What is the Hamming window for?

Question

I\'m working with some code that does a Fourier transform (to calculate the cepstrum of an audio sample). Before it computes the Fourier transform, it applies a Hamming window t

Answer 1

The FT of a finite length segment of sinusoid convolves the Fourier transform of the window against the sinusoid's frequency peak, since a property of the FFT is that vector multiplication in one domain is convolution in the other. The FT of a rectangular window (which is what any unmodified finite length of samples in an FFT implies) is the messy looking Sinc function which splatters any signal that is not exactly periodic in the window over the entire frequency spectrum.

The FT of a Hamming shaped window concentrates this "splatter" much nearer to the frequency peak after the convolution (than a Sinc function), resulting in a fatter but smoother frequency peak, but a lot less splatter across frequencies far from the frequency peak. This results in not only a cleaner looking spectrum, but also less interference from far away frequencies on any signal of interest.

This interpretation (as opposed to the "infinitely repeating" interpretation) makes it more clear why differently shaped windows than Hamming may give you better results with even less "leakage". In particular, a Hamming window will reduce the size of the first Sinc side lobe of "leakage" right next to the frequency peak in exchange for actually more "leakage" (or convolution splatter) far from the frequency of interest. Other windows may be more appropriate if you wish a different trade-off. The Harris paper (pdf here) linked in another answer above gives several examples of these different windows.