mohatta2015.pdf

THE MATH 43
to the medium response (multiplying by the conjugate of the single path, medium
coefficient).
If timing is not ideal (to φ To), then the single-path medium response must be
modeled by an equivalent, multi-tap medium response with tap delays corresponding
to the sample times. The subsequent filtering effectively interpolates to the ideal
sampling time. From a Nyquist sampling point of view, the samples must be fractionally
spaced when the signal has excess bandwidth (usually the case). Whether
fractionally spaced or symbol-spaced, the noise samples may be correlated depending
on the pulse shape used. However, with MF we do not need to account for this
correlation as long as the noise was originally white, the front-end filter was truly
matched to the pulse shape, and we use medium response coefficients to complete
the matching operation.
The story is similar for a dispersive channel. If the paths are symbol-spaced, the
receive filter perfectly matched to the pulse shape, and the filter output is sampled
at the path delays (perfect timing), then symbol-spaced MF is sufficient. Symbolspaced
MF can also be used when there is zero excess bandwidth. Otherwise,
fractionally spaced MF is needed.
2.3.7 Whitened MF
Another front end that allows discrete-time formulations is whitened matched filtering
(WMF). The idea is to perform a matched filter front end. This produces
one "sample" per symbol. However, the noise that was white at the input is often
correlated across samples. Such noise correlation can be accounted for in the receiver
design, but the design process is usually simpler if the noise is white. This
can be achieved (though not always easily) by whitening the samples.
If the pulse shape is root-Nyquist and a symbol-spaced channel model is used,
then performing partial MF to the pulse shape and sampling once per symbol period
gives uncorrelated noise samples. Further matching to the symbol-spaced medium
response would simply be undone by the whitening filter. Thus, in this specific
example, partial MF and whitened MF are equivalent. (We will use this fact in
Chapter 6 to relate the direct-form and Forney-form processing metrics.)
However, in general PMF and WMF are not equivalent. For fractionally spaced
path delays, matching to the medium response requires fractionally spaced PMF
samples. The subsequent whitening operation is on symbol-spaced results, so that
the symbol-spaced whitening does not undo the fractionally spaced matching.
The advantage of the WMF is that only one sample per symbol is needed for further
processing and the noise samples are uncorrelated. The disadvantage is that it
requires accurate medium response estimation, which typically involves partial MF
anyway. Also, computation of the WMF introduces a certain amount of additional
complexity. When the symbol waveform is time-varying, these computations can
be more complex. Use with CDM systems complicated things further.
In the remainder of this book, we will focus on the partial MF front end. In the
reference section of Chapter 6, references are given which provide more details on
the design of the WMF. While we will not use this front end directly, it is good to
be aware of it, particularly when reading the literature.