WIGNER-TYPE AND TIME-FREQUENCY ANALYSIS ON · WIGNER-TYPE a-b AND TIME-FREQUENCY ANALYSIS BASED ON CONJUGATE OPERATORS* Franz Hlawatsch, Teresa Twaroch, and Helmut Bolcskei INTHFT,

WIGNER-TYPE a-b AND TIME-FREQUENCY ANALYSIS BASED ON CONJUGATE OPERATORS*

Franz Hlawatsch, Teresa Twaroch, and Helmut Bolcskei

INTHFT, Vienna University of Technology, Gusshausstrasse 25/389, A-1040 Vienna, Austria email address: fhlawat s@email. tuwien. ac . at

Abstract-We extend the Wigner distribution (WD) to conjugate unitary operators A, and Bp. The resulting “AB- WD” is defined both as an a-b representation and as a time- frequency representation. Important properties and relations of the WD are generalized to the AB-WD.

1 INTRODUCTION A N D OUTLINE Recently, general frameworks for joint a-b representations

and time-frequency representations based on airs of unitary operators A, and Bg have been proposed [1]-{3]. In particular, the time shift operator T, and the frequency shift operator F, under1 ing Cohen’s class and the Wigner distribution (WD) [5], [147-[17] have been generalized to the concept of conjugate (or dual) operators A, and Bg [6, 8, 10, 12, 131. Classes of a-b or time-frequency (TF) representations based on conjugate operators retain the structure of Cohen’s class. Hence, each such class contains a central “AB-WD.”

This paper introduces and discusses the AB-WD. Section 2 summarizes the theory of conjugate operators. Sections 3 and 4 introduce the AB-WD as an a-b and TF representation, respectively. A special case is considered in Section 5.

Cohen’s class and WD. Let z(t) E ,&(Et) be a signal with Fourier transform’ X ( f ) = st z ( t ) e - j l f f f t dt. Cohen’s class of quadratic TF representations (QTFRs) [5], [14]-[17] consists of all QTFRs Cz(t, f) that are covariant to the time shift operator T, and frequency shift operator F, defined as (T, z ) ( t ) = z ( t - r ) and (F, z ) ( t ) = z(t) ej2n”t,

CF,T,z(t, f) = cz(t - 7, f - v ) . (1) Any QTFR of Cohen’s class can be written as

C, (t ,f) =J J z (t 1) z* ( t 2 ) h* (t 1 - t , t 2 - t ) e-j2nf(tl --t,)dt 1 dt2

with a 2-D kernel function h(t1, t2 ) . The central QTFR‘Ek Cohen’s class is the WD [5], [14]-[17]

t l t2

= l X ( f + i ) X * ( f - i ) e ” 2 “ t ” d v , (4)

from which any Cohen’s class QTFR can be derived as

Cz(t, f) = 1, J, +(t - t’, f - f’) WZ(t‘, f‘) dt’df’ (5)

with the kernel +(t, f) = s, h*(- t + $, -t - $) e - j z n f T d r .

‘Funding by FWF grant P10531-OPH. ‘Integrals extend from -co to 00 unless specified otherwise.

2 CONJUGATE OPERATORS

Generalizin the shift operators T,, F, underlying Co- hen’s class a n i the WD, we now consider two linear operators A, and Bp indexed by parameters a E 9 and ,f3 E Q with 4 E R. These operators are assumed to be unitary on a linear signal space X C L2(R), and to satisfy identical compo- sition properties AazAuq = A,,.,, and B p z B p I = Bpl.pz, where (Q,.) is a commutative group [l, 6, 9, 181. The eigenvalues A$ and eigenfunctions u f ( t ) of A,, defined by (A, u t ) ( t ) = u t ( t ) , are indexed by a “dual” parameter b. The A-Fourier transform (A-FT) [l, 5, 131 is defined as

XA(b) ( z , ~ : ) = / z ( t ) a f * ( t ) d t . t (6)

Similar definitions apply to A;,,, u t ( t ) and the B-FT X g ( a ) . Two operators A, and B p as defined above are called

conjugate [6, 10, 121 if a E Q, b E Q, and if application of one operator to an ei enfunction of the other operator merely produces a shift of the eigenfunction parameter, i.e.,

values of conjugate operators can be written as [6, 10, 121 (Bp u f ) ( t ) = zLfea(t) and (A, U, B ) ( t ) = ut.,(t). The eigen-

A;,, = e r j 2 ” ~ ( B ) ~ ( a ) = A* Ap,a > (7) where p ( g ) E R maps (G,.) onto (a,+) in the sense that P(91.92) = P(91) f p(g2) , k 7 0 ) = 0, and f 4 g - l ) = - d g ) with go the identity element in Q and g--’ the group-inverse of g . Due to (71, we shall write A& = Aa,b and A;,, = A;,, in the following. Conjugate operators commute up to a phase factor, A,Bp = A,,p BOA,. Their eigenfunctions are related as ( U , ~ , U : ) = A,+, ~ , ~ , 8 ( t ) A ~ , , d p ( a ) = u t ( t ) , and sG d ( t ) Ao,b = ut( t ) , where d d g ) = b ’ ( g ) I d!? if p ( g ) is differentiable. The A-FT and B-FT are related as X B (a) = J, X A ( ~ ) dp(b) and X A ( ~ ) = Jg X B ( ~ ) Ab,, dp(a) (Cf. the equivalent concept of “dual operators” in [8, 131).

Examples. The shift operators T,, F, underlying Co- hen’s class and the WD are conjugate with (9, .) = (R, +),

uT(t) = e j 2 f f f t , U;@‘) = 6( t ’ - t ) , X T ( ~ ) = X(f), and X F ( t ) = z(t) . The operators T,, F, are conjugate since (F, uT)(t) = UT+,(t) and (T, $) ( t ‘ ) = uf+,(t’). They

The operators underlying the hyperbolic QTFR class [19, 201 are conjugate as well, but the operators underlying the afine class and the power classes [21]-[24] are not conjugate.

= e f j 2 n P(,) p ( b )

p ( g ) = g , b = f, a = t , = e - j 2 n T f , A c t = ej2”ut ,

commute up to a phase factor, T,F, = e-j””‘” F u T r .

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3 AB-WD AS a-b REPRESENTATION

We now introduce' the AB- WD as an extension of the WD in (3), (4) to arbitrary conjugate operators A, and Bp:

W,""(a, b) e L X A ( b 0 P"') x,* ( b 0 P-"') A* a,p ~ P ( P ) 1

where X A ( b ) is defined in (6) and pl/' is defined by P112 0

pl/' = p. The AB-WD is a function of the parameter a of ; f ( t ) and the parameter b of u f ( t ) (see Section 4 for a time-frequency version of the AB-WD). It can be equivalently expressed in terms of the B-FT,

For A, = T, and Bp = F,, these two expressions reduce to (4 and (3), respectively, so that W:"(a,b) = W,(t,f). On t h e other hand, the AB-WD can be formally obtained from the WD by a unitary signal transformation and a parameter transformation (t ,f) + (a,b) [1O]-[12]: If X a , b = e-jZff p ( a ) p ( b ) (minus sign in the exponent)] then

where tr > 0 is a fixed reference time and ,U-'(.) is the function inverse to ,U(.). If X, ,b = ejZa p ( a ) p ( b ) , then

Interference formula:

[WtB(a , b)]' = /lz W,"" (a 0 a'/', b 0 pl/')

. W p ( a . a-, b 0 p--1/') dp(a) dp(P) .

Relation with AB-AF: We next introduce the AB-ambiguity function (AB-AF) as

A X (a , P ) = (Bp-1/2Aa-1/2 2, Bpi/zA,i/z z) AB A

and

Properties of the AB-WD. The pro erties of the AB- WD generalize the properties of the WD E], [14]-[17]:

Real-valued: W i B ( a , b) E R. Covariance property:

WBApBA@((a,b) = W,AB(a.a-',bOP-'). (8)

Marginal properties:

'While only the auto AB-WD will be considered for simplicity, we note that extension to the cross AB-WD is straightforward.

. W p ( a .a a-1/', b 0 p--1'2) &(a) d p ( b ) .

Uncertainty relations: We define the A-spread U:, B-spread

with bo #"6. These quantities are related to the AB-WD as

and there exist the bounds (uxxcataAty relations)

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Generalized instantaneous frequency property:

Covariant a-b representations. The class of all quadratic a-b representations &,(a, b) that are covariant to conjugate operators A,, B p as (cf. (8),(1)) QB~A,,(u, b) = &,(a a-', b 0 p-') can be formulated as [6, 10, 121

&,(a,b) = (2, H!,fz) = (11) l, / ~ , - I B ~ - I z) ( t l ) (A,-1Ba-1z)*(t2) h'(t1, t 2 ) dtldt2

with Hhqt = BbA,HA,-lBb-l, where H is an arbitrary linear operator with kernel h(t1, t 2 ) . Equivalently,

where the kernel u(a, P) is related to h(t1, t z ) [6,10,12]. The covariant class {&,(a, b)} is the extension of Cohen's class in (2) to arbitrary conjugate operators A,, Bp. In particular, the AB-WD is obtained with !P(a,P) E 1.

It can be shown that the covariant class {&=(a, b)} can be derived from the AB-WD W,""(a, b) as (cf. (5))

Q,(a,b) = // $(a*a- ' ,b*P- ' ) W:" (~,P)( ICL(~)~P(P)

(13) 9 2

where the kernel $(a, b) is related to the kernel !P(a, P ) in (12) as $(a, a) = J&z *(a, P) $,,A,& dd.1 dP(P).

AB-spectrogram. Setting h(t1,tz) = g(t1) g 8 ( t 2 ) in (11) yields the AB-spectrogram

S,AB(a,b) = IL$B(a,b)12 with L t B ( a , b ) = (cc ,BDA,~) .

Here, $(a,b) = W,"B(a-l,b-') so that S,A"(u,b) = JJG2 W,"" (a a-l, P b - l ) W,ABb, PI d P ( a ) 4.43.

4 AB-WD AS TF REPRESENTATION The AB-WD can be re-formulated as a quadratic time-

frequency (TF) representation. Let u t ( t ) denote the instantaneous frequency of the eigenfunction uf( t ) , and let ~:(f) denote the group delay of the eigenfunction ut ( t ) . For any ( a , b ) E g2, the corresponding functions u t ( t ) and ~:(f) are assumed3 to intersect in a unique TF point (t, f ) . Hence, there is a one-to-one correspondence (t, f) = Z(a,b), a, b) = Z-'(t, f) where Z(., e ) will be called the localization 6 nction of the operator pair A,, Bp. The localization

function is constructed by solving the system of eauations v q ) = f, ~:(f) = t for he TF version of the

All properties of the a-b version of the AB-WD (see Section 3) can be re-formulated for the TF version of the AB-WD. For example, the TF version of the covariance (8) reads

A where (a, b) 0 (a, P) = (U 0 a, b P). The marginal properties (9), (10) become

/v."B(t,ui?(t)) t d f i 2 ( t ; b ) =: Ix~(b)l~, (16)

where df i l ( f ;a ) and dfiZ(t;b) follow from .E(.), ut( . ) , , and dp(.l; All other properties and relations listed in Section 3 can e transferred to the TF domain as well.

5 ANEXAMPLE We shall finally consider an example. Let the operators

A, and Bp be defined on the space X = &(E+) as

4 t ) ( B ~ x) ( t ) -- &2- 1 4 W t / b )

where t , a , p > 0 with t , > 0 fixed. Since A,,A,, = A,,, and Bp,Bp, = Bplpz, the underlying group is the mug tiplicative group, (Q, 0 ) = (R+, .), with identity element go = 1 and inverse elements 9-l =: l/g. The eigenvalues/functions of A, and Bp are At ,a = e - j 2 r r ' n a 1 n b ,

B = j 2 z l n p l n o = u;(t) = 5 ej2-1nb W t / t , )

-& &(ln$ - lna). Note that p ( g ) = lng and d p ( g ) =

$. The A-FT is the Mellin-type transform X A ( ~ ) = z ( t ) e - j 2 * l n b l n ( t / t r ) $ and the B-FT is XB(U) = 6 z( t ,a) . The operators A, and B p are conjugate, ( B ~ u t ) ( t ) = u&(t) and (A, u t ) ( t ) -- u&(t). They com-

D IS the a-b, time-domain version of the &-distribution [19, 25, 261

w:B(~ , b) = t r u / z ( t , a e u / 2 ) z*(tra e-ulz) e - j z * ( l n b ) u d u

31n certain cases, this assumption holds if one uses the group delay of uf ( t ) and the instantaneous frequency of u f ( t ) ; here, an analogous theory can be formulated.

, and &3,r e

So00

mute up to a phase factor, A,$ = e-J2- l n a l n P B PA,. The a-b version of the AB-

00

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with a, b > 0 . It satisfies the covariance property

the marginal properties

db b LW W,””(a, b) - = tra lz(t,a)I2

and other properties (cf. Section 3). The covariant class (11) is the a-b, time-domain version of the hyperbolic class [19, 201

Qz(a ,b ) =

i t can be derived from W k B ( a , b) as (see (13))

With v t ( t ) = ( Inb) / t and ~:(f) E t r a , the localization function is obtained as ( t , f ) = Z(a, b) = ( h a , e) with inverse (a ,b ) = Z-*(t , f ) = ( & e t f ) . The TF version of the AB-WD is then [19, 25, 261

W

- z(teY/2) $*(t e - 3 2 n t f u du - t J _ W

for t > 0; i t satisfies the covariance property (cf. (14))

and the marginal properties (cf. (15), (16)) W

The class of QTFRs satisfying the covariance (17) is [19, 201

Q z ( t , f ) = Q x ( $ , e ” ) =

i t can be derived from %bB(t,f) as

Q z ( t , f ) = Sm Jm t , !J ($ , e t f - t ‘ f ’ ) z i B ( t ‘ , f ’ ) dt‘df’ t ’=O f’=-m

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WIGNER-TYPE AND TIME-FREQUENCY ANALYSIS ON · WIGNER-TYPE a-b AND TIME-FREQUENCY ANALYSIS BASED ON CONJUGATE OPERATORS* Franz Hlawatsch, Teresa Twaroch, and Helmut Bolcskei INTHFT,