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.