Generalized Hamming Window Family

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The aliased sinc function is obtained from the sinc function by applying the aliasing operator. It arises in digital audio as the spectrum of a sampled rectangular pulse. — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window.html

The Fourier transform of a signal is a frequency-domain function — Click for https://ccrma.stanford.edu/~jos/mdft/

The main lobe in the Fourier transform of a window function, such as the Chebyshev window, is the central local maximum in the transform magnitude (response about dc). Outside of the main lobe are usually side lobes (ripples in the magnitude spectrum). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.html

A side lobe in the Fourier transform of a window function, such as the Chebyshev window, is any local maximum in the transform magnitude outside of the main lobe (response about dc). — Click for https://ccrma.stanford.edu/~jos/sasp/Rectangular_Window_Side_Lobes.html

A periodic signal is a signal that forever repeats itself. — Click for http://en.wikibooks.org/wiki/Signals_and_Systems/Periodic_Signals

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Generalized Hamming Window Family

The generalized Hamming window family is constructed by multiplying a rectangular window by one period of a cosine. The benefit of the cosine tapering 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 and Hamming windows, defined below.

The basic idea of the generalized Hamming family can be seen in the frequency-domain picture of Fig.3.8. 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 scaled 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 see that

**Figure 3.8:** 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 $W_R(\omega) = M\cdot\hbox{asinc}_M(\omega)$ (the zero-phase, unit-amplitude case), this can be written as

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

(4.15)

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

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

(4.16)

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

(4.17)

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

Generalized Hamming Window Family

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University