Kernels and Multiple Windows for Estimation of the Wigner-Ville ...

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Kernels and Multiple Windows for Estimation of the Wigner-Ville Spectrum of Gaussian Locally Stationary Processes Patrik Wahlberg and Maria Hansson, Member, IEEE

Abstract—This paper treats estimation of the Wigner-Ville spectrum (WVS) of Gaussian continuous-time stochastic processes using Cohen’s class of time-frequency representations of random signals. We study the minimum mean square error estimation kernel for locally stationary processes in Silverman’s sense, and two modifications where we first allow chirp multiplication and then allow nonnegative linear combinations of covariances of the first kind. We also treat the equivalent multitaper estimation formulation and the associated problem of eigenvalue-eigenfunction decomposition of a certain Hermitian function. For a certain family of locally stationary processes which parametrizes the transition from stationarity to nonstationarity, the optimal windows are approximately dilated Hermite functions. We determine the optimal coefficients and the dilation factor for these functions as a function of the process family parameter. Index Terms—Locally stationary circular symmetric processes, multitaper spectral estimation, optimal estimation, time-frequency analysis, Wigner-Ville spectrum (WVS).

I. INTRODUCTION HIS paper treats estimation of the Wigner-Ville spectrum (WVS) of zero mean real- and complex-valued Gaussian stochastic processes defined on a continuous time axis. We study estimators in Cohen’s class of time-frequency representations that are optimal in the sense of giving minimum mean square error. A member of Cohen’s class is defined as the convolution where is the Wigner-Ville distribution (WVD) of a random signal and is a fixed deterministic kernel. The mean square optimal kernel was first derived by Sayeed and Jones [1]. We are interested in the relation between the covariance of a nonstationary process and the optimal kernel. First, we study locally stationary processes (LSP) in Silverman’s sense [2], [3] where the covariance function has the simple form . (There exists several alternative definitions of a locally stationary process [4]–[6].) This is a special case of the class of underspread processes studied by Hlawatsch et al. [7]–[11]. First, we compute the optimal ambiguity domain

T

Manuscript received November 29, 2004; revised February 18, 2006. This work was supported in part by the Swedish Research Council and by the The Swedish Foundation for International Cooperation in Research and Higher Education (STINT). The associate editor coordinating the review of this paper and approving it for publication was Dr. Hongya Ge. P. Wahlberg is with the School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, NSW 2308, Australia (e-mail: [email protected]). M. Hansson is with the Centre for Mathematical Sciences, Lund University, SE-221 00 Lund, Sweden (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2006.882076

kernel of a Gaussian LSP, which can be expressed in a manageable form in terms of the functions and . If the process, furthermore, is complex-valued circular symmetric the optimal kernel is even more tractable. The optimal kernel then turns out to be invariant to translation of and modulation of . In general, the real-valued case yields more cumbersome formulas and some results we present hold only for the complex-valued circular symmetric case. Second, we compute the optimal kernel for complex-valued circular symmetric Gaussian LSPs generalized in two stages: (i) (metaplectic) transformation by multiplication with chirps of linearly increasing frequency and (ii) nonnegative linear combinations of covariances of type (i). The theory of metaplectic transformations [12]–[14] implies a rotation of the time-frequency kernel determined by the chirp constant. The class (ii) is a quite general class which can be used to model many naturally occurring nonstationary signals. We show a few examples of how the optimal kernel works, one of which reveals that if the components of the sum of covariances of type (i) are isolated in time or frequency, the performance can be improved using kernels that are local in time or frequency. Thus, the generally spread principle “cut the signal into locally stationary pieces” should be replaced by “cut it into pieces with locally constant chirp parameter.” Finally, we discuss transformation of the time-frequency convolution into a computationally more efficient equivalent Thomson multitaper spectrogram estimator [15]–[20]. The estimate of the WVS is then a weighted sum of the spectrograms defined by the eigenfunctions of a certain Hermitian kernel determined by the covariance. In particular, we study a of complex-valued circularly family parametrized by and symmetric Gaussian LSPs where . The family parametrizes the transition the between stationarity and nonstationarity. When function decreases quicker than and we approach a sta. The oppostite extreme is the tionary process as maximal nonstationarity within the family. We show that the optimal Hermitian kernel can be approximated with small error by a kernel which allows Hermite functions as eigenfunctions. This means that Hermite functions can be used as window functions in the multitaper estimation. We compute the eigenvalues, which are the coefficients in the weighted spectrogram of the optimal estimator, and obtain their dependence of . The Hermite functions turn out to have a common dilation factor, . which depends on according to The results of this paper are true when the covariance funcof a Gaussian process and the estimation kernel tion

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have certain regularity properties, meaning a certain amount of asymptotic decay and a certain amount of smoothness. The smoothness is measured by asymptotic decay on the Fourier transform side. A sufficient condition is , which denotes Feichtinger’s algebra [21], [22]. Under this condition the WVD, the ambiguity function and Cohen’s class of time-frequency respresentations are all second-order processes [23]. Furthermore, the estimation performance criterion introduced in [1] consisting of the integral of the mean square over is finite [23]–[25]. The error of WVD is well defined more generally for every harmonizable is bigger than the Gaussian process [26]. The space of rapidly decreasing smooth functions, Schwartz space , i.e., an function and is contained in is integrable in both the time and frequency domains. The excludes stationary processes. condition

weakly stationary process, we can write and (1) translates to , in which case is said to be nonnegative definite, denoted . A process is called harmonizable [27], [29] if its covariance can be written as a Fourier-Stieltjes integral with respect to a measure of bounded variation (2)

The measure is called the spectral covariance. If is weakly has support on the diagonal and is nonnegstationary then ative. A process is harmonizable if and only if it can be reprewhere is the spectral measented by [27], [29]. sure which fulfills Definition 1: A locally stationary process (LSP) [2], [3], [30] has a covariance function of the form

A. Notation and Organization of the Paper the set of finite variance complex-valued We denote by . The stochastic variables over a probability space Fourier transform is defined and denoted by

(3) where and are complex-valued functions and coordinate transformation

denotes the

(4) For a function of several variables we denote partial Fourier , by or transform with respect to variables indexed by . The paper is organized as follows: Section II concerns the background on harmonizable, locally stationary, circularly symmetric Gaussian processes. Section III treats the relation between circular symmetric and analytic processes. Section IV treats estimation of the WVS from the WVD of a random LSP signal, and Section V treats briefly the case of real-valued Gaussian LSPs. In Section VI and VII we generalize to locally stationary processes in two steps and give some examples, and in Section VIII we discuss the multitaper formulation of the time-frequency convolution. Finally, in Section IX we derive an approximation of the Hermitian kernel for a family of LSPs and compute its eigenvalues and eigenfunctions.

The function must then have constant sign which we assume [31], and without to be nonnegative, loss of generality. In the Appendix we give a result (Lemma 4) . on conditions on and which are sufficient for For an LSP, (2) takes the form

(5)

Definition 2: For a circularly symmetric or proper process [32], [33] , the processes are identically distributed for all . According to Grettenberg’s theorem [28], [34] circular symand metry is equivalent to

II. HARMONIZABLE, LOCALLY STATIONARY, AND CIRCULARLY SYMMETRIC GAUSSIAN PROCESSES A second-order continuous-time stochastic process is a map . We assume that all processes have zero mean. The (auto-)covariance function is defined by and is said to be of nonnegative definite type, denoted , meaning that

(6) The following fourth-order moment function will be needed later on. Definition 3:

(1) The formula for arbitrary finite sequences , and integer . A function belongs to if and only if it is the covariance function of a stochastic process, and the set of covariances is closed under pointwise multiplication [27], [28]. For a

(7)

WAHLBERG AND HANSSON: WVS OF GAUSSIAN LOCALLY STATIONARY PROCESSES

sometimes called Isserlis’ theorem, is valid for any zero mean complex-valued Gaussian stochastic variables , , , [35]. For real-valued Gaussian processes we thus, have

(8) and for circularly symmetric Gaussian processes

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Hence, there exists LSPs which are approximately analytic. We do, however, not know if there exists proper LSPs (i.e., ) which are exactly excluding the weakly stationary case analytic. In this paper, we are interested in how the covariance function influences the optimal estimation of time-variable spectra, and in order to simplify the analysis we will assume that complex-valued processes are circularly symmetric LSPs. IV. THE WVS AND ESTIMATION USING COHEN’S CLASS The WVS of a process with covariance function by [1], [31], [36], [37]

is defined

(9)

III. CIRCULAR SYMMETRY AND ANALYTIC SIGNALS In time-frequency signal processing a real-valued harmonizable signal is often transformed to a complex-valued analytic where denotes signal defined by . the Hilbert transform Then , i.e., the spectral measure of has support in which makes it useful for time-frequency analysis, since interference terms between signal components are attenuated [31]. The spectral covariance of is the spectral covariance of cut off to the first quadrant and amplified

There exists also another complexification of a real-valued (or more generally any complex-valued) Gaussian process, namely a circular symmetric Gaussian process with covariance [28]. In fact, there exists a Gaussian process function such that and each have covariance and , and . For our problem of computation of the optimal ambiguity domain WVS estimation kernel, it is convenient to assume circular symmetry because it simplifies the analysis of the optimal estimation kernel. The two complexifications rarely give equal results [33], although the analytic signal corresponding to a weakly stationary real-valued process is circularly symmetric. A complex-valued process with LSP covariance defined by (3), and , is analytic if and only if has support in the region [33]. In the weakly stationary case , which implies , so the support condition above is fulfilled if is any positive function with support in . If has support in a bounded interval in and does and can be modified not fulfill the support condition, according to , with . Then fulfills the support condition approximately. Moreover, , is multiplied by an i.e., function. The result is a function in , and by (5) we get an covariance also in the time domain.

The expected ambiguity function (EAF) is defined by

(10) The WVS of an LSP is , and the . Both and can be said to be EAF is (it does not oscilconcentrated around the origin, since late), and . Therefore, the set of LSPs is a special case of underspread processes, developed and investigated in a series of papers by Hlawatsch, Kozek, Matz et al. [7]–[11], and this theory is, thus, applicable to LSPs. A process is underspread if the EAF has essential support in a rectangle around the . The WVS of an underspread process has a origin of area small amount of interference terms between signal components [11]. Of particular relevance for our problem is [8] where a general theory of time-varying spectral estimation of underspread processes is developed, and the error is separated into bias and variance. In our approach instead the total mean square error is minimized, and we are interested in the special case of LSPs and generalizations, and how parameters of LSPs affect the optimal estimator kernel and windows. We shall use the following results from [1], [23], [24], [36], be a zero mean real-valued, or complex-valued [37]. Let circularly symmetric, Gaussian stochastic process. Then, the stochastic Riemann integrals [27]

(11) exist for all argument values. Thus, and are second-order . We call the stochastic processes defined on the ambiguity process. We have WVD process and where is a partial Fourier transform of the function of Definition 3. The convolution stochastic Riemann integral (12)

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, i.e., is a second-order stochastic process on . The convolution (12) is by definition a member of Cohen’s class of time-frequency representations [31], [38]. The process can be represented by the stochastic integral

where

(19) (13) In the case of real-valued processes we get, using (8), an extra term, , where (20) where We want to use Cohen’s class for estimation of , and measure the quality of the estimate by the mean square error integral [1] In [25] we have studied the optimal kernel when the process is an LSP. Insertion of (3) into (19) gives for a circularly symmetric Gaussian LSP which is finite if , representation (13), to equal

Minimization of

. It can be shown, using the

(21)

(14)

Under certain conditions on and [25], which we assume are fulfilled, . Thus, by (15) and (18) the optimal ambiguity domain kernel is for a circularly symmetric Gaussian LSP

now gives [1], [23]

(22) (15)

denotes the indicator function for the open set . The optimal time-frequency kernel is computed by where

(16) for all , is We have is real-valued and reflection invariant [23]. For continuous, the value of the integral (14) is (17)

However, we have no guarantee that as required in the derivation of (14). It is nevertheless possible to regularize into (the set of smooth functions with compact and support) such that for an arbitrary [24]. In this paper, we will not go into detail about this regularization, but focus which we denote “optimal” although it is not on the kernel in which we a working. guaranteed to belong to the space In the circularly symmetric Gaussian case we obtain using (9) and (10)

(18)

From (22), we can see that is invariant to modulation of and translation of . Thus, the class of processes that have optimal kernel contains all LSPs where is modulated to arbitrary center frequency, i.e., has bandpass character. V. THE OPTIMAL KERNEL FOR REAL-VALUED GAUSSIAN LSPS For a real-valued Gaussian LSP we have to take into account of the denominator of (15). It is the third term (23) is the ambiwhere guity function of [12], [31]. Since is invariant to translation by (20) and (15). of , the same conclusion also holds for Modulation of , which in the real-valued case has to be defined by multiplication of a cosine instead of a complex exponential, has the following effects.

Thus, by (21) and (23), and are not invariant to modulaand . tion, and the same conclusion is true for

WAHLBERG AND HANSSON: WVS OF GAUSSIAN LOCALLY STATIONARY PROCESSES

VI. LOCALLY STATIONARY CIRCULARLY SYMMETRIC GAUSSIAN CHIRP PROCESSES

If

In order to generalize the LSP definition we introduce a chirp modulation factor. Definition 4: A locally stationary chirp process (LSCP) [39] is a Gaussian circularly symmetric process with covariance function of the form

(24)

where

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,

and (28)

then EAF

[21]. An MLSP which fulfills (28) has the and the WVS . Restricting to the circularly symmetric case and inserting the definition (27) into (19) and (18) we compute the optimal kernel (15)

is defined by (25)

As stated in Lemma 5 in the Appendix is in. If has covarivariant under multiplication by then , i.e., ance equals times a chirp, has covariance . We have which shows that is modulated by the chirp , whose frequency varies linearly with a delayed time axis . The LSPC concept is a special case of a so called metaplectic transformation [12]–[14]. If one multiplies a signal with a chirp according to , , then the WVD is coordinate [12]. The next theorem transformed as is proved in the Appendix and states that the optimal kernel for an LSCP is obtained by a coordinate transformation of the kernel of the corresponding LSP, which means that the optimal kernel transforms according to the metaplectic theory. define a circular symmetric Theorem 1: Suppose , Gaussian LSP with covariance given by (3). The optimal time-frequency kernel for a LSCP process with chirp parame, is then ters and , denoted

Generalization to the case where each term are LSCP covariand (acronym MLSCP), ances with individual constants i.e., (29)

gives, inserting (29) into (10),

and, inserting (29) into (19)

(26) where

is the optimal kernel of the corresponding LSP (i.e., ), defined by (16) and (22).

VII. MULTICOMPONENT LOCALLY STATIONARY CIRCULARLY SYMMETRIC GAUSSIAN PROCESSES In order to discuss more general covariances than Definition 1, we introduce the following definition. Definition 5: A multicomponent LSP (MLSP) is a process whose covariance has the form

(27)

where each term LSP.

where

An MLSCP has a covariance that is a sum of covariances with chirp behavior of various localization in time and chirp constants. This class of processes is quite general and adequate to model various phenomena within biology and physics. A. Discrete-Time Examples Here we study Gaussian circularly symmetric processes with covariance

is the covariance function of a (30)

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Fig. 1. W

(t; 2) (above) and 8 (t; 2).

which is a variant of the model (29) with synchronized to the chirp factor. be defined by (30) with Example 1: Let , and parameter values , , , , and . The time axis was . In Fig. 1 we show chosen as the integers and for this process, and in images of Fig. 2 one realization of the stochastic processes , , respectively, and the average of over ten realizations. Example 2: With a time axis identical to Example 1, we set , , , , , and . The WVS is shown in Fig. 3. The optimal integrated square error (17) is , while the empirical integrated square error is computed from 400 realizations. (This 20% excess we believe is due to discretisation effects.)

Fig. 2. W (t; 2 ) (above), W 3 8 (t; 2 ) (middle), and average of W 3 8 (t; 2) over 10 realizations (bottom).

The WVS of a process of this kind, which has two structures separated in time, is however estimated more efficiently by optimization of the kernel to each structure individually. This can

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respectively, a slight improvement from the global case. Similarly, there is some performance to gain if the components are separated on the frequency axis. VIII. MULTIPLE WINDOWS FORMULATION In this section we restrict to complex-valued circularly symmetric Gaussian processes. The time-frequency convolution (12) can be computed more efficiently using the Thomson multitaper method [15], [16]. By a windowed spectrogram of a process with respect to a window function we mean

In the Appendix, we prove the following lemma which says that the windowed spectrogram can be written as a stochastic double integral. is circularly symmetric, and Lemma 1: If then we have the equality in ,

(31) Next we give conditions which implies that the time-frequency convolution defined by Cohen’s class can be replaced by a weighted sum of windowed spectrograms, i.e., the Thomson multitaper method [20]. First, we define

(32)

using

. We have the equivalences

(33)

Fig. 3. W (t; 2 ) of the process of Example 2 (above), estimate using a global kernel (middle), and estimate using local kernels (below).

and is real-valued, from the inIf variance properties of [21], and by (33) is Hermitian. Thus, [22], is the kernel of a compact, since [40]. There exists, therefore, an self-adjoint operator on orthonormal set of eigenfunctions and eigenvalues such that (34)

be seen in Fig. 3 where the bottom subfigure shows the result of using separate kernels optimized to the first and the latter time interval halves, respectively (one realization). The sum of the and (400 realizations), errors were

Since , we also have for all [21], [22]. The next theorem, proved in the Appendix, shows that

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under these circumstances Cohen’s class can be replaced by a weighted sum of spectrograms. Theorem 2: Suppose is a complex-valued circularly symmetric Gaussian process with . Let and real-valued, let be defined by (32) and fulfill (34). Then we have the equality in

and (32), the effect of such a transformation on the eigenfunc, while tions is chirp multiplication the eigenvalues remain the same. In order to find an approximate solution of (37) we consider the Hermite polynomials [40]

and the Hermite functions where (38)

(35)

is an orthonormal basis for . The sequence is even/odd according to its index . One can prove that the are eigenfunctions of the Fourier transform, functions (39)

IX. APPROXIMATION OF THE EIGENFUNCTION-EIGENVALUE PROBLEM Here, we study a family of circularly symmetric Gaussian , with and LSPs indexed by . The following lemma says that is the interesting parameter range. The proof is as usual found in the Appendix. is a covariance Lemma 2: . As aforementioned, the parameter parametrizes the tranto stationarity sition from maximal nonstationarity . Using (22) we can compute the optimal kernel in the ambiguity domain as

(36)

More generally we will work with an orthonormal basis of dilated Hermite functions (40) Since for fixed clearly any the series expansion

, we have for

(41) where (42)

Since is even in both variables only evenly indexed Hermite functions appear in the expansion (41). Using and (39), we compute from (41)

where kernel

. The is computed from (32) and (36). If then and thus, . From now on we re. According to Theorem 2 the optimal WVS esstrict to timate is obtained using a Thomson multitaper estimator where the window functions and weights, respectively, are the eigenconsidered as an integral operfunctions and eigenvalues of ator. Therefore we want to solve (37) where the notation , indicates that the eigenvalues and eigenfunctions depend on . We are interested in this dependence. Notice that we assume that the eigenfunction index, denoted in (34), is constant (and arbitrary) in the following. The index is not visible in the notation for simplification. The following analysis is also valid when we modify multiplicatively as in (24) and (25). In fact, according to Thm. 1 by a chirp

Using (36) and again

we have

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The kernel to be eigendecomposed is thus, by (32)

In order to simplify, we introduce

(43) Then the eigendecomposition (37) is

Fig. 4. The function  ().

(44)

where the kernel

depends on

and ,

(45) We choose the parameter in the expansion (42) adapted , which depends only on the radius to the function . Since the function (where chosen such ), which is used in defined by (38) that , a reasonable strategy is to choose and (40), has variance as the variance of (normalized to integrate to one). Thus, we set Fig. 5. The relative squared approximation error J () as a function of .

Thus, the kernel (45) of the problem (44) can be approximated with where

(47) and the resulting eigenfunction-eigenvalue problem is

In Fig. 4 we show a plot of . In Fig. 5 we show the relative squared approximation error in (41) when the series is truncated after only one term, i.e., we plot (46) . Since almost all energy of is captured in the using for a wide range of values, we may for approxcoefficient imate analysis truncate the series expansion after the first term.

(48)

and eigenfunctions We want to determine the eigenvalues and then we can temporarily replace (48) by (49)

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where and . Then since . The index has been dropped for convenience. The next lemma, proved in the Appendix, shows that dilated Hermite functions solves (49). the dilated Hermite functions solves Lemma 3: If (49) with eigenvalues

where tion

(50) and depend on Referring back to (48), where we now let the eigenfunction index , we thus, obtain, using (43), our final theorem. the solution to (48) is given by Theorem 3: If

is a nonnegative bounded measure defined by a func. Suppose (51) converges for all , and

for and , where , , denotes the weight function . Suppose where and denotes the Sobolev space . Define of tempered distributions such that and where . is the covariance function Then of a locally stationary stochastic process, and . Lemma 5: Suppose , are defined as in Lemma 4. Let , and be defined by (25). Then (52)

where

, and

for each integer . Under the approximation (47) of the kernel (45), we have, thus, found a relation between the nonstationarity parameter and the dilation of the Hermite window functions of the optimal multitaper estimator.

Proof of Thm. 1: By insertion of (52) into (10) we have . By insertion of (52) into . Thus, it follows (19), from (18), (15), and (22) that the optimal ambiguity domain , fulkernel for the locally stationary chirp process, denoted fills where is the optimal kernel . Now (26) follows of the LSP corresponding to from (16) and the computation

X. CONCLUSION For complex-valued locally stationary Gaussian circularly symmetric processes, the minimum mean square error kernel for estimation of the WVS has a simple structure in the ambiguity domain and obeys a natural metaplectic transformation rule when the signal is multiplied by a chirp. In order to treat more general processes we have extended the method to processes that have covariance that is a nonnegative linear combination of LSP covariances. We have also shown how the time-frequency convolution can be replaced by the Thomson multitaper method which is computationally more efficient. Finally, we have studied a simple class of LSPs where the two functions which determine the covariance are Gaussians. This which parametrizes the transition class is indexed by to stationarity . We from nonstationarity have computed approximate optimal window functions, which turned out to be dilated Hermite functions, and weights. The . dilation factor depends on according to

Proof of Lemma 1: The result is proved by showing (53) We have where the short-time Fourier transform (STFT) is a finite-variance circularly symmeric Gaussian process. This follows from

, , the invariance properties of [21], and the assumed circular symmetry of . Using (6), (7) one verifies by straightforward computations that (54)

APPENDIX The following lemma [25] gives sufficient conditions on and such that (3) is a covariance and . Lemma 4: Suppose

Similarly, it is also possible to show

(55) (51) Now (53) follows from (54) and (55).

WAHLBERG AND HANSSON: WVS OF GAUSSIAN LOCALLY STATIONARY PROCESSES

Proof of Thm. 2: We prove the theorem by showing

In [23], the formula (56) is proved, where

Using Lemma 1, Definition 3, (32) and obtain from (35)

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. In fact, the inequality (59) is trivially true if , since by Bochner’s theorem It is also true if , and the latter statement is clearly true because the Fourier transform of a Gaussian is a Gaussian. Thus, by (59) and it is, hence, a covariance. Proof of Lemma 3: We use Mehler’s Formula [13] which in our notation is the identity

we where then

fulfills since

. If we choose , and, furthermore

Finally, a change of variables gives

(60) Since , is a square summable because sequence, the series, hence, converges in is an ONB. Thus, (60) is the eigenvalue-eigenfunction decomposition of the left-hand side kernel. (57) Likewise, for the cross-term we have, using (12) and (11) (58) Since (56), (57), and (58) are all equal the theorem is proved. Proof of Lemma 2: Suppose is a covariance. Inserting and into (3) and using the Cauchy-Schwarz inequality gives

Suppose on the other hand . Let be arbitrary. Then integer

,

and the

(59)

ACKNOWLEDGMENT The first author (P. Wahlberg) would like to express his gratitude to H. Feichtinger for the hospitality experienced at a very stimulating stay at NuHAG, Faculty of Mathematics, University of Vienna, during the winter 2004–2005. REFERENCES [1] A. M. Sayeed and D. L. Jones, “Optimal kernels for nonstationary spectral estimation,” IEEE Trans. Signal Process., vol. 43, pp. 478–491, 1995. [2] R. A. Silverman, “Locally stationary random processes,” IRE Trans. Inf. Theory, vol. 3, pp. 182–187, 1957. [3] ——, “A matching theorem for locally stationary random processes,” Comm. Pure Appl. Math., vol. 12, pp. 373–383, 1959. [4] D. Donoho, S. Mallat, and R. von Sachs, “Estimating covariances of locally stationary covariances: consistency of best basis methods,” in Proc. IEEE-SP Int. Symp. on Time-Frequency and Time-Scale Analysis, 1996, pp. 337–340. [5] R. Dahihaus, “Fitting time series models to non-stationary processes,” Ann. Statist., vol. 25, no. 1, 1997. [6] M. E. Oxley, T. F. Reid, and B. W. Suter, “Locally stationary processes,” in Proc. 10th IEEE Workshop on Stat Signal and Array Process., 2000, pp. 257–261. [7] G. Matz and F. Hlawatsch, “Time-varying spectra for underspread and overspead nonstationary processes,” in Proc. 32nd Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, 1998, pp. 282–286. [8] W. Kozek and K. Riedel, “Quadratic time-varying spectral estimation for underspread processes,” in Proc. IEEE-SP Int. Symp. Time-Frequency Time-Scale Anal., Philadelphia, PA, 1994, pp. 460–463.

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Patrik Wahlberg received the M.Sc. degree in electrical engineering from Lund University, Sweden, in 1991, and the Ph.D. degree from the same university in 1999. During 2000–2001, he was Research Fellow and Assistant Professor with the Department of Electroscience, Lund University, and from 2001 to 2006 he has been an Assistant Professor in Signal Processing at Malmö University, Sweden. During autumn 2002, he visited Nanyang Technological University, Singapore, and during 2004–2005 he visited the Numerical Harmonic Analysis Group, Vienna University, Austria, as a Postdoctoral Fellow. His research interests includes time-frequency analysis, Gabor analysis, stochastic process theory, and applications within biomedical signal processing.

Maria Hansson (M’96) was born in Sweden 1966. She received the M.Sc. degree in electrical engineering in 1989 and the Ph.D. degree in signal processing in 1996, both from Lund University, Sweden. Currently, she is an Associate Professor in Mathematical Statistics with the Centre for Mathematical Sciences, Lund University. Her current research interests include multiple window spectrum analysis and time-frequency analysis of stochastic processes with application to electroencephalogram signals, heart rate variability signals, and speech signals.