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:

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

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