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Time–Frequency Formulation, Design, and Implementation of Time-Varying Optimal Filters for Signal Estimation Franz Hlawatsch, Member, IEEE, Gerald Matz, Student Member, IEEE, Heinrich Kirchauer, and Werner Kozek, Member, IEEE
Abstract—This paper presents a time–frequency framework for optimal linear filters (signal estimators) in nonstationary environments. We develop time–frequency formulations for the optimal linear filter (time-varying Wiener filter) and the optimal linear time-varying filter under a projection side constraint. These time–frequency formulations extend the simple and intuitive spectral representations that are valid in the stationary case to the practically important case of underspread nonstationary processes. Furthermore, we propose an approximate time–frequency design of both optimal filters, and we present bounds that show that for underspread processes, the time-frequency designed filters are nearly optimal. We also introduce extended filter design schemes using a weighted error criterion, and we discuss an efficient time–frequency implementation of optimal filters using multiwindow short-time Fourier transforms. Our theoretical results 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
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HE enhancement or estimation of signals corrupted by noise or interference is important in many signal processing applications. For stationary random processes, the mean-square error optimal linear estimator is the (time-invariant) Wiener filter [1]–[7], whose transfer function is given by (1) and denote the power spectral densities where of the signal and noise processes, respectively. The extension of 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 work was supported by FWF Grants P10012-ÖPH and P11904-TEC. The associate editor coordinating the review of this paper and approving it for publication was Prof. P. C. Ching. F. Hlawatsch and G. Matz are with the Institute of Communications and Radio-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 formulation can be obtained by introducing an explicit time dependence, i.e., whether in a joint time-frequency (TF) domain the time-varying Wiener filter can be formulated as (2) and are suitably defined. Such where a TF formulation would greatly facilitate the interpretation, analysis, design, and implementation of time-varying Wiener filters. In this paper, we provide an answer to this and several other questions of theoretical and practical importance. • We show that for underspread [8]–[12] nonstationary processes, the TF formulation in (2) is approximately valid and are chosen as the Weyl if symbol [13]–[16] and the Wigner–Ville spectrum (WVS) [12], [17]–[19], respectively. We present upper bounds on the associated approximation errors. • We propose an efficient, intuitive, and nearly optimal TF design of signal estimators that is easily adapted to modified (weighted) error criteria. • We discuss an efficient TF implementation of optimal filters using the multiwindow short-time Fourier transform (STFT) [8], [20]. • We also consider the TF formulation, approximate TF design, and TF implementation of an optimal projection filter. Previous work on the TF formulation of time-varying Wiener filters [21]–[24] has mostly used Zadeh's time-varying transfer and the evolutionary spectrum function [25], [26] for and . A Wiener filtering [12], [27]–[29] for procedure using a time-varying spectrum based on subband AR models has been proposed in [30]. A somewhat different approach is the TF implementation of signal enhancement by means of a TF weighting applied to a linear TF signal representation such as the STFT, the wavelet transform, or a filterbank/subband representation. In [31] and [32], the STFT is multiplied by a TF weighting function similar to (2), where are chosen as the physical spectrum [18], [19], [33]. Wiener filter-type modifications of the STFT and subband signals have long been used for speech enhancement [34]–[37]. Methods that perform a Wiener-type weighting of wavelet transform coefficients are described in [38]–[41].
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The TF formulations proposed in this paper differ from the above-mentioned work on several points: • While, usually, Zadeh's time-varying transfer function and the evolutionary spectrum or physical spectrum are and , respectively, chosen for our development is based on the Weyl symbol and the WVS. This has important advantages, especially regarding TF resolution. Specifically, the Weyl symbol and the WVS are much better suited to systems and processes with chirp-like characteristics. Furthermore, contrary to the evolutionary spectrum and physical spectrum, the WVS is in one-to-one correspondence with the correlation function. • Our TF formulations are not merely heuristic but are shown to yield approximations to the optimal filters featuring nearly optimal performance in the case of jointly underspread signal and noise processes. We present bounds on the associated approximation errors, and we discuss explicit conditions for the validity of our TF formulations. In particular, we show that the usual quasi-stationarity assumption is neither a sufficient nor a necessary condition. • Contrary to the conventional heuristic STFT enhancement schemes, the multiwindow STFT implementation discussed here is based on a theoretical analysis that shows how to choose the STFT window functions and how well the multiwindow STFT filter approximates the optimal filter. The paper is organized as follows. Section II reviews the time-varying Wiener filter and the optimal time-varying projection filter. Section III reviews some TF representations and the concept of underspread systems and processes. In Section IV, a TF formulation of optimal filters is developed. Based on this TF formulation, Section V introduces simple and intuitive TF design methods for optimal filters. In Section VI, a multiwindow STFT implementation of optimal filters is discussed. Finally, Section VII presents numerical simulations that compare the performance of the various filters and verify the theoretical results. II. OPTIMAL FILTERS This section reviews the theory of the time-varying Wiener filter [2], [4]–[7] and a recently proposed “projection-constrained” optimal filter [42]. For stationary processes,there exist simple frequency-domain formulations of both optimal filters. These will be generalized to the nonstationary case in Section IV. A. Time-Varying Wiener Filter be a zero-mean, nonstationary, circular complex or Let real random process with known correlation function E or, equivalently, correlation operator1 . This
R
correlation operator of a random process x(t) is the self-adjoint and positive (semi-)definite linear operator [43] whose kernel is the correlation function r (t; t ) = E x(t) x (t ) . In a discrete-time setting, would be a matrix. 1The
f
g
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signal process is contaminated by additive zero-mean, nonstawith known tionary, circular complex or real random noise . Signal and noise are assumed correlation operator . We form an estito be uncorrelated, i.e., E of the signal from the noisy observation mate using a linear, generally time-varying system (linear ,2 operator [43]) with impulse response (kernel)
Our performance index will be the mean-square error (MSE), E E i.e., the expected energy tr of the estimation error , which can be shown to be given by
tr
tr
(3)
denotes the trace of an operator, denotes the adHere, tr and are the expected energies of joint of [43], and and the signal distortion the residual noise , respectively. The linear system minican be shown [2], [4]–[7] to satisfy mizing the MSE (4) The solution of (4) with minimal rank3 is the “time-varying Wiener filter” (5) denotes the (pseudo-)inverse of on its range . The minimum MSE achieved by the Wiener filter is given by
where
tr tr
tr
(6)
, let In order to interpret the time-varying Wiener filter range and the noise us define the signal space range . Since , the obspace . We note that4 servation space is given by , and . It can now be shown that the optimal signal estimate is an element of the signal space , . Since and , we also have i.e., . Since it can also be shown that , we get . These results are intuitive since any contribufrom outside would unnecessarily increase the tion to and noise space resulting error. Furthermore, if signal space are disjoint, , then the Wiener filter performs and obtains perfect reconan oblique projection [44] onto , i.e., . struction of the signal 2Integrals
01 1
are from to unless stated otherwise. general solution of (4) is + , where denotes the minimal rank solution in (5), is an arbitrary operator, and is the orthogonal projection operator on , the orthogonal complement space of = range with = + . (For = L (IR) [where L (IR) denotes the space of square-integrable functions], becomes the unique solution of (4). 4Since the signals s(t) etc. are random, relations like s(t) are to be understood to hold with probability 1. 3The
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In some applications, e.g., if signal distortions are less acceptable than residual noise, we might consider replacing the MSE in (3) by a “weighted MSE”
between expected signal energy and expected noise energy in span is E
E tr tr
(7)
. This amounts to replacing by and so that the resulting modified Wiener filter is
with by
B. Optimal Time-Varying Projection Filter using a linear, timeWe next consider the estimation of varying filter that is an orthogonal projection operator, i.e., , and self-adjoint, [43]. it is idempotent, Although the projection constraint generally results in a larger MSE, it leads to an estimator that is robust in a specific sense [45]. Further advantages regarding a TF design are discussed in Subsection V-B. and denote the eigenvalues and normalized Let , eigenfunctions, respectively, of the operator . It is shown in [42] that the mini.e., imum-rank orthogonal projection operator minimizing the MSE in (3) is given by
where denotes the rank-one orthogonal projection with operator defined by , and is the index set corresponding to positive eigenvalues. Thus, the performs a projection onto the optimal projection filter spanned by all eigenfunctions of space corresponding to positive eigenvalues. The MSE obtained with is given by tr
tr
(8)
Thus, if (and only if) , then in span , the expected signal energy is larger than the expected noise energy. E It is then easily shown that E for any signal , i.e., in , the expected signal energy is larger than the expected noise energy. Equivalently, E , which shows that the optimal E projects onto the space where, projection filter on average, there is more signal energy than noise energy. This is intuitively reasonable. If we use the “weighted MSE” (7) instead of the MSE (3), the is constructed as before resulting optimal projection filter replaced by . with C. Stationary Processes The special cases of stationary processes and nonstationary white processes allow particularly simple frequency-domain and time-domain formulations, respectively, of both optimal filters. These formulations will motivate the development of TF formulations of optimal filters in Section IV. and are wide-sense stationary processes, the corIf and are time-invariant, responding correlation operators i.e., of convolution type, and the optimal system becomes timeinvariant as well. The MSE in (3) does not exist and must be re. For any linear, placed by the “instantaneous MSE” E , the intime-invariant system with transfer function stantaneous MSE can be shown to be E (9) and are the power spectral densities (PSD’s) where and , respectively. The optimal system minimizing of [2]–[7]. A “min(9) satisfies imal” solution of this equation is
For an interpretation of this result, let us partition the obserinto three orthogonal subspaces corresponding vation space to positive, zero, and negative eigenvalues of : span span span so that
and , where are the orthogonal projection operators on , and , respectively. Note that the spaces . We now consider the expected energies of the signal and . The expected energy of any noise processes in the space in the one-dimensional (1-D) space span process is given by E . Hence, the difference
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(10) denotes the set of where frequencies where the PSD of the observation is positive. The minimum instantaneous MSE is given by E
(11)
, and
The frequency-domain formulations (10) and (11) allow an intuitively appealing interpretation of the time-invariant and Wiener filter (see Fig. 1). Let denote the sets of frequencies where and , respectively, are positive. Then, (10) shows
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III. TIME-FREQUENCY REPRESENTATIONS AND UNDERSPREAD PROCESSES In this section, we review some fundamentals of TF analysis that will be used in Sections IV and V for the TF formulation and TF design of optimal filters. A. Time–Frequency Representations Fig. 1.
Frequency-domain interpretation of the time-invariant Wiener filter.
that for , i.e., the Wiener filter passes all observation components in the frequency bands where only signal energy is present, which is intuitively reasonable. for , i.e., outside , the Furthermore, observation is suppressed, which is again reasonable since outside , there is only noise energy. In the frequency ranges , where both signal and noise are present, there is with the values of depending on the and at the respective frequency. relative values of The MSE in (9) can again be replaced by a weighted MSE. Here, the weights can even be frequency dependent, which results in the weighted MSE
We first review four TF representations on which our TF formulations will be based. • The Weyl symbol (WS) [13]–[16] of a linear operator with kernel (impulse (linear, time-varying system) is defined as response) (15) where and denote time and frequency, respectively. For underspread systems (see Section III-B), the WS can be considered to be a “time-varying transfer function” [8], [11], [46], [47]. • The Wigner–Ville Spectrum (WVS) [17]–[19] of a nonstais defined as the WS of the tionary random process correlation operator
(12)
(16)
. The optimal filter minimizing this with reweighted MSE is then simply given by (10) with and replaced by . placed by . For staWe next consider the optimal projection filter is time-invariant with tionary processes, it can be shown that the zero-one valued transfer function
For underspread processes (see Section III-B), the WVS can be interpreted as a “time-varying PSD” or expected [8], [9], [12]. TF energy distribution of • The spreading function [8], [13], [16], [26], [46], [48], [49] of a linear operator is defined as
(13) is the indicator function of , which is the set of frequencies where the signal PSD is an idealized bandpass is larger than the noise PSD. Thus, can be filter with one or several pass bands. Note that by a rounding operation, i.e., obtained from round . The instantaneous MSE obtained with can be shown to be given by where
E
(14)
and . If the weighted instantaneous MSE (12) is used, then the optimal projection filter is the idealized bandpass filter with pass. band(s) and is The case of nonstationary white processes dual to the stationary case discussed above. Here, the timevarying Wiener filter and projection filter are “frequency-invariant,” i.e., simple time-domain multiplications. All results and interpretations are dual to the stationary case. where
where and denote time lag and frequency lag, respectively. The spreading function is the 2-D Fourier transadmits the folform of the WS. Any linear operator defined as lowing expansion into TF shift operators [8], [13], [16], [49]: (17) Hence, the spreading function describes the TF shifts introduced by (see Section III-B). • The expected ambiguity function [8]–[12] of a nonstais defined as the spreading function tionary process of the correlation operator
(18) The expected ambiguity function is the 2-D Fourier transform of the WVS. It can be interpreted as a TF correlation function in the sense that it describes the correlation of
HLAWATSCH et al.: TIME-FREQUENCY FORMULATION, DESIGN, AND IMPLEMENTATION OF TIME-VARYING SIGNAL ESTIMATORS
process components whose TF locations are separated by in time and by in frequency [8]–[12]. B. Underspread Systems and Processes Since our TF formulation of optimal filters will be valid for the class of underspread nonstationary processes, we will now review the definition of underspread systems and processes [8]–[12], [46], [47]. According to the expansion (17), the effective support of the describes the TF shifts caused by spreading function a linear time-varying system . A global description of the TF is given by the “displacement spread” shift behavior of [8], [11], [46], which is defined as the area of the smallest recplane and possibly tangle [centered about the origin of the with oblique orientation] that contains the effective support of . A system is called underspread [8], [11], [46] if , which means that is highly concentrated plane and, hence, that causes about the origin of the is the 2-D Fourier transonly small TF shifts. Since , this implies that is a smooth funcform of and are called jointly underspread tion. Two systems if their spreading functions are effectively supported within the . same small rectangle of area Let us next consider a nonstationary, zero-mean random in (18) process. The expected ambiguity function describes the correlation of process components whose TF locations are separated by in time and by in frequency [8]–[12]. Since the expected ambiguity function is the spreading function , i.e., , of the correlation operator of a process it is natural to define the correlation spread as the displacement spread of its correlation operator, . A nonstationary random process is then i.e., [8]–[11]. Since in this case the called underspread if expected ambiguity function is highly concentrated about the plane, process components at different (i.e., origin of the not too close) TF locations will be effectively uncorrelated. is a smooth, effectively This also implies that the WVS of and non-negative function [8], [12]. Two processes are called jointly underspread if their expected ambiguity functions are effectively supported within the same small . rectangle of area Two special cases of underspread processes are “quasistationary” processes with limited temporal correlation width and “nearly white” processes with limited time variation of their statistics. We emphasize that quasistationarity alone is neither necessary nor sufficient for the underspread property.
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that similar simplifications can be developed for underspread nonstationary processes [50]. A. Time–Frequency Formulation of the Time-Varying Wiener Filter We assume that and are jointly underspread processes; specifically, their expected ambiguity functions are assumed to be exactly zero outside a rectangle (centered about plane) of area . The Wiener the origin of the , where the underfilter can then be split up as is defined by spread part (19) is the indicator function of , and the overspread is defined by . That is, and lie inside and outside , respectively. Since the spreading function is the 2-D Fourier transform of the Weyl symbol, (19) implies ; here, denotes 2-D convois the 2-D Fourier transform of and, lution and is a hence, a 2-D lowpass function. This means that . smoothed version of is characterized by the We now recall that the Wiener filter . A first step toward a TF relation (4), i.e., is to notice that removing the overspread part formulation of from does not greatly affect the validity of this relation, i.e., where part
Indeed, it is shown in Appendix A that the Hilbert-Schmidt (HS) incurred by this norm [43] of the error approximation is upper bounded as
(20) and, furthermore, that the “excess MSE” resulting from using instead of is bounded as
(21) IV. TIME–FREQUENCY FORMULATION OF OPTIMAL FILTERS In Section II, we saw that the time-varying Wiener filter and projection filter are described by equations involving linear operators and signal spaces. This description is rather abstract and leads to computationally intensive design procedures and implementations. However, for stationary or white processes, there exist simple frequency-domain or time-domain formulations that involve functions instead of operators and intervals instead of signal spaces and thus allow a substantially simplified design and implementation. We will now show
denotes the MSE obtained with . Hence, if is Here, and are jointly underspread, small, i.e., if and . This shows that the underspread is nearly optimal. part This being the case, it suffices to develop a TF formulation . Intuitively, we could conjecture that a natural TF for formulation of the operator relation would be . Indeed, norm of the error it is shown in Appendix B that the
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incurred by this approximation is upper bounded as
(22) can furthermore be formuAn upper bound involving norm) as well. Hence, lated for the error magnitude ( if is small, and are jointly underspread. Defining the TF i.e., if is effectively region where (with some5 small positive by ), we finally obtain the following TF formulation for :
(23)
tr The minimum MSE in (6) is given by which, for underspread, can be shown to equal . Inserting (23), we obtain the approximate TF formulation (24)
The relations (23) and (24) constitute a TF formulation of the time-varying Wiener filter that extends the frequency-domain formulation (10), (11) that is valid for stationary processes and the analogous time-domain formulation that is valid for nonstationary white processes to the much broader class of underspread nonstationary processes. This TF formulation of the time-varying Wiener filter allows a simple and intuitively appealing interpretation. Let us define the TF support and are effectively regions where the WVS of and positive by . The TF regions and correspond to the signal spaces and underlying the respective processes [42], [51]. With these definitions at hand, the Wiener filter can be interpreted in the TF domain as follows (see Fig. 2). , i.e., in the TF re• In the “signal only” TF region gion where only signal energy is present, it follows from (23) that the WS of the Wiener filter is approximately one:
Fig. 2. TF interpretation of the time-varying Wiener filter.
• In the “no signal” TF region, where no signal energy is , it follows from (23) that the present, i.e., for WS of the Wiener filter is approximately zero:
This “stop region” of the Wiener filter consists of two , where only parts: i) the “noise only” TF region noise energy is present and ii) the outside (complement) , i.e., the TF region of the observation TF region where neither signal nor noise energy is present; here, since (23) corresponds to the minimal Wiener filter. , where both • In the “signal plus noise” TF region signal energy and noise energy are present, the WS of the Wiener filter assumes values approximately between 0 and 1
Here, performs a TF weighting that depends on the relative signal and noise energy at the respective TF point. and Indeed, it follows from (23) that the ratio of is given by the “local signal-to-noise ratio” SNR
:
SNR
In particular, for equal signal and noise energy, i.e., , there is . SNR This TF interpretation of the time-varying Wiener filter extends the frequency-domain interpretation valid in the stationary case (see Fig. 1) to the broader class of underspread processes. B. Time–Frequency Formulation of the Time-Varying Projection Filter
Thus, passes all “noise-free” observation components without attenuation or distortion. 5An analysis of the WS of positive operators ([8] and [47]) suggests that we = A . use A
k k
k k
We recall from Section II-B that the optimal projection filter performs a projection onto the space span spanned by all eigenfunctions of the operator associated with positive eigenvalues. In order to obtain a TF formulation of this result, we consider the following three
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TF regions on which the signal energy is, respectively, larger than, equal to, or smaller than the noise energy:
It can finally be shown that the MSE obtained with (8)] is approximately given by
[cf.
With
which extends the relation (14) to general underspread processes. , these TF regions can alternatively be written as V. APPROXIMATE TIME–FREQUENCY DESIGN OF OPTIMAL FILTERS
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 underspread system. It is here known [8], [11], [47] that TF shifted versions of a well-TF localized function are approximate eigenfunctions of , and the WS values are the corresponding approximate eigenvalues, i.e.,
In the previous section, we have shown that the time-varying Wiener and projection filters can be reformulated in the TF doand are main if the nonstationary random processes jointly underspread. We will now show that this TF formulation of optimal filters leads to simple design procedures that operate in the TF domain and yield nearly optimal filters. A. Time–Frequency Pseudo-Wiener Filter We recall that for jointly underspread processes, an approximate expression for the Wiener filter's WS is given by (23). Let by setting us now define another linear, time-varying filter its WS equal to the right-hand side of (23) [50]:
(26) Hence, the optimal pass space (spanned by all eigenfuncwith ) corresponds to the TF region tions (comprising the TF locations of all such that ). The action of —passing signals in and suppressing signals in —can therefore be reformulated in the TF plane as passing signals in and suppressing signals outside . Thus, the TF region passes signal components in the TF rewe conclude that gion where the local TF SNR is larger than one and suppresses signal components in the complementary TF region, which is intuitively reasonable. performs a projection onto The optimal projection filter the space ; for jointly underspread signal and noise, this space is a “simple” space as defined in [42]. The WS of the projection operator on a simple space essentially assumes the values 1 (on ) and 0 (on the TF stop region) the TF pass region, in our case can be approximated as6 [42]. Hence, the WS of (25) is the indicator function of . This extends where the relation (13) to general underspread processes. Note that the only allows to pass or suppress projection property of ; no intermediate type of attenuation (which components of between 0 and 1) is would correspond to values of possible.
L
6Experiments show that this approximation
is valid apart from oscillations of
(t; f ) about the values 0 or 1. Hence, the approximation can be improved by a slight smoothing of L (t; f ), which corresponds to the removal of over-
spread components from
P
(cf. Section IV-A).
as the TF pseudo-Wiener filter. For jointly underWe refer to , , where (23) is a good approximation, spread processes , a combination of (26) and (23) yields . Hence, the TF pseudo-Wiener filter is and thus, of the Wiener a close approximation to the underspread part and, therefore, is nearly optimal; furthermore, the TF filter (see Section IV-A) applies equally well to interpretation of . The deviation from optimality is characterized by the error bounds in (20)–(22). For processes that are not jointly undermust be expected to perform poorly. Note spread, however, is a self-adjoint operator since according to (26), the that is real-valued. WS of For jointly strictly underspread processes, i.e., underspread processes whose support rectangle is oriented parallel to the and axes [8], [29], the WVS occurring on the right-hand side of (26) can be replaced by the generalized WVS [17]–[19], the (generalized) evolutionary spectrum [23], [24], [28], [29], or the physical spectrum (expected spectrogram) using an appropriate analysis window [18], [19], [33]. This is possible since in the strictly underspread case, these spectra become nearly equivalent [8], [9], [12], [19], [29]. is defined in the TF While the TF pseudo-Wiener filter domain, the actual calculation of the signal estimate can be performed directly in the time domain according to
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where , which is the impulse response of obtained from the WS in (26) as [cf. (15)]
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, can be
(27) An efficient approximate multiwindow STFT implementation will be discussed in Section VI. , the TF pseudo-Wiener Compared with the Wiener filter possesses two practical advantages: filter • Modified a priori knowledge: Ideally, the calculation requires knowledge of the correlation (design) of and [see (5)], whereas the design of operators requires knowledge of the WVS and [see (26)]. Although correlation operators and WVS are mathematically equivalent due to the one-to-one mapping (16), the WVS are much easier and more intuitive to handle than the correlation operators or correlation functions since WVS have a more immediate physical significance and interpretation. In practice, the quantities constituting the a priori knowledge (correlation operators or WVS) are usually replaced by estimated or schematic/idealized versions, which are much more easily controlled or designed in the TF domain than in the correlation domain. For example, an approximate or partial knowledge of the WVS will often suffice for a reasonable filter design. • Reduced computation: The calculation (design) of requires a computationally intensive and potentially unstable operator inversion (or, in a discrete-time setting, a matrix inversion). In the TF design (26), this operator inversion is replaced by simple and easily controllable pointwise divisions of functions. Assuming discrete-time signals of length , the computational cost of the degrows with , whereas that of (using sign of . divisions and FFT’s) grows only with The TF pseudo-Wiener filter satisfies an approximate “TF opjointly underspread, it can be shown that timality.” For the MSE obtained with any underspread system is approximately given by
(28) (assuming a minimal solution, i.e., Minimizing for ) then yields the in (26) and, thus, the . This interpretation suggests an exTF pseudo-Wiener filter tended TF design that is based on the following weighted MSE [extending (12)] with smooth TF weighting function
(29) . The filter minimizing this with reweighted TF error is given by (26) with and replaced by placed by .
B. Time–Frequency Pseudo-Projection Filter Next, we consider a TF filter design that is motivated by the approximate expression (25) for the WS of . Let us define a new linear system by setting its WS equal to the right-hand side 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 noise energy and 0 elsewhere. It is interesting that the WS of is a rounded version of the WS of the TF pseudo-Wiener filter in (26), i.e., round , which is consisdiffers from tent with the stationary case. Furthermore, only on the signal-plus-noise TF region ; in , we have the TF pass region , and in the TF stop region outside , there is . In general, is not exactly an orthogonal projection operator. and , where However, for jointly underspread processes , (25) is a good approximation, there is . Hence, the TF-designed filter is a reasonand thus, , and we able approximation to the optimal projection filter will thus call it TF pseudo-projection filter. However, for processes that are not jointly underspread, must be expected to perform poorly and to be very different from an orthogonal projection operator. Although is defined in the TF domain, the actual calculation of the signal estimate can be done in the time domain using as (cf. the impulse response of that is derived from (27)) (30) An efficient approximate multiwindow STFT implementation will be discussed in Section VI. , the TF Compared with the optimal projection filter pseudo-projection filter has two advantages: • Modified/reduced a priori knowledge: The design of requires knowledge of the space that is spanned by the eigenfunctions of the operator corresponding to positive eigenvalues (see Section II-B), merely presupposes that we whereas the design of in which the signal dominates know the TF region the noise. This knowledge is of a much simpler and more intuitive form, thus facilitating the use of approximate or partial information about the WVS. requires the so• Reduced computation: The design of lution of an eigenvalue problem, which is computationally intensive. In contrast, the proposed TF design only requires an inverse WS transformation [see (30)]. Assuming discrete-time signals of length , the computational cost grows with , whereas that of of the design of (using FFT’s) grows with . is furthermore advantaThe TF pseudo-projection filter or since geous as compared with the Wiener-type filters
HLAWATSCH et al.: TIME-FREQUENCY FORMULATION, DESIGN, AND IMPLEMENTATION OF TIME-VARYING SIGNAL ESTIMATORS
its design is less expensive and more robust (especially with respect to errors in estimating the a priori knowledge and with respect to potential correlations of signal and noise). Similar to the TF pseudo-Wiener filter, the TF pseudo-projection filter satisfies a “TF optimality” in that it minimizes the in (28) under the constraint of a -valued WS, TF MSE . Minimizing the TF weighted MSE i.e., in (29) rather than , again under the constraint , results in a generalized TF pseudo-projection filter that is given by , where . C. Time–Frequency Projection Filter For jointly underspread processes, the TF pseudo-projection approximates the optimal projection filter , but it filter is not exactly an orthogonal projection operator. If an orthogonal projection operator is desired, we may calculate the orthogonal projection operator that optimally approximates in the sense of minimizing . This can be shown to be equivalent to minimizing [42], [51]. That is, the TF designed projection filter (hereafter denoted by ) is the orthogonal projection op. erator whose WS is closest to the indicator function is given by the orthogonal It is shown in [42] and [51] that span projection operator on the space spanned by all eigenfunctions of with associated eigen, i.e., values with For jointly underspread processes where is close to an oris close to and, hence, also thogonal projection operator, close to the optimal projection filter . However, due to the inmay lead to a larger MSE than herent projection constraint, . In addition, the derivation of from requires the solution of an eigenproblem. Therefore, the TF projection filter appears to be less attractive than the TF pseudo-projection filter . VI. MULTIWINDOW STFT IMPLEMENTATION Following [8] and [20], we now discuss a TF implementation of the Wiener and projection filters that is based on the multiwindow short-time Fourier transform (STFT) and that is valid in the underspread case. A simple TF filter method is an STFT filter that consists of the following steps [8], [13], [20], [52]–[57]: • STFT analysis: Calculation of the STFT [19], [54], [55], [58] of the input signal
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 to STFT in a least-squares sense [57]. These steps implement a linear, time-varying filter (hereafter and . denoted ) that depends on We recall from Section IV-A that for jointly underspread proof the Wiener filter is nearly cesses, the underspread part for , it follows that optimal. Since
where is any real-valued function that is 1 on [a “minis the indicator function , cf. imal” choice for (19)]. Let denote the linear system defined by . If is chosen such that , is a self-adjoint operator with real-valued eigenvalues and . It is shown in [8] and [20] orthonormal eigenfunctions can be expanded as that (31) are STFT filters with TF weighting function and STFT windows . Thus, is represented as a weighted sum of STFT filters with identical and orthonormal windows TF weighting function . In practice, the eigenvalues often decay quickly so that it suffices to use only a few terms of (31) corresponding to the largest eigenvalues (an example will be presented in are arranged in Section VII). Assuming that the eigenvalues nonincreasing order, we then obtain where the
This yields the approximate multiwindow STFT implementation that is depicted in Fig. 3. It can be shown [8] that the of is bounded as approximation error
STFT Here, where is a normalized window. • STFT weighting: Multiplication of the STFT by a weighting function STFT
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STFT
which can be used to estimate the appropriate order . Since and are assumed jointly underspread, (23) holds, and
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we obtain the following simple approximation to the STFT : weighting function (32) A particularly efficient discrete-time/discrete-frequency version of the multiwindow STFT implementation that uses filter banks is discussed in [8]. Furthermore, it is shown in [8] and , the STFT’s used in the [20] that in the case of white noise multiwindow STFT implementation can additionally be used to , which is required to calculate estimate the WVS of according to (32). A heuristic, approximate TF implementation of time-varying Wiener filters proposed in [31] and [32] uses a single STFT filter ) with TF weighting function (i.e.,
where and are the physical spectra [18], [19], and , respectively. Compared with the mul[33] of tiwindow STFT implementation discussed above, this suffers from 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 (which ). Here, the TF approximates the optimal projection filter weighting function is
Since is not exactly underspread, the function defining the windows and the coefficients must be , and chosen such that it covers the effective support of hence (33) can be constructed from Subsequently, the and as explained above. The resulting multiwindow STFT filter leads to an approximation error that can be bounded as
with . This bound is a measure of the exand will thus be small in the underspread tension of even for since case. However, (33) is only an approximation. VII. NUMERICAL SIMULATIONS This section presents numerical simulations that illustrate and corroborate our theoretical results. Using the TF synthesis tech-
Fig. 4. Second-order statistics of signal and noise. (a) WVS of s(t). (b) WVS of n(t). (c) Expected ambiguity function of s(t). (d) Expected ambiguity function of n(t). The rectangles in parts (c) and (d) have area 1 and thus allow assessment of the underspread property of s(t) and n(t). The signal length is 128 samples.
nique introduced in [59], we synthesized random processes and with expected energies and , respectively. Thus, the mean input SNR is SNR dB. The WVS and expected ambiguity functions of and are shown in Fig. 4. From the expected ambiguity functions, it is seen that the processes are jointly underspread. The , its underspread part , and the WS’s of the Wiener filter are shown in Fig. 5(a)–(c). The TF TF pseudo-Wiener filter pass, stop, and transition regions of the filters are easily recogis a smoothed version of nized. It is verified that the WS of and that the WS of closely approximates that the WS of . Fig. 5(d)–(f) compare the WS’s of the optimal projection of filter , TF pseudo-projection filter , and TF projection filter . It is seen that the WS’s of these filters are similar, except for small-scale oscillations, and that the WS of is a rounded . version of the WS of SNR SNR The mean SNR improvements SNR (where SNR with ) achieved with the Wiener-type and projection-type filters are listed in is Table I. The performance of the TF pseudo-Wiener filter . Similarly, seen to be very close to that of the Wiener filter the performance of the TF pseudo-projection filter is close . In fact, performs to that of the optimal projection filter and the TF projection filter , even slightly better than both
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H
Fig. 5. WS’s of (a) Wiener filter . (b) Underspread part pseudo-projection filter ~ . (f) TF projection filter .
P
P
TABLE I MEAN SNR IMPROVEMENT ACHIEVED BY WIENER-TYPE AND PROJECTION-TYPE FILTERS
H
of Wiener filter. (c) TF pseudo-Wiener filter
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H . (d) Optimal projection filter P . (e) TF
which can be attributed to the orthogonal projection constraint and (see Section V-C). underlying The multiwindow STFT implementation discussed in Secapproximated according to (32), is contion VI, with sidered in Fig. 6. The signal and noise processes are as before. Fig. 6(a)–(c) show the WS’s of the multiwindow STFT using filter order and . For growing filters becomes increasingly similar to the Wiener filter or the TF pseudo-Wiener filter [cf. Fig. 5(a)–(c)]. Fig. 6(d) shows depends how the mean SNR improvement achieved with is seen on . Although the single-window STFT filter
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Fig. 6. Multiwindow STFT implementation of the Wiener filter. (a)–(c) WS’s of multiwindow STFT filters with filter order K improvement versus STFT filter order K .
=1 4 ;
;
and 10. (d) Mean SNR
H
Fig. 7. Filtering experiment involving an overspread noise process. (a) WVS of s(t). (b) WVS of n(t). (c) WS of Wiener filter . (d) Expected ambiguity function of s(t). (e) Expected ambiguity function of n(t). (f) WS of TF pseudo-Wiener filter . The rectangles in parts (d) and (e) have area 1 and thus allow assessment of the under-/overspread property of s(t) and n(t). (In particular, part (e) shows that n(t) is overspread.) The signal length is 128 samples.
H
to result in a significant performance loss, a modest filter order already yields practically optimal performance. An experiment in which the underspread assumption is viis again underspread olated is shown in Fig. 7. The signal (and, in addition, reasonably quasistationary), but the noise is not underspread—it is reasonably quasistationary, but its temporal 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 pseudois significantly different from the Wiener filter Wiener filter . The heavy oscillations of the WS of in Fig. 7(c) inis far from being an underspread system. The dicate that and were constructed such that they lie in processes is an linearly independent (disjoint) signal spaces. Here, , oblique projection operator [44] that perfectly reconstructs thereby achieving zero MSE and infinite SNR improvement. On the other hand, due to the significant overlap of the WVS of
and , the TF pseudo-Wiener filter merely achieves an SNR improvement of 4.79 dB. This example is important and is since it shows that mere quasistationarity of not sufficient as an assumption permitting a TF design of optimal filters. It also shows that in certain cases—specifically, when signal and noise have significantly overlapping WVS but belong to linearly independent signal spaces—the TF designed filter performs poorly, even though the optimal Wiener filter achieves perfect signal reconstruction. We stress that this situation implies that the processes are overspread (i.e., not underspread [12]), thereby prohibiting beforehand the successful use of TF filter methods. Finally, we present the results of a real-data experiment illustrating the application of the TF pseudo-Wiener filter to speech enhancement. The speech signal used consists of 246 pitch periods of the German vowel “a” spoken by a male speaker. This
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Fig. 8. Speech enhancement experiment using TF pseudo-Wiener filter. (a) WS of TF pseudo-Wiener filter . (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) Wigner distributions [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 emphasize higher frequency content. The resulting discrete-time signal was split into 123 segments of length 256 samples (corresponding to 2.32 ms), each of which contains two pitch periods. Sixty four of these segments were considered as individual realizations of a nonstationary process of length 256 samples and were used to estimate the WVS of this process. Furthermore, we used a stationary AR noise process that provides a reasonable model for car noise. Due to stationarity, the noise WVS equals the PSD for which a closed-form expression exists; hence, there was no need to estimate the noise WVS. Using the estimated signal WVS and the exact noise WVS, the TF pseudo-Wiener was then designed according to (26). The WS of filter is shown in Fig. 8(a). The remaining 59 speech signal segments were corrupted by 59 different noise realizations and input to the TF . The input SNR (averaged over the pseudo-Wiener filter dB, and the average SNR of 59 realizations) was SNR the enhanced signal segments obtained at the filter output was dB, corresponding to an average SNR improveSNR ment of about 6.7 dB. Results for one particular realization are shown in Fig. 8(b)–(d). VIII. CONCLUSION We have developed a time-frequency (TF) formulation and an approximate TF design of optimal filters that is valid for underspread, nonstationary random processes, i.e., nonstationary processes with limited TF correlation. We considered two types of optimal filters: the classical time-varying Wiener filter and a projection-constrained optimal filter. The latter will produce a larger estimation error but has practically important advantages concerning robustness and a priori knowledge. The TF formulation of optimal filters was seen to allow an intuitively appealing interpretation of optimal filters in terms of passing, attenuating, or suppressing signal components located in different TF regions. The approximate TF design of optimal filters is practically attractive since it is computation-
ally efficient and uses a more intuitive form of a priori knowledge. We furthermore discussed an efficient TF implementation of time-varying optimal filters using multiwindow STFT filters. Our TF formulation and design was based on the Weyl symbol (WS) of linear, time-varying systems and the WVS of nonstationary random processes. However, if the processes satisfy a more restrictive type of underspread property (if they are strictly underspread [8], [29]), then our results can be extended in that the WS can be replaced by other members of the class of generalized WSs [48] (such as Zadeh's time-varying transfer function [25]), and the WVS can be replaced by other members of the class of generalized WVS [18], [19] or generalized evolutionary spectra [29]. This extension is possible since for strictly underspread systems, all generalized WS’s are essentially equivalent and for strictly underspread processes, all generalized WVS and generalized evolutionary spectra are essentially equivalent [8], [9], [12], [19], [29], [47]. APPENDIX A UNDERSPREAD APPROXIMATION OF THE WIENER FILTER: PROOF OF THE BOUNDS (20) AND (21) and We assume that the expected ambiguity functions of are exactly zero outside the same (possibly obliquely oriented) centered rectangle with area (joint correlation spread) . We first prove the bound (20). Using and , the approximation error can be developed as
(34)
We will now show that that is oriented as rectangle
is zero outside the centered but has sides that are twice as
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long. Using
, it can be shown [13] that , where
Furthermore, fore obtain
tr
according to (6). We there-
tr tr
(35) and is known as the twisted convolution of [13]. In the domain, corresponds , which can be rewritten as to . With , we obtain (36)
tr
tr tr
tr tr
The first term is zero, i.e., tr , since outside . The second term can be bounded by applying Schwarz' inequality: tr Using (38), the upper bound in (21) follows.
The twisted convolution [cf. (35)] implies that
APPENDIX B TF FORMULATION OF THE WIENER FILTER: PROOF OF THE BOUND (22) We consider the approximation error and, hence, that the support of is confined to [recall that both and are confined to ]. is confined to The crucial observation now is that since and is confined to , (36) implies that must also be confined to since any contrioutside cannot be canceled by butions of and, thus, would contradict (36). Hence, we can write (34) as
Subtracting and adding
, this can be rewritten as with
According to the triangle inequality, the norm of is . It is shown in [8], [11], and bounded as is bounded as [46] that
(37) Applying the Schwarz inequality to (35) yields the bound . is Inserting this into (37) and using the fact that the area of (i.e., four times the area of ), we obtain (20):
where we have used bounded as
. Furthermore,
is
(38) We next prove the bound (21). The lower bound is trivial since is the minimum possible MSE. Let us concan sider the upper bound. With (3), the MSE achieved by be written as tr tr tr tr
tr
where we have used (38). Inserting these two bounds into gives (22). REFERENCES [1] N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Cambridge, MA: MIT Press, 1949. [2] T. Kailath, Lectures on Wiener and Kalman Filtering. Wien, Austria: Springer, 1981. [3] A. Papoulis, Probability, Random Variables, and Stochastic Processes. New York: McGraw-Hill, 1984. [4] L. L. Scharf, Statistical Signal Processing. Reading, MA: Addison Wesley, 1991. [5] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1992.
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[6] H. V. Poor, An Introduction to Signal Detection and Estimation. New York: Springer, 1988. [7] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I: Detection, Estimation, and Linear Modulation Theory. New York: Wiley, 1968. [8] W. Kozek, “Matched Weyl-Heisenberg Expansions of Nonstationary Environments,” Ph.D. dissertation, Vienna Univ. Technol., Vienna, Austria, Mar. 1997. [9] W. Kozek, F. Hlawatsch, H. Kirchauer, and U. Trautwein, “Correlative time-frequency analysis and classification of nonstationary random processes,” in Proc. IEEE-SP Int. Symp. Time-Frequency Time-Scale Anal., Philadelphia, PA, Oct. 1994, pp. 417–420. [10] W. Kozek, “On the underspread/overspread classification of nonstationary random processes,” in Proc. Int. Conf. Ind. Appl. Math., K. Kirchgässner, O. Mahrenholtz, and R. Mennicken, Eds. Berlin, Germany, 1996, pp. 63–66. , “Adaptation of Weyl-Heisenberg frames to underspread environ[11] ments,” in Gabor Analysis and Algorithms: Theory and Applications, H. G. Feichtinger and T. Strohmer, Eds. Boston, MA: Birkhäuser, 1998, ch. 10, pp. 323–352. [12] G. Matz and F. Hlawatsch, “Time-varying spectra for underspread and overspread nonstationary processes,” in Proc. 32nd Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, Nov. 1998, pp. 282–286. [13] G. B. Folland, Harmonic Analysis in Phase Space. Princeton, NJ: Princeton Univ. Press, 1989. [14] A. J. E. M. Janssen, “Wigner weight functions and Weyl symbols of non-negative definite linear operators,” Philips J. Res., vol. 44, pp. 7–42, 1989. [15] W. Kozek, “Time-frequency signal processing based on the Wigner-Weyl framework,” Signal Process., vol. 29, pp. 77–92, Oct. 1992. [16] R. G. Shenoy and T. W. Parks, “The Weyl correspondence and time-frequency analysis,” IEEE Trans. Signal Processing, vol. 42, pp. 318–331, Feb. 1994. [17] W. Martin and P. Flandrin, “Wigner–Ville spectral analysis of nonstationary processes,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-33, pp. 1461–1470, Dec. 1985. [18] P. Flandrin and W. Martin, “The Wigner–Ville spectrum of nonstationary random signals,” in The Wigner Distribution—Theory and Applications in Signal Processing, W. Mecklenbräuker and F. Hlawatsch, Eds. Amsterdam, The Netherlands: Elsevier, 1997, pp. 211–267. [19] P. Flandrin, Time-Frequency/Time-Scale Analysis. San Diego, CA: Academic, 1999. [20] W. Kozek, H. G. Feichtinger, and J. Scharinger, “Matched multiwindow methods for the estimation and filtering of nonstationary processes,” in Proc. IEEE ISCAS, Atlanta, GA, May 1996, pp. 509–512. [21] N. A. Abdrabbo and M. B. Priestley, “Filtering non-stationary signals,” J. R. Stat. Soc. Ser. B., vol. 31, pp. 150–159, 1969. [22] J. A. Sills, “Nonstationary signal modeling, filtering, and parameterization,” Ph.D. dissertation, Georgia Inst. Technol., Atlanta, Mar. 1995. [23] J. A. Sills and E. W. Kamen, “Wiener filtering of nonstationary signals based on spectral density functions,” in Proc. 34th IEEE Conf. Decision Contr., Kobe, Japan, Dec. 1995, pp. 2521–2526. [24] H. A. Khan and L. F. Chaparro, “Nonstationary Wiener filtering based on evolutionary spectral theory,” in Proc. IEEE ICASSP, Munich, Germany, May 1997, pp. 3677–3680. [25] L. A. Zadeh, “Frequency analysis of variable networks,” Proc. IRE, vol. 76, pp. 291–299, Mar. 1950. [26] P. A. Bello, “Characterization of randomly time-variant linear channels,” IEEE Trans. Commun. Syst., vol. COMM-11, pp. 360–393, 1963. [27] M. B. Priestley, “Evolutionary spectra and nonstationary processes,” J. R. Stat. Soc. Ser. B., vol. 27, no. 2, pp. 204–237, 1965. , Spectral Analysis and Time Series—Part II. London, U.K.: Aca[28] demic, 1981. [29] G. Matz, F. Hlawatsch, and W. Kozek, “Generalized evolutionary spectral analysis and the Weyl spectrum of nonstationary random processes,” IEEE Trans. Signal Processing, vol. 45, pp. 1520–1534, June 1997. [30] A. A. Beex and M. Xie, “Time-varying filtering via multiresolution parametric spectral estimation,” in Proc. IEEE ICASSP, Detroit, MI, May 1995, pp. 1565–1568. [31] P. Lander and E. J. Berbari, “Enhanced ensemble averaging using the time-frequency plane,” in Proc. IEEE-SP Int. Symp. Time-Frequency Time-Scale Anal., Philadelphia, PA, Oct. 1994, pp. 241–243. [32] A. M. Sayeed, P. Lander, and D. L. Jones, “Improved time-frequency filtering of signal-averaged electrocardiograms,” J. Electrocardiol., vol. 28, pp. 53–58, 1995.
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[58] F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic timefrequency signal representations,” IEEE Signal Processing Mag., vol. 9, pp. 21–67, Apr. 1992. [59] F. Hlawatsch and W. Kozek, “Second-order time-frequency synthesis of nonstationary random processes,” IEEE Trans. Inform. Theory, vol. 41, pp. 255–267, Jan. 1995. [60] T. A. C. M. Claasen and W. F. G. Mecklenbräuker, “The Wigner distribution—A tool for time-frequency signal analysis; Parts I–III,” Philips J. Res., vol. 35, pp. 217–250; 276–300; 372–389, 1980. [61] W. Mecklenbräuker and F. Hlawatsch, Eds., The Wigner Distribution—Theory and Applications in Signal Processing. Amsterdam, The Netherlands: Elsevier, 1997.
Franz Hlawatsch (S’85–M’88) received the Dipl.-Ing., Dr.techn., and Univ.-Dozent degrees in electrical engineering communications engineering/signal processing from the Vienna University of Technology, Vienna, Austria, in 1983, 1988, and 1996, respectively. Since 1983, he has been with the Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology. From 1991 to 1992, he spent a sabbatical year with the Department of Electrical Engineering, University of Rhode Island, Kingston. He authored the book Time-Frequency Analysis and Synthesis of Linear Signal Spaces—Time-Frequency Filters, Signal Detection and Estimation, and Range-Doppler Estimation (Boston, MA: Kluwer, 1998) and co-edited the book The Wigner Distribution—Theory and Applications in Signal Processing (Amsterdam, The Netherlands: Elsevier, 1997). His research interests are in signal processing with emphasis on time-frequency methods and communications applications.
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Gerald Matz (S’95) received the Dipl.-Ing. degree in electrical engineering from the Vienna University of Technology, Vienna, Austria, in 1994. Since 1995, he has been with the Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology. His research interests include the application of time-frequency methods to statistical signal processing and wireless communications.
Heinrich Kirchauer was born in Vienna, Austria, in 1969. He received the Dip.-Ing. degree in communications engineering and the doctoral degree in microelectronics engineering from the Vienna University of Technology, Vienna, Austria, in 1994 and 1998, respectively. In the summer of 1997 , he held a visiting research position at LSI Logic, Milpitas, CA. In 1998, he joined Intel Corporation, Santa Clara, CA. His current interests are modeling and simulation of problems for microelectronics engineering with special emphasis on lithography simulation.
Werner Kozek (M’94) received the Dipl.-Ing. and Ph.D. degrees in electrical engineering from the Vienna University of Technology, Vienna, Austria, in 1990 and 1997, respectively. From 1990 to 1994 he was with the Institute of Communications and Radio-Frequency Engineering, Vienna University of Technology, and from 1994 to 1998, 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 theoretic aspects of digital communication systems.