Adaptive Filters and Applications - Digital Signal Processing: Fundamentals and Applications - page 461

dJ
dw ¼ 0:96 ð0:032Þ¼0:992
Finally, substituting w z w 3 ¼ 0:992 into the MSE function, we get the minimum J min as
J min z 40 20w þ 10w 2 w¼0:992 ¼ 40 20 0:992 þ 10 0:992 2 ¼ 30:0006
As we can see, after three iterations, the filter coefficient and minimumMSE values are very close to the theoretical
values obtained in Example 10.1.

w 3 ¼ w 2 m

Application of the steepest descent algorithm still needs an estimation of the derivative of the MSE
function that could include statistical calculation of a block of data. To change the algorithm to do
sample-based processing, an LMS algorithm must be used. To develop the LMS algorithm in terms of
sample-based processing, we take the statistical expectation out of J and then take the derivative to
obtain an approximation of dJ
dw

, that is,

2

2

J ¼ e ðnÞ¼ðdðnÞwxðnÞÞ (10.9)

dw ¼ 2 ðdðnÞwxðnÞÞ dðdðnÞwxðnÞÞ

dJ

¼ 2 eðnÞxðnÞ (10.10)

dw

Substituting dJ
dw

into the steepest descent algorithm in Equation (10.8) , we achieve the LMS algorithm
for updating a single-weight case as

w nþ 1 ¼ w n þ 2 meðnÞxðnÞ (10.11)

where m is the convergence parameter controlling speed of convergence. For example, let us choose
2 m ¼ 0 : 01. In general, with an adaptive FIR filter of length N , we extend the single-tap LMS algo-
rithm without going through derivation, as shown in the following equations:

yðnÞ¼w n ð 0 ÞxðnÞþw n ð 1 Þxðn 1 Þþ / þ w n ðN 1 Þxðn N þ 1 Þ (10.12)

for i ¼ 0 ; / ; N 1
w nþ 1 ðiÞ¼w n ðiÞþ 2 meðnÞxðn iÞ

(10.13)

The convergence factor is chosen to be

1
NP x

0 < m <

(10.14)

where P x is the input signal power. In practice, if the ADC has 16-bit data, the maximum signal
amplitude should be A ¼ 2 15 . Then the maximum input power must be less than

P x < 2 15 2

¼ 2 30