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This example shows how to use an 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 example is equivalent to the Simulink® model 'rlsdemo' provided.

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)');
Hs = dsp.SignalSource(signal,'SamplesPerFrame',100,...
'SignalEndAction','Cyclic repetition');

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
Hn = dsp.SignalSource(noise,'SamplesPerFrame',100,...
'SignalEndAction','Cyclic repetition');

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'. Initialize the filter that operates on the noise.

Hd = dsp.FIRFilter('Numerator',fir1(31,0.5));% Low pass FIR filter


Set and initialize RLS adaptive filter parameters and values:

M      = 32;                 % Filter order
delta  = 0.1;                % Initial input covariance estimate
P0     = (1/delta)*eye(M,M); % Initial setting for the P matrix


Running the RLS adaptive filter for 1000 iterations. As the adaptive filter converges, the filtered noise should be completely subtracted from the "signal + noise". Also the error, 'e', should contain only the original signal.

Hts = dsp.TimeScope('TimeSpan',1000,'YLimits',[-2,2]);
for k = 1:10
n = step(Hn); % Noise
s = step(Hs);
d = step(Hd,n) + s;
step(Hts,[s,e]);
end


The plot shows the convergence of the adaptive filter response to the response of the FIR filter.

H  = abs(freqz(Hadapt.Coefficients,1,64));
H1 = abs(freqz(Hd.Numerator,1,64));

wf = linspace(0,1,64);

plot(wf,H,wf,H1);