Digital Signal Processing Reference
In-Depth Information
E
[
e
2
(
n
)]
min
0
B/C
w
(0)
FIGURE 7.9.
One-weight performance curve.
Taking the mean of the general squared-error function, (7.17), results in a general
mean-squared-error performance function:
[
]
=
[
]
-
[
]
[
]
()
()
() ()
() ()
2
2
T
T
T
Ee n
Ed n
2
Edn
XWWXXW
n
+
E
n
n
(7.24)
Again notice that the mean value of any sum is the sum of the mean values. The
product values of
d
and
X
and
X
with
X
T
cannot be further reduced since the mean
value of a product is the product of mean values only when the two variables are
statistically independent;
d
and
X
are generally not independent. This is still the
same second-order performance surface as before, but now it is not fluctuating with
d
and
X
but is rigid. However, if
d
and
X
are statistically time varying, the error
surface will wiggle as the statistics of
d
and
X
change.
7.5 SEARCHING FOR THE MINIMUM
In this section we deal with how the weights should be adjusted to find the minimum
in a reasonably efficient fashion. Of course, the weights could be adjusted randomly,
but life is too short. Since we will be dealing with real-time events and changes that
must be tracked, we need a relatively fast way of reaching the minimum.
Consider the one-weight system again to get an idea of how this search can be
conducted. Initially, the weight will equal some arbitrary value
w
(0,
n
), and it will
be adjusted in a stepwise fashion until the minimum is reached (Figure 7.10), The
size and direction of the step are the two things that must be chosen when making
a step. Each step will consist of adding an increment to
w
(0,
n
). Notice that if the
current value of
w
(0,
n
) is to the right of the minimum, the step must be negative
(but the derivative of the curve is positive); similarly, if the current value is to the
left of the minimum, the increment must be positive (but the derivative is negative).
This observation leads to the conclusion that the negation of the derivative indi-
cates the proper direction of the increment. Since the derivative vanishes at the
minimum, it can also be used to adjust the step size. With these observations we
conclude that the step size and direction can be made proportional to the negative
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