DSP Lab Experiments - Digital Signal Processing: An Introduction with MATLAB and Applications - page 342

Estimate of x ( n )
^
x ( n )

Observed
signal Channel

distortion/
noise

Adaptive
Filter

x ( n )

y ( n )

Error, e ( n )

d ( n )
Reference signal

Fig. D.15 Adaptive filter configuration

algorithm, based on minimizing the MSE error e mse ¼Ef½ e ð n Þ 2 g¼Ef½ d ð n Þ
x ð n Þ 2 g at every time instant n as we have shown earlier.

Define:
h ð k Þ¼½ h o ð k Þ h 1 ð k Þ h 2 ð k Þ h M ð k Þ , the filter coefficients at the kth instant.
y ð k Þ¼½ y ð k Þ y ð k 1 Þ y ð k 2 Þ y ð k M Þ , observed signal vector at kth
instant.The algorithm can be described in vector form (MATLAB-like code)
as follows:

h(0) = 0; % Initialize the filter coefficients.
for n = 1 : N % N = length(y);
x ð n Þ¼ h ð n 1 Þ y T ð n Þ ; % Filter output (this is matrix multiplication).
e ð n Þ¼ d ð n Þ x ð n Þ ;
h(n) = h(n - 1) + l * e(n)y(n); % l is the step-size.
end

The choice of l will affect the estimation accuracy and the convergence speed
of the algorithm. Small values of l will give better accuracy but slower
convergence. Large values will do the contrary. Very small or very large values
for l will cause significant errors. Hence, a compromise would be optimal.
Larger filter length M + 1 gives better estimation, but more delay.
Application 1: Noise reduction in narrowband signals: For estimation of
narrowband signals (like single-tone sinusoids) with known frequency band, we
can use a normal LPF for noise reduction, but for unknown frequency band, we use
adaptive Wiener filter with d(n) = y(n) and input sequence {y(n - 1)}, as shown
in Fig. D.16 .
Application 2: Channel estimation: In mobile communications, a ''training
sequence'' is sent before transmission of data. The receiver knows this signal and
utilizes a copy of it as the desired signal d(n). The adaptive Wiener filter can
arrange its optimal coefficients during the short period of transmitting the training
sequence before actual data are transmitted. Hardware or software implementation
of the above algorithm is possible.