[sdiy] samp. freq for ADSR ?

[Magnus Danielson]

From: Robert Holmström <robertholmstrom at hotmail.com>
Subject: Re: [sdiy] samp. freq for ADSR ?
Date: Sun, 04 Apr 2004 21:58:30 +0000
Message-ID: <BAY1-F254NkHsIZMU3U00027339 at hotmail.com>

Hej Robert,

> > > > Creating a 500Hz sampling frequency. Nyquist's theory is that the
> > > > maximum sampled frequency is half the sample rate, leaving the Buchla
> > > > MARF with a maximum of 250Hz (real low for audio).
> > >
> 
> I am not an expert in this so maybe somebody can explain it (simple) but 
> this is how I understand it.

I may fail on the simple part for parts of my discussion, but maybe you can
survive by skiping some of the text (theory corner) on the first reading.

> As written above the maximum sampled frequency is half the sample rate. This 
> means that if I want a 1Hz (low, just as an example) signal sampled, the 
> minimum samplingfrequency should be 2Hz.

That's correct.

> That sampled signal would be a 1Hz "square waveform" because only two 
> samples are taken in one full wave cycle. If I need any precision in the 
> sampled signal (like if the source was a saw waveform), I need higher 
> samplerate.

Actually, it wouln't be a squarewave, since we already established that you
need an anti-aliasing filter, so the overtones would be wiped out and you would
end up with a single sinewave anyway.

<theory corner>

However, this assumes a key thing, namely that the sample clock (this is the
signal establishing the signal rate) is not synchronous with the incomming
signal and if it is _exactly_ half the sampling rate, it is infact synchronous,
being of same clock (which is what synchronous breaks up into when looking at
it's original greek words, as explained in ITU-T Rec. G.701). When they are
synchronous, the phase relationship between half the sampling clock (i.e. the
Nyquist frequency) and the incomming sine will determine the amplitude
responce as being the cosine of the phase differance. Near frequency hit will
create a wobble in amplitude as the phase differance changes with the
difference in frequency unless the anti-aliasing filter is able to completely
remove the mirror frequency. Sufficient damping removes the wobbeling.

Now, most of that is theoretically correct, but usually not much of a problem.
There is another flaw with the above discussion which is also apparent in the
original Nyquist frequency discussion, since it assumes constant sines.
Actually, the assumptions show themselfs when looking at the proof provided
by B.M.Oliver, J.R.Pierce and C.E.Shannon in their article "The Philosophy of
PCM" from 1948, the proof is in the Appendix I. Besides being a very good
article which makes it points very clearly and IMHO is still valid reading for
many today dealing with PCM and sampling, it touches on the subject we are
discussing here. The thing is, this proof only discusses a function f(t) being
bandwidth limited by the bandwidth W0 (i.e. the Nyquist frequency) and uses
the Fourier transform. However, the Fourier transform isn't very good in
handling impulse responce type of signals (the Fourier transform is greatly
missused for doing it, even if it is part of the full solution). So, my point
here is that not all transients survive as well in a too tigh bandwidth limited
system. It takes time to build up the full picture.

If you translate the function into thinking in the LaPlacian domain, we see
that not only the frequency is relevant, but also the allowed range of
amplitude acceleration, which is the real part of the LaPlacian variable s.
Converting the signal into exponential functions of time only (similar to only
study the sine waveforms) we now must ask us how quick and slow rises we can
handle. And, for a bandwidth limited system there is indeed an upper limit to
the maximum speed that we can replicate over such a system! This means that
any signal having such acceleration would also be incorrectly transfered over
the system, i.e. loss of information.

</theory corner>

So, the conclusion is that transient functions also suffer information loss in
a bandwidth limited system, such as a sampled system. Additional "headroom" is
needed both for frequency as such and for transients as such or distorsion of
signal (i.e. loss of information) will occur.

For transient responses time will not repair impairments as they will with
constant amplitude signals such as sine (or a cluster of sines). This means
that the impairments needs to be reduced when the signal actually apears in
time.

> If the above is a correct explanation (?) that would explain why bass sounds 
> sound "good" on lower sample rates while high frequencys sounds "metallic". 
> The bass frequency waveforms are so long that a lower sample rate still 
> gives us an "accurate" sampled wave.

More or less, yes. For bassier sounds, you use up less of the given bandwidth
and transient responce of the system. This means that there is alot of
redundancy in there from a theoretical point of view. You may lower the
sampling rate to remove that redundancy for the benefit of better usage of
bits, without greatly reducing the actual information of the sound. When you
are trying to cut the corners too tight, you start to suffer more and more.

> Maybe I went a bit outside the ADSR subject, but I have wanted to ask this 
> for soo long that I thought it was good.

It's only until you try to design something that you learn how little you know
and what could possibly be an issue. This motivates your question very well and
you are correct in asking it. Hell, that's really a good road to increasing
ones knowledge!

Cheers,
Magnus - who is gonna toss himself in bed for a finite amount of time