Blind Deconvolution of Multi-Input Single-Output Systems Using the Distribution of Point Distances

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1 J Sign Process Syst (2) 65: DOI.7/s Blind Deconvolution of Multi-Input Single-Output Systems Using the Distribution of Point Distances Konstantinos Diamantaras Theophilos Papadimitriou Received: 22 January 2 / Revised: 27 October 2 / Accepted: 27 October 2 / Published online: 3 November 2 Springer Science+Business Media, LLC 2 Abstract In this paper we present a new blind identification and source separation method for linear Multi Input Single Output (MISO) convolutive systems driven by PAM sources. Our method is based on the estimation of the output difference distribution for pairs of outputs. We show that the most likely differences (not counting the zero difference) are the ones corresponding to the columns of the mixing matrix (upto a sign). The columns can be arranged in the correct order by using the block-toeplitz property of the transfer matrix. Thus the problem is transformed into the density estimation problem. The method is conceptually simple and can work with relatively small data sets although it is exponentially complex with the channel length or the number of input signals. Keywords Blind deconvolution Finite alphabet sources Combinatorial methods Introduction Blind Source Separation (BSS) refers to the recovery of n unknown signals, using m recorded mixtures. K. Diamantaras (B) Department of Informatics, TEI of Thessaloniki, Sindos, 574, Greece kdiamant@it.teithe.gr Th. Papadimitriou Department of International Economic Relations & Development, Democritus University of Thrace, Komotini, 69, Greece papadimi@ierd.duth.gr In BSS we try to develop methods needing the less possible aprioriknowledge on the system; the systems are never totally blind. In linear BSS, the mixing can be memoryless or convolutive and each case is treated quite differently. The mixing operator in memoryless systems is a constant matrix and there is no time shift of the source signals. However the instantaneous model cannot describe correctly the mixing procedure in an echoic environment where the same source arrives in the sensors with various delays. These cases are treated using the Multi-Input Multi-Output (MIMO) convolutive model. So the researcher must know if the mixture signals he is working with belong to the former (treated with instantaneous BSS methods) or the latter group (treated with multichannel blind deconvolution methods). Fortunately, the mixing procedure can be assumed according to the nature of the system and the desired accuracy of the recovered signals. For example, we do know that in recording studios the mixtures taken by a set of microphones are convolutive. However if the microphones are close to the sound sources, we can assume that wall distortions and echo are negligible and use an instantaneous BSS method yielding lower though acceptable accuracy. The instantaneous problem has been extensively studied in the past using Second-Order Statistics (AMUSE [, SOBI [2, OPCA [3) and Higher-Order Statistics (JADE [4, ICA [5). The convolutive problem is much more complex and is still an active research topic. The first methods aimed at disintegrating the multichannel system into a set of simpler SIMO (Single- Input Multi-Output) systems, which are easier to solve [6, 7. The special case of i.i.d. has been studied in [8, by exploiting the structure of cumulants. Colored sources

2 526 J Sign Process Syst (2) 65: have been treated in the past using both second order [9, and higher order statistics [. There are some very useful properties in the output sequence when the system is driven by finite alphabet sources. Typically such methods exploit the combinatorial analysis and the geometric properties of the observations constellation. Diamantaras e.a. studied the memoryless MISO (Multi-Input Single-Output) system driven by binary antipodal sources in [2. The system filter is recursively deflated, estimating eventually the filter and the source signals. We extended the basic idea to the convolutive case in [3. Richer finite alphabet sources, i.e. Pulse Amplitude Modulated (PAM) and Quadrature Amplitude Modulated (QAM) sources were investigated in [4. The proposed method exploits the distribution of the distances between the observations cluster centers. The mixing vectors can be estimated through a combinatorial analysis of the distances histogram. When the mixtures are more than the sources the system is called overdetermined (m > n); the opposite case system where the sensors are less than the sources is underdetermined (n > m). In the BSS framework there are several established and well-known methods treating systems of the former case, and much fewer treating the more demanding problem described in the latter case. One of the most difficult BSS problems is formed when many sources are mixed through convolution in just one mixture signal. The problem is described by the Multi-Input Single-Output model (MISO) and is unsolved in its general form. We present in this paper a new method for the blind deconvolution of PAM driven MISO systems. The basic idea relates to the probability distribution of the output differences. The second most probable differences are related to the columns of the system transfer matrix and they can be used so we can estimate the mixing filters. The paper is organized as follows: Section 2 describes the problem formulation, Section 3 gives the basic idea and the proposed method including transfer matrix estimation, source recovery and dealing with noise. Section 4 presents simulation examples for noiseless and noisy cases, while Section 5 discusses computational complexity issues. We finally conclude in Section 6. 2 Problem Formulation A Multi-Input Single-Output (MISO) convolutive system with n discrete inputs s i (k), i =,, n, and equal number of L-tap real filters h i (l), i =,, n, l =,, L, is described by Eq. x(k) = n L h i (l)s i (k l). () i= l= The input signals s i, the number of sources n, and the filters h i are unknown, but for the sake of simplicity let us assume that the filter length, L, is given. The estimation of L is treated in Section 3.. Eq. can be rewritten as L x(k) = h(l) T s(k l) (2) where l= h(l) =[h (l),, h n (l) T, l =,,, L, s(k) =[s (k),, s n (k) T. Suppose that the inputs are Pulse Amplitude Modulated (PAM) with M levels so s i (k) {v + mt m =,, M, v IR}. Without loss of generality, assume that the gap between successive levels is T =, so the discrete input symbol set is V s ={v m = v + m m =,, M } (3) Let us define the range of x(k) as follows V x ={x(k) from Eq. s i V s }. (4) Obviously, the cardinality of V x is M nl provided that there are no repeated values in V x. Taking a timewindow of length W starting at time k and stacking the previous outputs into a vector x(k) we obtain x(k) =[x(k),, x(k W + ) T = Hs(k), (5) where h() T h(l ) T h() T h(l ) T H=......, h() T h(l ) T s(k) =[ s(k) T, s(k ) T,, s(k W L + 2) T T. (6) (7) The dimensions of matrix H and vector s are (W Q) and (Q ), respectively, where Q = n(w + L ). (8)

3 J Sign Process Syst (2) 65: As before, we can define the sets of values for s and x as follows V s ={s(k) from Eq. 7 s i V s }=V Q s (9) V x ={x(k) from Eq. 5 s V s }. () The cardinalities are V s = V x =M Q, assuming that there are no repeated output values. We assume that the sources are independent white sequences, i.e., they satisfy the following conditions: A. The signals s i are independent to each other A.2 The samples s i (k) are i.i.d. We also assume that the symbols are emitted with equal probabilities, i.e. A.3 The symbol transmission probabilities are uniform: p s = Prob(s i (k) = v m ) = /M, k, i, m. () Assumption [A.3 is instrumental in the subsequent development of our method which is based on the probability distribution of the output differences. This probability distribution is shown to have interesting properties revealing the structure of the mixing system, provided that the source symbol distribution if uniform. 3 Output Difference Distribution Let us consider the difference between any two output vectors x(k), x(k ), at different time instances k = k, d(k, k ) = x(k) x(k ) = Hu(k, k ), (2) u(k, k ) = s(k) s(k ). (3) The distribution of d obviously depends on the distribution of u and u(k, k ) =[ū(k, k ) T,, ū(k W L + 2, where k W L + 2) T T (4) ū(k, k ) = s(k) s(k ) =[δ (k, k ),,δ n (k, k ) T (5) with δ i being the input differences δ i (k, k ) = s i (k) s i (k ). (6) Clearly, the range of values for δ i is the set U ={ (M ), (M 2),,(M 2), (M )}. (7) Then from Eq. 5 we have u U Q. In the sequel, we shall drop the indexes k, k, due to input stationarity. All δ i have the same distribution, call it p δ (), forall i, but this distribution is no longer uniform. Since the input symbols are transmitted with equal probability p s = /M, p δ is triangular. Indeed, we have: p δ (β) = Prob(δ i = β) = (M β )/M 2, β U. (8) So, the most probable difference is δ i =, with probability p δ () = M/M 2 = /M. The second most probable difference is δ i =±, with probability p δ (±) = (M )/M 2, the third most probable difference is δ i =±2, with probability p δ (±2) = (M 2)/M 2, andsoon. Based on the above, we can compute the distribution of the vector u given that the elements of u are statistically independent p u ([u,, u Q T ) = Prob(u =[u,, u Q T ) = Q i= p δ(u i ) (9) We make the following assumption relating the input and the output difference vectors, A.4 For all u, u U Q,ifu = u then Hu = Hu. If a matrix H satisfies [A.4 then we shall say that its columns h i are Delta-independent, i.e. they satisfy the following property: Q α i h i =, α i α i =, i (2) i= where ={δ δ 2 δ,δ 2 U} ={ 2(M ),, 2(M )}. (2) Property 2 is like linear independence except that the coefficients α i are constrained to fall into the discrete set. If the elements of H are random reals then

4 528 J Sign Process Syst (2) 65: Delta-independence holds with probability and so assumption [A.4 is satisfied. The most important consequence of [A.4 is that the probability of u propagates to the probability of d in the following sense Prob(d = Hu) = Prob(u). (22) Based on Eq. 9 and due to the triangular distribution of δ i the most likely vector u is u =[,, T = with probability p u () = /M Q.AccordingtoEq.22 the corresponding most likely output vector is d = and Prob(d = ) = /M Q. (23) The second most likely set of input vectors are those belonging to the set U ={u =[,,, u i,,, T u i =±, for some i : i Q} (24) All vectors in U have the same probability which, basedoneq.9,is p u (u U ) =[/M Q (M )/M 2 = (M )/M Q+. (25) There are 2Q such vectors. The output vectors d corresponding to those u U form the set D ={d = Hu u U } (26) and can be expressed as d =±h i, i =, 2,, Q (27) where h i is the i-th column of H. AccordingtoEq.22 these output differences have the same probability as the expression in Eq. 25: p d (d D ) =[/M Q (M )/M 2 = (M )/M Q+. (28) It is not difficult to see that all other input differences (except for u = and u U ) have smaller probabilities. Therefore, the corresponding output differences also have smaller probabilities. If we have the output difference distribution p d then simple sorting of the probabilities will reveal the columns of the mixing matrix H, upto a sign and without the correct order. In summary, the most likely output difference is d = with probability /M Q, the second 2Q most likely output differences are the vectors d =±h i,fori =, 2,, Q, each with probability (M )/M Q+. 3. Estimating Filter Length The filter length L can be readily estimated using the output covariance function r x (l) = E{x(k)x(k l)} which is obviously zero for l > L. Since r(l ) = h(l ) T h, if the product h(l ) T h is non-zero then L can be estimated by finding the smallest l > for which r x (l) =. 3.2 Recovering Column Order From the analysis in the previous sections it follows that the second 2Q most likely output differences d i, i =,, 2Q, form the set {h,, h Q, h,, h Q }. According to Eq. 6 the matrix H has a block-toepitz structure. Therefore, the first L columns forming the W L submatrix h() T. are filled with zeros everywhere from row 2 to row W. Similarly, the last L rows are filled with zeros everywhere from row to row W. Since we have estimates of the columns, even though without proper ordering, we can identify those vectors d i which are filled with zeros everywhere from row 2 to W. They should correspond to the first L columns of H and their negatives, so there should exist 2L such vectors. So far we know that the vectors d i estimate the columns of H (and their negatives) but without any prescribed order. If we want to estimate H we need to recover this ordering information. To that end, we exploit again the block-toeplitz structure of H, which is expressed by the following property h i, j = h i+q, j+lq for all q (29) We start with the vectors with zeros everywhere from row 2 to row W since we know that they belong to the first L columns. Then, following Eq. 29 we arrange the vectors d i so that d i, j = d i+q, j+lq,forallq. This will recover the mixing vectors h(l) upto a permutation π of their elements and a sign σ i, ie. we shall obtain ĥ(l) =[σ h π() (l), σ 2 h π(2) (l),,σ n h π(n) (l) T, for all l =,, L. (3)

5 J Sign Process Syst (2) 65: where σ i =±. We can tolerate this permutation and sign issue however, since it corresponds to a permutation and sign of the source signals, and it is well known that the order and the sign of the source signals cannot be recovered in a blind source separation scenario. The following example clarifies how H is estimated by arranging d i as described above. Example Consider the following system (n = 2, L = 2, W = 2,soQ = 6): [ x(k) = s(k) Assume binary inputs (M = 2), then the difference vector d = is the most frequent one with probability /M Q =.56 the next 2Q = 2 most frequent difference vectors d, each with probability (M )/M Q+ =.78, are the following (in no particular order) d = d 4 = d 7 = d = [.66 [.2.43 [ [.43 d 2 = d 5 = d 8 = [.66 d = [.29 [ [.2 d 3 = [.2.43 [ d 6 =.29 d 9 = [.43 [ d 2 =.2 Start with vector d which has a zero in row 2: this is column. We then arrange d 8 and d 6 in columns + L = 3 and + 2L = 5 since they satisfy the Toeplitz property [d =[d 8 2 and [d 8 =[d 6 2, respectively. So far, our estimate of the mixing matrix H is [ We repeat the procedure in an entirely analogous way starting with vector d 9 which has a zero in row 2: this is now column 2. (Vector d 2 is not eligible, although it also has a zero in row 2, because d 2 = d ). We then arrange d 4 and d 2 in columns 2 + L = 4 and 2 + 2L = 6 since they satisfy the Toeplitz property [d =[d 4 2 and [d 4 =[d 2 2, respectively. Our final estimate of the mixing matrix H is Ĥ =. [ Now we can form the estimates of the filter taps (mixing vectors) h(l) from the first row of Ĥ, namely, ĥ() =[ĥ,, ĥ,2 T =[.66,.43 T ĥ() =[ĥ,3, ĥ,4 T =[.29,.2 T. Notice that the order of the parameters is swapped compared to the true filter taps: h() =[.43,.66 T h() =[.2,.29 T. The parameters corresponding to the second source (.66,.29) are put first in our estimate while the parameters corresponding to the first source (.43,.2) are put second. At the same time the sign of the parameters corresponding to the first source is flipped. This simply means that when we shall extract the sources, the first source will be extracted second and with a flipped sign, while the second source will be extracted first. This is the typical sign and order ambiguity encountered in all blind source separation problems. The preceding analysis leads to the following algorithm for blind MISO channel identification: Algorithm Blind MISO channel identification Compute the output differences d(k, k ) for all pairs of indexes k, k Estimate the distribution p d of d Sort the probabilities ˆp d in decreasing order. Let d i, i =,, 2Q, be the values of d with probabilities ranging from the 2nd highest to the (2Q + )-th highest probability. Arrange the vectors d i into a block-toeplitz matrix as described above by forcing the satisfaction of property 29. From the elements of the submatrices we obtain our estimates of the mixing filters. 3.3 Recovering the Sources Given the estimates of the filter taps h(l) we can estimate the sources by minimizing [ŝ(k),, ŝ(k L + ) =arg min s,, s L V n L x(k) ĥ(l) s l (3) l=

6 53 J Sign Process Syst (2) 65: Since the input alphabet, V, is discrete the simplest way of minimizing Eq. 3 is by exhaustive search in the input space. Multiple estimates ŝ(k) of s(k) will be obtained by performing the above minimization for x(k), x(k + ),..., x(k + L ). These estimates are averaged out to yield the final estimate of s(k). 3.4 The Addition of Noise In the presence of additive noise e(k), the model becomes x(k) = Hs(k) + e(k) (32) where e(k) =[e(k),, e(k m + ) T. Similarly, the difference model also contains an additive noise term d(k, k ) = Hu(k, k ) + d e (k, k ) (33) where d e (k, k ) = e(k) e(k ). Using the Bayes rule we have p(d) = u = u p(d u)p(u) p de (d Hu)p(u) (34) So the distribution of d is a mixture of the distributions p de centered at the points Hu andscaledbythefactors p(u). If, for example, the noise is zero-mean, Gaussian, then p(d) is a Gaussian mixture. Our preceding analysis is still valid regarding p(u). Therefore, the tallest Gaussian bell is the one associated with u =, while the next 2Q tallest bells are the ones associated with u U. If we have an accurate estimate of the bell centers then we can apply Algorithm directly and obtain an estimate of the mixing matrix. The estimation of the distribution p(d) using a parametric method, such as the EM algorithm [5, has the advantage that it can produce good estimates of the centers even if the Gaussian bells merge with each other. The disadvantage is the fact that there can be, potentially, a very large number of bells (M Q ) while we are really interested in only (2Q) of them. The alternative is the use of nonparametric methods such as Parzen window smoothing [6, chapter 4. This approach is computationally more efficient although it can be quite demanding in memory especially for m > 4. In this work this will be our method of choice. 4Results We conducted various sets of experiments following noiseless and noisy MISO scenarios. We will present results on both cases..6 x p(d) Figure (Left) The 3 highest estimated probabilities for the vectors d arranged in decreasing order. The probabilities ranking from 2 to 7 (dark bullets) correspond to the vectors estimating the columns h(i) of H. (Right) The vectors d corresponding to the 3 highest probabilities. As in the left plot, the dark bullets correspond to the vectors d with probabilities ranking from 2 to 7. The crosses + indicate the position of the true vectors ±h(i).

7 J Sign Process Syst (2) 65: Noiseless Scenario 4.. Case The first noiseless experiment featured n = 2 independent sources with M = 3 PAM levels. The L = 3 filtertaps are: [.4792 h() =.6623 [.2838 h(2) =.923, h() = [.749,.3657 The value W = 2 was used for the size of the output vector and the probability distribution p d () was estimated using a simple frequency histogram on the output vector differences based on N = 2,5 data points. We have Q = 2(L + ) = 8 so we are looking for the difference vectors with frequencies from the 2nd highest to the (2Q + )-th = 7-th highest. After rearranging the vectors as described above we obtain the following filter estimation: ĥ() = [.48.66, ĥ() = [.7.37 [.28, ĥ(2) =.9 In Fig. (right) we show the vectors with the 3 highest estimated probabilities and their probability values (left). There is an obvious gap between the st and 2nd probabilities and also between the 7th and the 8th marking the beginning and the end of the probability values we are interested in. Ideally, the highest probability should be equal to /M Q = , and the probabilities from 2 to 7 should be equal to (M )/M Q+ =.6 4.Wecanseethatthe true and estimated clusters (the black dots and the crosses in Fig., right) practically coincide yielding very good estimates of the vectors ±h(i). The source reconstruction according to (3) gives Symbol Error Rate: SER = Case 2 In the second noiseless experiment we used n = 3 independent sources with M = 2 PAM levels. The filter taps (L = 4) were: h() =.3444, h() =.7, h(2) = , h(3) = The output vector had W = 2 size and following the presented method we estimated the probability distribution p d () using a simple frequency histogram on the output vector differences based on N = 6, data points. We have Q = 3(L + ) = 5 so we are looking for the difference vectors with frequencies from the 2nd highest to the (2Q + )-th = 3-st highest. 4 x p(d) Figure 2 (Left) The 5 highest estimated probabilities for the vectors d arranged in decreasing order. The probabilities ranking from 2 to 3 (dark bullets) correspond to the vectors estimating the columns h(i) of H. (Right) The vectors d corresponding to the 5 highest probabilities. As in the left plot, the dark bullets correspond to the vectors d with probabilities ranking from 2 to 3. The crosses + indicate the position of the true vectors ±h(i).

8 532 J Sign Process Syst (2) 65: Figure 3 (Left) Density estimation using kernel smoothing (Right) Contour of the density function indicating the highest peaks (larger circles correspond to higher peaks). Using our method we obtained the following tap estimates:.59 ĥ() =.34,.95 ĥ() =.7, ĥ(2) =.87, ĥ(3) = leading to perfect signal reconstruction (SER = ). In Fig. 2 we show the vectors with the 5 highest estimated probabilities. Again there is an obvious gap between the st and 2nd probabilities and also between the 3st and the 32nd marking the beginning and the end of the probability values we are interested in. Again the cluster estimates (black dots in Fig. 2, right) and the true values (crosses) practically coincide thus obtaining perfect source reconstruction. 4.2 Noisy Case We then tested the method in a noisy scenario with n = 2 independent binary sources (M = 2) and output timewindow size W = 2. The Signal-to-noise ratio (SNR) was set at 3dB. The mixing filters of length L = 2 were chosen randomly and they are shown below: h() = [.332.2, h() = [ x -4.5 x p(d).8 p(d) Figure 4 The 3 highest estimated probabilities for the vectors d arranged in decreasing order for W = 2 (left) andw = 3 (right).

9 J Sign Process Syst (2) 65: The density was estimated using N =,5 data samples and their pairwise differences. The non-parametric smoothing method was used with Gaussian kernels of width σ =.. We then counted the highest peaks of the estimated density and using the proposed methods we obtained the following estimates ĥ() = [.33.2, ĥ() = [ Subsequently, the source reconstruction by Eq. 3 yields SER =.27. Figure 3 shows the estimated disribution and the locations of its highest peaks. 5 Discussion A. Computational Issues Since the probability of the vectors we are looking for is (M )/M Q+, it follows that we need at least O(M Q+ ) difference samples for a reliable estimation of the probability density. Luckily, a data set {x(k)} with N samples yields N(N ) = O(N 2 ) differences, so the size of the original data set can be only O(M (Q+)/2 ). Still, since Q = n(w + L ), we conclude that the method has exponential complexity with respect to W, n, andl. B. The Window Size W Clearly the choice of the parameter W affects the performance of the algorithm. The increase of W gives more dimensions to the observation vector x but this does not lead to performance increase. The reason is that the length of the input vector s is Q = n(w + L ) and it also increases linearly with W byafactorn. So, for example, in a problem with M = 2, n = 3, andl = 4, the size of s is Q = 5, forw = 2, while for W = 3 we have Q = 8. So, although the output space dimension has increased from 2 to 3, the input space dimension has increased from 5 to 8 and the cardinality of the output state set V x has increased from 2 5 to 2 8. Therefore, the output space has more clusters and more data samples are needed in order to achieve the same performance. Figure 4 shows the sorted probabilities for experiment Case in Section 4.. with W = 2 and W = 3 for N = 25 samples. We can see that the probabilities are not clearly separated for W = 3 (right), as opposed to the case W = 2 (left). C. The Ef fect of Noise The method proposed in this paper is quite sensitive to noise. The reason is that it depends heavily on the correct clustering of the output values and the accurate estimation of the difference probabilities. In the case of low SNR the clustering becomes difficult as clusters of noisy observations tend to merge together into bigger clouds. In a future work we shall consider applying post-processing methods to. improve performance by imposing specific constraints on the estimated vectors. However, this is beyond the scope of the present paper. 6 Conclusions We presented a new blind deconvolution method for real, linear Multi Input Single Output (MISO) convolutive systems when the sources are M-level PAM signals. Our method is based on the observation that the most likely input vector differences (except for zero) are the vectors having all zeros except for one non-zero element equal to ±. Thus by estimating the distribution of the differences of the output vectors we are able to obtain the columns of the system transfer matrix without correct order and upto a sign. Fortunately, the correct order can be recovered taking advantage of the Toeplitz structure of the matrix. Once the system is estimated the sources can be recovered by a straightforward minimization procedure. References. Tong, L., Liu, R., Soon, V. C., & Huang, Y. F. (99). Indeterminacy and identifiability of blind identification. IEEE Transactions on Circuits and Systems I, 38(5), Belouchrani, A., Abed-Meraim, K., Cardoso, J.-F., & Moulines, E. (997). A Blind Source Separation Technique Using Second-Order Statistics. IEEE Transactions on Signal Processing, 45(2), Diamantaras, K. I., & Papadimitriou, Th. (23). Blind signal separation using oriented PCA neural models. In Proc. IEEE Int. Conference on Acoustic, Speech and Signal Processing (ICASSP-3), Hong Kong. 4. Cardoso, J.-F., & Souloumiac, A. (993). Blind beamforming for non Gaussian signals. IEE Proceesings-F, 4(6), Hyvärinen, A. (999) Fast and robust fixed-point algorithms for independent component analysis. IEEE Transactions on Neural Networks, (3), Inouye, Y., & Hirano, K. (997). Cumulant-based blind identification of linear multi-input multi-output systems driven by colored systems. IEEE Transactions on Signal Processing, 45(6), Tugnait, J. K. (998). On blind separation of convolutive mixtures of independent linear signals in unknown additive noise. IEEE Transactions on Signal Processing, 46(), Chen, B., & Petropulu, A. P. (2). Frequency domain blind MIMO system identification based on second- and higherorder statistics. IEEE Transactions on Signal Processing, 49(8), Diamantaras, K. I., Petropulu, A. P., & Chen, B. (2). Blind two-input two-output FIR channel identification based on second-order statistics. IEEE Transactions on Signal Processing, 48(2), An, S., Hua, Y., Manton, J. H., & Fang, Z. (25). Group decorrelation enhanced subspace method for identifying FIR

10 534 J Sign Process Syst (2) 65: MIMO channels driven by unknown uncorrelated colored sources. IEEE Transactions on Signal Processing, 53(2), Shamsunder, S., & Giannakis, G. B. (997). Multichannel blind signal separation and reconstruction. IEEE Transactions on Speech and Audio Processing, 5, Diamantaras, K. I., & Chassioti, E. (2). Blind separation of n binary sources from one observation: A deterministic approach. In Proc. Second Int. Workshop on ICA and BSS (pp ). Helsinki, Finland. 3. Diamantaras, K., & Papadimitriou, T. (26). Blind deconvolution of multi-input single-output systems with binary sources. IEEE Transactions on Signal Processing, 54(), Diamantaras, K. I., & Papadimitriou, Th. (29). Histogram based blind identification and source separation from linear instantaneous mixtures. In Lecture Notes in Computer Science Proceedings of the 8th Int. Conference on Independent Component Analysis and Signal Separation (Vol. 544, pp ). Paraty, Brazil. 5. Bishop, C. M. (995). Neural netowrks for pattern recognition. Oxford University Press. 6. Duda, R. O., Hart, P. E., & Stork, D. G. (2). Pattern classif ication. Wiley, NY. editor for the IEEE Transactions on Signal Processing, IEEE Signal Processing Letters and the Springer Journal of Signal Processing Systems. In 997, he was a co-recipient of the IEEE Best Paper Award in the area of Neural Networks for Signal Processing. He is the author of the book Principal Component Neural Networks: Theory and Applications, co-authored with S.Y.Kung (New York: Wiley, 996). He is currently chairman of the IEEE Machine Learning for Signal Processing (MLSP) Technical Committee and from 25 to 28 he was a member of the IEEE Signal Processing Theory and Methods (SPTM) TC. He has been the chairman of the 27 IEEE Int. Workshop on Machine Learning for Signal Processing (MLSP27) and the 2th International Conference on Artificial Neural Networks (ICANN2) both held in Thessaloniki, Greece, and has also served as technical committee member for various international signal processing and neural networks conferences. He is a member of the Technical Chamber of Greece. Konstantinos Diamantaras was born in Athens, Greece. He received his Diploma in Electrical Engineering from the National Technical University of Athens in 987 and the Ph.D. degree, also in electrical engineering, from Princeton University, Princeton, NJ, in 992. Subsequently, he joined Siemens Corp. Research, Princeton, as a post-doctoral researcher, and in 995, he worked as a researcher with the Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece. Since 998, he has been with the Department of Informatics, Technological Education Institute of Thessaloniki, where he currently holds the position of professor. His research interests include signal processing, neural networks, image processing, and parallel processing. Dr. Diamantaras is a Senior Member of IEEE and also serves as associate Theophilos Papadimitriou was born in Thessaloniki, Greece, in 972. He received the Diploma degree in mathematics from the Aristotle University of Thessaloniki, Greece, and the D.E.A. A.R.A.V.I.S (Automatique, Robotique, Algorithmique, Vision, Image, Signale) degree from the University of Nice-Sophia Antipolis, France, both in 996 and the Ph.D. degree in electrical engineering from the Aristotle University of Thessaloniki in 2. In 2, he joined the Department of International Economic Relations and Development, Democritos University of Thrace, Komotini, Greece, where, he served as a lecturer (22 28). Currently he holds the position of Assistant Professor in the same department. Dr. Papadimitriou served as a reviewer for various publications and as a member to scientific committees member for Conferences and workshops. In 27 he was a member of the organizing committee of the IEEE Workshop on Machine Learning for Signal Processing held in Thessaloniki, Greece. Theophilos Papadimitriou current research interests include digital signal and image processing, data analysis, and neural networks.