TURBO EQUALIZATION - Adaptive Signal Processing: Next Generation Solutions - page 164

Figure 3.9 Turbo equalizer using a linear interference canceler in place of the inner
decoder.

unit delay operator convention, we observe that

H ( z ) d i ¼ X h l z l d i ¼ X

L L

h l d il

l¼ 0 l¼ 0

which correctly captures the convolution operation of the channel.
The feedforward filter P ( z ) is recognized as the “matched filter” [9] to the channel,
that is, having an impulse response which is a time-reversed version of the channel
impulse response, but shifted to be causal. The feedback filter Q ( z ) has an impulse
response that behaves as the autocorrelation function of the channel impulse response
minus the center term, but again shifted to be causal. The impulse response terms fq k g
of Q ( z ) are thus derived from the channel autocorrelation terms according to
q kþL ¼ r k ¼ X
i

h i h iþk , k ¼+ 1, + 2, ... , +L:

With d i denoting the estimated channel symbols produced by the decoder, the output
of the interference canceler becomes

v i ¼ P ( z ) H ( z ) d i Q ( z ) d i þP ( z ) b i
¼ z L H ( z 1 ) H ( z ) d i [ H ( z 1 ) H ( z 1 ) r 0 ] d i þP ( z ) b i
¼ r 0 d iL þ X
k¼ 0

2 L

q k ( d ik d ik ) þ X

L

p k b ik :

k¼ 0

k L

=

The first term is the desired channel symbol delayed by L samples and scaled by r 0 , the
next term represents the residual intersymbol interference, and the final term is the
filtered noise. If the channel decoder correctly estimates the channel symbols, then
d i ¼ d i and the intersymbol interference terms vanish.