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196 Chapter 5 THE DISCRETE FOURIER TRANSFORM
compute their FFTs, and save the computation time in an array. Finally, we
will plot this execution time versus N.
MATLAB script:
>> Nmax = 2048; fft_time=zeros(1,Nmax);
>> for n=1:1:Nmax
>> x=rand(1,n);
>> t=clock;fft(x);fft_time(n)=etime(clock,t);
>> end
>> n=[1:1:Nmax]; plot(n,fft_time,’.’)
>> xlabel(’N’);ylabel(’Time in Sec.’) title(’FFT execution times’)
The plot of the execution times is shown in Figure 5.21. This plot is very
informative. The points in the plot do not show one clear function but appear
to group themselves into various trends. The uppermost group depicts a o(N 2 )
dependence on N, which means that these values must be prime numbers between
1 and 2048 for which the FFT algorithm defaults to the DFT algorithm.
Similarly, there are groups corresponding to the o ( N 2 /2 ) , o ( N 2 /3 ) , o ( N 2 /4 ) ,
and so on, dependencies for which the number N has fewer decompositions.
The last group shows the (almost linear) o (N log N) dependence, which is for
50
FFT execution times
45
o(N*N)
40
35
Time in Sec.
30
25
20
o(N*N/2)
15
10
o(N*N/4)
5
o(N*logN)
0
0 500 1000 1500 2000 2500
N
FIGURE 5.21 FFT execution times for 1