The generalized Hamming window family is constructed by multiplying a rectangular window by one period of a cosine. The benefit of the cosine tapering is lower side-lobes. The price for this benefit is that the main-lobe doubles in width. Two well known members of the generalized Hamming family are the Hann and Hamming windows, defined below.
The basic idea of the generalized Hamming family can be seen in the frequency-domain picture of Fig.3.8. The center dotted waveform is the aliased sinc function (scaled rectangular window transform). The other two dotted waveforms are scaled shifts of the same function, . The sum of all three dotted waveforms gives the solid line. We see that
In terms of the rectangular window transform (the zero-phase, unit-amplitude case), this can be written as
Using the shift theorem (§2.3.4), we can take the inverse transform of the above equation to obtain
Choosing various parameters for and result in different windows in the generalized Hamming family, some of which have names.