1、Rec. ITU-R TF.538-3 81 Characterization of sources and time scales formation RECOMMENDATION ITU-R TF.538-3 MEASURES FOR RANDOM INSTABILITIES IN FREQUENCY AND TIME(PHASE) (Question I7J-R 104/7) (1978-1990-1992-1994) The IT Radiocommunication Assembly, considering a) that there is a need for an adequa
2、te language with which to communicate the instability characteristics of standard frequency and time sources and measurement systems; b) that the classical variance does not converge for some of the kinds of random time and frequency instabilities; c) that major laboratories, observatories, industri
3、es and genera1 users have already adopted some of the Recommendations of the Sub-committee on Frequency Stability of the Technical Committee on Frequency and Time of the IEEE Society on Instrumentation and Measurement and the existence of the IEEE Standard No. 1139-1988 on “IEEE Standard Definitions
4、 of Physical Quantities for Fundamental Frequency and Time Metrology”; d) that frequency and time instability measures should be based on sound theoretical principles, conveniently usable, and directly interpretable; e) that it is desirable to have frequency and time instability measures obtainable
5、with simple instrumentation; f) that there is no accepted and appropriate measure for time-domain time instability in clocks and in measurement, comparison, and dissemination systems; g) that a time instability measure for random variations has been found which satisfies the inadequacy both for the
6、telecommunications industry as well as for time and frequency measurement, comparison and dissemination systems and for clocks, recommends 1. that the random instabilities of standard frequency and time signals should be characterized by the statistical measures Sy(f), S,(n or S,(f) in the frequency
7、-domain, and oy(z), Mod. oy(z) and o,(z) in the time-domain as defined below: 1.1 the measureof the normalized frequency instabilities y(r) in the frequency domain is Sy(f); i.e. the one-sided spectral density (O = 1/2 denotes an infinite time average. The measure written in equation (7) is often ca
8、lled the Allan variance. The xk and Xk + 1 are time residual measurements made at fk and fk + 1 = fk + z, k = 1,2, ., and l/z is the fixed sampling rate which gives zero dead time between frequency measurements. By “residual” it is understood that the known systematic effects have been removed. ITU-
9、R ITU-R TF-538-3 94 4855212 0522b44 546 84 Rec. ITU-R TF.538-3 Slope characteristics of log-log plot If the initial sampling rate is specified as lho, then in general one may obtain a more efficient estimate of CTy(z) using what is called an “overlapping estimate”. This estimate is obtained by compu
10、ting equation (8). 2 2 oy(.c Mod.o,(T) P P 1 1 O O -1 -1 -2 -2 -2 -3 where N is the number of original time departure measurements spaced by TO (N = M + 1, where M is the number of original frequency measurements of sample time, TO) and z = n 70. 2 oJr) rl 3 2 1 O -1 I If dead time exists between th
11、e frequency departure measurements and this is ignored in computing equation (7). it has been shown that the resulting stability values (which are no longer the Allan variances), will be biased (except for the white frequency noise) as the frequency measurements are regrouped to estimate the stabili
12、ty for n TO (n 1). This bias has been studied and some tables for its correction published. a -2 -1 O 1 2 If there is no dead time, then the original 2s can be combined to create a set of 7,s: =a-2 -4 -3 -2 -1 O i=k TABLE 1 The functional characteristics of five independent noise processes for frequ
13、ency instability of oscillators Description of noise process Random walk frequency Flicker frequency White frequency Flicker phase White phase I S,(f) = it depends on the particular frequency standard and is not fully understood in its physical basis. Examples of probable causes for the flicker “flo
14、or” are power supply voltage fluctuations, magnetic field fluctuations, changes in components of the standard, and microwave power changes. Finally the curve shows a deterioration of the stability with increasing averaging time. This occurs typically at times ranging from hours to days, depending on
15、 the particular kind of standard. 2 A “modified Allan variance”, Mod. oY(z), has been developed which has the property of yielding different dependences on T for white phase noise and flicker phase noise. The dependences for Mod. cry) are ,r3” and r1 respectively. Mod. oY(z) is estimated using the f
16、ollowing equation: where: N: original number of time variation measurements spaced by TO z = n TO the sample time of choice. Properties and confidence of the estimate are discussed in the technical literature. Maximum likelihood methods of estimating oJ2) for the specific models of white frequency n
17、oise and random walk frequency noise have been developed. These two models have been shown to be useful for sample times longer than a few seconds for caesium beam standards. The time instability in the time-domain for the above five power-law spectra may be measured using the 2 second-difference of
18、 adjacent time averages. This measure is also related to Mod. oy(z). where (dddr) = y and T = n TO. Therefore, x is the time deviation; the brackets ” denote an infinite time average. The bar “-” over the x denotes an average over an interval T. Hence, x is an optimum estimate of the time deviation
19、over the interval T if the deviations have a white spectrum. The three averages used in the second difference equation above are adjacent. Therefore, for a given value of k in the second difference above, these averages occupy a space of 37. The spectral density and time-domain relationships are as
20、follows: 2 o#) - ZTi =-q-i Since the usual types of measurement noise are centred around q = O, this gives a near-zero dependence on z (a desirable trait for a good measure). Other useful characteristics of this measure are: - it is equal to the classical standard deviation of the time difference me
21、asurements for T = TO, for white- noise PM; - it equals the standard deviation of the mean of the time difference measurements for 7 = N TO (the data length), for white-noise PM; Rec. ITU-R TF.538-3 87 - it is convergent and well behaved for the random processes commonly encountered in time and freq
22、uency metrology; - the z dependence indicates the power-law spectral-density model appropriate for the data; - the amplitude of GAZ) at a particular value of 2, along with assumption of one of the five power-law spectral-density models ( = -4. -3, -2, -1, O), provides enough information to estimate
23、the corresponding level in the frequency domain for any of the recommended IEEE-standard spectral-density measures. The problem of estimating the stability of individual clocks from comparison measurements has been studied proposing a general and consistent model to deal with signal difference measu
24、rements, without any a priori assumption of lack of correlation between clocks. 4. Conversion between frequency and time domains In general, if the spectral density of the normalized frequency fluctuations S,(n is known, the two-sample variance can be computed as follows: O O and: O Specifically, fo
25、r the power law model given by equation (6), the time-domain measure also follows the power law as derived from equations (6) and (1 1). The values of ha are characteristics of oscillator frequency noise. One may note for integer values (as often seems to be the case) that p=-a-l for -35a51 p - -2 f
26、or a21 where: These conversions have been verified experimentally and by computation. Table 2 gives the coefficients of the translation among the frequency stability measures from time domain to frequency domain and from frequency domain to time domain. 88 Rec. ITU-R TF.538-3 TABLE 2 Translation of
27、frequency stability measures from spectral densities in frequency domain to variance in time domain and vice versa (for 2n fh z = 1) Description of noise processes Random walk frequency Flicker frequency White frequency Flicker phase White phase 2 ay (FI = sy (f 1 = 2 cf-2 z 0;(z) 4x2 6 B = 210g,2 c
28、 = 112 A =- The slope Characteristics of the five independent noise processes are plotted in the frequency and time domains in Fig. 1 (log-log scale). 5. Confidence limits of time domain measurements To estimate the confidence interval or error bar for a Gaussian type of noise of a particular value
29、by(r) obtained from a finite number of samples, the following expression may be used (with non-overlapping estimates): Confidence Interval la = oy(z) . - K“2 for M 10 (18) where: A4 : CI : total number of data points used in the estimate as defined in the previous section Ki= Ki =0.99 KO= 0.87 = 0.7
30、7 2 = 0.75. ITU-R ITU-R TF-538-3 94 = 4855232 0522bY9 O28 Rec. ITU-R TF.538-3 89 As an example of the Gaussian model with M = 100, a = -1 (flicker frequency noise) and oy(z = i s) = 10-12, one may write: which gives: OJZ = 1 S) = (1 kO.08) X (20) A modified estimation procedure including dead time b
31、etween pairs of measurements has also been developed, The above confidence intervals apply to “non-overlapping estimates”. In the case of “overlapping estimates” The bias resulting from the application of two-sample variance to time intervals obtained by linking several successive measures with dead
32、 time has been determined as a function of noise type. This bias may be significant. The effect af the nature of the analogue filtering which limits the noise power of the signal in question about its nominal frequency has been determined, particularly for the use of a low pass filter instead of a b
33、and-pass filter centred on the nominal frequency. The degrees of freedom (d. f.) for overlapping estimates have been calculated. They are theoretically derived and plotted for power-law spectra for estimation of the confidence interval of the two-sample standard deviation. The confidence interval fo
34、r the two-sample standard deviation oy(z) is: showing the influence of auto-correlation of frequency fluctuations. the confidence interval is smaller and can be calculated. where: : the hat “*” : For a = +2 the improvement of the d.f. is nearly n times better than with respect to the non-overlapping
35、 estimate case. Significant improvement is also gained for o( = +1. For cc = O the ratio of the degrees of freedom is 2; for a = -1 it is 1.3; and for a = -2 it is 1-04. percentile values for the chi-square distribution estimate or the measured two-sample variance from finite set. XP, and XP, 6. Con
36、clusion The statistical methods for describing frequency and phase instability and the corresponding power law spectral density model described are sufficient for describing oscillator instability on the short term. Equations (14-16) show that the spectral density can be unambiguously transformed in
37、to the time domain measure. The converse is not true in all cases but is true for the power law spectra often used to model precision oscillators. Non-random variations are not covered by the model described. These can be either periodic or monotonic. Periodic variations are to be analysed by means of known methods of harmonic analysis. Monotonic variations are described by linear or higher order drift terms.
