Generalized Hamming Window Family

Consider the following picture in the frequency domain:

We have added 2 extra aliased sinc functions (shifted), which results in the following behavior:

• There is some cancellation of the side lobes
• The width of the main lobe is doubled

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

$$W_H( \omega ) \mathrel{\stackrel{\Delta}{=}}\alpha W_R( \omega ) + \beta W_R( \omega - \Omega_M ) + \beta W_R( \omega + \Omega_M )$$

Using the Shift Theorem dual, we can take the inverse transform of the above equation:

$$\begin{eqnarray*} w_H &=& \alpha w_R(n) + \beta e^{-j\Omega_M n}w_R(n) + \beta e^{j \Omega_M n} w_R(n) \\ &=& w_R(n) \left[ \alpha + 2 \beta \cos \left( \frac{2 \pi n}{M} \right) \right] \\ \end{eqnarray*}$$

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