mohatta2015.pdf

86 LINEAR EQUALIZATION
There are many similarities with fractionally spaced MF. Consider the case of
a nondispersive channel (one path), a receive filter perfectly matched to the pulse
shape, and a receive filter sampled at the correct time (perfect timing). Unlike the
MF case, we also need the pulse shape to be root-Nyquist for a one-tap LE to be
sufficient. In this case, LE becomes equivalent to MF. If the pulse shape is not
root-Nyquist, then multi-tap LE is needed. A fractionally spaced LE is needed if
there is excess signal bandwidth. Unlike MF, possible noise sample correlation due
to pulse MF and sampling needs to be accounted for.
The story is similar for a dispersive channel. If the paths are symbol-spaced, the
receive filter is root-Nyquist and perfectly matched to the pulse shape, and the filter
output is sampled at the path delays (perfect timing), then symbol-spaced LE is
sufficient. Symbol-spaced LE can also be used when there is zero excess bandwidth.
Otherwise, fractionally spaced MF is needed. If the excess bandwidth is small, the
loss due to symbol-spaced LE may be acceptable.
4.3.6 Performance results
Results were generated for QPSK with root-Nyquist pulse shaping. In Fig. 4.5,
BER vs. Eb/Nu is shown for the two-tap, symbol-spaced channel with relative
path strengths 0 and —1 dB and angles 0 and 90 degrees (TwoTS). Results are
provided for the matched filter, the analytical matched filter bound (REF), MISI
linear equalization, and MMSE linear equalization. The LE results correspond to
31 symbol-spaced taps centered on the first signal path.
Observe the following.
1. MMSE LE performs better than MISI LE as expected. At high SNR, the
performance becomes similar, as ISI dominates and MMSE focuses more and
more on ISI.
2. At low SNR,, MMSE LE, MF, and the MFB become similar, as noise dominates.
3. At low SNR, MISI LE performs worse than the MF, because MISI LE focuses
on ISI when noise is the real problem.
Results for fractionally spaced equalization and for fading channels are given in
Chapter 6.
4.4 MORE MATH
In this section we consider the extended system model. We briefly discuss full
zero-forcing, which is not always possible. Then, we focus on the MMSE and ML
solutions. In the CDM case, this leads to equalization weights that depend on the
spreading codes, which change every symbol period. A simpler solution is considered
based on averaging out the dependency of certain quantities on the spreading
codes. More approximate models of ISI are examined as a way of simplifying linear
equalizer design. Finally, the ideas of block, sub-block, and group linear equalization
are examined.