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Characterization of frequency stability

mmi Bureau ot ^mmm

NOV 2 3 1970

NBS TECHNICAL NOTE 394

Characterization

of Frequency Stability

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NBS TECHNICAL NOTE 394
ISSUED OCTOBER 1970

Nat. Bur. Stand. (U.S.), Tech. Note 394, 50 pages (Oct. 1970)
CODEN: NBTNA

Chorocterization of Frequancy Sfability

J. A. Barnes

A. R. Chi

L. S. Cutler

D.J. Healey

D. B. Leeson

T. E. McGunigai

J. A. Mullen

W. L.Smith

R. Sydnor

R. F. C. Vessot

G. M. R.Winkler

*The authors of this paper are members of the Subcom-
mittee on Frequency Stability of the Technical Committee
on Frequency and Time of the IEEE Group on Instrumen-
tation & Measurement. See page ii.

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(Order by SD Catalog No. 013.46:394), Price 60 cents

Members
Subcorainittee on Frequency Stability

of the Technical Comraittee on Frequency and Time
of the

Institute of Electrical & Electronics Engineers
Group on Instrumentation & Measurement

J. A. Barnes, Chairman
Time and Frequency Division
Institute for Basic Standards
National Bureau of Standards

A. R. Chi
National Aeronautical and Space

Greenbelt, Maryland 20771

L. S. Cutler

Hewlett-Packard Company
Palo Alto, California 94304

D. J. Healey
Westinghouse Electric Corporation
Baltimore, Maryland 2 1203

D. B. Leeson
California Microwave
Sunnyvale, California 94086

T. E. McGunigal
National Aeronautical and Space

Greenbelt, Maryland 20771

J. A. Mullen
Raytheon Company
Waltham, Massachusetts 02154

W. L. Smith
Bell Telephone Laboratories
Allentown, Pennsylvania 18103

R. Sydnor
Jet Propulsion Laboratory

R. F.C. Vessot
Smithsonian Astrophysical Observatory
Cambridge, Massachusetts 01922

G.M.R. Winkler
Time Service Division
U. S. Naval Observatory
Washington, D. C. 20390

11

Page

Glossary of Symbols v

Abstract ix

I. Introduction 1

II. Stateinent of the Problem . .^ 4

IIIj Background and Definitions 5

TV. The Definition of Measures of Frequency
Stability (Second Moment Type) 6

V. Translations Among Frequency Stability Measures .... 13

VI. Applications of Stability Measures 20

VII. Measurement Techniques for Frequency Stability .... 22

VIII. Conclusions 31

Appendix A 33

Appendix B 41

111

GLOSSARY OF SYMBOLS

B^(N, r, ^), B^{r,ll)

a

c , c
o 1

c(t)

D^(T)
X

_ w
277

g(t)

h

i, j, k, m, n

M

N

n(t)

R (t;
y

Bias function for variances based on finite
saraples of a process with a power -law
spectral density. (See [13].)

A real constant defined by (A15).

Real constants.

A real, deterministic function of time.

Expected value of the squared second
difference of x(t) with lag time t. See

(B8).

Fourier frequency variable.

High frequency cutoff of an idealized infinitely
sharp cutoff, low -pass filter.

Low frequency cutoff of an idealized infinitely
sharp cutoff, high -pass filter.

A real function of time.

rOi .
Positive, real coefficient of f in a power
series expansion of the spectral density of
the function y(t).

Integers, often a dumray index of summation.

Positive integer giving the number of cycles
averaged.

Positive integer giving the number of data
points used in obtaining a sample variance.

A non-deterministic function of time.

Autocovariance function of y(t). See (A3).

Positive, real number defined by r = T/t.

S An intermediate term, used in deriving (23).
The definition of S is given by (A9).

S (f) One-sided (power) spectral density on a per
hertz basis of the pure real function g(t).
The dimensions of S (f) are the diraensions
ofg^(t)/f. §

S (f) A definition for the raea sure of frequency stability,
One-sided (power) spectral density of y(t)
on a per hertz basis. The dimensions of
S (f) are Hz"^.
y

T Tirae interval between the beginnings of
two successive measurements of average
frequency.

t Time variable.

t An arbitrary, fixed instant of time.
o

t The time coordinate of the beginning of the
k-th measurement of average frequency.
By definition, t, = t, + T, k = 0, 1, 2* • • .

k+1 k

u Dummy variable of integration; u = 77 f t.

V(t) Instantaneous output voltage of signal

generator. See (2).

V . Nominal peak amplitude of signal generator
output. See (2).

V (t) Instantaneous voltage of reference signal.
^

. See (40).

V Peak amplitude of reference signal. See (40).
or

v(t) Voltage output of ideal product detector.

v (t) Low -pass filtered output of product detector.

x(t) Real function of time related to the phase of
the signal V(t) by the equation

x(t) = .

o

vi

x{t) . A predicted value for x(t).

y(t) Fractional frequency offset of V(t) from
the norainal frequency. See (7).

y Average fractional frequency offset during
the k-th measureraent interval. See (9),

(y) The saraple average of N successive values
of y . See (B4).

z (t) Non-deterministic (noise) function with
n

y

(power) spectral density given by (2 5).

0! Exponent of f for a power -law spectral
density.

y Positive, real constant.

6 (r-1) The Kronecker 6-function defined by
K,

6^(r-l) .
1, if r = 1

0, if otherwise

.

C(t) Amplitude fluctuations of signal. See (2).

/LI Exponent of t. See (2 9).

V{t) Instantaneous frequency of V(t). Defined by

V Norainal (constant) frequency of V(t).
o

X(t) The Fourier transforra of n(t).

|CT (N, T, T) Sample variance of N averages of y(t).
each of duration t, and spaced every T
units of time. See (10).

(a (N, T, T)) Average value of the
saraple variance <7 (N, T, t).

la (T) A second choice of the definition for the measure
of frequency stability. Defined by

O-^(T) = <cr^(N =2, T = T, T)> .

vii

U (T) Time stability measure defined by

X y

T Duration of averaging period of y(t) to
obtain y . See (9).

<^(t) Instantaneous phase of V(t). Defined by
*(t) = 2TTV t + (Pit).

o

(p(t) Instantaneous phase fluctuations about the
ideal phase, 2771^ t. See (2).

ij) (T, T) Mean square time error for Doppler radar,
^

See (BIO) .

CO = 2'n'f Angular Fourier frequency variable.

vm

CHARACTERIZATION OF FREQUENCY STABILITY

by
.

.

J. A. Barnes, A. R. Chi, L. S. Cutler,

D. J. Healey, D. B. Leeson, T. E. McGunigal,

J. A. Mullen, W. L. Smith, R. Sydnor,
'

R.F.C. Vessot, andG.M.R. Winkler

ABSTRACT

Consider a signal generator whose instantaneous output voltage V(t)

may be written as

V(t) = [V + €(t)] sin [27ri^ t + (p(t)]

where V and V are the nominal amplitude and frequency respectively
o o

T

d(p
of the output. Provided that €(t) and (p (t) = -7— are sufficiently small

for all time t, one raay define the fractional instantaneous frequency

deviation from nominal by the relation

O

A proposed definition for the measure of frequency stability is the

spectral density S (f) of the function y(t) where the spectrum is con-

sidered to be one-sided on a per hertz basis.

An alternative definition for the measure of stability is the infinite

time average of the sample variance of two adjacent averages of y(t);

that is, if
(.

'.!
k+T

where t is the averaging period, t , = t + T, k = 0, 1, 2* * ' , t is

arbitrary, and T is the time interval between the beginnings of two

successive measurements of average frequency; then the second measure

IX

of stability is

where ( ) denotes infinite time average and where T = T .

In practice, data records are of finite length and the infinite time

averages implied in the definitions are normally not available; thus estimates f

the two measures must be used. Estimates of S (f) would be obtained
y

from suitable averages either in the time domain or the frequency domain.
2An obvious estimate for 0" (t) is
y

m (v, - y, )

Parameters of the nn.easuring system and estimating procedure are

of critical importance in the specification of frequency stability. In

practice, one should experimentally establish confidence limits for an

estimate of frequency stability by repeated trials.

Key words: Allan variance; frequency; frequency stability; sample variance;
spectral density; variance.

CHARACTERIZATION OF FREQUENCY STABILITY

I. Introduction

The measurement of frequency and fluctuations in frequency has

received such great attention for so raany years that it is surprising that

the concept of frequency stability does not have a universally accepte<^

definition. At least part of the reason has been that some uses are most

readily described in the frequency domain and other uses in the time

doinain, as well as in combinations of the two. This situation is further

complicated by the fact that only recently have noise raodels been presented

which both adequately describe performance and allow a translation between

the time and frequency donaains. Indeed, only recently has it been recog-

nized that there can be a wide discrepancy between commonly-used time

domain measures themselves. Following the NASA-IEEE Symposium on

Short-Term Stability in 1964 and the Special Issue on Frequency Stability

of the Proc. IEEE of February 1966, it now seems reasonable to propose

a definition of frequency stability. The present paper is presented as

technical background for an eventual IEEE standard definition.

This paper attempts to present (as concisely as practical) adequate,

self -consistent definitions of frequency stability. Since more than one

definition of frequency stability is presented, an important part of this

paper (perhaps the most iraportant part) deals with translations among the

suggested definitions of frequency stability. The applicability of these

definitions to the more comraon noise models is demonstrated.

Consistent with an attempt to be concise, the references cited have

been selected on the basis of being of most value to the reader rather than

on the basis of being exhaustive. An exhaustive reference list covering

the subject of frequency stability would itself be a voluminous publication. .

Almost any signal generator is influenced to some extent by its

environment. Thus observed frequency instabilities raay be traced, for

exaraple, to changes in arabient temperature, supply voltages, magnetic

field, baroraetric pressure, humidity, physical vibration, or even ouput

loading to mention the more obvious. While these environmental influences

may be extremely important for many applications, the definition of fre-

quency stability presented here is independent of these causal factors.

In effect, we cannot hope to present an exhaustive list of environraental

factors and a prescription for handling each even though, in some cases,

these environmental factors may be by far the most important. Given a

particular signal generator in a particular environment, one can obtain its

frequency stability with the measures presented below, but one should

not then expect an accurate prediction of frequency stability in a new

environraent.

It is natural to expect any definition of stability to involve various

statistical considerations such as stationarity, ergodicity, average,

variance, spectral density, etc. There often exist fundamental difficulties

in rigorous attempts to bring these concepts into the laboratory. It is

worth considering, specifically, the concept of stationarity since it is

a concept at the root of many statistical discussions.

A random, process is mathematically defined as stationary if every

translation of the time coordinate maps the ensemble onto itself. As a

necessary condition, if one looks at the ensemble at one instant of tirae,

t, the distribution in values within the ensemble is exactly the same as

at any other instant of time, t . This is not to imply that the elements

of the ensemble are constant in time, but, as one element changes value

with time, other elements of the ensemble assume the previous values.

Looking at it in another way, by observing the ensemble at some instant

of time, one can deduce no information as to when the particular instant

was chosen. This same sort of invariance of the joint distribution holds

for any set of times t, , t^, . . . , t and its translation t, + t, t^ + t, . . . ,
1 2 n 12

t + T.
n

It is apparent that any ensemble that has a finite past as well as a

finite future cannot be stationary, and this neatly excludes the real world

and anything of practical interest. The concept of stationarity does

violence to concepts of causality since we iraplicitly feel that current

performance (i.e., the applicability of stationary statistics) cannot be

logically dependent upon future events (i.e., if the process is terminated

sometime in the distant future). Also, the verification of stationarity

would involve hypothetical measurements which are not experimentally

feasible, and therefore the concept of stationarity is not directly relevant

to experimentation. '

Actually the utility of statistics is in the formation of idealized

models which reasonably describe significant observables of real systems.

One may, for example, consider a hypothetical ensemble of noises with

certain properties (such as stationarity) as a model for a particular real

device. If a model is to be acceptable, it should have at least two properties

First, the raodel should be tractable; that is, one should be able to easily

arrive at estimates for the elements of the model; and, second, the model

should be consistent with observables derived frora the real device which

it is simulating.

Notice that one does not need to know that the device was selected

from a stationary ensemble, but only that the observables derived from

the device are consistent with, say, elements of a hypothetically stationary

ensemble. Notice also that the actual model used may depend upon how

clever the experimenter -theorist is in generating models.

It is worth noting, however, that while some texts on statistics give

"tests for stationarity, " these "tests" are almost always inadequate.

Typically, these "tests"' determine only if there is a substantial fraction

of the noise power in Fourier frequencies whose periods are of the same

order as the data length or longer. While this may be very iraportant, it

is not logically essential to the concept of stationarity. If a non-stationary

raodel actually becomes common, it will almost surely be because it is

useful or convenient and not because the process is "actually non-stationary.

Indeed, the phrase "actually non- stationary" appears to have no meaning

in an operational sense. In short, stationarity (or non-stationarity) is a

property of models not a property of data [l].

Fortunately, many statistical raodels exist which adequately describe

most present-day signal generators; many of these models are considered

below. It is obvious that one cannot guarantee that all signal generators

are adequately described by these models, but the authors do feel they

are adequate for the description of most signal generators presently

encountered.

II. Statement of the Problem

To be useful, a measure of frequency stability must allow one to

predict performance of signal generators used in a wide variety of situations

as well as allow one to make meaningful relative comparisons among signal

generators. One must be able to predict perforinance in devices which

may iTiost easily be described either in the time domain, or in the frequency

domain, or in a combination of the two. This prediction of performance

raay involve actual distribution functions, and thus second raonaent measures

(such as power spectra and variances) are not totally adequate.

Two common types of equipment used to evaluate the performance

of a frequency source are (analog) spectrum, analyzers (frequency domain)

and digital, electronic counters (time domain). On occasion the digital

counter data are converted to power spectra by computers. One must

realize that any piece of equipraent siraultaneously has certain aspects

raost easily described in the time domain and other aspects most easily

described in the frequency doinain. For example, an electronic counter

has a high frequency limitation, and experimental spectra are determined

with finite time averages.

Research has established that ordinary oscillators demonstrate

noise which appears to be a superposition of causally generated signals

and random, non-deterrainistic noises. The random noises include

thermal noise, shot noise, noises of undeternjined origin (such as flicker

noise), and integrals of these noises.

One might well expect that for the more general cases one would)

need to use a non-stationary model (not stationary even in the wide sense,

i.e., the covariance sense). Non-stationarity would, however, introduce

significant difficulties in the passage between the frequency and time

domains. It is interesting to note that, so far, experimenters have seldom

found a non (covariance) stationary model useful in describing actual

oscillators.

In what follows, an attempt has been made to separate general

statements which hold for any noise or perturbation frora the stateraents

which apply only to specific models. It is important that these distinctions

be kept in mind.

III. Background and Definitions

To discuss the concept of frequency stability immediately implies

that frequency can change with time and thus one is not considering

Fourier frequencies (at least at this point). The conventional definition of

instantaneous (angular) frequency is the time rate of change of phase;

that is,

ZTTVit) = ^^^ H i(t) , • (1)

where ^ (t) is the instantaneous phase of the oscillator. This paper uses

the convention that time dependent frequencies of oscillators are denoted

by T^{t) (cycle frequency, hertz), and Fourier frequencies are denoted by

60 (angular frequency) or f (cycle frequency, hertz) where

to = ZTTf .

In order for (1) to have meaning, the phase ^(t) must be a well-

defined function. This restriction immediately eliminates some "non-

sinusoidal" signals such as a pure, random, uncorrelated ("white") noise.

For most real signal generators, the concept of phase is reasonably

amenable to an operational definition and this restriction is not serious.

Of great importance to this paper is the concept of spectral density,

S (f). The notation S (f) is to represent the one-sided spectral density
g g

of the (pure real) function g(t) on a per hertz basis; that is, the total

"power" or mean square value of g(t) is given by

/
S (f)df .

Since the spectral density is such an important concept to what

follows, it is worthwhile to present some important references on spectrum

estiraation. There are raany references on the estimation of spectra

from data records, but worthy of special note are [2 - 5].

IV. The Definition of Measures of Frequency Stability (Second Moment Type) !

A. General . Consider a signal generator whose instantaneous

output voltage, V(t), may be written as
'

'

- I

V(t) = [V + e(t)] sin [2T71/ t + (p(t)], (2)

where V and V are the nominal amplitude and frequency respectively
o o

of the output and it is assuraed that

€(t)

V
< < 1

and

(P(t)

27TI/'
< < 1

(3)

(4)

for substantially all time t. Making use of (1) and (2) one sees that

^(t) = ZTTV t + (Pit) ,
o

(5)

and

l^(t) =
^o

+ ^ ^(t) . (6)
Equations (3) and (4) are essential in order that (p(t) may be defined

conveniently and unambiguously (see measurement section).

Since (4) must be valid even to speak of an instantaneous frequency,

there is no real need to distinguish stability measures from instability

measures. That is, any fractional frequency stability measure will be

far from unity, and the chance of confusion is slight. It is true that in

a very strict sense people usually measure instability and speak of stability.

Because the chances of confusion are so slight, the authors have chosen

to continue in the custom of measuring "instability" and speaking of stability

(a number always much less than unity).

Of significant interest to many people is the rf (radio frequency)

spectral density, S (f). This is of direct concern in spectroscopy and

radar. However, this is not a good primary measure of frequency stability

for two reasons: First, fluctuations in the amplitude, €(t), contribute

directly to S (f); and second, for many cases when €(t) is insignificant,

the rf spectrum, S (f), is not uniquely related to the frequency fluctuations

[6].

B. General: First definition of the measure of frequency stability-

-

frequency domain .

By definition, let

o

where <P{t) and V are as in (2). Thus y(t) is the instantaneous fractional
o

frequency deviation from the norainal frequency V . A proposed definition

of frequency stability is the spectral density S (f) of the instantaneous

fractional frequency fluctuations y(t). The function S (f) has the

dimensions of Hz

One can show [7] that if S (f) is the spectral density of the phase

fluctuations, then

* °' (8)
/ 1 \^

= -^ f^S (f).
v

I ^
\ o I

Thus a knowledge of the spectral density of the phase fluctuations, S (f),
(p

allows a knowledge of the spectral density of the frequency fluctuations,

S (f), --the first definition of frequency stability. Of course, S (f) cannot

be perfectly measured- -this is the case for any physical quantity; useful

estimates of S (f) are, however, easily obtainable.

C. General: Second definition of the measure of frequency stability --
time domain .

The second definition is based on the sample variance of the fractional

frequency fluctuations. In order to present this measure of frequency

stability, define y by the relation

v = - y(t)dt = , (9)
'

t
°

k

where t.
, ,

= t. + T , k = 0, 1, 2, . . . , T is the repetition interval for
k+1 k

measurements of duration t, and t is arbitrary. Conventional frequency

counters measure the number of cycles in a period T; that is, they measure

V T (1 + y ). When t is one second they count the number V (1 + y, ).ok ok
The second measure of frequency stability, then, is defined in analogy to

the saraple variance by the relation

N / , N '2
(N, T.T))= (^

n=l ' k=l

where (g) denotes the infinite time average of g. This measure of

frequency stability is dimensionless.

In many situations it would be wrong to assume that (10) converges

to a meaningful limit as N -* °°. First, of course, one cannot practically

let N approach infinity and, second, it is known that some actual noise

processes contain substantial fractions of the total noise power in the

Fourier frequency range below one cycle per year. In order to iraprove

comparability of data, it is important to specify particular N and T ,

For the preferred definition we recommend choosing N = 2 and T = t

(i.e., no dead time between measurements). Writing (O" (N = 2, T = t, t))

as (T (T), the Allan variance [8], the proposed measure of frequency

stability in the time domain may be written as

for T = T

.

Of course, the experimental estimate of (T (t) must be obtained
y

from finite samples of data, and one can never obtain perfect confidence

in the estimate --the true time average is not realizable in a real

situation. One estimates 0" (t) frora a finite number (say, ra) of values
y

2 2 .
of (J (2, T, T) and averages to obtain an estimate of 0" (t). Appendix A

shows that the ensemble average of (J (2, t, t ) is convergent (i.e., as

3-Q - 00) even for noise processes that do not have convergent ((J (N, t, t))

as N -* =° . Therefore, (J (t) has greater utility as an idealization than

does (cr (°°, t, t)) even though both involve assumptions of infinite averages.

In effect, increasing N causes 0" (N, T, t) to be more sensitive to the low

frequency components of S (f). In practice, one must distinguish between

an experimental estimate of a quantity (say, of (J (t)) and its idealized

value. It is reasonable to believe that extensions to the concept of statistical

("quality") control [9] may prove useful here. One should, of course,

specify the actual number m of independent saraples used for an estimate
of a"(T). . ,

In sumraary, therefore, S (f) is the proposed measure of (instan-

taneous) frequency stability in the (Fourier) frequency doraain and (7 (t)

is the proposed measure of frequency stability in the time domain.

D. Distributions . It is natural that people first become involved

with second moment measures of statistical quantities and only later with

actual distributions. This is certainly true with frequency stability.

While one can specify the arguraent of a distribution function to be, say,

(y^ - y, ), it makes sense to postpone such a specification until a real
k+1 k

use has inaterialized for a particular distribution function. This paper

does not attempt to specify a preferred distribution function for frequency

fluctuations.

E. Treatraent of Systematic Variations .

1. General . The definition of frequency stability 0" (t) in the

tirae domain is useful for raany situations. However, some oscillators,

for example, exhibit an aging or almost linear drift of frequency with

tirae. For some applications, this trend may be calculated and should be

removed [8] before estiraating 0" (t).

In general, a systematic trend is perfectly deterministic (i.e.,

predictable) while the noise is non-deterministic. Consider a function,

g(t), which may be written in the form

g(t) = c(t) + n(t) (12)

where c(t) is some deterrainistic function of time and n(t), the noise, is

a non -deterrainistic function of tirae. We will define c(t) to be the

systeraatic trend to the function g(t). A problem of significance here is

to determine when and in what sense c(t) is measurable.

2. Specific Case --Linear Drift . As an exaraple, if we consider

a typical quartz crystal oscillator whose fractional frequency deviation is

y(t), we may let

g(t) = ^y(t). (13)

10

With these conditions, c(t) is the drift rate of the oscillator (e.g., 10

•per day) and n(t) is related to the frequency "noise" of the oscillator by

a time derivative. One sees that the time average of g(t) becomes

^t + T
j_ r

o

T I
1

t +T
r o

g(t)dt = c^ + Y j
n(t)dt 14)

where c(t) = c is assumed to be the constant drift rate of the oscillator,

In order for c to be an observable, it is natural to expect the average

of the noise term to vanish, that is, converge to zero.

It is instructive to assume [8, 10] that in addition to a linear

drift, the oscillator is perturbed by a flicker noise, i.e..

y
s (f ) = / h i \ < f ^ 1

-1

0, i>{^ ,

where h is a constant (see Sec. V.A.2) and thus,

S (f) = ({Zrrfh f, ^ f ^ f,
n \ -1 h

:i5)

(16)

0, f>f^,

for the oscillator we are considering. "With these assumptions, it is seen

that

Lim 1
T - 00 Ta

t^+T

n(t) dt = X(0) = 17'

and that

VanL
'P

-» 00
ance

T

t, +T
k

n(t)dt
J

= 0, (18)

where X (f ) is the Fourier transform of n(t). Since S (0) = 0, X(0) must

also vanish both in probability and in mean square. Thus, not only does

n(t) average to zero, but one may obtain arbitrarily good confidence on

the result by longer averages.

11

Having shown that one can reliably estimate the drift rate, c ,

of this (comraon) oscillator, it is instructive to atterapt to fit a straight

line to the frequency aging. That is, let

g(t) = y(t) (19)

and, thus,

g(t) = c^ + c^(t - t^) + n'(t) (20)

where c is the frequency intercept at t = t and c is the drift rate
o o 1

previously determined. A problem arises here because

S , (f) = S (f) (21)
n y

and

Lira

'J'
-> 00

Variance
1
rVT

,-
I

n (t) d t

^t
k

(22)

for the noise model we have assumed. This follows from the fact that the

(infinite N) variance of a flicker noise process is infinite [7, 8, lO]. Thus,

c cannot be measured with any realistic precision- -at least, in an absolute
o

sense.

We may interpret these results as follows: After experimenting

with the oscillator for a period of time one can fit an empirical equation

to y(t) of the forra

y(t) - c + tc + n'(t),
o 1

where n (t) is non-deterrainistic. At some later time it is possible to

reevaluate the coefficients c and c . According to what has been said,
o 1

the drift rate c should be reproducible to within the confidence estimates

of the experiment regardless of when it is reevaluated. For c , however,

this is not true. In fact, the more one attempts to evaluate c , the larger

the fluctuations are in the result.

12

Depending on the spectral density of the noise term, it may be

possible to predict future mea sureraents of c and to place realistic

confidence limits on the prediction [ll]. For the case considered here,

however, these confidence limits tend to infinity when the prediction

interval is increased. Thus, in a certain sense, c is "raeasurable"
o

but it is not in statistical control (to use the language of the quality
i

control engineer [9]).

V. Translations Among Frequency Stability Measures •

A. Frequency Domain to Time Domain .

1. General . It is of value to define r = T/T; that is, r is the

ratio of the tirae interval between successive measurements to the duration

of the averaging period. Cutler has shown (see Appendix A) that

K^'''^''^^~-W^i "'y'"
[sin^(77f T)]

(TTfrf

sin (TTrfNT)

N^ sin^ (TTrf T)
(23)

Equation (23) in principle allows one to calculate the time domain stability

(cr (N, T, t)) from the frequency domain stability S (f).

2. Specific raodel . A model which has been found useful [7, 9,

10, 11, 12] consists of a set of five independent noise processes, z (t), n =
n

-2, -1, 0, 1, 2, such that , ; .

y(t) = z (t) + z ^ (t) + z (t) + z (t) + z (t),
-2 -1 o 1 2

(24)

and the spectral density of z is given by
n

S (f)
z
n

J
h f"^, ^ f ^ f,

= < n 1

0, f >f- , n = -2, -1, 0, 1, 2,
h .

(25;

where the h are constants. Thus, S (f) becomes
n y

.-2 -1
S(f) = h^f" + hf"+h +hf + h^f^

,

y -2 -1 o 1 2
(26)

13

for ^ f ^ f, and S (f) is assumed to be negligible beyond this range,
h y

In effect, each z contributes to both S (f) and <CT (N.T, t)) independently
n y y

of the other z . The contributions of the z to ((J (N, T, t)) are tabulated
n n y

in Appendix B.

Any electronic device has a finite bandwidth and this certainly

applies to frequency measuring equipment also. For fractional frequency

fluctuations, y(t), whose spectral density varies as

S (f) ~f^, a ^ -1, (27)

for the higher Fourier components, one sees (from Appendix A) that

(CT (N, T, T)) may depend on the exact shape of the frequency cutoff. This

is true because a substantial fraction of the noise "power" raay be in

these higher Fourier components. As a simplifying assumption, this

paper assuines a sharp cutoff in noise "power" at the frequency f for

the noise models. It is apparent from the tables of Appendix B that the

time domain measure of frequency stability may depend on f in a very

important way, and, in some practical cases, the actual shape of the

frequency cutoff may be very important [?]. On the other hand, there are

many practical measurements where the value of f. has little or no effect.

Good practice, however, dictates that the system noise bandwidth, f,
,

h
should be specified with any results.

In actual practice, the model of (24), (25), and (26) seems to fit

almost all real frequency sources. Typically, only two or three of the

h-coefficients are actually significant for a real device and the others can

be neglected. Because of its applicability, this raodel is used in much of

what follows. Since the z are assumed to be independent noises, it is
n -^

normally sufficient to compute the effects for a general z and recognize

that the superposition can be accoraplished by simple additions for- their

contributions to S (f) or <CT^ (N, T, t) >.

14

B. Time Domain to Frequency Domain

1. General . For general (g (N, T, t)) no simple prescription

is available for translation into the frequency domain. For this reason,

one might prefer S (f) as a general measure of frequency stability. This

is especially true for theoretical work.

Z. Specific model . Equations (24), (25), and (26) formi a i

realistic model which fits the random, non-deterministic noises found on

most signal generators. Obviously, if this is a good nnodel, then the

tables in Appendix B may be used (in reverse) to translate into the fre-

quency doraain.

Allan [8] and Vessot [12] showed that if

S (f)
y

where 0^ is a constant, then

.cxV ' ^ f ^ ^h
0, i > t ,

(28)

<0-^(N, T, T)> ~ ItI^, ZTTTi >> 1
y h

(29)

T
for N and r = — held constant. The constant (I is related to 0^ by
the mapping shown''' in Fig. 1. If (28) and (29) hold over a reasonable range

for a signal generator, then (28) can be substituted into (23) and evaluated

to deterraine the constant h from measurements of (o" (N, T, t)). It
C y

should be noted that the model of (28) and (29) may be easily extended to a

superposition of similar noises as in (26).

It should be noted that in Allan [8], the exponent, 0^, corresponds to the
spectrum of phase fluctuations while variances are taken over average
frequency fluctuations. In the present paper, ^ is identical to the exponent
a + 2 in [8].

15

9

FIG. I f^-a MAPPING

16

C. Translations Among the Time Domain Measures

1. General. Since <0-^(N, T, t)) is a function of N, T, and t
y

(for some types of noise f is also important), it is very desirable to be

able to translate among different sets of N, T, and t (f held constant);

this is, however, not possible in general.

2. Specific model . It is useful to restrict consideration tq a

case described by (28) and (29). Superpositions of independent noises

with different power -law types of spectral densities (i.e., different Qf's)

can also be treated by this technique, e.g. , (26). One may define two

"bias functions, " B and B by the relations [13]:

<(T^(N, T, T)>

B (N, r, iU) = —I : (30)
1 <a^(2, T, T))

and -

<a^(2, T, T))

B (r, M) = -^t-———T (31)
2 <Cr (2, T, T))

where r = T/t and fl is related to 0? by the mapping of Fig. 1. In words,

B is the ratio of the average variance for N samples to the average

variance for 2 samples (everything else held constant); while B is the

ratio of the average variance with dead time between measurements

(r ^ 1 ) to that of no dead time (r = 1 and with N = 2 and t held constant).

These functions are tabulated in [13], Figs. 2 and 3 show a computer plot

of B^(N, r=l, jLl) and B^(r, H).

Suppose one has an experimental estimate of ((7 (N , T '^,)^ ^-^d

its spectral type is known- -that is, (2 8) and (2 9) form a good raodel and jLt

is known. Suppose also that one wishes to know the variance at some other

set of measurement parameters, N , T , t An unbiased estimate of

(O" (N T T )) raay be calculated by the equation:y22 2

17

m

I-
o

X
I-

cvj

iZ

%

Fig. 3 THE BIAS FUNCTION, Bg (r,^^)

<%^(^'^2'^2^^=(^)

B^(N^,r^,^)B2(r^,^)

B^(N^,r^,jy)B^(r^,^)
<rr^CN^, T^, T^)), (32)

T T
where r = 1/'^, ^-i^d r = 2/t

3. General . While it is true that the concept of the bias

functions, B and B , could be extended to other processes besides those
1 2

^

with the power-law types of spectral densities, this generalization has not

been done. Indeed, spectra of the forra given in (2 8) (or superpositions

of such spectra as in (26)) seem to be the most common types of non-

deterministic noises encountered in signal generators and associated

equipment. For other types of fluctuations (such as causally generated

perturbations), translations must be handled on an individual basis.

VI. Applications of Stability Measures

Obviously, if one of the stability measures is exactly the iraportant

parameter in the use of a signal generator, the stability measure's

application is trivial. Some non-trivial applications arise when one is

interested in a different parameter, such as in the use of an oscillator in

Doppler radar raeasurements or in clocks.

A. Doppler Radar .

1. General . From its transmitted signal, a Doppler radar

receives from a moving target a frequency-shifted return signal in the

presence of other large signals. These large signals can include clutter

(ground return) and transraitter leakage into the receiver (spillover).

Instabilities of radar signals result in noise energy on the clutter return,

on spillover, and on local oscillators in the equipment.

The limitations of sub-clutter visibility (SCV) rejections due to

the radar signals themselves are related to the rf power spectral density,

S^^(f). The quantity typically referred to is the carrier-to-noise ratio and

can be mathematically approximated by the quantity

20

S (f)
V '

f"^

The effects of coherence of target return and other radar parameters are

amply considered in the literature [14-17].

2. Special Case . Because FM effects generally predorainate
over AM effects, this carrier-to-noise ratio is approximately given by [6]

S (f)
-

S (f')df'

for many signal sources provided |f - y | is sufficiently greater than zero.

(The factor of — arises from the fact that S (f) is a one-sided spectrum.
)

Thus, if f - V is a frequency separation from the carrier, the carrier

-

o

to -noise ratio at that point is approximately

IV \
\S (|f -V |)=-(t^— ) S (|f -y l) . (34)2 (p\' o 7 2 yf - y / y V o V

B. Clock Errors .

1. General . A clock is a device which counts the cycles of a

periodic phenoraenon. Thus, the reading error x(t) of a clock run from

the signal given by (2) is •

x(t) = , (35)

o

and the diraensions of x(t) are seconds.

H this clock is a secondary standard, then one could have

available sorae past history of x(t), the tirae error relative to the standard

.' clock. It often occurs that one is interested in predicting the clock error

x(t) for some future date, say t + t, where t is the present date. Obviously,

this is a problera in pure prediction and can be handled by conventional

methods [3]. ,

21

2. Special Case . Although one could handle the prediction of

clock errors by the rigorous methods of prediction theory, it is more

common to use simpler prediction methods[lO, ll]. In particular, one

often predicts a clock error for the future by adding to the present error

a correction which is derived from the current rate of gain (or loss) of

time. That is, the predicted error x(t + T) is related to the past history

of x(t) by the equation
rx(t ) - x(t -T>,

x(to+ T) = x(tj +T [
J

. (36)

It is typical to let T = t.

Thus, the mean square error of prediction for T = t becomes

<[x(tQ+ T) - ^{t^+ r)f) = <[x(t^+ T) - 2x(t^+x(tQ- T)p), (37)

which, with the aid of (11), can be written in the forra

. . <[x(t + T) - x(t + T)]2) = ZT^CT^T) . (38)
^ O O y

One can define a tirae stability raeasure, O" (t), by the equation

CT^(T) = T^ (T^(T) . (39)
x y

Clearly, however, the actual errors of prediction of clock readings are

dependent on the prediction algorithm used and the utility of such a definition

as rr (t) is not great. Caution should be used in employing this definition.

VII. Measurement Techniques for Frequency Stability

A. Heterodyne Techniques (general) It is possible for oscillators

to be very stable and values of 0" (t) can be as small as 10 in some
y

state of the art equipraent. Thus, one often needs raeasuring techniques

capable of resolving very sraall fluctuations in y(t). One of the most

comraon techniques is a heterodyne or beat frequency technique. In this

method, the signal frora the oscillator to be tested is mixed with a reference

signal of almost the sarae frequency as the test oscillator in order that one

is left with a lower average frequency for analysis without reducing the

22

frequency (or phase) flucutations themselves. Following Vessot, et. al.

[18], consider an ideal reference oscillator whose output signal is

V (t) = V sin ZttV t, (40)
r or o

and a second oscillator whose output voltage V(t) is given by (2):

V(t) = [V + C(t)] sin [27TU t + <p(t)]. Let these two signals be mixed in a

product detector; that is, the output of the product detector v(t) is eqxial

to the product 'yV(t) XV (t) , where y is a constant (see Fig. 4).
r

Let v(t), in turn, be processed by a sharp, low-pass filter with

cutoff frequency f, such that

<f, < f ' < y . (41)
h h o

One may write

y V(t) • V (t) = y V (V + e) [sin Zrrv t] IsiniZtrv t + (p)]
r or o ^ o o

(42)

V V
\ or o/ / C \ r T

v(t) = y ::; 1 + r— [cos CO - cos (4 Try t ^ CD)]
2 \ V / o

o

Assume that cos[cp(t)] has essentially no power in Fourier frequencies f

in the region f ^ f . The effect of the low -pass filter then is to remove

the second term on the extreme right of (42); that is,

V V
v'(t) =: y-^|-^ (1 + ^) cos cp(t). (43)

o

(P(t)

ZttV
o

'< < 1This separation of terms by the filter is correct only if

for all t (see (4)).

The following two cases are of interest:

Case I

The relative phase of the oscillators is adjusted so that |cp(t)
|

< < 1

(in-phase condition) during the period of measurement. Under these conditions

2 3

r
(D

1
^, ^

,£3

u

1
i

4_)

^ >
w
I)

fl

>^
Td

a ^ u
O OJ <u
^ u

(U

^ '{: 4-) .^^ 4->
?H ^^ "X. -^w--

>
-/VV- > •

1

1
k

^^
,^

1

^ ^ 1

[x^

1 1

t
1

rence llator

CO)

roduc
etecto

O

1

ft C •4-1
1—

1

(U •-1 > I—

'

•r-l

P^ O o

24

since cos cp (t) '^ 1. That is to say one detects the amplitude noise € (t)

of the signal.

Case II

The relative phase of the oscillators is adjusted to be in

approximate quadrature; that is ' •

(p'(t) = (p(t) +
j (45)

where |<p(t)
|

<< 1. Under these conditions,

and

If it is true that

cos (Pit) = sin ip (t) «(p (t)

v'(t) =^ V V co'(t) + ^ V (p'(t)€(t)
Z or o Z or

g(t)

V
<< 1 for all t (see (3)), then (47) becomes

y
V (t) V V (p(t)

Z or o

(46)

(47)

(48)

that is, V (t) is proportional to the phase fluctuations. Thus, in order to

observe <p (t) by this method, (3) and (4) must be valid. For different

average phase values, mixtures of amplitude and phase noise are observed.

In order to maintain the two signals in quadrature for long

observational periods, the reference oscillator can be a voltage -controlled

oscillator (VCO) and one may feed back the phase error voltage (as defined

in (48)) to control the frequency of the VCO [19]. In this condition of the

phase-locked oscillator, the voltage v (t) is the analog of the phase

fluctuations for Fourier frequencies above the loop cutoff frequency of the

locked loop. For Fourier frequencies below the loop cutoff frequency of

the loop, V (t) is the analog of frequency fluctuations. In practice, one

should measure the complete servo loop response.

Z5

B. Period Measurement. Assume one has an oscillator whose
f(fc)

voltage output may be represented by (2). H

the total phase

^(t) = 277^ t + (D{t)
o

V
< < 1 for all t and

(5)

is a monotonia function of time (that is,
ZTTU

between successive positive going zero crossings of V(t) is related to the

<P(t)
^ 1), then the time t

average frequency during the interval t ; specifically,

1- = y (1 +y )To n (49)
If one lets t be the time between a positive going zero crossing of V(t)

and the M-th successive positive going zero crossing, then

M
, - .

T o^ ^n
(50)

If the variations ^ T of the period are sraall compared to the average

period t , Cutler and Searle [?] have shown that one may make a reasonable

approximation to (cr^(N, T, t )) using period measurements.
y o

C. Period Measurement with Heterodyning . Suppose that <p{t) is

a monotonic function of time. The output of the filter of Sec. VII, A, (43)

becomes
V V

(51)

V V
/. ^ or o

V (t) «^ y — cos (p(t)
if

V
.o

<< 1. Then one raay measure the period t of two successive

positive zero crossings of v (t). Thus

1 I- I- = V y
T on

and for the M-th positive crossover

M I-— = V y
T o ' n

(52:

(53)

26

The magnitude bars appear because cos (p(t) is an even function

of (p(t). It is impossible to determine by this method alone whether cp is

increasing with time or decreasing with time. Since y may be very
n

small (~ 1 or 10 for very good oscillators), t may be quite long

and thus measurable with a good relative precision.

If the phase, (p(t), is not monotonic, the true y may be n^ar

zero but one could still have many zeros of cos (p(t) and thus (52) and (53)

would not be valid.

D. Frequency Counters . Assume the phase (either ^ or (/?) is a

monotonic function of time. If one counts the nuraber M of positive
going zero crossings in a period of time t , then the average frequency

M
of the signal is — . If we assume that the signal is V(t) (as defined in
(2)), then

V = ^ (1 + y ) . (54)Ton
If we assume that the signal is v (t) (as defined in (48)), then

— = I^ |y
I

. (55)
T o

'

^n'

Again, one raeasures only positive frequencies.

E. Frequency Discriminators . A frequency discriminator is a

device which converts frequency fluctuations into an analog voltage by

means of a dispersive element. For example, by slightly detuning a resonant

circuit from the signal V(t) the frequency fluctuations -— ^P (t) are con-
verted to amplitude fluctuations of the output signal. Provided the input

amplitude fluctuations ~— are insignificant, the output amplitude
fluctuations can be a good measure of the frequency fluctuations. Obviously,

more sophisticated frequency discriminators exist (e.g., the cesium beam).

From the analog voltage one may use analog spectrura analyzers

to determine S (f ) , the frequency stability. By converting to digital data,

other analyses are possible on a computer. ..

27

F. Cororaon Hazards .

1

.

Errors caused by signal processing equipment . The intent

of most frequency stability measurements is to evaluate the source and not

the measuring equipment. Thus, one must know the perforraance of the

measuring system. Of obvious importance are such aspects of the measuring,

equipment as noise level, dynamic range, resolution (dead time), and fre- I

quency range.

It has been pointed out that the noise bandwidth f is very

essential for the mathematical convergence of certain expressions. Insofar

as one wants to measure the signal source, one must know that the measuring
1

system is not limiting the frequency response. At the very least, one must

recognize that the frequency limit of the measuring system may be a very
|

2
important, implicit parameter for either 0" (t) or S (f). Indeed, one

y y
must account for any deviations of the measuring systein from ideality

such as a "non-flat" frequency response of the spectrum analyzer itself.

Almost any electronic circuit which processes a signal will,

to some extent, convert amplitude fluctuations at the input terminals into

phase fluctuations at the output. Thus, AM noise at the input will cause
a tirae -varying phase (or FM noise) at the output. This can impose inaportant
constraints on limiters and autoinatic gain control (AGC) circuits when good

frequency stability is needed. Similarly, this imposes constraints on equip-

ment used for frequency stability measurements.

2. Analog spectruin analyzers (Frequency Domain) . Typical

analog spectrura analyzers are very similar in design to radio receivers

of the superheterodyne type, and thus certain design features are quite

similar. For example, image rejection (related to predetection bandwidth)

is very important. Similarly, the actual shape of the analyzer's frequency

window is important since this affects spectral resolution. As with receivers,

dynamic range can be critical for the analysis of weak signals in the presence

of substantial power in relatively narrow bandwidths (e.g., 60 Hz).

28

The slewing rate of the analyzer raust be consistent with the

analyzer's frequency window and the post -detection bandwidth. If one has

a frequency window of 1 hertz, one cannot reliably estimate the intensity

of a bright line unless the slewing rate is much slower than 1 hertz/ second.

Additional post-detection filtering will further reduce the maximum usable

slewing rate.

3. Spectral density estimation from time doraain data . It is

beyond the scope of this paper to present a comprehensive list of hazards

for spectral density estimation; one should consult the literature [2 - 5].

There are a few points, however, which are worthy of special notice:

a. Data aliasing (similar to predetection bandwidth

problems).

b. Spectral resolution. i

c. Confidence of the estimate.

4. Variances of frequency fluctuations , (J (t). It is not un-

common to have discrete frequency modulation of a source such as that

associated with the power supply frequencies. The existence of discrete

frequencies in S (f) can cause (j (t) to be a very rapidly changing function

of T. An interesting situation results when t is an exact multiple of the

period of the modulation frequency (e.g., one makes t = 1 second and

there exists 60 Hz frequency modulation on the signal). In this situation,

cr (T= 1 second) can be very optimistic relative to values with slightly

different values of t.

One also must be concerned with the convergence properties of

0" (T) since not all noise processes will have finite limits to the estimates

of a (T) (see Appendix A). One must be as critically aware of any "dead

time" in the measurement process as of the systera bandwidth.

5. Signal source and loading . Inraeasuring frequency stability

one should specify the exact location in the circuit from which the signal

is obtained and the nature of the load used. It is obvious that the transfer

29

characteristics of the device being specified will depend on the load and

that the measured frequency stability might be affected. If the load itself

is not constant during the measurements, one expects large effects on

frequency stability.

6. Confidence of the estimate . As with any measurement in

science, one wants to know the confidence to assign to numerical results.

Thus, when one measures S (f) or 0" (t), it is important to know the
y y

accuracies of these estimates.

a. The Allan Variance . It is apparent that a single sample

variance, cr (2, T, t), does not have good confidence, but, by averaging

many independent samples, one can improve the accuracy of the estimate

greatly. There is a key point in this statement --"independent samples."

For this arguraent to be true, it is important that one sample variance be

independent of the next. Since (T (2, t, t) is related to the first difference
y

of the frequency (see (11)), it is sixfficient that the noise perturbing y(t)

have "independent increments, " i.e., that y(t) be a random walk. In

other words, it is sufficient that S (f) '^ f for low frequencies. One
y

can show that for noise processes which are more divergent at low fre-
—p

quencies than f , it is difficult (or in-ipos sible) to gain good confidence

p
on estimates of 0" (T). For noise processes which are less divergent

— p
than f , no problem exists.

It is worth noting that if we were interested in Cr (N = °°, t, t ),

then the linait noise would become S (f ) ~ f instead of f as it is for

C (2, T, T). Since most real signal generators possess low frequency

divergent noises, (a (2, t, t) ) is raore useful than CT (N = «, t, t).
y y

p
Although the sample variances, CT (2, t, t), will not be normally

distributed, the variance of the average of m independent (non-overlapping)
samples of a (2, t, t) (i.e., the variance of the Allan Variance) will decrease

as 1/m provided the conditions on low frequency divergence are met. For

sufficiently large ra, the distribution of the m-sample -averages of

30

cr (2, T, t) will tend toward normal (central liniit theorem). It is, thus,
y

•possible to estimate confidence intervals based on the normal distribution.

As always, one may be interested in t -values approaching the

limits of available data. Clearly, when one is interested in T-values of

the order of a year, one is severely limited in the size of m, the number

of samples of CT (2, t, t). Unfortunately, there seeras to be no substitute
y

for many samples and one extends t at the expense of confidence in the

results. "Truth in packaging" dictates that the sample size m be stated
with the results.

b. Spectral Density. As before, one is referred to the

literature for discussions of spectrum estimation [2-5]. It is worth pointing

out, however, that for S (f) there are basically two different types of
y

averaging which can be employed: Sample averaging of independent estimates

of S (f), and frequency averaging where the resolution bandwidth is made

much greater than the reciprocal data length.

VIII. Conclusions

A good measure of frequency stability is the spectral density, S (f),

of fractional frequency fluctuations, y(t). An alternative is the expected

variance of N sample averages of y(t) taken over a duration t . With

the beginning of successive sample periods spaced every T units of time,

the variance is denoted by (7 (N, T, t). The stability measure, then, is the

2
expected value of many measurements of 0" (N, T, t) with N = 2 and T = T;

p
that is, G (t). For all real experiraents one has a finite bandwidth. In

y
general, the time domain measure of frequency stability, 0" (t), is

dependent on the noise bandwidth of the system. Thus, there are four

important parameters to the time domain ineasure of frequency stability:

31

N, the number of saraple averages (N = 2 for preferred raeasure);

T, the repetition time for successive sample averages (T = T for

preferred measure);

T, the duration of each saraple average; and

f , the system noise bandwidth.

Translations among the various stability measures for common

noise types are possible, but there are significant reasons for choosing

N = 2 and T = t for the preferred measure of frequency stability in the

tirae domain. This measure, the Allan Variance, (N = 2) has been

referenced by [12, 20-22], and more.

Although S (f) appears to be a function of the single variable f,

actual experiraental estiraation procedures for the spectral density involve

a great many parameters. Indeed, its experimental estimation can be at

least as involved as the estimation of cr (t).
y

32

APPENDIX A

We want to derive (Z3) in the text. Starting from (10) in the text,

we have
/ N N ,

N-1

N
E
n=l

n/ N

N N
Z <yO-i S E

1-1 j=i \ ' J

(N-l)T'=

N t^+^ }n^^
2/" dt" f dt' <y(t')y(t'')
n=l t. tn n

N N t- + T t. + T

^1 1 f dt/ dt' /y<t',y(t"X
i=l j=l t t. \ /

(Al)

where (9) has been used. Now

y(t)y(t")\ = R^(t - t") (A2)

where R (t) is the autocorrelation function of y(t) and is the Fourier
y

transform of S (f), the power spectral density of y(t). Equation (A2) is

true provided that y(t) is stationary (at least in the wide or covariance

sense), and that the average exists. If we assurae the power spectral

density of y(t), S (f), has low and high frequency cutoffs f , and f (if
y jii h

necessary) so that
00

/ S (f) df exists
• *6 y

then if y is a random variable, the average does exist and we may safely

assume stationarity.

33

In practice, the high frequency cutoff, f , is alvyays present either

in the device being measured or in the measuring equipment itself. YtHien

the high frequency cutoff is necessary for convergence of integrals of

S (f) (or is too low in frequency), the stability measure will depend on f .

The latter case can occur when the measuring equipment is too narrow-

band. In fact, a useful type of spectral analysis may be done by varying

f purposefully [l8].

The low frequency cutoff f may be taken to be much smaller than

the reciprocal of the longest time of interest. The results of calculations

as well as measurements will be meaningful if they are independnt of f

as f . approaches zero. The range of exponents in power law spectral

densities for which this is true will be discussed and are given in Fig. 1.

To continue, the derivation requires the Fourier transform relation-

ships between the autocorrelation function and the power spectral density:

O)

S (f) = 4 / R (T) cos 27Tf TdT,
y ^ y

R (T) = r S (f) cos ZTTf Tdf . (A3)
y ^ y

34

Using (A3) and (A2) in (Al) gives

<a^(N, T, T)> :=
y

N

(N-l)T^ ^ / dfS {i)fd?' f
tn+>VT

dt' cos ZTfiit'-t")

n n

N N^ a, t.+T I

1=1 j^i-A) y ^t. -4.
N

1

.t,+T ^t. + T

dfS (f)/ dt"/ dt' COS 27Tf(t'-t'')

1
J

(N-1)T'=

N r ^'°

2/
ln=l

N N

i=lj-l

,^^ ,^, sin"^ 7Tf T
dfS f g

y (TTif

S (f) /
df JL

'o (27t£)
2 cos 277fT(j-i) - cos 27Tf[T(j-i) + t]

cos 277f[T(j-i) - t] (A4)

(The interchanges in order of integration are permissible here since the

integrals are uniformly convergent with the given restrictions on S (f).
)

The first suramation in the curly brackets is independent of the summation

index n and thus gives just

sin Tlf T

y"' m]Njf
df S_(f) ^7^^ (A5)

The kernel in the second term in the curly brackets may be further

simplified:

2 cos 27rfT(j-i) - cos 27Tf (T(j-i) + tJ - cos 27Tf (T(j-i) - TJ

(A6)

= 4 sin 77f T cos 2TrfT(j-i),

35

The second term is then

r°° S (f) ^i N
^— sin'^ 77f t) ) cos 277fT(j-i) . (A7)

(TTf) 1^
J
= l

(The interchange of summation and integration is justified. ) We must

now do the double sum. Let

j - i = k,

277fT= X. (A8)

Changing summation indices from i and j to i and k gives for the

sum
.

; N N N N-i
.

S = V y cosx(j-i)= ^ V coskx. (A 9)
1=1 j=l i=l k=l-i

The region of sumnaation over the discrete variables i and k is

shown in Fig. 5 for N = 4.

The summand is independent of i so one may interchange the

order of suramation and sum over i first. The sumraand is even in

k and the contributions for k < are equal to those for k > 0, and so

we raay pull out the term for k = separately and \vrite:

/N-1 N-k \ N
S = 2

j
^ cos kx ^ 1 + £ 1

Vk=l i=l / i=l

N-1 \

= 2 I > (N-k) cos kx

k=l

+ N . (AlO)

N-1
1 d 1V ikx

This may be written as

S=N+2Re f N - - -r-iy e'^"" (All)
L 1 dx j Z^

where Re [U] means the real part of U and d/dx is the differential

operator. The series is a simple geometric series and raay be summed

easily, giving

36

Fig. 5 Region of Summation for i and k for N = 4

37

S = N + 2Re < !N
1 d e - e

I

"

i dx i , IX
1 - e

iNx ^^,, ix, '
^ , 1 - e - N 1 - e >

= N + 2Re ( ^ ' )
4 sin x/2

I

sin Nx/2

sm x/2
(A12) 1

Combining everything ^ve get, after some rearrangement,

N
<ay(N,T,T)>

^_^ J
[ dfS (f)

y

sin TTf T

{TTirf

1 - 7 Y^""" I (A13)
N sin IT r f T

where r = T/t. This is the result given in (23).

We can determine a number of things very easily from this

equation. First let us change variables. Let 77f t = u, then

«^y(N,T,T)> = ^^_•1)77T J

2 r • 2
^ , u , sin u ,' sin Nru

du S (— ) <1 - ,
y ttt' 2 ) ^,2 . 2 ju ^ N sm ru

(A14)

The kernel behaves like u as u -* and like u as u -* '».

Therefore (d (N, T, t)) is convergent for power law spectral densities,
(y

S (f ) = h f , -without any low or high frequency cutoffs for - 3 < C < 1.
Y ry

o 1 /
^

Using (A14) for power law spectral densities we find

<CJ^(N, T, T)) = T
-QC-l

h C^ for -3<a<l

and

C =
CL

where \i = -« -1

u \

(N

N f . <y sin u /, sm Nru Vdu u { 1 - /
1)77 ^ u I N sxn ru -•

(A15)

This is the basis for the plot in Fig. 1 in the text of jLt vs. a. For

a ^ 1 we raust include the high frequency cutoff f .

38

For N = 2 and r = 1 the results are particularly simple. We

have ^

<cr^(2, T, T)> = t"^"^ h^ -|^
I

duu*^"^ sin^u (A16)

for power law spectral densities. For N = 2 and general r we get

«, fi 1 o , cos2u(r4-l) , cos2u(r-l) \
/=° 1 -cos2u-cos2ru+

^^ ' + ^ l

du S (^^)
i ^ ?^

/. 3 .2, o / "^ \ sin u sm ru ,^ ,^.du S (— . A17
3

U

The first form in (A17) is particularly simple and is also useful for r = 1

in place of (A 16).

Let us discuss the case for ry ^ 1 in a little more detail. As

mentioned above we must include the high frequency cutoff, f , for

convergence. The general behavior can be seen most easily from (A13).
-2

After placing the factor t outside the integral and combining the factor

f with S (f) we find that the remaining part of the kernel consists of
y

some constants and some oscillatory terms. If 2Trf-i^T > > 1 it is apparent

that the rapidly oscillating terms contribute very little to the integral.

Most of the contribution comes from the integral over the constant term
-3

causing the major portion of the t dependence to be the t factor

outside the integral. This is the reason for the vertical slope at U = -2

in the fl vs. Ot plot in Fig. 1 in the text. - "

One other point deserves some mention. The constant term of

the kernel discussed in the preceding paragraph is different for r = 1

from the value for r 7^ 1 . This is readily seen from (A17) for N = 2;

for r = 1 the constant term is 3/2 while for r 7^ 1 it is 1. This is

the reason that 6 (r-1) which appears in some of the results of

Appendix B. In practice, 6 (r-1) does not have zero width but is

39

smeared out over a width of approximately (ZTT^f^ t) . If there must be

dead time, r ?^ 1, it is wise to choose (r-1) >> ZTTL t) '' or
h

(r-1) <<(2 7Tf, T) • but M'ith ZTTf T >> 1. In the latter case, one may
h h

assume r » 1

.

40

APPENDIX B

Let y(t) be a sample function of a random noise process with a

spectral density S (f). The function y(t) is assuraed to be pure real

and S (f) is a one-sided spectral density relative to a cycle frequency

(i.e., the dimensions of S (f) are that of y per hertz). (For additional

information see Appendix A, [7, 8, 18]. )

Let x(t) be defined by the equation

x(t) = ^ = y(t). (Bi)

Define: t is arbitrary instant of tirae and
o

n
N

n=l

and let f be a high (angular) frequency cutoff (infinitely sharp) with

27Tf, T > > 1.
h

Special Case

Special Case

y y \ 2T

[x(t^+2T) - 2x(t^+T) + x(t^)]'

2~

)

t ^^ = t + T, n = 0, 1, 2, .. ., (B2)n+1 n

t^+T X(t +T) -x(t )

y^ = 7 /
y(t) dt =

,
(B3)

1 r n

:b5;

!B6'

0=(T)^ (a^2,T,T)>= < 2 ___i^ ^_ ) • ,B7)

41

Definition:

D ^ (T) = < [x(t +2T) - 2x(t + T) + x(t )f ) . (B8)
X \ o o o /

Consequency of Definitions

D^(T) = 2T^ cr^(T) = 2CT^(T) . (B9)
X y X

Definition:

^(T, T) = ([x(t +T+ T) - x(t +T) - x(t + T) + x(t )f\ . (BIO)X \o o o o/
Consequency of Definitions:

J/)^(T, T) = 2T^ <a^(2, T, T)) . (BID
...

X y

Special Case:

0''(T, T) := D^(T). (B12)
X X

42

RANDOM WALK y

S (f) = -^
y f

-2
s (f) =

T
T h

Quantity Relation

<cr^(N, T, T)>

2

h_2 • ^^•Ti-^[r(N+ 1) - 1], r > 1

<rT^(N, T, T)>
y ^a-^^rf^---

a^ir) h^- i^4^. N=Z. r=l

D®(T) = 2a^ (T)
X X

J.

.
2{2Trf T

=

-2 6

^ S 3

j^ lilLIli-(3r _ 1), for r > 1
-2 D

^ o * "^

'T
—^ - 1 ' foi- I- ^ 1-2 6 r

(B13)

(B14)

(B15)

(B16:

(B17)

43

FLICKER, y

^y^^) - T S (f ) =—~-^

r=T/T, O^f^f,

Quantity

<CJ^(N, T, T))
y

<cr"(N, T, T)>
y

O-^(T)
y

D^(T) ^ 2CT^(T)
X X

Relation

N

^r wbrS'N-"'
n=l

-2(nr) In(nr)

+ (nr+l)"^ ln(nr+l) + (nr-1)^ Inlnr-l

h
N In N

1
' N-1 ,

(r = 1)

h • 2 In 2, (N = 2, r = 1)

h ' 4 T-^ In 2

-2r^ln r + (r + 1)^ ln(r + l)

+ (r-ir ln|r-l

h • 2T'' (2 + In r), for r > > 1

h • 2T'' (2 - In r), for r < < 1

(B18)

(B19)

(B20)

(B21)

fB22:

(B23)

44

WHITE y (Random Walk x)

S (f) = h
y o X (Zirfr

r=T/T, ^i <£

Quantity Relation

<CT^(N, T, T))

h

Y"
• ^ "^ for r ^ 1

h • 7- r(N+l) T "^, for Nr ^ 1
6

<(J^(N, T, T))
,

h

; • T -\ r = 1

y

h

^ • T "\ N = 2, r = 1

D^(T) = 2CT^(T)
X X

h • T

0^(T, T)
-A.

h • T , for r ^ 1

h • T, for r ^ 1

(B24)

B25)

(B26)

(B27)

(B28)

45

FLICKER X

S (f) = h |f|
y 1

1
S (f) =

Quantity

r = T/T, 277 1 T > > 1, 27TL T >> 1, ^ f ^ f

,

h h h

Relation

<(7^ (N, T, T)>
y

<CT^(N, T, T))
h

y

D^(T) =: 2cr^(T)
X X

(T, T)
X

h
1 (277 T)^

In

N-1

' n=l

2 2
n r
2 2 ,

n r -1

(B29)

, for r >> 1

2(N+1)

'l NT^(277)^
2 + ln(277f T) - %^

h N^ -

1

, r = 1 (B30)

2,.^ _x2
1 T^(277)

3[2 + ln(277f T)] - In 2 N=2, r=l (B31)

(2 77)

• 2 3[2 + ln(277f T)] - In 2 (B32)

; [2 + ln(277L T)], r >> 1
(277)^ h

h
(2 77)'

4

3[2 + ln(277f T)] - In 2 , r = 1 (B33)

"7 [2 + ln(277f, T)], r << 1
(277)^ h

46

WHITE X

s {£) = h r
y 2 V^X^^^ " (ZTlfy

r = T/T; 6 (r-1) =
k

1 if r = 1

otherwise

27Tf T>> 1, ^ f ^ f^
h n

Quantity Relation

<a^ (N, T, T))
y

N + 6 (r-1) "h
"2 N(277)^ T^

<CT^(N, T, T)>
y

h •
2

N + 1
N(27T)^ ,3

;'- = i

/-! /tX ^-
'^h

; N = 2, r = 1
y (2 77)^ T^

D^ (T) :r 2cr^(T)
^^2

(277)=

J/)^(T, T)

'^2-[^^\'-^'],a.>

:B34)

(B35)

(B36)

(B37;

(B38)

47

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[4] R. B. Blackman and J. W. Tukey, The measurement of power
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Dec. 1967.

48

[12] R.F.C. Vessot, et al. , "An intercomparison of hydrogen and cesiiira
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[18] R.F.C. Vessot, L. Mueller, and J. Vanier, "The specification of
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[19] Floyd M. Gardner, Phaselock techniques, New York: Wiley, 1966,

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[21] A. G. Mungall, D. Morris, H. Daams, and R. Bailey, "Atomic
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49

ACKNOWLEDGMENT

During the process of writing this manuscript and its many revisions,

numerous people have added comraents and encourageraent for the authors.

The authors are particularly indebted to Mr, D. W. Allan, Drs. D. Halford,

S. Jarvis, and J. J. Filliben of the National Bureau of Standards. The

authors are also indebted to Mrs. Carol Wright for her secretarial skills

and patience in preparing the many revised copies of this paper.

50

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