Digital Signal Processing Reference
In-Depth Information
The second of the above assumptions is usual in digital communication,
where scrambling procedures are normally carried out at the transmitters.
As a consequence of this hypothesis, the spectrum of
s
is considered to
be flat (white), as statistical independence implies decorrelation. However,
the converse is not true, which means that the use of a whitening filter as an
equalizer is not sufficient to guarantee the recovery of the transmitted i.i.d.
sequence. Nevertheless, if the channel is minimum phase, the PEF, which
works as a whitening filter, is capable of providing a scaled version of
s
(
n
)
(
n
)
,
in a procedure similar to that of predictive deconvolution.
In fact, in the absence of a priori knowledge about the phase-response
behavior of the channel, the PEF or any other whitening filtering can
only ensure magnitude equalization. The phase distortion remains to be
compensated, which is classically carried out using an all-pass structure. Pre-
dictive equalization can be developed based on this principle by a nonlinear
configuration that combines the use of a PEF with a blind phase equalizer.
Such an approach, originally proposed by da Rocha et al. [49, 81, 195], is
further discussed in Chapter 7.
The above remarks clearly reveal the limitations of the predictive decon-
volution principle in the blind equalization problem, which is important for
well understanding the sequence of results in this chapter. Before addressing
them, let us summarize the key points of the discussion so far:
•
Awhiteningproceduredealsonlywithsecond-orderstatisticsandcan
onlyprovidemagnitudeequalization; itdoesdecorrelatethereceived
signal but does not ensure the recovery of the original i.i.d. sequence.
•
If the phase response of the channel is known a priori, it is possi-
ble to perform blind equalization by dealing only with second-order
statistics. In the specific case of a minimum-phase channel, the PEF
is the whitening filter that works as the optimum equalizer.
•
In the absence of such additional information, the recovery of an
i.i.d. signal requires more than second-order whitening. As dis-
cussed in Chapter 2, the joint cumulant of independent random
variables is null. This means that the concept of whitening must be,
in a way, extended to encompass all higher-order statistics when we
search for the optimum equalizer in an unsupervised mode.
These points can now be further investigated with the aid of two
theorems that constitute the theoretical foundation of blind equalization.
4.2 Fundamental Theorems
As commented in our brief historical, Sato's algorithm was a first practical
solution for blind channel equalization. Nevertheless, this proposition was
