[time-nuts] uncertainty/SNR of IQ measurements

[Lux, Jim]

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.