Discrete-time representations - Signal Processing and Optimization for Transceiver Systems - page 142

1

0.5

0

-0.5

-1

0 200 400 600 800 1000

n

Figure 4.26 . Performance of the FSE when the channel has zeros outside the unit
circle. The transmitted and reconstructed signals are shown (upper and lower traces,
respectively; vertical shifting is for clarity). The channel C 1 ( z ) is second-order FIR
with zeros at 1 . 01 e ± 0 . 2 jπ ,so1 /C 1 ( z )is unstable . The SSE method yields unbounded
results for s ( n ) , but the FSE method works well, as the above plot shows.

constellation has 64 symbols, as shown in Fig. 4.27 (top). If such a symbol
stream is transmitted over the noisy channel then the received signal

s ( n )after
equalization will contain distorted versions of the constellation symbols because
some of the channel noise still gets through the equalizers.
Assume that s ( n ) is a symbol stream with uncorrelated symbols. If we plot
the locations of the received noisy symbols in the complex plane for a large
number of received samples, we get a plot like the one shown in Fig. 4.27
(middle). In this example, the channel is as in Eq. (4.52) and the equalizer
1 /C 1 ( z ) (SSE) is assumed to be used. The noise is assumed to be zero-mean
complex with independent real and imaginary parts, each having a variance of
0.00004. We see that the constellation symbols are spread out, or scattered, due
to the random noise from the channel. The plot in the figure is called a scatter
diagram . It is similar to the noise cloud diagram shown in Fig. 2.6, Chap. 2.
From the scatter diagram we see that, if the noise is large enough, a received
symbol can be mistaken for a neighboring symbol.
Instead of using the SSE, if we use the fractionally spaced equalizer (4.54),
then the scatter diagram is as shown in the bottom plot of Fig. 4.27. The scatter
diagram for the FSE is clearly much better than that for the SSE.
For reference, Fig. 4.28 shows what happens to the constellation if the chan-
nel noise is added directly to the input constellation (bottom). Clearly, the effect
of noise acting directly on the constellation is negligible. But when the noise is
amplified by the equalizers, its effect on the received symbols is more noticeable,
as seen from Fig. 4.27. Still, the use of FSE produces a much better scatter
diagram than SSE in this example.