Geoscience Reference
In-Depth Information
with
N
elements, containing the same desired signal overlain by dif erent,
uncorrelated noise. h e i rst record is used as the primary input
S
and the second record as the reference input
X
.
As demonstrated by Hattingh (1988), the desired noise-free signal can be
extracted by i ltering the reference input
X
using the primary input
S
as the
desired response
d
. h e minimum error optimization problem is solved by
the least-mean-square norm. h e mean-squared error
e
i
2
is a second-order
function of the weights in the nonrecursive i lter. h e dependence of
e
i
2
on
the unknown weights
W
may be seen as a multidimensional paraboloid
with a uniquely dei ned minimum point. h e weights corresponding to
the minimum point on this error performance surface dei ne the optimal
Wiener solution (Wiener 1945). h e value computed for the weight vector
W
using the LMS algorithm represents an estimator whose expected value
approaches the Wiener solution as the number of iterations approaches
ini nity (Haykin 1991). Gradient methods are used to reach the minimum
point on the error performance surface. To simplify the optimization
problem, Widrow and Hof (1960) developed an approximation for the
required gradient function that can be computed directly from the data. h is
leads to a simple relationship for updating the i lter-weight vector
W
.
h e new parameter estimate
W
i+
1
is based on the previous set of i lter weights
W
i
plus a term that is the product of a bounded step size
u
, a function of the
input state
X
i
and a function of the error
e
i
. In other words, error
e
i
calculated
from the previous step is fed back into the system to update i lter coei cients
for the next step (Fig. 6.6). h e i xed convergence factor
u
regulates the
speed and stability of adaption. A low value of
u
ensures a higher level of
accuracy, but more data are needed to enable the i lter to reach the optimum
solution. In the modii ed version of the LMS algorithm by Hattingh (1988),
this problem is overcome by feeding the data back so that the i lter can have
another chance to improve its own coei cients and adapt to the changes in
the data.
