Summary of Generalized Hamming Windows

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Definition:

$\displaystyle w_H(n) = w_R(n) \left[ \alpha + 2 \beta \cos \left( \Omega_M n \right) \right], \quad n \in {\bf Z}, \; \Omega_M \isdef \frac{2 \pi}{M}$

(4.20)

where

$\displaystyle w_R(n) \isdefs \left\{\begin{array}{ll} 1, & \left\vert n\right\vert\leq\frac{M-1}{2} \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$

(4.21)

Transform:

$\displaystyle W_H( \omega ) \isdefs \alpha W_R( \omega ) + \beta W_R( \omega - \Omega_M ) + \beta W_R( \omega + \Omega_M ), \quad \omega\in[-\pi,\pi)$

(4.22)

where

$\displaystyle W_R(\omega) = M\cdot \hbox{asinc}_M(\omega) \isdefs \frac{\sin\left(M\frac{\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)}$

(4.23)

Common Properties

Rectangular + scaled-cosine window
Cosine has one period across the window
Symmetric ( $\Rightarrow$ zero or linear phase)
Positive (by convention on $\alpha$ and $\beta$ )
Main lobe is $4\Omega_M$ radians per sample wide, where $\Omega_M\isdef 2\pi/M$
Zero-crossings (``notches'') in window transform at intervals of $\Omega_M$ outside of main lobe

Figure 3.12 compares the window transforms for the rectangular, Hann, and Hamming windows. Note how the Hann window has the fastest roll-off while the Hamming window is closest to being equal-ripple. The rectangular window has the narrowest main lobe.