[time-nuts] Noise of digital frequency circuits

[Attila Kinali]

On Thu, 20 Sep 2018 23:45:27 +0200
Achim Gratz <Stromeko at nexgo.de> wrote:

> Attila Kinali writes:
> > 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.

Exactly. But sofar nobody has properly specified what the other
term of the multiplication is.
 
> > 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"?

Not really, but if you want to argue about the noise folding
process as being a sub-harmonic mixing process, then you need
to specify what the second signal is that does the mixing. 
Which in turn is again specifying the two terms of the multiplication
process as above.

> > 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.

This is exactly one of the things that made me stumble when I first
went through the relevant literature. A sinc pulse-train in time
domain becomes a rectangular pulse-train in the frequency domain,
whose amplitude decays with 1/f. This means, the folded down noise
is a sum of terms decaying with 1/f. But this sum does not converge,
ie it goes to infinity. One has to add an addtional filter of some
sort that increases the rate of decay to 1/f^2 for the sum to converge.
 

> 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

Oh.. thanks! I somehow missed this paper. The results look indeed
interesting. But I have to spend some time to work through the math
in order to fully understand it.


			Attila Kinali


-- 
Science is made up of so many things that appear obvious 
after they are explained. -- Pardot Kynes