Filter Design Toolbox 4.6

Adaptive Noise Cancellation Using RLS Adaptive Filtering

This demo illustrates the ability of the RLS filter to extract useful information from a noisy signal. The information bearing signal is a sine wave that is corrupted by additive white gaussian noise.

The adaptive noise cancellation system assumes the use of two microphones. A primary microphone picks up the noisy input signal, while a secondary microphone receives noise that is uncorrelated to the information bearing signal, but is correlated to the noise picked up by the primary microphone.

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

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

The model illustrates the ability of the Adaptive RLS filter to extract useful information from a noisy signal.

priv_drawrlsdemo
axis off

The information bearing signal is a sine wave of 0.055 cycles/sample.

signal = sin(2*pi*0.055*(0:1000-1)');

plot(0:199,signal(1:200));
grid; axis([0 200 -2 2]);
title('The information bearing signal');

The noise picked up by the secondary microphone is the input for the RLS adaptive filter. The noise that corrupts the sine wave is a lowpass filtered version of (correlated to) this noise. The sum of the filtered noise and the information bearing signal is the desired signal for the adaptive filter.

nvar  = 1.0;                  % Noise variance
noise = randn(1000,1)*nvar;   % White noise

plot(0:999,noise);
title('Noise picked up by the secondary microphone');
grid; axis([0 1000 -4 4]);

The noise corrupting the information bearing signal is a filtered version of 'noise':

nfilt  = fir1(31,0.5);             % 31st order Low pass FIR filter
fnoise = filter(nfilt,1,noise);    % Filtering the noise

"Desired signal" for the adaptive filter (sine wave + filtered noise):

d  = signal+fnoise;
plot(0:199,d(1:200));
grid; axis([0 200 -4 4]);
title('Desired input to the Adaptive Filter = Signal + Filtered Noise');

Set and initialize RLS adaptive filter parameters and values:

M = 32;                    % Filter order
lam = 1;                   % Exponential weighting factor
delta = 0.1;               % Initial input covariance estimate
w0 = zeros(M,1);           % Initial tap weight vector
P0 = (1/delta)*eye(M,M);   % Initial setting for the P matrix
Zi = zeros(M-1,1);         % FIR filter initial states

Running the RLS adaptive filter for 1000 iterations. The plot shows the convergence of the adaptive filter response to the response of the FIR filter.

Hadapt = adaptfilt.rls(M,lam,P0,w0,Zi);
Hadapt.PersistentMemory = true;
[y,e] = filter(Hadapt,noise,d);
H = abs(freqz(Hadapt,1,64));
H1 = abs(freqz(nfilt,1,64));

wf = linspace(0,1,64);
plot(wf,H,wf,H1);
xlabel('Normalized Frequency  (\times\pi rad/sample)');
ylabel('Magnitude');
legend('Adaptive Filter Response','Required Filter Response');
grid;
axis([0 1 0 2]);

As the adaptive filter converges, the filtered noise should be completely subtracted from the "signal + noise" signal and the error signal 'e' should contain only the original signal.

plot(0:499,signal(1:500),0:499,e(1:500)); grid;
axis([0 500 -4 4]);
title('Original information bearing signal and the error signal');
legend('Original Signal','Error Signal');