In (9.19) of the previous section, we derived that the FBS reconstruction sum gives
where . From this we see that if (where is the window length and is the DFT size), as is normally the case, then for . This is the Fourier dual of the ``strong COLA constraint'' for OLA (see §8.3.2). When it holds, we have
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 ( ), we can still obtain perfect reconstruction provided
When this holds, we say the window is . (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 . 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 samples, thereby yielding an ideal channel filter with bandwidth . Figure 9.16 compares example Hamming and Portnoff windows.
Duality of COLA and Nyquist Conditions
FBS Window Constraints for R=1