Discrete-time representations - Signal Processing and Optimization for Transceiver Systems - page 153

In this section we describe how MMSE equalizers are designed. We will
first address the case of a MIMO channel and then consider the special case of
channels with single input and output. Thus, we begin with the equalization
of a discrete-time channel (Fig. 4.37) with transfer matrix C ( z ) , possibly rect-
angular. This matrix is presumably obtained by discretizing a continuous-time
channel as in Sec. 4.3 (the notation H d ( z ) is replaced with the more convenient
C ( z ) here). The equalizer G ( z )isassumedtohavetheform

G ( z )= ∞ g ( m ) z −m , (4 . 67)

m = −∞

even though in practice it is desirable to make it a causal and FIR system. We
make the assumption that the input s ( n )andthenoise q ( n ) are jointly wide
sense stationary processes (Appendix E). So all the signals involved are WSS,
including the reconstruction error

e ( n )= s ( n ) − s ( n ) . (4 . 68)

Readers not familiar with Wiener filtering may want to review Appendix F at
this time, especially the orthogonality principle (Sec. F.2 in Appendix F). As
for other background material, the discussions in this section depend heavily on
the language of random processes (power spectra, autocorrelations, and so forth)
reviewed in Appendix E.

y ( n )

v ( n )

s ( n ) s ( n )

+

C ( z ) G ( z )

channel equalizer

q ( n )

Figure 4.37 . A channel with transfer function C ( z ) and additive noise q ( n ),

4.10.1 MMSE equalizers based on Wiener filtering

The reconstructed signal s ( n ) is a linear combination of samples of the received

noisy signal y ( n ), that is,

s ( n )= ∞ g ( m ) y ( n − m ) , (4 . 69)

m = −∞

where the received signal is

∞

y ( n )= c ( m ) s ( n − m )+ q ( n ) . (4 . 70)

m = −∞