NisrUNITED STATES DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

N«T L INST OF STAND «, TECH R.I.C_

A111D3 3T2Sbl

REFERENCE

NIST

PUBLICATIONS

NIST Technical Note 1337

-QC

100

.U5753

#1337

1990
NATIONAL INSTITUTE OF STANDARDS &

TECHNOLOGY

Research Informatk)n Center

Gaithersburg, MD 20899
Characterization of

Clocics and Oscillators

Edited by

D.B. Sullivan

D.W. Allan

D.A. Howe

F.L Walls

Time and Frequency Division

Center for Atomic, Molecular, and Optical Physics

National Measurement Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

This publication covers portions of NBS Monograph 140 published in 1974.

The purpose of this document is to replace this outdated monograph in the areas of methods for characterizing clocks and oscillators and definitions

and standards relating to such characterization.

*''>«rcs o«

*"

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director

Issued March 1990
National Institute of Standards and Technology Technical Note 1337

Natl. Inst. Stand. Technol., Tech. Note 1337, 352 pages (Jan. 1990)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1990

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
PREFACE

For many years following its publication in 1974, "TIME AND FREQUENCY: Theory

and Fundamentals," a volume edited by Byron E. Blair and published as NBS Monograph 140,

served as a common reference for those engaged in the characterization of very stable clocks and

oscillators. Monograph 140 has gradually become outdated, and with the recent issuance of a new military specification, MIL-0-55310B, which covers general specifications for crystal oscillators, it has become especially clear that Monograph 140 no longer meets the needs it so ably

served in earlier years. During development of the new military specification, a process involving

discussion and input from many quarters, a key author of the specification, John Vig of the US

Army Electronics Technology and Devices, urged the National Bureau of Standards (now the

National Institute of Standards and Technology, NIST) to issue a revised pubHcation to serve as reference for the characterization of clocks and oscillators. With NIST having agreed to this

task, the framers of the military specification used the nomenclature "N^S Monograph 140R" in

their document, anticipating a revised (R) volume which had not yet been prepared.

Considering the availability of a number of newer books in the time and frequency field,

the rewriting of a major volume like Monograph 140 seemed inappropriate. The real need has

not been for rework of everything in Monograph 140, but only for those parts which provide

reference to definitions and methods for measurement and characterization of clocks and oscillators, subjects which are fully covered in a number of papers distributed through a variety of

conference proceedings, books, and journals. For the near term, we concluded that the most

effective procedure would be to collect a representative set of these papers into one reference source with introductory comments which permit the reader to quickly access material required

to meet particular needs. Thus, we arrived at this particular collection. The editors' challenge

has been to select representative papers, to organize them in a convenient manner, and to deal

with errata and notation inconsistencies in a reasonable manner. In the longer term, the material in this volume needs to be more completely integrated. This task would profitably await

further developments in the area of phase noise measurements.

Donald B. Sullivan

David W. Allan

David A. Howe

Fred L. Walls

Boulder, Colorado

February 28, 1990

m

CONTENTS Page

PREFACE iii TOPICAL INDEX TO ALL PAPERS x ABSTRACT TN-1

A. INTRODUCTION

A.l OVERVIEW TN-1

A.2 COMMENTS ON INTRODUCTORY AND TUTORIAL PAPERS TN-2

A.3 COMMENTS ON PAPERS ON STANDARDS AND DEFINITIONS TN-3

A.4 COMMENTS ON SUPPORTING PAPERS TN-3

A.5 GUIDE TO SELECTION OF MEASUREMENT METHODS TN-5

A.6 RELATIONSHIP OF THE MODIFIED ALLAN VARIANCE TO THE ALLAN VARIANCE TN-9

A.7 SUPPLEMENTARY READING LIST TN-13

B. INTRODUCTORY AND TUTORIAL PAPERS

B.l Properties of Signal Sources and Measurement Methods

DA. Howe, D.W. Allan, and JA. Barnes

35th Annual Symposium on Frequency Control, 1981 Proceedings TN-14

I. The Sine Wave and Stability TN-14

II. Measurement Methods Comparison TN-20

III. Characterization TN-22

IV. Analysis of Time Domain Data TN-23

V. Confidence of the Estimate and Overlapping Samples TN-26

VI. Maximal Use of the Data and Determination of the Degrees of Freedom TN-28

VII. Example of Time Domain Signal Processing and Analysis TN-30

VIII. Spectrum Analysis TN-32

IX. Power-Law Noise Processes TN-38

X. PitfaUs in Digitizing the Data TN-39

XI. Translation From Frequency Domain Stabihty Measurement to Time Domain Stabihty

Measurement and Vice-Versa TN-48

XII. Causes of Noise Properties in a Signal Source TN-50

XIII. Conclusion TN-55

BJ Frequency and Time - Their Measurement and Characterization

Samuel R. Stein

Precision Frequency Control, Volume 2, Chapter 12, 1985 TN-61

12.1 Concepts, Definitions, and Measures of Stability TN-63

12.1.1 Relationship Between the Power Spectrum and the Phase Spectrum TN-65

12.1.2 The IEEE Recommended Measures of Frequency Stability TN-65

12.1.3 The Concepts of the Frequency Domain and the Time Domain TN-73

12.1.4 Translation Between the Spectral Density of Frequency and the Allan Variance TN-73

12.1.5 The Modified Allan Variance TN-75

12.1.6 Determination of the Mean Frequency and Frequency Drift of an Oscillator TN-77

12.1.7 Confidence of the Estimate and Overlapping Samples TN-80

12.1.8 Efficient Use of the Data and Determination of the Degrees of Freedom TN-83

12.1.9 Separating the Variances of an Oscillator and the Reference TN-86
Page

12.2 Direct Digital Measurement TN-87

12.2.1 Time-Interval Measurements TN-87

12.2.2 Frequency Measurements TN-89

12.23 Period Measurements TN-89

12.3 Sensitivity-Enhancement Methods TN-90

123.1 Heterodyne Techniques TN-90

123.2 Homodyne Techniques TN-91

123.3 Multiple Conversion Methods TN-97

12.4 Conclusion TN-101

B3 Time and Frequency (Time-Domain) Characterization, Estimation, and Prediction of Precision Clocks and Oscillators David W. Allan IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1987 TN-121

1. Introduaion TN-121

2. Systematic Models for Clocks and Oscillators TN-121

3. Random Models for Clocks and Oscillators TN-123

4. Time-Domain Signal Characterization TN-123

5. Time and Frequency Estimation and Prediction TN-126

6. Conclusions TN-127

B.4 Extending the Range and Accuracy of Phase Noise Measurements

F.L. Walls, A.J.D. Clements, CM. Felton, M.A. Lombardi and M.D. Vanek

42nd Annual Symposium on Frequency Control, 1988 Proceedings TN-129

I. Introduction TN-129

II. Model of a Noisy Signal TN-129

III. Two Oscillator Method TN-130

IV. The New NBS Phase Noise Measurement Systems TN-135

V. Conclusion TN-137

C. PAPERS ON STANDARDS AND DEFINITIONS

CI Standard Terminology for Fundamental Frequency and Time Metrology

David Allan, Helmut Hellwig, Peter Kartaschoff, Jacques Vanier, John Vig, Gemot M.R. Winkler,

and Nicholas Yannoni

42nd Annual Symposium on Frequency Control, 1988 Proceedings TN-139

1. Introduction TN-139

2. Measures of Frequency and Phase Instability TN-139

3. Characterization of Frequency and Phase Instabilities TN-139

4. Confidence Limits of Measurements TN-140

5. Recommendations for Characterizing or Reporting Measurements of Frequency and

Phase Instabilities TN-141

VI
Page

C2 Characterization of Frequency Stability James A. Barnes, Andrew R. Chi, Leonard S. Cutler, Daniel J. Healey, David B. Leeson,

Thomas E. McGunigal, James A. Mullen, Jr., Warren L. Smith, Richard L. Sydnor,

Robert F.C. Vessot, and Gemot M.R. Winkler

IEEE Transactions on Instrumentation and Measurement, 1971 TN-146

I. Introduction TN-147

n. Statement of the Problem TN-148

ni. Background and Definitions TN-149

IV. Definition of Measures of Frequency Stability TN-149

V. Translations Among Frequency Stability Measures TN-151

VI. Applications of Stability Measures TN-153

VII. Measurement Techniques for Frequency Stability TN-154

VIII. Conclusions TN-157

C3 Characterization of Frequency and Phase Noise

Comit6 Consultatif International Des Radiocommunications (CCIR)

Report 580 of the CCIR, 1986 TN-162

1. Introduction TN-162

2. Fourier Frequency Domain TN-163

3. Time Domain TN-163

4. Conversion Between Frequency and Time Domains TN-164

5. Measurement Techniques TN-166

6. Confidence Limits of Time Domain Measurements TN-166

7. Conclusion TN-167

D. SUPPORTING PAPERS

D.l Characterization and Measurement of Time and Frequency Stability

P. Lesage and C. Audoin

Radio Science, 1979 TN-171

1. Introdurtion TN-171

2. Definitions: Model of Frequency Fluctuations TN-171

3. Noise Processes in Frequency Generators TN-173

4. Measurements in the Frequency Domain TN-174

5. Measurements in the Time Domain TN-177

6. Characterization of Frequency Stability in the Time Domain TN-178

7. Characterization of Frequency Stability via Filtering or Frequency Noise TN-182

8. Spectral Analysis Inferred From Time Domain Measurements TN-183

9. Structure Functions of Oscillator Fractional Phase and Frequency Fluctuations TN-184

10. Power Spectral Density of Stable Frequency Sources TN-185

11. Conclusion TN-187

vu
Page

D2 Phase Noise and AM Noise Measurements in the Frequency Domain

Algie L. Lance, Wendell D. Seal, and Frederik Labaar

Infrared and Millimeter Waves, Vol. 11, 1984 TN-190

I. Introduction TN-190

II. Fundamental Concepts TN-191

A. Noise Sidebands TN-194

B. Spectral Density TN-194

C. Spectral Densities of Phase Fluctuations in the Frequency Domain TN-197 ^ D. Modulation Theory and Spectral Density Relationships TN-199

E. Noise Processes TN-202

F. Integrated Phase Noise TN-203

G. AM Noise in the Frequency Domain TN-205

III. Phase-Noise Measurements Using the Two-OsciUator Technique TN-206

A. Two Noisy Oscillators TN-208

B. Automated Phase-Noise Measurements Using the Two-Oscillator Technique TN-209

C. Cahbration and Measurements Using the Two-Oscillator System TN-210

IV. Single-Oscillator Phase-Noise Measurement Systems and Techniques TN-218

A. The Delay Line as an FM Discriminator TN-219

B. Calibration and Measurements Using the Delay Line as an FM Discriminator TN-225

C. Dual Delay-Line Discriminator TN-232

D. Millimeter-Wave Phase-Noise Measurements TN-234

V. Conclusion TN-238

DJ Performance of an Automated High Accuracy Phase Measurement System

S. Stein, D. Glaze, J. Levine, J. Gray, D. Hilliard, D. Howe, and L. Erb

36th Annual Symposium on Frequency Control, 1982 Proceedings TN-241

1. Review of the Dual Mixer Time Difference Technique TN-241

2. Extended Dual Mixer Time Difference Measurement Technique TN-242

3. Hardware Implementation TN-242

4. Conclusions TN-243

D.4 Biases and Variances of Several FFT Spectral Estimators as a Function of Noise Type and

Number of Samples

F.L. WaUs, D.B. Percival and W.R. Irelan 43rd Annual Symposium on Frequency Control, 1989 Proceedings TN-248

I. Introduction TN-248

II. Spectrum Analyzer Basics TN-248

III. Expected Value and Bias of Spectral Estimates TN-249

IV. Variances of Spectral Estimates TN-251

V. Conclusions TN-252

vm
Page

D^ A Modified "Allan Variance" with Increased Oscillator Characterization Ability

David W. Allan and James A. Barnes

35th Annual Symposium on Frequency Control, 1981 Proceedings TN-254

1. Introduction TN-254

2. Definition of "Allan Variance" and Related Concepts TN-254

3. Development of the Modified Allan Variance TN-255

4. Comparisons, Tests, and Examples of Usage of the Modified Allan Variance TN-256

5. Conclusion TN-257

D.6 Characterization of Frequency Stability: Analysis of the Modified Allan Variance and

Properties of Its Estimate

Paul Lesage and Th^ophane Ayi

IEEE Transactions on Instrumentation and Measurement, 1984 TN-259

I. Introduction TN-260

II. Background and Definitions TN-260

III. The Modified Allan Variance TN-260

IV. Uncertainty on the Estimate of the Modified Allan Variance TN-262

V. Conclusion TN-263

D.7 The Measurement of Linear Frequency Drift in Oscillators James A. Barnes

15th Annual PTTI Meeting, 1983 Proceedings TN-264

I. Introduction TN-264

II. Least Squares Regression of Phase on a Quadratic TN-265

III. Example TN-266

IV. Drift and Random Walk FM TN-270

V. Summary of Tests TN-270

VI. Discussion TN-276

VII. Conclusions TN-276

D.8 Variances Based on Data With Dead Time Between the Measurements

James A. Barnes and David W. Allan NIST Technical Note 1318, 1990 TN-296

1. Introduction TN-297

2. The Allan Variance TN-298

3. The Bias Function Bj(N,r,/i) TN-299

4. The Bias Function B2(r,/i) TN-300

5. The Bias Function B3(N,M,r,M) TN-300

6. The Bias Functions TN-302

7. Examples of the Use of the Bias Functions TN-303

8. Conclusion TN-304

Appendix TN-312

E. APPENDIX - Notes and Errata TN-336

IX
TOPICAL INDEX TO ALL PAPERS

The following index is designed for quick location of substantial discussion of a particular topic. The page numbers are those where the topic is first cited in the reference.

Aliasing TN-40

Allan Variance TN-24, TN-71, TN-123, TN-140, TN-150, TN-163, TN-179, TN-254,

TN-260, TN-298

Amplitude Modulation Noise TN-194, TN-205, TN-235

Bias Functions TN-162, TN-164, TN-299

Clock-Time Prediction TN-126, TN-140

Common Measurement Problems TN-156

Components Other than Oscillators (Methods) . . TN-37, TN-131, TN-176

Confidence of the Estimate TN-26, TN-80, TN-140, TN-156, TN-166, TN-180, TN-262

^Conversion Between Time and Frequen<y Domain . TN-48, TN-73, TN-142, TN-151, TN-164

Correlation Method TN-132, TN-176, TN-216

Dead Time TN-140, TN-164, TN-180, TN-296

Degrees of Freedom TN-28, TN-83

Digitization of Data TN-39

Direct Measurements

o Measurements at the Fundamental Frequency TN-15, TN-87

o Measurements after Multiplication/Division . . TN-101, TN-135

Drift TN-30, TN-77, TN-122, TN-151, TN-184, TN-264

Efficient Use of Data TN-28, TN-83

Filter Method TN-182

Frequency Synthesis TN-97

Heterodyne or Beat-Frequency Methods .... TN-17, TN-90, TN-154, TN-177

o Single-Conversion Methods TN-17, TN-90, TN-154, TN-177

o Multiple-Conversion Methods TN-97

Time-Difference (Time-Interval-Counter) . . . TN-20, TN-87, TN-177

- Dual-Mixer, Time-Difference Method . . . TN-17, TN-99, TN-177, TN-241

HomodjTie Measurements

o Phase-Lock-Loop Methods TN-18 TN-34, TN-93, TN-130, TN-176, TN-209

Discriminator Methods TN-92, TN-133, TN-174, TN-218

- Cavity-Discriminator Method TN-133, TN-174, TN-219

- Delay-Line Method TN-92, TN-134, TN-176, TN-219

Least Squares (Regression) TN-31, TN-77, TN-265

Modified Allan Variance TN-75, TN-143, TN-164, TN-255, TN-260

Overlapping Samples TN-26, TN-80, TN-140

Pickett-Fence Effect TN-46

Power Law Spectra (Random Noise) TN-38, TN-73, TN-123, TN-141, TN-167, TN-173, TN-200

Quantization Uncertainty TN-40, TN-89

Reference-Phase-Modulation Technique .... TN-133

Separating Oscillator and Reference Variances . . TN-86

Spectral Density TN-32, TN-43, TN-65, TN-130, TN-139, TN-150, TN-163, TN-172,

TN-194

Systematic Fluctuations (non-Random-Noise) . . . TN-22, TN-121, TN-150

Window Function (Leakage) TN-44
CHARACTERIZATION OF CLOCKS AND OSCILLATORS

D.B. Sullivan, D.W. Allan, DA. Howe and F.L. Walls, Editors

Time and Frequency Division

National Institute of Standards and Technology

Boulder, Colorado 80303

This is a collection of published papers assembled as a reference for those in- volved in characterizing and specifying high-performance clocks and oscillators. It

is an interim replacement for NBS Monograph 140, Time and Frequency: Theory

and Fundamentals, an older volume of papers edited by Byron E. Blair. This

current volume includes tutorial papers, papers on standards and definitions, and

a collection of papers detailing specific measurement and analysis techniques.

The discussion in the introduction to the volume provides a guide to the content

of the papers, and tables and graphs provide further help in organizing methods

described in the papers.

Key words: Allan variance, clocks, frequency, oscillators, phase noise, spectral

density, time, two-sample variance.

A. INTRODUCTION

A.1 OVERVIEW

The papers in this volume are organized into three groups: Introductory and Tutorial

Papers, Papers on Definitions and Standards, and Supporting Papers. The three sections (A.2,

A.3, and A.4) immediately following this introduction provide overviews of each of the three

groups of papers with comments on each paper. The arrangement of the papers in this particular order is somewhat arbitrary, since, for example, the first two papers under Supporting Papers

could be included with the Introductory and Tutorial Papers, while paper B.4 could easily be

placed with the Supporting Papers. Our rationale for the first group of papers (discussed in

more detail in section A.2) is that, taken as a group, they provide reasonably complete coverage

of the concepts used in characterizing clocks and oscillators. Several of the papers, taken individually, are good introductory papers, but, for this publication, need to be complemented with

additional material to provide coverage of an appropriate range of topics.

The second group (section C) of three papers discussed in section A.3 were specifically

written to address definitions and standards. This is a particularly important section, since

consistency in specification of performance can only be achieved if manufacturers and users refer

to the same measurement and characterization parameters.

The Supporting Papers in section D provide additional discussion of topics introduced in

the first group. The first papers in this group (D.l and D.2) also provide good introductory

material which might be used with section B to gain a better understanding of the concepts.

Section A.5 provides a table and graph designed to help the reader select a measurement

method to meet a particular need. To make this useful, it was kept simple and must therefore

be used with care. Such tabular information can never be arranged well enough to anticipate all

TN-1
of the wide range of measurement situations which might be encountered. However, it can serve

as a starting point for the decision-making process.

Section A.6 contains some new material which should be helpful in understanding the

relationship between the Allan variance and the modified Allan variance. Since these ideas are

unpublished, we include them here rather than with the papers on those topics. This section is followed by a reading list with references to major articles and books which can be used as supplementary resources. Because some of the papers include extensive reference lists, we have

limited our list to works which are either very comprehensive or only recently published. Particularly extensive reference lists are included with papers B.l, B.2, C.l, C.3, D.l, and D.2.

Since notation and definitions have changed over the period bridged by these papers, we

have highlighted problem areas on the papers with an asterisk (*). A note directs the reader to the Appendix where the particular problem is discussed. We have also used this device to

highlight inconsistencies and the usual typographic and other errors which creep into the litera- ture. The page numbers of the original publications are retained, but we have also used a continuous page numbering to simplify location of items in the volume.

The topical index on page xi organizes much of the material in the papers under a few

key subject headings. This index provides a shortcut to locating material on a particular topic.

A2 COMMENTS ON INTRODUCTORY AND TUTORIAL PAPERS

Paper B.l in this section, by Howe, Allan, and Barnes, was originally prepared and

presented as a tutorial paper and has been used with success as an introductory paper in our

annual Time and Frequency Seminar. This paper is now 9 years old, so there are a substantial

number of notes which relate to updates in notation. The paper is nevertheless highly readable

and introduces many of the key measurement methods, providing circuit diagrams with enough

specific detail to be useful in real laboratory situations. Furthermore, it includes discussion and

examples on handling of data which are useful for practical application of the concepts. The

paper presents a particularly useful discussion of the pitfalls encountered in digitizing data, a problem which is often overlooked.

The second paper (B.2) by Stein is more advanced and those familiar with the general

concepts may find it a better starting point. This and other papers in this collection cite earlier

IEEE recommendations on measures of frequency stability and, while much of this has not

changed, there is a new IEEE standard (paper C.l). In general, the reader should consult the

overview and papers of section C if there is any question concerning definitions or terminology.

Paper B.2 is quite comprehensive, introducing topics (not covered in paper B.l) such as the

modified Allan variance, the delay-Une-phase-noise-measurement system, and the use of fre- quency synthesis to reach frequencies far from normally available reference frequencies.

The materials in papers B.l and B.2, aside from differences in level of presentation, are organized in quite different ways. The Howe-Allan-Bames paper goes directly to the measure- ment concepts and then describes the means for analyzing the output data and understanding the

confidence of the measurements. On the other hand, the Stein paper carefully lays out the

theoretical background needed to analyze the data before introducing the measurement concepts.

Both papers cover time-domain and frequency-domain measurements.

Paper B.3 by Allan reviews the concepts of the two-sample or Allan variance and the

modified Allan variance showing how classical statistical methods fail to usefully describe the

time-domain performance of good oscillators. The Allan variance concept is also introduced in

TN-2
papers B. 1 and B.2, and the modified Allan variance is described further in papers D.4 and D.5.

The presentation in paper B.3 is particularly useful in that it discusses general aspects of performance of different types of oscillators (quartz, rubidium, hydrogen, and cesium) providing the

basis for prediction of time errors, a topic which may prove useful to those who must develop

system specifications.

Paper B.4 by Walls, Clements, Felton, Lombardi, and Vanek adds to the discussion of

frequency-domain measurements providing information on methods which can be used to in- crease the dynamic range for both carrier frequency and for Fourier frequencies up to 10 percent

from the carrier. With some aerospace hardware now carrying phase-noise specifications, this is an important addition to the literature.

A3 COMMENTS ON PAPERS ON STANDARDS AND DEFINITIONS

The first paper in this group (paper C.l) outlines the standard terminology now used for

fundamental frequency and time metrology. This document was widely circulated for comment

during the draft stage and, with its acceptance by IEEE as a standard, supersedes the earlier

reference (paper C.2) which had served as the foundation for characterization of frequency

stability. This latter paper is included because it is so widely cited, and the reader will probably

be confronted with specifications based on its recommendations. Paper C.2 contains additional

material on applications of stability measures and measurement techniques including a useful

discussion of some of the common hazards in measurements. Paper C.l restricts itself to very

concise statements of the definitions.

The reader will note that the updated terminology in the first paper (C.l) varies in a number of minor ways from the earHer paper (C.2). A notable addition to definitions is the

introduction of script "ell", £(f), which has become an important measure of phase noise. This

quantity was previously defined as the ratio of the power in one sideband, due to phase modulation to the total signal power. For Fourier frequencies far from the carrier, this quantity can be simply related to the usual spectral densities which are the quantities that are generally measured, but the relation breaks down in the important region near the carrier. To resolve this

problem, the new standard defines the approximate relation between !e{f) and spectral density as being exact and applicable for any Fourier frequency.

The third paper in this group (paper C.3), from the 1986 report of the International

Radio Consultative Committee (CCIR), presents the definitions and terminology which have

been accepted for international use by this body. The material in this particularly readable

document is fairly consistent with the IEEE standard and would be useful to those involved in

specification of performance for international trade. A number of minor changes to this document have been recommended by different delegations to the CCIR and these will likely be

made in their next publication.

A.4 COMMENTS ON SUPPORTING PAPERS

Papers D.l and D.2 are included in this collection for a number of reasons. First, they

provide alternative introductions to the general topic of oscillator characterization. And second,

they include material not fuUy covered by introductory papers, B.l and B.2.

TN-3
The first of these (D.l) by Lesage and Audoin covers most of the same ground as papers

B.l and B.2, but, in addition, includes a nice discussion of characterization of frequency stability

via filtering of phase or frequency noise, a method which may be especially useful for rapid,

automated measurements where high accuracy is not required. Furthermore, this paper discusses

characterization of stable laser sources, a topic not covered in any of the other reference papers.

Paper D.2 by Lance, Seal, and Labaar limits itself to discussion of the measurement of

phase noise and ampHtude-modulation (AM) noise. The paper presents a detailed discussion of

delay-line measurement methods which can be used if a second reference oscillator is not

available. The delay-line concept is also introduced in B.2 and B.4, but in much less detail. While the delay-line method is less sensitive (for lower Fourier frequency) than two-oscillator

methods, it is easier to implement. The paper contains many good examples which the reader

will find useful.

A complete discussion of AM noise is beyond the scope of this volume. AM noise is usually ignored in the measurement and specification of phase noise in sources under the as- sumption that the AM noise is always less than the phase noise. This assumption is generally

true only for Fourier frequencies close to the carrier. At larger Fourier frequencies the normalized AM noise can be the same order of magnitude as the phase noise. In systems with active

amplitude leveling, the normalized AM noise can be higher than the phase noise. Under this

condition the AM to PM conversion in the rest of the system may degrade the overall phase

noise performance. For these reasons, we cannot ignore amplitude noise altogether. Paper D.2

provides a useful discussion of ampUtude noise. Note 1 in appendix E provides further information on definitions, notation and, in particular, the specification of added phase noise and

amplitude noise for signal-handling components.

The next contribution (D.3) provides substantially more detail on the extension of the

time-domain, dual-mixer concept for highly accurate time and time-interval measurements. The

basic dual-mixer ideas are included in papers B.l and B.2.

Paper D.4, published recently, provides the first quantitative treatment of confidence

estimates for phase-noise measurements. To the best of our knowledge, this is the only available

treatment of this important subject. We expect to see additional papers on this topic in the

future.

Paper D.5 discusses the modified Allan variance in more detail than the introductory

papers B.l, B.2, and B.3. It is followed by the Lesage-Ayi paper (D.6) which provides analytical

expressions for the standard set of power-law noise types and also includes discussion of the

uncertainty of the estimate of the modified Allan variance.

Linear frequency drift in oscillators is treated by Barnes in paper D.7. As noted in this

paper, even with correction for drift, the magnitude of drift error eventually dominates all time

uncertainties in clock models. Drift is particularly important in certain oscillators (e.g., quartz

oscillators) and a proper measure and treatment of drift is essential. As with other topics

treated by this group of papers, introductory papers B.l, B.2, and B.3 present some discussion of

frequency drift, but D.7 is included because it contains a much more comprehensive discussion of

the subject.

The final paper (D.8) by Barnes and Allan contains the most recent treatment of measurements made with dead times between them. Paper C.2 introduced the use of bias functions,

Bj and B2, which can be used to predict the Allan variance for one set of parameters based on

another set (for the power-law noise models). This last paper extends those ideas, introducing a

third bias function, B3, which can be used to translate the Allan variance between cases where

TN-4
dead time is accumulated at the end and where dead time is distributed between measurements,

a useful process for many data-acquisition situations.

A^ GUTOE TO SELECTION OF MEASUREMENT METHODS

The table and graph in this section are quick guides to performance limits of the different

methods as well as indications of advantages and disadvantages of each. The ideas presented

here are drawn from papers B.l and B.4, but we have modified and expanded the material to make it more comprehensive. The table has been kept simple, so the suggested methods should

be viewed as starting points only. They cannot possibly cover all measurement situations.

For a given measurement task, it is often best to start with a quick, simple measurement

which will then help to define the problem. For example, faced with the need to characterize an

oscillator, a good starting point might be to feed the output of that oscillator along with the

output of a similar, but more stable, oscillator into a good mixer and then look at the output. If the two can be brought into quadrature by tuning one of the oscillators or by using a phaselocked loop, then the output can be fed to a spectrum analyzer to get an immediate, at least

qualitative idea, of the performance of the oscillator. This mixing process, which brings the

fluctuations to baseband where measurement is much more straightforward, is basic to many of

the measurement methods. A large number of measurement problems can probably be resolved

with this simple, single-conversion, heterodyne arrangement. If the simplest approach is insufficient, then some of the more advanced methods outlined below can be used.

There are many ways to go about categorizing the various measurement methods. Since

this volume is aimed at practical measurements, we choose to use the characteristics of the

measurement circuit as the basis for sorting. In this arrangement we have (1) direct measure- ments where no signal mixers are used, (2) heterodyne measurements where two unequal fre- quencies are involved, and (3) homodyne, measurements where two equal frequencies are in- volved. These methods are listed below.

I. Direct Measurements

1. Measurements at the Fundamental Frequency

2. Measurements after Multiplication/Division n. Heterodyne Measurements

1. Single-Conversion Methods

2. Multiple-Conversion Methods

3. Time-Difference Method

a. Dual-Mixer, Time-Difference Method

in. Homodyne Measurements

1. Phase-Lock-Loop Methods (two oscillators)

a. Loose-Phase-Lock-Loop Method

b. Tight-Phase-Lock-Loop Method

2. Discriminator Methods (single oscillator)

a. Cavity-Discriminator Method

b. Delay-Line Method

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frequency-domain measurement methods.

Curve A. The noise limit (resolution), S0(f), of typical double-balanced

mixer systems at carrier frequencies from 0. 1 MHz to 26 GHz.

Curve B. The noise Umit, S0(f), for a high-level mixer.

Curve C. The correlated component of S0(f) between two channels

using high-level mixers.

Curve D. The equivalent noise limit, S0(f), of a 5 to 25 MHz frequency

multiplier.

Curve E. Approximate phase noise limit for a typical delay-line system

which uses a 500 ns delay line. Curve F. Approximate phase noise limit for a delay-line system which

achieves a 1 ms delay through encoding the signal on an

optical carrier and transmitting it across a long optical fiber to a detector.

TN-8
Table 1 organizes the measurement methods in the above manner giving performance

limits, advantages and disadvantages for each. Figure 1 following the table provides limitations

for phase noise measurements as a function of Fourier frequency. The reader is again reminded

that the table is highly simplified giving nominal levels that can be achieved. Exceptions can be

found to almost every entry.

A.6 RELATIONSHIP OF THE MODIFIED ALLAN VARIANCE TO THE ALLAN VARIANCE

In sorting through this set of papers and other published literature on the Allan Variance,

we were stimulated to further consider the relationship between the modified Allan variance and

the Allan variance. The ideas which were developed in this process have not been published, so we include them here. Paper D.6, "Characterization of Frequency Stability: Analysis of the

Modified Allan Variance and Properties of Its Estimate," by Lesage and Ayi adds new insights

and augments the Allan and Barnes paper (D.5), "A Modified Allan Variance with Increased

Oscillator Characterization Ability." In this section we extend the ideas presented in these two

papers and provide further clarification of the relationship between the two variances, both of

which are sometimes referred to as two-sample variances.

Figure 2 shows the ratio [mod o {T)/a {T)f as a function of n, the number of time or phase samples averaged together to calculate mod o {r). This ratio is shown for power-law noise

spectra (indexed by the value of a) running from f to f^. These corrected results have a some- what different shape for a = -1 than those presented in either paper D.5 or paper D.6. Furthermore, this figure also shows the dependence on bandwidth for the case where a = 1. For all other values of a shown, there is no dependence on bandwidth. Table 2 gives explicit values for

the ratio as a function of n for low n as well as the asymptotic limit for large n. For a = 1, the

asymptotic limit of the ratio is considerably simplified from that given in papers D.5 and D.6.

With these results it is possible to easily convert between mod a (t) and a (t) for any of the

common power-law spectra.

The information in figure 2 and table 2 was obtained directly from the basic definitions of

a (t) and mod oAt) using numerical techniques. The results of the numerical calculations were

checked against those obtained analytically by Allan and Barnes (D.5) and Lesage and Ayi (D.6).

For a = 2, and -2 the results agree exactly. For a = 1 and -1 the analytical expressions are

really obtained as approximations. The numerical calculations are obviously more reliable.

Details of our calculations can be found in a NIST report [1]. A useful integral expression (not

commonly found in the literature) for moday(r) is

-i 2 r s^(f)sin\TZT:f)

modolix) ^— f^!^ —^df.

Figure 2 and table 2 show that, for the fractional frequency fluctuations, mod 0^(7)

always yields a lower value than o (t). In the presence of frequency modulation (FM) noise

with a > 0, the improvement is very significant for large n. This condition has been examined in

detail by Bernier [2]. For white FM noise, a = 0, the optimum estimator for time interval r is to use the value of the times or phases separated by t to determine frequency. This is analogous to the algorithm for calculating o (t) which yields the one-sigma uncertainty in the estimate

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TN-11
of the frequency measured in this manner over the interval t. The estimate of frequency,

obtained by averaging the phase or time data, is degraded by about 10 percent from that obtained by using just the end points [3]. This is analogous to the algorithm for calculating mod

ay(r) which, in this case, underestimates the uncertainty in measuring frequency by J2. For

white phase modulation (PM) noise, a = 2, the optimum estimator for frequency is obtained by

averaging the time or phase data over the interval r. This is analogous to the algorithm for

calculating mod cr (r) which yields the one-sigma uncertainty in the estimate for frequency

measured in this manner. This estimate for frequency is Jn better than that provided by o {r).

Based on these considerations, it is our opinion that mod o {t) can be profitably used

much more often than it is now. The presence of significant high-frequency FM or PM noise in

the measurement system, in an oscillator, or in an oscillator slaved to a frequency reference, is very common. The use of mod a {t) in such circumstances allows one to more quickly assess systematic errors and long-term frequency stability. In other words, a much more precise value

for the frequency or the time of a signal (for a given measurement interval) can be derived using

mod cr (t) when n is large.

The primary reasons for using a (t) are that it is well known, it is simple to calculate, it

is the most efficient estimator for FM noise (a < 0), and it has a unique value for all t. The

advantages of mod a (r) are cited in the above paragraph. There are some situations where a study of both cr (t) and mod oAt) can be even more revealing than either one. The disadvantages of using mod 0(7) are that it is more complex to calculate and thus requires more computer time and it has not been commonly used in the literature, so interpretation of the results is more difficult to reconcile with published information. Another concern sometimes raised is that

mod aS"^) does not have a unique value in regions dominated by FM noise (a > 0). With

rapidly increasing computer speeds, the computational disadvantage is disappearing The correct- ed and expanded information presented in figure 2 and table 2 addresses the concern about

uniqueness.

The primary disadvantage of using cr (t) is that the results can be too conservative. That

is, if the level of high-frequency FM noise is high, then the results are biased high, and it can

take much longer (often orders of magnitude longer) to characterize the underlying low-frequency performance of the signal under test.

References to Section A.6

[1] Walls, F. L., John Gary, Abbie O'Gallagher, Roland Sweet and Linda Sweet, "Time

Domain Frquency Stability Calculated from the Frequency Domain Description: Use of

the SIGINT Software Package to Calculate Time Domain Frequency Stability from the

Frequency Domain," NISTIR 89-3916, 1989.

[2] Bernier, L.G., Theoretical Analysis of the Modified Allan Variance, Proceedings of the

41st Frequency Control Symposium, IEEE Catalogue No. 87CH2427-3, 116, 1987.

[3] Allan, D.W. and J.E. Gray, Comments on the October 1970 Metrologia Paper "The U.S.

Naval Observatory Time Reference and Performance of a Sample of Atomic Clocks,"

International Journal of Scientific Metrology, 7, 79, 1971.

TN-12
A.7 SUPPLEMENTARY READING LIST

Books

Gerber, Eduard A. and Arthur Ballato (editors), Precision Frequency Control, Volumes 1 and 2 (Academic Press, New York, 1985).

Jespersen, James and Jane Fitz-Randolph, From Sundials to Atomic Clocks, (Dover

Publications, New York, 1982).

Kartaschoff, P., Frequency and Time, (Academic Press, New York, 1978).

Kruppa, Venceslav F. (editor). Frequency Stability: Fundamentals and Measurement, (IEEE

Press, New York, 1983).

Randall, R.B., Frequency Analysis, (Briiel and Kjaer, Denmark, 1987).

Vanier, J. and C. Audoin, The Quantum Physics of Atomic Frequency Standards, Volumes

1 and 2 (Adam Hilger, Bristol, England, 1989).

Special Issues of Journals devoted to Time and Frequency Topics

Proceedings of the IEEE, Vol. 54, February, 1966.

Proceedings of the IEEE, Vol. 55, June, 1967.

Proceedings of the IEEE, Vol. 74, January, 1986.

IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. UFFC-34,

November, 1987.

Regular Publications dealing with Time and Frequency Topics

IEEE Transactions on Ultrasonics, Ferrolectrics and Frequency Control.

Proceedings of the Annual Symposium on Frequency Control, IEEE.

Proceedings of the Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, available from the U.S. Naval Observatory.

Proceedings of the European Frequency and Time Forum.

TN-13
From: Proceedings of the 35th Annual Symposium on Frequency Control, 1981.

PROPERTIES OF SIGNAL SOURCES AND MEASUREMENT METHODS

D. A. Howe, 0. W. Allan, and J. A. Barnes

Time and Frequency Division

National Bureau of Standards

Boulder, Colorado 80303

Summary

This paper is a review of frequency stability measurement techniques and of noise properties of

frequency sources. First, a historical development of the usefulness of spectrum analysis and time domain measurements win be presented. Then the rationale will

be stated for the use of the two-sample (Allan)

variance rather than the classical variance.

Next, a range of measurement procedures will be

outlined with the trade-offs given for the various

techniques employed. Methods of interpreting the measurement results will be given. In particular,

the five commonly used noise models (white PM, flicker PM, white FM, flicker FM, and random walk

FM) and their causes will be discussed. Methods

of characterizing systematics will also be given.

Confidence intervals on the various measures will

be discussed. In addition, we will point out methods of improving this confidence interval for

a fixed number of data points.

Topics will be treated in conceptual detail.

Only light (fundamental) mathematical treatment

will be given.

Although traditional concepts, will be de- tailed, two new topics will be introduced in this

paper: (1) accuracy limitations of digital and

computer-based analysis and (2) optimizing the

results from a fixed set of input data. The final section will be devoted to fundamental (physical) causes of noise in commonly used

frequency standards. Also transforms from time to frequency domain and vice-versa will be given.

Key Words . Frequency stability; Oscillator noise

modeling; Power law spectrum; Time-domain sta- bility; Frequency-domain stability; White noise;

Fl icker noise.

Introduction

Precision oscillators play an Important role

in high speed communications, navigation, space

tracking, deep space probes and in numerous other

important applications. In this paper, we will

review some precision methods of measuring the

frequency and frequency stability of precision

oscillators. Development of topics does not rely

heavily on mathematics. The equipment and set-up

for stability measurements are outlined. Examples

and typical results are presented. Physical

interpretations of common noise processes are discussed. A table is provided by which typical

frequency domain stability characteristics may be

translated to time domain stability characteristics

and vice-versa.

I. THE SINE WAVE AND STABILITY

A sine wave signal generator produces a voltage that changes in time in a sinusoidal way

as shown in figure 1.1. The signal is an oscil -

lating signal because the sine wave repeats itself.

A cycle of the oscillation is produced in one period "T". The phase is the angle "" within a cycle corresonding to a particular time "t".

TIEFIGURE 1.1

It is convenient for us to express angles in

radians rather than in units of degrees, and

positive zero-crossings will occur at even multiples of 7i-rad1ans. The frequency "w" is the

number of cycles in one second, which is the

reciprocal of period (seconds per cycle). The

expression describing the voltage "V" out of a sine wave signal generator is given by V(t) = V sin C4>(t)] where V is the peak voltage amplitude.

Equivalent expressions are

V(t) = VpSin (zn ^)

TN-14
and

V(t) = V sin (27n>t).

Consider figure 1.2. Let's assume tnat the maximum

value of "V" equals 1, hence "V " = 1. We say that the voltage "V(t)" is normalized to unity.

If we know the frequency of a signal and if the

signal is a sine wave , then we can determine the

incremental change in the period "T" (denoted by

At) at a particular angle of phase.

"Frequency stability is the degree to which

an oscillating signal produces the same value

of frequency for any interval, At, throughout

a specified period of time".

Let's examine the two waveforms shown in figure 1.3. Frequency stability depends on the amount of time involved in a measurement. Of the two oscillating signals, it is evident that "2" is more stable than "1" from time t^ to t, assuming

the horizontal scales are linear in time.

V2

FIGURE 1.2 Note that no matter how big or small At may be, we can determine AV. Let us look at this from another

point of view. Suppose we can measure AV and At. From this, there is a sine wave at a unique minimum

frequency corresponding to the given AV and At. For infinitesimally small At, this frequency is called the instantaneous frequency at this t. The

smaller the interval At, the better the approximation of instantaneous frequency at t. When we speak of oscillators and the signals

they produce, we recognize that an oscillator has some nominal frequency at which it operates. The

"frequency stability" of an oscillator is a term

used to characterize the frequency fluctuations of

the oscillator signal. There is no formal definition for "frequency stability". However, one usually refers to frequency stability when com- paring one oscillator with another. As we shall see later, we can define particular aspects of an oscillator's output then draw conclusions about

its relative frequency stability. In general

terms.

1 4.

5I2B1£ ^^SDLETCt'

FIGURE 1.3 From time tj to tj, there may be some question as to which of the two signals is more stable, but

it's clear that from time tj to t, , signal "1" is at a different frequency from that in interval ti to tj.If we want an oscillator to produce a particular frequency Vj,, then we're correct in stating

that if the oscillator signal frequency deviates

from V. over any interval, this is a result of something which is undesirable. In the design of an oscillator, it is important to consider the sources of mechanisms which degrade the oscillator's frequency stability. All undesirable

mechanisms cause random (noise) or systematic

processes to exist along with the sine wave signal

of the oscillator. To account for the noise

components at the output of a sine wave signal

generator, we can express the output as

V(t) = [Vg + £(t)] sin [2nvQt * (P(t)]. (1.1)

TN-15
where V^ = nominal peak voltage aoplltude,

£(t) = deviation of amplitude from nominal,

Vp, = nominal fundamental frequency,

0(t) = deviation of phase from nominal.

Ideally "e" and "i>" should equal zero for all time. However, in the real world there are no perfect oscillators. To determine the extent of

the noise components "e" and "(>", we shall turn our attention to measurement techniques.

The typical precision oscillator, of course,

has a very stable sinusoidal voltage output with a frequency o and a period of oscillation T, which

is the reciprocal of the frequency (y = 1/T). One

goal is to measure the frequency and/or the fre- quency stability of the sinusoid. Instability is actually measured, but with little confusion it is often called stability in the literature. Naturally, fluctuations in frequency correspond to fluctuations in the period. Almost all frequency

measurements, with very few exceptions, are mea- surements of phase or of the period fluctuations

in an oscillator, not of frequency, even though

the frequency may be the readout. As an example, most frequency counters sense the zero (or near

zero) crossing of the sinusoidal voltage, which is the point at which the voltage is the most sensitive to phase fluctuations.

One must also realize that any frequency measurement involves two oscillators. In some instances, one oscillator is in the counter. It

is impossible to purely measure only one oscillator. In some instances one oscillator may be enough better than the other that the fluctuations

measured may be considered essentially those of

the latter. However, in general because frequency measurements are always dual, it is useful to define:

y (t) (1.2)

as the fractional frequency difference or deviation

of oscillator one, y^, with respect to a reference

oscillator y divided by the nominal frequency y . Now, y(t) is a dimensionless quantity and useful

in describing oscillator and clock performance;

e.g., the time deviation, x(t), of an oscillator over a period of time t, is simply given by:

x(t) = ; y(t')dt' (1.3)

Since it is impossible to measure instantaneous

frequency, any frequency or fractional frequency measurement always involves some sample time, <lt or "i" —some time window through which the oscillators are observed; whether it's a picosecond, a second, or a day, there is always some sample

time. So when determining a fractional frequency.

y(t), in fact what is happening is that the time

deviation is being measured say starting at some time t and again at a later time, t + x. The

difference in these two time deviations, divided

by t gives the average fractional frequency over

that period t:

y(t) = ^^^ * ^^ ' '^^^^ (1.4)

Tau, I, may be called the sample time or averaging

time; e.g., it may be determined by the gate time

of a counter. What happens in many cases is that one samples

a number of cycles of an oscillation during the

preset gate time of a counter; after the gate time

has elapsed, the counter latches the value of the

number of cycles so that it can be read out,

printed, or stored in some other way. Then there

is a delay time for such processing of the data

before the counter arms and starts again on the next cycle of the oscillation. During the delay

time (or process time), information is lost. We have chosen to call it dead time and in some instances it becomes a problem. Unfortunately for

data processing in typical oscillators the effects

of dead time often hurt most when it is the hardest

to avoid. In other words, for times that are short compared to a second when it is very difficult to avoid dead time, that is usually where

dead time can make a significant difference in the

data analysis. Typically for many oscillators, if

TN-16
the sample time is long compared to a second, the

dead time makes little difference in the data

analysis, unless it is excessive. New equipment

or techniques are now available which contribute

2 zero or negligible dead time.

In reality, of course, the sinusoidal output

of an oscillator is not pure, but it contains

noise fluctuations as well. This section deals

with the measurement of these fluctuations to determine the quality of a precision signal source. We will describe five different methods of measuring the frequency fluctuations in precision

oscillators.

1. 1 Common Methods of Measuring Frequency Sta- bility

A. Beat frequency method

The first system is called a heterodyne

frequency measuring method or beat frequency

method. The signal from two independent oscillators are fed into the two ports of a double

balanced mixer as illustrated in figure 1.4.

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^b ' •

F»EOU£IICT COUNTEt rimni

FIGURE 1.4

The difference frequency or the beat frequency,

v., is obtained as the output of a low pass filter

which follows the mixer. This beat frequency is then amplified and fed to a frequency counter and

printer or to some recording device. The fractional frequency is obtained by dividing v. by the

nominal carrier frequency v . This system has

excellent precision; one can measure essentially

all state-of-the-art oscillators.

B. Dual mixer time difference (DTMD) system

This system shows some significant promise and

has just begun to be exploited. A block diagram is shown is figure 1. 5.

DIML HDCX TIK DIFFODCE SYSTDI

l», • v,l I.J - -,l

atil) l"^ —^ '—Bt(i-il

Tii« eirrticKU: id) • tj • t, • iii^ - j^

r«tTio..i niQutucT: ,(M • i*^ • "'^"V

<!''

FIGURE 1.5

To preface the remarks on the DMTD, it should be

mentioned that if the time or the time fluctuations can be measured directly, an advantage is obtained over just measuring the frequency. The reason is that ona can calculate the frequency

from the time without dead time as well as know

the time behavior. The reason, in the past, that

frequency has not been inferred from the time (for

sample times of the order of several seconds and

shorter) is that the time difference between a pair of oscillators operating as clocks could not

be measured with sufficient precision (commercially

the best that is available is 10-^^ seconds). The

system described in this section demonstrates a precision of 10-^^ seconds. Such precision opens the door to making time measurements as well as frequency and frequency stability measuements for

sample times as short as a few milliseconds and

longer, all without dead time.

In figure 1.5, oscillator 1 could be con- sidered under test and oscillator 2 could be

considered the reference oscillator. These signals

go to the ports of a pair of double balanced

mixers. Another oscillator with separate symmetric

buffered outputs is fed to the remaining other two

TN-17
ports of the pair of double balanced mixers. This common oscillator's frequency is offset by a desired amount from the other two oscillators.

Then two different beat frequencies come out of

the two mixers as shown. These two beat frequencies will be out of phase by an amount proportional

to the time difference between oscillator 1 and

2—excluding the differential phase shift that may

be inserted. Further, the beat frequencies differ

in frequency by an amount equal to the frequency

difference between oscillators 1 and 2. This measurement technique is very useful

where one has oscillator 1 and oscillator 2 on the

same frequency. This is typical for atomic stan- dards (cesium, rubidium, and hydrogen frequency

standards).

Illustrated at the bottom of figure 1.5 is what might represent the beat frequencies out of

the two mixers. A phase shifter may be inserted

as illustrated to adjust the phase so that the two beat rates are nominally in phase; this adjustment

sets up the nice condition that the noise of the common oscillator tends to cancel (for certain

types of noise) when the time difference is determined. After amplifying these beat signals, the start port of a time interval counter is triggered

with the positive zero crossing of one beat and

the stop port with the positive zero crossing of

the other beat. Taking the time difference be- tween the zero crossings of these beat frequencies,

one measures the time difference between oscillator

1 and oscillator 2, but with a precision which has been amplified by the ratio of the carrier frequency to the beat frequency (over that normally

achievable with this same time interval counter).

The time difference x(i) for the i measurement

between oscillators 1 and 2 is given by eq (1.5).

x(i) = ^im. * .JL.

^b^^

%

(1.5)

where At(i) is the i time difference as read on the counter, t^. is the beat period, v is the

nominal carrier frequency, is the phase delay in radians added to the signal of oscillator 1, and k is an integer to be determined in order to remove

the cycle ambiguity. It is only important to know

k if the absolute time difference is desired; for

frequency and frequency stability measurements and

for time fluctuation measurements, k may be assumed

zero unless one goes through a cycle during a set of measurements. The fractional frequency can be

derived in the normal way from the time fluctuations.

w^(i, t) - W2(i, x)

^1,2^^' O = <

V

x(i K 1) - x(i)

I

At(i *1) - /lt(i)

(1.6)

v^ ^b^ \

In eqs (1.5) and (1.6), assumptions are made

that the transfer (or common) oscillator is set at

a lower frequency than oscillators 1 and 2, and

that the voltage zero crossing of the beat vj - u starts and that V2 ' "r stops the time interval counter. The fractional frequency difference may

be averaged over any integer multiple of i^:

^1,2^^"' """^b^ _ x(i * m) - x(i)

mx. (1-7)

where m is any positive integer. If needed, t^^ can be made to be very small by having very high

beat frequencies. The transfer (or common) oscillator may be replaced with a low phase-noise

frequency synthesizer, which derives its basic

reference frequency from oscillator 2. In this

set-up the nominal beat frequencies are simply

given by the amount that the output frequency of

the synthesizer is offset from ^2- Sample times

as short as a few milliseconds are easiliy ob- tained. Logging the data at such a rate can be a problem without special equipment. The latest NBS time scale measurement system is based on the OMTD and is yielding an excellent cost benefit ratio.

C. Loose phase lock loop method

This first type of phase lock loop method is illustrated in figure 1.6. The signal from an oscillator under test is fed into one port of a

TN-18
mixer. The signal from a reference oscillator is fed into the other port of this mixer. The signals

are in quadrature, that is, they are 90 degrees out

VOLTtSE COaTROL

OF FRCQU£NCT

FIGURE 1.6

of phase so that the average voltage out of the

mixer is nominally zero, and the instantaneous

voltage fluctuations correspond to phase fluctuations rather than to amplitude fluctuations

between the two signals. The mixer is a key

element in the system. The advent of the Schottky

barrier diode was a significant breakthrough in making low noise precision stability measurements. The output of this mixer is fed through a low pass filter and then amplified in a feedback loop,

causing the voltage controlled oscillator (reference) to be phase locked to the test oscillator.

The attack time of the loop is adjusted such that

a very loose phase lock (long time constant)

condition exists. This is discussed later in section VIII. The attack time is the time it takes the servo system to make 70% of its ultimate correction

after being slightly disturbed. The attack time

is equal to I/ttw. , where w. is the servo bandwidth.

If the attack time of the loop is about a second

then the voltage fluctuations will be proportional

to the phase fluctuations for sample tines shorter

than the attack time. Depending on the coefficient of the tuning capacitor and the quality of

the oscillators involved, the amplification used may vary significantly but may typically range from 40 to 80 dB via a good low noise amplifier.

In turn this signal can be fed to a specturm

analyzer to measure the Fourier components of the

phase fluctuations. This system of frequencydomain analysis is discussed in sections VIII to X.

It is of particular use for sample times shorter

than one second (for Fourier frequencies greater

than 1 Hz) in analyzing the characteristics of an oscillator. It is specifically very useful if one

has discrete side bands such as 60Hz or detailed structure in the spectrum. How to characterize

precision oscillators using this technique will be treated in detail later in section IX and XI. One may also take the output voltage from the

above amplifier and feed it to an A/D converter. This digital output becomes an extremely sensitive measure of the short term time or phase fluctuations between the two oscillators. Precisions of

the order of a picosecond are easily achievable.

D. Tight phase lock loop method

The second type of phase lock loop method

(shown in figure 1.7) is essentially the same as the first in figure 1.5 except that in this case

the loop is in a tight phase lock condition; i.e.,

the attack time of the loop should be of the order

of a few milliseconds. In such a case, the phase

fluctuations are being integrated so that the

voltage output is proportional to the frequency

MCIllATOI

UlOfl TIST

"1 "0

»tt*StO • (.ociedI 0SC!L1.*T0I 1

'

' i 70 Ml

COIF ( I LA nss FILIO lU] 101

TIGHT PHASE-LOCK LOOP wrrHoo OF measuring

FREQUENCY STABILITY

»OLT««t TO FltOUE«CT coiixirci

COCF c

1

riCOUCNCT PlIiTtI

COUI Tt» .

FIGURE 1.7

fluctuations between the two oscillators and is no longer proportional to the phase fluctuations for

sample times longer than the attack time of the

loop. A bias box is used to adjust the voltage on the varicap to a tuning point that is fairly

linear and of a reasonable value. The voltage

TN-19
fluctuations prior to the bias box (biased slightly

away from zero) may be fed to a voltage to frequency converter which in turn is fed to a frequency counter where one may read out the frequency

fluctuations with great amplification of the

instabilities between this pair of oscillators.

The frequency counter data are logged with a data

logging device. The coefficient of the varicap

and the coefficient of the voltage to frequency converter are used to determine the fractional

frequency fluctuations, y., between the oscillators, where i denotes the i measurement as shown in figure 1.7. It is not difficult to achieve a sensitivity of a part in 10^* per Hz resolution of the frequency counter, so one has excellent precision capabilities with this system.

E. Time difference method

The last measurement method we will illustrate

is very commonly used, but typically does not have

the measurement precision more readily available

in the first four methods illustrated above. This

method is called the time difference method, and

is shown in figure 1.3. Because of the wide

conversion, or multiplication factors. Caution

should be exercised in using this technique even

if adequate measurement precision is available

because it is not uncommon to have significant

instabilities in the frequency dividers shown in figure l.S—of the order of several nanoseconds.

The technology exists to build better frequency

dividers than are commonly available, but manufacturers have not yet availed themselves of state-ofthe-art techniques in a cost beneficial manner. A

triclc to by-pass divider problems is to feed the

oscillator signals directly into the time interval counter and' observe the zero voltage crossing into

a well matched impedance. (In fact, in all of the

above methods one needs to pay attention to impedance matching, caPle lengths and types, and con- nectors). The divided signal can be used to resolve cycle ambiguity of the carrier, otherwise

the carrier phase at zero volts may be used as the

time reference. The slope of the signal at zero volts is 2nV /ti, where Ti = l/vJi (the period of

oscillation). For V = 1 volt and a 5 MHz signal,

this slope is 3m volts/ns, which is a very good

sensitivity.

1

j| 099.362.188 us | SORt, StBOti j

i

(

1 c

i > 5 1

X'~V. (^^1 -N ^

f^^. L^2

FIGURE 1.8

bandwidth needed to measure fast rise-time pulses,

this method is limited in signal-to-noise ratio.

However, some counters are commercially available

allowing one to do signal averaging or to do

precision rise- time comparison (precision of time

difference measurements in the range of 10 ns to 10 ps are now available). Such a method yields a direct measurement of x(t) without any translation.

II. MEASUREMENT METHODS COMPARISON

When making measurements between a pair of

frequency standards or clocks, it is desirable to have less noise in the measurement system than the

composite noise in the pair of standards being

measured. This places stringent requirements on measurement systems as the state-of-the-art of

precision frequency and time standards has advanced

to its current level. As will be shown, perhaps one of the greatest areas of disparity between

measurement system noise and the noise in current

standards is in the area of time difference mea- surements. Commercial equipment can measure time

differences to at best 10-^^s, but the time fluctuations second to second of state-of-the-art

standards is as good as 10- ^^s. The disparity is unfortunate because if time

differences between two standards could be measured

with adequate precision then one may also know the

time fluctuations, the frequency differences, and

the frequency fluctuations. In fact, one can set

TN-20
Hltrachy SUtus

Exaapic of MtaturMwnt

H«ttK>d tt;

OaU d»duc1blt froa tht MMurMMnt

6TI1M Fr«q.

H«t«rodyn« or b«at

fr«qu*ncy

can't aaatur* can't Matura |y(t,t)| = Vat

iFreq.

1 Dual Mixar TIm Oi'ff. or Tim Inttrval Counttr

«<^) • f^

6 x(t,0 =

X (t x)-x(t) y{t.t) '

I

i y(t.T)=« y(t*T,i)-y(t.t)

2 Loose Phasa-lockad rafartnca oaclllator can't Masura

4 <t(t,T) =• u

H

Tight phase* lockad rafaranca oscillator

can't aaasure can't aaasura can't aaasura 6 y(t,T)

c -av

TABLE 2.1

T(M TiM Fr«flu»ncy frequency Mv«nUg«i

Ntttiod KCuricy susillty accuracy

(1 My)

(tiAiHty

01s4dvintigat Aoprojilaiu

Cost (kS)

Ouil Mlur TIm Olfftrcflca -lOOpi O.lpl -io-«*

No Mid tlM; Hiry cnooi* •WU tiM (la* to at lar^t a< MfirM); oicillator* aiy 0* at zare Daat -1.10-" ar diffirwtt; MtlunaanC

X f' bana>idtA taaily changed; Maunawic noise typically boija oscillators rwisa for tai^Xt ana longer; Matures

tl-MS. tiM staeillty,

frequency, and frequency,

ttabllity.

More cDMlai tnan otner eetnedt,

and hence li Mre suceotaole to axtranious signal pickup, a.g. . ground loops; ute tiM difference

is eodulo tne Met period, a. g.

,

200na I SHiz. -5

to 30

TiM Interval Counter -lOOpi lOpa -lO-i*

wide bandvidu input allows

a variety of tignals; siMl* ta use; cycle aa«i- -1.10*" guity aleott nn^r a pro* X f^ biM; Matures tiM. tisw ttebility, frequency, and frequency staeillty.

Measuraaent noise typicaly

in txcess of oscillator instaamtlet for tau valuet

up to and of the order of several Uioutand seconds:

nence. typically not useful

for ihort term staslllty

analysis.

1 to 7

Lmm ^hairlock reference* aacH later

UMful for snort ters tiM

staaility analysis at aell -O.loa depanua on -1.10*" at spectr^a analysis; calibration of a t-^ excellent pnate sentlti- »ar1cao curve vity—typically Mtter Uian nt etc411atDr>.

Calibration dependi on appro- priate reference otclllator

varlcap curve cAarecterittict.

Nttaredyne or iNt Frequaney -10-»*

NMtureaent noiaa can typically be Mde lett tnan

-1.10-** oaelllator inataollitiet

k T-' for tau «U and longer.

Hiniaua tau deterained by period of beet frequencytypical ly not adjuttable; can

not coopare oiclllatart n»»r

zero Mat; cannot tell anlcn otclllator it hign or loo in frequency: dead tiM often attociated xitn these aeatureeents.

0.2 to 1

Tignt RM*elocked reference* atcinitar

Meatureaent noise typically

lett than oscillator intta- -l.lO-'> bintlee for tau«li and

X T*^ longer: good aeetureaent syttM bano-vidth control; dead tiee can M aaM taall or negligible.

Need voltage controlled

reference oscillator; frequency sensitivity is a function of varlcap <1 tuning curve, nence not conducive to eaauring afitolute frequency dlfferencet.

' AtiuMt reference otclllator is aveilaole to the Mtrologist. T«U 2.2

TN-21
up an interesting hierarchy of kinds of measurement

systems: 1) those that can measure time, x(t); 2)

those that can measure changes in time or time

fluctuations 6x(t); 3) those that can measure

frequency, \>(y 5 (v-v )/v ); and 4) those that can measure changes in frequency or frequency fluctuations, 6v (6y 5 6u/v ). As depicted in table 2.1,

if a measurement system is of status 1 in this

hierarchy, i.e., it can measure time, then time

fluctuations, frequency and frequency fluctuations

can be deduced. However, if a measurement system

is only capable of measuring time fluctuations

(status 2 - table 2.1), then time cannot be deduced, but frequency and frequency fluctuations

can. If frequency is being measured (status 3

-

table 2.1), then neither time nor time fluctuations

may be deduced with fidelity because essentially

all commercial frequency measuring devices have

"dead time" (technology is at a point where that

is changing with fast data processing speeds that

are now available). Dead time in a frequency measurement destroys the opportunity of integrating

the fractional frequency to get to "true" time

fluctuations. Of course, if frequency can be measured, then trivially one may deduce the fre- quency fluctuations. Finally, if a- system can only measure frequency fluctuations (status 4

- table 2.1), then neither time, nor time fluctuations, nor frequency can be deduced from the data.

If the frequency stability is the primary concern

then one may be perfectly happy to employ such a measurement system, and similarly for the other

statuses in this measurement hierarchy. Obviously,

if a measurement method of Status 1 could be

employed with state-of-the-art precision, this

would provide the greatest flexibility in data

processing. From section 1, the dual mixer time

difference system is purported to be such a method.

Table 2.2 is a comparison of these different measurement methods. The values entered are nominal; there may be unique situations where

significant departures are observed. The time and

frequency stabilities listed are the nominal

second to second rms values. The accuracies

listed are taken in an absolute sense. The costs listed are nominal estimates in 1981 dollars.

Figure 2.1 is a diagram indicating the

sample time regions over which the various meUiods

FMcnaui nuuuci ,,.ij ITMUm '• « HI

e*Arnf.

FIGURE 2.1

are most appropriately applied. The large diagonally oriented area indicates the typical noise

limits of the measurement technique (at particular

values of sample time indicated on the horizontal

scale).

III. CHARACTERIZATION

Given a set of data of the fractional frequency or time fluctuations between a pair of

oscillators, it is useful to characterize these

fluctuations with reasonable and tractable models

of performance. In so doing for many kinds of

oscillators, it is useful to consider the flucuations as those that are random (may only be pre- dicted statistically) and those that are non- random (e.g. , systematics— those that are environmentally induced or those that have a causal

effect that can be determined and in many cases can be predicted).

3.1 Non-random Fluctuations

Non-random fluctuations are usually the main

cause of departure from "true" time or "true" frequency.

TN-22
If, for exaoiple, one has the values of the

frequency over a period of time and a frequency

offset from nominal is observed, one may calculate

directly that the phase error will accumulate as a ramp. If the frequency values show some linear

drift then the time fluctuations will depart as a quadratic. In almost all oscillators, the above

systematics, as they are sometimes called, are the

primary cause of time and/or frequency departure.

A useful approach to determine the value of the

frequency offset is to calculate the simple mean

of the set, or for determining the value of the

frequency drift by calculating a linear least squares fit to the frequency. A least squares quadratic fit to the phase or to the time derivative is typically not as efficient an estimator of

the frequency drift for most oscillators.

3. 2 Random Fluctuations

After calculating or estimating the systematic

or non-random effects of a data set, these may be subtracted from the data leaving the residual

random fluctuations. These can usually be best

characterized statistically. It is often the case

for precision oscillators that these random fluctuations may be well modeled with power law spectral densities. This topic is discussed later in sections VIII to X. We have

Sy(0 = h/. (3.1)

where S (f) is the one-sided spectral density of

the fractional frequency fluctuations, f is the

Fourier frequency at which the density is taken,

h 1s the intensity coefficient, and a is a number

amodeling the most appropriate power law for the

data. It has been shown ' that in the time

domain one can nicely represent a power law spectral density process using a well defined timedomain stability measure, a (t), to be explained

in the next section. For example, if one observes

from a log o *(t) versus t diagram a particular

slope (call it m) over certain regions of sample

time, T, this slope has a correspondence to a power law spectral density or a set of the sane with some amplitude coefficient h . In particular.

|j = -a -1 for -3 < a <1 and m s -2 for 1 < a. Further a correspondence exists between h and the

coefficient for a (x). These coefficients have

been calculated and appear in section XI. The

transformations for some of the more common power

law spectral densities have been tabulated making

it quite easy to transform the frequency stability

modeled in the time-domain over to the frequency

domain and vice versa. Examples of some power-law

spectra that have been simulated by computer are shown in figure 3.1. In descending order these

POWER LAW SPECTRA

k.1-' f'^^VH/W^VtSvS**^'^

FIGURE 3.1

have been named white noise, flicker noise, random

walk, and flicker walk (the lu in fig. 3.1 is angular Fourier frequency, u = 2nf).

Once the noise characteristics have been

determined, one is often able to deduce whether

the oscillators are performing properly or not and

whether they are meeting either the design specifications or the manufacturers specifications.

For example a cesium beam frequency standard or a rubidium gas cell frequency standard when working

properly should exhibit white frequency noise,

which is the same as random walk phase (or time)

for tau values of the order of a few seconds to several thousand seconds (see also sec. XI).

IV. ANALYSIS OF TIME DOMAIN DATA Suppose now that one is given the time or frequency fluctuations between a pair of precision

oscillators measured, for example, by one of the

techniques outlined in section I, and a stability

analysis is desired. Let this comparison be depicted by figure 4.1. The minimum sample time

10

TN-23
is determined by the measurement system. If the

time difference or the time fluctuations are available then the frequency or the fractional

frequency fluctuations may be calculatetl froB one period of sampling to the next over tt»e data

length as indiciated in figure 4.1. Suppose

further there are M values of the fractional

frequency y,. Now there are many ways to analyze

these data. Historically, people have typically

used the standard deviation equation shown in figure 4.1, a

^^ ^ (x), where y is the average

fractional frequency over the data set and is subtracted from each value of y. before squaring,

summing and dividing by the number of values minus

one, (M-1), and taking the square root to get the

standard deviation. At NBS, we have studied what

happens to the standard deviation when the data

set may be characterized by power law spectra

which are more dispersive than classical white

noise frequency fluctuations. In ottier words, if the fluctuations are characterized by flicker

noise or any other non-white-noise frequency

deviations, what happens to the standard deviation

for that data set? One can show that the standard

deviation is a function of the number of data

points in the set; it is also a function of the

dead time and of the measurement system bandwidth.

For example, using flicker noise frequency modulation as a model, as the number of data points

increases, the standard deviation raonotonically

increases without limit. Some statistical measures

have been developed which do not depend upon the

data length and which are readily usable for

characterizing the random fluctuations in precision

oscillators. An IEEE subcomnittee on frequency

stability has recommended what has come to be

known as the "Allan variance" taken from the set

of useful variances developed, and an experimental

estimation of the square root of the Allan vari- ance is shown as the bottom right equation in figure 4.1. This equation is very easy to imple- ment experimentally as one simply need add up the

squares of the differences between adjacent values

0^^ y< » divide by the number of them and by two, and

take the square root. One then has the quantity

which the IEEE subcommittee has recomnended for

specifixation of stability in the time domaii

denoted by.Tj (x).

"yb., =4 Cj(t+t) - y(t))2 >^' (4.1)

where ttre brackets "<>" denote infinite time

average, in practice this is easily estimated

from a^fririte data set as follows:

V^) =

1 ^"^

2(M-1) "^ (^i^l ' ^i)^

^\

(4.2)

where :the y. are the discrete frequency averages

as illustrated in figure 4.1.

»(ti

•m Oft J '" V-sJr ,?, "i • ^' •, '»> 'VliTTT .?, '"•'•'''

FIGURE 4.

1

A simulated plot of the time fluctuations, x(t)

between a pair of oscillators and of the corres- ponding fractional frequencies calculated from the

time fluctuations earti averaged over a sample time

T. At the bottom are the equations for the stan- dard deviation (left) and for the time-domain measure of frequency stability as recommended by

the IEEE subcomnittee on frequency stability

(right).

One would like to know how o (x) varies with

the sample time, x. A simple trick that one can

use that is very useful if there is no dead time,

is to average the previous values for y, and y- and call that a new y., averaged over 2x, similarly

average the previous values for y. and y. and call that a new y. averaged over 2x etc., and finally

apply the same equation as before to get a (2x). One can repeat this process for other desired

integer multiples of x and from the same data set

11

TN-24
be able to generate values for a (mt) as a function

of mt fron which one may be able to infer a model

for the process that is characteristic of this

pair of oscillators. If one has dead time in the measurements adjacent pairs cannot be averaged in an unambiguous way to simply increase the sample

time. One has to retake the data for each new sample time—often a very time consuming task. This is another instance where dead time can be a problem. How the classical variance (standard deviation

squared) depends on the number of samples is shown

in figure 4.2. Plotted is the ratio of the stan- dard deviation squared for N samples to the stan- dard deviation squared for 2 samples; <o2(2,i)> is the same as the Allan variance, a^{x). One can

see the dependence of the standard deviation upon the number of samples for various kinds of power

10' l»i UMUI or SIMPUS U)

FIGURE 4.2

The ratio of the time average of the standard

deviation squared for N samples over the time

average of a two sample standard deviation squared

as a function of the number of smaples, N. The

ratio is plotted for various power law spectral

densities that commonly occur in precision oscillators. The figure illustrates one reason why the

standard deviation is not a convenient measure of

frequency stability; i.e. it may be very important

to specify how many data points are in a data set

if you use the standard deviation.

law spectral densities commonly encountered as reasonable models for many important precision

oscillators. Note, a ^(i) has the same value as

the classical variance for the classical noise case (white noise FM). One main point of figure

4.2 is simply to show that with the increasing

data length the standard deviation of the common

classical variance is not well behaved for the

kinds of noise processes that are very often

encountered in most of the precision oscillators

of interest. One may combine eq (1.4) and eq (4.1), which

yields an equation for a (x) in terms of the time

difference or time deviation measurements.

v^^ =<^ (X(t+2T) - 2x(t+T) * X(t))2y^

(4.3)

which for N discrete time readings may be estimated

as,

o/x) = 2(N-2)t2

N-2 ih

x,)2

(4.4)

where the i denotes the number of the reading in the set of N and the nominal spacing between

readings is t. If there is no dead time in the

data and the original data were taken with the x's spaced by x , we can pick x in eq (4.4) to be any integer multiple of x , i.e. , x = mx :

•^yCM^O^ = N-2m

2(N-2m)m2x2 Ji^'^i+Zm

- 2x i+m x.)2

(4.5)

Equation (4.5) has some interesting consequences

because of the efficient data usage in terms of

the confidence of the estimate as will be explained

in the next section. EXAMPLE : Find the Allan variance, cr ^(i), of the

following sequence of fractional frequency fluctuation values y. , each value

averaged over one second.

y^ =4.36 x lO-s = 4.47 X 10- ^5

y2 = 4.61x10-3 yg = 3.96xl0-s

yj = 3. 19 X 10-s yj = 4. 10 x 10-s

y^ = 4.21 x 10-» yg = 3.08 x 10-^

(assume no dead-time in measurement of averages)

12

TN-25
since each average of the fractional frequency

fluctuation values is for one second, then the

first variance calculation will be at i » Is. We

are given H = 8 (eight values); therefore, the

number of pairs in sequence is M-1 - 7. We have:

Data iialMi r\rtt dlfftrtfiett Ftrtt dlfftrane* touartd y^ (» 10-») (y^.^ • yj) (» XO-*) ''t*!'

• y^)« (« 10-")

*.it •«*• ,,

,

*.n e.zs O.Oi

i.lS •1.42 2.02 4.21 1.02 1.04 4.«7 0.2f 0.07 ].M -0.$1 0.2«

4.10 0.14 0.02

3.oa •1.02 —t5S

v. confidence of the estimate and overlapping

samples

'^

One can imagine taking three phase or time measurements of one oscillator relative to another

at equally spaced intervals of time. From this

phase data one can obtain two, adjacent values of average frequency. From these two frequency mea- surements, one can calculate a single sample Allan

(or two-sample) variance (see fig. 5.1). Of course this variance does not have high precision

or confidence since it is based on only one frequency difference.

M-1

Z (yk.i-yJ' = *-5ixio-*o

k=l

...+^- '^

l^a-><

Therefore,

a^(ls)^ ^-^^,^yf = 3.2xl0-xx

and

t^yC^) = [«Jj(1s)]^ = [3.2 X 10-"]** = 5.6 X 10-6

Using the same data, one can calculate the

variance for i = 2s by averaging pairs of adjacent

values and using these new averages as data values

for the same procedure as above. For three second

averages (x = 3s) take adjacent threesomes and

find their averages and proceed in a siniliar

manner. More data must be acquired for longer

averaging times. One sees that with large numbers of data

values, it is helpful to use a computer or programmable calculator. The confidence of the estimate

on a (i) improves nominally as the square root of

the number of data values used. In this example,

^^8 and the confidence can be expressed as being

no better the l/J^ x 100% ' 35%. This then is the

allowable error in our estimate for the i = Is average. The next section shows methods of com- puting and improving the confidence interval.

FIGURE 5.1

Statisticians have considered this problem of

quantifying the variability of quantities like the

Allan Variance. Conceptually, one could imagine

repeating the above experiment (of taking the

three phase points and calculating the Allan

Variance), many times and even calculating the

distribution of the values. For the above cited experiment we know that

the results are distributed like the statistician's

chi-square distribution with one degree of freedom.

That is, we know that for most common oscillators

the first difference of the frequency is a normally

distributed variable with the typical bell-shaped

curve and zero mean. However, the square of a normally distributed variable is NOT normally

distributed. That is easy to see since the square

Is always positive and the normal curve is com- pletely synmetric and negative values are as likely as positive. The resulting distribution is called a chi-square distribution, and it has ONE "degree of freedom" since the distribution was obtained by considering the squares of individual

(i.e., one independent sample), normally distributed variables.

In contrast, if we took five phase values,

then we could calculate four consecutive frequency

values, as in figure 5.2. We could then take the

13

TN-26
AT

t-> ^+4—.—

M— «Ti«rT»R—t4-*Bfl»«amiR-*)

FIGURE 5.2

first pair and calculate a sample Allan Variance,

and we could calculate a second sample Allan

Variance from the second pair (i.e., the third and

fourth frequency measurements). The average of

these two sample Allan Variances provides an improved estimate of the "true" Allan Variance,

and we would expect it to have a tighter confidence

interval than in the previous example. This could

be expressed with the aid of the chi-square distribution with TWO degrees of freedom.

However, there is another option. We could

also consider the sample Allan Variance obtained

from the second and third frequency measurements.

That is the middle sample variance. Now, however, we're in trouble because clearly this last sample

Allan Variance is NOT independent of the other

two. Indeed, it is made up of parts of each of

the other two. This does NOT mean that we can't

use it for improving our estimate of the "true" Allan Variance, but it does mean that we can't

just assume that the new average of three sample

Allan Variances is distributed as chi-square with

three degrees of freedom. Indeed, we will en- counter chi-square distributions with fractional

degrees of freedom. And as one might expect, the

number of degrees of freedom will depend upon the

underlying noise type, that is, white FM, flicker

FM, or whatever.

Before going on with this, it is of value to review some concepts of the chi-square distribution. Sample variances (like sample Allan

Variances) are distributed according to the equation:

(5.1)

chi-square, d. f . is the number of degrees of

freedom (possibly not an integer), and ct^ is the "true" Allan Variance we're all interested in knowing—but can only estimate imperfectly.

Chi-square is a random variable and its distribution has been studied extensively. For some reason, chi-square is defined so that d. f .

, the

number of degrees of freedom, appears explicitly

in eq (5.1). Still, x^ is a (implicit)

function of d. f . , also. The probability density for the chi-square

distribution is given by the relation

where r( 1 j is the gamma function, defined by

the integral

r (t) = ;^ x^"^ e"'' dx (5.3)

Chi-square distributions are useful in determining specified confidence intervals for variances

and standard deviations. Here is an example.

Suppose we have a sample variance s^ = 3.0 and we know that this variance has 10 degrees of freedom.

(Just how we can know the degrees of freedom will

be discussed shortly.) Suppose also that we want

to know a range around our sample value of s^ = 3.0

which "probably" contains the true value, a^ . The

desired confidence is, say, 90%. That is, 10% of

the time the true value will actually fall outside

of the stated bounds. The usual way to proceed is to allocate 5% to the low end and 5% to the high

end for errors, leaving our 90% in the middle.

This is arbitrary and a specific problem might

dictate a different allocation. We now resort to tables of the chi-square distribution and find

that for 10 degrees of freedom the 5% and 95%

points correspond to:

where S^ is the sample Allan Variance, x* is

X^(.QS) = 3.94

X2(.95) = 18.3

for d.f. = 10 (5.4)

14

TN-27
Thus, with 90% probability the calculated sample

variance, s^, satisfies the inequality:

3.94 < (d.f

^

<18.3 (5.5)

and this inequality can be rearranged in the form

1.54 < a2 < 7.61

or, taking square roots:

1.28 < o < 2.76

(5.6)

(5.7)

Now someone might object to the form of eq (5.7) since it seems to be saying that

the true sigma falls within two limits with 90%

probability. Of course, this is either true or not and is not subject to a probabilistic interpretation. Actually eq (5.7) is based on the idea that the true sigma is not known and we estimate it with the square root of a sample

variance, s^. This sample variance is a random

variable and is properly the subject of probability, and its value (which happened to be 3.0 in the example) will conform to eq (5.7) nine

times out of ten. Typically, the sample variance is calculated

from a data sample using the relation:

1 N

x)^ (5.8)

where it is implicitly assumed that the x 's are random and uncorrelated (i.e., white) and where x

is the sample mean calculated from the same data

set. If all of this is true, then s^ is chi-square

distributed and has N-1 degrees of freedom.

Thus, for the case of white x , and a conventional sample variance (i.e., eq (5.8)), the

number of degrees of freedom are given by the

equation:

d.f. = N-1 (5.9)

The problem of interest here is to obtain the

corresponding equations for Allan Variances using

overlapping estimates on various types of noise

(i.e., white FM, flicker FM, etc.).

Other authors (Lesage and Audoin, and

Yoshimura) have considered the question of the

variance of the Allan Variances without regard to the distributions. This is, of course, a closely

related problem and use will be made of their

results. These authors considered a more restrictive set of overlapping estimates than will be considered here, however.

VI. MAXIMAL USE OF THE DATA AND DETERMINATION OF THE DEGREES OF FREEDOM .

6.1 Use of Data Consider the case of two oscillators being

compared in phase and exactly N values of the

phase difference are obtained. Assume that the

data are taken at equally spaced intervals, t . From these N phase values, one can obtain N-1

consecutive values of average frequency and from

these one can compute N-2 individual, sample Allan

Variances (not all independent) for i = t . These

N-2 values can be averaged to obtain an estimate

of the Allan Variance at i = t . The variance of

this variance has been calculated by the above

cited authors.

Using the same set of data, it is also possible to estimate the Allan Variances for integer

multiples of the base sampling interval, x = nt . Now the possibilities for overlapping sample Allan

Variances are even greater. For a phase data set

of N points one can obtain exactly N-2n sample

Allan Variances for t = nx . Of course only a fraction of these are generally independent.

Still the use of ALL of the data is well justified

(see fig. 6.1).

Consider the case of' an experiment extending

for several weeks in duration with the aim of

getting estimates of the Allan Variance for tau values equal to a week or more. As always the

purpose is to estimate reliably the "true" Allan

Variance as well as possible—that is, with as tight an uncertainty as possible. Thus one wants

15

TN-28
>KrA I

I I I

I .

N*S7 ...•... * . . +

h '

I I I

I I h I

I I I

I

I I I

I

I I I

I

)

The approach used is based on three equations

relating to the chi -square distribution:

'^'^ ^ H-.3-H '^tlt^

M2-4 ^fS-^

l»-«.-4 •-ft-**

FIGURE 6.1

to use the data as efficiently as possible since

obtaining more data can be very expensive. The

most efficient use is to average all possible

sample Allan Variances of a given tau value ttiat one can compute from the data. The problem comes in estimating how tigtit the

confidence intervals really are—that is, in estimating the number of degrees of freedom. Clearly,

if one estimates the confidence intervals pessimistically, then more data is needed to reach a specified tolerance, and that can be expensive.

The other error of over-confidence in a questionable value can be even more expensive. Ideally,

one has realistic confidence estimates for the most efficient use of the data, which is the

intent of this writing.

6.2 Determining the Degrees of Freedom

In principle, it should be possible to determine analytically the equations corresponding to eq (5.9) for all cases of interest. Unfortunately the analysis becomes quite complicated. Exact computer algorithms were devised for the

cases of white phase noise, white frequency modulation and random walk FM. For the two flicker

cases (i.e., flicker FM and PM) a completely

empirical approach was used. Due to the complexity

of the computer programs, empirical fits were

devised for all five noise types.

(dTM o^

ECx2] = d.f.

VarCx2] = 2(d.f.)

(6.1)

(6.2)

(6.3)

where the expression E[x*] means the "expectation,"

or average value of x. Var[x^] is the variance of

x^, and d.f. is the number of degrees of freedom.

A computer was ^used to simulate phase data

sets of some length, N, and then Allan Variances

with T = nx were calculated for all possible

samples. This "experiment" was repeated at least

1000 times using new simulated data sets of the same spectral type, and always of the same length,

N. Since the data were simulated on a computer,

the "true" Allan Variance, a^ , was known for many

of the noise models and could be substituted into

eq (5.1). From the 1000 values of s^/a^, distributions and sample variances were obtained. The

"experimental" distributions were compared with

theoretical distributions to verify that the

observed distributions truely conformed to the

chi-square distribution.

The actual calculation of the degrees of

freedom were made using the relation:

Is^ (6.4)

which can be deduced from eqs (6.1), (6.2),

and (6.3). The Var(s2) was estimated by the

sample variance of the 1000 values of the average

Allan Variances, each obtained from a phase data set of length N. Of course this had to be repeated for various

values of N and n, as well as for each of the five common noise types: white PM, flicker PM, white

FM, flicker FM, and random walk FM. Fortunately,

certain limiting values are known and these can be

used as checks on the method. For example, when

(N-l)/2=n, only one Allan Variance is obtained

from each data set and one should get about one degree of freedom for eq (6.4), which was observed

16

TN-29
in fact. Also for n=l the "experimental" conditions correspond to those used by Lesage and

Audoin, and by Yoshimura. Indeed, the method also

was tested by verifying that it gave results

consistent with eq (5.9) when applied to the

conventional sample variance. Thus, combining eq (6.4) with the equations for the variance of the

Allan Variances from Lesage and Audoin and

Yoshimura, one obtains:

White PM d. f. _ 18(N-2)2

" 35N-88 , for N > 4

Flicker PM d. f. = ?

White FM d. f. _ 2(N-2)2

" 3N-7 (6.

Flicker FM d.f. - 2(N-2)2

2.3N-4.9

Random Walk FM d. f. = N-2

for n=l. Unfortunately, their results are not totally consistent with each other. Where inconsistency arose the value in best agreement with

the "experimental" results was chosen.

The empirical equations which were fit to the

"experimental" data and the known values are summarized below:

White PM

Flicker PM

White FM

d.f. =. (Ni-l)(N-2n)

= 2(N-n)

d.f. .exp(lnfli in i^nilliNlil)

(6.6)

d.f. =

Flicker FM d.f. s

3(N-1) 2(N-2)

L 2n N _

4n2

4n2 * i

2.3N - 4.9*

^°'^ n=l

5N2 4n(N - 3n) ' ^o'" " 1 2

Random Walk FM d.f. s^t^ (N-l)2-3n(N-l).4n2

n (.N-j)''

The figures in Appendix I demonstrate the fit to the "experimental" data.

It is appropriate to give some estimate of

just how well these empirical equations approach

the "true" values. The equations have approximately (a few percent) the correct assymptotic

behavior at n=l and n=(N-l)/2. In between, the

values were tested (using the simulation results) over the range of N=5 to N=1025 for n=l to n=(N-l)/2 changing by octaves. In general, the

fit was good to within a few percent. We must acknowledge that distributional problems with the

random number generators can cause problems,

although there were several known values which

should have revealed these problems if they are present. Also for three of the noise types the exact number of degrees of freedom were calculated

for many values of N and n and compared with the "Monte Carlo" calculations. The results were all very good.

Appendix I presents the data in graphical

form. All values are thought to be accurate to within one percent or better for the cases of

white PM, white FM, and random walk FM. A larger

tolerence should be allowed for the flicker cases.

VII. EXAMPLE OF TIME-DOMAIN SIGNAL PROCESSING AND ANALYSIS

We will analyze in some detail a commercial

portable clock, Serial No. 102. This cesium was measured against another commercial cesium whose

stability was well documented and verified to be

better than the one under test. Plotted in figure

7.1 are the residual time deviations after removing

FTS PC VSCSU31

100m .

FIGURE 7.1

17

TN-30
a mean frequency of 4.01 parts in 10^^. Applying

the methods described in section IV and section V, we generated the c; (i) diagram shown in figure

7.2.

FTS TC VS C&001

;d'a

CjCt) (0i-i - » » - I -

;o"

/O"'

W K)* Kf 10'

FIGURE 7.2 One observes that the last two points are proportional to T and one is suspicious of a signifi- cant frequency drift.

If one calculates the drift knowing that

a (t) is equal to the drift times js a drift of

1.22 X 10-^* per day is obtained. A linear least squares to the frequency was removed and sections

IV and V were applied again. The linear least

squares fit showed a drift of 1.23 x 10-^* per

day, which is in excellent agreement with the

previous calculated value obtained from a (x).

Typically, the linear least squares will give a much better estimate of the linear frequency drift

than will the estimate from cj (t) being propor-

+1 ^

tional to T . Figure 7.3 gives the plot of the time residuals after removing the linear least squares and

figure 7.4 is the corresponding a (z) vs. i diagram. From the 33 days of data, we have used the

90% confidence interval to bracket the stability

estimates and one sees a reasonable fit corres- ponding to white noise frequency modulation at a level of 4.4 X 10-^^ x . This seemed excessive

FTS W V9 C<Oi

90* -

.^.

FIGURE 7.3

«)'V.

nnmsywEb

I

10

44

I I II r

• • • i I

.^-.«c'*r'*

»" K^ IC^ W'

FIGURE 7.4

CS "R: 102 VS NBS-i

£tZ

6fU^

iH

•'§,

-i^.H6"

uio^x^

FIGURE 7.5

18

TN-31
in terms of the typical performance of this par- ticular cesium and in as much as we were doing

some other testing within the environment, such as working on power supplies and charging and discharging batteries, we did some later tests. Figure 7.5 is a plot of a (t) after the standard

had been left alone in a quiet environment and had

been allowed to age for about a week. One ob- serves that the white noise frequency modulation

level is more than a factor of 4 improved over the

previous data. This led us to do some studies on the effects of the power supply on the cesium frequency as one is charging and discharging batteries, which proved to be significant. One

notices in figure 7.4 that the ct (i) values plotted

are consistent within the error bars with flicker

noise frequency modulation. This is more typical

of the kind of noise one would expect due to such

environmental perturbations as discussed above.

Careful time- and/or frequency-domain analyses can lead to significant insights into problems

and their solutions and is highly recommended by

the authors. The frequency- domain techniques will

be next approached.

VIII. SPECTRUM ANALYSIS

Another method of characterizing the noise in

a signal source is by means of spectrum analysis. To understand this approach, let's examine the

waveform shown in figure 8.1.

1 4.

FIGURE 8.1

Here we have a sine wave which is perturbed

for short instances by noise. Some loosely refer

to these types of noises as "glitches". The

waveform has a nominal frequency over one cycle

which we'll call "v^" (Vg = y). At times, noise causes the instantaneous frequency to differ

markedly from the nominal frequency. If a pure sine wave signal of frequency u. is subtracted

from this waveform, the remainder is the sum of the noise components. These components are of a variety of frequencies and the sum of their amplitudes is nearly zero except for the intervals

during each glitch when their amplitudes momen- tarily reinforce each other. This is shown graphically in figure 8.2.

1 -I- f

^

^ ^^

^^ H -r'

FIGURE 8.2 One can plot a graph showing rms power vs. frequency for a given signal. This kind of plot

is called the power spectrum . For the waveform of figure 8.1 the power spectrum will have a high

value at Uq and will have lower values for the

signals produced by the glitches. Closer analysis

reveals that there is a recognizable, somewhat constant repetition rate associated with the

glitches. In fact, we can deduce that there is a significant amount of power in another signal

whose period is the period of the glitches as shown in figure 8.2. Let's call the frequency of

the glitches v . Since this is the case, we will

observe a noticeable amount of power in the spec- trum at v with an amplitude which is related to the characteristics of the glitches. The power

spectrum shown in figure 8.3 has this feature. A predominant v component has been depicted, but

other harmonics also exist.

"R^CR

i_

^0

FIGURE 8.3

19

TN-32
Some noise will cause the instantaneous fre- quency to "jitter" around Uq, with probability of

being higher or lower than u-. We thus usually

find a "pedestal" associated with u^ as shown in figure 8.4.

FIGURE 8.4

The process of breaking down a signal into

all of its various components of frequency is called Fourier expansion (see sec. X). In other words, the addition of all the frequency

components, called Fourier frequency components,

produces the original signal. The value of a Fourier frequency is the difference between the

frequency component and the fundamental frequency.

The power spectrum can be normalized to unity such

that the total area under the curve equals one. The power spectrum normalized in this way is the

power spectral density . The power spectrum, often called the RF spectrum, of V(t) is very useful in many applications. Unfortunately, if one is given the RF spectrum, it is impossible to determine whether

the power at different Fourier frequencies is a result of amplitude fluctuations "e(t)" or phase

fluctuations "0(t). The RF spectrum can be separ- ated into two independent spectra, one being the

spectral density of "£(t)" often called the AM power spectral density and the other being the

spectral density of "(l>(t)" . For the purposes here, the phase-fluctuation

components are the ones of interest. The spectral

density of phase fluctuations is denoted by S.(f)

where "f" is Fourier frequency. For the frequently encountered case where the AM power spectral density is negligibly small and the total

modulation of the phase fluctuations is small

(mean-square value is much less than one rad^),

the RF spectrum has approximately the same shape

as the phase spectral density. However, a main

difference in the representation is that the RF spectrum includes the fund2unental signal (carrier), and the phase spectral density does not. Another major difference is that the RF spectrum

is a power spectral density and is measured in units of watts/hertz. The phase spectral density

involves no "power" measurement of the electrical

signal. The units are radians^/hertz. It is tempting to think of S (f) as a "power" spectral

density because in practice it is measured by

passing V(t) through a phase detector and measuring

the detector's output power spectrum. The measure- ment technique makes use of the relation that for

small deviations (60 << 1 radian),

s.(0 •(H^y (8.3)

where V (f) is the root-mean-square noise. voltage

per VHz at a Fourier frequency "f", and V is the

sensitivity (volts per radian) at the phase quadra- ture output of a phase detector which is comparing

the two oscillators. In the next section, we will

look at a scheme for directly measuring S (f). One question we might ask is, "How do frequency changes relate to phase fluctuations?"

After all it's the frequency stability of an oscillator that is a major consideration in many

applications. The frequency is equal to a rate of

change in the phase of a sine wave. This tells us that fluctuations in an oscillator's output fre- quency are related to phase fluctuations since we must change the rate of "(p(t)" to accomplish a shift in "w(t)", the frequency at time t. A rate of change of total ".j.(t)" is denoted by "«t»j(t)". We have then

27tv(t) = .^(t) (8.4)

The dot denotes the mathematical operation of

differentiation on the function j with respect to

its independent variable t.* From eq (8.4)

* As an analogy, the same operation relates the

position of an object with its velocity.

20

TN-33
and eq (1.1) we get

2m>(t) = y(t) = 2nvQ * i(t)

Rearranging, we have

27iv(t) - 2nvQ = kt)

or

^'(t) - «o = ^(8.5)

The quantity \>(t) - w- can be more conveniently

denoted as 5v(t), a change in frequency at time t. Equation (8.5) tells us that if. we differentiate

the phase fluctuations $(t) and divide by 2n, we

will have calculated the frequency fluctuation

6u(t). Rather than specifying a frequency fluctuation in terms of shift in frequency, it is useful to denote 6v(t) with respect to the nominal

frequency u^,. The quantity ^^ ' is called the

fractional frequency fluctuation** at time t and

is signified by the variable y(t). We have

y(t) = ^= im (8.6)

The fractional frequency fluctuation y(t) is

a dimensionless quantity. When talking about fre- quency stability, its appropriateness becomes

clearer if we consider the following example.

Suppose in two oscillators 6u(t) is consistently

equal to -^ 1 Hz and we have sampled this value for

many times t. Are the two oscillators equal in their ability to produce their desired output

frequencies? Not If one oscillator is operating

at 10 Hz and the other at 10 MHz. In one case, the average value of the fractional frequency

fluctuation is 1/10, and in the second case is 1/10,000,000 or 1 X 10-'. The 10 MHz oscillator

is then more precise. If frequencies are multiplied or divided using ideal electronics, the

fractional stability is not changed.

In the frequency domain, we can measure the

.spectrum of frequency fluctuations y(t). The

** Some international recommendations replace

"fractional" by "normalized".

spectral density of frequency fluctuations is denoted by S (f) and is obtained by passing the

signal from an oscillator through an ideal FM detector and performing spectral analysis on the

resultant output voltage. S (f) has dimensions of

<fractional frequency)2/Hz or Hz-^. Differentiation of (>(t) corresponds to multiplication by —

in terms of spectral densities. With further calculation, one can derive that

s,(f) --(^y s.(0 (8.7)

We will address ourselves primarily to S (f), that

is, the spectral density of phase fluctuations.

For noise-measurement purposes, S (f) can be measured with a straightforward, easily duplicated

equipment set-up. Whether one measures phase or frequency spectral densities is of minor Importance

since they bear a direct relationship. It is important, however, to make the distinction and to use eq (8.7) if necessary.

8.1 The Loose Phase-Locked Loop

Section I, 1.1, C described a method of

measuring phase fluctuations between two phaselocked oscillators. Now we will detail the pro- cedure for measuring S.(f).

Suppose we have a noisy oscillator. We wish

to measure the oscillator's phase fluctuations

relative to nominal phase. One can do this by

phaselocking another oscillator (called the re- ference oscillator) to the test oscillator and

mixing the two oscillator signals 90' out of phase

(phase quadrature). This is shown schematically

in figure 8.9. The two oscillators are at the same frequency in long term as guaranteed by the

phase-lock loop (PLL). A low-pass filter (to

filter the R.F. sum component) is used after the

mixer since the difference (baseband) signal is the one of interest. By holding the two signals

at a relative phase difference of 90", short-term

phase fluctuations between the test and reference

oscillators will appear as voltage fluctuations

out of the mixer.

21

TN-34
isirxc coat'Oi. or FxguEnci

FIGURE 8.9

With a PLL, if we can make the servo time constant very long, then the PLL bandwidth as a filter will be small. This may be done by lowering

the gain A of the loop amplifier. We want to translate the phase modulation spectrum to baseband spectrum so that it is easily measured on a low frequency spectrum analyzer. With a PLL filter, we must keep in mind that the reference

oscillator should be as good or better than the test oscillator. This is because the output of the PLL represents the noise from both oscillators,

and if not properly chosen, the reference can have

noise masking the noise from the test oscillator.

Often, the reference and test oscillators are of

the same type and have, therefore, approximately

the same noise. We can acquire a meaningful measurement by noting that the noise we measure is from two oscillators. Many times a good approximation is to assume that the noise power is twice

that which is associated with one oscillator.

S (f) is general notation depicting spectral density on a per hertz basis. A PLL filter output

necessarily yields noise from two oscillators.

The output of the PLL filter at Fourier

frequencies above the loop bandwidth is a voltage

representing phase fluctuations between reference

and test oscillator. It is necessary to make the

time-constant of the loop long compared with the

inverse of the lowest Fourier frequency we wish to measure. That '^' \ > hi f (lowest)'

This means

that if we want to measure S^(f) down to 1 Hz, the

loop time-constant must be greater than j- seconds. One can measure the time-constant by

perturbing the loop (momentarily disconnecting the

battery is convenient) and noting the time it takes for the control voltage to reach 70% of its

final value. The signal from the mixer can then

be inserted into a spectrum analyzer. A preamp

may be necessary before the spectrum analyzer.

• See Appendix Note # 3

The analyzer determines the mean square volts that

pass through the analyzer's bandwidth centered

around a pre-chosen Fourier frequency f. It is desireable to normalize results to a 1 Hz bandwidth. Assuming white phase noise (white PM),

this can be done by dividing the mean square voltage by the analyzer bandwidth in Hz. One may have to approximate for other noise processes.

(The phase noise sideband levels will usually be

indicated in rms volts-per-root-Hertz on most

analyzers.

)

8.2 Equipment for Frequency Domain Stability Measurements

(1) Low-noise mixer

This should be a high quality, doublebalanced type, but single-ended types

may be used. The oscillators should

have well-buffered outputs to be able to isolate the coupling between the two input RF ports of the mixer. Results **

that are too good may be obtained if the two oscillators couple tightly via

signal injection through the input

ports. We want the PLL to control

locking. One should read the specifications in order to prevent exceeding

the maximum allowable input power to the

mixer. It is best to operate near the

maximum for best signal-to-noise out of

the IF port of the mixer and, in some cases, it is possible to drive the mixer

into saturation without burning out the

device.

B

~1

J

B

FIGURE 8.10

(2) Low-noise DC amplifier

The amount of gain A needed in the loop

amplifier will depend on the amplitude

of the mixer output and the degree of •* See Appendix Note # 4

22

TN-35
varactor control in the reference oscillator. We may need only a small anount

of gain to acquire lock. On the other

hand, it may be necessary to add as much

as 30 dB of gain. Good low-noise DC amplifiers are available from a nuoiber of sources, and with cascading stages of

amplification, each contributing noise,

it will be the noise of the first stage

which will add most significantly to the

noise being measured. If a suitable

low-noise first-stage amplifier is not readily available, a schematic of an amplifier with 40 dB of gain is shown in figure 8.11 which will serve nicely for

the first stage. Amplifiers with very

low equivalent input noise performance

are also available from many manufacturers. The response of the amplifier

should be flat from OC to the highest

Fourier frequency one wishes to measure. The loop time-constant is inversely

related to the gain A and the determination of A is best made by experimentation knowing that t^ < ^^ (lowest )'

:ilDlC<TO»f(T[\,»3;7 L«l

UM WIS )m.\na

FIGURE 8.11

(3) Voltaqe-contrpned reference quartz

oscillator

This oscillator should be a good one with specifications available on its

frequency domain stability. The refer- ence should be no worse than the test

oscillator. The varactor control should

be sufficient to maintain phase-lock of

the reference. In general, low quality

test oscillators nay have varactor

control of as much as 1 x 10-* fractional

frequency change per volt. Some provision should be available on the reference

oscillator for tuning the mean frequency over a frequency range that will enable

phase- lock. Many factors enter into the

choice of the reference oscillator, and

often it is convenient to simply use two test oscillators phase- locked together.

In this way, one can assume that the

noise out of the PLL filter is no worse

than 3 dB greater than the noise from

each oscillator. If it is uncertain

that both oscillators are contributing

approximately equal noise, then one should perform measurements on three

oscillators taking two at a time. The

noisier-than-average oscillator will

reveal itself.

(4) Spectrum analyzer

The signal analyzer typically should be

capable of measuring the noise in rms volts in a narrow bandwidth from near

1 Hz to the highest Fourier frequency

of interest. This may be 50 kHz for

carrier frequencies of 10 MHz or lower. For voltage measuring analyzers, it is typical to use units of "volts per VHz". The spectrum analyzer and any associated

input amplifier will exhibit high-frequency roll off. The Fourier frequency

at which the voltage has dropped by 3 dB

is the measurement system bandwidth f^, or u)v = ^'^hi- ^^^^ f^*" ^^ measured

directly with a variable signal generator.

Section X describes how analysis can be

performed using a discrete fourier transform

analyser. Expanding digital technology has made

the use of fast-fourier transform analysis affordable and compact.

* See Appendix Note # 5

23

TN.36
Rather than measure the spectral density of

phase fluctuations between two oscillators, it is possible to measure the phase fluctuations introduced by a device such as an active filter or amplifier. Only a slight modification of the

existing PLL filter equipment set up is needed.

The scheme is shown in figure 8.12.

-JUMPEil TO NCASUKE INTimSIC NOISI OF SET-UP FIGURE 8.12

Figure 8.12 is a differential phase noise measurement set-up. The output of the reference

oscillator is split so that part of the signal

passes through the device under test. We want the two signals going to the mixer to be 90° out of

phase, thus, phase fluctuations between the two input ports cause voltage fluctuations at the

output. The voltage fluctuations then can be

measured at various Fourier frequencies on a spectrum analyzer.

To estimate the noise inherent in the test

set-up, one can in principle bypass the device

under test and compensate for any change in amplitude and phase at the mixer. The PLL filter

technique must be converted to a differential

phase noise technique in order to measure inherent

test equipment noise. It is a good practice to measure the system noise before proceeding to measurement of device noise.

A frequency domain measurement set-up is shown schematically in figure 3.13. The component

VAKACTSR

values for the low-pass filter out of the mixer

are suitable for oscillators operating at around

5 MHz.The active gain element (a ) of the loop is a DC amplifier with flat frequency response. One may replace this element by an integrator to achieve high gain near DC and hence, maintain

better lock of the reference oscillator in long

term. Otherwise long-term drift between the

reference and test oscillators might require

manual re-adjustment of the frequency of one or the other oscillator.

8.3 Procedure and Example ^

At the input to the spectrum analyzer, the

voltage varies as the phase fluctuations in short- term

S.(0 .(^)

V is the phase sensitivity of the mixer in volts

per radian. Using the previously described equip- ment set-up, V can be measured by disconnecting

the feedback loop to the varactor of the reference

oscillator. The peak voltage swing is equal to V in units of volts/rad if the resultant beat note

is a sine wave. This may not be the case for

state-of-the-art S^(f) measurements where one must

drive the mixer very hard to achieve low mixer

noise levels. Hence, the output will not be a sine wave, and the volts/rad sensitivity must be

estimated by the slew-rate (through zero volts) of

the resultant square-wave out of the mixer/amplifier.

The value for the measured S.(f) in decibels

9

Is given by:

V^gVoltage at f S^(f) = 20 log Vgfull-scale <t^detector voltage

EXAMPLE : Given a PLL with two oscillators such

that, at the mixer output:

FIGURE 8.13 Vg = 1 volt/rad

• See Appendix Note # 6

24

TN-37
V (45 Hz) = 100 nV per root hertz

nns solve for S.(45 Hz).

S<« "') ' i^'^^h^J - (^/ rad2 Hz-i

= 10-14 ia£ ^^ Hz

In decibels,

S^(45 HZ) = 20 log

^^ = 20 log

^

= 20 (-7) = -140 dB at 45 Hz

In the example, note that the mean frequency

of the oscillators in the PLL was not essential to computing S.(f). However, in the application of

S (f), the mean frequency v is necessary information. Along with an S (f), one should always

attach u In the example afiove Vq = 5 MHz, so we

have

S^ (45 HZ) = 10-^" ^. % = 5 MHZ.

From eq (8.7), S (f) can be computed as

\^''''^-[rTW

45

10,.14 n^Hz

S (45 Hz) = 8.1 X 10-25 Hz-i, V = 5 MHz. y

IX. POWER-LAW NOISE PROCESSES

Power-law noise processes are models of

precision oscillator noise that produce a particular slope on a spectral density plot. We often

classify these noise processes into one of five

categories. For plots of S (f), they are:

1. Random walk FM (random walk of frequency), S^ plot goes down as 1/f*.

2. Flicker FM (flicker of frequency), S. plot goes down as l/f^.

3. White FM (white of frequency), S plot

goes down as l/f*.

4. Flicker PM (flicker of phase), S^ plot

goes down as 1/f

.

5. White PM (white of phase), S. plot is flat.

Power law noise processes are characterized by

their functional dependence on Fourier frequency.

Equation 8.7 relates S^(f) to S (f). the spectral

9 y density of frequency fluctuations. Translation of

S (f) to time-domain data a (x) for the five model

noise processes is covered later in section XI. The spectral density plot of a typical oscillator's output usually is a combination of different power- law noise processes. It is very

useful and meaningful to categorize the noise processes. The first job in evaluating a spectral

density plot is to determine which type of noise

exists for a particular range of Fourier frequencies. It is possible to have all five noise

processes being generated from a single oscillator,

but, in general, only two or three noise processes

are dominant. Figure 9.1 is a graph of S (f)

showing the five noise processes on a log-log

scale. Figure 9.2 shows the spectral density of

phase fluctuations for a typical high-quality

oscillator.

SPECTRAL DENSITY OF PHASE

s,l»)««

M Vq » b MHz

to V

70 \

to

lie

IM ^^riiem ph IM

170 ^^•(ITt « ^^^p

10» lo' 10^ 10^ rouiici FicgucRCT (f| FIGURE 9.1

io'»i

SPECTRAL DENSm OF PHASE

Uiri<0

- Vq » 5 MHz \ ^-t{.lCU» fH ^

- >s,^ ^^-fLicui m ^^^^

^-•HiTE m

10" 10' 10* lo-" FOUIIEI FtEQUEIICT (f)

FIGURE 9.2

25

TN-38
X. PITFALLS IN DIGITIZING THE DATA

The advent and prolific use of digital com- puters has changed the manner in which processing

of analog signals takes place if a computer is used. This section addresses the most common problems In such analyses.

10.1 Oiscrete-Continuous Processes

Digital processing implies that data must be

presented to a computer or other processor as an array of numbers whether in a batch or in a time

series. If the data are not already in this form

(it usually is not when considering frequency

stability measurements), then it is necessary to transform to this format by digitizing. Usually,

the signal available for analysis is a voltage

which varies with frequency or phase difference

between two oscillators.

10.2 Digitizing the Data Digitizing the data is the process of conver- ting a continuous waveform into discrete numbers.

The process is completed in real time using an analog-to-digital converter (ADC). Three considerations in the ADC are of importance here:

1. Conversion time

2. Resolution (quantization uncertainty)

3. Linearity

An ADC "looks at" an incoming waveform at equispaced intervals of time T. Ideally, the output

of the ADC is the waveform (denoted by y(t))

multiplied by a series of infinitely narrow sampling intervals of unit height as in figure

10.1. We have at t = T

y-^W = y(t)6(t-T) « y(T)6(t-T) (10.1)

where 6(t-T) is a delta function. If y(t) is continuous at t = nT and n = 0, ±1, ±2,. . . , then

y^Ct) = £ y("T) 6 (t-nT)

i = integer

The delta function respresentation of a sampled

waveform eq (10.2) is useful when a subsequent

continuous integration is performed using it.

FIGURE 10.1

In ADC's, the input signal is sampled during an aperture time and held for conversion to a digital

number, usually binary. Sampling and processing

takes time which is specified as the conversion

time . This is the total time required for a complete measurement at one sample to achieve a given level of accuracy. If yp(t) is the ideal

discrete-time representation of continuous process

y(t), then the ADC output denoted by yUt) is:

y'z = y£-d ^ '^^t^ (10.3)

(10.2)

where "d" is the conversion time and c • -t the

at accuracy tolerance at "d" as a function of rate-ofchange in y(t). In general a trade-off exists

between d and e. For example, for a commonly

available, high-quality 10-bit ADC, a conversion

time of d = 10 ps yields a maximum error of 3%. Whereas given a 30 |js conversion time, we can obtain Q.1X maximum error. The error due to conversion time "d" is many

times negligible since processing in digital

filters and spectrum analysis takes place after

the converter. Conversion time delay can be of critical concern, however, where real-time proces- sing at speeds of the order of "d" become important

26

TN-39
such as in digital servo loops where corrections

are needed for fast changing errors.

A portion of the conversion-time error is a function of the rate of change ^ of the process

if the sample- and- ho Id portion of the AOC relies

on the charging of a capacitor during an aperture

time. This is true because the charge cycle will

have a finite time-constant and because of aperture

time uncertainty. For example, if the time-con- stant is 0.1 ns (given by say a 0.1 Q source resistance charging a 0.001 \if(i capacitor), then

a O.IX nominal error will exist for slope r^ =

IV/ps due to charging. With good design, this error can be reduced. The sampling circuit (before charge) is usually the dominant source of error and logic gate-delay jitter creates an aperture time uncertainty. The jitter typically

is between 2-5 ns which means an applied signal

slewing at, say, 1 V/ps produces an uncertainty

of 2-5 mV. Since c^ is directly proportional

to signal slewing rate, it can be anticipated that

high-level, high-frequency components of y(t) will

have the greatest error in conversion. For typical

ADC's, less than 0.1% error can be achieved by

holding ^ to less than 0.2 V/(js.

The continuous process y(t) is partitioned

into 2 discrete ranges for n-bit conversion. All analog values within a given range are represented

by the same digital code, usually assigned to the

nominal midrange value. There is, therefore, an inherent quantization uncertainty of ± ^ leastsignificant bit (LSB), in addition to other conversion errors. For example, a 10-bit ADC has a total

of 1024 discrete ranges with a lowest order bit

then representing about O.IX of full scale and

quantization uncertainty of ± 0.05%. We define the dynamic range of a digital

system as the ratio between the maximum allowable

value of the process (prior to any overflow condition) and the minimum discernable value. The

dynamic range when digitizing the data is set by

the quantizing uncertainty, or resolution, and

is the ratio of 2" to h LSB. (If additive noise

makes coding ambiguous to the h LSB level, then

the dynamic range is the ratio of 2 to the noise

uncertainty, but this is usually not the case.)

For example, the dynamic range of a 10-bit system

is 2" = 1024 to H, or 2048 to 1. Expressed in dB's, this is

20 log 2048 - 66.2 dS

if referring to a voltage-to-code converter.

The converter linearity specifies the degree

to which the voltage-to-code transfer approximates

a straight line. The nonlinearity is the deviation

from a straight line drawn between the end points

(all zeros to all ones code). It is usually not acceptable to have nonlinearity greater than H LSB which means that the sum of the positive errors or the sum of the negative errors of the individual

bits must not exceed h LSB (or ± h LSB). The

linearity specification used in this context

includes all effects such as temperature errors under expected operating temperature extremes and

power supply sensitivity errors under expected

operating supply variations.

10.3 Aliasing

Figure 10.1 illustrates equi spaced sampling

of continuous process y(t). It is important to have a sufficient number of samples/second to prop- erly describe information in the high frequencies.

On the other hand, sampling at too high a rate may unnecessarily increase the processing labor. As we reduce the rate, we see that sample values

could represent low or high frequencies in y(t).

This property is called aliasing and constitutes a source of error similiar to "imaging" which occurs

in analog frequency mixing schemes (i.e., in the

multiplication of two different signals).

If the time between samples (k) is T seconds,

then the sampling rate is w samples per second.

' 1 Then useful data in y(t) will be from to |y Hz and frequencies higher than ^ Hz will be folded

into the lower range from to ^ Hz and confused

with data in this lower range. The cutoff fre- quency is then given by

u-h (10.4)

and is sometimes called the "Nyquist frequency."

27

TN-40
We can use the convolution theorem to simply

illustrate the existence of aliases. This theorem

states that multiplication in the time domain

corresponds to convolution in the frequency domain,

and the time domain and frequency domain represen- tations are Fourier transform pairs.' The

Fourier transform of y(t) in figure 10.1(a) is denoted by Y(f); thus:

Y(f) convolved with Af is denoted by Y(f)*

A(f) and is shown in figure 10.2(c). We see that

the transform Y(f) is repeated with origins at

J.

f = Conversely, high frequency data with infor-

_ n mation around f = = will fold into the data around

the origin between -f and -t-f . In the computation

of power spectra, we encounter errors as shown in figure (10.3).

and

Y(f) = / y(t)e'J'^^^dt

y(t)

7

(10.5)

Y(f)eJ'^^V (10.5)

S(n

J\~-..

j!j> Ufin.*& nasaxMcr

The function Y(f) is depicted in figure 10.2(a).

The Fourier transform of A(t) is shown in figure

10.2(b) and is given by A(f) where applying the

discrete transform yields:

A(f) = i f 6(f - ^), n=-»

recalling that

A(t) = 2 6(t -nT),

n=-o»

from eq (10.2).

iYir>»A(n

(10.7)

(10.8)

iAcn

Cw)

FIGURE 10.3

Aliased power spectra due to folding, (a) True

Spectra, (b) Aliased Spectra. Two pioneers in information theory, Harold

Nyquist and Claude Shannon, developed design

criteria for discrete-continuous processing systems. Given a specified accuracy, we can convey

time-domain process y(t) through a finite bandwidth whose upper limit fj^ is the highest signifi- cant spectral component of y(t). For discretecontinuous process yi,(t), ideally the input signal

spectrum should not extend beyond f , or

'uiU (10.9)

FIGURE 10.2

where f is given by eq (10.4). Equation (10.9)

is refered to as the "Shannon limit."

In practice, there is never a case in which

there is absolutely no signal or noise component

above fj^. Filters are used before the ADC in order to suppress components above f^. which fold

into the lower bandwidth of interest. This so- called anti-aliasing filter usually must be quite

sophisticated in order to have low ripple in the

passband, constant phase delay in the passband,

and steep rolloff characteristics. In examining

the rolloff requirements of the anti-aliasing

filter, we can apply a fundamental filter property

that the output spectrum is equal to the input

28

TN-41
spectrum multiplied by the square of the frequency

response function; that is,

S(f) [H(f)]2 = S(f) out

(10.10)

The filter response must be flat to f^. and at

n tenuate aliased noise components at v ± f =

2nf ± f. In digitizing the data, the observed

spectra will be the sum of the baseband spectrum

(to f|^) and all spectra which are folded into the

baseband spectrum

S(f) = Sg(f) * S.i (2fj - f) S^a {2f^ * f)

observed

* S.2 (4fg - f )....* S. (2 i (fj * I f))

(10.11)

M = S^(f) ^ S. (2 i (f, * f))

° i=-M ^ *

where M is an appropriate finite limit. For a given rejection at an upper frequency,

clearly the cutoff frequency f for the antialiasing filter should be as low as possible to relax the rolloff requirements. Recall that an nth order low-pass filter has frequency response

function

H(f) =

* Kir

and output spectrum

S(f) = out

S(f)

(10.12)

(10.13)

and after sampling, we have (applying eq (10.11))

S(f)

observed 2n i=-M 2 i (f, + if)^"

l-(^\ 1 * 1—

11)

(10.14)

If f is chosen to be higher than f^, then the

first term (baseband spectrum) is negligibly

affected by the filter, which is our hope. It is

the second term (the sum of the folded in spectra)

which causes an error.

As an example of the rolloff requirement,

consider the measurement of noise process n(t) at

f = 400 Hz in a 1 Hz bandwidth on a digital spec- trum analyzer. Suppose n(t) is white; that is,

^n^^) = ^

k^ = constant

(10.15)

Suppose further that we wish to only measure the

noise from 10 Hz to 1 kHz; thus ^^^ = 1 kHz. Let

us assume a sampling frequency of f = 2f|^ or

2 kHz. If we impose a 1 dB error limit in

S , , and have 60 dB of dynamic range, then we can tolerate an error limit of 10-^ due to aliasing

effects in this measurement, and the second term in eq (10.14) must be reduced to this level. We can choose f = 1.5kHz and obtain

S(f)

observed

= k. Mz

i=-M 1+

2 i (f. if) 2"

(10.16)

The term in the series which contributes most is at i = -1, the nearest fold-in. The denominator

must be 10* or more to realize the allowable error

limit and at n > 8 this condition is met. The next most contributing term is i = +1 at which the error is < 10-^ for n = 8, a negligible contribution. The error increases as f increases for a fixed n because the nearest fold-in (i = -1) is coming down in frequency (note fig. 10.2(c)) and

power there is filtered less by the anti-aliasing

filter. Let us look at the worst case (f = IkHz) to determine a design criteria for this example. At f = 1 kHz, we must have n > 10. Thus the requirement in this example is for a 10-pole low-pass filter (50 dB/octave rolloff).

10.4 Some History of Spectrum Analysis Leading to the Fast Fourier Transform

Newton in his Principia (1587) documented the

first mathematical treatment of wave motion a1-

29

TN-42
though the concept of harmonics in nature was pointed out by Pythagoras, Kepler, and Galileo.

However, it was the work of Joseph Fourier in 1807

which showed that almost any function of a real variable could be represented as the sum of sines

and cosines. The theory was rigorously treated in

a document in 1822.

In using Fourier's technique, the periodic

nature of a process or signal is analyzed. Fourier

analysis assumes we can apply fixed amplitudes,

frequencies, and phases to the signal.

In the early 1900' s two relatively independent

developments took place: (1) radio electronics

and electric power hardware were fast growing

technologies; and (2) statistical analysis of events or processes which were not periodic became

increasingly understood. The radio engineer

explored signal and noise properties of a voltage

or current into a load by means of the spectrum

analyzer and measurement of the power spectrum. On the other hand, statisticians explored deterministic and stochastic properties of a process by means of the variance and self-correlation pro- perties of the process at different times. Wiener

(1930) showed that the variance spectrum (i.e.,

the breakdown of the variance with Fourier frequency) was the Fourier transform of the autocor- relation function of the process. He also theorized that the variance spectrum was the same as the power spectrum normalized to unit area. Tukey

(1949) advocated the use of the variance spectrum

in the statistical treatment of all processes

because (1) it is more easily interpreted than

correlation-type functions; and (2) it fortuitously

is readily measureable by the radio engineer.

The 1950 's saw rigorous application of statistics to communication theory. Parallel to this was the rapid advancement of digital computer

hardware. Blackman and Tukey (1959) and Welch

(1961) elaborated on other useful methods of

deriving an estimate for the variance spectrum by

taking the ensemble time-average sampled, discrete

line spectra. The approach assumes the random

process is ergodic. Some digital approaches

estimate the variance spectrum using Wiener's

theorem if correlation-type functions are useful

in the analysis, but in general the time-averaged,

sample spectrum is the approach taken since its

implementation is direct and straightforward. Most always, ergodicity can be assumed. °''

The variance of process y(t) is related to the total power spectrum by

a2[y(t)] = ; S (f) df.

-OB J

Since

-^cy(t)] = ji; It / y2(t) dt

-T

(10.17)

(10.18)

we see that if y(t) is a voltage or current into a 1-ohm load, then the mean power of y(t) is the

integral of S (f) with respect to frequency over

the entire range of frequencies (-«,»). S (f) is,

therefore, the power spectrum of process y(t).

The power spectrum curve shows how the variance is distributed with frequency and should be expressed

in units of watts per unit of frequency, or volts

squared per unit of frequency when the load is not considered.

Direct estimation of power spectra has been

carried out for many years through the use of

analog instruments. These have variously been

referred to as sweep spectrum analyzers, harmonic

analyzers, filter banks, and wave analyzers.

These devices make use of the fact that the spec- trum of the output of a linear system (analog

filter) is the spectrum of the input multiplied by

the square of the system's frequency response

function (real part of the transfer characteristic). Note eq (10.10). If y(t) has spectrum

S (f) feeding a filter with frequency response

y

function H(f), then its output is

S(f)

filtered = [H(f)]2 S ff) (10.19)

If H(f) is rectangular in shape with width Af,

then we can measure the contribution to the total

power spectrum due to S (f ± x-).

The development of the fast Fourier transform

(FFT) in 1965 made digital methods of spectrum

30

TN-43
estimation increasingly attractive. Today the

choice between digital or analog nethods depends more on the objectives of the analysis rather than

on technical limitations. However, many aspects

of digital spectrum analysis are not well known by

the casual user in the laboratory while the analog

analysis methods and their limitations are understood to a greater extent. Digital spectrum analysis is realized using

the discrete Fourier transform (DFT), a modified

version of the continuous transform depicted in eqs (10.5) and (10.6). By saopling the input

waveform y(t) at discrete intervals of time t. = At representing the sampled waveform by eq (10.2)

and integrating eq (10.5) yields

Y(f) = Z y(2t)e'J^^^^ (10.20)

Equation (10.20) is a Fourier series expansion. Because f(t) is specified as being bandlimited,

the Fourier transform as calculated by eq (10.20)

is as accurate as eq (10.5); however, it cannot extend beyond the Nyquist frequency, eq (10.4).

In practice we cannot compute the Fourier

transform to an infinite extent, and we are re- stricted to some observation time T consisting of

n^t intervals. This produces a spectrum which is not continuous in f but rather is computed with

resolution Af where

^"^t- I

(10.21)

With this change, we get the discrete finite

transform

N-1

Y(mAf) = E y,(t)e"J2^^"*

n=0 * (10.22)

The DFT computes a sampled Fourier series,

and eq (10.22) assumes that the function y(t)

repeats itself with period T. Y(mAf) is called

the "line spectrum." A comparison of the OFT with

the continuous Fourier transform is shown later in part 10.7.

The fast Fourier transform (FFT) is an algorithm which efficiently computes the line spectrum

by reducing the number of adds and multiplies

involved in eq (10.22). If we choose TMt to equal a rational power of 2, then a symmetric

matrix can be derived through which y»(t) passes

and quickly yields Y(m6f). An N-point transformation by the direct method requires a processing

time proportional to N^ whereas the FFT requires a time proportional to N logj N. The approximate

ratio of FFT to direct computing time is given by

N logj N log, N -w N = 1

N (10.23)

where N = 2^. For example, if H = Z^° , the FFT requires less than 1/100 of the normal processing

time. We must calculate both the magnitude and

phase of a frequency in the line spectrum, i.e.,

the real and imaginary part at the given frequency.

N points in the time domain allow N/2 complex

quantities in the frequency domain.

The power spectrum of y(t) is computed by

squaring the real and imaginary components, adding

the two together and dividing by the total time T. We have

S^(^,) = R[V(^f)?^> I[Y(mAf)]^ (,0 24)

This quantity is the sampled power spectrum

and again assumes periodicity in process y(t) with

total period T.^0

10.5 Leakage

Sampled digital spectrum analysis always

involves transforming a finite block of data. Continuous process y(t) is "looked at" for T time

through a data window which can functionally be described by

y'(t) = w(t)-y(t) (10.25)

where w(t) is the time domain window. The timediscrete counterpart to eq (10.25) is

y;(t) = w^(t)-y^(t) (10.26)

31

TN-44
and ^^At) is now the sampled version of w(t)

derived similaply to eq (10.2). Equation (10.26)

is equivalent to convolution in the frequency

domain, or

Y'(mAf) = W(nuif)*Y(m£kf) (10.27)

Y'(m^f) is called the "modified" line spectrum due

to convolution of the original line spectrum with

the Fourier transform of the time-domain window

function.

Suppose the window function is rectangular,

and

w^(t) = 1, = 0,

:!< t<I

2 - |,|- 2 ' H2I

(10.28)

This window is shown in figure 10.4(a). The

Fourier transform of this window is

W(nuif) = T Sinn niAf NT

TtmAfNT (10.29)

The transform process (eq 10.22) treats the

sample signal as if it were periodically extended.

Discontinuities usually occur at the ends of the

window function in the extended version of the

sampled waveform as in figure 10.5(c). Sample

spectra thus represent a periodically extended

sampled waveform, complete with discontinuites at

its ends, rather than the original waveform.

(a) vwwwwwvwww

(b) ww^

*->,

(c) T H

and is shown in figure 10.4(b). If y(t) is a sine wave, we convolve the spectrum of the sinusoid, a delta function, with W(inAf).

yiii^

rn

(.) r/2

-»>-i

v(»dn

/t

(W

T".. r=r^f

FIGURE 10.4

FIGURE 10.5

Spurious components appear near the sinusoid

spectrum and this is referred to as "leakage."

Leakage results from discontinuites in the periodically extended sample waveform.

Leakage cannot be eliminated entirely, but one can choose an appropriate window function w(t)

in order to minimize its effect. This is usually

done at the expense of resolution in the frequency

domain. An optimum window for most cases is the

Hanning window given by:

w(t) = [i - i cos i^^)T (10.30)

for < t < T and "a" designates the number of

times the window is implemented. Figure 10.6(a)

shows the window function and 10.6(b) shows the

Hanning line shape in the frequency domain for

various numbers of "Hanns." Note that this window

32

TN-45
UMNM&VIO^^

VO.[^.i«.^]

HMMINC UIC SMPR

Ok^ >tf 4^ AAt

FIGURE 10.5

eliminates discontinuities due to the ends of

sample length T. Each time the Manning window is applied, the

sidelobes in the transform are attenuated by }2

dB/octave, and the main lobe is widened by 2Af. The amplitude uncertainty of an arbitrary sine wave input is reduced as we increase the number of Hanns; however, we trade off resolution in fre- quency.The effective noise bandwidth indicates the

departure of the filter response away from a true rectangularly shaped filtered response (frequency

domain). Table lO-I lists equivalent noise bandwidth corrections for up to three applications of

the Manning window. ^^

Equivalent Noise

Number of Hanns Bandwidth

1 1.5 Af

2 1.92 Af

3

TABLE 10.

2. 31 Af

I

10.6 Picket-Fence Effect

The effect of leakage discussed in the pre- vious section gives rise to a sidelobe type re- sponse that can be tailored according to the

time-window function through which the analyzed

signal passes as a block to be transformed to the

frequency domain. Using the Manning window diminishes the amplitudes of the sidelobes, however, it increases the effective bandwidth of the passband

around the center frequency. This is because the

effective time-domain window length is shorter

than a perfect rectangular window. Directly

related to the leakage (or sidelobe) effect is one called the "picket- fence" effect. This is because

the sidelobes themselves resemble a frequency

response which has geometry much like a picket

fence.

The existence of both sidelobe leakage and

the resultant picket- fence effect are an artifact

of the way in which the FFT analysis is performed.

Frequency-domain analysis using analog filters

involves a continuous signal in and a continuous

signal out. On the other hand, FFT analysis

involves a continuous signal in, but the transform

to the frequency domain is performed on blocks of

data. In order to get discrete frequency information from a block, the assumption is made that the

block represents one period of a periodic signal.

The picket-fence effect is a direct consequence of

this assumption. For example, consider a sinewave

signal which is transformed from a time-varying

voltage to a frequency-domain representation

through an FFT. The block of data to be transformed will be length, T, in time. Let's say that the

block, T, represents only 4% cycles of the input

sinewave as in figure 10.5. Artificial sidebands

will be created in the transform to the frequency

domain, whose frequency spacing equals y, or the

reciprocal of the block length. This represents

a worst-case condition for sidelobe generation

and creates a large number of spurious discrete

frequency components as shown in figure 10.7(b).

If, on the other hand, one changes the block

time, T, so that the representation is an integral

number of cycles of the input sinewave, then the

transform will not contain sidelobe leakage compo33

TN-46
U)

(ti

SvuDsenw

."> <• »« A A i*^,

.' \ ; : I ; 'I ! fi " .' ; •, .". .<• , ^M/i/i/iiiin I M ;; ;: I; • 'J ; • ; J s L^iii^'^V v • t ! !• i < •' UL

'?</ il

FIGURE 10.7 nents and the artificial sideband frequency components disappear. In practice, when looking at complex input signals, the block time, T, is not synchronous with any component of the transformed

part of the signal. As a result, discrete frequency components in the frequency domain have

associated with them sidebands which come and go depending on the phase of the time window, T, relative to the sine components of the incoming

signal. The effect is much like looking through a picket fence at the sidebands. 12

An analogy to the sidelobe leakage and picketfence effect is to record the incoming time- varying

signal on a tape loop, which has a length of time,

T. The loop of tape then repeats itself with a period of T. This repeating signal is then coupled

to a scanning or filter-type spectrum analyzer.

The phase discontinuity between the end of one passage of the loop and the beginning of the loop

on itself again represents a phase-modulation

component, which gives rise to artificial sidebands

in the spectrum analysis. A word of caution —

this is not what actually happens in a FFT analyzer

(i.e., there is no recirculating memory). However,

the Fourier transform treats the incoming block as if this were happening.

10. 7 Time Domain-Frequency Domain Transforms

A. Integral transform

Figure 10.8 shows the well-known integral

transform, which transforms a continuous timedomain signal extending over all time into a

N1B3Ul_1QMSOR»\

,(0./3Y(nc^ 41

1" /N tk:

Yin Z'

\. wooer

J l-m

FIGURE 10.8

continuous frequency-domain signal extending over

all frequency. This is the ideal transform. In practice, however, one deals with finite times and

bandwidths. The integral transform then, at best,

is an estimate of the transform and is so for only

short, well-behaved signals. That is, the signal goes to zero at infinite time and at infinitely

high frequency.

B. Fourier series

The Fourier-series transform assumes periodicity in the time-domain signal for all time.

Only one period of the signal (for time T) is required for this kind of transform. The Fourier

series treats the incoming signal as periodic with

period, T, and continuous. The transformed spec- trum is then discrete with infinite harmonic

components with frequency spacing of y. This is shown in figure 10.9.

FIGURE 10.9

34

TN-47
XI. TRANSLATION FROM FREQUENCY DOMAIN STABILITY

MEASUREMENT TO TIME DOMAIN STABILITY MEASURE- MENT AND VICE- VERSA.

^HFLEDTUOONS •n^

Ylfl.l ^i.V-'^Ji

i(g.tjf;:^w

I***., -

FIGURE 10. 10 Figure 10.10 shows the transform from a sam- pled time-domain signal to the frequency domain. Note that in the frequency domain, ttw result is repetitive in frequency. This effect, commonly

called aliasing, is discussed -earlier in section

10.3. Figure 10.9 and figure 10.10 show the sym- metry between the time- and frequency-donain transforms of discrete lines.

U-l Procedure

Knowing how to measure S.(f) or S (f) for a y pair of oscillators, let us see how to translate

the power-law noise process to a plot of a 2(x).

First, convert the spectrum data to S (f), the

spectral density of frequency fluctuations (see

sections III and VIII). There are two quantities

wtiictj completely specify S (f) for a particular

power- law noise process: (1) the slope on a log-log plot for a given range of f and (2) the

amplitude. The slope we shall denote by "a"; therefore f^ is the straight line (on log-log

scale) which relates S (f) to f. The amplitude

will be denoted "h "; it is simply the coefficient

of f for a range of f. When we examine a plot of

spectral density of frequency fluctuations, we are looking at a representation of the addition of all

ttie power- law processes (see sec. IX). We have

C. Discrete fourier transform

Finally, we have the sampled, periodic

(assumed) time-domain signal which is transformed

to a discrete and repetitive (aliased) frequencydomain representation. This is sfto*<n in figure

10.11

Sy(f) = E h f^ (11.1)

In section IX, five power- law noise processes were outlined witti respect to S^(f). These five are the common ones encountered with precision

oscillators. Equation (8.7) relates these noise

processes to S (f). One obtains

>SC«ir TOUBCX-mMSCKM

II ll'l". ,1*1

1 1 1 1 1 1 1 1 1 '. ^ —

III!

I II Ml

lilt

' " I " " " -? 'V^

FIGURE 10.11

r

«Baocr

I til

i< >i nil

liLii.

Va

^

with respect to

1. Random Walk FM (f-^) . . . a = -2

2. Flicker FM (f-M . . . a = -1

3. White FH (1) . ..0=0

4. Flicker ^ (f) . . . o = 1

5. White 4iM (f2) . . . a = 2 slope on

log- log

paper

35

Table U. 1 is a list of coefficients for

translation from a ^(x) to S (f) and from S.(f) to y y <p

TN-48
a 2(t). In the table, the left column is the

designator for the power-law process. Using the

middle column, we can solve fo by computing the coefficientmiddle column, we can solve for the value of S (f)

a" and using the

measured time domain data a ^(t). The rightmost

column yields a solution for a 2(t) given frequency

domain data S.(f) and a calculation of the appro- priate "b" coefficient.

EXAMPLE : In the phase spectral density plot of figure

11.1, there are two power- law noise processes for

oscillators being compared at 1 MHz. For region

1, we see that when f increases by one decade

(that is, from 10 Hz to 100 Hz), S^(f) goes down

by three decades (that is, from 10- ^^ to 10-^*).

Thus, S (f) goes as l/f = f-^. For region 1, we s/n = h/ Sy(n = a aj(x)

[1

IJ(T) = b S^(f)

2 (2n)2 t2 f2 3^h

(2n)2 t2 f

3^

(white phase)

1 .038 * 3 ln(w^x)]f

(flicker noise) 1.038 + 3 In(uj^T)

2 X

1

(2n)2 t2 vJ

f2 (white frequency)

-1

2 X u2

2 In (2) • f3 (flicker frequency)

-2

2 ln(2) • f

6

"^0

(27T)2 X f* (random walk frequency) (2n)2 T f2 5

^}i

TABLE 11.1

Conversion table from time domain to frequency domain and from frequency

domain to time domain for common kinds of interger power law spectral densities;

f. (= ui. /2n) is the measurement system bandwidth. Measurement reponse should be wTthin 3 dB from O.C. to f^ (3 dB down high-frequency cutoff is at f.).

v^^ = -^ v^

SFdOL tJBKrnr OFPwasE

identify this noise process as flicker FM. The

rightmost column of table 11.1 relates c;2(x) to

S (f). The row designating flicker frequency

noise yields:

One can pick (arbitrarily) a convenient Fourier

frequency f and determine the corresponding values

of S.(f) given by the plot of figure 11.1. Say,

f = 10, thus S^(IO) = 10-11

given Vq ~ ^ '^^» *** obtain:

Solving for o2(x),

o2(x) = 1.39 X 10-20

FIGURE 11.1

therefore, a (x) = 1.18 x 10-i°. For region 2, we have white PM. The relationship between a^{z) and

36

TN-49
S^(f) for white PM is:

3f. '^V'^

= T5??-TrT-.s,(f)

Again, we choose a Fourier frequency, say f = 100,

and see that S (100) = 10-^*. Assuming f^^ = 10*

Hz, we thus obtain:

ajd) = 7.59 X 10-2*. ^2

therefore,

Oyd) = 2.75 X 10-" i.

The resultant time domain characterization is shown in figure 11.2.

Cf> r

l.» I 10'' ——"-^ <

2.TI.X 0"°t

Ptfi rrl

FIGURE 11.2

^

"«*>

The translation of S (f) of figure 11.1 yields this

a^d) plot. ^

df^S

.^*

'^v

TW

Sf<«)

B-ff .

tf-" .

\ WHDOt« SAICOT<

. \'^ ^^" 6 1> X \0'^^

\ nicisi Nnwm

\. r

u^ - If * iO-» >y <wMTC NOBK m ^\ ^'.?.<.' 10-%

- N. t« \

, , ,

I0-* O^ O'' O'i «J^ tf< 1 O

FIGURE 11.4

Figures 11.3 and 11.4 show plots of timedomain stability and a translation to frequency

domain. Since table 11.1 has the coefficients

which connect both the frequency and time domains,

it may be used for translation to and from either

domain.

XII. CAUSES OF NOISE PROPERTIES IN A SIGNAL SOURCE

12. 1 Power-law Noise Processes Section IX pointed out the five commonly used

power-law models of noise. With respect to S^(f), one can estimate a staight line slope (on a log-log

scale) which corresponds to a particular noise

type. This is shown in figure 12.1 (also fig. 9.1).

SPECTRAL DENSm OF PHASE

• UIOOII MM.( FN

\Wt

Hicul i>n

rOURIEK FtCauCKCT If)

FIGURE 11.3 FIGURE 12.1

37

TN-50
We can make the following general remarks about

power- law noise processes:

1. Random walk FM (1/f*) noise is difficult

to measure since it is usually very close to the carrier. Random walk FM usually relates to the OSCILLATOR'S

PHYSICAL ENVIRONMENT. If random walk FM

is a predominant feature of the spectral

density plot then MECHANICAL SHOCK,

VIBRATION, TEMPERATURE, or other environmental effects may be causing "random"

shifts in the carrier frequency.

2. Flicker FM (1/f^) is a noise whose

physical cause is usually not fully

understood but may typically be related

to the PHYSICAL RESONANCE MECHANISM OF AN ACTIVE OSCILLATOR or the DESIGN OR CHOICE OF PARTS USED FOR THE ELECTRONICS,

or ENVIRONMENTAL PROPERTIES. Flicker FM

is common in high-quality oscillators,

but may be masked by white FM (1/f^) or flicker PM (1/f) in lower-quality oscillators.

3. White FM (1/f^) noise is a common type

found in PASSIVE-RESONATOR FREQUENCY

STANDARDS. These contain a slave oscillator, often quartz, which is locked to

a resonance feature of another device

which behaves much like a h1gh-Q filter.

Cesium and rubidium standards have white

FM noise characteristics.

4. Flicker PM (1/f) noise may relate to a physical resonance mechanism in an oscillator, but it usually is added by

NOISY ELECTRONICS. This type of noise

is common, even in the highest quality

oscillators, because in order to bring

the signal amplitude up to a usable

level, amplifiers are used after the

signal source. Flicker PM noise may be

Introduced in these stages. It may also

be introduced in a frequency multiplier.

Flicker PM can be reduced with good

low-noise amplifier design (e.g., using

rf negative feedback) and hand-selecting

transistors and other electronic com- ponents.

White PM (f ) noise is broadband phase

noise and has little to do with the resonance mechanism. It is probably

produced by similar phenomena as flicker

PM (1/f) noise. STAGES OF AMPLIFICATION

are usually responsible for white PM noise. This noise can be kept at a very

low value with good amplifier design,

hand-selected components, the addition

of narrowband filtering at the output,

or increasing, if feasible, the power of the primary frequency source. ^-^

12.2 Other types of noise

A commonly encountered type of noise from a signal source or measurement apparatus is the

presence of 60 Hz A.C. line noise. Shown in figure 12.2 is a constant white PM noise source with 60 Hz, 120 Hz and 180 Hz components added.

This kind of noise is usually caused by AC power

getting into the measurement system or the. source under test. In the plot of S (f), one observes

discrete line spectra. Although S.(f) is a measure

of spectral density, one can interpret the line

spectra with no loss of generality, although one usually does not refer to spectral densities when

characterizing discrete lines. Figure 12.3 is the

time domain representation of the same white phase

modulation level with 60 Hz noise. Note that the

amplitude of a (t) varies up and down depending on sampling time. This is because in the time domain

the sensitivity to a periodic wave varies directly

as the sampling interval. This effect (which is an alias effect) is a very powerful tool for

filtering out a periodic wave imposed on a signal source. By sampling in the time domain at integer

periods, one is virtually insensitive to the

periodic (discrete line) term.

* See Appendix Note # 7 38

TN-51
JB

-fo «'•

•tx> «"-

•I10 «"-

«"-.

«'*.. -w

•1 10"

-150 ,0^..

1(jO ,o*--

t < I -

FIGURE 12.2

FM behavior masks the white PM (with the super- imposed vibration characteristic) for long aver- aging times.

B.

fO »'•• no

110 «"•

120 fiO

Vfl

,6"..

•NO ib": 150 ««•

ftO rb*--

naaLSNcr dctvun VMTE #>^ vrtH venanoK, ^^susnc sxynvrrr

FIGURE 12.4

Oit) f

TTCOOntM

ISK 10SE£ 1CX>3S XX30BC tZXX7s sttnE^Mc,TC»^

FIGURE 12.3 For example, diurnal variations in data due to day

to day temperature, pressure, and other environmental effects can be eliminated by sampling the

data once per day. This approach is useful for

data with only one periodic term. Figure 12.4 shows the kind of plot one might

see of S^(f) with vibration and acoustic sen- sitivity in the signal source with the device

under vibration. Figure 12.5 shows the translation to the time domain of this effect. Also

noted in figure 12.4 is a (typical) flicker FM behavior in the low frequency region. In the

translation to time domain (fig. 12.5), the flicker

"i

w

10 -

«'*.. V

>-«Tt *ri WITH VTBWmOM, Accusnc ssMsrnvTrr

-I- ^SK lac IQssc lOOsK tXOsc iQCCDsc

FIGURE 12.5

Figure 12.6 shows examples of plots of two power law processes (S.(f)) with a change in the

flicker FM level. (Example 1 is identical to the

example given in sec. XI.) Figure 12.7 indi- cates the effect of a lower flicker FM level as translated to the time domain. Note again the

existence of both power law noise processes. However for a given averaging time (or Fourier

frequency) one noise process may dominate over the

other.

39

TN-52
FIGURE 12.5a

XX3«r JOOCXs

atahRr 1 : TVMSWtnON toT»«.bOiM»«

VId*..

nio«»'m, i.««w

\ N

N

W^ \.^ lOiK. tOCH iBnx-n«c,-a*)

FIGURE 12.7a

CCO%. C,OCQfc.

ID 1d^

ICO »».. im Kf-.

t'X) S^ic'^--

itC o^..

r¥D w".. oo t(3».. K*0 KS".

niosR Pt 10 --

(3^^!^ o»-

ic;=i.

Q0V1Pt£ 2:T«*NUAI0H-W'n^tSHflM

TlOSR'ni

I I tao. 10«c lOOv 'OOO*. yapOOK.

FIGURE 12.6b FIGURE 12.7b

s>

•fO ,/..

loo ,,/'.. no «^--

•no S^0.6«..

ftO KJ-^*.. «0 ,6"..

fSO iflft..

•IWJ icf..

>w«i«D«<r "w^- to"" Wi

TUOO PI

aaWFVCi-. TWMKUSncX-WTKbCPUUN

» ) 1 1 t I ^,^ 1^ lOs:, oo». i«ao«- io,ooo»

FIGURE 12.5c FIGURE 12.7c

40

TN-53
-*3 to •

-roo ro*".

-tio fO

»o kT.

-f!0 c"-

-1(^ .o*^-

A».-

n"-- ,W

J&

-TO

-fOO

-f/0

-f20

•fiO -ho

•fW

©

•

3*(i) "^

10^+

10 •

»^*^"^*""r^*s*f"^"'*'"'^

H > t <

FIGURE 12.8

Ljvn_

FIGURE 12.9

NOSEY SVKiE .t jiMPUnCAmON ,

t.

NOST

POORER TKEO(JDCT,f

FIGURE 12.10

Excess device noise from transistors, capa- citors, resistors, and the like can introduce a low frequency noise which has been referred to as "popcorn" noise because of its sonic qualities.

Figure 12.8 shows a plot of S (f) from a signal source having such excess low frequency noise.

Figure 12.9 is the time domain representation.

The rise in amplitude of o for long averaging

times is particularly aggravating. The solution

to this kind of problem if it is introduced by

devices is to carefully grade the devices in the

assembly and testing process. Stages of amplification following a signal source many times rely on local degenerate or overall negative feedback schemes in order to minimize the excess noise from active gain elements

(such as transistors). This is the recommended

design approach. However, phase shift in the negative feedback circuit or poor bandwidth in the

gain elements can result in poor high frequency

noise behavior. Figure 12.10 shows a kind of result one might see as a gradual rise in S (f)

because of insufficient negative feed back at high Fourier frequencies.

Section X discussed aliasing in the frequency

domain. Figure 12.11 shows the resultant measure- ment anomaly due to digital sampling of a poorly

bandlimited (anti-aliased) white noise source. Noise voltage above the sampling frequency f is folded into the analysis region of interest. Note also that the stopband ripple characteristics are folded into the high-frequency portion of the passband. For a given sampling frequency, a compromise exists between increasing the high-

-JO -KO

110

to ••

-120

-fiO -W

V"'

12 10 4-

tcT..

liT- ^\

10 •

HI6W-rqECtENCY NaSE SMCVS UP ^ ANAL-fSlS

FIGURE 12.11

41

TN-54
frequency extent of the analysis band and improving

the anti-aliasing filter's stopband rejection.

Section X has an example of the filter requirements

for a particular case.

XIII. CONCLUSION

This writing highlights major aspects of time-domain and frequency-domain oscillator signal measurements. The contents are patterned after

lectures presented by the authors. The authors

have tried to be general in the treatment of topics, and bibliography is attached for readers

who would like details about specific items.

References

1. J. A. Barnes, Andrew R. Chi, Leonard S. Cutler, Daniel J. Healey, David B. Leeson,

Thomas E. McGunigal, James A. Mullen, Jr., Warren L. Smith, Richard L. Sydnor, Robert F. C. Vessot, and Gemot M. R. Winkler, "Characterization of Frequency Stability," NBS

Tech. Note 394 (1970); Proc. IEEE Trans, on Instrum. and Meas. IM-20 , 105 (1971).

2. D. W. Allan, "The Measurement of Frequency

and Frequency Stability of Precision Oscillators," NBS Tech. Note 669 (1975); Proc. 6th

PTTI Planning Meeting, 109 (1975).

3. D. W. Allan, "Statistics of Atomic Frequency

Standards," Proc. IEEE 54, 221 (1966).

4. Sections V and VI are from notes to be published by J. A. Barnes.

5. S. R. Stein, "An NBS Phase Noise Measurement

System Built for Frequency Domain Measurements Associated with the Global Positioning System, NBSIR 76-846 (1976). See also F. L. Walls

and S. R. Stein, "Servo Technique in Oscillators and Measurement Systems," NBS Tech. Note

692 (1976).

6. C. D. McGillen and G. R. Cooper, Continous and

Discrete Signal and System Analysis , Holt,

Rinehart, and Winston, Inc. (1974).

7. E. 0. Brigham, The Fast Fourier Transform . Prentice-Hall, Inc. , New Jersey (1974).

8. Private communication with Norris Nahman, June 1981.

9. G. M. Jenkins and D. G. Watts, Spectral

Analysis and Its Application , Holden-Day, San Francisco, 1968.

10. P. F. Panter, Modulation, Noise, and Spectral

Analysis , McGraw-Hill, New York, 1965, pp. 137-141.

11. D. Babitch and J. Oliverio, "Phase Noise of Various Oscillators at Very Low Frequencies,

Proc. 28th Annual Symp. on Frequency Control

,

May 1974. U. S. Army Electronics Command,

Ft. Monmouth, NJ.

12. N. Thrane, "The Discrete Fourier Transform

and FFT Analyzers," Bruel and Kjaer Technical

Review, No. 1 (1979).

13. Much of the material presented here is con- tained in NBS Tech. Note 679, "Frequency

Domain Stability Measurements: A Tutorial

Introduction," D. A. Howe (1976).

Bibliography

General References

1. November or December of even-numbered years,

IEEE Transactions on Instrumentation and Measurement (Conference on Precision Electromagnetic Measurements, held every two years).

2. Proceedings of the IEEE . Special issue on frequency stability, vol. 54, February 1966.

3. Proceedings of the IEEE . Special issue on time and frequency, vol. 60, no. 5, May 1972.

4. Proceedings of the IEEE-NASA Symposium on the Definition and Measurement of Short-Term

Frequency Stabi lity at Goddard Space Flight

Center, Greenbelt, MD, November 23-24, 1964,

Prepared by Goddard Space Flight Center (Scientific and Technical Information

Division, National Aeronautics and Space

Administration, Washington, DC, 1965).

Copies available for $1.75 from the US Govern- ment Printing Office, Washington, DC 20402.

5. Proceedings of the Annual Symposium on Frequency Control . Prepared by the US Army

Electronics Command, Fort Monmouth, NJ. Copies from Electronics Industry Association,

2001 Eye Street, N. W. , Washington, DC 20006, starting with 32nd Annual or call (201)

544-4878. Price per copy, $20.00.

6. J. A. Barnes, A. R. Chi, L. S. Cutler, et

al . , "Characterization of Frequency Stability," IEEE Trans, on Intsrum. and Meas. IM-20 . no. 2, pp. 105-120 (May 1971). Pre- pared by the Subcommittee on Frequency Sta- bility of the Institute of Electrical and

Electronics Engineers.

7. Time and Frequency: Theory and Fundamentals , Byron E. Blair, Editor, NBS Monograph 140,

May 1974.

8. P. Kartaschoff, Frequency and Time . London,

England: Academic Press, 1978.

9. G. M. R. Winkler, Timekeeping and Its Appli - cation, Advances in Electonics and Electron

Physics . Vol. 44, New York: Academic Press,

1977.

42

TN-55
10. F. M. Gardner, Phaselock Techniques , New

York: Wiley i Sons. 1956.

11. E. A. Gerber, "Precision Frequency Control

and Selection, A Bibliography," Proc. 33rd

Annual Symposium on Frequency Control, pp. 569-728, 1979. Available from Electronics

Industries Association, 2001 Eye St. NW,

Washington, DC 20006.

12. J. Rutjaan, "Characterization of Phase and

Frequency Instabilities in Precision Fre- quency Sources: Fifteen Years of Progress,"

Proc. IEEE, vol. 66, no. 9, September 1978,

pp. 1048-1075.

Additional Specific References (selected)

1. 0. W. Allan, "Statistics of Atomic Frequency

Standards," Proc. IEEE, vol. 54, pp. 221-

230, February 1966.

2. D. W. Allan, "The Measurement of Frequency

and Frequency Stability of Precision Oscillators," NBS Tech. Note 659, July 1975.

3. J. R. Ashley, C. B. Searles, and F. M. Palka,

"The Measurement of Oscillator Noise at Microwave Frequencies," IEEE Trans, on Micro- wave Theory and Techniques MTT-16 , pp. 753-

760, September 1963.

4. E. J. Baghdady, R. 0. Lincoln, and B. 0. Nelin, "Short-Term Frequency Stability:

Theory, Measurement, and Status," Proc. IEEE,

vol. 53, pp. 704-722. 2110-2111, 1965.

5. J. A. Barnes, "Models for the Interpretation

of Frequency Stability Measurements," NBS Technical Note 683, August 1976.

6. J. A. Barnes, " A Review of Methods of Analyzing Frequency Stability," Proc. 9th Annual

Precise Time and Time Interval (PTTI) Applications and Planning Meeting, Nov. 1977, pp. 61-84.

7. J. A. Barnes, "Atomic Timekeeping and the

Statistics of Precision Signal Generators,"

Proc. IEEE, vol. 54, pp. 207-220, February

1966.

8. J. A. Barnes, "Tables of Bias Functions, Bi and Bj, for Variances Based on Finite Samples

of Processes with Power Law Spectral Densities," NBS Tech. Note 375, January 1969.

9. J. A. Barnes and 0. W. Allan, "A Statistical

Model of Flicker Noise," Proc. IEEE, vol. 43,

pp. 176-178, February 1966.

10. J. A. Barnes and R. C. Moclcler, "The Power Spectrum and its Importance in Precise Fre- quency Measurements," IRE Trans, on Instrumen- tation 1^9, pp. 149-155, September 1960.

11. W. R. Bennett, "Methods of Solving Noise

Problems," Proc. IRE, May 1956, pp. 609-637.

12. W. R. Bennett, Electrical Noise . New York: McGraw-Hill. 1960.

13. C. Bingham. M. D. Godfrey, and J. W. Tukey,

"Modern Techniques of Power Spectrum Estimation," IEEE Trans. AU, vol. 15, pp. 56-66, June 1967.

14. R. B. Blackman and J. W. Tukey, The Measure- ment of Power Spectra . New York: Dover.

1358.

15. E. 0. Brigham and R. E. Morrow. "The Fast Fourier Transform," IEEE Spectrum, vol. 4,

no. 12, pp. 63-70, December 1967.

16. J. C. Burgess, "On Digital Spectrum Analysis

of Periodic Signals," J. Acoust. Soc. Am. , vol. 58, no. 3, September 1975, pp. 556-557.

17. R. A. Campbell, "Stability Measurement Techniques in the Frequency Domain," Proc. IEEE- NASA Symposium on the Definition and Measure- ment of Short-Term Frequency Stability, NASA

SP-80, pp. 231-235, 1965.

18. L. S. Cutler and C. L. Searle, "Some Aspects

of the Theory and Measurement of Frequency

Fluctuations in Frequency Standards," Proc. IEEE, vol. 54. pp. 136-154, February 1966.

19. W. A. Edson, "Noise in Oscillators," Proc. IRE. vol. 48. pp. 1454-1466, August 1960.

20. C. H. Grauling, Jr. and D. J. Healey, III,

"Instrumentation for Measurement of the

Short-Term Frequency Stability of Microwave

Sources," Proc. IEEE, vol. 54, pp. 249-257,

February 1956.

21. D. Hal ford, "A General Mechanical Model

for \f\° Spectral Density Random Noise with

Special Reference to Flicker Noise l/|fi,"

Proc. of the IEEE, vol. 56, no. 3, March

1968. pp. 251-258.

22. H, Hellwig, "Atomic Frequency Standards: A Survey." Proc. IEEE. vol. 63, pp. 212-229,

February 1975.

23. H. Hellwig, "A Review of Precision Oscillators," NBS Tech. Note 662, February 1975.

24. H. Hellwig, "Frequency Standards and Clocks: A Tutorial Introduction," NBS Tech. Note 616-R, March 1974.

25. Hewlett-Packard, "Precision Frequency Meas- urements," Application Note 116, July 1969.

26. G. .M. Jenkins and 0. G. Watts, Spectral

Analysis and its Applications , San Francisco:

Hoi den- Day, 1968.

27. S. L. Johnson, B. H. Smith, and D. A. Calder,

"Noise Spectrum Characteristics of Low-Noise

Microwave Tubes and Sol id- State Devices."

Proc. IEEE. vol. 54, pp. 258-265, February

1966.

43

TN-56
28. P. Kartaschoff and J. Barnes, "Standard Time

and Frequency Generation," Proc. IEEE, vol. 60, pp. 493-501, May 1972.

29. A. L. Lance, W. D. Seal, F. G. Mendoza, and

N. W. Hudson, "Automating Phase Noise Measurements in the Frequency Domain," Proc. 31st

Annual Symposium on Frequency Control, 1977,

pp. 347-358.

30. 0. B. Leeson, "A Simple Model of Feedback

Oscillator Noise Spectrum," Proc. IEEE L. , vol. 54, pp. 329-330, February 1966.

31. P. Lesage and C. Audoin, "A Time Domain

Method for Measurement of the Spectral Density

of Frequency Fluctuations at Low Fourier

Frequencies," Proc. 29th Annual Symposium on Frequency Control . 1975, pp. 394-403.

32. P. Lesage and C. Audoin, "Estimation of the

Two-Sample Variance with Limited Number of

Data," Proc. 31st Annual Symposium on Fre- quency Control, 1977, pp. 311-318.

33. P. Lesage and C. Audoin, "Characterization of

Frequency Stability: Uncertainty due to Finite Number of Measurements," IEEE Trans on Instrum. and Meas. IM-22 . pp. 157-151, June

1973.

34. W. C. Lindsey and C. M. Chie, "Specification

and Measurement of Oscillator Phase Noise

Instability," Proc. 31st Annual Symposium on Frequency Control, 1977, pp. 302-310.

35. 0. G. Meyer, "A Test Set for Measurement of

Phase Noise on High Quality Signal Sources,"

Proc. IEEE Trans on Instrum. and Meas. IM-19 , No. 4, pp. 215-227, November 1970.

36. C. D. Motchenbacher and F. C. Fitchen,

Low-Noise Electronic Design , New York: John

Wiley L Sons, 1973.

37. D. Percival, "Prediction Error Analysis of

Atomic Frequency Standards," Proc. 31st

Annual Symposium on Frequency Control, 1977,

pp. 319-326.

38. D. Percival, "A Heuristic Model of Long-Term

Clock Behavior," Proc. 30th Annual Symposium

on Frequency Control, 1976, pp. 414-419.

39. J. Shoaf, "Specification and Measurement of

Frequency Stability," NBS Internal Report

74-396, November 1974.

40. J. L. Stewart, "Frequency Modulation Noise in Oscillators," Proc. IRE, vol. 44, pp. 372-

376, March 1956.

41. R. F. C. Vessot, L. Mueller, and J. Vanier,

"The Specification of Oscillator Characteristics from Measurements in the Frequency

Domain," Proc. IEEE, pp. 199-206, February

1965.

42. F. L. Walls, S. R. Stein, J. E. Gray, and D. J. Glaze, "Design Considerations in State-ofthe-Art Signal Processing and Phase Noise Measurement Systems," Proc. 30th Annual

Symposium on Frequency Control, 1976, pp. 259-274.

43. F. Walls and A. Wainwright, "Measurement of the Short-Term Stability of Quartz Crystal Resonators and the Implications for Quartz

Oscillator Design and Applications," IEEE Trans, on Instrum. and Meas. , pp. 15-20,

March 1975.

44. G. M. R. Winkler, "A Brief Review of Frequency

Stability Measures," Proc. 8th Annual Precise

Time and Time Interval (PTTI) Applications

and Planning Meeting, Nov. 1975, pp. 489-527.

Appendix

DEGREES OF FREEDOM FOR ALLAN VARIANCE

Co^vlaan of wptrleoi aid anot far Vhita W

Tau (in units of data spacing)

44

TN-57
DEGREES OF FREEDOM FOR ALLAN VARIANCE

CoapviMn af M^lrloel vW tmmet far Vhlt* FM T

Tou (in units of data spacing)

1

DEGREES OF FREEDOM FOR ALLAN VARIANCES

WKit* Phoss Noiss

r 1 — 1 "TTiTrr 1—I—I I I M

r -I T ri ITT" —

E

O

•

L

9

E3

Tou (in Lniit* of data spacing)

DEGREES OF FREEDOM FOR ALLAN VARIANCE

laon af a^tPiHl md ' Valk FM

J

Tau (in units of dota spacing)

EO

-a

4

DEGREES OF FREEDOM FOR ALLAN VARIANCES

VKits Fr«^usncy Nojss

Tou (in units of dota spocing)

45

TN-58
DEGREES OF FREEDOM FOR ALLAN VARIANCES

Flicker PKa«« Modulation

t 1 —I—r 1 I 1 1 11 r—T- 1 nTTTrO

S

O

L

ao

Tau (in unjts of data •pocing)

DEGREES OF FREEDOM FOR ALLAN VARlAfiCES

Rondo* Volk FH

1T1TO

i

Tou (in units of doto •pccinq)

DEGREES OF FREEDOM FOR ALLAN VARIANCES

Flickar Fr«qu»ncy Modul-ation

ITTT] r III

o

a

saa

i^

CONFIDENCE FACTORS FROM CHI-SOUARE

90Z Intarvolt

Tou (in units of data apocing)

NuRib*r of Q»ar—* of Fr*«doni

46

TN-59
t 1—

r

LOWER CONFIDENCE FACTORS

TT 1 -111 rTTTTUPPER CONFIDENCE FACTORS

i--rrTrTa

3

Number of D»gr»»« of Fr«»doin

Nuitib«r of D«gr»«« of Fr««doiti

Dt*«rlbu«lon of Ra^le of Obaarvd Voriane* to AetJjal

100

z9

L 50

Q.

TK*er-«t.leal Rat. to of

' CKi—Squaf** to 0a3r*«« of *^

For- d.f. -n.8

Chi Sauanm/D•qn9* of Fre«dc

47

TN-60
From: Precision Frequency Control. Vol. 2, edited by EA. Gerber and A. Ballato

(Academic Press, New York, 1985), pp. 191-416.

12 Frequency and Timeand Characterization

-Their Measurement

Samuel R. Stein

Time and Frequency Division National Bureau of Standards

Boulder, Colorado

121

12.2

12.3

124

List of Symbols 192

Concepts, Definitions, and Measures of Stability 193

1 2.1 .1 Relationship between the Power Spectrum and the Phase Spectrum 195

12.1.2 The IEEE Recommended Measures of Frequency

Stability 195

12. 1.3 The Concepts of the Frequency Domain and the Time Domain 203

12.1.4 Translation between the Spectral Density of Frequency and the Allan Variance 203

12.1.5 The Modified Allan Variance 205

12.1.6 Determination of the Mean Frequency and

Frequency Drift of an Oscillator 207

12.1.7 Confidence of the Estimate and Overlapping

Samples 210

1 2.1 .8 Efficient Use of the Data and Determination of the Degrees of Freedom 213

1 2 1 .9 Separating the Variances of an Oscillator from the Reference 216

Direct Digital Measurement 2*7

12.2.1 Time-Interval Measurements 217

12.2.2 Frequency Measurements 219

12 2.3 Period Measurements 219

Sensitivity-Enhancement Methods 220

12.3.1 Heterodyne Techniques 220 12.32 Homodyne Techniques 221 12 32.1 Discriminator and Delay Line 222

12.3 22 Phase-Locked Loop 223 12 3.3 Multiple Conversion VIethods 227 1233 1 Frequency Synthesis 227

1 2.3 3 2 The Dual-Mixer Time-Difference Technique 229

12 33 3 Frequency Multiplication 231 Conclusion 231

PRECISION FREOUESCY CONTROL

Ov.-.ilj(ors anJ Si;inJirJi isBN')-i:.:^c«o:-4

191

TN-61
192 SAMUEL R. STEIN

LIST OF SYMBOLS

df Number or degrees of freedom / Fourier (cycle) frequency

G„{j<o) Open-loop transfer function of a phase-locked loop

F(s) Transfer function of the loop filter of a phase-locked loop

H{f — /o) Transfer function of a filter Ki Sensitivity of a linear phase detector in volts per radian P Number of counts accumulated by a time-interval counter Q Quality factor of a resonance, equal to the ratio of the stored energy to the energy lost in one cycle

s Independent variable of the Laplace transform

Sy Estimate of the two-sample or Allan variance S^f) One-sided power spectral density of the fractional-frequency deviations S^(/) One-sided power spectral density of the phase deviations S"(/) Two-sided power spectral density of the phase deviations

Sj*c(/) Two-sided power spectral density of the phase deviations of the carrier

^I^p</) Two-sided power spectral density of the phase deviations of the pedestal

^v^(/) Two-sided power spectral density of the instantaneous voltage

Vq Peak voltage of a signal generator

Vi Output voltage of a phase detector

V, Voltage applied to the tuning element of the voltage-controlled oscillator of a phase-locked loop

Vit) Instantaneous voltage of a signal generator or other device

x(t) Time deviation required by a signal generator operating at nominal frequency v, to accumulate phase equal to </K0- X{f) Fourier transform of xtf)

\\t) Instantaneous fractional-frequency offset from nominal

y\ Mean fractional-frequency offset over the kth interval a Exponent of/ for a power-law spectral density

Avq Frequency uncertainty due to the quantization of measurements

C Damping constant of a phase-locked loop H Exponent of t for a power-law Allan variance

v(r) Instantaneous frequency

vq Nominal frequency of a signal generator

v(r,; fj) Mean frequency over the interval t, £ t S tj

v'h Heterodyne frequency or difference frequency between two oscillators

p(X') Probability density of the chi-squared distribution

ffJliV, r. t) Sample variance of N fractional-frequency deviations, each averaged over a period T spaced at intervals T

ff;(T) The two-sample or Allan variance of the fractional-frequency deviations mod (T^(T) Modified Allan variance

T Averaging time

Tj Period of the time base of a counter

(p(t) Instantaneous phase deviation

X^ Chi-squared distribution

0) Fourier (angular) frequency

(o„ Natural frequency of a phase-locked loop

TN-62
12 FREQUENCY AND TIME MEASUREMENT 193

12.1 CONCEPTS, DEFINITIONS, AND MEASURES

OF STABILITY

This chapter deals with the measurement of the frequency or time stabiUty

of precision oscillators. It is assumed that the average output frequency is determined by a narrow-band circuit so that the signal is very nearly a sine wave. To be specific, it is also assumed that the output is a voltage, which is conventionally (Barnes et a/., 1971) represented by the expression

y(t) = IVo + £(0] sin[27rvot + ml (12-1)

where Vq is the nominal peak voltage amplitude, s{c) the deviation of

amplitude from nominal, Vq the nominal fundamental frequency, and <t>{t) the deviation of phase from nominal.

When either specifying or measuring the noise in an oscillator, one must

consider the nature of the reference. This may be either a passive circuit such

as a narrow-band filter, another similar oscillator, or a set of oscillators,

synthesizers, and other signal-generating equipment. A reference with lower

noise than the device under test may be available, and in this case the expressions developed in this chapter describe the noise in the oscillator alone. However, a state-of-the-art device will have lower noise than any

available reference. In this case all the expressions below refer to the sum of

device and reference noise. The most common approach to solving this problem is to compare two or more nearly identical devices. Under most

circumstances it is then reasonable to assume that each oscillator contributes

half of the measured noise. The most direct and intuitive method of characterizing the properties of a signal is to determine the two-sided spectrum of V{[), which is denoted

SVif) (Rutman, 1978). The variable/ is called a Fourier frequency and is very closely related to the concept of a modulation frequency. A positive

/ indicates a frequency above the carrier frequency Vq, while a negative /

indicates a frequency lower than the carrier. Since the noise can in theory

modulate the carrier at all possible frequencies, a continuous function is required to describe the modulation of V{t). S is called a spectral density and

SV(f) is the mean-square voltage < V^(r)> in a unit bandwidth centered at/. It

is proportional to the rf power per unit bandwidth delivered by the oscillator to a matched load. The total signal power is proportional to the mean square

voltage, which is also called the variance of the signal since the mean value of

V(t) is zero. The variance is therefore equal to the two-sided spectral density

integrated over all frequencies.

The two-sided spectrum is usually measured by an rf spectrum analyzer, a device that functions like a bandpass filter followed by a bolometer, as shown

TN-63
19A SAMUEL R. STEIN & BANDPASS FILTER

M(f-lo)

Iv'(t) MEAN -SQUARE

METER

OSCILLATOR

V(l)

FIG. 12-1 An rf spectrum anal>'zer. The device produces an output proportional to the mean-square value of the signal passing through a tunable narrow-band filter centered at frequency/oin Fig. 12-1. The spectrum of the filtered voltage V'{t) is equal to the square

of the magnitude of the filter transfer function H{f — /q) multiplied by the

spectrum of the input signal (Cutler and Searle, 1966). The variance of the

filtered voltage is obtained from ParsevaPs theorem:

d ifo)= r \H{f-fo)\'Sj^if)df. (12-2)

J — XI

If the bandpass filter is sufficiently narrow, so that Sj^if) changes negligibly over its bandwidth, then Eq. (12-2) may be inverted. With this assumption,

the power spectrum is estimated from the measurement using Eq. (12-3):

SVifo) = cUfol'B, (12-3)

where B = Jf ^ l^(/' -/o)l^ 4f' is the noise bandwidth of the filter and/o its center frequency. Figure 12-2 shows a typical two-sided rf spectrum. For

many oscillators the spectrum has a Lorentzian shape, that is,

SVif) =

2<K2>/7rA/3,B

(12-4)

1 + (f/Wy<i„2)f

•

The Lorentzian lineshape is completely described by the mean square voltage

<V-> and the full width at half maximum Afi^BS\^{\)

W-Afc

f'f

y"^^ y N----^ s ^**N

• I 1 N

^ 1

1 \

• ' N. / 1 1 \

FIG. 12-2 The rf spectrum of a signal. It is often useful to divide the spectrum into the

carrier and the noise pedestal. The spectral density of the carrier exceeds that of the noise

pedestal for Fourier frequencies smaller in magnitude than/j.

TN.64
12 FREQUENCY AND TIME MEASUREMENT 195

12.1.1 Relationship between the Power Spectrum and the

Phase Spectrum

The power spectrum differs from a delta function 5(f) due to the presence of the amplitude- and phase-noise terms, e{t) and <f>{t), respectively, included

in Eq. (12-1). Usually the noise modulation separates into two distinct components, so that one observes a very narrow feature called the carrier above the level of a relatively broad pedestal. The frequency that separates

the carrier and pedestal is denoted f^. Below this frequency the spectral

density of the carrier exceeds that of the pedestal. Assuming that the amplitude noise is negligible compared to the phase noise and that the phase

modulation is small, the relationship between the power and phase spectra

is eiven by (Walls and DeMarchi, 1975)

5"(/) is the two-sided spectrum of the phase fluctuations, which divides

into a carrier component Sjfj/) and a pedestal component Sjfp(/); lift) 'S given by

/(/c) = rs^jf)df^rs^{f)df (12-6)

since above/^ the pedestal dominates the noise sp)ectrum. The variance of the

carrier is equal to {V^/2)e~'^'\ with the remaining variance in the pedestal. If Avp is the width of the pedestal and Av^ the width of the carrier, then the power density in the carrier is equal to that in the pedestal when /(/g) =

ln(Avp,/Avj). For the pedestal, one may use the 3-dB linewidth for Avp

provided that |/^ S^{f)df< In 2. Otherwise the pedestal width is estimated

from Jav/2^<>(/)^/= 1^2. For the carrier, the linewidth is estimated by

calculating J^,^,2 S^Jf) J/ = In 2. The foregoing analysis makes it possible to draw certain conclusions

concerning detection of the carrier. We use Jp df to denote the integral over

the phase noise pedestal. If Jp S,(/) df < In 2, then the carrier may be resolved

irrespective of detector bandwidth. When ln(Avp,/AvJ > Jp S^(/) <i/ > In 2, the carrier may be resolved by restricting the detection bandwidth. But when

Jp S^if) df > ln(Avp/AvJ, the carrier can no longer be distinguished from the pedestal since its spectral density is smaller.

12.1.2 The IEEE Recommended Measures of Frequency Stabilit}

By the mid-1960s the problem of the specification of precision oscillators had become e.xtremely important, but there was very little uniformity among

manufacturers, metrologists, and applications engineers in the methods of

TN-65
196 SAMUEL R. STEIN

performing measurements or the description of measurement results. This

situation was complicated by the difficulty of comparing the various

descriptions. A measure of stability is often used to summarize some

important feature of the performance of the standard. It may therefore not be

possible to translate from one measure to another even though the respective

measurement processes are fully described and all relevant parameters are

given. This situation resulted in a strong pressure to achieve a higher degree

of uniformity. In order to reduce the difficulty of comparing devices measured in separate

laboratories, the IEEE convened a committee to recommend uniform measures of frequency stability. The recommendations made by the committee are based on the rigorous statistical treatment of ideal oscillators that obey a certain model (Barnes et aU 1971). Most importantly, these oscillators are assumed to be elements of a stationary ensemble. A random process is stationary if no translation of the time coordinate changes the probability

distribution of the process. That is, if one looks at the ensemble at one instant

of time, then the distribution in values for a process within the ensemble is exactly the same as the distribution at any other instant of time. The elements

of the ensemble are not constant in time, but as one element changes value

other elements of the ensemble assume previous values. Thus, it is not

possible to determine the particular time when the measurement was made.

The stationar>' noise model has been adopted because many theoretical

results, particularly those related to spectral densities, are valid only for this case. It is important for the statistician to exercise considerable care since

experimentally one may measure quantities approximately equal to either the instantaneous frequency of the oscillator or the instantaneous phase. But the

ideal quantities approximated by these measurements may not both be

stationary. The instantaneous angular frequency is conventionally defined as the time derivative of the total oscillator phase. Thus,

d

w(r) = - [27rvot + 0(r)], (12-7)

and the instantaneous frequency is written

v(r) = Vo-^— -f

.

(12-8)

In dt

For precision oscillators, the second term on the right-hand side is quite

small, and it is useful to define the fractional frequency

^„=;i^i^=J-f = ^, (12-9)

Vfl 2rtVo di dt

TN-66
12 FREQUENCY AND TIME MEASUREMENT 197

where

x{t) = 0(r)/27rvo (12-10)

is the phase expressed in units of time. Alternatively, the phase could be

written as the integral of the frequency of the oscillator: m = 00 + (2nlv{e)-Vo]dd. (12-11) Jo However, the integral of a stationary process is generally not stationary.

Thus, indiscriminate use of Eqs. (12-7) and (12-11) may violate the assumptions of the statistical model. This contradiction is avoided when one

accounts for the finite bandwidth of the measurement process. Although a more detailed consideration of the statistics goes beyond the scope of this treatment, it is very important to keep in mind the assumption that lie behind

the statistical analysis of oscillators. In order to analyze the behavior of real oscillators, it is necessary to adopt a model of their performance. The model

must be consistent with observations of the device being simulated. To make

it easier to estimate the device parameters, the models usually include certain predictable features of the oscillator performance, such as a linear frequency

drift. A statistical analysis is useful in estimating such parameters to remove

their effect from the data. It is just these procedures for estimating the deterministic model parameters that have proved to be the most intractable. A substantial fraction of the total noise power often occurs at Fourier

frequencies whose periods are of the same order as the data length or longer.

Thus, the process of estimating parameters may bias the noise residuals by

reducing the noise power at low Fourier frequencies. A general technique for minimizing this problem in the case of oscillators actually observed in the laboratory is discussed below.

It has been suggested that measurement techniques for frequency and time

constitute a hierarchy (Allan and Daams, 1975), with the measurement of the

total phase of the oscillator at the peak. Although more difficult to measure

with high precision than other quantities, the total phase has this status owing to the fact that all other quantities can be derived from it. Furthermore, missing measurements produce the least deleterious effect on a

time series consisting of samples of the total phase. Gaps in the data affect the computation of various time-dependent quantities for times equal to or

shorter than the gap length, but have a negligible effect for times much longer

than the gap length. The lower levels of the hierarchy consist of the time

interval, frequency, and frequency fluctuation. When one measures a quantity somewhere in this hierarchy and wishes to obtain a higher quantity, it is necessary to integrate one or more times. In this case the problem of missing

data is quite serious. For example, if frequency is measured and one wants to

TN-67
^^® SAMUEL R. STEIN

know the time of a clock, one needs to perform the integration in Eq. (12-1 1). The missing frequency measurements must be bridged by estimating the

average frequency over the gap, resulting in a time error that is propagated

forever. Thus, it is preferable to always make measurements at a level of the

measurement hierarchy equal to or above the level corresponding to the

quantity of principle interest. In the past this was rather difficult to do.

Measurement systems constructed from simple commercial equipment suf- fered from dead time, that is, they were inactive for a period after performing

a measurement. To make matters worse, methods for measuring time or phase had considerably worse noise performance than methods for measuring frequency. As a result, many powerful statistical techniques were

developed to cope with these problems (Barnes, 1969; Allan, 1966). The effect of dead time on the statistical analysis has been determined (Lesage and

Audoin, 1979b). Other techniques have been developed to combine short

data sets so that the parameters of clocks over long periods of time could be

estimated despite missing data (Lesage, 1983). The rationale for these

approaches is considerably diminished today. Low-noise techniques for the

measurement of oscillator phase have been developed. Now, commercial

equipment is capable of measuring the time or the total phase of an oscillator with very high precision. Other equipment exists for measuring the time

interval. These devices use the same techniques that were previously

employed for the measurement of frequency and are very competitive in performance.

The proliferation of microcomputers and microprocessors has had an

equally profound effect on the field of time and frequency measurement.

There has been a dramatic increase in the ability of the metrologist to acquire

and process digital data. Many instruments are available with suitable

standard interfaces such as IEEE-583 or CAMAC (IEEE, 1975) and

IEEE-488 (IEEE, 1978). As a result, there has been a dramatic change in direction away from analog signal processing toward the digital, and this process is accelerating daily. Techniques once used only by national

standards laboratories and other major centers of clock development and

analysis are now widespread. Consequently, this chapter will focus first on

the peak of the measurement hierarchy and the use of digital signal

processing. But the analysis is directed toward estimating the traditional measures of frequency stability. Considerable attention will be paid to problems associated with estimating the confidence of these stability measures and obtaining the maximum information from available data. The IEEE has recommended as its first measure of frequency stability the

one-sided spectral density Sy(f) of the instantaneous fractional-frequency

fluctuations yit). It is simply related to the spectral density of phase fluctua- tions since differentiation of the time-dependent functions is equivalent to

TN-68
12 FREQUENCY AND TIME MEASUREMENT I99

multiplication of their Fourier transforms byyw:

S,(f) = {f/yo)%if) = (2nfysjf). (12-12)

Section 12.1.1 on the relationship between the power spectrum and the phase spectrum described the analog method for the measurement of a

spectral density. If a voltage I^is the output of the oscillator, then the result of

the measurement is proportional to the rf power spectral density. But if the voltage were proportional to the frequency or phase of the oscillator, then the

result of the measurement would be proportional to the spectral density of

the frequency or phase. The most common units of S^(/) are radians squared

per hertz. Alternatively, the spectral density can be obtained by digital analysis of the

signal. For example, the quantity SJf) can be calculated from the Fourier

transform of x(r). The relevant continuous Fourier-transform pair is defined

as follows: w)=j; x(t)e-J^''^'dt (12-13)

and

.x(r) = ^J" X(f)e^-''^'df. (12-14)

However, one does not generally have continuous knowledge of the phase of

the oscillator. Since it is relatively easy to measure .x(f) at equally spaced time

intervals, we assume the existence of the series x,, where x, = x(/t) for integer

values of /. The discrete Fourier transform is defined by analogy to the •continuous transform (Cochran et al., 1967):

X{f)= t x(/)^-^-2»^'. (12-15)

J = -x

In practice the time series has finite length T consisting of N intervals of

length T, and it is not possible to compute the infinite sum. Nevertheless, it remains possible to compute a spectrum that is not continuous in/ but rather has resolution A/ where

A/= 1/7= i/iVr. (12-16)

The need to sum over all values of the index / is removed by assuming that the function .x(f) repeats itself with period T. The resulting spectrum contains no

information on the spectrum at Fourier frequencies less than l,'T. Truncation

of the time series also introduces spurious effects due to the turn-on and turnoff transients. These problems can be minimized through the use of a window

function. The computed spectrum is actually the square of the magnitude of

TN-69
200 SAMUEL R. STEIN

the window function multiplied by the desired spectrum. The use of a window

function reduces the variance of the spectrum estimate at the expense of

smearing out the spectrum to a small degree. With these changes but no window function, we arrive at the discrete finite transform

1 N-l

X(n^f) = i- y x(JtT)e--''2''-^^'". (12-17) ^ * = The spectral density of x(r) is computed from Eq. (12-17) by squaring the real and imaginary components, adding the two together, and dividing by the

total time T:

The digital method of estimating spectral densities has many advantages

over analog signal processing. Most important is the fact that it may be

computed from any set of equally spaced samples of a time series. As a result, the technique is compatible with other methods of characterizing the signal, that is, the sampled data can be stored and processed using a variety of

algorithms. In addition, each record of length Tproduces a single estimate of

the sp>ectrum for each of the N frequencies Af,2^f, . ..,N A/. It is therefore possible to estimate the entire spectrum much more quickly using the digital

technique than it would be using analog methods. The fast Fourier

transform, a very efficient algorithm for the computation of the discrete finite transform, has opened the way to versatile self-contained, commercial

spectrum analysis. It is also very straightforward to compute the spectrum

from data acquired by computerized digital data acquisition systems.

A result of the finite sampling rate is that the upper frequency limit of the

digital spectrum analysis is 1/2t, called the Nyquist frequency (Jenkins and

Watts, 1968). Power in the signal being analyzed that is at frequencies higher

than the Nyquist frequency affects the spectrum estimate for lower frequencies. This problem is called aliasing. The out-of-band signal is rejected by

only approximately 6dB per octave above the Nyquist frequency. Thus,

when significant out-of-band signals exist, they must be reduced by analog

filtering. One or more low-pass filters are usually sufficient for this purpose. As its second measure of frequency stability, the IEEE recommended the sample variance c7y(T) of the fractional-frequency fluctuations. It is a measure

of the variability of the average frequency of an oscillator between two

adjacent measurement intervals. The average fractional-frequency deviation

\\ over the time interval from r^ to r^ -I- t is defined as

1 r'"*' h = -\ >it)dt, (12-19)

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12 FREQUENCY AND TIME MEASUREMENT 291

x(t)

|-T-I|

f-T-l

TIME

FIG. 12-3 Measurement process for the computation of the sample variance. The phase

difference between two oscillators is plotted on the ordinate. The measurement yields a set of

frequencies averaged over equal intervals r separated by dead time T- i.

from which it follows that

yk =

x{t^ + t) - X(fJ

(12-20)

The quality i is often referred to as the sampling time or the averaging time. Equations (12-19) and (12-20) are not the only way to define mean frequency,

but they are the simplest. Other definitions lead to alternative measures of

stability that may have desirable properties.

Suppose that one has measured the time or frequency fluctuations between

a pair of precision oscillators and a stability analysis is desired. The process is illustrated in Fig. 12-3. These are N values of the fractional frequency y,. Each one is measured over a time t, and measurements are repeated after intervals of time T. If the measurement repetition time exceeds the averaging

time, then there is a dead time equal to T— t between each frequency

measurement, during which there is no information available. There are many ways to analyze these data. A fairly general approach is the N-sample variance defined by the relation

<<t;(.V,T.t)> =

1 N T.?,

1 ^ ^^

(12-21)

where the angle brackets denote the infinite time average. Frequently, Eq.

(12-21) does not converge as N - x, since some noise processes in oscillators diverge rapidly at low Fourier frequencies. This implies that the precision

with which one estimates the variance does not improve simply as the sample

size is increased. For this reason, the two-sample variance with no dead time

is preferred. Also called the Allan variance, it converges for all the major

noise types observed in precision oscillators. It may be written as

<^;(T) = <if;*-i-yJ'>. (12-22)

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202 SAMUEL R. STEIN

10'=.

ho»s6

10 'I ' ' ' 'I'l'l I '

I "I'll ' ' ' I'll!

1 10' 10* io' NUMBER OF SAMPLES N

FIG. 12-4 ;V -sample variance versus Allan variance. The two-sample variance converges for the important types of noise observed in frequency standards but the ratio of the traditional variance to the two-sample variance is an increasing function of sample size for flicker frequency

noise and random-walk frequency noise.

The dependence of the classical variance on the number of samples is shown

in Fig. 12-4 for the case of no dead time. The quantity plotted is the ratio of

the iV-sample variance to the Allan variance. Note that <t^(t) has the same

value as the classical variance for the white-noise frequency modulation.

However, the classical variance grows without bound for flicker-frequency

and random-walk-frequency noises. One may combine Eqs. (12-20) and (12-22) to obtain an equation for Gy(r) in terms of the time-difference or time-deviation measurements:

<7;(T) = <it--[.v(f + 2t) - 2.x(r -H T) + x{t)yy.

N discrete time readings mav be used to estimate the variance

.V-2

(12-23)

(12-24)

where f denotes the number of the measurement in the set of N and the nominal spacing between measurements is r. Since it has been assumed that there is no dead time between measurements, one can write x in Eq. (12-24) as an integer multiple of Tq. that is, r = /jitq, where Xq is the smallest spacing of

the data. In this case

ahrnxo) s

1 S-2m

2{N - 2m)m-Xo n .t'l I (.x,.,„-2.x,,„-K.x,)=. (12-25)

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12 FREQUENCY AND TIME MEASUREMENT 203

12.1.3 The Concepts of the Frequency Domain and the Time Domain

Spectral densities are measures of frequency stability in what is called the frequency domain since they are functions of Fourier frequency. The Allan

variance, on the other hand, is an example of a time-domain measure. In a

strict mathematical sense, these two descriptions are connected by Fourier

transform relationships (Cutler and Searle, 1966). However, for many years the inadequacy of measurement equipment created artificial barriers between

these two characterizations of the same noise process. As a result, many

specialized techniques have been developed to translate between the various measures of stability (Allan, 1966; Burgoon and Fischer, 1978). The preceding

sections have demonstrated how easily both types of stabiUty measures can be computed from the same data provided that the measurement process provides complete information. For example, both (7J(inzo) and Sj.(in A/) can be computed from evenly spaced samples of x{t). However, incomplete

information can result from either measurement dead time or interruptions in the data acquisition process. In these cases translation techniques remain

valuable. Both the spectral density and the Allan Variance are second-moment

measures of the time series .x(f). However, it is only possible to translate unambiguously from the spectral density to the Allan variance, not the

reverse. To calculate the spectral density it is necessary to use the autocorrelation function of the phase. The following discussion on power-law noise processes further demonstrates this dichotomy. As we shall see, the Allan

variance for a fi.xed measurement bandwidth does not distinguish between all of the noise processes that are commonly observed in precision oscillators.

12.1.4 Translation between the Spectral Density of Frequency

and the Allan Variance

The power-law model is most frequently used for describing oscillator phase noise. It assumes that the spectral density of frequency fluctuations is equal to the sum of terms, each of which varies as an integer power of

frequency. Thus, there are two quantities that completely specify Sy(f) for a

particular power-law noise process: the slope on a log-log plot for a given

range of/ and the amplitude. The slope is denoted by a and therefore/* is the

straight line on a log-log plot that relates Sy{f) to/. The amplitude is denoted

hj. When we examine a plot of the spectral density of frequency fluctuations, we represent it by the addition of all the power-law processes (Allan, 1966; Vessot et ai, 1966) with the appropriate coefficients:

5,(/)= i hj\ (12-26)

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204 SAMUEL R. STEIN

TABLE 12-1

Correspondence between Common Power-Law Spectral Densities and the Allan Variance*

Noise type sw <rJ(T)

White phase hzP 3/»M2n)^^

Flicker phase hj [1.038-f31n(23^»T)] ^ 1

White frequency K iWi/T)

Flicker frequency h.,n 21n(2)A., Random-walk frequency h.J-' i(2n)»fc.,T

' Where necessary for convergence the spectral density has been assumed to be zero for frequencies greater than the cutoff frequency /,.

This technique is most valuable when only a few terms in Eq. (12-26) are

required to describe the observed noise and each term dominates over several decades of frequency. This situation often prevails. Five power-law noise

processes (Allan, 1966: Vessot et ai, 1966) are common with precision

oscillators:

(1) random-walk frequency modulation a = — 2

(2) flicker frequency modulation a = — 1

(3) white frequency modulation a =

(4) flicker phase modulation a = 1 (5) white phase modulation a = 2 The spectral density of frequency is an unambiguous description of the

oscillator noise. Thus, the spectrum can be used to compute the Allan

variance (Barnes et ai, 1971):

<t;(t) = —^

rs^{f)sin*(nh)df. (12-27)

(jTVol)- Jo However, Eq. (12-27) shows that the .\llan variance is very sensitive to the

high frequency dependence of the spectral density of phase, thereby necessitating a detailed knowledge of the bandwidth-limiting elements in the measurement setup. The integral has been computed for each of the powerlaw noise processes, and the results are summarized in Table 12-1 (Barnes et ai, 1971). For a in the range — 2 < a S 0, the Allan variance is proportional

to t", where /i = - a - 1 . When the log of the Allan variance is plotted as a

function of the log of the averaging time, the graph also consists of straightline segments with integer slopes. However, Table 12-1 also shows that even if

* See Appendix Note # 8

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12 FREQUENCY AND TIME MEASUREMENT 205

the oscillator is reasonably modeled by power-law spectra, it is not practical to distinguish between white phase noise and flicker phase noise from the dependence of the Allan variance on x. In both cases aj s: 1/t".

12.1.5 The Modified Allan Variance

Table 12-1 also shows that the Allan variance has very different bandwidth

dependence for white phase noise and flicker phase noise. Therefore, these noise types have been distinguished by varying the bandwidth of the measurement system. If x(r) were measured, the noise type could be identified by computing the spectrum. However, both the approach of making

measurements as a function of bandwidth and the computation of the spectrum can be avoided by calculating a modified version of the Allan

variance. The algorithm for this variance has the effect of changing the bandwidth inversely in proportion to the averaging time (Snyder, 1981 ; Allan

and Barnes, 1981).

Each reading of the time deviation .x^ has associated with it a measurementsystem bandwidth/h . Similarly, we can define a software bandwidth/, =fiJn,

which is \!n times narrower than the hardware bandwidth. It can be realized by averaging n adjacent .Xj's. Based on this idea it is possible to define a modified Allan variance that allows the reciprocal software bandwidth to change linearly with the sample time t:

mod olix) = ^/ i X (.x,,i„ - 2.x,,, + x,)j'\

,

(12-28)

where T = otq. Equation (12-28) reduces to Eq. (12-23) for n = 1. One can see that mod ct^ (t) is the second difference of three time values, each of which is a nonoverlapping average of n of the x,'s. As n increases the software

bandwidth decreases zsfjn.

For a finite data set of N readings of x, (/ = 1 to A'), mod (T^(t) can be

estimated from the expression

"°'">^^^ = 2xV(.V-3.^I) Mr "X^'^'" " '•'"- "" '^^''

(12-29)

which is easy to program but takes more time to compute than the corresponding equation (12-24) for g]{z).

Table 12-2 gives the relationship between the time-domain measure mod <7^(t) and its power-law spectral counterpart. In the right-hand column

are the asymptotic values of the ratio of the modified Allan variance to the Allan variance. It is clear from the table that mod a](-:) is very useful for white

• See Appendix Note # 9

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206 SAMUEL R. STEIN

TABLE 12-2

Correspondence beiween Common Power>Law Spectral Densities and the Modified Allan Variance*

Noise type SAf) mod ^(t) mod ffi/ff?

White phase

Flicker phase

White Trequency

Flicker frequency

Raadom-walk frequency

ho

h.2r*

3/« 1

' n(2>t)* t»

[1.038 + 31ii(2j^»t)] 1

(2r)»

fc., (0.936)

*.j(5.42)t

I

0.5

0.674

0.824

* Where necessar>- the spectral density has been assumed to be zero for frequencies greater than

the cutoff frequency / . The constant n ir the number of adjacent phase values that are averaged

to produce the bandwidth reduction. The values in the last two columns are for the asymptotic

limit n -• x. In practice, n only needs to be 10 or larger before the asymptoticlimit is approached

within a few percent. When n > I the ratio in the last column is 1 in all cases.

phase modulation and flicker phase modulation, but for at < 1 the conventional .\llan variance gives both an easier-to-interpret and an easier-to- calcutate measure of stability.

It is interesting to make a graph of a versus n for both the ordinary Allan

variance and the modified Allan variance, such as the one shown in Fig. 12-5.

a 0-

PIG. 12-5 Relationship between a power-law spectral density whose slope on a log-log plot

is 2 and the corresponding sample variance whose slope on a log-log plot is fi. The solid line describes the behavior of the .^llan variance, while the dashed line shows the advantage of the modified Allan variance for white phase noise and flicker phase noise.

TN.76
12 FREQUENCY AND TIME MEASUREMENT 207

This graph allows one to determine power-law spectra for noninteger as well as integer values of a. In the asymtotic limit the equation relating /x and a for the modified Allan variance is

a = -/i - 1 for -3<a<3. (12-30)

12.1.6 Determination of the Mean Frequency and Frequency Drift

of an Oscillator

Before the techniques of the previous four sections can be meaningfully

applied to practical measurements, it is necessary to separate the deter- ministic and random components of the time deviation x(t). Suppose, for example, that an oscillator has significant drift, such as might be the case for a quartz crystal oscillator. With no additional signal processing, the Allan

variance would be proportional to t". The variance of the Allan variance

would be very small, further demonstrating that deterministic behavior has

been improperly described in statistical terms and the oscillator's predictability is much better than the Allan variance indicated. Unfortunately, it

is difficult to estimate the oscillator's deterministic behavior without intro- ducing a bias in the noise at Fourier frequencies comparable to the inverse

of the record length. In practice, it has been sufficient to consider two

deterministic terms in x{t):

x(t} = .xo + (Av/vo)f + iDf- -I- .Xi(f) (12-31)

The first term on the right-hand side is the synchronization error. The second term is due to imperfect knowledge of the mean frequency and is sometimes

called syntonization error. The quadratic term, which results from frequency

drift, is the most difficult problem for the statistical analysis because the Allan

variance is insensitive to both synchronization and syntonization errors. For white noise, the optimum estimate of the process is the mean.

Therefore, a general statistical procedure that can be followed is to filter the data until the residuals are white (Allan et ai, 1974: Barnes and Allan, 1966).

For example, at short times the frequency fluctuations of atomic clocks are

usually white. Taking the first difference of Eq. (12-31), we find that

;<,) = ^^D,+ '''"-^ "-"'"'

. (12-32)

vo T

and a linear least square fit to the frequency data yields the optimum estimate

of Av. However, the drift in atomic clocks is generally so small that the value

obtained for D will not be statistically significant when r is small enough to

TN-77
208 SAMUEL R. STEIN

satisfy the assumption of white frequency noise. Thus, we are led to consider

the first difference of the frequency,

yjt + t) - v(f) ^^ ^ xi(f + 2t) - 2xi(f + t) + x.fQ

Many atomic clocks are dominated by random walk of frequency noise for long averaging times. Thus, the first difference of the frequency data (the

second difference of the phase data) is white, and the optimum estimate of the

drift is just the simple mean. If instead, a linear least square fit were removed

from the frequency data in this region oft, then the random-walk residuals

would be biased, and it is likely that an optimistic estimate of <7j,(t) would be

obtained.

The optimum procedure would be different if the dominant noise type were flicker of frequency, rather than random walk. But there is no simple

prescription that can be followed to estimate the drift in that case. Fortunately, a ma.ximum likelihood estimate of the prameters for some

typical cases has shown that the mean second diff'erence of phase is still a good estimator of frequency drift in the sense that it introduces negligible bias

in the Allan variance. Thus, in practice there is a simple prescription for computing the Allan variance in the presence of significant drift. Starting

with the phase data, one forms the second difference and uses the simple

average to estimate the mean. The value oft chosen for creating these second

differences must be long enough so that the predominant noise process is random-walk frequency modulation. After subtracting this estimate, the

second-difference data is integrated twice to recover phase data with drift removed, and further analysis, including the computation of the Allan

variance, may proceed. Figures 12-6 through 12-10 illustrate the estimation

of drift. The quadratic dependence of the phase data in Fig. 12-6 nearly

obscures the noise. The first difference of this data produces the nearly linear frequency dependence shown in Fig. 12-7, and the second difference produces

the residuals shown in Fig. 12-8, which appear to be nearly white. Rigorous

statistical analysis of this data indicates that the first difference of the frequency is indeed white with 90% confidence. Next, the mean frequency

difference is subtracted. Then the residuals of Fig. 12-8 are integrated twice,

and the result is the estimate of the phase deviation with drift removed shown

in Fig. 12-9. Fig. 12-10 illustrates the Allan variance of this data calculated by

three techniques. The squares were computed from the data of Fig. 12-6,

while the open circles were computed following the recommended procedure

for estimating the drift. The validity of the approach is illustrated by the black

dots, which are the result of a statistically optimum parameter estimation

procedure.

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12 FREQUENCY AND TIME MEASUREMENT 209

140 TIME (DAYS)

FIG. 12-6 Measured phase difference between a frequency standard and a reference during a 140-day experiment. The nearly quadratic form of the data effectively obscures the noise.

'o>o

z

UJD

o

UJ

a.

u.

139 TIME (DAYS)

FIG. 12-7 One-day frequency averages obtained by taking the first differences of the data in Fig. 12-6. The ordinate is the fractional difference of the daily frequency from a nominal value. The nearly linear change in frequency with time is apparent, although the random deviations are

visible.

UJo

z

UJ

q:UJ-~

^^2

=)o

u. 138 TIME (DAYS)

FIG. 12-8 Second difference of the data in Fig. 12-6. The second difference operation has removed the nonrandom behavior and the residuals appear to be nearly white.

TN-79
210 SAMUEL R. STEIN

140 TIME (DAYS)

FIG. 12-9 Phase variations of the frequency standard due to the residuals, obtained by

performing two integrations on the data of Fig. 12-8. The ordinate scale is expanded

approximately 10 times compared to Fig. 12-6.

-11

»- -12

-13

o • • •

t B • 12 3 4 5 6 7 log T(sec)

FIG. 12-10 Logarithm of the square root of the Allan variance as a function of the logarithm of the averaging time for three different computation methods. The squares were computed from the data of Fig. 12-6 and show the effect of the drift. The open circles were computed from the data of Fig. 12-9. The closed circles were computed using an optimumparameter estimation procedure.

12.1.7 Confidence of the Estimate and Overlapping Samples

Consider three phase or time measurements of one oscillator relative to another at equally spaced intervals of time. From this phase data one can

obtain two adjacent values of average frequency and one can calculate a

single sample Allan variance (see Fig. 12-11). Of course, this estimate does not have high precision or confidence, since it is based on only one frequency

difference. For most commonly encountered oscillators, the first difference of the

frequency is a normally distributed variable with zero mean. However, the square of a normally distributed variable is not normally distributed. This is so because the square is always positive and the normal distribution is

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12 FREQUENCY AND TIME MEASUREMENT 211

TIME

FIG. 1 2-11 Calculation of two average frequencies y, and >2 by measuring the phase of an

oscillator x{t) at times fi, (j. and tj.

completely symmetric, with negative values being as likely as positive ones. The resulting distribution is called a chi-squared distribution, and it has one

"degree of freedom" since the distribution was obtained by considering the squares of individual (i.e., one independent sample), normally distributed

variables (Jenkins and Watts, 1968). In contrast, from five phase values four consecutive frequency values can be calculated, as shown in Fig. 12-12. It is possible to take the first pair and

calculate a sample Allan variance. A second sample Allan variance can be

calculated from the second pair (i.e., the third and fourth frequency

measurements). The average of these two sample Allan variances provides an improved estimate of the true Allan variance, and one would expect it to have

a tighter confidence interval than in the previous example. This could be

expressed with the aid of the chi-squared distribution with two degrees of

freedom.

However, there is another option. One could also consider the sample

Allan variance obtained from the second and third frequency measurements,

that is, the middle sample variance. This last sample Allan variance is not independent of the other two, since it is made up of parts of each of the others. But this does not mean that it cannot be used to improve the estimate of the true Allan variance. It does mean that the new average of three sample Allan

variances is not distributed as chi squared with three degrees of freedom. The

TIME

FIG. 12-12 Calculation of four frequency values y,. y,, y^, and y^ from five phase measurements at times f,, rj, fj, r,. and r,. The sample variance formed from y, and pi and the one formed from y^ and y^, are independent. The sample variance formed from ^2 and y^ is not independent of the other tv^ro but does contain some additional information useful in estimating

the true sample variance.

TN-81
212 SAMUEL R. STEIN

number of degrees of freedom depends on the underlying noise type, that is, white frequency, flicker frequency, etc., and may have a fractional value. Sample Allan variances are distributed as chi square according to the equation

t" = ms]la], (12-34)

where s] is the sample Allan variance, df the number of degrees of freedom

(possibly not an integer), and a] the true Allan variance, which we are

interested in knowing but can only estimate imperfectly.

The probability density for the chi-squared distribution is given by the

relation (Jenkins and Watts, 1968)

2\ = Pix")'

1 /y2^df/2-l.-z'/2

2««f/2r(df/2)

where r(df/2) is the gamma function, defined by the integral

T(t)=

I* x'-'e-'Jx.

Jo

(12-35)

(12-36)

A typical distribution is shown in Fig. (12-13).

Chi-squared distributions are useful in determining confidence intervals for variances and standard deviations, as shown in the following example.

Suppose one has a sample variance s^ = 3.0 and it is known that this variance has 10 degrees of freedom. The object is to calculate a range around

the sample value of s^ = 3.0 that probably contains the true value a]. The

desired confidence is, say, 90%. That is, 10^^ of the time the true value will actually fall outside of the stated bounds. The usual way to proceed is to allocate 5% to the low end and 5% to the high end for errors, leaving 90% in the middle. This is arbitrary and a specific problem might dictate a different

POC)

3.94 18.3

FIG. 12-13 Approximate form of a typical chi-squared distribution. For 10 degrees or

Treedom. S?^ of the area under the curve corresponds to values of x' less than 3.94, and an

additional 5% corresponds to values of x^ greater than 18.3.

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12 FREQUENCY AND TIME MEASUREMENT 213

allocation. By referring to tables of the chi-squared distribution, one finds that for 10 degrees of freedom (df = 10) the 5% and 95% points correspond to

Z^(0.05) = 3.94, z^(0.95) = 18.3. (12-37)

Thus, with 90% probability the calculated sample variance Sy = 3 satisfies the

inequality

3.94 < {d()sl/al < 18.3, (12-38)

and this inequality can be rearranged in the form

1.64 < (T^ < 7.61. (12-39)

The estimate s^ = 3 is a point estimate. The estimate 1.64 < aj < 7.61 is an interval estimate and should be interpreted to mean that 90% of the time

the interval calculated in this manner will contain the true aj.

12.1.8 Efficient Use of the Data and Determination of the

Degrees of Freedom

Typically, the sample variance is calculated from a data set using the

relation

^n3T „?/'" -^"^^'

where it is implicitly assumed that the r„'s are random and uncorrelated (i.e., white) and where f is the sample mean calculated from the same data set. If all of this is true, then 5^ is chi-squared distributed and has N — I degrees of

freedom.

Consider the case of two oscillators being compared in phase with N values of the phase difference obtained at equally spaced intervals Tq. From these N

phase values one obtains N — 1 consecutive values of average frequency, and

from these one can compute N — 2 individual sample Allan variances (not all independent) for t = Tq. These N — 2 values can be averaged to obtain an

estimate of the Allan variance at r = Tq.

The variance of this Allan variance has been calculated (Lesage and

Audoin, 1973; Yoshimura, 1978). This approach is less versatile than the method of the previous section since it yields only symmetric error limits. However, it is simple and easy to use. Let &{N) be the relative difference between the sample .Allan variance and the true value. Thus,

sj = [1 + A(iV)](T,2(T). (12-41)

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214 SAMUEL R. STEIN

TABLE 12-3

Variance of the Relative Difference between the Sample

Allan Variance and the True Value {CJN)*

Noise type a C,

White phase 2 3.88 Flicker phase 1 3.88 White frequency 2.99 Flicker frequency -1 2.31 Random-walk frequency —2 2.25

' N is the number of phase measurements. The result is accurate to better than 10 *„ for N larger than 10.

The quantity A(N) has mean zero. For N larger than 10, the variance of A is approximately

(7^(A) = CJN. (12-42)

Table 12-3 gives the constant C, for the five major noise types.

Using the same set of data it is also possible to estimate the Allan variances

for integer multiples of the base sampling interval t = mtQ. Now the

possibilities for overlapping sample Allan variances are even greater. For a data set of N phase points, one can obtain a maximum of exactly N — 2m

sample Allan variances for t = mxQ. Of course only {N - I) 2m of these are generally independent. Still, the use of all of the data is well justified since the confidence of the estimate is always improved by so doing. Consider the case

of an experiment extending for several weeks in duration with the aim of

getting estimates of the Allan variance for t values equal to a week or more. As always, the purpose is to estimate the "true" Allan variance as well as

possible, that is, with as tight an uncertainty as possible. Thus, one wants to use the data as efficiently as possible. The most efficient use is to average all possible sample Allan variances of a given t value that one can compute from

the data. This procedure is illustrated in Fig. 12-14. In order to calculate confidence intervals for a sample variance, it is necessary to know the number of degrees of freedom. This has been done by

both analytical and Monte Carlo techniques, and empirical equations have

been found that are accurate to 1% for white phase, white frequency, and

random-walk frequency modulation. The tolerance is somewhat larger for

flicker frequency and phase modulation (Howe et al., 1981). The empirical

equations for the degrees of freedom are given in Table 12-4. Table 12-5 gives the degrees of freedom for selected values of N, the total number of phase

values, and m, the number of intervals averaged. Figure 12-15 illustrates the number of degrees of freedom for all noise processes as a function of t for the case of 101 total phase measurements.

TN-84
12 FREQUENCY AND TIME MEASUREMENT 215

x(t)

-jH FULLY OVERLAPPING 7

' ,7

TIME

FIG. 12-14 Illustration of the case of t = 4to, for which the ratio of the number of fully overlapping to nonoverlapping estimates of the variance is more than 8 for the 57 phase points shown. When the averaging time for the computation of mean frequencies t exceeds the sampling time lo. the number of fully overlapping mean frequencies is far larger than the number

of nonoverlapping frequencies. In general, for large N approximately 2m times as many

estimates of the sample variances can be computed using the fully overlapping technique.

TABLE 12-4 Number of Degrees of Freedom for Calculation of the Confidence of the Estimate of a Sample Allan Variance'

Noise type df

White phase

Flicker phase

White frequency

Flicker frequency

Random-walk frequency

{S -HX.V - 2m)

2(.V - m)

r3(.V - 1) _ 2(.V - 2) 1 4m^

L 2m ~ N J 4m^ +

2{N - 2) 2.3N - 4.9

5:V=

for m = 1

for m > 2 4mi,V -- 3m I

.V - 2 (;V - 1)^ - 3m(.V - I) -I- 4m^ "^ (.V - 3)-

° For T = mtg from .V phase points spaced r^ apart.

TN-85
216 SAMUEL R. STEIN

TABLE 12-5 Number of Degrees of Freedom for Calculation of the Confidence of the Estimate of a Sample

Allan Variance for the Major Noise Types*

White Flicker White Flicker Random-walk

N m phase phase frequency frequency frequency

9 1 3.665 4.835 4.900 6.202 7.000 2 3.237 3.537 3.448 3.375 2.866 4 1.000 1.000 1.000 1.000 0.999

129 1 65.579 79.015 84.889 110.548 127.000

2 64.819 66.284 71.642 77.041 62.524 4 63.304 52.586 42.695 36.881 29.822

8 60.310 37.306 21.608 16.994 13.567 16 54.509 22.347 9.982 7.345 5.631 32 44.761 9.986 4.026 2.889 2.047 64 1.000 1.000 1.000 1.000 1.000

1025 1 526.373 625.071 682.222 889.675 1023.000

2 525.615 543.863 583.622 636.896 510.502 4 524.088 459.041 354.322 316.605 253.755

8 521.038 366.113 186.363 156.492 125.39S

16 514.952 269.849 93.547 76.495 61.241 32 502.839 179.680 45.947 36.610 29.210 64 478.886 104.743 21.997 16.861 13.288 128 432.509 50.487 10.003 7.281 5.516 256 354.914 17.429 4.003 2.861 2.005 512 1.000 1.000 1.000 1.000 1.000

* S is the number of equally spaced phase points that are taken m at a time to form the averaging

time.

12.1.9 Separating the Variances of the Oscillator and the Reference

A measured variance contains noise contributions from both the oscillator under test and the reference. The individual contributions are easily separated if it is known a priori that the reference is much less noisy than the device under test or equal to it in performance. Otherwise, the individual

contributions can be estimated by comparing three devices (Barnes, 1966).

The three possible joint variances are denoted by afj, cj^, and cf,,, while the individual device variances are ef, a], and al. The joint variances are composed of the sum of the individual contributions under the assumption

that the oscillators are independent:

^2 _ _2 , _2

Oij - CT, + Gj,

/-2 — ,-2 , _2

/t2 _ --2 I ^2

(12-43)

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12 FREQUENCY AND TIME MEASUREMENT 217

100 IT100 T (IN UNITS OF DATA SPACING)

FIG. 1 2-1 5 Number of degrees of freedom as a function of averaging time for the case of 101 phase measurements: The heavy broken line is for random-walk frequency noise, the light broken line is for flicker frequency noise, the dotted line is for white frequency noise, the heavy

solid line is for flicker phase noise, and the light solid line is for white phase noise.

An expression for each individual variance is obtained by adding two joint

variances and subtracting the third:

(12-44)

This method works best if the three devices are comparable in performance.

Caution must be exercised since Eqs. (12-44) may give a negative sample

Allan variance despite the fact that the true Allan variance is positive definite. This is possible because the confidence interval of the estimate is sufficiently

large to include negative variances. Such a result is an indication that the confidence intervals of the sample Allan variances are too large and that more data is required.

12.2 DIRECT DIGITAL MEASUREMENT

12.2.1 Time-Interval Measurements

A common technique for measuring the phase difference between oscil- lators having nearly equal nominal frequencies is the use of direct timeinterval measurements. In this section and those that follow, the symbols v,o and v,o are used to indicate the nominal values of Vi and Vj, respectively. In

the simplest form of this technique, a time-interval counter is started on some

TN-87
218 SAMUEL R. STEIN

ACTUAL PHASE

DIFFERENCE f

START SIGNAL

TIME

^MEASURED PHASE DIFFERENCE STOP

SIGNAL

FIG. 12-16 The phase difference measured by a time-interval counter is the phase difference between the start signal and the stop signal modulo the period of the stop signal.

arbitrarily selected positive-going zero crossing of the signal from one

oscillator (started on Vjq at time fj and stopped in the next positive-going zero crossing of the second oscillator (stopped on Vjo at time ti). The

measured time difference is

^2(^1) - xi(fi) ^ -Pt,[1 + (V20 - Vio)/v,o]. (12-45)

where P is the reading of the time-interval counter and t^ the period of its time base (Allan ei al., 1974). The units of the time difference is seconds of

oscillator number 1. Equation (12-45) demonstrates an important characteristic of both time- and phase-difference measurements. Because of distortion the phase of an oscillator is generally not well known except at zero crossings.

Thus, the quantity usually measured is X2(t2) — x^(t^). However, all analysis

techniques require the phase difference at the same time, and the translation

requires a correction that takes into account the difference in frequency

between the two oscillators. This correction is the reason for the second term

in the brackets on the right-hand side of Eq. (12-45).

The simple scheme described above measures a maximum accumulated

phase difference of one cycle of the signal. When the phase difference exceeds one cycle the counter reading is periodic, as shown in Fig. 12-16. This

ambiguity can be reduced by dividing the signals from each oscillator before

the time-interval measurement. The complete system is shown in Fig. 12-17. The effect of the divider is to increase the time interval before an ambiguity

DIVIDER G>

OSCILLATOR

1 DIVIDER G>

OSCILLATOR

2

tn

STARTTIME- INTERVAL COUNTER

FIG. 12-17 Schematic diagram of the dividers used in conjunction with a time-interval counter to increase the maximum measurable phase difference to S cycles of the stop signal.

TN-88
12 FREQUENCY AND TIME MEASUREMENT 219

occurs to AV^iO' where N is the divisor. Such measurement systems are used

at many standards laboratories for the long-term measurement of atomic

clocks, whose output is usually divided down to 1 pulse/sec. Since timeinterval counters with resolution better than O.lnsec are available, this measurement scheme is suitable for long-term performance monitoring,

yielding frequency-measurement precision of 10"'"* for 1-day averages.

12.2.2 Frequency Measurements

Average frequency is measured most directly using a frequency counter. Used this way, the counter determines the number of whole cycles M

occurring during a time interval t given by the counter's time base. Thus

v(0; r) = (M -I- AM)/x ^ M/-c, (12-46)

where virj; f,) denotes the average frequency over the interval from tj to ti and A:V/, the fractional cycle, is not measured by the counter. The starting

time is arbitrarily called f = 0. Thus, the quantization error is given by

Avg/<v> < 1/M. (12-47)

12,2.3 Period Measurements

For low frequencies, the number of cycles counted may be small and the quantization error can be very large. By measuring the period instead of the frequency, it is possible to decrease the error without increasing the duration

of the measurement. A period counter measures the duration of M whole

cycles of the signal as N cycles of the time base t^ . The fraction of a cycle AN

is not measured. Thus, we have

M = v(0: M/voX-V + AiV)T„ (12-48)

and therefore

v(0; iW/vo) ^ A//iVT, (12-49)

and the quantization error is

Avp/<v> < i/N. (12-50)

Frequency measurements are almost never used to characterize precision

oscillators, but period measurements are very common. A straightforward

extension of this method eliminates the bias potentially introduced by the quantization error and permits the measurement of accumulated phase. The

counter must be capable of being read without halting the counting process.

TN-89
220 SAMUEL R. STEIN

OSCILLATOR

TIME BASE

FIG. 12-18 Two-counter system to eliminate dead time in period measurements. The two

counters alternately count the number of cycles of the time base in N periods of the oscillator under test.

Alternatively, a second counter may be used to begin counting the same time

base when the first counter stops. The second approach is illustrated in Fig.

12-18. This type of measurement system is sometimes called a chronograph.

12.3 SENSITIVITY-ENHANCEMENT METHODS

12.3.1 Heterodyne Techniques

It is possible for oscillators to be very stable, and values ofo-j.it) can be as small as 10"^* in some state-of-the-art standards. Thus, one often needs

measuring techniques capable of resolving very small fluctuations in y{t). One

of the most common techniques is the heterodune or beat-frequency

technique. In this method the signal from the oscillator under test is mixed

with a reference signal of almost the same frequency so that one is left with a lower average frequency for analysis without reducing the frequency (or

phase) fluctuations themselves.

In principle, it is possible to analyze the most general measurement case, where no restrictions are placed on the average frequency or phase difference between the two oscillators under test. Equation 12-1 can be inverted as

27rvof + 0(f) = arcsin[F(:)/Vo] (12-51)

and used to obtain the series 4>{m-) by sampling the voltage at regular time

intervals. This direct technique is not used, because it requires unobtainable

mixer performance characteristics. The high-level rf signals that are required

for low-noise phase measurements produce significant harmonic distortion, so that the output of the phase detector deviates significantly from a sine wave. Furthermore, the distortions are generally sensitive to level and

environmental perturbations. However, the phase relationships among the various harmonics are very stable, so it is possible to use the repetition of one

TN-90
12 FREQUENCY AND TIME MEASUREMENT 221

point on the waveform in order to count cycles. The positive-going zero

crossings are normally chosen in order to provide immunity from changes in both the amplitude and symmetry of the waveform.

Consider two signals whose frequency difference is much less than the frequency of either oscillator:

^i(f) = KioSin[27rviof + (piU) + <t>yo] and (12-52)

Vzit) = V20 sin[27rVio^ + (pzi^) + <l>2ol where |vio - Vjol < Vio and the constants 0io and 02o represent the nominal

phases of the two signals.

Suppose that the two signals are mixed in a linear product detector and

filtered so that the signal at the sum frequency Vjo + V20 is highly attenuated. The result is

V(t) S FoCOS[27t(v,o - V2o)r + 010 - </>20 + (Plit) - 02(0], (12-53)

which may be characterized by any of the measurement techniques discussed

in Section 12.2. The amplitude Vq of the mixer output is a function of the mixer design, the input amplitudes, and the output termination (Walls et ai,

1976). Using the definition (12-10), we find that for the heterodyned signal

XH(O = (l/27rvH)A0(r), (12-54)

where

Vh = |vio - V20I (12-55)

and

A0(r) = 0,(0 - 02(f). (12-56)

Equation (12-54) may be rewritten as

XhU) = (vo/vH)x(r), (12-57)

from which we conclude that a given phase change corresponds to a larger

time deviation for the heterodyne signal than for the original signal. As a

result, the quantization error for the period measurement technique is reduced by the factor Vh/vq .

12.3.2 Homodyne Techniques

The limit of the heterodyne method, called homodyne, occurs when

Vio = ^20- In this case the output of the phase detector is given by

V{t) S Vo cos[0io - 020 + 0i(n - 02(0]- (12-58)

• See Appendix Note # 6

TN-91
222 SAMUEL R. STEIN

The analysis of phase noise is accompHshed by arranging that ^10 - 020 = "A which can be achieved with a phase shifter. Then,

V(t) ^-Vo sin[0i(O - </.j(f)] S Ko[02(f) - <^,(0]. (12-59)

There are various methods by which one can control the signal V2(t) so that

v,„ = vjo without producing significant correlation between 4>2(t) and <pi{t).

When any one of these methods is used, it is possible to use VU) as a measure

of ^r). Two methods, delay lines and phase-locked loops (Gardner, 1966), are described below.

12.3.2.1 DISCRIMINATOR AND DELAY LINE

The circuit of a discriminator or delay-line system for measuring phase

noise is illustrated in Fig. 12-19. The delayed signal is given by

Vzit) = ^i(t - h) = 1^20 sin[27:v,o(r - r^) -I- <^,(t - t^) + <^,o + 4>sl-(12-60)

When the phase shifter is set for quadrature, 0, — 2n\\Qt^ = n'2 and

^'2(0 = Vio sin[27tViof -I- <t>i{t - fd) + <^io + ".'2]. (12-61)

The output of the phase detector is given by

Vit) = Vol4>,{t - g - </>,(n]. (12-62)

Substituting Eq. (12-62) into Eq. (12-20), we obtain

y{t - u: t) = - V(t)/2nx'oVoti (12-63)

and we see that the delay-line method can be used to produce samples of

.v(mTo) by varying the delay time. However, the technique is used more

frequently with a fixed delay by restricting its application to the region of t much greater than the delay time, so that y(t — r^: r) is a good appro.ximation

for the instantaneous frequency. Under this assumption spectrum analysis of

PHASE SHIFTER V,(t) DISCRIMINATOR OR

DELAY LINE

Vj(t)

^NVoCt)

(X)

^"' OSCILLATOR V' PHASE

DETECTOR

FIG. 12-19 A delay-line phase-noise-measurement system. When the phase shifter is adjusted so that r2(r) is in phase quadrature with VgC). the output of the phase detector is approximately equal to the instantaneous frequency deviation of the oscillator. The spectral density of the source may be estimated for Fourier frequencies small compared to the inverse of

the delay time.

TN-92
12 FREQUENCY AND TIME MEASUREMENT 223

the signal from the mixer can be used to estimate the spectral density of the frequency fluctuations:

S,(/)^—-^Sk,ko(/) for /<ll/7if,. (12-64)

Frequency discriminators are applied in an analogous fashion. A resonant

circuit is often used to provide discrimination since it produces a phase shift proportional to the frequency deviation from the resonant frequency. For

example, the phase shift on reflection from a resonance with loaded quality

factor Q is = arctan(2ey) ^ 2Qy, (12-65)

provided that the frequency deviation is small compared to the bandwidth of

the resonance and the applied signal is nearly at the center frequency of the discriminator. This can be accomplished either manually or with a frequencylocked loop. The design of such a loop is similar to the phase-locked loop of

the next section. Once again, one can spectrum analyze the signal from the mixer to obtain

W)-(2^Vko(/) for f<Vo/Q- (12-66)

The noise floor for measurements made with either a delay hne or discrimi- nator normally results from white voltage noise in the analysis circuitry and

is independent of the Fourier frequency. We denote the noise floor Sy y^ (minimum) and find the noise floor for frequency or phase measurements by

S^(noise limit) = -rj S/noise limit) = ., ^ Syiyj{rmnimum).

(12-67)

Consequently, the discriminator or delay-line technique is limited in sensitivity since the output voltage is proportional to the frequency deviations.

Greater sensitivity is possible using two oscillators in a phase-locked loop.

The noise in the reference is an important consideration, even though the reference is passive in the case of a discriminator or a delay line. If the

oscillator has sufficiently low noise, then the circuits described measure the variations of the discriminator center frequency or the delay variations in the delay line.

12.3.2.2 PHASE-LOCKED LOOP

The block diagram for the most general phase-locked loop that will be

considered here is shown in Fig. 12-20. The noise voltage summed into the loop is a schematic way of representing (t>„{i), the open-loop phase noise of the

TN-93
224 SAMUEL R. STEIN

PHASE DETECTOR

I

!Qt

FH.TEN

F(«)

OSCILLATOR , UNDER TEST ' _

t NOISE VOLTAGE <1>^

FIG. 12-20 Block diagram or a phase-locked loop. The order of the loop is determined by

the filter transfer function. For convenience, noise in the oscillator under test is introduced at the summing junction.

oscillator under test. Phase noise in the reference oscillator is denoted by

The purpose of using a phase-locked loop is simply to guarantee that the two oscillators are, on the average, in phase quadrature. When the oscillators are near quadrature, the voltage output of the phase detector is proportional

to the difference in phase between the two output signals.

Analysis of the phase-locked loop yields the result

where G^^is) is the open-loop transfer function defined by

G.,(5) = :^^^1^ (12-69)

and <^„(5) and <f>,tf(s) are the Laplace transforms of the corresponding timevarying quantities. We can also calculate the voltage output of the phase

detector,

1 + G„{s)

as well as the feedback voltage to the varactor,

Vf(s) = F{s)V,(5) = ^ialfl i^^js) - 0.(5)]. (12-71)

Assuming that the phase noise of the two oscillators is not correlated.

TN-94
12 FREQUENCY AND TIME MEASUREMENT 225

INPUT o- -y^

T

-o OUTPUT

FIG. \1-1\ Circuit diagram of the most common loop filter for a second-order phase- locked loop. Resistor R^ is required for stable operation. Capacitor C provides the low- frequency gain needed to reduce the phase errors of the first-order loop.

Tj/T,

-SdBOCTAVE

FIG. 1 2-22 Bode plot for the loop filter of Fig. 12-21.

Thus, if we know the behavior of G^^jd), then we can relate the measured

spectrum of the voltage at the output of the phase detector or at the varactor tuner to the sum of the spectral densities of the phase noise of the two

oscillators. The loop filter is often chosen to be a pure gain. The resulting first-order loop has a significant drawback: the two oscillators are offset from quadrature by a phase shift proportional to their open-loop frequency difference. In order to maintain system calibration, the operator must remove the fre- quency offset from time to time. This problem can be eliminated by using a second-order loop. Figure 12-21 illustrates one loop filter that can be used to achieve the desired frequency response. The transfer function of this filter is

F{S) = (1 + 5T2)/5T„ (12-74)

where r, = /?2C and Tj = R^C. Figure 12-22 shows the Bode plot of the frequency-response function of this filter. Substitution of Eq. (12-74) into Eq.

(12-69) yields the open-loop frequency-response function

-2 ¥ 2K(o„a)

where

and

O^Jjco)

(o:

O)'

co. = LKoKJ,,r'

S = 2T2^n-

(12-75)

(12-76)

(12-77)

TN-95
226 SAMUEL R. STEIN

100 1-

1/T, lu.

ClJ FIG. 1 2-23 Bode plot of the open-loop frequency-response function for a phase-locked loop

having the loop filter of Fig. 12-21. Parameters were chosen to illustrate a stable condition.

The first requirement to be satisfied by the loop parameters is that the closed loop be stable. Since the transfer function G^^is) has no poles or zeros

for s > 0, a sufficient requirement for the phase-locked loop to be stable is that the slope of the Bode plot of |GeqOc^)| be less steep than - 12 dB/octave at the point where \Gg^{j(o)\ = 1. The Bode plot of IG^qfJaj)! is shown in Fig. 12-23 for a case where the loop operation is stable.

It is desirable for the loop to be nearly critically damped, that is, C = 1- At

critical damping the natural frequency of the loop is related to t^ by

co„.: = , = 2/T2. (12-78)

Under the same conditions the unity gain frequency is

">!.{ = ! =4.12/T,. (12-79)

The second requirement to be satisfied by the phase-locked loop is related to the accuracy with which spectral-density measurements can be made.

Substitution of Eq. (12-75) into Eq. (12-72) yields

SyAio) =

(o,^ - oj'-y- + 4:'-co'oj'„ [S^„» + 5^»]. (12-80)

Since the proportionality factor has a high pass response, it is possible to use an essentially constant calibration to relate Syjoj) and S^(oj). For example, if we require that

SyJ(^) ^ KllS^^Jc}) + S^Joj)-] (12-81)

with no more than lO"-; error for all Fourier frequencies greater than

2n rad/sec, then for the critically damped loop the requirement on t, is

T-. > 1.4 sec.

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12 FREQUENCY AND TIME MEASUREMENT 227

The third requirement on loop performance is that the frequency offset between the two oscillators produce negligible phase shift of the oscillators from quadrature. In the ideal loop the phase error for a frequency error Av

introduced at time f = is

However, in the actual circuit there is a finite phase error due to the limited loop gain of the amplifier of Fig. 12-21. Nevertheless, the phase error is reduced by 10^ compared to its value for a first-order loop. Typically, the error is less than the residual phase error due to the voltage offset at the mixer

output and should be much less than 1°. The feedback loop reduces the sensitivity of the system for measurements

of the phase spectral density for Fourier frequencies less than the unity-gain

frequency of the phase-locked loop. One way to avoid this problem is to

utilize the feedback voltage V(. Substituting Eq. (12-75) into Eq. (12-73), we

find that

Svci(^) = -r-i ^.^ ,4-^2 2 [^>rJ^) + 5y„(^)]- (12-82)

[or - (D-y + A<^-(D„a)'- For this case, the proportionality factor has a low pass response and a constant calibration factor may be used to relate Sv^io^) to S^{aj).

12.3.3 Multiple Conversion Methods

Quite often the beat frequency between the signal under test and the laboratory reference is unsuitable or inconvenient for frequency-stability

measurements. The frequency may be too high for the available counters or

the heterodyne factor may be too small to yield the required noise enhance- ment. Under these circumstances a second mixing stage in series with the first can be used to produce the desired beat frequency. On the other hand, the

direct beat frequency between two oscillators may be too small. For example,

the frequencies of commercial cesium-beam frequency standards are usually so close together that the beat frequency between two devices would be near

1 cycle day, making it impossible to observe the stability at shorter times. This limitation can be overcome by the use of two parallel mixing stages.

12.3.3.1 FREQUENCY SYNTHESIS

A commercial frequency synthesizer is usually the most convenient way to produce arbitrary reference frequencies for stability measurements. A mixing

stage preceding the synthesizer can be used both to bring the signal into the

TN-97
228 SAMUEL R. STEIN

9.3 GHz

OSCILLATOR

G>—

UNDER

MHz

Li kHz

TEST

SMHz

<: (~)-H X1836 COUNTER 9

EFERENCE

FREQUENCY

SYNTHESIZER 119.999 MHz

FIG. 12-24 Use of frequency synthesis lo measure oscillators whose frequency differs significantly from the available low-noise reference. It may be necessary to use a frequency

multiplier to bring the signal into the range of the available synthesizer or to overcome the

synthesizer's phase noise.

appropriate range and to enhance the oscillator noise compared to the short- term phase noise of the synthesizer. Figure 12-24 demonstrates both aspects

of the technique.

The initial mixing stage from the microwave frequency to the rf results in a

substantial heterodyne factor, 77.5 for the example chosen. The output of the

first conversion stage lies within the range of low-noise commercial frequency

synthesizers, which makes it possible to obtain a fixed, low beat frequency

over a wide range of input frequencies. The initial mixing stage also reduces

the frequency synthesizer's contribution to the measurement-system noise. Figure 12-25 shows the typical phase excursions of a high-quality commercial

synthesizer operated near 5 MHz.

Under some circumstances a frequency divider may be used to provide

the signal for the second mixing stage, as shown in Fig. 12-26. This technique

has the disadvantage of requiring a custom divider but results in much

lower measurement noise than the direct use of a synthesizer with a single

heterodyne stage.

X(t)

-10 80,000 TIME (SEC)

FIG. 1 2-25 Typical phase excursions of a commercial frequency synthesizer.

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12 FREQUENCY AND TIME MEASUREMENT 229

5.00688 MHz

G> f 726

OSCILLATOR

UNDER

TEST

SMHz

& H>

7 Hz COUNTER

REFERENCE

FIG 1 2-26 Use of a simple divider as a substitute Tor a commercial frequency synthesizer in a heterodyne measurement system. Better noise performance can result from the initial mixing

stage.

12.3.3.2 THE DUAL-MIXER TIME-DIFFERENCE

TECHNIQUE

There is no best answer to the question of how to make frequency-stability

measurements. However, by combining versatility with low-noise performance, the dual-mixer time-difference technique (Cutler and Searle, 1966: Allan

and Daams, 1975) shown in Fig. 12-27 comes close to the ideal. The original

motivation for this method was to use a transfer oscillator and two mixers in parallel to permit short-term frequency-stability measurements between

oscillators that have an inconveniently small frequency difference. The

transfer oscillator is most easily realized with a frequency synthesizer locked

to one of the oscillators, designated oscillator 1 in Fig. 12-27. By convention

the frequency of the synthesizer is set low compared to the oscillator under

test, so we write the frequency of the synthesizer as

V, = vi(l - l/R). (12-83)

The constant R is equal to the heterodyne factor, which can be seen by

calculating the beat frequency between oscillator 1 and the synthesizer:

Vbi = vi - V, = yJR. (12-84)

OSCILLATOR 1

MIXER

FREQUENCY

SYNTHESIZER

(=>

MIXER

ZEROCROSSING

DETECTOR

SCALER

1 ^^

ZEROCROSSING

DETECTOR

START TIME-INTERVAL COUNTER

STOP >

l_ SCALER --^„

OSCILLATOR 2

FIG. 12-27 A dual-mixer measurement system. The scalars measure the number of whole

cycles of elapsed phase, while the time-interval counter measures the fractional cycle.

TN-99
230 SAMUEL R. STEIN

The combination of oscillator, frequency synthesizer, and mixer functions as a divider and scaler 1 functions as the system clock, recording elapsed time in units of cycles of oscillator 1

.

The signals from oscillators 1 and 2 are represented according to Eq.

(12-52) with 4>io = 020 = 0. and the signal from the synthesizer is written

KU)= Kocos[2rtv,or + 0.(0]. [12-85)

The phase of the synthesizer retards nearly linearly in time compared to the

phase of oscillators 1 and 2. At time t;^ the synthesizer reaches phase

quadrature with oscillator 1 and the beat signal crosses zero (in the positive

direction), producing a pulse from the zero-crossing detector and starting the time-interval counter. At a later time tf^ the continued sweep of the

synthesizer has brought it into quadrature with oscillator 2, and a pulse is produced that stops the time-interval counter. The phase difference between

the oscillators can be written in terms of the three counter readings:

02(fM) - 0i(^«) = 2{N - M)n - Ini^^.Usi- f^)]tc/', (12-86)

where N is the reading of scaler 2, M the reading of scaler 1, P the reading of

the time-interval counter, and Tj the period of its time base (Stein ei ai, 1983).

Comparison with Eq. (12-45) for direct time-interval measurements reveals

that the role of the scalers is to accumulate the coarse phase difference between the oscillators, while the time-interval counter provides fine-grain

resolution of the fractional cycle. This process is illustrated in Fig. 12-28. The

advantage of the technique over direct time-interval measurements is that the noise performance is improved by the large heterodyne factor, allowing time

resolution of 0.1 psec to be obtained. The synthesizer degrades the noise

performance very little since it contributes to the noise only over the interval

x(t)

TIME

FIG. 12-28 Total elapsed phase measured b> ihe dual-mixer system of Fig. 12-27 (solid

line). This phase measurement consists of two components: the number of full cycles that have

elapsed is the step function plotted as a dashed line: the fractional cycle is the saw-tooth function

plotted as open circles.

TN-100
12 FREQUENCY AND TIME MEASUREMENT 23f

* The average beat frequency VB2(^vr- h) cannot be known exactly, but it may

be estimated with sufficient precision if it changes slowly compared to the interval between measurements. If the primed and unprimed variables represent two independent measurements, then

VB2(f.w: h) = if^' - N)/[/?(M' - MVv^o + t,(P' - P)]. (12-87)

12.3.3.3 FREQUENCY MULTIPLICATION

A frequency multiplier produces n full cycles of the output signal for each

cycle of the input signal, where n is an integer determined by the design of the device. Such a device is also a phase multiplier, that is, the total phase

accumulation of the output signal is n times as great as the phase accumulation of the input signal:

<D„„, = 27rv„„,f -I- (f)^Jt) = 27c(nvjr + n(f>Jt). (12-88)

It follows that the spectral density of the output signal is enhanced by a factor of n^ compared to the input signal,

making it easier to perform the necessary noise measurements. Similarly, it is also easier to make Allan-variance measurements. If the oscillator under test and the reference are both multiplied by the same factor, the beat frequency

will be n times larger than with no multiplication but the heterodyne factor

will be the same. The zero crossings that must be detected by the counter have n times higher slope and more easily overcome the voltage noise in the counter trigger circuits. The ability to measure frequency stability is only

enhanced if the multiphers have extremely low phase noise themselves. This is the case for many modem multipliers that are triggered by the zero crossings

of the input signal. As a result, the use of multipliers can reduce the performance requirements on the phase detector and the following low-noise

amplifiers.

12.4 CONCLUSION

The IEEE recommendations have achieved the goal of introducing

substantial uniformity in the specification of oscillator performance. The

Allan variance and the one-sided power spectral density of phase have proved

sufficient to evaluate oscillators for all common applications. In a few cases more specialized measures are helpful in relating performance to the specific

application. For example, the rms time-prediction error is helpful in judging a

clock's ability to keep time over long intervals (Allan and Hellwig, 1978).

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232 SAMUEL R. STEIN

However, the specialized performance measures are generally calculable in terms of the IEEE recommended measures.

Significant progress has been made during the last 15 years in measure- ment techniques and data processing. These advances have obscured the dividing line between the frequency domain and the time domain. Today the

spectral density and the variance are most often computed from the identical input data set, the equally spaced time series of the phase deviations. The

choice of a specific measurement setup can be made mostly on a cost versus performance basis. Perhaps the biggest advance in commercially available

equipment is the introduction of heterodyne measurement techniques for time-domain (counter-based) measurements. As a result, the noise performance of these systems has improved dramatically.

One recommendation that should be made is to perform measurements as high up in the measurement hierarchy as possible. Direct measurement of the

phase deviation is most desirable. This approach places the largest share of

the burden on the measurement equipment, minimizes long-term errors, and

maximizes data processing flexibility.

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399

Chapter 12

Ablowich. D.. Jr (1966) Some observations concerning frequency and time sundardization

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4"'0-4"3

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TN-117
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TN-118
CHAPTER 12 415

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• See Appendix Note # 10

TN-119
416 CHAPTER BIBLIOGRAPHIES

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Zelentosov. V. 1 . Suiakshin. A. S.. and Fedorov. N. P. (1976). Meter for measuring the carrier frequency of single ultrahigh-frequency pulses. Instrum. Exp Tech. (Engl. Transl.) 19. 476-477,

Zhilin. N. S.. and Subbotin. L. S. ( 1977). Devices for reproducing reference time intervals. Meas.

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investigation of the metrological characteristics of frequency meters with digital positive feedback and measures to improve them. Otbor Peredacha Inf. 50. 1 10-1 15 In Russian

TN-120
IEEE TRANSACTIONS ON L'LTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, VOL. UFFC-34. NO. 6. NOVEMBER 1987 647 Time and Frequency (Time-Domain) Characterization,

Estimation, and Prediction of Precision

Clocks and Oscillators

Invited Paper

DAVID W. ALLAN

Abstract— \ tutorial review of some time-domain methods of characterizing the performance of precision clocks and oscillators is presented. Characterizing both the systematic and random deviations is considered. The .Allan variance and the modified Allan variance are

defined, and methods of utilizing them are presented along m ith ranges and areas of applicability. The standard deviation is contrasted and

shov^n not to be, in general, a good measure for precision clocks and

oscillators. Once a proper characterization model has been developed, then optimum estimation and prediction techniques can be employed.

Some important cases are illustrated. As precision clocks and oscilla- tors become increasingly important in society, communication of their characteristics and specifications among the vendors, manufacturers,

design engineers, managers, and metroiogists of this equipment be- comes increasingly important.

i

4' Introduction

WHAT THEN." asked St. Augustine, "is time?

If no one asks me. I know what it is. If I wish

to explain it to him who asks me, I do not know." Though

Einstein and others have taught us a lot since St. Augustine, there are still many unanswered questions. In particular, can time be measured? It seems that it cannot; what

is measured is the time difference between two clocks. The time of an event with reference to a particular clock can be measured. If time cannot be measured, is it physical, an abstraction, or is it an anifact?

We conceptualize some of the laws of physics with time

as the independent variable. We attempt to approximate

our conceptualized ideal time by invening these laws so

that time is the dependent variable. The fact is that time

as we now generate it is dependent upon defined origins,

a defined resonance in the cesium atom, interrogating

electronics, induced biases, timescale algorithms, and

random penurbations from the ideal. Hence, at a signifi- cant level, time—as man generates it by the best means

available to him— is an artifact. Corollaries to this are that every clock disagrees with every other clock essentially

always, and no clock keeps ideal or ""true" time in an

abstract sense except as we may choose to define it. Fre- quency or time interval, on the other hand, is fundamental

to nature; hence the definition of the second can approach

Manuscnpi received .Slay 11. 1987: revised June 15. 1987. The author is with the Time and Frequency Division. National Bureau

of Standards, 3:.^ Broadwa% . Boulder. CO 80303. IEEE Lou Number 8716461.

the ideal—down to some accuracy limit. Noise in nature

is also fundamental. Characterizing the random variations

of a clock opens the door to optimum estimation of en- vironmental influences and to the design of optimum combining algorithms for the generation of uniform time and

for providing a stable and accurate frequency reference. Let us define V{t) as the sine-wave voltage output of a

precision oscillator:

K(/) = Fosin*(r) (1)

where $(f ) is the abstract but actual total time-dependent

accumulated phase starting from some arbitrary origin ^(r = 0) =0. We assume that the amplitude fluctuations are negligible around Vq. Cases exist in which this assumption is not valid, but we will not treat those in the context of this paper. This lack of treatment has no impact on the development or the conclusions in this paper. Since infi- nite bandwidth measurement equipment is not available

to us, we cannot measure instantaneous frequency; there- fore v{t) = ( l/27r) d^/dt is not measurable. We can

rewrite this equation with vq being a constant nominal fre- quency and place all of the deviations in a residual phase

K(r) = |/osin(27r.of + <^(0)- (2)*

We then define a quantity y(t) = (i>(t) - vq)/vq, which

is dimensionless and which is the fractional or normalized

frequency deviation of vit) from its nominal value. In- tegrating _v(r) yields the time deviation x{t), which has

the dimensions of time

.x(t Jo y{t')dt: (3)

From this, the time deviation of a clock can be written as

a function of the phase deviation:

2-i'o (4)

Syste.m.atic Models for Clocks and Oscillators

The next question one may ask is why does a clock

deviate from the ideal? We conceptualize two categories

• See Appendix Note # 11 TN-121
648 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRJCS. AND FREQUENCY CONTROL, VOL UFFC-34. NO 6, NOVEMBER 1987

y(t)

FREOUfNCY OFFSET

NEGATIVE FREQUENCT DRIFT

(a)

x(t)

.v(r) = Xq + vof + 1/2 Dr + e{t]

OSCILLATOR

^^t QU RB H H<po«) CS

TEXPERATUSE SEmSITIvITt m/ d.g K

Fig. 2. Nominal values for temperature coefficient for frequency stan- dards: QU = quartz crystal, RB = rubidium gas cell, H = active hy- drogen maser, H(pas) = passive hydrogen maser, and CS = cesium beam.

OSCILLATOR

..., OU RB H H(po») CS

Fig. 1. Frequency .v(f) and time .r(/) deviations due to frequency offset and to frequency drift in clock, (a) Fractional frequency error versus

time, (b) Time error versus time.

of reasons, the first being systematics such as frequency

drift (D). frequency offset ( Vq), and time offset (.to). In addition, there are systematic deviations that are often en- vironmentally induced. The second category is the ran- dom deviations e(f). which are usually not thought to be

deterministic. In general, we mav write

-•CNE'ic f;E^o SENSiT:viTY / c Fig. 3. Nominal values for magnetic field sensitivity for frequency stan- dards: QU = quartz crystal. RB = rubidium gas cell, H = active hydrogen maser, H(pas) = passive hydrogen maser, and CS = cesium

beam.

(5)*

GSCIL _ATGR

»-' °^ RB H HCpos) CS

1 w«

-It

--.i

.^ Though generally useful, the model in (5) does not apply

in all cases; e.g., some oscillators have significant fre- quency-modulation sidebands, and in others the fre- quency drift D is not constant. In some clocks and oscillators, e.g.. cesium-beam standards, setting D = is usually a better model.

Note that the quadratic D term occurs because x(t) is the integral of \{r), the fractional frequency, and is often

the predominant cause of time deviation. In Fig. 1 we

have simulated two systematic-error cases: a clock with

frequency offset, and a clock with negative frequency

drift. Figs. 2-6 summarize some of the important systematic influences on precision clocks and oscillators. In addition to Figs. 1-6, important systematic deviations may

include modulation sidebands, e.g., 60 Hz, 120 Hz, daily,

and annual dependences, which can be manifestations of

environmental effects such as deviations induced by vi- brations, shock, radiation, humidity, and temperature.

• See Appendix Note # 12

8E=5:cjc:bil:tt

Fig. 4. Nominal capability of frequency standard to reproduce same fre- quency after period of time for standards: QU = quartz crystal, RB =

rubidium gas cell. H = active hydrogen maser, H(pas) = passive hy- drogen maser. and CS = cesium beam. OSCILLATOR

QU RB H H (po«) CS

.1 I.

ABSOLJTE »CCUBACf Fig. 5. Nominal capability for frequency standard to produce frequency determined by fundamental constants of nature for standards: QU =

quartz crystal, RB = rubidium gas cell, H = active hydrogen maser, H ( pas ) = passive hydrogen maser. and CS = cesium beam.

TN-122
ALLAN: PRECISION CLOCKS AND OSCILLATORS 649

TABLE 1 Applicable Oscillators and Range of Applicability

Typical Noise Types a Name Cs H-Active H-Passive Qu Rb

2 white-noise PM slOOs si ms

1 flicker-noise PM £l s white -noise FM a 10 s 100s s T a lO^s 21 s 21 s -1 flicker-noise FM a days alO^s a days 21 s 2 10* -2 random-walk FM a weeks 2 weeks a weeks 2h 2 days

„ QU —I

If-"— IOSCILLATOR

RB H H(po«) CS

'/^^/A/V^'wvM'^

then the average fractional frequency for the /th measure- ment interval is

(6)

Fig. 6. Nominal values (ignonng sign) for frequency drift for frequency

standards: QU = quartz crystal, RB = rubidium gas cell, H = active hydrogen maser, H(pas) = passive hydrogen maser, and CS = cesium beam.

POWER UW SPECTRA

where — Tq over y, denotes the average over an interval tq. We can thus construct a set of discrete frequency values

from such a time-difference data set. If the standard deviation is calculated for this set of values, one can show

that for some kinds of power-law spectra encountered in precision oscillators the standard deviation is divergent

[1], [2], [5], i.e., it does not converge to a well-defined

value and is a function of data length. Hence the standard

deviation is seldom useful and can be misleading in characterizing clocks. An IEEE subcommittee has recom- mended Syif) in the frequency domain and a measure

a^yir) in the time domain [1]. Syif) is the one-sided

spectral density of y as a function of Fourier frequency/.

The latter is often called the Allan variance or two-sample

variance. The convergence of avCr) has been verified [1]-

[4] for the power law spectra of interest in precision clocks

and oscillators. The measure aj.{T) is defined as [1]

1 -.r<^'

Fig. 7. Simulated random processes commonly occurring in output signal of atomic clocks. Power law spectra 5{/) are proportional to u to some

exponent, where / is Fourier frequency ((w = 2t/) and 5^(/) =

Random Models for Clocks and Oscillators

The random-frequency deviations of precision clocks

and oscillators can often be characterized by power-law

spectra 5v(/) - /", where /is the Fourier frequency and a typically takes on integer values, i.e., —2, —1,0, 1,2

[l]-[4]. Fig. 7 shows noise samples corresponding to these different power law spectra, and Table I shows the nominal range of applicability of these power-law models.

Time-Domain Signal Characterization

Given a discrete set of time deviations x, taken in se- quence for the measurable time difference between a pair

of clocks or between a clock and some primary reference,

and given that the nominal spacing between adjacent time

difference measurements is tq (see Fig. 8 for an example),

* See Appendix Note # 13

ol{T) = -{{AyY) (7)

where AV is the difference between adjacent fractional frequency measurements, each sampled over an interval

T, and the brackets < > indicate an infinite time average

or expectation value. A pictorial description is shown in Fig. 9 for a finite data set. A data set of the order of 100

points is more than adequate for convergence of 0,(7),

though of course the confidence of the estimate will typically improve as the data length increases [6]. Given a discrete set of stored evenly spaced data, the value of T can be varied in the software [7]. If tq is the minimum data spacing for the original stored data set yj°,

then one can change the sampling time to r = htq by av- eraging n adjacent values of yf^ to obtain a new fractional frequency estimate y[, with sample time t as input to (7). Note this is different from averaging adjacent values of x. Hence in a very convenient way one can calculate <Tv(t) as a function of t, which will be shown to be very useful. For a finite data set of M values of y^, (7) for general t becomes (see Fig. 8 for an example computation of 0,(7 *• See Appendix Note # 14 TN-123

**
650 IEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRICS. AND FREQUENCY CONTROL. VOL. UFFC-34, NO 6. NOVEMBER 1987

x{t)

i + r

y(t)

'1 ^4

'STD OEV y (t) V H-1

MS (y. - 7)2 (t) -/. 1 2(M-1)

M-1

i = l (y

i + l yj'

Fig. 8. Simulated time deviation plot x(t) with indicated sample time t over which each adjacent fractional frequency y, is measured. Equations are for standard deviation and for estimate of <7,(t) for finite data set of M frequency measurements.

Often standard deviation diverges as data length increases when measuring long-term frequency stability of precision oscillators, whereas a,(T) converges.

dl 'f«r-»<tc» in alop* ^y = y^ ~ ^

,r\^

S I' a I

r «t« f *t

TIME

Fig. 9. Pictorial of computation of Allan variance. Simulated time variations plotted are random walk. At set sample time r. Ay = (.vj - lx~ -

jt,)/T is computed. With time of measurement of jCj t ahead of Jti and

that o( Xi T ahead of x,, all possible values of Ay a re computed. Each Ay is squared and average squared value determined. (Ay)': taking I /2

of this yields two-sample or Allan variance for that value of t. Value of

r can then be changed either in hardware or software to determine Allan

variance for another value of r. = To), i.e., n = 1)

o]\t) =

1 W - 2/1 + I 2 {?,.„-?,)' (8)

where .r, is taken from the set of M + 1 = N discrete time

deviation measurements between a pair of clocks or oscillators, / = 1 to Af + 1:

2{M -In ¥ \) *=i

where yl^-n and y[ are still adjacent fractional frequencies (i.e., no dead time exists between the measurements), each averaged over t = ntQ, and

t+/i-i

-r 1 V -TO ^k-n ~ ^k n i = k T

Alternately, one may write (see Fig. 9 for an example)

1 o:\r) =

2r^{M - 2/1 -t- I)

M - 2n + I Z (x.^2n-it.>„ +ar,)* (10)

i-t • See Appendix Note # 15

k ^k+\ - tq ^ y]

1= 1 -l-.r,. (11)

Equation (8) is obtained from a first difference on fre- quency, and (10) from the second difference on the time;

they are mathematically identical, yielding the option of

using frequency or time (phase) data. For power-law spectra the following proportionality applies: al(T) ~ t", where ^i is typically constant for a

particular value of a. A simple and elegant relationship

exists between the spectral density exponent a (in the re- lationship Syif) - /°) and //. i.e., fj. = -a - 1 (-3

< a < 1 ) and /i = -2 (a > 1 ) [8]. For example, for a significant range of t values, 0,(7) - r"" is proportional to T~" "' for cesium, rubidium, and passive hydrogen maser frequency standards. Therefore n has the value of - 1, and hence a has the value of (white-noise

frequency modulation). This is the classical noise exhibited by an important set of atomic clocks for t's beyond

a few seconds. In this case. o,{tq) is equal to the standard

deviation. Fortunately, for most cases with precision

clocks and oscillators where t > 1 s, the simple relation- ship /x = -a — 1 is applicable. It is convenient to plot

log Oyir) versus log t to estimate the value of /x and to

let /J = 2', / = 0, 1,2, • • (t = riTo). An ambiguity exists at ^ = -2; one cannot conveniently tell whether the noise process is flicker-noise phase

modulation (PM), a = -I- 1 , or white-noise PM. a = +2.

This ambiguity can be resolved by .realizing that for these cases Oyir) depends on the measurement bandwidth [2],

[3]. One can construct a variable software bandwidth /, by realizing the following [9]. [10]. In any measurement

system a hardware bandwidth/^ exists through which we

TN-124
ALLAN: PRECISION CLOCKS AND OSCILLATORS 651

TABLE II'

Typical Noise Types a Name al(T) =

Classical Standard

Deviation of x

Classical Sundard

Deviation of y

aoT-'

a-2-r

T a,(T)/>^ (constant) ~T • a,(T) VlnM/ln2

To- V,{To)>/(M + l)/6

undefined

undefined

2 white-noise PM

1 fiicker-noise PM

white-noise FM

- 1 flicker-noise FM

-2 random-walk FM

a,(T) V2(\ + \)/lN

-<7,(T) V2(A/ + i/3yv ^(To)

a,(T) >JN\nN/(l(N - 1) In 2) a,(r) V/V/2

* Note r is a general averaging time and Tq is the initial averaging time ( t = /itq, where n is an integer). Also note that the last four entries in the fourth column and the last two entries in the fifth column go to

infinity as A/ or A/ go to infinity. M is the initial number of frequency difference measurements and N the number of phase or time difference measuTeinents N = M + 1. If the spectral density is given by S,(f) = h^f, then

ao

(1.038 + 3 1og,(2T/,T)/..)

a., =2 1og,(2)ft-,

a., =

g

(2tA.,.

'Note this equality assumes use of modified aJ(T) = al(T).

measure the phase difference or the time difference between a pair of oscillators or clocks and we define t^ =

1 //ft. In other words, t/, is the sample time period through

which the time or phase date are observed or averaged.

Averaging n time or phase readings increases the sample

time window to ht/, = t^. Let t^ = 1//^; then/j = /,/«,

i.e., the software bandwidth is narrowed to/j. In other

words, f; = fh/n decreases as we average more values;

i.e., increase n (t = mtq). One can therefore construct a second difference composed of time deviations so-aver- aged and then define a modified OyiT) = a'yir) that will remove the ambiguity through bandwidth variation:

-2

ments between them, then it is possible to estimate a var- iance for each oscillator or clock. Often there is a refer- ence to which the rest are periodically measured at a sampling rate I/tq. If at each measurement the time or

frequency differences between the clocks are measured at nominally the same time, then the time difference or fre- quency difference can usually be estimated or calculated

between every possible pair in the set of oscillators or

clocks. Given a series of measurements, variances sjj can

be calculated on the time or frequency data between all pairs. It has been shown [13] that the individual clock

variances can be estimated using the following equations:

oUr) -

2T-n'{N - In + \)

Z ( Z (.t,^2„ - 2x,^„ + X,) 7=1 \ i=j

(12)

where N = M + 1 , the number of time-deviation measurements available from the data set. Now if ffJ(T) ~

t"', then ^' = -a - 1 (1 < a < 3) [10], [11]. Thus

Oyir) is typically employed as a subroutine to remove the ambiguity if av(T) - t~'. This is because the /x' = -a

— 1 relationship is valid as an asymptotic limit for large n and a < 1 and is not valid in general; however, there

is evidence that ajir) may be a better measure [12]. Specifically, for a = 2 and 1, n' /2 equals -3/2 and -1,

respectively, providing a clean differentiation between

white-noise PM and flicker-noise PM.

If three or more independent oscillators or clocks are available along with time (phase) or frequency measure-

• See Appendix Note # 16

where

ol^

B =

-^Jl'sl-B

m - 2 \; = i

^ y ,2

;;;—r ^^-j' m — I Kj (13)

m is the number of clocks available in the set, and si =

0. If the variance measures used are a^Cr) or aJ(T), then

(13) can be used to estimate the individual variances as a

function of t. Table II illustrates why one should not use the standard

deviation to characterize clocks. For the different kinds of

noise processes we list the standard deviation of the time

deviations and of the fractional frequency deviations as a

function of a^(T). The divergent nature of the standard

deviation is apparent. Even for classical white-noise FM

the standard deviation of the time diverges as the square

root of the data length, i.e., the number of samples N [2].

TN-125
652 IEEE TRANSACTIONS ON ULTRASONICS. FERROELECTRICS. AND FREQUENCY CONTROL. VOL. UFFC-34. NO 6. NOVEMBER 1987

TABLE III

Optimum

Typical Noise Types Prediction Time Error: a Name Jc(Tp) rms" Asymptotic Fonn

2 white-noise PM r. •a.(r.)/>/3 constant

1 flicker-noise PM ~r, • a,(T^) Vln r,/2 In Tq Vlnr,

while-noise FM ^P • »»(%) r/' — 1 flicker-noise FM ^p • ff,(T,)/>/ln2 T, -2 tandom-walk FM Tp a,(r,) rV'

'i, is the prediction interval.

Time and Frequency Estimation and Prediction

Using Oyi^i), a^(T), Sy{f), or S^if), one can characterize typical power law processes. Once characterized,

this opens the opportunity for determining optimum esti- mates of values by employing the statistical theorem that the optimum estimate of a white-noise process is the simple mean.

For example, consider the very common and very important case of white-noise FM typically found on the sig- nals from cesium standards, rubidium standards, and passive hydrogen masers. The optimum estimate of the frequency is the simple mean frequency, which is equivalent to (xs — X\)/Mtq. It is still all too common within our discipline to see our colleagues erroneously determining the frequency for these kinds of oscillators by calculating the slope from a linear least-squares fit to the time

deviations and quoting the standard deviation around that

fit as a measure of the clock performance. There are three problems in proceeding this way. First, the frequency es- timate is not optimum in a mean-square-error sense. It is equivalent to throwing away about 20 percent of the data

and thereby increasing the cost in the case of a calibration. Second, the standard deviation diverges as the square

root of the data length. Third, the standard deviation is significantly dependent on the filter form, e.g., linear least squares, as well as the clock deviations. On the other

hand, such a filter is sometimes useful for assessing outliers. The optimum "end-point" method outlined earlier has the risk that if either of the points is abnormal, (i.e., the model fails), the result will of course be adversely

effected. Therefore such a filter is useful to assess whether

there are outliers—paying special attention to the end

points. Also, if the measurement noise exceeds the combined noise in the clocks, then the end points will be adversely affected. The key message is that the end-point

method for estimating frequency is only optimum if the

noise is pure white FM, which is easy to determine from

a log ay(T) versus log t plot.

There are other useful, and maybe not so obvious, optimum estimators appropriate for time-difference data sets.

1) Given white-noise PM, the best time estimate is the simple mean of the time deviations; the frequency

estimate then is the slope from a linear least-squares

fit to the time deviations, and the frequency drift D

is determined from a quadratic least-squares fit to the time deviations per (1).

2) Given white-noise FM, the optimum estimate of the time is the last value; the optimum-frequency esti- mate is outlined in the previous paragraph, and the optimum-frequency-drift estimate is derived from a

linear least-squares fit to the frequency.

3) Given random-walk FM, the current optimum time

estimate is the last value plus the last slope (clock

rate) times the time since the last value; the optimum-frequency estimate is obtained from the last slope of the time deviations; and the optimum-frequency-drift estimate is calculated from the mean

second difference of the time deviations. Caution

needs to be exercised here, for typically there will be higher frequency component noise in a real data

stream, such as white-noise FM, along with ran- dom-walk FM, and this can significantly contaminate the drift estimate from a mean second differ- ence. If random-walk FM is the predominant long- term power-law process, which is often the case, then the effect of high-frequency noise can be re- duced by calculating the second difference from the

first, middle, and end-time deviation points of the data set.

The flicker-noise cases are significantly more complicated, though filters can be designed to approximate optimum estimation [14]-[16]. As the data length increases

without limit, time is not defined for flicker-noise PM,

and frequency is not defined for flicker-noise FM. This

has some philosophical implications for the definitions of

time and frequency, unless some low-frequency cutoff

limits exist. If significant frequency drift exists in the data,

it should be optimally subtracted from the data or it will bias the long-term values of a^(T):

a,{T) =

>/2' (14)

Once the power-law spectra are deduced for a pair of

oscillators or clocks, then one can also develop an optimum predictor. Table III gives both the optimum prediction uncertainty values for the various relevant pure

power-law spectra as well as their 'asymptotic forms. Special forecasting techniques must be used for optimal prediction when combinations of these processes are present

[17]. To illustrate how these concepts relate to real devices. Fig. 10 shows a <t^( r) diagram for some interesting

TN-126
ALLAN: PRECISION CLOCKS AND OSCILLATORS 6S3

-10(—

RbGASCEU

3 4 LOG TAU (seconds)

Fig. 10. Square root of Allan variance for variety of state-of-the-art precision oscillators including NBS-6, NBS primary frequency Standard.

RMS TIME DEVIATION

LOG TAUp (seconds)

Fig. 11. From frequency stability characterization shown in Fig. 10, op- timum prediction algorithms to minimize time error can be obtained. Based on optimum prediction procedures rms time prediction error for prediction interval t^ can be calculated for each oscillator shown in Fig. 10 and corresponding values are plotted in Fig. 11.

State-of-the-art oscillators, and Fig. 1 1 shows the rms time

prediction errors for the same set of oscillators.

Conclusion

In conclusion, it is clear that classical statistics do not allow characterization of common kinds of random signal

variations found in precision oscillators. The two-sample

Allan variance provides a valuable and convergent measure of the power-law spectral-density models useful in characterizing random deviations for most oscillators and

clocks. Once characterized, we can calculate optimum

time and frequency estimates as well as predicted values. Characterizing the random variations also provides near- optimum estimation of systematic effects, which often cause the predominant time and frequency deviations. For

example, if we wanted to optimally determine the static temperature dependence with the temperature set at two

different values, we would stabilize the oscillator at one

temperature and measure the frequency against a refer- ence for a time t„, corresponding to the t for the nominal

minimum Oyir) value. We would then change the temperature to the other value and repeat the measurement

with the same criteria and note the Ay'" between the two

optimally determined frequency values. If these two steps are repeated several times, an arbitrarily good precision

for the temperature coefficient is achieved if it is linear. The uncertainty is approximately given by Oy{T„)/y/p,

where P is the number of Ay^" values obtained from

switching back and forth. Knowing the characteristics of

both the random and the systematic deviations of precision clocks and oscillators clearly is useful to the designer, the manufacturer, the planner, and the user as well

as the vendor of these devices.

The aforementioned procedures usually work well if the

clocks or oscillators are in a reasonable environment. If the environment is adverse, other procedures and analysis

methods may have to be employed. As a general rule it is often useful to analyze the data in the frequency domain

as well as the time domain. The frequency domain is es- pecially useful if there are bright lines, i.e., sidebands to the carrier frequency. The effect of a modulation sideband

f„on Oyir) can be calculated, and is given by [18]

^pp Oyir) = — sm^{irf„T) (15)

where Xpp is equal to the peak-to-peak or twice the amplitude of the time-deviation modulation.

If one is trying to estimate the power-law spectral behavior between a pair of oscillators or clocks using Oyir),

it is apparent from (15; that if significant modulation

sidebands are present on the signal, these can seriously

contaminate that estimate. However, if a Oyir) plot dis- plays a character as given by (15), then the amplitude and

frequency of that modulation sideband can be estimated

from this time-domain analysis technique. In practice, this approach is often used, but these modulation sidebands can be more efficiently estimated in the frequency domain. If the measurement sampling rate I/tq is set equal

to /„, then the modulation sideband is aliased away and

has no effect on Oyir).

The best rule in all analysis is to use common sense. Very often the most revealing information may be in a

plot of the raw time (phase) difference or frequency-difference residuals after some trend has been removed. Such

a plot is usually the first thing to look at when characterizing clocks and oscillators. Caution here is also important as a pure random walk on the time residuals (white

FM) may be visually interpreted as having frequency

steps. This is especially true for flicker FM as often seen

in quartz-crystal oscillators. Following the time-residual

plot with a Oyir) analysis often answers the question as

to whether or not such steps are statistically significant.

Acknowledgment

Because the paper is mainly a review, the author is in- debted to a large number of people—many of whom are

reflected in the references. Specifically, sincere appreciation is expressed to Dr. James A. Barnes, Mr. Dick D.

TN-127
634 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRJCS, AND FREQUENCY CONTROL, VOL. UFFC-34, NO. 6, NOVEMBER 1987

Davis, Dr. John Vig, Dr. Donald B. Sullivan, Dr. Fred

L. Walls, and Dr. Marc A. Weiss for their extremely

helpful comments and suggestions.

References

[1] J. A. Bames etal. , "Characterization of frequency stability," in IEEE

Trans. Instntm. Meas., vol. IM-20, p. 105, NBS Tech. Note 394,

1971.

[2] D. W. Allan, "Statistics of atomic frequency standards," in Proc.

/£££•, vol. 54. 1966. p. 221.

[3] R. Vessot, L. Mueller, and J. Vanier, "The specification of oscillator

characteristics from measurements made in the frequency domain."

in Proc. IEEE Special Issue on Frequency Stability, vol. 54, Feb.

1966, pp. 199-207.

[4] P. Lesage and C. Audoin, "Characterization and measurement of time and frequency stability," /?a<fio5d., vol. 14, pp. 521-539, July/Aug.

1979.

[5] J. A. Barnes, "Atomic timekeeping and the statistics of precision sig- nal generators," in Proc. IEEE Special Issue on Frequency Stability, vol. 54, Feb. 1966, pp. 207-220.

[6] D. A. Howe, D. W. Allan, and J. A. Barnes, "Properties of signal sources and measurement methods," in Proc. 35th Ann. Symp. Fre- quency Control (SFC), May 1981, pp. 669, A1-A47.

[7] D. W. Allan, "The measurement of frequency and frequency stability of precision oscillator," Nat. Bur. Standards, Boulder, CO, NBS

Tech. Note 669, May 1975.

[8] M. J. Lighthill, "Introduction to Fourier analysis and generalized

functions." in Cambridge Monographs on Mechanics and Applied

Mathematics, G. K. Batchelor and J. W. Miles, Eds. London. England: Cambridge Univ. Press, 1964.

[9] J. J. Snyder, "An ultra-high resolution frequency meter," in Proc. 35th Ann. Frequency Control Symp., USAERADCOM, Ft. Monmouth, NJ, May 1981, pp. 464-469.

[10] D. W. Allan and J. A. Barnes, "A modified 'Allan variance' with

increased oscillator characterization ability," in Proc. 35th Ann. Fre- quency Control Symp., USAERADCOM, Ft. Monmouth, NJ, May

1981. pp. 470-475.

[11] P. Lesage and T. Ayi, "Characterization of frequency stability: Analysis of the modified Allan variance and properties of its estimate," IEEE Trans. Instrum. Meas., vol. IM-33, no. 4, pp. 332-336, Dec.

1984.

[12] L. G. Bemier. "Theoretical analysis of the modified Allan variance,"

in Proc. 41st Ann. Frequency Control Symp., USAERADCOM, Ft. Monmouth, NJ, May 1987.

[13] J. A. Barnes, Notes from NBS 2nd Sem. Atomic Time Scale Algorithms, June 1982, Time and Frequency Division. Nat. Bur. Standards. Boulder, CO 80303.

[14] J. A. Barnes and S. Jarvis, Jr., "Efficient numerical and analog modeling of flicker noise processes," Nat. Bur. Standards, Boulder, CO,

NBS Tech. Note 604, June 1971.

[15] J. A. Barnes, "The measurement of linear frequency drift in oscilla- tors," in Proc. 15th Ann. Precise Time and Time Interval (PTTl)

Applications and Planning Meeting, Naval Res. Lab., Washington,

DC, Dec. 1983, pp. 551-582.

[16] D. W. Allan et al. , "Performance, modeling, and simulation of some

cesium beam clocks," in Proc. 27th Ann. Symp. Frequency Control, 1972, p. 309, AD 771 042.

[17] D. W. Allan and H. Hellwig, "Time deviation and the prediction

error for clock specification, characterization, and application," in Proc. Position Location and Navigation Symp. (PLANS). 1978, p. 29.

[18] S. R. Stein, private communication.

David W. Allan for a photograph and biography, please see page 571 of

this Transactions.

TN-128
From: Proceedings of the 42nd Annual Symposium on Frequency Control, 1988.

EXTENDING THE RANGE AND ACCURACY OF PHASE NOISE MEASUREMENTS

F.L. Walls, A.J.D. Clements, CM. Felton, M.A. Lombardi, and M.D. Vanek

Time and Frequencv Division

National Institute of Standards and Technology

Boulder, Colorado 80303

?"r"T'i'ry

This paper describes recent progress in extending

high accuracy measurements of phase noise in

oscillators and other devices for carrier frequencies

from the rf to the millimeter region and Fourier

frequencies up to 10% of the carrier (or a maximum of

about 1 GHz). A brief survey of traditional

precision techniques for measuring phase noise is included as a basis for comparing their relative

performance and limitations. The single oscillator

techniques, although conceptually simple, require a set of 5 to 10 references to adequately measure the phase noise from 1 Hz to 1 GHz from the carrier. The

two oscillator technique yields excellent noise

floors if, for oscillator measurements, one has a comparable or better oscillator for the reference

and, for other devices, either pairs of devices or a reference oscillator with comparable or better noise.

We have developed several new calibration techniques

which, when combined with previous two oscillator

techniques, permits one to calibrate all factors

affecting the measurements of phase noise of

oscillator pairs to an accuracy which typically

exceeds 1 dB and in favorable cases can approach 0.4

dB. In order to illustrate this expanded two oscillator approach, measurements at 5 MHz and 10 GHz

are described in detail. At 5 MHz we achieved

accuracies of about ±0.6 dB for phase noise measurements from 20 Hz to 100 kHz from the carrier. At 10 GHz we achieved an accuracy of ±0.6 dB for phase noise measurements a few kHz from the carrier

degrading to about ±1.5 dB, 1 GHz from the carrier.

I. Introduction

This paper describes recent progress at the National

Bureau of Standards (NBS) in extending high accuracy

measurements of phase noise in oscillators,

amplifiers, frequency synthesizers, and passive

components at carrier frequencies from the rf to the millimeter region and Fourier frequencies up to 10% of the carrier (or a maximum of about 1 GHz) . An

examination of existing techniques for precision

phase noise measurements of oscillators[ 1-11 ] showed

that present approaches which don't require a second

"reference" oscillator have good resolution or noise

floor for Fourier frequencies extending over only 1 or 2 decades. [7-10] Consequently, these approaches

require a set of 5 to 10 references, either delay

lines or high Q factor cavities, in order to adequately measure the phase noise from 1 Hz to 1 GHz

from the carrier. Using the cavity approach would

require a entire set of reference cavities for each

carrier frequency measured. Similar considerations

also apply to phase noise measurements of the other

devices unless one can measure pairs of devices

The limitations of the single oscillator techniques

led us to adopt a two oscillator method for all measurements. This approach yields good resolution

or noise floor from essentially dc to the bandwidth

of the mixer if, for oscillator measurements, one has

a comparable or better oscillator for the reference

and, for other devices, either pairs of devices or a reference oscillator with comparable or better noise.

The major limitations in the accuracy of the two oscillator method (which also apply to the single

oscillator methods) are the calibration of: the mixer

phase-to-voltage conversion factor, the amplifier

gain versus Fourier frequency, and the accuracy of

the spectrum analyzer. We have developed several new calibration techniques which, when combined with

previous techniques, allow us to address each of these limitations. The net result is the development

of a complete measurement concept that permits one to calibrate all factors affecting the measurements of

phase noise of oscillator pairs to an accuracy which

typically exceeds 1 dB and in favorable cases can approach 0.4 dB. For other types of devices the limitations are similar if the noise of the reference

oscillator can be neglected. The ultimate accuracy

that can be easily achieved with this approach is now

limited by the accuracy of the attenuators in available spectrum analyzers.

Measurements at 5 MHz and 10 GHz are described in detail, in order to illustrate this expanded two oscillator approach. Specifically we measure the mixer phase-to-voltage conversion factor multiplied

by amplifier gain on all channels versus Fourier

frequency using a new ultra-wideband phase modulator.

We then measure the absolute mixer conversion factor

multiplied by the amplifier gain at one Fourier

frequency, the effect of the phase -lock loop on the measured noise voltage, and spectral density function

of the spectrum analyzers. These measurements are used to normalize the relative gains of the noise measurements on the various channels. At 5 MHz we achieved accuracies of about ±0,6 dB for phase noise measurements from 20 Hz to 100 kHz from the carrier. At 10 GHz we achieved an accuracy of ±0.6 dB for phase noise measurements a few kHz to 500 MHz from

the carrier. The accuracy degrades to about ±1.5 dB,

1 GHz from the carrier.

II. Model of a Noisy Signal

The output of an oscillator can be expressed as

V(t) = [V„ + e(t)] sin(2»ri/,t + (*(t)), (1)

where V^ is the nominal peak output voltage, and v^ is the nominal frequency of the oscillator. The time

variations of amplitude have been incorporated into

«(t) and the time variations of the instantaneous

frequency, i'(t), have been incorporated into ^(t).

The instantaneous frequency is

«'(t) = i/„ d[*(t);

2>rdt (2)

The fractional frequency deviation is defined as

Contribution of the U.S. copyright.Government, not subject to

v{t)

y(t)

d[.*(t)]

2jrf„dt (3)

432

TN-129
Power spectral analysis of the output signal V(t)

combines the power In the carrier u^ with the power

in <(t) and ^(t) and therefore is not a good method

to characterize «(t) or ^(t).

Since in many precision sources understanding the variations in <i(t) or y(t) is of primary importance,

we will confine the following discussion to frequency-domain measures of y(t), neglecting «(t)

except in cases where it sets limits on the measurement of y(t) . The amplitude fluctuations,

e(t), can be reduced using limiters whereas ^(t) can

be reduced in some cases by the use of narrow band

filters.

Spectral (Fourier) analysis of y(t) is often

expressed in terms of S^(f), the spectral density of

phase fluctuations in units of radians squared per Hz bandwidth at Fourier frequency f from the carrier i/^

.

Alternately Sy(f), the spectral density of fractional

frequency fluctuations in a 1 Hz bandwidth at Fourier frequency f from the carrier i/^ can be used

[1]. These are related as

S^(f) = — Sy(f)

f2 id^/Hz 0<f<«>

S^(f) can be intuitively understood as,

S^(f) =

W

(5)

increasingly important as users require the specification of phase noise near the carrier where

the phase excursions are large compared to 1 radian, or at Fourier frequencies which exceed the bandwidth

of typical circuit elements.

The above measures provide the most powerful (and

detailed) analysis for evaluating types and levels of

fundamental noise and spectral density structure in precision oscillators and signal handling equipment

as it allows one to examine individual Fourier

components of residual phase (or frequency)

modulation.

III. A. Two Oscillator Method *

Fig. 1 shows the block diagram for a typical scheme

used to measure the phase noise of a precision source using a double balanced mixer and a reference source.

Fig. 2 illustrates a similar technique for measuring

only the phase noise added in multipliers, dividers,

amplifiers, and passive components. It is very

important that the substitution oscillator be at the same drive level, impedance, and at the equivalent

electrical length from the mixer as the signal coming

from the reference oscillator. This is dramatically

illustrated in Fig. 3, discussed below. The output

voltage of the mixer as a function of phase

deviation, t^ , between the two inputs to the mixer is nominally given by

V„ = K cos A^ (8) BU

where A^(f) is the rms phase deviation measured at Fourier frequency f from the carrier in a bandwidth

BU. It should be noted that these are single-sided

spectral density measures containing the phase or frequency fluctuations from both sides of the carrier. The mean squared phase modulation in a measurement bandwidth, BW, is given by

Near quadrature (90* phase difference) this can be

approximated by

2n-l

= Kjii^, where S4> = [A^ - ir] <0.1 (9)

t^Hf) =

+BU/2

S*(f) df

-BW/2

(6)

Other measures sometimes encountered are Jf(f), dBC/Hz, and S^^,(f). These are related by [1-3]

where n is the integer to make S<^ ~ 0. The voltage to phase conversion ratio sensitivity, Kj , is dependent

on the frequency, the drive level, and impedance of

both input signals, and the IF termination of the mixer [4] . The combined spectral density of phase

noise of both input signals, the noise of the mixer,

and the amplitude noise from the IF amplifiers is

Si.(f) = ^oS,(f) Hz^/Hz

S^(f) = /(i/o-f) + /(i/o+f) = 2if(f) (7)

dBC/Hz = 10 log if(f)

!t(f) and dBC/Hz are single sideband measures of phase

noise. These are the revised definitions of it(f) and

dBC as per the most recent recommendation of the Standards Coordinating Committee for Time and

Frequency of the IEEE [3]. With these revised

definitions /(f) and dBC/Hz are now defined for arbitrarily large phase modulation, whereas

previously they were restricted to small angle

modulation. In some situations, especially where f becomes a sizeable fraction of i/g , the phase noise

spectrum is asymmetric about the carrier. In these

cases one should specify whether the upper sideband

noise Jt(i/|j+f) or the lower sideband noise i;(i/Q-f) is being referenced. These distinctions are becoming

* See Appendix Note # 6

Measurement oJ S (f) Between Two Oscillators

<S^

^

OSM

Tmelnunal

Counwt ^•(j^Js

f_ \ /

L-

| cattoaiaKT]

Spacmitn

Stop* K,^(t) voUs/radian

I I uRaduns Fig. 1. Precision phase measurement system using a spectrum analyzer. Calibration requires a recording

device to measure the slope at 'the zero crossing.

The accuracy is better than 0.4 dB from dc to 0.1 i/^ Fourier frequency offset from the carrier u^ . Carrier frequencies from a few Hz to 10^^ Hz can be accommodated with this type of measurement system

[4].

TN-130

433
given by

S*(f) = v„(f)

G(f)Kj

1

BU

(10)

Measurement of S(p(f ) for Two Amplifiefs

REF

^^,:^^SX

Fig. 2. Precision phase measurement system featuring

self calibration to 0.4 dB accuracy from do to 0.1 i/^ Fourier frequency offset from carrier. This system

is suitable for measuring signal handling equipment,

multipliers, dividers, frequency synthesizers, as well as passive components [4]

.

0.8 o<0 0,7 ^

^

0.6 > ns H

(/) 2 0.4 LU(A

UJ 0.3 V)<

I 0.2 Q. 01

T r

lOmW MIXER DRIVE

I I I I L

r 1 1

\

1

2mW MIXER DRIVE - -.——--fc,^ O.OIjiF ^V "^

- 0.02hf\.

- IkQ

- 50C - -—j- 1 1 10 20 50 12 5 FREQUENCY (kHz)

10 20 50 im

(often more than 20 dB) . Termination of the mixer IF port with 50 n maximizes the IF bandwidth, however,

termination with reactive loads can reduce the mixer

noise by ~ 6 dB, and increase K^ by 3 to 6 dB as shown in Fig. 3 [4]. Accurate determination of K^ can be achieved by measuring the slope of the zero crossing In volts/radian with an oscilloscope or other recording device when the two oscillators are beating slowly. For some applications the digitizer

in the spectrum analyzer can be used to measure both

the beat period and the slope in V/s at the zero crossing. The time axis is easily calibrated since

one beat period equals 2)r radians. The slope in volts/radian Is then calculated with a typical accuracy of 0.2 dB. Estimates of K^ obtained from measurements of the peak to peak output voltage

Induced can introduce errors as large as 6 dB in S^(f) even If the amplitude of the other harmonics is measured unless the phase relationship is also taken

into account [4]. S^(f) can be made independent of

the accuracy of the spectrum analyzer voltage

reference by comparing the level of an externally IF signal (a pure tone is best), on the spectrum

analyzer used to measure V„ with the level recorded

on the device used to measure Kj

.

The noise bandwidth of the spectrum analyzer also

needs to be verified. This calibration procedure is sufficient for small Fourier frequencies but looses

precision and accuracy due to the problems

illustrated in Fig. 3 and the variations of amplifier

gain with Fourier frequency. If measurements need to be made at Fourier frequencies near or below the phase- lock- loop bandwidth, a probe signal can be

injected inside the phase -lock- loop and the attenuation measured versus Fourier frequency.

Some care is necessary to assure that the spectrum

analyzer is not saturated by spurious signals such as the power line frequency and its multiples.

Sometimes aliasing in the spectnom analyzer is a problem. If narrow spectral features are to be

measured it is usually recommended that a flat top

window function (in the spectrum analyzer) be used.

In the region where the measured noise is changing

rapidly with Fourier frequency, the noise bandwidth

should be much smaller than the measurement

frequency. The approximate level of the noise floor

of the measurement system should be measured in order

to verify that it does not significantly bias the measurements or. If necessary, to subtract its effect

from the results

.

Fig. 3. Double -balanced mixer phase sensitivity at 5 MHz as a function of Fourier frequency for various

output terminations . The curves on the left were obtained with 10 mW drive while those on the right were obtained with 2 mW drive. The data demonstrate

a clear choice between constant, but low sensitivity

or much higher, but frequency dependent sensitivity

where V„(f) Is the RMS noise voltage at Fourier

frequency f from the carrier measured after IF gain

G(f) in a noise bandwidth BU. Obviously BU must be

small compared to f. This is very Important where

S^(f) is changing rapidly with f, e.g., S^(f) often

varies as f"' near the carrier. In Fig. 1, the output of the second amplifier following the mixer

contains contributions from the phase noise of the oscillators, the noise of the mixers, and the post

amplifiers for Fourier frequencies much larger than

the phase -lock loop bandwidth. In Fig. 2, the phase

noise of the oscillator cancels out to a high degree

Typical best performance for various measurement

techniques is shown in Fig. 4. The two oscillator

approach exceeds the performance of almost all available oscillators from below 0.1 MHz to over 100 GHz and is generally the technique of first choice

because of its versatility and simplicity. Figs. 10

- 12 give some examples. Phase noise measurements on pairs of signal sources can be made with an absolute accuracy better than 1 dB using the above calibration

procedure. Such accuracy is not always attainable

when the phase noise of the source exceeds that of

the added noise of the components under test (see

Fig. 2). The use of specialized high level mixers

with multiple diodes per leg increases the phase to voltage conversion sensitivity, K^ and therefore

reduces the contribution of IF amplifier noise [5] as shown in Fig. 4. Phase noise measurements can generally be made at Fourier frequencies from

approximately dc to 1/2 the source frequency. The

TN-131

434
major difficulty is designing a mixer terminations to remove the source frequency from the output signal,

which would generally saturate the low noise

amplifiers following the mixer, without degrading the

signal-to-noise ratio. As mentioned earlier the phase noise spectrvun is quite likely asymmetric when

f exceeds the bandwidth of the tuned circuits in the

device under test. For example one expects that the

phase noise at 1/2i/q is different than the phase

noise at 3/2i/o .

Comparison of Noise Floor

for Different Techniques

iiiiii| I iiiiiii{ I iiiiiii| I iiiiiii| I iiiiiii| I iiiiiii| I

mixer z

Fig. 4. Curve A.

Curve B.

Curve C.

Curve D.

Curve E.

Curve F.

10' 10^ 10-^ 10" 10^ 10^

Fourier Frequency (Hz)

The noise floor S^(f) (resolution) of

typical double balanced mixer systems (e.g.

Fig. 1 and Fig. 2) at carrier frequencies

from 0.1 MHz to 26 GHz. Similar

performance possible to 100 GHz [5]. The noise floor, S^(f), for a high level

mixer [5]

.

The correlated component of S^(f) between

two channels using high level mixers [5]. The equivalent noise floor S^(f) of a 5 to

25 MHz frequency multiplier.

Approximate phase noise floor of Fig. 8 using a 500 ns delay line. Approximate phase noise floor of Fig. 8 where a 1 ms delay has been achieved by

encoding the signal on an optical carrier

and transmitted it across a long optical

fiber to a detector.

reference source is also higher at the multiplied

frequency as shown in Section III.F below.

III. B. Enhanced Performance Usiny Correlation

Techniques

The resolution of the many systems can be greatly

enhanced (typically 20 dB) by using correlation

techniques to separate the phase noise due to the device under test from the noise in the mixer and IF amplifier [5, 11].

For purposes of illustration, consider the scheme

shown in Fig. 5. At the output of each double

balanced mixer there is a signal which is proportional to the phase difference, A^, between the two oscillators and a noise term, V, , due to contributions from the mixer and amplifier. The

voltages at the input of each bandpass filter are

VjCBP filter input) = Gj A^(t) + CiV^i(t) (11)

VjCBP filter input) = Gj A^(t) + C2V„2(t),

where Vgi(t) and V^j^*^) ^^^^ substantially

uncorrelated and Cj and Cj are constants. Each

bandpass filter produces a narrow band noise function

around its center frequency f:

Vi(BP filter output) = Gi[S^(f))* Bj* cos [2irft +

*(t)]

+ Ci[SvHi(f)l* Bi* cos [2.rft + ni(t)] (12)

V2(BP filter output) = G2[S^(f)]* B^* cos [2)rft +

V-Ct)]

+ C2[Sv„2(f)]* B2* cos [2>rft + n2(t)]

/^>—-(^)—<^ z -1

Pass yBand

FAarail y—>^ 1 tr-L\ MulBplMr W UMPass <> V_^

PanBwd

meralt

1

Loop

1 / ZG2\ A z ~l

ato1>—-(^)-^ h

HEF

Most double balanced mixers have a substantial non- linearity that can be exploited to make phase

comparison between the reference source and odd

multiples of the reference frequency. Some mixers

even feature internal even harmonic generation. The measurement block diagram looks identical to that of

In Fig. 1, except that the source under test is at an odd (even) harmonic of the reference source. This

method is relatively efficient (as long as the harmonics fall within the bandwidth of the mixer) for multiples up to xS although multiples as high as 25 have been used. The noise floor is approximately

degraded by the amount of reduction in the phase

sensitivity of the mixer. The phase noise of the

Fig. 5. Correlation phase noise measurement system.

where B^ and Bj are the equivalent noise bandwidths

of filters 1 and 2 respectively. Both channels are bandpass filtered in order to help eliminate aliasing

and dynamic range problems. The phases <Kt), ni(t)

and n2(t) take on all values between and 2n with

equal likelihood. They vary slowly compared to 1/f

and are substantially uncorrelated. When these two voltages are multiplied together and low pass

filtered, only one term has finite average value.

TN-132

435
The output voltage is

V^t = ^ GiGjS^Cf) Bi HBz'' + Di<cos[V«(t)-ni(t)]> (13)

+ D2<cos[V>(t) - n2(t)]> + D3<cos[ni(t)-n2(t)]>.

so that S^(f) is given by

(2)V2(f)

S^(f) = GiGjVBjBj

(14)

For times long compared to Bi"*B2'* the noise terms

Dj , Dj and D3 tend towards zero as Jt. Limits in the reduction of these terms are usually associated with

harmonics of 60 Hz pickup, dc offset drifts, and non- linearities in the multiplier. Also if the isolation

amplifiers have input current noise, they will pump

current through the source resistance. The resulting

noise voltage will appear coherently on both channels

and cannot be distinguished from real phase noise

between the two oscillators. One half of the noise power appears in amplitude and one half in phase

modulation.

Obviously the simple single frequency correlator used

in this example can be replaced by a fast digital

system which simultaneously computes the correlated

phase noise for a large band of Fourier frequencies.

Typical results show a reduction in noise floor of

order 20 dB over the noise floor of a single channel

(See Fig. 4). The great power of this technique is that it can be applied at any carrier frequency where

are available double balanced mixers. The primary

limitations come from the bandwidth and non- linearities in the cross correlator [5,11].

III. C. Reference Phase Modulation Method

Another method of determining S^(f) uses phase

modulation of the reference oscillator by a known

amount. The ratio of the reference phase modulation

to the rest of the spectrum then can be used for a relative calibration. This approach can save an enormous amount of time for measurements which are repeated a great many times. An adaptation of this

approach is utilized in the new NBS phase noise

system described in section IV below.

III. D. Frequency Discriminator Methods

It is sometimes convenient to use a high-Q resonance directly as a frequency discriminator as shown in

Fig. 6. The oscillator can be tuned 1/2 linewidth

(»'o/2Q) away from line center yielding a detected

amplitude signal of the form

V„„, = G(f)k^Qdy(f) [V +e(t)] (15)

This approach mixes frequency fluctuations between

the oscillator and reference resonance with the amplitude noise of the transmitted signal. By using

amplitude control (e.g. by processing to normalize

the data) , one can reduce the effect of amplitude

noise. [5] The measured noise at the detector is then related to the phase fluctuation of the reference resonance by

S^(f) = "oVHCf)

fQkiG(f) BU

for f< — . (16)

2Q

o

3

<

_l<

Z

o

CO<

10 r + 0.5

LU>

0.0 I

LUo

ob

-0.5 VCO n^QUENCY

Fig. 6. High-Q resonance used as a frequency

discriminator. Note that the peak response is displaced from the center of the resonance by about

the half bandwidth.

This approach normally has the limitations that f must be small compared to the linewidth of the cavity, and the effect of residual amplitude noise is difficult to remove; however, no reference source is needed. The calibration factors G(f)KdQ can be measured even for Fourier frequencies larger than

''o/2Q by stepping the source frequency an amount dy

(which is small compared to 1/2Q) and measuring the output voltage versus the modulation frequency, f. Differential techniques can be used to measure the inherent frequency (phase) fluctuations of two high-Q

resonators as shown in Fig. 7 [7]. The output

voltage is of the form V^„^ = 2QKdG(f) dy(f). The

phase noise spectrum of the resonators is then

obtained using Eq . 4.

S*(f)

''cVN(f)

2QfK.G(f)

1 "0 f< —

BU 2Q

(17)

Measurement of the Inherent Phase Noise in High-Q Resonators

I t

Fig. 7. Differential frequency discriminator using a pair of high-Q resonators. In this approach the phase noise of the source tends to cancel out.

TN-133

436
The phase noise In the source can cancel out by 20 to 40 dB depending on the similarity of resonant frequencies Q's and the transmission properties of

the two resonators. This approach was first used to demonstrate that the Inherent frequency stability of

precision quartz resonators exceeds the performance

of most quartz crystal controlled oscillators [7].

If only one resonance Is used, the output Includes

the phase fluctuations of both the source and the resonator. The calibration is accomplished by

stepping the frequency of the source and measuring

the output voltage, i.e., AV = G(f)K^(f) tu^. From

this measurement the phase spectrum can be calculated

as

S*(f) = VnCf)

f Ki(f)

r 1

BU

(18)

Fig. 8A shows one method of implementing this

approach at X-band. The cavity has a loaded quality

factor of order 25,000. Fig. 8B shows the measured

frequency discriminator curve. Note that K^(f) is constant for f<»'/(2Q) and decreases at values of f larger than the half bandwidth of the resonance as

K^f)

1

2Qf 2

1 + (

)

(19)

The frequency dependence shown in eq. (19) can be accurately determined by measuring AV when stepping

the source an amount Aj/ , which is small compared to

''o/Q- versus the modulation frequency, f. This

approach has poor resolution near the carrier and

limited high frequency response. Therefore the measurement of phase from close to the carrier out to 10% of the carrier could require a large set of

cavities with different Q factors. In order to achieve the required Q factor for close in measurements it may even be necessary to use cryogenic techniques

.

III. E. Delay Line Method

Another different approach uses a delay line to make

a pseudo reference which is retarded relative to the Incoming signal [7-10] as shown in Fig. 9.

The mixer output is of the form

Vout = 2irtjKd./„dy ,

and the input phase noise is given by

S^(f) =

VH(f)

2irfrjG(f)Ka

1

w

f<

1

(20)

(21)

This approach is often used at microwave frequencies

when only one oscillator is available. In this

technique the ability to resolve phase noise close to the carrier depends on the delay time. For example,

if f = 1 Hz and Tj = 500 ns , then, (Iwfr^)^ ~ 10"^'. The noise floor is 110 dB higher at f = 1 Hz than

that of the two oscillator method, decreasing as

1/f^ . Recent advances make it possible to encode the

rf signal on an optical signal which then can be

transmitted down an optical fiber to achieve delays

up to the order of 10"^ s with an Increase in the noise floor to approximately -140 dB relative to 1 rad^/Hz. The noise floor can be reduced by -20 to 40

dB using the correlation techniques described above

[11]. Note that the range of Fourier frequencies is usually limited to less than - l/r ^ . This technique

normally has good resolution over IH to 2 decodes in Fourier frequency. Therefore, measurements of phase

noise from close to the carrier out to 10% of the carrier require a large set of different delay lines

and hardware including optical delay lines,

associated lasers, modulators, and detectors.

Measurement of Phase Noise using a High-Q Cavity

10.6 GHz OSC

S,(f) =

4V . C(f)K»(f) tu^ OBM,

K'(f) - 1* (^/

Output

S, j.'Wl mU

DBM

Output

Volta(

<

o ^

[^

-500 kHz 10.6 GHz +500 KHz

Frequency

Fig. 8. A. block diagram of a high Q resonator used

as a frequency discriminator. B. Frequency

discriminator curve for the scheme shown in A used at X-band with a cavity having a loaded quality factor

of approximately 25,000.

Measurement of S^ (f ) Using a Delay Line

S»(0--^ SJt) &

S^(l) V^ - 2«Xd'<dVo0y '

r^y-l •Ni-I DeteyUne [-J ^

1

Spectrum

Ana>yzef

CaHxate

Slope K^G(f) volts/rackan T

K Radians

Fig. 9. Delay line frequency discriminator.

437

TN-134
either increase or decrease the phase sensitivity of

the mixer system. Fig. 4 shows the noise of a

Phase Noise of Quartz Oscillators Added Phase Noise by Amplifiers

-80

N

I -90

T3 -100

(0 -110 ^

O

-120 _ -130

0) -140

cuo -150

c

-160

>'^. "^ -170

</) -180

\ MESFET 106 GHz MESFET ^4 GHz

,D Si up to 1 GHz

12 3 4

Log Fourier Frequency (Hz)

Fig. 10. Phase noise performance of selected quartz

oscillators. The LN 5 MHz oscillator is driven at a high level to reduce the wideband noise while the LD

5 MHz oscillator is driven at a lower level to obtain

low phase noise close to the carrier.

Phase Noise at X Band

of Multiplied Quartz Qscillators

N

I

•a

CO

o

^. CD

_C

Log Fourier Frequency (Hz)

Fig. 11. Phase noise of the oscillators of Fig. 10 if multiplied to X-band in a perfect frequency

multiplier.

III. F. Multiplication/Division

The use of perfect frequency multipliers (or

dividers) between the signal source and the double

balanced mixer increases (decreases) the phase noise

level [12] as

S^i'i(f) = S^i'i(f) (22)

where u^ is the initial carrier frequency and I'j is the final carrier frequency. This can be used to

1 2 3 4 5 6 Log Fourier Frequency (Hz)

Fig. 12. Curves A and B show the phase noise added

by selected GaAs MESFET amplifiers. Curve C shows

the phase noise added by a typical common emitter

silicon bipolar transistor with a "good" rf bypass on

the emitter lead. Curve D shows the typical

performance of the same amplifier with a small

unbypassed impedance (approximately

1/transconductance) in the emitter lead. The added

phase noise is generally independent of frequency

over a very large range.

specialized 5 to 25 MHz multiplier referred to the 5 MHz input. A potential problem with the use of the multiplier approach comes from exceeding the linear

range of the mixer. Once the phase excursion, i(>

,

exceeds about 0.1 radian, non-linearities start to become important and at A^ ~ 1 radian, the measurement is no longer valid [12]. An additional

practical problem is that low noise multipliers are usually narrowband devices. Each significantly new

frequency generally requires a new set of frequency

multipliers

.

IV. A. The New NBS Phase Noise Measurement Systems

The new NBS phase noise measurement systems are a combination of the traditional two oscillator

approach shown in Figs. 1 and 2 plus the reference

phase modulation technique mentioned in section IIIC. The complete block diagram is sketched in Fig. 13. This approach yields the widest possible bandwidth

and the lowest phase noise of a single channel

system. It does, however, require the use of two sources for oscillator measurements. From hardware

considerations we generally use 3 different phase

noise measurement systems. Test set A accepts

carrier frequencies from 5 to 1300 MHz and can measure the phase noise from 1 Hz to about 10% of the carrier or a maximum of 100 MHz. Test set B accepts

carrier frequencies from 1 GHz to 26 GHz and can measure the phase noise from 0.01 Hz to about 500 MHz from the carrier. Test set C accepts carrier

frequencies from about 2 to 26 GHz and can measure

the phase noise from 0.01 Hz to 1 GHz from the carrier. Test set D accepts carrier frequencies from

33 to 50 GHz in WR22 waveguide and can measure the phase noise from 0.01 Hz to about 1.3 GHz from the carrier

.

The construction of the phase modulator between the reference source and the mixer will be described in

TN-135

438
detail elsewhere [12]. The low pass filter section

is used in order not to saturate the antplifiers with

the carrier feedthrough signal from the mixer. One

dc amplifier is used for phase noise measurements

from dc to 100 kHz from the carrier. One ac amplifier is used for phase noise measurements from

50 kHz to 32 MHz. This range is well matched for one of our spectrum analyzers. The wideband ac amplifier

has a bandwidth of 50 kHz to 1 . 3 GHz and is used when

the desired measurement bandwidth exceeds 32 MHz. The wide bandwidth spectrum analyzer also provides a convenient way to observe the gross features of the output phase noise and to identify any major spurious

outputs if present. In order to obtain the most

accurate measurement of the phase noise it is,

however, necessary to measure the amplitude of the first IF signal at about 21 MHz in order to avoid the variations in gains of the log amplifiers with

various environmental factors.

the beat period and pretrigerred at about -2V in order to accurately determine the slope of the output

in volts per radian. This calculation, shown in eq. (23) below, is typically accurate ± 2% or 0.2 dB.

OTTl I CTOW

Fig. 13. Generalized block diagram of the new NBS phase noise measurement systems. The phase noise of

carrier frequencies from 1 MHz to 100 GHz can be measured by varying the components in the phase

shifters and mixers. Dedicated measurement systems

covering 5 MHz to 50 GHz are described in the text.

IV. B. Measurement Sequence

1) The output power of the two sources to be

measured is typically set to between +5 and + 13 dBm

at the mixer. This takes into account the insertion

loss of the phase modulator. If the oscillators

don't posses sufficient internal isolation to prevent

unwanted frequency pulling, isolators (or isolation

amplifiers) are generally inserted between the sources and the mixer.

2) The absolute sensitivity of the mixer and the dc amplifier for converting small changes in phase to voltage changes is determined in a way similar to the traditional method, namely by allowing the two oscillators to slowly beat. The output of the dc amplifier is recorded by the digitizer in the FFT connected to the dc amplifier in order to accurately

determine the period of the beat. In test sets C and

D an additional 50 MHz digitizer is used to average

and record the beat frequency. The time scale of the digitizer is then expanded to approximately 10% of

K = Volts/Second) (Feriod/(2>r) (23)

3) The two sources to be measured are phase locked

together with sufficient bandwidth that the phase

excursions at the mixer are less than 0.1 radian.

The necessary phase lock gain is calculated using an estimate for the noise of the oscillators and the tuning rate for the reference oscillator. This is then verified by noting the peak to peak excursions

of the dc amplifier and using the measured conversion

sensitivity measured in step 2 above. If the peak

phase excursions are in excess of 0.1 radian, then

the phase lock loop bandwidth is Increased (if

possible) in order to satisfy this condition.

4) The modulator is driven by a reference frequency

(typically at +7 to +10 dBm) which steps through the Fourier frequencies of interest and the detected RMS voltage recorded on the appropriate spectrum

analyzer. This approach accurately yields the relative gains of each amplifier and its respective

spectrum analyzer since it automatically accounts for

the effect of the phase lock loop and residual

frequency pulling as well as the termination of the mixer and the variations in gain of the various

amplifiers with Fourier frequency. The amplitude of

the phase modulation on the carrier is constant in amplitude to better than ± 1 . 5 dB (typically ±0.2dB

for f < 500 MHz) for reference frequencies dc to about 10% of the carrier frequency or a maximum of 1 GHz. Initial measurements of the prototype

modulator are shown in Fig. 14.

This measurement is then combined with the measurement of the absolute mixer sensitivity

multiplied by the gain of the dc amplifier described

in step 2 above. The absolute gain of all the amplifiers shown in Fig. 13 can generally be

determined to an accuracy of ± 0.3 dB (1.5 dB for Fourier frequencies from 500 MHz to 1 GHz) over the Fourier frequencies of interest.

5) Next the spectral density function of the FFT is verified. The level of the noise determined by the FFT for the input of the noise source amplifier

sequentially shorted to ground, connected to ground

through a 100 kfl metal film resister, and connected

to ground through 200 pF, is then recorded. From

these data one can determine the Inherent noise

voltage and noise current of the noise source amplifier plus the FFT as well as the noise of the resistor to about ± 0.25 dB which is the stated accuracy of the FFT. This primary calibration of the FFT can be carried out from about 20 Hz to over 50

kHz. Above 50 kHz, the noise gain of the amplifier

we used contributes a significant amount of noise.

With some compensation the noise is flat to within

±0.2 dB from 20 Hz to 100 kHz. Next the noise source

is switched into the output of the mixer and the relative noise spectrum of all the spectrum analyzers

is calibrated by knowing their relative gains. This

procedure verifies the voltage references and noise

bandwidths of the various spectrum analyzers.

6) The noise voltage is recorded on the three spectrum analyzers over the Fourier frequencies of

interest, generally the same one used in step 4

TN-136

439
above. The measured noise voltages are scaled using

the measured gains and the spectral density of phase

noise calculated. The overlapping ranges of the various spectnim analyzers allows one the opportunity

to compare the measurements on the three spectrxim

analyzers. Typically one can obtain S^(f) of the oscillator pair to an accuracy of about ± 0.6 dB at Fourier frequency above 100 Hz. The agreement with

repeat measurements is often of order ± 0.2 dB as shown in Fig. 15.

7) The noise floor of the system is determined by

driving both sides of the measurement system with one oscillator having a similar power and impedance level

as that used in these measurements. If the noise

floor is within 13 dB of that measured in step 7 above, corrections are made to the measurement data

to remove the bias generated by the noise floor.

Measurements of the amplitude accuracy of the phase

modulation side bands generated by the prototype

phase modulators are summarized in Fig. IS. The same modulator was used for carrier frequencies from 5 to 300 MHz. The error in the phase modulation amplitude

is less than 0.5 dB for modulation frequencies from

dc to 10% of the carrier. The 10 GHz modulator also

maintains an accuracy of better than 0.5 dB out to 500 MHz from the carrier. At 1 GHz the modulation

amplitude is 1.5 dB high. Once this is measured it can be taken into account in the calibration

procedure. The 45 GHz modulator results shown in

Fig. 15 should be attainable over the entire WR22

wavequide bandwidth.

The performance that can be obtained with this measurement technique is illustrated by actual phase

noise data on oscillator pairs shown in Figs. 15 and

16. Typical accuracies are ±0.6 dB with a noise

floor at about - 175 dB relative to 1 radian^ /Hz.

The corrections applied to the raw data at 10 GHz are shown in Fig. 17. At low frequencies the effect of

the phase- locked loop is apparent while at the higher

frequencies the roll-off of the amplifiers are important. These effects have been emphasized here

in order to examine the ability of the calibration

process to correct for instrumentation gain

variations

.

V. Conclusion

We have analyzed several traditional approaches to making phase measurements and found that they all

lacked some element necessary for making phase noise measurements from essentially dc out to 10% of the carrier frequency with good phase noise floors and an accuracy of order 1 dB. By combining several of the techniques and adding a phase modulator which is exceptionally flat from dc to about 10% of the carrier frequency, we have been able to achieve

excellent phase noise floors, bandwidths of at least

10% of the carrier, and accuracies of order ±0.6 dB.

15

o

o 1.0

S 05

i(0

-0.5

2 -1.0

Q.

-1.5

1 1—

r

-I—I—r-

~>—

r

10 GHz

0.001 0.01 01 1 10 100 Modulation Frequency (f) in % of Carrier Frequency

Fig. 14 Measurement of the amplitude error of

modulation signal versus Fourier frequencies f, for these prototype phase modulators. Curves labeled 5, 100, and 300 MHz were obtained with the modulator

used in 5 to 1,300 MHz test set. The curve labeled

10 GHz was obtained with the modulator for the 2 to

26 GHz test set. The curve labeled 45 GHz was obtained with the WR22 test set.

N

I

cnT" a01

o

T3

CO

Two Oscillator Calibration Test @ 5MHz

136 - -

-

• System A

-

- ^v

X System B

-

- X A System B + A/4 -

138 - T X. -

-

IdB ^^

•^^^'^=—r X

-

140 -

1 1 ..i...J _X _1_1_1

-

10 Hz 100 Hz 1kHz Log Fourier Frequency (Hz)

Fig. 15 Demonstration of calibration accuracy for

two oscillator concept. The curve labeled System A

shows the measured phase noise of a pair of 5 MHz oscillators using the test set shown in Fig. 13. The curve labeled System B shows the measured phase noise

of the same pair of 5 MHz oscillators using a totally

separate measurement system with the oscillators held

in phase quadrature with the measurement test set of

A. The curve labeled System B + X/4 shows the phase

noise of the same pair of oscillators using test set

B with and extra cable length of A/4 inserted into

each signal path. The agreement between the three curves is in the worse case ± 0.15 dB.

TN-137

440
Phase Noise at X Band

T r

3 4 5 6 7 Log Fourier Frequency (Hz)

Fig. 16 Phase noise measurement on a pair of 10.6 GHz sources using the new NBS measurement technique.

CD

•ac

<nc

o

o§

oo

co

(0

Calibration Correction

(0 O

4 5 6 7 8

|jog Fourier Frequency (Hz)

Fig. 17 Correction factor applied to the measurement

data made to obtain the results of Fig. 16.

Acknowledgements

The authors are grateful to many colleagues,

especially David W. Allan, James C. Bergqulst, Andrea

DeHarchl, David J. Glaze, James E. Gray, David A. Howe, John P. Lowe, Samuel R. Stein and Charles Stone

for many fruitful discussions on this topic and the Calibration Coordination Group for the funding to Improve the accuracy and bandwidth of phase noise

metrology.

References

1. J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunlgal. J. A. Mullen,

Jr., W. L. Smith R. L. Sydnor, R. F. C. Vessot, G. M. Winkler, Characterization of Frequency Stability,

Proc. IEEE Trans, on I & M 2fi, 105-120 (1971).

3. D.W. Allan, H. Hellwig, P. Kartaschoff, J. Vanler, J. Vlg, G.M.R. Winkler, and N.F. Yannonl,

Standard Terminology for Fundamental Frequency and

Time Metrology, to be published In the Proc. of the 42nd Symposium on Frequency Control, Baltimore, MD, June 1-4, 1988.

4. F. L. Walls, and S. R. Stein, Accurate Measurements of Spectral Density of Phase Noise In Devices, Proc. of 31st SFC, 335-343, (1977).

(National Technical Information Service, Sills

Building, 5825 Port Royal Road, Springfield, VA

22161).

5. F. L. Walls, S. R. Stein. J. E. Gray, and D. J. Glaze, Design Considerations in State-of-the-Art

Signal Processing and Phase Noise Measurement

Systems, Proc. 30th Ann. SFC, 269-274 (1976).

(National Technical Information Service, Sills

Building, 5285 Port Royal Road. Springfield. VA

22161).

6. R. L. . Barger, M. S. Soren, and J. L. Hall,

Frequency Stabilization of a cw Dye Laser, Appl.

Phys. Lett. 22, 573 (1973).

7. F. L. Walls and A. E. Walnwrlght, Measurement of

the Short-Term Stability of Quartz Crystal Resonators

and the Implications for Crystal Oscillator Design

and Applications, IEEE Trans, on I & M 24, 15-20

(1975).

8. A. S. Risley, J. H. Shoaf, and J. R. Ashley,

Frequency Stabilization of X-Band Sources for Use in Frequency Synthesis into the Infrared. IEEE Trans, on

I 6i M. 21, 187-195 (1974).

9. J. R. Ashley, T. A. Barley, and G. J. Rast, The Measurement of Noise in Microwave Transmitters, IEEE

Trans, on Microwave Theory and Techniques, Special

Issue on Low Noise Technology, (1977).

10. A. L. Lance, W. D. Seal, F. G. Mendoza, and N. W. Hudson, Automating Phase Noise Measurements in the Frequency Domain, Proc. 31st Ann. Symp . on Freq.

Control, 347-358 (1977).

11. A. L. Lance and W. D. Seal, Phase Noise and AM

Noise Measurements in the Frequency Domain at Millimeter Wave Frequencies, from Infrared and

Millimeter Waves . Ken Button Ed., Academic Press, NY

1985.

12. F. L. Walls and A. DeMarchl, RF Spectrum of a Signal After Frequency Multiplication Measurement and Comparison with a Simple Calculation, IEEE Trans, on

I & M 24. 210-217 (1975).

13. F.L. Walls, A New Phase Modulator for Wideband

Phase Noise Measurement Systems, to be submitted to IEEE Transactions on Ultrasonics, Ferroelectrics and

Frequency Control.

2. J. H. Shoaf, D. Halford, and A. S. Risley,

Frequency Stability Specifications and Measurement,

NBS Technical Note 632, (1973). Document available

from US Government printing office. Order SD at

»C13. 46:632.

TN-138

441
42nd Annual Frequency Control Symooslum - 1988 STANDARD TERMINOLOGY FOR

FUNDA.HENTAL FREQUENCY AND TIME METROLOGY

David Allan, National Bureau of Standards, Boulder, CO 30303

HelaiuC HelLwlg, Nadonal Bureau of Standards,

Calchersburg, HD 20899 Pecer Kartaschoff, Swiss PTT, RiD. CH 3000 Bern 29, Switzerland

Jacques Vanler, National Research Council, Ottawa, Canada KlA 0R6 John Vlg, U.S. Amy Electronics Technology and Devices Laboratory,

Fort Monmouth, NJ 07703 Cernot M.R. Winkler, U.S. Naval Observatory, Washington. DC 20390 Nicholas F. Yannonl , Rome Air Development Center, Hanscoo AFB, Bedford, MA 01731

1. Introduction

Techniques to characterize and to measure the frequency

and phase instabilities in frequency and time devices and in received radio signals are of fundamental impor- tance to all manufacturers and users of frequency and

time technology.

In 1964, a subcoounittee on frequency stability was form- ed within the Institute of Electrical and Electronics

Engineers (IEEE) Standards Committee 14 and, later (in

1966), in the Technical Committee on Frequency and Time

within the Society of Instrumentation and Measurement

(SIM) , to prepare an IEEE standard on frequency stability. In 1963, this subcommittee completed a document proposing definitions for measures on frequency and phase stabilities (Barnes, et al., 1971). These recom- mended measures of instabilities in frequency generators

have gained general acceptance among frequency and time users throughout the world.

In this paper, measures in the time and in the frequency

domains are reviewed. The particular choice as to which

domain is used depends on the application. However, the users are reminded that conversions using mathematical

formulations (see Appendix I) from one domain to the other can present problems.

Most of the major manufacturers now specify Instability

characteristics of their standards in terms of these recommended measures. This paper thus defines and formalizes the general practice of more than a decade.

2. Measures of Frequency and Phase Instability

Frequency and phase instabilities shall be measured in terms of the instantaneous, normalized frequency departure y(t) from the nominal frequency f, and/or by phase

departure ^(t), in radians, from the nominal phase Ini/^t as follows:

y(t) =

x(t)

1 2-0 dt - 2,.Q

i(%^

S,(f) of y(t)

S^(f) of *(t)

s;(f) of ^(c)

S,(f) of x(t).

These spectral densities are related by the equations:

Si(f) = (2)rf)'' S^(f) n r

S (f) X

(2..o)

1 %(f>

2irt/,

A device or signal shall be characterized by a plot

of spectral density vs. Fourier frequency or by

tabulating discrete values or by equivalent means

such as a statement of power lav(s) (Appendix I).

According to the conventional definition

(Kartaschoff. 1978) of £(£) (pronounced "script

ell"), Z(f) is the ratio of the power in one side- band due to phase modulation by noise (for a 1 Hz bandwidth) to the total signal power (carrier plus

sidebands), that is,

„,„ _ POicr density, one phase-npi.se modulation sideband/Hz Total si^val po>«r

The conventional definition of Jf(f) is related to S,(f) by

/(f) 3 SS^(f)

only if the mean squared phase deviation, <^^(f)> =

the integral of S^(f) from f to «. is much smaller

than one radian. In other words, this relationship

is valid only for Fourier frequencies f far enough

from the carrier frequency and is always violated

near the carrier.

where x(t) is the phase departure expressed in units of

time

.

3. Characterization of Freouencv and Phase InstabtUties

a. Frequency Domain :

In the frequency domain, frequency and phase in- stability is defined by any of the following one- sided spectral densities (the Fourier frequency

ranges from to •)

;

* See Appendix Note # 17 419

Since S^(f) is the quantity that is generally mea- sured in frequency standards metrology, and JS(f) has become the prevailing measure of phase noise

among manufacturers and users of frequency standards, £{€) is redefined as

/(f) HS^(f) .

This redefinition is Intended Co avoid erroneous

use of f(f) in situations where the small angle

approximation Is not valid. In other words, S^(f)

TN-139
Is ch« pr*f*rred aeasure, since, unaoblguously, Ic slways c«n b« measured.

b. TlM-D^waln:

In Che clae domain, frequency Inacablilcy shall be defined by Che cvo-saaple devlaclon 9^(r) which is Che square rooc of che Cvo-saaple variance a^'(f).

This variance, ffy*(r), has no dead- Clae becveen che frequency saaples and is also called Che Allan

variance. For Che sampling clae r, we wrlce:

y 2 \<yv*i - y^)^

'k+l -'Vc' where ^Jc, + f X y(c)dc » — h>l " \

The symbol < > denotes an inflnlce Clme average. In practice, Che requirement of infinite time average is never fulfilled; Che use of che foregoing Cerms shall be permitted for finite time averages. The x, and x. . are cime residual mea- suremencs made at t, and c. . = C. +f , k=l,2,3,..., and 1/r is Che nominal fixed sampling rate which

gives zero dead cime becveen frequency measuremencs. "Residual" implies che known systematic

effects have been removed.

If dead time exists between the frequency departure

measurements and this is ignored in Che compucacion

of ffy(r), resulcing inscabilicy values vill be biased (except for vhite frequency noise). Some of

Che biases have been studied and some correction

cables published [Barnes, 1969; Lesage, 1983; Barnes and Allan, 1988]. Therefore, che cerm ff^Cf) shall noc be used Co describe such biased measurements. Rather, if biased instability measures are

made, Che information in the references should be used CO reporc an unbiased escimate.

If the initial sampling rate is specified as 1/r,

,

Chen it has been shown that, in general, we may obtain a more efficient estimate of <r^(r) using

what is called 'overlapping estimaces.' This

escimate is obtained by computing

y A.

2(N-2a)r'

N-2m

1=1

, ,. -2x. +x.) l+2m i+m i'

where N is the number of original Clme residual measuremencs spaced by fj(N=M+l, where M is the number of original frequency measurements of sample

time f,) and r = mr,

.

From the above equation, we see that ffy*(r) acts

like a second-difference operator on the time deviation residuals- -providing a stationary measure

of che stochastic behavior even for nonstationary

processes. Additional variances, which may be used

CO describe frequency instabilities, are defined in Appendix It.

c. Clock-TliTie Prediction

The variation of the clae difference ber^een a real clock and an ideal uniform cime scale, also known

as Clae Interval error, TIE, observed over a Cime Incar/al starting at time c,, and ending ac Cj+c shall be defined as:

TIE(c) - x(Cq+c) - X(Cq)

i:

c, + c y(e')dc'

For fairly simple models, regression analysis can

provide efflcienc escimaces of Che TIE (Draper and

Smith, 1966; CCIR. 1936). In general, there are many estimators possible for any statistical quancity. Ideally, we would like an efflcienc and unbiased estimator. Using the cime domain aeasure

(Ty*(f) defined in (b) , Che following escimace of

Che standard deviation (RMS) of TIE and its associated systematic departure due Co a linear frequency drifc (or ics uncercalncy) can be used Co predicc a probable clae Incerval error of a clock

synchronized ac c^Ca^O and left free running thereafter:

x(t.) RXSTIE = c esc

C* + <T + <7 (r = c) + (- ^>1^

where "a" is che normalized linear frequency drift

per unit of time (aging) or Che uncercalncy in che drift estimate, a^ the Cwo-saople deviation of

Che initial frequeftcy adjustaenc, », (r) che cwo- sample devlaclon describing Che random frequency

inscability of the clock at c=r , and x(t|,) is che inicial synchronization uncertainty. The third

Cerm in che brackets provides an optimum and un- biased estimate (under the condition of an opcimua

(RXS) prediction method) in the cases of white

noise FM and/or random walk FM. The third cerm is Coo optimistic, by about a factor of 1.^, for flicker noise FM, and too pessiaistic, by about a factor of 3, for white noise PM.

This estimate is a useful and fairly simple approximation. In general, a more complete error analysis becomes difficult; if carried out, such an

analysis needs to Include the methods of time prediction, the uncertainties of che clock paramecers, using che confidence limits of measurements defined

below, che detailed clock noise models, systematic

effects, etc.

&. Confidence Limits of Measurements

An estimate for <7^(r) can be made from a finite data set with H measurements of y^ as follows:

^y(r)

M-1

2(M-i) ^ ^-^j+i ^y 1

or, if the data are time readings x^ : M-1 a (r) y r (M-1) ._^ J J

i J

2 H

The 68 percent confidence interval (or error bar) , I,

,

for Gaussian noise of a particular value a^^r) obtained

from a finite number of saaples can be estimated as follows

:

where:

<r^(r)S-"<''

tocal number of data points used in Che estimate

,

an integer as defined in Appendix I,

«, =0.99,

420 TN-140
"0 0.87,

0.77,

0.75.

Aj «n axaopl* of Ch« C«uaslan model wleh M«100, a « -1 (£Iiclc«r frequency noise) *nd Ojif * 1 second) « 10'^*, we iB«y vrlce:

<7y(f). (0.77). (100)-* = <7y(f). (0.077)

which gives:

<7,(f - 1 second) » (I ± 0.08) x

10*^^.

If H is small, Chen the plus and minus confidence incervals become asymaecrlc and the k, coefficlencs are noc

valid; however, these confidence intervals can be calcu- lated (Lesage and Audoin, 1973).

If 'overlapping" estimates are used, as outlined above, then the confidence interval of the estimate can be shown to be less than or equal to I, as given above

(Howe, Allan. Barnes, 1981).

5. Reeomnendations for Characterizing or ReOQrtln£

Measurements of Frequency and Phase Instabilities

a. Nonrandom phenomena should be recognized, for

example:

any observed time dependency of the sta- tistical measures should be stated;

o the method of modeling systematic

behavior should be specified (for ex- ample, an estimate of the linear fre- quency drift was obtained from the coefficients of a linear least-squares

regression to M frequency measurements, each with a specified averaging or sample

time r and measurement bandwidth f^ )

o the environmental sensitivities should be

stated (for example, the dependence of frequency and/or phase on temperature,

magnetic field, barometric pressure, vibration, etc.);

b. Relevant measurement or specification

parameters should be given:

Che environment during measurement;

if a passive element, such as a crystal

filter, is being measured in contrast to

a frequency and/or time generator.

6. References

Barnes, J. A., Tables of bias functions, Bi and Bj . for variances based on finite samples of processes with

power law spectral densities, NBS, Washington, DC, Tech. Note 375, (Jan. 1969).

Barnes, J. A. and Allan, D.U. , "Variances Based on Data

with Dead Time Between the Measurements: Theory

and Tables," NBS Tech Note 1318 (1983).

Barnes. J. A.. Chi. A.R. . Cutler, L.S., Healey, D.J., Leeson, D.B. , McGunlgal, T.E., Mullen, J. A., Smith, W.L., Sydnor, R. , Vessot, R.F. andUinlcler,

C.M.R., Characterization of frequency stability, IEEE Trans. Instr. andMeas., Vol. IM-20 . 105-120,

(May 1971).

CCIR Report 898, Dubrovnik, (1986).

Draper, N.R. and Smith, H.. Applied Regression Analysis,

John Wiley and Sons, (1966).

Howe. D. A., Allan. D. W. , and Barnes, J. A., Properties

of signal sources and measurement methods. Proc. 35th Annual Symposium on Frequency Control, 669-

717. (May 1981).

Kartaschoff. P., Frequency and Time. Academic Press. New

York (1978).

Lesage. P.. and Audoin. C. Characterization of frequency

stability: uncertainty due to the finite number of

measurements. IEEE Trans. Instr. and Meas.. Vol.

IM-22 . 157-161. (June 1973).

Lesage. P.. Characterization of Frequency Stability:

Bias due to the Juxtaposition of time interval measurements. IEEE Trans, on Instr. and Meas, IM22. 204-207. (1983).

Stein. S. R. Frequency and Time: Their measurement and

characterization. Precision Frequency Control.

Vol. 2. Academic Press. New York. 19l'-232, (1985).

the method of measurements;

the characteristics of the reference

signal;

the nominal signal frequency fg

the measurement system bandwidth f,, and

the corresponding low pass filter

response;

the total measurement time and the number

of measurements M;

the calculation techniques (for example,

details of the window f'onctlon when estimating power spectral densities from

time domain data, or the assumptions

about effects of dead- time when estimating the two-sample deviation a^(r));

the confidence of the estimate (or error

bar) and its statistical probability

(e.g. " three - s Igma* )

APPENDIX I

1. Power-Law Spectral Densities

Power-law spectral densities are often employed as rea- sonable and accurate models of the random fluctuations

in precision oscillators. In practice, these random fluctuations can often be represented by the sum of five

independent noise processes, and hence:

Sy(f)

+2

5" h f° for < f < f. a=-2

for f > f^

where h, 's are constants, a's are integers, ar.d f^ is the high frequency cut-off of a low pass filter. High

frequency divergence is eliminated by the restrictions on f in this equation. The identification and charac- terization of the five noise processes are given in Table 1. and shown in Fig. 1.

421 TN-141
2. Convtrilon BeevcTi Fr«auancv and Ttma Domain

Th« op«r«clon of eh« eouncar, averaging th« fraquancy

for a Claa r, may b« thoughc of a* a f Hearing oparatlon. Tha transfer funcclon, H(f), of Chls aqulvalenc

fllcar Is Chan Cha Fourier cransfom of the Impulse response of the filter. The time domain frequency

Instability Is then given by

c/^ (M.T.r) / S,(f) I H(f) |» df.

where Sy(f) Is the spectral density of frequency fluctuations. l/T Is the measurement rata (T-r Is tha daad

time between measurements) . In the case of the Cwo- sanple variance |H(f)|» Is 2(»ln *«rf)/(irrf)» . The two- sample variance can thus be computed from

y

s (f) *i^»-^ df . (»'f)

Specifically, for the power law model given, the time

domain measure also follows a power law.

TABLE 1 - The functional characterlsclcs of Che Independenc noise processes

\ised In modeling frequency Inscablllcj of osellletors

Description of

Noise Process

< Slope characteristics of log log plot

Frequency domain Time-domain

S,(f) or

S;(f)

S^(f) or

S,(f) «^(0 <Ty(r) Mod.a,(r)

a P p p/2 m' Random Walk Freq Hodulatlon -2 -4 1 1/2 1

Flicker Frequency Modulation -1 -3

White Frequency Modulation -2 -1 -1/2 -1

Flicker Phase Modulation 1 -1 -2 -1 -2

Uhlte Phase Modulation 2 -2 -1 -3

S^(f) - i'lh.f -» - v\h^t» (^ 0-2)

s.(f)-2:irh,f-« -5^h,f' Mod.ffy(r) - |r|'

XA2LE 2 - Translation of frequency Instability measures from spectral densities In frequency domain to variances In time domain and vice versa (For 2)rf^r » 1)

Description of noise process <T2(r) - Sy(f) - S^(f) =

Random Walk Frequency

Modulation

Flicker Frequency Modulation

White Frequency Modulation

Flicker Phase Modulation

\ White Phase Modulation

1

A|f2 S,(f)|rl

B|f Sy(f)|f»

Cir" S,(f)lr-l

D|rlS,(f)|r-»

E|r*S,(f)lf-*

^(rO aj(r)]f-l

^[r^ <^^(r)]f«

4,r*

21og.2

C - 1/2

1.038 + 3 log,(2irf^r)

V 3f.

kx^

422 TN-142
y

2 2"r" ' * ^1 2108.2 * \ t7

h

1.038 3 log (2)rf.r) 3f.

(2»)

2 2 + h. (2,)^2

This lopUelcly assumes thac ch« random driving a«eh- anlsa for each term Is Independent of che ochers. In

addition, there Is the lopUclt assuaptlon that the mechanlsa Is valid over all Fourier frequencies, vhlch

laay not always be true.

The values of h, are characteristic models of oscillator

frequency noise. For Integer values (as often seems to be the case for reasonable models), >i = -a - 1, for -3 i a i 1, and ;» « -2 for ail, where ffy*(r)-f».

Table 2 gives the coefficients of the translation among

the frequency stability measures from time domain to frequency domain and from frequency domain to time domain.

The slope characteristics of the five Independent noise

processes are plotted In the frequency and time domains

In Fig. 1 (log-log scale).

APPENDIX n

ADDITIONAL VARIANCeS THAT MAY BE USP.P TO PgSCTtlBg

FREQUENCY TNSTAHTT. ITIgS IH THg TIME DOMAIH

1. Modified Allan, or Modified Two-Saaole. Variance

Mod g^»(f)

Instead of the use of <r '(r), a 'Modified Variance' Mod

9y'(r) may be used to cnaracterlze frequency Instabilities (Stein, 198S; Allan, 1987). It has the property of

yielding different dependence on r for white phase noise

and flicker phase noise. The dependence for Mod 9^(r)

Is f""* and f'^ respectively. Mod (7y*(r) Is defined

as: N-3»+l Mod <; (r) = —-—-

y 2r2 .2 (i,

' 1

-3m+l) ^—

j=l

UH-J-l

- 2x. +x,) +2a l+m I

where N Is the original number of time measurements spaced by r, and r = mr, the sample time of choice

((I=M-t-l). A device or signal shall be characterized by a plot of (Ty(f) or Oy^'t^r) or Mod <7y(r) or Mod <7y*(r) vs. sampling time r, or by tabulating discrete values or by

equivalent means such as a statement of power laws (Appendix I)

LogS.(f) N

-2 f2/1

k. / \f-1 f^ /

l\ fO /

1 1 1 1 Log Fourier Frequencr/, f

2. Other Variances

Several other variances have been Introduced by workers

In this field. In particular, before the Introduction

of the r^o-sample variance. It was standard practice to use the sample variance, s^ , defined as

f^ S,(f) (-ii2_l£i)2 df

LogS9i(f)

\

I V-^ 1 \

1

\ \-3

\ 1

1 \ ,-'2 1

1 \

1 \vH 1

1 1 K

fo

1 1 i 1 1

1 Log Fourier Frequency, f

In practice It may be obtained from a set of measure- ments of the frequency of the oscillator as 2=1

^ 1=1 ^ y)^

The sample variance diverges for some types of noise

and, therefore. Is not generally useful.

Other variances based on the structure function approach

can also be defined (Llndsey and Chi, 1976). For example, there are the Hadamard variance, the three- sample variance and the high pass variance (Rutaan

1978). They are occasionally used in research and scientific works for specific purposes, such as diffarantiating between different t;/pes of noise and for dealing with systematica and sidebands In the spectrum.

APPgNPIX III

Leg 0-y(T

^r-}

\ \i r-^ 1

J ^1 1

tO|

1

1

1 >

Log Sample Time, r

Slope characteristics of the five independent noise processes.

Fians 1

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Research Laboratory, Washington, D.C., (Dec 1976).

Winkler, G. H. R. . Hall, R. G., and Perclval, D. B., The

U.S. Naval Observatory clock ctoM reference and th« performance of a aanple of atoaic clocks. Matrologla. Vol. 4, 126-134, (Oct 1970).

Yoshlaura, K. , Characterization of frequency stability:

Uncertainty due to the autocorrelation of Che fre- quency fluctuations. IEEE Trans. Instr. and Meas.,

IM-27 . 1-7. (Mar 1978).

The authors are aeabera of the Technical Coaalttee TC-3,

Tiae and Frequency, of the IEEE Instrunentation and

Measurement Society. This paper is part of this Coaalctee's effort to develop an IEEE Standard. To this end,

the IEEE authorized a Standards Coordinating Coaalttee,

sec 21. and the development of the IEEE Standard under

PAR-P-1139. sec 21 is formed out of the I6M. UFFC and

MTT Societies; the authors are also members of SCC 21.

425 TN-145
lEtE TK',N_-.\CTIONS ON . NSTl L'M ENTATIO.-' AND MEASUREMENT. VOL. IM-20, NO. 2, M \Y 1971 Characterization of Frequency Stability

105

JAMES A. BARNES, senior member, ieee, ANDREW R. CHI, senior member, ieee, LEONARD S. CUTLER,

MEMBER, IEEE, DANIEL J. HEALEY, member, IEEE, DAVID B. LEESON, senior member, ieee, THOMAS E. McGUNIGAL, member, ieee, JAMES A. MULLEN, JR., senior member, ieee, WARREN L. SMITH, senior

MEMBER, IEEE, RICHARD L. SYDNOR, member, ieee, ROBERT F. C. VESSOT, and GERNOT M. R. WINKLER,

member, ieee

Abstract—Consider a signal generator whose instantaneous

output voltage VU) may be written as

V{1) = [Fo + i(l)] sin [2irvot + <p{t)]

where Fo and po are the nominal amplitude and frequency, respectively, of the output. Provided that e(0 and ^(0 = {d<p/(dt) are sufficiently small for all time t, one may define the fractional instantaneous frequency deviation from nominal by the relation

y(t) = ^•

2-irvo A proposed definition for the measure of frequency stability is the spectral density S,(/) of the function y(() where the spectrum

is considered to be one sided on a per hertz basis. An alternative definition for the measure of stability is the

infinite time average of the sample variance of two adjacent averages

of y(0; that is, if

1 f"*'

y* = - / y(t) T J,, dt

where t is the averaging period, (k+i = Ik + T,k = 0, 1, 2 • • •

, to is arbitrary, and T is the time interval between the beginnings of two successive measurements of average frequency; then the second measure of stability is

V \ _ /(yt^i - y*)'\

where ( ) denotes infinite time average and where T - r. In practice, data records are of finite length and the infinite

Manuscript received December 1, 1970. The authors of this paper are members of the Subcommittee on Frequency Stability of the Technical Committee on Frequency and Time of the IEEE

Group on Instrumentation and Measurement.

J. A. Barnes is with the Time and Frequency Division, Insti- tute lor Basic Standards, NBS, Boulder, Colo. 80302. A. R. Chi ia with the Head Timing Systems Section, NASA

Goddard Space Flight Center, Greenbelt, Md. 20771.

L. S. Cutler is with the Physical Research Laboratory, Hewlett- Packard Company, Palo Alto, Calif. 94304. D. J. Healey is with the Aerospace Division, Westinghouse

Electric Corporation, Baltimore, Md. 21203. D. B. Leeson is with California Microwave, Inc., Sunnyvale,

Calif. 94086. T. E. McGunigal is with the RF and Quantum Techniques

Section, NASA Goddard Space Flight Center, Greenbelt, Md.

21715.

J. A. Mullen is with the Research Division, Raytheon Company,

Waltham, Mass. 02154. W. L. Smith is with Bell Telephone Laboratories, Allentown,

Pa. 18108. R. L. Sydnor is with the Telecommunications Department,

Jet Propulsion Laboratory, Pasadena, Calif. 91103. G. M. R. Winkler is with the Time Service Division, U. S. Naval Observatory, Washington, D. C. 20390. R. F. C. Vessot ia with the Smithsonian Astrophysical Observatory, Cambridge, Mass. 02138.

time averages implied in the definitions are normally not available

;

thus estimates for the two measures must be used. Estimates of

Sfif) would be obtained from suitable averages either in the time domain or the frequency domain. An obvious estimate for iry(r) is

<tI(t) m v4l 2

Parameters of the measuring system and estimating procedure

are of critical importance in the specification of frequency stability. In practice, one should experimentally establish confidence limits for an estimate of frequency stability by repeated trials.

B,{r, m)

Co, Ci c(0

Dlir)

/ = «/2t

/.

git)

i, j, k, m, n M

N

n(0

/e,(r)

« See Appendix Note # 18

Glossary of Symbols

Bias function for variances based on finite samples of a process with a power-law spectral density.

(See [13].) A real constant defined by (70). Real constants. A real, deterministic ftmction of time.

Expected value of the squared

second difference of x{t) with lag

time r. See (80). Fourier frequency variable. High-frequency cutoff of an ideal- ized infinitely sharp cutoff low-pass

filter. Low-frequency cutoff of an ideal- ized infinitely sharp cutoff, highpass filter. A real ftmction of time.

Positive real coeflScient of /" in a power series expansion of the spectral density of the function y{t).

Integers, often a dummy index of summation.

Positive integer giving the number

of cycles averaged.

Positive integer giving the number

of data points used in obtaining a

sample variance. A nondetcrministic function of time.

Autocovariance function of y{t). See (58). Positive real number defined by

r = T/t.

TN-146
106 aXB TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, MAY 1971

S.(J)

S,U)

I

^1

Vit)

V,

Vr(t)

V(t)

V(t)

X(t)

vit)

9k

y

5.(r - 1)

«(0

An intermediate term used in deriving (23). The definition of <S

is given by (64). One-sided (power) spectral density on a per hertz basis of the pure real function git). The dimensions of

5,(/) are the dimensions of g'(t)/f. A definition for the measure of fre- quency stability. One-sided (power)

spectral density of y(t) on a per

hertz basis. The dimensions of

S,(J) are Hz"*.

Time interval between the beginnings of two successive measure- ments of average frequency.

Time variable. An arbitrary fixed instant of time.

The time coordinate of the beginning of the fcth measurement of average frequency. By definition, ^+1 = t, + T,k = 0,1,2 .

Dummy variable of integration;

tt s rfr. Instantaneous output voltage of

signal generator. See (2). Nominal peak amplitude of signal

generator output. See (2). Instantaneous voltage of reference

signal. See (40).

Peak amplitude of reference signal. See (40). Voltage output of ideal product

detector. Low-pass filtered output of product

detector. Real function of time related to the

phase of the signal V(t) by x(t) a

y(.t)]/(2TVo).

A predicted value for x(t). Fractional frequency offset of 7(0

from the nominal frequency. See (7). Average fractional frequency offset during the A:th measurement in- terval. See (9). The sample average of N successive

values of ?». See (76). Nondeterministic (noise) function

with (power) spectral density given

by (25). Exponent of / for a power-law

spectral density.

Positive real constant.

Ilie Kronecker 6 function defined

fl, if r = 1 by «.(r - 1) a

i^ ,^ . (0, otherwise.

Amplitude fluctiiations of signal.

See (2).

Exponent of r. See (29). Instantaneous frequency of V(t). Defined by

^(')-l^ !*(')•

Vo Nominal (constant) frequency of

Vit).

Kit) The Fourier transform of nit). <rliN, T, r) Sample variance of A^ averages

of yit), each of duration t, and

spaced every T units of time. See (10). Wli^, T, r)) Average value of the sample vari- ance irliN, T, r).

<^lir) \ second choice of the definition for the measure of frequency stability.

Defined by <7j(r) = (aliN = 2, T = T, r)). tUt) Time stability measure defined by

•pit)

f,iT, r)

w s 2ir/

fflir) = T fflir). Duration of averaging period of

yit) to obtain y^. See (9). Instantaneous phase of Vit). Defined by <t)(0 = 2Tvot + <pit). Instantaneous phase fluctuations

about the ideal phase 2Tv,}t. See (2). Mean-square time error for Doppler

radar. See (82). Angular Fourier frequency variable.

I. Introduction

THE measurement of frequency and fluctuations in frequency has received such great attention for so many years that it is surprising that the concept of frequency stability does not have a universally

accepted definition. At least part of the reason has been

that some uses are most readily described in the fre- quency domain and other uses in the time domain, as well as in combinations of the two. This situation is further complicated by the fact that only recently have

noise models been presented that both adequately describe performance and allow a translation between the time and frequency domains. Indeed, only recently has

it been recognized that there can be a wide discrepancy

between commonly used time domain measures themselves. Following the NASA-IEEE Symposium on ShortTerm Stability in 1964 and the Special Issue on Fre- quency Stability in the Peoceedings of the IEEE,

February 1966, it now seems reasonable to propose a

definition of frequency stability. The present paper is presented as technical background for an eventual IEEE

standard definition. This paper attempts to present (as concisely as practical) adequate, self-consistent definitions of frequency

stability. Since more than one definition of frequency

stability is presented, an important part of this paper

TN-147
BARNES eL al : CH.'KR.^CTKRIZ.ATION of rREQCENCY STABILITY 107

(perhaps the most important part) deals with translations among the suggested definitions of frequency stability. The applicability of these definitions to the more common noise models is demonstrated.

Consistent with an attempt to be concise, the refer- ences cited have been selected on the basis of being of most value to the reader rather than on the basis of being exhaustive. An exhaustive reference list covering

the subject of frequency stability would itself be a voluminous publication.

Almost any signal generator is influenced to some ex- tent by its environment. Thus observed frequency in- stabilities may be traced, for example, to changes in ambient temperature, supply voltages, magnetic field, barometric pressure, humidity, physical vibration, or even output loading, to mention the more obvious. While

these environmental influences may be extremely important for many applications, the definition of fre- quency stability presented here is independent of these causal factors. In effect, we cannot hope to present an

exhaustive list of environmental factors and a prescription for handling each even though, in some cases, these environmental factors may be by far the most important. Given a particular signal generator in a partic- ular environment, one can obtain its frequency stability

with the measures presented below, but one should not then expect an accurate prediction of frequency stability

in a new environment.

It is natural to expect any definition of stability to involve various statistical considerations such as sta- tionarity, ergodicity, average, variance, spectral density,

etc. There often exist fundamental difBculties in rigorous

attempts to bring these concepts into the laboratory. It

is worth considering, specifically, the concept of sta- tionarity since it is a concept at the root of many statistical discussions. A random process is mathematically defined as sta- tionary if every translation of the time coordinate maps

the ensemble onto itself. As a necessary condition, if one

looks at the ensemble at one instant of time t, the dis- tribution in values within the ensemble is exactly the same as at any other instant of time f. This is not to imply that the elements of the ensemble are constant

in time, but, as one element changes value with time,

other elements of the ensemble assume the previous values. Looking at it in another way, by observing the

ensemble at some instant of time, one can deduce no

information as to when the particular instant was chosen. This same sort of invariance of the joint distribution holds for any set of times h, tz, • ••, t, and its translation tl + T, ti + T, • • •

, tn + T. It is apparent that any ensemble that has a finite past as well as a finite future cannot be stationary, and

this neatly excludes the real world and anything of practical interest. The concept of stationarity does vio- lence to concepts of causality since we implicitly feel that current performance (i.e., the applicability of stationary statistics) cannot be logically dependent upon

future events (i.e., if the process is terminated some time

in the distant future). Also, the verification of station- arity would involve hypothetical measurements that are not experimentally feasible, and therefore the concept of stationarity is not directly relevant to experimentation.

Actually the utility of statistics is in the formation

of idealized models that reasonably describe significant

observables of real systems. One may, for example, consider a hypothetical ensemble of noises with certain properties (such as stationarity) as a model for a particular real device. If a model is to be acceptable, it should have at least two properties: first, the model

should be tractable ; that is, one should be able to easily

arrive at estimates for the elements of the models; and

second, the model should be consistent with obaervablet

derived from the real device that it is simulating.

Notice that one does not need to know that the device was selected from a stationary ensemble, but only that

the observables derived from the device are consistent

with, say, elements of a hypothetically stationary ensemble. Notice also that the actual model used may

depend upon how clever the experimenter-theorist is in generating models.

It is worth noting, however, that while some texta on statistics give "tests for stationarity," these tests are almost always inadequate. Typically, these tests determine only if there is a substantial fraction of the

noise power in Fourier frequencies whose periods are of the same order as the data length or longer. While this may be very important, it is not logically essential to the concept of stationarity. If a nonstationary model

actually becomes common, it will almost surely be be- cause it is useful or convenient and not because the

process is "actually nonstationary." Indeed, the phrase

"actually nonstationary" appears to have no meaning

in an operational sense. In short, stationarity (or nonstationarity) is a property of models, not a property of data [1]. Fortunately, many statistical models exist that adequately describe most present-day signal generators;

many of these models are considered below. It is obvious

that one cannot guarantee that all signal generators are adequately described by these models, but the authors

do feel they are adequate for the description of most

signal generators presently encountered.

II. Statement of the Problem

To be useful, a measure of frequency stability must

allow one to predict performance of signal generators

used in a wide variety of situations as well as allow one to make meaningful relative comparisons among

signal generators. One must be able to predict performance in devices that may most easily be described either

in the time domain, or in the frequency domain, or in a combination of the two. This prediction of performance may involve actual distribution functions, and thus

TN-148
108 second moment measures (such as power spectra and

variances) are not totally adequate.

Two common types of equipment used to evaluate the performance of a frequency source are (analog) spectrum

analyzers (frequency domain) and digital electronic counters (time domain). On occasion the digital counter data are converted to power spectra by computers. One

must realize that any piece of equipment simultaneously

has certain aspects most easily described in the time

domain and other aspects most easily described in the frequency domain. For example, an electronic counter has a high-frequency limitation, an experimental spectra

are determined with finite time averages. Research has established that ordinary oscillators demonstrate noise, which appears to be a superposition of causally generated signals and random nondeterministic

noises. The random noises include thermal noise, shot

noise, noises of undetermined origin (such as flicker noise), and integrals of these noises. One might well expect that for the more general cases one would need to use a nonstationary model (not sta- tionary even in the wide sense, i.e., the covariance sense).

Nonstationarity would, however, introduce significant difficulties in the passage between the frequency and time

domains. It is interesting to note that, so far, experi- menters have seldom found a nonstationary (covariance)

model useful in describing actual oscillators. In what follows, an attempt has been made to separate

general statements that hold for any noise or perturbation from the statements that apply only to specific models. It is important that these distinctions be kept in mind.

III. Background and Definitions

To discuss the concept of frequency stability immediately implies that frequency can change with time and

thus one is not considering Fourier frequencies (at least at this point). The conventional definition of instantan- eous (angular) frequency is the time rate of change of phase; that is

IKKH Tm.V.-\( riCVS ON I.N-STRUMENT»,TIO.S \.NI) MrA.-LllKM l-.S I' \M V l',l71 sent the one-sided spectral density of the (pure reali function g{t) on a per hertz basis; that is, the total "power" or mean-square value of git) is given by

2^(0 = ^- Mt) (1)

where *(0 is the instantaneous phase of the oscillator. This paper uses the convention that time-dependent

frequencies of oscillators are denoted by vit) (cycle fre- quency, hertz), and Fourier frequencies are denoted by

<i> (angular frequency) or/ (cycle frequency, hertz) where

Q> = 2wf. In order for (1) to have meaning, the phase *(0

must be a well-defined function. This restriction immediately eliminates some "nonsinusoidal" signals such as a pure random uncorrected ("white") noise. For most

real signal generators, the concept of phase is reasonably

amenable to an operational definition and this restric- tion is not serious. Of great importance to this paper is the concept of spectral density, SgiJ). The notation S, (/) is to repre-

/ SM dj.

Since the spectral density is such an important concept to what follows, it is worthwhile to present some

important references on spectrum estimation. There are many references on the estimation of spectra from data

records, but worthy of special note are [2]-[5].

IV. Definition of Measures of Frequency Stability

(Second-Moment Type)

.4. General

Consider a signal generator whose instantaneous out- put voltage V(0 may be written as

^(0 = Fo + «(0] sin [2itv,t + ^(01 (2) where Vq and v„ are the nominal amplitude and fre- quency, respectively, of the output and it is assumed

that

f «

.

and

\2m>o

« 1

(3)

(4j

for substantially all time t. Making use of (1) and (2) one sees that

and

*(0 = 2tv„< -h ,^(0

v{l) = .0 + ;r^(')- /it

(5)

(r.)

Equations (3) and (4) are essential in order that ifit] may be defined conveniently and unambiguously (sec measurement section).

Since (4) must be valid even to speak of an instantan- eous frequency, there is no real need to distinguish

stability measures from instability measures. That is. any fractional frequency stability measure will be far from unity, and the chance of confusion is slight. It is true that in a very strict sense people usually measure

instability and speak of stability. Because the chances of confusion are so slight, the authors have chosen to continue in the custom of measuring "instability" and speaking of stability (a number always much less than unity).

Of significant interest to many people is the radio fre- quency (RF) spectral density Sv(j). This is of direct concern in spectroscopy and radar. However, this is not a good primary measure of frequency stability for two

reasons. First, fluctuations in the amplitude £(0 contrib- ute directly to Sv(j) ; and second, for many cases when

TN-149
BARNES el al.: charactebization of frequency stability 109

((t) is insignificant, the RF spectrum Sy{f) is not uniquely related to the frequency fluctuations [6].

B. General-First Definition of the Measure of Frequency

Stability—Frequency Domain

By definition, let v<«-a' (7)

where <f{t) and vo are as in (2). Thus y{t) is the in- stantaneous fractional frequency deviation from the nominal frequency vo- A proposed definition of frequency

stability is the spectral density Sy (/) of the instantaneous

fractional frequency fluctuations y{t) . The function Sy{f)

has the dimensions of Hz"*. One can show [7] that if S^{f) is the spectral density

of the phase fluctuations, then

S,{f)

fS,{f). (8)

Thus a knowledge of the spectral density of the phase

fluctuations S^(/) allows a knowledge of the spectral

density of the frequency fluctuations Sy (/) , the first definition of frequency stability. Of course, Sy{f) cannot

be perfectly measured—this is the case for any physical

quantity; useful estimates of Sy{f) are, however, easily

obtainable.

C. General: Second Definition of the Measure of Fre- quency Stability—Time Domain

The second defirdtion is based on the sample variance of the fractional frequency fluctuations. In order to present

this measure of frequency stability, define ^t by the

relation

(9)

where <»+i = t^ + T, k = 0, 1,2, , T is the repetition

interval for measurements of duration r, and fo is arbitrary.

Conventional frequency counters measure the number of cycles in a period r; that is, they measure vor{l + y»). When T is 1 s they count the number of vo(l -|- y*). The second measure of frequency stability, then, is defined in analogy to the sample variance by the relation «(iv,r,,»-(^|:(s.-lts.)'). (10)

where (g) denotes the infinite time average of g. This measure of frequency stability is dimensionless. In many situations it would be wrong to assume that

(10) converges to a meaningful limit as N —* a>. First, of course, one carmot preictically let N approach infinity

and, second, it is known that some actual noise processes contain substantial fractions of the total noise power in

the Fourier frequency range below one cycle per year. In order to improve comparability of data, it is important

to specify particular N and T. For the preferred definition we recommend choosing N = 2 and T = r (i.e., no dead

time between measurements). Writing {<t\{N = 2, T = t,t))

as a\{T), the Allan variance [8], the proposed measure of frequency stability in the time domain may be written as

<r\{.r) =

(

^y*- ~ y*^'

)

(11)

for T = r. Of course, the experimental estimate of a\{T) must be

obtained from finite samples of data, and one can never

obtain perfect confidence in the estimate; the true time

average is not realizable in « real situation. One estimates

c\{t) from a finite number (say, m) of values of (rj(2, t, t) and averages to obtain an estimate of «tJ(t). Appendix I shows that the ensemble average of <rj(2, t, t) is convergent

(i.e., as m —» 00 ) even for noise processes that do not have

convergent {<r\{N, t, r)) aa N -* ». Therefore, (rj(r) has

greater utility as an idealization than does {ali'*>, r, r)) even though both involve ass\miptions of infinite averages. In effect, increasing N causes al{N, T, t) to become more

sensitive to the low-frequency components of 5,(/). In

practice, one must distinguish between an experimental

estimate of a quantity (say, of <t\(j)) and its idealized value. It is reasonable to believe that extensions to the

concept of statistical ("quality") control [9] may prove

useful here. One should, of course, specify the actual number m of independent samples used for an estimate

of <T\{r). In summary, therefore, /S,(/) is the proposed measiire of (instantaneous) frequency stability in the (Fourier)

frequency domain and <rj(r) is the proposed measure of frequency stability in the time domain.

D. Distributuma

It is natural that people first become involved with

second moment measures of statistical quantities and only

later with actual distributions. This is certainly true with

frequency stability. While one can specify the argument

of a distribution function to be, say (y*+i — y»), it makes

sense to postpone such a specfication until a real use has

materialized for a particular distribution fimction. This

paper does not attempt to specify a preferred distribution

function for frequency fluctuations.

E. Treatment of Systematic Variations

1) General: The definition of frequency stability <t\(j) in the time domain is useful for many situations. However, some oscillators, for example, exhibit an aging or almost

linear drift of frequency with time. For some applications,

this trend may be calculated and should be removed [8] before estimating <r\{T). In general, a systematic trend is perfectly deterministic

(i.e., predictable) while the noise is nondeterministic.

Consider a fimction g(J), which may be written in the form

TN-150
no

g{l) = cil) + nit) where c(t) is some deterministic function of time and n{t),

the noise, is a nondeterministic function of time. We will define c(l) to be the systematic trend to the function g{t).

A problem of significance here is to determine when and

in what sense c(t) is measurable.

S) Specific Case—Linear Drift: As an example, if we

consider a typical quartz crystal oscillator whose fractional frequency deviation is y(t), we may let

I£EB TIUNSACriONS ON INSTRUMENTATION AND MEASUMMENT, MAY 1971

(12) where Co is the frequency intercept at t = to and Cj is the drift rate previously determined. A problem arises here because

git) = ^<2/(0- (13)

With these conditions, c{t) is the drift rate of the oscil- lator (e.g., lO-^Vday) and n(0 is related to the fre- quency "noise" of the oscillator by a time derivative. One sees that the time average of g{t) becomes

ji I

^(0 dt = c,+jij^ n(t) dt (14)

where c(<) = Ci is assumed to be the constant drift rate of the oscillator. In order for Ci to be an observable,

it is natural to expect the average of the noise term to vanish, that is, converge to zero.

It is instructive to assume [8], [10] that in addition

to a linear drift, the oscillator is perturbed by a flicker noise, i.e.,

S.(J) = N-\ 0<f<U (15)

io, , / > u

where h.i is a constant (see Section V-A-2) and thus,

5.(/) =

1(2'^)'''-^' « ^ ^ ^ /* (16)

.0, i>U

for the oscillator we are considering. With these assumptions, it is seen that

and that

lim I; f

* nit) dt = k{0) =0 (17)

lim < variance U; f n(0 dH [ = (18)

where «(/) is the fourier transform of n(t). Since <Sn(0) = 0, »c(0) must also vanish both in probability and in mean square. Thus, not only does n{t) average to zero, but one may obtain arbitrarily good confidence on the

result by longer averages. Having shown that one can reliably estimate the drift rate Ci of this (common) oscillator, it is instructive to attempt to fit a straight line to the frequency aging.

That is, let

and thus

git) = yit)

git) = Co + c,(i - <o) + n'it)

(19)

(20)

S,.(J) = S,(J) and

lim s variance [Hr»'(0 dt

(21)

(22)

for the noise model we have assumed. This follows from

the fact that the (infinite A'') variance of a flicker noise process is infinite [7], [8], [10]. Thus, Co cannot be

measured with any realistic precision, at least, in an

absolute sense. We may interpret these results as follows. After ex- perimenting with the oscillator for a period of time one can fit an empirical equation to y(t) of the form

yit) = Co + tc, + n'it), where n'{t) is nondeterministic. At some later time it is possible to reevaluate the coefficients Co and Ci. According to what has been said, the drift rate Ci should be

reproducible to within the confidence estimates of the experiment regardless of when it is reevaluated. For Co, however, this is not true. In fact, the more one attempts

to evaluate Cq, the larger the fluctuations are in the

result. Depending on the spectral density of the noise term,

it may be possible to predict future measurements of

Co and to place realistic confidence limits on the prediction [11]. For the case considered here, however, these confidence limits tend to infinity when the prediction

interval is increased. Thus, in a certain sense, Cq it; "measurable" but it is not in statistical control (to use the language of the quality control engineer [9] )

V. Translations Among Frequency Stability

Measures

A. Frequency Domain to Time Domain

1) General: It is of value to define r = T/r\ that is, r is the ratio of the time interval between successive

measurements to the duration of the averaging period.

Cutler has shown (see Appendix I) that

<<^:(A^, T, r)) * N r „ c ,.^ [sin' (t/t)] /, sin' irriNr)

iN ^I*^-»^;^{'- N'sm'iiTTiT)

(23)

Equation (23) in principle allows one to calculate the

time-domain stability (oliN, T, t)) from the frequencydomain stability iS,(/). 2) Specific Model: A model that has been found useful [8], [10]-[13] consists of a set of five independent

noise processes 2«(0, n = —2, —1, 0, 1, 2, such that * See Appendix Note # 19 TN-151
BARNES et al. : chakacterization of fbwjcinct btabilitt 111

yit) = z.tit) + z.,{t) + Zo{t) + zi{t) + Z2{t) (24) Man [8] and Vessot [12] showed that if and the spectral density of z„ is given by

S..(f) = Kf, 0<f <U (25)

10, i > j„n = -2. -1,0,1,2,

where the hn are constants. Thus, Sy(f) becomes

S,(j) = h.,r' + h.J-' + ho + hri + h^\ (26)

for < / < /t and S,(/) is assumed to be negligible beyond

this range. In effect, each z, contributes to both S^if) and

(aliN, T, t)) independently of the other «,. The contributions of the «, to (<rJ(iV, T, t)) are tabulated in Appendix II. Any electronic device has a finite bandwidth and this certainly applies to frequency-measuring equipment also. For fractional frequency fluctuations yit) whose spectral

density varies as

S,(j) - /' > -1 (27)

for the higher Fourier components, one sees (from

Appendix I) that {al{N, T, t)) may depend on the exact shape of the frequency cutoff. This is true because a

substantial fraction of the noise "power" may be in these higher Fourier components. As a simplifying assumption,

this paper assumes a sharp cutoff in noise "power" at the

frequency /» for the noise models. It is apparent from the

tables of Appendix II that the time domain measure of frequency stability may depend on /» in a very important

way, and, in some practical cases, the actual shape of the

frequency cutoff may be very important [7]. On the

other hand, there are many practical measurements

where the value of /» has little or no effect. Good practice,

however, dictates that the system noise bandwidth /« should be specified with any results. In actual practice, the model of (24)-(26) seems to fit almost all real frequency sources. Typically, only two or three of the ^-coefficients are actually significant for a

real device and the others can be neglected. Because of

its applicability, this model is used in much of what

follows. Since the z, are assumed to be independent noises,

it is normally sufficient to compute the effects for a

general Zn and recognize that the superposition can be

accomplished by simple additions for their contributions

to S,(j) or (aliN, T, r)).

B. Time Domain to Frequency Domain

1) General: For general {<j\{N, T, t)) no simple pre- scription is available for translation into the frequency

domain. For this reason, one might prefer S»(/) as a general measure of frequency stability. This is especially true for theoretical work.

2) Specific Model: Equations (24)-(26) form a realistic model that fits the random nondeterministic noises found on most signal generators. Obviously, if this is a good

model, then the tables in Appendix II may be used

(in reverse) to translate into the frequency domain.

S,(f) =

0,

where a is a constant, then

WliN, T, r)) -

< / < /»

2Tr/» » 1

(28>

(29)

for N and r = T/r held constant. The constant n is- related to a by the mapping shown' in Fig. 1. If (28)

and (29) hold over a reasonable range for a signAl generator, then (28) can be substituted into (23) and evaluated

to determine the constant h^ from measurements of

(ffliN, T, t)). It should be noted that the model of (28)

and (29) may be easily extended to a superposition of

similar noises as in (26). C Translations Among the Time-Domain Measures

1) General: Since {a\{N, T, r)) is a function of N, T,

and r (for some types of noise /» is also important), it is- very desirable to be able to translate among different

sets of A^, T, and t (/» held constant). This is, however,

not possible in general.

2) Specific Model: It is useful to restrict consideration

to a case described by (28) and (29). Superpositions of

independent noises with different power-law types of

spectral densities (i.e., different a) can also be treated by

this technique, e.g., (26). One may define two "bias

functions," Bi and B, by the relations [13]

and

/f.(^.r,M) -(^«(2, r. r))

R(r N - (<^'(2. T, t))

^^('•' "^ = 11(2, r, r))

(30>

(31)

where r s T/t and m is related to a by the mapping of

Fig. 1. In words, Bi is the ratio of the average variancefor N samples to the average variance for two samples

(everything else held constant), while B, is the ratio of

the average variance with dead time between measure- ments (r ?<s 1) to that of no dead time (r = 1 and with ^ = 2 and t held constant). These functions are tabulated

in [13]. Figs. 2 and 3 show a computer plot of

B,{N, r = 1, /i) and 5,(r, n). Suppose one, has an experimental estimate of {<T\{Ni,

Ti, r,)) and its spectral type is known, i.e., (28) and (29)

form a good model and /x is known. Suppose also that one

wishes to know the variance at some other set of measure- ment parameters N,, T,, rj. An unbiased estimate of

{cli^i, Ti, Tj)) may be calculated by

1 It should be noted that in Allan [8], the exponent a cor- respionds to the spectrum of phase fluctuations while variances

are taken over average frequency fluctuations. In the present

paper, a is identical to the exponent a -f- 2 in [81.

TN-152
112 IKKK TKA.S>\CTIO.\S ON INSTKfMtNT VT1(;\ \ Ml \l K V->1 l(K M K N r \1 \ ^ l(l7i

Z-

^—^v 1 1

2 -X I 2 M

-3 ^

Fig. 1. /I — a mapping.

Fig. 2. Function Bi(iV, r = 1, ^).

<«^:(iV., T„ Tr» =

fe)'

.(A^,,r,,M)B2('y,.M)J

(<T:(iV,,r,,r,)), (32)

where ri = Ti/ti and r2 = T^hiS) General: While it is true that the concept of the

Ijias functions Bi and 82 could be extended to other

processes besides those with the power-law types of spectral densities, this generalization has not been done.

Indeed, spectra of the form given in (28) [or superpositions of such spectra as in (26)] seem to be the

most common types of nondeterministic noises encountered in signal generators and associated equipment. For

other types of fluctuations (such as causally generated

perturbations), translations must be handled on an in- -dividual basis.

VI. Applications or Stability Measures

Obviously, if one of the stability measures is exactly

the important parameter in the use of a signal generator,

the stability measure's application is trivial. Some non-

.trivial applications arise when one is interested in a difFig. 3. Bias function Bjfr, /t).

ferent parameter, such as in the use of an oscillator in Doppler radar measurements or in clocks.

A. Doppler Radar

1) General: From its transmitted signal, a Doppler

radar receives from a moving target a frequency-shifted

return signal in the presence of other large signals. These

large signals can include clutter (ground return) and

transmitter leakage into the receiver (spillover). Instabilities of radar signals result in noise energy on the

clutter return, on spillover, and on local oscillators in the equipment.

The limitations of subclutter visibility (SCV) rejec- tions due to the radar signals themselves are related to the RF power spectral density Svif). The quantity typically referred to is the carrier-to-noise ratio and can be

mathematically approximated by the quantity

Svif)

fJo Sy(J')dr

The effects of coherence of target return and other radar parameters are amplv considered in the literature [14]-[17].

2) Special Case: Because FM effects generally pre- dominate over AM effects, this carrier-to-noise ratio is approximately given by [6]

Svif)

fJn

hSMf - "oD. (33) Sy(j') dr

for many signal sources provided |/ — ^o| is sufficiently

greater than zero. (The factor of \ arises from the fact that S^(/) is a one-sided spectrum.) Thus, if / — i-o is

TN-153
/ nl. IUIt\CTLHlZ\TIO.\ OF FRK«UEN"CY STABILITY li:i

;i frecjuency .reparation fiom the carrier, the carrier-to- noise ratio at that point is approximately

^.^,(1/

B. Clock Errors

2 \f - J

SMI - -o\ (34)

1) General: A clock is a device that counts the cycles of a periodic phenomenon. Thus, the reading error x{t) of a clock run from the signal given by (2) is

y{t}. One of the most common techniques is a heterodyne

or beat frequency technique. In this method, the .signal from the oscillator to be tested is mixed with a reference

signal of almost the same frequency as the test oscillator

in order that one is left with a lower average frequency

for analysis without reducing the frequency (or phase)

fluctuations themselves. Following Vessot et al. [18],

consider an ideal reference oscillator whose output signal

is

Vr{t) = Ko, sin 2irvot (40)

(35)

and the dimensions of x{t) are seconds.

If this clock is a secondary standard, then one could

have available some past history of x{t), the time error

relative to the standard clock. It often occurs that one

is interested in predicting the clock error x{t) for some

future date, say U, + t, where ^o is the present date. Obviously, this is a problem in pure prediction and can be handled by conventional methods [3]. 2) Special Case: Although one could handle the prediction of clock errors by the rigorous methods of prediction

theory, it is more common to use simpler prediction

methods [10], [11]. In particular, one often predicts a clock

error for the future by adding to the present error a

correction that is derived from the current rate of gain

(or loss) of time. That is, the predicted error f (<o + r)

is related to the past history of x{t) by

i(/„ + r) = x(/„) + T

j(/o) - j(/n - T)

(36)*

and a second oscillator whose output voltage V{t) is given by (2): V{t) = [Vo + e(0] sin [2TVot + (f>{t)]. Let

these two signals be mixed in a product detector; that is, the output of the product detector v(t) is equal to the

product yV{t) X V,{t), where 7 is a constant (see Fig. 4). Let v{t), in turn, be processed by a sharp low-pass filter with cutoff frequency ji, such that

<j,<n< "n. (41) One may write

yV{l)-\\{l) = 7^o,(Vo + t)[sin 27rvo/][sin (2tvo/ + ^)]

= v{l) = y ^%^ (1 + -^)[cos^ - cos (4tvo/ + v)]. (42) Assume that cos Mt)] has essentially no power in Fourier

frequencies / in the region f > fi The effect of the low-pass

filter then is to remove the second term on the extreme

right of (42); that is

It is typical to let T = t. Thus, the mean-square error of prediction for T = r becomes

<[j:(/o + r) - xito + r)]') = {[xito + r) - 2x(/o) + x{to - r)]\ (37)

which, with the aid of (11), can be written in the form

([x(/o + r) - xito + t)]') = 2T'aliT). (38)

v'it) = y ¥^(^ + t) cos ifiit)

.

(43)

This separation of terms by the filter is correct only if

\[vit)/2TUo]\ « 1 for aU t (4). The following two cases are of interest. Case I: The relative phase of the oscillators is adjusted so that {fit)] ^ 1 (in-phase condition) during

the period of measurement. Under these conditions

One can define a time stability measure (tUt) by

(rlir) = T^fflir). (39)

v'it) ^IVo^'o +

I VoAO, (44)

Clearly, however, the actual errors of prediction of clock

readings are dependent on the prediction algorithm used

and the utility of such a definition as alir) is not great.

Caution should be used in employing this definition.

VII. Mea.surement Techniqxjes for Freqxjency

Stability

.4. Heterodyne Techniques (General)

It is possible for oscillators to be very stable and

values of <TuiT) can be as small as 10"" in some state-of- the-art equipment. Thus, one often needs measuring tech- niques callable of resolving very small fluctuations in « See Appendix Note # 20

since cos fit) =r 1. That is to say one detects the amplitude noise tit) of the signal.

Case II: The relative phase of the oscillators is adjusted to be in approximate quadrature; that is

^'(/) = At) + \

(45)

where 1/(^1 <K 1. Under these conditions,

cos v?(0 = sin v?'(0 ~ ¥''(0 (46)

and

v'it) = ^ VorlWit) +

I

Vor<p'it)^it). (47)

TN-154
114 IEEE TaANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, MAY 1971

Fig. 4. Heterodyne scheme.

If it is true that

(47) becomes

\l*{t)/Vo]\ <J 1 for all t (3), then

v'if) IVorV^'if); (48)

that is, t/(0 is proportional to the phase fluctuations. Thus, in order to observe f{t) by this method, (3) and

(4) must be valid. For different average phase values,

mixtures of amplitude and phase noise are observed. In order to maintain the two signals in quadrature for long observational periods, the reference oscillator can be a voltage-controlled oscillator (VCO) and one may

feed back the phase error voltage as defined in (48) to control the frequency of the VCO [19]. In this condition

of the phase-locked oscillator, the voltage v'it) is the

analog of the phase fluctuations for Fourier frequencies

above the loop cutoff frequency of the locked loop. For

Fourier frequencies below the loop cutoff frequency of ihe loop, i/it) is the analog of jrequency fluctuations. In practice, one should measure the complete servo-loop

response.

B. Period Meagurement

Assume one has an oscillator whose voltage output

may be represented by (2). If |[«(t)/^o]| < 1 for all t and the total phase

(0 « 2rr,i -I- ^(0 (5)

18 a monotonia function of time (that is, |[^(0/2«to]| < 1), then the time t between successive positive going zero

crossingB of V(jt) is related to the average frequency during

the interval r. Specifically

; = >'o(1 + 50. (49)

If one lets r be the time between a positive going zero crossing of V{t) and the 3fth successive positive going zero crossing, then

M = rod + 5.). (50)

If the variations Ar of the period are small compared to the average period ro. Cutler and Searle [7] have shown

that one may make a reasonable approximation to

{<rl{N, T, To)) using period measurements.

C. Period Measurement With Heterodyning

Suppose that <pit) is a monotonic function of time. The output of the filter of Section VII-A (43) becomes

v'it) ' 7 —7i COS <p{l) (51)

if |[«(<)/Vo]| <K 1. Then one may measure the period

r of two successive positive zero crossings of v'{t). Thus

; = "0 |y,|

and for the Jlfth positive crossover — = "o |y.|.

(52)

(53)

The magnitude bars appear because cos <p{t) is an even

function of fit). It is impossible to determine by this method alone whether v> is increasing with time or de- creasing with time. Since y» may be very small (~10-"

or 10"^' for very good oscillators), t may be quite long

and thus measurable with a good relative precision.

If the phase fit) \s not monotonic, the true y, may be

near zero but one could still have many zeros of cos fit)

and thus (52) and (53) would not be valid.

D. Frequency Counters

Assume the phase (either « or f) is a montonic function of time. If one counts the number M of positive going

zero crossings in a period of time t, then the average fre- quency of the signal is M/r. If we assume that the signal

is 7(0 as defined in (2), then

M = Vo(l + 5.). (54)

If we assume that the signal is v'it) as defined in (48),

then —

M ... = "0 ly.lAgain, one measures only positive frequencies.

(55)

E. Frequency Discriminators

A frequency discriminator is a device that converts frequency fluctuations into an analog voltage by means

of a dispersive element. For example, by slightly detuning a resonant circuit from the signal 7(0 the frequency

fluctuations (l/2r)]^(0 are converted to amplitude fluc- tuations of the output signal. Provided the input amplitude

fluctuations (e(0]/7o are insignificant, the output amplitude fluctuations can be a good measxire of the frequency

fluctuations. Obviously, more sophisticated frequency

discriminators exist (e.g., the cesium beam).

TN-155
RABNES et al.: ch.\ractkhiz.\tion or raEquENCT stabilitt From the analog voltage one may use analog spectrum

analyzers to determine Sy[j) , the frequency stability. By

converting to digital data, other analyses are possible on a computer.

F. Common Hazards

1) Errors Caused by Signal-Processing Equipment: The

intent of most frequency stability measurements is to evaluate the source and not the measuring equipment.

Thus, one must know the performance of the measuring

system. Of obvious importance are such aspects of the

measuring equipment as noise level, dynamic range, resolution (dead time), and frequency range.

It has been pointed out that the noise bandwidth /» is very essential for the mathematical convergence of certain expressions. Insofar as one wants to measure the signal

source, one must know that the measuring system is not

limiting the frequency response. At the very least, one must recognize that the frequency limit of the measuring

system may be a very important, implicit parameter for either (t*(0 or S,(j). Indeed, one must account for any

deviations of the meas\iring system form ideality such as a "nonflat" frequency response of the spectrum analyzer

itself. Almost any electronic circuit that processes a signal

will, to some extent, convert amplitude fluctuations at the input terminals into phase fluctuations at the output.

Thus, AM noise at the input will cause a time-varying

phase (or FM noise) at the output. This can impose important constraints on limiters and automatic gain control (AGO circuits when good frequency stabihty is needed.

Similarly, this imposes constraints on equipment used for frequency stability measurements.

S) Analog Spectrum Analyzers {Frequency Domain)

Typical analog spectrum analyzers are very similar in design to radio receivers of the superheterodyne type, and

thus certain design features are quite similar. For example, image rejection (related to predetection bandwidth)

is very important. Similarly, the actual shape of the analyzer's frequency window is important since this af- fects spectral resolution. As with receivers, dynamic

range can be critical for the analysis of weak signals in the presence of substantial power in relatively narrow bandwidths (e.g., 60 Hz).

The slewing rate of the analyzer must be consistent

with the analyzer's frequency window and the post-detection bandwidth. If one has a frequency window of 1 Hz,

one cannot reliably estimate the intensity of a bright

line unless the slewing rate is much slower than 1 Hz/s.

Additional post-detection filtering will further reduce the maximum usable slewing rate. 5) Spectral Density Estimation from Time Domain

Data: It is beyond the scope of this paper to present a comprehensive list of hazards for spectral density estimation; one should consult the literature [2]-[5]. There

115

are a few points, however, which are worthy of special

notice: a) data aliasing (similar to predetection bandwidth problems); b) spectral resolution; and c) confidence of the estimate.

4) Variances of Frequency Fluctuations <t\{t): It is not uncommon to have discrete frequency modulation of a

source such as that associated with the ix)wer supply

frequencies. The existence of discrete frequencies in 5,(/) can cause (i\{t) to be a very rapidly changing function

of T. An interesting situation results when r is an exact

multiple of the period of the modulation frequency (e.g., one makes t = 1 s and there exists 60-Hz frequency

modulation on the signal). In this situation, (r\{T = 1 s) can be very optimistic relative to values with slightly

different values of r. One also must be concerned with the convergence

prof)erties of a\{r) since not all noise processes will have

finite limits to the estimates of <i\{t) (see Appendix I). One must be as critically aware of any "dead time" in the

measurement process as of the system bandwidth.

5) Signed Source and Loading: In measuring frequency

stability one should specify the exact location in the

circuit from which the signal is obtained and the nature

of the load used. It is obvious that the transfer characteristics of the device being specified will depend on the load

and that the measured frequency stability might be

affected. If the load itself is not constant during the

measurements, one expects large effects on frequency

stability.

6) Confidence of the Estimate: As with any measurement

in science, one wants to know the confidence to assign to numerical results. Thus, when one measures S,(j) or <rj(r),

it is important to know the accuracies of these estimates.

a) The Allan Variance: It is apparent that a single

sample variance <r'(4, t, t) does not have good confidence,

but, by averaging many independent samples, one can improve the accuracy of the estimate greatly. There is a

key point in this statement, "independent samples." For

this argument to be true, it is important that one sample

variance be independent of the next. Since o-J(2, r, r) is related to the first difference of the frequency (11),

it is sufficient that the noise perturbing y{t) have "independent increments," i.e., that yit) be a random walk. In other words, it is sufficient that S,(j) -^ /"' for low

frequencies. One can show that for noise processes that

are more divergent at low frequencies than /"*, it is difficult (or impossible) to gain good confidence on

estimates of alir). For noise processes that are less divergent than /"', no problem exists. It is worth noting that if we were interested in

ff'^(M = °>, T, t), then the limit noise would become

S,(f) ~ f instead of /"' as it is for <rj(2, r, t). Since most

real signal generators possess low-frequency divergent

noises, ((^'(2, t, t)) is more useful than al{N = », r, r). Although the sample variances <rj(2, t, t) will not be

normally distributed, the variance of the average of m

TN-156
116 IEEE TH.^NSACTIO.NS ON INSTRUMENTATION \ND MEASUREMKM, M\V 1971 independent (nonoverlapping) samples of <7^(2, r, r)

(i.e., the variance of the Allan variance) will decrease as 1/m provided the conditions on low-frequency divergence

are met. For sufficiently large m, the distribution of the

m sample averages of (7^(2, t, t) will tend toward normal

(central limit theorem). It is thus p)ossible to estimate

confidence intervals based on the normal distribution. As always, one may be interested in t values approaching the limits of available data. Clearly, when one is interested in r values of the order of a year, one is severely

limited in the size of m, the number of samples of <r'(2, t, t). Unfortunately, there seems to be no substitute for many

samples and one extends r at the expense of confidence in the results. "Truth in packaging" dictates that the sample

size m be stated with the results.

b) Spectral Density: As before, one is referred to the

literature for discussions of spectrum estimation [2]-[5]. It is worth pointing out, however, that for S,(f) there are

basically two different types of averaging that can be

employed: sample averaging of independent estimates

of S,(f), and frequency averaging where the resolution

bandwidth is made much greater than the reciprocal data

length.

VIII. Conclusions

A good measure of frequency stability is the spectral

density St(f) of fractional frequency fluctuations y(jt). An alternative is the expected variance of N sample

sveragee of y{t) taken over a duration r. With the be^-

ning of successive sample periods spaced every T units

of time, the variance is denoted by al(N, T, r). The

stability measure, then, is the expected value of many

measurements of al{N, T, r) with ^ = 2 and T = r; that is, ff*(r). For all real experiments one has a finite bandwidth. In general, the time domain measure of frequency stability oKt) is dependent on the noise bandwidth of the system. Thus, there are four important

parameters to the time domain measure of frequency

stability.

N Number of sample averages (N = 2 for preferred

measure).

T Repetition time for successive sample averages (r = T for preferred measure),

r Duration of each sample average.

/» System noise bandwidth.

Translations among the various stability measures for common noise types are possible, but there are significant reasons for choosing N = 2 and T = r for the preferred measure of frequency stability in the time domain. This

measure, the Allan variance, (N = 2) has been referenced

by [12], [20]-[22] and more. Although St(j) appears to be a fxmction of the single

variable /, actual experimental estimation procedures

for the spectral density involve a great many parameters.

Indeed, its experimental estimation can be at least as involved as the estimation of ^'(t).

Appendix I We want to derive (23) in the text. Starting from (10

1

we have

{<rl(N, T, r))

if" , N N ^

-hZZ f' dt" f

' dt' {y{t')y{t")yi (56)

where (9) has been used. Now

m')yit")) = R.{i' - n (57)

where Ryir) is the autocorrelation function of y{t) and

is the Fourier transform of Syif), the power spectral

density of y{t). Equation (57) is true provided that

y{t) is stationary (at least in the wide or covariance

sense), and that the average exists. If we assume the power spectral density of y{t), Sy{f) has low and high

frequency cutoffs /j and /» (if necessary) so that

•'0 S,{f)df

exists, then if y is a random variable, the average does

exist and we may safely assume stationarity. In practice, the high-frequency cutoff /* is always

present either in the device being measured or in the measuring equipment itself. When the high-frequency

cutoff is necessary for convergence of integrals of Sy(f}

(or is too low in frequency), the stability measure will depend on /». The latter case can occur when the measuring equipment is too narrow-band. In fact, a useful type of spectral analysis may be done by varying /» purposefully [18].

The low-frequency cutoff /i may be taken to be much

smaller than the reciprocal of the longest time of interest. The results of calculations as well as measurements

will be meaningful if they are independent of /i as ft approaches zero. The range of exponents in power law

spectral densities for which this is true will be discussed

and are given in Fig. 1. To continue, the derivation requires the Fourier trans- form relationships between the autocorrelation function

and the power spectral density

S,(f) = 4 f

/e,(r) cos 2T/r di

Jo

R.ir) = f S,(f) COS 2T/r df.

•>o (58) Using (58) and (57) in (56) gives

TN-157
BvRNKS et al.: cr\racterization of frequency stability

!" ' df COS 2^1(1' - t") - T? Z L f dl S.(J)

J I, iy i-\ i-i Jo

f'

' dl" J'

df cos 2t/(/' - t")\

iir

{N ^AtK df S,(f) sin' rj

(irf) ^]

4((

31 >

I

-0 l O Qn ^i 2 -httlrdj ^, (2 COS 2T/T(i - i) N fz'x tn Un (2t/) - cos2t/[7'0' - i) + r] - cos2irnT(i - i) - r])J|.

(59) (The interchanges in order of integration are permissible

here since the integrals are uniformly convergent with

the given restrictions on Sy{f).) The first summation in the curly brackets is independent of the summation in- dex n and thus gives just

o o o o o o o\ o o o o o

o o\ o o o o \

s o o o^> o o o

3 Of—oO vp o o

o o ^\o o

o o o ^^o

5 6 o o

o o

o o

Fig. 5. Region of summation for i and k iot N = 4.

This may be written as

A^ [d,s,n^^. (60) S = N^2R.[n-\I-]% (66)

The kernel in the second term in the curly brackets

may be further simplified

2 cos 2ir/7'(i - - cos 2T/(r(; - i) + r) - cos 2TJ{T{i - r) - r) = 4 sin' r/r cos 2irjT{i - t). (61) The second term is then

- ^ [[ dl 1^ sin' x/r E Z COS 2rmi - {)) (62)

(The interchange of summation and integration is justified.) We must now do the double sum. Let

j — i = k

2tjT = X. (63) Changing summation indices from i and ; to t and k

gives for the sum

N N S N-i S s E Z cos x{i — t) = 53 2Z cos kx. (64)

i-i 1-1 1-1 t-l-i , The region of summation over the discrete variables i and k is shown in Fig. 5 for iV = 4. The summand is independent of t so that one may inter- change the order of summation and sum over i first. The summand is even in k and the contributions for

fc < are equal to those for fc > 0, and so we may pull out the term for fc = separately and write

Ecosfcx i; i) 4- El

= 2( E (A^ - fc) cos fcx) + A^. (65)

where Re[r/] means the real part of U and d/dx

is the differential operator. The series is a simple geo- metric series and may be summed easily, giving

4 sin"" i/2

sinVYi/2_

sin' x/2 (67)

Combining everything we get, after some rearrangement,

<«r^(Ar. T, r)) -JT^Jo '^^'^'^IW^ L^ ~ AT' sin' rr/rJ

^^^

where r = T/t. This is the result given in (23). We can determine a number of things very easily from

this equation. First let us change variables. Let rfr => u, then

{<rliN, T. r))

(AT - l)rT Jo \ttI u \ N sm ru)

(69)

The kernel behaves like u* as u — and like u"* as u —» »

Therefore {<r\{N, T, r)) is convergent for power law

spectral densities, S,(J) = hj', without any low- or highfrequency cutoffs for —3 < a ,< 1. Using (69) for power

law spectral densities we find

{<tI(N, T, r)) = T-'-'haCa, -3 < a < 1 = r'KCa, M = -a - 1

TN-158
118

and

.V

(-V - 1)7

r , . sin w /, sin ^ru\ —'*' / au u —2— SI — x'S 2—f .'o u [ A sm ru)

This is the basis for the plot in Fig. 1 in the text of ft versus a. For a > 1 we must include the high-frequency

cuto£F /». For .V = 2 and r = 1 the results are particularly simple. We have

<(r*(2, r. t)) = r-'-'A. -^

f

du u"' sin* u (71)

for power law sf)ectral densities. For N = 2 and general r we get

<<r*(2, T. r)>

IEEE TR.\NSACTIONS ON INSTKr.MEXTATION AND .MEASLTJlilENT, M\Y 1J7'. Appendix ii Let yU) be a sample function of a random noise process with a spectral density Sy{f). The function yU) is assumed to be pure real and Sy{f) is a one-sided spectral

('0) density relative to a cycle frequency (i.e., the dimensions

of 5„(/) are that of y^ per hertz). (For additional infor- mation see Appendix I, [7], [8], [18].) Let x(t) be defined by the equation

".-^/."""^.fe)

1 — cos 2u — cos 2ru+

cos2u(r-H) cos2ti(r— 1) 2

XT Jn XXT/ sm u sm ru

u'

(72)

The first form in (72) is particularly simple and is also useful for r = 1 in place of (71).

Let us discuss the case for « > 1 in a little more detail. As mentioned above we must include the high-frequency cutoflf /fc for convergence. The general behavior can be seen most easily from (68). After placing the

factor T-2 outside the integral and combining the factor

/"* with S„(/) we find that the remaining part of the

kernel consists of some constants and some oscillatory terms. If Tntjxr » 1 it is apparent that the rapidly oscil- lating terms contribute very little to the integral. Most

of the contribution comes from the integral over the constant term causing the major portion of the r dependence to be the t"* factor outside the integral. This is the reason for the vertical slope at /t = —2 in the

(k versus a plot in Fig. 1 in the text. One other point deserves some mention. The constant term of the kernel discussed in the preceding paragraph

is different for r = 1 from the value ior r ¥= 1. This is readily seen from (72) for N' = 2; for r = 1 the constant term is 3/2 while for r ^^ 1 it is 1. This is the reason

for 8k (r — 1), which appears in some of the results of Appendix II. In practice, 8i(r — 1) does not have zero width but is smeared out over a width of approximately

(2x/»t)-^ If there must be dead time r ^^ 1, it is wise to choose (r - 1) » (2^/»t)-» or (r - 1) « (2^/»r)-i but

with 2w/»T » 1. In the latter case, one may assume

r s: 1.

i(0 - ^ - 2/(0. (73)

Define the following, io is arbitrary instant of time and

<•*. 3 <. + r, n = 0, 1, 2, . • • , (74)

y. = - y(0 dt = ^^ (7o)

T J,, T

1 "

(y)v = ^ ]C y.. (76)

and let /» be a high-frequency cutoflf (infinitely sharp)

with 2ir/*r » 1.

Definition:

idiN, T, r)} ^ {-j^i S

^^" " ^^^-^r

(^"^

Special Case:

(.1(2, T, r)) = (^^^^^)- (78)

Special Case:

cl{r) ^ {c\{2, r, r)) ^ Axjt, + 2r) - 2x(/o + r) + X(QY\

^^^^

Definition:

DI(t) a {[xito + 2r) - 2xito + r) + l(<o))*). (80)

Consequence of Definitions:

DI(t) = 2r'aUr) ^ 2al(r). (81)

Definition:

4'\{T, r) s {[x{l, -t- r + r) - J«o + T) - x(to + r) + 7(0)'). (82)

Consequence of Definitions: ^KT, r) = 2r\ali2, T, r)). (83)

Special Case:

il(r, r) = D](r). (84)

Random Walk y

r = - . < / < /» r TN-159
BvRNES et al.: ch.\r\cteriz.\tion of frequency stability

Quantity Relation

(2t)

= {cr]{N,T,r)} h.y

{<tI(N, r, r)) h.,-

12 (ML

12 (Ml

6

2(2^)'

i^KT, h-2'

/l-2 •

(2x)=

(2x)

[r(A- + 1) - 1], r > 1

^'(T. r) Ao-|r|, r > 1

/lo-T. r < 1.

119

(100)

•iV, r = 1

(85) Flicker x

(86)

S«(/) = ^> i/l(s«(/) = (^)

AT = 2. r = 1 (87)

r = r/r, 27r/,r » 1, 2irfJ » 1, < / < /,.

Quantity Relation

^^^^ {,rl(N, T, r)) hr-^, ^2 + In (2T/.r)

(3r - 1) , r > 1

6 ^

(^ - 1 ) . ' < 1 • (89)

Flicker y

SM) = '-^ (s.(/) = ^)

r = r/r, < / < /» .

Quantity Relation

wiiN, i\ r)> ^-' •:v(A^^=^ S

^-^^ " ""^

+ Ar(Ar3-[) S

(-V - n)

•^"[nv'- i Jl' '^^^ (101)**

{^\{N, r, r)^ ^'• "yy»(t)'^ [2 + 1" (2W.r) - ~^^ ,

r = 1 (102)* *

<rlir) K :^2^2 13[2 + In (2TAr)] - In 2j.

y = 2,r = I (103)* *

• [-2(7ir)- In (nr) + {nr + if In (nr + 1) + {nr - If In \nr - 1|] (90) 'A'(^'

''

D;(r) = 2^]{r) (^-2^3(2 + In C-V/.r)] - In 2} (104) * *

<cr:(A', r,T)) h.,

N In A' (r= 1) (91)

iV - 1

'

<T'(r) ;i.r2 In 2, (.V = 2, r = 1) (92)

Dlir) = 2<rliT) h.rir' In 2 (93)

^;(7', r) h.rr'[-2r In r + (r + 1)= In (r + 1) + (r - l)Mn |r - 1|] (94)

~;i-,-2t'(2 + Inr), r » 1

~;i_,-2T'(2 - In r), r « 1 . (95) *

^.(^[2+ ln(2W^r)]. r»l

^.•(-^{3[2 + ln(2T/.r)] - In 2},

r = 1 (105)**

/i.-^[2 + ln(2W.r)]. r«l.

PF/iite X

H'^Atte y {Random Walk x)

S,{f) = h,f [s,{f) = ^)

r = r/r; 5»(r - 1) =

S.{f) = /lo 5.(/) =

Ao

(2t)^

1, if r = 1

0, otherwise

r = T/r, 0<f<U.

Quantity Relation

{a\{N,T,r)) ^-Irl-, r>l

2,r/,r»l, < / < /»

Qiuintity Relation

(<r,(A^, T, r)) h, ^;^^, -^

{al{N,r,r)) h,-^^^,-'^ , r=l

;io-ir(A'+ 1) \A~\ Nr < 1 (96)

{'r\{N,r,r)) ^j-\r\-\ r=\

^n r 1-1

D\{r) = 2<r;(r) h,-

.V = 2. r = 1

(97)

(98)

(99)

<rl{r)

X{2t)' f-

(106)

(107)

2 /o \2 3 (2t)

D;(r) = 2<7-(r) A,-|^

(2ir)

.V = 2, r = 1 (108)

^'(T. •2U h,-[2+ S,{r- l)](f^

(109)

(110)

« See Appendix Note # 21 •See Appendix Note # 22 TN-160
120 Acknowledgment

The authore are particularly indebted to D. W. Allan,

Dr. D. Halford, Dr. S. Janis, and Dr. J. J. Filliben of the National Bureau of Standards. The authors are also indebted to Mrs. Carol Wright for preparing the many

revised copies of this paper. Refebences

[1] E. T. Jayues, "Information theory and atatistical mechanics," Pkya. Rev., vol. 108, aec. 15, Oct. 1957, pp. 171-190.

(2] C. Bingham, M. D. Godfrey, and J. W. Tukey, "Modem

techniques of power spectrum estimation," lEJSE Trans. Audio Blectroeoust., vol. AU-15, June 1967, pp. 56-66.

[3] R. B. Blackman, Ldnear Data SmooihinQ and Prediclion in Theory and Practice. Reading, Mass.: Addison-Wesley,

1965.

[4] R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra. New York: Dover, 1958.

[5] E. 0. Brigham and R. E. Morrow, "The fast Fourier

transform," IEEE Spectrum, vol. 4, Dec. 1967, pp. 63-70.

[6] E. J. Baghdady, R. D. Lincob, and B. D. Nelin. "Short- term frequency stability: theory.-, measurement, and status," Proc. IEEE-NASA Symp. on Short-Term Frequency Slability (NASA SP-80), Nov. 1964, pp. 65-87; also in Proc. IEEE, vol. 53, JuJv 1965, pp. 704-722 and Proc. IEEE (Corresp.), vol 53, Dec. 1965. pp. 2110-2111.

17] L. Cutler and C. Searle, "Some aspects of the theorj- and measurement of frequencv fluctuations in frequency standards." Proc. IEEE, vol. 54, Feb. 1966, pp. 136-154.

IS] D. W. Allan. "Statistics of atomic frequency standards,"

Proc. IEEE, vol. 54, Feb. 1966, pp. 221-230.

19] N. A. Shewart, Economic Control of Quality of Manufactured Product. Princeton, N. J.: Van Nostrand, 1931,

p. 146.

TTCT! TR^NS.^CTIONS ON I^•STRt'ME^^AT10N AND MEASCKEMEKT, MAY 1971

[10] J. A. Barnes, "Atomic timekeeping and the statistics of

precision signal generator*," Proc. IEEE, vol. 54, Feb. 1966, pp. 207-220.

[11] J. A. Barnes and D. W. Allan, "An approach to the prediction of coordinated univenial time," Frequency, vol.

5, Nov.-Dec. 1967, pp. 15-20. [12] R. F. C. Veasot et al., "An intercomparison of hydrogen

and cesium frequency standards," IEEE Tram. Instrum.

Meas., vol. IM-15, Dec. 1966, pp. 165-176.

[13] J. A. Barnes, "Tables of bias functions, Bs and Bi, for variances based on finite sainples of processes with power

law spectra] densities," NBS, Washington, D. C, Tech. Note

375, Jan. 1969.

[14] D. B. Leeson and G. F. Johnson, "Short-term stability for a Doppler radar: requirements, measurements, and techniques,'' Proc. IEEE, vol. 64, Feb. 1966, pp. 244-248

[15] W. K. Saunders in Radar Handbook, M. I. Skolnik, Ed. New York: McGraw Hill, 1970, ch. 16. [16] R. S. Raven, "Requirements on master oscillators for co- herent radar," Proc. IEEE, vol. 54, Feb. 1966, pp. 237-243

[17] D. B. Leeson, "A simple model of feedback oscillator noise spectrum," Proc. IEEE (Lett.), vol. 54, Feb. 1966, pp. 329-

330. [18] R. F. C. Vessot, L. Mueller, and J. Vanier, "The specifica- tion of oscillator characteristics from measurements made

in the frequenrv domain," Proc. IEEE, vol. 54, Feb. 1966. pp. 199-207.

[19] Floyd M. Gardner, Phasclock Techniquei. New York:

Wiley. 1966. [20] C. Menoud. J. Racine, and P. Kartaschoff, ".Atomic hydrogen maser work at L5.R.H.. Neuchatel. Switzerland " Proc.

SJst Ann. Symp. Frequency Control, .^pr. 1967. pp. 543-567.

[21] A. G. Mungall, D. Morris. H. Daams, and R. Bailey. "Atomic

hydrogen maser development at the National Research

Council of Canada," Metrologia. vol. 4. July 196S, pp 87-94

[22] R. F. C. Vessot, "Atomic hydrogen masers. an introduction and progress report," Hewlett-Packard J., vol. 20, Oct.

1968, pp. 15-20.

Reprinted bv permission from IEEE TRIlNS ACTIONS ON

INSTRLMENT.^TION .\ND MEASUREMENT

Vol. lM-20, No. 2, Mav 1971 Conyriebt © 1971, by thf Institute of Electrical and Electronics Engineers, Inc PRINTED IN THE U.S.

A

TN-161
142 Rep. 580-2

Reprinted, with permission, from Report 580 of the International Radio Consultative

Committee (C.C.I.R.), pp. 142-150, 1986.

CHARACTERIZATION OF FREQUENCY AND PHASE NOISE *

(Study Programme 3B/7)

(1974-1978-1986)

1. Introductioo

Techniques to characterize and to measure the frequency and phase instabilities in frequency generators

and received radio signals are of fundamental importance to users of frequency and time standards.

In 1964 a subcommittee on frequency stability was formed, within the Institute of Electrical and Electronic Engineers (IEEE) Standards Committee 14 and later (in 1966) in the Technical Committee on Frequency and

Time within the Society of Instrumentation and Measurement (SIM), to prepare an IEEE standard on frequency

stability. In 1969, this subcommittee completed a document proposing definitions for measures on frequency and

phase stabilities. These recommended measures of stabilities in frequency generators have gained general

acceptance among frequency and time users throughout the world. Some of the major manufacturers now specify

stability characteristics of their standards in terms of these recommended measures.

Models of the instabilities may include both stationary and non-stationary random processes as well as systematic processes. Concerning the apparently random processes, considerable progress has been made

[IEEE-NASA, 1964; IEEE, 1972] in characterizing these processes with reasonable sutistical models. In contrast, the presence of systematic changes of frequencies such as drifts should not b^ modelled statistically, but should be

described in some reasonable analytic way as measured with respect to an adequate reference standard, e.g., linear regression to determine a model for linear frequency drift. The separation between systematic and random parts however is not always easy or obvious. The systematic effects generally become predominant in the long term, and

thus it is extremely important to specify them in order to give a full characterization of a signal's stability. This

Report presents some methods of characterizing the random processes and some important types of systematic

processes.

Since then, additional significant work has been accomplished. For example, Baugh [1971] illustrated the properties of the Hadamard variance - a time-domain method of estimating discrete frequency modulation

sidebands - particularly appropriate for Fourier frequencies less than about 10 Hz; a mathematical analysis of

this technique has been made by Sauvage and Rutman [1973]; Rutman [1972] has suggested some alternative time-domain measures while still giving general support to the subcommittee's recommendations; De Prins et al. [1969] and De Prins and Comelissen, [1971] have proposed alternatives for the measure of frequency stability in the frequency domain with specific emphasis on sample averages of discrete spectra. A National Bureau of

Standards Monograph ^devotes Chapter 8 to the "Statistics of time and frequency data analysis" [Blair, 1974]. This

chapter contains some measurement methods, and applications of both frequency-domain and time-domain measures of frequency/phase instabilities. It also describes methods of conversion among various time-domain

measures of frequency stability, as well as conversion relationships from frequency-domain measures to time- domain measures and vice versa. The effect of a finite number of measurements on the accuracy with which the two-sample variance is determined has been specified [Lesage and Audoin, 1973, 1974 and 1976; Yoshimura, 1978]. Box-Jenkins-type models have been applied for the interpretation of frequency subility measurements [Barnes, 1976; Percival, 1976] and reviewed by Winkler [1976).

Lindsey and Chie [1976] have generalized the r.m.s. fractional frequency deviation and the two-sample

variance in the sense of providing a larger class of time-domain oscillator subility measures. They have developed measures which characterize the random time-domain phase stability and the frequency stability of an oscillator's signal by the use of Kolmogorov structure funaions. These measures are connected to the frequency-domain

stability measure Sy{f) via the Meilin transform. In this theory, polynominal type drifts are included and some

theoretical convergence problems due to power-law type spectra are alleviated. They also show the close relationship of these measures to the r.m.s. fractional deviation [Cutler and Searle, 1966] and to the two-sample

variance [Allan, 1966]. And finally, they show that other members from the set of stability measures developed are important in specifying performance and writing system specifications for applications such as radar, communications, and tracking system engineering work.

Other forms of limited sample variances have been discussed [Baugh, 1971; Lesage and Audoin, 1975; Boileau and Picinbono, 1976] and a review of the classical and new approaches has been published

(Rutman, 1978J.

* See Appendix Note # 23 TN-162
Rep. 580-2 I43

Frequency and phase instabilities may be characterized by random processes that can be represented

statistically in either the Fourier frequency domain or in the time domain [Blackman and Tukey, 1959J. The

instantaneous, normalized frequency departure y(t) from the nominal frequency vq is related to the instantaneous- phase fluctuation <p(r) about the nominal phase 2nvorby:

1 d<p(0 «>(/) y(0 " ^ - ^ (1) 2nvo dr 2jrvo m - f^2nvo

where x(t) is the phase variation expressed in units of time.

2. Fourier frequency domain

In the Fourier frequency domain, frequency stability may be defmed by several one-sided (the Fourier

frequency ranges from to 00) spectral densities such as:

Sy(f) of y{t), S^if) of <p(0. S^if) of (iKr), 5,(/) of jc(r), etc.

These spectral densities are related by the equations:

Syif) - 4 ^*(/) (2)

5,(/) - 4n2/2 s^(J) (3)

(2^0 r

Power-law spectral densities are often employed as reasonable models of the random fluctuations in precision oscillators. In practice, it has been recognized that these random fluctuations are the sum of five independent noise processes and hence: y *„/» for 0</<^

Syif)=r =

~'

(5) {•

for />A

where A„'s are constants, a's are integers, and /* is the high frequency cut-off of a low pass filter. Equations (2), (3) and (4) are correct and consistent for stationary noises including phase noise. High frequency divergence is eliminated by the restriaions on / in equation (5). The identification and characterization of the five noise processes are given in Table I, and shown in Fig. 1. In practice, only two or three noise processes are sufficient to describe the random frequency fluctuations in a specific oscillator; the others may be neglected.

3. Time-domain

Random frequency instability in the time-domain may be defined by several sample variances. The

recommended measure is the two-sample standard deviation which is the square root of the two-sample zero dead-time variance Oy^ix) [von Neumann et al., 1941; Allan, 1966; Barnes et al., 1971] defined as:

OyHx) = <^^^^tJ_I^^ (6)

where

Xl o. I ~Xi y{t)At = -^^^t-J '^ and /^ ^ , = r^ -I- t (adjacent samples)

< > denotes an infinite time average. The Xk and Xk^.\ are time residual measurements made at /^ and

tk-¥\ " h + Xy k = Q, \y2, .. .^ and 1/t is the fixed sampling rate which gives zero dead time between frequency measurements. By "residual" it is understood that the known systematic effects have been removed.

TN-163
144 Rep. 580-2

If the initial sampling rate is specified as 1/to, then it has been shown [Howe el al, 1981] that in general one may obtain a more efficient estimate of o^(t) using what is called "overlapping estimates". This estimate is obtained by computing equation (7).

. N-lm

2(N - 2/n) T^ ,.i

where N is the number of original time departure measurements spaced by to, (N " A/ + 1, where M is the number of original frequency measurements of sample time, Tq) and t -= mto. The corresponding confidence

intervals [Howe et al., 1981], discussed in § 6, are smaller than those obtained by using equation (12), and the estimate is still unbiased.

If dead time exists between the frequency departure measurements and this is ignored in computing

equation (6), it has been shown that the resulting stability values (which are no longer the Allan variances), will be

biased (except for the white frequency noise) as the frequency measurements are regrouped to estimate the stability for mto (m > 1). This bias has been studied and some tables for its correction published [Barnes, 1969; Usage, 1983]. A plot of Oyix) versus t for a frequency standard typically shows a behaviour consisting of elements as shown in Fig. 1. The first part, with cfy(x) ~ t"'^^ (white frequency noise) and/or Oyit) ~ t~' (white or flicker phase noise) refiects the fundamental noise properties of the standard. In the case where o^(t) ~ t~', it is not

practical to decide whether the oscillator is perturbed by white phase noise or by flicker phase noise. Alternative techniques are suggested below. This is a limitation of the usefulness of 0^(1) when one wishes to study the nature of the existing noise sources in the oscillator. A frequency-domain analysis is typically more adequate for Fourier

frequencies greater than about 1 Hz. This t"' and/or t"''^ law continues with increasing averaging time until the

so-called flicker "floor" is reached, where Oy(T) is independent of the averaging time t. This behaviour is found in almost all frequency standards; it depends on the particular frequency standard and is not fully understood in its physical basis. Examples of probable causes for the flicker "floor" are power supply voltage fluctuations, magnetic

field fluctuations, changes in components of the standard, and microwave power changes. Finally the curve shows

a deterioration of the stability with increasing averaging time. This occurs typically at times ranging from hours to days, depending on the particular kind of standard. A "modified Allan variance", MOD a^Jt), has been developed [Allan and Barnes, 1981] which has the property of yielding different dependences on t for white phase noise and flicker phase noise. The dependences

for MOD <3y{x) are t"^^^ and t"' resf)ectively. The relationships between a^(T) and MODOyii) are also explained

in [Allan and Barnes, 1981; IEEE 198S, Lesage and Ayi, 1984]. MODay{x) is estimated using the following

equation:

MQDa^(T)- J £ [ I (a:.>2. - 2x„„ -h x.)f (8)

where A^ is the original number of time measurements spaced by Tq, and t = mto the sample time of choice. Properties and confidence of the estimate are discussed in Lesage and Ayi [1984]. Jones and Tryon [1983] and

Barnes et al. [1982] have developed maximum likelihood methods of estimating Oy{i) for the specific models of

white frequency noise and random walk frequency noise, which has been shown to be a good model for observation times longer than a few seconds for caesium beam standards.

4. CoDversion between frequency and time domains

In general, if the spectral density of the normalized frequency fluctuations Sy{f) is known, the two-sample

variance can be computed [Barnes et al^ 1971; Rutman, 1972]:

o,2(T) = 2/*^(/)^^d/ (9)

TN-164
Rep. 580-2 145

S^if)

\\

\\v

\\

\A

V, a

1 \N

s.

V

v;

1 s \\^„# 1 V. ^ f'

(T)

Fourier frequency

1 s

1 ^

• \ 1 ^\\ • S /I ^l'> y

V N r» ^

^

Sampling time

FIGURE 1 - Slopecharacteristicsof the five independent noise processes

(log scale)

Specifically, for the power law model given by equation (5), the time-domain measure also follows the power law as derived by Cutler from equations (5) and (9).

2/ ^ I. (2«)^ I -, , -, I. 1 ^ A 1038 + 3log«(2n/AT) 3/^ oAx) - A.. — T + /._, 2log.2 + /, - + A, ^^-^, + h ^^-^ (10)

Note. - The factor 1.038 in the fourth term of equation (10) is different from the value given in most previous

publications.

TN-165
146 Rep. S80-2

The values of h^ are characteristics of oscillator frequency noise. One may note for integer values (as often seems to be the case) that n - -a - 1, for — 3 :S a :S 1, and p,a -2 for a ^ 1 where aJ(T) ~ i''.

These conversions have been verified experimentally [Brandenberger ei al., 1971] and by computation

[Chi, 1977]. Table II gives the coefficients of the translation among the frequency stability measures from time

domain to frequency domain and from frequency domain to time domain.

The slope characteristics of the five independent noise processes are plotted in the frequency and time

domains in Fig. 1 (log log scale).

5. Measurement techniques

The spearal density of phase fiuctuations S^{f) may be approximately measured using a phase-locked

loop and a low frequency wave analyzer [Meyer, 1970; Walls et al., 1976]. A double-balanced mixer is used as the phase detector in a lightly coupled phase lock loop. The measuring system uses available state-of-the-art electronic components; also a very high quality oscillator is used as the reference. For very low Fourier frequencies (well below 1 Hz), digital techniques have been used [Atkinson et al., 1963; E>e Prins ei al., 1969; Babitch and

Oliverio, 1974]. New methods of measuring time (phase) and frequency stabilities have been introduced with

picosecond time precision [Allan and Daams, 1975], and of measuring the Fourier frequencies of phase noise with 30 dB more sensitivity than the previous state of the art [Walls et al., 1976].

Several measurement systems using frequency counters have been used to determine time-domain stability with or without measurement dead time [Allan, 1974; Allan and Daams, 1975). A system without any counter has

also been developed [Rutman, 1974; Rutman and Sauvage, 1974]. Frequency measurements without dead time can

be made by sampling time intervals instead of measuring frequency directly. Problems encountered when dead

time exists between adjacent frequency measurements have also been discussed and solutions recommended

[Blair, 1974; Allan and Daams, 1975; Ricci and Peregrino, 1976]. Discrete spectra have been measured by

Groslambert et al. [1974].

6. Confidence limits of time domain measurements A method of data acquisition is to measure time variations x, at intervals Tq. Then a^(T) can be estimated

for any t » nto (n is any positive integer) since one may use those Xj values for which j is equal to nk. An

estimate for o^(t) can be made from a data set with A/ measurements of Jy as follows:

dy (ntQ) = dy (t) —

or equivalent

M-l

iW^T) L ^^r^^'

7 = 1

'' (U,

Sy (t) S

1 '*'"'

(12)

Thus, one can ascertain the dependence of a^(T) as a function of t from a single data set in a very simple way. For a given data set, M of course decreases as n increases. To estimate the confidence interval or error bar for a Gaussian type of noise of a particular value o^(t)

obtained from a finite number of samples [Lesage and Audoin, 1973] have shown that:

Confidence Interval /„ * a^(T) • ic„ • A/-'''^ for A/ > 10 (13)

where: M: total number of data points used in the estimate,

a: as defined in the previous section,

Kj - K, - 0.99,

Ko - 0.87.

K_, - 0.77,

ic_2 - 0.75.

TN-166
Rep. 580-2 147

As an example of the Gaussian model with Af - 100, o - - 1 (flicker frequency noise) and

o^(t - 1 second) - 10" '^ one may write: 4 « Oyix) • <<. • A/ -'^2 - o,(T) • (0.77) • (100)

-1/2 Oyix) (0.077), (14)

which gives:

Oj,(T - 1 second) - (1 ± 0.08) x 10" '^ (15) A modified estimation procedure including dead-time between pairs of measurements has also been

developed [Yoshimura, 1978], showing the influence of frequency fluctuations auto-correlation.

7. Conclusion

The statistical methods for describing frequency and phase instability and the corresponding power law

spectral density model described are sufficient for describing oscillator instability on the short term. Equation (9) shows that the spectral density can be unambiguously transformed into the time-domain measure. The converse is not true in all cases but is true for the power law spectra often used to model precision oscillators. Non-random variations are not covered by the model described. These can be either periodic or monotonia Periodic variations are to be analyzed by means of known methods of harmonic analysis. Monotonic

variations are described by linear or higher order drift terms.

TABLE I — The fidftciional characteristics ofJive independent noise processes for frequency instability of oscillators

Description of noise process

Slope characteristics of log log plot

Frequency-domaine Time-domaine

Syif) S,(/)or S,(/) a^(T) 0(T)

a P-a - 2 ^ Vi/2

Random walk frequency -2 -4 1 W

Flicker frequency -1 -3

White frequency -2 -1 ->/i

Flicker phase I -1 -2 -1

White phase 2 -2 -1

S,{f)~ Aa/»

5; (/) - vo^ *<./°-' 'Wh^f^ (P - a - 2) S.if).±h.r-^.±h.f^

o'ir) - III" o(x) ~ itr

TN-167
148 Rep. 580-2

TABLE II - Translation offrequency stability measures from spectral densities in frequency domain to variance in time domain and vice versa (for 2nfi,x > I)

Description of noise process 0,2 (T) - Syif)' S,(/)-

Random walk frequency A pSy(f)^} ^ T-'oJ(t)/-^ % r-'o](x)]f-*

Flicker frequency B fS,(f)rf> ^

xoaj(x)]/- 4 ^«Jw/-'

White frequency c rsy(f)x-^ ^[T'aJ(T)/' ^ faj(x)/-^

Flicker phase D\f-' Sy(f)X-' ^ ^<^JW/' T ^"'H^"'

White phase E f-'SAf)r-' -^ T^a^(x)/^ 4 ^<y](^)r

A -

4n' 1.038 + 3 log, (Inf^x) 4n2

B - 2log«2

4n^ C- 1/2

REFERENCES

ALLAN, D. W. (February, 1966) Statistics of atomic frequency standards. Proc. IEEE, Vol. 54, 221-230. ALLAN, D. W. [December, 1974] The measurement of frequency and frequency stability of precision oscillators. Proc. 6th Annual Precise Time and Time Interval (PTTI) Planning Meeting NASA/IX)D (US Naval Research Laboratory,

Washington, DC), 109-142. ALLAN, D. W. and BARNES, J. A. [May, 1981] A modified "Allan Variance" with increased oscillator characterization ability. Proc. 35th Annual Symposium on Frequency Control. Philadelphia, PA, USA (US Army Electronics Command,

Ft. Monmouth, NJ 07703), 470-476 (Electronic Industries Association, Washington, DC 20006, USA).

ALLAN, D. W. and DAAMS, H. [May, 1975) Picosecond time difference measurement system. Proc. of the 29th Annual Symposium on Frequency Control, 404-411. ATKINSON, W. K.., FEY, L. and NEWMAN, J. [February, 1963) Spectrum analysis of extremely low-frequency variations of

quartz oscillators. Proc. IEEE, Vol. 51,2, 379. BABITCH, D. and OLIVERIO, J. [1974) Phase noise of various oscillators at very low Fourier frequencies. Proc. of the 28th Annual Symposium on Frequency Control, 150-159. BARNES, J. A. [January, 1969) Tables of bias functions, B| and B2, for variances based on finite samples of processes' with power law spectral densities. Tech. Note No. 375, National Bureau of Standards, Washington, DC, USA. BARNES, J. A. [August, 1976) Models for the interpretation of frequency stability measurements. NBS Technical Note 683. BARNES, J. A., CHI, A. R., CUTLER, L. S., HEALEY, D. J., LEESON, D. B., McGUNIGAL, T. E.. MULLEN, J. A.. SMITH, W. L., SYDNOR, R., VESSOT, R. F. and WINKLER. G. M. R. [May, 1971) Characterization of frequency

stability. IEEE Trans. Instr. Meas., Vol. IM-20, 105-120. BARNES, J. A., JONES, R. H., TRYON, P. V. and ALLAN, D. W. [December, 1982) Noise models for atomic clocks. Proc.

14th Annual Precise Time and Time Interval (PTTI) Applications and Planning Meeting, Washington, DC, USA,

295-307. BAUGH, R. A. [1971) Frequency modulation analysis with the Hadamard variance. Proc. 2Sth Annual Symposium on Frequency

Control, 222-225. BLACKMAN, R. B. and TUKEY, J. M. (1959) The measurement of power spectra. (Dover Publication, Inc., New York, N.Y.).

TN-168
Rep. 580-2 149

BLAIR, B. E. (Ed.) [May, 1974) Time and frequency: theory and fundamentals. NBS Monograph No. 140. (US Government

Printing Office, Washington, D.C. 20402). BOILEAU, E. and PICINBONO, B. (March, 1976] Statistical study of phase fluctuations and oscillator stability. IEEE Trans.

Instr. Meas., Vol. 25, 1, 66-75. BRANDENBERGER, H.. HADORN, F., HALFORD, D. and SHOAF, J. H. (1971) High quality quartz crystal oscillators: frequency-domain and time-domain stability. Proc. 25th Annual Symposium on Frequency Control, 226-230. CHI, A. R. (December, 1977) The mechanics of translation of frequency stability measures between frequency and time-domain

measurements. Proc. 9th Annual Precise Time and Time Interval (PTTl) Applications and Planning Meeting, Greenbelt, MD, USA. CUTLER, L. S. and SEARLE, C. L. (February. 1966) Some aspects of the theory and measurement of frequency fluauations in frequency standards. Proc. IEEE, Vol. 54, 136-154. DE PRINS, J. and CORNELISSEN, G. (October, 1971) Analyse spectrale discrete. Eurocon (Lausanne, Swiuerland).

DE PRINS, J., DESCORNET, G., GORSKI, M. and TAMINE, J. (December, 1969] Frequency-domain interpretation of

oscillator phase stability. IEEE Trans. Instr. and Meas., Vol. IM-18, 251-261. GROSLAMBERT, J., OLIVIER, M. and UEBERSFELD, J. [December, 1974] Spectral and short-term stability measurements. IEEE Trans. Instr. Meas., Vol. IM-23 4, 518-521. HOWE, DA., ALLAN, D. W. and BARNES, J. A. (May, 1981] Properties of signal sources and measurement. Proc. 35th Annual Symposium on Frequency Control, Philadelphia, PA, USA (US Army Electronics Command,

Ft. Monmouth, NJ 07703), 669-717 (Electronics Industries Association, Washington, DC 20006, USA). IEEE (May, 1972] Special issue on time and frequency. Proc. IEEE Vol. 60, 5. IEEE [1983] Frequency Stability: Fundamentals and Measurement, 77-80. Ed. V. F. Kroupa. IEEE Press Books. IEEE-NASA (1964) Proceedings of Symposium on short-term frequency stability. NASA Publication SP 80. JONES, R. H. and TRYON, P. V. (January- February, 1983] Estimating time from atomic clocks. NBS Res., Vol. 88, 1, 17-24. LESAGE, P. [1983] Charaaerization of frequency stability: Bias due to the juxtaposition of time interval measurements. IEEE

Trans. Instr. Meas., Vol. IM-32, 1, 204-207. LESAGE, P. and AUDOIN, F. (June, 1973) Characterization of frequency stability: uncertainty due to the finite number of measurements. IEEE Trans. Instr. Meas., Vol. IM-22, 1, 103. LESAGE, P. and AUDOIN, C. [March, 1974) Correction to: Characterization of frequency stability: uncertainty due to finite number of measurements. IEEE Trans. Instr. Meas., Vol. IM-23, I, 103. LESAGE, P. and AUDOIN, F. (May, 1975) A time domain method for measurement of the spectral density of frequency

fluctuations at low Fourier frequencies. Proc. of the 29th Annual Symposium on Frequency Control. 394-403. LESAGE, P. and AUDOIN, C. (September, 1976) Correction to: Characterization of frequency stability: uncertainty due to the

finite number of measurements. IEEE Trans. Instr. Meas., Vol. IM-25, 3. LESAGE, P. and AYI, T. [December, 1984] Characterization of frequency stability: analysis of modified Allan variance and

properties of its estimate. IEEE Trans. Instr. Meas., Vol. IM-33, 4, 332-336. LINDSEY, W. C. and CHIE, CM. (December, 1976] Theory of oscillator instability based upon structure functions.

Proc. IEEE, Vol. 64. 1662-1666. MEYER. D. G. [November. 1970) A test set for the accurate measurements of phase noise on high-quality signal sources. IEEE

Trans. Instr. Meas., Vol. IM-19. 215-227. PERCIVAL. D. B. (June. 1976] A heuristic model of long-term atomic clock behavior. Proc. of the 30th Annual Symposium on

Frequency Control. RICCI, D. W. and PEREGRINO, L. [June, 1976] Phase noise measurement using a high resolution counter with on-line data

processing. Proc. of the 30th Annual Symposium on Frequency Control. RUTMAN, J. [February, 1972) Comment on characterization of frequency stability. IEEE Trans. Instr. Meas., Vol. IM-21, 2, 85. RUTMAN, J. [March, 1974] Characterization of frequency stability: a transfer funaion approach and its application to measurements via filtering of phase noise. IEEE Trans. Instr. Meas., Vol. IM-23, I, 40-48. RUTMAN, J. (September, 1978) Characterization of phase and frequency instabilities in precision frequency sources: fifteen years of progress. Proc. IEEE, Vol. 66. 9, 1048-1075. RUTMAN, J. and SAUVAGE, G. (December, 1974) Measurement of frequency stability in the time and frequency domains via

filtering of phase noise. IEEE Trans. Instr. Meas., Vol. IM-23, 4, 515, 518. SAUVAGE. G. and RUTMAN, J. (July-August, 1973) Analyse spectrale du bruit de frequence des oscillateurs par la variance de Hadamard. Ann. des Telecom., Vol. 28, 7-8, 304-314. VON NEUMANN, J., KENT, R. H., BELLINSON, H. R. and HART, B. I. [1941] The mean square successive difference. Ann. Math. Stat., 12, 153-162. WALLS, F. L, STEIN, S. R., GRAY, J. E. and GLAZE, D. J. (June, 1976) Design considerations in state-of-the-art signal processing and phase noise measurement systems. Proc. of the 30th Annual Symposium on Frequency Control. WINKLER, G. M. R. [December, 1976) A brief review of frequency stability measures. Proc. 8th Annual Precise Time and Time

Interval (PTTI) Applications and Planning Meeting (US Naval Research Laboratory, Washington, DC), 489-528. YOSHIMURA, K. (March, 1978] Characterization of frequency stability: Uncertainty due to the autocorrelation of the frequency

Huctuations. IEEE Trans. Instr. Meas., lM-27 I, 1-7.

TN-169
150 Rep. 580-2, 738-2

BIBLIOGRAPHY

ALLAN, D. W., et al. (1973] Performance, modelling, and simulation of some caesium beam clocks. Proc. 27th Annual Symposium on Frequency Control, 334-346. FISCHER, M. C. [December, 1976] Frequency stability measurement procedures. Proc. 8th Annual Precise Time and Time

Interval (PTTI) Applications and Planning Meeting (US Naval Research Laboratory, Washington, DC), S7S-618. HALFORD, D. [March, 1968] A general mechanical model for |/| " spectral density noise with special reference to flicker noise

1/|/|. Proc. IEEE (Cones.), Vol. 56, 3, 251-258. MANDELBROT, B. [April, 1967] Some noises with l//sp>ectrum, a bridge between direct current and white noise. IEEE Trans.

Inf. Theory, Vol. IT- 13. 2, 289-298. RUTMAN, J. and UEBERSFELD, J. [February, 1972] A model for flicker frequency noise of oscillators. Proc. IEEE, Vol. 60, 2, 233-235. VANIER, J. and TETU, M. [1978] Time domain measurement of frequency stability: a tutorial approach. Proc. 10th Annual

Precise Time and Time Interval (PTTI) Applications and Planning Meeting, Washington, DC, USA, 247-291. VESSOT, R. [1976] Frequency and Time Sundards. Chap. 5.4 Methods of Experimenul Physics, Academic Press, 198-227. VESSOT, R., MUELLER, L. and VANIER, J. [February, 1966) The specification of oscillator charaaeristics from measurements made in the frequency domain. Proc. IEEE Vol. 54, 2, 199-207. WINKLER. G. M. R., HALL. R. G. and PERCIVAL. D. B. [October, 1970] The US Naval Observatory clock time reference and

the performance of a sample of atomic clocks. Meirologia, Vol. 6, 4. 126-134.

TN-170
Reprinted, with permission, from P. Lesage and C. Audoin in Radio Science. Vol. 14,

No. 4, pp. 521-539, 1979, Copyright ® by the American Geophysical Union.

Characterization and measurement of time and frequency stability *

P. Lesage and C. Audoin

Laboratoire de I'Horloge Atomique, Equipe de Recherche du CNRS, Universile Paris-Sud, Orsay, France

(Received February 1, 1979.)

The roles which spectral density of fractional frequency fluctuations, two-sample variance, and power spectra play in different parts of the electromagnetic spectrum are introduced. Their relationship

is discussed. Data acquisition in the frequency and the time domain is considered, and examples

are given throughout the spectrum. Recently proposed methods for the characterization of a single high-quaUty frequency source are briefly described. Possible difficulties and limitations in the

interpretation of measurement results are specified, mostly in the presence of a dead time between

measurements. The link between past developments in the field, such as two-sample variance and

spectral analysis from time domain measurement, and recently introduced structure fimctions is emphasized.

1. INTRODUCTION

Progress in the characterization of time and

frequency stability has been initiated owing to the

work of the various authors of papers delivered

at the IEEE-NASA Symposium on Short Term

Frequency Stability [1964] and of articles published

in a special issue of the Proceedings of the IEEE

[1966]. Presently widespread definitions of fre- quency stability have been given by Barnes et al. [1971]. Many of the most impoitant articles on

the subject of time and frequency have been gathered in the NBS Monograph 140 [1974] . Since that time, many papers have been published which

outline different aspects of the field. Owing to the extent of the subject, they will be only partly

reviewed here. We will emphasize recently

proposed principles of measurements and recent developments in the time domain characterization

of frequency stability. The subject of time predic-*

tion and modeling as well as its use for estimation

of the spectrum of frequency fluctuations [Percival,

1978] are beyond the scope of this paper. Recent

reviews which outline several different aspects of

the field of time and frequency characterization

have been published [Barnes, 1976; Winkler, 1976; Barnes, 1977; Rutman, 1978; Kartaschoff, 1978]

.

2. DEFINITIONS: MODEL OF FREQUENCY

FLUCTUATIONS

The instantaneous output voltage of a frequency

generator can be written as

Copyri(lit O 1979 by the American Geophyiic*! Unioa.

v(/) = [Ko -h AK(r)] COS [l-nv^t -t- «p(r)] (1) where V^ and v^ are constants which represent the

nominal amplitude and frequency, respectively.

AF'(0 and (p(r) denote time-dependent voltage and

phase variations.

Fractional amplitude fluctuations are defmed by

e(r)=AK(/)/Ko (2) A power spectral density of fractional amplitude

fluctuations S.(/) can be introduced if amplitude

fluctuations are random and stationary in the wide

sense. Usually, for high-quality frequency sources, one has

|£(0I « 1 (3) and amplitude fluctuations are neglected. However,

it is known that amplitude fluctuations can be

converted into phase fluctuations in electronic cir- cuits used for frequency metrology [Barillet and

Audoin, 1976; Bava et al., 1977a] and that they

may perturb measurement of phase fluctuations

[Brendelet al., 1977] . It is then likely that amplitude

fluctuations will become the subject of more detailed analysis in the future. According to the conventional definition of in- stantaneous frequency we have

v(/)= Vo + (l/2Tr)<f(/) (4)

In a stable frequency generator the condition

|«p(r)|/2irvo<sc 1 (5)

is generally satisfied. We will use the following notations [Barnes et

* See Appendix Note # 24 521

TN-171
322 LESAGE AND AUDOIN

fl/., 1971]: "S,(f)

x(/) =

2'rrvn and y{l) =

P(0

2-TTV„ (6)

where x(t) and >»(/) are the fractional phase and

frequency fluctuations, respectively. The quantity

x(t) represents the fluctuation in the time deflned

by the generator considered as a clock. At first, we will make the following assumptions:

1. The quantities jc(/) and>'(r) are random functions of time with zero mean values, which implies

that systematic trends are removed [Barnes et al., 1971] . They might be due to ageing or to imperfect

decoupling from environmental changes such as temperature, pressure, acceleration, or voltage.

Characterization of drifts will be considered in section 8.

2. The statistical properties of the stable fre- quency generators are described by a model which

is stationary of order 2. This point has been fully

discussed in the Uterature [Barnes et al, 1971; Boileau and Picinbono, 1976; Barnes, 1976] . This

assumption allows one to derive useful results and

to define simple data processing for the characterization of frequency stabiUty.

Actual experimental practice shows that, besides

long-term frequency drifts, the frequency of a high-quahty frequency source can be perturbed by

a superposition of independent noise processes, which can be adequately represented by random

fluctuations having the following one-sided power

spectral density of fractional frequency fluctuations:

Sy(J) = 2 ^''^' (7)

Syif) is depicted in Figure 1. Its dimensions are Hz~'. Lower values of a may be present in the

spectral density of frequency fluctuations. They

have not been clearly identified yet because of

experimental difficulties related to very long term

data acquisition and to control of experimental

conditions for long times. Moreover, the related noise processes may be difficult to distinguish from

systematic drifts. Finite duration of measurements introduces a low-frequency cutoff which prevents one from

obtaining information at Fourier frequencies smaller

Fig. 1. Asymptotic log-log plot OS 5, (/) for commonly encountered noise processes.

than 1/6, approximately, where 6 is the total duration of the measurement [Cutler and Searle, 1966]

.

Alternatively, this made it possible to invoke physical arguments to remove some possible mathematical difficulties related to the divergence of S^iJ)

Asf-* for a < 0. Furthermore, high p&ss filtering is always present

in the measuring instruments or in the frequency

generator to be characterized. It insures conver- gence conditions at the higher-frequency side of

the power spectra for a > 0. The spectral density of fractional phase fluctua- tions is also often considered. From (6), one can

write, at least formally,

5.(/) = (i/4^y')5m (8)

The dimensions of 5,(/) are s^ Hz"'. Similarly,

the spectral density of phase fluctuations <p(0 >s such as

5,(/) = (2irv„)^5,(/) (9)

It is expressed in (rad)^ Hz"'. The quantity Jf(f) [Halford et al, 1973] is sometimes considered to characterize phase fluc- tuations. If phase fluctuations at frequencies >/

are small compared with 1 rad, one has

-^(/) = t5,(/) (10) where S^{f) is the spectral density of phase fluc- tuations of the frequency generator considered. The

definition of Jf{f) implies a connection with the radio frequency spectrum, and its use is not recom- mended.

Since the class of noise processes for which >*(/)

is stationary is broader than that for which the

TN-172
TIME AND FREQUENCY 523

TABLE 1. Designation of noise processes

Designation Class of Stationarity

2 white noise of phase

1 flicker noise of phase

white noise of frequency

stationary phase

fluctuations

stationary first-order phase increments

stationary fust-order phase increments — 1 flicker noise of frequency stationary second-order phase increments —2 random walk of frequency stationary second-order phase increments

phase is stationary [Boileau and Picinbono, 1976]

,

S (/) should preferably be used in mathematical

analysis. However, it is true that many experimental

setups transduce <p(0 into voltage fluctuations and

aUow one to experimentally determine an estimate

of its power spectral density 5^(/).

Table 1 shows the designations of the noise

processes considered. It also indicates the class of stationarity to which they pertain, as will be

justified later on. S if) is one of the recommended definitions of

frequency stabiUty [Barnes et al, 1971]. It gives

the widest information on frequency deviations >>(() within the limits stated previously.

3. NOISE PROCESSES IN FREQUENCY GENERATORS

The white phase noise (a = 2) predominates for / large enough. It is the result of Jhe additive ther- mal (for the lower part of the electromagnetic

spectrum, including microwaves) or quantum (for

optical frequencies) noise which is unavoidably ^ superimposed on the signal generated in the oscilla- tor [Cutler andSearle, 1966] . It leads to a one-sided

spectral density S^(J) of the form FkTf^ / v\P ox Fhvof^ /v\P, depending on the frequency range, where k is Boltzmann's constant, h is Planck's

constant, T is the absolute temperature, F is the

noise figure of the components under consideration,

and P is the power delivered by atoms. The flicker phase noise (a = 1) is generated mainly

in transistors, where this noise modulates the cur- rent [Halfordetal, 196%; Healey, 1972] . The theory

of this noise is not yet very well understood.

Diffusion processes across junctions of semiconductor devices may produce this noise.

White noise of frequency (a = 0) is present in oscillators. It is the result of the noise perturbation

in the generation of the oscillation which is due

to white noise within the bandwidth of the fre- quency-determining element of the oscUlator [Blaquiere, \953a, b].his often masked by other types

of noise but has been observed in lasers [Siegman

and Arrathoon, 1968] and more recently in masers

( Vessot et al, 1977] . The one-sided spectral density

of fractional frequency fluctuations is then kT/PQ^

of hv^/PQ^ depending on the frequency range, as

stated above. Q is the quahty factor of the fre- quency-determining element.

White noise of frequency is typical of passive

frequency standards such as cesium beam tube and

rubidium cell devices as well as stabilized lasers.

It is related to the shot noise in the detection of

the resonance to which an oscillator is slaved [Cutler

andSearle, 1966].

FUcker noise of frequency and the random walk

frequency noise for which a = - 1 and —2, respectively, are sources of limitation in the long-term

frequency stabihty of frequency sources. They are observed in active devices as well as passive ones. For instance, flicker and random walk frequency

noises have been observed in quartz crystal resonators [Wainwright et al, 1974] and rubidium masers

[ Vanier et al., 1977] . The origin is not well understood yet. It might be connected, in the first case, with fluctuations in the phonon energy density

[Musha, 1975].

Figures 3 and 5 show, for the purpose of illustra- tion, S^(J) for a hydrogen maser for 10"* £ / s

3 Hz [Vessot et al., 1977] and for an iodinestabihzed He-Nc laser for 10"^ :S/£ 100 Hz [Cerez

et al, 1978] . In both cases, S^{f) is derived from

the results of time domain frequency measurements

aO,(T)

K)

,-iJ

10- T[.] w- »' 10' 10'

Fig. 2. Frequency stability, characterized by the root mean

square of the two-sample variance of fractional frequency

fluctuation, of a hydrogen maser. The part of the graph with

a slope of -\ is typical of white noise of frequency.

TN-173
524 LESAGE AND AUDOIN

ts(f)[H.-!l »-

K)r» \ /

K)-

f[H.]

10'

ts,(rW

10-a

s-4t

W' F[h, «'

Fig. 3. Spectral density of fractional frequency fluctuations of a hydrogen maser. The parts of the graph with slopes and 2 coincide with theoretical expectations.

(see section 6.4), as shown in Figures 2 and 4 but

is very close to theoretical limits specified above.

It is worth pointing out here that in systems where

a frequency source is frequency slaved to a fre- quency reference or phase locked to another fre- quency generator the different kinds of noise in- volved are filtered in the system [Cutler andSearle,

1966; C. Audoin, unpublished manuscript, 1976]

.

In these cases, at the output of the system, one

can find noise contributions pertaining to the model

(7) but appearing on the Fourier frequency scale

in an order different than that shown in Figure

1. This is depicted in Figures 6 and 7 for the case of a cesium beam frequency standard consisting

of a good quartz crystal oscillator which is frequency

controlled by a cesium beam tube resonator. The model for the frequency fluctuations is more

useful if the noise processes can be assumed to be gaussian ones (in particular, momenta of all orders can then be expressed with the help of

momenta of second order). The deviation of the

frequency being the result of a number of elementary perturbations, this assumption seems a reasonable one. Furthermore, the normal distribution of

Fig. S. Spectral density of fractional frequency fluctuations of a He-Ne iodine-stabilized laser. The solid line represents

the spectral density of fractional frequency fluctuations corre- sponding to experimental results, and the dotted line represents

the expected value of S (J).

y, the mean value of frequency fluctuations

averaged over time interval t as defmed in (16),

has been experimentally checked for a = 2, 1, 0,

and -1 [Lesage and Audoin, 1973, 1977}. This

is shown in Figure 8 for white noise of frequency,

for instance.

4. MEASUREMENTS IN THE FREQUENCY DOMAIN *

Measurement of power spectral density of fre- quency and phase fluctuations can be performed

in the frequency domain for Fourier frequencies

greater than a few 10 ~^ Hz owing to the availability

of good low-frequency si>ectrum analyzers.

4.1. Use of a frequency discriminator

Frequency discriminators are of current use to characterize radio frequency and microwave generators. A resonant device such as a tuned circuit or a microwave cavity acts as a transducer which

10^.

T[«]

10- 10'

1 'S,(f)[H.-:

w-'' \ /

10-"

- \^ /

»-» f'

10-* to-' 1 10' «*'

Fig. 4. Frequency stability, characterized by the root mean

square of the two-sample variance of fractional frequency Fig. 6. Spectral density of fractional frequency fluctuations of

fluctuations, of a He-Ne iodine-stabilized laser. a good quartz crystal oscillator.

* See Appendix Note # 6

TN-174
TIME AND FREQUENCY 525

Kr'

10-

10-".

s,(r)[H.-']

10- K)- 10'

F[H.j

10'

Fig. 7. Spectral density of fractional frequency fluctuations of the same quartz crystal oscillator as in Figure 6 but frequency

controlled by a Cs beam tube resonance.

transforms frequency to voltage fluctuations. This

method can be applied to optical frequency sources

too, as shown in Figures 9 and 10. Here the source

is, for instance, a CW dye laser, and the frequency

selective device is a Fabry-Perot etalon. The second

light pass allows one to compensate for the effects

of amplitude fluctuations and to adjust to a null the mean value of the output voltage. The slope

of this frequency discriminator equals 1 V MHz"',

typically, with a good Fabry-Perot etalon in the

visible.

(^anultt II Prtbtkililf

1

9S

. yy

^

9 >y

A f ^ <f^

7 ./

6

5 Xy IV T| ^

4 \ ^ 40 60 80 W) fi

^ sk

2 \,

,1 "^t

5 \,

1

1

1

1 N\VN

Fig. 8. Distribution of counting time results for white fre- quency noise (cesium beam frequency standards, t = 10 s) in Galtonian coordinates. Circles represent the cumulative probability corresponding to (t^ - t| with t = (t, ). Solid lines correspond to the normal distribution of the same width.

Fig. 9. Principle of frequency to voltage transfer in a frequency

discriminator.

4.2. Use of a phase detector

This technique is well suited for the study of

frequency sources in the radio frequency domain

0.2 MHz < Vq< 500 MHz, in a range where very

low noise balanced-diode mixers which utilize Schottky barrier diodes are available. This tech- nique has mainly been promoted by the National

Bureau of Standards \Shoqf, 1971; Walls and Stein,

1977]

.

Figure 1 1 shows the principle of the determination

of the phase fluctuations in frequency multipUers,

for instance. The two frequency multipliers are driven by the same source. A phase shifter is adjusted in order to satisfy the quadrature condition.

One then has

v(/) = D[<p,(0-<Pj(01 (11) *

where Z> is a constant and <p, and tpj ^^^ ^^^ phase

fluctuations introduced in the devices under test.

It is assumed that the mixer is properly used to allow a balance of the phase and amphtude fluctua- tions of the frequency source. This technique is often used to characterize phase

fluctuations of two separate frequency sources of

the same frequency. The quadrature condition is

F. P etalon Laser

Source -^^^

4:

dirrerential Amplifier

>-

Photocells

Fig. 10. Principle of frequency noise analysis of a dye laser.

* See Appendix Note # 25

TN-175
526 LESAGE AND AUDOIN

Outrtz Crjrstil Frequtncjf Source

Friquencf

Multiplier

X n

Phase

Shifter

Frequency

Multiplier

I n

Phiie

Comparator

Filler -». output

Fig. 1 1 . Principle of measurement of phase noise with a high-quality balanced mixer used as a phase comparator.

insured by phase locking the reference oscillator,

number 2 of Figure 12, to the oscillator under test. Fluctuations of the output voltage u(t) at frequencies/larger than the frequency cutoff of the phase

loop are proportional to phase fluctuations of

oscillator number 1. On the contrary, components

of u(0 at frequencies smaller than the above fre- quency cutoff are representative of frequency fluc- tuations of oscillator number 1

.

The requirement of having a reference oscillator

of the same quaUty as the oscillator to be tested

may be inconvenient. It has recently been shown

that the phase noise of a single oscillator can be

measured by using the mixer technique, but with

a delay line [Lance et al., 1977] . Figxire 13 shows

a schematic of the setup. The signal from the

frequency source is spht into two channels. The

reference channel includes a phase shifter for the

purpose of adjustment. It feeds one of the mixer

inputs. The other chaimel delays th^ signal before

it is applied to the second mixer input. It can be

seen that the power spectra density of the mixer

output is proportional to (2ir/T^)^5^(/), where t^

is the delay. The sensitivity of this technique is then reduced for low Fourier frequencies. However,

some signal to noise enhancement can be achieved

in a more elaborate configuration with two differential delay line systems in which cross-spectrum

analysis is performed on the signal output from

the two delay Une systems [Lance et al., 1978]

.

Another method has been proposed to determine

the power spectrum of fractional frequency fluctua- tions of a single high-quality frequency source

[Groslambert, 1977] . It is shown in Figure 14. Two

auxihary oscillators, wliich do not need to be of

the same quality as the oscillator under test, are used. They are phase locked to the frequency

generator to be characterized. The control voltages

v,(/) and Vj(r) are appropriately filtered in order

to obtain, at their outputs, a voltage v\{t) = K^{^f^ - (po)and V2(r)= K^{(^ - <ipo), respectively, where

Ky and A^j ^^ constants and the subscripts 0, 1,. and 2 refer to the oscillator under test, oscillator number 1, and oscillator number 2, respectively.

It can be shown that the cross-correlation function

of v\ and Vj is proportional to the autocorrelation

function of the frequency fluctuations of the

oscillator under test. Its spectral density of fractional frequency fluctuations can then be obtained

via Fourier transform.

Frequency Source n*1

Frequency Source n'2

Phase

Comparator

V

'(•)

-C^>—»• output

Frequency Control

Fig. 12. Principle of phase noise measurement of oscillators. A phase lock loop insures the phase quadrature of the two

phase-compared signals.

4.3. Precision of measurement in the frequency

domain

A spectrum analyzer includes a filter of bandwidth

A/, centered at frequency /, a nonlinear device

which measures the power in the filtered signal,

and a low pass filter which integrates the output

signal for the time T. The integration time T is not infinite, and the filter bandwidth A/ is not extremely narrow. Only an estimate S(f, A/, T)

of S(f) can then be obtained. Well-known results show that the precision p in the measurement of

the power spectral density of a gaussian process

is given by

TN-176
TIME AND FREQUENCY 527

Pkati

thiTter i

I Frtqutncy

lourct

Phase

comparator

Low-past

fitter

1 Delay

lint t

Output

Fig. 13. Principle of phase noise measurement of a single-frequency source with a delay line.

-1/2 P='{T Av)

No attempt has been made, to our knowledge, to specify p more accurately for the different noise

processes which can be encountered in frequency

metrology of stable sources.

5. MEASUREMENTS IN THE TIME DOMAIN

Time and frequency counting techniques are well

known [Cutler and Searle, 1966] . They are the

easiest to implement to provide information on the

low-frequency content (/ s 1 Hz) of the power

spectra of fractional frequency fluctuations.

5.1. The beat frequency method

A beat note at frequency v, is obtained from

two frequency sources under test, with frequencies

Vo and v'q, respectively, such that v^, ^ v' If Av„

(12) * and Avq denote the frequency fluctuations of the

two sources and Av, the frequency fluctuations of

the beat note, one has

Osc.1

Frequency

Generator

»,(«)

»,(t)

Filter

•,(t)

Correlator

Otc.2

Filter »;ui ^

Fig. 14. Principle of phase noise measurement of a single highquality frequency source with a correlator.

Av,

>'i =

AVn Av^

(13)

The fractional frequency fluctuations of the beat note are then proportional to those of the frequencies Vq and Vq but multiplied by the factor Vq/v,

which is much larger than unity.

With stable generators at frequencies lower than

approximately 100 GHz the frequency fluctuations are small enough that the beat note can be at low

frequency. The counter is then used as a period

meter, and a high precision in the measurement

is achievable.

Optical frequency standards show larger fre- quency fluctuations in absolute value. For instance, a laser stabilized at 500 THz (X = 0.6 jim) with

a fractional frequency stability of 1 x 10 ""exhibits

frequency fluctuations of 50 Hz. They can be easily

measured if the beat note is at 50 MHz, say, when

the counter is used as a frequency meter. In the case of iodine-stabilized He-Ne lasers the beat note

is easily obtained by locking the two lasers to different hyperfine components of the considered

iodine transition. Otherwise, the frequency offset technique is used [Barger and Hall, 1969]

.

5.2. Tlie time difference method

The time difference method [A Han and Daams,

1975] must be used with time standards which

deliver pulses as time scale marks. Distant time

comparison and synchronization by TV pulses, Ught

pulses, Loran-C pulses, for instance, pertain to this category. It provides information on the relative phases of the two clocks under test. Time interval measurements being very precise

* See Appendix Note # 26

TN-177
528 LESAGE AND AUDOIN

Frtquency Sourtt Phm

Shifter

Miitr Miitr

Auiiliarj

Frequencjf Sourtt ^U

Low Pits

Filter

Frequencj Source n»2

Low Pats

Filter i

Time

Interiil Counter

Fig. 15. Principle of the time difference method.

(a precision of 10-100 ns is typical, depending on

the class of the counter), they are also used with

c.w. frequency generators, as shown in Figure 15. This method is then well suited to the case in which

the frequency sources under test have the same

nominal frequency v,,, e.g., atomic frequency stan- dards. An auxihary frequency source, such as a frequency synthesizer, with frequency Vq allows one to obtain, at the output of the mixers, two

beat notes at the desired frequency v, = |vo -

Vo|. After amplification the zero crossing of one

of the beat notes starts the time interval counter,

and the zero crossing of the other beat note stops

it. One has

^1 =(*'o/^iX->:o-^o) (14) where x , is the fractional phase fluctuation of the beat note and x^ and x'^ that of the two frequency

standards. For instance, with Vg = 5 MHz, v, =

0.5 Hz, and a precision in the time interval measure- ment of 0. 1 |xs a precision of 10" '* s at the nominal

frequency Vq is achieved.

6. CHARACTERIZATION OF FREQUENCY STABILITY

IN THE TIME DOMAIN

6.1. Significance of experimental data

It is well estabUshed that measurement in the

time domain with an electronic counter samples

phase increments and gives A^<p{t,^) defined as ^M't) = *('* + t) - 9('J (15) The phase increment A,(p(/^) is related to y,^, the

average over time interval [i^, t^ + t] of fractional

frequency fluctuations. We have

— 1 f'*^ 1 y, = -\ y{t')dt' =- A,«p(r,)

T

J,» 2-rrv,T (16)

where v, is the mean frequency of the processed

signal. _

Samples of y,^ can be combined in many different

ways. Some of those which have been considered

will be reviewed here. On the other hand, the

number of samples is finite, and the question arises as to the related uncertainty in the characterization

of frequency stability and of the best use of the

data.

6.2. N-sample variance

The sequence of measurement is as shown in Figure 16. The mean duration of each measurement

is T, and T is the time interval between the beginnings of them.

In statistical estimation it is common to consider

sample variance [Papoulis, 1965] . The A^-sample

variance of y^^ is define,d as

cjUN, T, t) = N-\^A''~N^/') (17)

where the factor N/(N - 1) removes bias in the

estimation.

The dependence of the expectation value of the

A^-sample variance on the number N of samples,

the sample time t, and the power spectral density

has been considered by Allan [1966] . We will only

consider special cases in the following.

It can be shown that computation of the average

of the A^-sample variance introduces a filtering of

the power spectral density S^{f) [Barnes et al., 1971]. We have

{ahN, T, T)> • 1>- {f)\H(J)\'df (18) H(/) is the transfer function of a linear filter which

has the following expression:

\H{f)\' = N

N- 1

in-rr/T 1' f I" sin-nfNT ?

|

-^/t J 1 iNsin-nfT J J

(19)

! T _! T , ! T _

K

Fig. 16. Sequence of time domain measurement.

TN-178
TIME AND FREQUENCY 529

For/ such that and we have

we have

\H(J)V

T:fT<i 1

A^(7V + 1)

3 (tt/D^

(20)

(21)

Equation (21) shows that for finite A^ the integral

in (18) will converge at the lower limit for o =

-1 and o = -2, as well as for a = 2, 1, and

0. One sees that very low frequency components

of 5^(/) are best eliminated for small values of

A^.

6.3. Variance of time-averagedfrequency

fluctuations

When N goes to infinity, (<t^((», t, t)) becomes

o^iy,,), the variance of time-averaged frequency

fluctuations or of the first difference of phase

fluctuations [Cutler and Searle, 1966] as given by

<y'(>^)=((>\)') (22) where angle brackets denote mathematical expectation. In the presence of a single-pole low-pass filter with cutoff frequency/, we have the following re- lation between o^CVt) and S (/):

<''(>'*)

with

i: SM)

1

1 + if'/fl)

\fi,if)\'dr (23)

\n,(fr =

siwnfj

tt/t (24>

Equation (24) shows that a (yj converges for a = 2, 1,0 but diverges for fiicker noise of frequency

(a = — 1) and random walk of frequency (a = —2).

The variance (r^iy^) is no longer used to characterize

frequency stability. However, it is useful to relate the RF power spectral density to S^if) [Rutman,

1974a]

.

6.4. Two-sample variance

For the special case N = 2, (17) gives

<a^(2.r.T))= { S,(f)

\ \H,{f)\'df

Jo 1 "• \J IJc)

with

\H2if)\' = 2 sinir/T

it/t

(sin TifTf

(26)

(27)

For small/, such that -rr/T <: 1, \H^{f)\^ varies

as/^. The integral in (26) is thus defined for flicker noise of frequency (a = -1) and random walk of

frequency (a = -2), as well as for a = 2, 1, and

0. It is easy to show that the quantity (/t^., —

y^) represents a second-order difference of phase

fluctuations. It follows that second-order phase

increments are stationary for a = -1 and -2 (as

specified in Table 1).

6.4.1. Two-sample variance without dead time.

The two-sample variance (Allan variance) without

dead time, for T = t, is now generally accepted

as the measure of frequency stability in the time

domain. One sets

a^(T) = {<jI{2, t. t)> (28)

Table 2 gives asymptotic expressions of <t^(t) in the cases 2it/^t » 1 and 2-it/].t <: 1. Expressions

of a^(T) in the presence of a sharp high-frequency

cutoff/^ have been given by Barnes et al. [1971]

for the case 2it/^t »• 1. One sees in Table 2 that <t^(t) has a characteristic

T dependence for each type of noise considered,

TABLE 2. Asymptotic expressions of the two-sample variance

for the noise processes considered

g,'(T)

5,(/) 2Tt/^T » 1 2it/,t <: 1

<aj(2, 7',T)>=i<(y,,,-j;j'> (25)

K

8'jrr 3A,lD(27r/,T)

Ao2t

2>i.,ln2

2it'

It 2A,/>2

2tT'h_jy

* See Appendix Note # 27

TN-179
S30 LESAGE AND AUDOIN

such as <t^(t) = Jt/r". This is specified as follows

for 2-nf^i » 1:

2 2

1 2

1 -I-2 -1

For a = -I the <t^(t) graph is a horizontal line,

which justifies the designation 'flicker floor' for

that part of the graph.

Usually, under experimental conditions the rela- tion 2'Tr/^T » 1 is satisfied. The noise processes

which perturb the oscillation can then be identified

from a a^(T) graph if it is assumed that the aforementioned model of frequency fluctuations is valid. Table 2 shows in which cases ct^(t) depends on

the frequency cutoff. The latter must then be

specified.

The case l-ir/.r <: 1 is useful for the analysis

of the effect of frequency of phase servocontrol

loops where the frequency fluctuations of the fre- quency reference are low-pass filtered. Bias functions have been given to relate (1) the two-sample

variance with and without dead time and (2) the

two-sample variance to the A'^-sample variance

[Barnes et al, 1971].

6.4.2. Two-sample variance with dead time.

General expressions for the A^-sample variance with

dead time have been given [Barnes et al., 1971)

for useful values of a if the condition Ittf^i :»

1 is satisfied. The case of the two-ifeample variance

with dead time has not been emphasized enough

yet. Table 3 compares the two-sample variance with

and without dead time when the condition 2Tr/^T >> 1 is fulfilled. The condition of negligible dead

TABLE 3. Comparison of the two-sample variance with and

without dead time for 2ir/.T » 1

<^'(2. T, T)

Kf

"0

2-nMT - T) « 1 2<(r - T) » 1

3A

8irr

In {2nM

Ao/2t

2A_,ln2

2ir

A,ln(2iT/,T)

, ln(7"/T);r» T

3

Fig. 17. The solid line represents the variation of A_|<a,(2,

T, t)) versus t/iT - t) for the flicker noise of frequency.

The dotted line represents the asymptotic value for (T - t) •C T.

time is then l-nf^iT- t) <: 1. Results for l-rtf^r <: 1 are also available tP- Lesage and C. Audoin,

private communication, 1978). Table 3 shows that

in the presence of -dead time, i.e., iTtf^T - t) » 1, the expression for the two-sample variance

is noticeably modified for a = —1 and —2.

The case of the flicker noise of frequency is particularly interesting. Figure 17 shows the varia- tion of the two-sample variance with dead time

as a function of r/(T - t). The flicker floor does

not exist anymore if the value of this ratio is modified when the sampling time t is changed. The

identification of the noise process which perturbs

the oscillation might then be wrong if the effect of dead time is not taken into accoimt.

6.5. Precision in the estimation of the two-sample

variance

Measurements are always of finite duration, and

therefore the number of available values of y^ is finite. We are then faced with the problem of the precision in the estimation of the time domain

frequency stability measurement. This is an important one because successive characterizations of

the frequency stability of a given device allow one

to get information on the stationarity of the pro- cesses involved in the perturbation of its frequency

but within the limits of the precision of the characterization. Precision in the estimation of the fre- quency stability of individual oscillators of a set of/> frequency generators (/> > 2) [Gray and Allan,

* See Appendix Note # 28 ** See Appendix Note # 29

TN-180
TIME AND FREQUENCY 531

1974] criticaUy depends on the precision of the pip ~ 0/2 frequency comparisons which can be

performed by arranging oscillators in pairs. Fur- thermore, the uncertainty in the determination of

a^(T) translates directly into the imccrtainty in determining the A„ coefficients if the frequency

generator is perturbed by noise processes modeled

by (7). The precision in the estimation of time domain

measurements of frequency stability has been considered by several authors [Tausworthe, 1972; Le- sage and Audoin, 1973; Yoshimura, 1978]. It has

been determined for most of the experimental

situations which can be encountered in the two- sample variance characterization of frequency

stabihty, with or without dead time (P. Lesage and

C. Audoin, private commimication, 1978).

Calculation of the expectation value of the two- sample variance according to (25) requires an infmite

number of data. But, in practice, only m counting

results are available, and one calculates the estimated average of the two-sample variance as follows:

*'(2, r,T,m) =

1 2(m - 1) ,t- ^(.y,..-y.y (29)

One can easily show that the expectation value

of ^^(2, 7, T, m) equals the averaged two-sample

variance with dead time. Thus the fmite number

of meastirements does not introduce bias in the estimation of the two-sample variance.

The estimated averaged two-sample variance

(EATSV) being a random function of m, we need

to characterize the uncertainty*' in the estimation.

We thus introduce the variance of the EATSV,

according to the common understanding of i

variance. We set

a' (^^(2, r, T, m)\ = ( (^^(2, r, T, m) - (a;(2. T,t))]') (30) With the expression (29) of the EATSV we get

aMd'(2, r. T. m)J

[. ~12 /m-l m-l \

with

^/ = (//*.- >-/)'- 2 (a;(2.r,T)> (32) The classical law of large numbers [Papoulis,

196S] which states that the true variance of a sum

of (m - 1) uncorrelated random variables decreases

as \/(m - 1), even for small values of (m - 1), does not apply here. We are considering the quantities 3^ which are correlated because two adjacent

differences (j7,,, - y,) and (j?,,^ - y,^,) are obviously not independent.

Equation (31) can also be written as

o' [&l(2, r, T, m)] = m- 1 4

2(/n

with

r»=o,3<_*>

(33)

(34)

r^, which does not depend on m, represents the

autocorrelation coefficient of p^ and P,_^. Since

the same data are used in two adjacent pairs, the

autocorrelation coefficient F, , and possibly others,

differs from zero. Equation (33) then shows that the l/(m - 1) dependence also occurs for the

random variables considered, but asymptotically for

large enough values of (m — 1). The variance of the EATSV can be related to Syif) if it is assumed that the quantities y^ are normally distributed. This is a reasonable assumption, as shown in section 2.

It is useftil to introduce A(m), the fractional

deviation of ^1(2, T t, m) defmed as

A(m) =

ifl(2,T,r,m)-{al (2,7,7))

{al(2, T, T))

(35)

The standard deviation a (A(m)] defmes the precision in the estimation of the two-sample variance.

Expressions for a [A(/?i)] which are vahd for m

> 2 have been established for all possible values

of 2-n/^T and 2-nf^{T - t) but will not be given

here.

In practice, the time domain frequency stability

of a frequency source is characterized by the

standard deviation (^^(2, T, t, m)] '^^ We there- fore consider 8 defmed as

8 =

[hU2, r,T,m)]"'-<a;(2, r,T)> 1/2

<aj(2, r,T)> 1/2 (36)

a(8) specifies the precision in the estimation of

the time domain frequency stability measurement

TN-181
532 LESAGE AND AUDOIN

h ()

Fig. 18. Variation of /T, as a function of r = T/t for the commonly encountered noise processes and for 2'ir/,T = 10.

from a limited number of data and allows one to draw error bars on a frequency stability graph.

For 2it/.t » 1 and m :» 1 we have

a(8) => K^m' (37) The values of K^ are given as follows subject to the condition that the dead time is negligible, i.e., l-nf^iT- t) «c 1:

2 0.99

1 0.99 0.87 -1 0.77 -2 0.75

In the presence of dead time the values of K^

depend on the noise process considered as well as the values of 2-nf^r and r = T/t. Figure 18, vaUd for litf^j = 10 shows that the dependence

of K^ with dead time is especially pronounced in the vicinity of r = 1 for a = 1 and 2.

7. CHARACTERIZATION OF FREQUENCY STABILITY

VIA FILTERING OF PHASE OR FREQUENCY NOISE

Equations (27) and (28) show that the defmition

of the time domain measurement of frequency

stability a^(T) involves a filtering of S^{f) in a linear

filter. Figure 19 shows the impulse response of this

filter, which represents the sequence of measureFig. 19. Impulse response of a linear filter which represents computation of two-sample variance.

ment for T = r, and Figure 20 depicts the related

transfer function. One can also consider the effect of filtering a voltage proportional to y(t) or x(t)

in a physically realizable analog filter. A high pass filter of cutoff frequency l/irr has

been considered [Rutman, 1974Z>; Rutman and

Sauvage, 1974] . Its input receives a voltage proportional to x(t). It is provided by a mixer used as

a phase comparator. The rms value of the filtered signal is measured. Wben the frequency cutoff is changed, this rms value shows the same \i versus a dependence as shown in section 6.4.1. More

interesting is a bandpass filter centered at the

variable frequency/ = 1/2t but with a fixed value

of its quality factor. It allows one to distinguish

white and flicker noise of phase, as it gives p. =

3 for a = 2; the ft versus a dependence being

otherwise unchanged for a = 1, 0, -1, and -2.

Similarly, a frequency discriminator, giving an

output proportional to y{t), followed by two cas- caded resistance-capacitance (RC) filters and a rms

voltmeter allows one to obtain a useful approximation of the two-sample variance. The filters insure low-pass and high-pass filtering with RC =

t/2 [Wiley, 1977].

Fig. 20. Transfer function, of the linear filter with impulse

response shown in Figure 19.

TN-182
TIME AND FREQUENCY 533

8. SPECTRAL ANALYSIS INFERRED FROM TIME

DOMAIN MEASUREMENTS

The methods of time domain characterization of

frequency stability reviewed above allow one to identify noise process if they are described by

Syif) = 2 Kf (38)

This may not be the case. Furthermore, it is of

interest to determine the power spectral density

of fractional frequency fluctuations for Fourier

frequencies lower than 1 Hz. In this region, time

domain measurements are the most convenient, and

the question arises as to their best use for spectral

analysis.

8. 1 . Selective numericalJiltering

Equations (24) and (27) show that calculation of

the variance of the second difference of phase

fluctuations (the two-sample variance) involves a more selective Altering than calculation of the variance of the first difference of phase fluctuations.

One can then consider higher-order differences

[Barnes, 1966; Lesage and Audoin, 1975a, b]. The

nth-order difference of phase fluctuations is denoted as "A7.^<p(/t), where t and T have the same

meaning as in section 6.1. and 6.2. This nth dif- ference is deflned by the following recursive equation:

"Ar..«p(/J = '-"A^„9(^ + r) - '"-"A^.,9('.) (39)

which introduces binomial coefficients C^_,. We

have

/-o -«pU, + (/i-i-i)ri) (40) The transfer function //„(/, T, t) of the linear filter which represents the calculation of the variance

of the nth difference of phase fluctuations is given

by

l^,(/. T, t)| = (simr/r)-' sinir/T (41)

-tt/t

It should be pointed out that for/r •« 1, one

has

V"

< -J

19"

•IOt '^

-c:19

i-r--^«

tf^+10T

Fig. 21. Impulse response of a linear filter which represents computation of the variance of the 20th difference of phase

fluctuations. C'j represents binomial coefficients.

Figure 21 shows the impulse response of the linear

filter which represents the calculation of the

variance of the 20th difference of phase fluctuations, and Figure 22 shows the related transfer

function. A selective filtering is then involved

around freqoency 1/2t.

Such a variance is also known as a modified

Hadamard variance [Baugh, 1971]. The spurious

responses at frequencies (2/ -i- 1)/2t, where / is an integer, can be eliminated by a proper weighting

i

' H2o("'^-^)

3i10'

i

2 1 10'

10» •|l , A , A - 2VT

1 3 5 7

|/^,(/. 7',T)|«(27r/r)— /t<:I (42)

Fig. 22. Transfer function of the linear filter considered in Figure

21.

TN-183
534 LESAGE AND AUDOIN

of measurement results or by filtering with the help

of an analog filter [Groslambert, 1976].

Such a technique of linear filtering has been used

to show that good quartz crystal oscillators exhibit

flicker noise of frequency for Fourier frequencies

as low as 10"' Hz [Lesage and Audoin, 19756].

Furthermore, it is well suited to the design of

automated measurement setups [Peregrino and

Ricci, 1976; Groslambert, 1977).

The best use of experimental time domain data

for selective filtering has been considered by Boileau [1976].

8.2. High-pass filtering

If frequency fluctuations y{t) are filtered in an

ideal high-pass filter with transfer function (/(,(/,

,

/) such that

Go(/. ..-[

/</,

its output z{t) is such that

o1= ( G,(/,A)S,(f)'^= [ S,(J)<V

(43)

(44)

Equation (44) shows that the derivative of o^ is -S^{f), and spectral analysis, and therefore

characterization of frequency stabihty, are possible,

in principle, by high-pass filtering.

Possible realization of the high-pass filter by

techniques of digital data processing have been

specified, such as the method of funte-time variance

and the method of finite-time frequency control. Processing of finite-time data is aimed to properly

deal with the nonintegrable singularity of the power

spectral density at v = [Boileau, 1975; Boileau

and Picinbono, 1976] . The method is well suited

to the analysis of drifts or slow frequency changes.

Practical use of this method has not been reported

yet.

8.3. Use of the sample spectral density

It has been shown in section 8.1. that spectral

analysis from the Hadamard variance or its modified

forms requires a series of measurements at time

interval t in order to specify the spectral density

at frequency 1 /2t. Another point of view has been

considered [Boileau and Lecourtier, 1977] . From

a set of measurements oi y^, sampled at frequency

1/t, it allows one to obtain an estimation of the

spectral density for discrete values of the Fourier

frequency.

9. STRUCTURE FUNCTIONS OF OSCILLATOR

FRACTIONAL PHASE AND FREQUENCY

FLUCTUATIONS

Interest in the variance of /ith-order difference

of phase fluctuations was recognized early in the

fleld of time keeping (see for instance, Barnes

[1966]). This can be easily understood from (43),

which shows that an efflcient filtering of lowfrequency components of frequency fluctuations

is then introduced. It allows one to deal properly

with frequency drifts, which will now be considered,

and poles of S^{f) of order 2(/i - 1) at the origin.

It is equivalent to saying that the nth difference

of phase fluctuations allows one to consider random

processes with stationary nth-order phase increments.

This question has been formalized by Lindsey

and Chie [1976, 1977], who introduce structure functions of oscillator phase fluctuations. The nth

order structure function of phase fluctuations is nothing else but the variance of the nth difference

of phase fluctuations, as considered in section 8. Then, by definition, the nth-order structure function

of fractional phase fluctuations is given by

Z)r(T) = £{["A,,.x(/J]'} (45) where E { • ) means expectation value. The fractional

phase (or the clock reading) at time t^ is x{t^).

We assume 7 = t. Let us consider an oscillator, the phase (p'(/)

of which is of the following form except for an

additive constant:

9'(r) = yn»..- + <p(0 (46)

where H^ is a random variable modeling the ithorder frequency drift and <p(/) represents random

phase fluctuations. We then have

''('> = 2 ^-i^-'^^')

and

k-2

l-\ ,k

(47)

(48)

where d^ = Cl^/2'tTVQ is the normalized drift coeffiTN-184
TIME AND FREQUENCY 535

cient and x(t) and y(t) have the same meaning as

in the preceding sections. The notations x' and>''

refer to an oscillator with drift.

It can then be shown that we have

D['-\r) = 7^E[dlA

+ 2^ I sm ^-4-^df i. il-nf)'

and

D(-)(t) = 2^ sin^(ir/T)

l=n

l>n

(49)

(50)

If one applies (49) to the case of an oscillator without drift, one can easily show that the following

equations are satisfied:

and

<^'(n) = (l/T')Z>i'>(T)

<r^rT) = (l/2T^)Z)<V)

(51)

(52) This is indeed not surprising because apart from more or less complicated mathematical formalism

the defmitions of the considered variances and

structure functions are closely related, as has been

emphasized here.

Relations between sample variance and structure functions have been given by Lindsey and Chie

[1976], whereas the relation between structure functions and several different approaches of fre- quency stability characterization has been analyzed

hy Rutman [1977, 1978].

For the generaUy accepted noise model defmed

by (7) the t dependence of higher-order structure functions is the same as the two-sample variance, as shown in Table 4 [Lindsey and Chie, 1978*]

.

As stated above, structure functions of order n

allow one to consider spectral densities which vary

as/" at the origin with a & -2(/i - 1). For instance,

for n = 3 it is possible to characterize frequency

fluctuations of an oscillator with a power spectral

density of fractional frequency fluctuations given

by SM) = 2L-4 h. f. This oscillator exhibits

stationary third-order increments of phase fluctuations. In the presence of a frequency drift described

by a polynomial of degree / - 1 , structure functions

of degree /i < / are meaningless: their computation

yields a time-dependent result. For n = / the /th structure fimction shows a long-term t dependence

proportional to t^. This dependence disappears for n > I. Although a power spectral density of the

form/~^^~'> would also give the structure functions

a variation of the form t", this variation does not depend on n, provided that the function is meaningful. It is then possible, at least in principle, to identify frequency drifts and to specify their order. This is illustrated in Figure 23 according to Lindsey

and Chie (1978*). However, there are not yet

experimental proofs that such a characterization

is achievable in practice.

10. POWER SPECTRAL DENSITY OF STABLE

FREQUENCY SOURCES

The power emitted by a source of time-dependent

voltage v{t) given by (1) is S^(v) </v in the frequency

range (v, v + dv] , where SJy) is the power spectral

density of the source. The dimensions of S^(v) are V^ Hz"'. The main interest of power spectral

density, in frequency metrology, is related to highorder frequency multiplication. We will only intro- duce the subject by giving the relations between

S^(y) and S (/) and stating present problems in the Held.

TABLE 4. Structure functions of orders 1, 2, 3, and 4 for fractional phase fluctuations of commonly encountered noise processes

for 2ir/»T » 1 SAf) Z)i"(T) ^l^'(T) /)'"(T) /).<*>( T) hj'

h.J-'

21' '^'

2I' '-'IT

1

, A,ln(w/»T)

2ir

3

, A,ln(n/.T)

2*TT

5— A,ln(iT/.T)

IT iV

4A_,T'ln2

3AoT

6.75 A. ,t' -n'h ,t'

3 2ir'A.3T'

35

h^ In (ir/,T)

20.7 A _,t' JV A.jT

From Lindsey and Chie (1978fc]

TN-185
536 LESAGE AND AUDOIN

10 .

10'

i/o^^

a, (T)

Hrifl)

(riicliirj

T[S]

Fig. 23. (Solid line) Two-sample variance of an oscillator showing a linear frequency drift of 10"'° per day and flicker noise of frequency given by S^(J) = 7.2 x 10""/''. (Dotted

line) The third difference of fractional phase fluctuation is independent of the drift (according to Lindsey and Chie [19786]

.

Negligible amplitude noise and gaussian phase

fluctuations being assumed, it is well known that the autocorrelation function of v{t) is R^if) given

by

Ki-') = Y

cos2irv„TCxp [-^(2irv„)V (>-,)! (53)

As o'^iy^) is only defmed for stationary phase

fluctuations and for phase fluctuations with station- ary first increments, the same is true for R^{t) and

therefore S^(f).

10.1. White noise offrequency

This is the simplest to deal with. If the frequency

of the source is perturbed by a broadband white

noise of frequency, one has S^(J) = h^^ and o^{y^) = {hjl-r). Whence

yi

/?„(t) = —^ COS l-nv^T cxp

I j (54)

where t^ is the coherence time of the signal. We

have

\ = i^^lKy (55)

The one-sided power spectral density is then repre- sented by a Lorentzian given by

5„(v) = V\

217AV

(2TTAv)'-t- [2ir(v- vo)] (56)

where Av is the half width at half maximum of

the power spectra defined as

2Av = irvoAo (57) We obviously have ZttAv • t^ = 1. If oscillators are considered, one has h^ = kT/PQ^ in the radiofrequency and microwave domain and h^ =

hv^/PQ^ in the optical frequency domain, where

P is the power delivered by the oscillator and Q

the quality factor of the frequency-determining

element. Table 5 gives theoretical values of co- herence time and linewidth of good oscillators. It

is only intended to illustrate a comparison, often

made, of the spectral purity of oscillators. It must

be pointed out that, in practice, other noise pro- cesses exist which modify these results. Even any

meaning of t^ and Av is removed if ^„(t) is not

defined.

Multiplication of the frequency by n multipUes

Av and divides t^ by the factor n^.

10.2. White noise ofphase

Presently available good quartz oscillators are affected by white noise of phase. It is easy to show

from the definition (16) of y^^ that the following

equation is satisfied:

H2iTVoTa(>',)p = /?,(0)-/?,(T) (58) where R^{t) denotes the autocorrelation function

of the stationary phase fluctuations <p(/). The expression of the one-sided 5„(v) then fol- lows [Rutman, \91Aa\ Lindsey and Chie, 1978a]:

5„(v) = —e-'-'"* 2 [8(v-Vo)

+ 5,(v - vo) -h \ 5,(v) * 5, (v) -(-•••

] (59) where the asterisk denotes convolution and the bracket contains an infinite set of multiple-convolution products of S^(v) by itself. Such an equation

is not easily tractable. It is the reason why the

TABLE 5. Theoretical values of correlation time and power

spectrum linewidth for various oscillators

Oscillator Wo, Hz K, Hz-' T,, s 2Ai'. Hz

5-MHz

quartz xtal H maser He-Ne laser

5 X 10*

1.4 X \Q\

5 X 10'*

4 X lO-'' 4 X 10-"

3 X 10-"

10" 3 X 10""

10' 3 X 10"' 10-' 30

TN-186
TIME AND FREQUENCY 537

approximation of small-phase fluctuations is often

made. If R^(0) = <p^ <: 1, one has

V^ —

5„(v) = —^ *-•'

(8 (V - vo) + 5,(v - vo)] (60) 2

In this approximation the power spectrum consists

of a carrier at frequency Vq around which the spectrum of the phase fluctuations is translated.

If the frequency of the signal is multiplied by

n, the mean squared frequency fluctuation becomes

«^«p^. If «^(p^ <: 1, the power spectral density

is then given by

SM) = Y

'"^ lS(v - nvo) + n'S,(v - v^)] (61)

The power in the carrier decreases, and the power

in the pedestal increases. The relative powers in the carrier P^ and in the pedestal P^ are then given

by

P,= l-e-

(62)

(63)

respectively, where <J)^ represents the mean squared

value of phase fluctuations at the signal frequency

considered. It has been proved that (62) and (63) are valid, even if the condition <t>^ <^ 1 is not

satisfled (F. Clerc, private communication, 1977).

A Rttitiii Powtr (dB)

-50.

1 B W'

Fig. 24. Variation of the relative power in the carrier P^ and

the pedestal P, as a function of <)>', the mean squared phase

fluctuations.

Figure 24 shows the variation of P^ and P^ as

a function of <{) ^

. One easily understands that the

carrier may disappear if the multiplicative factor

is high enough. This has been theoretically and

experimentally investigated by Walls and de Marchi

(1975], Bava et al. (1977*), and Godone et al. [1978] . A signal has been synthesized at 761 GHz,

starting from a 5-MHz quartz oscillator, which

verifies theoretical conclusions.

10.3. Other noise processes

Much work remains to be done to analyze properly the effect of noise such as the flicker noise of

frequency or the random walk of frequency which

contributes power very close to the carrier. The

very interesting semiempirical approach by Halford

[ 197 1] has not yet been justified either theoretically or experimentally in a convincing maimer.

11. CONCLUSION

Widely used theoretical and experimental methods for the characterization of frequency stability in the time and frequency domain have been outlined. Recently used or proposed experimental

methods have been reviewed. The effect of dead

time on the interpretation of time domain measurements, as well as on their precision has been

emphasized. Recently introduced structure functions have been considered as well as their interest for the elimination of frequency drifts. The problems in the relation between the radiofrequency

power spectral density and the power spectral

density of phase fluctuations have been briefly

summarized.

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TN-187
538 LESAGE AND AUDOIN

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L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler (1971), Characterization of frequency stability, IEEE Trans. Instrum. Meas., IM-20{2), 105-120. Baugh, R. A. (1971), Frequency modulation analysis with the Hadamard variance. Proceedings ofthe 25th Annual Frequency

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Boileau, E. (1975), Elimination optimale des derives lentes dans

les mesures de fluctuation, Ann. Telecommun., 30(5-6), 163-

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Boileau, E., and Y. Lecourtier (1977), Description statistique de la stabilitc des oscillateurs. Proceedings of the Conference on Measurement in Telecommunications, pp. 231-236, Centre

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Boileau, E., and B. Picinbono (1976), Sutistical study of phase

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IM-26(2), 98-102. Cerez, P., S. J. Bennett, and C. Audoin (1978), Stabilitc de

frequence du laser helium-neon comportant une cuve d'iode

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Cutler, L. S., and C. L. Searle (1966), Some aspects of the theory and measurement of frequency fluctuations in fre- quency standards, Proc. IEEE, 54(2), 136-154. Godone, A., C. O. Weiss, and G. Kramer (1978). FM noise measurements on an optically pumped FIR laser, IEEE J. Quantum Electron, QE-14(5), 339-342. Gray, J. E., and D. W. Allan (1974), A method for estimating

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Groslambert, J. (1976), A new speanmi analyzer using both

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Halford, D. (1971), Infrared-microwave frequency synthesis

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1320-1 IM.

Lesage, P., and C. Audoin (1973), Charaaerization of frequency

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Lesage, P., and C. Audoin (19756), Insubilite de frequence des oscillateurs: Mesure dans le domaine frequence par une mcthodc temporelle, L'onde Elec, 55(2), 82-89. Lesage, P., and C. Audoin (1977), Estimation of the two-sample

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Lindsey, W. C, and C. M. Chie (1976), Theory of oscillator

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TN-188
TIME AND FREQUENCY 539

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TN-189
Copyright ® 1984 Academic Press. Reprinted, with permission, from Infrared and

Millimeter Waves. Vol. 11, pp. 239-289, 1984.

INFRARED AND MILLIMETER WAVES, VOL. II

CHAPTER 7

Phase Noise and AM Noise Measurements

in the Frequency Domain

Algie L. Lance, Wendell D. Seat, and Frederik Labaar

TR IV Operations and Support Group One Space Park Redondo Beach. California

I. Introduction 239

n. Fundamental Concepts 240

A. Noise Sidebands 243

B. Spectral Density 243

C. Spectral Densities of Phase Fluctuations in the Frequency Domain 246

D. Modulation Theory and Spectral Density Relationships 248

E. Noise Processes 251

F. Integrated Phase Noise 252 G. AM Noise in the Frequency Domain 254

III. Phase-Noise Measurements Using the TWO-OSCILLATOR TECHNIQUE 255

A. Two Noisy Oscillators 257

B. Automated Phase-Noise Measurements Using the Two-Oscillator Technique -58

C. Calibration and Measurements Using the Two-Oscillator System 259

IV. SINGLE-OSCILLATOR PHASE-NoISE MEASUREMENT

Systems and Techniques 267 A. The Delay Line as an FM Discriminator 268

B. Calibration and Measurements Using the Delay Line

as an FM Discriminator 274

C. Dual Delay-Line Discriminator 28

1

D. Millimeter- Wave Phase-Noise Measurements 283 References 287

I. Introduction

Frequency sources contain noise that appears to be a superposition of

causally generated signals and random, nondeterministic noises. The random

noises include thermal noise, shot noise, and noises of undetermined origin

(such as flicker noise). The end result is time-dependent phase and amplitude

fluctuations. Measurements of these fluctuations characterize the frequency

source in terms of amplitude modulation (AM) and phase modulation

(PM) noise (frequency stability).

239

Copynghl © 1984 by Academic Press, Inc

All nghls of reproduction in any form reserved. ISBN 0-12-147711-8

TN-190
240 A. L. LANCE, W. D. SEAL, AND F. LABAAR

The term frequency stability encompasses the concepts of random noise,

intended and incidental modulation, and any other fluctuations of the output

frequency of a device. In general, frequency stability is the degree to which

an oscillating source produces the same frequency value throughout a

specified period of time. It is implicit in this general definition of frequency

stability that the stability of a given frequency decreases if anything except

a perfect sine function is the signal wave shape.

Phase noise is the term most widely used to describe the characteristic randomness of frequency stability. The term spectral purity refers to the

ratio of signal power to phase-noise sideband power. Measurements of phase

noise and AM noise are performed in the frequency domain using a spectrum

analyzer that provides & frequency window following the detector (doublebalanced mixer). Frequency stability can also be measured in the time

domain with a gated counter that provides a time window following the

detector. Long-term stability is usually expressed in terms of parts per million per

hour, day, week, month, or year. This stability represents phenomena

caused by the aging process of circuit elements and of the material used in the frequency-determining element. Short-term stability relates to frequency

changes of less than a few seconds duration about the nominal frequency.

Automated measurement systems have been developed for measuring the combined phase noise of two signal sources (the two-oscillator technique)

and a single signal source (the single-oscillator technique), as reported by

Lance et al. (1977) and Seal and Lance (1981). When two source signals are applied in quadrature to a phase-sensitive detector (double-balanced

mixer), the voltage fluctuations analogous to phasefluctuations are measured

at the detector output. The single-oscillator measurement system is usually

designed using a frequency cavity or a delay line as an FM discriminator.

Voltage fluctuations analogous to frequency fluctuations are measured at the detector output.

The integrated phase noise can be calculated for any selected range of

Fourier frequencies. A representation of fluctuations in the frequency

domain is called spectral density graph. This graph is the distribution of

power variance versus frequency.

II. Fundamental Concepts

In this presentation we shall attempt to conform to the definitions,

symbols, and terminology set forth by Barnes et al. (1970). The Greek

letter v represents frequency for carrier-related measures. Modulationrelated frequencies are designated /. If the carrier is considered as dc, the frequencies measured with respect to the carrier are referred to as baseband,

offiset from the carrier, modulation, noise, or Fourier frequencies.

TN-191
7. PHASE NOISE AND AM NOISE MEASUREMENTS 241

(al

180°

270° ANGULAR FREQUENCY

(b)

Fig. 1 Sine wave characteristics: (a) voltage V changes with time t as (b) amplitude

changes with phase angle 4>.

A sine wave generator produces a voltage that changes in time t as the amplitude V changes with the phase angle 0. shown in Fig. 1. Phase is measured from a zero crossing, as illustrated by plotting the phase angle as the radius vector rotates at a constant angular rate determined by the frequency.

The ideal (perfect) sine-wave-related parameters are as follows: vq, average (nominal) frequency of the signal; \{t), instantaneous frequency of a signal

I d(b (1)

Vq, nominal peak amplitude of a signal source output; x, period of an oscilla- :ion (I/vq); Q, signal (carrier) angular frequency (rate of change of phase

with time) in radians

n = 27rvo

;

(2)

Qf, instantaneous angular frequency; V{t), instantaneous output voltage

of a signal. For the ideal sine wave signal of Fig. 1, in volts.

K(f)= l/osin(27rvof). (3) The basic relationship between phase 0, frequency vq, and time interval

T of the ideal sine wave is given in radians by the following: = 2rtVoT, (4)

where 4>{t) is the instantaneous phase of the signal voltage, V{t), defined for the ideal sine wave in radians as

0(0 = Inv^t. (5)

TN-192
242 A. L. LANCE. WD. SEAL, AND F. LABAAR

The instantaneous phase </>(r) of V(t) for the noisy signal is

0(f) = 27rvof + (/)(t), (6) where 4>(t) is the instantaneous phase fluctuation about the ideal phase

Invot of Eq. (4). The simphfied illustration in Fig. 1 shows the sine-wave signal perturbed

for a short instant by noise. In the perturbed area, the At; and At relation- ships correspond to other frequencies, as shown by the dashed-line waveforms. In this sense, frequency variations (phase noise) occur for a given

instant within the cycle.

The instantaneous output voltage V{t) of a signal generator or oscillator

is now

V(t) = [Fo + £(r)] sin[2rtvof + (/)(0], (7)

where Fq and vq are the nominal amplitude and frequency, respectively, and

£(r) and 0(r) are the instantaneous amplitude and phase fluctuations of the

signal.

It is assumed in Eq. (7) that

£(f)/Fo « 1 and — « 1 for all (t), 0(f) = d<t>/dt. (8) Vo

Equation (7) can also be expressed as

ViT) = iVo + <5£(f)] sin[27rvof + <i>o + «5<A(f)]. (9)

where 0o is a constant, 6 is the fluctuations operator, and deit) and (50(f) represent the fluctuations of signal amplitude and phase, respectively.

Frequency fluctuations ^v are related to phase fluctuations 6<l), in hertz, by

Sv = — = -— , (10)

271 2n at

i.e., radian frequency deviation is equal to the rate of change of phase deviation (the first-time derivative of the instantaneous phase deviation).

The fluctuations of time interval Sx are related to fluctuations of phase

(30, in radians, by

S4> = {2nvo)Sz. (11)

In the following, y is defined as thefractionalfrequencyfluctuation or fractional frequency deviation. It is the dimensionless value of ^v normalized to the average (nominal) signal frequency vq,

V = 5v/vo, (12)

TN-193
7. PHASE NOISE AND AM NOISE MEASUREMENTS 243

where >(f) is the instantaneous fractional frequency deviation from the nominal frequency Vq .

A. Noise Sidebands

Noise sidebands can be thought of as arising from a composite of lowfrequency signals. Each of these signals modulate the carrier-producing

components in both sidebands separated by the modulation frequency, as

illustrated in Fig. 2. The signal is represented by a pair of symmetrical

sidebands (pure AM) and a pair of antisymmetrical sidebands (pure FM).

The basis of measurement is that when noise modulation indices are

small, correlation noise can be neglected. Two signals are uncorrelated if their phase and amplitudes have different time distributions so that they do

not cancel in a phase detector. The separation of the AM and FM components

are illustrated as a modulation phenomenon in Fig. 3. Amplitude fluctuations can be measured with a simple detector such as a crystal. Phase or frequency

fluctuations can be detected with a discriminator. Frequency modulation

(FM) noise or rms frequency deviation can also be measured with an amplitude (AM) detection system after the FM variations are converted to AM variations, as shown in Fig. 3a. The FM-AM conversion is obtained

by applying two signals in phase quadrature (90°) at the inputs to a balanced

mixer (detector). This is illustrated in Fig. 3 by the 90° phase advances of

the carrier.

B. Spectral Density

Stability in the frequency domain is commonly specified in terms of spectral

densities. There are several different, but closely related, spectral densities that are relevant to the specification and measurement of stability of the frequency, phase, period, amplitude, and power of signals. Concise, tutorial

RADIAN FREQUENCY

la) radian frequency

(b)

CARRIER

v,y„

v. Vn RADIAN FREQUENCY

(c)

Fig. 2 (a) Carrier and single upper sideband signals; (b) symmetrical sidebands (pure AM); (c) an antisymmetrical pair of sidebands (pure FM).

TN-194
244 A. L. LANCE, W. D. SEAL, A>fD F. LABAAR

h

-FM VARIATION*

V UPPER SIDEBAND

UPPER I LOWER

SIDEBAND I SIDEBAND AM MEASURED

WITH AM DETECTOR

Vq (CARRIER)

(a)

; Vn(FM)|

(b)

U- CONVERTED -H

I TO FM

I

90° ADVANCE

90° ADVANCE

ADVANCED)

UPPER SIDEBAND

LOWER SIDEBAND

Vq (ADVANCED)

(c) (d)

Fig. 3 (a) Relationships of the FM signal to the carrier; (b) relationship of the AM signal

to the carrier ; (c) carrier advanced 90° to obtain FM-AM conversion ; (d ) AM-FM conversion

.

descriptions of twelve defined spectral densities and the relationships among

them were given by Shoaf ef al. (1973) and Halford et al. (1973). Recall that in the perturbed area of the sine wave in Fig. 1 the frequencies

are being produced for a given instant of time. This amount of time the signal

spends in producing another frequency is referred to as the probability

density of the generated frequencies relative to Vq. The frequency domain

plot is illustrated in Fig. 4. A graph of these probability densities over a period

of time produces a continuous line and is called the Power spectral density.

TN-195
7. PHASE NOISE AND AM NOISE MEASUREMENTS 245

Fig. 4 A power density plot

The spectral density is the distribution of total variance over frequency. The

units of power spectral density are power per hertz : therefore, a plot of power

spectral density obtained from amplitude (voltage) measurements requires

that the voltage measurements be squared.

The spectral density of power versus frequency, shown in Fig. 4, is a fwosided spectral density because the range of Fourier frequencies / is from

minus infinity to plus infinity.

The notation S,(/) represents the two-sided spectral density of fluctations of any specified time-dependent quantity ^(0- Because the frequency band

is defined by the two limit frequencies of minus infinity and plus infinity, the total mean-square fluctuation of that quantity is defined by

'sideband S,{f)df. (13)

Two-sided spectral densities are useful mainly in pure mathematical analysis

involving Fourier transformations.

Similarly, for the one-sided spectral density,

G.ideb.nd = S,(/) df. (14)

(15)

The two-sided and one-sided spectral densities are related as follows:

/•+ao .«+ao /•^oo S,,df = 2\ S,,df=\ S,,df,

J - 3> Jo Jo where 9, indicates one-sided and Qi two-sided spectral densities. It is noted

that the one-sided density is twice as large as the corresponding two-sided

TN-196
246 A. L. LANCE, W. D. SEAL, AND F. LABAAR

spectral density. The terminology for single-sideband versus double-sideband

signals is totally distinct from the one-sided spectral density versus twosided spectral terminology. They are totally different concepts. The defini- tions and concepts of spectral density are set forth in NBS Technical Note

632 (Shoaf et a/., 1973).

C. Spectral Densities of Phase Fluctuations in the

Frequency Domain

The spectral density Sy{f) of the instantaneous fractional frequency

fluctuations y(f) is defined as a measure of frequency stability, as set forth by Barnes et al. (1970). Sy{f) is the one-sided spectral density of frequency

fluctuations on a "per hertz" basis, i.e., the dimensionality is Hz"'. The

range of Fourier frequency / is from zero to infinity. S^vC/X in hertz squared

per hertz, is the spectral density offi-equency fluctuations Sv. It is calculated

as

''' bandwidth used in the measurement of ^v^^

The range of the Fourier frequency / is from zero to infinity. The spectral density of phasefluctuations is a normalized frequency domain

measure of phase fluctuation sidebands. S^Jif), in radians squared per

hertz, is the one-sided spectral density of the phase fluctuations on a "per

hertz" basis:

5 (/) =

^'Prmi /jyx

** bandwidth used in the measurement of ^0rms

The power spectral densities of phase and frequency fluctuation are related by

5,^(/) = (vi'/)5,(/). (18)

The range of the Fourier frequency / is from zero to infinity. ^snif)- '" radians squared Hertz squared per hertz is the spectral density

of angular frequency fluctuations SCI:

Sioif) = {2nySM)- (19)

The defined spectral densities have the following relationships:

SsAf) = V5S,,(/) = (l/27r)^S,n(/) = f'S.^if)- (20)

SM) = ihcofS,aif) = (vo//)^S,(/) = ISUfVf'l (21)

Note that Eq. (20) is hertz squared per hertz, whereas Eq. (21) is in radians

squared per hertz. The term S^,7fp(v), in watts per hertz, is the spectral density of the (square

* See Appendix Note # 30

TN-197
7. PHASE NOISE AND AM NOISE MEASUREMENTS 247

root oO radio frequency power P. The power of a signal is dispersed over the frequency spectrum owing to noise, instability, and modulation. This concept is similar to the concept of spectral density of voltage fluctuations

Sivif)- Typically, S^^if) is more convenient for characterizing a baseband

signal where voltage, rather than power, is relevant. S^/^Cv) is typically more convenient for characterizing the dispersion of the signal power in the vicinity of the nominal carrier frequency v^. To relate the two spectral

densities, it is necessary to specify the impedance associated with the signal.

A definition of frequency stability that relates the actual sideband power

of phase fluctuations with respect to the carrier power level, discussed by

Glaze (1970), is called if(/). For a signal with PM and with no AM, ^(/)

is the normalized version of S^r^{v), with its frequency parameter / refer- enced to the signal's average frequency vg as the origin such that / equals

V - vq. If the signal also has AM, ^{f) is the normalized version of those

portions of S^ffp(v) that are phase-modulation sidebands. Because / is the Fourier frequency difference (v - vg), the range of/ is from minus Vq to plus infinity. Since ^if) is a normalized density (phase

noise sideband power).

J. "if(/)4/- = l. (22) *

if(/) is defined as the ratio of the power in one sideband, referred to the input carrier frequency on a per hertz of bandwidth spectral density basis, to the total signal power, at Fourier frequency difference / from the carrier, per one device. It is a normalized frequency domain measure of phase fluctua- tion sidebands, expressed in decibels relative to the carrier per hertz:

power density (one phase modulation sideband)

carrier power

For the types of signals under consideration, by definition the two phasenoise sidebands (lower sideband and upper sideband, at -/ and / from vq, respectively) of a signal are approximately coherent with each other, and

they are of approximately equal intensity.

It was previously show that the measurement of phase fluctuations (phase

noise) required driving a double-balanced mixer with two signals in phase

quadrature so the FM-AM conversion resulted in voltage fluctuations at the mixer output that were analogous to the phase fluctuations. The operation of the mixer when it is driven at quadrature is such that the amplitudes

of the two phase sidebands are added linearly in the output of the mixer,

resulting in four times as much power in the output as would be present if only one of the phase sidebands were allowed to contribute to the output

* See Appendix Note #31

TN-198
248 A. L. LANCE, W. D. SEAL, AND F. LABAAR

of the mixer. Hence, for | / 1 < vq , and considering only the phase modulation

portion of the spectral density of the (square root of) power, we obtam

SivilfWiV^,)' S 4[S^Vo + /)]/(P,o,) (24)

and, using the definition of if(/),

^if) = [S^( vo + /)]/(P.o.) = iS,^(| / 1). (25)

Therefore, for the condition that the phase fluctuations occurring at rates (/) and faster are small compared to one radian, a good approximation in radians squared per hertz for one unit is ^(/) = iSUf). (26)

If the small angle condition is not met, Bessel-function algebra must be used

to relate ^(f) to S,^(/).

The NBS-defined spectral density is usually expressed in decibels relative to the carrier per hertz and is calculated for one unit as ^{f) = 10 log[i5,^(/)]. (27)

It is very important to note that the theory, definitions, and equations previously

set forth relate to a single device.

D. Modulation Theory and Spectral Density

Relationships

Applying a sinusoidal frequency modulation /„, to a sinusoidal carrier frequency vq produces a wave that is sinusoidally advanced and retarded

in phase as a function of times. The instantaneous voltage is expressed as,

F(f) = Vq sin(27rvot + A0 sin Inf^t), (28)

where A0 is the peak phase deviation caused by the modulation signal.

The first term inside the parentheses represents the linearly progressing

phase of the carrier. The second term is the phase variation (advancing and

retarded) from the linearly progressing wave. The effects of modulation can

be expressed as residual /„, noise or as single-sideband phase noise. For

modulation by a single sinusoidal signal, the peak-frequency deviation of

the carrier (vq) is Avo = A(/)/„, (29)

A0 = Avo/7„, (30)

where [^ is the modulation frequency. This ratio of peak frequency deviation

to modulation frequency is called modulation index m so that A0 = m and

m = Avo;/„. (31)

TN-199
7. PHASE NOISE AND AM NOISE MEASUREMENTS 249

The frequency spectrum of the modulated carrier contains frequency

components (sidebands) other than the carrier. Fbr small values of modulation mdex (m 4 1), as is the case with random phase noise, only the carrier and first upper and lower sidebands are significantly high in energy. The

j-atio of the amplitude of either single sideband to the amplitude of the

carrier is K./K = ma. (32)

This ratio is expressed in decibels below the carrier and is referred to as dBc

for the given bandwidth B: VJVo = 20 log(m/2) = 20 log(Avo/2/„) = 10 log(m/2)^ = 10 log(Avo/2/J'. (33)

If the frequency deviation is given in terms of its rms value, then

Av,„3 = Avo/v2. (34)

Equation (33) now becomes

Kb/^'o = ^(/) = 20 log(Av,^v 2^)

,35. = 101og(Av,„^v2/m)'-

The ratio of single sideband to carrier power in decibels (carrier) per hertz is

if(/) = 20 log(Av,„^//J - 3 (36)

and, in decibels relative to one squared radian per hertz,

S,^if) = 20 log(Av,^/„). (37)

The interrelationships of modulation index, peak frequency deviation, rms frequency, and spectral density of phase fluctuations can be found from

the following:

im = Avo/2/„ = Av,„,/V2i;;, (38) = 10exp(i?(/)/10) = iS,^(/); (39)

or

and

im = Av,^/vX = v 10 exp(if(/)/ 10) = Vi5,*(/), (40)

m = Avo//TM = 2Av,„^v'^ = 2VlOexp(if(/)/10) = 2v^W7)- (41)

The basic relationships are plotted in Fig. 5.

TN-200
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TN-201
7. PHASE NOISE AND AM NOISE MEASUREMENTS 251

E. Noise Processes The spectral density plot of a typical oscillator's output is usually a combination of different noise processes. It is very useful and meaningful to categorize these processes because the first job in evaluating a spectral

density plot is to determine which type of noise exists for the particular range of Fourier frequencies.

The two basic categories are the discrete-frequency noise and the power-law

noise process. Discrete-frequency noise is a type of noise in which there is a dominant observable probability, i.e., deterministic in that they can usually

be related to the mean frequency, power-line frequency, vibration frequencies, or ac magnetic fields, or to Fourier components of the nominal frequency.

Discrete-frequency noise is illustrated in the frequency domain plot of Fig. 6. These frequencies can have their own spectral density plots, which can be

defined as noise on noise. Power-law noise processes are types of noise that produce a certain slope on the one-sided spectral density plot. They are characterized by their dependence on frequency. The spectral density plot of a typical oscillator output is usually a combination of the various power-law processes.

In general, we can classify the power-law noise processes into five categories. These five processes are illustrated in Fig. 5, which can be referred to with

respect to the following description of each process.

(1) Random walk FM (random walk of frequency). The plot goes down as

1//*. This noise is usually very close to the carrier and is difficult to measure.

It is usually related to the oscillator's physical environment (mechanical

shock, vibration, temperature, or other environmental effects).

Frequency

Flu. 6 A basic discrete-frequency signal display.

TN-202
252 A. L. LANCE, W. D. SEAL, AND F. LABAAR

(2) Flicker FM (flicker of frequency). The plot goes down as 1//^. This

noise is typically related to the physical resonance mechanism of the active

oscillator or the design or choice of parts used for the electronic or power

supply, or even environmental properties. The time domain frequency

stability over extended periods is constant. In high-quality oscillators, this noise may be marked by white FM (1//^) or flicker phase modulation

0Af (1,/). It may be masked by drift in low-quality oscillators.

(3) White FM (white frequency, random walk of phase). The plot goes down as 1//^ A common type of noise found in passive-resonator frequency

standards. Cesium and rubidium frequency standards have white FM noise

characteristic because the oscillator (usually quartz) is locked to the reso- nance feature of these devices. This noise gets better as a function of time

until it (usually) becomes flicker FM (1//^) noise.

(4) Flicker (pM (flicker modulation of phase). The plot goes down as

\/f. This noise may relate to the physical resonance mechanism in an oscillator. It is common in the highest-quality oscillators. This noise can be

introduced by noisy electronics— amphfiers necessary to bring the signal

amplitude up to a usable level —and frequency multipliers. This noise can

be reduced by careful design and by hand-selecting all components.

(5) White (f)M (white phase). White phase noise plot is flat/°. Broadband

phase noise is generally produced in the same way as flicker <pM (1//). Late

stages of amplification are usually responsible. This noise can be kept low

by careful selection of components and by narrow-band filtering at the output

The power-law processes are illustrated in Fig. 5.

F. Integrated Phase Noise

The integrated phase noise is a measure of the phase-noise contribution

(rms radians, rms degrees) over a designated range of Fourier frequencies.

The integration is a process of summation that must be performed on the measured spectral density within the actual IF bandwidth (B) used in the measurement of S(p,„^. Therefore, the spectral density S,^(/) must be un- normalized to the particular bandwidth used in the measurement. Define

Su(/), m radians squared, as the unnormalized spectral density:

S„(/) = 2[10 exp(if(/) + 10 log B)/10]. (42)

Then, the integrated phase noise over the band of Fourier frequencies (/, to/„)

where measurements are performed using a constant IF bandwidth, in radians

squared, is

Sb(/. to/„)= f"s„(/)d/, (43)

TN-203
7. PHASE NOISE AND AM NOISE MEASUREMENTS 253

or, in rms radians,

Sb(/i to /J = vT 5u(/.) + 5„(/2) = • • • = S„(/J, (44)

and the integrated phase noise in rms degrees is calculated as

SB(360/2rt). (45)

The integrated phase noise in decibels relative to the carrier is calculated as

5b = 10 log(i Si). (46)

The previous calculations correspond to the illustration in Fig. 7, which

includes two bandwidths (Bl and B2) over two ranges of Fourier frequencies.

In the measurement program, different IF bandwidths are used as set forth by Lance et al. (1977). The total integrated phase noise over the differ- ent ranges of Fourier frequencies, which are measured at constant bandwidths as illustrated, is calculated in rms radians as follows:

^Blol — V (^Bl) + (^82) + + (^Bn) 1 (47)

where it is recalled that the summation is performed in terms of radians

squared.

CONSTANT BANDWIDTH

B2

Fig. 7 Integrated phase noise over Fourier frequency ranges at which measurements

were performed using constant bandwidth. '

TN-204
254 A. L. LANCE, W. D. SEAL, AND F. LABAAR

G. AM Noise in the Frequency Domain

The spectral density of AM fluctuations of a signal follows the same

general derivation previously given for the spectral density of phase fluctations. Amplitude fluctuations Se of the signal under test produces voltage

fluctuations SA at the output of the mixer. Interpretating the mean-square

fluctuations de and d in spectal density fashion, we obtain Ss^fX the spectral

density of amphtude fluctuations Se of a signal in volts squared per hertz:

SsAf) = (i K)Ks»AmA,^n (48)

The term ««{/) is the normalized version of the amplitude modulation (AM)

portion of Sj-^(\'X with its frequency parameter / referenced to the signal's

average frequency vq, taken as the origin such that the diff"erence frequency

/ equals v - vq . The range of Fourier frequency difference / is from minus

Vq to plus infinity.

The term Mf) is defined as the ratio of the spectral density of one amplitudemodulated sideband to the total signal power, at Fourier frequency difference /from the signal's average frequency Vq, for a single specified signal or device. The dimensionality is per hertz. if(/) and ««(/) are similar functions; the former is a measure of phase-modulated (PM) sidebands, the later is a cor- responding measure of amplitude-modulated (AM) sidebands. We introduce

the symbol ^{f) to have useful terminology for the important concept of

normalized AM sideband power.

For the types of signals under consideration, by definition the two amplitude-fluctuation sidebands (lower sideband and upp)er sideband, at -/

/ from Vo, respectively) of a signal are coherent with each other. Also, they

are of equal intensity. The operation of the mixer when it is driven at colinear

phase is such that the amplitudes of the two AM sidebands are added linearly

in the output of the mixer, resulting in four times as much power in the output

as would be present if only one of the AM sidebands were allowed to contri- bute to the output of the mixer. Hence, for |/| < Vq,

Sa(I / l)/(ATM,)^ = 4[S,7f?(vo + /)]/-P,o., (49)

and, using the definitionMf) = [5,rfF(vo + /)]/^.o., (50)

we find, in decibels (carrier) per hertz,

^(/) = (l/2Fo^)S,,(|/|). (51)

TN-205
PHASE NOISE AND AM NOISE MEASUREMENTS 255

III. Phase-Noise Measurements Using the Two-Oscillator

Technique

A functional block diagram of the two-oscillator system for measuring

phase noise is shown in Fig. 8. NBS has performed phase noise measurements

since 1967 using this basic system. The signal level and sideband levels can

be measured in terms of voltage or power. The low-pass filter prevents local oscillator leakage power from overloading the spectrum analyzer when

baseband measurements are performed at the Fourier (offset) frequencies

of interest. Leakage signals will interfere with autoranging and with the dynamic range of the spectrum analyzer.

The low-noise, high-gain preamplifier provides additional system sensitivity by amplyfying the noise signals to be measured. Also, because spectrum

analyzers usually have high values of noise figure, this amplifier is very desirable. As an example, if the high-gam preamplifier had a noise figure of

3 dB and the spectrum analyzer had a noise figure of 18 dB, the system sensitivity at this point has been improved by 15 dB. The overall system sensitivity

would not necessarily be improved 15 dB in all cases, because the limiting

sensitivity could have been imposed by a noisy mixer.

NOISY SIGNAL

PHASE SHIFTER

SMALL

FLUCTUATIONS

(THIS IS THE QUANTITY

TO BE MEASURED)

'tMMMHWMMVM'tMl^

NOISE ONLY

S«o('l

A^yA LOW-PASS _J ^""^S *

MIXER \/\j^ FILTER |^

Q-^

SPECTRUM ANALYZER

LOW-NOISE

AMPLIFIER

REFERENCE NO NOISE OSCILLATOR

Fig. 8 The two-oscillator technique for measuring phase noise. Small fluctuations from nommal voltage are equivalent to phase variations. The phase shifter adjusts the two signals

to quadrature in the mixer, which cancels carriers and converts phase noise to fluctuating dc

voltage.

* See Appendix Note # 6

TN-206
256 A. L. LANCE, W. D. SEAL, AND F. LABAAR

Assume that the reference oscillator is perfect (no phase noise), and that it can be adjusted in frequency. Also, assume that both oscillators are extremely

stable, so that phase quadrature can be maintained without the use of an

external phase-locked loop or reference. The double-balanced mixer acts as a phase detector so that when two input signals are identical in frequency

and are in phase quadrature, the output is a small fluctuating voltage. This

represents the phase-modulated (PM) sideband component of the signal

because, due to the quadrature of the signals at the mixer input, the mixer

converts the amplitude-modulated (AM) sideband components to FM, and

at the same time it converts the PM sideband components to AM. These

AM components can be detected with an amplitude detector, as shown in Fig. 3. If the two oscillator signals applied to the double-balanced mixer of Fig. 8 are slightly out of zero beat, a slow sinusoidal voltage with a peak-to-peak

voltage Kp,p can be measured at the mixer output. If these same signals are returned to zero beat and adjusted for phase quadrature, the output of

the mixer is a small fluctuating voltage (Sv) centered at zero volts. If the

fluctuating voltage is small compared to ^ Kp,p, the phase quadrature condition is being closely maintained and the "small angle" condition is being

met. Phase fluctuations in radians between the test and reference signals

(phases) are

S4> = 6{(t>, - 4>r). (52)

These phase fluctuations produce voltage fluctuations at the output of the mixer,

where phase angles are in radian measure and sin 50 = 54) for small b<i> {S4> <^ 1 rad). Solvmg for <30, squarmg both sides, and taking a time average

gives

iiS<t>y} = 4<(5r)^>/(Kp,p)2, (54)

where the angle brackets represent the time average. For the sinusoidal beat signal,

(^p,p)' = 8(FTMJ^ (55)

The mean-square fluctuations of phase 64> and voltage Sv interpreted in a spectral density fashion gives the following in radians squared per hertz: SM) = SM)/2{Kms)'- (56)

Here, Si,.(/), in volts squared pier hertz, is the spectral density of the voltage

fluctuations at the mixer output. Because the spectrum analyzer measures

TN-207
7. PHASE NOISE AND AM NOISE MEASUREMENTS 257

rms voltage, the noise voltage is in units of volts per square root hertz, which means volts per square root bandwidth. Therefore,

5J/) = [^tV.s,vfi] = iSv.n.s)'.B, (57)

where B is the noise power bandwidth used in the measurement.

Because it was assumed that the reference oscillator did not contribute any

noise, the voltage fluctuations f^ms represent the oscillator under test, and

the spectral density of the phase fluctuations in terms of the voltage measure- ments performed with the spectrum analyzer, in radians squared per hertz, is Ss/f) = i[(5t;_)^'B(I/„J^]. (58)

Equation (46) is sometimes expressed as W/) = 5,,(/)/K^ (59)

where K is the calibration factor in volts per radian. For sinusoidal beat

signals, the peak voltage of the signal equals the slope of the zero crossing

in volts per radian. Therefore, (Vp)^ = 2{V,„^)^, which is the same as the denominator in Eq. (56). The term S^^if) can be expressed in decibels relative to one square radian

per hertz by calculating 10 log S^^if) of the previous equation:

S,^(/) = 20 log(^i;,„3) - 20 log(K,„3) - 10 log(fi) - 3 . (60)

A correction of 2.5 is required for the tracking spectrum analyzer used in these measurement systems, if(/) differs by 3 dB and is expressed in decibels

(carrier) per hertz as ^if) = 20 logiSv,^,) - 20 log(K^) - 10 log(B) - 6. (61)

A. Two Noisy Oscillators

The measurement system of Fig. 6 yields the output noise from both

oscillators. If the reference oscillator is superior in performance as assumed

in the previous discussions, then one obtains a direct measure of the noise

characteristics of the oscillator under test.

If the reference and test oscillators are the same type, a useful approximation is to assume that the measured noise power is twice that associated

with one noisy oscillator. This approximation is in error by no more than

3 dB for the noisier oscillator, even if one oscillator is the major source of

noise. The equation for the spectral density of measured phase fluctuations

in radians squared per hertz is W/) + s,Jf) V(/)| -2(1/^)^^ rms/

|(iwo devices) 2Ss,if)(onedevicel (62)

TN-208 I
258 A. L. LANCE, W. D. SEAL. AND F. LABAAR

The measured value is therefore divided by two to obtain the value for the

single oscillator. A determination of the noise of each oscillator can be made

if one has three oscillators that can be measured in all pair combinations.

The phase noise of each source 1, 2, and 3 is calculated as follows:

<£,{f) = 10 logQ(10^'^'^"° + io^.3(/);io _ iQif23(/)/iO)-]^ (63)

^\f) = 10 log[i(10^'^<^"° + 10^"'^"'° - 10^''*^''°)], (64)

S^Hf) = 10 log[i(10^'^'^' '° + io^23(/)/io _ io^.^»/)"0)]. (65)

B. Automated Phase-Noise Measurements Using the

Two-Oscillator Technique

The automated phase-noise measurement system is shown in Fig. 9. It is controlled by a programmable calculator. Each step of the calibration and

measurement sequence is included in the program. The software program

controls frequency slection, bandwidth settings, settling time, amplitude

ranging, measurements, calculations, graphics, and data plotting. Normally,

the system is used to obtain a direct plot of if(/). The integrated phase noise can be calculated for any selected range of Fourier frequencies.

A quasi-continuous plot of phase noise performance £^if) is obtained

by performing measurements at Fourier frequencies separated by the IF bandwidth of the spectrum analyzer used during the measurement. Plots of

other defined parameters can be obtained and plotted as desired. The IF bandwidth settings for the Fourier (offset) frequency-range

selections are shown in the following tabulation:

IF Fourier IF Fourier bandwidth frequency bandwidth frequency (Hz) (kHz) (kHz) (kHz)

3 0.001-0.4 1 40-100

10 0.4-1 3 100-400

30 1-4 10 400-1300

100 4-10

200 10-^

The particular range of Fourier frequencies is limited by the particular

spectrum analyzer used in the system. A fast Fourier analyzer (FFT) is also incorporated in the system to measure phase noise from submillihertz to 25 kHz.

High-quality sources can be measured without multiplication to enhance

the phase noise prior to downconverting and measuring at baseband fre- quencies. The measurements are not completely automated because the calibration sequence requires several manual operations.

TN-209
7. PHASE NOISE AND AM NOISE MEASUREMENTS 259

OSCILLATOR UNDER TEST

_/^ A VARIABL ~V ^~^J ATTENUA ENUATOR

'reference'

'freQuencyI

I _ - - -I

DOUB

BALA

MIXERr°0^

oscilloscope

reference oscillator ^variable attenuator

LOW-PASS

-TER

PREAMPLIFIER

PRECISION

STEP ATTENUATOR LOW NOISE HIGH GAIN

PLOTTER

L <.'

PHASE-LOCKED LOOP * s. '

FFT ANALYZER

TRACKING SPECTRUM ANALYZER

CALCULATOR AND

PRINTER

c.

Fig. 9 An automated phase-noise measurement system.

Calibration and Measurements Using the

Two-Oscillator System

if(/) is a normalized frequency domain measure of phase-fluctuation side- band power. The noise power is measured relative to the carrier power

level. Correction must be applied because of the type of measurement and

the characteristics of the measurement equipment. The general procedure

for the calibration and measurement sequence includes the following:

measuring the noise power bandwidth for each IF bandwidth setting on the Tracking Spectrum Analyzer (Section III.C.l); establishing a carrier refer- ence power level referenced to the output of the mixer (Section III.C.2); obtaining phase quadrature of the two signals applied to the mixer (Section

III.C.3); measuring the noise power at the selected Fourier frequencies

(Section III.C.4); performing the calculations and plotting the data (Section

III.C.5); and measuring the system noise floor characteristics, usually re- ferred to as the system sensitivity.

1 . Noise-Power Bandwidth

Approximations of analyzer-noise bandwidths are not adequate for phase

noise measurements and calculations. The IF noise-power bandwidth of the tracking spectrum analyzer must be known and used in the calculations of

phase noise parameters. Figure 10 shows the results of measurements per- formed using automated techniques. For example, with a 1-MHz signal

input to the tracking spectrum analyzer, the desired incremental frequency

changes covering the IF bandwidth are set by calculator control.

TN-210
260 A. L. LANCE, W. D. SEAL, AND F. LABAAR

dB

FREQUENCY

Fig. 10 Plot of automated noise-power bandwidth.

The spectrum analyzer power output is recorded for each frequency setting over the range, as illustrated in Fig. 10. The 40-dB level and the 100 incre- ments in frequency are not the minimum permissible values. The recorded can be plotted for each IF bandwidth, as illustrated, and the noise-power

bandwidth is calculated in hertz as

noise power bandwidth =

(Pi + P2 + ^3 + •• + P'°°)A/

peak power reading

(66)

where A/ is the frequency increment in hertz and the peak power is the maximum measured point obtained during the measurements. All power values

are in watts.

2. Setting the Carrier Power Reference Level

Recall from Section III.6 that for sinusoidal signals the peak voltage of

the signal equals the slope of the zero crossing, in volts per radian. A frequency

offset is established, and the peak pov/er of the difference frequency is measured as the carrier-power reference level; this establishes the calibration

factor of the mixer in volts per radian. Because the precision IF attenuator is used in the calibration process, one must be aware that the impedance looking back into the mixer should be

50 Q. Also, the mixer output signal should be sinusoidal. Fischer (1978)

discussed the mixer as the "critical element" in the measurement system.

It is advisable to drive the mixer so that the sinusoidal signal is obtained

at the mixer output. In most of the TRW systems, the mixer drive levels are 10 dBm for the reference signal and about zero dBm for the unit under test.

TN-211
7. PHASE NOISE AND AM NOISE MEASUREMENTS 261

System sensitivity can be increased by driving the mixer with high-level

signals that lower the mixer output impedance to a few ohms. This presents

a problem in establishing the calibration factor of the mixer, because it might be necessary to calibrate the mixer for different Fourier frequency

ranges. The equation sensitivity = slope = beat-note amplitude does not hold

if the output of the mixer is not a sine wave. The Hewlett-Packard 3047

automated phase noise measurement system allows accurate calibration

of the phase-detector sensitivity even with high-level inputs by using the derivative of the Fourier representation of the signal (the fundamental and

its harmonics). The slope at s radians is given by

A sm(t> — B sin 30 -f- C sin 5<t> = A cos 4> - 'iB cos 30 + 5C cos 50 = A - 3B + 5C +

.

(67)

Referring to Fig. 9, the carrier-power reference level is obtained as follows.

(1) The precision IF step attenuator is set to a high value to prevent

overloading the spectrum analyzer (assume 50 dB as our example).

(2) The reference and test signals at the mixer inputs are set to approximately 10 dBm and dBm, as previously discussed.

(3) If the frequency of one of the oscillators can be adjusted, adjust its frequency for an IF output frequency in the range of 10 to 20 kHz. If neither

oscillator is adjustable, replace the oscillator under test with one that can

be adjusted as required and that can be set to the identical power level of the

oscillator under test. (4) The resulting IF power level is measured by the spectrum analyzer,

and the measured value is corrected for the attentuator setting, which was assumed to be 50 dB. The correction is necessary because this attenuator

will be set to its zero decibel indication during the measurements of noise

power. Assuming a spectrum analyzer reading of -40 dBm, the carrier- power reference level is calculated as

carrier power reference level = 50 dB — 40 dBm = 10 dBm. (68)

3. Phase Quadrature of the Mixer Input Signals

After the carrier-power reference has been established, the oscillator under

test and the reference oscillator are tuned to the same frequency, and the

original reference levels that were used during calibration are reestablished. The quadrature adjustment depends on the type of system used. Three

possibilities, illustrated in Fig. 9, are described here.

( I) If the oscillators are very stable, have high-resolution tuning, and are not phase-locked, the frequency of one oscillator is adjusted for zero dc

TN-212
262 A. L. LANCE, W. D. SEAL, AND F. LABAAR

voltage output of the mixer as indicated by the sensitive oscilloscope.

Note: Experience has shown that the quadrature setting is not critical if the sources have low AM noise characteristics. As an example, experiments

performed using two HP 3335 synthesizers showed that degradation of the phase-noise measurement became noticeable with a phase-quadrature

offset of 16 degrees.

(2) If the common reference frequency is used, as illustrated in Fig. 9, then

it is necessary to include a phase shifter in the hne between one of the oscil- lators and the mixer (preferably between the attenuator and mixer). The

phase shifter is adjusted to obtain and maintain zero volts dc at the mixer

output. A correction for a nonzero dc value can be applied as exemplified by

the HP 3047 automated phase-noise measurement system.

(3) If one oscillator is phase-locked using a phase-locked loop, as shown

dotted in on Fig. 9, the frequency of the unit under test is adjusted for zero dc output of the mixer as indicated on the oscilloscope.

A phase-locked loop is a feedback system whose function is to force a

voltage-controlled oscillator (VCO) to be coherent with a certain frequency,

i.e., it is highly correlated in both frequency and phase. The phase detector

is a mixer circuit that mixes the input signal with the VCO signal. The mixer

output is \\ ± Vq. when the loop is locked, the VCO duplicates the input

frequency so that the difference frequency is zero, and the output is a dc

voltage proportional to the phase difference. The low-pass filter removes

the sum frequency component but passes the dc component to control the VCO. The time constant of the loop can be adjusted as needed by varying

amplifier gain and RC filtering within the loop.

A loose phase-locked loop is characterized by the following.

(1) The correction voltage varies as phase (in the short term) and phase

variations are therefore observed directly.

(2) The bandwidth of the servo response is small compared with the Fourier frequency to be measured.

(3) The response time is very slow. A tight phase-locked loop is characterized by the following.

( 1

)

The correction voltage of the servo loop varies as frequency.

(2) The bandwidth of the servo response is relatively large.

(3) The response time is much smaller than the smallest time interval t

at which measurements are performed.

Figure 1 1 shows the phase-noise characteristics of the H.P. 8640B synthesizer measured at 512 MHz. The phase-locked-loop attenuation characteristics extend to 10 kHz. The internal-oscillator-source characteristics are

plotted at Fourier frequencies beyond the loop-bandwidth cutoff at 10 kHz.

TN-213
PHASE NOISE AND AM NOISE MEASUREMENTS 263

512-MHz OSCILLATOR

FOURIER FREQUENCY (Hz)

Fig. 1 1 Phase-locked-loop characteristics of the H.P. 8640B signal generator, showing

the normahzed phase-noise sideband power spectral density.

4. Measurements, Calculations, and Data Plots The measurement sequence is automated except for the case where manual

adjustments are required to mamtam phase quadrature of the signals. After phase quadrature of the signals into the mixer is established, the IF atten- uator is returned to the zero-decibel reference setting. This attenuator is set to a high value [assumed to be 50 dB in Eq. (65)] to prevent saturation

of the spectrum analyzer during the calibration process. The automated measurements are executed, and the direct measurement

and data plot of S^{f) is obtained in decibels (carrier) per hertz using the equation

S£{f) = — [carrier power level — (noise power level — 6 + 2.5 - 10 log B - 3)]. (69)

The noise power (dBm) is measured relative to the carrier-power level (dBm), and the remaining terms of the equation represent corrections that must be applied because of the type of measurement and the characteristics of the measurement equipment, as follows.

(1) The measurement of noise sidebands with the signals in phase

quadrature requires the -6-dB correction that is noted in Eq. (69). (2) The nonhnearity of the spectrum analyzer's logarithmic IF amplifier

results in compression of the noise peaks which, when average-detected,

require the 2.5-dB correction.

(3) The bandwidth correction is required because the spectrum analyzer

measurements of random or white noise are a function of the particular

bandwidth used in the measurement.

TN-214 i
264 A. L. LANCE, W. D. SEAL, AND F. LABAAR

(4) The - 3-dB correction is required because this is a direct measure of

if(/) of two oscillators, assuming that the oscillators are of a similar type

and that the noise contribution is the same for each oscillator. If one oscil- lator is sufficiently superior to the other, this correction is not required.

Other defined spectral densities can be calculated and plotted as desired. The plotted or stored value of the spectral density of phase fluctuations

in decibels relative to one square radian (dBc rad^/Hz) is calculated as SM) = ^(/) + 3. (70)

The spectral density of phase fluctuations, in radians squared per hertz, is calculated as S,^(/)= 10exp(S,^(/)/10), (71)

The spectral density of frequency fluctuations, in hertz squared per hertz, is

SsXf) = PSs^if). (72)

where S^^iF) is in decibels with respect to 1 radian.

5. System Noise Floor Verification A plot of the system noise floor (sensitivity) is obtained by repeating the automated measurement procedures with the system modified as shown in Fig. 12. Accurate measurements can be obtained using the configuration

shown in Fig. 12a. The reference source supplies 10 dBm to one side of the mixer and dBm to the other mixer input through equal path lengths;

phase quadrature is maintained with the phase shifter. G- © 50-

n

TERMINATION

OdBm

10 dBm

PHASE

SHIFTER &

nOdSn

(b)

Fig. 12 System configurations for measuring the system noise floor (sensitivity): (a) configuration used for accurate measurements; (b) alternate configuration sometimes used.

TN-215
7. PHASE NOISE AND AM NOISE MEASUREMENTS 265

The configuration shown in Fig. 12b is sometimes used and does not

greatly degrade the noise floor because the reference signal of 10 dBm is larger than the signal frequency. See Sections IV. B and IV. C. 4 for additional

discussions related to system sensitivity and recommended system evaluation.

Proper selection of drive and output termination of the double-balanced

mixer can result in improvement by 15 to 25 dB in the performance of phasenoise measurements, as discussed by Walls et al. (1976). The beat frequency

between the two oscillators can be a sine wave, as previously mentioned,

with proper low drive levels. This requires a proper terminating impedance

for the mixer. With high drive levels, the mixer output waveform will be

clipped. The slope of the clipped waveform at the zero crossings, illustrated by Walls et al. (1976), is twice the slope of the sine wave and therefore improves the noise floor sensitivity by 6 dB, i.e., the output signal, proportional

to the phase fluctuations, increases with drive level. This condition of clipping

requires characterization over the Fourier frequency range, as previously

mentioned for the Hewlett-Packard 3047 phase noise measurement system.

An amplifier can be used to increase the mixer drive levels for devices that have insufficient output power to drive the double-balanced mixers.

Lower noise floors can be achieved using high-level mixers when available

drive levels are sufficient. A step-up transformer can be used to increase the mixer drive voltage because the signal and noise power increase in the same ratio, and the spectral density of phase of the device under test is un- changed, but the noise floor of the measurement system is reduced.

Walls et al. (1976) used a correlation technique that consisted primarily

of two phase-noise measurement systems. At TRW the technique is used as shown in Fig. 13. The cross spectrum is obtained with the fast Fourier

transform (FFT) analyzer that performs the product of the Fourier trans-- form of one signal and the complex conjugate of the Fourier transform of

Fig. 13 Cross-spectrum measurement using the two-oscillator technique.

TN-216
266 A. L. LANCE, W. D. SEAL, AND F. LABAAR

the second signal. This cross spectrum, which is a phase-sensitive characteristic, gives a phase and amplitude sensitivity measure directly. A signalto-noise enhancement greater than 20 dB can be achieved.

If the double-balanced phase noise measurement system does not provide

a noise floor suflScient for measuring a high-quality source, frequency

multiplier chains can be used if their inherent noise is 10-20 dB below the measurement system noise. In frequency multiplication the noise increases

according to

10 log(final frequency/original frequency). (73) *

o

a.O

z

<

ill ^

-140 -

-160 -

NOISE

-180 - FLOOR

FOURIER FREQUENCY (Hz)

lal

= -120 -

Fig. 14 Data plots of the automated phase-noise measurement system: (a) a high quality 5-MHz quartz oscillator; (b) combmed noise of two H.P. 8662A synthesizers (subtract 3 dB

for a single unit).

* See Appendix Note # 32

TN-217
7. PHASE NOISE AND AM NOISE MEASUREMENTS 267

The following equation is used to correct for noise-floor contribution

Pnf, in dBc/Hz, if desired or necessary:

^(/Xcorrected) = -^{f) + 10 log (74)

The correction for noise-floor contribution can also be obtained by using

the measurement oiS^^ij) of Eq. (57). Measurement of Si,.(y ) of the oscillator plus floor is obtained, then S^Xf) 'S obtained for the noise floor only. Then,

SJ/) = SM) - SsSf)

(05c + nf

)

(75)

Figure I4a shows a phase noise plot of a very high-quality (5-MHz)

quartz oscillator, measured by the two-oscillator technique. The sharp

peaks below 1000 Hz represent the 60-Hz line frequency of the power supply

and its harmonics and are not part of the oscillator phase noise. Figure 14b

shows measurements to 0.02 Hz of the carrier at a frequency of 20 MHz.

IV. Single-Oscillator Phase-Noise Measurement Systems and

Techniques

The phase-noise measurements of a single-oscillator are based on the measurement of frequency fluctuations using discriminator techniques. The

practical discriminator acts as a filter with finite bandwidth that suppresses

the carrier and the sidebands on both sides of the carrier. The ideal carrier- suppression filter would provide infinite attentuation of the carrier and

zero attenutation of all other frequencies. The effective Q of the practical

discriminator determines how much the signals are attenuated. Frequency discrimination at very high frequencies (VHF) has been obtained using slope detectors and ratio detectors, by use of lumped circuit elements of inductance and capacitance. At ultrahigh frequencies (UHF)

between the VHF and microwave regions, measurements can be performed

by beating, or heterodyning, the UHF signal with a local oscillator to obtain a VHF signal that is analyzed with a discriminator in the VHF frequency

range. Those techniques provide a means for rejecting residual amplitudemodulated (AM) noise on the signal under test. The VHF discriminators

usually employ a limiter or ratio detector. Ashley et al. (1968) and Ondria (1968) have discussed the microwave

cavity discriminator that rejects AM noise, suppresses the carrier so that the input level can be increased, and provides a high discriminated output to improve the signal-to-noise floor ratio. The delay line used as an FM dis- criminator has been discussed by Tykulsky (1966), Halford (1975), and

* See Appendix Note # 6

TN-218
268 A. L. LANCE, W. D. SEAL, AND F. LABAAR

fl

TUNER

CIRCU LATOR T

VARIABLE SHORT

CIRCUIT

DELAY LINE

la) (bl

DELAY LINE

OSCILLATOR UNDER TEST

LOW-NOISE AMPLIFIER

DIGITAL SIGNAL ANALYZER

IFFT)

Vjy DIRECTIONAL XL/~~VcS' l^ll-TER PHASE MIXER

SHIFTER

SPECTRUM ANALYZER

PLOTTER PRINTER COMPUTER

COUPLER

(c)

Fig. 15 Single-oscillator phase-noise measurement techniques: (a) cavity discriminator;

(b) reflective-type delay-line discriminator; (c) one-way delay line.

Ashley ex al. (1968). Ashley et al (1968) proposed the reflective-type delayline discriminator shown in Fig. 15b. The cavity can also be used to replace

delay line. The one-way delay line shown in Fig. 15c is implemented in the TRW measurement systems. The theory and applications set forth in this section are based on a system of this particular type.

A. The Delay Line as an FM Discriminator

1 . The Single-Oscillator Measurement System

The single-oscillator signal is split into two channels in the system shown

in Fig. 15. One channel is called the nondelay or reference channel. It is also referred to as the local-oscillator (LO) channel because the signal in this channel drives the mixer at the prescribed impedance level (the usual LO

drive). The signal in the second channel arrives at the mixer through a delay

line. The two signals are adjusted for phase quadrature with the phase

shifter, and the output of the mixer is a fluctuating voltage, analogous to the frequency fluctuations of the source, centered on approximately zero dc

volts. The delay line yields a phase shift by the time the signal arrives at the balanced mixer. The phase shift depends on the instantaneous frequency of

TN-219
7. PHASE NOISE AND AM NOISE MEASUREMENTS 269

the signal. The presence of frequency modulation (FM) on the signal gives

rise to differential phase modulation (PM) at the output of the differential delay and its associated (nondelay) reference line. This relationship is linear

if the delay t^ is nondispersive. This is the property that allows the delay line to be used as an FM discrimmator. In general, the conversion factors are a

function of the delay (tj) and the Fourier frequency / but not of the carrier frequency.

The differential phase shift of the nominal frequency vq caused by the delay

line is A0 = 27rvoTd, (76)

where t^ is the time delay.

The phase fluctuations at the mixer are related to the frequency fluctuations

(at the rate/) by

(50 = Inr, dv(f).

The spectral density relationships are W/) = (iKX^)' SM)

and

Then,

Ssdf) = f S,^if).

Ss.,if) = ilnfr,)' S,^(/)

dim

(77)

(78)

(79)

(80)

where the subscript dim indicates delay-line method. From Eq. (56), the

spectral density of phase for the two-oscillator technique, in radians squared

per hertz, is

Ss,(f) = 4 SM) 5,,(/)

(V )^ 2(V )^ .2(K„,)B.

(81)

because

and

per hertz.

(^;.p)' = ^(v,^)' = 4(^'p)' = 4[2(t.v„J^]

if(/) = 2(SJ/)/(V;,p)^) = (<3i-TMs)'/4(KTM)'B (82)

TN-220
270 A. L. LANCE, W. D. SEAL. AND F. LABAAR

The sensitivity (noise floor) of the two-oscillator measurement system

includes the thermal and shot noise of the mixer and the noise of the base- band preamplifier (referred to its input). This noise floor is measured with

the oscillator under test inoperative. The measurement system sensitivity of

the two-oscillator system, on a per hertz density basis (dBc/Hz) is nfU= 101og[2(^t;„)V(Fp,p)^]. (83)

where Sv„ is the rms noise voltage measured in a one-hertz bandwidth.

The two-oscillator system therefore yields the output noise from both

oscillators. If the reference oscillator is superior m performance, as assumed

in the previous discussions, then one obtains a direct measure of the noise

characteristics of the oscillator under test. If the reference and test oscillators are the same type, a useful approximation is to assume that the measured

noise power is twice that associated with one noisy oscillator. This approximation is in error by no more than 3 dB for the noisier oscillator. Substituting

in Eq. (80) and using the relationships in Eq. (56), we have, per hertz,

nf) = 2[(<3t.v.s)V(l^p.p)'](27t/Td)^ (84)

dim Examination of this equation reveals the following.

(1) The term in the brackets represents the two-oscillator response. Note that this term represents the noise floor of the two-oscillator method.

Therefore, adoption of the delay-line method results in a higher noise by the

factor {Infx^)^ when compared with the two-oscillator measurement method.

The sensitivity (noise floor) for delay lines with different values of time delay

are illustrated in Fig. 17. (2) Equation (84) also indicates that the measured value of ^(f) is periodic in co = 2nf. This is shown in Fig. 21. The first null in the responses

is at the Fourier frequency / = I/Tj. The periodicity indicates that the cali- bration range of the discriminator is limited and that valid measurements

occur only in the indicated range, as verified by the discriminator slope shown

in Fig. 16. (See Fig. 23.) (3) The maximum value of {Infr^y can be greater than unity (it is 4 at / = 1 2Td). This 6-dB advantage is utilized in the noise-floor measurement.

However, it is beyond the valid calibration range of the delay-line system.

The 6-dB advantage is offset by the line attenuation at microwave frequencies,

as discussed by Halford (1975).

The delay-line discriminator system has been analyzed in terms of a powerlimited system (a particular idealized system in which the choice of power

oscillator voltage, the attenuator of the delay line, and the conversion loss

TN-221
7. PHASE NOISE AND AM NOISE MEASUREMENTS 271

of the mixer are limited by the capability of the mixer) by Tykulsky (1966),

Halford (1975), and Ashley et al. (1977). For this particular case, Eq. (83) indicates that an increase m the length of the delay line (to increase t^ for decorreiation of Fourier frequencies closer to the carrier) results in an

increase in attenuation of the line, which causes a corresponding decrease

in Vpip. The optimum length occurs where t^ is such that the decrease in K , is approximately compensated by the increase in {Infr^}, i.e., where

d 2nfT,

dTiA = 0. (85)

ptp This condition occurs where the attenuation of the delay line is I Np

(8.686 dB). However, when the system is not power limited, the attenuation

of the delay line is not limited, because the input power to the delay line can

be adjusted to maintain Kp,p at the desired value. The optimum delay-line

length is determined at a particular selectable frequency. However, since the attenuation varies slowly (approximately proportional to the square root

of frequency), this characteristic allows near-optimum operation over a considerable frequency range without appreciable degradation in the measurements. A practical view of the time delay {x^) and Fourier-frequency functional

relationship can be obtained by reviewing the basic concepts of the dualchannel time-delay measurement system discussed by Lance (1964). If the

differential delay between the two channels is zero, there is no phase differ- ence at the detector output when a swept-frequency cw signal is applied

to the system. Figure 16 shows the detected output interference display when

a swept-frequency cw signal (zero to 4 MHz) is applied to a system that has

Tj = 1 500 n$

f =2MHz

f = 4 MHz

Fig. 16 Swept-frequency interference display at the output of a dual-channel system

with a differential delay of 500 nsec.

TN-222
272 A. L. LANCE, W. D. SEAL, AND F. LABAAR

a differential delay of 500 nsec between the two channels. The signal amplitudes are assumed to be almost equal, thus producing the familiar voltagestanding-wave pattern or interference display. Because this is a two-channel

system, there is a null every 360°, as shown.

2. System Sensitivity {Noise Floor) When Using the Differential Delay-Line Technique

Halford (1975) has shown that the sensitivity (noise floor) of the single- oscillator differential delay-line technique is reduced relative to the twooscillator techniques. The sensitivity is modified by the factor

Sd = 2(1 - cos 2nfxa). (86)

For cDTj = IndT^ <^ 1 a good approximation is

Sj = 2(1 - cos 2nfr,) = (cotJ^CI - i^(a;Td)^] = (Infr^f = 6-, (87)

where d is the phase delay of the differential delay line evaluated at the frequency / Figure 17 shows the relative sensitivity (noise floor) of the twooscillator technique and the single-oscillator technique with different delay-line lengths. The f~^ slope is noted at Fourier frequencies beyond

SINGLE OSCILLATOR

FOURIER FREQUENCY

Fig. 17 Relative sensitivity (noise floor) of single-oscillator and two-oscillator phasenoise measurement systems.

TN.223
PHASE NOISE AND AM NOISE MEASUREMENTS 273

about 1 kHz. For Fourier frequencies closer to the carrier, the slope is /"\

i.e., the sum of the /

" ^ slope of Eq. (87) and the /~ ' flicker noise. Phase-locked sources have phase-noise characteristics that cannot be

measured at close-in Fourier frequencies using this basic system. The

relative sensitivity of the system can be improved by using a dual (twochannel) delay-line system and performing cross-spectrum analysis, which

will be presented in this chapter.

Labaar (1982) developed the delay-line rf bridge configuration shown in Fig. 18. At microwave frequencies where a high-gain amplifier is available,

suppression of the carrier by the rf bridge allows amplification of the noise

going into the mixer. A relative sensitivity improvement of 35 dB has been

obtained without difficulty. The limitations of the technique depend on the available rf power and the carrier suppression by the bridge. Naturally, if the rf input to the bridge is high one must use the technique with adequate

precautions to prevent mixer damage that can occur by an accidental bridge

unbalance. Labaar (1982) indicated the added advantage of using the rf bridge carrier-suppression technique when attempting to measure phase

noise close to the carrier when AM noise is present. Figure 19 shows the

DELAY-LINE , rf BRIDGE

PHASE

SHIFTER ATTENUATOR ^

DELAY LINE

Y AMPLIFIER

UUT * u

POWER

SPLITTER ^

MIXER

PHASE

SHIFTER

Fig. 18 Carrier suppression using an rf bridge to increase relative sensitivity. (Courtesy Instrument Society of America.)

TN-224
274 A. L. LANCE, W. D. SEAL, AND F. LABAAR

1

X ^^SIN()rrf) /

>

'_ ^

> ^/ AM LEAKAGE

/<=° Frequency (Hz)

Fig. 19 Phase detector output (AM-PM crossover); t, delay time.

mixer output for phase (PM) and amplitude (AM) noise in the single- oscillator delay-line FM discriminator system. It is noted that the phase

noise and AM noise intersect and that the AM will therefore Hmit the measurement accuracy near the carrier. Even though AM noise is much lower

than phase noise in most sources, and even though the AM is normally

suppressed about 20 dB, there is still AM at the mixer output. This output

is AM leakage and is caused by the finite isolation between the mixer ports.

The two-oscillator technique does not experience this problem to this extent because the phase noise and AM noise maintain their relative rela- tionships at the mixer output independent of the offset frequency from the

carrier.

B. Calibration and Measurements Using the Delay

Line as an FM Discriminator

The block diagram of a practical single-oscillator phase noise measure- ment system is shown in Fig. 20. The signals in the delay-line channel of the system experience the one-way delay of the line. With adequate source power, the system is not limited to the optimum 1 Np (8.686 dB) previously

discussed for a power-limited system. Measurements are performed using

the followmg operational procedures.

(1) Measure the tracking spectrum analyzer IF bandwidths as set forth

in Section III.C.l. (2) Establish the system power levels (Section IV.B.l).

(3) Establish the discriminator calibration factor (Section 1V.B.2).

(4) Measure and plot the oscillator characteristics in the automatic

system used (Section IV. B. 3). (5) Measure the system noise floor (sensitivity) (Section IV.B.4).

TN-225
7. PHASE NOISE AND AM NOISE MEASUREMENTS 275

SOURCE a

COUPLER NO t

A • SPECTRUM j

' analyzer'

LOW-PASS

FILTER

ATTEN. NO I h^^H- ISOLATOR

ATTEN NO 2 -H>^ dcBLOCK POWER

AMP

DELAY LINE

COUPLER

ISOLATOR

SIGNAL GENERATOR

(FREQUENCY- MODULATION

CAPABILITY!

ATTEN NO 3 MIXER MgH HIGH GAIN LOW NOISE

1 TO AUTOMAT

SYSTEM

2 HP5420A

(SU8MI

3 HP 5390A FREQUENCY STABILITY ANALYZER (0.01 Hz TO 10 kHz)

QMATIC MEASUREMENT y^ IFATTEi

" "*-C H—

)A DIGITAL SIGNAL ANALYZER ^sj ATTEN

LLIHERTZ TO 25kHz| ^^ ^n 4

LOW-PASS

FILTER

PHASE

SHIFTER

OSCILLOSCOPE

(dc COUPLED!

AMP ^"0 4

Fig. 20 Single-oscillator phase noise measurement system using the delay line as an FM discriminator. (From Lance et al.. 1977a.)

1. System Power Levels The system power levels are set using attenuators, as shown in Fig. 20. Because the characteristic impedance of attenuator No. 4 is 50 Q. mismatch

errors will occur if the mixer output impedance is not 50 Q. As previously

discussed, the mixer drive levels are set so that the mixer output signal, as observed during calibration, is sinusoidal. This has been accomplished in TRW systems with a reference (LO) signal level of 10 dBm and a mixer

input level of about dBm from the delay line. A power amplifier can be used to increase the source signal to the measure- ment system. This amplifier must not contribute appreciable additional

noise to the signal.

2. Discriminator Calibration

The discriminator characteristics are measured as a function of frequency

and voltage. The hertz-per-volt sensitivity of the discriminator is defined

as the calibration factor (CF). The calibration process involves measuring

the effects of intentional modulation of the source (carrier) frequency. A

known modulation index must be obtained to calculate the calibration

factors of the discriminator. The modulation index is obtained by using

amplitude modulation to establish the carrier-to-sideband ratio when there

is considerable instability of the source or when the source cannot be fre- quency modulated.

TN-226
276 A. L. LANCE, W. D. SEAL, AND F. LABAAR

It is convenient to consider the system equations and calibration techniques

in terms of frequency modulation of stable sources. If the source to be

measured cannot be frequency modulated, it must be replaced, during the

calibration process, with a modulatable source. The calibration process

will be described using a modulatable source and a 20-kHz modulation

frequency. However, other modulation frequencies can be used. The cali- bration factor of this type discriminator has been found to be constant over

the usable Fourier frequency range, within the resolution of the measuring

technique. The calibration factor of the discriminator is established after the system power levels have been set with the unit under test as the source. The discriminator calibration procedures are as follows.

( 1

)

Set attenuator No. 4 (Fig. 20) to 50 dB.

(2) Replace the oscillator under test with a signal generator or oscillator that can be frequency modulated. The power output and operating frequency

of the generator must be set to the same precise frequency and amplitude

values that the oscillator under test will present to the system during the measurement process.

(3) Select a modulation frequency of 20 kHz and increase the modulation

until the carrier is reduced to the first Bessel null, as indicated on the spectrum

analyzer connected to coupler No. 1. This establishes a modulation index

(m = 2.405).

(4) Adjust the phase shifter for zero volts dc at the output of the mixer,

as indicated on the oscilloscope connected as shown in Fig. 20. This establishes the quadrature condition for the two inputs to the mixer. This quadrature

condition is continuously monitored and is adjusted if necessary.

(5) Tune the tracking spectrum analyzer to the modulation frequency

of 20 kHz. The power reading at this frequency is recorded in the program

and is corrected for the 50-dB setting of attenuator No. 4, which will be set to zero decibel indication during the automated measurements. P(dBm) = ( - dBm power reading) + 50 dB (88)

This power level is converted to the equivalent rms voltage that the spectrum

analyzer would have read if the total signal had been applied

Kms = V

10''''°/1000 + R. (89)

(6) The discriminator calibration factor can now be calculated because

this power in dBm can be converted to the corresponding rms voltage using

the following equation:

Kms = VdO'' '°/1000) X R, (90)

where i? = 50 Q in this system.

TN-227
7. PHASE NOISE AND AM NOISE MEASUREMENTS 277

(7) The discriminator calibration factor is calculated in hertz per volt as CF = mfj^ 2 K„, = 2.405/^^-2 K^,. (91)

The modulation index m for the first Bessel null as used in this technique

is 2.405. The modulation frequency is /„.

3. Measurement and Data Plotting

After the discriminator is calibrated, the modulated signal source is re- placed with the frequency source to be measured. Quadrature of the signals

into the mixer is reestablished, attenuator No. 4 (Fig. 20) is set to dB, and

the measurement process can begin.

The measurements, calculations, and data plotting are completely auto- mated. The calculator program selects the Fourier frequency, performs

autoranging, and sets the bandwidth, and measurements of Fourier frequency

power are performed by the tracking spectrum analyzer. Each Fourier

frequency noise-power reading P„ (dBm) is converted to the corresponding rms voltage by

t^irTMs = Vl0"''"''""°/1000x R. (92)

The rms frequency fluctuations are calculated as

<5Vrms = t-'lrms ^ CF. (93)

The spectral density of frequency fluctuations in hertz squared per hertz is calculated as SM) = (dv,TM)VB, (94)

where B is the measured IF noise-power bandwidth of the spectrum analyzer.

The spectral density of phase fluctuations in radians squared per hertz is calculated as SM) = SUfW- (95)

The NBS-designated spectral density in decibels (carrier) per hertz is cal- culated as

i^(/)dB= 101ogiS,^(/). (96)

Spectral density is plotted in real time in our program. However, the data can be stored and the desired spectral density can be plotted in other forms. Integrated phase noise can be obtained as desired.

TN-228
278 A. L. LANCE, W. D. SEAL, AND F. LABAAR

4. Noise Floor Measurements

The relative sensitivity (noise floor) of the single-oscillator measurement

system is measured as shown in Fig. 12a for the two-oscillator technique.

The delay line must be removed and equal channel lengths constructed,

as in Fig. (12a). The same power levels used in the original calibration and

measurements are reestablished, and the noise floor is measured at specific

Fourier frequencies, using the same calibration-measurement technique, or by repeating the automated measurement sequence. A correction for the noise floor requires a measurement of the rms voltage

of the oscillator (firms) ^nd a measurement of the noise floor rms voltage

(t'"2rms)- These voltages are used in the following equation to obtain the corrected value:

t-rms = Vd^'lrms)' " {v^,^,)^ (97)

The value v,^^ is then used in the calculation of frequency fluctuations.

If adequate memory is available, each value of 1)1^^5 can be stored and used

after the other set of measurements are performed at the same Fourier

frequencies.

The following technique was developed by Labaar (1982). Carrier suppression is obtamed using the rf bridge illustrated m Fig. 18. One can easily

improve sensitivity more than 40 dB. At 2.0 and 3.0 GHz 70-dB earner

suppression was realized. In general, the improvement in sensitivity will depend on the availability of an amplifier or adequate input power.

Figure 21 shows the different noise floors in a delay-line bridge discriminator. It is good measurement discipline to always determine these noise

floors; also, the measurements, displayed in Fig. 21, give a quick understanding of the physical process involved. The first trace is obtained by terminating the input of the baseband spectrum analyzer. The measured output

noise power is then a direct measure of the spectrum analyzer's noise figure

(NF). The input noise is thermal noise and is usually indicated by "KTB,"

which is short for "the thermal noise power at absolute temperature of T

degrees K(elvin) per one hertz bandwidth (B). This KTB number is, at 18°C,

about -174dBm/Hz.

Figure 21a shows that trace number 1 for frequencies above about 1 kHz is

level with a value of about -ISOdBm = -(174-24) dBm, which means that the spectrum analyzer has an NF of 24 dB. At 20 Hz the NF has gone up to about 48 dB. To improve the NF, a low-noise (NF, 2dB). low-frequency

(10 Hz- 10 MHz) amplifier is inserted as a preamplifier. Terminating its input now results in trace number 2. At the high frequency end, the measured

power goes up by about 12-13 dB, and the amplifiers gain is 34 dB. This means that the NF is improved by 34 - 12-13 = 21-22 dB, which is an

TN-229
PHASE NOISE AND AM NOISE MEASUREMENTS 279

Fig. 21 (a) Noise contribution analysis: (b) phase noise test setup using a delay-line rf bridge discriminator (rf = 2.8 GHz; O. termination points. NF, noise figure: LF. low frequency. (From Seal and Lance. 1981.) NF :i: 2-3 dB as expected, i.e., the first stage noise predominates. The lowfrequency end at 20 Hz gives an NF of 26 dB, which overall is quite an improvement.

In trace number 3 the mixer is included with it's rf (signal) port terminated.

It is clear from this trace that certainly up to 100 kHz, the noise generated

by the mixer diodes being "pumped" by the LO signal dominates. This case

represents the "classic" delay-line discriminator. The last trace (number 4)

includes the low-noise, high-gain rf amplifier that can be used because the

carrier is suppressed in the delay-line rf bridge discriminator, in contrast

to the classic delay-line discriminator case. This trace shows that from I kHz

on up the measured output power is flat, representing a 2-3-dB NF.

At about 20-40 Hz, trace numbers 3 and 4 begin nearing their cross- over floor. In this particular case, which is discussed in full by Labaar (1982), the measurement systems noise floor (resolution) has been improved by

40 dB.

Figure 22 shows plots of phase noise as measured at two frequencies

using delay lines of different lengths. The delay line used measure at 600 MHz

TN-230
280 A. L. LANCE, W. D. SEAL, AND F. LABAAR

£ -120 -

-140 -

-190

LINE HARMONICS

40 100 FOURIER FREQUENCY {Hz)

Fig. 22 Phase noise of 600-MHz oscillator multiplied to 2.4 GHz (From Lance et ai. 1977a.)

was about 500 nsec long, as noted by the first null, i.e., the reciprocal of the Fourier frequency of 2 MHz is the approximate differential time delay.

Note that a shorter delay line (approximately 250 nsec differential) is used

to measure the higher frequency because the delay-line discriminator

calibration is valid only to a Fourier frequency at approximately 35% of

the Fourier frequency at which the first null occurs, if a linear transfer function is assumed.

The actual transfer function of a delay-line discriminator (classic and rf bridge types) is sinusoidal, as shown in Fig. 23a. The baseband spectrum

analyzer measures power in a finite bandwidth, and as a consequence it is possible to measure through a transfer-function null if the noise power does

not change substantially over a spectrum-analyzer bandwidth. The following

power relations then hold

/'mcas(co) = \|^0J

*> oj — P{w) doi' ^ '

0} — Acj 2 Ao)

/•u-f 4(u 2 dcj' = P{w). (98)

TN-231
PHASE NOISE AND AM NOISE MEASUREMENTS 281

^-;(t)

z /^V'^ ^N )— f\r 2jT

r

UJ

(al

CORRECT

(SINUSOIDAL) TRANSFER

FUNCTION

-160 -

10 100 1000 10.000 FREQUENCY (kHz)

(b)

Fig. 23 Transfer functions for a delay-line rf bridge discriminator: (a) actual; (b) approximate (linear) and "correct" (sinusoidal). Phase noise: H.P. 8672A at 2.4 GHz.

Figure 23b shows the results using a linear approximation and the "correct" transfer function for a delay-line rf bridge discriminator. The correct

transfer function breaks down close to the null because the signal level drops below the system's noise floor, as explained by Labaar (1982).

Using the sinusoidal transfer function in the calculator software gives correct results barring frequency intervals of 5 to 10 spectrum analyzer's

bandwidths (10 x 30 = 300 kHz) centered at the transfer function nulls. These particular data were selected to illustrate the characteristics of the system. Recall that one can easily make the noise floor 40 dB lower using

the rf bridge shown in Fig. 18.

C. Dual Delay-Line Discriminator

1 . Phase Noise Measurements

The dual delay-line discriminator is shown in Fig. 24. This system was

suggested by Halford (1975) as a technique for lowering the noise floor of the delay-line phase noise measurement system. The system consists of

TN-232
282 A. L. LANCE, W. D. SEAL, AND F. LABAAR

SIGNAL GENERATOR

(FREQ.-MOD.

CAPABILITY)

ATTEN. :yr NO. 2 -^

ORTEC AMP.

9431

L_|r?fl 4

HEWLETT-PACKARD

5420A DIGITAL SIGNAL ANALYZER

ATTEN NO.

3

^^HM Ml I Cl^. ^

DUAL- CHANNEL

dcSCOPE

ORTEC AMR

943FECAMP __ I '^

^•'^''^

I LOW-PASS 1

1

I /Ov

"S. JFILTER j

l ^M (X/ '^'^"

PHASE

SHIFTER

PHASE

SHIFTER

ATTEN. NO, 4

_|_ 3-dB COUPLER

COUPLER

DELAY LINE

Fig. 24 A dual delay-line phase noise measurement system. (Courtesy Instrument Society of America.)

two differential delay-line systems. The single-oscillator signal is applied

to both systems and cross-spectrum analysis is performed on the signal

output from the two delay-line systems. Signal processing is performed with

the Hewlett-Packard 5420A digital signal analyzer. The cross spectrum is obtained by taking the product of the Fourier transform of one signal and

the complex conjugate of the Fourier transform of a second signal. It is a phase-sensitive characteristic resulting in a complex product that serves

as a measurement of the relative phase of two signals. Cross spectrum gives

a phase- and amplitude-sensitive measurement directly. By performmg

the product Sy{f) Sx{f)*, a certam signal-to-noise enhancement is achieved.

TN-233
7. PHASE NOISE AND AM NOISE MEASUREMENTS 283

The low-noise amplifiers preceding the digital signal analyzer are used

when performing measurements at Fourier frequencies from 1 Hz to 25 kHz.

The amplifiers are not used when performing measurements below the Fourier frequency of 1 Hz.

2. Calibrating the Dual Delay-Line System

Each delay line in the system is calibrated separately following the same

basic procedure set forth m Section IV.B. The Hewlett-Packard 5420 measures the one-sided spectral density of frequency fluctuations in hertz squared per hertz. The spectral density of phase fluctuation in radians

squared per hertz can be calculated as

Ss^if) = Sa/)//', (99)

and

^if) = Ss.if)/2f\ (100)

per hertz. The Hewlett-Packard 5420 measurement ofSf^if) in Hz^/Hz must,

therefore, be corrected by 1/2/^ However, the/^ correction must be entered

in terms of radian frequency (co = 2nf). This conversion is accomplished by

^if) = SM)(l/2f'){4n'/4n') = [27r^S,,(/)]/(co)^ (101)

per hertz since Eq. (100) can be stated in the following terms:

Signal-to-noise enhancement greater than 20 dB has been obtained using

the dual-channel delay-line system.

D. Millimeter-Wave Phase-Noise Measurements

1 . Spectral Density of Phase Fluctuations

The delay line used as an FM discriminator is based, in principle, on a

nondispersive delay line. However, a waveguide can be used as the delay

line because the Fourier frequency range of interest is a small percentage

of the operating bandwidth (seldom over 100 MHz), and the dispersion can

be considered negligible.

The calibration and measurement are performed as set forth in Section

IV.B. The modulation index m is usually established using the carrier-to- sideband ratio that uses amplitude modulation because millimeter sources

are either unstable or cannot be modulated. The two approaches to measure- ments at millimeter frequencies are shown in Figs. 25 and 26. Figure 25

shows the direct measurement using a waveguide delay line. This system

offers improved sensitivity if adequate input power is available. The rf bridge

and delay-line portion of the system differs from Fig. 18 because pre- and

TN-234
284 A. L. LANCE, W. D. SEAL, AND F. LABAAR

LOW

NOISE

Fig. 25 Millimeter-wave phase noise measurements using a waveguide delay line. (From

Seal and Lance. 1981.)

post-bridge amplifiers with appropriate gam are not available, so the sensitivity can equal the amount of carrier suppression.

Figure 26 shows the use of a harmonic mixer to downconvert to the convenient lower frequency where post-bridge amplifiers are available. The relatively low sensitivity to frequency drift that is characteristic of delayline discriminators becomes an advantage here. A separate calibration

generator is required, as shown in Fig. 25, and a power meter is used to assure proper power levels during the calibration process.

2. Spectral Density of Amplitude Fluctuations

AM noise measurements require equal electrical length in the two channels

that supply the signals to the mixer. The delay line must be replaced with the necessary length of transmission line to establish the equal-length condition

when the systems shown in Figs. 25 and 26 are used. The AM noise measure- ment system is calibrated and the noise measurements are performed directly

in units of power for a direct measurement of »?(/) in dBc/Hz. *n{f) is the

spectral density of one modulation sideband divided by the total signal

power at a Fourier frequency difference /from the signal's average frequency

vq . The system calibration establishes the detection characteristics in terms

of total power output at the IF port of the mixer (detector).

TN-235
7. PHASE NOISE AND AM NOISE MEASUREMENTS !85

20dBm

CALIBRATION GENERATOR

I

dBm

r

POWER METER

SPECTRUM ANALYZER

LOW-PASS

FILTER

\ X LOW-NOISE Y SOLID-STATE -- AMPLIFIER

PHASE

SHIFTER

t

rATTEN. DELAY

LINE PHASE

SHIFTER AMP i

LOW-PASS

FILTER

LOW-NOISE

.PREAMP. TO AUTOMATIC

PHASE/AM NOISE MEASUREMENT

SYSTEM

Fig. 26 A millimeter-wave hybrid phase noise measurement system that produces IF

frequency and uses a delay-line discriminator at IF frequency. (From Seal and Lance, 1981.)

The AM noise measurements are performed according to the following.

(1) A known AM modulation (carrier-sideband ratio) must be established to calibrate this detector in terms of total power output at the IF port.

The modulation must be low enough so that the sidebands are at least 20 dB below the carrier. This is to keep the total added power due to the modulation small enough to cause an insignificant change in the detector

characteristics.

(2) The rf power levels are adjusted for levels of approximately 10 dBm

at the reference port and dBm at the test port of the mixer.

TN-236
286 A. L. LANCE, W. D. SEAL, AND F. LABAAR

(3) Approximately 40 dB is set in the precision IF attenuator. The system

is adjusted for an out-of-phase quadrature condition.

(4) The modulation frequency and power level are measured by the automatic baseband spectrum analyzer. The total carrier-power reference

level is measured power, plus the carrier-sideband modulation ratio, plus

the IF attenuator setting. (5) The AM modulation is removed, the IF attenuator set to dB, and

the system re-checked to verify the out-of-phase quadrature (maximum dc

output from the mixer IF port). Noise (!/„) is measured at the selected Fourier

frequencies. A direct calculation o{m{f) in dBc/Hz is

m{f) = [{modulation power (dBm) -I- carrier-sideband ratio (dB)

-I- IF attenuation (dB) - noise power (dBm) -I- 2.5 dB

-lOlog(BW)]. (102)

Figure 27 illustrates the measurements of AM and phase noise of two

GUNN oscillators that were offset in frequency by 1 GHz. The measurements

were performed using the coaxial delay-line system.

-20 r

FOURIER FREQUENCY (Hz)

Fig. 27 Phase noise and AM noise of 40- and 41-GHz Gunn oscillators. (From Seal and Lance. 1981.)

TN-237
7. PHASE NOISE AND AM NOISE MEASUREMENTS 287

V. Conclusion

The fundamentals and techniques for measurement of phase noise have

been set forth for two basic systems. The two-oscillator technique provides

the capability for measuring high-performance cw sources. The system

sensitivity is superior to the single-oscillator technique for measuring phase

noise very close to the carrier. High-stability sources such as those used in frequency standards applications can be measured without using phase-locked loops. However, most

microwave sources exhibit frequency instability that requires phase-locked

loops to maintain the necessary quadrature conditions. The characteristics of the phase-locked loops must be evaluated to obtain the source phase

noise characteristics. Also, in principle, one must have three sources at the same frequency to characterize a given source. If three sources are not available, one must assume that either one source is superior in performance or

that they have equal phase noise contributions.

The single-oscillator technique employing the delay line as an FM dis- criminator has adequate sensitivity for measuring most microwave sources. The economic advantages of using this system include the fact that only one source is required, phase-locked loops are not required, system configuration is relatively inexpensive, and the system is inherently insensitive to oscillator frequency drift. The single-oscillator technique using the delay-line discriminator can

be adapted to measure the phase noise of pulsed sources. Pulsed sources have been measured at 94 GHz by F. Labaar at TRW, Redondo Beach,

California. Acknowledgments

Our initial preparations for developing a phase noise measurement capability were the result of discussions with Dr. Jorg Raue of TRW. Our first phase noise development effort was

assisting in the evaluation of phase noise measurement systems designed and developed by

Bill Hook of TRW (Hook. 1973). The efforts of Don Leavitt of TRW were vital in initiating the measurement program. We are very grateful to Dr. Donald Halford of the National Bureau of Standards in Boulder, Colorado, for his interest, consultations, and valuable suggestions during the development of

the phase noise measurement systems at TRW.

We appreciate the measurement cross-checks performed by C. Reynolds, J. Oliverio, and

H. Cole of the Hewlett-Packard Company. Dr. J. Robcn Ashley of the University of Colorado, Colorado Springs, and G. Rast of the U.S. Army Missile Command, Redstone Arsenal, HuntsviUe, Alabama.

References

Allen, D. W. (1966). Proc. IEEE 54. 221-230.

Ashley, J. R.. Searles, C. B., and Palka, F. M. (1968). IEEE Trans. Microwave Theory Tech. [VnT-l6(9), 753-760.

4

TN-238
288 A. L. LANCE, W. D. SEAL, AND F. LABAAR

Ashley. J. R.. Barley, T. A., and Rast, G. J. (1977). IEEE Trans. Microwave Theory Tech. MTT-25t4), 294-318. Baghdady, E. J., Lincoln, R. N., and Nelin, B. D. (1965). Proc. IEEES3, 704-722.

Barnes, J. A. (1969). "Tables of Bias Functions, fl, and fl,, for Variances Based on Finite Samples of Processes with Power Law Spectral Densities," National Bureau of Standards

Technical Note 375. Barnes, J. A., Chie, A, R., and Cutler, L. S. (1970). "Characterization of Frequency Stability," National Bureau of Standards Technical Note 394; also published in IEEE Trans. Instrum.

A^ea5. IIVl-20(2). 105-120(1971). Brandenberger. H., Hadorn, F., Halford, D., and Shoaf, J. H. (1971). Proc. 25th Ann. Symp.

Freq. Control, Fort Monmouth, New Jersey, pp. 226-230.

Culter, L. S., and Searle, C. L. (1966). Proc. IEEE 54, 136-154.

Fisher, M. C. (1978). "Frequency Domain Measurement Systems," Pajjer presented at the 10th Annual Precise Time and Time Interval Applications and Planning Meeting, Goddard

Space Flight Center, Greenbelt, Maryland.

Halford, D. (1968). Proc. IEEE 56(2), 251-258.

Halford, D. (1975). "The Delay Line Discriminator," National Bureau of Standards Technical Note 10, pp. 19-38. Halford, D., Wainwright, A. E., and Barnes, J. A. (1968). Proc. 22nd Ann. Symp. Freq. Control, Fort Monmouth, New Jersey, 340-341.

Halford. D., Shoaf, J. H., and Risley, A. S. (1973). Proc. 27th Ann. Symp. Freq. Control, Cherry

Hill. New Jersey, pp. 421^30.

Hook. W. R. (1973). "Phase Noise Measurement Techniques for Sources Having Extremely Low Phase Noise." TRW Defense and Space Systems Group Internal Document, Redondo

Beach, California. Hewlett-Packard Company (1965). "Frequency and Time Standards," Application Note 52. Hewlett-Packard Co., Palo Alto, California. Hewlett-Packard Company (1970). "Computing Counter Applications Library," Application Notes 7, 22, 27, and 29. Hewlett-Packard Co., Palo Alto, California. Hewlett-Packard Company (1976). "Understanding and Measuring Phase Noise in the Fre- quency Domam." Application Note 207. Hewlett-Packard Co., Loveland, Colorado

Labaar. F. (1982). " New Discriminator Boosts Phase Noise Testing." Microwaves 21(3), 65-69. Lance, A. L. (1964). "Introduction to Microwave Theory and Measurements." McGraw-Hill,

New York. Unce, A. L., Seal, W. D., Mendoza, F. G., and Hudson, N. W. (1977a). Microwave J. 20(6), 87-103. Lance, A. L., Seal, W. D., Mendoza, F. G., and Hudson. N. W (1977b). Proc. 31st Annu. Symp.

Freq. Control, Atlantic City, New Jersey, pp. 463-483. Unce, A. L.. Seal, W. D., Halford, D., Hudson, N., and Mendoza. F. (1978). "Phase Noise Measurements Using Cross-Spectrum Analysis." Paper presented at the IEEE Conference on

Electromagnetic Measurements, Ottawa, Canada. Unce. A. L., Seal. W. D., and Labaar, F. (1982). ISA Trans. 21(4), 37^W.

Meyer, D. G. (1970). IEEE Trans. IM-19(4), 215-227. Ondria, J. (1968). IEEE Trans. Microwave Theory Tech. IV1TT.16(9), 767-781. Ondria, J. G. (1980). lEEE-MTT-S Int. Microwave Symp. Dig., pp. 24-25. Payne, J. B., Ill (1976). Microwave System News 6(2), 118-128.

Scherer, D. (1979). " Design Principles and Measurement Low Phase Noise RF and Microwave

Sources." Hewlett-Packard Co.. Palo Alto. California.

Seal. W. D.. and Lance. A. L. (1981) Microwave System .\ews 11(7), 54-61. Shoaf. J. H.. Halford, D., and Risley, A. S. (1973). "Frequency Stability Specifications and Measurement." National Bureau of Standards Technical Note 632.

* See Appendix Note # 33

TN-239
7. PHASE NOISE AND AM NOISE MEASUREMENTS 289

Tykulsky. A. (1966). Proc. IEEE 54(2). 306. Van Duzer, V. (1965). Proc. IEEE-NASA Symp. Definition Meas. Short-Term Freq. Stability. NASA SP-80, 269-272.

Walls, F. L., and Stem, S. R. (1977). Proc. J 1st Ann. Symp. Freq. Control. Atlantic City, New

Jersey, pp. 335-343.

Walls, F. L., Stein, S. R., Gray, J. E,, Glaze, D. J., and Allen, D. W. (1976). Proc. 30th Ann. Symp. Freq. Control. Atlantic City, New Jersey, p. 269.

TN-240
36th Annual Frequency Control Symposium 1982

PERFORMANCE OF AN AUTOMATED HIGH ACCURACY

PHASE MEASUREMENT SYSTEM

S. Stein, D. Glaze, J. Levine, J. Gray, D. Hilliard, D. Howe

Time and Frequency Division

National Bureau of Standards

Boulder, Colorado 80303

and

L. Erb ERBTEC Engineering Inc. Boulder, Colorado

Summary

A fully automated measurement system has been developed that combines many properties

previously realized with separate techniques.

This system is an extension of the dual mixer

time difference technique, and maintains its

important features: zero dead time, absolute

phase difference measurement, very high precision, the ability to measure oscillators of equal frequency and the ability to make mea.sure- ments at the time of the operator's choice. For

one set of design parameters, the theoretical

resolution is 0.2 ps, the measurement noise is 2 ps rms and measurements may be made within 0.1 s of any selected time. The dual mixer technique

has been extended by adding scalers which remove

the cycle ambiguity experienced in previous

realizations. In this respect, the system

functions like a divider plus clock, storing the epoch of each device under test in hardware.

The automation is based on „the ANSI/IEEE583 (CAMAC) interface standard. Each measurement channel consists of a mixer, zero-crossing

detector, scaler and time interval counter. Four channels fit in a double width CAMAC module

which in turn is installed in a standard CAMAC

crate. Controllers are available to interface

with a wide variety of computers as well as any IEEE-488 compatible device. Two systems have

been in operation for several months. One operates 24 hours a day, taking data from 15 clocks for the NBS time scale, and the other is used for short duration laboratory experiments.

Review of the Dual Mixer

Time Difference Technique

It is advantageous to measure time directly

rather than time fluctuations, frequency or frequency fluctuationns. These measurements

constitute a hierarchy in which the subsequently

listed quantities may always be calculated from

the previous ones. However, the reverse is not true when there are gaps in the measurements.

In the past, frequency was usually not derived

from time measurements for short sample times

because time interval measurements could not be performed with adequate precision. The dual

mixer technique, illustrated in Figure 1, made

it possible to realize the precision of the beat

frequency technique in time interval measurements.

The signals from two oscillators (clocks) are applied to two ports of a pair of double balanced mixers. Another signal synthesized from one

of the oscillators is applied to the remaining

two ports of the mixer pair. The input signals may be represented in the usual fashion

V^(t) =V^Q sin [27tv^Qt + (j)^(t)],

V2(t) = V2Q sin [2nv2Qt + if^^t)] and

V3(t) = V^^ cos [2nv^^t > $3(t)]

where v = u,q(1-1/R) and R is a constant

usually called trie heterodyne factor.

The low passed outputs of the two mixers

are

^B1=^B10 '^^ t*i(t) -*s(t)] and

Vg2 = Vg2o sin [*2(t)-<i>g(t)] where

*(t) = 27TU^t + (t)(t).

The time interval counter starts at time tj^ when

Vpi crosses zero in the positive direction and stops at time t|^, the time of the very next positive zero crossing of Vg2- Thus

*^(tf.,)

- *^(t^,) = 2M7I and

*9(tJ - '^AK) = 2Nn where

'2''N N^

N and M are integers.

Subtracting the two equations in order to com- pare the phases of oscillators 1 and 2, one obtains

*2^^n)'*i(^m)

= *s(tN)-*s(tM)*2(N-M)n N-" "s' M' The phase of an oscillator at time t^j may

M

I me Lo be written in terms of its phase at t^ and its

314 TN-241
average frequency over the Interval t^. < t^,.

*(t,^) = i>it^) + 2n[G(t,^;t,^)](t,^-t^) and

when we apply this equation to both <^2 ^"^ *s

we find

*2(tM)-*l(tM) = 2(N-M)n

-27i[.g2(tM;t^)](t^-t„)

where v B2 v^-

Since M and N are not measureable with the

equipment in Figure 1, the dual mixer technique

has heretofore only been used to measure the

phase difference between two oscillators modulo

271. We denote the period of the time interval counter time base by t and the number of counts recorded in a measuremfent by P. Then the phase

difference between the two oscillators is given

by

[*2(tM)-*i(tM)]mod 271 = -27:[ve2(t„;t^)]x^P

Figure 2 illustrates the output of the measurement system over a period of time. If a measurement begins and ends without the time

interval counter making a transition between

zero and its maximum value, e.g. , t "^ ^m "^ ^m '^ t, , then the phase difference can Be caTcula?ed

from the data. If t < tw < t, < t^. < t , then

the data must be corrected by 27i "o calculate

the phase difference. Experience has shown that

there are many measurement situations for which

the number of transitions of the time interval counter which occur between t„ and tj. cannot be

known. For this reason, a mooification has been

developed which removes the ambiguity by measuring M and N.

Extended Dual Mixer Time

Difference Measurement Technique

In order to configure the system to acquire

complete phase information, two scalers are added to count the zero crossings of each mixer.

Figure 3 is the block diagram of a two channel

system. It is constructed from identical circuit modules and therefore contains an unused

time interval counter. However, this design

permits very straightforward, and inexpensive

extension to the comparison of an arbitrarily

large number of oscillators with no need for switching any signals.

The counter outputs are combined to form

the phase difference between oscillators.

*2(tM)-*i(tM) 2(Nq-M^)7i + 2(N-M)7T

-27t[v„(t„;tJ]T P

The first term is a constant which represents

the choice of the time origin and can be ig- nored. The last two terms and their sum are plotted in Figure 4.

The average beat frequency Ugo (^mI^n^ cannot be known exactly. However, ft miy Be estimated with sufficient precision from the previous pair of measurements designated ' and

. The average frequency is approximately

'B2 (t„;t^) = (N"-N')/[R(M"-M')/v^Q + Tj.(P"-P')]

provided that it changes sufficiently slowly

compared to the interval t^<t|^. A typical value

for this error will be grven in the following

section.

Hardware Implementation

All measurement channels consist of a mixer, zero-crossing detector, scaler and time

interval counter. Four such circuits can be built in a double width CAMAC module. The

system is easily expanded to compare many oscillators and a complete system for making phase

comparisons among four clocks is shown in Figure

5. We have chosen parameters which are reason- able for comparing state-of-the-art atomic standards. Thus, the synthesizer is offsat lOHz

below oscillator # 1 and R = 5 x 10 . The

outputs from both mixers are approximately lOHz. The noise bandwidth is 100 Hz. The time interval counter is twice the frequency of oscillator

#1 or approximatel^g 10 MHz. The quantization

error is 1/2R = 10 cycle or 0.2ps which is a factor of ten smaller than the measurement

noise. As stated earlier, an error will result

from frequency changes which violate the con- stancy assumption .UjWd to estimate u no. A change in V2 ^V "^^ during the interval be- tween two nreasurements will result in a time

deviation error of lOps. Thus, one must make more closely spaced measurements for oscillators

which have large dynamic frequency changes than

for more stable devices. Two other sources of inaccuracy are the sensitivities to the amplitude and phase of the common oscillator. Figure

6 shows the measured value of x = (j)/27xu as a function of the amplitude of the input signal

and the phase of the synthesizer.

The new measurement system has many desirable features and properties:

(1) It has very high resolution, limited by the internal counters to 0.2 ps and by noise to approximately 2 ps.

(2) It has much lower noise than divider based measurement systems. However compromises

made to achieve low cost, low power, small size and automatic operation degrade the performance compared to state-of-the-art

systems for comparing 2 oscillators.

(3) The operation is fully automatic.

(4) NBS has developed a detailed operating

manual for the equipment and software.

315 TN-242
(5) All oscillators in the range of 5 MHz ± 5 Hz may be compared. Other carrier frequencies such as 1 MHz, 5.115 MHz, 10 MHz and

10.23 MHz are also usable. However, different carrier frequencies may not be mixed

on the same system. The system has been

successfully tested with an oscillator offset 4.6 Hz from nominal 5MHz. Measurements

were made at intervals of 2 hours between

which the system had to accumulate approximately 2 X 10 7T. The system has also been

tested with an oscillator offset 4 x 10 , and no errors were detected during a period

of 40 days.

(6) All sampling times in the range of 1 second

to 15 days with a resolution of 0.1 second

are possible. Measurements may be made on command or in a preprogrammed sequence.

(7) Measurements are synchronized precisely,

i.e. at the picosecond level, with the reference clock. They may therefore be synchronized with important user system

events, such as the switching times of a FSK or PSK system.

(8) All oscillators are compared synchronously

and all measurements are performed within a maximum interval of 0.1 second. As a re- sult, the phase of any oscillator needs to

be interpolated to the chosen measurement

time for an interval of 0.1 second maximum.

This capability, which is not present in either single heterodyne measurement sy- stems or switched measurement systems

eliminates a source of "measurement" error which is generally much larger than the

noise induced errors. For example, interpolation of the phase gf a high performance

Cs clock (a ~ 10 /i^) over a period of 3 hours woula produce approximately 1.5 ns phase uncertainty. To maintain 4 ps accuracy requires measurements simultaneous to

0.1s.

(9) There are no phase errors due to the switching of rf signals since there is no switching anywhere in the analog measurement system.

(10) No appreciable phase errors are introduced

when it is necessary to change the reference clock since, as shown in Figure 6, the peak error due to changes in synthesizer

phase is 20 ps.

(11) The measurement system is capable of mea- suring its own phase noise when the same

signal is applied to two input ports.

Figure 7 shows the phase deviations between

two such channels over a period of 75,000

seconds and Figure 8 is the corresponding

Allan variance plot. Figure 9 shows the phase deviations between 2 input channels

over a oeriod of 40 day^.

(12) Since the IEEE-583 (CAMAC) interface stan- dard has been followed for all the custom

hardware, the system may be easily interfaced to almost any instrument controller.

NBS has already tested the system using a large minicomputer, a small minicomputer

and a desk top calculator. Interfaces

between IEEE-583 and IEEE-488 controllers

are available and have been used successfully.

(13) The system is capable of camparing a very

large number of oscillators at a reasonable

cost per device.

There are also disadvantages to this mea- surement system. The most important are:

(1) The complexity of the hardware is greater

than for some systems. It is possible

that this will reduce reliability.

(2) A high level of redundancy is difficult to achieve. The system design stresses size, power, convenience and cost, resulting in an increase in the number of possible

single point failure mechanisms compared to some other techniques. For example, a CAMAC power supply failure will result in a loss of data for all devices being measured.

(3) A substantial committment is required in both specialized hardware and software.

(4) If an oscillator under test experiences a phase jump which exceeds 1 cycle, the measurement system records a jump with

incorrect absolute magnitude. As a result,

it may not be applicable to signals which

are frequency modulated with discontinuous

phase steps larger than 2n.

Conclusions

We have demonstrated a new phase measure- ment system with very desirable properties: All oscillators in the range of 5MHz ± 5Hz may be measured directly. The sampling times are only

restricted by the requirement that they exceed

one-,^econd. The noise floor is a (2,t) = 3 x 10 /t in short term and the tinre deviations

are less than 100 ps. All circuitry is designed

as modules which allows expansion at modest

cost. Compatibility with a variety of computers

is insured through the use of the IEEE-583

interface and adapters are available to permit

use with an IEEE-488 controller. The system

makes it feasible to make completely automated

phase measurements at predetermined times on large numbers of atomic clocks. It's own noise

is one-hundred times less than the state-of-theart in clock performance. It will be used in the near future to make all measurement needed

to compute NBS atomic time, but it will also be very valuable for any laboratory which uses three or more atomic clocks.

References

1. D. W. Allan, "The measurement of frequency

316 TN-243
and frequency stability of precision

oscillators," NBS Tech. Note 669

(1975). "CAMAC Instrumentation and Interface stan- dards," Institute of Electrical and Electronic Engineers, Inc., 345 E. 47th St. New York, NY 10017.

D. J. Glaze and S. R. Stein, "Picosecond

time difference measurements utilizing CA- MAC based ANSI/IEEE -488 data acquisition

hardware", NBS Tech Note 1056 (in preparation). V = V - V

ZERO

CROSSING

DETECTOR

STOP

START

TIME INTERVAL COUNTER

«-

ZERO

CROSSING

DETECTOR

-»

(2(t^)-*j(t^)]Md 2n = -2''[''b2<S(-'n'^'c''

Figure 1. Dual Mixer Time Difference Measurement System

Figure 2. Dual Mixer Data

*2

G>

FREQUENCY

SYNTH.

*1

Figure 3. Extended Dual Mixer Time Difference Measurement

System

317 TN-244
VS('"*1<S<' ' ^^Vo'" ' 2(N-M)n -2"[iB2<S(''^>^'c''

2(N-M)n —^—— Sum

Figure 4. Extended Dual Mixer Data

Synthesizer

H>

• 0—i>-^

Clock *4 [>

Offset

5 MHz

r Start "rr-rr

PULSE DISTRIBUTION MODULE

TT—n—IT

Start Start 10 MHz Time Base

Output '"P"* MEASUREMENT MODULE

Stop Inputs

6X4 = 24 clocks maximum

requires one pulse distribution module and

six measurement modules maximum

Figure 5. System Blocic Diagram

DATA

x(ps)

1000 - • e 800 - e e 600

1

- e

e 400 - e e INPUT AMPLITUDE

200 e

e

_L_ 1 1

OdB KlVrms

. 1 . . 1 . -3 -9dB

x(ps)

SYNTHESIZER PHASE

Figure 6. Measured Time Difference vs. Input Amplitude and

Synthesizer Phase

318 TN-245
PHASE PLOT Cl««k N*. 4- 8 Lm% »^f Siap* mf -1. 990200*- 17/S Ramovad

nb«4« -nb«4b

OW 0S mmy 82

1«-11

-l.-ll

T- 74700 SECONDS

Figure 7. Raw Phase Data for Two Channels Driven from the Same Source

B«i laaHa. CleoH N«. 4- B

Figure 8. Noise Floor of Measurement System

319 TN-246
CLOCKS u

19.58 29. 2S DOYS

I ,X0K;.-4S947. 19-nSR-82

Figure 9. Raw Phase Data for Two Channels Driven from the Same Source

320 TN-247
BIASES AND VARIA.N'CES OF SEVER.\L FFT SPECTRAL ESTIMATORS

AS A FUNCTION OF NOISE TYPE AND NUMBER OF SAMPLES

F. L. Walls, Time asid Frequency Division,

National Institute of Standards and Technology, Boulder, CO 80303

D. B. PerciviJ, Applied Physics Laboratory,

University of Washington, Seattle, WA 98106 From: Proceedings of the

W. R. Irelan, Irelan Electronics,

'^^''^ Annual Symposium on

412 Janet, Tahlequah, OK 74464 Frequency Control, 1989.

Abstract

We theoretically and experimentally investigate the biases and the variances of Fast Fourier transform (FFT) spectral es- timates with different windows (data tapers) when used to analyze power-law noise types /", /~^, /"' and /"*. There is a wide body of literature for white noise but virtually no investiga- tion of biases and variances of spectral estimates for power-law

noise spectra commonly seen in oscillators, amplifiers, mixers,

etc. Biases (errors) in some cases exceed 30 dB. The experimental techniques introduced here permit one to analyze the performance of virtually any window for any power-law noise. This makes it possible to determine the level of a particular

noise type to a specified statistical accuracy for a particular

window.

I. Introduction

Fast Fourier transform (FFT) spectrum analyzers are very commonly used to estimate the spectral density of noise. These

instruments often have several different windows (data tapers)

available for analyzing different types of spectra. For example,

in some applications spectral resolution is important; in others, the precise amplitude of a widely resolved line is important; and

in still other applications, noise analysis is important. These

diverse applications require different types of windows.

We theoretically and experimentally investigate the biases and variances of FFT spectral estimates with different windows

when used to analyze a number of common power-law noise

types. There is a wide body of literature for white noise but vir- tually no investigation of these effects for the types of p>ower-law

noise spectra commonly seen in oscillators, amplifiers, mixers,

etc. Specifically, we present theoretical results for the biaaes associated with two common windows — the uniform and Hanning windows — when applied to power-law spectra varying

as /°, /"' and /"*. We then introduce experimental tech- niques for accurately determining the biases of any window and

use them to evaluate the biaMS of three different windows for power-law spectra varying as /*, /"', /"' and /"*. As an ex- ample we find with /"* noise th&t the uniform window can have

errors ranging from a few dB to over 30 dB, depending on the

length of span of the /"'* noise. We have also theoretically investigated the variances of FFT spectral estimates with the uniform and Banning windows

(confidence of the estimates) as a function of the {>ower-law

noise type and as a function of the amoxmt ai data. We in- troduce experimental techniques that make it relatively easy to independently determine the variance of the spectral estimate

for virtually any window on any FFT spectrum analyzer. The

variance that is realized on a particular instrument depends not

only on the window but on the specific implementation in both

hardware and software. We find that the variance erf the spectral density estimates for white noise, /", is very similar for three specific windows available on one instrument and almost

identical to that obtained by standard statistical analysis. The

variances for spectral density estimates of /"* noise are only

4% higher than that of /° noise for two of the windows studied. The third window — the uniform window — does not yield usable results for either /"' or f~* noise. Based on this work it is now possible to determine the minimum number of samples necessary to determme the level of a particular noise type to a specified statistical accuracy as a

function of the window. To our knowledge this was previously

possible only for white noise — although the traditional results are generally valid for noise that varied as /~^, where 3 was

equal to or less than 4.

II. Spectrum Analyzer Basics

The spectrum analyzer which was used in the experimental work reported here is fairly typical of a number a( such in- stnoments cturently available from various manufacturers. The

basic measurement process generally consists of taking a string a[ Nt — 1024 digital samples of the input wave form, which we

represent here by A'l, X2, , Xs, The basic measurement

period was 4 ms. This yields a sampling time At = 3.90625 ^s. Associated with the FFT of a time series with .V, data points, there are usually {N,/2) + I = 513 frequencies

/; =

N.At'

J =0,1 .V./2.

The fundamental frequency /j is 250 Hz, and the Nyquist fre- quency fs./i is 128 kHz. Since the spectrum anadyzer uses an anti-aliasing filter which significantly distorts the high fre- quency portion of the spectrum, the instrument only displays the measured spectrum for the lowest 400 nonzero frequencies,

namely, /i = 250 H2, /j = 500 Hz /400 = 100 kHz. The exact details of bow the spectrum analyzer estimates the spectrum for Xi Xs, are unfonunately not provided

in the documentation supphed by the manufacturer, so the fol- lowing must be regarded only as a reasonable guess on our part

aa to its operation (see [1] for a good discussion on the basic ideas behind a spectrum analyzer; two good general references

for s}>ectr&l analysis are [2] and [4j). The sample mean.

^ = -pr^^''

is subtracted from each of the samples, and each of these "de- meaned" samples is multipled by a window ht (sometimes called a data taper) to produce

A-<*> = /i. {X, - X)

.

The spectral estimate.

5,(/;) = At

N. Y^ v-(*)g-t2»/, (A(

<zl

J =0.1. -V./2.

TN.248
is then computed using an FFT algonthm.

The subscnpt "'1" on S\if,) indicates that this is the spectral estimate formed from the first block of .V, samples. A

similar spectral estimate Sjl/j) is then formed from the second

block of contiguous data Xn,+i , -^"^,+2, . • •, ^jn, • In all, there

are Ni, different spectral estimates from Nt contiguous blocks,

and the spectrum analyzer averages these together to form

N, S{f:)^jryS,{f,). (1)

It is the statistical properties of S(/;) with which we are con- cerned in this paper. Unfortunately some important aspects of the windows are not provided in the doc\imentation for the instrument. One im- portant detail is the manner in which the window is normalized. There are two common normalizations:

(=1 * tml and

N, Y^hi^i. (2)

The first of these is common in engineering appUcations because

it ensures that the power in the windowed samples X, is the same as in the original demeaned samples: the second is equivalent to the first in expectation and is computationaly more

convenient, but it can result in small discrepancies in power

levels. Either normalization affects only the level of the spectral estimate and not its shape.

There are three windows built into the spectrum analyzer used here. The first is the uniform (rectangular, default) win- dow h[ ' = 1/\/N,. The second is the Hanning data window,

for which there are several slightly different definitions in the

literature. In lieu of specific details, we aasxime the following

symmetric definition:

/.<"> = c<'^'(i-cosHl<Iz£i)). cos

N.

N. + l<t<N,;

i<.<^.

here C^^' is a constant which forces the normalization in Equation (2). The third window is a proprietary 'fattened peak"

window, about which little specific information is available (it

is evidently designed to accurately measure the heights of peaks

in a spectrum).

III. Expected Value and Biai of Spectral Eftimates

III.A. Theoretical Analysis

We need to ass\une a noise model for the Xt's in order

to determine the statistical properties of 5(//) in Equation (1). We consider three different models, each of which is represented

in terms of a Gaussian white noise process Ci with mean zero and variance <t^. The second-order properties of each model

are given by a spectral density function S() defined over the

interval [-l/(2A<),l/(2At)] in cycles/At. The first model is a

discrete ptirameter, white noise process (/" noise):

Xt = «, and S(/) = <7?A<

.

The second model is a discrete-parameter, random- walk process (nominally f~^ noise):

X, = Y,t, and 5(/) =

^ Af

4 sin'' |7r/Af)

The third model is a discrete-parameter, random-run process (nominaJly /"* noisej:

^• = EE*' *°^ ^(^) =

(7?A<

16sm*(7r/Ai)

Continuous parameter versions of these three models have been

used extensively in the literature as models for noise commonly

seen in oscillators. For each of the three models we have derived expressions

for E{S{f)}, the expected value of S(f). These expressions

depend on the window /»,, the number of samples .V, in each

block and — in the case of a random-run process — the number of blocks Ni. The details behind these calculations will be

reported elsewhere [3]; here we merely summarize our conclusions for the three models in combination with the uniform and

Hanning windows and N, = 1024.

First, for a white noise process,

E{S{f,)}^s{f,: ; = 1,2, ,512, when the uniform window is used. For the Hanning window,

the above equality also holds to a ver>' good approximation for 2 < J < 511 and to within 0.8 dB for ; = 1 and 512 (the latter

is d no practical importance since the highest frequency index

given by the spectrum analyzer is ; = 400). These theoretical calculations agree with our experimental data except at /i (see Table 1). Second, for a random-walk process,

£{5(/;)}=2S(/,), ;=1,2 512,

when the uniform window is used, i.e., the expected value is trmce what it should be at all frequencies. This theoretical re- sult has been verified by Monte Carlo simulations, but it does

not agree with our experimental data, which shows no signifi- cant level shift in the estimated spectrum. The source of this discrepancy is currently under investigation, but it may be due

to either (a) factors in the experimental data which effectively

make it band-limited, random-walk noise, i.e., its spectral shape

is markedly different from f~^ for, say, < / < /i or (b) an

incorrect guess on our part as to how the spectral estimate is normalized by the spectrum analyzer. For the Hanning window, we found that

E{S{fj)] =

1.08S(/j) j = l; 1.485(/,) J =2;

1.155(/;) J =3;

1.075(/;) J =4;

1.04S(/,) ;=5;

S{f}) 6 < ; < 511 to within 3%.

i.e., S{fj) is essentially an unbiased spectral estimate except

for the lowest few frequencies. This theoretical result has been

verified by Monte Carlo simulations and also agrees in general

with our experimental data. Third, for a random-run process. E{S{fj)]^CsJ- 1 < ; < 400,

TN-249
10 a good appoximation when the uniform window is used. where C.v, is a constant which depends on the number of blocks

.\'t, and increases as .Vi increases. Thus the shape of E{S(fj)}

follows that of a random- walk process (/~^) rather than that

of a random- run process (/"*). This shape has been verified experimentally (see the next subsection), but the dependence

of the level on .Vj has not. The increase in level of £{S(/j)} as

.Vi increases is due to the fact that the expected value of the sample variance of a block of ,V, samples increases with time — by contrast, it is constant with time for the white noise and

random-walk cases. For the Harming window, we found that

£{5(/;)} = C;v-.5(/;), 4<j<400,

to a good approximation, where again CJ^^ is a constant —

different from C.v, — which depends on the number of blocks

.Vj eind increases as Ni increases. For frequencies less than

/< the theoretical results indicate significant (greater than 4%)

distortion in the shape, but these do not agree in detail with the expenmental values reported in Table 1. For /; > /« the shape

has been verified experimentally, but the dependence of the level on iVj has not. The discrepancy in level between the theoretical and experimental results is yet to be resolved, but it is probably

due to a mismatch between the assumed random-run model and

the true spectrum for the data (possibly band-limited randomnm).

1 ) V«rlf7 Spectral DmMity Pmrtion & Voltag* R*faivtca —m—

2) MMSun Spsctnn R*lAtlv« to NdIm Soure* Raqulr*« Multiple Scam

lAJ«(HUUTt WITT

I wm wa.il

I

i"^^"T |^

"""i *

3) Haasur* Flltar Tranafar Ptxtctlon h(f)

^

1 1"^ ^T i^

m4) Calculata Biaaaa

B(f ) - Sv^f) - h2(f) Svn(f)

Figure 1. Outline of measurement procedure for determining

the biases in spectral estimators.

III.B. Experimental Determination

The following proced\ire can be used to experimentally de- termine the bias in the spectral estimate of any noise spectrum

using any window in a particular instrument. The basic concept

is to implement a filter that, when applied to white noise, mimics the approximate noise spectra of interest and then measures

the level of the white noise and the filter transfer function in a way which has high precision and accuracy as illustrated in Fig- ure 1. First, the level of a known white noise is measured over

a convement range. The higher the frequency span the faster that this is accomplished. Obviously, the chosen range must

be one over which the noise source is accurate. To obtain a

precision of order 0.2 dB generally requires 1000 samples This measurement verifies that the spectral density function and the

internal reference voltage of the FFT are accurately calibrated and working properly. Virtually all of the windows accurately

determine the value of white noise if the first few channels are ignored as explained above. Figure 2 shows the measurement

of a noise source, which has been independently determined to have a noise spectral density of 99.8 dBV/Hz by the three win- dows. (Appendix A shows the circuit diagram for this noise source which has an accuracy of better than 0.2 dB for frequencies from 20 to 20 kHz.)

(»«S. 1000 -95 dB(V>

/OIV

Figure 2. Spectral estimation of a white noise standard us- ing the uniform, Hanning and the proprietary 'fattened peak"

windows.

Second, an approximately flat spectrum is measured over

the frequency range of interest. It is not important if there are small variations in the level that change slowly over the fre- quency span. Third, the transfer function of the filter is deter- mined for the frequencies of interest using a very narrow spectral source (typically an audio oscillator is sufficient). The very nar- row source is accurately measured by the window since there is no problem with either high frequency or low frequency noise biasing the estimate. The use of a window with a flattened peak

response is helpful but not necessity if the frequency source is sufficiently stable. This transfer function is then applied to the measured white noise spectrum in step two above. This yields a very accurate value for the "true" spectral density of the white

noise source as measured through the filter. This "true value" is then compared to that obtained by the FFT analyzer. The dif- ference between that measured in steps two and three and that measured directly with the FFT is the bias in the spectral esti- mate for that particular window and noise type. The accuracy

of this approach comes from the fact that the calibration has been broken up into steps that can individually be determined

with high precision and very small bias. The primary assumption is that the FFT analyzer is linear. Even this assumption

can be checked by using precision attenuators. If the known

white noise in step one does not extend to the frequencies of

interest, then there is an additional assumption that the FFT

is flat with frequency. This assumption is nearly always good

except perhaps near the last few channels where the effect of the antialiasing filter might cause small inaccuracies. Figures 3a and 3b show the "true" spectral estimate and

the estimates as measured on a particular instrument using the

uniform Banning, and the instrument's propnetar^' "flattened peak" windows for noise that varies as f~* over much of the

TN-250
BMS. 1000 Table 1. .Approximate Biases m FFT 5pec trai Esf.riates

-SB \\. ; de<v> \^__J\.mttoc noise type f° X^ rnjcnotit HMt channel # uniform Hannan? flattened peak

1 19.6 dB 19.6 dB 20.1 dB

2 small small 16.7 dB

5de \

3 i i 7,2 dB

/OtV \

4

5

small

i

noise type /"*

VVtWFOIH channel # uniform Hanning flattened peak

-M ^\

STAUT, 2S0 HZ a*t 954. 85 Hz STOP. 100 000 Hs 1 unusable 8.6 dB 10.0 dB

2 0.4 dB 9.1 dB

RMS. 1000 3

4

0.4 dB

small

4.0 dB

-52

d8<v> ^--^tWMW; 1.2 dB

,i_!^\;,\<;;n>TIB« RMC

5 i 1.1 dB

^uorar*'*-^^^:*;,^^

. 6 1.1 dB

7 1.0 dB

8 O.S dB

12 9 0.6 dB

/OIV iT^'-'-^.^OOKW 10

111213

14

0.6 dB

0.5 dB

0.4 dB

0.4 dB

small

-148

START. 15 250 Mz B«i 375 MI STOP. 1 00 000 Hz *

Figures 3a (top) and 3b (bottom). Difference between the true spectrum (top) which varies approximately as /"* and that estimated by the uniform, Hanmng and proprietary ''flattened peak" windows (bottom).

range from 1 kHz to 100 kHz. The scan is to 100 kHz. and

1000 samples were taken for all curves. Note the considerable

difference between the spectral estimates for chaimels 1 to 3 for the Hanning and proprietary "flattened f>eak'' windows. These

results conflnn the theoretical calculations above showing that,

for the Hanning window, the first 3 channels should be ignored. For the "flattened peak" window, the first 14 rh^nn.*!* should be ignored. For both /"' and /"* noise, the uniform window

does not yield usable spectral estimates over any portion of the

scan. Note in this example that at frequencies above 80 kHz

there is a small step is the spectral estimates. This is due

to digitizing errors of the signal due to quantization. If the

digitizer had more bits, these errors would not occur. This

problem of dynajnic range is common whenever the spectrum of

interest covers many decades. The usual solution is to use filters to divide the spectrum into various frequency range segments

which are sxiitable for the dynamic range of the FFT.

Table 1 summarizes the measured experimental biases in the spectral estimates of a particiilar instrument with three dif- ferent windows for power-law noise types varying from /° to f~* . This covers most of the random types of noise found in oscillators and signal processing equipment. We do not advocate

using the biases reported in this table to correct data — they

only indicate which channels should not be relied upon for data

analysis.

rv. Variances of Spectral Estimates

rV.A. Theoretical Analysis

We have derived expressions for var{S(f)] — the variance

of 5(7) — for each of the three models considered in Section

III. A. These expressions depend primarily on the number of blocks N^. Again, the details behind these calculations wiU be

reported elsewhere [3]. First, for a white noise process, the uniform window yields

var{S(fj)] = S'{fj)/Nt, 1<;<511.

while the Hanning window yields

f 0.695^ (/j)AV», ;=1;

var{Sif,)}={ S^(/;)/;V4, 2 < j < 510; {l.03S^{f,)/Ni, ;=511.

These results are consistent with our experimental results and

with standard statistical theory.

Second, for a random-walk process, the uniform window

yields

var{S{f,)} = 55'(/;)/.V4. 1 < ; < 511.

TN-251
while the Haniung window yields

1.3052(/;)/.V4,

2.20S^(/;)/.Vk,

1.3lS^(/,)/A'»,

l.l5S\f,)/N,,

l,06S»(/;)/N».

1.045'(/;)/Af».

i S'ifj)/N,,

ror{S(/;)} = ,

; = 2; ; = 3; ;=4;

;=5;

;=6;

J =7;

8 <; < 511 to within 3%.

Except for the few lowest frequencies, the resiiJts for the Hiui- ning window agree with our experimental results and with stan- dard statistical theory; however, the factor of five in the variance

for the uniform window disagrees with oui experiments and with

standard theory (although it has been verified by Monte Carlo

techniques). The cause of this discrepancy is under investigation, but we think it is due to the band-limited nature of the experimental data. Third, for a random-run process, the variance computations are not useful since the variance is dominated by the fact that the expected value of the sample variance for each block of samples increases with time. The agreement which we found be- tween standard statistical theory and our experimental results on the l/Ni rate of decrease of variance is undoubtedly due

to the band-limited nature of the experimental data. We will attempt to verify these conclusions in the future using Monte

Carlo techniques.

SORT (MAC? / BW) -73 decv)

10 OB /Olv

-153

STARTi

^

\^1

2S0 Hi XSTOPi 100 000 Ha

Figure 4. Comparison of the spectral estimate of /"* powerlaw noise with 1000 samples with that obtained with 32 samples. The text explains how these two curves are used to obtfun

the fractional RMS confidence of the spectral estimate for 32

samples.

rV.B. Experimental Determination

The following procediire can be used to experimentally de- termine the variance of the spectral estimates of virtually any

type of noise spectrum with any type of window for a particular instrument. Since the si>ectral density of interest is in general

nonwhite, we must determine both the "^rue value" and a way

to normalize the fractional error of the estimate as a fimction

of the number of samples. This can be done by making use

of the above theoretical analysis that shows that the variance should decrease as the square root of the nimiber d samples

since they are approximately statistically independent (in fact, exactly so in the cases of white and random-walk noise). As an

example we have chooen to take .Vj = 1000 blocks of the -.-ar.ous power-law noise types examined in III.B above and compare the value of the spectral estimate with that obtained from .V;, = 32

blocks (see Figure 4). Since the variance of the 1000 block data

is about 32 times smaller than that c^ the 32 block data, it can

serve as an accurate estimate of the "true value." Let Siooo(/;

)

represent this quantity at the;-th channel (frequency). By subtracting the 1000 block data from the 32 block data at the ;-th channel, we then have one estimate of the error for the 32 block

data; by repeating this procedure over Nc different channels and Nr different replications, we can obtain accurate estimates

of the variance for the 32 block data. Let S32,(/j ) represent the

spectral estimate for the 32 block data at the ;-th channel and

the t-th replication. To compensate for the variation in the level of the spectral estimates with channel, it is necessary to divide the error at the j-th channel by the "true value" Siooo(/j)- The mean square fractional error of the 32 block data for the noise type under study is given by

rl,-

1 ^r^c S\OOo(f}) ^ var{S32,{fj)}

It is assumed that all channels with bias — as indicated in Table 1 — have been excluded in the summation over j. It is edso important that the changes in the spectral density not exceed

the dynamic range of the digitizer because imder this condition

the quantization errors — in addition to causing biases in the

spectral estimates as discussed earlier — can lead to situations where the variance does not improve as N^ increases. These val- ues can be scaled to any number of blocks N^ if care is taken to avoid these quantization errors. Upper and lower approx:mate

67% confidence limits for S{fj) — the true spectral density at channel j — using Banning, uniform and the proprietary "'flat- tened peak" windows for N^ approximately independent blocks

are given by

S(/;)(l±V(a,;V»))

where S{fj) is the spectral estimate given by Equation (1) and

V{a,Ni) is the fractional variance given in Table 2 for /" and a = 0, -2, -3 and -4 (these results were obtained by averaging

over NrNe = 1200 channels). The variances obtained are very

close to those obtained from standard statistical analysis for white noise, i.e., S(l•i^^w^

Table 2. Confidence Intervals for FFT Spectnd Estimates

power law window

noise type iinifnrm Hanning flattened peak

r 1.02/v^ 0.98/v^ 0.98/ v^

r' 1.02/v^ l.(A/y/Ki 1.04/v^

/-' unusable l.Oi/s/TT^ 1.04/^7^

r* unusable 1.04/v^ 1.04/v^

V. Conclusions

We have introduced experimental techniques to evaluate the statistical properties of FFT spectral estimates for common

noise types found in oscillators, amplifiers, mixers and similar

TN-252

I
dev-.ces. and we have corrpared these »nth •.heoreticai cajcuations We have used these lechmques to study the biases and

variances of FFT spectral estimates using the uniform. Han- mng. and & propnetar>' "flattened f)«ak" window. The theoretical ac&iysis was greatly hampered because the instrument

manufacturer does not disclose the exact form of the "flattened peak" window or the normalization procedure for the other win- dows. Nevertheless, we obtained fair agreement between the

theoretical and the experimental analysis. The vanances of the spectral estimation were virtually identical to a few per- cent for p to /"* noise except for the uniform window which

is incapable of measuring noise which falls aR faster than f~^

.

There was a very large difference in the biases d the first few

channels for the three windows. The Hanning window showed

significant biases in the first 3 channels while the proprietary

"flattened p)eak" window showed large biaaes for /** noise even up to channel 13 The Hanning window therefore yields useful information over three times wider frequency range than the

proprietary "fattened peak" window. In the particular instru- ment studied, the proprietary "flattened peak" window is the

best choice for estimating the height of a narrow band source, while the Hanning window is by far the best choice for spectral

analysis of common noise types found in oscillators, amplifiers,

mixers, etc. We have also shown that the 67% confidence levels for spectral estimation as a ftmction of the number of contiguous

nonoverlapping blocks, .Vj, is approximaieiy given by

^ = ^"" = ^J

for white noise (/°) and by

S = S^ 1 ±

1.04 v.v»

for noise types /"' to /"*. This agrees to withm 47c of that found by standard statistical analysis for white noise Using

this data one can now determine the number of samples neeessarj- to estimate — to a given level of statistical uncertainty — the spectrum of the various noise types commonly found in oscillators, amplifiers, mixers, etc.

References

[1] D. Babitch and J Oliveno, "Phase Noise of Various Oscil- lators at Ver>' Low Fourier Frequencies." in Proceedings of the tSth Annual Symposium on PrtqMeney Control. 1974. pp. 150-159.

[2] A. V. Oppeaheim and R. W. Schafer, Dtsertte-Ttme Signal

Proeesiing. Englewood Cliffs. New Jersey: Prentice Hall. 1989, ch. 11, pp. 695-767.

[3] D. B. Percival and F. L. Walls. "Spectral Analysis of Power- Law Processes with a Spectrum Analyzer," manuscript m

preparation, 1989.

[4] M. B. Priestley, Spectral Analyaxa and Time Series London: Academic Press, 1981. ch. 7, pp. 502-612. ^

-V in li.-Kr 7^_k

«i> It

lu

W OUT

^^

NOZK II I iw*!! nil cofocxrcm :R'R:

C4ir 1241

Figure 5. Circuit diagram d a precision noise source.

Appendix. Precuion Nob* Source

Figure 5 shows the circuit diagram of a precinon noise source whoae spectral density can be determined £rom first principles to ±02 dB over the frequency range from 20 Hz to 20 kHz.

The spectral density is basically given by the Johnson noise of the 10* ohm resistor, V,' = AkTR, where T is in Kelvin, and k is Boltzmann's constant. Corrections due to the input noise volt- age and noise current d the amplifier amount to about 02 dB

for the circuit elements shown. All resistors are precision iTc metal film resistors. The output level can be switched from

-100 dBV/Hz to -80 dBV/Hz. By adjusting the noise-gair. ca- pacitors one can make the noise spectrum flat to wuhin 3 dB

out to 200 kHz. There is also provision to measure the input

noise voltage of the amplifier by shorting the input to ground

or the combined nosse voltage and noise current by s^ntching a 220 pF capaaior into the input instead of the 10* ohm noise

resistor.

TN-253
Proc. 35th Ann. Freq. Control Symposium, USAERADCOfI, Ft. Monmouth, NJ 07703, Hay 1981

A MODIFIED "ALLAN VARIANCE" WITH INCREASED

OSCILLATOR CHARACTERIZATION ABILITY

David W. Allan and James A. Barnes

Summary

Heretofore, the "Allan Variance," a ^(x), has become the de facto standard for measuring oscillator instability in the time-domain. Often

oscillator frequency instabilities are resonably

model able with a power law spectrum: S (f) ~ f"^ where y is the normalized frequency, ^f is the Fourier frequency, and a is a constant over some range of Fourier frequencies. It has been shown

that for power law spectrum a ^(t) ~ t*^, and that

p = -a-1 for -3<c(< + 1, where t is the nomimal

sample time over which each value of y is measured.

The modified "Allan Variance" developed in this

paper yields p = -a-1 for al] a in the range -3<a,

which removes the previous ambiguity: p = -2 for

+l<a. In other words, with the modified "Allan

Variance" one can easily distinguish between white

phase noise (a = +2) and flicker phase noise

(a = +1) — commonly occurring for the short term instabilities of quartz crystal oscillators and

active hydrogen masers.

Key Words . Flicker Noise; Frequency Stability;

Oscillator Noise Modeling; Power Law Spectrum;

Time-Domain Stability; White Noise.

Introduction

The random fluctuations in precision oscillators may often be characterized by a power law

spectrum:

Time and Frequency Division

National Bureau of Standards

Boulder, Colorado 80303

the same, i.e

SyCf) = h^ f^ , (1)

where y is the normalized frequency deviation, f is the Fourier frequency, h is the intensity of

the particular noise process, and a is constant

over some range of f. The typical values of a

are: +2 (white noise phase modulation, PM); +1

(flicker noise PM); (white noise frequency

modulation, FM); -1 (flicker noise FM); and -2 (random walk FM). The Allan variance, as it has come to be known, has been demonstrated as a very

useful statistical tool for characterizing these

various random processes with the exception that

if a = +1 or +2, the dependence on x is nominally

~ T-2. It is not at all unconfflon for white PM and flicker PM to occur in precision

oscillators for t of the order of one second and

shorter. The modified Allan variance, as developed in this paper, depends as x-^ for a = +1 and as

t-3 for a = +2. This yields a clear distinction in the time domain between these heretofore somewhat

ambiguous processes.

Definition of "Allan Variance"

and Related Concepts

Define y, the normalized frequency deviation,

as

y(t) = v(t)

(2)

where v(t) is the output frequency of the oscillator being studied, and v is nominally the same frequency, but of a reference oscillator assumed

for the moment without loss of generality to be

better than the test oscillator. The time deviation from some arbitrary origin (t =0) is the integral of the frequency deviations (from that

origin):

x(t)

,th

= /y(t' )

• dt' (3)

The i— average frequency deviation over an interval , X, is

yi =

^i+i (4)

where the assumption is made that the time deviation measurements are nominally spaced x apart.

470

TN-254
The "Allan Variance" Is defined as:

o/(0 = i <(y^^i -y^)^> (5)

where the brackets "< >" denote Infinite time

average. Using equation (4), one may write:

o/ii) = ^

<(Xi,2 2^1+1 * ^i)'> (6)

It has been shown that typically o *(t)

u ^12

varies as x^, and that \i = -a-1 for -3 < a < +1. '

Hence, we see one of the dimensions of usefulness

of a 2(x); i.e., ascertaining the dependence on x allows an estimate of a (the power law spectral

type of noise). However, if a > +1, then tj s -2,

and the x dependence becomes somewhat ambiguous as to the type of noise 1n this region. It is interesting to note that in the region a > +1, o *(x)

Is bandwidth (f.) dependent; i.e., the bandwidth

of the measurement system will affect the value of o (x), and furthermore, one may use the bandwidth

y 3 dependence to determine the value of a (see also

Appendix Ref. 2).

Development of the Modified Allan Variance

One may also write o^iz)

generalized autocovariance function:

One may also write o^iz) in terms of a

.^(^) = ^ C4U^(x) U^(2x)]

where

U^(x) = 2CR^(0) - R^(x)],

(7)

(8)

U^(x) ~ a(a)

-a+1 -a+1

(10)

Hence, one notes that by changing the reciprocal

bandwidth as well as x, one affects a ^(x) in similar ways, depending on the value of a. From

this, one should be able to deduce the value of a, since the bandwith dependence becomes stronger for a moving positive from +1, and the x dependence

becomes stronger as a moves negative from +1. One

can change the bandwidth in the hardware or in the

software. In the past, it has typically been done

3 4

in the hardware. James Snyder has shown that it

is relatively easy to change the bandwidth in the

data processing by a clever technique and we have

followed his lead. In particular, we have chosen

a new variance analysis scheme which coincides

with the Allan variance at the minimum sample

time, X , (i.e., minimum data spacing), but which

changes the bandwidth in the software as the

sample time, x, is changed.

Each reading of the time deviation, x., has associated with it an intrinsic nominal (hardware) measurement system bandwidth, f. . Define

X. = XTT-; and similarly we may define a software

h bandwidtn, f = fi,/n, which is 1/n times narrower

than the hardware bandwidth. This software bandwidth can be realized by averaging n adjacent

x.'s; X = nxu, where x = l/'f-- We have defined

a modified Allan variance which allows the reciprocal software bandwidth to change linearly with the

sample time, x:

and where

\M = <x(t+x) x(t)> (9)

the classical autocovariance function of x(t).

Using the Fourier transforms of generalized functions, one may determine the coefficients relating

the power spectral density to o ^(x). Ref. 1 gives these relationships. It is of interest to note that U (x) has the following approximate form

in the region a > + 1 (see Appendix Ref. 2):

Mod c^Ht) = 2Tj([^X)^x.,2, - 2x.,^ * X.)

\ 1=1

where x = nx.

>(11)

. . Eq. 11 clearly coincides with Eq.

6 for n = 1. One can see that, in general, we

have formed a second difference of three time

readings with each of the three being an average

of n of the x.'s (with non-overlapping averages).

As n increases, the (software) bandwidth decreases

and this bandwidth varies just as f^ = f^^/n- For a finite data set of N ireadings of x^

(i = 1 to N), we may write an estimate:

471

TN-255
Mod a^Hz) =

2xin2(N-3n-Hl) '

(12)

N-3n+l

E

j=i

n+j-l

L i=J

T 2

Eq. 12 is easy to program, but takes more time to compute than for o (x). This is only of signifi- cance for the smaller computer or handheld calculator.

Comparisons, Tests, and Examples of Usage

of the Modified Allan Variance

We simulated various power law noise processes, and applied Eq. 12. Shown in Fig. 1 are the

resulting i-dependences of the modified Allan

variances for a = -2, -1, 0, +1, and +2. The

solid lines drawn are the anticipated or theoretical slopes for the particular noise process. One

sees excellent agreement for white noise PM and

for flicker noise PM, and nominal agreement for

the others. One can express Eq. 11 in terms of the gen- eralized autocovariance function:

Mod ay2(T) = j^ {C4U^(1:) "x^^T)] • n

n-1

+ E (n-i)[-6U^(iT^) + 4U^((n+i)T^)

i-1

(13)

+ 4U^((n-i)i^) - U^((2n+i)t^j)

- U^((2n-i)Tjj)]

In the range ^3 < a < +1, one may write:

U^(t) = a(a) • x'°*'^ , (14)

which when substituted in Eq. 13, and using Eq. 7,

yields

Mod a. Hz) = o 2(1) (l * —^

y I" n2 4n °*^ - (2n]

1 E (n-i) . [- i=l '-

y

n-1

-a+1

-6i-°*^*4(n+i)-"*l (15)

ilnH)-"*^ + 4(n-i)-"*^ - (2n-i)"°*l

Since we know that o^{x) is well behaved in this

range and p = -a -1, it is of interest to look at the ratio:

R(n) = Mod Oy2(t)/Oy2(x)

As stated before, at n = 1 (t = t ) the ratio is unity. One can evaluate Eq. 15 with a computer. A reasonable empirical fit may be formed, which is good to 0.5% or better of Eq. 15:

R(n) = q"P"VP

qn

(16)

which approaches p/q asymptotically as n ap- proaches infinity, and is within IX of p/q for

n > 8. Listed in Table 1 are the empirical values

of p, q, and E and the quality of fit for the

appropriate power law noise processes.

T*BLE 1

No 1st Type

P q E fit

Nod o„(T)/o„(t) Nu« '"'o

Whitt FM 1 2

Flicker FK -1 99.9 1*8

Rando* Walk FM -Z 33 40

Flicker Walk FM -3 1 1

2 perfect .707

2.35 W .S21

2.35 <W .908

-- perfect 1

The results of Table 1 are in reasonable

agreement with simulated results of Fig. 1(a)

through 1(e). The last row in Table 1, "flicker

walk" frequency modulation, is out of the range of

applicability of a, but the ratio, R(n), is still

convergent.

472

TN-256
The U (t) function for flicker noise PM is extremely complicated and has not been developed,

but one can arrive at an empirical value for it. The U (t) function is derivable for the other

power law spectral processes. Table 2 gives the

relationships between the time domain measure

Mod o^(x) and its power law spectral counterpart,

given in Eq. 1. Also listed in the right hand

column of Table 2 are the asymptotic values

of R(n):

Noise Tyi>« Nod Oy»(x) CoMcnt <(M Nue a

Whit* W -2 "•>

"2' Exict 1 (2n)J n t*

Flicktr PM 1

1.038 31n(u.^T) Upfrical 1 "l (2n)i iS

Whitt FH Exict 0.5

FUeker FM -1 h.j an(2) R(n) Eapipical; Exact Available0,674

Randoa Walk FH -2 h.,. i^ • «o) Eapirical; Exact Available

0.824

It is clear from Table 2 that Mod a ^ix) is very useful for white PM and flicker noise PM, but

for a < +1 the conventional Allan variance, a^ix),

gives both an easier-to-interpret and an easier-tocalculate measure of stability.

It is interesting to make a graph of a versus

\i for both the ordinary Allan variance and the

modified Allan variance. Shown in Fig. 2 is such

a graph. This graph allows one to determine power

law spectra for non-interger as well as interger

values of a. The dashed line for the modified

Allan variance has been intentionally moved to the

left in Fig. 2 because for small values of n the value of [1 will appear to be slightly more negative

that for o ^(t), even though for large n, they

both approach the same slope (i.e., the same values of p). In fact, in the asymtotic limit,

the equation relating p and a for the modified

Allan variance is

a = -p -1, for -3 < a < +3 . (17)

The value of p = -4 for a = +3 was verified empirically with simulated data, and it appears that

for a > +3, p remains at -4. A direct application for using the modified

Allan variance recently arose in the analysis of

atomic clock data as received from a Global

Positioning System (GPS) satellite. We were

interested in knowing the short-term characteristics of the newly developed, high-accuracy NBS/GPS

receiver, as well as the propagation fluctuations.

Fig. 3 shows both o^d) and Mod o^ix) for comparison. Using Mod a ^(i), we can tell that the

fundamental limiting noise process involved in the

system is white noise PM with the exciting result

that averaging for four minutes can allow one to ascertain time difference to better than one nanosecond excluding other systematic effects.

Conclusion

We have developed a supplemental measure, the

"Modified Allan Variance" (Mod o 2(x)), which has very useful properties when analyzing oscillator

or signal stability in the presence of white noise

phase modulation or flicker noise phase modulation.

It also works reasonably well as a stability

measure for other commonly occuring noise processes

in precision oscillators.

We would recommend that for most time domain

analysis, a ^(x) should be the first choice. If a 2(t) depends on t as x-'-, then the modified

Allan variance can be used as a substitute to help

remove the ambiguity as to the noise processes.

Acknowledgments

The authors are deeply grateful for stimulating discussions with Dr. James J. Snyder, and

for the useful comments of Dr. Robert Kamper and

Mr. David A. Howe.

References

1. James A. Barnes, Andrew R. Chi, Leonard S. Cutler, Daniel J. Healey, David B. Leeson, Tomas E. McGunical, James A. Mullen, Jr., Warren L. Smith, Richard L. Sydnor, Robert F.

C. Vessot, and Gemot M. R. Winkler, Proc. IEEE Trans. Instrum. Meas. IM-20 , 105 (1971);

also published as NBS Tech. Note 394 (1971).

2. David W. Allan, Proc. IEEE 54, 221(1966).

3. R. F. C. Vessot, Proc. IEEE 54, 199 (1966).

4. James J. Snyder, Proc. 35th Annual Symp. on

Freq. Control (1981), to be published.

• See Appendix Note # 34

473

TN-257
SIMULATED NOISE

0.36 r

OO

2 -2

10

1 10 10' SAMPLE TIME, T

(d)

"• •" 0.22 tO

J

1 10 10' SAMPLE TIME, T

o

o

2 -2

10

FLrCKERFM

X J

1 10 10^ SAMPLE TIME, t

I

(e)

10

RANDOM WALK FM

.0.2r1/2

SAMPLE TIME.T

Fig la-e. Mod a (x) using Eq. 12 was calculated for different sample times for

independently generated and simulated noise processes, which were white phase noise, flicker phase noise, white frequency noise,

flicker frequency noise, and random walk frequency noise, respectively. Mod a (x) was computed for 399 data points in each case. One sees the excellent fit to the theory for white phase noise and

flicker phase noise, an important new contribution in the ability to characterize oscillators having these noise processes.

474

TN-258
332 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 4, DECEMBER 1984

Copyright ® 1984 IEEE. Reprinted, with permission, from IEEE Transactions on

Instrumentation and Measurements. Vol. IM-33, No. 4, pp. 332-336, December 1984.

Characterization of Frequency Stabflity: Analysis of

the Modified Allan Variance and Properties of

Its Estimate

PAUL LESAGE and THEOPHANE AYl

Abstnct-An inalytical expression for the modified Allan variance is given for each component of the model usually considered to describe the frequency or phase fluctuations in frequency standards. The rela- tion between the Allan variance and the modifled Allan variance is specified and ccmipaied with that of a previously published analysis. The uncertainty on the estimate of the modified Allan variance calcu- lated from a finite set of measurement data is discussed.

Manuscript received December 8, 1983; revised February 17, 1984. The authors are with the Laboratoire de rHorloge Atomique, Equipe

de Recherche du CNRS, associee a I'Universite Paris-Sud, 91405 Orsay,

France.

TN-259
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 4, DECEMBER 1984 333

I. Introduction

Many works [ 1 1-[5] have been devoted to the characterization of the frequency stability of ultrastable frequency sources and have shown that the frequency noise of a generator can be

easily characterized by means of the "two-sample variance"

[2] of frequency fluctuations, which is also known as the "Allan variance" [2] in the special case where the dead time

between samples is zero. An algorithm for frequency measurements has been devel- oped by J. J. Snyder [6], [7]. It increases the resolution of frequency meters, in the presence of white phase noise. It has been considered in detail by D. W. Allan and J. A. Barnes (8).

They have defined a function called the "modified Allan

variance" and they have analyzed its properties for the com- monly encountered components of phase or frequency fluctu- ations (3). For that purpose, the authors of [81 have expressed

the modified Allan variance in terms of the autocorrelation of

the phase fluctuations. For each noise component, they have

computed the modified Allan variance and deduced an empirical expression for the ratio between the modified Allan variance and the Allan variance. In this paper, we show that the analytical expression of this ratio can be obtained directly, even for the noise components

for which the autocorrelation of phase functions is not defined

from the mathematical point of view. We give the theoretical expressions and compare them with those published in [8]

.

The precision of the estimate of the modified Allan variance

is discussed and results related to white phase and white fre- quency noises are presented.

II. Background and Definitions

In the time domain, the characterization of frequency stability

is currently achieved by means of the two-sample variance

[2] (ay (2, T, t)> of fractional frequency fluctuations. It is defined as <a^(2, r,r)> = i<(7kM -yk?> (1) where the quantity yi^ is the average value of the fractional fre- quency fluctuations y{t) over the time interval {t,(, t/f +r)

such that

yk

1 r''*'

y(t)dt. (2)

In (2), t^ represents the moment at which the Jtth observation

time interval starts. We have

t^ =to +kT, T>T (3) where Tq is an arbitrary time origin, * is a positive integer, and

T is the time interval between the beginning of two successive observations.

In all the following, we assume that the dead time between

samples is zero. We then have

T = T. (4) In this special case, the two-sample variance is well known as

,y,., .^yy.,.,.„. (5)

the Allan variance Oy (r)

al(T) = {ali2,T,T)).

The relation between the Allan variance and>'(r) can be ex- pressed as

(^'(r) = p-{^jj yiOdt-j'" y(t)dtj). (6)

Equation (6) shows that Oyir) is proportional to the true vari- ance of the output of a linear filter with input signal ><(/) and

impulse response hi(t) in Fig. 1.

2T

Fig. 1. Variations with time of the linear filter impulse response which

represents the signal processing for the Allan variance calculation.

T T

T, "* " "

Fig. 2. Illustration of the algorithm considered for the measurement of

periodic signal frequency.

The fractional frequency fluctuations j'(r) are actually well described by a conventional model which consists of a set of

five independent noise processes [2]. Taking into account the

finite bandwidth of the processed signal and assuming a single

pole filter, the one-sided power spectral density Sy{f) of yit) can be written as Syin=h^ r

1 + (^)

(7)

where coefficients h^ do not depend on /. The integer a equals

2, 1, 0, - 1, and -2. /^ is the 3-dB bandwidth of the hardware

filter.

III. The Modified Allan Variance

The main property of the algorithm developed by J. J. Snyder is to increase the precision on the measure of periodic

signal frequency, in presence of white phase noise (7) . It consists in dividing a time interval r into n cycles of clock period

To such as

r = «To. (8)

Therefore, from a given observation time interval of duration

2t, n overlapping time intervals of duration r can be obtained,

as depicted in Fig. 2. Another property of this algorithm is to reduce the total observation time by a factor n/2. Following this way, Allan and Barnes have introduced the "modified Allan variance" [8] such as

•ro + (/+2rt)To Mod

1 /ri « ( r'o*('*^'')To

2t \Ln ,.il^,o*(<+«)ro

-J

yit)dt]]). (9)

It can be easily seen from (9) that the calculation of each

statistical sample involved in the definition of Mod a^ (r) requires a signal observation of duration 3r. The impulse response /»„(?) of the equivalent linear filter consists in finite sum of n shifted impulse responses h^{t).

We have

(10)

TN-260
334 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33. NO. 4, DECEMBER 1984

NOISE TYPE

WHITE P M

FLICKER P M

WHITE F M

TABLE I Analytical Expression for the Modified Allan Variance within Condition lirfcTa » I

a Mod a (-) y

3h3f^

h r n-l , 2

. , , , 3nen(2TTf t) * :(n-k)4en(^ - 1) - m <^ ')]

n^^l

2n'

FUCKER F M

2h_ en2

4n - 3n + 1 -J

X I (n-k) x|^ (k +2n)en(k +2n)-(k - 2n)en(2n - k )]

+

-J (k.fn)(k-2n)tn(k*n) +1 (k - n)(k +2n)iln| k - n; + 3k^lnk - k[(n + 2k)tn(k +i)

(n -2k)£n|k -,

RANDOM WALK F M

40 8n^ 20n

In order to illustrate (10), variations with time of the shifted functions /ii(r- ztq) and of the impulse response ^„(r) are represented in Fig. 3(a) and (b), respectively, for n = 10. For n = \, the Allan variance and the modified one are equal.

We have = „2, Mod a^(T) = a; (t) (11)

One can express (9) in terms of the spectral density Sy(f).

We have

Mod4(T) = ";;i77r*j"j -T5^(/)sin'' (7r/nTo)d/

"^' r 1 Sy(f) cos (InfkTo)

k = l

sin* {iffnro)df (12)

It should be noted that the integrals involved in ( 1 2) are convergent for each noise component. The analytical expression

for the modified Allan variance can therefore be deduced

directly from this equation.

In the following, it is assumed that the condition InfcTo » 1 is fulfilled. This means that the hardware bandwidth of the measurement system must be much larger than the reference clock frequency.

We have calculated the modified Allan variance for each

noise component. Results are reported in Table I. It appears

that the analytical expression for Mod Oyir) is relatively simple

except for flicker phase and flicker frequency noises where it is given as a finite sum of functions depending on n. In order

to compare the AUan variance with the modified one, we

z 1

= 2

= 3

r4

= 5

= 7

= e

= 9

= 10

it^JM

1

(a)

'.2nT„

Fig. 3. (a) The impulse response h„it), associated with the modified

Allan variance calculation, represented as a sum of n shifted impulse response hyit). It is assumed n = 10. (b) Variations with time of

the impulse response h„(t), in the special case where n = 10.

* See Appendix Note # 35

TN-261
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 4, DECEMBER 1984 33S

TABLE II Analytical Expressions and Asymptotical Valus for R(n)

(Results Are Valid within Condition Itt/cTo » I)

a R(n) lim R(n)

n — ^

2

In

1

1 1 "-1 J iJ )'

n f „ . ._„ , If " ( r\ L-\ A On { '1 'n/ .^^

3 ir\(Z-f^nzJ k = l k' k' 1

2n^ 0.5

. 1

." - Jn * 1 ^ __1 , :{„.,()" (k +2n)£n(k +2n)-(k -2n)tn(2n -k)

n L 2 n'^zn'^ k = l ^ '

+ -i (k +n)(k - 2n)Zn(k n) * i( k - n){k 2n )in k - n + 3k^ ink - k (n 2k) Cn(k » j)

- (n -2k)in '.y.- J

0.787

- 2 23 , 1 * 1 40 8n 20n* 0.825

consider the ratio R (n) defined in [8] as

/?(/!) = Mod 4{r)/a^(T). (13)

The analytical expressions for Rin), deduced from Table I, are reported in Table II. One can see that Rin) does not depend on the product /^To, except for flicker phase noise modulation. The asymptotic values of R(n) are also Listed in Table II. Fig. 4 depicts the variations of Rin) with n. It shows that, for large values of n, white phase and flicker phase noise modulations have different dependences. As outlined in [8],

this gives a means to easily distinguish these two noise processes, in the time domain. For large n, and for a = 0, - 1 , - 2, Rin) remains a constant. Consequently the Allan variance can

be deduced from the modified one, for these noise processes. A comparison with results of [8] shows a good agreement

for a = 2, and - 2. But, for a = 1 and - 1 , our expressions for the modified AUan variance and ratio /?(n) disagree, especially

for flicker phase noise modulation. This discrepancy might

be due to the fact that in [8] , Mod Oyir) is expressed in terms

of the autocorrelation function of phase fluctuations which

is not defined fora = 1.

iv. uncertainty on the estimate of the

Modified Allan Variance

Equation (9) shows that the definition of the modified Allan

variance theoretically impUes an infinite set of time intervals. Practically, one can only estimate this quantity from a finite set of m successive cycles similar to the one depicted in

Fig. 3(b). Let Mod a^(r) be the estimated modified Allan variance (EMAV) such as Mod a^(T) =

,R(n)

1 iT^n^I m I n — y y

k » 1 I «= 1 ' (14)

Fig. 4. Variations with n of the ratio R («), for fractional frequency

fluctuations with power law spectrum Sy(f) = h^ (1/1 + (f/fc)'^)f°, within condition litfcTQ » 1. ('For a = +1, ^ (/») is a function of ^ and tq. The reported variations are for Inf^TQ = 10*.)

where

ro + (/ 2/i)To (* - 1 )3m'o

''fO + (i + /i)To + (k-l

yit)dt

I

)3rtTo

fO + 'To+('f-l)3'>To

* See Appendix Note # 35

yit)dt.

TN-262

(15)
336 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-33, NO. 4, DECEMBER 1984

The EMAV is a random function of m. Its calculation requires an observation time of duration 3mT.

We consider e, the fractional deviation of the EMAV relative to the modified Allan variance defined as foUows:

€ = Moda^(T)- Modff^(T)

Mod aJ(T)

(16)

The standard deviation o(e) of e defines the relative uncertainty on the measurement of the modified Allan variance, due to the

finite number of averaging cycles. We have

a(e) =

1 Mod aJ(T)

{(t" [Mod oUt)]}"^ (17)

where a^ [Mod dyir)] denotes the true variance of the EMAV

such as aMModa^(T)] =<[Mod a^(T)]'>- [Mod aJ(T)]'. (18)

We assume that the fluctuations >»(/) axe normally distributed [10]. One can therefore express ([Mod Oyir)]^) as m2<[Moda^(T)]'>

= (m2 +2m) [Mod aj(r)]2

+ 4 ^ (m

p = i p) j 2 "£'(«-/)/„+ n/o

/-I (19)

where /„ are integrals which depend on n and on the noise process. We have

SrrV^ Syif) r

cos 6iinpfTo

X {6 cos 27r/To / - 4 cos Itt/toO + n)

- 4 cos 2nfTo(i - n) + cos InfToU + 2n) + cos 27r/To(/ - 2n)} df. (20) For each noise component, the expression for a(e) can be deduced from the calculation of integrals involved in (20). These expressions are generally lengthy and complicated except

for white phase and white frequency noise modulations, where

integrals /„ equal zero. We have limited the present analysis to these two noise components. We get for a(e)

2 a(e) = —, for a = 2 and 0. m

(21)

We now compare (21) with previously published results related to the estimate of the Allan variance [5). For a given time

observation of duration "imr, it can be easily deduced from

[5) that the relative uncertainty on the estimate of the AUan

variance varies asymptotically as 1.14 m'^l'^ and 1.0 m"^!'^ for = 2 and 0, respectively. For these two noise components, the uncertainty on the EMAV is larger than the uncertainty on the estimated Allan variance, but of the same order of magnitude.

V. Conclusion

We have calculated the analytical expression for the modified Allan variance for each component of the model usually

considered to characterize random frequency fluctuations in precision oscillators. These expressions have been compared

with previously published results and the link between the Allan variance and the modified Allan variance has been

specified.

The uncertainty on the estimate of the modified Allan variance has been studied and numerical values have been reported

for white phase and white frequency noise modulations.

In conclusion, the modified AUan variance appears to be well suited for removing the ambiguity between white and flicker phase noise modulation. Nevertheless, the calculation of the modified Allan variance requires signal processing which is complicated, compared to the Allan variance. In the presence

of white or flicker phase noise, the Allan variance cannot be

easily deduced from the modified Allan variance. Further- more, for a given source exhibiting different noise components,

the determination of the AUan variance from the modified one is difficult to perform. For most of time-domain measurements, the use of the AUan variance is preferred.

ACKNOWLEDGMENT

The authors would like to express their thanks to Dr. Claude

Audoin for constructive discussions and valuable comments on

the manuscript.

References

[ 1 1 D. W. Allan, "Statistics of atomic frequency standards," Proc. IEEE, vol 54, pp. 221-230, Feb. 1966.

[2J J. A. Barnes et al., "Characterization of frequency stability," IEEE Trans. Instrum. Meas., vol IM-20, pp. 105-120, May 1971.

[3] L. S. Cutler and C. L. Searle, "Some aspects of the theory and measurement of frequency fluctuations in frequency standards," Proc IEEE, vol 54, pp. 136-154, Feb. 1966.

[4] J. Rutman, "Characterization of phase and frequency instabilities in precision frequency sources: Fifteen years of progress," Proc. IEEE, vol 66, pp. 1048-1075, Sept 1978.

[5) P. Lesage and C. Audoin, "Effect of dead-time on the estimation of the two-sample variance," IEEE Trans. Instrum. Meas., vol

IM-28, pp. 6-10, Mar. 1979.

[6] J. J. Snyder, "Algorithm for fast digital analysis of interference

fringes," /4ppi Opt., vol 19, pp. 1223-1225, Apr. 1980.

[7] —, "An ultra-high resolution frequency meter," in Proc. 35th Annu. Symp. Frequency Control (Fort Monmouth, NJ), 1981,

pp. 464-469.

[8] D. W. Allan and J. A. Barnes, "A modified Allan variance with

increased oscillator characterization ability," in />oc. 35th Annu. Symp. Frequency Control (Fort Monmouth, NJ), 1981, pp. 470-

474.

[9] B. Picinbono, "Processus de diffusion et stationnarite," C.R. Acad. Sci.,\6l 271, pp. 661-664, OcL 1970.

[10] P. Lesage and C. Audoin, "Characterization of frequency stability: Uncertainty due to the finite number of measurements," IEEE Trans. Instrum. Meas., voL IM-22, pp. 157-161. June 1973.

0018-9456/84/0900-0336$01.00 © 1984 IEEE

TN-263
From: Proceedings of the 15th Annual PTTI Meeting, 1983.

THE MEASUREMENT OF LINEAR FREQUENCY DRIFT IN OSCILLATORS

James A. Barnes

Austron, Inc., Austin, Texas

ABSTRACT

A linear drift in frequency is an important element in most

stochastic models of oscillator performance. Quartz crystal

oscillators often have drifts in excess of a part in ten to

the tenth power per day. Even commercial cesium beam devices

often show drifts of a few parts in ten to the thirteenth per

year. There are many ways to estimate the drift rates from

data samples (e.g., regress the phase on a quadratic; regress

the frequency on a linear; compute the simple mean of the

first difference of frequency; use Kalman filters with a drift term as one element in the state vector; and others).

Although most of these estimators are unbiased, they vary in

efficiency (i.e., confidence intervals). Further, the estimation of confidence intervals using the standard analysis

of variance (typically associated with the specific estimation technique) can give amazingly optimistic results. The

source of these problems is not an error in, say, the re- gressions techniques, but rather the problems arise from

correlations within the residuals. That is, the oscillator

model is often not consistent with constraints on the analysis technique or, in other words, some specific analysis

techniques are often inappropriate for the task at hand.

The appropriateness of a specific analysis technique is critically dependent on the oscillator model and can often be

checked with a simple "whiteness" test on the residuals.

Following a brief review of linear regression techniques,

the paper provides guidelines for appropriate drift estimation for various oscillator models, including estimation of

realistic confidence intervals for the drift.

I . INTRODUCTION

Almost all oscillators display a superposition of random and deterministic

variations in frequency and phase. The most typical model used isl^J:

X(t) = a + b*t + Dr«t2/2 + (j)(t) (1) ^

where X(t) is the time (phase) error of the oscillator (or clock) relative to some standard; a, b, and Dr are constants for the particular clock; and q)(t) is

the random part. X(t) is a random variable by virtue of its dependence on (j)(t).

* See Appendix Note # 36

551

TN-264
Even though one cannot predict future values of X(t) exactly, there are often

significant autocorrelations within the random parts of the model. These cor- relations allow forecasts which can significantly reduce clock errors. Errors

in each element of the model (Eq. 1) contribute their own uncertainties to the

prediction. These time uncertainties depend on the duration of the forecast

interval, t, as shown below in Table 1:

TABLE 1. GROWTH OF

MODEL ELEMENT CLOCK

NAME PARAMETER

Initial Time Error a

Initial Freq Error b

Frequency Drift Dr

Random Variations <j)(t)

RMS TIME

ERROR

Constant

- T

- t2

- t3/2*

*The growth of time uncertainties due to the random component, (()(t), can have various time dependencies. The three-halves power-law

shown here is a "worst case" model. i^J

One of the most significant points provided by Table 1 is that eventually, the

linear drift term in the model over-powers all other uncertainties for sufficiently long forecast intervals! While one can certainly measure (i.e., estimate) the drift coefficient, Dr, and make corrections, there must always remain

some uncertainty in the value used. That is, the effect of a drift correction

based on a measurement of Dr, is to reduce (hopefully!) the magnitude of the

drift error, but not remove it. Thus, even with drift corrections, the drift

term eventually dominates all time uncertainties in the model.

As with any random process, one wants not only the point estimate of a parameter, but one also wants the confidence interval. For example, one might be

happy to know that a particular value (e.g., clock time error) can be estimated

without bias, he may still want to know how large an error range he should

expect. Clearly, an error in the drift estimate (see Eq. 1) leads directly to a time error and hence the drift confidence interval leads directly to a confidence interval for the forecast time.

II. LEAST SQUARES REGRESSION OF PHASE ON A QUADRATIC

A conventional least squares regression of oscillator phase data on a quadratic

function reveals a great deal about the general problems. A slight modification of Eq. 1 provides a conventional model used in regression analysis '-^J :

X(t) b't + C't2 + ,^(t) (2)

552

TN-265
where c = Dr/2. In regression analysis. It is customary to use the symbol "Y"

as the dependent variable and "X" as the independent variable. This is in conflict with usage in time and frequency where "X" and "Y" (time error and frequency error, respectively) are dependent on a coarse measure of time, t, the

independent variable. This paper will follow the time and frequency custom even

though this may cause some confusion in the use of regression analysis text books.

The model given by Eq. 2 is complete if the random component, (}»(t), is a white

noise (i.e., random, uncorrelated, normally distributed, zero mean, and finite

variance).

III. EXAMPLE

One must emphasize here that ALL results regarding parameter error magnitudes

and their distributions are totally dependent on the adequacy of the model. A

primary source of errors is often autocorrelation of the residuals (contrary to

the explicit model assumptions). While simple visual inspection of the residuals is often sufficient to recognize the autocorrelation problem, "whiteness

tests" can be more objective and precise.

This section analyzes a set of 94 hourly values of the time difference between

two oscillators. Figure 1 is a plot of the time difference (measured in microseconds) between the two oscillators. The general curve of the data along with

the general expectation of frequency drift in crystal oscillators leads one to

try the quadratic behavior (Model #1; models #2 and #3 discussed below). While

it is not common to find white phase noise on oscillators at levels indicated on the plot, that assumption will be made temporarily. The results of the regression are summarized in a conventional Analysis of Variance, Table 2.

TABLE 2. ANALYSIS OF VARIANCE QUADEIATIC FIT TO PHASE

(Units: seconds squared)

SOURCE SUM OF SQUARES

2.32E-9

4.26E-12

2.329E-9

Regression

Residuals

Total

Coefficient of simple determination 0.99713

Parameters:

a = 1.10795E-5 (seconds) t-ratio = 161.98

b = 1.4034E-10 (sec/sec) t-ratio = 152.02

c = -3.7534E-16 (sec/sec^) t-ratio = -143.51

d.f. MEAN SQUARE

3

91 4.68E-14

94 2.48E-11

553

TN-266
u

c

0)u

<u

•H

Q)

CO

(0

•H

— o

VI

554

TN-267
Dr = 2c - -7.507E-16 (about -6.5E-11 per day)

Std. Error - 0.05231E-16

(Note: a is the value estimated for the a-parameter, etc.)

The Analysis of Variance, Table 2, above suggests an impressive fit of the data

to a quadratic function, with 99.71% of the variations in the data "explained"

by the regression. The estimated drift coefficient, Dr, is -7.507E-16 (sec/sec^)

or about -6.5E-11 per day 143 times the indicated standard error of the

estimate. However, Figure 2, a plot of the residuals, reveals significant autocorrelations even visually and without sensitive tests. (The autocorrelations

can be recognized by the essentially smooth variations in the plot. See Fig. 5 as an example of a more nearly white data set.) It is true that the regression

reduced the peak-to-peak deviations from about 18 microseconds to less than one microsecond. It is also true that the drift rate is an unbiased estimate of the

actual drift rate, but the model assumptions are NOT consistent with the autocorrelation visible in Fig. 2. This means that the confidence intervals for the

parameters are not reliable. In fact, the analysis to follow will show just how

extremely optimistic these intervals really are.

At this point we can consider at least two other simple analysis schemes which

might provide more realistic estimates of the drift rate and its variance. Each

of the two analysis schemes has its own implicit model; they are:

(2) Regress the beat frequency on a straight line.

(Model: Linear frequency drift and white FM.)

(3) Remove a simple average from the second difference of the phase.

(Model: Linear frequency drift and random walk FM.)

Continuing with scheme 2, above, the (average) frequency, Y(t), is the first

difference of the phase data divided by the time interval between successive

data points. The regression model is:

Y(t) = b + Dr't + e(t) (3) "

where e(t) = [<^(t + Tq) - 4>(t)]/To« Following standard regression procedures as before, the results are summarized in another Analysis of Variance Table, Table

3.

TABLE 3. ANALYSIS OF VARIANCE LINEAR FIT TO FREQUENCY

(Units: sec^/sec^)

SOURCE SUM OF SQUARES d.f„ MEAN SQUARE

Regression 1.879E-18 2

Residuals 2.084E-20 91 2.29E-22

Total 1.899E-18 93 2.042E-20

(Note: Taking the first differences of the original data set reduces the

number of data points from 94 to 93.)

* See Appendix Note # 37

555

TN-268
V

CO

(C x:a

+j

+j

*"^ •H

in M-4 a:

Z)

O o •H r 4J

(d Ui3H s

cr

O2 s —Z vh Z «M D

(C CO iH

rd3

'0•H

CO

(U «

CM

•H

UJIoX

m* o -^X o

UJ

COIOX

(£> 5rI

556

TN-269
Coefficient of simple determination 0.9605

Parameters:

b = 1.049E-10 (sec/sec) t-ratio » 3.32

Dr = -7.635E-16 (sec/sec2) t-ratio » -4.70

Std. Error = 0.1624E-16 (sec/sec^)

While the drift rate estimates for the two regressions are comparable in value

(-7.507E-16 and -7.635E-16), the standard errors of the drift estimates have

gone from 0.052E-16 to 0.162E-16 (a factor of 3). The linear regression's coefficient of simple determination is 96.05% compared to 99.17% for the quadratic

fit. Figure 3 shows the residuals from the linear fit and they appear more nearly white. A cumulative perlodograml^l is a more objective test of whiteness, however. The periodogram. Fig. 4, does not find the residuals acceptable

at all.

IV. DRIFT AND RANDOM WALK FM

In the absence of noise, the second difference of the phase would be a constant,

Dr''^o2. If one assumes that the second difference of the noise part is white,

then one has the classic problem of estimating a constant (the drift term), in

the presence of white noise (the second difference of the phase noise). Of

course, the optimum estimate of the drift term is just the simple mean of the

second difference divided by Tq • The results are summarized below, Table 4:

TABLE 4. SIMPLE MEAN OF SECOND DIFFERENCE PHASE

Simple mean Dr = -6.709E-16 t-ratio = -2.45

Degrees of Freedom = 91

Standard Deviation s = 26.2405E-16

Standard Deviation of the Mean = 2.7358E-16

Figure 5 shows the second difference of the phase after the mean was subtracted.

Visually, the data appear reasonably white, and the periodogram, Fig. 6, cannot

reject the null hypothesis of whiteness. Now the standard error of the drift

term is 2.735E-16, 52 times larger than that computed for the quadratic fit!

Indeed, the estimated drift term is only 2.45 times its standard error.

V. SUMMARY OF TESTS

The analyses reported above were all performed on a single data set. In order

to verify any conclusions, all three analyses used above were performed a total

of four times on four different data sets from the same pair of oscillators.

Table 5 summarizes the results:

557

TN-270
at

a:3O

X

UJ2

o

z

z

z

-a•H

(0

0)

>i'c

C .H

(U I 3 «W

CPM-i

Q) -H

+J

I (0

i-i n ^^

•H

c

>• X

rO

I

558

TN-271
5

•H

•Om

•H

I

M

559

TN-272
(Q10 x:ao

IM•H

fO

•Ocou

0) wI m

•H

IT)K

CO

CO

u

V.u

CO

lOKI

560

TN-273
m . ^

lO

eg

KX 00 N. > o (d >•o 1 ""^

M-l > tM t) •H z

UJ •o o ou u 0) oc M

u. 1 NO• 00

•H

P^

$5Oo

55o

561

TN-274
TABLE 5. SUMMARY OF DRIFT ESTIMATES

(Units of l.E-16 sec/sec^)

ESTIMATION

PROCEDURE

& MODEL

Quad Fit

(White PM

and Drift)

DRIFT

ESTIMATE

-7.507

-8.746

-6.479

-6.468

COMPUTED PASS

STANDARD WHITENESS

ERROR TESTS?

.0523 No

.0493 No

.0645 No

.0880 No

.162 No

.206 No

.192 No

.295 No

2.736 Yes

9.335 No

3.424 Yes

3.543 Yes

1st Difference

Linear Fit

(White FM

and Drift)

-7.635

-8.558

-6.443

6.253

Second Difference

Less Mean

(Random Walk

FM and Drift)

•6.710

-7.462

-6.870

6.412

One can calculate the sample means and variances of the drift estimates for each

of the three procedures listed in Table 5, and compare these "external" estimates

with those values listed in the table under "Computed Standard Error," the

"internal" estimates. Of course the sample size is small and we do not expect

high precision in the results, but some conclusions can be drawn. The comparisons are shown in Table 6.

It is clear that the quadratic fit to the data displays a very optimistic internal estimate for the standard deviation of the drift rate. Other conclusions

are not so clear cut, but still some things can be said. Considering Table 5,

the "2nd Diff - Mean" residuals passed the whiteness test three times out of

four. The external estimate of the drift standard deviation lies between the

internal estimates based on the first and second differences. Since the oscillators under test were crystal oscillators, one expects flicker FM to be present

at some level. One also expects the flicker FM behavior to lie between white FM

and random walk FM. This may be the explanation of the observed standard deviations, noted in Table 6.

562

TN-275
TABLE 6. STANDARD DEVIATIONS

PROCEDURE

(Model)

Quad Fit (White PM)

1st Diff - Lin

(White FM)

2nd Diff - Mean

(Rand Wlk FM)

EXTERNAL ESTIMATE

(Std. Dev. of Drift

Estimates from Col.

if 2, Table 5)

1.08

1.08

0.44

INTERNAL ESTIMATE

(RMS Computed Std.

Dev. Col. #3,

Table 5)

0.065

0.203

5.45

VI. DISCUSSION

In all three of the analysis procedures used above, more parameters than just

the frequency drift rate were estimated. Indeed, this is generally the case. The estimated parameters Included the drift rate, the variance of the random

(white) noise component, and other parameters appropriate to the specific model

(e.g., the initial frequency offset for the first two models). If these other

parameters could be known precisely by some other means, then methods exist to

exploit this knowledge and get even better estimates of the drift rate. The

real problems, however, seem to require the estimate of several parameters in

addition to the drift rate, and it is not appropriate to just ignore unknown

model parameters.

To this point, we have considered only three, rather ideal oscillator models,

and seldom does one encounter such simplicity. Typical models for commercial

cesium beam frequency standards include white FM, random walks FM, and frequency

drift. Unfortunately, none of the three estimation routines discussed above are appropriate to such a model. This problem has been solved in some of the recent

work of Jones and Tryonl^J , [6] , Their estimation routines are based on Kalman

Filters and maximum likelihood estimators and these methods are appropriate for

the more complex models. For details, the reader is referred to the works of

Jones and Tryon.

Still left untreated are the models which, in addition to drift and other noises,

incorporate flicker noises, either in PM or FM or both. In principle, the

methods of Jones and Tryon could be applied to Kalman Filters which incorporate

empirical flicker modelsl'J. To the author's knowledgti, however, no such analyses have been reported.

VII. CONCLUSIONS

There are two primary conclusions to be drawn:

(1) The estimation of the linear frequency drift rates of oscillators

and the inclusion of realistic confidence intervals for these

563

TN-276
estimates are critically dependent on the adequacy of the model

used and, hence the adequacy of the analysis procedures

t

(2) The estimation of the drift rate must be carried along with the

estimation of any and all other model parameters which are not

known precisely from other considerations (e.g., Initial frequency and time offsets, phase noise types, etc.)

More and more, scientists and engineers require clocks which can be relied on to

maintain accuracy relative to some master clock. Not only Is It Important to

know that on the average the clock runs well, but It Is essential to have some measure of time Imprecision as the clock ages. For example, the uncertainties

might be expressed as, say,, 90% certain that the clock will be within 5 microseconds of the master two weeks after synchronization. Such measures are what

statisticians call "Interval estimates" (In contrast to point estimates) and

their estimations require Interval estimates of the clock's model parameters.

Clearly, the parameter estimation routines must be reliable and based on sound

measurement practices. Some Inappropriate estimation routines can be applied

to clocks and oscillators and give dangerously optimistic forecasts of

performance.

564

TN-277
APPENDIX A

REGRESSION ANALYSIS

(Equally Spaced Data)

We begin with the continuous model equation:

X(t) = a + b»t + c't2 + 4,(t) (Al)

We assume that the data is in the form of discrete readings of the dependent

variable X(t) at the regular intervals given by:

t = nxo

Equation (Al) can then be written in the obvious form:

Xj^ = a + Tq b*n + Tq^ cn^ + (j>(nTo)

for n= 1, 2, 3 •••, N.

Next, we define the matrices:

1 1 1

N =

2 4

3 9

ti ft

1 N n2

X =

Xl

X2

X3

_ Xn _,

(A2)

(A3)

T = 1

To

To2.

(NT)'X =

N

I Xn

1

N

^o I Xn n

1

N

To2

I Xn n2

1

N

I Xn

1

N

I Xn n

1

N

I Xn n2

1

= T 'nx

'nnx

565

TN-278
where four quantities must be calculated from the data:

N

I n=l

Sx - Xn

N

^nnx "^ Z ^n ^ n=l

N

Snx ~ I ^n ^ n=l

N

^xx ~ 2, ^

n=l

Define

B =

With these definitions, Eq. A3 can be rewritten In the matrix form:

X = Nil + £ <A4)

and the coefficients, B, which minimize the squared errors are given by:

a

B = 6 = T • ( N'N )"1 • ( N'X ) (A5)

The advantage of evenly spaced data for these regressions Is that, with a bit of

algebra, the matrix, ( N'N )~^, can be written down In closed form:

( N'N )-l =

where

A B C

B D E

C E F

1 / G (A6)

A = 3 [3 (N + 1) + 2]

B = -18 (2N + 1)

C = 30

D = 12 (2N + 1) (8N + 11) / [(N +1) (N + 2)]

E = -180 / (N + 2)

F = 180 / [(N + 1) (N + 2)]

566

TN-279
and

G - N (N - 1) (N - 2)

Also, the Inverse of T is just:

1 '

r-1 1/to

1/to2_i

The complete solution for the regression parameters can be summarized as follows;

There are four quantities which must calculate from the data:

N

Sx = I Xn

n=l

N

'nnx ~ 1 ^n n^

n=l

'nx

N

I Xn n n=l'XX

N

I Xn2

n»l

for n = 1, 2, 3, ..., N. Based on these four quantities, the regression parameters are calculated from the seven following equations:

a = (A Sx + To B Snx + Tq C S^nx) / G

b = (B Sx + To D Snx + Tq E S^nx) / (G Tq)

c = (C Sx + To E Snx + Tq^ F S^nx) / (G Tq^)

r2 = (Sxx - a Sx - To b S^x " Xo^ c S^nx) / (N - 3)

2 = a2 A / G

<^h' r2 D / (G To2)

.2 = a2 F / (G To^)

567

TN-280
whe/e the coefficients A, B, C, etc., are given by:

A = 3 [3N (N + 1) + 2]

B = -18 (2N + 1)

C = 30

D = 12 (2N + 1) (8N + 11) / [(N + 1) (N + 2)]

E = -180 / (N + 2)

F = 180 / [(N + 1) (N + 2)]

and

G = N (N - 1) (N - 2).

In matrix form, the error variance for forecast values is:

Var (X^) = a2 [1 + ^' (N'N)"^ N^l

where Nj^' = [1 nQ nQ] and Tq tiq Is the date for the forecast point, Xj^. That

Is, nQ = N + K and K Is the number of lags past the last data point at lag N.

568

TN-281
APPENDIX B

REGRESSIONS ON LINEAR AND CUBIC FUNCTIONS

The matrixes ( N*N)~^ for the linear fit and cubic fit, which correspond to Eq,

A6 in Appendix A are as follows:

For the linear fit:

(N'N)-l = A

B

where

and

A = 2 (2N + 1)

B = -6

C = 12 / (N + 1)

D = N (N - 1)

For the cubic fit:

1 / D

(N'N) -1 = B

C

where

F

H

1 / K

A = 8 (2N + 1) (N + N + 3)

B = -20 (6n2 + 6N + 5)

C = 120 (2N + 1)

D

'= -140

E = 200 (6N^ + 27 n3 + 42 n2 + 30 N + 11) / L

F = -300 (N + 1) (3N + 5) (3N + 2) / L

G = 280 (6n2 + 15N + 11) / L

569

TN-282
and

H - 360 (2N + 1) (9N + 13) / L

I - -4200 (N + 1) / L

J - 2800 / L

K - N (N - 1) (N - 2) (N - 3)

L - (N + 1) (N + 2) (N + 3).

The restrictions on these equations are that the data Is evenly spaced beginning with n « 1 to n N, and no missing values. For error estimates (and their

distributions) to be valid, the residuals must be random, uncorrelated, (I.e.,

white).

570

TN-283
REFERENCES

1, D.W. Allan, "The Measurement of Frequency and Frequency Stability of Precision Oscillators," Proc. 6th PTTI Planning Meeting, pl09 (1975).

2. J. A. Barnes, et al., "Characterization of Frequency Stability," Proc IEEE

Trans, on Inst, and Meas., IM-20, pl05 (1971).

3* N.R. Draper and H. Smith, "Applied Regression Analysis," John Wiley and

Sons, N.Y., (1966).

4. G.E.P. Box and G.M. Jenkins, "Time Series Analysis," Holden-Day, San

Francisco, p294, (1970).

5. P.V. Tryon and R.H. Jones, "Estimation of Parameters in Models for Cesium

Beam Atomic Clocks," NBS Journal of Research, Vol. 88, No. 1, Jan-Feb. 1983.

6. R.H. Jones and P.V. Tryon, "Estimating Time from Atomic Clocks," NBS Journal

of Research, Vol. 88, No. 1, Jan-Feb. 1983.

7. J. A. Barnes and S. Jarvis, Jr., "Efficient Numerical and Analog Modeling of

Flicker Noise Processes," NBS Tech. Note 604, (1971).

571

TN-284
0) ^_^ u

</i c

cc QJ z> M

o

X

•H

UJ 'O 2 Q) )- tn

(T3 o j:: 2 Cl< 2 1 2

3 iH

(X

• 0^ •H

fa

X

572

TN-285
a, pH

V) VW

(T3 u O •H r +J

(0

J- 3

cr oz e —z s^ z U-l D

a: OT iH

<as

-0•H

(0

(1) tf

CM

-H

fa

U

i/i IQM

in o ^ U

toI

573

TN-286
inO

sidua n) z 0) -H *^ H H

UJ 1 2

>iM-l

«4-l \- C -H

(U "O o s z CP-JJ "—

0) (0 z ^< H

z fa ^

o

q: 1 ro

•

•H

fa

K

I

574

TN-287
m

lO ^ u

"s. c O •H >- r-^ o*-• 1 >- <4-l a

z

•H Ul ~. g

llJ

0} •H cc U* u.

1

-a-

• 60

iH

P:^

575

TN-288
n

s:aO

•H

-occu

0) WI in

•H

lOK

00

CMUu

«>

inI

576

TN-289
m

lO

M

q:X

o

> > o

>- M-l o «M z

bJ

•H 3 •t3 O c

ou

cr 0) u. OT

•H

$5

o

o

55

o

577

TN-290
2

Q.

UJt ^ X o z > < g t o — q:

q: < o o

>-oz

UJ

ID3

Ll < VS^\^

>v ^s^ 1- ^v ^s^ cn >^ ^V <

^V >w^ o ^V >v UJ ^^ >V a: \ ^V O \ >. u. \ \

>- \ \ H H \ Z 5 1 < ^

•-t GC 0)0: UJ <Ui O oo z

ccz 3 03u. 1 11 __ lO CVi / // / H / / cn / ^ < y y/^ O y^ ^^ X.

^''^ ^/^ o / y^ <

y'

^^ CO ^O^ '^ ooocvjI o

578

TN-291
UJH to

iiQ < z > < uJ s §

UJou

U.

in <o

inoI

579

TN-292
QUESTIONS AND ANSWERS

MR. ALLAN:

We have found the second difference drift estimator to be useful for

the random walk FM process. One has to be careful when you apply it,

because if you use overlapping second differences, for example, if

you have a cesium beam, and you have white noise FM out to several

days and then random walk FM beyond that point, and you are making

hourly or 2 hour measurements, if you use the mean of the second

differences, you can show that all the middle terms cancel, and in

fact you are looking at the frequency at the beginning of the run and

the frequency at the end of the run to compute the frequency drift and

that's very poor.

DR. BARNES:

My comment would be: There again the problem is in the model, and not

in the arithmatic. The models applied here were 3 very simple models,

very simple, simpler than you will run into in life. It was pure random

walk plus a drift or pure frequency noise plus a drift, or pure random

walk of frequency noise plus a drift and it did not approach at all any

of the noise complex models where you would have both white frequency and

random walk frequency and a drift. That's got to be handled separately,

I'm not even totally sure how to perform it in all cases at this point.

MR. McCASKILL:

I would like to know if you would comment on the value of the sample time

and the reason why, of course, is that the mean second difference does

depend on the sample time? For instance, if you wanted to estimate the

aging rate or change in linear change of frequency, what value of sample

time would you use in order to make that correction. So, really, the

question is, how does the sample time enter into your calculations?

DR. BARNES:

At least I will try to answer in part. I don't know in all cases, I'm

sure, but if you have a complex noise process where you have at short

term different noise behavior than in long term, it may benefit you to

take a longer sampling time and effectively not look at the short term,

and then one of the simple models might apply. I honestly haven't

looked in great detail at how to choose the sample time. It is an interesting question.

580

TN-293
MR. McCASKILL:

Well, let me go further, because we had the benefit of being out at

NBS and talking with Dr. Allan earlier and he suggested that we use

the mean second difference, and the only problem is if we want to

calculate or correct for our aging rate at a tau of five days or ten

days, and you calculate the mean second difference, you come up with

exactly the Allan Variance. What appears is that in order to come up

with the number for the aging rate, you have to calculate the mean

second difference using a sample time whenever you take your differences

of longer than, let's say, ten days sample time. So we use, of course,

the regression model, but we use on the order of two or three weeks in

order to calculate an aging rate correction for, say, something like the

rubidium in the NAVSTAR 3 clock. It looks like in order to come up

with a valid value for the Allan Variance of five or ten day sample

time you have to calculate the aging rate at a longer, maybe two or three times longer sample time.

DR. BARNES:

Dave Allan, do you think you can answer that?

MR. ALLAN:

Not to go into details, but if you assume that in the longer term you

have random walk frequency modulation as the predominant noise process

in a clock, which seems to be true for rubidium, cesium and hydrogen,

you can do a very simple thing. You can take the full data length

and take the time at the beginning, the time in the middle and the

time in the end and construct a second difference, and that's your

drift.

There is still the issue of the confidence interval on that. If you

really want to verify your confidence interval, you have to have enough

data to do a regression. You need enough data to test to be sure the

model is good.

DR. WINKLER:

That argument is fourteen years old, because we have been critized

here. I still believe that for a practical case where you depend on measurements which are contaminated and maybe even contaminated by

arbitrarily large errors if they are digital. These, theoretical

advantages that you have outlined may not be as important as the

benefits which you get when you make a least square regression. In

this case you can immediately identify the wrong data. Otherwise, if

you put the data into an algorithm you may not know how much your data

is contaminated. So for practical applications, the first model even though theoretically it is poor, it still gives reasonable estimates

of the drift and you have residuals which let you identify wrong phase

values immediately.

581

TN-294
DR. BARNES:

I think that's true and you may have different reasons to do regression

analysis, if your purpose is to measure a drift and understand the

confidence intervals, then I think what has been presented is reasonable,

if you have as your purpose to look—to see if there are indications of

funny behavior in a curve that has such strong curvature or drift that

you can't get it on graph paper without doing that, I think it is a very

reasonable thing to do. I think looking at the data is one of the

healthiest things any analyst can do.

582

TN-295
NIST Technical Note 1318, 1990.

VARIANCES BASED ON DATA WITH DEAD TIME BETWEEN THE MEASXJREMENTS

James A. Barnes

Austron, Inc.

Boulder, Colorado 80301

and

David W. Allan

Time and Frequency Division

National Institute of Standards and Technology

Boulder, Colorado 80303

The accepted definition of frequency stability in the time domain is

the two-sample variance (or Allan variance) . It is based on the

measurement of average frequencies over adjacent time intervals,

with no "dead time" between the intervals. The primary advantages

of the Allan variance are that (1) it is convergent for many

encountered noise models for which the conventional variance is

divergent; (2) it can distinguish between many important and

different spectral noise types; (3) the two-sample approach relates

to many practical implementations; for example, the rms change of an oscillator's frequency from one period to the next; and (4) Allan

variances can be easily estimated at integer multiples of the sample

interval

.

In 1974 a table of bias functions which related variance estimates

with various configurations of number of samples and dead time to

the Allan variance was published [1]. The tables were based on noises with pure power-law spectral densities.

Often situations occur that unavoidably have dead time between

measurements, but still the conventional variances are not

convergent. Some of these applications are outside of the time-andfrequency field. Also, the dead times are often distributed

throughout a given average, and this distributed dead time is not

treated in the 1974 tables.

This paper reviews the bias functions B^,^ (N, r ,/i) , and B2(r,/i) and

introduces a new bias function, B3 (2 ,M, r ,//) , to handle the commonly

occurring cases of the effect of distributed dead time on the

computed variances. Some convenient and easy- to-interpret

asymptotic limits are reported. A set of tables for the bias

functions are included at the end of this paper.

Key words: Allan variance; bias functions; data sampling and dead

time; dead time between the measurement; definition of frequency

stability; distributed dead time; two-sample variance

TN-296
1 . Introduction

The sample mean and variance indicate respectively the approximate magnitude

of a quantity and its uncertainty. For many situations a continuous function

of time is sampled, or measured, at fairly regular intervals. Sampling is not

always instantaneous. It takes a finite time and provides an "average

reading." If the underlying process (or noise) is random and uncorrelated in

time, then the fluctuations are said to be "white" noise. In this situation,

the sample mean and variance calculated by the conventional formulas,

m

_ j_ N _

n=l

(1)

1 ^ - n=l

provide the needed information. The "bar" over the y in eq (1) above denotes

the average over a finite time interval. In time and frequency work, y is

defined as the average fractional (or normalized) frequency deviation from

nominal over an interval r and at some specified measurement time. As in

science generally, the physical model determines the appropriate mathematical

model. For the white noise model, the sample mean and variance are the

mainstays of most analyses.

Although white noise is a common model for many physical processes, more general noise models are being identified and used. In precise time and

frequency measurement, for example, there are two quantities of great

interest: instantaneous frequency and phase. These two quantities by

definition are exactly related by a differential, (We are NOT considering

Fourier frequencies at this point.) That is, the instantaneous frequency is

the time rate of change of phase. Thus, if we were employing a model of white

frequency-modulation (white FM) noise, then the phase noise is the integral of

the white FM noise, commonly called a Brownian motion or random walk.

Therefore, depending on whether we are currently interested in phase or frequency, the sample mean and variance may or may not be appropriate.

2

TN-297
By definition, white noise has a power spectral density (PSD) that is constant

with Fourier frequency. Since random walk noise is the integral of white

noise, the power spectral density of a random walk varies as 1/f^ (where f is

the Fourier frequency) [2]. We encounter noise models whose power spectral

densities are various power laws of their Fourier frequencies. Flicker noise

is very common and is defined as a noise whose power spectral density varies

as 1/f over a relevant spectral range. If an oscillator's instantaneous

frequency is well modeled by flicker noise, then its phase would be the

integral of the flicker noise. It would have a PSD which varied as 1/f^

.

Noise models whose PSD's are power laws of the Fourier frequency but not

integer exponents are possible as well but not as common. This paper

considers power-law PSD's of a quantity y(t); y(t) is a continuous sample

function which can be measured at regular intervals. For noises whose PSD's

vary as f* with a < -1 at low frequencies, the conventional sample mean and

variance given in eq (1) do not converge as N gets large [2, 3]. This lack of

convergence renders the sample mean and variance ineffective and often

misleading in some situations.

Although the sample mean and variance have limitations, other time-domain

statistics can be convergent and quite useful. The quantities that we consider in this paper depend significantly on the details of the sampling

procedures. Indeed, each sampling scheme has its own bias, and this is the

motivation for the bias functions discussed in this paper.

2. The Allan Variance

Recognizing that for particular types of noise, the conventional sample

variance fails to converge as the number of samples, N, grows, Allan suggested

that we set N = 2 and average many of these two-sample variances to get a convergent and stable measure of the spread of the quantity in question [3].

This is what has come to be called the Allan variance.

TN-298
More specifically let us consider a sample function of time as indicated in

figure 1. A measurement consists of averaging y(t) over the interval r. The

next measurement begins at a time T after the beginning of the previous

measurement interval. There is no logical reason why T must be as large as r or larger- -if T < r, then the second measurement begins before the first is

completed, which is unusual but possible. When T = r, there is no dead time

between measurements

.

The accepted definition of the Allan variance is the expected value of a twosample variance with no dead time between successive measurements. In

symbols, the Allan variance is given by

a^ir) = ^E[(y„^i-y,)2], (2)

where there is no dead time between the two sample averages for the Allan

variance and the E[»] denotes the expectation operator.

3. The Bias Function B^CN.r,/*)

Define N to be the number of sample averages of y(t) used in eq (1) to

estimate a sample variance (N = 2 for an Allan variance) . Also define r to be

the ratio of T to r (r = 1 when there is no dead time between measurements)

.

The parameter /i is related to the exponent of the power law of the PSD of the

process y(t) , If a is the exponent in the power-law spectrum for y(t) , then

the Allan variance varies as r raised to the fj. power, where a and n are related as shown in figure 2 [2-4] . We can use estimates of /i to infer a, the

spectral type. The ambiguity in a for /i = -2 has been resolved by using a modified o^(t) [5-7] .

Often data cannot be taken without dead time between sample averages, and it

is useful to consider other than two-sample variances. We will define the

bias function B^CN.r./i) by the ratio,

B,(N,r.^) = y^^y^r) ' ^^^

TN-299
where a^(N,T,r) is the expected sample variance given in eq (1) and based on N

measurements at intervals T and averaged over a time r and r = T/t . In words,

Bj(N,r,/i) is the ratio of the expected variance for N measurements to the

expected variance for two samples (everything else held constant) . The

variances on the right in eq (3) depend implicitly on the noise type even

though fi or a are not shown as independent variables. The noise- type

parameter, /*, is shown as an independent variable for all of the bias

functions in this paper, because the values of the ratio of these variances

explicitly depend on /x as will be derived later in the paper, Allan showed

that if N and r are held constant, then the a, n relationship shown in figure

2 is the same; that is, we can still infer the spectral type from the t dependence using the equation a = -^-1, -2 < /x < 2 [3].

4. The Bias Function B2(r,/x)

The bias function B2(r,/i) is defined in [1] by the relation,

u f^ „N - <^^(2.T,r ) _ a^(2,T,r)

^^(^'^) - aH2,r,r) " aj(r) " ^^^

In words, B2(r,/i) is the ratio of the expected two-sample variance with dead

time to that without dead time (with N = 2 and r the same for both variances)

.

A plot of the B2(r,/i) function is shown in figure 3. The bias functions Bj^ and B2 represent biases relative to N = 2 rather than infinity; that is, the

ratio of the N sample variance (with or without dead time) to the Allan

variance and the ratio of the two- sample dead- time variance to the Allan

variance respectively. Xi

5. The Bias Function BgCN.M.r./i)

Consider the case where a great many measurements are available with dead time

between each pair of measurements (T^ > t^) . The measurements are averaged

over the time interval t^ , the spacing between the beginning of one measurement to the next is Tg , and it may not be convenient to retake the

data. We might want to estimate the Allan variance at, say, multiples M of

5

TN-300
the averaging time r^ . If we average groups of the measurements of y(t) , then

the dead times between the original measurements are distributed periodically

throughout the new average measurements (see figure 4) . Define

M+i-1

1 f

- yt = -^ IVn' (5)

n=i

where y^ are the raw or original measurements based on dead time Tq-Tq.

Also define the two-sample variance with distributed dead time as

a2(2,M,T,r) = hE[(y,-y,^^)^ ]

,

(6)

with r = Mtq and T = MTq .

We can now define B3 as the ratio of the N-sample variance with distributed

dead time to the N-sample variance with dead time accumulated at the end as in

figure 1

D /M M N g^(N,M,T,r) ,_, ^3(N.M.r,M) =

.^(N,T,r) " ^^^

Although B3(N,M,r,/i) is defined for general N, the tables in the Appendix

confine treatment to the case where N = 2. There is little value in extending

the tables to include general N. Though the variances on the right in eq (7)

depend explicitly on N, T and r, the ratio BgCN.M.r./i) depends on the ratio

r = T/r , and on fi as developed later in this paper.

In words, Ej (2 ,M.,r: ,fi) is the ratio of the expected two-sample variance with

periodically distributed dead time, as shown in figure 4, to the expected twosample variance with all the dead time grouped together as shown in figure 1. Both the numerator and the denominator have the same total averaging time and

dead time, but they are apportioned differently. The product B2(r,/i) •

B3(2,M,r,/i) is the distributed dead-time variance over the Allan variance for

a particular T, r, M and fi.

TN-301
Some useful asymptotic forms of B3 can be found. In the case of large M and

M > r, we may write that

B3 « ^-^, 1^/1^2,

*3 2 in(r) + 3' '^ "'

(8)

One simple and Important conclusion from these two equations Is that for the cases of flicker FM noise and random-walk FM noise, the r" dependence for

large r Is the same whether or not there Is periodically distributed dead

time. The values of the variances differ only by a constant, and in the

latter case the constant is 1. This conclusion Is also true for white FM

noise, and in this case the constant Is also 1.

In the cases r » 1 and -2 < /* < -1, we may write for the asymptotic behavior

of B3

B3 = M«, a = -M-1 , (9)

as was determined empirically. In this region of power- law spectrum the B3

function has an M* dependence for an f* spectrum.

6. The Bias Functions

The bias functions can be written fairly simply by first defining the

function,

F(A) = 2A'**2 . (A + 1)M*2 - |(A - l)!''*^, (10)

The bias functions become

N-1 N-n

1 + I

• F(nr)

- ,.- . n=l N(N-l) ,,,.

TN-302
1 + HFCr")

BzCr.p) = 2(1-2') ' (12)

as given in [ 1 ]

, and

M-1

2M + M-F(Mr) - X (M-n) [2F(nr) -F( (M+n)r) -F((M-n)r)]

B3(2,M,r,/i) = mM^l^n^.^^ ^ oi . (13)

(M'' + 2)[F(r) + 2]

as indicated in the appendix.

For n = 0, eqs (11), (12), and (13) are the indeterminate form 0/0 and must be

evaluated by I'Hopital's rule. Special attention must also be given when

expressions of the form 0° arise. We verified a random sampling of the table

entries using noise simulation and Monte Carlo techniques. No errors were

detected. The results in this paper differ some from those in [8], which

suggests that there may be some mistakes. Tables for the three bias functions

are listed at the end of the paper (note that the computer print-out did not

have a symbol for Greek mu = /i) .

7. Examples of the Use of the Bias Functions

The spectral type, that is, the value of /x, may be inferred by varying t, the

sample time. However, another useful way of determining the value of fi is by

using Bi(N,r,)Lt) as follows: calculate an estimate of ay(N,T,T) and ay(2,T,r)

and hence Bi(N,r,/i); then use the tables to infer the value of fi

.

Suppose one has an experimental value for ay (Nj^ , Tj^ , t^) and its spectral

type is known, that is, n is known. Suppose also that one wishes to know the

variance at some other set of measurement parameters, N2 , T2 , rg . An unbiased

estimate of ay(N2,T2,r2) may be calculated by the equation:

where rj^ = T^/t^ and r2 = l^/"^!-

TN-303
Since the time -domain definition for frequency stability is the Allan

variance, it behooves us, where possible, to relate other variances to the

Allan variance. If we have an N- sample variance on data with dead- time T-r

and we know the power- law spectral type (the value of n) , then we may write

2.. c7;(N,T,r)

""y^^^ - B,(N!r,M) B,(r.M)

' ^^'^

If we have an N- sample variance where each data entry is an average of M

samples with distributed dead time, then we may write

„2/^>, = aa(N,M,T,r)

""y^^^ Bi(N,r,/i) B2(r,/i) B3 (N,M, r ,/i)

' ^^°^

8. Conclusion

For some important power- law spectral density models often used in

characterizing precision oscillators (S (f) ~ f**

, a = -2, -1, 0, +1, +2), we have studied the effects on variances when there is dead time between the

frequency samples , and the frequency samples are averaged to increase the

integration time. Since dead time between measurements is a common problem

throughout metrology, the analysis here has broader applicability than just to

time and frequency. Specifically, this kind of analysis has been used with

gage blocks and standard volt cells- -showing that the classical variance may

be non- convergent in some cases [9].

Heretofore, the Allan variance has been shown to have some convenient

theoretical properties in relation to power- law spectra as the integration or sample time is varied (if o^ (t) ~ r*^ , then a = -/x -1, -2 < /x < 2) . Since

Oy(T), by definition, is estimated from data with no dead time, the sample or integration time can be unambiguously changed to investigate the r dependence.

From our analysis, we have concluded that for the asymptotic limit of several

samples being averaged with dead time present in the data, the t dependence of

the variances is the same. The a = -p. -1 relationship still remains valid for

white FM noise (/z = -1, a = 0), flicker FM noise (/i = 0, a = -1), and for

9

TN-304
random-walk FM noise (/x = +1 , a = -2). The asymptotic limit is approached as the product of number of samples averaged and the initial data sample time,

Tq , becomes larger than the dead time (M > r) . The variances so obtained

differ only by a constant, which can be calculated as given in this paper.

A knowledge of the appropriate power- law spectral model is required to

translate a distributed dead- time variance to the corresponding value of the

Allan variance. In principle, the power- law spectral model can be estimated

from the t^ dependence, using the variance analysis on the data as outlined

above

.

9. References

[1] J. A. Barnes, "Tables of Bias Functions, Bj^ and B2 , for Variances Based on Finite Samples of Processes with Power Law Spectral Densities," NBS Tech.

Note 375 (1969)

.

[2] J. A. Barnes, "Atomic Timekeeping and the Statistics of Precision

Signal Generation," IEEE Proc . 54, No. 2, pp. 207-220, Feb. 1966.

[3] D.W. Allan, "Statistics of Atomic Frequency Standards, "IEEE Proc.

54, No. 2, pp. 221-230, Feb. 1966.

[4] RFC Vessot, L. Mueller, and J. Vanier, "The Specification of

Oscillator Characteristics from Measurements Made in the Frequency

Domain," IEEE Proc. 54, No. 2, pp. 199-207, Feb. 1966.

[5] D.W. Allan and J. A. Barnes, "A Modified "Allan Variance" With

Increased Oscillator Characterization Ability," Proc. 35th Annual

Frequency Control Symposium, USAERADCOM, Ft. Monmouth, NJ , May 1981,

pp. 470-475.

[6] D.W. Allan, "Time and Frequency (Time-Domain) Characterization,

Estimation, and Prediction of Precision Clocks and Oscillators," IEEE

Transactions on UFFC , November 1987.

[7] P. Lesage and T. Ayi , "Characterization of Frequency Stability: Analysis

of the Modified Allan Variance and Properties of its Estimate," IEEE

Trans. Instrum. Meas .

, IM-33 . no. 4, pp. 332-336, Dec. 1984.

[8] N.D. Faulkner and E.V.I. Mestre, "Time-Domain Analysis of Frequency

10

TN-305
Stability Using Non-zero Dead-Time Counter Techniques," IEEE Trans, on

Instrum. &Meas., IM-34 . 144-151 (1985).

9] D.W. Allan, "Should the Classical Variance Be Used as a Basic Measure in

Standards Metrology?" IEEE Trans, on Instrumentation and Measurement,

IM-36, 646-654, 1987.

11

TN-306
Table 1 . Table of some bias function identities

Bi(2,r,M) = 1 Bi(N,r,2) = (N(N+l))/6

Bi(N,l,l) = N/2

Bi(N,1,m) = (N(l-N'^))/[(2(N-l)(l-2'^)] for ^^0

= N ln(N)/[2(N-l) ln(2)] for /i=0

Bi(N,1,m) = 1 for /i <

for n >

N-1 = [2/(N(N-l)] I (N-n)

n=l

Bi(N,r,-l) = 1 if r > 1 Bi(N,r,-2) = 1 if r ?i 1 or B2(0,M) =

B2(l,/i) = 1 B2(r,2) = r2

B2(r,l) = (3r - l)/2 if r > 1 B2(r,-1) = rifO<r<l

= 1 if r > 1 B2(r,-2) = if r=0

= 1 if r=l = 2/3 otherwise

B3(2,M,1,M) = 1 B3(2,M,r,-2) = M

B3(2.r,/i) = 1 63(2, M,r, 2) = 1 B3(2,M,r,-l) = 1 for r > 1

12

TN-307
())A )U9iu8jnse9^ ^

UJsQ5

fc<O

CO

Ld

w

a>

•-^U 0)u

a>

C

Xc

a> CO 2 t

a> 0) ^^- -H • C H CO - U

4) 4) a >

•r4

O

CO bO

0)> ^o

CQ CO V 0) -^a - ii3 « c

0) >> B • U 4) V C H 4)

<u :} (0 3 «

a* <« CO

4) 4)

u

W

B CO (H T3 >>

-H

O <0

CO c C 4)

B

O 3 o ^ cr u

4-) 4) (4-1 O >-i CO tw 4) w U

(f4 iJ c c CO 4> •H > *J W

iJ (0 CO

••-> >

«w -O O CO 4) ^ C «w a o o B

tM CO iJ i-> U

CO 4) 1 V4 n o

4J >

to CO 4J 3

'-H (M CO ^ O M 4) O 4J > CO • 4J ^^

«-l 3

CO o

4> -H ^ U (0 3 CO u bO^

s^ o

13

TN-308
Oy (T) ~ T

White PM V | 2+

Flicker PM

White FM

Flicker FM

FM

FM

Sv(0-t'

Random

Walk

Flicker

Walk

Mod. Oy^(T)~T^'

Figure 2. A plot of the relationship between the frequency- domain power- law

spectral-density exponent a and the time-domain two-sample Allan variance

exponent /i (a = -/i-1, -2 < fi < 2 and q > 1 for n = -2). Also shown is the

similar relationship between a and the modified Allan variance with exponent

on r of fi' (q = -/i'-l, -4 < /i' < 2) . The pointing arrows indicate the mu- alpha relationship (a vs. fi or /i') for which the particular variance applies.

14

TN-309
THE BIAS FUNCTION, B2 (r,/i)

Figure 3. A three dimensional plot of the bias function BjCr./i), where

r = T/r , and the dead time is T - r. The "fin" at r = 1 and ^i = -2 approaches

zero width as the measurement bandwidth approaches infinity (see appendix ref

.

[3]).

15

TN-310
(;)A S)U9UJ9JnSB9|/\|

16

CH C( c • 4J 1-1 X

4) -<} 4J iJ > X! iH

(t II 4J c > X •0 3 T 4; (4-1 4.) JC 2 3 u y di c to X •»-( to Ui CO x: 0) to CQ 1-1 4>

4-1

u 4)

x E M e 1-1 c <U v> •r^ 4) 4J M bO •<-! 4-> X to 4= u T3 U iJ •o CO . 0) CO u 4) W > c q; •d

4-> to 1-1 "O 14-1 c u

(U o Oi •o

p E > V c o

<1>V

C CO

u 4)

U 1-1 to E 3 V (0 to (.1 1-1

CO ^ •< 4) 4; 4J

(Vw 1-1 3 4)

B , 1 V c M o 4) 4) s 3 w H ^^ x: to O a u CO (M ' II a

to 1 -o

V^ c ^-v CO 1 4) a> H e 4-) > ^ T3 4) 3 o (U 4) X e 1 4J c •r-l c 1-1 3 to }-l 4) ij XI 1-1 iJ

4«S

H 1-1 >-l

M

(TJ T3 s-^ S-i to 1-1 iJ to4) -^

to > -o

X •o (V 1-i 4) O s "O E c

«0 to 1-1 iH

0) 4-) jr 4->

X •o iJ 1 iJ » c "O 1-1 T3 •H w to to 3 to >

(V 4) 4> bO o T3 w T3 4) rt w V u

>-i •o bO-O c

0) (1> (U CO

C CO > e 4J >-i to 1-1 to 1-1 3 4) >-l u X! > 4) CO o 1-1 CO B > > v )-l iH

4J—

u

U 4J 4)

a

u CO 1 -H

bO e 1-1 ^ 4)

a c to •o 1-1 -H E

1-1 (0 E a CO c CO 1-1 E (0 •H CO CO to 1 s •o to • t^ <D c 4-1 3 *- (V > CO 4) -o 4J u to to V 1 <u jn - iJ CO T5 w CO 3 H to X CO tw u II <4-l 1-t •H 0) o to o }-l E

-0 4J P5T-I c t> O to CQ 4-1 o ^-1 -^ > 1-1 'r^ CO iJ 73 Vj T5

4-) c 0) CO to 1-1 e 4) 4) U-l 4; V4 bO •r-l V4 —t

•0

u •H iJ CO

a

U

w )-i E T5 3 o 4) CO to 4-1 C

^-4 —I 4) CO to to ^^ <0 a bO 1 c

f-l s. e CO •H u CO >-i ? E w c CO 4) 4-) s < c > c o o CO CO (0 II >l-l « T3 • iJ 4J O 4> »- •J- CO 3 > iJ 4)

5-1 XI 4-» CO ,C 73

0) 4J tH -1

HC

)-l (0

M 4} 3 to 3 3 4-1 CO CJ hO^ « 4) '-t

T-l ^H 1-1 ^

tiH ^ -O

H

CO oH

CQ 2

TN-311
Appendix

With reference to figure 1 , the frequency sampling window has an equivalent

phase sampling window. The intent is to evaluate the variance, S(M) , of the

sampled phase function in terms of the phase autocorrelation function, R(t) . The process here is to correctly account for terms and cross -terms coming from

squaring and averaging the samples for each M. The B3(2,M,r,/i) function can

then be obtained from the relation,

B3(2,M.r,M)

=^

SCD-M" + 2

for appropriate M, r, and /x. The denominator is just the two-sample variance

with dead time for MT and Mr (in accordance with the definition of

E^(2 ,M.,v ,iJ,)) . The factors common to the numerator and denominator are ignored

in the following.

For M = 1, the variance S(l) is just

S(l) = 4-R(0) - 4-R(r) - 4-R(T) + 2-R(T+t) + 2R(T-r)

,

where use has been made of the definition of the autocorrelation function,

R(T) = E[^(t) • .^(t+T)].

It is convenient to define a function G(T) as

G(T) = 2-R(T) - R(T+r) - R(T-r).

Similarly, S(2) can now be written in the form,

S(2) = 8-R(0) - 8-R(r) + 2-G(T) - 4-G(2T) - 2-G(3T).

Following this procedure, we can verify that the general S(M) is just

17

TN-312
S(M) = 4-M-R(0) - 4-M-R(r) - 2-M-G(MT)

M-1

+2 I (M-n)[2-G(nT) - G((M+n)T) - G((M-n)T)]

n=l

Following the work of Barnes and Allan [2,3], we can define the function U(r)

by the relation,

U(r) = 2-R(0) - 2-R(r),

and also define

F(nr) = G(nT)/U(r),

where r = T/r . The function U(t) for power-law power spectral densities has

the form,

U(r) = , 4_2M+2

which yields

F(nr) = 2-(nr)''^2 _ (nr+l)'' + 2 -|nr-l|''^2

Finally, the working relation can be written as

M-1

2-M + M-F(Mr) -^ (M-n) [2-F(nr) - F((M+n)r) - F((M-n)r)

B3(2,M,r,M) =

"=^

[2 + F(r)] • M** + 2

18

TN-313
Mu \ N=

Bl(N,r,*u) for r = .01

8 16 32 64 128 256 512 1024 INF

-2 l.OOOE-KK) l.OOOE+00 l.OOOE+00 l.OOOE+00 i.oooe+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 1.0006+00 1.0006+00

-1.8 1.091E+00 1.210E+00 1.360E+00 1.545E+00 1.772E+00 2.0e9E+00 2.349E+O0 2.4566+00 2.4946+00 2.512E+00

-1.6 1.199E+00 1.487E+00 i.ewt+oo 2.455E+00 3.239E+00 4.440E+00 5.505E+00 6.002E+O0 6.1986+00 6.3096+00

-1.4 1.328E+00 l.fi56f+00 2.6fifiF+00 3.991E+00 6.054E+O0 9.500E+O0 1.282t+01 1.4566+01 1.5326+01 1.5856+01 -1.2 1.482E+00 2.346E+O0 3.880E+00 6.598E-K)0 1.145E+01 2.026E+01 2.947E+01 3.4866+01 3.7546+01 3.9806+01

-1 1.667E+00 3.000E+00 5.667E+O0 l.lOOE+01 2.167E+01 4.2»k+01 6.628E+01 8.1906+01 9.0646+01 1.0006+02

-.8 1.884E+00 3.860E+00 8.303E+00 l.«2«b>01 4.041E-K)1 8.691E+01 1.443E+02 1.86flF+02 2.1376+02 2.5196+02

-.6 2.134E+00 4.959E+00 1.205E+01 2.977E+01 7.296E+01 1.697E+02 2.995E+02 4.0766+02 4.851E+02 6.4236+02

-.4 2.407E+00 6.279E+00 1.703E+01 4.65SF+01 1.247E+02 3.106E+O2 5.814E+02 8.»/b+02 1.0436+03 1.7156+03

-.2 2.677E+00 7.714E+00 2.296E+01 6.836E+01 1.976E+02 5.220E+O2 1.036E+03 1.5806+03 2.0646+03 5.581E+03 2.912E+00 9.075E+00 2.909E+01 9.282E+01 2.857E+02 7.951E+02 1.672E+03 2.7196+03 3.8266+03

.2 3.0e9E+00 1.018E+01 3.448E+01 1.162E+02 3.7636+02 1.097E+03 2. 4466+03 4.2636+03 6.4536+03

.4 3.203E+00 1.095E+01 3.857E+01 1.354E+02 4.572E+02 1.390E+03 3.287E+03 6. 1726+03 1.0146+04

.6 3.268E+00 1.143E+01 4.133e+01 1.495E+02 5.221E+02 1.649E+03 4.1426+03 8.4196+03 1.5146+04

.8 3.302E+00 1.170E+01 4.303E-K)1 1.590E+02 5.708E+02 1.869E+03 4.991E+03 1.1046+04 2.1896+04

1 3.319E+00 1.185E+01 4.403E+01 1.653E+02 6.066E+02 2.055E+03 5.842E+03 1.4126+04 3.1096+04

1.2 3.327E+00 1.192E+01 4.461E+01 1.693E+02 6.330E+02 2.215E+03 6.717E+03 1.7826+04 4.;«*:+04

1.4 3.330E+00 1.196E+01 4.494E+01 1.720E+02 6.530E+02 2.359E+03 7. 6406+03 2.2346+04 6.169E+04

1.6 3.332E-KI0 1.198E+01 4.514E-K)1 1.738E+02 6.688E+02 2.493E+03 8. 6376+03 2.7936+04 8.697E+04

1.8 3.XX^.*00 1.199E+01 4.526E+01 1.750E+02 6.819E+02 2.623E+03 9.737E+03 3.4936+04 1.2316+05

2 3.3336+00 1.200E+01 4.533F+C1 1.760E+02 6.933E+02 2.752E+03 1.097E+O4 4.3786+04 1.7496+05

hu \ N=

Bl(N,r,»u) for r = .03

16 32 64 128 256 512 1024 INF

-2 1.0006+00 1.0006+00 1.0006+00 l.OOOE+OC 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00

-1.8 1.091E+00 1.211E+00 1.366E+00 1.57:^+00 1.8276+00 1.9506+00 1.9956+00 2.0106+O0 2.0156+OO 2.0166+00

-1.6 1. 1996+00 1.491E+00 1.9106+00 2.5326+00 3. 3416+00 3.7826+00 3.96X+00 4.0306+00 4.0536+00 4.0646+00

-1.4 1.329E+00 1.863E+00 2.7196+00 4.1336+00 6.0606+00 7.2686+00 7.8156+00 8.044E+00 8.1356+00 8.191E+00

-1.2 1.4€4E+00 2.355E+00 3.9186+00 6.7686+00 1.0936+0! 1.37K+01 1.5256+01 1.5956+01 1.6266+01 1.651E+01

-1 1.6676+00 3.0006+00 5.6676+00 1.1006+01 1.9286+01 2.5606+01 2.9296+01 3.1276+01 3.2296+01 3.333E+01

-.8 1.8786+00 3.8256+00 8.1416+00 1.7536+01 3.304E+01 4.6266+01 5.4996+01 6.0296+01 6.3416+01 6.770E+01

-.6 2.114E+00 4.R38F+00 1.147E+01 2.7066+01 5.4416+01 8.0496+01 9.9976+01 1.1346+02 1.2256+02 1.4036+02

-.4 2.362E+00 6.005E+00 1.566E+01 3.9936+01 8.5246+01 1.3356+02 1.7446+02 2.0666+02 2.3146+02 3.0996+02

-.2 2.604E+O0 7.2426+00 2.047E+01 5.5816+01 1.2596+02 2.0946+02 2.8976+02 3.6196+02 4.2566+02 8.5716+02

2.ei9E+00 8.4306+00 2.5466+01 7.3486+01 1.7486+02 3.0966+02 4.5686+02 6.0826+02 7.6116+02

_2 2.992E+00 9. 4606+00 3.0136+01 9.1276+01 2.2866+02 4.324E+02 6.856F+02 9.8346+02 1.32WE+03

.4 3.1186+00 1.027E+01 3.4126+01 1.0776+02 2.R35F+02 5.750E+02 9.8686+02 1.5416+03 2.2776+03

.6 3.2036+00 1.087E+01 3.728E+OI 1.2206+02 3.3726+02 7.357E+02 1.3766+03 2.3646+03 3.8686+03

.6 3.257E+00 1.128E+01 3.9666+01 1.33Vt+02 3.887E+02 9. 152E+02 1.8796+03 3.5806+03 6.5546+03

1 3.2906+00 1.155E+01 4.141E+01 1.4376+02 4.3826+02 1.117E+03 2.5336+03 5.3956+03 1.1146+04

1.2 3.309E+00 1.1726+01 4.267E+01 1.5186+02 4.867E+02 1.3486+03 3.3926+03 8. 1286+03 1.9036+04

1.4 3.3206+00 1.184E+01 4.360E+O1 1.5886+02 5.3526+02 1.6156+03 4.535F+03 l.Z2«E+04 3.2766+04

1.6 3.327E+00 1.191E+01 4.431E+01 1.6506+02 5.8516+02 1.9306+03 6.0686+03 1.8666+04 5.6846+04

1.8 3.331E+00 1.1966+01 4. 4876+01 1.7066+02 6.3746+02 2.3046+03 8. 1426+03 2.8496+04 9.9366+04

2 3.33.3E+00 1.2006+01 4.533E+01 1.7606+02 6.9336+02 2.7526+03 1.0976+04 4.3786+04 1.7496+05

19

k^

TN-314
Bl(N,r,»u) for r = .1

Hu \ 14= 4 8 16 32 64 128 256 512 1024 INF

-2 l.OOOE+00 i.oooe+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 i.oooe+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00

-1.8 1. 09:^+00 1.226E+O0 1.438E+00 1.547E+00 1.579E+00 1.586E+00 1.586E+O0 l.Sftf)F+O0 1.584E+00 1.W3F+O0

-!.<> 1.205E+00 1.522E-K)0 2.021E+00 2.329E+00 2.449E+00 2.490E+O0 2.502E+O0 2.505E+00 2.505E+00 2.504E+00

-1.4 1.337E+O0 1,907E*00 2.832E+00 3.477E+00 3.772E+00 3.892E+00 3.938E+00 3.954E+O0 3.960E+O0 3.962E+00

-1.2 1.491E-K)0 2.396E-HX) 3.931E+O0 5.125E+00 5.751E+00 6.046E-HX) 6.177E+00 6.2-Ak+OO 6.260E+00 6.278E+O0

-1 1.667E-H)0 3.000E-K)0 5.375E+O0 7.429E+00 8.653E+00 9.312E+O0 9.652E+00 9.825E+O0 9.912E+00 l.OOOE+01

-.8 1.860e+00 3.720E+00 7.204E+00 1.055E+01 1.281E+01 1.419E-K)1 1.502E+01 1.550E+01 1.577E+01 1.616E+01

-.6 2.065E+00 4.540E+00 9.423E+O0 1.461E+01 1.859E+01 2.137E+01 2.326E+01 2.452E+01 2.537E+01 2.70^+01

-.4 2.273E+O0 5.430E+00 1.199E+01 1.971E+01 2.642E+01 3.177E+01 3.592£*01 3.910E+01 4.153E+01 4.922E+01

-.2 2.472E+00 6.344E+00 1.481E+01 2.S8SF+01 3.673E+01 4.664E+01 5.542E+01 6.314E+01 6.989E+0I 1.156E+02 2.653E+00 7.236E+00 1.779E+01 3.300E-K)1 5.003E+01 6.771E+01 8.566E+01 1.037E+02 1.219E+02

.2 2.810E-KK) 8.065E+00 2.080E+01 4.111E+01 6.690E+01 9.750E+01 1.331E+02 1.741E+02 2.214E+02

.4 2.940E+00 8.8O4E+O0 2.376E+01 5.015E+01 8.816E+01 1.397E+02 2.084E+O2 2.993E+02 4. 194E+02

.6 3.043E+00 9.445E+O0 2,W«+01 6.014E+01 1.149E+02 1.9yyk+02 3.295E+02 5.265E+02 8.254E+02

.8 3.123E-K)0 9.989E+00 2.938E-H)1 7.U8E+01 1.487E+02 2.862E-K)2 5.269E+02 9.466E+02 1.678E+03

1 3.184E+00 1.045E+01 3.204E+01 8.347E+01 1.918E+02 4.114E-K)2 8.523E+02 1.735E+03 3.500E+03

1.2 3.2306+00 1.084E-H)1 3.463E+01 9.72^+01 2.470E+02 5.943E-H)2 1.394E+03 3.232E+05 7.453E+03

1.4 3.265E+00 1,118E+01 3.721E+01 1.130E+02 3. 183E+02 8.(f»3Af+02 2.304E+03 6.103E+03 1.613E+04

1.6 S.^'it+OO 1.147E+01 3.982E+01 1.309E+02 4.113E+02 1.263E+03 3.843E+03 1.166E+04 3.535E+04

1.8 3.315E+00 1.174E+01 4.252E+01 1.517E+02 5.330E+02 1.85^+03 6.467E+03 2.250E+04 7.830E+04

2 3.333E+00 1.200E+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E-K)4 4.378E+04 1.749E+05

Bl(N,r,iiu) for r

Mu \ N= 16 32 64 128 256 512 1024 INF

-2 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO 1.000E+0() -1.8 1.124E+00 1.226E+O0 1.259E+00 1.265E+00 1.265E+O0 1.263E+00 1.262E+00 1.261E+O0 1.261E+00 1.260E+00

-1.6 1.254E+00 1.479E+00 1.564E+00 1.589E+00 1.594E+00 1.594E+O0 1.593E+00 1.591E+0() 1.591E+00 1.590E+0(i -1.4 1.388t+0() 1.759E+00 1.925E+00 1,988E+00 2.009E+00 2.014E+O0 2.015E+00 2.015E+00 2.014E+O0 2.013E+O0

-1.2 1.527E+00 2.068E+00 2.351E+00 2.479E+00 2.534E+O0 2.556E+00 2.5fc5£+(»0 2.569E+00 2.570E+O0 2.571E+00

-1 1.667E+00 2.405E+O0 2.850E+00 3.087E+O0 3.2O9E+O0 3.271E+00 3.302E+00 3.318E+00 3.326E+00 5.333E+O0

-.s 1.806E+00 2.769E+O0 3.435E+O0 3.847E+00 4.093E+00 4.2M+00 4.325E+00 4.376E+00 4.405E+00 4.447E+00

-.6 1.943E+0<:i 3. 160E+C'0 4.122E+00 4.804E+00 5.275E+O0 5.595E+00 5.812E+O0 5.958E+00 6.056E+00 t..252E+0(i -.4 2.076E+00 3.577E+00 4.927E+00 6.024E+00 6.889E+00 7.564E+00 8.086E+00 8.487E+00 8.795E+00 9.776E+00

-.2 2.203E+OO 4.021E+00 5.877E+O0 7.597E+00 9.147E+00 1.053E+01 1.175E+0I 1.282E+01 1.377E+01 2.017E+01 2.32SE+00 4.492E+00 7.000E+00 9.651E+00 1.238E+01 1.515E+01 1.796E+01 2.079E+01 2.36:^+01

.2 2.440E+00 4.994E+00 8.338E+00 1.236E+01 1.710E+01 2.260E+01 2.897E+01 3.632E+01 4.479E+01

.4 2.550E+00 5.530E+00 9,940E+00 1.599E+01 2.411E+01 3.493E+01 4.926E+01 6.823E+01 9.330E+01

.6 2.654E+00 6.1O6E+O0 1.187E+01 2.089E+01 3.472E+01 5.581E+01 8.786E+01 1.365E+02 2.103E+02

.8 2.754E+00 6.729E+00 1.422E+01 2.756E+01 5.098E+01 9.187E+01 1.632E+02 2.874E+02 5.037E+02 2.852E+00 7.406E+00 1.708E+O1 3.673E+01 7.618e+01 1.552E+02 3.132E+02 6.292E+02 1.261E+03

1.2 2.947E+O0 8. 147E+00 2.059E+01 4.942E+01 1.157E+02 2.678E+02 6.170E+02 1.419E+03 3.261E+03

1.4 3.042E+00 8.963E+00 2.493E+01 6.709E+01 1.781E+02 4.703E+02 1.241E+03 3.272E+03 8.631E+03

1.6 3. 137E+O0 9.86^+00 3.030E+01 9.185E+01 2.774E+02 8.381E+02 2.535E+03 7.6736+03 2.324E+04

1.8 3.234E+00 1.087E+01 3.699E+01 1.267E+02 4.366f+02 1.511E+03 5.245E+03 1.823E+04 6.343E+04

2 3..333E+00 1.200E+01 4.533f+01 1.760E+02 6.933E+02 2.752E+03 1.097E+04 4.378E+04 1.749E+05

20

TN-315
Mu \ N=

Bl(N,r,iu) for r = 1

8 16 32 M 128 256 512 1024 INF

-2 8.333E-01 7.5O0E-01 7.0836-01 6.875E-0I 6.771E-01 6.719E-01 6.693E-01 6.680E-01 6.673E-01 6.667E-01

-1.8 8.581E-01 7.827E-01 7.431E-01 7.226E-01 7.122E-01 7.068E-01 7.042E-01 7.028E-01 7.021E-01 7.014E-01 -1.6 8.866E-01 8.221E-01 7.864E-01 7.672E-01 7.570E-01 7.517E-01 7.490E-01 7.476E-01 7.468E-01 7.461E-01

-1.4 9.193E-01 8.700E-01 8.410e-01 8.245E-01 8.154E-01 8.105E-01 8.079E-01 8.06SF-01 8.058F-01 8.051E-01 -1.2 9.569E-01 9.284E-01 9.105E-01 8.997E-01 8.933E-01 8.897E-01 8.877E-01 8.866E-01 8.860E-01 8.854E-01

-1 1.00(€+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO i.oooe+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.8 1.050E+00 1.088E+00 l.llTE+OO 1.137E+O0 1.150E+00 1.160E+00 1.165E+O0 1.169E+00 1.171E+O0 1.175E+00

-.6 1.107E+00 1.197E+O0 1.271E+O0 1.327E+O0 1.370E+00 1.401E+00 1.422E+00 1.438E+O0 1.448E+00 1.470E+00

-.4 1.172E+00 1.333E-K)0 1.476E+O0 1.599E+00 1.700E+00 1.782E+00 1.847E+00 1.898E+00 1.9388+00 2.065E+00

-.2 1.247E+00 1.502E+00 1.754E-K)0 1.994E+O0 2.216E+O0 2.418E+O0 2.599E+O0 2.759E+00 2.900E+00 3.AS3E+00

1.333E+00 1.714E+O0 2.13X+00 2.581E+00 3.048E+00 3.528E+O0 4.016E+O0 4.509E+O0 5.005E+00

.2 1.432E+O0 1.982fc-H)0 2.658E+00 3.471E+00 4.432E+00 5.555E+00 6.858E+00 8.363E+00 i.oioe+01

.4 1.546E+00 2.320E+00 3.391E-K)0 4.846E+00 6.801E+00 9.407E+00 1.287E+01 1.744E+01 2.350E+01

.6 1.677E+O0 2.750E+00 4.424E+00 7.006E+O0 1.096E+01 1.698E+01 2.614E+01 4.005E+01 6.114E+01

.8 1.827E+00 3.299E+O0 5.894E+00 1.045E+01 1.841E+0I 3.230E+01 5.652E+01 9.872E+01 1.722E+02

1 2.000E+00 4.000E+00 8.000E+00 1.600E+01 3.200E+01 6.400E+01 1.280E+O2 2.560E+02 5. 120E+02

1.2 2.198E-K)0 4.900E+00 1.104E+01 2.5O6E+01 5.717E+01 1.308E+02 2.999E+02 6.881E+02 1.580E+03

1.4 2.426E-HXI 6,059E+00 1.546E+01 3.999E+01 1.044E+02 2. /3W:+02 7.202E+O2 1.897E+03 5.003E+03

1.6 2.688E+(X) 7.555E+00 2.191E+01 6.479E+01 1.938E+02 5.833E+02 1.762E+03 5.331E+03 1.615E+04

1.8 2.988E-MX) 9.490E+O0 3.i:»E+01 1.063E+02 3.646E+02 1.260E+03 4.372E+03 1.519E+04 5.286E+04

2 3.33:-ie+oo 1.200E+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E+04 4.378E+04 1.749E+05

Hu \ N=

BHN,r,iiu) for r = 1.01

16 32 64 128 256 512 1024 INF

-2 l.OOOE+OO l.OOOE+OO 1.000c +00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1.8 9.156t-01 8.682E-01 8.425E-01 8.288E-01 8.217E-01 8.181E-01 8.162E-01 8.153E-01 8. 148E-01 8.143E-01

-1.6 9.098E-01 8.566E-01 8.264E-01 8.098E-01 8.009E-01 7.963E-01 7.938t-01 7.926E-01 7.920E-01 7.913E-01

-1.4 9.286E-01 8.84OE-01 8.573E-01 8.419E-01 8.333E-01 8.287E-01 8.262E-01 8.249E-01 8.242E-01 8.235E-01

-1.2 9.599E-01 9..331E-01 9.160E-01 9.056E-01 8.995E-01 e.96OE-01 8.940E-01 8.929E-01 8.924E-01 8.917E-01

-1 1.000E+0(i 1.00(€+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.8 1.048£+0() 1.O85E+O0 1.113E+00 1.133E+00 1.147E+00 1.155E+O0 1.161E+00 1.165E+00 1.167E+00 1.170E+00

-.6 1.104E+00 1.193E+O0 1.265E+00 1.321E+00 1.362E+00 1.393E+00 1.414E+00 1.429E+00 1.440E+00 1.461E+<:)0 -.4 1.168E+00 1.327E+00 1.468E+00 1.589E+00 1.689E+00 1.770E+00 1.835E+O0 l.^S+00 1.924E+00 2.050E+00

-.2 1.243t+00 1.495E+00 1.743E+00 1.980E+00 2.200E+O0 2.4O0E+O0 2.579E+00 2.737E+O0 2.877E+O0 3.829E+00

1.329E+O0 1.706E+00 2.12OE+O0 2.563E+O0 3.025E+00 3.500E+O0 3.984E+00 4.472E+00 4.963E+00

^ 1.428E+W 1.972E+O0 2.642E+00 3.447E+00 4.400E+00 5.512E+00 6,8O4£+O0 8.296E+00 1.002E+01

.4 1.541E+00 2.309E+O0 3.371E+00 4.814E+00 6.754E+00 9.340E+00 1.277E+01 1.731E+01 2.332E+01

.6 1.672E+00 2.738E+00 4.400E+O0 6.963E+00 1.089E+01 1.687E+01 2.597E+01 3.978E+01 6.073E+01

.8 1.822E+00 3.285E+00 5.864E+00 1.039E+01 1.830E+01 3.212E+01 5.619E+01 9.814E+01 1.712E+02

1 1.995E+00 3.985E+00 7.966E+00 1.593E+01 3.185E+01 6.369E+01 1.274E+02 2.547E+02 5.095E+02

1.2 2. 194E+00 4.885E+00 l.lOOE+01 2.497E+01 5.695E+01 1.303E+02 2.987E+02 6.854E+02 1.573E+03

1.4 2.422E+00 6.045E+O0 1.542E+01 3.9^+01 1.041E+02 2.730E+02 7.ia)E+02 1.892E+03 4.988E+03

1.6 2.685E+00 7.542E+00 2.187E+01 6.466E+01 1.934E+02 5.822t+02 1.758E+03 5.321E+03 1.611E+04

1.8 2.986E+0(» 9.482E+00 3.135E+01 1.062E+02 3.643E+02 1.259E+03 4.367E+03 1.518E+04 5.280E+O4

2 3.333E+00 1.200E+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E+04 4.378E+04 1.749E+05

21

TN-316
Mu \ N=

Bl(N,r,»u) for r = 1.1

8 16 32 64 128 256 512 1024 INF

-2 : l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO

-1.8 9.527E-01 9.243E-01 9.082E-01 8.994E-01 8.948E-01 8.924E-01 8.911E-0I 8.905E-01 8.902E-01 8.898E-01 -1.6 : 9.415E-01 9.048E-01 8.831E-01 8.709E-01 8.642E-01 8.607E-01 8.588E-01 8.578E-01 8.574E-01 8.569E-01

-1.4 9.493E-01 9.157E-01 8.948E-01 8.825E-01 8./55fc-01 8.717E-01 8.696E-01 8.6«SF-01 8.680E-01 8.674E-01 -1.2 ! 9.696E-01 9.482E-01 9.341E-01 9.254E-01 9.201E-01 9.171E-01 9. 154E-01 9.145E-01 9.140E-01 9.134E-01

-1 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE-KK) l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.e 1.040E+00 1.072E+O0 1.096E-K)0 1.114E+00 1.126E+O0 1.134E+00 1.139E+00 1.142E+00 1.144E+00 1.147E+00

-.6 1.088E+00 1.166E+00 1.230e+00 1.281E+00 1.319E+O0 1.346E+00 1.366F+00 1.379E+00 1.389E+00 1.4O8E+O0

-.4 1.147E+00 1.288E+00 1.416E+00 1.526E+00 1.617E+00 1.692E-HX) 1.751E+O0 1.797E+O0 1.832E+00 1.948E+00

-.2 1.217E+00 1.444E+00 1.671E+00 1.889E+00 2.091E+00 2.275E+00 2.440E+00 2.586E+00 2.714E+00 3.593E+00

1.298E+00 1.643E+00 2.026E+O0 2.435E+00 2.B63F+00 3.304E+O0 3.752E+00 4.205E+00 4.661E+00

.2 1.393E+00 1.898E+00 2.521E+O0 3.272E+O0 4.160E+00 5.199E+00 6.404E+00 7.797E+O0 9.402E+O0

.4 1.504E+00 2.223E+00 3.219E+O0 4.575£-K)0 6.398E+00 8.828t+00 1.205E+01 1.632E+01 2. 197E+01

.6 1.633E+00 2.640E-HX) 4.213E+00 6.639E+00 1.035E+01 1.602E+01 2.4<O£+01 3.771E+01 5.754E+01

.8 1.78:^+00 3. 177E+00 5.637E-K)0 9.954E+00 1.750E+01 3.068E+01 5.365E+01 9.366F+01 1.634E+02

1 1.957E+00 3.870E+00 7.696E+00 1.535E+01 3.065EK)1 6.126E-K)1 1.225E+02 2.449E+02 4.89^+02

1.2 2.158E+O0 4.766E-HX) I.069E+01 2.422E+01 5.521E+01 1.263E+02 2.894E+02 6.640E+02 1.524E+03

1.4 2.391E-HX) 5.929E+00 1.508E+01 3.897E-K)1 1.016E+02 2.666E*02 7.011E+02 1.847E+03 4.870E+O3

1.6 2.66r»£+00 7.443E+00 2. 154E+01 6.366E+01 1.904E+02 5.730E+02 1.731E+03 5.237E+03 1.586E+04

1.8 . 2.972E+00 9,418£+00 3.111E+0I 1.053EK)2 3.614E-K)2 1.249E+03 4.333t+03 1.506E+04 5.238E+04

2 3.33:3£+00 1.200E+01 4.533E+01 1.760E+02 6.933E-H)2 2.752E+03 1.097E+04 4.378E+04 1.749E+05

Mu \ N=

Bl(N,r,(iiu) for r = 2

16 32 64 128 256. 512 1024 INF

-2 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO

-1.8 9.901E-01 9.836E-01 9.796E-01 9.774E-01 9.761E-01 9.755E-01 9.752E-01 9.750E-01 9.749E-01 9.748E-01

-1.6 9.845E-01 9. 737E-01 9.669E-01 9.630E-01 9.607E-01 9.595E-01 9.589E-01 9.586E-01 9.584E-01 9.582E-01

-1.4 9.837E-01 9.719E-01 9.641E-01 9.593E-01 9.565E-01 9.550E-01 9.541E-01 9.537E-01 9.534E-01 9.532E-0I

-1.2 9.886E-01 9.799E-01 9.738E-01 9.699E-01 9.675E-01 9.661E-01 9.653E-01 9.648E-01 9.646E-01 '?.643E-01

-1 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00

-.8 1.019£+0(i 1.035E+00 1.048E+O0 1.058E+00 1.064E+00 1.O69E+O0 1.072E+00 1.074E+00 1.075E+00 1.077E+00

-.6 1.046E+00 1.090E+0(i 1.126E+O0 1.156E+00 1.178E+00 1.195E+00 1.207E+O0 1.215E+00 1.221E+«» 1.233E+00

-.4 1.0e4E+00 1.16eE+00 1.246E+00 1.315E+00 1.373E+00 1.420E+00 1.457E+00 1.487E+00 1.509E+00 1.583E+00

-.2 1.133E+00 1.277E+0«) 1.425E+00 1.568E+00 1,702£+00 1.824E+00 1.934E+00 2.031E+00 2.117E+O0 2.704E+00

1.1»5E+0C. 1.427E+00 1.688E+00 1.971E+O0 2.267E+O0 2.573E+00 2.884E+00 3.198E+00 3.515E+O0

.2 1.273E+00 1.629E+0() 2.075E+00 2.615E+00 3.256E+00 4.005E+00 4.876E+O0 5.882E+00 7.042E+00

.4 1.369E+00 l.WlE+00 2.644E+O0 3.659E+O0 5.025E+00 6.847E+00 9.267E+00 1.247E+01 1.670E+01

.6 1.486E+f)0 2.264E+00 3.485E+O0 5.371E+00 8.262E+O0 1.267E+01 1.937E+01 2.955E+01 4.498E+01

.8 1.628E+00 2.750E+00 4.733e+O0 8.215E+00 1.431E+01 2.494E+01 4.347E+01 7.576E+01 1.32OE+02

1 1.800E+00 3.400E+00 6.600E+00 1.300E+01 2.580E+01 5.140E+01 1.026E+02 2.050E+02 4.098E+02

1.2 2.007E+00 4.271E+O0 9.409E+00 2.114E+01 4.800E+01 1.096E+02 2.510E+02 5.756E+02 1.321E+03

1.4 2.255E+00 5.439E+00 1.3ME+01 3.512E+01 9.142E+01 2.395E+02 6.299E+02 1.659E+03 4.374E+03

1.6 2.552E+O0 7.012E+O0 2.014E+01 5.y3bk+01 1.773E+02 5.334E+02 1.611E+03 4.874E+03 1.476E+04

1.8 2.908E+00 9.132E+00 3.006E+01 1.017E+02 3.487E+02 1.205E+03 4.179E+03 1.453E+04 5.053E+04

2 3.33.3t+00 1.200E+01 4.533E+01 1.76OE+02 6.933E+02 2.752E+03 1.097E+04 4.378E+04 1.749E+05

22

TN-317
BKN.r.mj) for r = 4

Mu \ N= 4 8 16 32 64 126 256 512 1024 irF

-2 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO

-1.8 9.974E-01 9.957E-01 9.946E-01 9.940e-01 9.936E-01 9.yj5k-01 9.934E-01 9.933E-01 9.yj3t-01 9.933E-01 -1.6 9.952E-01 9.918E-01 9.896E-01 9.884E-01 9.876E-01 9.873E-01 9.870E-01 9.869E-01 9.869E-0I 9.868E-01

-1.4 9.941E-01 9.898E-01 9.869E-01 9.851E-01 9.840E-01 9.834E-01 9.831E-01 9.82yk-01 9.829E-01 9.82«fc-01 -1.2 9.953E-01 9.916E-01 9.890E-01 9.873E-01 9.862E-01 9.856E-0I 9.8536-01 9.851E-01 9.850E-01 9.849E-01

-1 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+00

-.8 l.OlOE+00 1.019E+00 1.026E+00 1.031E+O0 1.035E+00 1.037E+O0 1.039E+00 1.040E+00 1.041E+00 1.042E+00

-.6 1.027E+00 1.053E+00 1.075E+00 1.093E+00 1.107E+O0 1.117E+00 1.124E+O0 1.129E+00 1.132E+O0 1.140E+O0

-.4 1.054E+00 1.109E+00 1.160E+00 1.205E+00 1.243E+00 1.274E+00 1.299E+00 1.318E+00 1.333E+00 1.382E+00

-.2 1.092E+00 1.194E+00 1.298E+00 1.399E+00 1.494E+00 1.580E+00 1.658E+00 1.727E+00 1.788E+00 2.204E+O0

1.144E+00 1.318E+O0 1.514E+00 1.726E+O0 1.949E+00 2. 179E+00 2.414E+00 2.651E+00 2.890E+00

.2 1.214E+00 1.495E+00 1.848E+00 2.275E+00 2.783E+00 3.377E+00 4.067E+00 4.865E+00 5,784E+00

.4 1.303E+0(i 1.742E+O0 2.356E+O0 3. 196E+00 4.326E+O0 5.834E+00 7.837E+00 1.049E+01 1.399E+01

.6 1.415E+00 2.082E+00 3.129E+00 4.748E+00 7.22Vt+0C l.lOlE+01 1.676E+01 2.550E+01 3.875E+01

.8 1.555E*00 2.548E+00 4.303E+O0 7.385E+00 1.278E+01 2.219E+01 3.860E+01 6.718E+01 1.170E+O2

1 1.727E+(X) 3.182E+00 6.091E+00 1.191E+01 2.355E+01 4.682E+01 9.336E+01 1.865E+02 3.726E+02

1.2 1.938E+00 4.045E-KK) 8.826E+O0 1.974E+01 4.472E+01 1.020E+02 2.336E+02 5.356f+02 1.229E+03

1.4 2.194EWXI 5.219E+00 1.303E+01 3.340E+01 8.686E+01 2.275E+02 5.981E+02 l.b/bt+03 4. 154E+03

1.6 2.504E*00 6.819E+00 1.952E+01 5.745E+01 1.715E+02 5. 160E+02 1.558E+03 4.714E+03 1.428E+04

1.8 2.879E+O0 9.006E+00 2.960E+01 l.OOOE+02 3.431E+02 1.186E+03 4.112E+03 1.429E+04 4.972E+04

2 5.333E+00 1.200E+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E+04 4.378E+04 1.749E+05

Bl(N,r,i»u) for r = 8

flu \ N= 4 8 16 32 64 128 256 512 1024 INF

-2 l.OOO£+<00 1.000E+(W l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO

-1.8 ^.?93E-01 9.988E-01 9.985E-01 9.983E-01 9.982E-01 9.981E-01 9.981E-01 9.981E-01 9.981E-01 9.981E-01

-1.6 9.984E-01 9.973E-01 9.966E-01 9.962E-01 9.960E-01 9.958E-01 9.958E-01 9.957E-01 9.957E-01 9.957E-01

-1.4 «.978E-01 9.961E-01 9.950E-01 9.944E-01 9.940E-01 9.937E-01 9.936E-01 9.936E-01 9.935E-01 9.935E-01 -1.2 9.'?«0E-01 9.963E-01 9.952E-01 9.945E-01 9.940E-01 9.938E-01 9.936E-01 9.935E-01 9.935E-01 9.934E-01

-1 1.00<)£+()0 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO 1.000E+0(» -.6 1.006f+00 l.OllE+00 1.014E+00 1.017E+00 1.020E+00 1.021E+00 1.022E+O0 1.022E+00 1.023E+00 1.023E+00

-.6 1.017£+(m:i 1.033E+00 1.047E+00 1.058E+O0 1.067E+O0 1.073E+O0 1.078E+00 1.081E+00 1.083E+00 1.088E+O0

-.4 1.037E+OO 1.075E+00 l.llOE+00 1.142E+O0 1.168E+00 1.189E+00 1.207E+00 1.220E+00 1.231E+00 1.264E+00

-.2 1.069E+00 1.145E+00 1.223E+00 1.299E+00 1.371E+O0 1.436E+O0 1.494E+00 1.546E+00 1.592E+O0 1.905E+00 l.n6E+00 1.255E+00 1.413E+00 1.584E+00 1.763E+00 1.949E+00 2.137E+00 2.328E+O0 2.520E+00

.2 i.ieiE+<H) 1.420E+00 1.720E+O0 2.083E+00 2.514E+O0 3.019E+O0 3.606E+00 4.284E+00 5.066E+00

.4 1.268£+(X) 1.657E+00 2.202E+00 2.946E+00 3.949E+O0 5.286E+00 7.062E+O0 9.414E+00 1.252E+01

.6 1.380E+<:»0 1.991E+00 2.950E+00 4.433E+O0 6.706E+00 1.017E+01 1.544E+01 2.345E+01 3.559E+01

.8 I.521E+00 2.453E+00 4.101E+00 6.996E+00 1.206E+01 2.090E+01 3.630E+01 6.315E+01 1.099E+02

1 1.696E+O0 3.087E+00 5.870E+O0 1.143E+01 2.257E+01 4.483E+01 8.935E+01 1.784E+02 3.565E+02

1.2 1.910E+00 3.953E+00 8.590E+00 1.917E+01 4.341E+01 9.898E+01 2.265E+02 5.195E+02 1.192E+03

1.4 2.170E+O0 5.135E+00 1.279E+01 3.276E+01 8.515E+01 2.230E+02 5.862E+02 1.544E+03 4.071E+O3

1.6 2.486E+O0 6.751E+O0 1.930E+01 5.678E+01 1.695E+02 5.099E+02 1.540E+03 4.658E+03 1.411E+04

1.8 2.870E+00 8.963E+00 2.945E+01 9.951E+01 3.413E+02 1.179E+03 4.090E+03 1.421E+04 4.945E+04

2 3.333E+O0 1.200E+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E+O4 4.378E+04 1.749E+05

23

TN-318
Hu \ N=

Bl(N,r,iu) for r = 16

8 \i 32 M 128 256 512 1024 INF

-2 ! l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+00

-1.8 9.998E-01 9.997E-01 9.996E-01 9.995E-01 9.995E-01 9.995E-01 9.995E-01 9.995E-01 9.995E-01 9.995E-01

-l.fc ! 9.995E-01 9,991E-01 9.98Vt-01 9.987E-01 9.987E-01 9.986E-01 9.986E-01 9.986E-01 9.986E-01 9.986E-01

-1.4 9.992-01 9.985E-01 9.981E-01 9.979E-01 9.977E-01 9.976E-01 9.976E-01 9.976E-01 9.975E-01 9.975E-01 -1.2 . 9.991E-01 9.984E-01 9.979E-01 9.976E-01 9.974E-01 9.973E-01 9.972E-01 9.972E-01 9.972E-01 9.971E-01

-1 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 -.8 1.003E+00 1.006E+00 1.008E+00 l.OlOE+00 l.OllE+00 1.012E+00 1.012E+O0 1.013E+00 1.013E+00 1.013E+00

-.6 l.OllE+00 1.021E+00 1.030E+00 1.037E+00 1.043E+O0 1.047E+O0 1.050E+00 1.052E+00 1.053E+00 1.056E+00

-.4 1.026E+00 1.053E+00 1.07^+00 I.IOIE+OO 1.119E+00 1.135E+00 1.147E+O0 1.157E+00 1.164E+O0 1.188E+O0

-.2 1.054E+00 1.113E+00 1.174E+O0 l.Zi3E+00 1.289E+O0 1.339E+O0 1.38SE+00 1.425E+O0 1.461E+00 1.705E+00

1.097E+00 1.214E+00 1.346E+O0 1.489E+00 1.639E+00 1.794E+O0 I.952E+00 2.112E+00 2.273E+O0

.2 1.160e+00 1.372E+00 1.637E+00 1.958E+00 2.340E+O0 2.787E+00 3.307E+00 3.907E+00 4.599E+O0

.4 1.247E+00 1.606E+00 2.108E+O0 2.794E+00 3.717E+00 4.950E+00 6.587E+00 8.754E+00 1.162E+01

.6 1.360E+00 1.939E+00 2.848E+00 4.254E+O0 6.409E+00 9.696E+O0 1.469E+01 2.228E+01 3.379e+01

.8 1.504E+00 2.404E+00 3.997E+00 6.794E+00 1.169E+01 2.023E+01 3.512E+01 6. 106E+01 1.062E+O2

1 1.681E-HX) 3.043E+00 5.766E+00 1.121E+01 2.211E+01 4.389E+01 8.747E+01 1.746E+02 3.489E+02

1.2 1.898E+00 3.914E+00 8.491E+O0 1.894E+01 4.285E+01 9.769E+01 2.236E+02 5. 127E+02 1.177E+03

1.4 . 2.161E+00 5.104E+00 1.270E+01 3.251E+01 8.450E+01 2.213E+02 5.817E+02 1.532E+03 4.039E+03

1.6 2.480e+00 6.727E+00 1.923E+01 5.654E+01 1.688E+02 5.077E+O2 1.533E+03 4.639E+03 1.405E+04

1.8 . 2.866E+00 8.949E+00 2.940E+01 9.934E+01 3.406E+02 1.177E+03 4.083E+03 1.419E+04 4.936E+04

2 3.333E+00 1.200E+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E+O4 4.378E+04 1.749E+05

Mu \ K=

Bl(N,r,Bu) for r = 32

8 16 32 64 128 256 512 1024 INF

-2 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO

-1.8 9.999E-01 9.999E-01 9.999E-01 9.999E-01 9.999E-01 9.9VWt-01 9.9yyt-oi 9.998E-01 9.998E-01 9.998E-01 -1.6 9.998E-01 9.997E-01 9.996E-01 9.996E-01 9.996E-01 9.995E-01 9.995E-01 9.995E-01 9.995E-01 9.995E-01

-1.4 9.997E-01 9.994E-01 9.993E-01 9.992E-0I 9.991E-01 9.991E-01 9.991E-01 9.991E-01 9.991E-01 9.991E-01 -1.2 9.996E-01 9.993E-01 9.991E-01 9.99OE-01 9.989t-01 9.988E-01 9.988E-01 9.988E-01 9.988E-01 9.988E-01

-1 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00

-.8 1.002E+0.:i 1.003E+00 1.005E+00 1.006E+00 1.006E+00 1.007E+00 1.007E+00 1.007E+00 1.007E+00 1.00^+00

-.6 l.OOTE+OCt 1.014E+00 1.019E+00 1.024E+O0 1,028E+00 1.030E+O0 1.032E+00 1.033E+00 1.034E+00 1.036E+00

-.4 1.019E+00 1.039E+00 1.057E+00 1.073E+O0 1.087E+00 1.098E+00 1.107E+00 1.114E+00 1.119E+00 1.136E+00

-.2 1.043E+00 1.090E+00 1.139E+00 1.186E+00 1.230E+00 1.271E+O0 1.307E+00 1.339E+00 1.368E+00 1.563E+00 1.08^+00 1.184E+00 1.297E+00 1.420E+00 1.550E+0t) 1.68^+00 1.819E+00 l.Vb/k+00 2.095E+00

'; 1.146E+00 1.338E+00 1.579E+00 1.871E+O0 2.218E+O0 2.625E+00 3.097E+00 3. 64^+00 4.272E+00

.4 1.233E+00 1.572E+00 2.046E+00 2.693E+00 3.565E+00 4.729E+O0 6.275E+00 8.320E+00 1.102+01

.6 1.348e+00 1.90^+00 2.787E+00 4. 147E+00 6.231E+00 9.408E+00 1.424E+01 2.158E+01 3.270E+01

.8 1.494E+00 2.378E+00 3.940E+00 6.685E+00 1.149E+01 1.986E+01 3.447E+01 5.992E+01 1.042E+02

1 1.674E+O0 3.021E+O0 5.716E+00 l.lllE+01 2.188E+01 4.344E+01 8.656E+01 1.728E+02 3.453E+02

1.2 1.893E+00 3.898E+00 8.44^+00 1.883E+01 4.261E+01 9.714E+01 2.223E+02 5.097E+02 1.170E+03

1.4 2. 1588+00 5.091E+00 1.266E+01 3.242E+01 8.425E+01 2.206E+02 5.799E+02 1.527E+03 4.027E+O3

1.6 2.478E+00 6.719E+00 1.9^20E+01 5.646E+01 1.685E+02 5.070E+02 1.531E+03 4.632E+03 1.403E+O4

1.8 2.865E+00 8.945E+O0 2.938E+01 9.928E+01 3.405E+02 1.176E+03 4.081E+03 1.418E+04 4.934E+04

2 3.33:S+00 1.200E+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E+04 4.378E+04 1.749E+05

24

TN-319
Mu \ N=

BKN.r.iu) for p = 64

8 16 32 64 128 256 512 1024 INF

-2 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO

-1.8 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1.6 9.999E-01 9.999E-01 9.999E-01 9.999E-01 9.999E-01 9.9V9t-01 9.998E-01 9.998E-01 9.998E-01 9.9988-01

-1.4 9.yvvt-oi 9.998E-01 9.997E-01 9.997E-01 9.997E-01 9.997E-01 9.997E-01 9.996E-01 9.996E-01 9.996E-01 -1.2 9.998E-01 9.997E-01 9.996E-01 9.995E-01 9.995E-01 9.995E-01 9.y9bl:-01 9.995E-01 9.995E-01 9.995E-01

-1 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00

-.6 l.OOlE+00 1.002E+00 1.003E+00 1.003E+00 1.004E+00 1.004E+00 1.004E+00 1.004E+00 1.004E+00 1.004E+00

-.6 1.005E+00 1.009E+00 1.013E+00 1.016E+00 1.018E+00 1.020E+00 1.021E+00 1.022E+00 1.022+00 1.024E+00

-.4 1.014E+00 1.028E+00 1.042E+00 1.054E+00 1.064E+00 1.072E+00 1.078E+00 1.0e3£+00 1.087E+00 l.lOOE+00

-.2 1.035E+00 1.073E+00 1.112E+00 1.151E+O0 1.187E+O0 1.220E+00 1.249E+00 1.275E+00 1.2y«E+00 1.456E+00

1.073£-K)0 1.161E+00 1.261E+00 1.369E+00 1.482E+00 1.600E+00 1.719E+00 1.840E+00 1.961E+00

.2 1.135E+00 1.313E+00 1.536E+00 1.808E+00 2.129E+00 2.506E+00 2.944E+00 3.450E+00 4.033E+OO

.4 1.224E-KK) 1.548E+00 2.003E+00 2.624E+00 3.461E+00 4.578E+00 6.060E+00 8.023E+O0 1.062E+01

.6 1.341E+00 1.889E+00 2.749E+00 4.080E+00 6.119E+00 9.229E+O0 1.396E+01 2.114E+01 3.203E+01

.8 1.489E+00 2,36.'«+00 3.909E+00 6.624E+O0 1.137E+01 1.966E+01 3.411E+01 5.y2Vt+01 1.031E+02

1 1.670E+O0 3.010E+00 5.691E+00 1.105E+01 2.177E+01 4.322t+01 8.611E+01 1.719E+02 3.435E+02

1.2 1.890E*0(i 3.891E+00 8.429E+00 1.879E+01 4.251E+01 9.690E+01 2.218E+02 5.085E+02 1.167E+03

1.4 2.156E+00 5.087E+00 1.265E+01 3.23tt:+01 8.415E+01 2.204E+02 5.7y3E+02 1.526E+03 4.022E+03

1.6 2.477E+00 6.716E+00 1.919E+01 5.644E+01 1.685E+02 5.067E+02 1.530E+03 4.630E+O3 1.402E+O4

1.8 2.865E-MX) 8.943E+00 2.938E+01 9.927E+01 3.404E+02 l,176E+03 4.080E+03 1.418E+04 4.933E+04

2 3.333E+00 1.2OOE+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E+04 4.378E+04 1.749E+05

Mu \ N=

Bl(N,r,iiu) for r = 128

8 16 32 64 128 256 512 1024 INF

-2 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 1.000E+0() l.OOOE+OO 1.00(€+00

-1.8 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO

-l.t l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 1.000E+0() 9.999E-01 9.999E-01 9.999E-01 9.999E-01

-1.4 IMK)E*00 9.999E-01 9.999E-01 9.9«9E-01 9.999E-01 9.999E-01 9.999E-01 9.999E-0I 9.999E-01 9.999E-01

-1.2 9.999E-01 9.999E-01 9.998E-01 9.998E-01 9.998E-01 9.998E-01 9.998E-01 9.998E-01 9.998E-01 9.998E-01

-1 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+^Xi l.OOOE+OO l.OOOE+OO

-.8 I.OOIE+OO I.OOIE+OO 1.002E+00 1.002E+00 1.002E+O0 1.002E+00 1.002E+00 1.002E+00 1.002E+00 1.002E+00

-.6 1.003E+0(' 1.006E+00 1.008E+00 I.OIOE+OO 1.012E+00 1.013E+00 1.014E+00 1.014E+O0 1.015E+O0 1.015E+00

-.4 I.OIOE+OO 1.021E+00 1.031E+00 1.040E+00 1.047E+00 1.05X+00 1.058E+00 1.062E+00 1.065E+00 1.074E+00

-, 2 1.029E+0() 1.060E+00 1.092E+00 1.124E+00 1.153E+O0 1.181E+00 1.205E+00 1.226E+00 1.245E+00 1.375£+'J0 1.065E+00 1.144E+00 1.232E+00 1.329E+00 1.430E+00 1.534E+00 1.640E+00 1.748E+00 1.856E+00

* ^ 1.127E+00 1.294E+00 1.504E+00 1.759E+O0 2.062E+O0 2.416E+00 2.827E+00 3.303E+00 3.851E+00

.4 1.217E+00 1.532E+O0 1.973E+O0 2.576E+00 3.:a^+oo 4.471E+O0 5.909E+00 7.813E+O0 1.033E+01

.6 1.336E+00 1.877E+O0 2.725E+00 4.037E+00 6.048E+O0 9.115E+O0 1.378E+01 2.086E+01 3. 160E+01

.8 1.4e6E+00 2.354E+00 3.891E+O0 6.589E+00 1.131E+01 1.955E+01 3.391E+01 5.893E+01 1.025E+02 1.6i«£+00 3.005E+0«) 5.679E+00 1.103E+01 2.172E+01 4.311E+01 8.589E+01 1.714E+02 3.426E+02

1.2 1.889E+00 3.887E+00 8.421E+00 1.877E+01 4.246E+01 9.680E+01 2.215E+02 5.079E+02 1.166E+03

1.4 2. 156€+(K3 5.085E+00 1.265E+01 3.237E+01 8.411E+01 2.203E+02 5.790E+02 1.525E+03 4.021E+03

1.6 2.477E+{>0 6.715E+00 1.919E+01 5.643E+01 1.684E+02 5.067E+02 1.530E+03 4.629E+03 1.402E+04

1.8 2.865E+00 8.943f+O0 2.938E+01 9.926E+01 3.404E+02 1.176E+03 4.080E+03 1.418E+04 4.932E+04

3. 333E+()0 1.200E+01 4.533E+01 1.760E+02 6.93^+02 2.752E+03 1.097E+04 4.37^+04 1.749E+05

25

TN-320
Nu \ N= 4

BKN.r.iu) for r = 256

8 16 32 64 126 256 512 1024 Itf

-2 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1.8 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 -1.6 i.oooe+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO

-1.4 l.OOOEvOO l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 9.999E-01 9.999E-01 9.999E-01 -1.2 l.OOOE-KW 9.999E-01 9.999E-01 9.999E-01 9.999E-01 9.999E-01 9.999E-01 9.yy9t-oi 9.999E-01 9.999E-01

-1 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO

-.8 l.OOOE+00 l.OOlE+00 l.OOlE+00 I.OOIE+OO I.OOIE+OO l.OOlE+00 l.OOlE+00 I.OOIE+OO l.OOlE+00 l.OOlE+00

-.6 1.002E+00 1.004E+0C 1.005E+00 1.007E+O0 1.008E+00 i.ooe£+oo 1.009E+00 1.009E+00 I.OIOE+OO l.OlOE+00

-.4 1.008E+00 1.016E+00 1.023E+00 1.029E+O0 l.OSE+OO 1.039E+00 1.043E+O0 1.046E+00 1.048E+00 1.055E+00

-.2 1.024E+00 1.050E+00 1.077E+00 1.1O3E+O0 1.127E+00 1.150E+00 1.170E+00 1.188E+00 1.204E+00 1.311E+00

1.059E+O0 1.130E+00 1.210E+00 1.296E+00 1.388E+00 1.482E+00 1.577E+00 1.674E+00 1.772E+00

.2 1.121E+00 1.280E+00 1.479E+00 1.722E+00 2.009E+O0 2.346E+00 2.737E+O0 3.189E+O0 3.710E+O0

.4 1.212E+00 1.520E+00 1.952E+00 2.541E+00 3.335E+00 4.394E+00 5.800E+00 7.662E+00 1.012E+01

.6 1.333E+00 1.869E+00 2.709E+00 4.010E+O0 6.003E+00 9.042E+00 1.366E+01 2.068E+01 3.132E+01

.8 1.484E+00 2.350E+00 3.881E+00 6.570E+00 1.127E+01 1.948E+01 3.380E+01 5.873E+01 1.022E+02

1 1.668E+00 3.003E+00 5.673E+00 l.lOlE+01 2. 169E+01 4.305E+01 8.578E+01 1.712E+02 3.421E+02

1.2 1.889£+00 3.886E+00 8.418E+00 1.876E+01 4.244E+01 9.675E+01 2.214E+02 5.077E+02 1.165E+03

1.4 2.156E+00 5.064E+00 1.264E+(il 3.236E+01 8.410E+O1 2.202E+02 5.789E+02 1.525E+03 4.020E+03

1.6 2.477E+00 6.715E+0C' l.«19E+0I 5.642E+01 1.684E+02 5.066E+02 1.530E+03 4.629E+03 1.402+04

1.8 2.865E+00 8.943E+00 2.938E+01 9.926E+01 3.404E+02 1.176E+03 4.08OE+O3 1.418E+04 4.932E+04

2 3. 333E+00 1.200E+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E+04 4.378E+04 1.749E+05

Bl(N,r,iiiu) for r = 512

Mu \ N= 4 8 16 32 64 128 256 512 1024 INF

-2 1.0(»0E+0(i l.OOOt+OO 1.00(€+00 1.000E+0() l.OOOE+OO l.OOOE+OO 1.000E+0() 1.00(€+00 l.OOOE+00 l.OOOE+00

-1.8 l.OOOE+W l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO 1.0CM)£+0<) 1.000£+0<j l.OOOE+00 1.000E+<X l.OOOE+00

-1.6 1.0()0E+(X1 1.000E+0(i l.OOOE+OO l.OOOE+OO l.OOOE+00 1.0(»OE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+00

-1.4 l.OOOE+OO l.OOOE+O') l.OOOE+OO i.oooE+o<:i l.OOOE+O*' l.OOOE+OO l.OOOE+OO 1.000E+<X 1.0<X€+00 i.(X>o£+o<:> -1.2 l.OOOt+OC' 1.0(iOE+00 1.000E+«) l.(X»0E+w i.oooE+0*:) i.oooe+00 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.iX<€+00

-1 l.OOOE+W l.OOOE+f-O 1.000£-^0<J 1.000£i<i0 l.OOOE+O*) l.OOOE+OO l.OOOE+Oi) l.OOOE+OO l.O0OE+<X' 1 . OOOE+X'

-.s 1.(J00E-H)() 1.000t+0(i l.OOlE+00 I.OOIE+OO I.OOIE+OO l.OOlE+00 I.OOIE+CK) I.OOIE+OO l.OOlE+00 I.OOIE+OO

-.6 l.OOlE-00 1.003£+rXi 1.004E+00 1.004E+0C' 1.005E+00 1.006E+00 1.006E+00 1.006E+00 1.006E+O0 1.007E+00

-.4 1.006E+0-:' 1.012E+00 1,017E+00 1.022E+00 1.026E+00 1.030E+00 1.032E+00 1.054E+00 1.036E+00 1.041E+00

-,2 1.02i)E+0<) 1.042E+0(:) 1.064E+00 1.086E+0<:i 1.107E+CKJ 1.125E+00 1.142E+00 1.157E+C*j 1.170E+OC' 1.261E+<)0

1.054E+0<0 I.II8E+1X' 1.191E+(Xi 1.270E+00 1..353E+00 1.438E+00 1.52frE+00 1.614E+O0 1 . 703E+00

^, 1.116E+<X) 1.26dE+<»0 1.460E+00 1.692E+00 1.967E+O0 2.290E+<Xi 2.665E+00 3.098E+0C. 3.5'»8E+00

.4 1 . 20«t+00 1.512E+0<:. 1.936E+00 2.516E+O0 3.296E+C»0 4.33^+00 5.721E+O0 7.553E+00 9.973£-f00

.6 1.331E+0() 1.864E+00 2.699E+()0 3.992E+00 5.973E+00 8.994E+00 1.359E+01 2.056E+01 3.114E+C'l

.8 1.48^?+<X) 2.347E+00 3.875£+0(i 6.558£+0() 1.125E+01 1.945E+01 3.373E+01 5.862E+01 1.020E+02

1.6t7E+0(' 3.001E+00 5.670E+00 l.lOlE+01 2. 168E+01 4.303E+O1 8.572E+01 1.711E+tX' 3.419E+02

it ^ 1.889E+W 3.e85E+<X) 8.416E+{k;) 1.87(i£+01 4.243E+01 9.67^+01 2.214E+02 5.076E+02 1.165E+03

1.4 2.156E+00 5.084t+0C. 1.264E+01 3.236E+01 8.410E+01 2.202E+02 5.789E+02 1.525E+0? 4.020E+03

1.6 2.477E+00 6.714E+0(i 1.919E+01 5.642E+01 1.684E+02 5.066E+02 1.530E+03 4.628£+0? 1.402E+04

1.8 2.8t5E+C)0 8.943E+O0 2.93eE+01 9.926E+01 3.4045+02 1.176E+03 4.080E+0? 1.418E+04 4.932E+04

2 : 3.33.3E+00 1.200E+01 4.533E+0i 1.76OE+02 6. 933E+02 2.752E+0.^ 1.097E+M 4.378£+04 1.749E+05

26

TN-321
Mu \ N=

Bl(N,r,iu) for r = 1024

e 16 32 (A 128 256 512 1024 INF

-2 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1.8 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO -1.6 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1.4 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO -1.2 i.oooe+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.8 1.000E-K)0 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.6 l.OOlE+00 1.002E+00 1.002E+00 1.003E+O0 1.003E+O0 1.004E+00 1.004E+00 1.004E+00 1.004E+00 1.004E+00

-.4 1.004E+00 1.009E+00 1.013E+00 1.017E+00 1.020E+00 1.022t+00 1.024E+00 1.026E+00 1.027E+O0 1.031E+00

-.2 1.017E+00 1.035E+00 1.054E+00 1.073E+00 1.090E+00 1.106E+O0 1.120E+00 1.132E+00 1.144E+00 1.220E+00

1.049E+00 1.108E+00 1.175E+00 1.248E+00 1.324E+00 1.402E+00 1.483E+00 1.564E+O0 1.645E+00

.2 1.112E-H)0 1.259E+00 1.444E+00 1.6«iE+00 1.934E+00 2.245E+O0 2.607E+00 3.025E+00 3.507E+O0

.4 1.206E+O0 1.505E+00 1.924E+00 2.497E+00 3.268E+00 4.297E+00 5.663E+00 7.472E+O0 9.862E+O0

.6 1.330E+O0 1.860E+00 2.693E+00 3.980E+00 5.953E+00 8.963E+00 1.354E+01 2.049E+O1 3.102E+O1

.8 1.482E+00 2.345E+00 3.872E+O0 6.552E+O0 1.124E+01 1.942E+01 3.369E+01 5.855t+01 1.018E+02

1 1.667E+00 3.001E+00 5.66flF+00 l.lOOE+01 2. 167E+01 4,301E+01 8.569E+01 1.711E+02 3.418E+02

1.2 1.889E+00 3.885E+00 8.416E+00 1.876E+01 4.243E+01 9.672E+01 2.213E+02 5.075E+02 1.165E+03

1.4 2.156E+00 5.084E+00 1.264E+01 3.236E+01 8.409E+01 2.202E+02 5.7e9E+02 1.525E+03 4.020E+O3

1.6 2.477E+00 6,714E+00 1.919E+01 5.642E+01 1.684E+02 5.066E+02 1.530E+03 4.628E+03 1.402E+04

1.8 2.865E+00 8.94:^+00 2.y3»:+01 9.926E+01 3.404E+02 1.176E+03 4.080E+03 1.418E+04 4.932E+04

2 3.333E+Ck:i 1.200E+01 A.smm 1.760E+02 6.933E+02 2.752E+03 1.097E+04 4.37K+04 1.749E+05

Mu \ H=

BKN.r.nu) for r - 2048

16 32 64 128 256 512 1024 INF

-2 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1.8 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO 1.00(€+00

-1.6 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1.4 1.000E+0(i l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1.2 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-1 l.OOOE+iXi l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.6 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.6 l.OOlE+00 l.OOlE+00 1.002E+00 1.002E+00 1.002E+00 1.002E+00 1.003E+00 1.003E+00 1.003E+00 1.003E+00

-.4 1.003E+00 1.007E+00 I.OIOE+OO 1.012E+00 1.015E+00 1.017E+00 1.018E+00 1.019E+00 1.020E+00 1.023E+00

-.2 1.014E+00 1.030E+O0 1.046E+00 1.061E+00 1.076E+00 1.089E+00 l.lOlE+00 1.112E+00 1.122E+O0 1.186E+00

1.045E+00 l.lOOE+00 1.162E+O0 1.229E+00 1.299E+00 1.372E+O0 1.446E+00 1.521E+00 1.596E+00

.2 1.108E+0(i 1.251E+00 1.431E+O0 1.648E+00 1.906E+00 2.209E+O0 2.560E+00 2.966E+00 3.434E+O0

.4 1.204E+00 1.501E+00 1.916E+O0 2.483E+00 3.247E+O0 4.266E+O0 5.620E+00 7.412E+O0 9.780E+00

.6 1.329E+0() 1.858E+00 2.WWF+00 3.972E+00 5.941E+00 8.942E+O0 1.351E+01 2.044E+01 3.095E+01

.8 1.4e2E+00 2.345E+00 3.870E+O0 6.548E+00 1.123E+01 1.941E+01 3.367E+01 5.851E+01 1.018E+02

1 1.667E+00 3.000E+00 5.667E+O0 l.lOOE+01 2.167E+01 4.301E+01 8.568E+01 1.710E+O2 3.417E+02

1.2 1.889E+00 3.885E+00 8.415E+00 1.875E+01 4.243E+01 9.672E+01 2.213E+02 5.075E+02 1.165E+03

1.4 2.156E+00 5.084E+00 1.264E+01 3.236E+01 8.409E+01 2.202E+02 5.789E+02 1.525E+03 4.020E+03

1.6 2.477E+00 6.714E+00 1.919E+01 5.642E+01 1.684E+02 5.066E+02 1.530E+03 4.628E+03 1.402E+04

1.8 2.86.5E+W 8.943E+O0 2.938E+01 9.926E+01 3.404E+02 1.176E+03 4.0eOE+03 1.418E+04 4.932E+04

2 3.333E+0':i 1.200E+01 4.533E+01 1.760E+02 6.933E+02 2.752E+03 1.097E+O4 4.378E+04 1.749E+05

27

TN-322
Bl(N,r,Bu) for r = ItflNITY

Mu \ N= 16 32 64 128 256 512 1024 INF

-2-1.8 -1.6

-1.4 -1.2

-1-.8 -.6

-.4

-.2

.2

.4.6

.8

11.2

1.4

1.6

1.8

2

l.OOOE+00

l.OOOE+00

l.OOOE-KiO l.OOOE+00

l.OOOE-KX l.OOOE+OO

l.OOOE+OO

l.OOOE+0*.-) 1.000t+0() l.OOOE+OC) l.OOOE+OO

1.091E+CK:)

1.198E+(:>0 1.327E+O0

1.482E+00

1.667E-K)0

1.889E-t-00 2.156E-K)0 2.477E+O0 2.865E+00

3.333E+00

l.OOOE+OO l.OOOE+OO

l.OOOE+00

l.OOOE+OO

l.OOOE+OO

l.OOOE+OO l.OOOE+OO

l.OOOE+OO

l.OOOE+OO

l.OOOE+00

i.oooE+o«:i 1.210E+00

1.487EtOG 1.854E+00 2.343E+00

3.000E+00

3.885E+00

5.084E+00 6.714E+00

8.943E+00

1.200E+01

l.OOOE+OO

l.OOOE+OO

l.OOOE+OO

l.OOOE+OO

l.OOOE+OO l.OOOE+00

l.OOOE+OO

l.OOOE+OO

l.OOOE+OO

l.OOOE+00

l.OOOE+00

1.360E+00

1.890E+00 2.680E+00

3.867E+00

5.667E+00

8.415E+00

1.264E+01

1.919E+01 2.938E+01

4.533E+01

l.OOOE+OO

l.OOOE+OO l.OOOE+00

l.OOOE+OO

l.OOOE+OO

l.OOOE+00

l.OOOE+OO

l.OOOE+00

l.OOOE+OO

l.OOOE+00 l.OOOE+00

;.541E+00 2.441E+00 3.958E+00

6.543E+00

l.lOOE+01

1.875E+01 3.236E+01

5.642E+01 9.926E+01

1.760E+02

l.OOOE+00

l.OOOE+00

l.OOOE+OO

l.OOOE+OO

l.OOOE+00

l.OOOE+OO

l.OOOE+00

l.OOOE+OO

l.OOOE+OO l.OOOE+OO

LOOOE+Ok' 1.757E+00 3.184E+00

5.916E+00

1.123E+01 2.167E+01 4.243E+01

8.409E+01

1.684E+02 3.404E+02

6.933E+02

l.OOOE+00

l.OOOE+OO

l.OOOE+00

l.OOOE+00

l.OOOE+00

l.OOOE+00

l.OOOE+OO l.OOOEHK)

l.OOOE+OO l.OOOE+00

l.OOOE+OO 2.009E+00 4.174E+00

8.903E+00

1.940E+01 4.300E+01

9.672E+01 2.202E+02

5.066E+02

1.176E+03 2.752E+03

l.OOOE+OO

l.OOOE+OO

l.OOOE+00

l.OOOE+OO

l.OOOE+00

l.OOOE+00

l.OOOE+00

l.OOOE+OO

l.OOOE+00

l.OOOE+00 l.OOOE+00 2.30^+00

5.489E+00

1.344E+01 3.364E+01 8.567E+01 2.213E+02

5.789E+02

1.530E+03 4.080E+03

1.097E+04

l.OOOE+OO

l.OOOE+OO

l.OOOE+00

l.OOOE+OO

l.OOOE+00 l.OOOE+OO

l.OOOE+00 l.OOOE+00

l.OOOE+tV i.OOOE+00 l.OOOE+00 2.642E+00

7.231E+00

2.034E+01

5.846E+01

1.710E+02 5.075E+02

l,525E+03 4.62^+03

1.418E+04 4.378E+04

l.OOOE+OO

l.OOOE+OO

l.OOOE+00 l.OOOE+00

l.OOOE+OO

l.OOOE+OO

l.OOOE+OO l.OOOE+OO

l.OOOE+00 l.OOOE+00

l.OOOE+OO 3.033E+O0

9.533E+00

3.080E+01 1.017E+02 3.417E+02

1.165E+03 4.020E+03

1.402E+O4 4.932E+04

1.749E+05

l.OOOE+OO l.OOOE+OO

l.OOOE+00 l.OOOE+OO

l.OOOE+00

l.OOOE+OO

l.OOOE+OO

l.OOOE+OO

l.OOOE+OO l.OOOE+00

28

TN-323
B2(r,mi)

HU \ r » .0001 .0003 .001 .003 .01 .03 .1 .3 .5 .7

-2 6.667E-01 6.667E-01 6.667E-01 6.667E-01 6.667E-01 6.667E-01 6.667E-01 6.667E-01 6.667E-01 6.667E-01

-1.8 1.112E-01 1.385E-01 1.762E-01 2.195E-01 2.793E-01 3.479E-01 4.431E-01 5.S66E-01 6.264E-01 6.889E-01 -1.6 1.874£-02 2.90eE-02 4.70eE-02 7.306E-02 1.183E-01 1.836E-01 2.979E-01 4.693E-01 5.901E-01 7.013E-01 -1.4 3.205E-O3 6.196E-03 1.276E-02 2.467E-02 5.0eiE-O2 9.828E-02 2.032E-01 3.999E-01 5.572E-01 7.061E-01 -1.2 5.Se6E-04 1.345E-03 3.525E-03 8.489E-03 2.225E-02 5.362E-02 1.410E-01 3.444E-01 5.273E-01 7.052E-01

-1 1.000e-04 3.000E-04 l.OOOE-03 3.000E-03 l.OOOE-02 3.000f-O2 l.OOOE-01 3.000E-01 5.000E-01 7.oooe-oi

-.8 1.862E-05 6.956E-05 2.949E-04 l.lOlE-03 4.662E-03 l./3!)t-02 7.271E-02 2.642E-01 4.749E-01 6.916E-01

-.6 3.687E-06 1.715E-05 9.231E-05 4.290E-04 2.288E-03 1.047E-02 5.438E-02 2.351E-01 4.517E-01 6.80eE-01

-.4 8.121E-07 4.678E-06 3.174E-05 1.809E-04 1.204E-03 6.664E-03 4.195E-02 2.112E-01 4.303E-01 6.683E-01

-.2 2.159E-07 1.510E-O6 1.260E-05 8.606E-05 6.921E-04 4.507E-03 3.340E-02 1.915E-01 4.103E-01 6.S46E-01 7.726E-0e 6.240E-07 6.065E-06 4.745E-05 4.404E-04 3.250C-03 2.742E-02 1.750E-01 3.915E-01 6.401E-01

.2 3.906E-08 3.397E-07 3.594E-06 3.048E-05 3.100E-04 2.494E-03 2.316E-02 1.611E-01 3.740E-01 6.251E-01

.4 2.590E-0e 2.311E-07 2.530E-06 2.228E-05 2.381E-04 2.020E-03 2.006E-02 1.492E-01 3.574E-01 6.097E-01

.6 2.013E-0e 1.80eE-07 2.001E-O6 1.788E-05 1.955E-04 1.70eE-O3 1.773E-02 1.388E-01 3.416E-01 5.943E-01

.8 i.7ooe-oe 1.529E-07 1.697E-06 1.524E-05 1.683E-04 1.493e-03 1.593E-02 1.2971-01 3.267E-01 5. /89t-01

1 1.500E-0e 1.350E-07 1.5O0E-06 1.349E-05 1.495E-04 1.337E-03 1.450E-02 1.215E-01 3.125E-01 5.635E-01

1.2 1.357E-0e 1.221E-07 1.356E-06 1.221E-05 l,;«*-04 1.216E-03 1.333t-02 1.141E-01 2.989E-01 5.483E-01

1.4 1.245E-0e 1.120E-07 1.245E-06 1.120E-O5 1.244E-04 1.118E-03 1.233t-02 1.074E-01 2.859E-01 5.333E-01

1.6 1.152£-0e 1.037E-07 1.152E-0t 1.037E-05 1.152E-04 1.036E-03 1.147E-02 1.012E-01 2.735E-01 5.186E-01

1.8 1.072E-08 9.645E-0e 1.072E-06 9.645E-06 1.072E-04 9.642E-04 1.070E-02 9.541E-02 2.615E-01 5.042E-01

2 i.oooE-oe 9.000E-08 l,000e-06 9.000E-06 l.OOOE-04 9.000E-04 l.OOOE-02 9.000E-02 2.500E-01 4.900E-01

29

TN-324
B2(p,iu)

m \ r = 1 1.01 1.1 2 4 8 16 32 64 128

-2 l.OOOE+00 6.667E-01 6.667E-01 6.6676-01 6. 6676-01 6.6676-01 6.6676-01 6.6676-01 6.6676-01 6.6676-01

-1.8 l.OOOE+00 8.614E-01 7.8«3t-01 7. 1966-01 7.0626-01 7.0286-01 7.0186-01 7.0156-01 7.0156-01 7.0146-01

-1.6 ! i.oooe-Kw 9.429E-01 8.7O8E-01 7.7876-01 7.5616-01 7.4946-01 7.4726-01 7.4656-01 7.4626-01 7.4626-01

-1.4 l.OOOE+00 9.776E-01 9.281E-01 8.4466-01 8.1926-01 8. 1036-01 8.0716-01 8.0586-01 8.0536-01 8.0526-01

-1.2 ! l.OOOE+OO 9.929E-01 9.6936-01 9. 181E-01 8.9906-01 8.9126-01 8.8796-01 8.8656-01 8.8596-01 8.8566-01

-1 i.oooe+00 l.OOOE+OO 1.0006+00 1. 0006+00 1. 0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00

-.e i.oooe+00 1.004E+00 1.0246+00 1.0916+00 1.1286+00 1.1486+00 1.1596+00 1.1666+00 1.1706+00 1.1726+00

-.6 l.OOOE+OO 1.0O6E+O0 1.0436+00 1.1926+00 1.2906+00 1,3516+00 1.3926+00 1.4186+00 1.4366+00 1.4476+00

-.4 l.OOOE+OO 1.007E+00 1.0606+00 1.3046+00 1.4946+00 1.633F+00 1.7386+00 1.8176+00 1.8776+00 i.y2-dt+oo -.2 l.OOOE+00 1.009E+00 1.0756+00 1. 4296+00 1.7526+00 2.02/k+OO 2.2656+00 2.4726+00 2.6526+00 2.809E+00

1.000E-H» l.OlOE+00 1.0696+00 1.5666+00 2.0786+00 2.5816+00 3.0826+00 3.5826+00 4.0626+00 4.5826+00

.2 l.OOOE+00 l.OllE+00 1.1026+00 1.7186+00 2.4896+00 3.3646+00 4.365f+00 5.5146+00 6.8346+00 8.3516+00

.4 l.OOOE+00 1.012E+00 1. 1146+00 1.8866+00 3.0076+00 4.4736+00 6.4046+00 8.951E+00 1.231E+01 1.6746+01

.6 l.OOOE+00 1.013E+00 1.1266+00 2. 0716+00 3.6SflF+00 6.0516+00 9.6736+00 1.5166+01 2.3486+01 3.6096+01

.6 l.OOOE+00 1.014E+00 1.1386+00 2.2756+00 4.4756+00 8.2976+00 1.4956+01 2.653F+01 4.6696+01 8.1796+01

1 l.OOOE+OO 1.015E+00 1.1506+00 2. 5006+00 5.5006+00 1.1506+01 2.3506+01 4.7506+01 9.5506+01 1.9156+02

1.2 l.OOOE+00 1.016E+00 1.1626+00 2.7476+00 6.7846+00 1.6076+01 3.7416+01 8.6446+01 1. 9916+02 4.5796+02

1.4 l.OOOE+OO 1.017E+00 1.1746+00 3.0136+00 8.38yk+00 2.2596+01 6.0086+01 1.5906+02 4.2016+02 1.1096+03

1.6 1. 0006*00 1.018E+00 1.1866+00 3.3166+00 1.0396+01 3.1886+01 9.706E+01 2.9476+02 8.9376+02 2. 7106+03

1.8 l.OOOE+00 1.019E+00 1.1986+00 3.6426+00 1.2896+01 4.5136+01 1.5746+02 5.485E+02 1. 9106+03 6.65^+03

2 l.OOOE+OO 1.020E+00 1.2106+00 4.0006+00 1.6006+01 6.4006+01 2.560E+02 1.024E+03 4.0<»6E+O3 1.6386+04

HU \ r = 256 512 1024 2048 4096 Iff

-2 6.667E-01 6.667E-01 6.667E-01 6.667E-01 6.667E-01 6.667E-01

-1.8 7.014E-01 7.014E-01 7.0146-01 7.014E-01 7.014E-01 7. 0146-01

-1.6 7.461E-01 7.461E-01 7.461E-01 7.461E-01 7.461E-01 7.461E-01

-1.4 8.051E-01 8.051E-01 e.051E-01 8.051E-01 8.051E-01 8.051E-01

-1.2 S.855E-01 8.854E-01 8.854E-01 8.854E-01 8.854E-01 8.854E-01

-1 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+'X) l.OOOE+OO

-.8 1.173£+*>j 1.174E+O0 1.174E+00 1.174E+00 1.174E+00 1.175E+00

-.6 1.455E+00 1.460E+00 1.46^+00 1.465E+00 1.467E+00 1.470E+00

-.4 1.957E+00 1.983E+00 2.003E+00 2.018E+00 2.029E+00 2.065E+O0

-.2 :.045E+0<) 3.064E+00 3. 167E+00 3.257E+00 3.336E+00 3.863E+00

5. 0826+00 5.582E+00 6.082E+00 6.582E+00 7.082E+O0

.2 1.00«E+01 1.2O9E+01 1.4396*01 1.703E+01 2.006E+01

.4 2.259E+01 3.031E+O1 4.050E+01 5.394E+01 7.167E+01

.6 5.521E+01 8.418E+01 1.281E+02 1.947E+02 2.955E+02

.d 1.429En)2 2.493E+02 4.346E+02 7.571E+02 1.319E+03

1 3.835E+02 7.675E+02 1.5.Vf+03 3.072E+03 6. 144E+03

1.2 1.0526+03 2.4186+03 5.556E+03 1.2776+04 2.933E+04

1.4 : 2.9286+03 7.727E+03 2.039E+04 5.3826+04 1.420E+05

1.6 ! 8.2156+03 2.490E+O4 7.549E+04 2.2886+05 6.937E+05

1.8 : 2.317E+04 8.067E+O4 2.8096+05 9. 7826+05 3.4066+06

2 ! 6.554E+04 2.6216+05 1.0496+06 4.1946+06 1.6786+07

30

TN-325
B3(2,H,r,iu) for p « .01

flu \ 1*= 1 2 4 8 16 32 (A 129 256 512 1024

-2 i.oooe+oo 2.000E+00 4.000E+00 8.000E+00 1.600E+01 3.200E+01 6.400E+O1 1.280E+02 2.560E+O2 5.120E+02 1.0246+03

-1.8 l.OOOE+00 2.214E+O0 5.842E+00 1.829E+01 6.502E+O1 2.536E+02 1.173E+03 3.308E+03 4.S36E+03 6.0906+03 7. 1456+03

-1.6 l.OOOE+00 2.419E+00 7.398e+O0 2.638e+01 1.018E+O2 4. 151E+02 1.8686+03 5.075E+03 7.772E+03 8.7486+03 9.804+03

-1.4 l.OOOE+00 2.617E+O0 8.745E+00 3.295E+01 1.3W)E+02 5.358E+02 2.342E+03 5.964E+03 8.263E+03 9.7006+03 1.063F+O4

-1.2 i.oooe-HX) 2.eilE+00 9.937E+O0 3.843E+01 1.534E+02 6.247E+02 2.595E+03 6.253E+C3 8.449E+03 9,671E+03 1.0366+04

-1 l.OOOE+00 3.oooe+oo l.lOOE+01 4.300E+01 1.710e+O2 6.830E+02 2.674E+03 6.094E+03 8.047E+O3 9.024E+O3 9.512E+03

-.8 l.OOOE+OO 3.180E+00 1.191E+01 4.651E+01 1.823fc+02 7.W>SE+02 2.5e8E+03 5.578E+03 7.208E+O3 7.950E+O3 8.280E+03

-.6 l.OOOE+00 3.337E+O0 1.258E+01 4.848E+01 1.851E+02 6.877E+02 2.342E+03 4.7;bt+03 6.045E+03 6.577E+03 6.791E+03

-.4 l.OOOE+OO 3.444E+00 1.285E+01 4.ei2£+01 1.766E+02 6.215E+02 l.Vi/t+03 3.774E+03 4.686E+C3 5.041E+O3 5.171E+03

. 1 l.OOOE+00 3.463E+00 1.252E+01 4.480E+01 1.557E+02 5. 131E+02 1.4a>E+03 2.712E+03 3.304E+03 3.521E+03 3.594E+03

l.OOOE+OO 3.368E+00 1.151E+01 3.866E+01 1.253E+02 3.826E+02 1.015E+03 1.753E+03 2.097E+O3 2.216E+03 2.254E+03

.2 l.OOOE+OO 3.163E+00 9.9y2E+00 3.090E+O1 9.213E+01 2.1)/9fc+02 6.2536+02 1.021E+03 1.200E+O3 1.2596+03 1.277E+03

.4 l.OOOE+00 2.8Kit+O0 8.251E+O0 2.314E+01 6.2«p£+01 1.595E+02 3.521E+02 5.442E+02 6.280E+02 6.549E+02 6.627E+02

.6 l.OOOE+00 2.574E+O0 6.b/!)fc+00 1.651E+01 4.018E+O1 9.228E+01 1.851E+02 2.709E+02 3.071E+O2 3. 1836+02 3.215E+02

.8 l.OOOE+OO 2.270E+00 5.123E+00 1.141E+01 2.475E+01 5.09«fc+01 9.280E+O1 1.285E+02 1.431E+02 1.476E+02 1.4886+02

1 l.OOOE+OO 1.98Vt+00 3.940E+00 7.728E+00 1.486E+01 2.731E+01 4.505E+01 5.908E+01 6.464E+01 6.629E+01 6. 6736+01

1.2 l.OOOE+OO 1.737E+O0 3.0O8E+O0 5.177E+00 8. 77bfc+00 1.435E+01 2.142E+01 2.660E+01 2.859E+01 2.917E+01 2.yi3t+01

1.4 l.OOOE+OO 1.514E+O0 2.289E+00 3.446E+00 5.133E+O0 7.440E+00 1.005E+01 1.182E+01 1.248E+01 1.267E+01 1. 2726+01

1.6 l.OOOE+OO 1.319E+00 1.738E+O0 2.285E+00 2.985E+O0 3.82at+00 4. 678E+O0 5.211E+O0 5.4O5E+00 5.459E+O0 5.4736+00

1.8 l.OOOE+00 1.149E+O0 1.319E+00 1.513E+O0 1.729E+00 1.960E+O0 2.166E+00 2.286E+O0 2.328E+00 2.34OE+O0 2.3436+00

2 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

B3(2,M,r,iu) for r = .03

Mu s M= 1 16 32 64 128 256 512 1024

_) l.OOOE+OO 2.000E+00 4.000E+00 8.000E+00 1.600E+01 3.200E+01 6.4006+01 1.2806+02 2.5606+02 5. 1206+02 1.024E+03

-1.8 l.OOOE+OO 2.216E+00 5.866E+00 1.865E+01 7.138E+01 2.8126+02 4.880E+02 6.534E+02 8. 0756+02 9.647E+02 1.143E+03

-1.6 l.OOOE+00 2.422E+O0 7.449E+00 2.705E+01 1. 1196+02 4.2796+02 7.195E+02 9.170E+02 1.065E+03 1.1806+03 1.274E+03

-1.4 l.OOOE+fX) 2.623E+00 8.817E+O0 3.378E+01 1.411E+02 5.073E+O2 8.285E+02 1.022E+O3 1.147E+03 1.2296+03 1.2836+03

-1.2 l.OOOt+00 2.816E+00 l.OOOE+01 3.910E+01 1.607E+02 5.386E+02 8.548E+02 1.027E+03 1.1256+03 1.1806+03 1.2126+03

-1 l.OOOE+OO 3.000E+00 l.lOOE+01 4.300E+01 1.710E+02 5.328E+02 8.221E+02 9. 6676+02 1.039E+03 1.0756+03 1.0936+03

-.8 l.OOOE+OO 3. 164E+00 1. 1756+01 4.523E+01 1.719E+02 4.971E+02 7.455E+02 8.6066+02 9. 1226+02 9.350E+02 9.451E+02

-.6 l.OOOE+OO 3.290E+00 1.217E+01 4.543E+01 1.6326+02 4.376E+02 6.3786+02 7.247E+02 7.600E+02 7.740E+02 7.794E+02

-.4 l.OOOE+OO 3.356E+O0 1.2136+01 4.331E+OI 1.455E+02 3.618E+02 5. 126E+02 5.7436+02 5.974E+02 6.057E+02 6.0866+02

-.2 l.OOOE+OO 3.339E+00 1.157E+01 3.896E+01 1.214E+02 2.797E+02 3.851E+02 4.2626+02 4.405E+02 4.4526+02 4.468E+02

l.OOOE+OO 5.22OE+00 1.052E+O1 3.298E+01 9.4626+01 2.0196+02 2. 7026+02 2.957E+02 3.0406+02 3.0666+02 3.074E+O2

.2 l.OOOE+00 3.036E+O0 9.146E+00 2.6.T5E+01 6.912E+01 1.366E+02 1.776E+02 1.9246+02 1.9706+02 1.9836+02 1.987E+02

.4 l.OOOE+OO 2.787E+O0 7.64OE+O0 2. 0026+01 4.774E+01 8.737E+01 1.1046+02 1.1846+02 1.2096+02 1.2156+02 1.217E+02

.6 1.0006+00 2.510E+00 6.1866+00 1.461E+01 3.1536+01 5.345E+01 6.5646+01 6.9756+01 7.0956+01 7.1286+01 7.1366+01

.8 l.OOOE+OO 2.2-^21+00 4.897E+00 1.035E+01 2.0136+01 3. 162E+01 3.7726+01 3.9736+01 4.0306+01 4.0456+01 4.049E+01 l.OOOE+00 1.967E+O0 3.818E+O0 7.179E+00 1.254E+01 1.8266+01 2.116E+01 2.2106+01 2.2366+01 2.2426+01 2.244E+01

1.2 l.OOOE+OO 1.725E+00 2.9466+00 4.907E+O0 7.6886+00 1.037E+01 1.1686+01 1.2096+01 1.2206+01 1.2236+01 1.224E+01

1.4 l.OOOE+00 1.5086+00 2.259E+O0 3.i22k+O0 4.658F+00 5.829t+O0 6.371E+O0 6.5386+00 6.5B3F+O0 6.5956+00 6.5986+00

1.6 l.OOOE+OO 1.316E+00 1.725E+O0 2.234E+00 2. 9016+00 3.251E+00 3.4506+00 3.5116+00 3.5276+00 3.5316+00 3.5326+00

1.8 l.OOOE+OO 1.1486+00 1.314E+00 1.497E+00 1.676E+O0 1.805E+O0 1.8606+00 1.8766+00 1. 8816+00 1.8826+00 1.8826+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 1.0006+00 l.OOOE+OO 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00

31

TN-326
Hu \ Nb 1

B3(2,fl,r,«j) for r = .1

2 4 8 16 32 M 128 256 512 1024

-2 l.OOOe-HX) 2.000E+00 4.000E+00 8.0006+00 1.600E+01 3.2006+01 6.4006+01 1. 2806+02 2.5606+02 5.1206+02 1.024E+03

-1.8 l.OOOE+00 2.229E+O0 6.118E+00 2.3276+01 4.4196+01 5. 75/E+Ol 6.871E+01 7.8316+01 8.6656+01 9.3936+01 1.003E+02

-1.6 l.OOOC+00 2.448E+00 7.866E+O0 3.1386+01 6.0976+01 7.9346+01 9.2676+01 1.0266+02 1.1016+02 1.1586+02 1.201E+02

-1.4 l.OOOE+00 2.654E+00 9.270E+00 3.6266+01 6.9196+01 8.8326+01 1.0056+02 1.0646+02 1.1366+02 1.1706+02 1.1936+02

-1.2 ! i.oooe*oo 2.e42E+O0 1.032E+01 3.8536+01 7.1336+01 8.911E+01 9.9096+01 1. 0486+02 1.0806+02 1.099E+02 1.1096+02

-1 i.oooe+00 3.000E+00 1.1006+01 3.8636+01 6.9066+01 8.4536+01 9.2276+01 9.6136+01 9.807E+01 9.9036+01 9. 9526+01

-.8 l.OOOE-HX) 3.117E+O0 1.127E+01 3.6946+01 6.3656+01 7.6456+01 8.722F+01 8.4786+01 8.591E+01 8.6406+01 8.6626+01

-.6 l.OOOE+00 3.180E+00 1.114E+01 3.385F+01 5.6176+01 6.6326+01 7.0496+01 7.2156+01 7.280E+01 7.3056+01 7.3146+01

-.4 i.oooe+oo 3. 179E+00 1.061E+O1 2.9816+01 4.7606+01 5.5326+01 5.8256+01 5.9316+01 5.9686+01 5.981E+01 5.9856+01

-.2 l.OOOE+00 3.111E+00 9.757E+00 2.5286+01 3. 8846+01 4.4476+01 4.6476+01 4.7146+01 4.7356+01 4.7426+01 4.7445+01

l.OOOE+OO 2.980E+00 8. 6836+00 2.0696+01 3.0596+01 3.4556+01 3.5876+01 3.6296+01 3.641E+01 3.6456+01 3.6466+01

.2 l.OOOE+00 2.799E+O0 7.502E+00 1.6426+01 2.XBF+01 2.6036+01 2.6896+01 2.7146+01 2.7216+01 2.7236+01 2.724E+01

.4 l.OOOE+OO 2.5e4E+O0 6.317E+O0 1.2696+01 1.7356+01 1. 9106+01 1.9646+01 1.9796+01 1.9846+01 1.9856+01 1.985E+01

.6 l.OOOE+00 2.352E+00 5.2086+00 9.591E+00 1.2626+01 1.3726+01 1.4056+01 1.4146+01 1.4166+01 1.4176+01 1.417E+01

.8 l.OOOE+OO 2.116E+O0 4.221E+00 7.1256+00 9.0136+00 9.6836+00 9.8786+00 9.931E+00 9.9446+00 9.9486+00 9.9496+00

1 l.OOOE+OO 1.888E+00 3.377E+00 5.2236+00 6.354E+00 6.7456+00 6.8576+00 6.8866+00 6.8946+00 6.8966+00 6.8966+00

1.2 l.OOOE+OO 1.674E+00 2.676E+O0 3.791E+00 4.435E+00 4.6536+00 4.7146+00 4.7306+00 4.7346+00 4.7356+00 4.735E+00

1.4 l.OOOE+OO 1.477E+00 2.105E+00 2.7326+00 3.074E+O0 3. 1876+00 3.2186+O0 3.2266+00 3.2286+00 3.2286+00 3.229E+00

1.6 l.OOOE+OO 1.299E+00 1.647E+00 1.960E+00 2.120E+O0 2. 1726+00 2.1866+00 2.1896+00 2.1906+00 2.191E+00 2.191E+00

1.8 l.OOOE+OO 1.141E+00 1. 2856+00 1.401E+00 1.457E+O0 1.4756+00 1.4806+00 1.481E+O0 1.481E+00 1.482E+O0 1.482E+O0

2 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

B3(2,H,r,iu) for r = .3

flu \ «= 1 2 4 8 16 32 64 128 256 512 1024

-2 l.OOOE+OO 2.000E+00 4.000E+00 8.0006+00 1.6006+01 3.2006+01 6.4006+01 1.2806+02 2.560E+02 5. 1206+02 1.024E+03

-1.8 ! l.OOOE+OO 2.3f«F+00 5.582E+00 9.024E+00 1.341E+01 1.980E+01 2.987E+01 4.649E+01 7.464E+01 1.230E+02 2.065E+02

-1.6 l.OOOE+OO 2.6696+00 6.503E+00 9.614E+00 1.237E+01 1.521E+01 1.854E+01 2.285E+01 2.882E+01 3.746E+01 5.0236+01

-1.4 l.OOOE+00 2.(&/b+00 6.937E+00 9.770E+00 1.171E+01 1.317E+01 1.440E+01 l.bb/fc+01 1.681E+01 1.825E+01 2.002E+01

-1.2 l.OOOE+OO 2.964E+O0 7.0136+00 9.562E+00 1.102E+01 1.188E+01 1.242E+01 1.278E+01 1.3056+01 1.3286+01 1.3486+01

-1 l.OOOE+OO 3.000E+00 6.8336+00 9.0836+00 1.021E+01 1.077E+01 1.105E+01 1.119E+01 1.1266+01 1.130E+01 1.132E+01

-.8 l.OOOE+00 2.976E+00 6.4776+00 8.422E+00 9.300E+00 9.686E+00 9.850E+00 9.918E+O0 9.944E+00 9.9526+00 9.953E+00

-.6 l.OOOE+OO 2.902E+00 6.0086+00 7.655E+00 8.339k+00 8.610E+O0 8.7136+00 8.750E+00 8.7636+00 8.7666+00 8.766E+00

-.4 l.OOOE+OO 2.789E+00 5.4786+00 6.844E+00 7.374E+O0 7.5666+00 7.633F+00 7.655E+00 7.662E+00 7.664E+00 7.664E+00

-.2 l.OOOE+00 2.648E+00 4.9236+00 6.0366+00 6.443E+O0 6.580E+00 6.6246+00 6.6386+00 6.6426+00 6.643E+O0 6. 6436+00

l.OOOE+OO 2.487E+00 4.3746+00 5.2646+00 5.5736+00 5.671E+O0 5.701E+00 5.7096+00 5.7126+00 5.712E+00 5.713E+00

.2 l.OOOE+OO 2.316E+O0 3.8506+00 4.5486+00 4.781E+O0 4.851E+O0 4.871E+O0 4.8766+00 4.878E+O0 4.8786+00 4.879E+O0

.4 l.OOOE+00 2.140E+00 3.3626+00 3. 9006+00 4.0736+00 4.1236+00 4.137E+O0 4.1406+00 4.141E+00 4.141E+O0 4.1426+00

.6 l.OOOE+00 1.967E+00 2.9196+00 3.324E+O0 3.4506+00 3.4866+00 3.4956+00 3.4986+00 3.4986+00 3. 4996+00 3.499E+O0

.8 l.OOOE+00 1.799E+O0 2.5226+00 2.820E+00 2. 9106+00 2. 9356+00 2.9426+00 2.9436+00 2.9446+00 2.9446+00 2.944E+00

1 l.OOOE+00 1.6396+00 2. 1716+00 2.3836+00 2.4466+00 2.4636+00 2.4686+00 2.4696+00 2.4696+00 2.469E+00 2.469E+00

1.2 l.OOOE+00 1.4896+00 1.8646+00 2.0096+00 2.0516+00 2.0626+00 2.0656+00 2.0666+00 2.0666+00 2.0666+00 2.066E+00

1.4 l.OOOE+00 1.3506+00 1.5976+00 1.6906+00 1.7166+00 1.7236+00 1.7256+00 1.7266+00 1.7266+00 1.7266+00 1.7266+00

1.6 l.OOOE+00 1.2236+00 1.367E+00 1.4206+00 1.4346+00 1.4386+00 1.439E+00 1.4406+00 1.4406+00 1.4406+00 1.440E+00

1.8 l.OOOE+OO 1.1066+00 1.1696+00 1.1926+00 1.1986+00 1.2006+00 1.2006+00 1.2006+00 1.2006+00 1.2006+00 1.200E+00

2 ! 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1. 0006+00 1.0006+00 1.0006+00 1. 0006+00 l.OOOE+00

32

TN-327
Hu \ H= 1

B3(2,H,r,«j) for r = 1

2 4 8 16 32 64 128 256 512 1024

-2-1.8 -1.6

-1.4 -1.2

-1-.8

-.6

-.4

-.2

.2

.4

.6.8

11.2

1.4 1.6

1.8

2

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1.00C€+00

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l.OOOE+OO l.OOOE+OO

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l.OOCE+OO l.OOOE+OO

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B3(2,n,r,i«u) for r = 1.01

flu \ r*= 1 16 32 64 128 256 512 1024

-2 l.OOOE+OO 2.0O0E+O0 4.000E+00 8.000E+00 1.600E+01 3.200E+01 6.400E+01 1.280E+02 2.560E+02 5.120E+02 1.024E+03

-1.8 l.OOOE+OO 1.300E+00 1.771E+00 2.545C+O0 3.852E+O0 6.095E+00 9.968E+00 1.669E+01 2.836E+01 4.867E+01 8.401E+01

-1.6 l.OOOE+OO 1.105E+00 1.234E+O0 1.407E+00 1.650E+00 2.004E+00 2.532E+O0 3.324E+O0 4.518E+O0 6.324E+00 9.057E+OO

-1.4 l.OOOE+OO 1.037E+00 1.073E+00 l.llOE+OO 1.155E+O0 1.209E+O0 1.278E+00 1.367E+00 1.484E+O0 1.637E+00 LeaSEHO

-1.2 1.000E+'X> l.OllE+00 1.019E+00 1.026E+00 1.032E+00 1.039E+00 1.046E+00 1.054E+00 1.063E+00 1.073E+O0 1.084E+00

-1 1 . 0<»E+O<J l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.8 l.OOOE+OO 9.956E-01 9.933E-01 9.921E-01 9.913E-01 9.908E-01 9.905E-01 9.902E-01 9.899E-01 9.897E-01 9.896E-01

-.6 l.OOOE+OO 9.938E-01 9.910E-01 9,898E-01 9.8VA-01 9.889E-01 9.887E-01 9.886E-01 9.885E-01 9.884E-01 9.884E-01

-.< l.CKOOE+OO 9.9-32E-01 ''.905E-01 9.895E-01 9.891E-01 9.889E-01 9.888E-01 9.888E-01 9.887E-01 9.887E-01 9.887E-01

. 1 l.OOOE^Xi 9.931E-01 9,907E-01 9.89^-01 9.896E-01 9.8«^-01 9.895E-01 9.894E-01 9.894E-01 9.894E-01 9.e94E-0l

l.OOOE+OO 9.934E-01 9.913E-01 9.906E-01 9.904E-01 9.903E-01 9.903E-01 9.903E-01 9.903E-01 9.903E-01 9.903E-01

l.OOOE+OO 9.938E-01 9.920E-01 9.914E-01 9.913E-01 9.912E-01 9.912E-01 9.912E-01 9.912E-01 9.912E-01 9.912E-01

.4 l.OOOE+OO 9.944E-01 9.yM-01 9.923E-01 9.y22t-0l 9.922E-01 9.922E-01 9.922E-01 9.922E-01 9.922E-01 9.922E-01

.6 l.OOOE+OO 9.950E-01 9.936E-01 9.9-^-^-01 9.ri2£-01 9.931E-01 9.931E-01 9.931E-01 9.931E-01 9.931E-01 9.931E-01

.8 l.OOOE+OO 9.<'56€-01 9.945E-01 9.942E-01 9.941E-01 9.941E-01 9.941E-01 9.941E-01 9.941E-01 9.941E-01 9.941E-01 l.OOOE+OO 9.963E-01 9.954E-01 9.952E-01 9.951E-01 9.951E-01 9.951E-01 9.951E-01 9.951E-01 9.951E-01 9.951E-01

1.2 l.OOOE+OO 9.970E-01 9.963E-01 9.961E-01 9.961E-01 9.961E-01 9.961E-01 9.961E-01 9.961E-01 9.961E-01 9.961E-01

1.4 l.OOOE+OO 9.978E-01 9.972E-01 9.971E-01 9.970E-01 9.970E-01 9.970E-01 9.970E-01 9.970E-01 9.970E-01 9.970E-01

1.6 l.OOOE+OO 9.985E-01 9.981E-01 9.981E-01 9.980E-01 9.980E-01 9.980E-01 9.980E-01 9.980E-01 9.980E-01 9.980E-01

1.8 l.OOOE+OO 9.992E-01 9.991E-01 9.990E-01 9.990E-01 9.990E-01 9.990E-01 9.990E-01 9.990E-01 9.990E-01 9.990E-01

2 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOCE+OO l.OOOE+OO l.OOOE+OO

33

s;:

TN-328
B3(2,H,r,«u) for P e 1.1

Hu \ 1^ 1 2 4 8 16 32 64 128 256 512 1024

-2 l.OOOE+00 2.000E+00 4.000E+00 8.000E+00 1.600E+01 3.200E+01 6.400E+01 1.280E+02 2.560E+02 5.120E+O2 1.024E+03

-1.8 l.OOOE+00 1.494E+00 2.286E+00 3.604E+00 5.845E+O0 9.700E+O0 1.637E+01 2.795E+01 4.808E+01 8.311E+01 1.441E+02

-1.6 i.oooe+00 1.250E+00 l.b/3t+00 2.017E+00 2.657E+00 3.601E+O0 5.011E+O0 7.134E+00 1.034E+01 1.519E+01 2.254E+01

-1.4 l.OOOE+00 1.119E+00 1.243E+O0 1.385E+00 1.5!Wif+00 1.772E+00 2.050E+00 2.412E+00 2.887E+00 3.512E+O0 4.336E+00

-1.2 l.OOOE+00 1.044E+00 i.oe2E+oo 1.117E+00 1.153E+O0 1.191E+00 1.234E+00 1.282E+00 1.336E+00 i.avat+oo 1.470E+00

-1 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO

-.8 l.OOOE+00 9.740E-01 9.5e4£-01 9.484E-01 9.414E-01 9.361E-01 9.318E-01 9.204-01 9.251E-01 9.225E-01 9.202E-01

-.6 l.OOOE+00 9.590E-01 9.380E-01 9.268E-01 9.202E-01 9.160E-01 9.132E-01 9.111E-01 9.096E-01 9.08SF-01 9.077E-01

-.4 l.OOOE+OO 9.510E-01 9.292E-01 9.193E-01 9.145E-01 9.119E-01 9.105E-01 9.096E-01 9.090E-01 9.087E-01 9.0e5E-01

-.2 l.OOOE+00 9.476E-01 9.270E-01 9.189E-01 9. 156E-01 9. 142E-01 9.135E-01 9.131E-01 9.129E-01 9.128E-01 9.128E-01

l.OOOE+OO 9.472E-01 9.287E-01 9.223E-01 9.200E-01 9.192E-01 9.189E-01 9. 187E-01 9. 187E-01 9.186E-01 9. 186E-01

.2 i.oooe+00 9.490e-01 9.327E-01 9.277E-01 9.261E-01 9.256E-01 9.254E-01 9.254E-01 9.253E-01 9.253E-01 9.2536-01

.4 l.OOOE+00 9.523E-01 9.382E-01 9.342E-01 9.331E-01 9.327E-01 9.326E-01 9.326E-01 9.326E-01 9.326E-01 9.326E-01

.6 l.OOOE+OO 9.566E-01 9.447E-01 9.415E-01 9.4066-01 9.404E-01 9.403E-01 9.403E-01 9.403E-01 9.403E-01 9.403E-01

.8 l.OOOE+OO 9.617E-01 9.517E-01 9.492E-01 9.485E-01 9.483E-01 9.483E-01 9.483E-01 9.483E-01 9.483E-01 9.4836-01

1 l.OOOE+OO 9.674E-01 9.592E-01 9.572E-01 9.567E-01 9.566E-01 9.565E-01 9.565E-01 9.565E-01 9.565E-01 9.565E-01

1.2 l.OOOE+OO 9.735E-01 9.670E-01 9.655E-01 9.651E-01 9.650E-01 9.649E-01 9.649E-01 9.649E-01 9.649£-C. <i.649E-01 1.4 l.OOOE+OO 9.798E-01 9.751E-01 «.739E-01 9.736E-01 9.735E-01 9.735E-01 9.735E-01 9.735E-01 •5. 35E-01 9.7356-01

1.6 i.oooe+'X> 9.864E-01 «'.832E-01 9.825E-01 9.823E-01 9.822E-01 «.;22E-01 9.822fc-01 9.822E-01 9.822E-01 9.822E-01

1.8 l.OOOE+OO 9.931E-01 9.916E-01 9.912E-01 9. :1E-01 9.911E-01 9.911E-01 9.911E-01 9.911E-01 9.911E-01 9.911E-01

I.OOOE+OO 1.00(€+00 l.CHJOE-^) l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO

B3(2,M,r,iu) for r = 2

16 32 64 128 256 512 1024

-2 l.OOOE+00 2.000E+00 4.000E+00 e.ooo£+oo 1.600E+01 3. 2006+01 6.4006+01 1.280E+O2 2.5606+02 5.1206+02 1.024E+O3

-1.8 l.OOOE+00 1.690E+00 2.862E+00 4.877E+O0 8.362E+00 1.441E+01 2.491E+01 4.3196+01 7.5006+01 1.3046+02 2.267E+02

-1.6 l.OOOE+OO 1.445E+O0 2.089E+O0 3.0376+00 4.4536+00 6.5836+00 9.7986+00 1.4666+01 2.2036+01 3.3196+01 5.010E+O1

-1.4 l.OOOE+OO 1.2i^+00 1.5686+00 1.9646+00 2.47X+00 3. 137E+O0 4.0066+00 5.1506+00 6.657E+O0 8.644E+O0 1.126E+01 -1.2 l.OOOE+OO 1.109E+00 1.224E+00 1.347E+00 1.484E+00 l.(S.'W.+O0 1.8136+00 2.014E+00 2.244E+00 2.5076+00 2.810E+00

-I l.OOOE+00 l.OOOE+00 1.0006+00 l.OOOE+00 1.0006+00 l.OOOE+OO 1.0006+00 1.0006+00 1.0006+00 l.OOOE+00 l.OOOE+00

-.8 l.OOOE+00 9.1986-01 8.591E-01 e.ll3E-01 7.7236-01 7.3956-01 7.1156-01 6.8736-01 6.6646-01 6.4836-01 6.325E-01

-.6 l.OOOE+00 8.632E-01 7.7466-01 7.151E-01 6.735E-01 6.4356-01 6.2136-01 6.0476-01 5.9226-01 5.8286-01 5./b;t-01

-.4 l.OOOE+OO 8.255E-01 7.ai3fc-01 6.727E-01 6.3946-01 6.1886-01 6.0576-01 5.9726-01 5.917E-01 5.8816-01 5.8576-01

.,2 l.OOOE+OO 8.0286-01 7.0806-01 6.614E-01 6.377E-01 6.2516-01 6. 1836-01 6.1446-01 6.1236-01 6.1106-01 6. 103E-01 1.0006+00 7.922E-01 7. 0526-01 6.6846-01 6.5236-01 6.4516-01 6.4176-01 6.4016-01 6.3936-01 6.3896-01 6.3876-01

.2 1.0006+00 7.912E-01 7. 1436-01 6.8606-01 6.7546-01 6.7136-01 6.6976-01 6.6906-01 6.6876-01 6.6866-01 6.686E-01

.4 l.OOOE+00 7.979E-01 7.3166-01 7.101E-01 7.031E-01 7.0086-01 7.0006-01 6.9976-01 6.9966-01 6.9966-01 6.9966-01

.6 l.OOOE+OO 8.1066-01 7.5476-01 7.3846-01 7.3376-01 7.3236-01 7.3196-01 7.3186-01 7.3186-01 7.3186-01 7.3186-01

.8 l.OOOE+OO 8.2836-01 7.8206-01 7.6966-01 7.6646-01 7.655F-01 7.6536-01 7.6526-01 7.6526-01 7.6526-01 7.6526-01

1 l.OOOE+00 8.5006-01 8.125E-01 8.0316-01 8.0086-01 8.0026-01 8.0006-01 8.0006-01 8.0006-01 8.0006-01 8.0006-01

1.2 l.OOOE+00 8.7506-01 8.4566-01 8.3866-01 8.3696-01 8.36??-01 8.3646-01 8.3646-01 8.363F-01 8.3636-01 8.3636-01

1.4 l.OOOE+OO 9.0296-01 8.8116-01 8.7606-01 8.7476-01 8.7456-01 8.7446-01 8.7446-01 8.7446-01 8.7446-01 8.7446-01

1.6 . l.OOOE+00 9.331E-01 9. 1866-01 9.1536-01 9.145E-01 9.1436-01 9.142E-01 9.1426-01 9.1426-01 9.1426-01 9.1426-01

1.8 l.OOOE+00 9.656E-01 9.5826-01 9.5666-01 9.5626-01 9.5616-01 9.5606-01 9.5606-01 9.5606-01 9.5606-01 9.5606-01

2 l.OOOE+OO l.OOOE+OO 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1. 0006+00 1.0006+00 1. 0006+00 1.0006+00

34

TN-329
B3(2,H,r,«i) for P « 4

Hu \ n= 1 2 4 8 16 32 64 128 256 512 1024

-2 l.OOOE+00 2.000E+00 4.000E+00 e.oooE+oo 1.600E+01 3.2O0E+O1 6.400E+01 1.280C+O2 2.560E+02 5.120e+02 1.024E+03

-1.8 l.OOOE+00 1.728E+O0 2.986E+O0 5.170E+O0 8.964E+O0 1.556E+01 2.705E+O1 4.704E+01 8.185E+01 1.425E+02 2.480E+02

-1.6 1.0006+00 1.494E+00 2.232E+00 3.341E+00 5.014E+00 7.544E+00 1.137E+01 1.718E+01 2.597E+01 3.929E+01 5.948E+01

-1.4 i.oooe+00 1.296E+00 1.678E+00 2.174E+00 2.823E+O0 3.676E+00 4.799E+00 6.279E+O0 8.232E+00 l.OeiE+01 1.4206+01 -1.2 l.OOOE+00 1.132E+O0 1.279E+00 1.444E+00 1.631E+00 1.844E+00 2.0e8E+O0 2.367E+00 2.688E+00 3.0b7t+O0 3.480E+00

-1 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO i.oooe+00 l.OOOE+OO l.OOOE+OO l.OOOE+00

-.8 i.oooe+oo 8.967E-01 8. 123E-01 7.420E-01 6.824E-01 6.313E-01 5.872E-01 5.489E-01 5.157E-01 4.868E-01 4.616E-01

-.6 l.OOOE+OO 8.198E-01 6.934E-01 6.031E-01 5.370E-01 4.880E-01 4.513E-01 4.237E-01 4.028E-01 3.870E-01 3.7506-01

-.4 l.OOOE+00 7.663E-01 6.256E-01 5.392t-01 4.849E-01 4.502E-O1 4.276E-01 4.128E-01 4.032E-01 3.968E-01 3.9266-01 -.2 l.OOOE+00 7.330E-01 5.946E-01 5.217E-01 4.S24E-01 4.607E-01 4.486E-01 4.417E-01 4.378E-01 4.3Mf;-01 4.342E-01 l.OOOE+OO 7.167E-01 5.89ee-01 5.323E-01 5.057E-01 4.932t-01 4.871E-01 4.841E-01 4.826E-01 4.819E-01 4.815E-01

,2 l.OOOE+OO 7.144E-01 6.030E-01 5.595E-01 5.422E-01 5.351E-01 5.322E-01 5.310E-01 5.305E-01 5.303E-01 5.X2E-01

.4 l.OOOE+OO 7.232E-01 6.285t-01 5.962E-01 5.851E-01 5.812E-01 5.798E-01 5.794E-01 5.7921-01 5.791E-01 5.791E-01

.6 l.OOOE+00 7.409E-01 6.621E-01 6.384E-01 6.312E-01 6.291E-01 6.284E-01 6.282E-01 6.281E-01 6.281E-01 6.281E-01

.8 l.OOOE+00 7.654E-01 7.013E-01 6.839E-01 6.792E-01 6.779E-01 6.775E-01 6.775E-01 6.774E-01 6.774E-01 6.7746-01

1 l.OOOE+OO 7.yWik-01 7.443E-01 7.315E-01 7.283E-01 7.275E-01 7.273E-01 7.273E-01 7.273E-01 7.273E-01 7.2736-01

1.2 l.OOOE+OO 8.298E-01 7.903E-01 7.810E-01 7.788E-01 7.782t-01 7.781E-C1 7.781E-01 7.781E-01 7.781E-01 7.781E-01

1.4 l.OOOE+OO 8.679E-01 8.3«9t-01 8.323E-01 8.307E-01 8.303E-01 8.303E-01 8.302E-01 8.302E-01 8.302E-01 8.302E-01

1.6 l.OOOE+OO 9.090E-01 8.899E-01 8.fi!V>f-01 8.846E-01 8.843E-01 8.843E-01 8.843E-01 8.e43E-01 e.843E-01 8.8436-01

1.8 l.OOOE+00 9.531E-01 9.435E-01 9.414E-01 9.40eE-01 9.4O7E-01 9.407E-01 9.407E-01 9.407E-01 9.407E-01 9,407E-01

2 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00

B3(2,n,r,iu) for r = 8

ftu \ N= 1 2 4 e 16 32 64 128 256 512 1024

-2 l.OOOE+00 2.000E+00 4.000E+00 8.000E+O0 1. 6006+01 3.200E+01 6.4O0E+O1 1.280E+02 2.560E+02 5. 120E+O2 1.024E+O3

-1.8 l.OOOE+OO 1.737E+00 3.019E+00 5.247E+00 9. 126E+00 1.58SE+01 2.763E+01 4.809E+01 8.371E+01 1.457E+02 2.537E+02

-1.6 l.OOOE+OO 1.509E+00 2.276E+O0 3.436E+00 5.191E+00 7.850E+O0 1.188E+01 1.798E+01 2.723E+01 4.126E+01 6.251E+01

-1.4 l.OOOE+00 1.311E+00 1.717E+O0 2.251E+00 2.952E+O0 3.877E+00 5.096E+00 6.704E+00 8.825E+00 1.162E+01 1.532E+01

-1.2 l.OOOE+OO 1.142E+O0 1.302E+00 1.4e4E+00 1.693E+O0 l.y/2E+00 2.206E+00 2.520E+O0 2.882E+O0 3.296E+O0 3.773E+00

-1 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO

-.8 l.OOOE+OO e.85lE-01 7.aft.1F-01 7.059E-01 6.351E-01 5.739E-01 5.208E-01 4.747E-01 4.3466-01 3.997E-01 3.693E-01

-.6 l.OOOE+OO 7.965E-01 6.489E-01 5.4O5E-01 4.600E-01 3.996E-01 3.541E-01 3.1986-01 2.9386-01 2.741E-01 2.592E-01

-.4 l.OOOE+OO 7.331E-01 5.666E-01 4.614E-01 3.939E-01 3.502E-01 3.216E-01 3.0286-01 2.904E-01 2.8236-01 2.769E-01

-.2 l.OOOE+OO 6.931E-01 5.284E-01 4.390E-01 3.896E-01 3.620E-01 3.4636-01 3.374E-01 3.323k-01 3.294E-01 3.277E-01

l.OOOE+00 6.738E-01 5.230E-01 4.526E-01 4.191E-01 4.030E-01 3.952E-01 3.913E-01 3.893E-01 3.884E-01 3.8796-01

,2 l.OOOE+OO 6.719E-01 5.405E-01 4.876E-01 4.661E-01 4.572E-01 4.534E-01 4.518E-01 4.511E-0I 4.508E-01 4.507E-01

,4 l.OOOE+OO 6.838E-01 5.732E-01 5.347E-01 5.2126-01 5. 164E-01 5.146E-01 5.140E-01 5.137E-01 5.136E-01 5.136E-01

.6 l.OOOE+OO 7.061E-01 6.155E-01 5.878E-01 5.793E-01 5.767E-01 5.7596-01 5.7566-01 5.7566-01 5.7bbt-01 5./5bt-01

.8 l.OOOE+00 7.361E-01 6.634E-01 6.4366-01 6.3826-01 6.367E-01 6.363E-01 6.362E-01 6.361E-01 6.361E-01 6.3616-01

1 l.OOOE+00 7.717E-01 7.147E-01 7.004E-01 6.9686-01 6.959E-01 6.9576-01 6.9576-01 6.V5/k-01 6.95/t-Ol 6. 95 /k -01 1.2 l.OOOE+OO 8.115E-01 7.680E-01 7.5786-01 7.554E-01 7.548E-01 7.546E-01 7.546E-01 7.5466-01 7.5466-01 7.5466-01

1.4 l.OOOE+OO 8.546E-01 8.230E-01 8. 159E-01 8. 142E-01 8.138E-01 8.137E-01 8. 1376-01 8. 1376-01 8. 1376-01 8.1376-01

1.6 l.OOOE+00 9.005E-01 8.797E-01 8.751E-01 8.741E-01 8.738E-01 8.7386-01 8.7376-01 8.737E-01 8.737E-01 8.7376-01

1.8 l.OOOE+00 9.489E-01 9.386E-01 9.3636-01 9.357E-01 9.356E-01 9.356E-01 9.356f-01 9.356f-01 9.356f-01 9.3566-01

2 l.OOOE+OO l.OOOE+OO l.OOOE+OO 1.0006+00 l.OOOE+OO l.OOOE+OO 1.0006+00 1.0006+00 l.OOOE+OO 1.0006+00 1.0006+00

35

TN-330
B3(2,H,r,«j) for r = 16

Hu \ 1^ 1 2 4 8 16 32 64 128 256 512 1024

-2 l.OOOE+00 2.000E+00 4.000E+00 B.OOOE+OO 1.6O0E+O1 3.200E+O1 6.4O0E+O1 1.290E+02 2.560E+02 5.120E+O2 1.024E+O3

-1.8 1.000E*00 1.740E+00 3.028E+00 5.269E+O0 9.171E+O0 1.596E+01 2.779E+01 4.83(*+01 8.424E+01 1.467E+02 2.K>.'*+02 -1.6 l.OOOE+OO 1.513E+O0 2.290E+O0 3.467E+O0 5.249E+O0 7.950E+00 1.204E+01 1.825E+01 2.765E+01 4.191E+01 6.351E+01

-1.4 i.oooe-KX) 1.316E+O0 1.7321+00 2.290E+O0 3.001E+00 3.953E+00 5.209E+00 6.865E+00 9.061E+O0 1.194E+01 1.574E+01 -1.2 i.oooe+00 1.I46E+00 1.312E+O0 1.502E-KK) 1.720E+00 1.970E+00 2.Z)/t+00 2.587E+O0 2.966E+O0 3.401E+00 3.901E+O0

-1 l.OOOE+00 i.oooe+00 i.oooe+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.8 1.00C€-K)0 8.788E-01 7.752E-01 6.860E-01 6.0e9E-01 5.420E-01 4.839E-01 4.334E-01 3.894E-01 3.511E-01 3. 178E-01 -.6 i.oooe+00 7.826E-01 6.220e-01 5.026E-01 4.132E-01 3.45eE-01 2.W9E-01 2.564E-01 2.273E-01 2.052E-01 1.885E-01

-.4 l.OOOE+00 7.119E-01 5.287E-01 4.112E-01 3.3!Wf-01 2.85aF-01 2.527E-01 2.313E-01 2.172E-01 2.078E-01 2.017E-01

-.2 i.oooe+00 6.66flF-01 4.845E-01 3.tt3%-01 3.2/bt-Ol 2.958E-01 2.V/7E-01 2.674E-01 2.615E-01 2.S81E-01 2.562E-01 l.OOOE+00 6.455E-01 4. 785t-01 3.992E-01 3.612E-01 3.426E-01 3.:W5F-01 3.290E-01 3.267E-01 3.256E-01 3.250E-01

.2 l.OOOE+OO 6.446E-01 5.001E-01 4.410E-O1 4. 167E-01 4.065E-01 4.021E-01 4.003E-01 3.995E-01 3.991E-01 3.990E-01

.4 i.oooe+00 6.597E-01 5.395E-01 4.970E-01 4.819E-01 4.765E-01 4.74S-01 4.738E-01 4.735E-01 4.734E-01 4.734E-01

.6 i.oooe+00 6.865E-01 5.891E-01 5.591E-01 5.498E-01 5.469E-01 5.460E-01 5.45eE-01 5.457E-01 5.456E-01 5.456E-01

.8 i.oooe+00 7.210E-01 6.439E-01 6.227E-01 6. 169E-01 6.154E-01 6. 149E-01 6.148E-01 6.148E-01 6.148E-01 6.148E-01

1 i.oooe*oo 7.606E-01 7.00eE-01 6.858E-01 6.821E-01 6.812E-01 6.809E-01 6.809E-01 6.809E-01 6.809E-01 6.809E-01

1.2 l.OOOE+OO 8.038E-01 7.586E-01 7.480E-01 7.455E-01 7.449E-01 7.447E-01 7.447E-01 7.447E-01 7.447E-01 7.447E-01

1.4 l.OOOE+OO 8.495E-01 8. 169E-01 8.096E-O1 8.079E-01 8.075E-01 8.074E-01 8.074E-01 8.074E-01 8.074E-01 8.074E-01

1.6 l.OOOE+OO 8.974E-01 8.762E-01 8.715E-01 8.704E-01 8.701E-01 8.701E-01 8.701E-01 8.701E-01 8.701E-01 8.701E-01

1.8 l.OOOE+OO 9.475E-01 9.370E-01 9.346E-01 9.341E-01 9.340E-01 9.339E-01 9.339E-01 9,339E-01 9.33yt-01 9.:f3Vt-01

2 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

B3(2,H,r,iu) for r = 32

Mu \ H= 1 2 4 8 16 32 64 128 256 512 1024

.-> l.OOOE+OO 2.000E+00 4.000E+00 8.000E+00 1.600E+01 3.200E+01 6.4O0E+01 1.280E+02 2.560E+02 5. 120E+02 1.024E+03 -1.8 • l.OOOE+OO 1.741E+00 3.030E+00 5.276E+00 9.184E-K)0 1.599E+01 2.784E+01 4.847E+01 8.439E+01 1.469E+02 2.!Wf+02

-1.6 l.OOOE+OO 1.515E+00 2.2Vbt+00 3.477E+00 5.269E+00 7.984E+00 1.210E+01 1.834E+01 2.779E+01 4.212E+01 6.384E+01

-1.4 l.OOOE+OO 1.318E+O0 1.738E+00 2.291E+00 3.02OE+O0 3.982E+00 5.252E+O0 6.927E+O0 9. 137E+O0 1.205E+O1 1.590E+01 -1.2 l.OOOE+OO 1.147E+O0 1.316E+O0 1.510E+O0 l./'/4;+00 1.987E+00 2.280E+00 2.616E+O0 3.003E+OO 3.447E+O0 S.Vb/b+OO

-1 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.8 l.OOOE+OO 8.753E-01 7.677E-01 6.747E-01 5.941E-01 5.240E-01 4.630E-01 4.100E-01 3.639E-01 3.237E-01 2.887E-01

-.6 l.OOOE+OO 7.738E-01 6.052E-01 4.789E-01 3.838E-01 3.120E-01 2.b/7t-01 2. 166E-01 1.855E-01 1.619E-01 1.441E-01

-.4 l.OOOE+OO 6.975E-01 5.030E-01 3.770E-01 2.950E-01 2.412E-01 2.059E-01 1.826E-01 1.673E-01 1.572E-01 1.505E-01

-.2 l.OOOE+OO 6.480E-01 4.532E-01 3.445E-01 2.833E-01 2.486E-01 2.28ak-01 2.175E-01 2.110E-01 2.073E-01 2.051E-01 l.OOOE+OO 6.251E-01 4.465E-01 3.609E-01 3.194E-01 2.991E-01 2.891E-01 2.841E-01 2.816E-01 2.904E-01 2.798E-01

.2 l.OOOE+OO 6.256E-01 4.718E-01 4.085E-01 3.821E-01 3.710E-01 3.663E-01 3.642E-01 3.634E-01 3.630E-01 3.62eE-01

.4 l.OOOE+OO 6.440E-01 5. 173E-01 4. r/Jk-01 4.561E-01 4.503E-01 4.4eiE-01 4.473E-01 4.471E-01 4.469E-01 4.469E-01

.6 l.OOOE+OO 6.747E-01 5.733E-01 5.418E-01 5.321E-01 5.290E-01 5.2e0£-01 5.277E-01 5.276E-01 5.276E-01 5.276E-01

.8 l.OOOE+OO 7.128E-01 6.333k-01 6.114E-01 6.054E-01 6.038E-01 6.033E-01 6.0321-01 6.032E-01 6.031E-01 6.031E-01

1 l.OOOE+OO 7.?5!5,T-01 6.941E-01 6.788E-01 6.750E-01 6.740E-01 6.738E-01 6.737E-01 6.737E-01 6.737E-01 6.737E-01

1.2 l.OOOE+OO 8.004E-01 7.545E-01 7.438E-01 7.412E-01 7.406E-01 7.405E-01 7.404E-01 7,404E-01 7.4O4E-01 7.4O4E-01

1.4 l.OOOE+OO 8.475E-01 8.146E-01 8.072E-O1 8.0M)t-01 8.051E-01 8.050E-01 8.050E-01 8.049E-O1 8.049E-01 8.049E-01

1.6 l.OOOE+OO 8.964E-01 8.750E-01 8.702E-01 e.691E-01 8.689E-01 8.688E-01 8.688E-01 8.688E-01 8.6e8E-01 8.688E-01

1.8 l.OOOE+OO 9.471E-01 9.365F-01 9.341E-01 9.,TV.f-01 9.334E-01 9.334E-01 9.334E-01 9.334E-01 9.334E-01 9.334E-01

2 . l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

36

TN-331
B3(2,n,r,au) for r = 64

fli \ N> 1 2 4 8 16 32 64 128 256 512 1024

-2 l.OOOE^OO 2.000E+00 4.000E+00 8.000E400 1.6O0E-K>l 3.200E+O1 6.400E+01 1.290E+O2 2.5606+02 5.120E+O2 1.0246+03 •1.8 l.OOOE^OO 1.741E+00 3.031E+00 5.277E+O0 9.188E+00 1.600e+01 2.785E+01 4.8496+01 8.4436+01 1.470E+O2 2.S9E+02

-1.6 l.OOOE-KX) 1.515E+00 2.297E+00 3.481E+00 5.275E+00 7.995E+00 1.212E+01 1.836E+01 2.7846+01 4.2196+01 6.395E+01

-1.4 l.OOOE^ 1.319E+00 1.740E+O0 2.295E+00 3.027E+00 3.993E+O0 5.268E+00 6.950E+00 9. 1706+00 1.2106+01 1.5966+01 -1.2 l.OOOE^OO 1.148E+00 1.318E+00 1.513E+00 1.737E+00 1.994E+00 2.290E-HX) 2.6296+00 3.0196+00 3.4676+00 3.9816+00

-1 l.OOOE^CO 1.000E*00 l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+00 1.0006+00 1.0006+00 1.0006+00 1.0006+CO

-.8 l.OOOE-KW 8.733E-01 7.635F-01 6.683E-01 5.«I>6f-01 5. 137E-01 4.512E-01 3.967E-01 3.4936-01 3.0816-01 2.7226-01 -.4 i.oooe*oo 7.6e3E-01 5.945E-01 4.637E-01 3.650E-01 2.904E-01 2.340E-01 1.9126-01 1.5886-01 1.3436-01 1.1576-01

-.4 l.OOOE+00 6.873E-01 4.849E-01 3.531E-01 2.668E-01 2.102E-01 1.730E-01 1. 4856-01 1.3236-01 1.2166-01 1.1466-01 -.2 l.O00E«O0 6.341E-01 4.299E-01 3.152E-01 2.504E-01 2.135E-01 1.924E-01 1.804E-01 1.7346-01 1.6956-01 1.6726-01 1.000E*00 6.098E-01 4.225E-01 3.320E-01 2.880E-01 2.663E-01 2.556E-01 2.5036-01 2.4746-01 2.4636-01 2.4566-01

.2 l.OO0E*OO 6.116E-01 4.512E-01 3.&46E-01 3.568E-01 3.450E-01 3.400E-01 3.3786-01 3.3696-01 3.3656-01 3.36.T-01

.4 1.000E*00 6.332^-01 5.021E-01 4.553E-01 4.384E-01 4.3236-01 4.300E-01 4.2926-01 4.2896-01 4.2886-01 4.2876-01

.6 i.oooe*oo 6.673E-01 5.634E-01 5.310E-01 5.210E-01 5.178E-01 5.168E-01 5. 1656-01 5. 1646-01 5. 1646-01 5.1646-01

.8 l.OOOE^ 7.0e2t-01 6.274E-01 6.051E-01 5.990E-01 5.973E-01 5.969E-01 5.9676-01 5.9676-01 5. 9676-01 5.9676-01

1 l.OOOE+00 7.526E-01 6.90eE-01 6.753e-01 6.714E-01 6.705E-01 6.702E-01 6.7026-01 6.7026-01 6.7026-01 6.7026-01

1.2 i.oooe+00 7.990E-01 7.527E-01 7.419E-01 7.394E-01 7.3e8E-01 7.3e6£-01 7.3866-01 7.3866-01 7.3866-01 7.3866-01

1.4 l.OOOE+00 8.468E-01 8. 137E-01 8.063E-01 8.045E-01 8.041E-01 8.0406-01 8.0406-01 8.0406-01 8.0406-01 8.0406-01

1.6 l.OOOE+00 8.960E-0I e.745E-01 8.698E-01 8.6e7E-01 8.684e-01 8.684E-01 8.684E-01 8.684E-01 8.6846-01 8.6846-01

1.8 l.OOOE+OO 9.470E-01 9.363e-01 9.340E-01 9.334E-01 9.»*:-01 9.332E-01 9.332E-01 9.3326-01 9.332E-01 9.232E-01

2 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 1.0006+00 l.OOOE+00 1.0006+00 l.OOOE+00

B3(2,n,r,Mj) for r = 128

Hu \ ft 1 2 4 8 16 32 64 128 256 512 1024

-2 l.OOOE+OO 2.0006+00 4.0006+00 8.0006+00 1.600E+01 3.2006+01 6.4006+01 1.2906+02 2.5606+02 5.1206+02 1. 0246+03

-1.8 l.OOOE+OO 1.741E+00 3.031E+00 5.278E+00 9.189E+00 1.600E+01 2.7866+01 4.8506+01 8.444E+01 1.4706+02 2.560E+02

-1.6 l.OOOE+OO 1.516E+O0 2.297E+00 3.4826+00 5.2776+00 7,9986+00 1.2126+01 1.8376+01 2.785E+01 4.221E+01 6.398E+01

-1.4 l.OOOE+OO 1.319E+00 1.741E+00 2.2966+00 3.030E+00 3.997E+O0 5. 2746+00 6. 9596+00 9.182E+00 1.2126+01 1,5996+01 -1.2 l.OOOt+OO 1.148E+00 1.319E+00 1.515E+00 1.739E+00 1.9986+00 2.294E+00 2.635E+00 3.026E+00 3.4756+00 3.yy2t+oo

-1 l.OOOE+OO l.OOOE+00 1.0006+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO

-.8 l.OOOE+OO 8.721E-01 7.611E-01 6.647E-01 5.d08E-01 5.079E-01 4.444E-01 3.891E-01 3.410E-01 2.9926-01 2.627E-01

-.6 l.OOOE+OO 7.647E-01 5.8756-01 4.539E-01 3.529E-01 2.764E-01 2.1866-01 1.747E-01 1.415E-01 1.164E-01 9.729E-02

-.4 l.OOOE+OO 6.803E-01 4.7206-01 3.3596-01 2.467E-01 1.881E-01 1. 4946-01 1.24OE-01 1.0726-01 9.6146-02 8.884E-02

.,2 l.OOOE+OO 6.235E-01 4.1216-01 2.928E-01 2.252E-01 l.(V)AE-01 1.646E-01 1.519E-01 1.447E-01 1.4056-01 1.381E-01 l.OOOE+00 5.9786-01 4.036E-01 3.094E-01 2.634E-01 2. 4076-01 2.294E-01 2.238E-01 2.210E-01 2.196E-01 2. 189E-01

.2 1.0006+00 6.0116-01 4.3W*:-01 3.665E-01 3.376E-01 3.253E-01 3.200E-01 3.177E-01 3.1686-01 3.1636-01 3.162E-01

.4 l.OOOE+OO 6.2566-01 4.9146-01 4.4336-01 4.259E-01 4. 1966-01 4.173E-01 4.1646-01 4.1616-01 4.1606-01 4.159E-01

.6 l.OOOE+00 6.626E-01 5.571E-01 5.2426-01 5.139E-01 5. 1076-01 5.097E-01 5.0946-01 5.0936-01 5.0926-01 5.092E-01

.8 1.0006+00 7.0566-01 6.240E-01 6.0166-01 5.9546-01 5.9376-01 5.9326-01 5.9316-01 5.9306-01 5.9306-01 5.930E-01

1 1.0006+00 7.5136-01 6.891E-01 6.7366-01 6.6976-01 6.6876-01 6.685E-01 6.6846-01 6.6846-01 6.684E-01 6.684E-01

1.2 1.0006+00 7.9836-01 7.5206-01 7.4116-01 7.3866-01 7.3806-01 7.3786-01 7.3786-01 7.3786-01 7.3786-01 7.3786-01

1.4 1.0006+00 8.4656-01 8. 1336-01 8.0596-01 8.0426-01 8.0386-01 8.0376-01 8.0376-01 8.0376-01 8.0376-01 8.0376-01

1.6 1.0006+00 8.9596-01 8.7446-01 8.6966-01 8.6866-01 8.6836-01 8.6826-01 8.6826-01 8.6826-01 8.6826-01 8.682F-01

1.8 1.0006+00 9.4696-01 9.363F-01 9.3396-01 9.3346-01 9.3326-01 9.3326-01 9.3326-01 9.3326-01 9.3326-01 9.3326-01

2 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1. 0006+00 1. 0006+00 1.0006+00 1.0006+00 1.0006+00 l.OOOE+OO 1.0006+00

37

TN-332
B3(2,n,r,w) for p s 256

Hu \ 1^ 1 2 4 e 16 32 64 128 256 512 1024

-2 : i.oooe^ 2.000E+00 4.0O0E+OO e.oooE+oo 1.600E+01 3.200E+O1 6.400E+O1 1.280E+O2 2.560E+O2 5. 120E+02 1.024E+03

-1.8 l.OOOE+00 1.741E+00 3.031E+00 5.278k+00 9.189E+00 1.600E+01 2.786E+01 4.850E+01 8.445E+01 1. 4706+02 2.560E+02

-1.6 . l.OOOE+00 1.516E+O0 2.297E+00 3.482E+00 5.278E+00 7.999E+00 1.212E+01 1.838E+01 2.786E+01 4.??7t+01 6.399E+01

-1.4 l.OOOE+OO 1.319E+00 1.741E+00 2.297E+00 3.O31E+O0 3.999E+00 5.277E+00 6.962E+O0 9.187E+00 1.212E+01 1.599E+01

-1.2 l.OOOE+00 1.149E+O0 1.319E+O0 1.515E+00 1.740E+00 1.999E+O0 2.296E+O0 2.637E+O0 3.029fc+00 3.479E+00 3.996E+00

-1 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

-.8 l.OOOE+00 8.714E-01 7.597E-01 6.626E-01 5.781E-01 5.045E-01 4.405E-01 3.&48E-01 3.363E-01 2.941E-01 2.573E-01

-.6 l.OOOE+OO 7.623E-01 5.830E-01 4.475E-01 3.450E-01 2.674E-01 2.0e6E-01 1.640E-01 1.3038-01 1,047E-01 8.533F-02

-.4 l.OOOE+OO 6.750E-01 4.626f-01 3.235E-01 2.321E-01 1.720E-O1 l.JZJt-Ol 1.062E-01 8.899E-02 7.763E-02 7.014E-02

-.2 l.OOOE+OO 6.152E-01 3.981E-01 2.753E-01 2.054E-01 1.6SSF-01 1.427E-01 1.296E-01 1.221E-01 1.178E-01 1.153E-01 l.OOOE+00 5.882fc-01 3.885E-01 2.913E-01 2.436E-01 2.201E-01 2.0e4E-01 2.026E-01 1.997E-01 1.982E-01 1.973E-01

.2 l.OOOE+OO 5.929E-01 4.Z)3t-01 3.525E-01 3.226E-01 3.099E-01 3.04S-01 3.022E-01 3.011E-01 3.007E-01 3.005E-01

.4 l.OOOE+00 6.202E-01 4.837E-01 4.347E-01 4.170E-01 4.105E-01 4.0eiE-01 4.072E-01 4.069E-01 4.068E-01 4.067E-01

.6 l.OOOE+00 6.596E-01 5.530E-01 5.198E-01 5.094E-01 5.061E-01 5.051E-01 5.047E-01 5.046E-01 5.046E-01 5.046E-01

.8 l.OOOE+00 7.042E-01 6.221E-01 5.995E-01 5.933E-01 5.916E-01 5.911E-01 5.910E-01 5.910E-01 5.910E-01 5.910E-01

1 l.OOOE+OO 7.507E-01 6.883E-01 6.727E-01 6.688E-01 6.679E-01 6.676E-01 6.676E-01 6.675E-01 6.675E-01 6.675E-01

1.2 l.OOOE+OO 7.981E-01 7.516E-01 7.4O8E-01 7.3e2E-01 7.376E-01 7.375E-01 7.374E-01 7.374E-01 7.374E-01 7.374E-01

1.4 l.OOOE+OO 8.464E-01 8. 132E-01 8.058E-01 8.041E-01 8.036f-01 8.036E-01 8.035E-01 8.035E-01 8.035E-01 8.035E-01

1.6 l.OOOE+OO 8.958E-01 e.743E-01 8.696E-01 8.685E-01 8.682E-01 8.682E-01 8.682E-01 8.682E-01 8.682E-01 8.682E-01

1.8 l.OOOE+OO 9.469E-01 9.362E-01 9.:£tVt-01 9.3iae-oi 9.332E-01 9.332E-01 9.332E-01 9.332k-01 9.332t-01 9.i32t-01

2 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO

Mu \ H=

B3(2,H,r,iu) for r = 512 12 4 8 16 32 64 128 256 512 1024

-2 l.OOOE+00 2.000E+00 4.000E+00 8.000E+00 1.600E+01 3.2O0E+O1 6.4O0E+O1 1.280E+02 2.560E+02 5. 120E+02 1.024E+03

-1.8 l.OOOE+00 l,741E+00 3.031E+O0 5.278E+O0 9.190E+O0 1.600E+01 2.786E+01 4.850E+01 8.445E+01 1.470E+02 2.560E+02

-1.6 l.OOOE+OO 1.516E+00 2.297E+00 3.482E+00 5.278E+00 8.000E+00 1.213E+01 l.RSaF+Ol 2.786E+01 4.222k+01 6.400E+01

-1.4 l.OOOE+OO 1.319E+O0 1.741E+O0 2.297E+00 3.031E+00 4.000E+00 5.277E+O0 6.964E+00 9.189E+00 1.212E+01 1.600E+01

-1.2 l.OOOE+OO 1.149E+00 1.319E+O0 1.516E+00 1.741E+00 2.000E+00 2.297E+00 2.638E+O0 3.03OE+O0 3.481E+00 3.998E+00

-1 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+00 l.OOOE+OO

-.8 l.OOOE+OO 8.711E-01 7.589E-01 6.614E-01 5.765E-01 5.026E-01 4.383E-01 3.823E-01 3.336E-01 2.911E-01 2.542E-01

-.6 l.OOOE+OO 7.608E-01 5.800E-01 4.433E-01 3.39a-01 2.614E-01 2.020E-01 1.570E-01 1.229E-01 9.709E-02 7.751E-02

-.4 l.OOOE+00 6.712E-01 4.l)b/t-01 3.143E-01 2.213E-01 1.601E-01 1.198E-01 9.316E-02 7.561E-02 6.403E-02 5.639E-02

-.2 l.OOOE+00 6.085E-01 3.870E-01 2.613E-01 1.897E-01 1.487E-01 1.2536-01 1.118E-01 1.041E-01 9.970E-02 9.716E-02

l.OOOE+00 5.803E-01 3.761E-01 2.764E-01 2.274E-01 2.031E-01 1.911E-01 1.851E-01 1.821E-01 1.806E-01 1.799E-01

.2 l.OOOE+OO 5.863E-01 4.136E-01 3.4136-01 3. lOeE-01 2.978E-01 2.922t-01 2.8V«t-01 2.887E-01 2.883E-01 2.881E-01

.4 l.OOOE+OO 6. 162E-01 4.781E-01 4.284E-01 4.104E-01 4.039E-01 4.014E-01 4.005E-01 4.002E-01 4.001E-01 4.000E-01

.6 l.OOOE+OO 6.576E-01 5.504E-01 5.169E-01 5.064E-01 5.031E-01 5.021E-01 5.0186-01 5.016E-01 5.016E-01 5.016E-01

.8 l.OOOE+00 7.033t-01 6.2UE-01 5.9e4E-01 5.921E-01 5.904E-01 5.899E-01 5.8986-01 5.8986-01 5.898E-01 5.898E-01

1 l.OOOE+00 7.503E-01 6.879E-01 6. 723t-01 6.684E-01 6.674E-01 6. 6726-01 6. 6716-01 6.671E-01 6.671E-01 6.671E-01

1.2 l.OOOE+00 7.980E-01 7.515E-01 7.406E-01 7.381E-01 7. 3756-01 7.3736-01 7.3736-01 7.3736-01 7.373E-01 7.373E-01

1.4 l.OOOE+OO 8.4636-01 8. 132E-01 8.057E-01 8.040E-01 8.036E-01 8.0356-01 8.035F-01 8.035E-01 8.035E-01 8.035E-01

1.6 l.OOOE+00 8.9586-01 8.743E-01 8.696E-01 8.685E-01 e.682E-01 8.6826-01 8.681E-01 8.681E-01 8.6eiE-01 8.681E-01

1.8 l.OOOE+00 9.469E-01 9.362E-01 9.3396-01 9. ax* -01 9.332E-01 9.332t-01 9.332t-0l 9.'332fc-01 9.332E-01 9.332k-01

2 ! l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO l.OOOE+OO 1.0006+00 1.0006+00 l.OOOE+OO l.OOOE+OO l.OOOE+00

38

TN-333
Nu \ H= 1

B3(2,H,r,«j) for r = 1024

2 4 8 16 32 64 12B 256 512 1024

-2 i.oooe+00 2.000E"H)0 4.0006+00 8.0006+00 1.6006+01 3.200E+O1 6.4006+01 1. 2806+02 2.5606+02 5. 1206+02 1.0246+03

-1.8 l.OOOE+OO 1.741E+O0 3. 0316+00 5.2786+00 9. 1906+00 1.6006+01 2.7866+01 4.8506+01 8.4456+01 1. 4706+02 2.5606+02

-1.6 l.OOOE+00 1.516E+00 2.2976*00 3.4826+00 5.2786+00 8.0006+00 1.2136+01 1.8386+01 2.7866+01 4.2226+01 6. 4006+01 -1.4 l.OOOE+00 1.319E+00 1.7416+00 2. 2976+00 3.0316+00 4. 0006+00 5.2786+00 6.9646+00 9.1896+00 1.2136+01 1.6006+01 -1.2 l.OOOE+OO 1.149E+00 1.319E+00 1.5166+00 1.7416+00 2.0006+00 2.2976+00 2.6396+00 3.0316+00 3.4826+00 3.9996+00

-1 i.oooe+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 l.OOOE+00 1. 0006+00 1. 0006+00 1.0006+00

-.8 i.oooe+00 8.708E-01 7.5656-01 6,6076-01 5.7566-01 5.0156-01 4.3706-01 3,8096-01 3.3206-01 2.8946-01 2.5246-01

-.6 l.OOOE+00 7.598E-01 5.7816-01 4.4066-01 3.3646-01 2.5756-01 1.9776-01 1.5246-01 1.1816-01 9.2106-02 7.2386-02

-.4 l.OOOE+00 6.683E-01 4.5066-01 3.0766-01 2.1346-01 1.5146-01 1.1056-01 8.3486-02 6.5696-02 5.3966-02 4.6226-02

-.2 l.OOOE+00 6.031E-O1 3.7806-01 2.4996-01 1.76ff-01 1.3516-01 1.1126-01 9.7446-02 8.9566-02 8.5046-02 8.2446-02

l.OOOE-HXl 5.737E-01 3.6586-01 2.6396-01 2. 1386-01 1.8906-01 1.7676-01 1.7056-01 1.6756-01 1.6596-01 1.6526-01

.2 i.oooe+00 5.811E-01 4.0596-01 3,3236-01 3.0126-01 2.8906-01 2.8236-01 2.7986-01 2.7886-01 2.7836-01 2. 7816-01

.4 i.oooe+00 6.133E-01 4.7406-01 4.2386-01 4.0566-01 3.9906-01 3.9656-01 3.9566-01 3.9536-01 3.951E-01 3.9516-01

.6 i.oooe+00 6.564E-01 5.4866-01 5. 1506-01 5.0456-01 5.0126-01 5.0016-01 4.9986-01 4.9976-01 4.9976-01 4.9966-01

.8 l.OOOE+00 7.0286-01 6.2046-01 5.9776-01 5.9156-01 5.8976-01 5.8936-01 5.8916-01 5.8916-01 5.8916-01 5.891E-01

1 l.OOOE+00 7.502E-01 6.8776-01 6.7216-01 6.6826-01 6.6726-01 6.6706-01 6.66^-01 6.6696-01 6.6696-01 6.6696-01

1.2 i.oooe-KX) 7.979E-01 7.5146-01 7.4O6E-01 7.3806-01 7.3746-01 7.3736-01 7.3/2E-01 7.3726-01 7.3726-01 7.3726-01

1.4 l.OOOE+00 8.4636-01 8. 1316-01 8. 05 /t -01 8.0406-01 8.0366-01 8.0356-01 8.0356-01 8.0356-01 8.0356-01 8.0356-01

1.6 l.OOOE-KK) 8.9586-01 8.7436-01 8.696E-01 8. 6856-01 8.682E-01 8. 6826-01 8.681E-01 8.681E-01 8.6816-01 8.6816-01

1.8 l.OOOCKW 9.4696-01 9,3626-01 9.3396-01 9.333F-01 9.332t-01 9.3326-01 9.3326-01 9.332t-01 9.332fc-01 9.332t-01

2 l.OOOE+OO 1.0006+00 1.0006+00 l.OOOE+00 l.OOOE+00 1.0006+00 1.0006+00 1.0006+00 1. 0006+00 1.0006+00 l.OOOE+00

Nu \ H= 1

B3(2,fl,r,iu) for r = 2048

2 4 8 16 32 64 128 256 512 1024

-I 1. 0006+00 2.0006+00 4.0006+00 8. 0006+00 1.6006+01 3.2006+01 6.4006+01 1.2806+02 2.5606+02 5. 1206+02 1.0246+03

-1.8 1.0006+00 1.741E+O0 3.0316+00 5.27K+O0 9.1906+00 1.6006+01 2.7866+01 4.8506+01 8.4456+01 1.4706+02 2.5606+02

-1.6 1.0006+00 1.5166+00 2.2976+00 3.4826+00 5.2786+00 8.0006+00 1.2136+01 l.K«t+01 2.7866+01 4.2226+01 6.4006+01

-1.4 1.0006+00 1.3206+00 1.7416+00 2.2976+00 3.031E+OO 4.0006+00 5.2786+00 6.9646+00 9.1896+00 1.2136+01 1.6006+01 -1.2 l.OOOE+iX) 1.149E+00 1.3196+00 1.5166+00 1.741E+O0 2.0006+00 2.297E+00 2.6396+00 3.0316+00 3.4826+00 4.0006+00

-1 l.OOOE+Oi) 1.0006+00 1.0006+00 i.oooe+00 1.0006+00 1.0006+00 1. 0006+00 1.0006+00 l.OOOE+00 1.0006+00 1.0006+00

-.e 1.0006+00 8.7076-01 7.5826-01 6.6036-01 5.751E-01 5.0096-01 4.3636-01 3.8006-01 3.3116-01 2.88bt-01 2.5146-01

-.6 1. 0006+00 7.5916-01 5.7686-01 4.3886-01 3.3426-01 2.5496-01 1.9496-01 1.4946-01 1.1496-01 8.881E-02 6.901E-02

-.4 l.OOOE+00 6.6626-01 4.4686-01 3.0256-01 2.0756-01 1.4486-01 1.0356-01 7.6286-02 5.8316-02 4.6466-02 3.8646-02

"•A. 1.000E+<X( 5.9876-01 3.7066-01 2.4076-01 1.6646-01 1.2406-01 9.9626-02 8. 5656-02 7.7636-02 7.3036-02 7.0386-02

1.0006+00 5.6816-01 3.5706-01 2.5346-01 2.0236-01 1.7706-01 1.6446-01 1.5826-01 1.5516-01 1.5356-01 1.5276-01

.2 1.0006+00 5.7686-01 3.9956-01 3.2506-01 2.9346-01 2.9006-01 2.7426-01 2.7176-01 2.7066-01 2.701E-01 2.6996-01

.4 1.0006+00 6.1116-01 4.7096-01 4.2046-01 4.0216-01 3.9536-01 3.9296-01 3.9206-01 3.9166-01 3.9156-01 3.9146-01

.6 1.0006+00 6.bi>bk-01 5.4756-01 5.1386-01 5.0326-01 4.9996-01 4.9886-01 4.9856-01 4.9846-01 4.9846-01 4.9846-01

.8 1.0006+00 7.0266-01 6.2016-01 5.9736-01 5.9116-01 5.8946-01 5.8896-01 5.8886-01 5.8876-01 5.8876-01 5.8876-01

1 1.0006+00 7.5016-01 6.8766-01 6.7206-01 6.6816-01 6.6716-01 6.6696-01 6.6686-01 6.6686-01 6.6686-01 6.6686-01

1.2 1.0006+00 7.9796-01 7.5146-01 7.4056-01 7.3806-01 7.3746-01 7.3726-01 7.3/2k-01 7.3726-01 7.3726-01 7.3726-01

1.4 1.0006+00 8.4636-01 8. 1316-01 8.0576-01 8.0406-01 8.0366-01 8.0356-01 8.0356-01 8.0346-01 8.0346-01 8.0346-01

1.6 1.0006+00 8.9586-01 8.7436-01 8.6966-01 8.6856-01 8.6626-01 8.6826-01 8.6816-01 8.6816-01 8.6816-01 8.6816-01

1.8 1.0006+00 9.4696-01 9.3626-01 9.3396-01 9.3336-01 9.3326-01 9.332t-01 9.3326-01 9.3326-01 9.3326-01 9.332E-01

2 1.0006+00 1.0006+00 1.0006+00 1. 0006+00 1.0006+00 1.0006+00 1.0006+00 1. 0006+00 1.0006+00 1. 0006+00 1.0006+00

39

TN-334
Hu \ tt= 1

B3(2,H,p,iu) for p » 4096

2 4 8 16 32 64 126 256 512 1024

-2 i.oooe+00 2.000C+00 4.000€-K)0 8.000E+O0 1.600E^1 3.2O0e^l 6.4006+01 1.2806*02 2.5606*02 5. 1206*02 1.0246*03

-1.8 l.OOOE+00 1.741E*O0 3.031E+00 5.278E+O0 9.190e+00 1.6006*01 2.7866*01 4.8506*01 8.445E*01 1.4706*02 2.5606*02 -1.6 l.OOOE+OO 1.516E+00 2.297E+00 3.4e2£+00 5.278E+O0 8.000E>O0 1.2136*01 1.8386*01 2.7846*01 4.//a:*0l 6.4006*01 -1.4 l.OOOE+00 1.320E+00 1.741E+00 2.297E+00 3.031E+00 4.0006^ 5.2786*00 6.9646*00 9.1906*00 1.2136*01 1.6006*01 -1.2 l.OOOE+00 1.149E+O0 1.319E+00 1.516£*00 1.741E+00 2.000E+00 2.2976*00 2.6396*00 3.0316*00 3.4826*00 4.0006*00

-1 l.OOOE+OO l.OOOE+OO l.OOOE+00 l.OOOE-KX) l.OOOE+00 l.OOOE+00 1.0006*00 1.0006*00 1.000E*O0 1.0O0E*OO 1.0006*00

-.8 l.OOOE+00 8.706€-01 7.581E-01 6.6O1E-01 5.748E-01 5.005E-01 4.3586-01 3.7966-01 3.3066-01 2.8796-01 2.5086-01

-.6 1.000E*00 7.5e7E-01 5.760e-01 4.376E-01 3.327E-01 2.5336-01 1.931E-01 1.4746-01 1.1296-01 8.6656-02 6.6796-02

-.4 l.OOOE+00 6.646E-01 4.440e-01 2.9e7E-01 2.030E-01 1. 3996-01 9.R3;#-02 7.0896-02 5.2796-02 4,0856-02 3.2976-02

-.2 l.OOOE+OO 5.950E-01 3.645E-01 2.330E-01 1.578E-01 1.1486-01 9.0076-02 7.5906-02 6.7776-02 6.3106-02 6.0426-02

l.OOOE+00 5.633E-01 3.494E-01 2.443e-01 1.925E-01 1.667E-01 1.5396-01 1.4766-01 1.4446-01 1.4286-01 1.4206-01

.2 l.OOOE^OO 5.733E-01 3.943E-01 3. 189E-01 2.87CE-01 2.734E-01 2.6756-01 2.6506-01 2.6396-01 2.6346-01 2.632E-01

.4 l.OOOE+00 6.095E-01 4.686E-01 4.178E-01 3.994E-01 3.9266-01 3. 9026-01 3.892-01 3.8896-01 3.8876-01 3.8876-01

.6 l.OOOE+00 6.550e-01 5.4686-01 5.130E-01 5.024E-01 4.991E-01 4. 9906-01 4.9776-01 4.9766-01 4.975E-01 4.9756-01

.8 l.OOOE+00 7.024E-01 6.199E-01 5.971E-01 5.909E-01 5.891E-01 5.887E-01 5.8856-01 5.8856-01 5.885E-01 5.8856-01

1 i.oooe+oo 7.500E-01 6.876E-01 6.719E-01 6.680E-01 6.6706-01 6.6686-01 6.6676-01 6.6676-01 6.667E-01 6.667E-01

1.2 l.OOOE+00 7.979E-01 7.514E-01 7.405E-01 7.380E-01 7.3746-01 7.372E-01 7.3726-01 7.3726-01 7.3726-01 7.3726-01

1.4 l.OOOE+00 8.463E-01 8. 131E-01 8.057E-01 8.O4OE-01 8.0366-01 8.035E-01 8. 0346-01 8.0346-01 8.034E-01 8. 0346-01

1.6 l.OOOE+00 8.958e-0I 8.743E-01 8.696E-01 8.685E-01 8.6826-01 8.682E-01 8.681E-01 8.681E-01 8.681E-01 8.681E-01

1.8 1.000€+00 9.469E-01 9.362E-01 9.339E-01 9.333e-01 9.332t-01 9.:»*-01 9.3326-01 9.3326-01 9.232E-01 9.332t-01

2 l.OOOE+OO l.OOOE-KK) l.OOOE+00 l.OOOE+00 l.OOOE-KXJ 1.0006*00 1.0006*00 1.0006*00 1.000E*00 1.0006*00 1.0006*00

flu \ 11= 1

B3(2,fl,r,iu) for r = 8192

2 4 8 16 32 64 128 256 512 1024

-2 l.OOOE+00 2.000E+00 4.000E+00 8.000E+00 1.600E+01 3.200E+OI 6.4006+01 1.2806+02 2.5606+02 5. 1206+02 1.024E+O3

-1.8 1.000E*00 1.741E+O0 3.031E+00 5.27^+00 9.190E+O0 1.600E+01 2.7866+01 4.850E+01 8. 4456+01 1.470E+O2 2.560E+02

-1.6 l.OOOE+00 1. 5166+00 2.2976+00 3.482E+00 5. 2786+00 8.0O0E+O0 1.2136+01 1.838E+01 2.7866+01 .Z/A+01 6.400E+01

-1.4 1.000E*00 1.320E+O0 1.741E+00 2.297E+00 3.031E+O0 4.000E+O0 5.2786+00 6.9646+00 9.1906+00 1.2136+01 1.600E+01

-1.2 1.0006+00 1.149E+00 1.320E+00 1.516E+00 1.741E+00 2.000E+00 2.297E+00 2.6396+00 3.031E+O0 3.4826+00 4.0OOE+O0

-1 l.OOOE+00 l.OOOE+OO l.OOOE+00 l.OOOE+OO l.OOOE+OO l.OOOE+OO 1.0006+00 1.0006+00 1.0006+00 1. 0006+00 l.OOOE+00

-.8 l.OOOE+00 8.706E-01 7.5eOE-01 6.599E-01 5.7466-01 5.0036-01 4.3566-01 3.7936-01 3.3036-01 2.876E-01 2.5O5E-01

-.6 l.OOOE+OO 7.584E-01 5.7546-01 4.,%fiF.-01 3.317E-01 2.5226-01 1.9186-01 1.461E-01 1.1156-01 8.5236-02 6.5336-02

-.4 l.OOOE+00 6.6346-01 4.4186-01 2.959E-01 1.997E-01 1.3636-01 9.4446-02 6.685E-02 4.8646-02 3.6636-02 2.871E-02

-.2 l.OOOE+OO 5.920E-01 3.5946-01 2.2666-01 1.506E-01 1.0716-01 8.2116-02 6.7786-02 5.9556-02 5.4836-02 5.212E-02

l.OOOE+OO 5.591E-01 3.4296-01 2.36S-01 1.839E-01 1.5786-01 1.4486-01 1.3836-01 1.351E-01 1.3356-01 1.327E-01

.2 l.OOOE+OO 5.704E-01 3.8996-01 3.1396-01 2.817E-01 2.6796-01 2.6206-01 2.5946-01 2.5836-01 2.57^-01 2.5766-01

.4 l.OOOE+00 6.0e3E-01 4.6696-01 4.1596-01 3.9746-01 3.9066-01 3.881E-01 3.872E-01 3. 8686-01 3.867E-01 3.867E-01

.6 l.OOOE+OO 6.546E-01 5.4636-01 5.1246-01 5.0186-01 4.985E-01 4.9756-01 4.9716-01 4.9706-01 4.9706-01 4.970E-01

.8 l.OOOE+OO 7.0236-01 6. 1986-01 5.9706-01 5.9076-01 5.890E-01 5.8856-01 5.8846-01 5.8846-01 5.884E-01 5.8846-01

1 1.0006+00 7.5006-01 6.8756-01 6.719E-01 6.6806-01 6.670E-01 6.6686-01 6.6676-01 6.667E-01 6.667E-01 6.667E-01

1.2 l.OOOE+OO 7.9796-01 7.5146-01 7.4056-01 7.3806-01 7.3746-01 7.3726-01 7.3726-01 7.372E-01 7.372E-01 7.3726-01

1.4 l.OOOE+OO 8.4636-01 8.1316-01 8.0576-01 8.0406-01 8. 0366-01 8.0356-01 8.0346-01 8.0346-01 8.0346-01 8.034E-01

1.6 1.0006+00 8.9586-01 8.7436-01 8.696E-01 8.685E-01 8.6826-01 8.6826-01 8.6816-01 8.6816-01 8.6816-01 8.681E-01

1.8 1.0006+00 9.4696-01 9.3626-01 9,33Vk-01 9.3336-01 9.3326-01 9.3326-01 9.3326-01 9.3326-01 9.3326-01 9.3326-01

2 l.OOOE+OO 1.0006+00 l.OOOE+OO l.OOOE+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 1.0006+00 l.OOOE+00

40

TN-335
E. APPENDIX - Notes and Errata

The notes listed below are included to bring attention to notation and other problems. The page

number provides reference to the location of the problem or comment.

1. page TN-4

AM noise is especially important in the measurement of the (residual) phase noise added

by amplifiers and other signal handling components. Often the driving source for the

measurements is a frequency synthesizer that has phase and/or amplitude noise that is comparable or larger than the added noise of the component under test. Most measure- ment systems are configured so that the phase noise of the source cancels out to a large

degree (at high Fourier frequencies decorrelation effects limit the cancellation). In such

measurement systems, the AM to PM conversion factor and the AM noise of the source may then set the noise floor. Discussion of the effect of AM noise and AM to PM

conversion factors on the accuracy and precision of phase noise measurements is found in

"Residual Phase Noise Measurements of vhf, uhf, and Microwave Components" by G. K.

Montress, T. E. Parker, and M. J. Loboda Proc. 43rd Annual Symposium on Frequency

Control, pp. 349-359 (1989) and in "Accuracy Model for Phase Noise Measurements by F. L. Walls, C. M. Felton, and A. J. D. Clements, 21st Annual Precision Time and Time

Interval Meeting (1989). The notation in these papers as well as that in other parts of the

literature differs from that given below. Our notation below is drawn from a modest level

of consensus among individuals responsible for setting standards. We expect that it will gradually be adopted within standards committees. The following comments are directed

specifically at the specification of noise performance.

The total power spectral density in a signal can be approximated by expanding eq 12-5 of

paper B.2 (by Stein) and extending it to include the spectral density of relative amplitude

fluctuations, SJT). The double-sideband density written in single-sideband form is given by

where

1(f) - \s^{f)df.

1(f) is the integrated phase modulation due to the pedestal and <S(f) represents the carrier

with frequency width ± f^. The effect of large S^(f) on power in the carrier has not, to our knowledge, been explored. The power spectral density of relative phase fluctuations, S^(f),

is normalized to one rad^/Hz and ^JJ) is normalized to the carrier voltage, but the total

power spectral density, Sy(f) is not normalized and has the units of V^/Hz. All of these are single-sided spectral densities. For most measurement purposes, we can disregard the

carrier and find that, away from the carrier, the above expression simplifies to

Syif) •= 4[W) + ^J<f)] for ^/^ 00.

TN-336
The more general expression is important only very near the carrier and in certain types

of frequency multiplication. The single-sideband, amplitude noise, normalized to the total

signal power is given simply by SJf) for < f< <». The measurement of added phase and

amplitude noise for amplifiers and other signal handling components should specify the

signal level since the AM noise level and the contribution due to AM-to-PM conversion

depend on the signal level.

2. page TN-6

For a direct measurement, time accuracy only has meaning when the phase of the timebase oscillator of the frequency counter is known with respect to some time standard.

Either it is phase locked or is calibrated with respect to that standard at the time of

measurement. The phase of the time-base oscillator can then be measured with respect

to the phase of the frequency standard being calibrated (accounting for cable delays, etc.).

Except for the cycle ambiguity of the carrier, the phase of the frequency standard being

measured can carry time information and have time accuracy. This technique is not

common, but is very useful and eliminates divider noise that typically occurs in going from

5 or 10 Mhz to 1 pulse per second. Caution must be exercised to assure that the phase

point measured in a sine wave is at a reproducible voltage and impedance so that the cycle

ambiguity is an exact integer.

3. page TN-35

A second-order servo loop provides substantially enhanced performance. See, for example,

F.L. Walls and S.R. Stein, "Servo techniques in oscillators and measurement systems," NBS

Tech. Note 692 (1976).

4. page TN-35

This error can be identified and corrected using the phase modulation scheme described

in paper B.4 on page TN-136.

5. page TN-36

Low-noise DC amplifiers have been substantially improved since publication of this paper.

6. pages TN-37. TN-91. TN-130. TN-174. TN-206 and TN-218

The reader is reminded that the discussions of frequency-domain measurements assume

incoherent noise processes. Often the phase noise spectrum of a signal will contain bright

spectral features (spurious lines) other than the carrier. Frequency-domain measurements

are often useful in identifying such features. But if these spurious lines are narrow compared to the measurement bandwidth, statistical measures such as S^(f) and i?(f) are not appropriate. It is better to specify the phase deviation in terms of the rms value of

the phase deviations, 0rms('"'^s radians), without reference to bandwidth. This specification in rms radians can be related to the Allan variance (see note # 8 below).

7. page TN-51

Humidity is often an important environmental factor. See, for example, J.E. Gray, H.E.

Machlan and D.W. Allan, "The Effect of humidity on commercial cesium beam atomic

clocks," 42nd Annual Symp. on Frequency Control, pp. 514-518 (1988) and F.L. Walls, "The

TN-337
Influence of Pressure and Humidity on the Medium and Long-Term Frequency Stability

of Quartz Oscillators," 42nd Annual Symp. on Frequency Control, pp. 279-283 (1988).

8. page TN-74

For a bright line (one which is narrow compared with the measurement bandwidth), the

solution of eq 12-27 simpUfies to

v/2* olx) - ^sin^(7rA),

where 0rmsi^ ^^e rms value of the phase deviations. The above relationship may be useful

where one is trying to determine the effect of a bright line in the time domain. If the

bright Line is the dominant factor, the plot of oJj) versus r has strong sin (Trfr) oscillations and it can be ambiguous. In that case, it is better to provide a specification in terms

of 0j^5 without reference to bandwidth. Statistical measures such as aJr) and S^(f) are not meant to be used to describe coherent signals. For further discussion see paper B.l,

section 12.2 (page TN-51).

page TN-75

A set of brackets, [ ], are missing in eq 12-29. The equation should read

moda (t) =< V

n+j'-l

i-J

10. page TN- 119

The reference (Walls and DeMarchi, 1975) is listed as being on pages 310-317. The page

numbers should be 210-217.

11. page TN-121

Most of the hterature uses the expression [Vq + e(t)] instead of Vq as in eq (2). Vq is the

peak voltage amplitude and e (t) is the voltage deviation of the amplitude from nominal.

12. page TN-122

In eq (5), most of the Hterature uses Xi(t) instead of e(t). e(t) is usually the voltage

deviation as described in note 11 above.

13. page TN-123

There is an error in the caption for figure 7. The last portion of that caption should read:

"where f is Fourier frequency ((w = lirf) and S (f) = u> Sx(f))."

14. page TN-123

In the right-hand column, last paragraph, 5th line, there is an extraneous minus sign. The

quantity y7^° should read y\° . Also, note that the use of supercript t and Tq with y is not consistent with the new IEEE standard definitions (see paper C.l, page TN-139).

TN-338
15. page TN-124

In eq 9 the subscript, k - n, should read k + n. That is, the equation should read

16. page TN-125

The quantity aJr) is now commonly known as modal t). This latter form has been

recently adopted by IEEE as the standard terminology.

17. page TN-139

IEEE Std 1139-1988, IEEE Standard Definitions of Physical Quantities for Fundamental

Frequency and Time Metrology, is an almost exact replica of this paper (D.l). The paper was published during the latter period of the development of the standard. The only

substantial difference is that, wherever it occurs, the word "departure" in the paper is replaced in the IEEE standard by "deviation."

18. page TN-146

This widely cited paper provided the de facto standards for terminology and oscillator

characterization until the recent adoption of the IEEE standard presented in paper C.l

(page TN-139). For terminology, the latest IEEE standard should always take precedence.

This paper (C.2) is fairly consistent with the IEEE standard. One exception is that, in this

paper, N denotes the number of frequency measurements. The symbol M in the IEEE

standard is the same as N in this paper. In the standard, the equation relating M and N

is M = N - 1.

19. page TN-151

Equation (23) can be substantially simplified as shown, for example, by Stein (page TN-74,

eq (12-27)), which is

eo o'(T) ^fS(f)sm\nfz)df.

20. page TN-154

In eq 36 the T outside the brackets should be a r. The equation should read

x(tQ)-x(tQ-T)

xitQ + x) - x(tQ) + T:

21. page TN-160

The 2 expressions listed as eq (95) are in error. They should read

~/i_jt2[3 + 21nr - l/(6r2)] r>l

~/i_i7^[3 - 21nr] r<l.

TN-339
22. page TN-160

In eqs (101), (102), (103), (104), and (105), a Greek 7 was mistakenly replaced by the

number 2. In each of these equations the quantity [2 + ln(27rf,,T)] should be replaced by

the quantity [7 + ln(27rf^T)]. 7, Euler's constant, has the value 0.5772156649

23. page TN-162

This paper is included in this collection because it presents the internationally accepted

terminology and definitions. There are no substantial inconsistencies with the new IEEE

standard (paper C.l, page TN-139), but the latest IEEE standard should is considered to be the most up-to-date authority. A new version of this international report should be

issued by the CCIR in 1990.

24. page TN-171

The definitions for symbols used in this paper are fairly consistent with those adopted in

the recent IEEE standard (C.l). One exception is that, in this paper, N denotes the

number of frequency measurements. The symbol M in the IEEE standard is the same as N in this paper. Another is that, while this paper uses ju as the exponent of t in describing the power-law noise processes, the paper adopts the opposite sign convention for fi. v(t) is used where many other papers use V(t) for instantaneous voltage. Some confusion

is generated when this small v is typeset in the equations to look almost identical to the

Greek 1/, a symbol which is used exclusively to represent frequency. For example. In equation (1) the left-hand quantity is voltage, whereas the i/(t) and i/^ in equation (4) are

clearly frequencies. Finally, the authors of this paper, in equation (2), define e(t) as the

normalized amplitude fluctuations, a very sound choice, but the reader should note that

most other papers have not normalized it.

25. page TN-175

For consistency with figure 12 and the text, u(i), the left-hand member of eq (11) should

probably be u(t).

26. page TN-177

Walls, Percival and Irelan (D.4) have recently addressed the more accurate specification

of the quantity p in eq (12).

27. page TN-179

It is important to note that the expressions in Table 2 in this paper are derived assuming

use of a single-pole fUter. The calculations can also be done using an ideal (infinitely

sharp) filter. The solutions in these two limits are useful because they define the range of

practical values (using n-pole filters) for the expressions. Table I in this section is an

expansion of Table 2 of Lesage and Audoin providing both the single-pole results as well

as the results for an infinitely sharp filter. There are discrepencies in several of the coefficients between terms in Table 2 in the paper and those in Table I on the next page.

28. page TN-180

Barnes and Allan (paper D.8) have recently completed further analysis of the effect of

dead time on measurements.

TN-340
c

a

)-( 3CO

0) JS

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3

hi

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CO

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C ftJQ O

I

CO cd 6

" ^ B

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11) C

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b

C/3

O

s

1)

3 .S

C/5

V D Ci0) 3-^

3 .C

A c/3 A ^

c

u.

i-i

J3

<N CO

00

«o

<N <N

CM

m <4-<

fete

m

te

te

fe- es

y.—

*^ y-"^ <N J=

fe- Jq

CM (N

te CM

00

eno

CM

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CM

0)

CO

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m

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feCM m

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TN-341
29. page TN-180

If the ratio of T/t is constant and greater than 1 (the usual case), the problem described

is eliminated. However, in taking data for a plot of CTy(T) versus t, it is difficult to achieve this in the hardware and not possible to do it with software processing alone. For

further discussion see paper D.8 (page TN-296).

30. page TN-197

The most recent definitions and concepts for spectral density are given in a new IEEE

standard (paper C.l on page TN-138). This new standard should be consulted as the latest authority on definitions and terminology.

31. page TN-198

The newly accepted definition of ie(f) is given in paper C.l. This new definition, £(f) =

y2S^{f), was always valid for Fourier frequencies far from the carrier. It has now been

extended to cover all frequencies.

32. page TN-217

Equation (73) should read 20 log (final frequency/original frequency).

33. page TN-239

On page TN-198 the authors refer to a paper by Glaze (1970). The reference, apparently

lost in printing, is: Glaze, D.J. (1970). "Improvements in Atomic Beam Frequency Standards at the National Bureau of Standards, "IEEE Trans. Instrum. Meas. IM-19(3), 156-160.

34. page TN-257

There are two errors in Table 2. Under R(n) the first entry should be 1/n rather than 1. The second item in the same column is not single valued (1), but takes on different values

for different measurement bandwidths. The reader is referred to section A.6 (page TN-9)

for a discussion of this topic.

35. pages TN-261 and TN-262

Subsequent work on modar(T) and R(n) is reported in section A.6 (page TN-9) of this

report. There are some differences between the results in A.6 and the ones reported in

Tables I and II and Figure 4 in this paper.

36. page TN-264

To be consistent with other papers in the literature, 0(t) in eq (1) should probably be

written as Xj(t).

37. page TN-268

The term e(t) in eq (3) is normally used to represent the amplitude fluctuations in the

output voltage of an oscillator. This term might be better designated Yi(t).

TN-342
NlST-1 14A U.S. DEPARTMENT OF COMMERCE

(REV. 3-89) NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

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2. PERFORMINQ ORGANIZATION REPORT NUMBER

3. PUBUCATION DATE March 1990

4. TITLE AND SUBTITLE

Characterization of Clocks and Oscillators

S. AUTHOR(S)

D.B. Sullivan, D. W. Allan, D. A. Howe, and F.L. Walls, Editors

6. PERFORMING ORGANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

GAITHERSBURG, MD 20899

7. CONTRACT/GRANT NUMBER

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DOCUMENT DESCRIBES A COMPUTER PROGRAM; SF-185, FIPS SOFTWARE SUMMARY, IS ATTACHED.

1 1 . ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBUOGRAPHY OR

LITERATURE SURVEY, MENTION IT HERE.)

This is a collection of published papers assembled as a reference for those in- volved in characterizing and specifying high-performance clocks and oscillators. It

is an interim replacement for NBS Monograph 140, Time and Frequency: Theory

and Fundamentals, an older volume of papers edited by Byron E. Blair. This

current volume includes tutorial papers, papers on standards and definitions, and

a collection of papers detailing specific measurement and analysis techniques.

The discussion in the introduction to the volume provides a guide to the content

of the papers, and tables and graphs provide further help in organizing methods

described in the papers.

12. KEY WORDS (6 TO 12 ENTRIES: ALPHABETICAL ORDER: CAPITAUZE ONLY PROPER NAMES: AND SEPARATE KEY WORDS BY SEMICOLONS)

Key words: Allan variance, clocks, frequency, oscillators, phase noise, spectral

density, time, two-sample variance.

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