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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000 1417

Time–Frequency Formulation, Design, andImplementation of Time-Varying Optimal Filters for

Signal EstimationFranz Hlawatsch, Member, IEEE, Gerald Matz, Student Member, IEEE, Heinrich Kirchauer, and

Werner Kozek, Member, IEEE

Abstract—This paper presents a time–frequency frameworkfor optimal linear filters (signal estimators) in nonstationaryenvironments. We developtime–frequency formulationsfor theoptimal linear filter (time-varying Wiener filter) and the optimallinear time-varying filter under a projection side constraint. Thesetime–frequency formulations extend the simple and intuitivespectral representations that are valid in the stationary case tothe practically important case of underspread nonstationary pro-cesses. Furthermore, we propose an approximatetime–frequencydesignof both optimal filters, and we present bounds that showthat for underspread processes, the time-frequency designedfilters are nearly optimal. We also introduce extended filter designschemes using a weighted error criterion, and we discuss anefficient time–frequency implementationof optimal filters usingmultiwindow short-time Fourier transforms. Our theoreticalresults are illustrated by numerical simulations.

Index Terms—Nonstationary random processes, optimal filters,signal enhancement, signal estimation, time-frequency analysis,time-varying systems, Wiener filters.

I. INTRODUCTION

T HE enhancement or estimation of signals corruptedby noise or interference is important in many signalprocessing applications. For stationary random processes, themean-square error optimal linear estimator is the (time-in-variant)Wiener filter [1]–[7], whose transfer function is givenby

(1)

where and denote the power spectral densitiesof the signal and noise processes, respectively. The extensionof the Wiener filter to nonstationary processes yields a linear,time-varying filter (“time-varying Wiener filter”) [2], [4]–[7]for which the simple frequency-domain formulation in (1) is

Manuscript received June 17, 1998; revised September 20, 1999. This workwas supported by FWF Grants P10012-ÖPH and P11904-TEC. The associateeditor coordinating the review of this paper and approving it for publication wasProf. P. C. Ching.

F. Hlawatsch and G. Matz are with the Institute of Communications andRadio-Frequency Engineering, Vienna University of Technology, Vienna,Austria.

H. Kirchauer is with Intel Corp., Santa Clara CA 95052 USA.W. Kozek is with Siemens AG, Munich, Germany.Publisher Item Identifier S 1053-587X(00)03289-X.

no longer valid. We may ask whether a similarly simple formu-lation can be obtained by introducing an explicit time depen-dence, i.e., whether in a joint time-frequency (TF) domain thetime-varying Wiener filter can be formulated as

(2)

where and are suitably defined. Sucha TF formulation would greatly facilitate the interpretation,analysis, design, and implementation of time-varying Wienerfilters.

In this paper, we provide an answer to this and several otherquestions of theoretical and practical importance.

• We show that forunderspread[8]–[12] nonstationary pro-cesses, the TF formulation in (2) is approximately validif and are chosen as the Weylsymbol [13]–[16] and the Wigner–Ville spectrum (WVS)[12], [17]–[19], respectively. We present upper bounds onthe associated approximation errors.

• We propose an efficient, intuitive, and nearly optimal TFdesign of signal estimators that is easily adapted to modi-fied (weighted) error criteria.

• We discuss an efficient TF implementation of optimal fil-ters using the multiwindow short-time Fourier transform(STFT) [8], [20].

• We also consider the TF formulation, approximate TFdesign, and TF implementation of an optimal projectionfilter.

Previous work on the TF formulation of time-varying Wienerfilters [21]–[24] has mostly used Zadeh's time-varying transferfunction [25], [26] for and the evolutionary spectrum[12], [27]–[29] for and . A Wiener filteringprocedure using a time-varying spectrum based on subbandAR models has been proposed in [30]. A somewhat differentapproach is the TF implementation of signal enhancementby means of a TF weighting applied to a linear TF signalrepresentation such as the STFT, the wavelet transform, or afilterbank/subband representation. In [31] and [32], the STFTis multiplied by a TF weighting function similar to (2), where

are chosen as thephysical spectrum[18],[19], [33]. Wiener filter-type modifications of the STFT andsubband signals have long been used for speech enhancement[34]–[37]. Methods that perform a Wiener-type weighting ofwavelet transform coefficients are described in [38]–[41].

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Copyright IEEE 2000

1418 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 5, MAY 2000

The TF formulations proposed in this paper differ from theabove-mentioned work on several points:

• While, usually, Zadeh's time-varying transfer functionand the evolutionary spectrum or physical spectrum arechosen for and , respectively,our development is based on the Weyl symbol and theWVS. This has important advantages, especially re-garding TF resolution. Specifically, the Weyl symbol andthe WVS are much better suited to systems and processeswith chirp-like characteristics. Furthermore, contrary tothe evolutionary spectrum and physical spectrum, theWVS is in one-to-one correspondence with the correlationfunction.

• Our TF formulations are not merely heuristic but areshown to yield approximations to the optimal filtersfeaturing nearly optimal performance in the case ofjointly underspread signal and noise processes. Wepresent bounds on the associated approximation errors,and we discuss explicit conditions for the validity of ourTF formulations. In particular, we show that the usualquasi-stationarity assumption is neither a sufficient nor anecessary condition.

• Contrary to the conventional heuristic STFT enhance-ment schemes, the multiwindow STFT implementationdiscussed here is based on a theoretical analysis thatshows how to choose the STFT window functions andhow well the multiwindow STFT filter approximates theoptimal filter.

The paper is organized as follows. Section II reviews thetime-varying Wiener filter and the optimal time-varying pro-jection filter. Section III reviews some TF representations andthe concept of underspread systems and processes. In SectionIV, a TF formulation of optimal filters is developed. Based onthis TF formulation, Section V introduces simple and intuitiveTF design methods for optimal filters. In Section VI, a multi-window STFT implementation of optimal filters is discussed.Finally, Section VII presents numerical simulations that com-pare the performance of the various filters and verify the theo-retical results.

II. OPTIMAL FILTERS

This section reviews the theory of the time-varying Wienerfilter [2], [4]–[7] and a recently proposed “projection-con-strained” optimal filter [42]. For stationary processes,thereexist simple frequency-domain formulations of both optimalfilters. These will be generalized to the nonstationary case inSection IV.

A. Time-Varying Wiener Filter

Let be a zero-mean, nonstationary, circular complex orreal random process with known correlation functionE or, equivalently, correlation operator1 . This

1The correlation operatorR of a random processx(t) is the self-adjointand positive (semi-)definite linear operator [43] whose kernel is the correlationfunctionr (t; t ) = Efx(t)x (t )g. In a discrete-time setting,R would bea matrix.

signal process is contaminated by additive zero-mean, nonsta-tionary, circular complex or real random noise with knowncorrelation operator . Signal and noise are assumedto be uncorrelated, i.e., E . We form an esti-mate of the signal from the noisy observation

using a linear, generally time-varying system (linearoperator [43]) with impulse response (kernel) ,2

Our performance index will be the mean-square error (MSE),i.e., the expected energy E Etr of the estimation error , which can beshown to be given by

tr tr (3)

Here, tr denotes the trace of an operator, denotes the ad-joint of [43], and and are the expected energies ofthe residual noise and the signal distortion

, respectively. The linear system mini-mizing the MSE can be shown [2], [4]–[7] to satisfy

(4)

The solution of (4) with minimal rank3 is the “time-varyingWiener filter”

(5)

where denotes the (pseudo-)inverse ofon its range . The minimum MSE

achieved by the Wiener filter is given by

tr

tr tr (6)

In order to interpret the time-varying Wiener filter , letus define thesignal space range and thenoisespace range . Since , the ob-servation spaceis given by . We note that4

, and . It can now be shown thatthe optimal signal estimate is an element of the signal space,i.e., . Since and , we also have

. Since it can also be shown that , we get. These results are intuitive since any contribu-

tion to from outside would unnecessarily increase theresulting error. Furthermore, if signal spaceand noise space

are disjoint, , then the Wiener filter performsan oblique projection [44] onto and obtains perfect recon-struction of the signal , i.e., .

2Integrals are from�1 to1 unless stated otherwise.3The general solution of (4) isH +XP , whereH denotes the minimal

rank solution in (5),X is an arbitrary operator, andP is the orthogonal pro-jection operator onS , the orthogonal complement space ofS = rangefR gwithR = R +R . (ForS = L (IR) [whereL (IR) denotes the space ofsquare-integrable functions],H becomes the unique solution of (4).

4Since the signalss(t) etc. are random, relations likes(t) 2 S are to beunderstood to hold with probability 1.

HLAWATSCH et al.: TIME-FREQUENCY FORMULATION, DESIGN, AND IMPLEMENTATION OF TIME-VARYING SIGNAL ESTIMATORS 1419

In some applications, e.g., if signal distortions are less accept-able than residual noise, we might consider replacing the MSEin (3) by a “weighted MSE”

tr

tr (7)

with . This amounts to replacing by andby so that the resulting modified Wiener filter is

B. Optimal Time-Varying Projection Filter

We next consider the estimation of using a linear, time-varying filter that is anorthogonal projection operator, i.e.,it is idempotent, , and self-adjoint, [43].Although the projection constraint generally results in a largerMSE, it leads to an estimator that is robust in a specific sense[45]. Further advantages regarding a TF design are discussed inSubsection V-B.

Let and denote the eigenvalues and normalizedeigenfunctions, respectively, of the operator ,i.e., . It is shown in [42] that the min-imum-rank orthogonal projection operator minimizing theMSE in (3) is given by

where denotes the rank-one orthogonal projectionoperator defined by with

, and isthe index set corresponding to positive eigenvalues. Thus, theoptimal projection filter performs a projection onto thespace spanned by all eigenfunctions ofcorresponding to positive eigenvalues. The MSE obtained with

is given by

tr tr (8)

For an interpretation of this result, let us partition the obser-vation space into three orthogonal subspaces correspondingto positive, zero, and negative eigenvalues of:

span

span

span

so that and , where, and are the orthogonal projection operators on

the spaces , and , respectively. Note that. We now consider the expected energies of the signal and

noise processes in the space. The expected energy of anyprocess in the one-dimensional (1-D) space spanis given by E . Hence, the difference

between expected signal energy and expected noise energy inspan is

E E

Thus, if (and only if) , then in span , the ex-pected signal energy is larger than the expected noise energy.It is then easily shown that E Efor any signal , i.e., in , the expected signalenergy is larger than the expected noise energy. Equivalently,E E , which shows that the optimalprojection filter projects onto the space where,on average, there is more signal energy than noise energy. Thisis intuitively reasonable.

If we use the “weighted MSE” (7) instead of the MSE (3), theresulting optimal projection filter is constructed as beforewith replaced by .

C. Stationary Processes

The special cases of stationary processes and nonstationarywhite processes allow particularly simple frequency-domainand time-domain formulations, respectively, of both optimalfilters. These formulations will motivate the development of TFformulations of optimal filters in Section IV.

If and are wide-sense stationary processes, the cor-responding correlation operators and are time-invariant,i.e., of convolution type, and the optimal system becomes time-invariant as well. The MSE in (3) does not exist and must be re-placed by the “instantaneous MSE” E . For any linear,time-invariant system with transfer function , the in-stantaneous MSE can be shown to be

E

(9)

where and are the power spectral densities (PSD’s)of and , respectively. The optimal system minimizing(9) satisfies [2]–[7]. A “min-imal” solution of this equation is

(10)

where denotes the set offrequencies where the PSD of the observation

is positive. The minimum instantaneous MSEis given by

E (11)

The frequency-domain formulations (10) and (11) allowan intuitively appealing interpretation of the time-invariantWiener filter (see Fig. 1). Let and

denote the sets of frequencies whereand , respectively, are positive. Then, (10) shows


Fig. 1. Frequency-domain interpretation of the time-invariant Wiener filter.

that for , i.e., the Wiener filter passesall observation components in the frequency bands whereonly signal energy is present, which is intuitively reasonable.Furthermore, for , i.e., outside , theobservation is suppressed, which is again reasonable sinceoutside , there is only noise energy. In the frequency ranges

, where both signal and noise are present, there iswith the values of depending on the

relative values of and at the respective frequency.The MSE in (9) can again be replaced by a weighted MSE.

Here, the weights can even be frequency dependent, which re-sults in the weighted MSE

(12)

with . The optimal filter minimizing thisweighted MSE is then simply given by (10) with re-placed by and replaced by .

We next consider the optimal projection filter . For sta-tionary processes, it can be shown thatis time-invariant withthe zero-one valued transfer function

(13)

where is the indicator function of, which is the set of frequencies where the signal PSD

is larger than the noise PSD. Thus, is an idealized bandpassfilter with one or several pass bands. Note that can beobtained from by a rounding operation, i.e.,round . The instantaneous MSE obtained with canbe shown to be given by

E (14)

where and.

If the weighted instantaneous MSE (12) is used, then the op-timal projection filter is the idealized bandpass filter with pass-band(s) .

The case of nonstationary white processes and isdual to the stationary case discussed above. Here, the time-varying Wiener filter and projection filter are “frequency-in-variant,” i.e., simple time-domain multiplications. All resultsand interpretations are dual to the stationary case.

III. T IME-FREQUENCYREPRESENTATIONS ANDUNDERSPREADPROCESSES

In this section, we review some fundamentals of TF analysisthat will be used in Sections IV and V for the TF formulationand TF design of optimal filters.

A. Time–Frequency Representations

We first review four TF representations on which our TF for-mulations will be based.

• The Weyl symbol(WS) [13]–[16] of a linear operator(linear, time-varying system) with kernel (impulseresponse) is defined as

(15)

where and denote time and frequency, respectively. Forunderspread systems (see Section III-B), the WS can beconsidered to be a “time-varying transfer function” [8],[11], [46], [47].

• TheWigner–Ville Spectrum(WVS) [17]–[19] of a nonsta-tionary random process is defined as the WS of thecorrelation operator

(16)

For underspread processes (see Section III-B), the WVScan be interpreted as a “time-varying PSD” or expectedTF energy distribution of [8], [9], [12].

• The spreading function[8], [13], [16], [26], [46], [48],[49] of a linear operator is defined as

where and denote time lag and frequency lag, respec-tively. The spreading function is the 2-D Fourier trans-form of the WS. Any linear operator admits the fol-lowing expansion into TF shift operators defined as

[8], [13], [16], [49]:

(17)

Hence, the spreading function describes the TF shifts in-troduced by (see Section III-B).

• The expected ambiguity function[8]–[12] of a nonsta-tionary process is defined as the spreading functionof the correlation operator

(18)

The expected ambiguity function is the 2-D Fourier trans-form of the WVS. It can be interpreted as a TF correlationfunction in the sense that it describes the correlation of


process components whose TF locations are separated byin time and by in frequency [8]–[12].

B. Underspread Systems and Processes

Since our TF formulation of optimal filters will be valid forthe class ofunderspreadnonstationary processes, we will nowreview the definition of underspread systems and processes[8]–[12], [46], [47].

According to the expansion (17), the effective support of thespreading function describes the TF shifts caused bya linear time-varying system . A global description of the TFshift behavior of is given by the “displacement spread”[8], [11], [46], which is defined as the area of the smallest rec-tangle [centered about the origin of the plane and possiblywith oblique orientation] that contains the effective support of

. A system is calledunderspread[8], [11], [46] if, which means that is highly concentrated

about the origin of the plane and, hence, that causesonly small TF shifts. Since is the 2-D Fourier trans-form of , this implies that is a smooth func-tion. Two systems and are calledjointly underspreadif their spreading functions are effectively supported within thesamesmall rectangle of area .

Let us next consider a nonstationary, zero-mean randomprocess. The expected ambiguity function in (18)describes the correlation of process components whose TF loca-tions are separated byin time and by in frequency [8]–[12].Since the expected ambiguity function is the spreading functionof the correlation operator , i.e., ,it is natural to define thecorrelation spread of a process

as the displacement spread of its correlation operator,

i.e., . A nonstationary random process is thencalledunderspreadif [8]–[11]. Since in this case theexpected ambiguity function is highly concentrated about theorigin of the plane, process components at different (i.e.,not too close) TF locations will be effectively uncorrelated.This also implies that the WVS of is a smooth, effectivelynon-negative function [8], [12]. Two processes andare called jointly underspreadif their expected ambiguityfunctions are effectively supported within thesame smallrectangle of area .

Two special cases of underspread processes are “quasista-tionary” processes with limited temporal correlation width and“nearly white” processes with limited time variation of their sta-tistics. We emphasize that quasistationarity alone is neither nec-essary nor sufficient for the underspread property.

IV. TIME–FREQUENCYFORMULATION OF OPTIMAL FILTERS

In Section II, we saw that the time-varying Wiener filterand projection filter are described by equations involvinglinear operators and signal spaces. This description is ratherabstract and leads to computationally intensive design proce-dures and implementations. However, for stationary or whiteprocesses, there exist simple frequency-domain or time-domainformulations that involve functions instead of operators andintervals instead of signal spaces and thus allow a substantiallysimplified design and implementation. We will now show

that similar simplifications can be developed forunderspreadnonstationary processes [50].

A. Time–Frequency Formulation of the Time-Varying WienerFilter

We assume that and are jointly underspread pro-cesses; specifically, their expected ambiguity functions are as-sumed to be exactly zero outside a rectangle(centered aboutthe origin of the plane) of area . The Wienerfilter can then be split up as , where theunder-spread part is defined by

(19)

where is the indicator function of , and theoverspreadpart is defined by

. That is, and lie in-side and outside , respectively. Since the spreading functionis the 2-D Fourier transform of the Weyl symbol, (19) implies

; here, denotes 2-D convo-lution and is the 2-D Fourier transform of and,hence, a 2-D lowpass function. This means that is asmoothed version of .

We now recall that the Wiener filter is characterized by therelation (4), i.e., . A first step toward a TFformulation of is to notice that removing the overspread part

from does not greatly affect the validity of this relation,i.e.,

Indeed, it is shown in Appendix A that the Hilbert-Schmidt (HS)norm [43] of the error incurred by thisapproximation is upper bounded as

(20)

and, furthermore, that the “excess MSE” resulting from usinginstead of is bounded as

(21)

Here, denotes the MSE obtained with . Hence, if issmall, i.e., if and are jointly underspread,

and . This shows thatthe underspreadpart is nearly optimal.

This being the case, it suffices to develop a TF formulationfor . Intuitively, we could conjecture that a natural TFformulation of the operator relationwould be . Indeed,it is shown in Appendix B that the norm of the error


incurred by thisapproximation is upper bounded as

(22)

An upper bound involving can furthermore be formu-lated for the error magnitude ( norm) as well. Hence,

if is small,i.e., if and are jointly underspread. Defining the TFregion where is effectivelypositive by (with some5 small

), we finally obtain the following TF formulation for :

(23)

The minimum MSE in (6) is given by trwhich, for underspread, can be shown to equal. Inserting (23), we obtain the approximate TF for-

mulation

(24)

The relations (23) and (24) constitute a TF formulation of thetime-varying Wiener filter that extends the frequency-domainformulation (10), (11) that is valid for stationary processesand the analogous time-domain formulation that is valid fornonstationary white processes to the much broader class ofunderspread nonstationary processes. This TF formulationof the time-varying Wiener filter allows a simple and intu-itively appealing interpretation. Let us define the TF supportregions where the WVS of and are effectivelypositive by and

. The TF regionsand correspond to the signal spaces and

underlying the respective processes [42], [51]. With thesedefinitions at hand, the Wiener filter can be interpreted in theTF domain as follows (see Fig. 2).

• In the “signal only” TF region , i.e., in the TF re-gion where only signal energy is present, it follows from(23) that the WS of the Wiener filter is approximately one:

Thus, passes all “noise-free” observation componentswithout attenuation or distortion.

5An analysis of the WS of positive operators ([8] and [47]) suggests that weuse� � k �A k � = k �A k � .

Fig. 2. TF interpretation of the time-varying Wiener filter.

• In the “no signal” TF region, where no signal energy ispresent, i.e., for , it follows from (23) that theWS of the Wiener filter is approximately zero:

This “stop region” of the Wiener filter consists of twoparts: i) the “noise only” TF region , where onlynoise energy is present and ii) the outside (complement)of the observation TF region , i.e., the TF regionwhere neither signal nor noise energy is present; here,

since (23) corresponds to theminimalWiener filter.

• In the “signal plus noise” TF region , where bothsignal energy and noise energy are present, the WS ofthe Wiener filter assumes values approximately between0 and 1

Here, performs a TF weighting that depends on therelative signal and noise energy at the respective TF point.Indeed, it follows from (23) that the ratio of and

is given by the “local signal-to-noise ratio”

SNR :

SNR

In particular, for equal signal and noise energy, i.e.,SNR , there is .

This TF interpretation of the time-varying Wiener filter ex-tends the frequency-domain interpretation valid in the stationarycase (see Fig. 1) to the broader class of underspread processes.

B. Time–Frequency Formulation of the Time-VaryingProjection Filter

We recall from Section II-B that the optimal projection filterperforms a projection onto the space span

spanned by all eigenfunctions of the operatorassociated with positive eigenvalues. In order to obtain

a TF formulation of this result, we consider the following three


TF regions on which the signal energy is, respectively, largerthan, equal to, or smaller than the noise energy:

With, these TF regions can alternatively be written as

Note that .There exists a correspondence between the “pass space”

span of and the TF region. Indeed, if and

are jointly underspread, then is an underspreadsystem. It is here known [8], [11], [47] that TF shifted versions

of a well-TF localized functionare approximate eigenfunctions of, and the WS values

are the corresponding approximate eigenvalues, i.e.,

Hence, the optimal pass space (spanned by all eigenfunc-tions with ) corresponds to the TF region

(comprising the TF locationsof all such that ). The action of

—passing signals in and suppressing signals in —cantherefore be reformulated in the TF plane as passing signals inthe TF region and suppressing signals outside . Thus,we conclude that passes signal components in the TF re-gion where the local TF SNR is larger than one and suppressessignal components in the complementary TF region, which isintuitively reasonable.

The optimal projection filter performs a projection ontothe space ; for jointly underspread signal and noise, this spaceis a “simple” space as defined in [42]. The WS of the projectionoperator on a simple space essentially assumes the values 1 (onthe TF pass region, in our case ) and 0 (on the TF stop region)[42]. Hence, the WS of can be approximated as6

(25)

where is the indicator function of . This extendsthe relation (13) to general underspread processes. Note that theprojection property of only allows to passor suppresscomponents of ; no intermediate type of attenuation (whichwould correspond to values of between 0 and 1) ispossible.

6Experiments show that this approximation is valid apart from oscillations ofL (t; f) about the values 0 or 1. Hence, the approximation can be improvedby a slight smoothing ofL (t; f), which corresponds to the removal of over-spread components fromP (cf. Section IV-A).

It can finally be shown that the MSE obtained with [cf.(8)] is approximately given by

which extends the relation (14) to general underspread pro-cesses.

V. APPROXIMATE TIME–FREQUENCYDESIGN OFOPTIMALFILTERS

In the previous section, we have shown that the time-varyingWiener and projection filters can be reformulated in the TF do-main if the nonstationary random processes and arejointly underspread. We will now show that this TF formulationof optimal filters leads to simpledesign proceduresthat operatein the TF domain and yield nearly optimal filters.

A. Time–Frequency Pseudo-Wiener Filter

We recall that for jointly underspread processes, an approxi-mate expression for the Wiener filter's WS is given by (23). Letus now define another linear, time-varying filter by settingits WS equal to the right-hand side of (23) [50]:

(26)

We refer to as theTF pseudo-Wiener filter. For jointly under-spread processes , , where (23) is a good approximation,a combination of (26) and (23) yields ,and thus, . Hence, the TF pseudo-Wiener filter isa close approximation to the underspread partof the Wienerfilter and, therefore, is nearly optimal; furthermore, the TFinterpretation of (see Section IV-A) applies equally well to

. The deviation from optimality is characterized by the errorbounds in (20)–(22). For processes that are not jointly under-spread, however, must be expected to perform poorly. Notethat is a self-adjoint operator since according to (26), theWS of is real-valued.

For jointly strictly underspread processes, i.e., underspreadprocesses whose support rectangleis oriented parallel to the

and axes [8], [29], the WVS occurring on the right-hand sideof (26) can be replaced by the generalized WVS [17]–[19], the(generalized) evolutionary spectrum [23], [24], [28], [29], or thephysical spectrum (expected spectrogram) using an appropriateanalysis window [18], [19], [33]. This is possible since in thestrictly underspread case, these spectra become nearly equiva-lent [8], [9], [12], [19], [29].

While the TF pseudo-Wiener filter is definedin the TFdomain, the actual calculation of the signal estimate can be per-formed directly in the time domain according to


where , which is the impulse response of , can beobtained from the WS in (26) as [cf. (15)]

(27)

An efficient approximate multiwindow STFT implementationwill be discussed in Section VI.

Compared with the Wiener filter , the TF pseudo-Wienerfilter possesses two practical advantages:

• Modified a priori knowledge:Ideally, the calculation(design) of requires knowledge of the correlationoperators and [see (5)], whereas the design ofrequires knowledge of the WVS and[see (26)]. Although correlation operators and WVS aremathematically equivalent due to the one-to-one map-ping (16), the WVS are much easier and more intuitiveto handle than the correlation operators or correlationfunctions since WVS have a more immediate physicalsignificance and interpretation. In practice, the quan-tities constituting thea priori knowledge (correlationoperators or WVS) are usually replaced by estimatedor schematic/idealized versions, which are much moreeasily controlled or designed in the TF domain than inthe correlation domain. For example, an approximate orpartial knowledge of the WVS will often suffice for areasonable filter design.

• Reduced computation:The calculation (design) ofrequires a computationally intensive and potentiallyunstable operator inversion (or, in a discrete-time setting,a matrix inversion). In the TF design (26), this operatorinversion is replaced by simple and easily controllablepointwise divisions of functions. Assuming discrete-timesignals of length , the computational cost of the de-sign of grows with , whereas that of (usingdivisions and FFT’s) grows only with .

The TF pseudo-Wiener filter satisfies an approximate “TF op-timality.” For jointly underspread, it can be shown thatthe MSE obtained withanyunderspread system is approxi-mately given by

(28)

Minimizing (assuming a minimal solution, i.e.,for ) then yields the in (26) and, thus, the

TF pseudo-Wiener filter . This interpretation suggests anex-tended TF designthat is based on the following weighted MSEwith smooth TF weighting function [extending (12)]

(29)

with . The filter minimizing thisweighted TF error is given by (26) with re-placed by and replaced by

.

B. Time–Frequency Pseudo-Projection Filter

Next, we consider a TF filter design that is motivated by theapproximate expression (25) for the WS of. Let us define anew linear system by setting its WS equal to the right-handside of (25):

Since is the indicator function of the TF region, the WS of is 1 for

those TF points where there is more signal energy than noiseenergy and 0 elsewhere. It is interesting that the WS ofis arounded version of the WS of the TF pseudo-Wiener filterin (26), i.e., round , which is consis-tent with the stationary case. Furthermore, differs from

only on the signal-plus-noise TF region ; inthe TF pass region , we have, and in the TF stop region outside , there is

.In general, isnotexactly an orthogonal projection operator.

However, for jointly underspread processes and , where(25) is a good approximation, there is ,and thus, . Hence, the TF-designed filter is a reason-able approximation to the optimal projection filter , and wewill thus call it TF pseudo-projection filter. However, for pro-cesses that are not jointly underspread,must be expected toperform poorly and to be very different from an orthogonal pro-jection operator.

Although is defined in the TF domain, the actual calcula-tion of the signal estimate can be done in the time domain usingthe impulse response of that is derived from as (cf.(27))

(30)

An efficient approximate multiwindow STFT implementationwill be discussed in Section VI.

Compared with the optimal projection filter , the TFpseudo-projection filter has two advantages:

• Modified/reduceda priori knowledge: The design ofrequires knowledge of the space that is spanned

by the eigenfunctions of the operatorcorresponding to positive eigenvalues (see Section II-B),whereas the design of merely presupposes that weknow the TF region in which the signal dominatesthe noise. This knowledge is of a much simpler and moreintuitive form, thus facilitating the use of approximate orpartial information about the WVS.

• Reduced computation:The design of requires the so-lution of an eigenvalue problem, which is computation-ally intensive. In contrast, the proposed TF design only re-quires an inverse WS transformation [see (30)]. Assumingdiscrete-time signals of length , the computational costof the design of grows with , whereas that of(using FFT’s) grows with .

The TF pseudo-projection filter is furthermore advanta-geous as compared with the Wiener-type filtersor since


its design is less expensive and more robust (especially with re-spect to errors in estimating thea priori knowledge and withrespect to potential correlations of signal and noise).

Similar to the TF pseudo-Wiener filter, the TF pseudo-pro-jection filter satisfies a “TF optimality” in that it minimizes theTF MSE in (28) under the constraint of a -valued WS,i.e., . Minimizing the TF weighted MSEin (29) rather than , again under the constraint

, results in a generalized TF pseudo-projection filterthat is given by , where

.

C. Time–Frequency Projection Filter

For jointly underspread processes, the TF pseudo-projectionfilter approximates the optimal projection filter , but itis not exactly an orthogonal projection operator. If an orthog-onal projection operator is desired, we may calculate the or-thogonal projection operator that optimally approximatesin the sense of minimizing . This can be shown tobe equivalent to minimizing

[42], [51]. That is, the TF designed projectionfilter (hereafter denoted by ) is the orthogonal projection op-erator whose WS is closest to the indicator function .It is shown in [42] and [51] that is given by the orthogonalprojection operator on the space spanspanned by all eigenfunctions of with associated eigen-values , i.e.,

with

For jointly underspread processes whereis close to an or-thogonal projection operator, is close to and, hence, alsoclose to the optimal projection filter . However, due to the in-herent projection constraint, may lead to a larger MSE than

. In addition, the derivation of from requires the solu-tion of an eigenproblem. Therefore, the TF projection filterappears to be less attractive than the TF pseudo-projection filter

.

VI. M ULTIWINDOW STFT IMPLEMENTATION

Following [8] and [20], we now discuss a TF implementationof the Wiener and projection filters that is based on the multi-window short-time Fourier transform (STFT) and that is validin the underspread case.

A simple TF filter method is anSTFT filter that consists ofthe following steps [8], [13], [20], [52]–[57]:

• STFT analysis: Calculation of the STFT [19], [54], [55],[58] of the input signal

STFT

Here, where is a normal-ized window.

• STFT weighting: Multiplication of the STFT by aweighting function

STFT STFT

Fig. 3. Multiwindow STFT implementation of the Wiener filter.

• STFT synthesis: The output signal is the inverse STFT[54], [55], [58] of the weighted STFT:

STFT

We note that is the signal whose STFT is closest toSTFT in a least-squares sense [57].

These steps implement a linear, time-varying filter (hereafterdenoted ) that depends on and .

We recall from Section IV-A that for jointly underspread pro-cesses, the underspread partof the Wiener filter is nearlyoptimal. Since for , it follows that

where is any real-valued function that is 1 on[a “min-imal” choice for is the indicator function , cf.

(19)]. Let denote the linear system defined by. If is chosen such that ,

is a self-adjoint operator with real-valued eigenvaluesandorthonormal eigenfunctions . It is shown in [8] and [20]that can be expanded as

(31)

where the are STFT filters with TF weighting functionand STFT windows . Thus, is

represented as a weighted sum of STFT filters with identicalTF weighting function and orthonormal windows

. In practice, the eigenvalues often decay quickly sothat it suffices to use only a few terms of (31) correspondingto the largest eigenvalues (an example will be presented inSection VII). Assuming that the eigenvaluesare arranged innonincreasing order, we then obtain

This yields the approximatemultiwindow STFT implementationof that is depicted in Fig. 3. It can be shown [8] that theapproximation error is bounded as

which can be used to estimate the appropriate order. Sinceand are assumed jointly underspread, (23) holds, and


we obtain the following simple approximation to the STFTweighting function :

(32)

A particularly efficient discrete-time/discrete-frequency ver-sion of the multiwindow STFT implementation that uses filterbanks is discussed in [8]. Furthermore, it is shown in [8] and[20] that in the case of white noise , the STFT’s used in themultiwindow STFT implementation can additionally be used toestimate the WVS of , which is required to calculateaccording to (32).

A heuristic, approximate TF implementation of time-varyingWiener filters proposed in [31] and [32] uses a single STFT filter(i.e., ) with TF weighting function

where and are thephysical spectra[18], [19],[33] of and , respectively. Compared with the mul-tiwindow STFT implementation discussed above, this suffersfrom a threefold performance loss since

i) Only one STFT filter is used.ii) The physical spectrum is a smoothed version of the WVS

[18], [33].iii) The STFT window is not chosen as the eigenfunction of

corresponding to the largest eigenvalue.An approximate multiwindow STFT implementation can

also be developed for the TF pseudo-projection filter(whichapproximates the optimal projection filter ). Here, the TFweighting function is

Since is not exactly underspread, the functiondefining the windows and the coefficients must bechosen such that it covers theeffectivesupport of , andhence

(33)

Subsequently, the and can be constructed fromas explained above. The resulting multiwindow STFT filter

leads to an approximation error that can bebounded as

with . This bound is a measure of the ex-tension of and will thus be small in the underspreadcase. However, even for since(33) is only an approximation.

VII. N UMERICAL SIMULATIONS

This section presents numerical simulations that illustrate andcorroborate our theoretical results. Using the TF synthesis tech-

Fig. 4. Second-order statistics of signal and noise. (a) WVS ofs(t). (b) WVSof n(t). (c) Expected ambiguity function ofs(t). (d) Expected ambiguityfunction ofn(t). The rectangles in parts (c) and (d) have area 1 and thus allowassessment of the underspread property ofs(t) andn(t). The signal length is128 samples.

nique introduced in [59], we synthesized random processesand with expected energies and ,respectively. Thus, the mean input SNR is SNR

dB. The WVS and expected ambiguity functions ofand are shown in Fig. 4. From the expected ambiguity func-tions, it is seen that the processes are jointly underspread. TheWS’s of the Wiener filter , its underspread part , and theTF pseudo-Wiener filter are shown in Fig. 5(a)–(c). The TFpass, stop, and transition regions of the filters are easily recog-nized. It is verified that the WS of is a smoothed version ofthe WS of and that the WS of closely approximates thatof . Fig. 5(d)–(f) compare the WS’s of the optimal projectionfilter , TF pseudo-projection filter , and TF projection filter

. It is seen that the WS’s of these filters are similar, exceptfor small-scale oscillations, and that the WS ofis a roundedversion of the WS of .

The mean SNR improvementsSNR SNR SNR(where SNR with ) achievedwith the Wiener-type and projection-type filters are listed inTable I. The performance of the TF pseudo-Wiener filter isseen to be very close to that of the Wiener filter. Similarly,the performance of the TF pseudo-projection filteris closeto that of the optimal projection filter . In fact, performseven slightly better than both and the TF projection filter ,


Fig. 5. WS’s of (a) Wiener filterH . (b) Underspread partH of Wiener filter. (c) TF pseudo-Wiener filterH . (d) Optimal projection filterP . (e) TFpseudo-projection filter~P. (f) TF projection filterP .

TABLE IMEAN SNR IMPROVEMENT ACHIEVED BY

WIENER-TYPE AND PROJECTION-TYPE FILTERS

which can be attributed to the orthogonal projection constraintunderlying and (see Section V-C).

The multiwindow STFT implementation discussed in Sec-tion VI, with approximated according to (32), is con-sidered in Fig. 6. The signal and noise processes are as be-fore. Fig. 6(a)–(c) show the WS’s of the multiwindow STFTfilters using filter order and . For growing

becomes increasingly similar to the Wiener filter orthe TF pseudo-Wiener filter [cf. Fig. 5(a)–(c)]. Fig. 6(d) showshow the mean SNR improvement achieved with dependson . Although the single-window STFT filter is seen


Fig. 6. Multiwindow STFT implementation of the Wiener filter. (a)–(c) WS’s of multiwindow STFT filters with filter orderK = 1; 4; and10. (d) Mean SNRimprovement versus STFT filter orderK .

Fig. 7. Filtering experiment involving an overspread noise process. (a) WVS ofs(t). (b) WVS ofn(t). (c) WS of Wiener filterH . (d) Expected ambiguityfunction ofs(t). (e) Expected ambiguity function ofn(t). (f) WS of TF pseudo-Wiener filterH . The rectangles in parts (d) and (e) have area 1 and thus allowassessment of the under-/overspread property ofs(t) andn(t). (In particular, part (e) shows thatn(t) is overspread.) The signal length is 128 samples.

to result in a significant performance loss, a modest filter orderalready yields practically optimal performance.An experiment in which the underspread assumption is vi-

olated is shown in Fig. 7. The signal is again underspread(and, in addition, reasonably quasistationary), but the noiseis notunderspread—it is reasonably quasistationary, but its tem-poral correlation width is too large, as can be seen from Fig. 7(e).A comparison of Fig. 7(c) and (f) shows that the TF pseudo-Wiener filter is significantly different from the Wiener filter

. The heavy oscillations of the WS of in Fig. 7(c) in-dicate that is far from being an underspread system. Theprocesses and were constructed such that they lie inlinearly independent (disjoint) signal spaces. Here, is anoblique projection operator [44] thatperfectlyreconstructs ,thereby achieving zero MSE and infinite SNR improvement. Onthe other hand, due to the significant overlap of the WVS of

and , the TF pseudo-Wiener filter merely achievesan SNR improvement of 4.79 dB. This example is importantsince it shows that mere quasistationarity of and isnot sufficient as an assumption permitting a TF design of op-timal filters. It also shows that in certain cases—specifically,when signal and noise have significantly overlapping WVS butbelong to linearly independent signal spaces—the TF designedfilter performs poorly, even though the optimal Wiener filterachieves perfect signal reconstruction. We stress that this situ-ation implies that the processes areoverspread(i.e., not under-spread [12]), thereby prohibiting beforehand the successful useof TF filter methods.

Finally, we present the results of a real-data experiment illus-trating the application of the TF pseudo-Wiener filter to speechenhancement. The speech signal used consists of 246 pitch pe-riods of the German vowel “a” spoken by a male speaker. This


Fig. 8. Speech enhancement experiment using TF pseudo-Wiener filter. (a) WS of TF pseudo-Wiener filterH . (b) Two pitch periods of noise-free speech signal.(c) Noise-corrupted speech signal. (d) Enhanced speech signal. All signals are represented by their time-domain waveforms and their (slightly smoothed) Wignerdistributions [19], [58], [60], [61]. Horizontal axis: time (signal duration is 256 signal samples, corresponding to 2.32 ms). Vertical axis: frequency in kilohertz.

signal was sampled at 11.03 kHz and prefiltered in order to em-phasize higher frequency content. The resulting discrete-timesignal was split into 123 segments of length 256 samples (corre-sponding to 2.32 ms), each of which contains two pitch periods.Sixty four of these segments were considered as individual re-alizations of a nonstationary process of length 256 samples andwere used to estimate the WVS of this process. Furthermore, weused a stationary AR noise process that provides a reasonablemodel for car noise. Due to stationarity, the noise WVS equalsthe PSD for which a closed-form expression exists; hence, therewas no need to estimate the noise WVS. Using the estimatedsignal WVS and the exact noise WVS, the TF pseudo-Wienerfilter was then designed according to (26). The WS ofis shown in Fig. 8(a).

The remaining 59 speech signal segments were cor-rupted by 59 different noise realizations and input to the TFpseudo-Wiener filter . The input SNR (averaged over the59 realizations) was SNR dB, and the average SNR ofthe enhanced signal segments obtained at the filter output wasSNR dB, corresponding to an average SNR improve-ment of about 6.7 dB. Results for one particular realization areshown in Fig. 8(b)–(d).

VIII. C ONCLUSION

We have developed a time-frequency (TF) formulation andan approximate TF design of optimal filters that is valid forun-derspread, nonstationary random processes, i.e., nonstationaryprocesses with limited TF correlation. We considered two typesof optimal filters: the classical time-varying Wiener filter and aprojection-constrained optimal filter. The latter will produce alarger estimation error but has practically important advantagesconcerning robustness anda priori knowledge.

The TF formulation of optimal filters was seen to allow anintuitively appealing interpretation of optimal filters in termsof passing, attenuating, or suppressing signal components lo-cated in different TF regions. The approximate TF design ofoptimal filters is practically attractive since it is computation-

ally efficient and uses a more intuitive form ofa priori knowl-edge. We furthermore discussed an efficient TF implementationof time-varying optimal filters using multiwindow STFT filters.

Our TF formulation and design was based on the Weylsymbol (WS) of linear, time-varying systems and the WVSof nonstationary random processes. However, if the processessatisfy a more restrictive type of underspread property (if theyare strictly underspread[8], [29]), then our results can beextended in that the WS can be replaced by other members ofthe class ofgeneralized WSs[48] (such as Zadeh's time-varyingtransfer function [25]), and the WVS can be replaced by othermembers of the class ofgeneralized WVS[18], [19] or gen-eralized evolutionary spectra[29]. This extension is possiblesince for strictly underspread systems, all generalized WS’s areessentially equivalent and for strictly underspread processes,all generalized WVS and generalized evolutionary spectra areessentially equivalent [8], [9], [12], [19], [29], [47].

APPENDIX AUNDERSPREADAPPROXIMATION OF THEWIENER FILTER:

PROOF OF THEBOUNDS (20) AND (21)

We assume that the expected ambiguity functions ofandare exactly zero outside the same (possibly obliquely ori-

ented) centered rectanglewith area (joint correlation spread).

We first prove the bound (20). Usingand , the approximation error

can be developed as

(34)

We will now show that is zero outside the centeredrectangle that is oriented as but has sides that are twice as


long. Using , it can be shown [13] that, where

(35)

is known as thetwisted convolutionof and[13]. In the domain, correspondsto , which can be rewritten as

. With , we obtain

(36)

The twisted convolution [cf. (35)] implies that

and, hence, that the support of is confined to[recall that both and are confined to ].

The crucial observation now is that since is confined toand is confined to , (36) implies that

must also be confined to since any contri-butions of outside cannot be canceled by

and, thus, would contradict (36). Hence, wecan write (34) as

(37)

Applying the Schwarz inequality to (35) yields the bound.

Inserting this into (37) and using the fact that the area ofis(i.e., four times the area of), we obtain (20):

(38)

We next prove the bound (21). The lower boundis trivial since is the minimum possible MSE. Let us con-

sider the upper bound. With (3), the MSE achieved by canbe written as

tr

tr

tr

tr tr

Furthermore, tr according to (6). We there-fore obtain

tr tr

tr

tr tr

tr tr

The first term is zero, i.e., tr , sinceoutside . The second term can be bounded by

applying Schwarz' inequality:

tr

Using (38), the upper bound in (21) follows.

APPENDIX BTF FORMULATION OF THE WIENER FILTER: PROOF OF THE

BOUND (22)

We consider the approximation error

Subtracting and adding , this can be rewritten aswith

According to the triangle inequality, the norm of isbounded as . It is shown in [8], [11], and[46] that is bounded as

where we have used . Furthermore, isbounded as

where we have used (38). Inserting these two bounds intogives (22).

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Franz Hlawatsch (S’85–M’88) received theDipl.-Ing., Dr.techn., and Univ.-Dozent degreesin electrical engineering communications en-gineering/signal processing from the ViennaUniversity of Technology, Vienna, Austria, in 1983,1988, and 1996, respectively.

Since 1983, he has been with the Institute ofCommunications and Radio-Frequency Engineering,Vienna University of Technology. From 1991 to1992, he spent a sabbatical year with the Departmentof Electrical Engineering, University of Rhode

Island, Kingston. He authored the bookTime-Frequency Analysis and Synthesisof Linear Signal Spaces—Time-Frequency Filters, Signal Detection andEstimation, and Range-Doppler Estimation(Boston, MA: Kluwer, 1998) andco-edited the bookThe Wigner Distribution—Theory and Applications inSignal Processing(Amsterdam, The Netherlands: Elsevier, 1997). His researchinterests are in signal processing with emphasis on time-frequency methodsand communications applications.

Gerald Matz (S’95) received the Dipl.-Ing. degreein electrical engineering from the Vienna Universityof Technology, Vienna, Austria, in 1994.

Since 1995, he has been with the Institute of Com-munications and Radio-Frequency Engineering, Vi-enna University of Technology. His research interestsinclude the application of time-frequency methods tostatistical signal processing and wireless communi-cations.

Heinrich Kirchauer was born in Vienna, Austria, in1969. He received the Dip.-Ing. degree in communi-cations engineering and the doctoral degree in micro-electronics engineering from the Vienna Universityof Technology, Vienna, Austria, in 1994 and 1998,respectively.

In the summer of 1997 , he held a visiting researchposition at LSI Logic, Milpitas, CA. In 1998, hejoined Intel Corporation, Santa Clara, CA. Hiscurrent interests are modeling and simulation ofproblems for microelectronics engineering with

special emphasis on lithography simulation.

Werner Kozek (M’94) received the Dipl.-Ing. andPh.D. degrees in electrical engineering from the Vi-enna University of Technology, Vienna, Austria, in1990 and 1997, respectively.

From 1990 to 1994 he was with the Institute ofCommunications and Radio-Frequency Engineering,Vienna University of Technology, and from 1994 to1998, he was with the Department of Mathematics,University of Vienna, both as a Research Assistant.Since 1998, he has been with Siemens AG, Munich,Germany, working on advanced xDSL technology.

His research interests include the signal processing and information theoreticaspects of digital communication systems.

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