On the Convergence of Steepest Descent and Least Mean-Square Algorithms for MIMO Systems Alberto Zanella IEIIT-BO/CNR, DEIS/University of Bologna 40136 Bologna, Italy Email: azanella@deis.unibo.it Marco Chiani IEIIT-BO/CNR, DEIS/University of Bologna 40136 Bologna, Italy Email: mchiani@deis.unibo.it Moe Z. Win LIDS, Mass. Instit. of Tech., Cambridge, MA 02139, USA Email: moewin@mit.edu Abstract—In this paper, we investigate convergence properties of both the steepest descent (SD) and least mean-square (LMS) algorithms applied to a multiple-input-multiple-output system in a Rayleigh fading environment with correlated fading. For a given value of the adaptation step, we evaluate the probability that the algorithms are stable. Then, we compare two stable strategies to choose the adaptation step of SD and analyze their speed of convergence. Our results are valid for an arbitrary number of transmit and receive antennas for the uncorrelated fading, and for an arbitrary number of receive antennas less or equal to the number of transmit antennas for the case of correlated fading. I. I NTRODUCTION In the past few years, the increasing demand for higher capacity has brought the proposal of several transmission schemes based on multiple antennas at both the transmitter and the receiver [1], [2]. They are called multiple-input-multiple- output (MIMO) systems and are able to provide a high spectral efficiency in a rich and quasi-static scattering environment. Owing to the large computational complexity required for this scheme, a simplified version, called V-BLAST (Vertical BLAST) has been proposed in [3]. A BLAST scheme [1], [3] is primarily based on the following three steps: a) interference nulling to reduce the effect of the other (interfering) signals on the desired one; b) ordering to select the substream with the largest signal-to-noise ratio (SNR); c) successive interference cancellation (SIC). Moreover, it was observed also in [1] that a mean square error criterion can be used instead of zero forcing for interference and noise mitigation. An analytical evaluation of the performance of MIMO systems with mean square error criterion and uncorrelated flat fading environment can be found in [4]. In BLAST systems, the computational bottleneck is due to the signal processing during the phases of nulling and ordering [5] of received signals; in particular, they require operations of matrix inversion with a consequent increase in terms of computational complexity. This motivates the investigation of robust low-complexity algorithms to cope with this problem that can affect real-time applications. In the literature, some efficient algorithms [5], [6] have been proposed to reduce the computational cost of the nulling and cancellation steps for BLAST systems. In this work, we consider some well-known iterative algorithms, such as the steepest descent (SD) or the least mean-square (LMS) algorithms, to perform ordering and nulling. We investigate the convergence properties of these algorithms when different values of parameters such as the adaptation step, the number of transmit and receive antennas, and signal-to-noise ratio are considered. Our results can be used in choosing the more suitable value of the adaptation step as a tradeoff between numerical stability and speed of convergence. Moreover, we consider two strategies for the choice of the adaptation step and investigate their speed of convergence. II. SYSTEM MODEL The MIMO system investigated in this work is characterized by N T transmit and N R receive antennas; the original data stream is divided into N T substreams which are simultaneously transmitted, by N T M -PSK modulators, to the propagation channel. The resulting spectral efficiency is therefore N T × log 2 M [bit/s/Hz]. At time k, the N R -dimensional signal z(k) at the output of the receiving antennas can be written as z(k)= Cb(k)+ n(k) (1) where b(k) accounts for the transmitted symbols with E {b(k)} = 0 and E  b(k)b(k) †  = I 1 , n(k) is the additive Gaussian noise vector with E  n(k) · n(k) †  = ν 2 I, and ν 2 = N 0 /E D , where E D is the mean (averaged over fading) received energy of the signal transmitted by a generic antenna and N 0 /2 is the two-sided thermal noise power spectral density per antenna element. The matrix C is the (N R × N T ) channel matrix: C =   | | | c 1 c 2 ... c NT | | |   , (2) with columns consisting of the propagation vector c j corre- sponding to the j th transmitted symbol. In this work, we consider a flat Rayleigh fading environment with correla- tion among the receiving antennas. The elements of C are modelled as Gaussian r.v’s with zero-mean, independent real and imaginary parts with equal variance, with correlation matrix indicated as Σ = E  c j c † j  for j =1,...,N T and E  |c i,j | 2  =1. In case of uncorrelated fading Σ = I. 1 The superscript † denotes conjugation and transposition. 0-7803-8533-0/04/$20.00 (c) 2004 IEEE IEEE Communications Society 677