CCA BASED ALGORITHMS FOR BLID EQUALIZATIO OF FIR MIMO SYSTEMS Javier Vía and Ignacio Santamaría Dept of Communications Engineering University of Cantabria 395 Santander, Cantabria, Spain E-mail: {jvia,nacho}@gtasdicomunicanes ABSTRACT In this work the problem of blind equalization of multiple-input multiple-output (MIMO) systems is formulated as a set of canonical correlation analysis (CCA) problems CCA is a classical tool that finds maximally correlated projections of several data sets, and it is typically solved using eigendecomposition techniques Recently, it has been shown that CCA can be alternatively viewed as a set of coupled least squares regression problems, which can be solved adaptively using a recursive least squares (RLS) PSfragalgo- rithm Unlike other MIMO blind equalization techniques based replacements on second-order statistics (SOS), the CCA-based algorithms does not require restrictive conditions on the spectral properties of the source signals Some simulation results are presented to demonstrate the potential of the proposed CCA-based algorithms h h M h P ITRODUCTIO Blind equalization of FIR MIMO channels is a common problem encountered in wireless and mobile communications Although some higher-order statistics methods have been successfully developed [], it is well known that, under mild assumptions on the source signals and the FIR channels, second-order statistics (SOS) are sufficient for blind equalization [2, 3] However, a common drawback of all these methods is their restrictive conditions on the spectral properties of the source signals In this paper we consider the application of canonical correlation analysis (CCA) to blind equalization of FIR MIMO channels CCA is a well-known technique in multivariate statistical analysis, which has been widely used in economics, meteorology, and in many modern signal processing problems [4] It was developed by H Hotelling [5] as a way of measuring the linear relationship between two multidimensional sets of variables and was later extended to several data sets [6] Recently, the reformulation of CCA as a set of coupled least squares (LS) regression problems has been exploited to derive an adaptive RLS-CCA algorithm [7,8] Furthermore, in [7,9] we have shown that maximizing the correlation among the outputs of the equalizers (ie CCA) provides a good equalization criterion for single-input multiple-output (SIMO) systems In this work, we extend these ideas to the blind equalization of FIR MIMO systems, which is reformulated as a set of CCA problems Provided that the lengths of the FIR channels are known, the source signals can be extracted unambiguously by a set of nested CCA problems The constraint on the lengths of the channels avoids the requirement of colored input signals with different spectra in [], which is very This work was supported by MCYT (Ministerio de Ciencia y Tecnología) under grant TEC24-645-C5-2 h MP Fig MIMO system restrictive in practical wireless systems Based on these ideas, we propose batch and adaptive equalization algorithms and evaluate their performance in comparison to other blind FIR-MIMO equalization techniques 2 BLID MIMO EQUALIZATIO Suppose the multiple-input multiple-output (MIMO) system shown in Fig, where the signals,,, are the outputs of an unknown M-input/P -output finite impulse response (FIR) system driven by M signals,, The channel input-output relationship can be expressed as x = H s = H[l] s[n l], () where x = [,, ] T, s = [,, ] T, h i = [h i,, h ip ] T, and H = [h h M ] is the channel response matrix, which is associated to the transfer function H(z) = H[l]z l l= Denoting the length of the subchannel h ij as L ij, and considering the M single-input multiple-output (SIMO) channel lengths l=
L i = max(l i,, L ip ), we can rewrite () as where x = x = T (h i)s i = T s, i= [ x T,, x T [n K + ]] T, s i = [s i,, s i[n K L i + 2]] T, [ T s = s T,, s T M ], h i[] h i[l i ] T (h i) =, h i[] h i[l i ] T = [T (h ) T (h M )], and K is a parameter determining the dimensions of vectors and matrices This formulation of the MIMO model, which is known as the slide-window formulation [3], can be used to develop a new equalization procedure based on the following conditions Condition (MIMO Channel) a) P M b) The SIMO channel lengths L,, L M are known c) H(z) is irreducible and column reduced Condition 2 (Equalizer length) The parameter K satisfies K L M P M with L = L i Condition 3 (Source signals) The source signals are uncorrelated and (K + )-th order persistently exciting Based on conditions and 2, it can be proved that T is a full column rank matrix [3], and then there exists a set of M matrices W i of size P K i, with i = K + L i, such that i= [W W M ] T T = W T T = ci, where c is some nonzero constant Denoting the k-th column of W i as w ik we can write, for i =,, M and k, l =,, i w T ikx[n + k ] = w T ilx[n + l ] (2) A good review of previously proposed methods for blind equalization of FIR MIMO systems can be found in [2, 3] Some examples are the MIMO constant modulus algorithm (MIMO-CMA) [], which is based on higher order statistics, and the column anchored zero forcing equalizers (CAZE) [2], which requires white source signals Furthermore, the blind identification via decorrelation procedure presented in [3, ] assumes source signals with different spectral properties to be able to work with less restrictive channel conditions The new equalization algorithms proposed in this paper avoid the restrictions on the spectral properties of the source signals, which are replaced by the condition b This condition requires the previous knowledge or estimation [] of the SIMO channel lengths Finally, we must point out that, under the proposed assumptions, multiple sources affected by channels of the same length can only be estimated up to a rotation (unitary) matrix, which is a common indeterminacy in blind MIMO equalization problems [2] This indeterminacy can be solved by exploiting the finite alphabet property of the input signals [2], using higher order statistics, or assuming source signals with different spectral properties [3] 3 CCA-BASED BLID MIMO EQUALIZATIO Let us start considering, without loss of generality, L L M, and defining the following data sets, for i =,, M, k =,, and j =,, i S i = [s i s i[n + ]] T, X k = [x[n + k ] x[n + k + 2]] T, Z ij = [z j z j[n L j + L i]], Z i = [Z i Z ii], where z i = [s i,, s i[n + ]] T is the desired output of the i-th equalizer Based on these definitions, Eq (2) can be rewritten as X k w ik = X l w il, k, l =,, i (3) In the Appendix we prove the following theorem Theorem If the source signals are uncorrelated and (K + )- th persistently exciting then, in the limiting case of, the solutions of (3) satisfy X k w ik = Z ia i, for i =,, M, (4) where a i is an arbitrary vector Taking into account the channel noise, the solution of (3) can be obtained by minimizing J i = k,l= subject to the nontrivial restriction i k= ẑ ik ẑ il 2, w H ikr kk w ik =, (5) where ẑ ik = X k w ik is the k-th estimate of z i, and R kl = X H k X l is an estimate of the crosscorrelation matrix Defining ẑ i = ẑ ik, i k= as the final estimate of z i, we can avoid the interference between different source signals by means of the additional orthogonality restrictions Ẑ H i ẑ i = [,,, β i], (6) where Ẑi denotes the estimate of Zi with the possible effect of a scale and unitary matrix indeterminacy, and β i = ẑ i 2 In this way, the above problem is equivalent to maximize β i = 2 i k,l= ẑ H ikẑ il,
Initialize i = and arrange L L M while i M do Obtain the number of signals to extract si Obtain the number of restrictions to apply i Form the CCA problem with X,, X i Consider the i main solutions [Ẑi Ẑi(i )] Obtain the next si solutions ẑ i,, ẑ i+si Update i = i + si end while PSfrag replacements Algorithm : Summary of the CCA procedure for blind equalization of FIR MIMO channels subject to (5) and (6) This problem is the generalization of canonical correlation analysis to several complex data sets proposed in [7], which is equivalent to the Maximum Variance (MAXVAR) generalization [6] Solving the CCA problem by the method of Lagrange multipliers we obtain the following GEV problem i R iw i = β id iw i, (7) h h P h M MSE (db) 2 3 4 5 6 7 Optimum (LS) Equalizer CCA based Equalizer 2 4 6 SR (db) Fig 2 Performance of the batch algorithm s 3 s 2 s where w i = [w T i,, w T i i ] T, R R i R i = R i R i i, D i = R R i i Then, the main CCA solution is obtained as the eigenvector associated to the largest eigenvalue of (7), and the remaining eigenvectors and eigenvalues are the subsequent solutions of the CCA problem Furthermore, by noting that R kk R kl = X k + X l, where X + k is the pseudoinverse of X k, the GEV problem (7) can be viewed as i coupled LS regression problems J ik = ẑ i β ix k w ik 2, k =,, i, and then, the CCA solution can be obtained by solving these LS regression problems iteratively [7], which is equivalent to the wellknown power method to extract the eigenvectors and eigenvalues of (7) In [7] we have developed an on-line procedure based on the direct application of the RLS algorithm to these i LS regression problems A direct consequence of (4) is that, in the noiseless case, Z i constitutes a basis for the subspace defined by the CCA solutions with canonical correlation equal to, of the CCA problem with data sets X k, for k =,, i Then, from this CCA problem we can extract, up to an unitary matrix, si source signals, where si is the number of inputs affected by a SIMO channel with length L i To this end, the previously extracted signals [Ẑi Ẑi(i )] are considered as the i main solutions of the CCA problem, where i i = (L j L i + ) is the number of orthogonality restrictions to apply, and Ẑij denotes the estimate of Z ij Then, the si source signals are obtained as the subsequent si CCA solutions Algorithm summarizes the batch procedure for the sequential extraction of the source signals It is interesting to point out that, in the limiting case of L = = L M =, the CCA based procedure is equivalent to a prewhitening step, which is typically used in Blind Source Separation (BSS) and Independent Component Analysis (ICA) algorithms Furthermore, the direct application of the RLS based adaptive CCA algorithm presented in [7] is used in the paper to develop an adaptive version of the MIMO blind equalization algorithm Finally, straightforward modifications of this procedure can be used to develop blind equalization algorithms robust to outliers or impulsive noise [8], or to exploit the finite alphabet property of the source signals by means of a soft decision procedure [9] 4 SIMULATIO RESULTS In this section the performance of the proposed algorithms is evaluated In all the simulations the results of 3 independent realizations are averaged We consider 6-QAM source signals distorted by a MIMO channel and corrupted by zero-mean white Gaussian noise The equalizer length parameter is selected as L M K = P M In the first example we compare the performance of the batch CCA procedure with the optimum equalizer, ie the informed (not blind) LS equalizer The MIMO channel is a 3-input/6-output system with lengths L = 6, L 2 = 5 and L 3 = 4; the channel impulse response is shown in Table The performance of the algorithms as a function of the signal to noise ratio (SR) is shown in Fig 2 We can see that the equalization procedure for the first signal is close to the optimum, whereas the remaining signals show a noise floor that is due to the strict orthogonality constraints among the extracted signals and their delayed versions This noise floor decreases when increases ( = in Fig 2) In the second example we compare the performance of the adaptive algorithm with the MIMO-CMA [] Here, a 2 4 MIMO
i n h i h i2 h i3 h i4 h i5 h i6 223 + 896j 482 77j 766 883j 96 + 45j 633 86j 855 + 76j 669 + 438j 58 + 53j 484 692j 747 28j 79 + 683j 9 + 32j 2 66 + 452j 8 249j 269 + 59j 6 923j 2j 287 453j 3 89 79j 453 + 939j 36 + 79j 688 74j 53 669j 36 + 248j 4 3 + 56j 26 43j 77 45j 658 + 547j 723 78j 22 + 833j 5 88 457j 233 67j 86 575j 734 749j 367 + 27j 989 57j 749 + 8j 527 + 23j 74 + 248j 856 + 585j + 678j 8 + 97j 33 + 699j 932 + 788j 487 + 6j 234 + 96j 268 587j 669 + 277j 2 2 325 42j 65 + 897j 49 + 377j 436 557j 983 52j 785 594j 3 794 + 269j 246 87j 489 83j 97 + 23j 2796 54j 629 9j 4 35 784j 644 + 3j 27 + 35j 6 + 475j 68 297j 878 267j 92 44j 2796 689j 88 + 3j 883 98j 742 + 49j 33 49j 224 79j 732 + 6j 65 772j 242 689j 99 + 94j 98 + 955j 3 2 769 + 69j 8 + 64j 775 892j 887 + 5j 772 368j 465 76j 3 359 + 45j 4 + 934j 257 83j 672 + 7j 53 456j 2762 + 276j Table Impulse responses of the MIMO channel used in the simulation examples First signal Second signal 5 5 acements h h P h M 5 PSfrag replacements h h P 5 5 h M 5 5 5 Fig 3 Performance of the adaptive algorithm in a 2 4 MIMO system Constellations of the equalized signals after 2 iterations SR=3dB, λ = 99 channel with L = 6 and L 2 = 5 is constructed from the two first source signals and four first observations of Table The RLS forgetting factor is λ = 99 and the signal to noise ratio is SR=3dB We consider separately the ISI and the co-channel interference, which can be added to obtain the total interference defined in [2] The simulations have shown the strong dependency of the initialization parameters for the MIMO-CMA, which have failed in the extraction of the second signal in 24 out of the 3 independent experiments Fig 3 shows the eye pattern of the proposed adaptive algorithm after 2 iterations and Fig 4 shows the convergence for the two source signals As we can see, the CCA-RLS converges faster than the MIMO-CMA, and, analogously to the batch case, the co-channel interference in the second signal depends on the RLS forgetting factor 5 COCLUSIOS In this paper, new batch and adaptive algorithms for FIR MIMO blind equalization have been presented The algorithms are based on the reformulation of the blind MIMO equalization procedure as a set of nested CCA problems These CCA problems can be rewritten as different sets of coupled LS problems, which can be solved adaptively by means of the RLS Unlike other SOS MIMO blind equalization methods, the CCA-based algorithms do not require strict conditions on the spectral properties of the source signals By means of some simulation results we have shown that the proposed CCA algorithms consistently outperform other blind FIR-MIMO equalization techniques
5 First signal ISI Co Channel Second signal ISI Co Channel which implies { aij[l k] if (l k) (L g ikj [l] = j L i), otherwise, replacements h h P h M Interference (db) 5 5 2 25 MIMO CMA CCA RLS CCA RLS MIMO CMA 3 2 3 4 5 2 3 4 5 Samples x 4 Samples x 4 Fig 4 Performance of the adaptive algorithm APPEDIX PROOF OF THEOREM In this appendix we show that, if the source signals are uncorrelated and (K + )-th order persistently exciting, in the limiting case of, the solutions of (3) will satisfy (4) as well Let us start rewriting (3) and (4), for i =,, M and k, l =,, i, as S j[n + k ]g ikj = S j[n + l ]g ilj, (8) S j[n + k ]g ikj = i Z ija ij, (9) respectively, where a i = [ a T i,, a T ii] T and gikj = T T (h j)w ik is the composite channel-equalizer response Considering that, for uncorrelated signals and i j lim SH i [n + k ]S j[n + l ] =, the equality (8) can be decoupled and rewritten as S j[n + k ]g ikj = S j[n + l ]g ilj, for i =,, M and k, l =,, i, or equivalently S j[n + k ]g ikj = S j[n + k]g i(k+)j, k =,, i Taking into account that a sequence s i is K-th order persistently exciting if and only if S i is of full column rank for some, it can be deduced that the K + L j different columns in S j[n + k ] and S j[n + k] are linearly independent and then, for i, j =,, M where a ij[l k] is some constant Finally, it is straightforward to prove that the above equation yields, for i =,, M, j =,, i and k =,, i S j[n + k ]g ikj = Z ija ij, where a ij = [a ij[],, a ij[l j L i]] T This implies (9) and concludes the proof 6 REFERECES [] Y Li and K J R Liu, Adaptive source separation and equalization for multiple-input/multiple-output systems, IEEE Trans Inform Theory, vol 44, pp 2864-2876, ov 998 [2] Z Ding and Y Li, Blind Equalization and Identification, ew York: Marcel Dekker, 2 [3] G B Giannakis, Y Hua, P Stoica and L Tong, Signal Processing Advances in Wireless and Mobile Communications - Volume I, Trends in Channel Estimation and Equalization, Prentice-Hall, 2 [4] A ehorai and A Dogandzic, Generalized multivariate analysis of variance: a unified framework for signal processing in correlated noise, IEEE Signal Proc Magazine, vol 2, pp 39-54, Sep 23 [5] H Hotelling, Relations between two sets of variates Biometrika, vol 28, pp 32-377, 936 [6] JR Kettenring, Canonical analysis of several sets of variables, Biometrika, vol 58, pp 433-45, 97 [7] J Vía, I Santamaría and J Pérez, Canonical Correlation Analysis (CCA) algorithms for multiple data sets: Application to blind SIMO equalization, European Signal Processing Conference (EUSIPCO 25), Antalya, Turky, Sept, 25 [8] J Vía, I Santamaría and J Pérez, A robust RLS algorithm for adaptive canonical correlation analysis, Proc of 25 IEEE International Conference on Acoustics, Speech and Signal Processing, Philadelphia, USA, March 25 [9] J Vía and I Santamaría, Adaptive Blind Equalization of SIMO Systems Based On Canonical Correlation Analysis, IEEE Workshop on Signal Processing Advances in Wireless Communications (SPAWC 25), ew York, USA, June 25 [] Y Hua, S An and Y Xi, Blind identification of FIR MIMO channels by decorrelating subchannels, IEEE Trans Signal Processing, vol 5, pp 43-55, May 23 [] A Gorokhov and Ph Loubaton, Subspace based techniques for second order blind separation of convolutive mixtures with temporally correlated sources, IEEE Trans on Circuit and Systems, vol 44, n 9, pp 83-82, September 997 g ikj [l] = g i(k+)j [l + ], g ikj [K + L j ] = g i(k+)j [] =,