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Restricting Aliasing to Stop-Bands

To eliminate the relatively heavy transition-band aliasing (when critically sampling the channel signals), we can define overlapping bands such that each band encompasses the transition bands on either side. However, unless a full $ 2\times\null$ oversampling is provided for each band (which is one easy solution), the bandwidth (in bins) is no longer a power of two, thereby thwarting use of radix-2 inverse-FFTs to compute the time-domain band signals.

To keep the channel bandwidths at powers of two while restricting aliasing to stop-band energy, the IFFT bands can be widened to include transition bands on either side. That is, the desired pass-band plus the two transition bands span a power-of-two bins. This results in overlapping channel IFFTs. Figure 10.38 shows how the example of Fig.10.34 is modified by this strategy.

Figure 10.38: Channel-frequency-response overlay for an octave filter bank designed using a length 127 Dolph-Chebyshev window (80 dB SBA) and length 256 FFT size.

The basic principle of filter-bank band allocation is to enclose each filter band plus its transition bands within a wider band that is a power-of-two bins wide.11.23 The band should roll off to reach its stop-band at the edge of the wider encompassing band. It is fine to have extra space in the wider band, and this may be filled with a continuation of the enclosed band's stop-band response (or some tapering of it--since we assume stop-band energy is negligible, the difference should be inconsequential). The desired bands may overlap each other by any amount, and may have any desired shape. The encompassing bands then overlap further to reach the next power of two (in width) for each overlapping extended band. (See the gammatone and gammachirp filter banks for examples of heavily overlapping bands in an audio filter bank [111].)

In this approach, pass-bands of arbitrary width are embedded in overlapping IFFT bands that are a power-of-2 wide. As a result of this flexibility, the frequency-rotation trick of §10.7.7 is no longer needed for real filter banks. Instead, we simply allocate any desired bands between dc and half the sampling rate, and then conjugate-symmetry dictates the rest. In addition to a left-over ``dc-Nyquist'' band, there is a similar residual ``Nyquist-limit'' band (a typically negligible band about half the sampling rate). In other words, since the pass-bands may be any width and the encompassing IFFT bands may overlap by any amount, they do not have to ``pack'' conveniently as power-of-two blocks.

The minimum channel bandwidth is defined as two transition bands plus one bin (i.e., the minimum pass-band width is zero, corresponding to one bin, or one spectral sample). For the Dolph-Chebyshev window, the transition bandwidth is known in closed form [155]. In our examples, we have a length 127 window with 80 dB stop-band attenuation in the lowpass prototype [chebwin(127,80)], corresponding to a transition width of 6.35 bins in a length 256 FFT, which was rounded up to 7 bins in software for simplicity of band allocation. Therefore, our minimum channel bandwidth is 15 bins (two transition bands plus one sample for the band center). The next highest power of two is 16, so that is our minimum encompassing IFFT length for any band.

The dc and Nyquist channels are combined into a single channel containing the left-over residual filter-bank response consisting of a low transition down from dc and a high-frequency transition up to the sampling rate (in the complex-signal case). When N is sufficiently large so that these bands contain no audible energy, they may be discarded. We include them in all examples here so as to preserve the (near) perfect reconstruction property of the filter bank. Thus, the 7-bin dc channel is combined with the 7-bin Nyquist channel to form a single 16-bin encompassing residual band that may be discarded in many audio applications (when the initial FFT size is sufficiently large for the sampling rate used).

In the example of Fig.10.38, the initial FFT size is 256, and the channel bandwidths (pass-bands only, excluding transitions), from top to bottom, are 121, 64, 32, 16, and 8 bins. The top band is reduced by 7 bins to leave a transition band to the sampling rate. Similarly, the lowest band lies above a transition band consisting of bins 0-6. The encompassing IFFTs (containing transitions) are lengths 256, 128, 64, 32, 32, for the interior bands, and a length 32 IFFT handles the dc and Nyquist bands (which are combined into a single 14-bin band about dc, which occupies 28 bins when the transition bands are appended). Letting [lo,hi] denote a band by its lower and upper bin limit, the non-overlapping adjacent pass-band edges in ``spectral samples''11.24 of the interior bands are [8, 15], [16, 31], [32, 63], [64, 127], and [128, 248]; the overlapping encompassing IFFT band edges are then [1, 32], [9, 40], [25, 88], [57, 184], [1, 256], i.e., they each contain a pass-band and two transition bands, and have a power-of-2 length. The downsampling factor for each channel can be computed as the initial FFT size (256) divided by the IFFT size ($ 32$ , $ 32$ , $ 64$ , $ 128$ , or $ 256$ ) for the channel.

Figure 10.39 shows the counterpart of Fig.10.35 for this example. In this case, the aliased signal energy comes only from channel-filter stop-bands. For narrow bands, the aliasing is suppressed by at least 80 dB (the side-lobe level of the chosen Dolph-Chebyshev window transform). For bands significantly wider than one bin (the minimum bandwidth in this example is the dc-Nyquist band at 14 bins), the stop-band consists of a sum of shifts of the window-transform side lobes, and these are found to be more than 80 dB down due to cancellation (more than 90 dB down in most bands of this example).

Figure: Same example as in Fig.10.38 but looking at the effects of aliasing due to channel-signal downsampling. Compare to Fig.10.35.

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.

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Real Filter Bank Example
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Aliasing on Downsampling