[time-nuts] Re: Simple simulation model for an OCXO?

[Magnus Danielson]

Matthias,

On 2022-05-02 17:12, Matthias Welwarsky wrote:
> Dear all,
>
> I'm trying to come up with a reasonably simple model for an OCXO that I can
> parametrize to experiment with a GPSDO simulator. For now I have the following
> matlab function that "somewhat" does what I think is reasonable, but I would
> like a reality check.
>
> This is the matlab code:
>
> function [phase] = synth_osc(samples,da,wn,fn)
> # aging
> phase = (((1:samples)/86400).^2)*da;
> # white noise
> phase += (rand(1,samples)-0.5)*wn;
> # flicker noise
> phase += cumsum(rand(1,samples)-0.5)*fn;
> end
>
> There are three components in the model, aging, white noise and flicker noise,
> with everything expressed in fractions of seconds.
>
> The first term basically creates a base vector that has a quadratic aging
> function. It can be parametrized e.g. from an OCXO datasheet, daily aging
> given in s/s per day.
>
> The second term models white noise. It's just a random number scaled to the
> desired 1-second uncertainty.
>
> The third term is supposed to model flicker noise. It's basically a random
> walk scaled to the desired magnitude.

What you have shown is 2 out of the standard 5 noise-types. The Leeson 
model describes how 4 noise-types can be generated, but in addition to 
that the Random Walk Frequency Modulation is included. For standard, see 
IEEE Std 1139. For Leeson model, see David Leeson articles from special 
issue of Feb 1966, as I believe you can find on IEEE UFFC site. The 
Wikipedia Allan Deviation article may also be of some guidance.

Noise-types:

White Phase Modulation - your thermal noise on phase

Flicker Phase Modulation - your 1/f flicker noise on phase

Because we have oscillators that integrate inside the resonators 
bandwidth, we also get their integrated equivalents

White Frequency Modulation - your thermal noise on frequency - forms a 
Random Walk Phase Modulation

Flicker Frequency Modulation - Your 1/f flicker noise on frequency

Random Walk Frequency Modulation - Observed on disturbed oscillators, a 
random walk in frequency, mostly analyzed for finding rare faults.

You have only modelled the first two. This may or may not be relevant 
depending on the bandwidth of your control loop and the dominant noise 
there. It may not be relevant. The phase-noise graph will illustrate 
well where the power of the various noise-types intercept and thus 
provide a range over frequency for which one or the other is dominant. 
You control loop bandwidth will high-pass filter this for your locked 
oscillator, and low-pass filter for your reference oscillator/source. 
The end result will be a composit of both. The Q-value / damping factor 
will control just how much jitter peaking occurrs at the cut-over 
frequency, and the basic recommendation is to keep it well damped at all 
times.

The ADEV variant of this is similar.

Now, to simulate flicker you have a basic problem, whatever processing 
you do will end up needing the full time of your simulation data to 
maintain the slope. You can naturally cheat and do a much reduced setup 
that only provides the 1/f slope of PSD at and about the area where it 
dominates. Notice that this can occurr both for the FPM and FFM cases. 
Also notice that if we respect the model, these should be completely 
independent.

Simulation wise, you can turn WPM into WFM by integration, and FPM into 
FFM by integration. Similarly the WFM becomes RWFM through a second 
integration. Just source each of these independent to respect the model.

The Leeson model for an oscillator does not have these fully 
independent, but for most uses, simulate fully independent, and you will 
not make more fool of yourself than anyone else usually does, present 
company included.

As you see, flicker and random-walk is not the same thing. Often "random 
walk" implies Random Walk Frequency Modulation.

Another thing. I think the rand function you use will give you a normal 
distribution rather than one being Gaussian or at least pseudo-Gaussian. 
A very quick-and-dirty trick to get pseudo-Gaussian noise is to take 12 
normal distribution random numbers, subtract them pair-wise and then add 
the six pairs. The subtraction removes any bias. The 12 samples will 
create a normalized deviation of 1.0, but the peak-to-peak limit is 
limited to be within +/- 12, so it may not be relevant for all noise 
simultations. Another approach is that of Box-Jenkins that creates much 
better shape, but comes at some cost in basic processing.

> As an example, the following function call would create a phase vector for a
> 10MHz oscillator with one day worth of samples, with an aging of about 5
> Millihertz per day, 10ps/s white noise and 10ns/s of flicker noise:
>
> phase = osc_synth(86400, -44e-6, 10e-12, 10e-9);
>
> What I'd like to know - is that a "reasonable" model or is it just too far off
> of reality to be useful? What could be changed or improved?

As I've stated above, I think it misses key noise-types. I really wonder 
if the flicker noise model you have is producing real flicker noise.

Using Bob's suggestion of using real data is not bad, noise simulation 
is hard and a research field in itself. My quick and dirty methods get 
you started at relatively low cost, except for flicker, flicker always hurt.

Cheers,
Magnus