dc

e
n
y
n
2E en 2E en 2Eenxnk 2 Rex
( k)
wk wk wk
R ( k)
E e x
ex n n k
• Optimality condition for minimum mean-square error
w
0 for k 0, 1,....,
N
• k Mean-square error is a second-order and a parabolic function of tap weights as a
multidimentional bowl-shaped surface
• Adaptive process is a successive adjustments of tap-weight seeking the bottom of the
bowl(minimum value )
• Steepest descent algorithm
– The successive adjustments to the tap-weight in direction opposite to the
vector of gradient )
– Recursive formular ( : step size parameter)
1
wk( n 1) wk( n) , k 0, 1,....,
N
2 w
k
w ( n) R ( k), k 0, 1,....,
N
k
ex
• Least-Mean-Square Algorithm
– Steepest-descent algorithm is not available in an unknown environment
– Approximation to the steepest descent algorithm using instantaneous estimate
R ( k)
e x
ex n nk
w ( n 1) w ( n)
e x
k k n nk
• LMS is a feedback system
• In the case of small , roughly similar to steepest descent algorithm