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Specific Windows

  • Recall that the rectangular window transform is $ \hbox{\sc Nyquist}(2\pi/M)$ , implying the rectangular window itself is $ \hbox{\sc Cola}(M)$ , which is obvious.

  • The window transform for the Hamming family is $ \hbox{\sc Nyquist}(4\pi/M)$ , implying that Hamming windows are $ \hbox{\sc Cola}(M/2)$ , which we also knew.

  • The rectangular window transform is also $ \hbox{\sc Nyquist}(K2\pi/M)$ for any integer $ 1\leq K\leq M/2$ , implying that all hop sizes given by $ R=M/K$ for $ K=1,2,\ldots,M/2$ are COLA.

  • Because its side lobes are the same width as the sinc side lobes, the Hamming window transform is also $ \hbox{\sc Nyquist}(K2\pi/M)$ ,for any integer $ 2\leq K\leq M/2$ , implying hop sizes $ R=M/K$ are good, for $ K=2,\ldots,M/2$ . Thus, the available hop sizes for the Hamming window family include all of those for the rectangular window except one ($ R=M$ ).

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Inverse DFT and the DFT Filter Bank Sum