A frequency domain approach for blind identification with filter bank precoders P. P. Vaidyanathan and Bojan Vrcelj Dept. Electrical Engineering, California Institute of Technology Pasadena, CA 91125, USA ppvnath@systems.caltech.edu, bojan@systems.caltech.edu Abstract. 1 It is well-known that filter bank precoders can be used for blind identification as well as equalization of FIR channels. In this paper we introduce a new blind iden- tification scheme which directly identifies the frequency do- main equalizer coefficients. The precoder redundancy re- quired for this is the same as in the earlier methods, but the proposed method offers simplicity. For example closed form formulas are involved rather than iterative computa- tion of annihilating eigenvectors as in earlier methods. I. INTRODUCTION Figure 1(a) shows a digital transmultiplexer structure used in communications. In recent years this structure has been studied in great depth [1]-[4]. Its usefulness in channel equalization and blind identification has been recognized [1], [4]. A tutorial overview of the theory and applications of this system is available in the companion paper [6]. In the system shown we can regard s k (n) as symbol streams from M users. In some applications these independent streams may have been derived from a single user (as in DMT systems) but this detail is not relevant in our dis- cussion here. In general the received signal  s k (n) suffers from interference from other users (s m (n),m = k) and also from distortion due to the noisy channel C(z). We will assume that the channel is FIR with order ≤ L, C(z)= L  n=0 c(n)z −n (1) We also assume P > M, so the transmultiplexer has re- dundancy. More specifically, we let P = M + L (2) as in [2], [3]. Writing the filters in polyphase form [5] H m (z)= P −1  k=0 z k E mk (z P ),F m (z)= P −1  k=0 z −k R km (z P ) (3) we can redraw Fig. 1(a) as in Fig. 1(b). The system shown in the box is the blocked version C b (z) of the channel. As in [2]–[4] we constrain E(z) and R(z) to be constants E and R. Then the filters F k (z) and H k (z) have order ≤ P − 1. We further assume as in [3] that R =  R 1 0  1 Work supported in part by the ONR grant N00014-99-1- 1002, USA. which is called the zero-padding constraint. Here R 1 is M × M , and is referred to as the precoder matrix. (a) C(z) channel s (n) 0 s (n) 1 s (n) M−1 transmitter filters P F (z) 0 P P F (z) M−1 F (z) 1 + noise e(n) receiver filters s (n) 0 s (n) 1 H (z) 1 H (z) 0 H (z) M−1 P P P s (n) M−1 z z z z −1 z −1 z −1 C(z) channel E(z) s (n) 0 s (n) 1 s (n) M−1 (b) s (n) 0 s (n) 1 s (n) M−1 R(z) vector s(n) vector s(n) P P P P P P C (z) b y(n) e(n) Fig. 1. (a) The M -user transmultiplexer, and (b) polyphase version. With filters restricted as above, we have (ignoring noise)  s(n)= EAR 1 s(n) (4) A represents the effect of the channel completely: A =            c(0) 0 ... 0 c(1) c(0) ... 0 . . . . . . . . . . . . c(L) 0 c(L) . . . . . . . . . 0 0 ... c(L)            (5) Aim of the paper. If the FIR channel C(z) is known, then by appropriate choice of FIR filters F k (z) and H k (z),