# The mean value of phase noise as a stochastic process

1. What is the mean value of phase noise as a stochastic process?
2. Where can I get a theoretical analysis of this topic?
• how much detail do you require? In circuits, at each node, the total phase noise contribution is the combined noise densities, scaled by a short-term constant. What models do you use for your own phase noise thinking, at the circuit level? at the system level? – analogsystemsrf May 17 at 2:11
• Just theoretical analysis level.@analogsystemsrf – wu yi May 17 at 2:15
• It is not necessary to consider the spectral characteristics, but only the mean value, because the spectral characteristics are not accurate in theory to describe phase noise well.@analogsystemsrf – wu yi May 17 at 2:22
• I don't use any model, just consider phase noise purely.@analogsystemsrf – wu yi May 17 at 2:32
• Phase noise in what? You've tagged "PLL", do you mean in a PLL circuit? – TimWescott May 17 at 3:48

Noise is a slippery concept that sort of wriggles out of the way as you try to define it at low frequencies, phase noise in particular.

Remember, noise is a model. A PLL is a physical system with its own behaviour. We can attempt to describe what the physical system is doing in terms of any given model, but sometimes the model doesn't fit well. At that point, we can try to force an interpretation of the behaviour of the system in terms of the model, and get strange answers. Or we can accept that the model isn't universal, and switch to a different model in different regions.

The model most of us use for noise is that it's an ergodic process, that the mean power in any given bandwidth is constant. When applied to oscillators and the like, we assign different offsets to different processes. Out at 10MHz offset, noise is often flat, and referred to as floor noise. In at 100Hz, we are usually well into the 1/f region. The spectral content of the noise is a crucial element of the model. The noise mechanisms at these offsets are well understood, Johnson noise, flicker noise etc.

The main motivation for a model is prediction, what will the signal be at some time in the future. With an ergodic process, we assume that what happened in the past will still be happening in the future. We assume that (for instance) the 20kHz offset rms noise in a 1Hz bandwidth will still have the same mean.

However, what happens when we get down to 0.001 Hz offset. What about $$\1\mu\$$Hz ? Over a period of minutes or days, the ergodicity assumption is no longer valid. There are mechanisms of warmup, drift, ageing, all of which affect the frequency, and are not ergodic.

When we want to move from phase noise spectral density to 'total phase excursion', which we get by summing over all frequencies, how many is 'all'? How long do we sum for. We get an explosion as we approach DC, a finite frequency error becomes unbounded in phase, and we have to truncate the low frequency spectrum somewhere.

For all of these reasons, many workers use Allen Variance instead of phase noise spectral density for very low offset frequencies.

Finally, phase noise of a complicated system like a PLL is the result of adding many noise contributions together, there's at least the reference oscillator, the output oscillator, the frequency processing chain (mixers, dividers, isolation amplifiers), the phase detector, and the components in the PLL. All of these have to be measured. Once you have measured them all, they can be combined, at any given offset frequency, by the standard PLL equations.

• Thanks for your reply, your reply gives me something essential that I need to learn. – wu yi May 17 at 8:11

stochastic process = random process

e.g. thermal noise added to signal causing phase noise measured near carrier in dBc/Hz at an offset from carrier, measured as single-sideband or 1/2 double-sideband values.

It is also known as random jitter in time domain.

• Thanks for your answer, but It seems that the mean value of phase noise can't derive from your answer.@Sunnyskyguy EE75 – wu yi May 17 at 2:26
• perhaps you really want the RMS value – analogsystemsrf May 17 at 2:49
• Thanks for your answer, but RMS is apparently not mean value.@analogsystemsrf – wu yi May 17 at 2:53
• mean value of a Gaussian distribution is ....... Zero. I believe Carl Gauss did some early work on this. Later, Alan Turing wrote his master's thesis on similar philosophical concerns. Read the Turing work. – analogsystemsrf May 17 at 4:48
• Thank you for your reply. Zero mean is the most probable result. I'm looking for an answer.@analogsystemsrf – wu yi May 17 at 5:25