I am in no means qualified to really discuss this... it seemed logical to me though. In what sense do you feel he was incorrect though? How would you draw it differently?
SometimeWarrior: (off-topic) You go to UCSB? I'm at Cal Poly, computer engineering...
Phony. The guy takes the sampling of a 41.5 kHz wave, and since it's like a 2.5kHz wave, he extrapolates backwards and says that's a problem at 21kHz, which it's not.
A little question: would the wave be accurately reconstructed after the 44kHz sampling, if correct interpolation were used (ie: sinc) ?
A brick wall low pass filter should ring. You see his first criticism - that the 21kHz sine wave is amplitude modulated? The ringing removes the amplitude modulation. It's the ringing that fills in the gaps between the sample points, and also maintains the 21kHz wave at the correct amplitude in the regions where the samples appear to be low in amplitude. Without the ringing, you'll get the amplitude modulation. Pick your poison!
I don't see any reason why the sampling process would add amplitude modulation to the original waveform.
Also, you talk about how "long" a filter will ring for. I'm not sure what you mean by this. Do you mean the bandwidth of frequencies which produce noticable ringing? It is my understanding that this ringing is in the frequency domain, not the time domain - however I am going to research this further.
As far as the click at a sudden cutoff, you may be right, but it is not discussed in any of the papers, books, or webpages I have read so far.
Since a true brick-wall filter would have a slope of infinity and therefore be non-differentiable at the cut-off point, practical ones do not exist and therefore my be a non-issue.
All practical filters have some slope. From what I've read the biggest problem with a very steep filter is the ringing effect. You are right in that if you can't increase the sampling rate some sort of softer filter is needed to reduce ringing. I don't think the answer is to have a filter transisition beyond the Nyquist limit, since this will introduce aliasing errors. My guess is that some sort of gentle filter starting around 20kHz would be good and would not audibly dull the sound.
My question is, what type of filters are typically being used these days for audio? Maybe we're talking about something so slight that it doesn't even make an audible difference. It would be nice to hear some samples.
Just a side note: This is not just a problem in audio, this is also a problem in digital imaging as well, and there are well known filters (Lanczos, cubic) to reduce the same ringing we've been talking about due to Gibbs Phenomenon.
Ringing in filters happens whenever there is an abrupt discontinuity in the frequency response of the filter. If there is sonic "content" in the original signal at the frequency this discontinuity exists, pre or post ringing will appear. The more abrupt the discontinuity (~ higher slope, or steeper filter), the greater the time-domain ringing.In CD brickwall filters, the discontinuity happens near 22 KHz. So, if there is any ringing, it will appear at these frequencies. Also, for this ringing to appear, the signal must have content at these frequencies.Thinking a little bit more about this, in a properly recorded cd signal, there should be no content at these frequencies, because it must have been filtered at the AD stage in order to avoid aliasing. However, this AD pre-filtering can also produce ringing, if there was any content at the filtering frequencies in the original signal. So, if there is any ringing, it is already present in the recorded signal, and (most likely) not produced by the cd player brickwall filter. I say most likely because if the CD player filters at a lower frequency than the AD filter, it will eliminate the AD filter ringing, but will introduce its own. However, I don't think this is likely to be the case. Also, in the case of same synthetic generated signals, there can be no AD stage, then the ringing would happen due to the player filter, but I also think this is not very common, or likely to happen in commercial CD's.So, there can be ringing in the cd signal, but this ringing will be near 22 KHz, which is inaudible. Also, using a proper not ultra-steep AD filtering, this ringing time duration can be minimized.Although 22 KHz is not audible, if there are nonlinearities in the playback path at this frequencies, this 22 KHz ringing might intermodulate with other signals and produce intermodulation products which fall into the audible band of frequencies, and in fact "became" audible. However, and as recapitulation, for this intermodulation products to happen and be audible (which is our "final" main concern now), some conditions have to be satisfied:- There must be signal content in the original signal at 22 KHz. For the ringing to have audible significance, this 22 KHz content must be of relatively high amplitude, and of "transient" type, that is, of very short attack or decay times.- The AD stage pre-filtering must be quite steep. I believe that with a relatively "soft-edge" filter, the ringing duration and amplitude can be quite minimized. However, I don't know how steep are actually the filters commonly used.- The playback chain (mostly speakers or headphones) must be able to reproduce this 22 KHz signals, and also be quite nonlinear a this frequencies. I think that good speakers of headphones capable of reproducing such high frequencies are not likely to be very nonlinear. However, I could be wrong.So, I think that this ringing in CD audio is very, very difficult to be noticeable, and even if it was a real problem, could be effectively adressed using proper AD pre-filtering.
Question 1: would that 21kHz wave be accurately reconstructed after the 44kHz sampling, if correct interpolation were used (ie: sinc) ? Nyquist says yes.
I'm not an expert in signal processing (yet), but i think the problem is cardinal (sinc) interpolation is practically impossible to do in real-time, since you should know the amplitude of samples which are after the ones you're processing (in the time domain).