Filter Design Toolbox 4.6

Linear Prediction Using the NLMS Adaptive Filter

This demo casts the normalized least mean squares adaptive filter in a linear prediction framework. The current sample of a noise corrupted sine wave is predicted using 32 past samples, i.e., a 32nd order normalized LMS adaptive filter is applied to the data.

Note: This demo is equivalent to the Simulink® model 'lmsadlp' provided in the Signal Processing Blockset™.

% Reference: S. Haykin, "Adaptive Filter Theory", 3rd Edition, Prentice Hall
,
% N.J., 1996.

The model shows a normalized least mean squares adaptive filter in a linear prediction framework.

priv_drawnlmsdemo;
axis off

The desired signal is a sine wave of 0.015 cycles/sample and a cosine of 0.008 cycles/sample.

N = 500;
sig = [sin(2*pi*0.015*(0:N-1)) 0.5*cos(2*pi*0.008*(0:N-1))];
plot(0:2*N-1,sig); grid;
title('Desired Input to the Adaptive Filter');

The input to the adaptive filter is a delayed version of the desired signal corrupted by white noise of variance 0.5.

nvar = 0.5;                   % Noise variance
noise  = nvar*randn(1,2*N);   % Noise
n = sig + noise;              % The noise corrupted sine wave.
x = [0 n];                    % Delayed input for linear prediction
d = [sig 0];                  % Desired signal to the adaptive filter
M = 32;                       % NLMS adaptive filter order
mu = 0.2;                     % Normalized LMS step size.

plot(0:2*N,x); grid;
title('Input to the Adaptive Filter');

Create and use the Normalized LMS adaptive filter object of length M, step size 0.2 and offset 1e-6

Hadapt = adaptfilt.nlms(M,mu,1,1e-6);
[y,e] = filter(Hadapt,x,d);
cla;
plot(0:1000,[d' y']);
grid on;
axis([0 1000 -2 2]);
title('Adaptive Linear Prediction');
legend('Actual Signal','Predicted Signal');

The autocorrelation of the prediction error shows that the input is white noise.

X = xcorr(e(50:end),'coeff');
[maxX idx] = max(X);
plot(X(idx:end));
grid;
title('Autocorrelation of the Prediction Error');