Subspace-based Channel Shortening for the Blind Separation of Convolutive Mixtures

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1 Subspace-based Channel Shortening for the Blind Separation of Convolutive Mixtures Konstantinos I. Diamantaras, Member, IEEE and Theophilos Papadimitriou, Member, IEEE IEEE Transactions on Signal Processing, vol. 54, no. 1, pp , October 26 Abstract A novel subspace-based channel shortening procedure is proposed based on the structure of the delayed auto-correlation matrices of the observation process. This purely second order approach applies to overdetermined MIMO channels with independent, white sources. The channel may be sparse and its length is assumed to be unknown. Through successive deflations the problem can be transformed into an instantaneous BSS problem which is simpler to solve using, for example, ICA techniques. The algorithm is computationally fast although it requires large input data sets. Such data can be acquired either through large numbers of sensors, or by using increased data sampling rate. When not enough data are available, the method can still be used for reducing the channel length thus simplifying the problem for subsequent treatment. 1 Introduction Blind Source Separation (BSS) is a very active research domain of the signal processing community mainly due to its usefulness in telecommunications. However, the application range of BSS is wider and includes medical signal processing [1], audio processing [2], image processing [3], economy [4], chemistry [5], and many more. BSS refers to the estimation of n unknown signals, given just a set of mixtures observed at m sensors. The term blind refers to our incomplete knowledge of the mixing operator. The BSS methods can be divided according to the mixing process into memoryless linear mixture BSS (also known as instantaneous BSS) and convolutive mixture BSS (also referred to as multichannel blind deconvolution/equalization). In the convolutive mixture case the term BSS is also used when filtered versions of the original source signals are extracted. In the case of instantaneous BSS the mixing operator is a constant matrix and there is no time shift of the source signals. The static nature of the instantaneous model makes it often inadequate for real world applications [6]. In most real-world applications, the signal K.I. Diamantaras is with the Department of Informatics, Technological Education Institute of Thessaloniki, GR-574, Sindos, Thessaloniki, Greece ( kdiamant@it.teithe.gr). Th. Papadimitriou is with the Department of Int. Economic Relations and Development, Democritus University of Thrace, GR-691, Komotini, Greece ( papadimi@ierd.duth.gr). This work has been supported by the EPEAEK Archimides-II Programme, project BSP-GRID, funded in part by the European Union (75%) and in part by the Greek Ministry of National Education and Religious Affairs (25%). 1

2 propagation through the transmission medium is not instantaneous, causing delays in the time of arrival between the sources. Instantaneous BSS cannot tolerate these delays. Moreover, in such applications the observation signals are not clean copies of the source signals, but multipath distorted copies caused by reflections from obstacles between the source and the receivers. Such situations are best simulated by dynamic Multi-Input Multi-Output (MIMO) systems and are solved through the convolutive BSS problem. In both instantaneous and convolutive BSS cases, the techniques proposed for the source separation are typically characterized by the type of signal statistics used. The Second-Order Statistics (SOS) methods use only the mean and the covariances (first- and second-order statistics) of the mixture signals, while Higher Order Statistics methods use cumulants and polyspectra. The SOS methods depend, in general, on the fact that the power spectral densities of the source signals are different (AMUSE [7], SOBI [8], OPCA-BSS [9]). Recently, Xerri et al. in [1] showed that even in cases that the source signals share the same power spectral density, they can be separated using second order conditional statistics in an iterative framework. There is a large variety of blind deconvolution problems treated using SOS: blind identification and equalization of FIR MIMO channels driven by colored signals are treated in [11, 12], multicarrier systems with cyclic prefix are discussed in [13], while quasistationary sources in a MIMO context are estimated in [14]. The estimation of the source signal in Single-Input Multi-Output (SIMO) systems has also been studied, in the past, using Second-Order Statistics (SOS) [15, 16, 17, 18, 19]. Following similar principles, the deconvolution of multiple signals has been achieved through SOS [2, 21, 22, 23, 24]. The use of Higher-Order Statistics and ICA has been also used for the blind separation of sources from convolutive mixtures [25, 26, 27, 28, 29, 3]. This paper extends our previous work presented in [31]. The proposed method is a recursive channel shortening algorithm (called channel deflation ) which applies to overdetermined MIMO-FIR channels. The deflation of the channel is achieved by subspace projections exploiting the structure of second order statistics of the signals observed at the receivers. We assume that (a) the sources are independent and temporally white (i.e. white in the time-domain), (b) the channel length is unknown, and (c) the channel may be sparse. Each deflation transformation removes one non-zero tap from the MIMO system equation. Thus after a number of successive deflations the system can be reduced to a single-tap system, and the original sources can be retrieved using instantaneous BSS methods based on higher order statistics, such as ICA. Section 2 describes the model and the assumptions. In Section 3 we present the proposed channel deflation method and simulation results are shown in Section 4. We conclude in Section 5. 2 Problem formulation and assumptions We consider a general linear, time-invariant MIMO system described as follows: x i (k) = or x(k) = L 1 l= j=1 n a ij (l)s j (k d(l)) + v i (k), i = 1,, m (1) L 1 A(l)s(k d(l)) + v(k) (2) l= 2

3 The delays d(),, d(l 1) are non-negative integers sorted in ascending order with d() = and d(l 1) = T 1. T is the channel length: T = d(l 1) + 1 L with equality if and only if d() =, d(1) = 1,..., d(l 1) = L 1, in which case (1) describes the usual convolution operation. However, the delays may not be consecutive numbers, allowing (1) to represent a sparse multi-channel. Our model involves n independent inputs (sources) s 1 (k),, s n (k), and m observed outputs x 1 (k),, x m (k). We assume that there are more observations than sources (m > n), so the multi-channel tap A(l), for any l, is a tall matrix of size m n. The sources and the channel may be complex, in general, as in a typical multipath situation in digital communications. The noise is modelled by m additive white processes v 1 (k),, v m (k). We use the obvious definitions for the source, observation, and noise vectors s(k), x(k), and v(k), respectively. If the channel is sparse, then A(),, A(L 1) is simply the sequence of the non-zero taps. For completeness, we shall denote by Ā the total MIMO channel including the zero taps, as follows { A(l), if l : τ = d(l) Ā(τ) = τ =, 1,, T 1 (3), otherwise so T 1 x(k) = Ā(τ)s(k τ) + v(k). (4) τ= Our goal is to extract the inputs directly (not just filtered versions of them) using only the system output data. Of course, we are constrained by the problem-inherent limitations, i.e., the presence of an indeterminable permutation and scaling. In the following discussion we shall use the time-delayed auto-correlation matrices for the signals involved. We define this matrix for any stationary signal z(k) and delay l as follows: Our assumptions are described next: 1. The sources are mutually uncorrelated R z (l) = E{z(k)z H (k l)} E{s i (k)s j(l)} = i j, any k, l (5) The superscript denotes the complex conjugate. The sources are also white in the timedomain: E{s i (k)s i (k l)} = any i, k, and l (6) 2. Without loss of generality, we assume the following normalization Thus, according to assumptions 1) and 2) we have: where δ(l) is the Kronecker delta function. E{ s i (k) 2 } = 1 i = 1...n. (7) R s (l) = δ(l) I (8) 3

4 3. The noise processes are mutually uncorrelated and white in the time-domain: E{v i (k)v j (l)} = i j, any k, l (9) E{v i (k)v i (k l)} = any i, k, and l (1) 4. The noise power of each sensor is σ 2. According to assumptions 3) and 4) we have: 5. The noise is independent to the sources: 3 Blind Source Separation R v (l) = δ(l) σ 2 I (11) E{s(k)v H (l)} =, any k, l. (12) Our proposed method is based on the properties of the time-delayed auto-correlation matrices R x (l) of the mixture process x. First we shall use the auto-correlations to estimate the channel length and then we shall recursively shorten the channel by removing the taps A(l) based on the subspace properties R x (l). 3.1 Estimating channel length Let us consider the series of auto-correlations R x (τ) of the mixture process x, for all possible delays τ =, 1,. Note that for any τ T the auto-correlation is zero Conversely for < τ < T : R x (τ) = τ T. R x (τ) = E{x(k)x H (k τ)} {[T 1 ] = E Ā(i)s(k i) + v(k) i= [T 1 ]} s H (k j τ)ā H (j) + v H (k τ) j= (13) so R x (τ) = T 1 Ā(i)E{s(k i)s H (k i)}ā H (i τ) i= T 1 (Ā(i)E{s(k + i)v H (k τ)} i= +E{v(k)s H (k i τ)}ā H (i) ) + R v (τ) (14) 4

5 Based on the assumptions 4 and 5, the second line of Eq. (14) is equal to zero. Moreover, Ā H (i τ) = for i < τ, and therefore: T 1 R x (τ) = Ā(i)R s ()Ā H (i τ) (15) i=τ The auto-correlation matrix for the delay τ = T 1 is R x (T 1) = E{x(k)x H (k T + 1))} = Ā(T 1)R s ()Ā H () = Ā(T 1)Ā H () = A(L 1)A H (). (16) Therefore, the channel length T can be estimated by the largest delay ˆl for which R x (ˆl) (see section 4.1). 3.2 Subspace structure of delayed auto-correlations The central idea of our method is to establish a connection between the subspace structure of the matrices R x (l) and the channel taps A(l). Let A(l) = U A (l)σ A (l)v H A (l) describe the economy-size SVD of A(l), l =,, L 1: Recall that m > n, so A(l) is a tall matrix and let A(l) have full column rank. In the economy-size SVD the sizes of the matrices U A (l), Σ A (l), and V A (l), are m n, n n, and n n, respectively. Σ A (l) is a n n diagonal matrix involving only the non-zero singular values, while U A (l) and V A (l) involve only those singular vectors associated with the non-zero singular values. Using the above notation and according to (16) we have R x (T 1) = A(L 1)A H () = U A (L 1)Σ A (L 1)V H A (L 1) V A ()Σ A ()U H A (). (17) We note that the matrices U A (L 1), U A () are tall (size m n) whereas the matrix [Σ A (L 1)V H A (L 1)V A()Σ A ()] is square n n. Therefore, the left null space of U A (L 1) is the left null space of R x (T 1). Similarly, the left null space of U A () is the right null space of R x (T 1). Indeed, for any z we have: The reverse is also true. We have z H U A (L 1) = z H R x (T 1) = (18) z H R x (T 1) = [ z H U A (L 1) ] Σ A (L 1)V H A (L 1) V A ()Σ A ()U H A () = (19) 5

6 Since the n m matrix Σ A (L 1)VA H(L 1)V A()Σ A ()U H A () has full rank it follows that Similarly, z H U A (L 1) =. z H U A () = R x (T 1)z = (2) Now let [ Σx R x (T 1) = [U x Ū x ] ] [V x V x ] H = U x Σ x V H x (21) be the economy-size SVD of R x (T 1). According to the above, the matrices U x, U A (L 1) have the same size (m n) and the same null space. The same is true for the pair of matrices V x, U A (). Therefore, 3.3 Left and right deflation span{u x } = span{u A (L 1)} = L T 1, (22) span{v x } = span{u A ()} = R T 1. (23) In this section we shall introduce the recursive channel deflation process, a series of linear, channel shortening transforms based on the subspace structure of the delayed auto-correlations R x (l). We have two options for selecting the linear projector that will shorten the channel by one tap: 1. The right deflation projector 2. The left deflation projector P = I V x V H x (24) Q = I U x U H x (25) where V x and U x are given in (21). Because of (23) the matrix-tap A() remains unchanged when projected on the subspace R T 1, i.e. A H ()V x Vx H = A() H. Consequently, the projection on the subspace orthogonal to R T 1 using the right deflation projector P, yields : The rest of the taps are transformed as follows PA() =. (26) A (1) (l) = PA(l), l = 1, 2,, L 1 (27) 6

7 We assume that P is not orthogonal to A(l), l = 1, 2,, L 1, thus A (1) (l). If taps are drawn independently from some random distribution this assumption is true with probability 1. Thus the following linear transformation x (1) (k) = Px(k) = A (1) (1)s(k 1) + +A (1) (L 1)s(k d(l 1)) + v (1) (k) (28) v (1) (k) = Pv(k) (29) will produce a new, shorter MIMO channel with the same inputs as the original system, yet, the number of non-zero taps in the new channel is L (1) = L 1, i.e. 1 less than the original MIMO channel. We will say that the channel has been deflated. The new channel length (including the zero taps) is T (1) = d(l 1) d(1) + 1 (3) Note that the difference T T (1) may be greater than 1. For example, if the delays are [d(), d(1), d(2), d(3)] = [, 2, 4, 7], then the original channel would be [A(),, A(2),, A(4),,, A(7)] with length T = 8, while the deflated channel would be [A(2),, A(4),,, A(7)], of length T (1) = 6. In an entirely analogous way, we may use the left deflation projector Q, which is now orthogonal to A(L 1), to obtain the transformation x (1) (k) = Qx(k) = A (1) ()s(k) + + A (1) (L 2)s(k d(l 2)) (31) +v (1) (k) where now v (1) (k) = Qv(k), QA(L 1) =, A (1) (l) = QA(l), l =, 1,, L 2. Again the channel is shortened but now the last tap, A(L 1), is annihilated. The new channel length is T (1) = d(l 2) + 1. Clearly, the above deflation idea can be used recursively to shorten the channel even further to any desired length. For the sake of simplicity, let us assume that we use the left deflation transform. Let us define and note that R (1) x (l) = QR x (l)q H (32) R (1) x (T (1) 1) = A (1) (L 2)A (1)H () (33) It is rather straightforward to see that T (1) 1 is the largest delay l for which R (1) x (l). This allows us to estimate T (1) even though the actual value d(l 2) may not be known. The SVD of R (1) x (T (1) 1) will allow us to construct another set of right and left deflation projectors. The corresponding linear transformation x (2) (k) = Qx (1) (k) or x (2) (k) = Px (1) (k) will annihilate either A (1) (L 2) or A (1) (), thus reducing the number of non-zero taps by one more. Generalizing for D deflations, we note that every projection kills n dimensions. Thus after D successive projections the data x (D) will lie on a (m Dn)-dimensional subspace of IR m. Obviously, there must be enough mixture signals so that m > Dn. Similarly, after D successive deflations the rank of the surviving taps A (D) (l) will be m 2Dn. 7

8 3.4 Reducing the convolutive system to an instantaneous one If enough mixture signals are available then we could reduce the convolutive MIMO system into an instantaneous (memoryless) system. Then the problem will be transformed into a simpler one, i.e. the blind separation of sources from instantaneous mixtures. This is a well studied problem which can be approached by numerous methods (see for example [32, 33]). To that end we need the following assumption: 5) The number of mixture signals is m Ln, (34) Depending on our choice of the series of deflation projectors we may end up on different final memoryless systems. For example, if we constantly use the right deflation transformation then, after (L 1) deflations, the final system will be x (L 1) (k) = A (L 1) ()s(k), while, if the left deflation transformation is constantly used then we ll obtain x (L 1) (k) = A (L 1) (L 1)s(k d(l 1)). In either case, the mixing matrix A (L 1) () (or A (L 1) (L 1)) will have size m n and rank n, i.e. it will have full rank. The proposed technique can be summarized in the following algorithm MIMO Deflation Algorithm 1. Estimate channel length 2. Set q, x () (k) = x(k) 3. While length > 1 do 4. End 3.5 Discussion (a) Calculate the left projector Q or right projector P according to (24) or (25), respectively. (b) Transform the received signal using the chosen projector to obtain x (q+1) (k) = Qx (q) (k) or x (q+1) (k) = Px (q) (k). (c) Estimate new channel length (d) Set q q + 1 The proposed deflation transformation is a linear transformation based on the signal second order statistics. Interestingly, as opposed to most SOS methods, the sources must be white. All the information needed for the channel shortening is not in the spectral shaping of the sources but rather in the subspace structure of the delayed autocorrelation matrices. On the other hand, 8

9 in contrast to most HOS methods, our approach does not rely on the non-gaussianity of the sources and in fact, our method can separate white Gaussian signals. The dimensionality of the output vector must be very rich since each deflation removes dimensions: it is actually a projection to a smaller subspace. Essentially, the method spends dimensions in order to acquire the benefit of reduced channel length. The large number of mixture signals is the main requirement of the method. It can be achieved either by using more sensors or by increasing the sampling rate of the current sensors. There is clearly a cost associated with having more sensors. In comparison, temporal oversampling becomes more attractive, especially with today s extremely high processing speeds and cheap memory costs. Of course, there is a limitation on the sampling rate dictated by the Shannon upper bound and related to the signal frequency content. The transformation of the filter taps according to Eq. (27) decreases the size of the taps since A (1) (l) F Q 2 A(l) F A(l) F. The amount of decrease depends on the angle between the subspace spanned by the columns of V x and A(l). For example, if the two subspaces are orthogonal, i.e. Vx T A(l) = then A (1) (l) = QA(l) = A(l) and so the tap remains the same after the transformation. On the other extreme, if the two subspaces are identical then V x Vx T A(l) = A(l), so A (1) (l) = QA(l) = and the tap is annihilated after the transformation. It is desired that after the final (L 1) th deflation, the leading tap is sufficiently larger than the other ones. In the noiseless case, this means that A (L 1) () F >. This would require that the subspace spanned by the columns of A() is not very close to any of the subspaces V x used in each deflation. Clearly, the proposed deflation procedure can be used in two ways for shortening the channel (a) to a length T > 1, so as to assist other MIMO methods whose performance is heavily influenced by the channel length, or (b) to a length T = 1, as a preprocessing step prior to instantaneous ICA/BSS methods. The second case is preferable since instantaneous ICA/BSS methods are less computationally demanding and more robust to noise. In general, the instantaneous BSS problem is much easier to solve and there is a large literature of related methods. However, in many applications there may not be enough dimensions to spend in order to achieve the desired channel reduction. Computationally, the method is very attractive due to its closed form: there is no iterative cost function optimization, except of course, the fast and numerically well-behaved SVD computation. In simulation almost all CPU time is consumed in the computation of the delayed auto-correlations and the estimation of the channel length, while in comparison, the SVD computation time is negligible. 4 Simulations In this section we demonstrate the performance of the method in terms of the quality of tap annihilation. Furthermore, we perform actual source separation by transforming a convolutive MIMO system into a memoryless one and then employing a standard BSS method to obtain the original sources. 4.1 Channel length estimation In our experiments, we use the following channel estimation procedure: (a) we compute the norm R x (l) F for all l T where T is a very loose upper bound of T, (b) once we find that 9

10 the norm R x (l + 1) F is within 1% of R x (l) F, for some l = l, we compute the mean, µ, and the standard deviation, σ, of the sequence R x (l) F for l = l,, T, (c) we estimate the channel length by ˆT = l for the first lag l such that R x (l) F < µ + 1σ. Our Monte Carlo experiments showed that this procedure achieves very good results, independently of the SNR: In 21, experiments with SNR ranging between 1dB and 4dB, there were 8 channel length misclassifications using this procedure (failure rate less than.4%). 4.2 Blind channel deflation First we performed a double experiment designed to study the effects of (a) different noise levels and (b) different sample sizes N. After L 1 consecutive right deflation transformations (thus obtaining a memoryless system) we measure the size of the transformed taps by their Frobenius norm A (L 1) (l) F, l =,, L 1. We performed two sets of simulations. The first simulation set shows the performance at difference SNR levels. The fixed parameters are as follows: number of sources n = 3, number of non-zero taps L = 7, number of mixtures m = 25 (m > Ln = 21), number of samples N = 5, filter delays {d(),, d(6)} = {, 1, 5, 9, 14, 16, 18}, complex matrix-taps generated by a random generator following the normal distribution, 16-QAM complex sources. The mixtures are contaminated with complex, additive, Gaussian noise. The SNR ranges between 1dB and 3dB. The filter length T is estimated by detecting a jump on the reverse Frobenius norm sequence R x (l) F, starting from an arbitrary upper bound l = T > T and going down to l = 1. The filter length estimation procedure is repeated after each deflation transformation, since the filter length has changed. Fig. 1 shows the mean and standard deviation of the final norm for each tap at each SNR level for 1 Monte Carlo experiments. The size of the leading tap, after the deflation, is between 25 and 35 times greater than the size of the second larger tap. This value depends on the SNR. In the second simulation set we use the same parameters as above except that the SNR is now fixed at 1dB while the sample size varies between N = 5 and N = 1. Fig. 2 shows the clear performance improvement as the number of samples increases. Next we show an actual source separation experiment involving two steps: (1) the application of L 1 consecutive channel deflations which lead to a memoryless system and (2) the application of JADE [34], a well known higher-order BSS method, which separates the sources. Figure 3 shows the constellation of the reconstructed 36-QAM sources obtained from N = 5 samples of m = 25 mixture signals. The complex channel has length T = 18 with L = 7 nonzero taps and SNR level at 3dB. The source-separation quality is measured by the normalized crosscorrelation matrix C = [c ij ] with elements c ij = k ŝ i (k)s j(k)/( k ŝ i(k) 2 k s j(k) 2 ) 1/2 : C = Figure 4 shows the performance of the source separation measured by the Bit Error Rate (BER) of the retrieved signals using the JADE algorithm. This experiment is the result of 1 Monte Carlo simulations of a system with n = 3 sources, m = 18 mixtures, filter length L = 18, and N = 1 data samples. The filter is dense and the SNR varies from 1dB to 4dB. Clearly, both the mean and the variance of the BER diminishes as the SNR increases and practically vanishes for SNR > 25dB. 1.

11 Figure 5(left) shows the change in the Frobenius norm of each tap in a system of length L = 6 after 5 successive deflation transformations (the SNR level is at 3dB). We note that after the q-th deflation step all tap sizes A(q)(p) F for p 5 q decrease somewhat, while for p > 5 q the taps practically vanish. Figure 5(right) displays the mean and the standard deviation of the final relative tap sizes A (L 1) (p) F / A (L 1) () F, p =,, 5, after 1 Monte Carlo simulations. 5 Conclusions This paper introduces a subspace method for recursive channel shortening applicable to m n convolutive MIMO channels. This so-called channel-deflation procedure can be used for blindly estimating the n independent white sources since the channel can be reduced down to a memoryless (instantaneous) system, which can then be solved using well known ICA/BSS methods. Furthermore, the channel may be sparse while the channel length can be estimated using the norms of the delayed auto-correlation matrices. The method is computationally very fast since no iterative, nonlinear optimization is involved. The main drawback is the large number of mixtures required. Nevertheless, channel deflation may still be useful, even when not enough observations are available, because the reduction of the channel size simplifies the problem and facilitates its subsequent treatment. References [1] S. Makeig, A. Bell, T.-P. Jung, and T. Sejnowski, Independent Component Analysis of electroencephalographic data, in Advances in Neural Information Processing Systems 8, M. M. et al., Ed. Cambridge, MA: MIT Press, [2] A. Bell and T. Sejnowski, Learning the High-Order structure of a natural sound, Network: Computation in Neural Systems, vol. 7, pp , [3] J. Liu, X. Zhang, J. Sun, and M. Lagunas, A Digital Watermarking Scheme Based on ICA Detection, in Proc. 4th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA23), Nara, Japan, 23, pp [4] K. Kiviluoto and E. Oja, Independent Component Analysis for Parallele Financial Time Series, in ICONIP 98 Proceedings, (S.Usui and T. Omori), Eds., vol. 2, Kitakyushu, Japan, 1998, pp [5] S. Triadaphillou, J. Morris, and E. Martin, Application of Idependent Component Analysis to Chemical Reactions, in Proc. 4th International Symposium on Independent Component Analysis and Blind Signal Separation (ICA23), Nara, Japan, 23, pp [6] A. Bell and T. Sejnowski, An information maximization approach to blind separation and blind deconvolution, Neural Computation, vol. 7, pp , [7] L. Tong, R. Liu, V. Soon, and Y. Huang, Indeterminacy and identifiability of blind identification, IEEE Trans. Circuits Syst. I, vol. 38, no. 5, pp ,

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14 7 SNR=1db 7 SNR=3db A (L 1) (p) F 4 3 A (L 1) (p) F p p Figure 1: The quality of annihilation of the channel taps is measured by the Frobenius norm of the final taps after L 1 deflations. The figure shows the mean and standard deviation of the final taps after 1 Monte-Carlo experiments for different SNRs. 8 N=5 8 N= A (L 1) (p) F A (L 1) (p) F p p Figure 2: Mean and standard deviation of the norm of the final taps after 1 Monte-Carlo experiments for different sample sizes N. 14

15 1.5 s 1 estimated 2 s 2 estimated s 3 estimated Figure 3: Reconstruction of three 36-QAM sources using a combination of exhaustive deflation and application of the JADE BSS method with 3dB noise level. The MIMO filter has length T = 18 with L = 7 nonzero taps. 15

16 n=3, m=18, L=6, N= BER SNR (db) Figure 4: Mean and standard deviation of the Bit Error Rate after 1 Monte Carlo experiments and for various values of the SNR A(p) F q= q=1 q=2 q=3 q=4 p=5 A(p) F / A() F tap index (p) tap index (p) Figure 5: Top figure: Evolution of the filter tap sizes (Frobenius norm) after q deflations (q =,, 5). Bottom figure: relative tap sizes A (L 1) (p) F / A (L 1) () F for p =,, 5, after all deflation steps are completed. 16