Digital Filters - Digital Signal Processing: A Practitioner's Approach - page 67

3.9.2 Step Size

In all the methods the choice of plays an important role. Various authors [5] have
adopted different techniques for the choice of . From the value of s k we can obtain
the incremental value of p. A sign change in the slope of a function indicates that a
minimum or maximum occurs in between these points. Hence to find the minimum
for the function given by (3.34), we need to find the sign change in the slope of
r J k ð p Þ E f e k s k g . Let us define a variable k such that

k ¼ sign ð E f e k s k gÞ ½ ; ð 3 : 40 Þ

which may take the value 1or þ 1 depending on r J k ð p Þ . We change the step size

to half its original value if we see any change in k . In other words, if

j k k 1 j¼ 2 ; ð 3 : 41 Þ

it implies that a sign change has occurred and the value of changes as

new ¼ old

2 ;

ð 3 : 42 Þ

otherwise

new ¼ old : ð 3 : 43 Þ

By changing the value of for a change in k , the value of p converges to the
precise value in fewer iterations.

3.9.3 Performance
A signal with additive noise such that the SNR of the signal is 10 dB is
implemented for different normalised frequencies of the incoming signal from
0.1 to 0.4 at SNR ¼ 10 dB. A few values are shown in Table 3.1. It shows the
performance of the algorithm expressed as the nominal mean and its deviation from
the true value as an error. The algorithm is now used to demodulate an FSK by

Table 3.1 Performance of the algorithm

Normalised frequency Mean Error

0.2000 0.2000 1 : 7762 10 6

0.2100 0.2098 2 : 4752 10 4

9443 10 5

0.2200 0.2200 2 :

4 : 4484 10 6

0.2300 0.2300