8
P. A. HUMBLET
Eiam{k))2
Eiam{k))2$id) + l
which yields a mean square error equal to
V Eiam(k))2
{ Eiam(k))2$id) + l
Note that $(0), and thus C(6), are constant if Nyquist's criterion is met. In that case only e(0) is
non zero. As can also be observed from (3) this technique yields good results as long as $(0) does
not have nulls. Fortunately this is the case on channels used for commercial data transmission.
A similar minimization can be done when c (*) is restricted to have only finitely many non zero
coefficients. It yields a system of linear equations that the filter coefficients must satisfy.
The previous theory is interesting as it allows to quantify the mean square error, but at first
sight it does not appear to be practically useful: it assumes that the channel response hit) is
known. If this were the case, we might as well design sit) to avoid intersymbol interference
altogether.
What makes it important, in fact what makes high data rate transmissions over telephone
channels possible, is the discovery in the 1960's that the filter coefficients ci£) can be adjusted by
the demodulator itself to compensate for the effects of the channel response. This is called
"adaptive equalization".
The basic observation is that the partial derivative of the mean square error with respect to
ci£) is equal to the expected value of the product lir'ik) am^k*) rik—£). r' can be measured.
a
m(k) j
s e
j
t n e r
known
a
priori if a training sequence is sent by the modulator, or can be estimated
if the filter coefficients are already such that the probability of error is small. The demodulator
can thus estimate a descent direction and update the filter coefficients while data is being
transmitted, thus continuously adjusting for variations in the channel response.
Techniques for adaptive equalization and the study of their rate of convergence and steady
state performances are still active topics of research, specially for channels that are rapidly varying
or for which $(0) has nulls. The previous discussion constitutes only an introduction to the
subject.
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