Handbook of Frequency Stability Analysis

1 Handbook of FrequencyStability Analysis RileyNIST Special Publication 1065 he 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 industry's competitiveness; advance science and engineering; and improve public health, safety, and the environment. One of the agency's basic functions is to develop, maintain, and retain custody of the national standards of measurement, and provide the means and methods for comparing standards used in science, engineering, manufacturing, commerce, industry, and education with the standards adopted or recognized by the Federal Government.

2 As an agency of the Commerce Department's Technology Administration, NIST conducts basic and applied research in the physical sciences and engineering, and develops measurement techniques, test methods, standards, and related services. The Institute does generic and precompetitive work on new and advanced technologies. NIST's research facilities are located at Gaithersburg, MD 20899, and at Boulder, CO 80305. Major technical operating units and their principal activities are listed below. For more information visit the NIST Website at , or contact the Publications and Program Inquiries Desk, 301-975-3058. Office of the Director National Quality Program International and Academic Affairs Technology Services Standards Services Technology Partnerships Measurement Services Information Services Weights and Measures Advanced Technology Program Economic Assessment Information Technology and Applications Chemistry and Life Sciences Electronics and Photonics Technology Manufacturing Extension Partnership Program Regional Programs National Programs Program Development Electronics and Electrical Engineering Laboratory Microelectronics Law Enforcement Standards Electricity Semiconductor Electronics Radio- Frequency Technology1 Electromagnetic Technology1 Optoelectronics1 Magnetic Technology1 Materials Science and Engineering Laboratory Intelligent Processing of Materials Ceramics Materials Reliability1 Polymers Metallurgy NIST Center for Neutron Research Chemical Science and Technology

3 Laboratory Biotechnology Process Measurements Surface and Microanalysis Science Physical and Chemical Properties2 Analytical Chemistry Physics Laboratory Electron and Optical Physics Atomic Physics Optical Technology Ionizing Radiation Time and Frequency1 Quantum Physics1 Manufacturing Engineering Laboratory Precision Engineering Manufacturing Metrology Intelligent Systems Fabrication Technology Manufacturing Systems Integration Building and Fire Research Laboratory Applied Economics Materials and Construction Research Building Environment Fire Research Information Technology Laboratory Mathematical and Computational Sciences2 Advanced Network Technologies Computer Security Information Access Convergent Information Systems Information Services and Computing Software Diagnostics and Conformance testing Statistical Engineering _____1At Boulder, CO 80305 2 Some elements at Boulder, CO T NIST Special Publication 1065 Handbook of Frequency Stability Analysis Riley Under contract with: Time and Frequency Division Physics Laboratory National Institute of Standards and Technology 325 Broadway Boulder, CO 80305 July 2008 Department of Commerce Carlos M.

4 Gutierrez, Secretary National Institute of Standards and Technology James M. 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: Phone: (202) 512-1800 Fax: (202) 512-2250 Mail: Stop SSOP, Washington, DC 20402-0001 iii Dedication This Handbook is dedicated to the memory of Dr.

5 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 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.

6 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 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: iv Preface I have had the great privilege of working in the time and Frequency field over the span of my career.

7 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 Barnes and Allan in establishing the fundamentals at NBS, and Howe in carrying on that tradition today at NIST.

8 Together with such people as Weiss and 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 . v Contents 1 INTRODUCTION .. 1 2 Frequency Stability Analysis .. 2 Background .. 2 3 DEFINITIONS AND TERMINOLOGY .. 5 Noise Model .. 5 Power Law Noise .. 5 Stability Measures .. 6 Differenced and Integrated Noise .. 6 Glossary .. 7 4 8 5 TIME DOMAIN Stability .. 9 SIGMA-TAU PLOTS .. 9 VARIANCES .. 10 Standard Variance .. 13 Allan Variance .. 14 Overlapping Samples .. 15 Overlapping Allan Variance .. 16 Modified Allan Variance .. 17 Time Variance.

9 18 Time Error Prediction .. 19 Hadamard Variance .. 20 Overlapping Hadamard Variance .. 20 Modified Hadamard Variance .. 21 Total Variance .. 23 Modified Total Variance .. 25 Time Total Variance .. 26 Hadamard Total Variance .. 26 Th o1 .. 29 Th oH .. 31 MTIE .. 33 TIE rms .. 34 Integrated Phase Jitter and Residual FM .. 34 Dynamic Stability .. 36 CONFIDENCE INTERVALS .. 37 Simple Confidence Intervals .. 37 Chi-Squared Confidence Intervals .. 38 DEGREES OF FREEDOM .. 38 AVAR, MVAR, TVAR, and HVAR EDF .. 38 TOTVAR EDF .. 41 MTOT EDF .. 41 Th o1 / Th oH EDF .. 42 NOISE IDENTIFICATION .. 43 Power Law Noise Identification .. 43 Noise Identification Using B1 and R(n) .. 44 The Autocorrelation Function .. 44 The Lag 1 Autocorrelation .. 45 Noise Identification Using r1 .. 45 Noise ID Algorithm .. 46 BIAS FUNCTIONS .. 48 B1 BIAS FUNCTION .. 48 B2 BIAS FUNCTION.

10 48 B3 BIAS FUNCTION .. 49 R(N) BIAS FUNCTION .. 49 TOTVAR BIAS FUNCTION .. 49 vi MTOT BIAS FUNCTION .. 49 TH O1 BIAS .. 50 TH OH BIAS .. 50 DEAD TIME .. 51 UNEVENLY SPACED DATA .. 54 55 Frequency OFFSET .. 56 Frequency DRIFT .. 57 DRIFT Analysis METHODS .. 57 PHASE DRIFT Analysis .. 57 Frequency DRIFT Analysis .. 58 ALL TAU .. 59 ENVIRONMENTAL SENSITIVITY .. 60 61 TRANSFER FUNCTIONS .. 64 6 Frequency DOMAIN Stability .. 67 NOISE SPECTRA .. 67 POWER SPECTRAL DENSITIES .. 68 PHASE NOISE PLOT .. 69 SPECTRAL Analysis .. 69 PSD WINDOWING .. 70 PSD AVERAGING .. 70 MULTITAPER PSD Analysis .. 70 PSD 71 7 DOMAIN CONVERSIONS .. 73 POWER LAW DOMAIN CONVERSIONS .. 73 EXAMPLE OF DOMAIN CONVERSIONS .. 74 8 NOISE SIMULATION .. 77 WHITE NOISE GENERATION .. 78 FLICKER NOISE GENERATION .. 78 FLICKER WALK AND RANDOM RUN NOISE GENERATION .. 78 Frequency OFFSET, DRIFT, AND SINUSOIDAL COMPONENTS.