Graphics Programs Reference
In-Depth Information
6.10 Adaptive Filtering
The fi xed fi lters used in the previous chapters make the basic assumption
that the signal degradation is known and it does not change with time. In
most applications, however, an
a priori
knowledge of the signal and noise
statistical characteristics is usually not available. In addition, both the noise
level and the variance of the genuine signal can be highly nonstationary with
time, e.g., stable isotope records during the glacial-interglacial transition.
Fixed fi lters thus cannot be used in an nonstationary environment without a
knowledge of the signal-to-noise ratio.
In contrast, adaptive fi lters widely used in telecommunication industry
could help to overcome these problems. An adaptive fi lter is an inverse
modeling process, which iteratively adjusts its own coeffi cients automati-
cally without requiring any
a priori
knowledge of signal and noise. The
operation of an adaptive fi lter includes, (1) a fi ltering process, the purpose
of which is to produce an output in response to a sequence of data, and (2)
an adaptive process providing a mechanism for the adaptive control of the
fi lters weights (Haykin 1991).
In most practical applications, the adaptive process is oriented towards
minimizing an error signal or cost function
e
. The estimation error
e
at an in-
stant
i
is defi ned by the difference between some desired response
d
i
and the
actual fi lter output
y
i
, that is the fi ltered version of a signal
x
i
, as shown by
where
i
=1, 2, …,
N
and
N
is the length of the input data vector. In the case
of a nonrecursive fi lter characterized by the vector of fi lter weights
W
with
f
elements, the fi lter output
y
i
is given by the inner product of vector
W
and
the input vector
X
i
.
The selection of the desired response
d
that is used in the adaptive process
depends on the nature of the application. Traditionally,
d
is a combined sig-
nal that contains a signal
s
and random noise
n
0
. The signal
x
contains a noise
n
1
uncorrelated with the signal
s
but correlated in some unknown way to the
noise
n
0
. In noise canceling systems, the practical objective is to produce a
system output
y
that is a best fi t in the least-squares sense to the signal
d
.
Different approaches have been developed to solve this multivariate min-
imum error optimization problem (e.g., Widrow and Hoff 1960, Widrow
et al. 1975, Haykin 1991). Selection of one algorithm over another is in-
fl uenced by various factors: the rate of convergence (number of adaptive
