[sdiy] Re: linear FM

[ASSI]

I've probably missed the real start of this discussion by several 
years, but my mindless ramblings on this anyway...

While negative frequencies are phase reversed equivalents of the 
positive frequency. Frequency is phase velocity and the integral of 
that gives the momentaneous phase. Folding the momentaneous phase back 
to any 2pi wide interval and mapping it appropriately to amplitude 
gives an oscillator. Triangle relaxation oscillators work like that, 
but there is an irony involved: they already employ a negative 
frequency for half of their period, essentially using a chopper to 
reverse the sign of frequency at pi intervals. Sawtooth doesn't, but 
the discontinuity at zero phase does not bode well for doing a phase 
reversal (or trying to hold frequency zero) near that point. Also the 
integrator is typically hard reset to zero, which is not what you want.

Linear-through-zero IMHO could function almost like a sawtooth 
oscillator: stubbornly just integrating frequency to get momentaneous 
phase. You will have to reset the integrator periodically, but not 
necessarily at the same point you reset the output amplitude. It should 
probably have around half a period worth of headroom on either side to 
allow for slight variation of the reset. The reset operation should set 
it back exactly +-2pi when it gets close to the limits, which woud be a 
crucial point of the design. For the actual output you need a folder 
that folds phase from -1.5pi...1.5pi to -1...1. This gives sawtooth 
and/or triangle(s with varying frequency) based on the number of folds 
and more waveforms when you integrate a shaper into it.

I'm not saying that these circuits will be easy to realize with 
discrete elements, but with some trimming and careful compensation a 
linear-through-zero VCO should be possible. Why should it work? The 
integrator is never reset to zero as in the sawtooth oscillator, but 
just advanced or set back 2pi. I've not been thinking it through 
completely yet, but you could probably use a double integrator scheme, 
where one is active while the other is being reset and then they are 
swapped. For the output you use the sum of both to go through the 
folder, which rids you of any discontinuities. A folder to sawtooth 
still has the problem of not liking to stop around the cliff, which you 
can probably work around with some mild form of hysteresis (triangle is 
no problem). It would have some droop near zero frequency, but that is 
probably not much of a problem.


Achim.
-- +<[ Q+ & WAVE#46 & microQkb Omega sonic heaven ]>+ --

SD adaptation for microQ OS2.14:
http://homepages.compuserve.de/Stromeko/microQ-ADA214R11.zip (Win)
http://homepages.compuserve.de/Stromeko/microQ-ADA214R11.sit (Mac)