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2004 IEEE Sensor Array and Multichannel Signal Processing Workshop E'ILTERBANK PRECODING FOR MIMO FREQUENCY SELECTIVE CHANNELS: MINIMUM REDUNDANCY AND EQUALIZER DESIGN Hjaya krishna. A, K. 1.! S.Hari vkrishna@protocol.ece.iisc.ernet.in, hari@ece.iisc.ernet.in, Dept. of ECE, Indian Institute of Science, Bangalore - 560012. ABSTRACT This paper deals with the problem of designing zero forcing finite impulse response (ZF-FIR) equalizers for multiple input multiple output (MIMO) FIR channels, in a polynomial matrix framework. When channel knowledge is available at the transmitter, a precoding operation, which introduces redundancy, can he performed to enable FIR equalization at the receiver. We derive the expression for the minimum rcdundancy required to render an arbitrary MlMO channel matrix polynomially invertible. The non-uniqueness of the FIR inverse can be uscd to design equalizers based on different criteria such as noise minimization, low delay and low complexity. We provide ;L solution for the equalizcr which minimizes the output noise powcr. Simulation results are provided to demonstrate the effectiveness of the proposed design. On the other hand, methods which perform FIR equalization at the receiver [3][5]in general require n/r > N and the channel matrix to be irreducible. 'A polynomial matrix H ( z ) is polynomially invertible if and only if it is irreducible, i.e, it has full rank for all z # 0, including z = 00. We present a filterbank precoding framework which enables FIR equalization for an arbitrary channel (which may even be singular) of any dimension. At the transmitter, the sourccc signal vector is processed by an FIR prccoder before being transmitted. The precoder is chosen in such a way that the precoded channel is FIR invertible. The minimum redundancy required to accomplish FIR invertibility is derived. It is shown that the design freedom available for the precodcr can be used to aid in the design of the noiseminimizing equalizer. 2. THE PRECODING FRAMEWORK 1. INTRODUCTION Multiple antenna communication systems have attracted a lot of attention due to the substantial increase in data rates that they facilitate. But in the presence of intersymbol interference (ISI), equalization becomes a challenging task. Space-time processing methods [ 1][2] tackle IS1 by converting the frequcncy sclcctivc MIMO channel into a set olparallel flat tiding channels. On the other hand, FIR equalizer designs [ 3 ] view the FIR channel as a polynomial matrix in the delay operator 2 - l . Availability of channel knowledge at the transmitter can also be exploited to combat IS1 and to improve system performance. Space-time precoding methods [ 11 use the channel information to allocate power along the singular modcs of the block-Sylvester matrix corresponding to the channel. But because of the implicit IJR nature of the decoding operation, there is a trade-off between the achievable rate and the performance of thc system. In [4],the knowledge of the 11.1x N channel H ( z ) is utilized to design a pre-equalizer, i.e, a polynomial matrix T ( z )such that H ( z ) T ( z )= I , at the transmitter. But this method requires hd 5 N , and assumes that the channel coefficient matrices are orthogonal. 0-7803-8545-4/04/$20.00 02004 IEEE Consider the MIMO signal model given by L1/-1 y(*n) = H(7L - k j s ( k ) +.(en) (1) k=O whcre x ( k > i s the N-length input vector, y ( n ) is the Mlength received signal vector, N ( k ) , k = O , l , ..., LH - 1 is the time domain representation of the M x N frequency selcctive MIMO channel, and v ( n ) is the M-length noise vector. In the absence of noise, we can write Y ( z )= H(zjX(z) 1 (2) in the z-domain. Now, an FIR equalizer, (left inverse) for H ( z ) exists if and only if il.12 N and W ( z ) is irreducible. Thereforc, given an arbitrary H ( z ) ,we wish to find an N x I - ( K 5 N ) precoder matrix G ( z ) , so that the 11.1 x Ir' composite channel matrix C(z) = W ( ; ) G ( z ) has an FIR left inverse. N - K is the redundancy introduced by the precoder. If the channel matrix N(z ) is already FIR invertible, then we can chose G ( z ) = I (N = K ) , so that no redundancy is introduced. The question we address here is: what is the maximum K (minimum redundancy N - IT) 692 for which G(z) is irreducible?. Before we tackle this question; we will look at Some properties of the Smith form of a polynomial matrix. We can always choose 2.1, Smith form so that g i ( z ) I c , ( t ) , 1 5 i 5 K . Since the minimum rank of H ( z ) is T, Given a p x q polynomial matrix A(z) with normal rank T , T 5 min(p, q), we can write [6] A(z) = E ( z ) B ( z ) F ( z ) (3) where E ( z ) and F ( z ) are unimodular matrices (square irreducible matrices) of sizesp x p and g x q respectively. B ( z ) is a y x 4 matrix of the form B ( z ) = diay(n~(zj,cr2(z), . . . ,a,(z), 0,. . - 0 ) such that c y i ( z ) I ni+l(z) (i.e, a i ( z ) divides cxi+l(z)).The m o n k polynomials cyi(z) arc called thc kvnrilrnt polyrromiUES of A ( z ) . Wc say that A ( z ) is equivalent to B ( r ) and write A ( z ) B ( z ) . Since E ( z ) and F ( z ) are unimodular matrices, it follows from Q. 3 that rank[A(z)] = rank[B(z)] Vz. Thercfore, if the minimum rank of A(B) ovcr all values o f z is r,, it can be shown that q ( z ) = Q ~ ( z = ) . I - = cyTnL(z)= 1. If an 'ri x k (n > k ) matrix P ( z ) is irreducible, then its Smith form is - P(z' [ IkXk O(n-k)xk 1 hl(Z) = (4) .... = h,,,*(z)= 1 and hr_+1(z), "'.', hr(z) where W ( z )is an arbitrary IC x Ii' unimodular matrix. Thcn, we have r IF,, X T r r ~ O(n.1- T,, T,,, ) X r,, 1 If A' 5 T , , ~, we have irrcducible. be the miiiimunr rank of the given M x H ( z ) . An N x K precoder G ( z ) which retzders the coniposite channel nlntrix C(Z) = H ( z ) G ( z ) poiynomially irivertible exists fatidonly (f K 5 rTn. Theorem 2 Let N FIR charinel - Now, since r m is the minimum rank of H(zz),the submatrix formed by the first T , columns ~ of D ( z ) is Therefore, C ( z ) 2.2. Minimum Redundancy 1 Now we will prove the if part. Let H ( z ) = U ( z ) D ( z ) V ( z be ) the Smith form decomposition of the channel matrix. Choose Proof follows directly from the Binet-Cauchy formula [61. # where T is the normal rank of H ( z ) . Hence, hi(t) I ~ ( z ) for 1 5 i 5 K only if K 5 T,. Therefore, C ( z ) is irreducible only if K 5 T ~ . The following result is stated in [ 6 ] . Theorem 1 Let Pi(z),,i = I,2, ~ L I W invciriunt polytzornid s {ai~(~), aia(t),....,~+~(.z)], i = 1,2, where ~i is the normal rank o f e ( z ) . Let P ( z ) = P1(z)P~(z) have nornial rank T , with ivivarianfpol~~ronzials {cq ( z ) , ., cyT(z)}.. Then, for 1 5 k 5 T a i d i = 1,2, crik(z) I a k ( z ) . we have - I K x hO(A1-K) x K I . Therefore, C ( z ) is a This result can be seen as a generalization to the MIMO case of the minimum redundancy results obtained for the multicarrier modulation case [7J. 2.3. Minimum rank and channel zeros Proofi We will first prove thc only i f part. Let { c i ( z ) } {hi(.)} , and (.gi(z)) denote the invariant polynomials ofC(z), H ( z )and G(a)respectivcly. Suppose C ( z )is irreducible. Then '"' - [ IKXK O(M-K)xK I Given an M x N channel H ( z ) with normal rank T , let A(z) denote the g.c.d of the minors of order T of H ( z ) (if H ( z ) is square and has full normal rank, then A(z) is its determinant). Expanding A(;) in term5 of its factors, we have 693 The rank of H ( z ) drops below the normal rank T only at the channel zeros 6,. z = 1.2,. . . , s. We now relate the minimum rank of H ( t ) to the maximum among the multiplicities of the channel zeros. Corollary 1 Let I- and r, be the normal rank and the mininziirn rank o f f H ( z ) respectively. Let A(z) be as in Eq. 5. Then we have r,,, 2 7’ - maxl{r,} A random non-square polynomial matrix is irreducible with probability 1 [S]. Therefore, given an arbitrary M x N channel H ( z ) with ~$1# iV, filterbank precoding utilizes all the degrees of freedom i.e, the throughput is min(M, N ) symbols per channel use. If AI < N , then the channel can be completely pre-equalized i.e, H ( z ) G ( z ) = 1. If h1 > N , FIR equalization can be accomplished without precoding. Compared to space-time processing methods [1][2], our scheme provides better throughput at a much lesser complexity. When M N , we need a redundancy of 1 i.e, K = N - 1. For large N , the rake of our scheme is in the same range as that of the space-time precoding method [ I ] . 3. EQUALIZER DESIGN cz=o .-I F ( L F - ~ )] Theorem3 r f C is tall and furl rattk arid the noise v ( n ) in Eq. I is white, the equalizer which minimizes the output noise power is given by F = J(CHC)-’CH. Proo$ The output of the equalizer can be written as Y ( z )= F ( Z ) C ( Z ) X ( Z )f F ( z ) V ( z ) (8) Using Eq. 6 and Eq. 7, we can rewrite Eq. 8 as Y(.) = FCX(n)-k rn({2) (9) where X ( n ) and U(71)are the blocked versions of z ( n )and v ( n ) , of dimensions K ( L c LF - 1) x 1 and M L p x 1 respectively. Thc covariance matrix of the noise component of y ( a )is given by + The design of thc equalizer can be based on different criteria such as low delay, low complexity and noise minimization. The polynomial matrix framework enables one to utilize many of the results in systems theory literature regarding the construction of inverse systems [9]. In this section, wc derive the noise-minimizing equalizer, using the design freedom available at thc precoder. Given the M x K irreduciblc precodcd channel C ( z ) , the equalizer design problem is to find a polynomial matrix L ,., - 1 F(r) = F ( i ) z p i such that F ( z ) C ( z ) = I . This can also be written as FC = 3,where F = \ F ( 0 ) F(1) rank if C(2 ) is column-reduced. Any polynomial matrix can be brought to colurnn-reduced form by multiplying it to the right by a suitable unimodular matrix [6]. From the proof of theorem 2. it follows that we have the freedom of choosing a K x A” unimodular factor of G(z). We can choose that unimodular matrix such that C ( z ) becomes column-reduced, resulting in a full rank C. In the following section, superscript H denotes Herrnitian transpose. M y = E[FUV”FH]= F F H (10) in view of the white noise assumption. The ooise-minimizing equalizer is obtained by minimizing t r [ F F Hsubjcct J to the constraint .FC = J . Using the Lagrange multiplier method, wc get 2 3 - CCH = 0 (1 1) where C is the K x K ( L c + L F - 1) Lagrange multiplier matrix. From Eq. 11, we have (6) 23c and 3 = [ I 0 . . . 0 1. 7 and J are K x M L F ,and C is M L F x K ( L c L F - 1).C is tall if L F 2 -.K ( L r - 1 ) C is full rank if and only if [ lo] = LCHC (12) From Eq. 11 and Eq. 13, we have + i C ( z ) is irreducible and column-reduced. L F 2 maxlSj5nr-K v$ - 1, where v$ are the Kronecker indices of the dual space of span(C(z)). The degrees of all the columns of C ( z )are equal. The third restriction can be overcome by deleting the null columns of C as explained in [ 3 ] . Therefore, given an irreducible C ( z ) and with L F chosen large enough, C is full 4. SIMULATION RESULTS Simulations were carried out with 3 x 2 channel coefficient matrices Hi with independent Gaussian distributed complex coefficients (Rayleigh fading) with an exponential power delay profile. LH was chosen to be 4.Source signals were generated from a BPSK constellation. The noise generated 1 I 694 the minimum redundancy required to enable FIR equalization of an arbitrary channel. We have shown that our scheme achieves better rates than block-based schemes Iike spacetime precoding, with lesser complexity. 6. REFERENCES A. Scaglione, P. Stoica, S. Barbarossa, G. B, Gian- nakis, and H. Sampath, “Optimal designs for spacetime linear precoders and decoders,” ZEEE Trans. Signal Processing, Vol. 50, pp. 1051-1063, May 2002. 5 10 15 m G . G. Raleigh and J. M. Cioffi, “Spatio-tempora1 coding for wireless communications,” IEEE Trans. Cummun., Vol. 46, pp. 357-366, Mar. 1998. I 25 30 SNU ldB1 5. K. Tugnait, “FIR inverses to MTMO rational trans- Fig. 1. BER performance of the MN-FIR equaher (without precoding) and the ST decoder (with precoding). fer functions with applications to blind equalization,” in Proc. 30th Asiloniur cot$ on Signals, S y s t e m arid Computers, vol. 1, pp. 295-299, 1997. was spatially and tcmporally white. Thc results were averaged over 100 random channels. For comparison purposes, we implemented the space-time (ST) prccoding and dccoding [ 1, Lemma 31. Since Hi wcre randomly generatcd and H,-,,,-l was full rank, the channel was both irreducible and column reduced. Hence, noprecoding was pcrforrned and the minimum noise FIR (MN-FIR) equalizer was used. ST precoding involved transmission along the eigcnmodes of ‘H“7-l where ‘H is the block-Sylvester channel matrix. The block size was chosen to be 32. H. Sampaih, H. Bolcskei, and A. J. Paulraj, “PreEqualization for MIMO wireless channels with delay spread,”in Proc. IEEE-VTS Full VTC-2000,vol. 3, pp. I 175- 1 I78.2000. As mentioned in scc.2.2, for non-square random chan- Y.-P. Lin and S.-M, Phoong, “Minimum redundancy for IS1 frec FIR filterbank transccivers,” lEEE Trans. Signul Processing, vol. 50, pp. 842-853, Apr. 2002. V. Pohl, V. Jungnickel, E. Jorswieck, and C. von Helmolt, “Zero forcing equahzing filter for MIMO channels with intersymbol interference," in Proc. IEEE PIMRC-2002,2002. T. Kailath, Linear systems, Englewood ClilTs, NJ: Prenticc-Hall, 1980. nels, our scheme is full rate, i.e, the throughput for [he 3 x 2 channel is 2 symbols per channel use (2 bitslseclHz). In the S T precoding scheme, the rate is limited by the conditioning o f X H R and by the zero padding used to avoid IBI. Fig. 1 shows the BER curves for the MN-FIR equalizer (with a rate of 2 b/s/Hz) and the ST dccoder (with rates of 1.70 b/s/Hz and I .76 b/s/Hz). X. -G. Xia, W. Su, and H. Liu, “Filterbank precoders for blind equalization: Polynomial Ambiguity Resistant Precoders (PARP),” IEEE Trans. Circuits arid Sysrcnts-l, vol. 48, pp. 193-209, Feb. 2001. B. Moore and L. Silverman, “A new characterization of feedforward delay-free inverses,” IEEE Trans. Info. Theory, vol. 19, pp.126- 129, Jan. 1973. The ST decoding method involves thc eigen decomposition of R H X ,of dimension N ( L c P - 1) x N ( L c + P - 1) with P >> Lc. Therefore its computational complexity and latency arc much higher than that of the MN-FIR equalizer. But it has the advantage o f trading off rate for performance. + P. Loubaton, E. Moulines, and P. Regalia, “Subspacc method for blind idcntification and deconvolution,” in Sigiinl processing advarices iri wireless comrnunicution‘s, Englewood Cliffs, NJ: Prentice-Hall, 2000. 5. CONCLUSION In this paper, we have presented a filterbank precoding scheme for MIMO frequency selective channels. We have derived 695