[gregorio]

hmm... let me see... well so for digital audio (16/44.1k) you band-limit the input (so removing all frequencies over 22k)... because sampling theorem only works for band-limited signals. we get perfect reconstruction of band-limited signals OK.

No, sampling theorem also works for signals that are not band-limited to the Nyquist Freq and there are practical applications for this (see “Undersampling”). The problem with audio is that the images created by the sampling process will alias back into “band of interest” and so perfect reconstruction of that band cannot be achieved. Hence why frequencies above Nyquist must be removed.

Now the idea is that the original soundwave and band-limited to 22k is non-distinguishable...

No, the idea is that the original sound wave up to the filter roll-off is indistinguishable from the band-limited digitised signal (after reconstruction). If the original sound wave contains frequencies above the filter roll-off (and cut-off) obviously it will be distinguishable, though not audibly of course assuming that roll-off is not significant within the audible band. Hence why a sampling rate significantly higher than 40k was chosen.

so first we band-limit it in a specific way

Not really, although it depends what you mean by “specific way”. In practice, first we band-limit in a quite simplistic way, IE. We sample at some massive oversampled rate, requiring a relatively simple analogue filter that only needs to band-limit to a Nyquist frequency in the several megaHertz range. However, we then (second) need to apply a quite specific “Decimation filter” in the digital domain, to achieve our desired output sample rate.

I’m not sure I understand the relevance/context of the rest of your first paragraph. After the decimation process there are no (or virtually no) encoded freqs above the Nyquist frequency, while on reconstruction images above the Nyquist frequency will again be generated, which is why an anti-imaging filter is required in the DAC process.

Again, I am not claiming there is audible difference between original and 22k band-limited signal but that there is more to it than just sampling theorem...

Up to the transition band of the filter (say roughly 20kHz) what difference do you expect that may or may not be audible? Obviously there will be a difference within and above the transition band but assuming a reasonably designed filter there would be no question of audibility (as the difference is above 20kHz). So how is “there more to it” than sampling theory?

G

 

[gregorio]

Looks like we are finally on the same page here.

Hopefully! As the context of my post was the magnitude of Gibbs effect relative to the “straight angles” of the sawtooth, square and triangle waveforms which are eliminated by the curved/“sinusoidal” angles in the sound produced by a violin.

G

 

[71 dB]

hmm... let me see... well so for digital audio (16/44.1k) you band-limit the input (so removing all frequencies over 22k)... because sampling theorem only works for band-limited signals. we get perfect reconstruction of band-limited signals OK. Now the idea is that the original soundwave and band-limited to 22k is non-distinguishable... so first we band-limit it in a specific way (say technically removing all Fourier coefficients/components over frequencies 22k.... but those Fourier sin, cos base waves have all uniformly spaced frequencies)... secondly even pure sine waves of frequencies that are not in the base of Fourier series will also be band-limited... [say sin2x will be perfectly reproduced in base of 1,sinx,cox,sin2x,cos2x,sin3x,cos3x but sin(2.1x) will not... but 2.1 is within the band-limit of 3]... so for me the question is whether we experience sounds as is or through components (i.e. Fourier coefficients) (but again those components are related to a very specific uniformly distributed base of Fourier series) ...

Again, I am not claiming there is audible difference between original and 22k band-limited signal but that there is more to it than just sampling theorem... and for my part I quite often prefer listening to YouTube music videos (= compressed) over my CD collection (since large large portion of new artists on YouTube is nowhere to be found on CDs)

PCM digital audio is stored in time space and the reconstruction of the data into analog signal approximates (audibly totally accurately) the mathematical process of summing time shifted sinc-functions scaled by the sample points.

Now, lets see what happens in frequency space:

Say we have PCM audio sampled at 44.1 kHz and we take a 2^15 = 32768 point FFT of it. That's about 0.74 seconds of audio. Our frequency resolution ∆F = 44100 / 32768 ≈ 1.346 Hz. So, for non-negative frequencies we have DC (0 Hz), ∆F, 2∆F, 3∆F,...,16383∆F. We have also the window function w(t) and its spectrum W(f) which gets convoluted with the signal spectrum S(f) smearing it:

spectrum = W(f) ∗ S(f).

Now, what happens if the audio has frequency 100.5 * ∆F ≈ 135.3 Hz? It falls "between" the frequency points 100∆F and 101∆F. Well, the resulting spectrum has "halved" components at both frequencies. If we have 100.1 * ∆F instead, about 90 % goes to 100∆F and 10 % to 101∆F. Because of windowing, frequencies get smearing more or less anyway. However, if we have only 0.74 seconds of audio, telling 100 ∆F and 100.1 ∆F from each other isn't "easy", because during the 0.74 seconds, they differ only 𝝅/5 radians from each other. If you double the FFT size to 65536 points (1.486 seconds of audio), those same frequences become 200 ∆F and 200.2 ∆F and they differ more, 2𝝅/5 radians from each other and so on. Nobody "forces" you to use small FFTs unless you want good temporal resolution.

 

[71 dB]

Hopefully! As the context of my post was the magnitude of Gibbs effect relative to the “straight angles” of the sawtooth, square and triangle waveforms which are eliminated by the curved/“sinusoidal” angles in the sound produced by a violin.

G

All I had an issue with was you calling the sound of violin sinusoidal. Well, luckily violins aren't that boring sounding!

 

[medon]

I was listening to Flac files, every time a new track starts you can listen the beginning of the track at a very very low volume, during maybe less than a second then the track starts at normal and audible volume, it's almost like an echo of the beginning or the track, i have encounter this phenomenon before, what is it? why it happens?

... maybe something like this?

At around 30 seconds into the track the bass/guitar starts.
Half seconds or so before you can already hear it, much more silent. Some sort of pre-echo.

 

[bigshot]

That sounds like print through on the master tape. Magnetic tape can transfer loud sound from one piece of tape to a piece overlapping it in the wrap. Master tapes are often kept "tails out" so print through occurs *after* the sound instead of before.

 

[bigshot]

With loud sounds, the magnetism transfers from one section of tape to another creating an echo. https://en.wikipedia.org/wiki/Print-through

 

[Ryokan]

Is that like molecular switching?

 

[bigshot]

Magnetic transfer. Like if you leave a magnet on metal long enough, it will become slightly magnetized.

 

[old tech]

Well, magnetic tape can store digital signals as well, eg DAT.

 

[Davesrose]

Well, magnetic tape can store digital signals as well, eg DAT.

Oh, you also had to go to that rabbit hole. Reminds me of also folks saying laser disc is still a medium to collect because it was the first to have digital audio with video formats. Never mind the fact that laser disc looks awful with current large screen TVs (especially compared to 4k HDR). DAT certainly weren't the first forms to store digital info on magnetic tape. In fact earliest home PCs (like Radio Shack TSR-80) first main storage was cassette tape. Makes sense given that age when you consider performance of single density floppy disks as well.

 

[medon]

Well, magnetic tape can store digital signals as well, eg DAT.

... well your cell phone operates in a "digital" network using electro-magnetic waves for transmission, but in the end the electromagnetic wave doesn't know if it's meant to be digital or analog.
Whacky metaphor maybe, but you get the idea.

There is nothing inherently "digital" in our analog world, "digital" is only an abstraction layer.

 

[bigshot]

I have digital underpants.

 

[medon]

I have digital underpants.

... exciting, tell us more! Like... so?

 

[gregorio]

Well, magnetic tape can store digital signals as well, eg DAT.

Sure, although DA Tape used a different formula. The fact that magnetic tape could be used to store digital data obviously doesn’t preclude it from also being used to store analogue signals. The same is true of some analogue optical media, like film and video tape but also obviously, the discussion was specifically about analogue tape, pre-echo caused by bleed through (which doesn’t occur with digital audio tape).

There is nothing inherently "digital" in our analog world, "digital" is only an abstraction layer.

Not sure what you mean? There is nothing inherently digital or analogue in our acoustic world. I’m also not sure how “digital is only an abstraction layer”, except maybe metaphorically.

G