Example 8.4: Let the system shown in Figure 8.1 be a second-order FIR filter. The
input x(n) is a zero-mean white noise given byExample 8.3, and the I/O equation
is expressed as
yn xn 3xn À 1 2xn À 2:
Find the mean m
y
and the autocorrelation function r
yy
k of the output y(n).
(a) m
y
E yn Exn 3Exn À 1 2Exn À 2 0.
(b) r
yy
k E yn kyn
14r
xx
k 9r
xx
k À 1 9r
xx
k 1 2r
xx
k À 2 2r
xx
k 2
14s
2
x
if k 0
9s
2
x
if k Æ 1
2s
2
x
if k Æ 2
0
otherwise.
V
b
b
b
`
b
b
b
X
8.2 Adaptive Filters
Manypractical applications involve the reduction of noise and distortion for extraction
of information from the received signal. The signal degradation in some physical
systems is time varying, unknown, or possibly both. Adaptive filters provide a useful
approach for these applications. Adaptive filters modifytheir characteristics to achieve
certain objectives and usuallyaccomplish the modification (adaptation) automatically.
For example, consider a high-speed modem for transmitting and receiving data over
telephone channels. It employs a filter called a channel equalizer to compensate for
the channel distortion. Since the dial-up communication channels have different char-
acteristics on each connection and are time varying, the channel equalizers must be
adaptive.
Adaptive filters have received considerable attention from manyresearchers over the
past 30 years. Many adaptive filter structures and adaptation algorithms have been
developed for different applications. This chapter presents the most widelyused adap-
tive filter based on the FIR filter with the LMS algorithm. Adaptive filters in this class
are relativelysimple to design and implement. Theyare well understood with regard to
convergence speed, steady-state performance, and finite-precision effects.
8.2.1 Introduction to Adaptive Filtering
An adaptive filter consists of two distinct parts ± a digital filter to perform the desired
signal processing, and an adaptive algorithm to adjust the coefficients (or weights) of
that filter. A general form of adaptive filter is illustrated in Figure 8.2, where d(n) is a
desired signal (or primaryinput signal), y(n) is the output of a digital filter driven bya
reference input signal x(n), and an error signal e(n) is the difference between d(n) and
y(n). The function of the adaptive algorithm is to adjust the digital filter coefficients to
ADAPTIVE FILTERS
359