# [sdiy] Re: linear FM

ASSI Stromeko at compuserve.de
Wed Feb 5 21:26:35 CET 2003

```On Wednesday 05 February 2003 10:40, jhaible at debitel.net wrote:
> Hmm, reversing the direction of current flow thru a capacitor doesn't
> mean reversing frequency.

A triangle oscillator can be viewed as a system that is based on
integrating the frequency input to phase and then mapping the phase to
amplitude in a linear-modulo-2pi fashion. Hence if you reverse the
current flow, you reverse the sign of the variable being integrated,
which is frequency.

> Actually, a triangle wave is a mix of many
> partials, all with their own frequencies (if you're using a fourier
> series). And while it's certainly convenient to speak of "phase" to
> describe a fraction of a triangle cycle (just as you would for a sine
> wave), I'm not sure if this can be backed up mathematically at all.
> I guess it's all just a matter of definitions. But does this equation
> f = d_phi/d_t make sense for any other waveform than a sine? (Other
> than referring to the _fundamental_ sine if you speak of the "phase
> of a triangle wave"?)

Phase and frequency are well defined concepts for all periodic
functions (and even beyond with some constraints). That's incidentally
why fourier analysis will yield an amplitude and a phase (or complex)
spectrum in general. Anyway, in the digital domain you'd use a phase
accumulator (which naturally wraps around modulo2 in a binary
implementation) and what I've been proposing is the analog equivalent
of that - a phase integrator.

The finite range of integration and the obvious deficit of not wrapping
around automatically are the main problem, the other one is smooth
mapping of phase to amplitude. Both would (relatively) easily be
solvable in an IC, producing it with discrete components is
challenging. For one, set-back of the integrator by 2pi must be
accomplished very precisely over some range of input voltages and in a
very short time. Failure to do so results in (potentially colored)
noise. The other problem is mapping of phase to amplitude, any error
here also produces noise or unwanted spectral components.

On Wednesday 05 February 2003 17:13, jhaible at debitel.net wrote:
> (But ... there is also a definition of phase for each harmonic, if
> you develop the periodic waveform into a fourier series.
> How do these two definitions fit together ? Especially when you apply
> linear FM to an oscillator with a waveform that's different from
> sine?

I hope you are familiar with the concept of a phasor as fear I can't
explain it fully in a few lines (you probably have seen them in
modulation diagrams). The length of that is the amplitude,
the rotational speed of the phasor is frequency and angle the phase.
One phasor is one sine component (cosine is just 90° away). If you add
another phasor (harmonic component) you add it to the tip of the first.
The resultant curve (somewhat resemblant of a cycloide) defines the
phasor (addition of the two in vector fashion) for the combined
waveform and in general can even have it's own frequency (harmonically
related phasors will share a fundamental frequency).

> And, if we use our oscillator waveform - triangle, saw etc. - as an
> input for a sine shaper, would it matter at all? (I don't think so).)

Working out what the spectrum is after applying an arbitrary waveshaper
to an arbitrary waveform is an entirely different matter, even though
it's highly interesting.

Achim.
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