DESIGN OF PRECODING AND EQUALIZATION FOR BROADBAND MIMO TRANSMISSION Chi Hieu Ta and Stephan Weiss Communications Research Group, School of Electronics and Computer Science University of Southampton, Southampton SO17 1BJ, UK Keywords: Precoding, equalization, broadband MIMO. tions in [8] and [7] are then extended for the MIMO case in [9]. Palomar et al. [3] generalize the results on joint design Abstract of linear precoding-equalization to several criteria, classified into Schur-concave and Schur-convex objective functions. In We propose a new approach to broadband MIMO precoding general, there the equalizer matrix is derived as MMSE filter and equalisation by the use of a broadband singular value de- and the precoder matrix is determined through the SVD of the composition to decouple the MIMO system matrix into inde- whitened channel. pendent subchannels. In a second step, ISI present in the sub- In this paper, we propose a method for precoding and equal- channels is eliminated using methods developed for SISO sys- ization for point-to-point broadband MIMO channels. Differ- tems. A numerical example is given. ent from block transmission, we utilise in a first step a recently proposed broadband singular value decomposition (BSVD) to 1 Introduction eliminate CCI. In a second step, the decoupled SISO channels are precoded and equalised using standard methods such as Multi-input multi-output (MIMO) systems have attracted much in [8]. attention recently due to their promise to significantly improve The paper is organised as follows. In Sec. 2, the overall the channel capacity and the link reliability [1]. The key rea- channel and system setup are laid out. Sec. 3 addresses the first son for this fact is that in MIMO systems aimed at capacity step in the proposed design, aiming at CCI cancellation, while increases, multiple data streams or signals can be transmitted Sec. 4 considers the minimisation of ISI. Finally, a numerical over the channel simultaneously, which is often referred to as example is provided in Sec. 5, while conclusions are drawn in spatial multiplexing. Sec. 6. In order to exploit a MIMO system for transmission with In our notation, we use lower- and uppercase boldface font increased capacity, one must eliminate co-channel interference for vector and matrix quantities, respectively. Vector quanti- (CCI) between independent data streams as well as eliminate ties in the z-domain are also denoted as underline variables, inter-symbol interference (ISI) in case of a frequency selective e.g. X(z). The operator {·}˜ denotes the parahermitian trans- channel. A considerable amount of research has focussed on pose, i.e. A˜ (z) = AH(z−1). the narrowband case, where e.g. [5] has shown that joint pre- coder and equaliser design can achieve much if deployed on 2 Channel Model and System Setup their own, provided that the channel state information (CSI) can be made available to both sides of the transmission link. In 2.1 Spatio-Temporal MIMO System Matrix the narrowband case the channel matrix can be diagonalised by means of mostly SVD-based operations. In the following, we assumed a stationary MIMO channel with R×T The demand of high transmission rate, however, requires transfer function C(z) ∈ C (z), such that the use of broadband systems which are affected by frequency L−1 selective channels, i.e. the link between any receiver and trans- X C(z) = C[l] z−l (1) mitter antenna pair is characterised by an impulse response. l=0 Therefore, broadband MIMO systems, in which each sub chan- nel between transmit and receive antennas is considered as a characterises a frequency selective MIMO system with T trans- frequency selective fading channel, has drawn great interest of mit and R receive antennas. The maximum support length of researchers. A method that is often used to remove CCI and the channel impulse responses (CIRs) between each pair of R×T ISI in broadband MIMO systems is to introduce redundancy to transmit and receive antennas is L. The matrix Cl ∈ C the data streams. In [8] and [7] optimal solutions to the joint contains the lth time slice of these CIRs. We further assume precoder- equalizer design problem for block transmission over that each of the T MIMO inputs emerges from a time multiplex single input single output (SISO) frequency selective channels of J input lines. Similarly, each of the R output can be demul- are discussed. The optimality criteria considered are the zero tiplexed into J signals. The resulting spatio-temporal MIMO forcing (ZF) and minimum mean square error (MMSE) crite- system with input S(z) ∈ CJT and output R(z) ∈ CJR can ria [8] and the maximization of mutual information [7]. Solu- be written as R(z) = H(z)S(z), whereby the spatio-temporal MIMO matrix, by appropriate ordering of S(z) and R(z), takes only first order in z, as noted earlier. Specifically, the block-pseudo-circulant form −1 H(z) = H0 + z H1 with (4) C (z) z−1C (z) ··· z−1C (z) 0 P −1 1 C0 0 ...... 0 C (z) C (z) ··· z−1C (z) 1 0 2 . .. H(z)= . . (2) . . .. H0 = CL−1 C0 (5) C (z) C (z) ··· C (z) P −1 P −2 0 .. .. . 0 The matrices Cp(z), p = 0, 1,...P − 1, are the P polyphase 0 CL−1 ... C0 components of C(z) such that 0 ... 0 CL−1 ... C1 .. P −1 . X P −p C(z) = Cp(z ) z (3) . . CL−1 p=0 H1 = (6) . . 0 C (z) = P C[nP + p] z−n or alternatively p n . . . 0 ······ 0 2.2 Precoder and Equaliser PT ×PM Since terms with z−1 are only located in the upper right tri- Fig. 1 indicates the positions of a precoder U(z) ∈ C PM×PR angle, two methods can be chosen to eliminate the polynomial and an equaliser B˜ (z) ∈ C in the transmitter and re- ceiver, respectively, whereby M ≤ min{T,R}. As hinted in order of the MIMO system matrix by suppressing H1 such that Sec. 1, the aim of U(z) and B˜ (z) is to diagonalise H(z), thus only a part of H0 in (5) is utilised: (i) transmitting L − 1 lead- eliminating CCI, and to reduce each polynomial on the main ing zeros in S(z), or (ii) discarding the first L − 1 in R(z). diagonal to a scalar element in order to suppress ISI. Thus eliminating the polynomial nature of H(z), the precoder In [5], a narrowband system with P = 1 and H(z) = and equaliser systems can be selected as non-polynomial ma- C[0] = constant is addressed. There, the precoder and equaliser trices whose design can be accomplished using standard linear are not of polynomial form and can be calculated by singular algebraic methods. value decomposition (SVD) of C[0], or a sequence of SVDs in case the received signals cannot be jointly processed, resulting 2.3 Proposed Design in a diagonalised system that therefore avoids CCI but does not The approach to MIMO precoding and equalisation proposed impose ISI because of its narrowband nature. in this paper is based on a broadband singular value decompo- In a broadband scenario for P = 1, the H(z) = C(z) is sition (BSVD) of H(z) as proposed in [2, 6], such that an R × T polynomial matrix of order L − 1. As the number of polyphase components P is increased, the matrix size in- Σ(z) 0 H(z) = B(z) U˜ (z) (7) creases, but the polynomial order reduces in accordance with 0 0 the shortening polyphase responses. Once P = L is reached, the polyphase components Cp(z) are constants with no depen- whereby Σ(z) = diag{Σ0(z), Σ1(z), ··· ΣPM (z)}. This per- dency on z. However, the block-pseudo-circulant form of (2) mits to take the polynomial nature of the MIMO channel trans- means for all P > L, the spatio-temporal MIMO system ma- fer function into account without resorting to block transmis- trix H(z) will be a first order polynomial. sion, giving greater flexibility in the choice of M and hence the In order to overcome the polynomial order, the block-trans- code rate and subsequently the coding gain. However, the di- mission based systems in [8] for T = R = 1 and in [9] for agonalisation according to (7) only eliminates CCI, while each arbitrary T and R rely on a time multiplex that is chosen longer of the PM subchannels is dispersive and causes ISI. than the channel support, i.e. P = M + L. As a result, the To cancel ISI, in a second step a precoder and equaliser can matrix H(z) in (2) is a sparse block-pseudo-circulant matrix of be designed for each subchannel, e.g. according to the SISO design discussed in [8]. Thus, block transmission is invoked, but only for small portion of the system design. In addition, the V(z) precoder and equaliser design can take the individual properties of each subchannel, such as the SNR, into account. ~ X(z) U(z) H(z) B(z) X(z) 3 MIMO System Decomposition Via BSVD MPTP RP MP 3.1 Broadband Singular Value Decomposition Fig. 1. MIMO channel H(z) with precoder U(z) and equaliser In the following we characterise the BSVD described in [2, ˜ B(z). 6], and the resulting properties of the subchannels Σm(z).A BSVD can be obtained via two broadband eigenvalue decom- very robust and stable in achieving both a diagonalisation and positions, whereby a parahermitian matrix R1(z) = H(z)H˜ (z) spectral majorisation of any given covariance matrix, whereby is decomposed such that the algorithm is stopped either after reaching a certain measure for suppressing off-diagonal terms or after exceeding a speci- ˜ R1(z) = B(z)Γ1(z)B(z) .