Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 63

3.8.3 Criteria
We have two parameters, r and p, in the filter. Suppose we need to adapt the filter so
its output is maximum. This may sound vague but in a radio receiver we tune, or
adjust, the frequency so the listening is best. We now know the sensitivity of the
filter - how the parameter affects the characteristics of the filter and to what degree.
We define a criteria function J ð p Þ , known as the objective function, as follows:

J ð p Þ¼ E ð e k Þ

!
;

X

N 1

1
N

¼ e i ð 3 : 26 Þ

i ¼ 0

where e k ¼ x k u k . Now we plot J ð p Þ against parameter p (Figure 3.23). Notice
that a minimum occurs at a single point. It is also essential that most of the time we
must know the slope of this function given as S ð p Þ¼r J ð p Þ .

0.5

0.4

0.3

Distinct minima

0.2

0.1

0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

p

Figure 3.23 Criteria function J ð p Þ

This function is depicted in Figure 3.24. Perhaps you are wondering how these
functions have been arrived at numerically. We describe this in greater detail in
Chapter 6, where we give applications and some problems for research. In brief, the
filter is excited by a pure sine wave and J ð p Þ is computed using (3.26). Computing
the slope of J ð p Þ requires deeper understanding.

3.8.4 Adaptation
The most popular algorithm for adaptation is the steepest gradient method, also
called the Newton-Raphson method. For incrementing the parameter p, we use the
gradient r J ð p Þ , defined as

d J ð p Þ
dp :

r J ð p Þ¼ ð 3 : 27 Þ