Geoscience Reference
In-Depth Information
In most practical applications the adaptive process is oriented towards
minimizing an estimation error
e
. h e estimation error
e
at an instant
i
is
dei ned by the dif erence between the desired response
d
i
and the actual i lter
output
y
i
, which is the i 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 i lter characterized by a vector of i lter weights
W
with
f
elements, the i lter output
y
i
is given by the inner product of the transposed
vector
W
and the input vector
X
i
.
h e choice of desired response
d
that is used in the adaptive process depends
on the application. Traditionally,
d
is a combination signal that is comprised
of a signal
s
and random noise
n
0
. h e signal
x
contains noise
n
1
that is
uncorrelated with the signal
s
but correlated in some unknown way with
the noise
n
0
. In noise canceling systems the practical objective is to produce
a system output
y
that is a best i t in the least-squares sense to the desired
response
d
.
Dif erent approaches have been developed to solve this multivariate
minimum error optimization problem (e.g., Widrow and Hof 1960, Widrow
et al. 1975, Haykin 1991). h e selection of one algorithm over another is
inl uenced by various factors including the rate of convergence (the number
of adaptive steps required for the algorithm to converge closely enough to
an optimum solution), the misadjustment (the measure of the amount by
which the i nal value of the mean-squared error deviates from the minimum
squared error of an optimal i lter, e.g., Wiener 1945, Kalman and Bucy
1961), and the tracking (the capability of the i lter to work in a nonstationary
environment, i.e., to track changing statistical characteristics of the input
signal) (Haykin 1991).
h e simplicity of the least-mean-squares (LMS) algorithm, originally
developed by Widrow and Hof (1960), has made it the benchmark against
which other adaptive i ltering algorithms are tested. For applications in
earth sciences we use this i lter to extract the noise from two signals
S
and
X
, both containing the same signal
s
, but with uncorrelated noise
n
1
and
n
2
(Hattingh 1988). As an example, consider a simple duplicate set of
measurements on the same material, e.g., two parallel stable isotope records
from the same foraminifera species. You would expect two time-series, each
