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o NBS TECHNICAL NOTE 683

MB3 U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

:

; '

Models for the Interpretation of

Frequency Stability Measurements

QC

100

. U5753

no. 683

c.2

1976
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards' was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology

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- Located at Boulder. Colorado 80302.
r 4 1976

Models for the Interpretation of

Frequency Stability Measurements

-^- \,ec^ <\\CcA Z-^^O, Q"

James A. Barnes

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

*o*T sk

V,u o* *

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

Edward 0. Vetter, Under Secretary - Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued August 1976
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 683

Nat. Bur. Stand. (U.S.), Tech Note 683, 44 pages (August 1976)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING Oh F ICE WASHINGTON: 197b

For sale by the Superintendent ot Documents. U.S. Government Printing Office . Washington. DC. ^0402

(Order by SD Catalog No. C13. 46:683). (Add 25 percent additional for other than U.S. mailing
Contents

Page

1. Introduction 1

2. Present Models 1

2.1 The measurement of frequency stability 1

2.2 Models and noise simulation 2

2.3 Model elements 3

2.3a Deterministic elements 4

2.3b Gaussian noise elements 4

2.3c Sporadic elements 4

2.4 Flicker noise models 6

3. Model Pathologies 7

3.1 The ultraviolet catastrophe 7

3.2 The infrared catastrophe and stationarity 8

4. Some Stationary and Convergent Models 12

5. The Adequacy of Present Models 17

5.1 Frequency steps and sporadic noise 17

5.2 Model implications 20

6. Conclusion 21

Appendix A. A Stationary, Convergent Model 23

Appendix B. Building an ARIMA Model to Fit a Given Spectrum 25

Appendix C. An Example of a Sporadic Noise with f~l Spectral Density 33

in
ACKNOWLEDGMENT

Several people had significant impact on the formulation of this paper. The author wishes to acknowledge

specifically many helpful discussions and exchanges of

correspondence with D. W. Allan, J. De Prins, S. Jarvis,

W. C. Lindsey, J. Rutman, and G.M.R. Winkler.

IV
Models for the Interpretation of Frequency Stability Measurements

James A. Barnes

The results of measurements of frequency stability are normally

interpreted in the terms of models. Typically, these models are expressed as a power-law for the power spectral density (e.g.,

Sy(f) = h_]f"'). The experiments which provide the basis for

tnese models are always limited in their range and include neither

zero nor infinite Fourier frequency. Some authors have neglected

the limits to the actual physical devices and the underlying data

and have encountered mathematical difficulties with the models at either zero or infinite frequency or both. These problems are associated with the model and not the actual oscillators being

modeled. By carefully taking into account actual limits of the

device and constraining discussions to be concerned only with

quantities which can be observed in a practical sense, one can avoid the problems of non-convergence and non-stationarity.

Even for stationary and convergent models, however, there is growing evidence that the typical Gaussian models may be inadequate.

There appear to be occasional (sporadic) steps in the frequency

of oscillators of a magnitude which is difficult to explain with

conventional models.

Key Words: Flicker noise; frequency; frequency stability; oscillators; phase noise; stationary models.

1. Introduction

This paper reviews models used in interpreting the measurements of frequency sta- bility. The purpose of this review is to identify models which are useful for the mea- surements. The reader is specifically referred to Slepian's paper [1], "On Bandwidth,"

and the discussion of "reality" (his "facet A") and models for reality (his "facet B").

In the literature one can find papers which propose many models for frequency

instabilities. Some of these models are simply expressed in mathematical terms but have

very questionable applicability to real devices. One often finds that these models ( not

the devices which they were intended to simulate) have severe pathologies such as non- existent instantaneous frequencies. This paper explores some of these model pathologies

and attempts to put them into perspective with the important goal: to adequately describe

real oscillators.

The paper also details the elements of conventional models currently in use, discusses

some of the experimental evidence supporting these models, and presents some results

suggesting an inadequacy of these models.

2. Present Models

2.1 The Measurement of Frequency Stability

The measurement of frequency stability has received a great deal of attention for

well over ten years. Some of the key events which have occurred recently are as follows:

a. In November 1964 NASA and IEEE sponsored a Symposium on Short-Term Frequency

Stability [2].

b. A special issue of the Proceedings of the IEEE was published [3] in February 1966

which was devoted to frequency stability.
c. Following this publication, a subcommittee of the IEEE was formed to define

precisely the term "frequency stability." To date, the only publication of

this subcommittee [4] appeared in October 1970 as an NBS Tech Note and was later published in May 1971. This document proposed the power spectral density

of the fractional frequency fluctuations and the "Two-Sample Variance" as the

recommended measures of frequency stability.

d. Additional papers have appeared [5,6,7,8,9,10], giving confidence intervals on the "Two-Sample Variance" and suggesting other variations of measurement methods

which lead to convergent results even in the presence of flicker FM. (While

there are numerous references to frequency stability, this list provides access

to a really vast field.

)

Insofar as oscillator models are concerned, nearly all current publications acknowledge the presence of fluctuations of frequency whose power spectral density varies

as the reciprocal Fourier frequency--i .e. , flicker FM--over substantial spectral

regions.

2.2 Models and noise simulation

Given a data set there are various means of analyzing the data and model building.

Once a model has been devised it is possible to use the model to simulate performance or predict future values of the noise in an optimal way. In use today there are three means

of data analysis: (1) Power spectral density estimation [11]; (2) Fitting of Auto-Regressive,

Integrated, Moving Average (ARIMA) models [12]; and (3) the use of the two-sample variances [4],

Oy(-r). Methods (1) and (3) have been recommended [4] as preferred measures of frequency

stabil ity.

It seems that the power spectral density is almost always the choice of the theoretician for use in models and design applications. For short-term stability measurements

(f > 1 Hz) an experimenter will often use an analog spectrum analyzer and obtain spectra

"on line." For longer term data (f < 1 Hz) one could make use of digital computers and

obtain spectrum estimates following any of a number of texts [11]. Unfortunately, the

process of spectrum estimation tends to be a rather specialized field, and many experimenters do not avail themselves of these techniques.

Another, more recent technique is the fitting of particular models (ARIMA models) to the data [12]. This has the very practical advantage that when this process is completed,

one has immediately a means of simulating the data on a computer, optimally predicting

future values, and estimating the power spectral density. Like direct spectrum estimation,

however, the analysis techniques tend to be specialized, and not many experimenters make

use of this technique. Of course, this may change in the future, but so far it is little

used for frequency-stability measurements.

Perhaps the most common means of analyzing frequency instabilities of oscillators

is the use of the two-sample variance, omt), and the use of tables as in Ref. [4] to infer the

associated spectral density. This common use of o-y(-r) probably arises from (1) the relative simplicity of its estimation from experimental data and (2) the common occurrence of

power-law- type spectral densities for the observed instabilities. In a certain practical
sense o

2 (t) allows a very rapid estimation of the spectral density S (f). In the sense of

obtaining the best confidence in the result, the spectrum estimated in this way probably

does not represent the most efficient use of the data, but it is a quick-and-easy method.

Since there exist three essentially equivalent methods of describing a given oscillator's instabilities, it is of value to be able to translate theoretically between these

2 different cases. Table 4.1 provides a translation between ay

(x) and S (f) for those few

special cases given (i.e., power-law spectral densities). While it is true that the cases given in Table 4.1 are yery commonly encountered in practice, they do not suffice for the

general problem. Cutler [4] has provided an equation for translating from the (one-sided)

power spectral density to the two-sample variance (also see Ref. [13]):

0|(T) /At c t*\ sin TTfr df Sy(f

> * -T^fT^ (1.1)

At present no simple, closed form of the inverse of this relation is known to the author;

however, Lindsey and Chie [6] have shown that S

y

(f) can be given in terms of the Mel Tin

transforms of the Kolmogorov structure functions [6,14], which in turn are closely related

to the two-sample variance.

In contrast, if one knows the parameters \§\ > °-jj' °

a for an ARIMA model [12], then

one can immediately estimate the spectrum by the relation [12]:

2 I 1 ^-t&rf . e^Ti^f -i2irqf 12

1 - <f>-,e -i2irf i4irf ,-i2Trpf|2

(1.2)

Now, however, the inference of the <j>'s and 0's from S

y

(f) is not so simple. Appendix B provides a means of estimating the ARIMA parameters from S y

(f) by a graphical technique.

Figure 2.1 provides a map for the translations discussed above.

There have been other suggested measures of frequency instability also, but these are

not in common use. They include the Hadamard Variance [15], the Kolmogorov structure functions [6], and others [5,7,16].

ARIMA

Model

Eq. (4.21 -

sy

(f)

Eg. (4.1

Table (4.1)

I -4 1

I 4

Appendix E3 I —

t J MSpecial Cases Only) y

Fig. 2.1 Model Translations

2.3 Model elements

In general, the elements of a model for the instabilities of an oscillator can be

divided into those caused by variations in the environment (e.g., temperature, acceleration,

3
magnetic field, etc.) and those intrinsic to the oscillator. Although the environmentally

dependent elements are very important (in some cases the most important elements), this

paper will treat only the intrinsic elements—the elements present for oscillators in

nearly ideal environments.

2.3a Deterministic elements

Deterministic elements are elements of the model which, in principal, can be

known with perfect confidence. For example, quartz crystal oscillators often display

a very stable, continuous, linear drift of frequency. The deterministic elements for

most oscillators include:

Periodic terms,

Phase (time) offset,

Frequency offset, and

Linear frequency drift.

While other deterministic elements are possible, these are the more common ones. There

are problems concerning how one estimates these quantities [4].

2.3b Gaussian noise elements

The typical models [4] used for the noise observed on oscillators can be expressed

as a specific functional form for the power spectral density of the fractional frequency

fluctuations, y(t). An often used functional form for the power spectral density is a power-law model

:

Sy(f) = h_

2 |f|" 2 + h^lfl" 1 + h Q

+ h^fl 1 + h 2

|f|

2

for (2.1)

o<.f

i £|f| <fh <-.

While other functional forms could certainly be treated within the model framework,

(2.1) is the most common form. To date, almost all models have been Gaussian in nature,

even if one does not assume a power-law dependence as in (2.1).

Typically, the behavior outside of the interval (f^, f

n

) is unspecified because it

hasn't been measured. One should note this explicit statement on the limits of the

Fourier frequencies, which ought to conform to the region of direct observation. Beyond

this range one is free to choose suitably convergent forms for the spectrum. While it

is true that one must assume some form for the spectrum outside of this range, one might just as well choose a convenient form which does not complicate life too much.

2.3c Sporadic elements

There have been suggestions that, on occasion, the frequency of an oscillator

suddenly takes on a new and permanent average value [17]. These changes have

limited documentation, and it is not clear whether these frequency steps are distinct and additional to the other parts of the model or just an unsuspected visual

aspect of the data. A third possibility is that the "flicker part" of the model

manifests itself as sudden changes in frequency [18]. Still, its spectrum could

be proportional to |f|~ . One possible model is sporadic flicker noise as postulated

by Mandelbrot [19]. Table 2.1 summarizes the model elements.

In terms of these model elements, the analysis scheme in general should follow

that of Fig. 2.2.
o

Ul •— o X> •— ro i/l 4-> L. CD ro o i/l F- o_ •—

(i) to o

4-> z

l/l T3 >- c

l/l ro < Q

I— o

nk ^uc

CD3

cr z

(U o

i_

Ll_ h-< L. o: o <

Qro

— on

03

(13 Q

T3O

CM

CM +

x:+

4-> C

cd

CD oO

+Jc

a)E

CD

CD

CDCO

•r-O *— c CD T3 c O

n3 E

10 »—

lO ro 3 D

n3 in CJ rs

l-1 CN1

— CM

CD103 Z

-t-> o ^ ro o -C r Q.

CD •—-• 13CT CD CD fc ^ 4-1 u- u CD <4- r i/> O c/i CD y- tr D "4- ,—

1_ cr O CD CD

Cl) l/l 4-> s_ > ro 4- u _c u c Q. • 

i_ CD

—"

T3

ro

D

O

a) cr CD •— c CD E 1_

T3

TO

1-o

Q. to

CL Q.

CD CD

4-> 4-1

1/1 l/l C C

CD CD T3 -a

"O TD 3 Z3

CM

CD

(0
2.4 Flicker noise models

During the same period of time considered above, various models of flicker noise have

also been developed.

a. A transfer function model was proposed [20], which generated flicker noise, but

its use was \/ery inefficient. (The discrete model was simply in error when the impulse

response was taken as proportional to x~*/3.)

b. Efficient analog and numerical approximations to flicker noise were presented in 1971 [21 ]. Although done independently of Ref. [12], these models are clearly ARIMA

models and directly amenable to the treatments of Ref. [12]. There is an important

difference between Refs. [12] and [21] in the sense that [21] is synthetic in its

treatment where [12] is analytic . That is, Ref. [21] considers the problem of synthesizing a (ARIMA) model which approximates a given power spectral density. In con- trast, Ref. [12] analyzes a specific noise sample by fitting ARIMA models to it. One

can then estimate the power spectral density of the sample noise by means of the

ARIMA parameters.

The noise model of Ref. [21] is the typical model used in current simulation

studies [22,23].

c. Independently and nearly concurrently to Ref. [21], Mandelbrot [24] developed

an efficient Gaussian noise generator which produces numbers which approximate a flicker process. This model is obtained by superposing several bandlimited, independent white noises in just such a way as to obtain an arbitrarily good approximation to

a flicker noise over a finite spectral region. Although there are many similarities

to Ref. [21], the model of Ref. [24] is not as useful in that it does not have a simple

inverse and the methods of Ref. [12] are not particularly useful for this model. This

is true even though the band-limiting filters used are of the Auto Regressive (AR) form.

d. It has been pointed out that diffusion processes [25] can give rise to flicker

noise. This is based on the recognition that Heavi side's [26] semi-infinite diffuser

is the physical analog of Abel's [27] half-order integral.

e. Halford [28] showed that there is a broad class of Poisson processes which

give rise to flicker noise. From the computer simulation point of view, these Poisson

noise samples are not nearly as easy to generate as the models given in Refs. [21]

and [24].

f

.

The models presented above are all similar in the sense that they usually generate

Gaussian noises. Unique among all of the models presently known to simulate flicker

noise is the sporadic model presented by Mandelbrot [19], which is non Gaussian, at least in orders higher than the first. A similar model has been presented by Rutman

and Ubersfeld [18]. The possible applicability of this sporadic model to real oscillators appears very interesting in view of recent results in several laboratories.

An example of a sporadic flicker model is presented in Appendix C.
3. Model Pathologies

Over the past many years people have become progressively more interested in the measurement of frequency stability. Today, oscillators exist whose frequencies might change

only a few parts in l(r 5 from one measurement to the next. The documentation of such outstanding performance in exact quantitative terms is indeed of interest.

The process of this documentation always involves the acquisition of data. It is significant to note that these data always have an upper frequency limit and a finite duration.

If one were to consider the Fourier transform of these data, it is clear that only a finite

range of Fourier frequencies could be resolved, which does not include zero Fourier frequency. If the measurement data spanned a duration T in time and had an upper frequency

limit fu, then it is clear that the Fourier transform is meaningful for Fourier frequencies

in the range from 1/T to f

n

. It is perhaps surprising (but true) that most model pathologies arise out of the

simple error of extrapolating experimental results beyond their range of applicability and

direct observation. By "model pathology" I specifically mean that the model (facet B) is pathologic, not the device or data (facet A) which were intended to be modeled.

These model pathologies fall naturally into two categories: the first pathology con- sidered arises from extrapolating behavior too high in frequency. This pathology is called

the "ultraviolet catastrophe." The other pathology arises from extrapolating behavior too

low in frequency and is called the "infrared catastrophe." In model pathology, a concomitant of the infrared catastrophe is the "non-stationarity bugaboo." I don't mean to imply

that all oscillators must of necessity be "stationary" in their performance; I only mean thatthe blind extrapolation of results to zero Fourier frequency should not be used as an indictment against the usefulness of stationary models . 3.1 The ultraviolet catastrophe

Many models of oscillators recognize the existence of "white noise" perturbing the

instantaneous frequency of the oscillator over some significant range of Fourier frequencies [29]. In long term this gives rise to a "Brownian Motion" of the phase fluctuations.

Now if one forgets about all of the band-limiting effects in the real oscillator,

amplifiers, etc., one can become concerned with the mathematically ideal Brownian Motion.

This Brownian Motion is characterized as having independent increments; that is, if <j>(t) is the Brownian Motion of phase, then

A<f>(t) = <f> (t + At) - 4 (t) (3-D

is independent (i.e., random, uncorrected) of A<f>(t') provided the two intervals do not overlap. The result is that the limit as Ax -> of the quantity A

^|

t ) does not exist [30].

That is, one cannot even speak of an instantaneous frequency defined by the relation [4].

Of course, the problem arises from the assumption that the differences, A<J>(t), are independent for all non-overlapping Ax > 0, even, say, Ax = lO" 2 ^ sec. If one looks at
real electronics, one knows that transistors and even the wires and connectors have very substantial frequency-limiting effects. The net result is that in real electronic circuits all orders of differentiation can be assumed to exist with complete safety [1]. One just does

not have to be concerned with the idea that coherent x-rays or y-rays might be coming out of

the output connector! (One should note that if the spectrum of the signal is band limited,*

then the spectrum of the frequency fluctuations is similarly band limited.) Thus, this is a pathology of the model and not of the device.

3.2 The infrared catastrophe and stationarity

Many experiments on oscillators reveal estimates of the power spectral density of the

frequency fluctuations varying as |f| a

for a ~ -1 over a significant but limited range of

the Fourier frequencies [29,31,32,33,34,35]. (This is normally called "Flicker Noise" frequency modulation.) If one now considers the "ideal" model of flicker noise with an assumed

spectral density varying as |f| for all f , then some significant problems arise.

First, it is obvious that the integral

/

f" | df - co

, (3,3)

f-j >

so one has problems with the "ultraviolet catastrophe." I assume that this aspect of model

pathology has been adequately handled above. Also, however, one has

/

f

1 >

fl-'dr... (3 - 4)

which imples an infinite amount of power in the very low frequencies--!' .e. , an "infrared

catastrophe." While some authors are not too bothered with either the infrared or the

ultraviolet catastrophe itself, they point out that Flicker noise implies a non-stationary

model [5,36], and hence one is hard put to even speak of a power spectral density at all.

Thus, the assumption of a power spectral density varying as |f| a for a f.-l and for all f is

not even self- consistent.

This problem is slightly more subtle than the ultraviolet catastrophe and perhaps

explains why many people become hung up on this issue. In order to put this problem into

perspective, it is of interest to consider a specific thought experiment.

Let us suppose that we have a rather large set of experimental data in the form of a single time series, x-s f° r 1 1 i IN. I assume that the xi are the result of measurements

at regular intervals x and that we have already carried out many experiments on this data

set. In particular, I assume that I have looked at the time series

* See Ref [1] for a discussion of band limits.
AXi = Xi+1

- Xi for 1 < i < N-l (3.5)

and discovered that the Ax,- are consistent with a model of random, uncorrected (i.e.

"white") numbers with, say, zero mean, a Gaussian distribution, and finite variance,

2 a , just to be specific.

In a loose use of the words, one might say that an acceptable model for the X;

is a Brownian Motion, but one shouldn't get trapped into either the ultraviolet or infrared

catastrophe too easily. One might, however, get concerned about the stationarity of the

X-j , since a Brownian Motion is nonstationary [5,36].

At this point it is of value to introduce another model which can also fit the observations. I will introduce this one model in two different but equivalent forms. First,

consider the apparatus of Fig. 3.1.

White

Gaussian

noise

generator

Fig. 3.1. Possible Model

< 4 < If one reads the meter at intervals t, one can obtain a time series y-j for 1 _ i _ N. Now, if one selects the filter time constant, RC, such that

RC » Nt, (3.6)

the yn

. would have essentially the same statistical properties as the x,- • Indeed, y. could

also be modeled well by a Brownian Motion.

An equivalent mathematical model would be to let [12]

Yj - *yi_i = a

1 (3.7)

where the a. are random, normal deviates with variance a?, and <)> is a constant chosen such

that

1-0 » N,

< 1 (3.8)

The point of significance is that the y-model js_ stationary [12]. This is easy to see from Fig. 1, since the y^ are the output of a linear, realizable filter processing a sta- tionary noise. Thus, the y-process has a spectral density, and it varies as

Sy(f)

-2

Nt

(3.9)

i

' ail If! 1.1

"2T
Now the question of significance is which model —the y-model or the Brownian Motion

model--is the best model for the x-j? I think that it is clear that the data themselves

cannot yield an answer to this question! One might have aesthetic preferences, but one

model is every bit as good as the other as far as objective tests are concerned. I should

mention that the same statements could have been made for flicker noises rather than

Brownian Motions, but the details are just a bit more involved.

There is another question which is not answerable from the data: Is the underlying

X-process stationary? The corollary to this is to note that even if the x-j were non- stationary (i.e., a Brownian Motion), the xi could be well modeled by a stationary process

(y-j) over a finite range (1,N). (Since the average frequency of an oscillator is normally

set by the manufacturer and even calibrated later, average values are of little consequence.)

The trivial nature of this stationarity issue is further emphasized by the example in Appendix B. On the basis of aesthetics (or "parsimony" [12]) one can object to the introduction

of parameters like RC or $, since, by construction, they are not observable. Indeed, if one predicts that a specific measurement depends upon one of these parameters in an important way, then he would conclude that the measurement requires more data than is available [1]. Thus, one is led to building "cutoff independent" [37] measures of frequency stability.

On the other hand, one can make a plausibility argument for the existence of an actual cutoff frequency, even though to measure it requires more data than is currently

available. One notes that the frequency of most quartz crystal oscillators is perturbed

by a flicker noise (|f|~ ), at least over a finite range of Fourier frequencies, not including zero frequency. If it did, in fact, include zero Fourier frequency, then its

output frequency now would almost surely be infinite (at least if one thinks of the

crystal as infinitely old). Actually, one would just be unable to bring its frequency

within specifications manually. But, since I would conclude that such an oscillator were malfunctioning (output frequency outside of specifications), I would not take data on it

in the first place. One could go further to the ridiculous and ask if the finite age of

the universe doesn't itself preclude the existence of low-frequency divergent Fourier frequency components down to zero frequency.

The point is that data like the x,- series (facet A) can be well modeled (facet B) by

either stationary, convergent models or non-stationary, divergent ones. In terms of

observation, neither model is more correct nor incorrect than the other. Which one a person chooses to use is a matter of convenience and simple prejudice. Most papers on frequency stability prefer the use of stationary statistics and the intuitively familiar

concepts like power spectral densities. Those who do not prefer this approach may not be

wrong, but they are dealing with a less familiar branch of statistics; and, based on experience, they seem to be prone to extrapolating results beyond the range of direct observation

with its attendant problems.

In the interests of completeness, I should note that there are other types of

non-stationarity besides low-frequency divergences. The conclusions presented here

10
are explicitly restricted to apparent low- frequency divergences only and not to other

types of non-stationarity.

In short, the infrared catastrophe is unobservable because of finite time limits,

and the ultraviolet catastrophe is not real because of finite pass bands of all real

electronics. The challenge is to develop useful, consistent models which reflect

these points rather than to be caught up into the mathematical complexities of unreal

models which ignore the real constraints of the equipment and the data-acquisition

process which were intended to be modeled.

In what follows, it will often be convenient to use an expression of the form,

"the noise has a |f| spectral density," for example. This does ncrt mean to imply that

this spectral form continues to either zero or infinite frequency. What it does mean is that there are many stationary and convergent models which are consistent with the avail-

-7 able data and which exhibit a predominantly |f| spectral character over the pertinent

range of Fourier frequencies.

It is true that many calculated quantities might depend on the detailed assumptions

of the spectral density of the model outside of the range of meaningful Fourier frequencies--

that is, outside of the range of direct observation. This, however, is not a weakness in

the model but rather serves to remind the theoretician that he is extrapolating beyond the

range of reliable data. A simple example is worthwhile. Consider a function x(t) which experimentally appears

i -2

to "have a |f| power spectral density." What one concludes from this is that the conven- tionally defined variance of x will depend critically on the low frequency limit of the

|f| behavior. Indeed, if experiment has not revealed this low-frequency cutoff, then

any estimate of this variance is questionable at best. (This is in contrast to Ref. [19].)

Clearly, the assumption that there is no lower cutoff frequency (i.e, zero Fourier frequency) is as unfounded as any other assumption. Also, even if it were known (by virtue

of other information) that x was "really" a Brownian Motion, it would still be possible to model x arbitrarily well over any finite spectral range (not including zero Fourier frequency)

i i-7 with a stationary process which "has a |f| spectral density."

As noted above, the goal is to develop good and useful models for the interpretation of

stability measurements of real oscillators. To the extent that the development of non- stationary and/or divergent models aids the realization of that goal, one should support

the development. To the extent that it is merely a mathematical exercise, it can be

appreciated for that, also. Otherwise, much of the discussions of non-stationarity and

divergences are counterproductive in that they raise false issues (e.g., the detailed

behavior at unobservably small Fourier frequencies). Ref. [1] is quite germane to the

discussions above.

It is probably true that several authors have erred in not being adequately explicit

and precise in their treatment of the infrared and ultraviolet catastrophes, although

their results seem to work. On the other hand, there have been authors who have carefully

and rigorously pursued these catastrophes to their illogical conclusions. The purpose of

11
this paper is to be a bit more careful and explicit regarding model assumptions and to

come to the accepted conclusions with (hopefully) some improvement in believability. The

results themselves are not new.

4. Some Stationary and Convergent Models

It is worthwhile to provide a specific example of a stationary, convergent model with

all orders of derivatives existing. The model presented also has application to some real

oscillators. The objective is to develop a mathematical model for an oscillator whose

frequency is thought to be perturbed by a (band limited) white noise.

Specifically, a model for the phase fluctuations will be developed which is stationary

and convergent. In particular, we will require the model to be a reasonable fit to the data

between the Fourier frequencies of, say, one cycle per one hundred years (~3 x 10" Hz)

and a few kilohertz--a range of about 14 decades. The upper frequency cuoff, of course, has

an obvious physical basis in bandwidths of narrow-band amplifiers and various processing

equipment. The low-frequency limit involves times which are reasonably longer in duration

than any manufacturer might be expected to investigate.

The model power spectral density of the phase is taken to be

-a2 f2

Vf) = h ov

o

2 J +

-2

' (4-D

where h is a constant, v is the nominal frequency of the oscillator, f

£ is the lower

cutoff frequency (f, 3 x 10" 10 Hz), a is a time constant associated with the upper

cutoff frequency (a ~ 10" 5 sec), and f is the Fourier frequency. Making use of Equation (8)

f Ref. [4], the power spectral density of the fractional frequency fluctuations,

for the model is just

, . 1 dd> y(t) = -*— ft

. (4. 2 ; 2 TMo

f

2 e -a

2 f2 V

f) = h ° f\ + f

Z (4.3)

" h„ for f . < f < - . o £ a

A plot of (4.3) is shown in Fig. 4.1, which indeed shows the "white" character of the frequency fluctuations over the requisite band of frequencies. It is probably worth noting

again that the detailed behavior of the actual oscillator may differ substantially from

the model outside of the specified range. However, meaningful measurements cannot reasonably depend on non-observables. Thus, our interest is to develop measures of frequency

stability which do not depend critically on these non-observables. Indeed, we will see

that one can define a measure of frequency stability which is quite insensitive to f r

12
too

CMC

cr

CD

SCM

CM 4- CM

1 CM 4- C 3

r— i CD +

CM CM <=?

>> 3

I/O

.*. <*D f O 1 E i

C 3 1

iS- o +-> 1

o

a; X

Qtn co

co

i i

II o ai

1

-o oj o 4- c r-

"""

^t" o

Ll_ CM

1—o

>-

I/O

X

13
Since, for the model, tf>(t) is stationary and has a power spectral density given by (4.1), we can obtain the autocorrelation function of the phase, R-Jt). The details of this calculation can be found in Appendix A. For the case where

f

£ a = 3 x 10" 15 « 1

one obtains

Ra(0)

2

TTV h 2f„

and if x » - - 3 x 10" sec, one obtains

R*(f):

2o o

1 - 2TTf £

T +

(27Tf £T) 2

(4.4)

(4.5)

We are now in a position to calculate the mean square of the change in average frequency over an interval t for the model. Specifically, one makes use of (4.2) to note that

^k s /

t. + T

k y(t)dt

d>(t k + x) - 4>(t

k

)

2tiv t (4.6)

where t, , , = t, + x.

k+1 k

Thus, the "two-sample variance," defined by the relation

,2s o

2 (t]

y

Ak+i ~hy

(4.7)

becomes

OJ(T) J /

2 2 2 \

*(t

k+2x;

2<|>(t k+T) + <|>(t k

)

l >

(4.8)

where the brackets denote infinite time average. Equation (4.8) can be expressed in

terms of the autocovariance functions, Ra(x), noting that

(UX) - <[<0(t

n+T).* (t

n )]> (4.9)

14
The result is

a

2 (x)

, 2 2 2

5TT V T 6^(0) - siy

+ 2Ra (2t)

(4.10)

Substitution of (4.4) and (4.5) into (4.10) yields

h c, s ,. o

o\,(t) ~ ~^- 2t L -d9(7Tf f T) (4.11)

However, by assumption

Trf^x « 1 ,

that is, all measurements involve averages of duration t, which are quite small com- pared to one hundred years, j- . Thus,

a

°; (t) - 2x (4.12]

in complete agreement with other results [4,29,31,32].

This same treatment can be extended to other power-law types of power spectral

densities. In particular, one can assume a stationary model for the phase such that

2-2

o .-a t S*(f) = h

Q v

g+2

(ff+r) 2

(4.13)

For this development one makes use of pair 569 in Campbell and Foster [38] instead of

444. The mathematics is a bit more involved than for white noise, but essentially the same outline can be followed. The detailed calculations will not be performed here,

but the results are provided in Table 4.1

15
Spectral Density* Spectral Density* Two-Sample Variance

Noise of fractional of phase fluctuations

Type frequency fluctuation

Sy(f > Vf > «!(t)

White

phase

noise

. 9 hJf| h v h

2

• °3fT h 2 o

(2TTT) 2

, 2

h

l v

o

|f| V - 2 3{[ Y +ln(27Tf

h T)]-ln 2}

(2ttt)

hv2

Ifl 2

h2R

Y = 0.577 . . .

-1

Ifl 3

h

-l

'

2 In 2

, 2

h -2v

o

Ifl 4

h -2-

(2tt) 2

| t | 6

Table 4.1

Flicker , , f, phase V 1

"

1

White . frequency o

Flicker , . _i

frequency -1 '

'

Random

walk ,

i f i-2

frequency -2'

'

It is perhaps worth emphasizing that for white frequency noise there is no problem

with stationarity of the frequency fluctuations. For the phase, however, a "convergence

factor," —~—~ , was used to obtain a stationary model for both phase and frequency. By

f +f

£

_ 2 f

2 explicitly displaying a high-frequency cutoff, e a

, we avoided all problems of infinite

variances and the non-existence of certain derivatives (e.g., frequency). Of course, the

high-frequency cutoff of the model has its physically observable counterpart in reality in

the form of numerous band-limiting devices.

It is also worthwhile making one other observation: The technique followed above is very close to the development of generalized functions [39] or the theory of distributions

[40]. Thus, it is not surprising that calculations based on the use of generalized functions have yielded the same results [31,32].

*Assumes that spectral density is valid between limits < f„ < f

< fh < °°, that f

£

T « 1, and that fur » 1. The high-frequency cutoff ,~fu,~is important for some noises and should be measured. n

16
5. The Adequacy of Present Models

5.1 Frequency steps and sporadic noise

If a high-quality oscillator were disturbed, it would not be surprising to note a sudden change in its output frequency. However, people have suggested [17], from time to time, that even the best clocks and oscillators display these sudden changes in frequency

even though they are in a nearly ideal environment. Although there is limited objective

documentation of these frequency steps ("F-steps") in the open literature [41,42], people

in the field often discuss their occurrences.

At the present time there appear to be three possible positions one might assume

relative to these F-steps: (1) It is a characteristic of the Gaussian noise processes

(i.e., Flicker FM), which cause the fluctuations to appear to undergo sudden steps in

frequency, but the Gaussian noise models are adequate to explain the appearance; (2) the

Gaussian noise models used to explain frequency and phase fluctuations need to be augmented

to include F-steps; and (3) there are apparent (non-Gaussian) F-steps, but they are a consequence of sporadic noise [19] , and this sporadic noise simultaneously explains both

F-steps and flicker FM--these are two facets of the same process.

These three possibilities are considered separately below:

(1) No F-steps . Figure 5.1 is the plot of the simulated phase of an oscillator

which is perturbed by a pure (Gaussian) flicker noise. This plot was obtained by

the use of an ARIMA model as discussed in Ref. [21]. Since this is a computersimulated noise, it is known that no special, ad hoc , additions to the model are needed to explain the changes in slope (e.g., at point A). Indeed, the

appearance of this plot should be typical of the phase plots of any oscillator

perturbed by ficker FM. The implication is that F-steps may just be a consequence

of the visual appearance of the data.

This hypothesis, however, can be tested by objective means and has been

tested to some extent [41,42]. Basically, one can define the problem to be one of detecting a "signal" (an F-step) in the presence of "noise" (all of the Gaussian

noises perturbing the oscillator). The optimum detector (in a signal-to-noise

ratio sense) is just a low-pass filter processing the first derivative of the frequency. Detection occurs if the output of the filter exceeds some preset con- dition of acceptability. Figure 5.2 shows the result of the use of such a detector

on actual clock data.

Since one has assumed that the "noise" part is Gaussian, it is a simple

matter to calculate the rate of false detection and compare this to the observed

rate of detection. In the test runs made to date on real oscillators (totaling

ten runs), one should have detected two false steps. In actuality, 12 to 15

F-steps were detected, which raises severe questions concerning the "no F-step"

option; that is, the pure Gaussian model (without steps) seems inadequate to explain the observed results. This is consistent with other observations.

17

(su) >]0y«3 3NI1 3AI1V13M

19
(2) Gaussian noise and F-steps . The F-step detector discussed above was designed

according to the assumptions that this model was correct; and in fact, F-steps were

detected. The observations are consistent with this model.

The ad hoc addition of F-steps to the Gaussian noise models appears adequate,

but still left unresolved are the very difficult questions of rates of occurrences

of F-steps, amplitude distributions, and correlations. This model has numerous parameters, only a few of which have been resolved by experiment. In the sense

of Ref. [12], this does not seem to be a "parsimonious" model--it may, however, be

a good model

.

(3) Sporadic models for flicker FM . Ref. [19] presents a class of non-Gaussian

models which are capable of generating flicker noise. That is, when noise samples

are subjected to typical tests (e.g., the determination of two-sample variances), the

results are totally consistent with the flicker assumption. Not only do samples of

this sporadic noise satisfy the flicker diagnostics, but the probability of detecting

"F-steps" with the step detector described above can be made similar to the actual

oscillator results.

Thus, at least superficially, one has two viable models for the description of

oscillator performance. There remain two significant questions: (a) is there any

real difference in the two models; and (b), if there is, which model is the best

model? The first of these questions is discussed in the next section.

5.2 Model implications

In the previous section, three models of oscillator noise were proposed:

1. No F-steps, just Gaussian noise,

2. Gaussian noise and F-steps,

3. Gaussian noise (short-term) and sporadic noise.

For the sake of argument, let's suppose that model 3 above represents "reality," and

that, incorrectly, one has chosen model 2 for use in designing a clock system.

It is important to note that the apparent F-steps in model 3 are eventually selfcorrecting ; no adjustments should be made in long term. There would be other corrections

which might be used to advantage if model 3 were accepted.

Since F-steps (model 2) are detectable with reasonable probability, one can design a system to detect and "remove" these steps. However, errors of detection and errors in quantifying the sizes of the steps will naturally occur. Since these errors will tend to

be random and uncorrelated , the resulting clock system will become perturbed by a random

walk of frequency.

While it is possible that model 3 ("reality") was not as divergent in long term as a random walk of frequency, the act of "correcting" the F-steps guarantees that the resulting

system will have random walk of frequency perturbations. Thus, the use of model 2 could

lead to an overall deterioration of the long-term stability of such a clock system

relative to a much simpler use of the clock data based on model 3. Indeed, one can ask

if this state of affairs is not a reality in some major time scales currently in widespread use.

20
While there may well be other implications to the use of models 2 and 3, the above

example at least shows that the differences are not trivial. Also, model 2 has been

applied to the "correction" of F-steps in actual clock systems [43] so it is a very real

problem.

6. Conclusion

In all theoretical developments of oscillator statistics, models are used. Typical

oscillator models assume a power-law dependence on the Fourier frequency for the power spectral density (e.g., S y (f) = h_-|f~') over a range of Fourier frequencies. Although the

experimental basis for these models involves only a finite range for the Fourier frequency

components not including zero or infinity, some authors extrapolate these models beyond the

range of observation. The error committed by extrapolating a non-integrable spectrum toward

the high-frequency end is called the "ultraviolet catastrophe" and toward the low end is called the "infrared catastrophe." A concomitant of the "infrared catastrophe" is the "non- stationarity bugaboo." These are problems of oscillator models, not of actual oscillators [1].

One can show that a properly constructed, stationary model with a {very, very) lowfrequency cutoff cannot practically be distinguished from a model with an "infrared

catastrophe." This is true provided (1) that average values are not important, and

(2) that one is not willing to perform experiments which require hundreds or thousands or

even more years for the answer. Both of these conditions are true for oscillator models,

since (1) the absolute phase is only observable modulo 2tt for phase noise models, and the

frequency of the oscillator is conditioned by the manufacturer to be within specifications

and hence average frequency is not important for frequency models; and (2) practical uses

of the stability measurements preclude extended experiments.

Some authors have objected to the use of models with low-frequency cutoffs because

important quantities (like some variances) depend critically on the exact cutoff frequency. However, the important point to note is that if a quantity depends critically on the power in spectral components which have not been measured, then one is on shaky ground

at best to estimate such a quantity at all. For example, if it happens to be true that

an experiment suggests that a substantial fraction of the total noise power might lie in

Fourier components below the reciprocal data length, then the experimenter can make no believable estimate of quantities which depend critically on the total noise power. This

is not a fault of models; it is a real problem caused by limited data. Thus, one can object to stationary models with low-frequency cutoffs only on aesthetic grounds or their

failure to model other aspects (not low-frequency dependent) of oscillator behavior.

As for the "ultraviolet catastrophe," all real electronic devices have very finite

frequency limits. In fact, one can safely assume that all orders of differentiation of

the phase exist as well. Thus, the "ultraviolet catastrophe" has its resolution in the

very real, measurable band limits of the oscillator circuitry.

While one can construct stationary, convergent Gaussian models for the noise

observed on real oscillators, there is growing evidence that these models might be

21
inadequate. It has been suggested that one might have to add to the typical noise models

the possibility of sudden steps in the instantaneous frequency of the oscillator. Rather

than this ad_ hoc approach, it might be possible to model both the flicker noise and the

steps with the sporadic noise models of Mandelbrot. It is not yet known which, if any,

of these models suggested above is best.

22
Appendix A. A Stationary, Convergent Model

Consider the two functions

A(f)=-T

] 2 (A_1)

A

+ f

and

B(f) = e -a

2

f

2 (A-2)

The Fourier transforms are

a( t) -f. e

- ZvfM (A-3)

I

and

2 b( t ) =^. e -(4) (A-4) a

respectively, where use has been made of transform pairs 444 and 710, respectively, from Campbell

and Foster [38].

If we now assume that the (one-sided) power spectral density of the phase is

Sjf) = v* h n A(f)B(f), (A-5)

tp u o

it is possible to calculate the autocovariance function, RA (t), as the Fourier transform of (A-5).

Or, more simply, R,(t) can be calculated from the convolution of a(t) with b(t). That is,

2 °°

( t ) = !^° f a(t -u)b(u)du, (A-6)

or, explicitly,

V ~~^h~2 |

e J

e\ a ) * du

-co

oo 2

+ e

2^ / e-(^) " 27Tf ^

U du ]

•

;a-7)
(The factor of 1/2 arises from the assumption that S^(f) is a one-sided density). By

changing variables in the integrals in (A-7), R.(t) can be written in the form

2. TTV h tytj - ,£* ,<'i«J e2 -v

[, + erf (^ . V)

where

+ e ^f£t

1 - erf (r + '*•)"

(A-8)

2 erf(x) = A /" e -X dx

as in Ref. [14].

For fo ~ 3 x 10" 10 Hz and a ~ 10" 5 sec, then f,a ~ 3 x 10" 15 « 1 Thus,

2, TTV h V0) ~Tff

(A-9)

:a-io)

and 2, (2TTf £

T)'

|j

" Zirftf + —2T :a-id

for — << t << l/f«. As noted in the text, RAt) diverges as f^ approaches zero; however,

for positive, non-zero values of f

? one can have a stationary and convergent model with

all derivatives defined.

24
Appendix B. Building an ARIMA Model to Fit a Given Spectrum

In general, ARIMA models can approximate a wide range of spectral shapes. This

appendix, however, will consider only a rather restricted set of these models. Also,

the objective of this appendix is to provide a means of building an ARIMA model to fit

a prescribed spectrum. Other techniques [12] emphasize the building of ARIMA models to fit a particular data set regardless of its spectrum.

One method of simulating a flicker noise with ARIMA models has already been published

[21]. The treatment here will be a graphical approach using Bode plots.

Consider a random, uncorrelated, Gaussianly distributed, discrete time series, a t

, with mean zero and variance a . Also consider a time series to^ deduced from the a

t by

the equation

u, t

- ^t

_ } -4> 2

co t

_ 2

- ...-<y>

t

_ p = a

t

- 8

1 a t

_ 1

2 t-2 q t-q

(B-l)

where the cj^-'s and G-'s are constants. We can define the index-lowering operator, B,

by the relation

Bx t -=X t _v (B-2)

This allows (B-l) to be rewritten in the form

(1 - ^B - *

2 B

2

- ... - cf, p

B p

) Ut = (1 - 0.B - *

2

B

2

- ... - B

q )a

t (B-3)

The expression in parentheses on the left side of (B-3) is called the Auto Regressive (AR)

operator of order p. That on the right is called the Moving Average (MA) operator of

order q. We can further define another time series, Z

t by the relation

A

d Z

t

= cot

, (B-4)

where AZ

t = (1-B)Z

t

= 1^ - Z

t

_-| . That is, Z

t is the d-fold (finite) integral of wt

. Thus,

Z. is an Auto Regressive, Integrated, Moving Average process defined by the <j>-. 's,6 n

- 's , oi,

and the d differences of (B-4). In particular it is an ARIMA (p,d,q) process.

In order for oj^ to be stationary and invertible, there are restrictions on the <j>'s and e's [12]. However, for the present discussions it will suffice to consider only the

following two processes

OJ t

= (1 - 6B)a

t (B-5)

(1 - 4>B)x t

= yt (B-6)

For this case it is sufficient that -1 < e < 1 and -1 < <\> < l

.

25
In these equations one sees that if one sets <)> = 9 and y^ = uu, then one obtains

Xf - a

t

. That is, if one considers (B-5) and (B-6) to define two digital filters,

then (B-5) and (B-6) are inverse filters for each other when $ - 0. Thus, it is a simple

matter to extend any understanding one gains about one filter to the other.

It is of value to consider the effect of a single MA filter of the form (B-5) when

is further restricted to the interval 0^9 < 1. Figure (B-l) shows the general behavior

of the transfer function (magnitude squared) of such a filter and the straight-line approximations to this transfer function. The "knee" in the approximation occurs at a frequency

f_ which is related to 6 by the empirical relationship

1-TTf

9 * v^r (B " 7) c

which will not be derived here.

Similarly, Fig. (B-2) is the transfer function of the AR filter with $ = 9 = 0.728 as

in Fig. (B-l). Hence, also,

1-TTf (B-8)

1+TTf

c

We can now construct an approximation to a process which is a mixture of flicker noise

and white noise. Figure (B-3) shows the Bode plot of the desired filter transfer function.

The frequency axis of Fig. (B-3) is in terms of cycles per data spacing. Thus, the Nyquist

frequency is just 1/2. For the sake of the example, we will assume that it is sufficient

to approximate the spectrum to a lower frequency of .002 (cycles per data spacing).

Figure (B-4) superimposes the Bode plot of the filter which will be used onto the

spectrum of Fig. (B-3). From each of the "knees" in the approximation one can determine

the appropriate <J> or 9 for a filter to be cascaded in series.

Thus, from "knees" at the frequencies of .0233 and .0033, one obtains <j)'s for the

AR filters (eq. B-6) of .8636 and .9795, respectively. These numbers have been calculated

from eq. (B-8). Similarly, from the "knees" at the frequencies of .062 and .0087, one obtains values of .6740 and .9468, respectively, for the MA filters. One decides to use an AR filter if the function turns downward in response with increasing frequency as

in Fig. (B-2). Correspondingly, one chooses a MA filter when the function turns upward as

in Fig. (B-l).

The final filter can be obtained by cascading the output of one filter to the input

of the next in the form

a), = (1 - .6740B)a

t

(1 - .8636B) x

t

= w

t

yt

- (i - .9468B)x

t

(1 - .9795B) z

t

= y t

26

(B-9)
10

o

QO)

.1

/

Slope / 6 db/octy

/ /

//

Z

t

= (1 - 0B)a

t /

6 = 0.728 //

y/ / ^

/

/> —^

/

f = .05

c

.001 ,01 1 Frequency

Fig. (B-l) Transfer Function of Simple MA-Filter

27
100

10

o

en

f = 0.05

c ^

\ \ \

(1 - *B)Z

t

= a t \A

<j) = 0.728 V^

Slope

\

\ \^

-6 db/oct. \

\

.001 .01 1 Frequency

Fig. (B-2) Transfer Function of Simple AR-Filter

28
100

10

>,

CD O

o

CDC-

\. Slope ^^ -3 db/oct.

.001 .01 .1 Frequency

Fig. (B-3) Bode Plot of Desired Spectrum

29
100

10

CD O

\

\

\N f = .0033

c \

\N \\ nSlope ^ \ -6 db/oct \ \

\\

\ \

\ f = .0233

\

c

f = .0087

c . \ Slope X \ -6 db/oct. \ \

\\

\\^

f = .062

c

001 .01 1 Frequency

Fig. (B-4) Bode Plot of Filter Cascade

30
where a+ is the input to the cascade and z t

is the output. One can eliminate co t

, xt

and y^ and write an equivalent expression in the form

(1 - *

1 -B - cj) 2 B

2 )z

t

= (1 - 9.,-B - 2 B

2 )a

t

, (B-10)

where

^

= 1.8431, e

1 = 1.6208

\> 2

= -.8459, 9 2

= -.6381

(B-ll)

Making use of equation (4.2) one can obtain the spectrum corresponding to (B-10).

Figure (B-5) is a plot of the spectrum superimposed on the Bode plot of Figure (B-3).

The parameter a

2

= 0.319 was selected to make the spectrum fit the Bode plot in amplitude.

In this example, one could let z t model the frequency of an oscillator, and it is clearly stationary [12]. To model the phase, one could define a parameter v-t by the

relation

(1 - <t>'B)\> t = z t (B-12)

where -, , , is large compared to any data sample to be tested.

In the current example fn was taken as .002. Thus, one could select <J>' ~ .99999

to be quite adequate. Note that v

t is also stationary.

By selecting <f>' = .99999 one obtains a stationary model for v

t if one so desires.

Of course, for actually generating a data set one could use <$>' = 1, since no observational difference in the simulated data would result--that is why <j>' was chosen as close to unity as it was. Thus, the entire difference between a stationary and a nonstationary model (v t

) for the phase is centered in whether one chooses .99999 or

unity for <j>'. Further, this choice is totally unobservable for short data sets

(N ~ 1/f^ = 500).

Clearly, these techniques could be used to develop an ARIMA model to simulate

data over a much broader range of frequencies than done here. In any event, equations

(B-10) and (B-12) allow one to simulate a stationary noise with the desired spectrum.

31
100

10

Q

S-

+-> o

CL

\

\

\

\

vs.

>

o

2

= 0.319

3

001 .002 .01 .1 Frequency

Fig. (B-5) Actual Spectrum of ARIMA (2,0,2) Model

.5

32
Appendix C. An Example of a Sporadic Noise with f" Spectral Density

Consider the identically distributed, independent, random variables t^ with Gaussian

distribution, zero mean, and unit variance. Also consider the independent, random variables r , with a rectangular distribution in the range e < r

n < 1 , where <_ e < 1. That is, the distributions are as indicated in Figure (C-l).

Next define mn

= —

.

ii r

n Since the r are rectangularly distributed,

n

3 J

p(r) 1

' e

< r < 1 1-e

otherwise,

(C-l)

then the distribution of m is given by

n

3 J

p' (m)dm = p(r)dr

1 1-e Vm

(C-2)

Thus, as shown in Fig. (C-2),

P'(m)

1 1-e ~1

m

, for 1 Imi

(C-3)

1

, otherwise.

One can obtain a sporadic noise, [19], x n

> whose spectral density is approximately

f (flicker noise) over the range e < f < 1/2, by following the procedure below:

Define:

= <

to , for 1 n £ m,

u)-| , for m, < n £ m-, + m~

w

2

, for m-j + m

2 < n 1 m, + nu + m

3 V for t

k < n < t k+1

(C-4)

where t

fc = t

|< _ ] + m^. An example of such a noise is presented in Fig. (C-3).

33
p(co)

Fig. (C-l)a. Distribution Density of

p(r)

Fig. (C-l)b. Distribution Density of r

34
1 1-e

Fig. (C-2) Distribution density of m = —

35
.CD

-a

reoc

O-E

X

36
References

[1] Slepian, D., On bandwidth, Proc. IEEE, Vol. 64, No. 3, pp. 292-300,

(March 1976).

[2] Proc. IEEE-NASA Symposium on Short-Term Frequency Stability, NASA SP-80,

Goddard Space FLight Center, Greenbelt, Md., November 23-24, 1964.

[3] Proc. IEEE , vol. 54, No. 2 (Feb. 1966).

[4] Barnes, J. A., et al . , Characterization of frequency stability, IEEE Trans.

Instrum. Measure, vol. IM-20, pp. 105-120 (May 1971).

[5] Boileau, E., Picinbono, B., The statistical study of phase fluctuations and

oscillator stability, IEEE Trans, on I&M, vol. IM-25, No. 1, pp. 66-75

(March 1976).

[6] Lindsey, W. C, Chie, C. M., Theory of oscillator instability based upon

structure functions. To be published.

[7] Rutman, J., Characterization of frequency stability: A transfer function

approach and its application to measurements via filtering of phase noise,

IEEE Trans, on I&M, IM-23, No. 1, pp. 40-48 (March 1974).

[8] Lesage, P., Audoin, C, Characterization of frequency stability: uncertainty

due to the finite number of measurements, IEEE Trans, on I&M, IM-22, No. 2, pp. 157-161 (June 1973).

[9] Lesage, P. Audoin, C, Correction to: characterization of frequency sta- bility: Uncertainty due to finite number of measurements, IEEE Trans, on

I&M, IM-23, No. 1, p. 103 (March 1974).

'10] Lesage, P., Audoin, C, Correction to 'characterization of frequency stability:

uncertainty due to the finite number of measurements.' To be published.

'11] Jenkins, G. M., Watts, D. G., Spectral analysis and its applications, Holden-Day,

San Francisco (1968).

12] Box, G.E.P., Jenkins, G. M., Time series analysis, Holden-Day, San Francisco,

Calif. (1970).

"13] Rutman, J., Comment on 'characterization of frequency stability,' Trans, on I&M,

IM-21, No. 1, p. 85 (Feb. 1972).

"14] Kolmogorov, A. N., Dissipation of energy in locally isotropic turbulence, Akad.

Nauk. USSR, vol. 32, No. 16 (1941).

"15] Baugh, R., Frequency modulation analysis with the Hadamard variance, Proc. 25th

Annual Symp. on Frequency Control, April 26-28, 1971.

|16] Cornelissen, G. (Ph.D. Dissertation), Brussels (1976).

"17] Winkler, G.M.R., private communication.

18] Rutman, J., Ubersfeld, J., A model for flicker frequency noise of oscillators,

Proc. IEEE, vol. 60, No. 2, pp. 233-235 (Feb 1972).

.19] Mandelbrot, B., Some noises with 1/f spectrum, a bridge between direct current

and white noise, IEEE Trans, on Info. Theory, vol. IT-13, No. 2, pp. 289-298

(April 1967).

"20] Barnes, J. A., Allan, D. W., A statistical model of flicker noise, Proc. IEEE,

vol. 54, No. 2, pp. 176-178 (Feb. 1966).

37
[21] Barnes, J. A., Jarvis, S., Efficient numerical and analog modeling of flicker

noise processes, NBS Tech. Note 604 (June 1971).

[22] Wilson, R. G., Prediction of phase (time) errors in oscillator-driven clocks,

Aerospace Report No. TOR, 0075 (5461-04)-! , The Aerospace Corp. (June 1975).

[23] Mealy, 6. L., Sensor data bank development: time standard error models and

data file, Tech. Report AFAL-TR-75-65, Air Force Avionics Lab. (June 1975).

[24] Mandelbrot, B. B., A fast fractional Gaussian noise generator, Water Resources

Research, vol. 7, No. 3 (June 1971).

[25] Barnes, J. A., The generation and recognition of flicker noise, NBS Tech.

Note 9284 (June 1967).

[26] Heaviside, 0., Electromagnetic theory, pp. 128-129, Dover Publications,

New York, N. Y. (1950).

[27] Courant, R., Differential and integral calculus, II, Chapt. IV, Sect. 7,

Interscience, New York, N. Y. (1957).

[28] Halford, D., A general mechanical model for |f| a spectral density noise with

special reference to flicker noise l/|f|, Proc. IEEE, vol. 56, No. 3, pp. 251-258

(March 1968).

[29] Cutler, L., and Searle, C, Some aspects of the theory and measurement of frequency fluctuations in frequency standards, Proc. IEEE, vol. 54, pp. 136-154

(Feb. 1966).

[30] Doob, J. L., The Brownian movement and stochastic equations, Annals of Mathematics, vol. 43, No. 2 (April 1942).

[31] Barnes, J. A., Atomic timekeeping and the statistics of precision signal generators, Proc. IEEE, vol. 54, pp. 207-220 (Feb. 1966).

[32] Allan, D. W., Statistics of atomic frequency standards, Proc. IEEE, vol. 54,

pp. 221-230 (Feb. 1966).

[33] Atkinson, W. R., Fey, L., Newman, J., Spectrum analysis of extremely low frequency variations of quartz oscillators, Proc. IEEE (Corres.), vol. 51, p. 379

(Feb. 1963).

[34] Fey, L., Atkinson, W. R., Newman, J., Obscurities of oscillator noise, Proc. IEEE (Corres.), vol. 52, pp. 104-105 (Jan. 1964).

[35] Troitsky, V., Certain problems of the theory of fluctuations in oscillators. The

influence of flicker noise, Isv. Vysshikh Uchebn. Zavedenii, Radiofiz., vol. 1,

pp. 20-33 (1958).

[36] DePrins, J., et al . , Frequency-domain interpretation of oscillator phase sta- bility, IEEE Trans, on IPM, vol. IM-18, No. 4, pp. 251-261 (Dec. 1969).

[37] Barnes, J. A., Allan, D. W., Effects of long-term stability on the definition and

measurement of short-term stability, IEEE Symposium on Short-Term Frequency Stability, Goddard Space Flight Center, Greenbelt, Maryland, pp. 119-123, NASA SP-80.

[38] Campbell, G. A., Foster, R. M., Fourier integrals for practical applications,

D. van Nostrand, N. Y., N. Y. (1948).

[39] Lighthill, M., Introduction to Fourier analysis and generalized functions,

Cambridge (1962).

38
[40] Zemanian, A. H., Distribution theory and transform analysis, McGraw-Hill, N. Y. (1965),

[41] Allan, D. W., et al . , Performance, modeling, and simulation of some cesium beam

clocks, Proc. of the 27th Annual Symposium on Frequency Control, pp. 334-346 (1973).

[42] Barnes, J. A., Unpublished lecture notes (August 1975).

[43] Winkler, G.M.R., Hall, R. G., Percival , D. B., The U. S. Naval Observatory clock time

reference and the performance of a sample of atomic clocks, Metrologia, vol. 6, No. 4,

pp. 126-134 (Oct. 1970).

39
MbVlliA.^ V. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR RKPORT NO.

NBS-TN-683

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

MODELS FOR THE INTERPRETATION OF FREQUENCY

STABILITY MEASUREMENTS

5. Publication Date August 1976

6. Performing Organization Code

277.00

7. AUTHOR(S)

James A. Barnes

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

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Same as item 9

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15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here. ) The results of measurements of frequency stability are normally interpreted in the

terms of models. Typically, these models are expressed as a power-law for the

power spectral density (e.g., S

y

(f) = h_-|f_1

). The experiments which provide the

basis for these models are always limited in their range and include neither zero norinfinite Fourier frequency. Some authors have neglected the limits to the actual

physical devices and the underlying data and have encountered mathematical difficulties

with the models at either zero or infinite frequency or both. These problems are associated with the model and not the actual oscillators being modeled. By carefully

taking into account actual limits of the device and constraining discussions to be

concerned only with quantities which can be observed in a practical sense, one can avoid the problems of non-convergence and non-stationarity.

Even for stationarity and convergent models, however, there is growing evidence that

the typical Gaussian models may be inadequate. There appear to be occasional (sporadic

steps in the frequency of oscillators of a magnitude which is difficult to explain

with conventional models.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Flicker noise; frequency; frequency stability; oscillators; phase noise; stationary

models.

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