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In Reply to: Re: Not quite right. posted by Ted Smith on February 14, 2007 at 11:17:27:
"Also a proper reconstruction filter is the same as a proper antialiasing filter: ideally it passes all freqs less than the Nyquist rate unchanged and eliminates all higher freqs."Actually, if you go back to Shannon's proof for the theorem you'll find that the reconstructor is unambiguously defined, and that the AA filter is not defined at all.
--
The reconstructor, for the theorem to hold, has to be our infinitely-wiggly f(r)iend Sinc(x). If applied in all its glory the wiggles cancel out completely, so it has no drawbacks in the time domain (despite contrary claims from, oh, the Wadias and the Todds).
Small detail: most silicon-based reconstructors aren't quite like Sinc, but are half-band low-pass FIRs, with only 3dB attenuation at
fs/2.--
The AA is not defined. It is up to humanity to come up with an AA filter that guarantees zero aliasing all the while being transparent to the human ear. If you use Sinc(x) or a wiggly FIR here you introduce wiggles into the recording. These are there to stay.
And again, most silicon-based AAs are half-band FIRs, even the ones in 'pro' gear.--
Much more interesting for AA duties are linear phase low-pass filters with a strictly monotonic frequency curve (no ripple) and a wide transition band. If you want a (more or less) flat extension to 20kHz then the latter can only be supported by using a higher sampling rate. In the case of 96kHz you would keep everything flat to 20kHz (or rather, slowly dropping to -0.1dB at 20kHz or so), and then slowly roll off to -140dB at 48kHz. Such filters still introduce time-domain wiggles, but they are much shorter/faster than conventional brick wall filters.
See Dunn, Craven, ...
--CD's Catch-22:
1) We can only hear out to 20kHz.
2) Nyquist/Shannon are right and fs > 40kHz perfectly captures a 20 kHz band-limited signal.
3) So 44.1kHz is fine.
3) There is no way for transparently limiting a signal to 20kHz while using a 44.1kHz sampling rate AND keeping the in-band frequency response ruler flat.
5) So either 44.1kHz is not sufficient OR the system's frequency response should be allowed to roll-off through the upper octaves.
Follow Ups:
HowdyThanks for the correction RE the anti-aliasing. I tend to think of the pure audio world and the simplification of both filters being the same works well, at least in the abstract. Obviously when you get to best practices things are different...
-Ted
P.S. Why didn't you jump in earlier when people were making a mockery of Nyquist rather than picking on me :)
Hi Ted,not picking on you, not at all.
And why not earlier? Well, I only visit PHP twice a year ;-)
Back to business. All too often the reconstructor is specified as a filter (any filter?) with flat in-band response and infinite rejection past fs/2, and the specification for the ideal (ideal according to whose requirements?) AA filter. But specifying the reconstructor like that is wrong, or at least misleading. And copying the spec for the AA filter is wrong, as it is good for Nyquist, but not necessarilly so for the application at hand.
bring bac k dynamic range
HowdyWell that was my point :)
The op claimed that you needed more samples and that you could only represent signals whose peaks got sampled exactly. My point was that there are counter intuitive (to the uninitiated) restrictions caused by the requirement for band limited input (e.g. between 1/4 and 1/2 Nyquist only sine waves can be represented, no triangle, square, ...) and even more so the fact that there are no perfect implementations of the required filters. I was pretty careful to talk about ideal vs. actual implementation being a problem.
Oh well, back to "The Nutcracker" in multichannel on SACD which doesn't have these particular problems :)
Ted and Werner - Excellent discussion! Thanks so much...
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