TRACKING ANALYSIS OF SOME SPACE-TIME BLIND EQUALIZATION ALGORITHMS Magno T. M. Silva and Maria D. Miranda Mackenzie Presbyterian University Electrical Engineering Postgraduate Program Rua da Consolac ¸˜ ao, 896, 01302-970, S˜ ao Paulo, SP, Brazil Emails:{magnotmsilva, mdm}@mackenzie.br ABSTRACT Although space-time blind equalization algorithms have been widely studied in the last decade, their convergence and tracking behavior are not yet completely characterized. In this paper, we address the tracking analysis of space-time blind quasi-Newton algorithms, using the energy conser- vation relation. We obtain an expression to adjust the al- gorithms to reach the same steady-state mean-square error. Assuming a low degree of nonstationarity and high signal- to-noise ratio, close agreement between analytical and sim- ulation results are observed. 1. INTRODUCTION Blind adaptive algorithms are usually employed for signal recovery in several digital communication applications. In nonstationary environments, it is important to study their ability to track channel changes. The literature contains some steady-state analysis for CMA (Constant Modulus Algorithm) based on the energy conservation relation [1, 2] and on the stochastic averaging method [3]. Although the energy conservation relation method tends to require stronger assumptions than the latter, it can bypass many of the difficulties encountered in the nonlinear algorithm anal- ysis [4]. The CMA tracking analysis of [2] was extended in [5] to blind quasi-Newton algorithms that minimize the Godard cost function [6]. It was also verified that the ratio between the minimum mean-square error for the Shalvi-Weinstein Algorithm (SWA) [7] and CMA is the same obtained be- tween the RLS and LMS algorithms [8]. In this paper, we extend these results to quasi-Newton space-time blind al- gorithms that minimize the multiuser Godard cost function [9]. We consider the energy conservation relation method and some assumptions of the MU-CMA (Multiuser-CMA) MSE analysis presented in [10]. In the sequel, data model is presented and some space- time blind algorithms are revisited. Then, we present the tracking analysis, followed by simulation results. We close the paper, making some concluding remarks. 2. DATA MODEL A MIMO system with N sources and with an antenna ar- ray which has L>N sensors is depicted in Fig. 1. The source sequences a i (n), i =1,...,N satisfy the circularity condition and are assumed independent and identically dis- tributed (i.i.d.), with the same statistics, independent from one another, non Gaussian, and zero-mean. The transmit- ted signals suffer intersymbol and co-channel interferences. The channel H ij (z) from the i th source to the j th sensor is modeled by an FIR filter with K c coefficients and η i , i =1, 2,...,L represent additive white Gaussian noise. The outputs of the L sensors are processed with N parallel space-time FIR equalizers, each with K t time diversity and M = LK t taps. The i th equalizer’s output can be written as y i (n)= u T (n)w i (n − 1), where u(n)=[ u T 1 (n) u T 2 (n) ··· u T L (n)] T , u p (n)=[ u p (n) u p (n−1) ··· u p (n−K t −1) ] T , with p =1, 2,...,L, and w i (n − 1) is the equalizer weight vector at time n − 1. () z H 1 () an 2 () a n () N a n 1 () n η 2 () n η () L n η () z W 1 () u n 2 () u n () L u n 1 () y n 2 () y n () N y n channel equalizer Fig. 1. MIMO equivalent system model. In the case of joint blind simultaneous recovery of all input signals, the multiuser Godard cost function is given by [9] J G = N  i=1   J Gi + α N  j=1,j=i δ1  δ=0 |r ij (δ)| 2   , (1)