[sdiy] Oscillator waveforms imperfections and formants

Richie Burnett rburnett at richieburnett.co.uk
Sun Jul 18 00:05:50 CEST 2021


Ok... A pure impulse (pulse with infinitesimally small width) has an infinite and flat spectrum. If you integrate a train of these you will get an ideal sawtooth waveform (with infinitesimally short reset time) and an infinite spectrum with a -6dB/Oct roll-off slope.

Now for what happens with real pulses that have finite width, and sawtooths that have a finite reset time...

A pulse (boxcar function) in the time domain has a sinc shaped spectrum in the frequency domain. That means it has a spectral roll-off above a certain frequency and it has deep notches at certain frequencies related to the pulse width. These are the "formants" you referred too.  If you integrate a train of these pulses you get a sawtooth waveform with a finite reset time. This has a spectrum that is just like an ideal sawtooth with -6db/Oct roll-off up to a certain frequency, then the rolloff rate increases and you get notches related to the reset time. "Formants" if you like.

Another way of thinking about this is to think of a real pulse as being the result of an ideal impulse through a boxcar FIR filter. And likewise a real sawtooth as being the result of an ideal sawtooth through a boxcar filter. The boxcar filter adds additional rolloff slope and contributes notches into the spectrum. (You can see where the additional roll off and comb-like notches come from, by concidering the boxcar filter as a combination of two processes: integration and comb filtering.)

Whilst it is true that non-zero reset time in a sawtooth oscillator leads to HF rolloff and notches in the spectrum, the reset time would have to be quite large before the point at which these started to occur dropped below 20kHz and became audible. For example, you can clearly hear the effect of altering "triangle symmetry" on a synth that allows the waveform to be swept from sawtooth, through triangle, to a ramp. But for an analog sawtooth oscillator with 10us reset time most of the interesting shaping happens above 100kHz.  By the time a saw core analog oscillator had sufficient reset time for it to interestingly shape the audible spectrum I think the pitch tracking would be well and truly wrecked!

I hope this helps answer your questions.

-Richie, 

Sent from my Xperia SP on O2

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