[sDIY](early) wavetable synthesis

[mbartkow at ET.PUT.Poznan.PL]

Paul,
sorry for replying so late:

> > When you sample any waveform, its spectrum is being replicated
> > at each multiple of the sampling frequency. For example, if you
> > sample a 1kHz sine using 8kHz sampling rate, the spectrum of the
> > sampled signal contains the original 1kHz partial (and a comple-
> > mentary -1kHz one) as well as all partials resulting from the
> > n*8kHz-1kHz, n*8kHz+1kHz formula, e.g. 7kHz, 9kHz, 15kHz, 17kHz
> > and so on.
> 
> Hmmm so there's more than the law of the Niquist frequency?

There is a whole sampling theory and the Shannon/Nyquist law is
just a mere conclusion from the one.

> But actually I didn't mean sampling but the other way around, putting a
> digitally generated wave through a DAC. Or this apply to this as well?
> Could you tell me more about the sampling law you're describing there?

When you digitally generate a wave in a form of a series of its discrete
values you are basically sampling its waveform inside the system. Only
having the original formula that describes the wave, and being sure its
theoretical bandwidth is limited to the half of the sampling rate you
may generate an alias-free physical signal.
If you reproduce the wave on the basis of some other sampled wave, you
are resampling a signal which already has been sampled, so its spectrum
is not limited. Lots of aliasing typically occur in such cases (the cha-
racteristical lo-fi sound of the Amiga/tracker music). Pro samplers 
usually apply interpolation to the waveform stored in the memory before 
the resampling process. Interpolation may be thought of as lowpass fil-
tering whose aim is to limit the spectrum (to have more smooth waveform.)

> > When you sample a complex signal, its whole spectrum is replicated
> > in the same way as in the example. Aliasing occurs, when the sampling
> > rate is too low and the reflected spectrum overlaps with the high end
> > of the base band.
> 
> But when is the samplingrate too low? The Niquist law say the sampleing
> freq. should be at least twice as high as the frequency you're sampling.

not the frequency, but the bandwidth. You cannot talk about frequency 
of a complex, nonperiodic signal.



> > > If yes, what happens with these partials when the pitch of the
> > > wavetabele is changed (by different methods used)?
> > 
> > All partials scale appropriatelly.
> > If you read out the samples from the wavetable faster than
> > originally sampled, the whole spectrum scales upwards, and
> > more partials fall outside of the "hearing range".
> 
> Well that technique isn't used very much. Instead the numbers of samples
> taken from a wavetable is reduced for a higher pitch or multiplied for a
> lower pitch. What happens than with partials created by sampling?

Well, AFAIK it IS used. You must not skip samples from the wavetable, if
the wave is not appropriately smoothed (i.e. its bandwidth is appropriately
limited), otherwise heavy aliasing occurs.
There is nothing mysterious in the partials created by the sampling (I assume
we are talking here about replicas of the original spectrum). They are carried
in the "gaps" between sample instances. Scalling the signal scales everything
accordingly.


I understand it might be confusing without a deep knowledge on the
Fourier transform properties. Try the excellent Bristow-Johnson tutor
on wavetable synthesis from the music-dsp site.

mb


--

Maciej Bartkowiak, PhD
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