[Fpga-synth] Signals and Sample Rates

[Eric Brombaugh]

On May 17, 2008, at 7:14 AM, Scott Gravenhorst wrote:

> I am trying to understand how to deal with signals and sample rates.
>
> For example, when a limit is given, such as for a digital state  
> variable filter being 1/6 of the sample rate, higher frequencies  
> causing instability, I think that this means a sine wave at 1/6 the  
> sample rate.  But most of our synth signals are not sinewaves and  
> thus can have frequency components much higher than the fundamental.
>
> How does one deal with the transients in sawtooth or rectangular  
> waves?
>
> Is it "OK" to use such a filter because the energy in the highest  
> harmonics tends to be diminished?
>
> I like the sound I get out of a digital SVF, but that's in a synth  
> with a sample rate of 1 MHz.  Seems to be enough headroom there.   
> However, I'm working on a synth that currently has a sample rate of  
> 125KHz.  That sets the limit much lower at 20.833 KHz.  Is that my  
> magic number (being just above what most of us can hear) ?  I.e., is  
> 125 KHz too low?
>
> What sorts of things will I need to look for to judge the quality of  
> my SVF at that sample rate?

The first thing you need to ask is "what is the effect of violating  
the 1/6 spec on the SVF?"

With most DSP the foremost issue is to avoid aliasing caused by  
violating the Nyquist criterion. That means you don't do anything that  
would generate frequency content beyond Fs/2. The consequence of  
violating Nyquist is that the signals beyond Fs/2 will fold back into  
the lower frequency range and more often than not will be  
anharmonically related to the signal you actually want to hear,  
resulting in noise and distortion.

Square, saw and triangle waves all have the potential to violate  
Nyquist - they are essentially infinite series of summed sinusoids  
with harmonic content that extends far beyond human hearing range.  
Using very high sample rates can mitigate aliasing when these  
waveforms are generated naively (using simple math that models the  
instantaneous voltage changes found in ideal waveforms). This works  
primarily because the ratio of sample rate to fundamental frequency is  
so high that the aliases which are created are at such a low level  
that they are not bothersome. But don't be fooled - they are still  
there!

So, getting back to the question at hand, what happens when you drive  
a signal beyond Fs/6 into your SVF model?  The only real way to tell  
is to do a full sweep of frequency (0 - Fs/2) and analyze the results.  
Chances are it will fall into one of the following categories:

1) The filter's outputs (HP, LP, BP, Notch) fail to perform as  
desired. Mainly this means that as frequency increases, the LP and BP  
states begin to exhibit less attenuation than at lower frequencies.
2) The filter becomes unstable. Perhaps stimulating the SVF with  
inputs beyond Fs/6 excites some undesirable behavior resulting in  
chaotic outputs.
3) Perhaps the Fs/6 limit applies not to the input frequency but to  
the corner frequency setting. It could be that you shouldn't attempt  
to configure a corner frequency > Fs/6, but input frequencies > Fs/6  
are OK and the filter action is still sane beyond that limit.

As you can see, there are a lot of potential answers to what happens -  
probably even more than listed above. If the answer turns out to be  
something like 1) or 2), then your only solution is to avoid  
generating any frequency content > Fs/6. You can either do this with  
band-limited waveform generation techniques, or by applying some sort  
of pre-filtering. If the answer is like 3) then the solution is simply  
to avoid setting the corner frequency too high.

For your present project with Fs = 125kHz, the 1/6 limit is not much  
of a constraint, especially considering that most musical content will  
be confined to fundamentals less than 8kHz or so. If your filter has  
problems 1) or 3) then you should be OK with this. If your filter's  
problem is 2) (chaotic behavior with inputs > Fs/6) then you'll need  
some more signal processing to protect it though.

You will find that with such a low Fs that naively generating  
waveforms with sharp transitions will result in noticeable aliasing in  
the higher registers. Beyond about 2kHz or so your square and sawtooth  
waves will begin to sound a little dirtier, and beyond 6kHz you can  
really hear those little birdies chirping in the background when you  
sweep the frequency around.

Eric