Sine hard sync
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- KVRAF
- 7579 posts since 17 Feb, 2005
I maybe don't understand PWM as it is applied to audio oscillators of arbitrary waveform? Is this like a phase modulation with the modulator controlling the phase of the oscillator to 2 points of 0 and 180 degrees?
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- KVRAF
- 8491 posts since 12 Feb, 2006 from Helsinki, Finland
It is whatever you define it to be, there's no commonly agreed upon definition for anything other than pulse waves.camsr wrote: Fri Nov 17, 2023 10:59 pm I maybe don't understand PWM as it is applied to audio oscillators of arbitrary waveform? Is this like a phase modulation with the modulator controlling the phase of the oscillator to 2 points of 0 and 180 degrees?
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- KVRAF
- 1607 posts since 12 Apr, 2002
I think I already mentioned this elsewhere a couple of times, but now we might have a very good context for the question, possibly sparkling some new discussion, so let me raise it once again.
So to antialias the sine sync we are adding BLEPs of the few first orders. If the sine frequency is not very high, the sine's derivatives are falling off quickly as the derivative order n grows and we can simply ignore the derivative discontinuities above a certain small n.
Now if the sine frequency is larger, derivatives are falling off slower and we need more BLEP orders for the same antialiasing quality.
At some frequency close to Nyquist (Fs/2pi or smth like this, didn't bother to to the exact math, but you get the idea) the derivatives stop falling off. Somewhere around this point is where we'll need infinitely many BLEPs. It's not necessarily exactly this point, because the amplitudes of "unit-magnitude" BLEP residuals of different orders are different. It could be that they are decaying a bit with n and the real critical point is exactly at Nyquist. So either at Nyquist, or at some close frequency we cannot any longer antialias sine sync by BLEPs, because we need to use infinitely many BLEPs per sync transition (or even "more than infinitely many", if the amplitudes of residuals grow with n).
If this critical point occurs exactly at Nyquist, this leads to the following conjecture: we can antialias discontinuities in bandlimited signals by BLEPs (the sum of BLEP residuals of growing order will ultimately converge), while for signals which are not bandlimited we cannot do it (the amplitudes of BLEP residuals won't decay with n).
This in turn would allow to ask the question of whether signals like exp(ft) is a bandlimited signal (which cannot be answered in terms of standard Fourier analysis). Generally, this proposes a new, more general definition of bandlimitedness (a signal is bandlimited if rectangular windowing of this signal can be antialiased with a converging sum of BLEP residuals). Practical application of this is e.g. generating an antialiased "exponential sawtooth", etc.
This definition of bandlimitedness may be in turn related to Paley-Wiener-Schwartz theorem, which effectively connects spectrum width to derivative decay rate.
So to antialias the sine sync we are adding BLEPs of the few first orders. If the sine frequency is not very high, the sine's derivatives are falling off quickly as the derivative order n grows and we can simply ignore the derivative discontinuities above a certain small n.
Now if the sine frequency is larger, derivatives are falling off slower and we need more BLEP orders for the same antialiasing quality.
At some frequency close to Nyquist (Fs/2pi or smth like this, didn't bother to to the exact math, but you get the idea) the derivatives stop falling off. Somewhere around this point is where we'll need infinitely many BLEPs. It's not necessarily exactly this point, because the amplitudes of "unit-magnitude" BLEP residuals of different orders are different. It could be that they are decaying a bit with n and the real critical point is exactly at Nyquist. So either at Nyquist, or at some close frequency we cannot any longer antialias sine sync by BLEPs, because we need to use infinitely many BLEPs per sync transition (or even "more than infinitely many", if the amplitudes of residuals grow with n).
If this critical point occurs exactly at Nyquist, this leads to the following conjecture: we can antialias discontinuities in bandlimited signals by BLEPs (the sum of BLEP residuals of growing order will ultimately converge), while for signals which are not bandlimited we cannot do it (the amplitudes of BLEP residuals won't decay with n).
This in turn would allow to ask the question of whether signals like exp(ft) is a bandlimited signal (which cannot be answered in terms of standard Fourier analysis). Generally, this proposes a new, more general definition of bandlimitedness (a signal is bandlimited if rectangular windowing of this signal can be antialiased with a converging sum of BLEP residuals). Practical application of this is e.g. generating an antialiased "exponential sawtooth", etc.
This definition of bandlimitedness may be in turn related to Paley-Wiener-Schwartz theorem, which effectively connects spectrum width to derivative decay rate.