Can Allan variance be generalized such that the "Oscillator model" is not presumed?

Question

The definition of Allan variance, $\sigma^2[ \tau ]$, which relates to "stability of clocks" is described on the Wikipedia page as being derived in terms of an "Oscillator model":

"The oscillator being analysed is assumed to follow the basic [wave form] model of $V_\text{osc}[~t~] := V_0 \operatorname{Sin}[ ~ \Phi_\text{osc}[~t~] ~ ]$"

where $V_0$ is constant (i.e. independent of the parameter $t$) and the "total phase" is also expressed as

$$ \Phi_\text{osc}[~t~] := \omega_\text{osc}~t + \phi_\text{osc}[~t~] $$

in terms of a (constant) "nominal angular frequency, $\omega_\text{osc}$" and a "fluctuating component, $\phi_\text{osc}[~t~]$", along with the parameter $t$.

My question:
Are there "generalizations of Allan variance" based on more general "wave form models"?

Such as perhaps based on

• "smooth amplitude modulated oscillation": $V_{AM}[~t~] := | V_{AM}[~t~] | \operatorname{Sin}[ ~ \Phi_{AM}[~t~] ~ ]$,
  where $| V_{AM}[~t~] | \gt 0$, and $\exists ~ \frac{d}{d~t} \left[~V_{AM}[~t~]~\right]$, or even

• "any sequence which changes sign often enough": $V_\text{alt}[~t~] := | V_\text{alt}[~t~] | \operatorname{Sin}[ ~ \Phi_\text{alt}[~t~] ~ ]$,
  where $| V_\text{alt}[~t~] | \gt 0$, but where differentiability (wrt. parameter $t$) or even continuity (wrt. parameter $t$) are not required of $V_\text{alt}[~t~]$, or even

• "any suitably wiggled sequence": $V_w[~t~]$,
  where $\small \forall t_k : \left( \exists t_j, t_p : (t_k - t_j) ~ (t_p - t_k) > 0 \text{ and } (V_w[~t_k~] - V_w[~t_j~]) ~ (V_w[~t_p~] - V_w[~t_k~]) > 0 \right) \implies $ $\small \left(\exists t_q : (t_p - t_j) ~ (t_q - t_p) > 0 \text{ and } (V_w[~t_q~] - V_w[~t_j~]) ~ (V_w[~t_p~] - V_w[~t_q~]) > 0 \right)$, or even

• "any strictly monotonous function": $V_m[~t~]$,
  where $\small \forall t_k : \left( \exists t_j, t_p : (t_k - t_j) ~ (t_p - t_k) > 0 \right) \implies \left( (V_m[~t_k~] - V_m[~t_j~]) ~ (V_m[~t_p~] - V_m[~t_k~]) > 0 \right)$

?

And if so, how are the corresponding "generalizations of Allan variance" called?

Answer 1

In general Allen variance is not a good way to characterize non-harmonic processes and processes which modulate the phase, frequency or amplitude of the oscillator in a non-stochastic manner for the same reason that the conventional variance is not a good measure for processes that do not, at least approximately, follow a normal distribution.

For practical purposes the Allen variance is just one way of expressing the timing/frequency uncertainty of an oscillator. Phase noise and jitter are others. The main problem with oscillator noise is, that it is caused by different physical mechanisms, which operate on very different time scales. Long term drift, for instance, changes oscillator frequency over hundreds and thousands of hours and is a non-linear function of oscillator lifetime. It will not even show up in an Allen variance measurement, which may last a couple of minutes or a week.

Flicker noise is characterized by a flicker timescale of a few (ten) Hz down to fractions of a mHz, but can change the frequency in almost discrete steps (popcorn noise caused by e.g. dust moving around in the cavity or on the crystal).

Jitter is a quantity that is measured period to period and is sometimes made very large on purpose to suppress amplitude noise effects and crosstalk in circuits, even though the long term stability of the oscillator(s) is excellent. In such a case the Allen variance would be high and contain no information about the actual performance.

Phase noise has a non-trivial spectrum that may fold high frequency noise and flicker noise into one frequency domain description, but may have very little to say about the absolute timing error introduced by a frequency jump. Depending on timescale and application the Allen variance may, or may not be a useful way of expressing oscillator noise.

In all applications that I had to deal with, we had to characterize our clocks in more than one way, or we would have missed very important error sources in our measurements.

Answer 2

The definition of Allan variance does not depend on that waveform model. The Allan variance is, in fact, a variance, like the standard variance, that can be used to characterize any random process with a certain degree of stationarity (I won't go into the details of this requirement, but they can be found in [1]). Given a random process $x(t)$, the Allan variance (or two-sample variance) is defined as

$$\sigma^2_{\bar{X}}(\tau) := \frac{1}{2}\mathrm{E}\{[\bar{X}_t(\tau)-\bar{X}_{t-\tau}(\tau)]^2\}\tag{1}$$

where $\mathrm{E}$ represents the expectation value and

$$\bar{X}_t(\tau) = \frac{1}{\tau}\int_{t-\tau}^t x(t)\,\mathrm{d} t$$

is the running average of the process over the time interval $\tau$. The factor $1/2$ in (1) is chosen so that for a white noise process the Allan variance coincides with the standard variance $\mathrm{E}\{[\bar{X}_t(\tau)-\mu_X]^2\}$ of the running average, $\mu_X$ being the process mean.

For instance, the Allan variance is commonly used to characterize the noise in a DC voltage measurement or the noise of an accelerometer. For instance, for a DC voltage measurement, the model is simply

$$v(t) = V_0 + v_\text{n}(t)$$

where $V_0$ is a DC constant voltage and $v_\text{n}$ is a voltage noise process that can be characterized by the Allan variance $\sigma_{\bar{v}_\text{n}}(\tau)$.

The waveform model given in your question is needed, in the case of sinusoidal oscillators, to define the relationship between the waveform generated by a real oscillator and an underlying random process suitable, from the point of view of the applications, to characterize the oscillator's noise. This random process is the phase noise $\Phi_\text{osc}(t)$.

If you have an oscillator generating a non-sinusoidal waveform, it is still possible to define a phase noise, but you have to use as "waveform model" the analytic signal associated to the waveform generated by the oscillator. The analytic signal allows one to define an instantaneous phase for a wide class of signals for which the Hilbert transform is defined (for more details see also [2]). You can then characterize this so defined phase noise with the Allan variance (actually, in the case of oscillators, one characterizes with the Allan variance the frequency noise which is related to the phase noise by a derivative).

[1] D. B. Percival, and A. T. Walden. Wavelet Methods for Time Series Analysis, Cambridge University Press, Cambridge, 1993.

[2] B. Picinbono, "On Instantaneous Amplitude and Phase of Signals", IEEE Trans. Sign. Proc., 45, n. 3, pp. 552-560, 1997.