Performance Analysis of GEVD-Based Source Separation With Second-Order Statistics

Transcription

1 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 59, O. 0, OCTOBER Performance Analysis of GEVD-Based Source Separation With Second-Order Statistics Arie Yeredor Abstract One of the simplest (and earliest) approaches to blind source separation is to estimate the mixing matrix from the generalized eigenvalue decomposition (GEVD), or Exact Joint Diagonalization, of two target-matrices. In a second-order statistics (SOS) framework, these target-matrices are two different correlation matrices (e.g., at different lags, taken over different time-intervals, etc.), attempting to capture the diversity of the sources (e.g., diverse spectra, different nonstationarity profiles, etc.). More generally, such matrix pairs can be constructed as generalized correlation matrices, whose structure is prescribed by two selected associationmatrices. In this paper, we provide a small-errors performance analysis of GEVD-based separation in such SOS frameworks. We derive explicit expressions for the resulting interference-to-source ratio (ISR) matrix in terms of the association-matrices and of the sources temporal covariance matrices. The validity of our analysis is illustrated in simulation. Index Terms Blind source separation, exact joint diagonalization, generalized eigenvalue decomposition, independent component analysis, matrix pencil, perturbation analysis. I. ITRODUCTIO AD PROBLEM FORMULATIO The blind source separation (BSS) problem consists of separating unobserved source signals from their observed mixtures. The basic BSS paradigm involves a static, linear, square and noiseless, real-valued mixture model X = AS, in which S =[s s sk ] T is a K matrix containing the K unobserved source signals (each of length ) as its rows; A is the unknown K K mixing matrix (assumed to be nonsingular); and X = [x x xk ] T is the K matrix of K observed mixtures. The key to separation is usually some general statistical or structural information on the sources, the most common of which is the assumption of statistical independence thereof. Several popular extensions of this basic model involve additive noise and/or nonsquare mixture matrices. Perhaps one of the most conceptually appealing and computationally simple approaches (proposed in different contexts, e.g., by Tong et al. in AMUSE [], [3], by Cardoso in FOBI [], by Yeredor in CHESS [5], by Tomé in [6], by Parra and Sajda in [7] and more recently by Ollila et al. in DOGMA [8]), is to base the estimation of A on exact joint diagonalization (EJD) of two target-matrices, constructed from the observed mixtures X. Such a framework is sometimes also called a matrix-pencil approach [6] or a generalized eigenvalue decomposition (GEVD) approach [7]. The two K K target-matrices, denoted and, are usually empirical estimates of some true (unknown) matrices R and R satisfying R = AD A T and R = AD A T () Manuscript received February 0, 0; revised June 07, 0; accepted June 09, 0. Date of publication June 7, 0; date of current version September, 0. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Jerome Idier. A preliminary, partial version of this work was presented at the International Conference on Independent Component Analysis and Source Separation (ICA), Paraty, Brazil, March 5 8, 009. The author is with the School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel ( arie@eng.tau.ac.il). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 0.09/TSP where D and D are diagonal by virtue of the sources statistical independence. R is usually [7] (but not necessarily) the observations zero-lag correlation matrix. R is selected according to the sources statistical model: For example, for separation based only on non-gaussianity of the marginal distributions of the sources (ignoring their temporal structure, if any), R can be taken as a linear combination of certain cumulants matrices of the observations (e.g., [7], [], and [9]) or as differently-weighted covariance matrices, such as in DOGMA [8] or in [5] (where R is taken to be the Hessian of the observations joint log-characteristic function away from the origin). In many other cases of interest, when the diversity of the sources is exhibited through different temporal-correlation structures, both R and R can be based on second-order statistics (SOS). To define a general framework in this context, let P and P denote some arbitrary matrices, which we term association-matrices," and let the empirical generalized correlation matrices and be given by = XP X T = XP X T () [we shall show in the next section that these and can be regarded as estimates of some R and R (respectively) of the form ()]. By proper choice of the association-matrices P and P, several particular classical choices of and may be identified. For example, when P = I; obviously becomes the observations empirical correlation matrix. Then: if P is taken as an all-zeros matrix with (0j`j) along its 6`th diagonals, becomes the unbiased, symmetrized estimate of the observations lagged correlation matrix at lag `, as used for stationary sources, e.g., in the Algorithm for Multiple Unknown Signals Extraction (AMUSE) [], [3], and in similar algorithms (e.g., [0] []); likewise, if P is a general Toeplitz matrix, can be regarded as a linear combination of estimated correlation matrices at different lags, or, alternatively, as the correlation matrix between linearly filtered versions of the observations, as used, e.g., in [6]; if P is taken as an all-zeros matrix with a sequence of M nonzero values of M somewhere along its main diagonal, becomes the empirical zero-lag correlation estimated over the respective time-segment, as used for non-stationary sources, e.g., in [7], [], and [3]; spectral matrices at certain frequencies, time-frequency matrices at certain time-frequency points [], or cyclic correlation matrices [5] can also be obtained by setting P to the appropriate (possibly complex-valued) transformation matrices. Performance analysis of such GEVD-based separation has not been addressed before in the context of the above-mentioned particular approaches, but would be instrumental in predicting the expected performance and possibly for choosing the association-matrices. Our goal in this work is to provide closed-form expressions for the resulting interference to source ratio (ISR) matrix for the general case, namely for GEVD-based separation relying on any chosen pair of generalized correlation matrices, namely on any chosen pair of association-matrices. The paper is structured as follows. In the following section, we provide the basics of the GEVD solution. Our error-analysis is presented in Section III, where we obtain explicit expressions for the resulting mean ISR in terms of the association-matrices and the sources temporal covariance matrices. In Section IV, we present some simulation results illustrating the validity of our analytic results, and providing empirical testing of their noise-sensitivity. Concluding remarks are provided in Section V X/$ IEEE

2 5078 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 59, O. 0, OCTOBER 0 II. THE MATRIX-PAIR GEVD SOLUTIO Consider the basic mixture model X = AS with zero-mean, statistically-independent sources, each having an temporal covariance matrix C k = E[s k s T k ]; (k =;...;K). Let P and P denote some association-matrices. Define D = E[SP S T ] D = E[SP S T ] (3) which are diagonal due to the sources independence (and zero-mean), regardless of the specific choice of P and P. Likewise, define R = E[XP X T ]=AD A T R = E[XP X T ]=AD A T : () The generalized eigenvalues and eigenvectors matrices of a matrixpair (pencil) (Q ;Q ) are (respectively) a diagonal matrix 3 and a matrix W satisfying Q W = Q W3. Assuming invertibility of Q ; 3 and W are also seen to be the (standard) eigenvalues and eigenvectors matrices of Q 0 Q. Assuming invertibility of both R and R, it is evident from (), that A is (up to arbitrary permutation and scaling of its columns) the generalized eigenvectors matrix of the matrix pencil (R 0 ;R0) (with 3 = D D 0 ), and therefore also the eigenvectors matrix of R R 0. ow let the matrices = XP X T and = XP X T denote estimates of R and R (respectively). ote that while the true R and R of () are symmetric with any choice of P and P (since D and D are diagonal with any P and P ), their estimates and may become nonsymmetric if P and P are nonsymmetric. We shall therefore restrict the discussion from now on to symmetric associationmatrices P and P, which would guarantee that and would be symmetric as well, just like R and R. Under some commonly met conditions (see, e.g., [6]), the closedform EJD of and can be readily obtained from the GEVD of 0 ; 0 ( ), or, equivalently, from the eigen-decomposition of ^Q = 0 : The eigenvectors matrix ^A which satisfies ^Q ^A = ^A ^D (where ^D is some diagonal matrix), also satisfies = ^A ^D ^A T and = ^A ^D ^A T, with some diagonal matrices ^D and ^D, such that ^D = 0 ^D ^D. When the estimates and are exact and the model is identifiable, the resulting ^A coincides with A (up to the inevitable scale and permutation ambiguities). aturally, however, departure of and from their true values inflicts errors on ^A. III. ERROR AALYSIS OF THE GEVD SOLUTIO A common measure of the estimation error (useful in the BSS context) is to consider the resulting overall mixing-unmixing matrix T = ^A 0 A. Assuming, just for simplicity of notations, that the scaling and permutation ambiguities have been resolved, T would ideally be the identity matrix. However, due to estimation errors in ^A, its off-diagonal elements T [k; `] (k 6= `) would not vanish, and would reflect a residual mixing per realization. The second moments of these elements, E[T [k; `]], are usually called the `-to-k ISR (denoted ISR k;`), under the assumption that all sources have equal energy (if the sources have different energies, then each ISR k;` should be normalized by the ratio between the energies of the `th and kth sources, so as to reflect the mean relative residual energy of the `th source in the reconstruction of the kth source). If the scaling and/or permutation ambiguities are not resolved, then the resulting T matrix would equal the nominal (ambiguity-free) T up to permutation and scaling of its rows. Still, ISR k;` of the nominal T would reflect the relative residual energy presence of the `th source in the reconstruction of the kth source, the only difference being that the kth source might be obtained with a different index in the reconstruction, according to the remaining permutation error. Before turning to the error analysis, we observe a very appealing and rather simplifying invariance property of T in this context: Lemma : Given a specific realization S of the sources, the same value of T would be obtained (in our framework) with any (nonsingular) A. Proof: Let us assume first that A = I, and denote the estimated and in this case as (I) (I) = XP X T = SP S T = XP X T = SP S T : (5) Likewise, denote ^Q (I) = 0 (I) (I), with eigenvectors and eigenvalues matrices ^A (I) and ^D (I) (respectively): ^Q (I) ^A (I) = ^A (I) ^D (I) : (6) Evidently, the matrix T in this case is given by T (I) ^A 0 (I)I = = 0 ^A (I)A = 0 ^A (I). ow consider a general mixing matrix A. We then have i = XP ix T = ASP is T A T = A i(i) A T ; i =; (7) and, consequently, ^Q = A ^Q (I) A 0. It is readily observed that the matrix A ^A (I) is the eigenvectors matrix of ^Q (with eigenvalues matrix ^D (I) ), since [using (6)] ^QA ^A (I) = A ^Q (I) A 0 A ^A (I) = A ^Q (I) ^A (I) = A ^A (I) ^D (I) ; (8) ^A 0 so ^A = A ^A (I), and therefore T = ^A 0 A = (I) = T (I), which establishes the invariance of T in A. ote that this property is in accordance with the well-known equivariance property (e.g., [7]), shared by several (but certainly not by all) BSS algorithms, as well as by the GEVD-based algorithm (at least in our SOS framework). For more general conditions for equivariance of GEVD-based separation, see [8]. We note further, that this property only holds in the noiseless case, but falls apart in the presence of additive noise. Our error analysis would therefore only be valid in the noiseless case, but, as we shall demonstrate empirically in simulation, would still serve as a reasonable approximation under high signal-to-noise ratio (SR) conditions. Thanks to this invariance property, we may analyze the perturbation in T under the conveniently simple nonmixing condition A = I, knowing that the same result would hold true with any other invertible mixing matrix A. In the following Section III-A we quantify (under the nonmixing condition) the effect of estimation errors in and on all T [k; `]. Our analysis relates T [k; `] to the estimation errors in and (per realization). Then, in Section III-B we exploit statistical properties of the sources to characterize (still under the nonmixing condition) the relevant statistical properties of the estimation errors in and, which would in turn lead us (using the results of Section III-A) to the second moments of all T [k; `], namely to all ISR k;`. A. A Small-Errors Perturbation Analysis for the GEVD Assuming the nonmixing condition A = I, wehaver = D and R = D, so we may denote (assuming that D is nonsingular) = D + E = D + E = D (I + D 0 E ) (9)

3 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 59, O. 0, OCTOBER where E D and E D are respective zero-mean estimation errors, assumed small in our framework of small-errors analysis. Defining D = D D 0 and considering the small-errors analysis, which neglects second (and higher) order terms in E ; E, we get ^Q = 0 =(D + E )(I + D 0 E ) 0 D 0 (D + E )(I 0 D 0 E )D 0 D + E D 0 0DE D 0 : (0) In the error-free case the eigenvectors matrix A of ^Q would be the true mixing-matrix, ^A = A = I, and the eigenvalues matrix ^D would equal D. Let us denote by E and the resulting respective errors in these matrices, namely ^A = I + E and ^D = D +, such that ^Q ^A = ^A ^D ) ^Q(I + E) =(I + E)(D + ). Substituting (0), we have (again, using the small-errors assumption) and ^Q(I + E) (D + E D 0 0 DE D 0 )(I + E) D + E D 0 0 DE D 0 + DE () (I + E)(D + ) D + ED + : () Equating these terms, we get DE 0ED = DE D 0 0E D 0 + : (3) Eventually, we would have T = ^A 0 A = ^A 0 I I 0E, and since we are only interested in the off-diagonal terms of T, we may ignore the unknown, which only interacts with the diagonal elements of E in (3). Denoting by d i [k] the [k; k]th element of D i (i = ; ), we have, for the off-diagonal terms in (3) (recalling that D = D D 0 ), condition and the diagonality of D ;D, we note that (for k 6= `) E i [k; `] =s T k P i s` for i = ;. Thus, the covariance of E i [k; `] and E j [k; `] for (i; j) f; g (and k 6= `) is given by Cov(E i [k; `]; E j [k; `]) = E s T k P i s`s T k P j s` = E[s k [p]p i[p; q]s`[q]s k [m]p j [m; n]s`[n]] p;q;m;n= = P i[p; q]pj [m; n]e[sk [p]s k [m]s`[q]s`[n]] pqmn = P i[p; q]pj [m; n]ck [p; m]c`[q; n] pqmn = P i[p; q]c`[q; n]p j [n; m]ck [m; p] pqmn =TrfP ic`p jc k g (7) (where we have used the statistical independence of the sources, as well as the symmetry of P j and of C k ). Therefore, defining the K K matrices Q ; ;Q ; and Q ; with elements Q i;j [k; `] =Cov(E i [k; `]; E j [k; `]) =TrfP ic`p jc k g; i;j =; (8) and using (5), we obtain for all k 6= ` K (assuming equalenergy sources) E[k; `] d [k] d 0 d [`] [k] d [`] = d [k] E[k; `] 0 E[k; `] d [k]d [`] d [`] k 6= ` K: () ISR k;` = E[T [k; `]] = d [k]q ; [k; `] +d [k]q ; [k; `]0d [k]d [k]q ; [k; `] (d [`]d [k]0d [k]d [`]) : (9) Applying some straightforward algebraic manipulations, we end up with the following expression for T [k; `] =0E[k; `]: T [k; `] = d[k]e[k; `] 0 d[k]e[k; `] d [`]d [k] 0 d [k]d [`] k 6= ` K (5) which establishes the explicit dependence of T (under the small-errors assumption) on the generalized-correlations (small) estimation errors E and E under the nonmixing condition A = I. B. Explicit Expressions for the ISR We now turn to calculate the second moments of the off-diagonal elements of T (namely, the ISRs), based on (5). We begin with expressions for the coefficients d [k] and d [k] (for k =; ;...;K): d i[k] =Di[k; k] =E[s T k P is k ]=TrfP ic k g; i =; : (6) ext, recall the zero-mean error-matrices E = 0 D ; E = 0 D (defined under the nonmixing condition X = S). For computing the ISR from (5) we need to know the variances and covariance of E [k; `] and E [k; `] (for all k 6= `). From the nonmixing By denoting ED, we imply that all eigenvalues of E are much smaller than the smallest eigenvalue of D; We shall also use the Taylor series expansion (I + E) = I 0E + o(e ) for E I. If the sources have different energies, then each ISR k;` has to be further multiplied by the energy-ratio of the `th and kth sources, namely by TrfC`g=TrfC k g. Considerable simplification occurs in the case of stationary sources, whenever Toeplitz matrices are used as the association-matrices. The simplification is based on the observation, that circulant Toeplitz matrices are diagonalized by the Fourier Transform matrix. Assuming that all sources have correlation sequences with some finite maximal effective-width L, their noncirculant Toeplitz covariance matrices can be approximated as circulant Toeplitz matrices for L (say > 0L). We may therefore approximate the respective terms as follows: d i[k] =TrfP ic k g L Trf(F H ~ P if )(F H ~ C k F )g =Trf ~ P i ~ C k g = n= ~p i[n]~ck [n] (0) where F is the (unitary) Fourier transform matrix, such that F [m; n] = p expf0j (m0)(n0) g, and P ~ i and C ~ k are diagonal matrices with the sequences ~p i[n] and ~ck [n] (respectively) along their diagonals. These sequences are the (real-valued) Discrete Fourier Transforms (DFTs) of the (symmetric) sequences p i [n] and c k [n], which are in turn the (symmetric) generating sequences of the Toeplitz matrices P i and C k, such that for all m; n [;]; p i [m 0 n +] = P i [m; n]

4 5080 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 59, O. 0, OCTOBER 0 TABLE I EMPIRICAL IVERSE-ISR VALUES ([DB]) AD THEIR AALYTICALLY-PREDICTED VALUES (I PARETHESES) and c k [m0n+] = C k [m; n] (the latter is simply the autocorrelation sequence of the kth source). amely, for n =;...; ; ~p i [n] = ~c k [n] = =0 m=0= =0 m=0= p i [m]e 0j ; c k [m]e 0j () (assuming that is even). Further approximation (still assuming L), allows conversion of the discrete-sum over the (sufficiently smooth ) product of DFTs into a frequency-domain integral over the product of discrete-time Fourier transforms (DTFTs), with n= ~p i[n]~ck [n] L 5 i (e j! ) = S k (e j! ) = 0 =0 m=0= =0 m=0= 5 i(e j! )Sk (e j! )d!; () p i [m]e 0j! (3) c k [m]e 0j! : () The latter is obviously the power-spectrum of the kth source. For the matrix terms Q i;j[k; `] we similarly obtain Q i;j [k; `] L L Trf ~ P i ~ C` ~P j ~ C k g 0 5 i (e j! )5 j (e j! )S k (e j! )S`(e j! )d!: (5) Thus, these approximations alleviate the need to compute the trace of a product of two (four, respectively) matrices for each d i [k] (Q i;j[k; `], respectively), an O( ) (O( 3 ), respectively) operation: In the stationary case these computations can be substituted with sufficiently fine linear integration (independent of ). For large (say 00) the difference in computational load can be quite significant. We note in passing, that in the nonsquare case of more mixtures than sources, a GEVD solution can still exist, since the rows of the unmixing matrix can still be estimated from the GEVD of and (attained using standard GEVD tools), followed by elimination of rows orthogonal to all columns of and (as such rows would essentially reconstruct nonexisting, all-zeros sources). As far as our performance analysis is concerned, in such cases the missing sources can simply be regarded as sources with zero-covariance matrices. Since each ISR k;` depends only on C k and on C` (and not on the other sources covariances), these fictitious sources will have no effect on the relevant ISR expressions for the true sources. IV. SIMULATIO RESULTS We present simulation results for three experiments. The first two experiments demonstrate the good match between the analytically-predicted and the empirically-obtained performance, with both nonstationary and stationary sources. The third experiment tests the sensitivity of the performance prediction to additive noise. In the first experiment, we consider a mixture of K = 5sources. The first source s [n] is a stationary zero-mean, unit variance Gaussian white noise process, and the other four are generated (for k =; 3; ; 5) as s k [n] = 3`=0 h k[`] ~w k [n 0 `], where fh k [`]g 3`=0 are fixed filter coefficients, and ~w k [n] are nonstationary uncorrelated sequences, each generated as ~w k [n] =(+0:5 cos(n=m k + k ))w k [n], in which w k [n] is a zero-mean unit variance Gaussian white noise sequence, independent of the other sequences. The specific values of the filters coefficients fh k [`]gl=0, 3 the modulation periods M k and the phases k are specified in Table I below. We used an observation length of = 50. The two association-matrices were chosen as P = I, and P a block-diagonal matrix with two blocks, each a symmetric 5 5 Toeplitz matrix. The first block had the value along its main diagonal and 3 along its two first sub-diagonals (and zeros elsewhere); the second block had the value along its first subdiagonals and 0 along its second (and zeros elsewhere). So, in effect (up to irrelevant scaling), is the observations sample zero-lag correlation matrix, whereas is the sum of several sample-correlation matrices taken over the two halves of the observation interval as follows: two times the zero-lag correlation over the first half; six times the lag-one correlations over the first half; eight times the lag-one correlations over the second half; and 0 times the lag-two correlations over the second half. Table I summarizes the resulting empirical ISR k;` values for each of the signal-pairs combinations, in terms of the inverse averaged ISR in [db], averaged over 000 independent trials. The mixing matrix elements were redrawn independently from a standard Gaussian distribution in each trial. The numbers in parentheses represent our analytically-predicted values, calculated in (9) [using the matrix-form expressions (6) and (8) and normalizing by the different energy-ratios of the sources]. The good match (usually to within [db]) is evident. ext, we turn to stationary sources, so as to enable exploitation of the frequency-domain expressions for long observation intervals. In the second experiment we mixed and separated K =3stationary sources. Each source is a Gaussian autoregressive moving-average (ARMA) process of orders (,), namely, each source was generated by passing a zero-mean unit-variance white Gaussian process through a linear, timeinvariant system with two poles and two zeros, as specified in Table II below, followed by power-normalization. We present results obtained with three different sets of Toeplitz-structured association-matrices, with generating-sequences p [m] and p [m] as specified in Table III for jmj 3 (for jmj > 3 all p i [m] were set to zeros). The results for the three cases are shown (versus the observation length ) in Fig. in terms of the empirically obtained ISR k;` values for all signal-pairs combinations, superimposed on the analytically-

5 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 59, O. 0, OCTOBER TABLE II POLES AD ZEROS TABLE III TOEPLITZ-GEERATIG SEQUECES FOR THE THREE CASES Fig.. Empirical ISRs versus mixing-matrix parameter in (6); Analytically predicted ISR for the noiseless case. Fig.. Empirical and analytically-predicted ISRs versus observation length. predicted plots, obtained by (9) [using the frequency-domain expressions (0) and (5)]. The results reflect averaging over 000 independent trials (with randomized mixing matrices). Once again, the good match is evident for all three cases. It is important to observe, that none of the cases dominates (or is dominated) by the others for all ISRs: Each of the three cases attains the best (or worst) performance among the three for at least one ISR k;`. In the last experiment, we test (empirically) the sensitivity of the performance to additive, spatially and spectrally white Gaussian noise at the mixtures. We used the same three signals from the second experiment, with case c association-matrices (the only case which ignores the zero-lag correlations, and might therefore be the least sensitive to temporally-white noise), using the longest observation length =0. Since the equivariance property does not hold in the presence of additive noise, the performance depends on the mixing matrix. To illustrate this dependence, we used parameterized mixing matrices generated as A = cos() sin() sin() 0 sin() cos() sin () cos() 0 sin() cos() sin() sin () cos() cos() cos() (6) followed by normalization of the rows, so as to obtain mixtures with equal power, enabling convenient definition of the SR as the ratio between the (equal) power of each mixture and the variance of the additive noise. These mixing matrices are all parameterized by a single parameter, such that for = 0we have A = I, and for different values of we get different (nonorthogonal) matrices, with different condition-numbers. the expression is obviously 360 -periodic. We present the empirically obtained ISR values versus the parameter [0; 360] for three different SR values: High (30 [db]), medium (0 [db]) and (relatively) low (0 [db]). ote that these values only reflect the separation quality, computed from ^A 0 A, and not the estimation quality of the sources, which also involves the inevitable noise-term (multiplied by ^A 0 ). We also present the analytically-predicted performance, which is obviously independent of. At the high SR the separation performance is clearly seen to still coincide uniformly with the predicted values (to within less than [db]). At the medium SR, notable (-dependent) deviation is observed, but may still be considered tolerable. However, as could be expected, at the lower SR very significant deviations (up to more than 0 [db] for some values of ) occur, reflecting the strong dependence of the separation performance on the mixing matrix in the presence of significant noise. V. COCLUSIO We considered the GEVD-based separation of independent sources with different temporal covariance structures, when the two target-matrices are based on SOS. We provided analytic expressions for the expected separation performance under a small-errors assumption (without any further assumptions on the sources distributions). Such an assumption can be justified in practice whenever all the predicted ISRs are sufficiently low (say, below 00 [db]) and the SR (if noise is present) is sufficiently high (say, above 0 [db]) as demonstrated in our simulations. In the general case the ISR expression involves manipulation of potentially prohibitively large ( ) matrices. However, we also derived approximate frequency-domain expressions for the stationary case (with Toeplitz association-matrices), which eliminate the need to

6 508 IEEE TRASACTIOS O SIGAL PROCESSIG, VOL. 59, O. 0, OCTOBER 0 manipulate such matrices. The good agreement between the theoretically-predicted and the empirically-obtained separation performance was demonstrated in simulation. A remaining key-question is, given the sources covariance matrices and our analytic expressions for the performance prediction, whether (and if so, how) the association-matrices can be chosen so as to optimize the performance (in some sense). This question will be addressed in future work. REFERECES [] A. Yeredor, On optimal selection of correlation matrices for matrixpencil-based separation, in Proc. 8th Int. Conf. Independent Component Anal. Source Separation (ICA), 009, pp [] L. Tong, V. C. Soon, Y.-F. Huang, and R. Liu, AMUSE: A new blind identification algorithm, in Proc. ISCAS, 990, pp [3] L. Tong and R. Liu, Blind estimation of correlated source signals, in Proc. th Asilomar Conf., 990, pp [] J. F. 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Laheld, Equivariant adaptive source separation, IEEE Trans. Signal Process., vol., no., pp , 996. A Class of Scaled Bessel Sampling Theorems Luc Knockaert Abstract Sampling theorems for a class of scaled Bessel unitary transforms are presented. The derivations are based on the properties of the generalized Laguerre functions. This class of scaled Bessel unitary transforms includes the classical sine and cosine transforms, but also novel chirp sine and modified Hankel transforms. The results for the sine and cosine transform can also be utilized to yield a sampling theorem, different from Shannon s, for the Fourier transform. Index Terms Bessel functions, chirp transform, Hankel transform, sampling theorems. I. ITRODUCTIO The classical Shannon [] sampling theorem is omnipresent in large areas of signal processing. umerous generalizations have been derived [] [5], while extensions of the sampling theorem have arisen in connection with wavelets [6], the fractional Fourier and quasi-fourier transforms [7] [0]. In this correspondence, we develop sampling theorems for a class of scaled Bessel unitary transforms in a constructive way, based on the generalized Laguerre functions. This class includes the sine and cosine transforms, but also novel chirp sine and modified Hankel transforms. The results for the sine and cosine transform can also be utilized to yield a sampling theorem, different from Shannon s, for the Fourier transform. ovel asymptotic truncation error expressions are derived and pertinent examples are presented. II. MAI RESULTS We begin by stating the following result due to Kramer [], [], [5]. Theorem : Let be a real unitary operator over L [0; ] with product kernel K(xt) such that the set fk(x k )g forms a complete orthogonal basis in L [0; ] for some countable set of strictly increasing positive constants f k g. Suppose further that f (t) is -bandlimited to B, i.e., the transform ^f (x) = 0 K(xt)f(t)dt is in L [0;B] such that ^f (x) =0for x>b. Then, f (t) admits the sampling representation f (t) = where the sampling kernel is k= and k = K(u 0 k) du. Proof: See [], [], and [5]. f k (v) = K(uv)K(u k )du k 0 k B k(bt) () X/$ IEEE Manuscript received February 5, 0; revised June, 0; accepted June, 0. Date of publication June 7, 0; date of current version September, 0. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Xiang-Gen Xia. This work was supported by a grant of the Research Foundation-Flanders (FWO-Vlaanderen). The author is with the ITEC-IBC-IBBT, Ghent University, B-9050 Gent, Belgium ( luc.knockaert@intec.ugent.be). Digital Object Identifier 0.09/TSP