IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-23, NO. 4, DECEMBER 1974

Measurement of Frequency Stability in Time and Frequency

Domains Via Filtering of Phase Noise

JACQUES RUTMAN, MEMBER, IEEE, AND GERARD SAUVAGE

Abstract-A recently developed theoretical analysis has shown s;that it is possible to measure the Allan variance (a time-domainmeasure of frequency instability) without any statistical treatmentof data from an electronic counter. The measurement is made viahigh-pass filtering of phase noise with a test set similar to the oneused for frequency-domain measurements. The unique test setdescribed in this paper relies on this principle and is capable ofmeasuring the short-term frequency instability of the best quartz-crystal oscillators in both time and frequency domains.

I. INTRODUCTION

T HE short-term frequency instability of high-qualityoscillators may be characterized either in the frequency

domain by the spectral density So (f) of phase noise¢(t), or in the time domain by the square root of theAllan variance o,2(r) of the relative frequency fluctuationy(t) averaged over a time interval r. (See [1] for completedefinitions of these parameters.) The phase noise spectraldensity is measured by making a low-frequency spectralanalysis of the output signal of a phase detector. Clas-sically, the Allan variance is measured by making a greatnumber of frequency (or period) measurements with adigital electronic counter, followed by a particular statis-tical treatment of the data [2].

In a recent paper [3], a theoretical analysis of therelations between time- and frequency-domain charac-terizations has shown that the Allan variance y2(r) couldalso be measured by filtering the output signal of a phasedetector with a high-pass filter having a cutoff frequencyfc= (irr- For all types of noise encountered in realoscillators, a simple relation has been demonstrated be-tween the rms value of the filter output voltage and thesquare root of the Allan variance. Moreover, a so-calledbandpass variance has been defined [3] which gives adifferent variation with r for white and flicker phasenoises, respectively, t-312 and r1- (whereas a,(r) '--1 inboth cases).The purpose of this paper is to describe the experimental

test set which has been implemented following this theo-retical analysis. This test set allows the measurement ofthe phase noise spectral density (or of the related scriptS(~f) '' 2S (f), [4]), of the Allan variance via high-passfiltering, and also of the bandpass variance which cannotbe measured conveniently with a digital electroniccounter. The performance of the test set, together with

Manuscript received June 19, 1974; revised August 22, 1974.The authors are with Adret Electronique, BP 33, 78190 Trappes,

France.

Fig. 1. Block diagram of the experimental test set: Phase detectortechnique with a very loose phase-lock loop. The cutoff frequencyfH is normally set by the adjustable filter. Otherwise, it is equal to200 kHz.

experimental results on a quartz oscillator, will bepresented.

II. EXPERIMENTAL TEST SET [5]

The test set relies on the well-known phase detectortechnique [4]. A mixer driven with two signals havingthe same nominal' frequency vO and placed in quadratureby phase locking, behaves as a phase detector (only verygood double-balanced mixers using low-noise Schottkydiodes can be used for this application).The block diagram of the test set is shown in Fig. 1.

With this principle of measurement, a very loose phase-lock loop is required, inasmuch as the voltage varies asphase for Fourier frequencies above the loop bandwidth(see Appendix I). The loop bandwidth B must be smallcompared to the lowest Fourier frequency of interest(typically: B < 1 Hz), and may be adjusted by varyingthe loop amplifier gain.The 300-kHz Butterworth low-pass filter removes the

components at vO and 2v0. As the voltage level at the outputof the phase detector is very low (a few ,uV) it is necessaryto use an amplifying chain with a very low-noise pre-amplifier. A low source-impedance value (100 £2) is usedto optimize the preamplifier noise, since a noise voltageof 1 ,V rms in a 100-kHz bandwidth is equal to the thermial

I The possible values of vo are limited by the input mixer. Forexample, 1 MHz < vo < 500 MHz.

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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, DECEMBER 1974

noise of a 6004. resistor. Bipolar transistors are usedbecause of this low source impedance. The study of noisein transistors leads to specific values of the bias currents,which give about 1.6 nV/(Hz)l"2 noise, referred to theinput, over the range 100 Hz-200 kHz. The overall amplify-ing chain has a variable gain G (5000, 10000, 20000, 30000,and 40000) and a 0.3 Hz-200 kHz 3-dB bandwidth. Unlessa narrower low-pass filter is inserted in the test set, thevalue of the cutoff frequency fH appearing in the Allanvariance for white and flicker phase noises [1] will beequal to 200 kHz. The amplifier output voltage is fedthrough an adjustable filter to a true rms voltmeter whichcovers the frequency range 0.5 Hz-500 kHz.There are three basic types of measurements possible,

according to the mode of the adjustable filter: spectraldensity of phase noise with a narrow bandpass filter;Allan variance with a high-pass filter with cutoff (7rr) ';and bandpass variance with constant Q and center fre-quency (2r)-l. As spectral density analysis with such atest set is classic [4], we study in the next section thetime-domain measurement via filtering of phase noise.

III. TIME-DOMAIN MEASUREMENTVIA FILTERING

In this section, the Allan variance is related to the rmsvoltage read on the voltmeter. First, let us recall thedefinition of the "high-pass variance" [3], as seen by

8 00

O-HP2(T) = 22 S0(f) IHHP() 12 df (1)

where HHP(f) is the response of a high-pass filter withcutoff frequency (7rr)-'. It has been demonstrated thata bias coefficient close to unity must be applied to oHP (r)to get the true o,(r), for the different kinds of noiseencountered in oscillators (Table I).

For a noise-free reference oscillator, the phase detectoroutput reads

vo (t) = koo (t) (2)

where ko, the phase detector constant (in V/rad), may bemeasured by shifting the frequency of one source, replacingthe preamplifier by a resistance equal to its input im-pedance (100 Q), and measuring the beat note amplitude.The value of ko is adjusted with the input variable at-tenuators (typically: 0.1 < ko < 0.3 V/rad for the sourcesunder test).The amplifier output reads

v,(t) = koG4, (t) (3)

where ol (t) is the low-pass filtered 4 (t) with fH = 200 kHz(Sol(f) = So (f) for f < fH; Sol(f) = 0 for f > fH).The voltage at the o'utput of the linear high-pass filter

with a response HHp(f) is given by

TABLE IBIAS COEFFICIENTS FOR POWER-LAW SPECTRAL DENSITIES AND

SECOND-ORDER BUTTERWORTH HIGH-PASS FILTER [3]

S,(f)

k0

ki7fk2f2

k3

UHP(T)e-ff(T)o-Y(r)1.63

fl (rrfH)see below

1.19

1.06

1.03

rTrfH 10 100 1000 10000 0

fl(7rTfH) 1.20 1.37 1.44 1.48 1.63

The rms value of v2 (t) is given by- fH 1/2

VRMS = [E ]1I2 = koGf So(f) HHHp(f) 12 df

(6)

The square root of the high-pass variance is then equal to

81/2-HP(T ) = VRMS.work,G (7)

When two equally noisy, statistically independent sourcesare used, they both contribute to VRMS, and the precedingequation2 must be divided by V2- to get the stability of onedevice. Except for a bias coefficient close to unity, OHP(7)possesses all the properties of the square root of the Allanvariance c,(r) [3] which is thus estimated without anyelectronic counter.

IV. EXPERIMENTAL RESULTSFirst, the residual noise of the test set itself has been

measured by driving the two inputs of the mixer with onlyone oscillator with a 7r/2 phase shift in one channel. Thephase noise of this oscillator is thus cancelled, and thevoltage indicated is due to the test set itself.

In the frequency domain, the residual script R (f) fora value k0 = 0.3 V/rad is found to be

1-14 ±5f (8)

In decibels, we can thus measure - 170 dB for Fourierfrequencies above 300 Hz, and -145 dB at 1 Hz (noisemeasured in a 1-Hz bandwidth).

In the time domain, the residual high-pass variance ina bandwidth fH = 2 kHz for ko = 0.3 V/rad, is given by

v2 (t) = k0G42 (t) (4)where E6]

S12 (f) = HHp(f) 12 S,1 (f) = HHP (f) 12 k2G2SOI (f).

TR(T) - 4.3 X 10-14VoT

PIO in MHz

r in s(9)

(5) 2 Equation (7) is also used for bandpass variance measurements.

516

517RUITMAN AND SAUVAGE: TIME AND FREQUENCY DOMAINS

5, (P) dS undcer )rdtilWz

Fig. 2. Spectral density of phase noise.

For instance, o-R (1 ms) = 4.3 X 10-12 for a 10-MHz os-

cillator.As an example, we report now the performance of

a 10-MHz quartz-crystal oscillator. The frequency-domainmeasurements shown in Fig. 2 exhibit white phase noise

(f° law), flicker phase noise (fl law), and flicker frequency

noise (f-3 law).Curve 3(a) shows the square root of the high-pass

variance which is an estimate of the Allan variance (fH =

20 kHz in this experiment): the r-1 law is due to white

and flicker phase noises; the r0 law is due to flicker frequency

noise.Curve 3 (b) shows the square root of a bandpass

variance with Q = 2: the white phase noise now resultsin a t-3/2 law for r < 4.10-4 s, and the fficker phase noise in

a --1 law.The experimental curves of Figs. 2, 3(a), and 3(b) may

be represented by (10), (11), and (12), respectively, as

2 X 10- 10-ll 110)SO (f) + ~~+ (10)

f3f

-HP(T) = 7 X 10-13 + 1.5 X 10-l1 (11)

2 X1015 101l3ffBp (r) = 3X12 +

1-1+ 10 11. ( 12)

3T +

Fig. 3. Time-domain stability (fH= 20 kHz). (a) High-pass vari-ance. (b) Bandpass variance: (Q = 2).

The time-domain data are in good agreement with thefrequency-domain data.

V. CONCLUSIONThe theoretical analysis developed in [3] leads to the

construction of a test set which allows frequency stabilitymeasurements to be made both in time and frequencydomains by the same experimental process, employingboth adequate filtering of phase noise and rms measure-ment. In particular, the time-domain measurement needsno statistical treatment of data from an electronic counter,the standard-deviation measurement being performed bythe operational principle of the true rms voltmeter. Thetest-set residual noise has been kept to a level such thatthe best quartz-crystal oscillators available today can bestudied.

APPENDIX I

In a classical time-domain measurement system suchas the one described in [4], a very tight phase-lock loophas to be used and the correction voltage at the oscillatorvaries as frequency, consequently, the loop bandwidth isrelatively large and the response time is much smallerthan the smallest time interval T at which we wish to

measure. The measurements are made with a frequencycounter following a voltage to frequency converter. Thedata are then analyzed by computer with a programdesigned to compute the Allan variance. With the principleused in this paper, a very loose phase-lock loop is used forboth time- and frequency-domain measurements, andhence, there is no need to change the loop parameterswhen one goes from one domain to the other.

APPENDIX II: THE EXPERIMENTAL TEST SET

The test set was devised in two modules as shown in

Fig. 4. The upper module includes the input attenuators,the low-noise mixer, the loop amplifier, and the low-noiseamplifying chain. When phase lock is achieved, the outputvoltage of this module is proportional to the phase noise

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. im-23, NO. 4, DECEMBER 1974

Fig. 4. The experimental test set.

of the source under test. The lower module includes threebanks of filters: the first one with ten bandpass filters forspectral-density measurements (0.5 Hz < f < 30 kHz);the second with six high-pass filters for the Allan variancemeasurements (10-5 s < T < 1 s); and the third with sixbandpass filters with constant Q for the bandpass variancemeasurements. Once a given ifiter has been selected, itsoutput voltage is fed to the true rms voltmeter. A com-mercially available adjustable filter may easily be usedwhen other values of f or r are needed. The entire test setis battery powered.

REFERENCES[1] J. A. Barnes et al., "Characterization of frequency stability,"

IEEE Trans. Instrum. Meas., vol. IM-20, pp. 105-120, May 1971.[21 D. W. Allan, "Statistics of atomic frequency standards," Proc.

IEEE, vol. 54 pp. 221-230, Feb. 1966.[3] J. Rutman, "Characterization of frequency stability: a transfer-

function approach and its application to measurements via6fitering of phase noise," IEEE Trans. Instrum. Meas., vol.IM-23, pp. 40-48, Mar. 1974.

[41 J. H. Shoaf et al., "Frequency stability specification and measure-ment: high-frequency and microwave signals," Nat. Bur. Stand.,Rep. No. 632, Jan. 1973.

[5] G. Sauvage, Thesis to be published in 1974, University of Paris,France.

[6] A. Papoulis, Probability, Random Variables, and StochasticProcesses. New York: McGraw-Hill, 1965.

Spectral and Short Term Stability MeasurementsJACQUES GROSLAMBERT, MARCEL OLIVIER, AND JEAN UEBERSFELD

Abstract-The noise performance of an oscillator can be giveneither in the spectral or in the time domain. Two types of apparatusare generally necessary to measure these noise characteristics,spectral analyzers and frequency counters. The system describeduses spectral density to time domain conversion and measures boththe short term frequency stability and the phase spectral densityof an oscillator. Bias functions, depending on the spectral density,are calculated. They are used to determine systematic errors intro-duced by the apparatus.

I. INTRODUCTION

THERE ARE two methods of specifying the frequencyTstability of an oscillator: we can use, in the timedomain, the variance of the frequency fluctuations aver-aged over a time T or, in the spectral domain, the spectraldensity of the phase or of the frequency fluctuations. Thedefinitions of these quantities and the relations betweenvariance and spectral densities are well described in [1]and [2]. A great number of methods can be used to meas-ure them. Frequency stability data are usually obtainedby measuring the time periods of zero crossings of thebeat between the signal to be analyzed and a referenceclock, or by analyzing, in the spectral or time domain, theinstantaneous phase difference between the oscillator and

Manuscript received July 3, 1974; revised September 5, 1974.The authors are with the LPMO, Bensacon, France.

a reference. These methods imply the use of a frequencycounter or a spectrum analyzer.We have realized a system which gives simultaneously

the spectral density of the phase fluctuations of an oscil-lator and the short term frequency stability I(T) of thefrequency fluctuations; the integrating time r may be ashigh as 30 s. In this apparatus a spectral density to timedomain conversion is used.

II. SYSTEM ANALYSISThe relations between time domain frequency stability

and spectral density may be written [1], [2] as

(I(r))2 = f22 S.(f) sin2 wfr df.

The Allan variance of the frequency fluctuations canalso be related to Ss ( f)

o0(a21(2,T,T) )= 22|So ( f) sin4 fT df.

In these relations, S, ( f) is the one sided phase spectraldensity.The practical use of these formulas is as follows. The

phase detector technique (Fig. 1) gives us a signal, pro-portional to the phase fluctuations of the oscillator to beanalyzed. This signal feeds a series of six-pole filters cen-

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