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Time-Frequency Localizationvia thie Weyl Correspondence

Jayakumar RamanathanPankaj Topiwala

September 1992

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Time-Frequency Localization via the Weyl Correspondence

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Jayakumar RamanathanPankaj Topiwala

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A technique of producing signals whose energy is concentrated in a given regionof the time-frequency plane is examined. The degree to which a particular signalis concentrated is measured by integrating the Wigner distribution over the givenregion. This procedure was put forward by Flandrin, and has been used for time-varying filtering in the recent work of Hlawatsch, Kozek, and Krattenthaler. Inthis paper, the associated operator is studied. Estimates for the eigenvaluedecay and the smoothness and decay of the eigenfunctions are established.

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Time-Frequency Localizationvia the Weyl Correspondence

Jayakumar Ramanathan

Pankaj Topiwala

September 1992

A Publication of MITRE'sMathematical Research Project

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Cover Illustration: The Wigner distribution of achirp signal in 0 dB noise after filtering the noiseby time-frequency localization techniques.

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ABSTRACT

A technique of producing signals whose energy is concentrated in a given regionof the time-frequency plane is examined. The degree to which a particular signal isconcentrated is measured by integrating the Wigner distribution over the given region.This procedure was put forward by Flandrin, and has been used for time-varying fil-tering in the recent work of Hlawatsch, Kozek, and Krattenthaler. In this paper, theassociated operator is studied. Estimates for the eigenvalue decay and the smoothnessand decay of the eigenfunctions are established.

iii

ACKNOWLEDGMENTS

We wish to thank the MITRE Sponsored Research program and Eastern MichiganUniversity for their support during various stages of this work. This paper will appearin the SIAM Journal of Mathematical Analysis, and we are especially grateful to thatjournal's referees for bringing references [5,10,12] to our attention and for suggesting aresult along the lines of Proposition 10.

iv

TABLE OF CONTENTS

SECTION PAGE

1 Introduction ................. ........................... 1

2 The Weyl Correspondence ............ ..................... 3

3 Localization via a Cut-Off ............ ..................... 5

4 Numerical Examples ........... ........................ 13

List of References ............... ............................ 21

V

LIST OF FIGURES

FIGURE PAGE

1 Localization Domains: a) Zigzag Domain, b) Disk of Area 50, c) 4x2Rectangle, and d) Parallelogram ...... ................... .. 14

2 Magnitude of Eigenfunctions 1-3 for Localization onto a Zigzag Do-main (in figure La), with a Plot of Eigenvalues Sorted According toDecreasing Magnitude .......... ....................... .15

3 Weyl Eigenvalue Plot for Localization onto a Disk of Area 50 (figure1b), Sorted According to Decreasing Magnitude ..... ............ 16

4 Prolate Spheroidal Wavefunctions 1-9 for Localization on a Rectangle(figure 1c), with Eigenvalue Plot ......... ................... 17

5 Weyl Eigenfunctions 1-9 for Localization on a Rectangle (figure 1c),and a Plot of Eigenvalues in Decreasing Magnitude .............. .. 18

6 Wigner Distribution Intensity Plot of (a) Chirp Signal, (b) Chirp in 0dB Gaussian White Noise, and (c) Filtered Noisy Signal ........... .. 19

7 Plots of the Real Part of (a) Chirp Signal, (b) Chirp Signal in 0 dBGaussian White Noise, and (c) Filtered Noisy Signal; Part (d) Is anEigenvalue Plot. The First Two Eigenfunctions, Weighted by theirEigenvalues, Are Used in the Filtering ...................... .20

vi

SECTION 1

INTRODUCTION

It is well known that the time-frequency characteristics of a square integrable signalcannot be arbitrary. For example, no such signal can be both time and band limited.The Heisenberg uncertainty principle provides another quantitative restriction on thejoint time-frequency behavior of a square integrable signal. These facts indicate that asignal cannot have all its energy concentrated in a finite region of the time-frequencyplane.

Nevertheless, in many applications it is important to use signals whose time-frequency characteristics are highly localized. Among other things, the work of Landau,Pollack, and Slepian [1,21 produced a rigorous development of band limited signals thatare as concentrated as possible within a prescribed timespan. More recently, there hasbeen interest in finding signals that are localized in general regions of phase space viamethods that keep time and frequency on an equal footing (see the papers of Daubechiesand Paul [3,4]).

In this paper, we study a localization technique that uses the Wigner distributionto measure the degree to which a signal is concentrated in a particular region. Thisleads to a self-adjoint localization operator that is easy to study in terms of the Weylcorrespondence. Under the Weyl correspondence, a function of two variables-calledthe symbol--is associated with an operator on functions of one variable. The symbolof the localization operator is simply the indicator function of the given region in thetime-frequency space. The eigenfunctions of this operator with large eigenvalues spana subspace that can be used to determine the component of a general signal thatis concentrated within the given region of the time-frequency plane: one computesthe projection (either orthogonal or weighted by the eigenvalues) of a general signalinto this subspace. This procedure was put forward by Flandrin [5], who derived anumber of useful results, including Lemma 4 below. It has since been developed inthe context of time-varying filtering (see for example the papers of Hlawatsch, Kozek,and Krattenthaler [6,71). This paper is devoted to the further study of the localizationoperator described above. The asymptotic properties of the eigenvalues are studied andit iq established that they are O(k 3 / 4 ). In addition, the eigenfunctions with nonzeroeigenvalues are shown to have faster than exponential decay in both the time andfrequency domains. This, of course, leads to a statement regarding the smoothness ofthese eigenfunctions. In particular, we show that they are analytic. In the last section,some numerical examples are provided.

I I I I I !1

2

SECTION 2

THE WEYL CORRESPONDENCE

The basic properties of the Weyl correspondence that we will need are collected in

this section; for a thorough treatment, the reader is referred to the book of Folland [8].

Let f,g E L2 (R). A general time-frequency shift of f is

p(T,a)f(t) = Je 2xiat f(t + T).

The cross-ambiguity function of f and g is

A(f,g)(r,o) = (p(7r,a)f,g)

= J irae2 f f(t + r)g(t)dt

= J e2 'iasf(s + r/2)g(s - T/2) ds.

The value of A(f,g) at a particular point is the cross-correlation between a particu-lar time-frequency shift of f against g. The ambiguity function is therefore a time-frequency cross-correlation between two functions. The Wigner distribution is thetwo-dimensional Fourier transform of the cross ambiguity function, thus giving it theinterpretation of a time-varying spectrum. The Wigner distribution can be written as

W (f, g) ( 1) = J e-21rirf(t + r/2)g(t - r/2) dT.

Several useful properties of the Wigner distribution are catalogued below.

THEOREM 1. Let f,g E L 2(R). Then

(i) W(f,g)(ý,t) E L2(R2) and IIW(f,g)jj 2 = -Ifj12 ugll 2,(ii) w(f,g) E Co(R2 ) and JiW(f,g)11o. < 11! 11 1gll,

(iii) W(g,f) = W(f,g),

(iv) W(f, )(Ct) = W(f,g)(t,-ý), and(v) W(p(a, b)f, p(a, b)g)(ý, t) = W(f, g)( -b, t + a).

The Weyl correspondence uses the Wigner distribution to define a correspondencebetween functions of two variables and operators on L2(R). It is defined, via duality,by

(Lsf,g) = /.(Lsf~g) S(ý, t) W(f , g)(ý, t) d~dt

where f, g E L2 (R) and S(ý, t) is a function with appropriate decay properties. S(ý, t)is the symbol of the operator Ls. The following theorem of Pool [91 is useful.

3

THEOREM 2. A symbol S(ý, t) E L2 (R2 ) gives rise to an operator Ls that is Hilbert-Schmidt on L 2 (R). Moreover, the mapping S • Ls is a unitary operator from L 2(R2 )to the Hilbert-Schmidt operators on L2 (R).

LEMMA 3. Ls is self-adjoint if S(ý,t) is real valued.

The following result was derived by P. Flandrin in [5], based on results of Janssen[10]. We will follow the development in [8].

LEMMA 4. The eigenfunctions of the operator Ls corresponding to a radially symmetricsymbol S are:

h ( 21/4 -1 - ) t2d. e -2r12

) (t VT _2,5'c'dt]

PROOF: Theorem 1.105 in [7] shows that

2(- 1 )k' .- (2!/T7rZ)J-LIk)(47rI z12). for j > k

W(hj,hk)( ,t) =

I r - 2, 9r zI12 2

(jL-k) ) 9

2(-1)J. 'IC- V J• (4r, lz-2), for k _>j

where z = t + i2 and L(') is the associated Laguerre polynomial. Set r = I and notethat

(Lshj, hk) = JfS(r)IV(hPhk)d~dt

0, forj kS(0) d2d

(r S,,, ) rdr, for j=k.(-1 ~ ~ ~ (47 r(r )~2 (1)

Hence

A, = (-1)J47 S(r) -2CrTL()(4r )rdr.

4

SECTION 3

LOCALIZATION VIA A CUT-OFF

The localization operator we are concerned with is Lx,, where X0 is the char-acteristic function of some bounded domain in the t - ý plane. We will assume that0 c [-B, B] x [-T, TJ. Note that

(Lx 0 f,g) - JjW(f,g).

Lx11 is self adjoint since X& is real valued. Pool's theorem implies that Lx0 is Hilbert-Schmidt. Hence there is an orthonormal basis 01,02,... of L 2(R) and real numbersA1, A2,... such that Lx0q~k = Akbk. The Hilbert-Schmidt norm of Lxn is (Ak 12

101, the measure of Q. We will assume that the eigenvalues are arranged in orderof decreasing absolute value. It is easy to check that the largest positive eigenvaluecorresponds to the maximum energy an L 2 function can have within the domain Q. Thecorresponding eigenfunction would then be a time function with energy as concentratedas possible within Q. Our principal aim in this paper is to study the decay of theeigenvalues of LX0 and the smoothness properties of the eigenfunctions.

Several properties of the associated kernel will be of importance.

LEMMA 5. The kernel of the operator Lx, is given by the equation

Ss+i,K (s, t) S +f(• t) e2,ri(s9-t)C dý

and has the properties

(i) K(s,t) =_O if Is+tŽ >2T and

(ii) if the cross-sections Ql of Q in the ý direction consist of at most Mintervals, then IK(s,t)( <_ CM

PROOF: The formula for the kernel is well known (see [8]). The kernel can be writtenas

s+t.K(s,t) =F

where

F(r1 , 1) = e

5

Item 1 follows from the observation that jt 0 if ItI > 2T. We now verify item 2.By assumption, the cross-section is the union of at most AM disjoint intervals: Qt =UL=[i, [#i] for some integer L < Al. Therefore one can estimate that

Ct Ze7ri(aCk +Ok )n sin(7r( 3k - ak)rl)< , sin(7r(#k-- ak)7r

z~ 7 77~2L 2M

77 1 + 1 7,71+_ <Tsince sin(Ax)/x < . The estimate only needs to be verified when Is + tI < 2T.In this case,

Is - t± + 2T > Is - tI + tI _2 21sl.

Using this and the estimate for F(r1 , t) yields

IK(s,t) I < CMis- ti+ 1

CM- 21sl + 1 - 2T

for all large s. The estimate in the theorem follows easily by adjusting the constant asnecessary.

For domains Q with piecewise C1 boundary, we can show that Ak is 1 Theproof is a modification of Weyi's classical work on the asymptotics of eigenvalues ofintegral equations. The following lemma contains some useful standard facts about theWeyl correspondence. Again we refer the reader to [8] for the proofs.

LEMMA 6.

(i) The operators corresponding to S(ý, t) and S(ý - ýo, t - to) are unitarilyequivalent.

(ii) Suppose the symbols S1(ý,t) and S2 (ý,t) are related by an orthogo-nal change of variables. Then the corresponding operators are unitarilyequivalent.

We will first prove that Ak is O(3-) for domains Q of the following form:

(,) Q = {(•,t) : a(t) 5 ý 5 /(t),t E [-T,TII

6

where a(t), 0(t) are C1 functions with vanishing endpoint values. The kernel associatedto the symbol Xn(s, t) is:

K(s,t) = e-i(s-t)[a+I•](• t ) sin(ir[a - #](2-•t )(s - t))r(s -t)

Moreover, it is easy to check that K is a Lipschitz function whose gradient DK existsa.e. and satisfies the inequality

IDKI < C a.e. (2)

LEMMA 7. If fS is of the form (*) then there are symmetric finite rank approximations

kN(s,t) ZaQXQ

where Q ranges over squares of the form

{(ý,t) : i/N < ý <_ (i + 1)/N,j/N < t < (j + 1)/N} - N 2 < ij < N 2 -1

with the property that

if IK(s,t) - kN(s,t)12 dsdt is 0(1/N). (3)

PROOF: Set

(Q- I K(s,t) dsdt.IQ-I J-

The symmetry of the kernel K forces symmetry of kN. Let R = UQ. Then

f IK(s, t) - kN(s,t)12 dsdt = f Ij I(s, t) -kN(s,t)12 dsdt

< ZIQI I IDKI2

using Poincare's inequality [11]. Since K(s, t) is supported within the strip Is+tI < 2T,equation (2) yields the estimate

R IK(s,t) - kN(s,t)12 dsdt < N (4)

7

The mean square error over the exterior of R can be handled as follows:

, 12 12 <

iI.2R IK - kNI = I,2KR JK dsdt <N (5)

in view of lemma 5. Putting equations (4) and (5) together yields the estimate inequation (3).

In view of Satz III of Weyl's paper [111, one has

A4 N2+1 +.- _ C/N.

We will apply this inequality to N/2 where N = [A/kTT-1/2]. (Here [ denotes thegreatest integer function.) This yields:

4(N/2)2- +1.. + A2 < 2C/N.

Since the eigenvalues have been arranged in decreasing order, the left-hand side of theabove inequality is greater than or equal to 3N 2A2. This yields an inequality of the

form A• < C/N 3 . Clearly 11N is 0(1/v"k). These remarks imply that Ak is O(k3-•4) fordomains with property (*). We now use a cut and paste argument to derive the resultfor general 0 with C1 boundary.

LEMMA 8. If K'(s, t) and K"(s, t) are two symmetric real kernels in L2 (R2 ) with

AkI<_C and I< l_

then the eigenvalues of K(s,t) = K'(s,t) + K"(s,t) must satisfy the same estimate:

tAkI 1 C for some constant C.

PROOF: Satz I of Weyl's paper [11] implies that

IA2k+1 I _< IAk+l I + JA1+ Ifor all positive integers k. From this it is straightforward to verify that the eigenvaluesof K(s, t) have the required decay property.

THEOREM 9. If Q is a bounded domain with piecewise C0 boundary, then Ak is 0( -1

PROOF: Clearly, such an Q can be decomposed into a finite union of nonoverlappingsub-domains fl = Uk 2 k where each 2k can be put into the form (*) after a rigid motionin the plane. By lemma 6, each L... has eigenvalues with the sought-after decayproperty. Lemma 8 then implies that L., also has the same property.

This estimate is in fact sharp, at least for annular regions.

8

PROPOSITION 10. Let 11 = {(•, t) : 6 < (ý2 + t2 )1/2 < R} where 0 < e < R. Then

0 < limsup k 3/4 Ak < 00.k-0oo

PROOF: According to lemma 4, the k-th eigenvalue is

R

Ak = (-1)k4r I exp(-2rr 2 )Lk(4rr 2 ) rdr.

The following classical asymptotic expansion for Laguerre polynomials, valid forx E [c', if] with 0 < c' <if, will be essential [12, Theorem 8.22.2]:

7i/ 2 x1 /4 kl/ 4 exp(-x/2) Lk(x) = cos(2(kx)1/2 - ir/4) (1 + a,(x)k-'/2 + O(k-'))

+ sin(2(kx) 2 - 7r/4) (B(x)k-1/2 + (k-')

Applying this formula with x = 4rr 2 yields

lAkI = k-'14 10 (k) + k- 3 / 41l(k) + 0(k-5/4) (6)

whereR

Io(k) = 23/27r1 /4 j r 1/2 cos(4(kr)1/ 2r - ir/4) dr

andR

II(k) = 23/ 2 irl/4 I r1/2Al(4 7rr2) cos(4(k~r)l/2r - ir/4) dr

+ 23/2r"1/4 I rl/2B1 (41rr2) sin(4(kwr)1/2r - 7r/4) dr.

Consider the behavior of Io(k). Using a double-angle formula, one has

1o(k) = 2 3 /2i"tr/4 r1/2 cos(4(kir)l/ 2 r) dr + 23/27r1/4 r1/ 2 sin(4(kwr) 1 /2r) dr.

These are the Fourier cosine and sine integrals of the smooth function ri/2 on theinterval [E, R], evaluated as 4(kr) 1/ 2. As such, both integrals are 0(k-1/ 2). A similarargument shows that the integrals in lI(k) are 0(k-1/ 2 ) as well.

9

Finally, with a little more care one can show that

lim sup k/ 2 Io(k) > 0.k-oo

In fact, write the function r 1 / 2 = e(r)+(r'/ 2-_(r)), where t(r) is the linear function thatinterpolates between the endpoint values of r 1/ 2 at c and R. The function r1/ 2 -1(r) is aLipschitz function on the real line with support in [E, R). Consequently, the Fourier sineand cosine integrals of this function evaluated at 4(kr)1/2 are O(k- 1). It is thereforeenough to show that the integral

Ao(k) = ? (eir/4 j t(y) exp(4i(kir)1 2 r) dr)

has the property thatlim sup k/ 2Ao(k) > 0.k-oo

This is easy to show directly.

For general domains, we obtain the weaker result that the sequence of eigenvaluesis not absolutely summable.

PROPOSITION 11. The series E Ak is not absolutely convergent.

PROOF: It is well known that W(Ok, 01) is an orthonormal basis for L 2(R2 ). Theequations fffl W(Ok, 0j) = 6WAk imply that Xn = , AjW(¢k, 0k) in L2 (R2). On theother hand, if E IAki < oo then E AkW(¢k, 0k) would have to converge uniformly toan element of Co(R 2) (see theorem 1 part 2). This is clearly a contradiction.

We now examine the smoothness and decay of the eigenfunctions. We assume that11 is an open set contained in a rectangle [-B, BI x [-T, TI of which all cross-sectionsin the ý and t directions consist of at most M intervals. We now examine an equationof the form L,.0 = A0, where A is a nonzero real number and 4 e L2 (R). Fubini'stheorem implies that

AO(s) = JK(st)¢(t)dt

holds for all s E R\Z, where Z is some set of measure zero. Lemma 5 yields the estimate

IAOs~l:5 C - .+2T 1()IIA¢(~l-<I + ISl I-f-2T

<_____ C'

ISI 1-- Is0 1

10

for all s E R\Z. This last estimate implies that 0 E Loc(R). Now define

E(S) = sup

Note that E'(s) is even and decreasing in Jsj. The quantity JA]$(s)I can now be estimatedas follows:

C + _-2TIAB(s] _< is +----i -s-2T [6(t)dt

4CTs +1(KsI - 2T)-Isl+1

for all s E (R\Z) n {s' : )s'j > 2T}. Therefore, given any b > 1, there is an so such thatif Isj _> Iso0 and s C R\Z then

E(IsI + 2T) < •E(Isl).

Iterating this estimate yields

E(IsI + 2nT) < (Is 1).

It is straightforward to check that

I1(s)l < Cb-IsI/(2T) Vs E R\Z. (7)

As a consequence, ý is smooth and has all its derivatives in L2.

Now, by theorem 1 part 4, we have that

JJ W(qk,¢j) = Jf W(Ok,4j) = Akbkt

where fl = {(•,t) : (-t, ) E f0}. Hence Lxj4?k = Akqk, and by the preceding discussione is smooth and has all its derivatives in L 2. Now for a given sj,d n 0f

dsn L,=,

Using the estimate analogous to equation (7) for € with b >> 1, one has I4(•)I <ce-4 "11. Then

I dn? < 2C ode-4 -f(21rý)n dn

n! dsn 183 - n! J< C F(rn) C

n!2n+l 2n+ln-Hence the power series of 4 at s = sl converges in some interval around sl. Because ofthe symmetry in the role of 4 and 4, the same observations hold for 4?. The precedingdiscussion is summarized in the following theorem.

11

THEOREM 12. Suppose Q is an open set contained in the interval [-B, B] x [-T, T]with the property that all its cross-sections in both the ý and t directions consist of atmost M intervals. Then:

(i) for any b > 0, there is a constant Cb such that

14(s)l < -CbebsJ Vs, andk i <5Cb-b", w

and(ii) 0 and ý are analytic and have all their derivatives in L2 .

Note that (1) actually implies (2) in theorem 12 by the Paley-Wiener theorems[13], although we have chosen to give the elementary argument.

It is instructive to consider the case when S1 is a ball centered at the origin. It iswell known that, in this case, the Ok are Hermite functions (see lemma 4). The Hermitefunctions h,(t) in lemma 4 will satisfy estimates of the form h,(t) < Ce-(')t2 forany c > 0. It is directly evident that they will then satisfy the weaker inequalities intheorem 12. This theorem states that this weaker decay statement holds for generalregions in the time-frequency plane. We do not know whether these estimates can beimproved.

12

SECTION 4

NUMERICAL EXAMPLES

For illustration purposes, we provide four numerical examples of time-frequencylocalization. These examples are obtained by discretizing the kernel in the integralrepresentation of the operator given in lemma 5. We consider localization on domainsof the form of a "zigzag," a disk, a rectangle, and a parallelogram, as indicated infigure 1. Note that while the operators here are all Hermitian, so that they have realeigenvalues, they do not generally have real eigenfunctions unless their kernels are real.This is true if the symbol S(ý, t) satisfies S(-ý, t) = S(ý, t). For clarity of eigenfunctionrepresentation, projection domains were chosen to allow this symmetry when possible.

Localization on the unusual zigzag domain (figure la) produces the unfamiliareigenfunctions in figure 2 (as these are complex, we have plotted their magnitude). Eightsamples per unit length are used, over the time domain [-8,8]. Most of the energy isconcentrated in the desired region [-2,2], although there is some leakage. Note that thefirst three eigenfunctions have one, two, and three peaks, respectively. For the case ofthe disk (figure lb), a plot of the eigenvalues is shown in figure 3. As noted in lemma 4,the eigenfunctions in this case are the well-known hermite functions. Next, localizationon the rectangle (figure 1c) allows comparison to the prolate spheroidal wavefunctions.The first nine solutions for the prolate spheroidal and Weyl operator cases are depictedin figures 4 and 5. Again, eight samples per unit length are used over the time domain[-8,81. While there is energy leakage outside the desired domain [-2,2], the amountdiffers in the two cases. The prolate spheroidal wavefunctions have 1/x decay whilethe Weyl eigenfunctions have exponential decay. This difference is visible when onecompares the last three eigenfunctions in each case. A plot of the eigenvalues is alsoincluded.

Finally, we provide a simple illustration of these ideas in the context of filteringGaussian noise from a corrupted linear FM (chirp) signal. Figures 6a and 7a show,respectively, the Wigner distribution intensity plot and the actual plot of the real partof a linear FM chirp centered at 0 frequency. (Although chirp signals used in radar arenot centered at 0 frequency, that is irrelevant for our purposes because of the covarianceof the Wigner distribution under time and frequency shifts (theorem 1, part 5).) Inthe discretization, 16 samples per unit length are used over the time domain [-2,2].No windowing is applied to remove the echo effect in these Wigner plots, although wehave found a simple cosine-squared window to be effective. Figures 6b and 7b showthis signal after 0 dB Gaussian white noise has been added. To filter the noise, we notethat theoretically, the Wigner distribution of a chirp signal is a measure concentratedalong a diagonal line corresponding to the slope of the chirp. In particular, a chirp canbe localized in any domain containing its time-frequency support.

13

Frequency

Frequency

-2 2

(a) (b)

Frequency FrequencyFrrequency

2x+1/8

2xx-1/8Time Time

(c)

Figure 1. Localization Domains: a) Zigzag Domain,b) Disk of Area 50, c) 4x2 Rectangle, and

d) Parallelogram

As an elementary example of time-frequency localization, the noisy chirp of figure7b is projected onto the first two eigenfunctions (weighted according to the eigenvalues)for the domain in figure Id. This results in the signal in figure 7c, with a Wignerdistribution as shown in figure 6c. A plot of the eigenvalues for this localization operatoris provided in figure 7d. In fact, the chirp in figure 7a is orthogonal to the secondeigenfunction, illustrating an interesting fact. Numerically, the chirp appears to be anexact solution to the problem of localizing onto an infinite diagonal band domain. Formore numerical examples of time-frequency localization, the reader is referred to [6,7].

14

0.15

0. 5.

0.1 0.

0,026 0.025

-7.6 -6 -2.6 2.6 5 7.5 7 ?5 -5 -2.5 2.5 7. 6

.1 . 1.

0.6

0 0.4

0 40.

0O. 2 A-7.5 -6 -2.5 2.5 5 7.5 -0.2

Figure 2. Magnitude of Eigenfunctions 1-3 for Localizationonto a Zigzag Domain (in figure la), with a Plot of Eigenvalues

Sorted According to Decreasing Magnitude

It is interesting at this point to remark on the conjecture of Flandrin [5] thatfor localization onto convex domains, the top eigenvalue is bounded above by 1. Thisseems to hold (at least numerically) in our examples, even for figure la, which is notconvex. However, something like convexity is certainly necessary in general, since wehave nonconvex numerical examples where the top eigenvalue exceeds 4.

15

I

0.8.

0.6-

0.42

0, 2

20 40 tV

Figure 3. Weyl Eigenvalue Plot for Localizationonto a Disk of Area 50 (figure 1b), Sorted

According to Decreasing Magnitude

16

00

0.1.0-22.5 7.6 -7.5 - -2. 5 6 7.-

0.

-7.5 - -2.., a 5 7.4 0

0.2 6.2

-0.-0

0.-0.

0.: 0.2

.1o 02 0.12

0ý

0 .1

170

0.8

0.6.

0.4-

0.2-

10 Is 20 2S

Figure 4. Prolate Spheroidal Wavefunctions 1-9 for Localizationon a Rectangle (figure 1c), with Eigenvalue Plot

17

0.26

-- :.s -6 2. - 1~ - s 7." -7- 2 7s -6-

0.0

0.1

0.

011

00

0.2

-71 41 Y1 ve1-.G -. -

-0. B.

0.06.

02.

0.22

Figure 5. Weyl Eigenfunctions 1-9 for Localizationon a Rectangle (figure lc), and a Plot of

Eigenvalues in Decreasing Magnitude

118

Frequency Frequency

00

Time -Time

2 2 -2 0 2

(a) (b)

Frequency

0

Time

-2 0 2

(C)

Figure 6. Wigner Distribution Intensity Plotof (a) Chirp Signal, (b) Chirp in 0 dB Gaussian

White Noise, and (c) Filtered Noisy Signal

19

tim J I I% N A Itn

(a) (b)

eigenvalue

01

0.6

t ~0.4,-4j 0. viý0. 4

0.2

4 6 8 i0

(c) (d)

Figure 7. Plots of the Real Part of (a) Chirp Signal,(b) Chirp Signal in 0 dB Gaussian White Noise, and

(c) Filtered Noisy Signal; Part (d) Is an Eigenvalue Plot.The First Two Eigenfunctions, Weighted by their

Eigenvalues, Are Used in the Filtering.

20

LIST OF REFERENCES

1. Slepian, D., and H. 0. Pollack, 1961, "Prolate Spheroidal Wavefunctions, FourierAnalysis and Uncertainty: I," Bell Syst. Tech. J., Vol. 40, pp. 43-64.

2. Landau, H. J., and H. 0. Pollack, 1961, 1962, "Prolate Spheroidal Wavefunctions,Fourier Analysis and Uncertainty: II, 1II," Bell Syst. Tech. J., Vols. 40, 41, pp.43-64, 1295-1336.

3. Daubechies, I., 1988, "Time-Frequency Localization Operators-a Geometric PhaseSpace Approach, I," IEEE Trans. Inf. Thry., Vol. 24, pp. 605-612.

4. Daubechies, I., and T. Paul, 1988, "Time-Frequency Localization Operators-aGeometric Phase Space Approach, II," Inverse Problems, Vol. 4, pp. 661-680.

5. Flandrin, P., 1988, "Maximal Signal Energy Concentration in a Time-FrequencyDomain," Proc. ICASSP, pp. 2176-2179.

6. Hlawatsch, F., W. Kozek, and W. Krattenthaler, 1990, "Time Frequency Subspacesand Their Application to Time-Varying Filtering," Proc. ICASSP, pp. 1609-1610.

7. Hlawatsch, F., and W. Kozek, 1991, "Time-Frequency Analysis of Linear SignalSubspaces," Proc. ICASSP, pp. 2045-2048.

8. Folland, G. B., 1989, Harmonic Analysis in Phase Space, Princeton, N.J.: PrincetonUniversity Press.

9. Pool, J., 1966, "Mathematical Aspects of the Weyl Correspondence," J. Math.Phys., Vol. 7, pp. 66-76.

10. Jansscn, A. J. E. M., 1981, "Positivity of Weighted Wigner Distributions," SIAMJ. Math. Anal., Vol. 12, pp. 752-758.

11. Weyl, H., 1911, "Das Asymptotische Verteilungsgesetz der Eigenwerte Linearer Par-tieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraum-strahlung)," Mathematische Annalen, Vol. 71, pp. 441--479.

12. Szego, G., 1959, Orthogonal Polynomials, Providence, RI: AMS Colloquium Publi-cations, pp. Vol. 23.

13. Katznelson, Y., 1968, An Introduction to Harmonic Analysis, New York: JohnWiley.

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