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Slower clocks mean lower power consumption, any other benefits?

Could the multirate filter be more sharper than the single-rate filter?

Previously, someone asked the similar question, but the answer is only about the power efficiency. Advantage of complex filtering in multirate applications

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    $\begingroup$ The higher frequency filter banks can respond more quickly than the broadbanded processing can. $\endgroup$ Commented Apr 23 at 14:45
  • $\begingroup$ Thanks a lot, could you please explain in more details? $\endgroup$
    – user71925
    Commented Apr 23 at 14:47
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    $\begingroup$ Have a look at well written books from Rabiner, Fliege, Vaidyanathan, and Akansu for Multirate Systems, Multiresolution Analysis, and Filterbanks. They are more complex to implement but they offer various advantages over a single filter. And some times the problem naturally lends itself into a filterbank approach, such as used in audio coding. $\endgroup$
    – Fat32
    Commented Apr 23 at 19:33

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I'm not exactly a wavelets and filter banks guy. And I couldn't find exactly the diagram I wanted on the web. But I found this:

enter image description here

This depicts how, in all four cases, the 1 (single) or 2 or 4 or 8 filters (in parallel) adds up to the same broad spectrum. So that single broad spectrum is first divided into to bands with complementary frequency response. This is called Half-band filtering.

enter image description here

Now that bottom band can be resampled at half the original sample rate. Then that half spectrum is this downsampled whole spectrum and this half-band filtering can be applied again.

Now in the first halfband filter, the result is the low half. You can subtract that LPF from the original signal (you have to line up the delays so that it's relatively "zero-phase". The result of that subtraction is the complementary high halfband output. That can be left alone. Just the low half is repeatedly halved.

So the length of the impulse response of these filters (which is a measure of time) are inversely proportional to the bandwidth of each of the filters. The high-band filter that is the top octave will have more precise envelope edges of whatever is coming out of that top half band.

The next filter has a quarter of the original bandwidth and will have an impulse response twice as long as the top octave filter.

So the frequency precision of the band centers stay roughly constant with the high-bands and the low bands. We might expect a comparable amount of features in each octave.

So if pink noise goes in, roughly the same amount of energy is in each band (except the very lowest band, which has more octaves of bandwidth than the others). White noise would have roughly the same amount of energy per unit frequency.

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