# [time-nuts] Simulating Oscillator Noise: Difficulties Simulating Flicker FM Noise

Kyle Wesson kyle.wesson at mail.utexas.edu
Thu Apr 22 17:06:48 EDT 2010

```Hello,

I am trying to simulate oscillator noise by following the procedure
outlined in James Barnes' paper: "Simulation of Oscillator Noise"
(1984) 28th Annual Frequency Control Symposium. In the paper, Barnes
explains the models of the five typical types of noise that occur in
oscillators and a method for their simulation.

I've followed the steps he presents in his paper and have been unable
to produce simulated output for flicker FM noise that leads to an flat
Allan variance graph (ie. all Allan variance values are nearly
constant for all tau values). Instead, the Allan variance values of my
simulated flicker FM noise start out constant at the Allan variance
value I desire but then tend upwards by two to three orders of
magnitude (nearly every simulation) about halfway through the range of
possible tau values. In short, it starts out flat and then increases
rapidly about halfway through the tau range. I believe there may be a
couple possibilities and am wondering if anyone else has come across
the same issues or knows of a solution.

1) To simulate flicker FM noise, Barnes uses a set of ARIMA
coefficients to model the noise. Is an updated set of coefficients
available that would have better accuracy or produce better simulation
results? Is the ARIMA method typically used with the availability of
today's higher computational power?

2) Barnes devotes a section of the paper to random number generation
and states that the random numbers to be used should be normally
distributed with zero mean and unit variance. I  used the built-in
Matlab command randn() to generate the random data but only achieved
an all-flat Allan variance plot when the random number generator was
seeded with a particular number. The majority of the time (using a
"random" seed), this method produced non-flat results as described
above. I then attempted the two methods Barnes presents in his paper
to generate the random numbers which provided similar non-flat
results.

Are random normally distributed random numbers optimal for these
simulations? Would another distribution produce results consistent
with the expectation of an all-flat (ie. constant) Allan variance for
flicker noise?

I appreciate any advice or ideas you or your colleagues can provide.
As needed I can provide individuals my generated Allan variance plots,
but I didn't want to send them to the whole mailing list.