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Hey Aleksey,
at 44100 Hz you could get a 22050 "sine" that looks like this, if you zoom in on it in a wave editor: '.'.'.'.
But you could also have this: .'.'.' which is out of phase with the first one. Am making some error here?
As I understand it, the lower you go from the Nyquist frequency, the more "resolution" you get for phase.
Perhaps this is why recordings at 192 kHz sound better than recordings at 44,1 kHz  better phase relations between harmonics..
This is why CD recordings are said to have a "digital" high end.
I have done some tests several months ago  I built a small app in MSP which FFT'd a sound, and then multiplied the "phase" of the individual components by 010 times. At 0 you get an interesting effect... everything sounds digital.. and this does remind me of this "coldness" of the CD high end.
What do you think of that?
This might put a new light on some things.
Cheers,
Damian
Out of phase example is not really a phase information  it is not 'stored'. It is like overall phase of the signal  you can't tell whether you have inverted phase or not, without comparing it to something else.
Phase information does not get more or less precise if you change sampling rate because phase precision is infinite between DC and Nyquist/2 frequencies, statistically speaking. Precision you get depends on the analysis.
You may see this way: if we are talking about signal reconstruction (digital to analog conversion), DAC transforms digital signal into analog signal with "infinite sampling rate", and in your terms phase precision becomes infinite.
Hi Aleksey!
Sorry, I didn't notice the replies, I thought the "next/last/first/previous" buttons are for topics, not for pages... silly me. Only found out about your reply after I combed the massive backlog of emails I don't have anymore (hooray!)
Aleksey, what do you mean by that "phase information is not stored"?
Do you mean that phase can be different if you take a different external signal as reference?
If so  let me clarify  all the time I was talking about relations of phase inside a single digital PCM sound, like a sample, a recording, or a CD track. The reference is the sound itself  you look at overtones' phase in relation to the fundamental, and then you have both the harmonic in question and the global phase reference in the same system, which only allows these relations a finite number of "constelations". Constelations, since every sound can have a lot, lot of overtones.. and every overtone can further have very many phase "positions". Those two numbers, multiplied, can give you a very big number indeed  but still a finite one. Do you see what I'm trying to aim at?
Thanks,
Damian
I'm not quite understanding what you are trying to say/ask.
As for the phase information of the Nyquist/2 and DC frequencies, it is not stored anywhere in comparison to other frequencies. But both Nyquist/2 and DC do have polarity information (negative or positive) stored together with their magnitude, so I was probably not very precise  but still this is not a full phase information, because it simply does not exist.
Aleksey  I think we misunderstood eachother. For me, the f_s/2 has two phases ("typical" and "reverse"), with the amplitude always being positive. For you, the amplitude can be negative too.
What I'm talking about:
Take a sound. One of its overtones is the fundamental. Take that fundamental, and compare the starting offset (where the sine of that fundamental starts out), with the starting offsets of other overtones. What you get is phase relation  relation of the phase of all overtones to the fundamental. With natural tones, a continuum of possibilities for phase relations exists.
With PCM, it's very limited. If a sound is in the top two octaves (with f_s=44.1k), then the sound's first (and onlyonly audible) harmonic can only be in about 2 of those 'relations' to the fundamental (how many depending what exact pitch the sound is): for example, with a pitch of 11025 Hz, the first harmonic being 22050 Hz (so at Nyquist frequency), the fundamental's minimum either falls on the first harmonic's minimum or maximum. Image: imageshack
With slightly lower pitches, you get just about the same resolution, enhanced a bit by time quantisation noise. This changes when you go to a pitch of 7350 Hz or less, when you get a third harmonic (3x7350 Hz = 22050 Hz).
Again, with the example of a note at 11025 Hz: if you keep the same pitch but double the sampling frequency, you only get 4 possibilities for phase relations, then 8, then 16, and so on. In the real world, this sound can have that first harmonic shifted by 0.001 pi radians from the fundamental (and our ear will probably be able to hear it). With PCM, at 44.1 kHz, you can only have it shifted by 0, or 1 pi radians.
If you lower the pitch of the sound one octave, the fundamental gets twice the "phase resolution" (because it has twice as many samples in a cycle that can serve as starting points). Thus, the relation of phase from the highest harmonic (which is somewhere between 11.025 and 22.05 kHz) can be described with precision two times higher.
Make the sound's pitch even lower, and you get even better precision.
This is why bass from CD can't be beat. A 44,1 Hz bass sound has, at 44.1 kHz, 1000 'starting points' to choose from for its fundamental. It has 500 'starting points' for the first harmonic, which is at 88.2 Hz. Then it has 333 of those for the third, and 250 'starting points' for the fourth harmonic. Even stopping here you can get very many different timbres: 1000x500x333x250 = 41625000000 different setups of "phase relations". This can record timbre quite well.
How does this compare to the number I got for the highest octave, namely 4? I think the timing resolution in the highest octaves is quite weak.
This is a limitation of PCM.
Does this sound interesting at all, Aleksey?
Your logic is a bit flawed here. You are missing the fact that sound perception (and sound analysis mathematics) has statistics aspect. It IS possible to represent phases near Nyquist/2 frequency precisely: speaking in your terms, this phase information is stored in several oscillations of higher frequency  not only in a single oscillation as you try to say.
And only DC and Nyquist/2 frequency itself contains two 'phases' only: (negative and positive)  in terms of Fourier analysis these states are represented as 'positive' and 'negative' magnitude.
Do you mean adding quantisation noise, which "stretches" and then again "shortens" some oscillations?
Well, I have seen a nonlinear phase shifter, and a free one too. It's the PhaseTone from Tritone Digital. It sounds good, but could be so much better. You can do a tool like that a thousand times better!
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