Characterization of 

Clocks and Oscillators 

Edited by 

D.B. Sullivan 

D.W. Allan 

D.A. Howe 

EL. 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The purpose of this document is to replace this outdated monograph in the 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. 

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 Monograph 140 has gradually become outdated, and with the recent issuance of a 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-O-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 publication 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 “NBS 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 

. . . 

ul 
NIST-114A U.S. DEPARTMENT OF COMMERCE 1. PUBUCATION OR REPORT NUMBER 

(REV. 3-89) NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY NIST/TN-1337 ’ 

2. PERFORMINQ ORQANIZATION REPORT NUMBER 

BIBLIOGRAPHIC DATA SHEET 

3. PUBUCATION DATE 

March 1990 

I. TITLE AND SUBTITLE 

Characterization of Clocks and Oscillators 

i. AUTHOR(S) 

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

1. PERFORMINQ ORQANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS) 7. CONTRACT/GRANT NUMBER 

U.S. DEPARTMENT OF COMMERCE 

NATIONAL INSTlTUTE OF STANDARDS AND TECHNOLOGY 

GAITHERSBURQ, MD 20899 8. TYPE OF REPORT AND PERIOD COVERED 

1. SPONSORINQ ORQANIZATION NAME AND COMPLETE ADDRESS (STREBT, CITY, STATE, ZIP) 

0. SUPPLEMENTARY NOTES 

DOCUMENT DESCRIBES A COMPUTER PROGRAM; SF-185, FIPS SOFTWARE SUMMARY, IS ATTACHED. 

UTERATURE SURVEY. MENTION IT HERE.) 

This is a collection of published papers assembled as a reference for those involved in characterizing and sp&iijCng 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. 

1. KEY WORDS 16 TO 12 ENTRIES: ALPHABETICAL ORDER: CAPITAUZE ONLY PROPER NAMES: AND SEPARATE KEY WORDS BY SEMICOLONS) 

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

density, time, two-sample variance. 

L AVAILABIUTV 14. NUMBER OF PRINTED PAQES 

x UNUMITED 

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVlCE (NTIS). 352 

ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. QOVERNMENT PRINTING OFFICE, 

15. PRICE 

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ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINQFIELD.VA 22161. 

ELECTRONIC FORM 

*U.S. GOVERNMENT PRINTING OFFICE: 1990 775-260/25098 
CONTENTS Page 

PREFACE ............................................................... 

TOPICAL INDEX TO ALL PAPERS ............................................ 

AI3sTIacr .............................................................. 

A. INTRODUCIlON 

A.1 OVERVIEW .......................................................... 

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

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

A.4 COMMENTS ON SUPPORTING PAPERS .................................... 

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

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

A.7 SUPPLEMENTARY READING LIST ........................................ 

B. INTRODUCTORY AND TUTORIAL PAPERS 

. . . 

. . . . . . . 111 

. . . . . . . . x 

. . . . . TN-l 

. . . . TN-l 

. . . . TN-2 

. . . TN-3 

. . . . TN-3 

. . . . TN-S 

. . . . TN-9 

. . . TN-13 

B.l Properties of Signal Sources and Measurement Methods 

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

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

I. 

II. 

III. 

IV. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

XI. 

XII. 

XIII. 

The Sine Wave and Stability ......................................... 

Measurement Methods Comparison .................................... 

Characterization .................................................. 

Analysis of Time Domain Data ....................................... 

Confidence of the Estimate and Overlapping Samples ....................... 

Maximal Use of the Data and Determination of the Degrees of Freedom ......... 

Example of Time Domain Signal Processing and Analysis ..................... 

Spectrum Analysis ................................................ 

Power-Law Noise Processes .......................................... 

Pitfalls in Digitizing the Data ........................................ 

Translation From Frequency Domain Stability Measurement to Time Domain Stability 

Measurement and Vice-Versa ...................................... 

Causes of Noise Properties in a Signal Source ............................. 

Conclusion ..................................................... 

. . TN-14 

. . TN-20 

. TN-22 

. . TN-23 

. . TN-26 

. . TN-28 

. . TN-30 

. . TN-32 

. . TN-38 

. . TN-39 

. TN-48 

. . TN-50 

. TN-55 

B.2 Frequency and Time - Their Measurement and Characterization 

Samuel R. Stein 

Ptvcision 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 
12.2 Direct Digital Measurement ................................................ TN-g7 

12.2.1 TimeInterval Measurements .............................................. TN-87 

12.2.2 Frequency Measurements ................................................ TN-89 

12.23 PeriodMeasurements ................................................... TN-89 

12.3 Sensitivity-Enhancement Methods ............................................ TN-99 

123.1 Heterodyne Techniques .................................................. TN-90 

123.2 Homodyne Techniques .................................................. ‘IN-91 

123.3 Multiple Conversion Methods ............................................. TN-97 

12.4 Conclusion ........................................................... TN-101 

B.3 Time and Frequency (Time-Domain) Characterization, Estimation, and Prediction of Precision 

Clocks and Oscillators 

David W. Allan 

IEEE Transactions on Ultmsonics, Fenvelecttics, and Frequency Control, 1987 , . . . . . . , . . . . . . . . . . . . TN-121 

1. Introduction .......................................................... TN-121 

2. Systematic Models for Clocks and Oscillators ................................... TN-121 

3. Random Models for Clocks and Oscillators ..................................... TN-123 

4. TimeDomain 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, C.M. 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-139 

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

V. Conclusion ........................................................... TN-137 

C. PAPERS ON STANDARDS AND DEFINITIONS 

C.l 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. 

2. 

3. 

4. 

5. 

Introduction .......................................................... TN-139 

Measures of Frequency and Phase Instability ................................... TN-139 

Characterization of Frequency and Phase Instabilities .............................. ‘IN-139 

Confidence Limits of Measurements .......................................... TN-149 

Recommendations for Characterizing or Reporting Measurements of Frequency and 

Phase Instabilities ..................................................... TN-141 

vi 
Page 

C.2 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 Gernot M.R. Winkler 

IEEE Tmsactions on Instmmentation and Measmment, 1971 . . . . . . . . . . , . . . , . . . . . . . . . . . . . . . . TN-146 

I. 

II. 

III. 

IV. 

V. 

VI. 

VII. 

VIII. 

Introduction .......................................................... TN-147 

Statement of the Problem ................................................. TN-148 

Background and Definitions ............................................... TN-149 

Definition of Measures of Frequency Stability ................................... TN-149 

Translations Among Frequency Stability Measures ................................ TN-151 

Applications of Stability Measures ........................................... TN-153 

Measurement Techniques for Frequency Stability ................................. TN-154 

Conclusions.. ......................................................... TN-lS7 

C3 Characterization of Frequency and Phase Noise 

Comitt Consultatif International Des Radiocommunications (CCIR) 

Report 580 of the CUR, 1986 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TN-162 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

Introduction ........................................................ 

Fourier Frequency Domain .............................................. 

TimeDomain ....................................................... 

Conversion Between Frequency and Time Domains ............................. 

Measurement Techniques ............................................... 

Confidence Limits of Time Domain Measurements ............................. 

Conclusion ......................................................... 

TN-162 

TN-163 

TN-163 

TN-164 

TN-166 

TN-166 

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. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

Introduction ................................................ 

Definitions: Model of Frequency Fluctuations ......................... 

Noise Processes in Frequency Generators ............................ 

Measurements in the Frequency Domain ............................ 

Measurements in the Time Domain ................................ 

Characterization of Frequency Stability in the Time Domain ............... 

Characterization of Frequency Stability via Filtering or Frequency Noise ...... 

Spectral Analysis Inferred From Time Domain Measurements ............. 

Structure Functions of Oscillator Fractional Phase and Frequency Fluctuations . . 

Power Spectral Density of Stable Frequency Sources .................... 

Conclusion ................................................. 

......... TN-171 

......... TN-171 

......... TN-173 

......... TN-174 

......... TN-177 

......... TN-178 

......... TN-182 

......... TN-183 

......... TN-184 

......... TN-185 

......... TN-187 

Vii 
D.2 Phase Noise and AM Noise Measurements in the Frequency Domain 

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

Infrared and Millimeter Waves, Vol. II, 1984 . . . . . . . . . . . . . . . . . , . . . . . . . . . , . . . . . . , . . . . . . . TN-198 

I. 

II. 

III. 

IV. 

V. 

Introduction .......................................................... TN-198 

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-282 

F. Integrated Phase Noise ............................................... TN-283 

G. AM Noise in the Frequency Domain ...................................... TN-285 

PhaseNoise Measurements Using the Two-Oscillator Technique ...................... TN-286 

A. Two Noisy Oscillators ................................................ TN-288 

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

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

Single-Oscillator PhaseNoise Measurement Systems and Techniques ................... TN-218 

A. The Delay Line as an FM D kriminator ................................... 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 PhaseNoise Measurements ................................ TN-234 

Conclusion ........................................................... TN-238 

D3 Performance of an Automated High Accuracy Phase Measurement System 

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

36th Annual Symposium on Frequency Con&, 1982 Pmceedings ............................. 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. Walls, 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 

. . . 

Vlll 
Page 

D.5 A Modified “Allan Variance” with Increased Oscillator Characterization Ability 

David W. Allan and James A. Barnes 

35th Annual Symposium on Frequency Con&, 1981 Pnxeedings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Th6ophane Ayi 

IEEE Tmnsactions on Instrumentation and Measumnent, 1984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ‘lW259 

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-Z% 

1. Introduction .......................................................... TN-297 

2. The AUan Variance ..................................................... TN-298 

3. The Bias Function Bt(N,r,l) ............................................... TN-299 

4. The Bias Function B#) ................................................. TN-300 

5. The Bias Function Bs(N,M,rp) ............................................. TN-300 

6. TheBias 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 
TOPICAL INDEX TO ALL PAPERS 

The foilowing index is designed for quick location of substantial discussion of a particular topic. The page numbers are 

those where the topic is fist cited in the reference. 

Aliasing ............... 

Ailan Variance ............ 

Amplitude Modulation Noise ........ 

Bias Functions ............ 

Clock-Time Prediction .......... 

Common Measurement Problems ...... 

Components Other than Oscillators (Methods) . . 

Confidence of the Estimate ........ 

Conversion Between Time and Frequency Domain . 

Correlation Method ........... 

Dead Time .............. 

Degrees of Freedom ........... 

Digitization of Data ........... 

Direct Measurements 

o Measurements at the Fundamental Frequency . 

o Measurements after Multiplication/Division . . 

Drift ............... 

Efficient Use of Data .......... 

Filter Method ............. 

Frequency Synthesis ........... 

Heterodyne or Beat-Frequenq Methods .... 

o Single-Conversion Methods ....... 

o MultipleConversion Methods ....... 

o Time-Difference (Time-Interval-Counter) ... 

- Dual-Mixer, TimeDifference Method ... 

Homodyne Measurements 

o PhaseLock-Loop Methods ........ 

o Discriminator Methods ......... 

- Cavity-Discriminator Method ...... 

- Delay-Line Method ......... 

Least Squares (Regression) ........ 

Modified Allan Variance ......... 

Overlapping Samples .......... 

Pickett-Fence Effect ........... 

Power Law Spectra (Random Noise) ..... 

Quantization Uncertainty ......... 

Reference-PhaseModulation Technique .... 

Separating Oscillator and Reference Variances . . 

Spectral Density ............ 

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

T-N-260, TN-298 

TN-194, TN-205, TN-235 

TN-162, TN-164, TN-299 

TN-126, TN-140 

TN-156 

TN-37, TN-131, TN-176 

TN-26, TN-80, TN-140, TN-156, TN-166, TN-180, TN-262 

TN-48, TN-73, TN-142, TN-151, TN-164 

TN-132, TN-176, TN-216 

TN-140, TN-164, TN-180, TN-296 

TN-28, TN-83 

TN-39 

TN-15 TN-87 

TN-101, TN-135 

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

TN-28, TN-83 

TN-182 

TN-97 

TN-17, TN-90, TN-154, TN-177 

TN-17, TN-90, TN-154, TN-177 

TN-97 

TN-20, TN-87, TN-177 

TN-17, TN-99, TN-177, TN-241 

TN-18 TN-34, TN-93, TN-130, TN-176, TN-209 

TN-92, TN-133, TN-174, TN-218 

TN-133, TN-174, TN-219 

TN-92, TN-134, TN-176, TN-219 

TN-31, TN-77, TN-265 

TN-75, TN-143, TN-164, TN-255, TN-260 

TN-26, TN-80, TN-140 

TN-46 

TN-38, TN-73, TN-123, TN-141, TN-167, TN-173, TN-200 

TN-40, TN-89 

TN-133 

TN-86 

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

TN-194 

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

Window Function (Leakage) . . . . . . . . TN-44 

TN-40 

X