[time-nuts] Phase Detectors/Mixers for DMTD and PN measurements

[Attila Kinali]

On Fri, 17 Jan 2020 11:23:36 +0100
Magnus Danielson <magnus at rubidium.se> wrote:

> > Yes, and that's exactly the problem.
> > To expand a little bit: Noise down-folding comes to be, when
> > a signal with harmonic content passes through a non-linear element.
> > Then, this element acts as kind of a mixer on the signal with itself,
> > thus the spikes, that the harmonics are, mix down the noise around
> > those spikes down to the fundamental frequency. (For a more thorough
> > mathematical explanation, see [1])
> 
> The main benefit of diodes is that they are non-linear. 

Yes. Otherwise they would make very bad mixers :-)

And yes, there is lots of stuff to know about mixers and their
behaviour. I have not yet scratched at the surface, much less
understood anything. 

> There is NIST reports about dividers doing odd divisions having
> increased noise, but dividing by 2 removes that. This is where it is
> very useful to understand how a PWM waveform spread it's energy into
> even and odd harmonics, as 50% has the even harmonics go to zero, and as
> a consequence not contributing to 1/f noise mix increase.

The even harmonics also give an explaintion why amplifiers of
almost the same specs can have vastly different 1/f performance. 
Linearity is key for 1/f noise, if you chain devices.

> > [2] The harmonics of a square wave decrease with 1/i, i being the
> > number of the harmonic. Passing it through a first order filter
> > adds a 1/f term, ie the harmonics then decay with 1/i^2.
> > Every increase in filter order adds another 1/f or 1/i term.
> > The sum of all 1/i for i=1,2,3,...infinity diverges (ie goes to infinity). 
> > The sum of all 1/i^2 for i=1,2,3,...infinity is ~1.645.
> > The sum of all 1/i^3 for i=1,2,3,...infinity is ~1.202.
> 
> You make an assumption about the first order lowpass filter frequency
> for those numbers. You would prefer to keep the bandwidth somewhat
> higher than the frequency of the square, since that will provide too
> strong AM-to-PM conversion, but with that the converged numbers will
> become somewhat larger.

Yes. I simplified here quite a bit for the sake of brevity.
There are more issues, like the sum being actually over 1/(2i-1)
and things like that. The math gets quite convoluted if you add
all the details, but does not give more insight.

			Attila Kinali

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