Generalized Hamming Window Family

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 $0.5\cdot W_R(\omega) = 0.5\cdot M\cdot\hbox{asinc}_M(\omega)$ (scaled rectangular window transform). The other two dotted waveforms are scaled shifts of the same function, $0.25\cdot W_R(\omega\pm\Omega_M)$ . The sum of all three dotted waveforms gives the solid line. We see that

• there is some cancellation of the side lobes, and
• the width of the main lobe is doubled.

Figure 3.8: Construction of the generalized Hamming window transform as a superposition of three shifted aliased sinc functions.

In terms of the rectangular window transform $W_R(\omega) = M\cdot\hbox{asinc}_M(\omega)$ (the zero-phase, unit-amplitude case), this can be written as

$$W_H( \omega ) \isdefs \alpha W_R( \omega ) + \beta W_R( \omega - \Omega_M ) + \beta W_R( \omega + \Omega_M )$$ (4.15)

where $\alpha=1/2$ , $\beta=1/4$ in the example of Fig.3.8.

Using the shift theorem (§2.3.4), we can take the inverse transform of the above equation to obtain

$$w_H = \alpha w_R(n) + \beta e^{-j\Omega_M n}w_R(n) + \beta e^{j \Omega_M n} w_R(n),$$ (4.16)

or,

$$\zbox {w_H(n) = w_R(n) \left[ \alpha + 2 \beta \cos \left( \frac{2 \pi n}{M} \right) \right].} \protect$$ (4.17)

Choosing various parameters for $\alpha$ and $\beta$ result in different windows in the generalized Hamming family, some of which have names.