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Nyquist(N) Windows

In (9.19) of the previous section, we derived that the FBS reconstruction sum gives

$\displaystyle \hat{x}(n) = N\sum_{r=-\infty}^\infty {\tilde w}(rN)x(n-rN)$ (10.20)

where $ {\tilde w}=\hbox{\sc Flip}(w)$ . From this we see that if $ M<N$ (where $ M$ is the window length and $ N$ is the DFT size), as is normally the case, then $ w(rN) = 0$ for $ \vert r\vert = 1,2, \dots\,$ . This is the Fourier dual of the ``strong COLA constraint'' for OLA (see §8.3.2). When it holds, we have

$\displaystyle \hat{x}(n) = N w(0) x(n).$ (10.21)

This is simply a gain term, and so we are able to recover the original signal exactly. (Zero-phase windows are appropriate here.)

If the window length is larger than the number of analysis frequencies ($ M>N$ ), we can still obtain perfect reconstruction provided

$\displaystyle w(rN) = 0, \hspace{1cm} \vert r\vert=1,2,\dots \qquad\hbox{[$w$\ is Nyquist($N$)]}$ (10.22)

When this holds, we say the window is $ \hbox{\sc Nyquist}(N)$ . (This is the dual of the weak COLA constraint introduced in §8.3.1.) Portnoff windows, discussed in §9.7, make use of this result; they are longer than the DFT size and therefore must be used in time-aliased form [62]. An advantage of Portnoff windows is that they give reduced overlap among the channel filter pass-bands. In the limit, a Portnoff window can approach a sinc function having its zero-crossings at all nonzero multiples of $ N$ samples, thereby yielding an ideal channel filter with bandwidth $ 2\pi/N$ . Figure 9.16 compares example Hamming and Portnoff windows.

Figure 3.35:
\includegraphics[width=\twidth]{eps/colawin}

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Duality of COLA and Nyquist Conditions
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FBS Window Constraints for R=1