1. Introduction to wavelets 2. Review of Fourier theory and filters 3. Orthonormal transforms of time series 4. The discrete wavelet transform 5. The maximal overlap discrete wavelet transform 6. The discrete wavelet packet transform 7. Random variables and stochastic processes 8. The wavelet variance 9. Analysis and synthesis of long memory processes 10. Wavelet-based signal estimation 11. Wavelet analysis of finite energy signals Appendix. Answers to embedded exercises References Author index Subject index.

/pdf/wavelet-methods-for-time-series-analysis-1m2hg2nm53.pdf

Wavelet Methods for Time Series Analysis

Atomic clocks have been instrumental in science and technology, leading to innovations such as global positioning, advanced communications, and tests of fundamental constant variation. Timekeeping precision at 1 part in 10 18 enables new timing applications in relativistic geodesy, enhanced Earth- and space-based navigation and telescopy, and new tests of physics beyond the standard model. Here, we describe the development and operation of two optical lattice clocks, both using spin-polarized, ultracold atomic ytterbium. A measurement comparing these systems demonstrates an unprecedented atomic clock instability of 1.6 × 10 –18 after only 7 hours of averaging.

/pdf/an-atomic-clock-with-10-18-instability-1b0wxe31pz.pdf

An atomic clock with $10^{-18}$ instability

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 U.S. 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. As an agency of the U.S. 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 http://www.nist.gov, or contact the Publications and Program Inquiries Desk, 301-975-3058.

/pdf/handbook-of-frequency-stability-analysis-548iski2e2.pdf

Handbook of frequency stability analysis

For the past 50 years, atomic standards based on the frequency of the cesium ground-state hyperfine transition have been the most accurate time pieces in the world. We now report a comparison between the cesium fountain standard NIST-F1, which has been evaluated with an inaccuracy of about $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$, and an optical frequency standard based on an ultraviolet transition in a single, laser-cooled mercury ion for which the fractional systematic frequency uncertainty was below $7.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}$. The absolute frequency of the transition was measured versus cesium to be 1 064 721 609 899 144.94 (97) Hz, with a statistically limited total fractional uncertainty of $9.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$, the most accurate absolute measurement of an optical frequency to date.

/pdf/single-atom-optical-clock-with-high-accuracy-24r5pinhss.pdf

Single-Atom Optical Clock with High Accuracy

Atomic frequency standards using trapped ions or cold atoms work intrinsically in a pulsed mode. Theoretically and experimentally, this mode of operation has been shown to lead to a degradation of the frequency stability due to the frequency noise of the interrogation oscillator. In this paper a physical analysis of this effect has been made by evaluating the response of a two-level atom to the interrogation oscillator phase noise in Ramsey and multi-Rabi interrogation schemes using a standard quantum mechanical approach. This response is then used to calculate the degradation of the frequency stability of a pulsed atomic frequency standard such as an atomic fountain or an ion trap standard. Comparison is made to an experimental evaluation of this effect in the LPTF Cs fountain frequency standard, showing excellent agreement.

Frequency stability degradation of an oscillator slaved to a periodically interrogated atomic resonator

Total variance is a statistical tool developed for improved estimates of frequency stability at averaging times up to one-half the test duration. As a descriptive statistic, total variance performs an exact decomposition of the sample variance of the frequency residuals into components associated with increasing averaging times. As an estimator of Allan variance, total variance has greater equivalent degrees of freedom and lesser mean square error than the standard unbiased estimator has.

/pdf/total-variance-an-estimator-of-long-term-frequency-stability-ut6bslaiqi.pdf

Total variance, an estimator of long-term frequency stability [standards]

The phase of a frequency standard that uses periodic interrogation and control of a local oscillator (LO) is degraded by a long-term random-walk component induced by downconversion of LO noise into the loop passband. The Dick formula for the noise level of this degradation is derived from an explicit solution of an LO control-loop model.

A derivation of the long-term degradation of a pulsed atomic frequency standard from a control-loop model

This paper reviews the author's previous work on free-running timescales based on Kalman filters that act upon clock comparisons. The natural Kalman clock ensemble algorithm tends to optimize long-term timescale stability at the expense of short-term stability. By subjecting each postmeasurement error covariance matrix to a non-transparent reduction operation, one obtains corrected clocks with improved short-term stability and little sacrifice of long-term stability. A new result on covariance matrix reduction is also stated.

A review of reduced kalman filters for clock ensembles

We give results of recent work on a newly developed frequency stability characterization, called Total variance, whose main advantages are improved confidence at and near the longest averaging time of half the data duration, and lower sensitivity to drift removal.

Total Variance: A Progress Report on a New Frequency Stability Characterization

An interval timer is used in a single-mixer frequency-stability measurement system in place of an event timer. The dead-time problem is avoided with the aid of a reference pulse train and an algorithm for ambiguity resolution. The noise floor test and the unfolding algorithm are described. Advantages of the technique are high precision, convenient interfacing, and low cost. A disadvantage is the vulnerability of the technique to missing data. >

C.A. Greenhall

Papers

Total variance, an estimator of long-term frequency stability [standards]

A derivation of the long-term degradation of a pulsed atomic frequency standard from a control-loop model

A review of reduced kalman filters for clock ensembles

Total Variance: A Progress Report on a New Frequency Stability Characterization

A method for using a time interval counter to measure frequency stability