Micro-programmed Adaptive Filtering Applications - Digital Design of Signal Processing Systems: A Practical Approach - page 456

e½n¼d½ny½n

T
n x n

y½n¼h

where h n ¼ h n [0], h n [1], h n [2], ... , h n ½N 1 and x n ¼ x[n], x[n 1], x[n 2], ... , x[n (N 1) ].

Using these expressions, the mean squared error is written as:

h i

2

T

xðhÞ¼ E ðd½nh

n x n Þ

Expanding this expression results in:

2

T
n

T T

n

xðhÞ¼ E ½d½n þh E ½x n x

n h n

2 h E ½d½nx n

The error is a quadratic function of the values of the coefficients of the adaptive filter and results in
a hyperboloid [1]. The minimum of the hyperboloid can be found by taking the derivative of ı x (h)
with respect to the values of the coefficients. This results in:

¼ R 1 h opt ¼ arg min

h

xðhÞ

x p

T

where E[d[n]x n ].

Computing the inverse of a matrix is computationally very expensive and is usually avoided.
There are a host of adaptive algorithms that recursively minimize the error signal. Some of these
algorithms are LMS, NLMS, RLS and the Kalman filter [1, 2].
The next sections present Least Mean Square (LMS) algorithms and a micro-programmed
processor designed around algorithms.

R x ¼ E ½x n x

n

and p ¼

11.3.2 Least Mean Square (LMS) Algorithm
This is one of the most widely used algorithms for adaptive filtering. The algorithm first computes
the output from the adaptive filter using the current coefficients from convolution summation of:
y½n¼ L 1
k¼

0 h n ½kx½nk

ð 11 : 1 Þ

The algorithm then computes the error using:

e½n¼d½ny½n ð 11 : 2 Þ

The coefficients are then updated by computing a new set of coefficients that are used to process
the next input sample. The LMS algorithm uses a steepest-decent error minimization criterion that
results in the following expression for updating the coefficients [1]:

h nþ 1 ½k¼ h n ½kme½nx½nk ð 11 : 3 Þ

result in fast
convergence but may result in instability, whereas small values slow down the convergence. There
are algorithms that use variable step size for

The factor m determines the convergence of the algorithm. Large values of m

m and compute its value based on some criterion for fast

conversion [5, 6].