[sdiy] Re: linear FM

[Magnus Danielson]

From: ASSI <Stromeko at compuserve.de>
Subject: [sdiy] Re: linear FM
Date: Tue, 04 Feb 2003 23:07:23 +0100

> 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.

Right, well almost... right at least...

For a waveform function f(x) (a waveform function is a function which reoccur
for every 2pi, i.e. f(x+2pi) = f(x)) we can write

y = f(Phi)

where Phi is the total (or absolute) phase.

Phi = theta + Phi0

gives a relative phase. The total relative phase theta is then

theta = omega * t + phi

where omega is the angular velocity and phi (notice, small phi, not the big phi
from above) is the relative phase. Then, for completeness

omega = 2 * pi * f

where f is the frequency in Hertz if t is the time in seconds.

The phase expressions given avoids detailing sources of modulations, which have
been deeply studied.

It can then be realized that a negative frequency (i.e. negative f) is equalent
to that of a negative t, since the sign multiplies and as long as omega times
t is negative the effect is the same regardless of it being f or t being
negative.

> Folding the momentaneous phase back to any 2pi wide interval and mapping it
> appropriately to amplitude gives an oscillator.

Well, no. An oscillator is a device is a device which generates a re-occuring
waveform. From the point of LaPlace analysis it is a device which has a pair
(or several pairs) of poles on the jw-axis.

In synthesizer design is the phase accumulator design (sawtooth) commonly used,
but it is not the only way to view it.

> 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.

That's only one way of viewing it. However, there is actually a contra-proof to
this, if you look carefull you will find the rising time being a little
different from the falling time. You can measure this with a counter by
measuring the positive and negative pulse withs. When they differ, you are not
reversing the frequency or the time, you just get a different waveform
function, which is an entierly different thing.

So, the actual phase is not entierly in the capacitors charge, but the state of
the schmitt-trigger also needs to be accounted for, those together with the
actual load current (which we assume to have a different amplitude and not only
different sign) will help to resolve the actual phase.

However, you can use it as an analogy to understand how a triangle core can be
used to represent negative frequency.

> 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.

What the sawtooth core does is that it gives a waveform simlar to that of the
2pi wrapped phase. It does not however give the wrapped phase, since the reset
period is a distorsion of the phase. Again, it helps to understand it as an
analogy, but it stays there.

Cheers,
Magnus