Summary of Generalized Hamming Windows

Definition:

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

where

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

Transform:

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

where

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

Common Properties

Window is a rectangular window plus a scaled cosine, where the 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 3.6 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 3.6:** Comparison of window transforms for the rectangular, Hann, and Hamming windows.
$\includegraphics[width=\twidth]{eps/RectHannHamm}$

Rectangular window properties:

Abrupt transition from 1 to 0 at the window endpoints.
Roll off is approximately -6 dB per octave.
First side lobe is dB relative to main lobe peak.

Hann window properties:

Smooth transition to zero at window endpoints.
Roll off is approximately -18 dB per octave (as $T\rightarrow 0$ ).
First side lobe is dB relative to main lobe peak.

Hamming window properties:

Discontinuous ``slam to zero'' at endpoints.
Asymptotic roll-off is approximately -6 dB per octave.
Side lobes are closer to ``equal ripple''.
First side lobe is dB down = dB better than Hann.^4.4

Previous: Matlab for the Hamming Window
Next: The MLT Sine Window

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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See Also

Spectral Audio Signal Processing
    Spectrum Analysis Windows
       Generalized Hamming Window Family
          Summary of Generalized Hamming Windows