Generalized Hamming Window Family

The generalized Hamming family of windows is constructed by adding one period of a cosine function to the rectangular window. The benefit of adding the cosine segment 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 window and the Hamming window.

The basic idea of generalized Hamming family can be seen in the frequency-domain picture of Fig.3.2. 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 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 can note the following results:

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

**Figure 3.2:** Construction of the generalized Hamming window transform as a superposition of three shifted aliased sinc functions.
$\includegraphics[width=3in]{eps/shiftedSincs}$

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

$\displaystyle W_H( \omega ) \isdef \alpha W_R( \omega ) + \beta W_R( \omega - \Omega_M ) + \beta W_R( \omega + \Omega_M )$

where $\alpha=\frac{1}{2}$ , $\beta=\frac{1}{4}$ in the example of Fig.3.2.

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

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

or,

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

(4.1)

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

Subsections

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written by 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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Spectral Audio Signal Processing
Spectrum Analysis Windows
Generalized Hamming Window Family