[sdiy] Re: linear FM

[jhaible]

> > I can assure you that you do not need to explain to us what phasors
> > are.
>
> Sorry, I didn't meant to insult or patronize anyone. However my
> difficulties to bring my original point across led me to believe that I
> had to explain the terminology I'm using in more detail.

Well at least for me it was good to be reminded of the phase relationships
of harmonics, by you and by Ian. Thanks again.

But back to the main point:

> On Wednesday 05 February 2003 22:03, Ian Fritz wrote:
> > The variable being integrated is current.  An electronic device
> > cannot integrate a frequency.  This is nonsense.
>
> You and I are talking about different things it appears, so
> interpreting what I'm saying indeed does not make sense interpreted in
> your context.
>
> I am talking about integrating frequency and I mean it. I'm perfectly
> aware that the frequency variable will be represented by a current and
> the integration product a voltage when I try to implement that as an
> analog system like the one you are talking about. That voltage variable
> (representing phase) maps to another voltage variable which represents
> the momentaneous amplitude of the oscillator.

My reaction to your mail was the same as Ian's: in a capacitor it's current
that's integrated over time to get a voltage, and nothing else.
But there is certainly more, because we surely can look at a VCO from
a different perspective (in PLL analysis it's common to describe the VCO
as an integrating phase-in / frequency-out building block), and generally
you can map any physical unit to an electrical analogy, if you do it
in a consistent way.
And given a certain ambiguity of triangle oscillator states (unlike a
saw oscillator, a triangle oscillator has no clear mapping of a voltage
to a phase), I think your proposal of actually _defining_ such a clear
mapping, at the expense of working with negative frequency, is
really interesting!
It's just that I don't know if it's consistent, for the reason I tried to
describe in my previous mail. So I will try it here again, referring
to your example:

>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.

My point is that the same could be said for a sine oscillator, implemented
as an oscillating filter (no comparators and no switching at all!):
The current in one (or more) capacitor(s) reverses direction (just not
resulting in constant ramps as the triangle oscillator), but no one would
think of calling this "reversing the frequency". So where is the difference?


>> 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

Let's try to look at this from a different perspective. I think the first,
initial
definition of an angle is that of a fraction of a circle. The origin of
phase
is in an angle, and the origin of frequency is the spinning wheel. This
can be represented by e**jPhi = e**jwt. The sine and cosine are just
projections of this, or mathematically the imaginary and real part.
So a sine wave is really closest to this original concept, even if
it's applicable to complex waveforms like a triangle (and thanks again
for clarifying this).
Therefore, in your phasor representation, you always have this e**jwt
in the origin, even if you add the representations for the harmonics at
the tip of this fundamental phasor. (I hope I get the words right - you
certainly get the idea.)

Now I would not describe a triangle wave with a fundamental phasor
that permanently changes direction; I would describe it with a rotating
fundamental and some rotating additions added-on, just as you
explained.

Which doesn't mean this must be the only way to describe a triangle
wave.
Are you proposing a description with a non-rotating (but oscillating
between two end positions) fundamental phasor, and a more complex
add-on on top of it, which would result in the same overall behaviour
as the usual description? Somehow you must make your mapping
of current and phase, voltage and frequency, consistent to the
phasor description ...

JH.