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Summary of Generalized Hamming Windows


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

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.

Figure 3.12: Comparison of window transforms for the rectangular, Hann, and Hamming windows.
\includegraphics[width=\twidth]{eps/RectHannHamm}


Rectangular window properties:


Hann window properties:


Hamming window properties:


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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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