Adaptive Filters and Applications - Digital Signal Processing: Fundamentals and Applications - page 460

dJ
dw

dJ
dw

dJ
dw

dJ
dw

Case 0 and 0

Case 0 and 0

J

J

J n
J n 1

J n
J n 1

J min

J min

w *

w n 1 w n w

w n w n 1 w

w *

FIGURE 10.6

Illustration of the steepest descent algorithm.

coefficient w nþ 1 will be decreased to approach the optimal value w . When dJ

dw ¼ 0, the best coef-

ficient w nþ 1 is reached.

EXAMPLE 10.2
Consider the following quadratic MSE function for the Wiener filter:
J ¼ 40 20w þ 10w 2
Use the steepest decent method with an initial guess of w 0 ¼ 0 and m ¼ 0:04 to find the optimal solution for w
and determine J min by iterating three times.
Solution:
Taking a derivative of the MSE function, we have
dJ
dw ¼20 þ 10 2w n

When n ¼ 0, we calculate

dw ¼ 0:04 ð20 þ 10 2w 0 Þ w 0 ¼0 ¼0:8
Applying the steepest decent algorithm, it follows that

dJ

m

dJ
dw ¼ 0 ð0:8Þ¼0:8

w 1 ¼ w 0 m

Similarly for n ¼ 1, we get

dw ¼ 0:04 ð20 þ 10 2w 1 Þ w 1 ¼0:8 ¼0:16

dJ

m

dJ
dw ¼ 0:8 ð0:16Þ¼0:96

w 2 ¼ w 1 m

and for n ¼ 2, it follows that

dw ¼ 0:04 ð20 þ 10 2w 2 Þ w 2 ¼0:96 ¼0:032

dJ

m