Example of Overlap-Add Convolution

Let's look now at a specific example of FFT convolution:

• Impulse-train test signal, 4000 Hz sampling-rate
• Length $L=31$ causal lowpass filter, 600 Hz cut-off
• Length $M\approx L$ rectangular window
• Hop size $R=M$ (no overlap)

We will work through the matlab for this example and display the results. First, the simulation parameters:

L = 31;         % FIR filter length in taps
fc = 600;       % lowpass cutoff frequency in Hz
fs = 4000;      % sampling rate in Hz 

Nsig = 150;     % signal length in samples
period = round(L/3); % signal period in samples

FFT processing parameters:

M = L;                  % nominal window length
Nfft = 2^(ceil(log2(M+L-1))); % FFT Length
M = Nfft-L+1            % efficient window length
R = M;                  % hop size for rectangular window
Nframes = 1+floor((Nsig-M)/R);  % no. complete frames

Generate the impulse-train test signal:

sig = zeros(1,Nsig);
sig(1:period:Nsig) = ones(size(1:period:Nsig));

Design the 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);
w = hamming(L); % FIR filter design by window method
h = w' .* hideal; % window the ideal impulse response

hzp = [h zeros(1,Nfft-L)];  % zero-pad h to FFT size
H = fft(hzp);               % filter frequency response

Carry out the overlap-add FFT processing:

y = zeros(1,Nsig + Nfft); % allocate output+'ringing' vector
for m = 0:(Nframes-1)
    index = m*R+1:min(m*R+M,Nsig); % indices for the mth frame
    xm = sig(index);  % windowed mth frame (rectangular window)
    xmzp = [xm zeros(1,Nfft-length(xm))]; % zero pad the signal
    Xm = fft(xmzp);
    Ym = Xm .* H;               % freq domain multiplication
    ym = real(ifft(Ym))         % inverse transform
    outindex = m*R+1:(m*R+Nfft); 
    y(outindex) = y(outindex) + ym; % overlap add
end

The time waveforms for the first three frames ($m=0,1,2$ ) are shown in Figures 8.12 through 8.14. Notice how the causal linear-phase filtering results in an overall signal delay by half the filter length. Also, note how frames 0 and 2 contain four impulses, while frame 1 only happens to catch three; this causes no difficulty, and the filtered result remains correct by superposition.

Figure 8.12: OLA Example, Frame 0.

Figure 8.13: OLA Example, Frame 1.

Figure 8.14: OLA Example, Frame 2.