[sdiy] Re: linear FM

[ASSI]

On Thursday 06 February 2003 13:54, jhaible at debitel.net wrote:
> > Then I haven't grasped the content in Achim's emails fully.
>
> Or I haven't. I'm still not so sure. Achim ?

I'm glad we have finally agreed on some important definitions and find 
some common ground for terminology, which should make the rest of the 
discussion much easier. :-)

Side remark: I can't help it, but for me analog synthesizers are not 
just musical instruments or electronic circuits, but direct descendants 
from analog computers. Their field of application is different and 
their operational parameters are optimized for frequency domain input 
and output, but they are nevertheless some of the most complex analog 
computers that have ever been built.

[I've tossed out the quotations as I'm trying to answer questions from 
several mails at once. I'm sorry, but it is still somewhat long-winded, 
I hope to make more sense this time around.]

As to the question of the model(s) being consistent: yes they are, both 
mathematically and as proven by real life examples of digital 
oscillators based on phase accumulation.

The idea of running an oscillator backwards in time to explain negative 
frequencies is nice from the standpoint of intuiton and actually works 
for most if not all of the math too as long as you're talking about 
ideal(ized) systems. I know for sure it doesn't work with lossy systems 
as they are non-reversible.

The proposed oscillator has precisely the advantage that a) the 
complete (wrapped, but to an interval larger than 2pi) phase 
information is explicitly in it's state, b) you can always tell the 
sign of frequency except of course at zero by looking at the state and 
it's derivative and c) the output is a direct mapping (aka function) 
from that state, but it is never used to feed back to it. It is these 
properties that make it practical to implement the linear-through-zero 
operation (at the expense of some rather hard problems to solve in an 
actual implementation - but these have nothing to do with negative 
frequency as such).

The system idea is roughly sketched as follows: the frequency input is 
integrated into phase state (lets for the moment assume unwrapped). You 
can tell the sign of frequency by the first derivative of that state 
(naturally), but you don't need too lok at that. You never need to use 
the output or derivatives of the state to advance it and no matter how 
wildly you modulate frequency, it is always continous.

Since a real integrator has a finite range, you must wrap the phase, 
drawing on the property that phases spaced 2pi apart are 
indistinguishable when looking at the output. That mucks up the first 
derivative, but we don't use it anyway, so no problem. It is precisely 
this operation that presents the largest difficulty in implementation, 
as you need to make it transparent to the state->output mapping, but I 
think it can be done.

Note that you can't just wrap to 2pi, as after each wrapping you end up 
at the other border of your range and you'd need to decide whether to 
wrap again using the sign of frequency. This is precisely where the 
Schmitt-Trigger discussion stranded - you simply can't make that 
decision at that very moment, besides you'd accumulate the wrapping 
error very fast when you try to stop the oscillator at this point by 
setting it's frequency to zero. So you got to go to some larger range 
which ensures that you never need to wrap twice in some finite interval 
of time and you never need to decide which direction to map to. A range 
of n*4pi is mightily convenient, as you can always map to zero (which 
solves quite some accuracy problems). I think that the other problems 
can be solved by ganging two or more integrators and resetting only one 
of them at a time while steering the integration current to the other 
ones, but I get ahead of myself here.

Last but not least you need to fold the wrapped state space into the 
output. This folder needs more accuracy than your ordinary waveshaper 
as you must make sure that at least two equivalent representations of 
any wrapped phase end up at the same output amplitude. The upshot is, 
if you get it right, it's relatively easy to provide many harmonically 
related frequencies from the same phase state simultaneously (sine 
banks anyone?). Also, as an unexpected bonus (I almost fell off my 
chair when I realized this) it is a piece of cake to add phase 
modulation once the other puzzle pieces are in place - you just add the 
(properly wrapped) PM input to the state space before the output 
conversion (that needs extra range to accommodate the PM). There you 
have it: an analog FM operator (which always was really a PM operator 
as implemented by Yamaha).

I'm not sure if sine oscillators and oscillating filters can be 
"reversed" in the above sense. I think not, as all the topologies I'm 
familiar with use (part of) their state for the output and/or feedback. 
More to the point they use the first and second derivatives of their 
state and that's going to be discontinous as the frequency becomes 
negative. I don't know of any way to handle that gracefully, I've toyed 
with the idea of misusing a quadrature oscillator and phase modulation, 
but drawn a blank.


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

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