Hadamard variance

The Hadamard variance (HVAR) is a measure of stability of clocks and oscillators. It uses 3-sample variance, not unlike the Allan variance, which uses 2-sample variance. But unlike the Allan variance, the Hadamard variance is able to converge a selected few types of phase noise for which the Allan variance calculation fails to converge. Furthermore, the HVAR is insensitive to drift, which makes it useful for certain oscillator types.

The Hadamard variance is named after the mathematician Jacques Hadamard (1867 – 1963; originator of the Hadamard transform). The HVAR definition is based on second-difference and third-difference calculations, analogous to the modified Allan variance definition. There are also variants of the HVAR definition which includes the modified Hadamard variance (utilizing an expanded number of averaging factors as compared to the single averaging factor used by the standard HVAR) and also includes the moving Hadamard variance, which utilizes all possible 3-sample combinations.

The Hadamard variance is defined as

: $operatorname\{HVAR\}(\; au)\; =\; frac16\; leftlangle\; (y\_\{n+2\}\; -\; 2y\_\{n+1\}\; +\; y\_n)^2\; ight\; angle$

where

: $y\_n\; =\; leftlangle\; frac\{Delta\; f\}\{f\}\; ight\; angle$

is the normalized frequency ("ƒ") errors, "n" is the average taken over sampling period , and "τ" is the time of each sample period.

Applications

Stability analysis of oscillators. Possibly the most common application of stability variance metrics, which includes the Hadamard Variance, is the characterization of precision oscillators used for timekeeping. The fidelity of even the most expensive oscillators ($200k, hydrogen maser) is not ideal and suffer from phase noise which perturbs the oscillating frequency in a way that is dependent on the type of phase noise involved. There are basically five phase noise types relevant to oscillator stability.

* White noise phase WFM
* Flicker noise phase FPM
* White noise frequency WFM
* Flicker noise frequency FFM
* Random walk frequency RWFM

Each of these noise types has a different unwanted modulation effect on the carrier frequency and can be identified on a time or frequency domain plot (spectral density plot) by the slope of the curve. For example, RWFM will have a 1/"ƒ" ⁴ effect on the HVAR [dBc/Hz] vs frequency log plot whereas FFM will have a 1/"ƒ" ³ effect, WFM will have a 1/"ƒ" ² effect, and FPM will have a 1/"ƒ" effect.

RWFM noise is difficult to measure in the frequency domain since it is so close to the carrier. This type of noise is associated with the physical (mechanical) environment, such as vibration, shock and temperature. FPM noise is associated with the electronic support peripheral components of the oscillator, such as the amplifier and other supporting components.

The reason why non-standard stability variances (such as HVAR) are used in oscillator stabilization analysis instead of the classical standard variances (such as traditional sample or population variance) is because these non-standard variances usually converge for values approaching infinity whereas the standard sample variance diverges for very large or very small times in the presence of these five noise types. Additionally, IEEE STD 1139 and DoD has adopted a finite-difference approach with respect to analytical metrics for the analysis of these instabilities.

References

1. "Relating The Hadamard Variance To MCS Kalman Filter Clock Estimation", Capt Steven Hutsell, USAF, Falcon AFB.

2. "The Science Of Timekeeping", David W. Allan, Neil Ashby, Clifford C. Hodge, HP app note 1289, 1997.

3. "A Multi-Variance Analysis In The Time Domain", Todd Walter, Stanford University.

4. "Properties of Oscillator Signals and Measurement Methods", D.A. Howe, D.W. Allan, J.A. Barnes, NIST Time and Frequency Division.

5. "IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology - Random Instabilities", IEEE STD 1139.

6. "Oscillator, Crystal Controlled, General Specification", Department of Defense, MIL-PRF-55310. [http://www.dscc.dla.mil/Downloads/MilSpec/Docs/MIL-PRF-55310/prf55310.pdf]