[sdiy] Analog bandwidth

[rburnett at richieburnett.co.uk]

> Has anyone actually measured the true signal bandwidth at the output
> of an analog synthesizer? Any make or model would do.

Yes, things like TR-808, TB-303, JX-3P, Alpha Juno... etc,

> Obviously in theory there's no need to have much happening above 20k.
> But I'm wondering how true that is in real circuits.

There is energy above 20kHz but not much.  Waveforms like Saw, Square 
and PWM inherent have 1/f rolloff (-6dB/oct), and triangles have 1/f^2 
(-12dB/oct).  Add to this the roll-off of the synth's filter of another 
-12dB/oct, -18dB/oct or -24dB/oct and there's not much energy left above 
20kHz, and what is there rolls off very quickly.  Many of the old 
Roland's have ceramic capacitors across their outputs (for RFI reasons) 
fed through resistors that put another pole in the response too, often 
around 15kHz or so.

> Motivation is to consider how much bandwidth DSP systems need to do a
> good simulation.

That really depends on the complexity of what the simulation is doing.  
It's a big subject, but 44.1kHz can be more than adequate for a lot of 
"Analogue Modelling" stuff, like modelling a TR-808's damped resonator 
drum sounds or some analogue synth sawtooth oscillators.  However, you 
might need to temporarily go up to 96kHz, 192kHz or higher to model 
things like guitar distortion with acceptable attenuation of aliasing.

> 44.1k is clearly a very poor representation of the real thing.

Everything you listen to on a CD is represented at Fs=44.1kHz, so I 
would be careful about saying that.  44.1kHz is perfectly adequate for 
representing the output from an analogue synth that you intend to listen 
too directly, but it might not be enough to deal with some extreme 
post-processing like a hard-clipping distortion for example.  (In this 
case you might choose to up-sample by saw 8 times, process the 
distortion, filter out all the ultrasonic hash that you can't hear 
anyway, and then decimate back to Fs=44.1kHz to listen again or burn to 
CD.)

> But I'm
> wondering how high you really need to go, and how careful you have to
> be about modelling features like limited opamp gain bandwidth product
> and possible single pole phase shifts (etc).

You only go as high in sample rate as you absolutely need too, and only 
for those processes that need it, otherwise you waste CPU resources.

Usually op-amps have GBP up in the MHz range, so unless the op-amp is 
being used with a gain up in the hundreds it's pole won't start 
dramatically to impact the audio response below 20kHz.  Again, it's in 
areas like guitar distortion where GBP might be pushed, but good 
analogue designers try to make their designs so that the behaviour isn't 
critically dependent on a particular op-amp parameter.

> If there are real differences the usual abstract models of (say) a
> four-pole filter aren't going to be all that close to the
> imperfections of the real thing.

An example you might want to look at is the analogue state-variable 
filter.  Without compensation the filter is slightly more apt to 
self-oscillate at high cutoff frequencies because of increased phase-lag 
in the feedback path at high frequencies.  If you look at some of the 
discrete analogue Roland SVF designs (one of the Jupiter service notes) 
you'll see little ceramic caps of a few pF connected across some 
resistors at the inputs of the summing amps in the feedback path to give 
lead-compensation and balance out the phase-lag in the feedback path.  
This makes the self-resonance more even across the audio range.  
Ironically, some digital filter designs have "similar" problems to 
contend with, but this is due to the unit-sample delay inherent in 
digital feedback systems.

It's a big subject, and this is a very general quick reply, so I hope it 
helps answer some of your questions,

-Richie,