisfying

~'kt = ~k~.

respectively. Then F is a legal WD (continuous or discrete, correspondingly) if and only if

(2.4) CkkCmn = CmkCkn ~

NOW, let {~bkt}, generated by (2.3), be an orthonormal basis of ~ 2 ( ~ 2 ) and let F e , ~ 2 ( f f ~ 2 ) be represented by

F(t, co)= ~ ck,~,~,(t,co).

181

(2.5)

* . CranCnm, Ckk ~ O,

Vm, n,k e Z.

(2.11)

The last result is of primary importance as it facilitates direct testing as to whether or not a prescribed finite energy function constitutes a legal WD.

k, leZ

Then F(t, co) is a WD function if an only if there exists a vector a e 12(R) whose elements a k satisfy

cu = aka*;

~ lakl2 < ~ .

(2.6)

keZ

Similar results pertinent to the discrete definition (2.2) are given in [20]. Let {~bk(n)}, (n,k = 0,1, .... N - 1) denote a complete orthonormal basis, then the set { Wkt} defined by Wu(n, p) 1 N-1

= 2--N ~ qbk(r)c~*(n -- r)exp[ -- jrcp(2r -- n)/N] r=0

(2.7) forms a complete orthonormal time-frequency basis, and any discrete 12(R2) function can be expressed as

~. CktWkt(n,p).

(2.8)

k,l=O

Let x(n) denote a discrete-time signal represented by N-1

N-1

x(n) = ~ ak~bk(n); ak = ~ x(n)qb*(n). k=0

(2.9)

n=0

Then the WD of x(n) (eq. (2.2)) takes the form N-1

Wx(n,p) =

~

Mixed time-frequency representations generate spaces in which time-variant processing (filtering) can be executed and readily interpreted. Most generally linear processing in time-frequency Wigner space entails the input-output relation [22]. ~gr(t, co) = I f n(t, co;z,~)Wx(r,~)drd~, ddR 2

aka~ Wu(n,p).

(2.10)

k,l=0

The following theorem [1] is equally applicable to both continuous and discrete versions of the WD. Let F be a finite energy time-frequency function represented by (2.5) or by (2.8), for the continuous or the discrete time-frequency functions,

(3.1)

where Oy(t, o9), the output time-frequency distribution, is not necessarily a legal WD. Wx(t, co) is the input WD associated with the input signal x(t), and H(t, co; z, ¢) is the system function or the Wigner space filter. The special case characterized by

n(t, co; r, ~) = B(t, co)6(t - z)6(co - ~)

(3.2)

or subsequently by

Or(t, co) = B(t, co) Wx(t, co)

N-1

F(n,p)=

3. Wigner space filtering

(3.3)

was intensively investigated in I-2, 7, 16, 22]. Generally, the Wigner space filter constitutes a four-dimensional function (of which two-dimensions are interpretable as time variables and the remaining two as frequency variables). Since we aim at representing results by an orthonormal basis expansion, an appropriate four dimensional basis is generated as follows. Let {~bk} be an orthonormal basis of .L~'2(R). Then the set {q~u; ~kt~--CkkC~,} constitutes an orthonormal basis of LP2(R2). The twodimensional cross WD, defined as l~x,y(t, co; z , ~ ) ~ f ~ J JR 2

x(t + t'/2,z + r'/2)

× y*(t - t'/2, r - z'/2) × exp1--j(t'co + r'~)] dt' dr' (3.4)

182

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

leads to the desired

~2(~4) basis {~kl,s}

~u~(t, to; z, ~) = Ws~,,C~,,(t, to; z, 4).

Use of Eq. (3.6) yields (3.5)

The set {q~kl,s} can be easily shown to be both orthonormal and complete in £ez(R'~). Furthermore, it satisfies the relations

~kl,s(t, 09; Z, ¢) = ~lkr(t,CO)~bls(r,4),

(3.6)

x ~ ava*~bpq(z,¢)dzd4

(3.13)

p, qeZ

or upon interchanging the order of summation and integration

where ~bklis defined in Eq. (2.3), and

~t,~( t, to; z, 4 ) = ~,~kt(t, to; Z, ~).

(3.7)

Or(t, to)=

~

bkt,s~k,(t, to) Z ava*

k,l,r,s~Z

p, qe•

The Wigner space filter can then be expanded on { ~klrs}, namely

H(t, to; z, 4)=

~

bkt,~kt,,(t, to; r, 4)

(3.8)

(3.14) Substituting Eq. (2.4) into (3.14) yields

k,l,r,s~Z

As the WD of an arbitrary signal is necessarily real, we focus exclusive attention on real-valued Wigner space filters. The presumed real-valuedness of H(t, ~; z, 4) implies

Z

bu,~k,(t, to) ~. apa*

k,l,r, s e Z

p, qEZ

Since {flu} is orthonormal, i.e., satisfying

bu,s = ( H, ~k,,~) 4 = ( H*, ~*~bk,,~) 4* * = (H,~rskt)* = brskl,

Or(t, to)=

(3.9)

where

= 6,.q&..

(3.16)

Eq. (3.15) yields

(A,B)4-~ Ill[

A(t, to;z,4)B*(t, to;z,~)dtdtodrd¢.

J J J JRa

Oy(t, to)=

(3.10) =

We next seek a direct relationship between the expansion coefficients {bu,~} representing the Wigner space filter, the expansion coefficients {a~a* } representing the input WD and the output time-frequency function represented by {Ck,}. With Wx(t, to)= ~. apa*~pq(t, to)

(3.11)

p,q~Z

and the Wigner space filter represented by Eq. (3.8), Eq. (3.1) takes the form

Or(t, to) = II 2 n(t, to; z, 4) W~(z, 4) dr d4 ddR = fR f 2

~

k,l.r.s~2r

p,q~Z

b**,~ ~ a~a*~,,,(t, to)b,,p6~,q p, qEZ

E CkrO k r ( t ' t o ) ' k,r, ~Z

(3.17)

where

Ck, = ~ albu,sa*.

(3.18)

l,s, ~£

Eq. (3.18) is a key result enabling the execution of Wigner space filtering by directly operating on the expansion coefficients. It also constitutes a starting point for obtaining explicit legality constraints on the filter coefficients (bkt,s). Since this representation is unique, it constrains the filter itself.

4. Legal Wigner space filters

bkt,~k,,~(t, to;z,~)

× ~ apa*~kp,(z, 4)dzd~.

Z k,l,r,s~Z

(3.12)

This section presents necessary and sufficient conditions on a Wigner space filter, that ensure its

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

legality, i.e., legality of the output time-frequency function. The conditions are obtained by applying the legality constraints of a time-frequency function, Eq. (2.11), to a filtered time-frequency function, Eq. (3.18). Next, the constraints specified in Eq. (2.11) are considered separately. Constraint I cm, = C*m.

(4.1)

Since we restrict ourselves to real-valued Wigner space filters, this constraint is automatically satisfied. With Eq. (3.9) in mind we always have =

Y,

bnlms asal* = *

l,s, eZ

~ bmsnlasa~ = c,,,. l,s, EZ

(4.2)

C o n s t r a i n t II

(4.3)

Substituting Eq. (3.18) into (4.3) yields CkR =

Z bktksala* ~ 0 I,s, eZ

V k • Z,

(4.4)

mtspq =- buksbr~p,,q - b,,,lksbkp,,q .

~k * >1 0; atbtsas

For any vector a e 12(•), condition (4.9) can now be rephrased as follows. Let J/=~ [mt~pq] be a fourdimensional array (tensor).//e 12(R¢), and let V denote outer product matrix V = aa ~, i.e., V is Hermitian matrix of rank 1. Eq. (4.9) then takes the form V.,/¢'. V ~ = 0

The proof of Eq. (4.12) is detailed in Appendix A. Substituting Eq. (4.10) into Eq. (4.12) leads to the following explicit constraint on the filter expansion coefficients

(4.5)

12(•).

(4.6)

Eq. (4.6) indicates that /~k is semi-positive definite for all k. C o n s t r a i n t III

(4.13)

5. Legal Wigner space filters: An interpretation

Applying Eqs. (3.18)-(4.7) yields Z bk'ksa* at ~., bramqa*ap Ls~Z p,qeZ

~, bmlk~asat * ~', bkm q aq* ap, LseZ

(4.8)

p, qeZ

which may be rewritten as ~_, (bklk, b,,,p,,q -- bmtksbkpnq)as* a t aq* ap l,s,p, qEJ~

=

~ I,s,p,q~Z

Eq. (4.13) together with Eq. (4.6) are necessary and sufficient constraints ensuring that a prescribed four-dimensional function constitutes a legal Wigner space filter.

(4.7)

CkkC,,. = C,,kCk,.

=

(4.12)

= bmlksbkpnq "k- bmpksbklnq "+" bmlkqbkpns -1- bmpkqbklns.

bksA bklks

or in a matrix form Va •

(4.11)

leading to the following conclusion. For an arbitrary vector a e l Z ( R ) , .J¢ satisfies Eq. (4.11) if and only if the elements of J / meet the constraint

I,s, eZ

a B k a + >>- 0

(4.10)

bktksbrapn q -1- bkpksbmlnq "4- bklkqbrapn s + bkpkqbralns

which can be rewritten as Z

where

mlspq -b- mpslq -1- mlqps -k- mpqls = 0 .

Ckk ~ O.

183

ata s* mlspqa q* ap = 0

Va•/2(R),

(4.9)

In this section, we propose a time-domain interpretation of the legality constraints (4.6) and (4.13). The special case of the vastly investigated time-frequency masking, Eq. (3.3), is specifically discussed. A crucial simplification of Eqs. (4.13) and (4.4) is achieved through a rather cumbersome algebraic sequence (Appendix B). These conditions are reduced to b mp.q = b.,pb .* ,

(5.1a)

bmp~q = b.pb*q,

(5.1b)

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

184

b,.p.q = b',.b'.* bpq; ~. apbpqa* >~ 0 p,q~Z

Va e lZ(R),

(5.2)

where bmp, b" and/~pq are defined in Appendix B. It is shown in Appendix B that a Wigner space filter is indeed legal if and only if either Eqs. (5.1a) or (5.1b) or (5.2) are satisfied. I m p o s i n g constraint (5.1a) on Eq. (3.18) leads to

c,,. = ~, a.b,.p.qa* = ~, b,,pav E b.qaq-* * = g,.~*. p, q E Z

pEZ

0

q~Z

(5.3) Consequently, the Wigner space operation takes the form

Fig. 1. Filter 1 - a time-invariant LP filter H(n,p; re, q) rearranged as a two-dimensional array 14(nN + p, mN+ q). The cut-off frequency is 150 Hz.

Wy(t,~o) = fR~ f gz~'~(t'°; r, ~)W=(r, - ~ ) d r d~, (5.4) where

~(t,r)= ~ ~,.,,4,,.(t)4,,,(~)

(5.5)

m, p E Z

and 1~.~ is defined by Eq. (3.4). T h e corresponding t i m e - d o m a i n filtering operation is

y(t) = fa B(t, ~)x(x) dr,

(5.6)

0 Fig. 2. Filter 2 a time-invariant LP filter H(n,p; re,q) rearranged as a two-dimensional array H(nN + p, rnN + q). The cut-off frequency varies from 55 to 445 Hz.

where

x(t) = ~ a p ~ ( t ) .

(5.7)

p ¢ ]r

Namely, imposition of Eq. (5.1 a) implies the generation of a linear time-variant filter. This result was p r o p o s e d as a sufficiency constraint in 1-14, 22]. A similar conclusion is reached in conjunction with Eq. (5.1b). T o exemplify this m e t h o d three design cases are illustrated. Fig. 1 depicts a time-variant low-pass (LP) filter (with cut-off frequency of 150Hz). A time-variant L P filter is depicted in Fig. 2. The cut-off frequency varies from 55 to 445 Hz. The last example is a time-variant band-pass (BP) filter (100-225 Hz). The filter is active within the interval 5 mS < t < 10 mS and is described by Fig. 3. T w o input signals and the corresponding outputs

Fig. 3. Filter 3 - a time-invariant band-pass (BP) filter H(n,p; m, q) rearranged as a two-dimensional array I4(nN + p, mN+ q). The pass region is (100 225 Hz). The filter is active within the interval 5 mS < t < 10 mS.

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

(a)

185

(b)

-2

-2 I

I

I

I

50

100

50

100

(c)

(d)

-2

-2 ,

i

I

50

100

0

50

100

(f)

(e)

-2

-2 I

I

I

I

50

100

5O

100

(g)

(h)

-2

-2 0

I

i

50

100

0

I

I

50

100

Fig. 4. Two input signals and the corresponding outputs associated with each of the above three filters: (a) input signal 1 - a stationary white Gaussian signal, (b) input signal 2, (c) filter 1 output to input 1, (d) filter 1 output to input 2, (e) filter 2 output to input 1, (f) filter 2 output to input 2, (g) filter 3 output to input 1 and (h) filter 3 output to input 2.

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

186

associated with each of the three filters are shown in Fig 4. Constraint (5.2) leads to an entirely different filter class. Combining Eqs. (3.18) and (5.2) leads to

Cm, = ~ aqb,pn~a* = b'mb'~* ~ a~bpqa*, p,q~Z

(5.8)

p,q~Z

which corresponds to a Wigner space operation describable by

V/r(t,~o) = Wb,(t, co)

B(z,~)Wx(Z,~)dzd~,

(5.9)

2

where Wb,(t,m)=

~ b'~b',*O.,.(t, co) m,,~z

(5.10)

and/~(z, ~) is a semi-positive-definite weight function (Eq. (5.2)) given by

B(z,~)=

~. ~)pq~kpq(z,~).

(5.11)

p,q ~ ,E

The corresponding time-domain operation is

y(t) = ~/2 b'(t),

(5.12)

where

and

b'(t) ~= ~ b'mqb,~(t).

(5.13b)

mE7

The time-frequency function/~(z, ~), defined by Eq. (5.11), corresponds to the Wigner weight function, detailed in [12]. /~(z,¢) may be viewed as a time-frequency energy matched filter, and ctx as a nonlinear (quadratic) operation which estimates the energy content of the input signal within a prescribed time-frequency interval. Since the function b'(t) is independent of the input signal, it may be selected as constant. We may now interpret the Wigner space operation defined by Eq. (5.13a), or by its time-domain analog

~x= fa~f ff(t,~)x*(t)x(r)dtdr; ff(t, z) = ~ /~p~tp~,(t)~b*(r) p, q e Z

(5.14)

as a nonlinear projection operation referred to as the energy matched filter. A straightforward comparison with its classical counterpart highlights two essential differences. First, while the classical linear 'matching' is carried out in either the time or frequency domains, here the 'matching' is accomplished in the Wigner time-frequency space. The potential advantages and expected efficiency in situations where strong time and frequency overlaps exists between the desirable and the undesirable signals are quite obvious. Second, unlike classical 'matching' which is applied directly to the signal, the Wigner space operation, owing to its well-known properties, is closely linked to the direct matching of the corresponding time-frequency energy distributions. Hence, the name. Fig. 5 shows a design example of a time-frequency energy matched filter. Fig. 5(a) and (b) depict the reference signals. Figs. 5(c) and (d) show the corresponding time-frequency matched filters (B(t,e~)) computed in accordance with Eq. (5.11). Fig. 6 depicts the filter's response to the detected signals versus the input SNR. As previously mentioned, the problem of characterizing a legal Wigner masking filter has been long standing (e.g., [7, 16, 22]). We readily conclude that the Wigner space masking (Eq. (3.3)) and the nonlinear filtering associated with Eq. (5.12) are distinct operations. (Note that the output of a nonlinear filter is a constant, Ctx,while the result of a masking operation is generally a time-frequency function.) Hence, presuming its existence, a legal Wigner masking filter belongs, necessarily, to the class of linear time-variant filters. It should be pointed out that the existence of legal masking filters has not been demonstrated. In fact, for specific examples of the input signal (e.g., an ideal chirp) it can be shown that no such filter function belonging to &e2(R2) exists.

6. Legal discrete Wigner filters For digital signal processing applications, a discrete version of the legal Wigner filter is necessary. Unfortunately, not all properties associated with the continuous WD are preserved by the discrete WD justifying a separate discussion. Herein,

D. Adler, S. Raz /Signal Processing 39 (1994) 179-191

(a) 0.4q

0.4

0.2

0.2

187

(b) rl

h

'

I'

t,

!~ ,'

,

i

~ h

I i

~

0

0

":~',,v, II lI I ",VI v

-0.2

"'

"',,/V ~,,

-0.2

-v

50

0

i

-0.4

i

-0.4

,J

50

100

(c)

(d)

N

BI(LoJ)

100

B 200~

0 I00

100 ' ~

t

t

Fig. 5. A design example of a time-frequency energy matched filter: (a) real parts of the signal sl (solid) and s2 (dashed). (b) imaginary parts of the signals sl (solid) and s2 (dashed), (c) the time-frequency matched filter corresponding to sl, (d) the time frequency matched filter corresponding to s2.

I00

>I-_1

\ lO-I

i \.

< G3 0 13:: O._ (~ 0

where H(n, p; m, q) and Oy(n, p) denote the Wigner space filter and the Wigner space output function, respectively. The discrete two-dimensional WD defined by

i

\

l~i,~(n,p;m,q) \

lO=Z

\

A

N-1 Y, f ( k l , k ~ ) . g * ( n --4N2k,,k~=O

\

!

n"

- k , , m - k~)

xexp[-j~((2kl - n)p + (2k2 - m)q)/N]

t~

t0-a -30

1

(6.2)

generates a four-dimensional basis -~5

-20

- 15

- 10

-5

SNR (db) Fig. 6. The p e r f o r m a n c e of the W i g n e r m a t c h e d filters e r r o r p r o b a b i l i t i e s versus the i n p u t S N R (the depicted p e r f o r m a n c e is b a s e d on c o m p u t i n g 10000 s a m p l e - f u n c t i o n s per S N R point).

~u,~(n,p; re, q) = ~'&,.&,(n,p; re, q)

(6.3)

satisfying

~kl,~(n,p; m, q) = ~k,(n,p)~,~(m, q)

(6.4)

and the discrete WD definition [13] (Eq. (2.2)) is adapted. By analogy to Eq. (3.1) we have

~*rs(n,p; re, q) = ~,~kz(n,p ; re, q),

where ~kr is defined in Eq. (2.7). The Wigner space filter can now be represented as N-1

N-1

Oy(n,p)=4N

~ m,q=O

(6.5)

H(n,p;m,q)Wx(m,q),

(6.1)

H(n,p;m,q)=

~ k.l,r,s=O

bu~kl,~(n,p;m,q).

(6.6)

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

188

Since only real-valued Wigner space filters are of interest, the expansion coefficients must obey

burs =

brskl. *

(6.7)

Eqs. (6.1) and (6.6) together with Eq. (2.10) lead to N-1

Or(n,p)=

~. CkrWkr(n,p),

(6.8)

k,r=O

where N-1

Ck, = ~

atbklrsa*.

(6.9)

l,s=O

Noting the identical forms of Eqs. (3.18) and (6.9), we may readily conclude that Eqs. (4.2), (4.4) and (4.13) remain equally valid for the discrete case. This in turn leads to concluding that the discrete Wigner space filter must obey one of the following legality constraints: N-1

~"

Wr(n, P) =

l,V~,~(n,p;m,q)Wx(m,-q)

(6.10a)

m,q=O or

N-1

Wr(n, p) =

14~.a(n,p; m, -- q) W,,(m,q) (6.lOb)

2 m,q=O

or

N-1

Wr(n,p) = Wb,(n,p) ~

B(m,q)Wx(m,q)

(5.1) or Eq. (5.2). These alternative constraints split the space of legal filters into two distinct categories. Eq. (5.6), stemming from Eq. (5.1), represents a general linear time-varying filter. On the other hand, Eq. (5.9), a consequence of Eq. (5.2), corresponds to a nonlinear (quadratic) time-invariant filter. Examination of the time-domain operator associated with the quadratic filter clearly indicates that only the amplitude of the filtered signal depends on the input signal. In fact, Eq. (5.9) leads to interpreting the operation as an estimator of the energy content of the input signal in the corresponding time-frequency intervals (namely, a time-frequency matched filter). Identical conclusions hold for discrete filters. The sole difference relates to the fact that discrete filters are invariably characterized by finite expansion coefficient matrices, and are thus numerically implementable. Since the output time-frequency distribution constitutes, by definition, a legal Wigner distribution, the corresponding filtering may be transformed and interpreted in the time domain. A direct time-domain execution reduces significantly the computational load.

(6.11)

ra, q=O

where N-1

[hsC~l(n)(a~(m)

/~(n, m) =

(6.12)

l,s=O N-1

/~(m,q)=

~

/~sffzs(m,q)

(6.13)

l,s=O

and l~.s is defined by Eq. (6.2). We showed that basically legal discrete Wigner filters have the same form as continuous filters. The main difference stems from the fact that discrete filters are characterized by a finite number of coefficients, enabling a direct numerical implementation of the filtering process.

Appendix A. The necessary and sufficient conditions ensuring the validity of Eq. (4.9) Here, we show that Eq. (4.9) is satisfied if and only if Eq. (4.12) is met. The left-hand side of Eq. (4.9) is recognized as a polynomial of rank four with the variables al,as* ,aq*, ap and the coefficients mlspq. Rearranging the coefficients of (4.9), in an increasing order of the indexes, such that l/> p and s >~ q, leads to ,

Z

7. Conclusions The legality of a Wigner space filter is ensured by satisfying the constraints imposed by either Eq.

•

alas mlspqaq ap l,s,p, qEZ (mlsp q + mpslq q- mlqps +

l,s,p, qe~l>p;s>q 2 * * (mlslq + mlqls)a I a s aq

+ s>q

mpqts)ala* a* ap

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

+

~

(m,,ps+ mp,,,)at(a*)2ap

bmpk~[b k ~ b k ~

I,s,p~Z I>p

(A.1)

= bm~qbkpks(bkpnqbkpks -- bkpnsbkpkq)

l,s~Z

For arbitrary {al}, the cited polynomial vanishes identically if and only if all its coefficients equal zero. Consequently, Eq. (A.1) is satisfied if the following four equations are satisfied:

+ bmpksbkpkq(bkpnsbk~q -- bkpnqbkpks) = [bmpkqbk~ks -- br~pk~bkpkj [bkv,,qbkpk~ -- bkp,~bk,k~].

(B.3) Eq. (B.3) has two solutions

rnl~z~= 0,

(A.2)

mt~z~ + mt~ = 0,

(A.3)

bkpk q ~- bkpk s = bmp k

m,~p~+ mp~ = 0,

(A.4)

or

bmpkq

and m~s~ + mpa~ + mt,ips + m~,s = 0.

(A.5)

Eqs. (A.2)-(A.4) constitute special cases of Eq. (A.5), hence Eq. (A.5) forms the necessary and sufficient condition on mts~ ensuring the validity of Eq. (4.9).

bkpnq -- bkpns ~ bkpn bkpkq bkpks

Eqs. (4.4) and (4.13) should hold for all k,m,n and l, s, p, q. To simplify these expressions, two special cases of Eq. (4.13) are examined:

Rewritten, (B.4) yields

Applying constraint (I) to Eq. (4.13) results in bmpnq : bmpkq bkpnq" bkpk q '

bkpkq

~ O.

(B.1)

Similarly, constraint (II) yields bmpnqbkpks "~- bmpnsbkpkq -- brapkqbkpns -- bmpksbkpnq : O.

(B.2) Substituting Eq. (B.1) into Eq. (B.2) yields 0 = bmpkqbklmq bmpksbkpns bkpk q bkpks ~ ~ bkpk bkpkq

or

b,p. = 1.

bk~]

(B.Sb)

bmpnq = bmpkbkpnq = bmpkbnqkp ^ , ^ ^, , = bmpkbnqkbkqkp ^ ^

= bmpkb.~kbkpkq

Vk e ~_

(B.6a)

or

(B.6b)

(bmpk -- bralk )(bkpnqbklks -- bklnqbkpks

+ bkp,~bk~kq -- bu.~bkpkq) = 0.

(B.7)

As the expansion coefficients must satisfy Eq. (3.9), Eq. (B.7) is further simplified via the following algebraic sequence: 0

(bmpk -- bm,k ^ )(b.qkpbk,k~ * * __ bnqktbkpks +

* * bn~kpbklkq -- b.~klbkpk~)

^ A, * * = (bmpk -- bmzg)[b.qk(bkqk,buks-bkqk~bkpk~) ^,

-- bmpksbkpnq

(B.5a)

Substituting Eq. (B.6a) into the left-hand side of Eq. (4.13) and Eq. (B.Sa) into the right-hand side of Eq. (4.13) results in

(II) p = l , q ~ s .

= b"k'Fb- bkpk,

bm~m= 1

bml,k~ = bmpkbkj,k~;

bmpnq = bkpn bmpkq : bkpnbkqmbkpkq. ~ ~*

(I) p = l , q = s ;

bmt~bkpns

(B.4b) "

Combining Eq. (B.1) with Eq. (B.5) leads to

Appendix B. Derivation of Eq. (5.1) and (5.2)

-

(B.4a)

bmpk~a ^

bkp.q = bkp.bkpkq;

-

bkp.q]

L b~

+

+ ~ mlasal2(a~) . 2 = 0.

189

,

*

+ b.~k(bkskpbklkq -- bksubkpkq)] : (bmpk -- bmlk)(b*qk -- b*sk)(b~k,bklks -- b*qklbkpks).

(B.8)

190

D. Adler, S. Raz / Signal Processing 39 (1994) 179-191

An identical result is achieved by using Eqs. (B.5b) and (B.6b). One solution of Eq. (B.8) is (B.9)

[~mpk = [~,n,k

and, therefore, b r a p k i s an exclusive function of the first and last integers, i.e., ^

(B.10)

b,.pk ~- b~k .

Substituting this result into Eq. (B.1) yields bmpnq

h' h ' * hUkpkq t.,mkUnk

Vk e 7/.

(B.11)

Recalling condition (4.4), bkpkq m u s t also satisfy apbkpkqa* >~ 0 p.q, e ~

'7'a ~ [2(~) and Vk 6 7/.

(B.12)

The second solution of Eq. (B.8) is (B. 13)

bkpkqbklks = bktkqbkpks,

which together with Eqs. (3.9) and (4.4) satisfies Eq. (2.11). This implies that bktks is an 'outer product' [1, Theorem 4] i.e., B" ~. /2(~2).

bklk s = UklUks h" h"*.,

(B.14)

Substitution of Eq. (B.14) into eq. (B.6) yields bmpnq ~ U~,m p k UF.* n q k t /~,, k p t . ,~,,* kq

(B.15a)

or bmpnq

=

~* 1,,, w,* .

(B.15b)

tJkpnUkqmtJkpUkq

Eqs. (B.11), (B.15a) and (B.15b) should hold for all k e Z. For a given k = ko, Wigner space filter is legal if and only if there exist a vector b' E 12 (~) and matrices B e 12(~2), n e 12([~2), such that the filter expansion coefficients are consistent with one of the following relations: b,,,p,,q = b,npbn*,

(B.16a)

bmpnq = bnpbmq, *

(B.16b)

bmpnq

h' ~ m ~h'* n

vpq

apbpqa q ~ 0

, p,q~

Va ~ /2(~),

(B.17)

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[2] G.F. Boudreaux-Bartels and T.W. Parks, "Time-varying filtering and signal estimation using Wigner distribution synthesis techniques", IEEE Trans. Acoust., Speech Signal Processing, Vol. ASSP-34, No. 3, 1986, pp. 442-451. [3] T.A.C.M. Claasen and W.F.G. Mecklenbrauker, "The Wigner distribution - a tool for time-frequency signal analysis. Part I. Continuous-time signals", Philips J. Res., Vol. 35, 1980, pp. 217-250. [4] T.A.C.M. Claasen and W.F.G. Mecklenbrauker, "The Wigner distribution a tool for time-frequency signal analysis. Part I1. Discrete-time signals", Philips J. Res., Vol. 35, 1980, pp. 276-300. [5] T.A.C.M. Claasen and W.F.G. Mecklenbrauker, "The Wigner distribution - a tool for time-frequency signal analysis. Part III. Relations with other time-frequency signal transformations", Philips J. Res., Vol. 35, 1980, pp. 372 389. [6] L. Cohen, "A critical review of the fundamental ideas of joint time frequency distributions", Proc. IEEE Internat. Conf. Systems and Circuits 1986, 1, 42, pp. 28-33. [7] L. Cohen, "Time frequency filtering", Proc. IEEE Internat. Conf. Acoust. Speech Sional Processing, 1988, pp. 2212 2215. [8] L. Cohen, "Time frequency distributions - A review", Proc. IEEE, Vol. 77, No. 7, July 1989, pp. 941-981. [9] P. Flandrin, "Maximum signal energy concentration in a time frequency domain", Proc. IEEE lnternat Conf. Acoust. Speech Signal Processin9 1988, pp. 2176-2179. [10] F. Hlawatsch and W. Kozek, "Time-frequency analysis of linear signal spaces", Proc. IEEE lnternat. Conf. Acoust. Speech Signal Processing, 1991, pp. 2045-2048. [11] L.D. Jacobson and H. Wechsler, "Joint spatial/spatialfrequency representation", Signal Processing, Vol. 14, No. 1, January 1988, pp. 37-68. [12] A.J.E.M. Janssen, "Wigner weight function and Weyl symbols of non-negative definite linear operators", Philips J. Res., Vol. 44, 1989, pp. 7-42. [13] F. Peyrin and R. Prost, "A unified definition for the discrete-time, discrete-frequency, and discrete-time/frequency Wigner distribution", IEEE Trans. Acoust. Speech Signal Processing, Vol. ASSP-34, No. 4, 1986, pp. 858-866. [14] S.C. Pei and T.Y. Wang, "The Wigner distribution of linear time-variant systems", IEEE Trans. Acoust. Speech Signal Processing, Vol. ASSP-36, No. 10, 1988, pp. 1681-1684. [15] S. Raz, "Synthesis of signals from Wigner distributions: Representation on biorthogonal bases", Signal Processing, Vol. 20, No. 4 August 1990, pp. 303-314. [16] B.E.A. Saleh and N.S. Subotic, "Time-variant filtering of signals in the mixed time-frequency domain", IEEE Trans. Acoust. Speech Signal Processing, Vol. ASSP-33, No. 6 1985, pp. 1479-1485. [17] S.M. Sussman, "Least squares synthesis of radar ambiguity functions", IRE Trans. Inform. Theory, Vol. IT8, No. 4, April 1962, pp. 246-354. [18] V.A. Topkar, S.K. Mullick and E.L. Titlebaum, "Realisability of Wigner distribution", Electronic Letters, Vol. 25, No. 11, May 1989, pp. 744 745.

D. Adler, S. Raz / Signal Processing 39 (1994) 179 191 [19] B.V.K. Vijaya Kumar, C.P. Neuman and K.J. DeVos, "Discrete Wigner synthesis", Signal Processing, Vol. 11, No. 3, October 1986, pp. 277-304. [20] J. Wexler and S. Raz, "Synthesis of discrete-time signals from distributions", Electronic Letters, Vol 25, No. 2, January 1989, pp. 93-95.

191

[21] J. Wexler, and S. Raz, "Wigner-space synthesis of discretetime periodic signals", submitted (also EE Publ. 734, November 1989, Technion liT). [22] K.-B. Yu, "Signal representation and processing in the mixed time-frequency domain", Proc. IEEE lnternat. Conf Acoust. Speech Signal Processing, 1987, pp. 1513 1516.