Adaptive Filters - Digital Signal Processing and Applications with the C6713 and C6416 DSK - page 251

which is the difference between the desired signal d ( n ) and the adaptive filter's
output y ( n ). The weights or coefficients w k ( n ) are adjusted such that a mean squared
error function is minimized. This mean squared error function is E [ e 2 ( n )], where E
represents the expected value. Since there are k weights or coefficients, a gradient
of the mean squared error function is required. An estimate can be found instead
using the gradient of e 2 ( n ), yielding

( ) = () + () - ( )

wn + 1 wn 2 b enxn k k = 0 1 , ,..., N - 1 (7.3)

k k

which represents the LMS algorithm [1-3]. Equation (7.3) provides a simple but
powerful and efficient means of updating the weights, or coefficients, without the
need for averaging or differentiating, and will be used for implementing adaptive
filters. The input to the adaptive filter is x ( n ), and the rate of convergence and accu-
racy of the adaptation process (adaptive step size) is

.
For each specific time n , each coefficient, or weight, w k ( n ) is updated or replaced
by a new coefficient, based on (7.3), unless the error signal e ( n ) is zero. After the
filter's output y ( n ), the error signal e ( n ) and each of the coefficients w k ( n ) are
updated for a specific time n , a new sample is acquired (from an ADC) and the
adaptation process is repeated for a different time. Note that from (7.3), the weights
are not updated when e ( n ) becomes zero.
The linear adaptive combiner is one of the most useful adaptive filter structures
and is an adjustable FIR filter. Whereas the coefficients of the frequency-selective
FIR filter discussed in Chapter 4 are fixed, the coefficients, or weights, of the adap-
tive FIR filter can be adjusted based on a changing environment such as an input
signal. Adaptive IIR filters (not discussed here) can also be used. A major problem
with an adaptive IIR filter is that its poles may be updated during the adaptation
process to values outside the unit circle, making the filter unstable.
The programming examples developed later will make use of equations
(7.1)-(7.3). In (7.3) we simply use the variable

b

b in lieu of 2 b .

7.2 ADAPTIVE STRUCTURES

A number of adaptive structures have been used for different applications in adap-
tive filtering.

1. For noise cancellation. Figure 7.2 shows the adaptive structure in Figure 7.1
modified for a noise cancellation application. The desired signal d is corrupted
by uncorrelated additive noise n . The input to the adaptive filter is a noise n

¢

that is correlated with the noise n . The noise n could come from the same

source as n but modified by the environment. The adaptive filter's output y is
adapted to the noise n . When this happens, the error signal approaches the
desired signal d . The overall output is this error signal and not the adaptive
filter's output y . If d is uncorrelated with n, the strategy is to minimize E ( e 2 ),

¢