%% 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.% Copyright 1999-2005 The MathWorks, Inc.% $Revision: 1.8.6.8 $  $Date: 2005/11/02 03:18:56 $%%% 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 variancenoise = randn(1000,1)*nvar;   % White noiseplot(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 filterfnoise = 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 orderlam = 1;                   % Exponential weighting factordelta = 0.1;               % Initial input covariance estimatew0 = zeros(M,1);           % Initial tap weight vectorP0 = (1/delta)*eye(M,M);   % Initial setting for the P matrixZi = 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');displayEndOfDemoMessage(mfilename)