CHARACTERIZATION TUTORIAL

David W. Allan and Frequency Division National Bureau of Standards Boulder, Colorado 30303

is a very simplistic picture or nominal model of ABSTRACT most. precision oscillators. The random fluctuations or deviations in precision Managers ar9 often required to make key program oscillators can often be characterized by power decisions based on the performance of some law spectra. In other words, if the time elements of a large system. This paper is residuals are examined, after removing the intended to assist the manager in this important systematic effects, one or more of the power law task in so far as it relates to the proper use of spectra shown in Figure 2 are typically observed. precise and accurate . An intuitive The meaning of power law spectra is that if a approach will be used to show how a clock’s Fourier analysis or spectral density analysis is stability is measured, why it is measured the way proportional to fB; B designates the particular it is, and why it is described the way it is. 4n power law process (8 = 0, -l,-2,-3,-4 and u = intuitive explanation of the meaning of time 2nf). The first process shown in Figure 2 is domain and frequency domain measures as well as called phase modulation (PM)., The why they are used will also be given. noise is typically observed in the short term fluctuations, for example, of an active hydrogen Explanations of when an “Allan ” plot maser for sample of from one second to should be used and when it should not be used about 100 seconds. This noise is also observed will also be given. The relationship of the rms In quartz crystal oscillators for sample times in time error of a clock to a CI~(T) diagram will the vicinity of a millisecond and shorter. The also be given. The environmental sensitivities PM, f-l, is the second line in of a clock are often the most important effects Figure 2. This kind of noise is often found for determining its performance. Typical environ- sample times of one millisecond to one second in mental parameters of concern and nomj.nsl quartz crystal oscillators. The f-* or random sensitivity values for commonly used clocks will walk PM indicated by the third line is what is be reviewed. observed for the time deviations of rubidium, cesium, or hydrogen frequency standards. If the SYSTEMATIC AND RANDOM DEVIATIONS IN CLOCKS first difference is taken of a series of discrete time readings from the third line, then the This paper is tutorial in nature with a minimum result is proportional to the frequency of mathematics -- the goal being to characterize deviations, which will be an f” process or a clock behavior. First, time deviations or white noise frequency modulation (FM) process. frequency deviations in clocks may be categorized In other words the time and the frequency are into two types: systematic deviations and random related through a derivative or an integral deviations. The systematic deviations come in a depending upon which way one goes. The variety of forms. Typical examples are frequency derivative of the time deviations yields the sidebands, diurnal or annual variations in a frequency deviations, and the integral of the clock’s behavior, time offset, frequency offset frequency deviations yields the time deviations. and frequency drift. Figure 1 illustrates some So, random walk time deviations result from white of these. noise Ft4. In general, the spectral density of the frequency fluctuations is w* times the If a clock has a frequency offset, the time spectral density of the time flu tuations. The deviations will appear as a ramp function. On fourth line in Figure 2 is an f-s process. If the other hand, if a clock has a frequency drift, this were representative of the time fluctdations then the resulting time deviations will appear as then the frequency would be an f-’ or a flicker a quadratic time function --the time deviations noise FM process. This process is typical of the will be proportional to the square of the running output of a quartz for sample time. There are many other systematic effects times longer than one second or the output of that are very important to consider in under- rubidium or cesium standards in the long term (on standing a clock’s characteristics and Figure 1 the order of a few hours, few days, or few weeks

249 depending upon which standard). We find thatbin the average over all time. In practice, finite very long term, most atomic clocks have an f data sets yield rapidly converging estimates. type behavior for the time fluctuations -- making The square root of the Allan variance is denoted the frequency fluctuations an fs2 process or oyw; Of is an efficient estimator of the random walk FM. These five power law processes power law spectra model for the data. MOW 0 (T) are very typical and one or more of them are changes with T indicates the exponent for thg appropriate models for essentially every power law process. In fact, in the case of precision oscillator. Characterizing the kind of random walk FM, Ok is statistically the most power-law process thus becomes an important part efficient estimator of this power law process. If of characterizing t’ne performance of a. clock (1). power law processes are good models for a clocks Once a clock has been characterized in terms of random fluctuations, which they typically are, its systematic and its random characteristics then the Allan variance analysis is faster and then a time deviation model can be developed. A more accurate in estimating the process than the very simple and useful model that is commonly Fast Fourier Transform. .Some virtues of the used is given by the following equation (2): Allan variance are: It is theoretically and straightforwardly relatable to the power law x(t) = x0 + y;t + 1/2D.t2 + e(t) (1) spectral type (6 =-p-3, -2

1 there Is a bandwidth employed in order to develop optimum prediction dependence (4). A software trick can be employed algorithms (3). to effectively vary the bandwidth rather than doing it with the hardware. Rather than THE CONCEPT OF AN ALLAN VARIANCE calculating Oy(T) from individual phase points the phase can be averaged over an interval T. Figure 4 illustrates a simulated random walk PM Hence the x,, x2, and x3 from Figure 4 become process. Suppose this process is the time phase or time difference av?rages. 4s T difference between two clocks or the time of a increases the effective measurement system clock with repsect to a perfect clock. 4gain bandwidth decreases. This technique removes the this process is typical of the time deviations ambiguity problem. We have called this the for rubidium, cesium, and passive hydrogen modified Allan variance or modified o (T) clocks. Choose a sample time ?, as indicated, analysis technique. Figure 6 shows tK e U, a and note the three time deviation readings (x,, mapping for the modified Allan variance. For x2’ and x ) indiaated by the circles and spaced white noise PM, u is equal to -3, and for flicker by the in2 erval T. The frequency deviation y, is noise PM, 11 is equal to -2. proportional to the slope between x, and x2 (y, = (x2 - X,)/T). Similarly y2 is proportional to the TIME PREDICTION ERROR OF A CLOCK slope between x2 and x3 (~2 = (x3 - x2)/~). The difference in slope, Ay, is a measure of the Another -concept which has become useful is the frequency change from the first T sample interval computation of the time error of prediction. In to the next adjacent T sample interval. With a the case of white noise FM and random walk noise fixed value of ‘I, imagine averaging through the FM, ‘I Ok is the optimum time error of entire data set for all possible readings of x,, predictlon. For white noise PM the optimum value x2’ and x3 displaced by T each yielding a By. achievable is T o,(r)/d3, and for flicker noise The average squared value of Ay divided by 2 is FM it iS T O,(T)/dln 2. called the “Allan variance”. In theory, it is

250 kpplying some of the above concepts to state of mental conditions will allow the manager to know the art frequency standards yields the frequency the best clock or clocks to buy and the best way stability plot shown in Figure 7 (51, and the to implement them. For example, a rubidium clock corresponding RMS time prediction error plot coupled to a GPS receiver (used in the com- shown in Figure 8. The RMS time prediction error mon-view mode with UTC(USN0 MC) or with UTC(NBS) plot is bassd on reference (2) and is a function would have better short term and better long term of the levels of noise in the clock and also the stability than any commercial cesium clock uncertainties associated with determining the available. The stability would be somewhat worse systematic deviations due to a finite data in the vicinity of T equal to one day. In length. practice, there are some problems with this idea, but it illustrates the point. ENVIRONMENTAL EFFORTS ON CLOCKS Lagtly, Figures 9 through 13 show nominal values From a management point of view, the characteris- for some important clock coefficients that tics of the various clocks should be related to managers and design engineers need to properly the needs of a particular program. It is assess when evaluating which clock or clocks will important to keep in mind that, in practice, best serve their needs. These are only nominal systematic and environmental effects often are values and there will be exceptions. A band of the predominant influence on the time and values is listed for these coefficients ranging frequency deviations of a clock. The reliability from nominal best performance available from of a clock is often a basic issue, and the laboratory-type standards through the range of manager should assure himself that adequate typical values observed and specified for reliability has been documented. The manager commercially available standards. also needs to ask the following questions in each application that he may have. What are the ACKNOWLEDGEMENTS environmental conditions, the rapidity of the temperature changes, the magnetic field The author wishes to express appreciation to Mr. conditions, the shock and vibration conditions, Roger Beehler and Dr. Richard Jones for their and the humidity conditions? How do these assistance in proofing t’ne text and to Dr. Lindon conditions affect the clock’s performance? All Lewis and Dr. Fred Walls for assistance with the clocks are affected at some level by changes in data in Figures 9-l 3. the above environmental parameters plus some others as well. Some clocks are affected by REFERENCES barometric pressure. Vibration can be extremely important. In some clocks the servos will (1 ) Barnes, J. A., Chi, 4.) Cutler, L. L., unlock, for example, if a 1 kHz vibration is Healey, D. J., Lesson, D. B., McCunigal, T., present. (6) Some clocks are accoustically Mullen, ,J. Jr., Smith, W.. , Sydnor, Q., sensitive. What is t’ne gravitational or g Vessot, R., and Winkler, G. M. “Characteri- sensitivity? What are the cost, size, weight and zation of Frequency Stability,” Proc. IEEE power requirements? Line voltage power fluctua- Trans. on Instrumental and Measurement tions can affect clocks. Changes in the dc power IM-20, 1971 p. 105. NBS Tech. Note 394 can affect some clocks. We have found that a 1971. good clock environment .can improve clock performance considerably, and we have provided a (2) Allan, D. W. and Hellwig, H., “Time highly controlled environment for the NBS clock Deviation and Time Prediction Error for ensemble to improve the performance over that Clock Specification, Characterization and obtained in typical laboratory environments. Application,” IEEE Position Location and Another very important question to ask is what is Navigation Symposium Proceedings, 1978 pp. a clock’s lifetime? Redundancy and/or multiple 29-35. clocks are sometimes necessary to overcome lifetime and reliability problems. It is (3) Box, G. E. P. and Jenkins, G. M., Time important to take a system’s approach in SeriesAnalysis Forcasting and Control, establishing the best clock(s) and clock(s) Holden-Day, 1970, Catalog No. 77-79534. configuration. In some cases the program needs are for synchroniztion to UTC, in other cases the (4) Vessot, R., Mueller, L., and Vanier, J., needs are for syntonization, i.e. the frequencies “The Specification of Oscillator Character- within a system need to agree. 9ften the need is istics from Measurements Made in tine for time or frequency self consistency within a Frequency Domain,” Proc. of the IEEE, Vol. program, e.g. GPS requires time consistency. It 54, 1966 pp. 199-207. seems many people are buying cesium standards as a panacea, when in fact they may not be solving (5) Winkler, C. Y. R., “Improved Master Clock the problem at hand. Buying a cesium standard Reference System at USNO,” PTTI 1983, to be does not guarantee synchronization. However, it published. does guarantee syntonization within some accuracy. All clocks will diverge and eventually (6) Walls, F. L., “Vibration and Acceleration depart from synchronization tolerance. It’s just Induced Timing Errors of Clocks and Clock a matter of time! Knowing a clock’s characteris- Systems, IT 36th Annual Symposium on Frequency tics, the system requirements, and the environ- Control , 1982 p. 37.

25 1 Figure 1. Frequency, y(t), and time, x(t) deviation due to frequency offset and to frequency drift in a clock.

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Figure 3. Simulated power law noise processes with their optimum predicted estimates.

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