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I am trying to find a mathematical model to understand the correlated phase noise.
Here, I see the reference phase noise is almost the same after up-conversion and then down-conversion. How does correlated noise get canceled as though the mixer has not added any noise at all?
I am looking for a signal processing angle to this problem and any math that supports the theory will be extremely helpful as well.
Many thanks!
The OP clarified that he understands the fundamental details of mixers, and how phase noise can be correlated between the input and output ports. The primary question is how phase noise cancellation occurs when the phase noise is correlated at the two input ports of the mixer.
This occurs in down-conversion only and not in up-conversion given the output in this case is the difference of the frequency AND phase of the input ports.
Since the mixer is a time domain multiplier, this is easily proven mathematically from the cosine product rule, showing that such a product of two sinusoids will result in the sum and difference of the two frequencies and phases:
$$\cos(\alpha)\cos(\beta) = \frac{1}{2}\cos(\alpha+\beta) + \frac{1}{2}\cos(\alpha-\beta) $$
$$\cos(\omega_1t+\phi_1)\cos(\omega_1t+\phi_2) = $$
$$\frac{1}{2}\cos((\omega_1+\omega_2)t+\phi_1+\phi_2) + \frac{1}{2}\cos((\omega_1-\omega_2)t+ \phi_1 - \phi_2) \label{1} \tag{1}$$
Upconversion or downconversion using a mixer is done with an appropriate filter to eliminate the upper or lower component at the mixer output. An example configuration as a down-converter is shown in the graphic below where a low pass filter (LPF) is used:
In the case of correlated phase noise, $\phi_1 = \phi_2 = \phi(t)$, that is the phase terms are both time varying and identical. In the down-converter, a low pass filter is used at the output of the mixer to eliminate the sum component, and equation \ref{1} becomes:
$$\frac{1}{2}\cos((\omega_1-\omega_2)t+ \phi_1 - \phi_2) = $$
$$\frac{1}{2}\cos((\omega_1-\omega_2)t+ \phi(t) - \phi(t)) = \frac{1}{2}\cos(\omega_1-\omega_2)t$$
Notice in this case the phase noise has completely cancelled, and how that can occur in the down-conversion process, but not up-conversion.
Decorrelation Through Time Delay
It should be noted that any subsequent time delay between the nodes prior to multiplying will serve to decorrelate the higher frequency offsets of the phase noise. Phase noise in general is depicted in the frequency domain showing the power spectral density of phase fluctuations, where we can see how much of the phase noise power is due to lower frequency fluctuations versus higher frequency fluctuations. It is intuitive to see how an extremely low phase noise fluctuation (the phase error is changing very slowly in time) will still be very highly correlated at the input and output of a relatively short delay element (meaning the slow phase hasn't changed quickly due to the short delay). In contrast, a component of the phase noise that is changing very quickly in time (the higher frequency offset components of phase noise), would be completely different at the input and output of a relatively long delay. Thus if we were to add any delay between the correlated phase noise sources prior to multiplying, the phase noise cancelation would appear as a high pass filter: the lower frequency components would be cancelled and the higher frequency components would add in power for a 3 dB increase in phase noise. This is well represented as a first order high pass filter (+6 dB/octave or doubling in frequency) with a cut-off of $1/T$ Hz, where $T$ represents the time delay in seconds. I have implemented transponders where the interrogator used a very noisy local oscillator, which was returned by a transponder with modulation that used constellations where demodulation wouldn't otherwise be possible with such a poor phase noise. The interrogator however would mix it's own local oscillator with the returned signal, cancelling out the poor phase noise and providing low error rate demodulation (and not possible by an adversarial monitor without also having the unmodulated interrogation signal)! The delay between interrogator and transponder limited the useful modulation bandwidth of such an approach.
