W. Kozek and H.G. Feichtinger. NUHAG, Dept. of Mathematics, University of Vienna. J. Scharinger. Altenberger Str. 69, A-4040 Linz, Austria ABSTRACT

Transcription

1 GABOR ANALYSIS AND LINEAR SYSTEM IDENTIFICATION 3 W. Kozek and H.G. Feichtinger NUHAG, Dept. of Mathematics, University of Vienna Strudlhofg.4, A{1090 Vienna, Austria ( kozek@tyche.mat.univie.ac.at) J. Scharinger Institute of Systems Science, Johannes Kepler University Linz Altenberger Str. 69, A-4040 Linz, Austria ( js@cast.uni-linz.ac.at) ABSTRACT Many practically important linear systems are slowly time{varying. In this paper we consider the frequency{domain identication of slowly time{varying systems with limited impulse response (underspread systems). We review time{frequency representations of LTV systems and its connection to the Gabor expansion of signals. It is shown how the matched multiwindow Gabor expansion can be used for the optimum frequency{domain identication and canonical realization of underspread LTV systems. 1. INTRODUCTION Nonparametric identication of linear time{varying (LTV) systems is a problem of particular relevance for communication engineering. For instance, LTV wave propagation channels are characterized by known partial dierential equations but highly variable and unstructured side constraints. Moreover, in control engineering a nonparametric approach may be appropriate in case of modelling uncertainty as a rst step towards model selection. Linear system theory is dominated by the treatment of the time{invariant case. For LTI systems the frequency domain characterization (transfer function) is matched in the sense that the "basis" functions of the Fourier transform are eigenfunctions of the system. The eigenfunctions of a general LTV system are a priori unknown, but in case of slow time{variation one may assume that windowed sinusoids are approximate eigenfunctions of the system. According to this assumption one characterizes the LTV system by a time{varying transfer function, i.e., a joint function of time and frequency. System identication can then 3 Funding by the FWF grant S7001-MAT and OENB grant 4913.

2 be performed by the observation of windowed versions of the input and output signal leading to point estimates of the transfer function. When the input signal is approximately white noise over some frequency band of interest, then one can obtain a global estimate of the transfer function for this band. Obviously, such a procedure cannot work for a general LTV systems with arbitrarily fast time{ variations. However, detailed theoretical results in the sense of a bias/variance analysis seem to exist only for the stationary case. In this paper we report about recent results on the frequency{domain identication of LTV systems. A brief review of nonparametric LTV system theory is followed by a discussion of the Gabor expansion and its application to the realization of underspread systems. We present explicit bias and variance results for the transfer function estimation and show that optimum estimators can be formulated in terms of Gabor expansions of the input/output signal. 2. LTV SYSTEM REPRESENTATION We consider LTV systems which act as integral operators on L 2 {signals. The input{output relation is given by 1 : (Hx) (t) := h(t; )x(t 0 )d = k H (t; s)x(s)ds (1) where k H (t; s) is the kernel (impulse response) and h(t; ) := k H (t; t 0 ) is the delay{spread function [10]. For engineering applications, the delay{spread function is often advantageous since it depends separately on absolute time t and time delay. I.e., for an LTI system h(t; ) is invariant w.r.t. t. Apart from limit case considerations we assume square{integrable kernels that correspond to Hilbert{Schmidt (HS) operators [7] with an inner product hg; Hi := k G (t; s)k H (t; s)dt ds (2) and associated HS norm khk 2 := hh; Hi (overline denotes complex conjugation). Spreading Function. The spreading function of an LTV system is dened as the Fourier transform of h(t; ) w.r.t. absolute time [10]: S H (; ) := h(t; )e 0j2t dt; (3) it is generally complex{valued even in case of a real{valued impulse response. Slowly time{varying systems are characterized by a limited {support of S H (; ). In a more symmetric way, we can interpret S H (; ) as time{frequency{ shifting representation of an LTV system in the sense that the output signal is 1 All sums and integrals go from 01 to 1 unless otherwise stated.

3 given by a superposition of time{shifted (delayed) and frequency{shifted (modulated) versions of the input signal: (Hx) (t) = S H (; )x(t 0 )e j2t d d; (4) For later use we introduce a time{shift operator (T x) (t) := x(t 0 ); and a frequency shift operator: (M x) (t) := x(t)e j2t : This allows to reformulate (3) in a signal free formulation H = S H (; )M T d d; (5) and we can formally dene the spreading function via an inner product S H (; ) = hh ; M T i : (6) The spreading function is a unique system representation since the map h(t; ) 7! S H (; ) is unitary. Time{Varying Transfer Function. The spreading function characterizes the energy preserving eects, time delay and frequency shift, in a pointwise clear manner. However, for the study of ltering eects of an LTV system it is more natural to represent the system by a time{varying transfer function. Such a transfer function can be dened by a Fourier transform w.r.t. the lag variable [11]: H (t; f) := h(t; )e 0j2f d; (7) which reduces to the usual time{independent transfer function in case of an LTI system. The time{varying transfer function is the symplectic Fourier transform of the spreading function: H (t; f) = S H (; )e 0j2(f 0t) d d: (8) In order to formulate the system identication problem in a compact manner we introduce time{frequency shifting of operators by P(t; f) := M f T t PT (0t) M (0f ) ; (9) i.e., P(t; f) is unitarily equivalent to P and its eigensignals are time{frequency{ shifted versions of that of P. With other words, when P selects signals temporally and spectrally localized about zero, then P(t; f) selects signals temporally and spectrally localized at (t; f). The transfer function of P(; ) gives a clear picture: P (;) (t; f) = P (t 0 ; f 0 ) (10) The kernel of P(; ) is given by: (P(; )) (t; s) = k P (t0; s0)e j2(t0s) ; when k P (t; s) is the kernel of P.

4 With this notational convention we can introduce a time{frequency{parametrized system representation dual to (5) (continuous Weyl{Heisenberg expansion of an LTV system) [4]: H = H (t; f)p 0 (t; f)dt df (11) where P 0 is a \prototype" operator dened by H (t; f) = hh; P 0 (t; f)i () P0 (t; f) = (t)(f): (12) The label time{varying transfer function suggests that H (t; f) reects time{ frequency{selective ltering in a manner just as the time{independent LTI system's transfer function characterizes frequency{selective ltering. However, this interpretation would require that the prototype operator P 0 (t; f) performs a perfect time{frequency localization in obvious contradiction to Heisenberg's uncertainty principle. Hence, one may already expect that for a general LTV system H (t; f) does not have much practical relevance. 3. UNDERSPREAD SYSTEMS Slowly time{varying systems with limited time delay can be characterized by a compactly supported spreading function ( [a;b] (:) denotes the indicator function of the interval [a; b] on the real line): S H (; ) = S H (; ) [00 ; 0 ]() [00 ; 0 ]() with H = ; (13) where 0 characterizes the maximum frequency shift (proportional to the maximum velocity of time{variations) and 0 characterizes the maximum time shift. Systems with H 1 will be called underspread [4]. An important practical example is the mobile radio propagation channel [8]. Transfer Function Calculus. LTI systems can be viewed as a limit case of underspread systems with 0! 0. Moreover, one can show that all of the well{ known properties of an LTI system's transfer function hold approximately for the time{varying transfer function of an underspread system [6]. Here, we mention only one important property of the transfer function for two jointly underspread (sub)systems G, H: the transfer function of the composite system is essentially given by the product of the subsystems' transfer functions 2 k GH 0 G H k 2 < c 3 H kgk2 khk 2 ; (14) where c is a system independent constant. Two LTI systems are jointly underspread with H = 0 such that (14) implies the well known identity: GH (f) = G (f) H (f): We emphasize that for overspread systems ( H > 1) the error bound in (14) gets dominant, and, more generally, the time{varying transfer 2 Jointly underspread means that both systems satisfy the same spreading constraint.

5 function loses all of the important properties of the LTI system's transfer function. Discrete Weyl{Heisenberg Expansion. According to (8) the transfer function of an underspread system is a 2D lowpass function. Hence, using the sampling theorem of 2D functions [1] and (10) one can formulate a discretized version of (11) based on a dierent prototype operator: H = 1 X H k X l H (kt; lf )P (kt; lf ) (15) with the sampling grid and the prototype operator matched to the underspread support as follows T F = 0 0 and T F = 1 H ; (16) S P (; ) = [00 ; 0 ]() [00 ; 0 ]() () H (t; f) = hh; P (t; f)i (17) The validity of a discrete expansion makes the transfer function calculus to a feasible framework for signal processing or control engineering problems where one needs to compute composite systems or inverse systems in a numerically ecient way. 4. GABOR EXPANSION The time{varying transfer function is a joint function of time and frequency. This suggests that time{frequency signal representations will play an important role in the realization of transfer function estimators that are based on input{output observations. In this section we give a brief review of linear time{frequency signal representations and their application to LTV system realization. Short{Time Fourier Transform. The short{time Fourier transform (STFT) of a signal x(t) is dened as [9, 3] (ST F T x) (t; f) := x(s)(s 0 t)e 0j2f s ds = hx; M f T t i ; where (t) is the analysis window. window and is highly redundant. The STFT depends signicantly on the Gabor Expansion. The Gabor expansion of a signal x(t) is dened as [3, 12] X X x(t) = (G x) [k; l] (M lf T kt g) (t) (18) k l where the coecients can be dened as a sampled STFT: (G x) [k; l] := (ST F T x) (kt; lf ); (19)

6 and g(t) is called synthesis window. Note that the validity of (18) imposes severe constraints on the choice of the analysis/synthesis windows [3, 12]. System Representation via Multiplicative Modication. Using our notation for time{frequency{shifted operators (see (9)) it is straightforward to write (18) in a signal{free formulation: I X X = P g; (kt; lf ); k l where I denotes the identity operator and (P g; ) (t; s) := g(t)(s). For a synthesis of LTV systems it is natural to consider multiplicative perturbations of this resolution of the identity [2]: H X X G = M G [k; l]p g; (kt; lf ); (20) k l where M G [k; l] is a multiplicator function that characterizes the time{frequency ltering behaviour of H G in a manner similar to a sampled transfer function. Eq.(20) is just a discrete version of multiplicative modication of the STFT [9] where the system is given by H ST F T = M(t; f)p g; (t; f)dt df: (21) Based upon appropriately selected analysis and synthesis windows one can show that any underspread system can be realized by such an STFT lter [6]. This does not hold true for the Gabor lters as dened by (20). One way to overcome the basic restrictions of Gabor lters is the introduction of multiwindow methods [5, 12]. Consider the singular value decomposition of the prototype operator in (15) 1X P = s m P um;vm (22) m=1 here s m are the singular values, u m (t); v m (t) are the singular signals, and (P um;vm ) (t; s) := u m (t)v m (s). Now, when we insert (22) in (15) comparison with (20) shows that we can realize any underspread system by multiplicative modication of a multiwindow Gabor expansion. The windows are the singular functions of the prototype operator thus determined by the spreading constraint (13), and the multiplicator function is given by the samples of H (t; f). A precise realization would require an innite number of windows. In practice, one has to truncate (22) in the sense of a nite{rank approximation leading to an approximation of (15) in the following way: NX X X fh = s m H (kt; lf )P um;vm (kt; lf ): (23) m=1 k l For a given N, this approximation is optimal (in a HS sense) when the indices of (22) correspond to nonincreasing order of the singular values.

7 5. TRANSFER FUNCTION ESTIMATION We now show that the above introduced prototype operator P also establishes an optimum estimator of the time{varying transfer function H (t; f) under convenient statistical assumptions. System Identication Setup. We assume an observation of (i) a circular complex, stationary white Gaussian input process and (ii) the corresponding output process contaminated by statistically independent, circular complex, stationary additive white Gaussian noise. We consider point estimation of the time{varying transfer function dened by its prototype operator P 0 in the form: H (t; f) = hh; P 0 (t; f)i For notational simplicity we assume an input process with normalized variance Efx(t)x(s)g = (t 0 s): According to the above assumptions, the output signal (Hx) (t) is a realization of a zero{mean, circular complex Gaussian nonstationary process. We assume a noiseless observation of the input signal x(t) and a noisy observation of the output signal: y(t) = (Hx) (t) + n(t); where n(t) is circular complex, zero{mean, stationary white Gaussian noise with covariance: Efn(t)n(s)g = n 2 (t 0 s): Estimator. The estimator is dened as a sesquilinear form based on a dierent prototype operator b P: b H (t; f) = D y; b P(t; f)xe Based on the statistical independence of x(t) and n(t) it is straightforward to show that the expectation of the estimate results in The bias is then given by: E n b H (t; f) o = D H; b P(t; f) E : B(t; f) := E n b H (t; f) o 0 H (t; f) = D H; e P(t; f) E ; where we have introduced a bias operator dened as: ep := b P 0 P 0 : There exist two essentially dierent strategies for unbiased estimation: (I) Unbiased estimators with P b = P 0. (II) Unbiased estimators with P b 6= P but nonoverlapping spreading functions of the bias operator and the system: S H (; )Se P (; ) 0: (24)

8 The following variance analysis will show that the latter is the practically relevant case. Variance. The variance of the estimator is dened as V 2 (t; f) := E b H (t; f) 2 0 E n bh (t; f) o 2 ; in general it depends on time and frequency. One can show [4] that the variance of the estimator is given by ( 3 denotes the adjoint operator): V 2 (t; f) = D HH 3 ; bp b P 3 (t; f) E + 2 nk b Pk 2 (25) Note that the constant term is proportional to the HS norm of the prototype operator. This shows that we can already exclude nite variance unbiased estimation of the time{varying transfer function of a general LTV system (without a priori knowledge) since the prototype operator P 0 (see (12)) is not HS. The variance is depends on time and frequency. However, in practice we would like to use one and the same point estimator for any point of the time{frequency plane. In order to optimize an unbiased time{frequency invariant estimator we have to consider global measures for the variance. First, we have a tight bound for the maximum variance: V 2 (t; f) k b Pk 2 khk2 + 2 n Second, we compute an integrated variance by integrating over the time{ frequency dependent term of (25): V 2 0 := t f V 2 (t; f) 0 2 nk b Pk 2 dt df = k b Pk 2 khk 2 Both the integrated and the maximum variance are proportional to the HS norm of b P. Hence, an optimum time{frequency independent unbiased estimator has to minimize the HS norm of its prototype operator without introducing bias. However, for underspread LTV systems with known spreading constraint this optimization is seen to be trivial when formulated in terms of Sb P (; ): The spreading function must be equal to one within the support of S P (; ) and arbitrary elsewhere (cf. (24)). Since the HS norm of b P is equal to the L 2 {norm of Sb P (; ), we have to minimize its support which just leads back to the prototype operator introduced in (15) bp opt = P : Note that our bias/variance analysis holds for arbitrary support of the system's spreading function. Estimator Realization via Gabor Analysis. For a practical (digital) implementation we have to reformulate the estimator in terms of the STFT of the :

9 input and observed output. To this end, note that our specic sesquilinear form can be written as a product of the STFTs of the output and input signal: hy; P g; (t; f)xi = hy; M f T t i hx; M f T t gi = (ST F T y) (t; f)(st F T g x) (t; f): Then, as discussed in the previous section we introduce a truncated singular value decomposition of the prototype operator such that a canonical estimator of the critically sampled transfer function is given by: b H (kt; lf ) = NX m=1 s m (G vm y) [k; l](g um x) [k; l]; i.e., up to a conjugation we have a product of the multiwindow Gabor coecients of the input signal and the output signal. In a digital implementation, this formulation is structurally simple and permits the use of FFTs. Observe that the estimator realization is structurally equivalent to the system realization via multiplicative modication of the Gabor expansion (see (23)). Consequently, one can use the Gabor coecients not only for the system identi- cation but also for a subsequent processing of the input or output signal. This makes our approach a candidate for adaptive signal processing matched to nonstationary environments. 6. CONCLUSIONS We have studied the frequency domain identication of linear time{varying (LTV) systems. The discussion has been restricted to underspread systems whose frequency{domain characterization achieves the essential properties of the frequency{domain characterization of LTI systems. We have pointed out the realization of such underspread systems via multiplicative modication of a multiwindow Gabor expansion. A canonical class of point estimators in terms of the input and the output signal has been introduced. Under the assumption of a white{noise input signal and observation of the output signal subject to additive white noise, one can derive closed form expressions for the bias and variance of the point estimate. We have then specialized to unbiased time{frequency{ invariant estimators and evaluated their performance in the sense of a maximum and integrated variance. The theoretically optimum estimator has been determined by its spreading function: It is an indicator function with the support given by the spreading constraint of the system at hand. These theoretical estimators can be approximately realized as the product of a matched multiwindow Gabor expansion of the known input signal and the observed output signal.

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