Contents

•	Contents

This help file for the EDF program contains the following topics:

•	Contents

•	About

•	Introduction

•	License

•	Program Operation

•	EDF

•	Confidence Intervals

•	Classic EDF Algorithms

•	Total & Thêo1 EDF Algorithms

•	Combined EDF Algorithms

•	Support

•	Acknowledgments

•	References


About

EDF.HLP

Version 1.5 of 10/17/03

Microsoft Windows® Help File For

  EDF

EDF Calculations for Frequency Stability Analysis

©2003 Hamilton Technical Services

195 Woodbury Street

S. Hamilton, MA 01982 USA

Phone: 978-468-3703

Fax: 978-626-3006

E-Mail: bill@wriley.com

Web: http://www.wriley.com


Introduction

•	Introduction

The EDF program is a small Microsoft Windows® application to calculate the number of equivalent chi-squared degrees of freedom for estimates of various statistical measures of frequency stability.  It can find the edf for the Allan, modified Allan, Hadamard, Total, modified Total, Hadamard Total, and Thêo1 variances, as applicable, for either overlapping or non-overlapping (short or long stride) samples by means of classicial emperical formulae or a new combined (full or simplified) algorithm.  Besides the choice of algorithmic method, variance type, and stride, the edf values depend on the power law noise type (white PM, α=2 through random run FM, α=-4), the number of (phase) data points, and the averaging factor.  The EDF program is intended mainly as a convenient way to become familiar with and compare these methods, but it can also be used to validate code for implementing these algorithms, and to determine edf values for an actual analysis.

•	Noise Types

The allowable noise types and α for each of the supported variance types are shown in the table below:

Variance Type/α	WH PM 2	FL PM 1	WH FM 0	FL FM -1	RW FM -2	FW FM -3	RR FM -4
Allan	•	•	•	•	•
Modified	•	•	•	•	•
Hadamard	•	•	•	•	•	•	•
Total	•	•	•	•	•
Mod Total	•	•	•	•	•
Had Total	•	•	•	•	•	•	•
Thêo1	•	•	•	•	•


License

•	License

A license is hereby granted for the use of this program.  EDF may be freely installed, used, and distributed to others as a service by its author to the time and frequency community.  No third-party licensed code is used.

This program has been extensively tested, but it is never possible to declare a program completely bug-free.  No warranty is made, nor is any liability assumed, in connection with the use of the program.


Program Operation

•	EDF Program

The EDF program has a single dialog box as shown in the following screen shot:

•	Controls

The EDFprogram dialog box contains the following controls:

Dialog Box	ALT	Control	Control

Control	Key	Type	Description

Variance Type	V	Combo	Choose the variance type

Noise Type	N	Combo	Choose noise type

# Data Points	D	Edit	Enter # phase data points

Averaging Factor	A	Edit	Enter averaging factor

Type	Group	EDF algorithm type

Original	L	Radiobutton	Original classic algorithms

Full	U	Radiobutton	Full combined algorithm

Simple	I	Radiobutton	Simplified combined algorithm

Stride	Group	Analysis stride type

Long	G	Radiobutton	Long stride (non-overlapping)

Short	T	Radiobutton	Short stride (overlapping)

Calc	C	Pushbutton	Calculate edf

Close	S	Pushbutton	Close program

Copy	O	Pushbutton	Copy results to clipboard

View	W	Pushbutton	View clipboard contents

Help	H	Pushbutton	Invoke this help file

EDF	Text	Display edf value

•	Operation

Select the desired Variance type , Noise type, algorithm Type, and Stride.  Enter the # of phase Data Points and Averaging Factor.  Press Calc to calculate the corresponding edf value.  Then, optionally, press Copy to copy the results to the Windows clipboard, and View to display the clipboard contents.  The clipboard viewer will remain open to display subsequent results that are copied to the clipboard.  Invoke this Help file as desired, and end the program with the Close button.

•	Limits

The edf calculations are subject to a number of restrictions and limits, most of which are automatically indicated by the program controls.  For example, fewer noise types are associated with the non-Hadamard variance types, and a certain minimum number of analysis points are required depending on the varianvce type, # of data points and averaging factor.  For programming reasons, the # of data points is limited to  999,999,999.

•	Errors

Errors in the edf calculations are indicated by an error message in the results box.


EDF

•	EDF

The equivalent # of chi-squared degrees of freedom (EDF) for a second-moment statistic such as the Allan variance is the basis for determining the confidence level of the particular variance estimate. The EDF depends on the statistic type (e.g. overlapping Allan variance, modified Allan variance, etc.), the # of data points (generally expressed as N, the # of phase data points), and the power law noise type (white FM noise, etc., generally expressed as α, the exponent of Sy(f), the spectral density of the fractional frequency fluctuations).

The EDF can be estimated either by an empirical formula or by a theoretical expression (often based on an autocorrelation).  These calculations are performed for each of the statistics as described in the Help topics below.  They are divided into two categories, original (classic) EDF algorithms and the new combined EDF algorithm.  The latter has two forms, full and simple.


Confidence Intervals

•	Confidence Intervals

Sample variances are distributed according to the expression:

where χ² is the Chi-square, s² is the sample variance, σ² is the true variance, and edf is the equivalent number of degrees of freedom (not necessarily an integer). The edf is determined by the number of analysis points and the noise type. The statistical measures used for time-domain frequency stability analysis often establish single or double-sided confidence intervals with a selectable confidence factor, based on χ² statistics. The general procedure is to choose a single or double-limited confidence factor, p, calculate the corresponding χ² value, determine the edf from the variance type, noise type and number of analysis points, and thereby set the statistical limit(s) on the variance. For a double-sided limits:


Classic EDF Algorithms

•	Classic EDF Algorithms

•	Overlapping Allan Variance EDF

The classic empirical formulae for the edf of the overlapping Allan variance, as given by D.A. Howe, D.W. Allan and J.A. Barnes in their paper "Properties of Signal Sources and Measurement Methods" at the 35th Annual Symposium on Frequency Control in May 1981 are as follows:

Noise	  Alpha	EDF Formula

W PM	   2

F PM	   1

W FM	   0

FFM	  -1

RW FM	  -2

•	Overlapping Hadamard Variance EDF

The edf for the fully overlapping Hadamard variance (HVAR) can be found by an algorithm developed by C.A. Greenhall based on its generalized autocovariance function.  The HVAR edf is found either as a summation (for small m cases with a small number of terms) or from a limiting form for large m where 1/edf = (1/p)(a0-a1/p) with the coefficients as follows:

Noise Type	a0	a1
W FM	0.78	0.50
F FM	1.00	0.62
RW FM	1.03	0.61
FW FM	1.06	0.53
RR FM	1.30	0.54


Total & Thêo1 EDF Algorithms

•	Total & Thêo1 EDF Algorithms

•	Total Variance EDF

The EDF for the Total variance is determined by the method described in the paper "Total Variance Explained" by D.A. Howe presented at the Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium in April 1999.  It applies to -2 ≤ α ≤ 0.  The edf for theTotal variance is given by the formula b(T/τ) -c, where T is the length of the data record, τ is the averaging time, and b & c are coefficients that depend on the noise type as shown in the following table:

Power Law Noise Type	b	c
White FM	1.50	0.00
Flicker FM	1.17	0.22
Random Walk FM	0.93	0.36
•	Modified Total Variance EDF

The EDF for the modified Total variance is determined by the method described by D.A. Howe and F. Vernotte in their paper "Generalization of  the Total Variance Approach to the Modified Allan Variance" presented at the 31th Precise Time and Time Interval Meeting in  December 1999, and private communications with D.A Howe in March and May 2000.  It applies to -2 ≤ α ≤ 2, and uses the same formula as the Total variance with different coefficients (subject to certain restrictions) as shown in the following table:

Power Law Noise Type	b	c
White PM	1.90	2.10
Flicker PM	1.20	1.40
White FM	1.10	1.20
Flicker FM	0.85	0.50
Random Walk FM	0.75	0.31
•	Hadamard Total Variance EDF

The EDF for the Hadamard Total variance is determined by the method described by D. Howe, R. Beard, C. Greenhall, F. Vernotte  and B. Riley their paper "A Total Estimator of the Hadamard Function Used for GPS Operations" presented at the 32nd Precise Time and Time Interval Meeting in  November 2000.  It applies to -4 ≤ α ≤ 0, and uses the formula (T/τ)/(b0+b1τ/T), where T is the length of the data record, τ is the averaging time, and b0 & b1 are coefficients that depend on the noise type as shown in the following table:

Power Law Noise Type	b0	b1
White FM	0.559	1.004
Flicker FM	0.868	1.140
Random Walk FM	0.938	1.696
Flicker Walk FM	0.974	2.554
Random Run FM	1.276	3.149
•	Thêo1 EDF

The EDF for the Thêo1 (Theoretical Variance #1) statistic is determined by the method described in the paper "Estimation of Very Long-Term Frequency Stability Using a Special-Purpose Statistic" by D.A. Howe and T.K. Peppler presented at the 2003 IEEE International Frequency Control Symposium in May 2003.   It applies to -2 ≤ α ≤ 2, and the edf for the Thêo1 variance is determined by the following approximation formulae for each power low noise type:

Noise	  Alpha	EDF Formula

W PM	   2

F PM	   1

W FM	   0

FFM	  -1

RW FM	  -2


Combined EDF Algorithm

•	Combined EDF Algorithm

The combined EDF algorithm uses an analytical method rather than empirical formulae to find the edf for modified and unmodified Allan and Hadamard variances for overlapping and non-overlapping samples for all applicable noise types.  It is based on modeling the phase as the first difference of a continuous-time pure power-law process.

The inputs to the combined edf algorithm are as follows:

Term	Description	Remarks
a	FM noise exponent	-4 ≤ α ≤ 2
d	Order of phase difference	1=1st difference 2=Allan variance 3=Hadamard variance
m	Averaging factor	τ/τ0
F	Filter factor	1=modified variance m=unmodified variance
S	Stride factor	1=long (non-overlapping) m=short (overlapping)
N	# phase data points	Sample period=τ0
•	Reference

C.A. Greenhall and W.J. Riley, "Uncertainty of Stability Variances Based on Finite Differences", Proceedings of the 35nd Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting , December 2003 (to be published).


Support

•	Support

Support for the EDF program is available from:

W. J. Riley	Telephone (non-business hours):

Hamilton Technical Services	978-468-3703

195 Woodbury Street	Fax (any time):

S. Hamilton, MA 01982 USA	978-626-3006

E-Mail:	bill@wriley.com

Hamilton Technical Services develops software for frequency stability analysis, including the popular Stable programs.  More information about those products can be found on the web at http://www.wriley.com.


Acknowledgment

•	Acknowledgment

This program is heavily based on the work of C.A Greenhall, whose contributions to the field of frequency stability analysis are gratefully acknowledged.


References

•	References

The following references apply to the EDF program:

1.	"IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology - Random Instabilities", IEEE Std 1139-1999 , July 1999.

2.	C.A. Greenhall, "Estimating the Modified Allan Variance", Proc. IEEE 1995 Freq. Contrl. Symp., pp. 346-353, May 1995.

3.	D.A. Howe & C.A. Greenhall, "Total Variance: A Progress Report on a New Frequency Stability Characterization", Proc. 1997 PTTI Meeting, pp. 39-48, December 1997.

4.	C.A. Greenhall, private communication, May 1999.

5.	D.A. Howe, private communication, March 2000.

6.	D.A. Howe, "Total Variance Explained", Proc. 1999 Joint Meeting of the European Freq. and Time Forum and the IEEE Freq. Contrl. Symp ., pp. 1093-1099, April 1999.

7.	D.A. Howe, private communication, March 2000.

8.	C.A. Greenhall, "Recipes for Degrees of Freedom of Frequency Stability Estimators", IEEE Trans. Instrum. Meas., Vol. 40, No. 6, pp. 994-999, December 1991.

9.	D.A. Howe, "Methods of Improving the Estimation of Long-Term Frequency Variance", Proc. 11th European Freq. and Time Forum , pp. 91-99, March 1997.

10.	J.A. Barnes and D.W. Allan, Variances Based on Data with Dead Time Between the Measurements", NIST Technical Note 1318 , 1990.

11.	C.A Greenhall, private communication, May 2000.

12.	D. Howe, R. Beard, C. Greenhall, F. Vernotte and W. Riley, "A Total Estimator of the Hadamard Function Used for GPS Operations", Proceedings of the 32nd Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting , pp. 255-268, November 2000.

13.	C.A. Greenhall, "The Generalized Autocovariance: A Tool for Clock Noise Statistics", TMO Progress Report 42-137 , Jet Propulsion Laboratory, Pasadena, CA 91109, May 1999.

14.  C.A. Greenhall and W.J. Riley, "Uncertainty of Stability Variances Based on Finite Differences", Proceedings of the 35nd Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting , December 2003 (to be published).