Simulation Results

To illustrate the advantage of the radix-2/4/3/5 FFT on the applicable number of sampling points, we do the following experiment: Within the range from 1 to 10000, we choose 50 groups of Ns evenly. The numbers of sampling points (Ns) and the data sequences are

obtained from the procedure as follows:

for (i=1:50)

Ns(i)=111+(i-1)*200;

for (j=1:Ns(i)) x(i,j)=j;

end end

We plot the required numbers of sampling points for which the FFT algorithms can be applied in terms of the number of sampling points for the single-radix FFT’s in Figure 4.5 and for the mixed-radix FFT’s in Figure 4.6.

Figure 4.5: The comparison of applicable number (Ni) of sampling points which can be applied by the single-radix FFT’s.

In the figure, ‘DFT’ means using discrete Fourier transform which can be applied with any number of sampling points. Thus ‘DFT’ is a straight line. The radix-2 FFT has the most numbers of sampling points than any other single-radix FFT as discussed in section 2.5 since

step-level is. Hence, it is not a good idea to use the single-radix FFT’s.

Figure 4.6: The comparison of applicable number (Ni) of sampling points which can be applied by the mixed-radix FFT’s.

Figure 4.6 shows the similar plots for the mixed-radix FFT’s. In the figure, ‘radix-2,3 FFT’ includes two algorithms, i.e., radix-2/3 FFT and radix-3/2 FFT, since they have the same number of re-sampled points. And so does for the ‘radix-2,5 FFT’ and ‘radix-2,3,5 FFT’.

From the figure, we can find that ‘radix-2,3 FFT’, ‘radix-2,5 FFT’ and ‘radix-2,3,5 FFT’ all follow the straight line of ‘DFT’, and as Ns is larger, the step-levels of ‘radix-2,3 FFT’ and

‘radix-2,5 FFT’ are larger too. As for the ‘radix-2,3,5 FFT’, it has the smallest step-level and approaches that of the ‘DFT’.

Figure 4.7: The comparison of ΔN (Ni - Ns) for different mixed-radix FFT’s.

Figure 4.7 shows the ΔN (Ni - Ns) plots which are derived from data from Figure 4.6 for each mixed-radix FFT. Obviously, ‘radix-2,5 FFT’ has the maximum amplitude and

‘radix-2,3,5 FFT’ has the minimum amplitude. To further demonstrate the above property is to plot the data in the form of 1 - |ΔN/Ns| versus the number of sampling points as shown in Figure 4.8.

Since interpolation modifies the original signal and introduces effectively noise to the signal, it is interesting to investigate how the S/N of the signal will be affected by the interpolation process. The 1 - |ΔN/Ns| figure reflects indirectly the S/N of the processed signal.

In the figure, it can be seen that the radix-2,3,5 FFT has the best S/N than other two algorithms. However, as for other two algorithms, their S/N’s are not too bad since their values of 1 - |ΔN/Ns| are all larger than 0.9.

Next, we will compare the run-time for different FFT algorithms. First we compare the run-times of the radix-2 FFT and the radix-2/4 FFT as shown in Figure 4.9 and then compare the run-times of the radix-2,3,5 FFT and the radix-2/4,3,5 FFT as shown in Figure 4.10 since they have the same numbers of re-sampled points, respectively.

Figure 4.9: The comparison of run-times of radix-2 FFT and radix-2/4 FFT.

Obviously, the radix-2/4 FFT is faster than radix-2 FFT and the run-time of radix-2/4 FFT is almost half of that of radix-2 FFT. This result is consistent with that of Table 2.3. In this experiment, for the radix-2/4 FFT, only the first stage is decimated by the factor 2, and all other stages are decimated by the factor 4. So, the radix-2/4 FFT is almost the same as the

radix-4 FFT, except that the number of sampling points that can be applied to radix-2/4 FFT is of the power of 2, not of the power of 4.

Figure 4.10: The comparison of run-times of radix-2,3,5 FFT and radix-2/4,3,5 FFT.

In this experiment, the radix-2/4,3,5 FFT was obtained by replacing the radix-2 algorithm of the radix-2,3,5 FFT with the radix-2/4 algorithm. From this figure, we can see that the radix-2/4,3,5 FFT is faster than the radix-2,3,5 FFT, but there are some positions that these two algorithms overlap. This is because the numbers of re-sampled points are (2×3^m3×5^m5) where the radix-2/4,3,5 FFT is the same as the radix-2,3,5 FFT.

The comparison of run-times of radix-2,3 FFT, radix-2,5 FFT and radix-2,3,5 FFT that have similar numbers of re-sampled points is shown in Figure 4.11 . Obviously, radix-2,3 FFT and radix-2,5 FFT have similar performance of computation speed but the radix-2,3,5 FFT has a better computation speed performance than the above two algorithms generally.

Figure 4.11: The comparison of run-times of radix-2,3 FFT, radix-2,5 FFT and radix-2,3,5 FFT.

And the comparison of run-times of radix-2,3 FFT, radix-2,5 FFT and radix-2/4,3,5 FFT is shown in Figure 4.12. Obviously, radix-2/4,3,5 FFT is much better than other two algorithms.

Figure 4.12: The comparison of run-times of radix-2,3 FFT, radix-2,5 FFT and radix-2/4,3,5 FFT.

As a conclusion, based on results of Figures 4.6, 4.7, 4.8 and 4.12, we can conclude that the radix-2/4,3,5 FFT cooperating with interpolation is the best algorithm in doing FFT which satisfy both the requirements of the performance of computation speed and the applicable numbers of sampling points.