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The Fast Fourier Transform 199
computing platform used to execute the MATLAB script. The plot in Figure 5.22
was obtained on a 33 MHz 486 computer. It shows that for low values of L
the linear convolution is faster. The crossover point appears to be L = 50,
beyond which the linear convolution time increases exponentially, while the
high-speed convolution time increases fairly linearly. Note that since N =2 ν ,
the high-speed convolution time is constant over a range on L.
□
5.6.5 HIGH-SPEED BLOCK CONVOLUTIONS
Earlier we discussed a block convolution algorithm called the overlap-andsave
method (and its companion the overlap-and-add method), which is
used to convolve a very large sequence with a relatively smaller sequence.
The MATLAB function ovrlpsav developed in that section uses the DFT
to implement the linear convolution. We can now replace the DFT by the
radix-2 FFT algorithm to obtain a high-speed overlap-and-save algorithm.
To further reduce the computations, the FFT of the shorter (fixed) sequence
can be computed only once. The following hsolpsav function
shows this algorithm.
function [y] = hsolpsav(x,h,N)
% High-speed Overlap-Save method of block convolutions using FFT
% --------------------------------------------------------------
% [y] = hsolpsav(x,h,N)
% y = output sequence
% x = input sequence
% h = impulse response
% N = block length (must be a power of two)
%
N = 2^(ceil(log10(N)/log10(2));
Lenx = length(x); M = length(h);
M1 = M-1; L = N-M1; h = fft(h,N);
%
x = [zeros(1,M1), x, zeros(1,N-1)];
K = floor((Lenx+M1-1)/(L)); % # of blocks
Y = zeros(K+1,N);
for k=0:K
xk = fft(x(k*L+1:k*L+N));
Y(k+1,:) = real(ifft(xk.*h));
end
Y = Y(:,M:N)’; y = (Y(:))’;
A similar modification can be done to the overlap-and-add algorithm.
MATLAB also provides the function fftfilt to implement the overlapand-add
algorithm.
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