Discrete-time representations - Signal Processing and Optimization for Transceiver Systems - page 141

we find

E 0 ( z )=0 . 2776 z − 1 , 2 . 3440 z − 1 +0 . 8959 z − 2 .

E 1 ( z )= −

Here we have inserted a delay z − 1

into each polyphase component so that the

FSE transfer function is causal:

G 2 ( z )= E 0 ( z 2 )+ zE 1 ( z 2 )= 2 . 3440 z − 1 +0 . 2776 z − 2 +0 . 8959 z − 3 .

− (4 . 54)

The reader can verify by explicit computation that

T ( z )=[ C 2 ( z ) G 2 ( z )] ↓ 2 = z − 1

and similarly R 0 ( z ) E 0 ( z )+ R 1 ( z ) E 1 ( z )= z − 1 . Thus 1) in the

absence of noise. If we had not inserted delays into E k ( z ) we would have the
noncausal equalizer G 2 ( z )= − 2 . 3440 z +0 . 2776 + 0 . 8959 z − 1 .

s ( n )= s ( n −

4.8.2.B Channel with zeros outside the unit circle

A variation of the example is obtained by changing ρ from 0.999 to 1.01. In this
case, the channel has zeros outside the unit circle. So 1 /C 1 ( z ) is unstable, and
the symbol spaced equalizer does not work at all. That is, the plot of s ( n )grows

unboundedly because the tiny channel noise is amplified by the unstable system
1 /C 1 ( z ). The fractionally spaced equalizer, however, works wonderfully, as seen
from the reconstructed signal shown in Fig. 4.26 (bottom trace).
In this example we have

1 . 6342 z − 1 +1 . 0201 z − 2 C 1 ( z )=1 . 0000 −

and

0 . 3171 z − 1 − 1 . 6342 z − 2 − 0 . 3071 z − 3 +1 . 0201 z − 4 . C 2 ( z )=1 . 0000 −

The polyphase components of the channel are

1 . 6342 z − 1 +1 . 0201 z − 2 , 0 . 3071 z − 1 . R 0 ( z )=1 . 0000 − R 1 ( z )= − 0 . 3171 −

Solving for the polyphase components E 0 ( z )and E 1 ( z ) of the equalizer, we find
E 0 ( z )=0 . 2649 z − 1 , 2 . 3183 z − 1 +0 . 8799 z − 2 . E 1 ( z )= −

Again, we have inserted a delay z − 1 into each polyphase component so that the
FSE transfer function is causal:

G 2 ( z )= E 0 ( z 2 )+ zE 1 ( z 2 )= 2 . 3183 z − 1 +0 . 2649 z − 2 +0 . 8799 z − 3 . −

The reader can verify by explicit calculation that T ( z )=[ C 2 ( z ) G 2 ( z )] ↓ 2 = z − 1 .

4.8.2.C Scatter diagrams

For the SSE and FSE systems in the above example, assume now that the in-
put s ( n )isactuallyasymbolstreamfroma6-bitQAMconstellation. This