Digital Signal Processing Reference
In-Depth Information
x(n)
d(n)
MODEL
y(n)
FILTER
FIGURE 7.8
Equation error adaptive bilinear filter.
estimates. Their greatest advantage is their inherent simplicity. In fact, the re-
cursive estimation problem can be converted into a two-channel nonrecursive
estimation problem, as shown in Figure 7.8. In practice, the delayed samples
of the reference signal
d
are used in Equation 7.14 in place of the delayed
samples of the actual output
y
(
n
)
(
n
)
. Equation 7.14 can then be written as
a
T
x
b
T
d
d
T
(
)
=
(
)
+
(
)
+
(
)
(
)
y
n
n
n
n
Cx
n
,
(7.15)
where
x
is the vector of the
N
2
past samples of the reference signal,
a
is the vector formed with the
N
1
coefficients of the nonrecursive part of the linear filter,
b
is the vector formed
with the
N
2
coefficients of the recursive part of the linear filter, and
C
is the
N
2
×
(
n
)
is the vector of the
N
1
past input samples,
d
(
n
)
N
1
matrix formed with the coefficients
c
i, j
of Equation 7.14. According
to the nonrecursive nature of the estimation, the stability problems of the re-
cursive structures are avoided. Moreover, the error surface is quadratic in the
coefficients, and thus it has a unique minimum, unless the autocorrelation
matrix of the input vector is singular. The drawback is that the estimation
procedure leads to a biased estimate of the optimal solution, thus limiting the
performances in the identification procedure.
7.3
Adaptive Algorithms for Nonlinear Acoustic Echo Cancelers
The most popular algorithms for the adaptation of linear filters are the well-
known LMS and NLMS algorithms.
18
,
43
However, LMS and NLMS algo-
rithms, when applied to acoustic echo cancelers, usually exhibit low con-
vergence and tracking rates because of the high degree of correlation of the
speech signal and the high number of coefficients needed for accurate mod-
eling of the echo path. To increase the convergence rate in the presence of
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