NATL INST OF STAND & TECH A111D7 DMIfia? NIST Special Publication 1065 Handbook of Frequency Stability Analysis W.J. Riley Nisr National Institute of Standards and Technology • U.S. Department of Commerce rhe National Institute of Standards and Technology was established in 1988 by Congress to "assist industry in the development of technology ... needed to improve product quality, to modernize manufacturing processes, to ensure product reliability ... and to facilitate rapid commercialization ... of products based on new scientific discoveries." NIST, originally founded as the National Institute of Standards in 1901, works to strengthen U.S. industry's competitiveness; advance science and engineering; and improve public health, safety, and the environment. 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Turner Deputy Director Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the entities, materials, or equipment are necessarily the best available for the purpose. National Institute of Standards and Technology Special Publication 1065 Natl. Inst. Stand. Technol. Spec. Publ. 1065, 136 pages (July 2008) CODEN: NSPUE2 U. S. GOVERNMENT PRINTING OFFICE Washington: 2008 For sale by the Superintendent of Documents, U. S. Government Printing office Internet: bookstore.gpo.gov - Phone: (202) 512-1800 - Fax: (202) 512-2250 Mail: Stop SSOP, Washington, DC 20402-0001 Dedication This handbook is dedicated to the memory of Dr. James A. Barnes (1933-2002), a pioneer in the statistics of frequency standards. James A. Barnes was born in 1933 in Denver, Colorado. He received a Bachelor's degree in engineering physics from the University of Colorado, a Masters degree from Stanford University, and in 1966 a Ph.D. in physics from the University of Colorado. Jim also received an MBA from the University of Denver. After graduating from Stanford, Jim joined the National Institute of Standards, now the National Institute of Standards and Technology (NIST). Jim was the first Chief of the Time and Frequency Division when it was created in 1967 and set the direction for this division in his 15 years of leadership. During his tenure at NIST Jim made many significant contributions to the development of statistical tools for clocks and frequency standards. Also, three primary frequency standards (NBS 4, 5, and 6) were developed under his leadership. While he was division chief, closed-captioning was developed (which received an Emmy award) and the speed of light was measured. Jim received the NBS Silver Medal in 1965 and the Gold Medal in 1975. In 1992, Jim received the I.I. Rabi Award from the IEEE Frequency Control Symposium "for contributions and leadership in the development of the statistical theory, simulation and practical understanding of clock noise, and the application of this understanding to the characterization of precision oscillators and atomic clocks." In 1995, he received the Distinguished PTTI Service Award. Jim was a Fellow of the IEEE. After retiring from NIST in 1982, Jim worked for Austron. Jim Barnes died Sunday, January 13, 2002, in Boulder, Colorado after a long struggle with Parkinson's disease. He was survived by a brother, three children, and two grandchildren. Note: This biography is published with permission and taken from his memoriam on the UFFC web site at: http://www.ieee-uffc.org/fcmain.asp?page=barnes. Ill Preface I have had the great privilege of working in the time and frequency field over the span of my career. I have seen atomic frequency standards shrink from racks of equipment to chip scale, and be manufactured by the tens of thousands, while primary standards and the time dissemination networks that support them have improved by several orders of magnitude. During the same period, significant advances have been made in our ability to measure and analyze the performance of those devices. This Handbook summarizes the techniques of frequency stability analysis, bringing together material that I hope will be useful to the scientists and engineers working in this field. I acknowledge the contributions of many colleagues in the Time and Frequency community who have contributed the analytical tools that are so vital to this field. In particular, I wish to recognize the seminal work of J. A. Barnes and D.W. Allan in establishing the fundamentals at NBS, and D.A. Howe in carrying on that tradition today at NIST. Together with such people as M.A. Weiss and C.A. Greenhall, the techniques of frequency stability analysis have advanced greatly during the last 45 years, supporting the orders-of-magnitude progress made on frequency standards and time dissemination. I especially thank David Howe and the other members of the NIST Time and Frequency Division for their support, encouragement, and review of this Handbook. iv Contents 1 INTRODUCTION 1 2 FREQUENCY STABILITY ANALYSIS 2 2. 1 Background 2 3 DEFINITIONS AND TERMINOLOGY 5 3.1. Noise Model 5 3.2. Power Law Noise J 3.3. Stability Measures 6 3.4. Differenced and Integrated Noise 6 3.5. Glossary 7 4 STANDARDS 8 5 TIME DOMAIN STABILITY 9 5.1. Sigma-Tau Plots 9 5.2 Variances 10 5.2.1. Standard Variance 13 5.2.2. Allan Variance 14 5. 2. 2. Overlapping Samples 15 5.2.4. Overlapping Allan Variance 16 5. 2. 5 Modified Allan Variance 17 5.2.6. Time Variance 18 5.2. 7. Time Error Prediction 19 5.2.8. Hadamard Variance 20 5.2.9. Overlapping Hadamard Variance 20 5.2. 10. Modified Hadamard I 'ariance 21 5.2. 1 1. Total Variance 23 5.2. 12. Modified Total Variance 25 5.2. 13. Time Total Variance 26 5.2. 1 4. Hadamard Total Variance 26 5.2.15. Theol 29 5.2.16 TheoH 31 5.2.17 MTIE 33 5.2.18. TlErms 34 5.2.19. Integrated Phase Jitter and Residual FM 34 5.2.20. Dynamic Stability 36 5.3. Confidence 1>4TERVALS 37 5.3. 1. Simple Confidence Intervals 37 5.3.2 Chi-Squared Confidence Intervals 38 5.4. Degrees OF Freedom 38 5. 4. 1. A VAR. MVAR. TVAR. and HVAR EDF 38 5.4.2. TOTVAREDF 41 5.4.3. MTOTEDF 41 5.4.4. Thiol / TheoH EDF 42 5.5. Noise Identification 43 5.5. 1. Power Law Noise Identification 43 5.5.2. Noise Identification Using B, and R(n) 44 5.5.3. The Autocorrelation Function 44 5.5.4. The Lag I Autocorrelation 45 5.5.5. Noise Identification Using ri 45 5.5.6. Noise ID A Igorithm 46 5.6.