Example 3: Overlap-Add Convolution

• Impulse-train signal, 4000 Hz sampling-rate
• Length 32 lowpass filter, 600 Hz cut-off
• Length $M=32$ Hanning overlap-add window (causal)
• Hop size = $R=M/2$ ($50$ % overlap)

Matlab code:

Fs = 4000;              % sampling rate in Hz
Nsig = 128;             % signal length in samples
L = 32;                 % filter length in taps
fc = 600;               % cutoff frequency in Hz
M = 32;                            % window length
Nfft = 2^(ceil(log2( M + L - 1))); % FFT Length
R = M/2;                           % Hop Size
Nframes = floor( Nsig/R );         % No. of frames            

% Generate a test signal containing five impulses
sig = zeros(1,Nsig); period = round(Nsig/5);
sig(1:period:Nsig) = ones(size(1:period:Nsig));

figure(1);clf
subplot(211); stem(sig); axis( [0 length(sig) -0.2 1.2]);
subplot(212); stem(h); grid; axis( [0 length(h) -0.2 0.5]);

% *** design a lowpass filter using the window method
epsilon = .0001; % avoids 0 / 0
nfilt = (-(L-1)/2:(L-1)/2) + epsilon;   
hideal = sin( 2*pi*fc*nfilt/Fs ) ./ (pi*nfilt); % sinc fn
h = hamming( L )' .* hideal; % windowed sinc fn

hzp = [h zeros(1,Nfft-L)];  % Zero-pad h to FFT size
H = fft(hzp);               % Filter frequency response
y = zeros(1,Nsig + Nfft);   % Allocate the output+'ringing' vector

% *** Overlap add loop, with plots
figure(2);
subplot(411); stem(sig); axis( [0 length(sig) -0.2 1.2]);
title('signal'); ylabel('amplitude');
window = hanning(M)';       % Signal window (could also be rect)
for m = 0:(Nframes-1)
    index = m*R+1:min(m*R+M,Nsig);      % index for the mth frame
    xm = sig(index) .* window(index-m*R); % windowed mth frame
    xmzp = [ xm zeros(1,Nfft-length(xm))];      % zero pad the signal
    Xm = fft( xmzp );
    Xfilt = Xm .* H;                    % freq domain multiplication
    xmfilt = real(ifft(Xfilt));         % inverse transform
    outindex = m*R+1:(m*R+Nfft);        %
    y(outindex) = y(outindex) + xmfilt; % overlap add

    xmplot = [ zeros(1,m*R) xm zeros(1,Nsig-(m*R+M)) ];
    winplot = [ zeros(1,m*R) window zeros(1,Nsig-(m*R+M)) ];
    xmfiltplot = [ zeros(1,m*R) xmfilt zeros(1,Nsig-(m*R+Nfft)) ];
    subplot(412);stem(xmplot); axis( [0 length(sig) -0.2 1.2]);
    hold on; plot(winplot,'--'); hold off
    title('windowed frame');
    subplot(413);plot(xmfiltplot); grid; axis( [0 length(sig) -0.1 0.4]);;
    title('windowed frame - filtered');
    subplot(414);plot(y); grid; axis( [0 length(sig) -0.1 0.4]);
    title('overlapp add buffer'); xlabel('samples');
    cmd = sprintf('print -deps ola%d',m); disp(cmd); eval(cmd);
end