[time-nuts] Re: uncertainty/SNR of IQ measurements

[Joseph Gwinn]

On Fri, 27 Aug 2021 03:30:28 -0400, time-nuts-request at lists.febo.com 
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> Date: Thu, 26 Aug 2021 10:36:46 -0700
> From: "Lux, Jim" <jim at luxfamily.com>
> Subject: [time-nuts] uncertainty/SNR of IQ measurements
> To: Discussion of precise time and frequency measurement
> 	<time-nuts at lists.febo.com>
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> This is sort of tangential to measuring time, really more about 
> measuring phase.
> 
> I'm looking for a simplified treatment of the uncertainty of I/Q 
> measurements.  Say you've got some input signal with a given SNR and you 
> run it into a I/Q demodulator - you get a series of I and Q measurements 
> (which might, later, be turned into mag and phase).
> 
> If the phase of the input happens to be 45 degrees relative to the LO 
> (and at the same frequency), then you get equal I and Q values, with 
> (presumably) equal SNRs.
> 
> But if the phase is 0 degrees, is the SNR of the I term the same as the 
> input (or perhaps, even, better), but what's the SNR of the Q term (or 
> alternately, the sd or variance) - Does the noise power in the input 
> divide evenly between the branches?  Is the contribution of the noise 
> from the LO equally divided? So the I is "input + noise/2" and Q is 
> "zero + noise/2"
> 
> If one looks at it as an ideal multiplier, you're multiplying some "cos 
> (omega t) + input noise" times "cos (omega t) + LO noise" - so the noise 
> in the output is input noise * LO + LO noise *input and a noise * noise 
> term.
> 
> I'm looking for a sort of not super quantitative and analytical 
> treatment that I can point folks to.

There are many treatments of phase-detector noise available outside 
of the time world.  The following may help.

.<https://en.wikipedia.org/wiki/Rice_distribution>

.<http://www.seas.ucla.edu/brweb/papers/Journals/HRTCASMar13.pdf>

By and large, these sources assume that the I and Q noise components 
are statistically independent, which may be only partially true in 
time-nut service.  A phase detector implements the pointwise multiply 
operation, yielding sum and difference terms.  The sought-for phase 
is in the difference term, in which noise in common will cancel out 
(in the voltage domain), while the noise not in common will add as 
power.

The easiest analytical approach to sorting this out is to draw the 
block diagram, and trace AM and PM phase noise expressed as 
band-limited zero-mean random functions of time through the block 
diagram.

Joe Gwinn