Digital Signal Processing Reference
In-Depth Information
h
i
T
h
opt
¼
R
1
R
yd
ð
3
:
52
Þ
yy
which is known as the Wiener-Hopf Equation.
3.5.2 Adaptive Filters
An adaptive filter is a programmable filter whose coefficients (i.e., impulse
response values, {h(k)}) are changed or adapted. Normally one seeks to adapt the
coefficients so as to give an optimal estimate x
ð
n
Þ
of the original signal {x(n)} at
the time instant n, and an adaptive feedback algorithm is used to update the filter
coefficients. The measure of optimality is typically based on the information
available for the given application. Illustrative examples will be presented later in
this
Sect. 3.5.7
.
Because of their ability to adapt the coefficients, adaptive filters cannot only be
used for ''homing in'' on the optimal Wiener solution for time-invariant scenarios,
they are also useful for applications in which conditions are continuously varying.
They can, for example, be used to compensate for time-varying channel condi-
tions. Figure
3.23
shows a generic adaptive filter structure.
3.5.3 Choice of the Desired Signal
In communications systems, a ''training sequence'' is sent before transmission of
data. The receiver knows this signal and utilizes a copy of it as the desired signal
d(n). As such the adaptive filter can adapt coefficients to the point of near opti-
mality during the short period in which the training sequence is transmitted.
In noise cancelers, d(n) can be selected to be the observed data y(n), or a
delayed version thereof. This signal differs from the true signal x(n) in that it
contains additive random noise. In such a scenario, the adaptive filter is usually
Estimate of
x
(
n
)
^
x
(
n
)
Observed
signal
Channel
distortion/
noise
Adaptive
Filter
x
(
n
)
y
(
n
)
Error,
e
(
n
)
d
(
n
)
Reference signal
Fig. 3.23
Generic adaptive filter configuration
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