1、 Recommendation ITU-R TF.538-4 (07/2017) Measures for random instabilities in frequency and time (phase) TF Series Time signals and frequency standards emissions ii Rec. ITU-R TF.538-4 Foreword The role of the Radiocommunication Sector is to ensure the rational, equitable, efficient and economical u
2、se of the radio-frequency spectrum by all radiocommunication services, including satellite services, and carry out studies without limit of frequency range on the basis of which Recommendations are adopted. The regulatory and policy functions of the Radiocommunication Sector are performed by World a
3、nd Regional Radiocommunication Conferences and Radiocommunication Assemblies supported by Study Groups. Policy on Intellectual Property Right (IPR) ITU-R policy on IPR is described in the Common Patent Policy for ITU-T/ITU-R/ISO/IEC referenced in Annex 1 of Resolution ITU-R 1. Forms to be used for t
4、he submission of patent statements and licensing declarations by patent holders are available from http:/www.itu.int/ITU-R/go/patents/en where the Guidelines for Implementation of the Common Patent Policy for ITU-T/ITU-R/ISO/IEC and the ITU-R patent information database can also be found. Series of
5、ITU-R Recommendations (Also available online at http:/www.itu.int/publ/R-REC/en) Series Title BO Satellite delivery BR Recording for production, archival and play-out; film for television BS Broadcasting service (sound) BT Broadcasting service (television) F Fixed service M Mobile, radiodeterminatio
6、n, amateur and related satellite services P Radiowave propagation RA Radio astronomy RS Remote sensing systems S Fixed-satellite service SA Space applications and meteorology SF Frequency sharing and coordination between fixed-satellite and fixed service systems SM Spectrum management SNG Satellite
7、news gathering TF Time signals and frequency standards emissions V Vocabulary and related subjects Note: This ITU-R Recommendation was approved in English under the procedure detailed in Resolution ITU-R 1. Electronic Publication Geneva, 2017 ITU 2017 All rights reserved. No part of this publication
8、 may be reproduced, by any means whatsoever, without written permission of ITU. Rec. ITU-R TF.538-4 1 RECOMMENDATION ITU-R TF.538-4 Measures for random instabilities in frequency and time (phase) (1978-1990-1992-1994-2017) Scope Frequency and phase instabilities may be characterized by random proces
9、ses that can be represented statistically in either the Fourier frequency domain or in the time domain. This Recommendation presents various methods and techniques for characterization of these frequency and phase instabilities. Keywords Random instabilities, Allan variance, time metrology, statisti
10、cal measures, phase, frequency The ITU Radiocommunication Assembly, considering a) that there is a need for an adequate 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
11、for some of the kinds of random time and frequency instabilities; c) that major laboratories, observatories, industries and general users have already adopted some of the Recommendations of the Technical Committee on Frequency and Time of the IEEE Society on Instrumentation and Measurement and the e
12、xistence of the IEEE Standard No. 1139-2008 on “IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology Random instabilities”; d) that frequency and time instability measures should be based on sound theoretical principles, conveniently usable, and directly inte
13、rpretable; e) that it is desirable to have frequency and time instability measures obtainable with simple instrumentation, recommends 1 that the random instabilities of standard frequency and time signals should be characterized by the statistical measures Sy( f ), S( f ) or Sx( f ) in the frequency
14、-domain, and y(), Mod. y(), x(), y(t,), and TheoBR in the time-domain as defined below: 1.1 the measure of the normalized frequency instabilities y(t) in the frequency domain is Sy( f ); i.e. the one-sided spectral density (0 f ) of the normalized frequency instabilities y(t) (t) 0)/0, where (t) is
15、the instantaneous carrier frequency, and 0 is the nominal frequency; 1.2 the measure of the phase instabilities (t) in the frequency domain is S( f ); i.e. the one-sided spectral density (0 f ) of the phase instabilities (t) at a Fourier frequency f ; 1.3 the measure of the phase instabilities expre
16、ssed in time units (phase-time) x(t) in the frequency domain is Sx( f ); i.e. the one-sided spectral density (0 f ) of phase-time instabilities x(t), where x(t) (t) / 2 0; x(t) being related to y(t) by y(t) dx(t) / dt; 1.4 the relationships of the above spectral densities are given below: 2 Rec. ITU
17、-R TF.538-4 Sy( f ) f 220 S( f ) 42 f 2 Sx( f ) (1) The dimensions of Sy( f ), S( f ) and Sx( f ) are respectively Hz1, Rad2 Hz1 and s2 Hz1; 1.5 the measure of the normalized frequency instabilities y(t) in the time domain is the two-sample standard deviation, y(), and the modified two-sample standa
18、rd deviation, Mod. y() and TheoBR variance as defined in Annex 1; 1.6 the measure of time instabilities in the time-domain is x() as defined in Annex 1; 1.7 the measure of the variations of the normalized frequency instabilities y(t) in the time domain is the two-sample standard deviation, y(t,) as
19、defined in Annex 1; 2 that, when stating statistical measures of frequency and time instability, non-random phenomena should be recognized, e.g.: 2.1 any observed time dependency of the statistical measures should be stated; 2.2 the method of measuring systematic behaviour should be specified (e.g.
20、an estimate of the linear frequency drift was obtained from the coefficients of a linear least squares regression to M frequency measurements, each with a specified averaging or sample time and bandwidth fh); 2.3 the environmental sensitivities should be stated (e.g. the dependence of frequency and/
21、or phase on temperature, magnetic field, barometric pressure, etc.); 3 that, when stating a measure of frequency and time instability, all relevant measurement parameters should also be specified: 3.1 the method of measurements; 3.2 the characteristics of the reference signal; 3.3 the nominal signal
22、 frequency v0; 3.4 the measurement system bandwidth fh and the corresponding low pass filter response; 3.5 the total measurement time or number of measurements M; 3.6 the calculation techniques (e.g. details of lag-windows when estimating power spectral densities from time domain data, or the assump
23、tion of the effect of dead-time in estimating the two-sample standard deviation y(); 3.7 the confidence of the estimate; 4 that a graphic illustration or an analytic expression of the measures of the frequency and time instabilities should be provided and should include confidence intervals when app
24、ropriate (i.e. Sy( f ), S( f ) and Sx( f ) as a function of f; y(), Mod. y() and x() as a function of ; and/or y(t,) as a function of t and ). Rec. ITU-R TF.538-4 3 Annex 1 Characterization of frequency and phase noise 1 Definition of terms Frequency and phase instabilities may be characterized by r
25、andom processes that can be represented statistically in either the Fourier frequency domain or in the time domain. The instantaneous, normalized frequency departure y(t) from the nominal frequency v0 is related to the instantaneous-phase fluctuation (t) about the nominal phase 2 0 t by: y(t) 12 0dt
26、dt .t)2 0 (2) () = ()2v0where x(t) is the phase variation expressed in units of time. 2 Fourier frequency domain In the Fourier frequency domain, frequency instability may be defined by several one-sided (the Fourier frequency ranges from 0 to ) spectral densities such as: Sy( f ) of y(t), S( f ) of
27、 (t), S .( f ) of .(t), Sx( f ) of x(t), etc. These spectral densities are related by the equations: Sy( f ) f 220 S( f ) (3) S .( f ) (2 f )2 S( f ) (4) Sx( f ) 1(2 0)2S( f ) (5) Power-law spectral densities are often employed as reasonable models of the random fluctuations in precision oscillators
28、. In practice, it has been recognized that for many oscillators these random fluctuations are the sum of five independent noise processes and, with few limitations, the following equation is representative: Sy( f ) 22h f for 0 f fh0 for f fh(6) where hs are constants, s are integers, and fh is the h
29、igh frequency cut-off of a low pass filter. Equations (3), (4) and (5) are correct and consistent for stationary noises including phase noise. High frequency divergence is eliminated by the restrictions on f in equation (6). The identification and characterization of the five noise processes are giv
30、en in Table 1, and shown in Fig. 1. In practice, only two or three noise processes are usually sufficient to describe the random frequency fluctuations in a specific oscillator; the others may be neglected. 4 Rec. ITU-R TF.538-4 3 Time domain Random frequency instability in the time domain may be de
31、fined by several sample variances. The square-root of a sample variance is termed a deviation, and is the statistic usually reported. A Allan deviation y() A measure of random frequency instability is the two-sample standard deviation which is the square root of the two-sample zero dead-time varianc
32、e 2y() defined as: 2y() 1/2 ( yk + 1 yk )2 (7) where: yk 1 tktk + y(t) dt xk + 1 xk and tk 1 tk (adjacent samples) denotes an infinite time average. The measure written in equation (7) is often called the Allan variance (AVAR). The xk and xk 1 are time residual measurements made at tk and tk + 1 tk
33、, k 1, 2, ., and 1/ 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. If the initial sampling rate is specified as 1/0, then in general one may obtain a more efficient estimate of
34、y() using what is called an “overlapping estimate”. This estimate is obtained by computing equation (8). 2y() 12(N 2n) 2 i 1N 2n(xi 2n 2xi n xi)2 (8) where N is the number of original time departure measurements spaced by 0 (N M 1, where M is the number of original frequency measurements of sample t
35、ime, 0) and n 0. If dead time exists between the 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 mea
36、surements are regrouped to estimate the stability for n 0 (n 1). This bias has been studied and some tables for its correction published. If there is no dead time, then the original yis can be combined to create a set of yks: yk 1n i = kk + n 1yi Rec. ITU-R TF.538-4 5 TABLE 1 The functional characte
37、ristics of five independent noise processes for frequency instability of oscillators Description of noise process Slope characteristics of log-log plot Frequency domain Time domain Sy( f ) S( f ) or Sx( f ) 2y() Mod. 2y() 2x() 2 Random walk frequency 2 4 1 1 3 Flicker frequency 1 3 0 0 2 White frequ
38、ency 0 2 1 1 1 Flicker phase 1 1 2 2 0 White phase 2 0 2 3 1 Sy( f ) h f = 1, 2 2 y() S( f ) v20 h f 2 v20 h f 2; Mod. y() Sx( f ) 142 h f 2 14 h f 1 x() 6 Rec. ITU-R TF.538-4 An “overlapping estimate” of y() can then be obtained: y() 12(M 2n + 1) k 1M 2n + 1(yk n yk )2 (9) Thus, one can ascertain t
39、he dependence of y() as a function of from a single data set in a very simple way. A plot of y() versus for a frequency standard typically shows a behaviour consisting of elements as shown in Fig. 1. The first part, with y() 1/2 (white frequency noise) and/or y() 1 (white or flicker phase noise) ref
40、lects the fundamental noise properties of the standard. In the case where y() 1, it is not practical to decide whether the oscillator is perturbed by white phase noise or by flicker phase noise. f 4f 3f 2f 1f0S(f)0 1 1 / 21 / 2 1y()D 0 1F o u r i e r f r e q u e n c yS a m p l i n g t i m eF I G U R
41、 E 1S l o p e c h a r a c t e r i s t i c s o f t h e f i v e i n d e p e n d e n t n o i s e p r o c e s s e s( l o g - l o g s c a l e )Rec. ITU-R TF.538-4 7 Alternative techniques are suggested below. This is a limitation of the usefulness of y() when one wishes to study the nature of the existin
42、g noise sources in the oscillator. A frequency-domain analysis is typically more adequate for Fourier frequencies greater than about 1 Hz. This 1 and/or 1/2 law continues with increasing averaging time until the so-called flicker “floor” is reached, where y() is independent of the averaging time . T
43、his behaviour is found in almost all frequency standards; it depends on the particular frequency standard and is not fully understood in its physical basis. Examples of probable causes for the flicker “floor” are power supply voltage fluctuations, magnetic field fluctuations, changes in components o
44、f 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 the particular kind of standard. B Modified Allan deviation Mod. y() A “modified Allan varianc
45、e (MVAR)”, Mod. 2y(), has been developed which has the property of yielding different dependences on for white phase noise and flicker phase noise. The dependences for “modified Allan deviation (MDEV)” Mod. y() are 3/2 and 1 respectively. Mod. y( is estimated by taking the square-root of the followi
46、ng equation: Mod. 2y() 12 2 n2 (N 3n 1) j = 1N 3n + 1xi i jn + j 1(xi 2n 2xi n xi)2(10) where: N : original number of time variation measurements spaced by 0 n 0 the sample time of choice. Properties and confidence of the estimate are discussed in the technical literature. Maximum likelihood methods
47、 of estimating y() for the specific models of white frequency noise 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. C Time deviation x() The time instability in the time-domain f
48、or the five independent noise processes shown in Fig. 1 may be measured using the second-difference of adjacent time averages. This measure is also related to Mod. 2y(). 2x() ( Mod y) / 3 (11) 2x() (1/6) (12) where (dx/dt) y and n 0. Therefore, x() is the time deviation (TDEV). The brackets “ ” denote an inf
