It rests on the observation that the maximum of the first K+l terms in the outer sum of (2),
over all sequences with miO), mil), mik—L) arbitrary and mik— Z,+l), ...miK) given, can
be expressed in terms of the maximum of the first K terms over all sequences with m(0), m(l),
...miK—L—1) arbitrary and miK—L), ...m(K—1) given. Thus by iterating on K one can find the
sequence maximizing (2) after an amount of computation proportional to 2 ( L + 1 . This is
much less than expected, but is still too large to be commonly used.
The technique that is usually chosen to combat intersymbol interference is called "linear
equalization". It is a heuristic method most easily explained when the sl(t) form a one
dimensional signal set, i.e. s1 (t) a1 s (t). In that case the demodulator computes rik) = rit),
sit—kT). Instead of processing the r(k) to maximize expression (2), the r(k) are passed
through a time invariant linear digital filter with coefficients ci£) to yield the sequence /*'(&),
r'ik) ~^r(k-t)c{e)
This is readily implementable by using high speed digital logic. The goal is to choose the cik) to
minimize the mean square error
and then to decide mik), such that
i s
closest to r'ik). If one defines
*(0) - 2 s(t), sit-tT) e-jMd
then the mean square error can be expressed as
/ d6iEiamik))2\Ci$)$ie) - 1|2 + \Cid)\2$i$))
if one assumes that the amk) are zero mean and linearly independent. The first term expresses
the effect of intersymbol interference while the second term is due to noise. It is readily found that
the optimum CiS) is
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