Theorem: For any
,
Proof:
This is perhaps the most important single
Fourier theorem of all. It
is the basis of a large number of
FFT applications. Since an FFT
provides a
fast Fourier transform, it also provides
fast
convolution, thanks to the convolution theorem. It turns out that
using an FFT to perform convolution is really more efficient in
practice only for reasonably long convolutions, such as
. For
much longer convolutions, the savings become enormous compared with
``direct'' convolution. This happens because direct convolution
requires on the order of
operations (multiplications and
additions), while FFTbased convolution requires on the order of
operations, where
denotes the logarithmbase2 of
(see §
A.1.2 for an explanation).
The simple
matlab example in Fig.
7.13 illustrates how much faster
convolution can be performed using an FFT.
^{7.16} We see that
for a length
convolution, the
fft function is
approximately 300 times faster in Octave, and 30 times faster in
Matlab. (The
conv routine is much faster in Matlab, even
though it is a builtin function in both cases.)
Figure 7.13:
Matlab/Octave program for comparing the
speed of direct convolution with that of FFT convolution.
N = 1024; % FFT much faster at this length
t = 0:N1; % [0,1,2,...,N1]
h = exp(t); % filter impulse reponse
H = fft(h); % filter frequency response
x = ones(1,N); % input = dc (any signal will do)
Nrep = 100; % number of trials to average
t0 = clock; % latch the current time
for i=1:Nrep, y = conv(x,h); end % Direct convolution
t1 = etime(clock,t0)*1000; % elapsed time in msec
t0 = clock;
for i=1:Nrep, y = ifft(fft(x) .* H); end % FFT convolution
t2 = etime(clock,t0)*1000;
disp(sprintf([...
'Average directconvolution time = %0.2f msec\n',...
'Average FFTconvolution time = %0.2f msec\n',...
'Ratio = %0.2f (Direct/FFT)'],...
t1/Nrep,t2/Nrep,t1/t2));
% =================== EXAMPLE RESULTS ===================
Octave:
Average directconvolution time = 69.49 msec
Average FFTconvolution time = 0.23 msec
Ratio = 296.40 (Direct/FFT)
Matlab:
Average directconvolution time = 15.73 msec
Average FFTconvolution time = 0.50 msec
Ratio = 31.46 (Direct/FFT)

A similar program produced the results for different FFT lengths shown
in Table
7.1.
^{7.17} In this software environment, the
fft function
is faster starting with length
, and it is never significantly
slower at short lengths, where ``calling overhead'' dominates.
Table 7.1:
Direct versus FFT convolution times in milliseconds
(convolution length = ) using Matlab 5.2 on an 800 MHz Athlon Windows PC.
M 
Direct 
FFT 
Ratio 
1 
0.07 
0.08 
0.91 
2 
0.08 
0.08 
0.92 
3 
0.08 
0.08 
0.94 
4 
0.09 
0.10 
0.97 
5 
0.12 
0.12 
0.96 
6 
0.18 
0.12 
1.44 
7 
0.39 
0.15 
2.67 
8 
1.10 
0.21 
5.10 
9 
3.83 
0.31 
12.26 
10 
15.80 
0.47 
33.72 
11 
50.39 
1.09 
46.07 
12 
177.75 
2.53 
70.22 
13 
709.75 
5.62 
126.18 
14 
4510.25 
17.50 
257.73 
15 
19050.00 
72.50 
262.76 
16 
316375.00 
440.50 
718.22 

A table similar to Table
7.1 in Strum and Kirk
[
79, p. 521], based on the number of real
multiplies, finds that the
fft is faster starting at length
,
and that direct convolution is significantly faster for very short
convolutions (
e.g., 16 operations for a direct length4 convolution,
versus 176 for the
fft function).
See Appendix
A for further discussion of FFT algorithms and their applications.
Next Section: Dual of the Convolution TheoremPrevious Section: Shift Theorem