Tightening the IFFTs

In this example the top band is not downsampled at all, and the interior bands are oversampled by approximately 2. This is because the desired pass-band widths started out at a power of 2, so that the addition of transition bands forced the next higher power of 2 for the IFFT size. Narrowing the width of the top band from 121 bins to $128-2\cdot 7 = 114$ bins would enable use of a length 128 IFFT for the top band, and similarly for the lower bands. In other words, when the desired spectral partition is that of an ideal octave filter bank, as sketched in Fig.10.31, narrowing each octave-band by twice the transition width of the lowpass prototype filter (and ``covering down'' to keep them adjacent) will produce a relatively ``tight'' FFT filterbank design in which the IFFT sizes remain the same length as in the heavily aliased case discussed above (Fig.10.35). When applied to the octave filter bank, the pass-bands become a little narrower than one octave. We may call this a quasi octave filter bank.