[Please move if appropriate]
Hi,
I've been following a very interesting thread elsewhere in this forum concerning the measurement of oscillator stability and it has struck me how very similar the concepts and goals are to the mainstay of my work that seems at first to be a wholly unrelated discipline. However, the more I read about characterizing oscillator stability, the more I realize just how close it is to what I do.
I'd like to know if there is an unrecognized interdisciplinary overlap that may be of use.
Some background:
I am a physical chemist, specifically a colloid chemist, which means I study the behavior of nanoparticles dispersed in a material continuum. My particular focus is the measurement of the motion of charged nanoparticles in a liquid due to the application of an external electric field (electrophoresis). If a laser light is incident on the particles then the phase of the light scattered by them is determined by their position and the temporal change in the phase manifests itself as a Doppler frequency shift of the light.
The scattered light is mixed at a detector with unscattered light ('reference') that is frequency-shifted relative to the incident light by an amount w
o (typ. 250Hz - 20kHz). For a perfect stationary scatterer, this leads to a sinusoidal detector signal with the same frequency as the difference between the reference and scattered light. This can be considered the ideal oscillator.
In a version of this experiment that I developed for my PhD (phase analysis light scattering (PALS)), three separable components to the phase are considered:
1. Random brownian motion due to particle diffusion (gaussian)
2. Linearly changing with time due to a constant velocity (e.g., settling)
3. Periodic oscillation due to particle motion in an alternating field
Prior to the advent of PALS, attempts to separate these components were unsuccessful. Autocorrelation was the standard method. Unfortunately, because the photodetector signal is proportional to the intensity of the light, the phase cannot be obtained directly. With PALS, it can.
The detector signal, E(t), is just like for an oscillator:
E(t) = A(t)exp(i.phi(t))
with the optical phase being:
phi(t) = w
o(t) + phi
linear(t) + phi
gaussian(t) + B(t)cos(w
p(t))
Demodulating the photodetector signal about wo leads an IQ pair from which are readily transformed to A(t) and phi(t).
In this representation of phi(t), B(t)cos(w
p(t)) is akin to period phase noise with a frequency w
p and amplitude B(t). phi
gaussian represents the stochastic process from which the diffusion coefficient and, hence, particle size are estimated.
Determination of the three components uses statistical methods that are very similar in concept to those used for estimating Allan variance etc. For PALS, there are two functions that are constructed from the experimental amplitude and phase data: phase structure function and phase difference function. Computationally these are easy to construct esp. compared to autocorrelation/FFT.
The structure function allows for estimation of the gaussian term, and the amplitude and frequency of the periodic phase variation. No
a priori knowledge of the frequency is needed. As with autocorrelation, the amplitude of the periodic phase variation can be dominated by the random noise. Unlike the autocorrelation function, the structure function does not decay and so coexisting fast and slow periodic phase variations can be revealed.
If the frequency is known then the phase difference function allows the amplitude and frequency of the periodic phase variation to be estimated in the absence of the random noise term.
In the realm of my experiments, I easily measure periodic phase variations of amplitudes of a few mrad in the presence of gaussian noise with sigma of the order of a few Hz. The periodic phase variation frequencies range from 1 to 1000Hz.
I hope you can see the parallels even though the scale of the numbers is doubtless quite different.
I'm very curious to see if this method can be used with IQ data obtained from oscillator stability experiments.
Here is a link to the pertinent and long chapter from my PhD thesis (file size too big to attach). The hard-core equations begin about half-way through but the first half should help show the similarities I have alluded to.
https://1drv.ms/b/s!AhmR5if7W0HCjccI0YLXWbEq5VTccAIn the light scattering world (which is very important in the pharmaceutical/food/cosmetic/paint/water treatment industries), PALS has replaced autocorrelation/FFT as the
de facto data analysis method.