Introduction to Audio Processing by Mads G. Christensen

Author:Mads G. Christensen
Language: eng
Format: epub
ISBN: 9783030117818
Publisher: Springer International Publishing

(7.7)

This is also illustrated in Fig. 7.5. By observing that the Fourier transform is given by , we can see that adding a number of zeros to this summation does not change the spectrum of x n. Hence, we can add as many zeros as we like. The result is that any time we wish to compute an F-point DFT of an N-sample signal where N < F, we can simply use zero-padding. Aside from the practical issue of a radix-2 algorithm needing a certain number of samples, there is another reason we would like to compute DFTs where F ≫ N, namely, that the underlying continuous spectrum X(ω) is sampled on a finer grid this way! Thus, we get a clearer picture of what X(ω) really looks like. In Fig. 7.6 the effect of zero-padding is illustrated for a signal whose original length is 100 samples. It is zero-padded to length 256 which is 28.

Fig. 7.5The process of zero-padding the data x(n), which has N samples, by augmenting it with F − N zeros to obtain F samples, from which the F-point Fourier transform can be computed

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