[time-nuts] Noise of digital frequency circuits

[Achim Gratz]

Attila Kinali writes:
> Yes. This effect has been known for a few decades at least.  What kind
> of puzzles me is, that I have not seen a mathematically sound
> explanation of it, so far.

I'm afraid I can't help with the rigor, but the fundamentals seem simple
enough to me.

> People talk of aliasing and sampling, but do not describe where the
> sampling happens in the first place.  After all, it's a
> time-continuous system and as such, there is no sampling.

That may be quibbling over terminology and definitions not actually
specified in those papers.  Localization in the frequency domain
requires periodicity in the time domain (by definition) and moving
spectral features around can be done by convolution of the noise
spectrum with a localized signal (not necessarily of compact support,
but assume for the moment it is so you get a clearly defined pivot
frequency).  That means you need to do multiplication in the time domain
with something periodic, so all you need to produce noise folding is for
instance a periodically varying NTF.  I guess we can tick that box in
all instances you've mentioned.

> One could look at it as a (sub-harmonic) mixing system, but
> even that analogy falls short, as there is no second input.

Does it even matter if you call it a "second input"?

Reading the Egan paper I guess the line of arguments that leads to
"sampling", "mixing" and "aliasing" getting used is that the
periodically varying NTF (or ISF if you like) looks and acts
sufficiently like a Dirac comb that you can use sampling theory to
interpret the results.  Or conversely, that you can take the results and
postulate a sampling process with a sampling aperture that happens to
look virtually identical to (one period of) your NTF.  This seems not
much different than what gets routinely done when reasoning about
real-world systems that do "proper" sampling, but of course do not sport
a perfect dirac pulse sampling aperture.

> It also fails at describing why there is not infinite energy being
> down-mixed, as the resulting harmonic sum does not converge.

The actual integral or sum to compute would likely be governed by
something sinc-like, so convergence would eventually still happen with
any physically realizable input.  That assumes you don't already need to
start with some generalization of the Fourier transform that has more
strictly defined convergence behaviour.

[…]
>> > If you divide by something that is not a power of 2, then it is important
>> > that each stage produces an output waveform with a 50% duty cycle. Otherwise
>> > flicker noise which has been up-mixed by a previous stage, will be down-mixed
>> > into the signal band, increasing the close-in phase-noise.
>> 
>> Wow, another thing I never knew.  
>
> I do not think that anyone was aware of this.

Funnily a paper I just read in TCAS-I (February 2017) by Pepe and
Andreani about phase noise in harmonic oscillators seems to mention this
(I think) as a known result w.r.t. flicker noise upconversion and
generalize (ref. eq. 90) on previous results for several particular
oscillator topologies which guarantee the necessary conditions.  There
is also lots of discussion about the relation to the ISF and results in
conjunction with it that goes right above my head.  The direction their
math is taking looks intriguing, so maybe you are able to glean
something from it to use.  Whether there's any pre-existing link of
those results specifically to frequency dividers I don't know.


Regards,
Achim.
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