Transceivers with decision feedback equalizers - Signal Processing and Optimization for Transceiver Systems - page 634

Example 19.4: Effect of increasing the block size
In Chap. 18 we observed that zero-padded optimal linear transceivers have the
property that the symbol error probability increases with block size M. A similar
behavior is also observed in the case of zero-padded optimized DFE transceivers
with or without zero forcing. This is demonstrated in Fig. 19.20 for channel
C 1 ( z ) used in earlier examples. To obtain these plots we have used 2-bit PAM
with σ s = 1 and channel noise σ q =0 . 01 . The channel input power per symbol
was p 0 /M =5 , and the channel energy is normalized to 4 in this example. Figure
19.21 shows similar plots for the channel C 2 ( z )=1+2 z − 1 + z − 2 with energy
normalized to 4 and power per symbol p 0 /M =1 . All other parameters are as
above. In these examples, the error probabilities for the linear transceivers are
very large, even for moderately large block sizes.
Note that the error probability increases monotonically with block sizes for
all four jointly optimal transceivers - linear as well as DFE transceivers, with
or without zero forcing. No theoretical proof of this monotonic behavior is
known. In Chap. 8 where we considered lazy zero-padded precoders with zero-
forcing linear equalizers, we formally proved that the error probability increases
monotonically with block size M (see Appendix 8.A of Chap. 8). For the case
of jointly optimized transceivers (linear or DFE), such as the ones shown in
Fig. 19.20, such a formal proof has not been reported in the literature, though
counter examples have not been known either.
Note finally that, even though the symbol error probability increases mono-
tonically with block size, for the case of DFE systems it tends to saturate at a
much lower value than for linear transceivers. Thus, even with C 2 ( z ) , which has
a double zero on the unit circle, the error probabiliy for the DFE system without
zero forcing continues to be excellent as M grows. For the linear transceivers,
however, the error probability becomes impractically large in this example.