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Blackman-Harris Window Family

The Blackman-Harris (BH) window family is a straightforward generalization of the Hamming family introduced in §3.2. Recall from that discussion that the generalized Hamming family was constructed using a summation of three shifted and scaled aliased-sinc-functions (shown in Fig.3.8). The Blackman-Harris family is obtained by adding still more shifted sinc functions:

$\displaystyle w_{BH}(n)= w_R(n)\sum_{l=0}^{L-1} \alpha_l \cos( l \Omega_M n), \protect$ (4.26)

where $ \Omega_M\isdef 2\pi/M$ , and $ w_R(n)$ is the length $ M$ zero-phase rectangular window (nonzero for $ n\in[-(M-1)/2,(M-1)/2]$ ). The corresponding window transform is given by

$\displaystyle W_{BH}(\omega) = \sum_{k=-(L-1)}^{L-1}\alpha_k W_R(\omega + k\Omega_M),$ (4.27)

where $ W_R(\omega) =
M\cdot\hbox{asinc}_M(\omega)$ denotes the rectangular-window transform, and $ \Omega_M = 2\pi/M$ as usual.

Note that for $ L=1$ , we obtain the rectangular window, and for $ L=2$ , the BH family specializes to the generalized Hamming family.

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``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