Deep Thought

Today (Wed., 3/30/11) I am taking notes and trying to figure out exactly how to *correctly* derive the expected Allan deviation given the quantization error and an oscillator model. I've decided I want to use a Gaussian white noise model for the inherent frequency fluctuations in the oscillator. (Other behavior should show up in the Allan deviations for larger window sizes.)

Key parameters are:

1. Standard deviation of oscillator frequency (when averaged over 1-s window sizes) in cycles per nominal second. (We can plot the result as a function of this, and use that to do a Bayesan back-calculation of the probabilities of different standard deviations given the raw deviation data. Actually, this dist would be sharply peaked, given the large amount of data available - so we can just do a direct reverse lookup of the standard deviation given the Allan deviation.)

2. Fractional part only of mean oscillator frequency, in [0,1] say, because the integer part is irrelevant to the quantization error.

And computational variables include:

3. The possible values of the quantized frequency deviation samples d1=x1-2x2+x3 for adjacent 1-second windows. These are small integers; a range [-10,+10] would be more than adequate. For this we scan through the range of possible values.

4. Given d1, always let x1=0, and then scan possible values of x2. These will always be integers in a range close to the integer part of mean oscillator frequency. For example, 10,000,012 +/- 10. Given d1 and x2, compute x3=d1+2*x2.

5. Phase error phi1 of first (x1) measurement in oscillator cycles, in the range [-1/2, 1/2]. This is uniformly distributed so we can just integrate over the range. The actual initial phase is Phi1=x1-phi1.

6. Actual frequencies F1=Phi2-Phi1 and F2=Phi3-Phi2 can then be integrated over the range of values that will yield the target values of x2 and x3. And, at each value, the probability density values can be looked up from the normal dist. Overall answer of the integrations over phi1, F2, and F3: A probability for the particular (d1,x2) configuration in question. Add these up for the possible x2s, and you get a probability for the given d1 value. Get the probs for all d1 values and you can compute the RMS value of that distribution, half of that is the sigma_y(1) Allan dev.

Assuming this is a good model, the sigma_y(1) Allan deviation value should tell us, quite accurately, what the most likely standard deviation of the 1s-window oscillator frequency F is. And similarly, looking at the y values should tell us the mean frequency (although we know this drifts over time).

I think my next step is to write a short C or Matlab or Mathematica program to actually do this calculation. Only advantage of Mathematica: Maybe we can use it to evaluate the integral analytically. If not, I can do it numerically.