fc^;= NBS TECHNICAL NOTE 632

U57S

U.S. EPARTMENT

OF

COMMERCE

National

Bureau

of

Standards

Frequency Stability Specification

and Measurement: High Frequency

and Microwave Signals
NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards1 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 and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measure- ment system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Center for Computer Sciences and

Technology, and the Office for Information Programs. THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consistent system of physical measurement; coordinates

that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. The Institute consists of a Center for Radiation Research, an Office of Measurement Services and the following divisions: Applied Mathematics—Electricity—Heat—Mechanics-—Optical Physics—Linac

Radiation2—Nuclear Radiation-—Applied Radiation-—Quantum Electronics3—

Electromagnetics3—Time and Frequency3—Laboratory Astrophysics3—Cryogenics3

. THE INSTITUTE FOR MATERIALS RESEARCH conducts materials research lead- ing to improved methods of measurement, standards, and data on the properties of

well-characterized materials needed by industry, commerce, educational institutions, and Government; provides advisory and research services to other Government agencies; and develops, produces, and distributes standard reference materials. The Institute consists of the Office of Standard Reference Materials and the following divisions: Analytical Chemistry—Polymers—Metallurgy—Inorganic Materials—Reactor

Radiation—Physical Chemistry.

THE INSTITUTE FOR APPLIED TECHNOLOGY provides technical services to pro- mote the use of available technology and to facilitate technological innovation in industry and Government; cooperates with public and private organizations leading to the development of technological standards (including mandatory safety standards), codes and methods of test; and provides technical advice and services to Government agencies upon request. The Institute also monitors NBS engineering standards activities and

provides liaison between NBS and national and international engineering standards

bodies. The Institute consists of a Center for Building Technology and the following

divisions and offices: Engineering Standards Services—Weights and Measures—Invention and Innovation—Product Evaluation Technology—Electronic Technology—Technical

Analysis—Measurement Engineering—Fire Technology—Housing Technology4 —Federal Building Technology4—Building Standards and Codes Services4—

Building Environment4—Structures, Materials and Life Safety4—Technical Evaluation and Application4

. THE CENTER FOR COMPUTER SCONCES AND TECHNOLOGY conducts re- search and provides technical services designed to aid Government agencies in improving cost effectiveness in the conduct of their programs through the selection, acquisition, and effective utilization of automatic data processing equipment; and serves as the principal focus within the executive branch for the development of Federal standards for automatic data processing equipment, techniques, and computer languages. The Center

consists of the following offices and divisions: Information Processing Standards—Computer Information—Computer Services —Systems Development—Information Processing Technology.

THE OFFICE FOR INFORMATION PROGRAMS promotes optimum dissemination and accessibility of scientific information generated within NBS and other agencies of

the Federal Government; promotes the development of the National Standard Reference Data System and a system of information analysis centers dealing with the broader

aspects of the National Measurement System; provides appropriate services to ensure

that the NBS staff has optimum accessibility to the scientific information of the world, and directs the public information activities of the Bureau. The Office consists of the following organizational units: Office of Standard Reference Data—Office of Technical Information and

Publications-—Library—Office of International Relations.

•Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D.C. 20234.

• Part of the Center for Radiation Research.

3 Located at Boulder, Colorado 80302.

* Part of the Center for Building Technology.
Frequency Stability Specification

and Measurement:

High Frequency and

Microwave Signals

J.H. Shoaf

D. Halford

A.S. Risley

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

NBS Technical notes are designed to supplement the

Bureau's regular publications program. They provide

a means for making available scientific data that are of transient or limited interest. Technical Notes may

be listed or referred to in the open literature.

w

U.S. DEPARTMENT OF COMMERCE, Peter G. Peterson, Secretary

NATIONAL BUREAU OF STANDARDS, Lawrence M. Kushner, Acting Director

Issued January 1973
National Bureau of Standards Technical Note 632

Nal. Bur. Stand.(U.S.), Tech. Note 632. 70 pages (January 1973) CODEN: NBTNAE

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D. C. 20402

(Order by SD Catalog No. C13.46:632).
CONTENTS

Page

1. INTRODUCTION AND BACKGROUND 1

2. TERMINOLOGY FOR SPECIFICATION OF FREQUENCY

STABILITY 2

3. OPERATIONAL SYSTEMS FOR MEASUREMENT OF

FREQUENCY STABILITY AT NBS (HIGH FREQUENCY

REGION) 5

3. 1. Frequency Domain Measurements 6 3.2. Time Domain Measurements 12

3. 3. Differential Phase Noise Measurements . 16

4. OPERATIONAL SYSTEMS FOR MEASUREMENT OF

FREQUENCY STABILITY AT NBS (MICROWAVE REGION). 18

4. 1. Discussion of the Measurement System 18

4. 2. Description of the Measurement System 22

4. 3. Calibration Procedure . . . 23

4. 4. Measurement Procedure 24

4. 5. Additional Techniques for Frequency Stability

Measurement Systems at X-Band 28

5. SUMMARY 28

6. REFERENCES 33

APPENDIX A, Glossary of Symbols . 35

APPENDIX B, Stability Measure Conversion Chart 38

APPENDIX C, Translation of Data from Frequency Domain into

Time Domain Performance 39

APPENDIX D, Spectral Densities: Frequency Domain

Measures of Stability 40

APPENDIX E, A Sample Calculation of Script £ 47

2 APPENDIX F, A Sample Calculation of Allan Variance, a (t) . 50

y

APPENDIX G, Computing Counter a (T) Program ....... 52

APPENDIX H, Some Important References. . 53

BIBLIOGRAPHY 5 6 TABLE 1. Worksheet for Calculation of S ,(f) 26

60

iii
LIST OF FIGURES

Page

Figure 1. Typical Frequency Domain Measurement of

Frequency Stability (Phase Sensitive Mode) .... 6 Figure 2. Stepped Gain Operational Amplifier 8 Figure 3. Low Noise Amplifier 8 Figure 4. Adjustable RC Filter 9 Figure 5. Battery Bias Box 9 Figure 6. Plot of Script a£ versus Frequency f 11 Figure 7. Typical Time Domain Measurement of Frequency

Stability (Frequency Sensitive Mode) 12

Figure 8. Plot of Sigma versus Tau 14

Figure 9. Frequency Stability Measurement Utilizing a Computing Counter 15 Figure 10. Differential Phase Noise Measurement 16

Figure 11. Adjustable Phase Shifter (5-MHz Delay Line) .... 16

Figure 12. Photograph of Single Oscillator Frequency Stability

Measurement System 20

Figure 13. Single Oscillator Frequency Stability Measurement

System 21

Figure 14. Frequency Domain Plot of X-Band Gunn Diode

Oscillator Signal Source (Single Oscillator Method). 27

Figure 15. Frequency Stability Measurement System (PhaseLock Servo Loop) 29

Figure 16. Frequency Stability Measurement System (OffsetFrequency Phase-Lock Servo Loop) 30

Figure 17. Frequency Stability Measurement System (Large

Frequency-Offset Phase-Lock Servo Loop) 31

Figure 18. Time Domain Plot of X-Band Gunn Diode Oscillator

Signal Source 32
FREQUENCY STABILITY SPECIFICATION AND MEASUREMENT:

HIGH FREQUENCY AND MICROWAVE SIGNALS*

John H. Shoaf, Donald Halford, and A. S. Risley

Atomic Frequency and Time Standards Program Areas

National Bureau of Standards

Boulder, Colorado 80302 USA

This report gives concise definitions for specifying

frequency stability for measurements in the frequency

domain and time domain. Standards of terminology and

of measurement techniques are recommended. Measurement systems in the high frequency and microwave regions are described in adequate detail so that the systems may

be duplicated.

Key Words: Allan variance; Frequency stability measurements; Measurement system description;

Phase noise; Spectral density; Stability

definitions; Terminology standards.

I. INTRODUCTION AND BACKGROUND

At the beginning of FY- 71 the Department of Defense Joint

Services Calibration Coordinating Group (DoD/CCG), J. L. Hayes

(Chairman), Metrology Engineering Center , Pomona, California,

contracted with the National Bureau of Standards (NBS) to write a paper pertaining to the specification and measurement

of frequency stability. The project was under the jurisdiction of the

DoD/CCG Time and Frequency Working Group. The current members

of this group are Peter Strucker (Chairman), Metrology Engineering

Center, Pomona, California; J. M. Rivamonte, U. S. Army Metrology

and Calibration Center, Redstone Arsenal, Alabama; and E. L.

Kirkpatrick, Aerospace Guidance and Metrology Center, Newark Air

Force Station, Newark, Ohio.

* Contribution of the National Bureau of Standards, not subject to copyright.
The first year's work (FY-71) culminated as NBS Report 9794 [1].

The second year's work (FY -72) extended the frequency range to include

a working measurement system at X-band. The results of work done in

both FY-71 and FY -72 are documented in this paper.

The purpose of the project is to establish recommended standards

of terminology and measurement techniques for frequency stability.

Emphasis is placed on details of useful working systems (apparatus)

that could be duplicated by others in the field of frequency stability

measurements. Uniformity of data presentation is stressed in order to

facilitate interpretation of stability specifications and to enable one to communicate and compare experimental results more readily. An

authoritative paper [2] by the Institute of Electrical and Electronics

Engineers (IEEE) Subcommittee on Frequency Stability is used extensively as the prime reference, in the preparation of this report. This

report presents the description and performance of frequency stability

measurement systems capable of precise measurements on state-of-theart sources. Sufficient theory of the measurement systems is given to enable a person to readily understand the principle of operation.

2. TERMINOLOGY FOR SPECIFICATION

OF FREQUENCY STABILITY

The term frequency stability encompasses the concepts of random

noise, intended and incidental modulation, and any other fluctuations of

the output frequency of a device. In this report we are mainly (but not

totally) concerned with random fluctuations corresponding to Fourier

frequencies in the 10 to 10 hertz range. The measurement of frequency stability can be accomplished in both the frequency domain (e. g. , spectrum analysis) and the time domain (e. g. , gated frequency counter).
In the aforementioned manuscripts [1], [2] the authors chose to use two independent definitions, each related to different useful methods of

measurement. (See Appendix A for a Glossary of Symbols. ) The frequency domain definition of frequency stability is the

one-sided spectral density of the fractional frequency fluctuations, S (f), where y = glV V • The fractional frequency fluctuation spectral density

S (f) is not to be confused with the radio frequency power spectral

density S / (v), nor with S (v) , which are not good primary meas-

\'RFP oV

jares of frequency stability [2], (There is some discussion of this in

Appendix D. ) Phase noise spectral density plots [i. e. , S , (f) versus f]

00

are a common alternative method of data presentation. The spectral

density of phase fluctuations is related to S (f) by

s *Jf > = (f

2

/ f2 )s (f) • (1) 60 o y

The time domain definition of frequency stability uses the type of

sample variance called the Allan variance [3] of y:

<*>T,r,f

h» . /^f ^.JL^^ . m

\ n=l k=l

The particular Allan variance with N = 2 and T = T is found to be

especially useful in practice. It is denoted by:

/(yk+1 - yk

> </(T) = < 0y (N = 2,T=T,r,fh

)> : ( ,

The bar over the y indicates that y has been averaged over a specified

time interval T. The angular brackets indicate an average of the quantity over time. (See example in Appendix F. )
In the time domain we are concerned with the measurement

2 ^

of Allan variances at different time intervals. Plots of [ex (T)] 2

versus T ("sigma versus tau") on a log-log scale are commonly used

for data presentation. A convenient chart which enables one to translate from frequency domain measures to time domain measures (and

often conversely) is found in Appendix B. An example of this translation is given in Appendix C.

Script ?£(f) is a frequency domain measure of phase fluctuations

(noise, instability, modulation) used at NBS. Script s£(f) is defined as

the ratio of the power in one phase noise sideband, referred to the input

carrier frequency, on a per hertz of bandwidth spectral density basis,

to the total signal power, at Fourier frequency f from the carrier, per

one device. (See Appendix D. ) Power density (one phase modulation sideband)

( ' = Power (total signal)

* ( )

For small 60,

S (f) = 2£(f) . (5)

A practical system for the measurement of Script £(i) or s

fi Jf ) wi11 be

described in detail later.

It seems appropriate here to discuss briefly the types of noise

that affect the output frequency of all known signal sources. The noises can be characterized by their frequency dependence.

Common Name

White Phase Noise

Flicker Phase Noise

White Frequency Noise

(Random walk of phase)

Flicker Frequency Noise

S

60

(f) Vf) =f S

o0

(f)

f° f+2

f"

1 f+1

f" 2

f°

f"

3 f"

1
3. OPERATIONAL SYSTEMS FOR MEASUREMENT

OF FREQUENCY STABILITY AT NBS

(HIGH FREQUENCY REGION)

John H. Shoaf and Donald Halford

Most conventional systems for measurement of frequency stability

until recently have utilized time domain techniques primarily. As

indicated previously, in order to have a comprehensive and sufficient

measure of frequency stability, it is preferred that the measurements

involve both frequency and time domain techniques.

Fortunately, frequency domain and time domain methods for

measuring frequency stability require similar apparatus except that:

to make measurements in the frequency domain you must have a frequency window (spectrum analyzer) following the detector; for the

time domain you must have a time window (gated counter) following the

detector.

It was the introduction of good double -balanced mixers that permitted measurement of frequency stability by improved techniques [4] -[7],

[11]. The double -balanced mixer, considered as a phase sensitive detector, makes possible meaningful frequency stability measurements of

high-quality signal sources in both the frequency domain and the time

domain. The results are quantitative and may be obtained from a measurement system which is reasonable in cost.

The frequency stability measurement systems described below

have been used at NBS since 1967. The functional block diagrams in

figures 1, 7, 9, and 10 are referred to in the detailed descriptions of the

3 9 particular systems. The carrier frequency range 10 Hz to 10 Hz is

easily covered with these techniques.
3. 1. Frequency Domain Measurements

(l)

OSCILLATOR

(4)

MIXER

(3)

PAD

(2)OSCILLATOR

750yH

50f2 O.OOlyF

VERY LOOSE PHASE LOCK

LOOP. VOLTAGE VARIES

AS PHASE IN SHORT TERM

(7)VOLT

BOX

IMS y^—

10yF

FREQUENCY DOMAIN

TIME DOMAIN

(OPTIONAL)

CHART RECORDER

COMPUTER

PRINTER

COUNTER

CONVERTER L

FIGURE 1; TYPICAL FREQUENCY DOMAIN MEASUREMENT OF FREQUENCY STABILITY

(PHASE SENSITIVE MODE) OSCILLATOR UNDER TEST REFERENCE OSCILLATOR OF HIGH SPECTRAL PURITY (VARACTOR TUNING) ADJUSTABLE ATTENUATOR (TYPICALLY lOdB) DOUBLE-BALANCED MIXER (SCHOTTKY BARRIER DIODES)

NBS LOW-NOISE dc AMPLIFIER (SEE FIGURE 3) OPERATIONAL AMPLIFIER (SEE FIGURE 2) BATTERY BIAS BOX (SEE FIGURE 5) SPECTRUM ANALYZER (LOW FREQUENCY, HIGH RESOLUTION)

(Figures 1, 7, and 10 show circuit values appropriate for

measuring high quality 5 -MHz quartz crystal oscillators. )

Figure 1 illustrates the measurement system typically used at NBS for frequency domain measurements. It should be noted that time

domain data can also be obtained simultaneously, although usually this

system is used only for frequency domain measurements. (For time

domain measurements it is often more convenient to use a slightly
modified measurement setup to be described later. ) In this frequencydomain setup the oscillator under test is fed into one side of a low-noise

double-balanced mixer which utilizes Schottky barrier diodes. The

reference oscillator is fed into the other side of the mixer through an

attenuator, typically 10 dB. The mixer acts as a phase sensitive

detector so that when the two signals are identical in frequency and are

in phase quadrature the output is approximately zero volts dc. When

this output is sent back to the reference oscillator via the varactor

tuning, phase lock is achieved. As with any feedback system, care must

be observed to avoid instabilities, e. g. , servo loop oscillations. The

phase-lock loop contains proper termination at the output of the mixer

followed by operational amplifiers with adjustable gain. The time

constant of the loop may be adjusted as needed by varying the amplifier

gain within the loop and by use of the RC filter (fig. 4) indicated in the

diagram. Finally, a battery bias box is included at the varactor input

in order to operate in a suitably linear portion of the varactor'

s

frequency versus voltage curve. A very loose phase-lock loop is indicated inasmuch as the voltage

varies as phase (in short term), and in this frequency domain measurement we are observing the small phase variations directly. By the

phrase very loose phase-lock loop , we mean that the bandwidth of the servo response is small compared to the lowest frequency f at which

we -wish to measure (i.e., the response time is very slow). Operational

amplifiers are arranged in a circuit as shown in figure 2 for convenience

of adjusting the gain and for self-contained battery supply voltage.

Special NBS low-noise dc amplifiers used in certain precision measurements are shown in figure 3. At NBS we have arranged in a small

chassis the adjustable RC or CR filters utilizing low-noise components,

with rotary switches for various combinations of R and C (see fig. 4).
OUTPUT

FRONT REAR e—

©

FIGURE 2: STEPPED GAIN OPERATIONAL AMPLIFIER

5k£2

2 EA *

IN270 ;?

DC OUT ZERO ADJ

2N4250

2.43kfi

OVERLOAD

INDICATOR -f f

lkii

49.9S2 vw '

-±r 1 .35V

2 EACH

D MERCURY

CELLS IN PARALLEL

49.9f2

o 2N221 9

-1- 150

MPS

6521

150pF

2.43kfi

* w> II \

#327 LAMP

^=10 U F ±:24V

4 EA 6V BATTERIES

IN SERIES

MERCURY BATTERY lOUTPUT

rH|l|i|l|l|"hr0 10.8V [

FROM: A. E. WAINWRIGHT

NOTEBOOK AUG. 1967, p. Ill

FIGURE 3: LOW NOISE AMPLIFIER
SIB

INPUT

RC 6 o A CR

S1A —[ So

RC6 o

a a o Q i a 6 6 6

• r

6 6 6 6 6 6

^l

RC6 °glC R A"'

±

rJl

6 6 6 6 6

FIGURE 4: ADJUSTABLE RC FILTER

FILTER MODE SWITCH (ROTARY, 3 WAFER) RESISTOR SWITCH (ROTARY, SHORTING TYPE) CAPACITOR SWITCH (ROTARY, PROGRESSIVE SHORTING)

l.OyF lkil

II «*

1 .OyF lkn

10 TURN HELIPOT 1 U(J

•]*

1 .4V MERCURY CE

"I 3"

NON-SHORTING

SWITCH FIGURE 5: BATTERY BIAS BOX

THE UNITS ARE BUILT ON SEPARATE CHASSIS AND CONNECTED IN SERIES

TO FACILITATE FINE ADJUSTMENT BETWEEN STEP VOLTAGES

FRONT

+ REAR »-©
The battery bias box is arranged with vernier as shown in figure 5,

facilitating fine frequency adjustments via the varactor frequency adjustment in one oscillator. A wave analyzer is used to obtain the noise plot

information relevant to stability (frequency domain). The phase noise

sideband levels are read out in rms volts on the analyzer set to certain

chosen values of frequency, f. For typical high quality signal sources

this corresponds to measuring only those phase noise sidebands which

are separated from the carrier by the various f intervals chosen.

Script =£(f) may be calculated with the assumption that both sources

contribute equally; however, if one source -were the major contributor,

then the noise of that source would be no worse than 3 dB greater than

the value of Script s£(f) so calculated. A typical plot is shown in figure 6. A sample calculation may be found in Appendix E.

10
C3

CO

en

<-3

11
3. 2. Time Domain Measurements

In figure 7 a measurement system typically used at NBS for

stability measurements in the time domain is shown. It will be noted

(1)

(U)

OSCILLATOR

[4) MIXER

(3)

750uH

son

_J(>.001yF

J

PAD

(2)

VERY TIGHT PHASE LOCK LOOP. VOLTAGE

VARIES AS FREQUENCY

7) OSCILLATOR

VOLT

BOX

(8)

SPECTRUM

ANALYZER

FREQUENCY

DOMAIN

(OPTIONAL)

i^lOy

COMPUTER

(13 PRINTER

COUNTER

' (12]

(11)

CONVERTER

i, (10)

49) CHART y RECORDER

TIME DOMAIN

FIGURE. 7: TYPICAL TIME DOMAIN MEASUREMENT OF FREQUENCY STABILITY

(FREQUENCY SENSITIVE MODE)

ITEMS (1) THROUGH (8) SAME AS FIGURE 1 (9) OPERATIONAL AMPLIFIER

(10) STRIP CHART RECORDER FOR QUALITATIVE OBSERVATION

(11) VOLTAGE-TO-FREQUENCY CONVERTER

(12) FREQUENCY COUNTER WITH LOW DEAD TIME

(13) DIGITAL RECORDER WITH FAST RECORDING SPEED (INHIBIT TIME COMPATIBLE WITH COUNTER DEAD TIME)

(14) COMPUTER (OPTIONAL METHOD OF DATA ANALYSIS)

that the principle of operation is similar to that used in the frequency

domain measurement wherein the reference oscillator is locked to the

test oscillator. However, for the time domain measurement we use a

12
very tight phase-lock loop and the correction voltage at the oscillator

varies as frequency. By the phrase very tight phase-lock loop , we mean

that the bandwidth of the servo response is relatively large (i. e. , the

response time is much smaller than the smallest time interval T at which we -wish to measure). Caution- -there are problems 'which are

potentially present in tight phase-lock loop systems. This is a very

convenient setup for observing frequency fluctuations in longer term.

However, with the time constant appropriately adjusted and the means

for taking sufficiently fast samples the system is readily used for short

term measurements, as well as for the longer term measurements, in

the time domain. For qualitative observations any suitable oscilloscope

or strip chart recorder may be used. For quantitative measurements

the system at NBS utilizes a voltage-to-frequency converter, a frequency

counter, and a printer capable of recording rapid samples of data with

very short dead time. The data are analyzed typically by computer via

a program designed to compute the appropriate Allan variance [2], [3].

In our computer program log cr versus log T along with the associated

confidence in the a are automatically plotted on microfilm. For small

batches of data a desk calculator could be used and the computer analysis

would not be necessary. An example of a specific Allan variance computation is shown in Appendix F. A typical plot of log a versus log T is shown in figure 8. The dashed lines indicate the slopes which are

characteristic of the types of noise indicated.

13
i

: 1 1 1 \— -

l

!"i

'.

1

i

- / - ~- -- - ... :

;

: ill

;

i

j /

L....L

.:.:. :-: '""

* / !

j

_::.

s- -

- Ku

•

/ I,'-

i / - /

i , / i 3

- cr- u*

-

a)

•

t

r

f y '- - -i— - 1 — / /

?r . j /

- L - -

i - r

/

/ < --H

..._. - - — /

Y

/ - -1-

i—H

<4-t ID

e 1 4

i.I. 4

o // o - S

4)

> —

•

' * <u qV '/

M

.<« /

/ / t- 2

1

!

H

-<

- qj

f

. 1 ;- ....j. r—

i

ni

: j -;--'--

- <5

f

'i~i Ij\r

:>

II

14-

Q

.i.. /;; //

t

:

; s

• <-<

~J — *l s H &

j

v i H y ...

-H . X v /—

" H •** —

.

- *? ^ a, • *v /I

i

j

i S -44^ ' i in -=f t / ....

/

1

1 T — ! /

r /

/

*

1

j- /

t

> I

1

...:._ -;-;

j ""/

/

1 1

• !

I

1

1

i

i - --p.—

/ i

j T

i-1

i.L.

;

|

: ....,_

1 1

1

!

!

...j ..

1

1

'

1 | -_:_L

X...L : n

! ! 1

. : 1 i.I. :

i

r

-- n A = A 8J3t{AV ' [(J,) X)]

"T '1 " C/ I

1 : q oi co r^ to in ^j

I O

<n cd r^ 
 in rr ji co r* uj in ^r en

GO

GO

GO

CO

LU

CD

14
The convenience of obtaining time domain data has been greatlyenhanced by utilizing recently developed counters [8] which are programable to automatically compute a versus T. A block diagram (fig. 9)

shows the measurement setup using a computing counter programed for

(T (t) . The program -which was used is given in Appendix G. Certain limitations of deadtime are inherent in the use of this time

domain method. However, in general (except for very short T),

frequency stability in the time domain may be measured quickly and

accurately using a computing counter.

SIGNAL

SOURCE

/

WIDE-BAND

MIXER

FILTER COMPUTING

COUNTER

PRINTER

i

SIGNAL

SOURCE

FIGURE 9: FREQUENCY STABILITY MEASUREMENT UTILIZING A COMPUTER COUNTER

15
3. 3. Differential Phase Noise Measurements

An additional useful system illustrated in figure 10 is used for

differential phase noise measurements of various discrete components

which are frequently used in stability measurement systems. In this

system only one frequency source is used. Its output is split so that

part of the signal passes through the component to be tested.

(i) (14) REFERENCE OSCILLATOR

(15)

(a:

MIXER

(3)

750pH

__rvv>'-\_

50Q

_Jo.001y

PAD

(16) DEVICE OR COMPONENT

UNDER TEST

AMPL,

« SHIFTER (8)SPECTRUM ANALYZER

FREQUENCY DOMAIN

FIGURE 10: DIFFERENTIAL PHASE NOISE MEASUREMENT

COMPUTER

(13) PRINTER COUNTER (12)

(ID

CONVERTER

1 (10)

(9) CHART RECORDER

TIME DOMAIN

ITEMS (1) THROUGH (14) SAME AS FIGURES 1 AND 7 (15) ANY DEVICE OR COMPONENT UPON WHICH NOISE MEASUREMENTS ARE DESIRED

(AMPLIFIERS, FILTERS, CAPACITORS, CABLES, PADS. ETC.)

(16) NBS ADJUSTABLE PHASE SHIFTER, 5MHz (SEE FIGURE 11) The signal is adjusted via a phase shifter (fig. 11) so that it is in phase

quadrature with the other part of the original signal and is downconverted in the Schottky barrier diode mixer as described in the other

INPUT SI S2 S3 S4 S5 S6 S7 S8 (^uuuuuuuu 1° 2° 4° 8° 16° 32° 64° 90°

S9

50fi

OUTPUT

FIGURE 11: ADJUSTABLE PHASE SHIFTER (5MHz DELAY LINE)

RG174/U CABLE WAS USED FOR EACH SEGMENT OF PHASE SHIFT CALCULATED AT ~ 1 cm PER DEGREE AT 5 MHz.

16
systems. The switchable 50f2 load in figure II is not essential but is included for convenience. A low-pass filter is included before the

signal is amplified in special low-noise, low-level dc amplifiers and

observed on the spectrum analyzer. Script :£(f) values are calculated

at various frequency values, f, and plotted. A sample calculation is shown in Appendix E. The measurement system noise level (e. g. , see fig. 6) is easily

evaluated. Using the differential phase noise measurement system

shown in figure 10, let the "device under test" be a short length of coaxial

cable (which is itself not a source of noise). The small amount of noise

observed on the spectrum analyzer represents the system noise, mainly

due to the mixer (4) or the first amplifier (5). The calculation of the

system noise is then the procedure given in Appendix E.

17
4. OPERATIONAL SYSTEMS FOR MEASUREMENT

OF FREQUENCY STABILITY AT NBS

(MICROWAVE REGION)

John H. Shoaf and A. S. Risley

Thorough investigation of stability measurement techniques in

the X-band region revealed that a method different from that described

above for HF measurements -was desirable in most cases. The recommended measurement system is described here in detail. Other

techniques of stability measurements in X-band will be discussed in

less detail later. The recommended system* is a single -oscillator

system as shown in the photograph (fig. 12) and in the block diagram

(fig. 13). Prime references are papers by Ashley et al. [9] and

Ondria [10].

4. 1. Discussion of the Measurement System

The single-oscillator frequency stability measurement system is basically a frequency modulation (FM) demodulator. That is, it can

retrieve from the modulated carrier the signal with which the carrier

was originally frequency modulated.

An important consideration when making measurements is that of

maintaining the quadrature condition- -a 90° average phase difference

between the signals in the reference channel and the signal channel as seen at the mixer. Unfortunately, there is a fairly high probability that

during the course of a measurement the average phase difference will

fluctuate a few degrees about the desired 90° setting. Therefore, it is recommended that an occasional check of the quadrature condition be

* The recommended system is discussed here as a frequency domain

measurement. However, time domain measurements can also be made.
made. (In a two- oscillator system of measurement discussed later the

quadrature condition- -in long term- -is established and maintained by

phase-locking one source to the other. A similar procedure could be

used here, but we consider it to be unnecessary in practice. ) In practice, there is a low frequency limit to the usefulness of

this method for the measurement of FM noise. We have seen limiting

values of f 'ranging from as high as 500 Hz to as low as 2. 5 Hz.

The single-oscillator system and the two-oscillator system each

has an upper frequency limit; i. e. , a value of f above which frequency

stability measurements cannot meaningfully be made. For the singleoscillator system this upper limit is f « W , where W is the 3-dB

c c resonance line-width of the loaded discriminator cavity. In the NBS

single-oscillator system, measurements were made at values of f as high as 100 kHz.

19
CO>-

CO

CO

PQ

I—

co>-

C2J LU

C3

COO

LU

_J

C£

CO

CM

LU

CJ3

20
a

O LU —< I

I— Q_

<_> =>

UJ CD ce <_> —I—

3_

<c

az

l CD

OO

OS

CD(—3CD

OO

1

'

"

as —1 CD

<C h- 1— c_> <n I 1 I >- 1—

ir hi

t_> ea T

UJhUJ Q 1—

i r-S CD

UD > .—1

as

a

oo>-

CD

UJOS

<_> oo

CD

!J3

c

as

ca uj

<C as

CQ aD

I CD X OO

21
4. 2. Description of the Measurement System

A description of the single -oscillator measuring system may

readily be followed by referring to the block diagram (fig. 13). The

X-band source under test is connected at the left-hand side of the system. The signal passes through an isolator {l} and variable attenuator

{2} before it is split via a 3-dB directional coupler {3}. (It should be

noted that isolators {l}, {4}, {7}, {ll}, and 114} are used at several

points throughout the system as a means of preventing any serious

reflections which might otherwise exist. ) Part of the signal enters the

reference channel (upper arm), passing through a phase shifter {6} via

a variable attenuator {5} and eventually through a 90° twist into the

balanced mixer {8}. The other part of the signal enters the signal

channel (lower arm), passing through a three-port circulator {9} con- nected to a discriminator cavity {10} at one port. A variable attenuator

[12} is also in the signal channel before the signal reaches the balanced

mixer. The output of the mixer goes to either of two spectrum analyzers

{17}. A 10-dB directional coupler {l3}is utilized in the signal channel

to facilitate detection of resonance tuning of the cavity. This is observed

via a detector {15} with a dc voltmeter readout {16}.

The only component in the system which is not readily available

commercially as a stock item is the discriminator cavity [10]. It is a TE right circular one-port cavity. The coarse tuning is accomplished

by means of a movable end wall. The fine tuner is a small diameter rod

which can be moved coaxially in the cavity. The diameter of the rod

should be such that the cavity frequency changes no more than 1. 5 MHz

for 0.05 inch (~ 1 millimeter) change in depth of insertion. At any

desired frequency the coupling of the cavity should be such that the

absorption is very nearly complete. For further discussion of coupling

see reference [4]. The cavity Q needs to be high enough for good

sensitivity but adequately low for sufficient bandwidth. The cavity used

22
in the frequency stability measurement system described here has an unloaded Q of approximately 20, 000. Additional details are available

upon request from the authors.

4. 3. Calibration Procedure

Initial calibration of the measurement system is necessary in

order to assign an absolute scale to the stability measurements. To

facilitate calibration, a sinusoi dally -modulated X-band source is used to drive the system. The frequency-modulated signal is observed on an RF

power spectral density analyzer and the modulation level is adjusted to a value sufficient to completely suppress the X-band carrier. For sinusoidal modulation, the first carrier null corresponds to a modulation

index of 2. 4. Modulation at 5 kHz was found to be convenient to use because of the particular dispersion and bandwidth settings which -were available on the particular spectrum analyzer used to display the carrier

suppression. The detailed procedure for obtaining the calibration factor

follows.

(a) With the discriminator cavity {10} far off resonance, set the

level of the X-band signal (as determined with a dc voltmeter {16} at the

detector {l5}) to a convenient value and record the value. The first

variable attenuator {2} should be used for this adjustment. Any convenient level may be chosen provided an equal amount of power also will

be available from the signal source which is to be evaluated.

(b) Adjust the cavity to resonance. The dc voltmeter at the

detector is used to determine resonance. Place the dc voltmeter at the

output of the mixer (8} and adjust the phase shifter {6} in the reference

channel until the dc output at the mixer is zero (phase quadrature).

Remove the dc voltmeter and connect the output of the mixer to a spectrum

analyzer {17} tuned to the modulation frequency, 5 kHz. Record the rms

voltage reading (V ) o- the spectrum analyzer.

23
(c) It is now possible to calculate the calibration factor K.

(A v) K =-7

-SS£ , (6) rms

where (A p) is the rms frequency deviation of the carrier due to rms

intentional frequency modulation. This deviation is calculated using the

equation

(Ay) = 0.707 (Av) ., (7) rms peak

where {Av) , is the product of the modulation index with the frequency

of sinusoidal modulation, i. e. , 2.4x5 kHz = 1Z. 025 kHz. Therefore the

calibration factor in our case is

v 0. 707(12.025 kHz) 8.51 kHz K = - = — . (8) rms rms

4.4 Measurement Procedure

The procedure for obtaining data for the spectral density plot is

quite similar to the calibration procedure except that the X-band carrier

is not subjected to intentional modulation.

(a) With the cavity far off resonance, set the level at the detector

to the same value obtained during calibration. Use the variable attenuator {2} for this adjustment. The other variable attenuators {5}, {12} are set to zero.

(b) Adjust the cavity to the resonant frequency of the X-band

source. Adjust the phase shifter so that the dc output at the mixer is zero. Attach the spectrum analyzer to the output of the mixer and

record rms voltage readings (v ) for various frequency settings of

the spectrum analyzer. A low noise amplifier may be necessary to obtain useful readings at large Fourier frequencies. A second reading

24
(v ) should be taken at each value of f with the signal strongly attenrms

uated in the signal channel. This is to record the residual additive

background noise not attributable to actual phase noise on the carrier.

This attenuation is accomplished by inserting all (> 20 dB) of the attenuation in the variable attenuator {l2}.

(c) In order to calculate values of S (f) for plotting at various

Fourier frequencies it is convenient to make a tabulation of results. An

example of some typical results is given below in Table 1. The following

relations are used:

vrms V(v' ) 2

- (v" ) 2

(9) rms rms

fiy = v X K (10) rms rms

(6k ) 2

V£) - -"it- <n)

where B is the bandwidth at which the readings were made on the

spectrum analyzer, and

S

60

(f) = -J- (12)

Values of S (f) which were calculated this way (Table 1) are plotted in

figure 14.

25
TABLE 1

Worksheet for Calculation of S .(f)

Calibration factor K

51 X 10 Hz

0. 88 V

= 9. 67 X 10 Hz/V

f(Hz)

f 2

(Hz

2

)

B

(Hz)

Vrms

(MV)

6vrms

(Hz)

(6i/rms ) 2

(Hz 2

)

S (f)

(Hz)

60

(dB)*

5 X 10

3 2.5 X 10

7

(74 dB)t

100 660 6. 38 40. 7 0.41

(-3.9 dB)J -77.9

2.5 X 10

3

6. 2 X 10

6

(68 dB)

100 7 80 7.54 56.9 0. 57 (-2.4 dB) -70.4

1 X 10

3 1.0 X 10

(60 dB)

100 1000 9. 67 93. 5 0.94

(-0. 3 dB)

-60. 3

640 4. 1 X 10

5

(56 dB)

10 1200 11. 6 135. 1.4

(1.5 dB) -54.5

320 1.0 X io

5

(50 dB)

10 4 60 4.45 19. 8 2.(3.0 dB) -47.0

210 4.4 x 10 4

(46 dB)

10 5 80 5. 61 31.5 3. 2

(5. 1 dB) -40.9

150 2. 2 X 10 4

(44 dB)

10 680 6.58 43. 3 4. 3 (6.3 dB)

-37. 7

90 8. 1 X 10

3

(39 dB)

1 230 2. 22 4.93 4.9

(6.9 dB)

-32. 1

40 1. 6 X 10

3

(32 dB)

1 300 2.90 8.41 8.4

(9. 3 dB) -22.7

20 400

(26 dB)

1 450 4. 35 18.9 19.0

(12. 8 dB) -13.2

10 100

(20 dB)

1 700 6.77 45. 8 46.

(16. 6 dB)

- 3.4

*S. .(f) is tabulated in deci

° 2

t dB relative to 1 Hz

J dB relative to 1 Hz 2 Hz" 1

2 -1 bels relative to 1 radian Hz

26
oo Oi o CD o 1— 1 31 1——1

CD

COo

LUaCD x-n — a o Q CD o nz o =r i— 1

^: i_u —> g~

CD

C£

a o ^ i

PQ -J 1 _l X —

CD

U_ CO

CD

O oo 1— LU 1

CD _J

_l CD

Q_ -^1 »—i ^ CO

1—« ^^

<c

s: Lu

CD CD

C=» CdD >- CD

CD CO

~z^ LU 1 o Z=> <c ' CD ^ LU CD

Qi i—* LU CO

CD

27
4.5. Additional Techniques for Frequency Stability Measurements

at X-Band

It has been found convenient and desirable, under certain

circumstances, to use other techniques for measuring frequency stability at X-band. Where two X-band sources are available, phase- or frequency-locking techniques similar to those used at HF can be used.

(See figs. 15, 16, and 17. ) Good wide-band double -balanced mixers

with coaxial connectors are available [11] which permit many of the

measurements to be performed -without use of waveguide components.

The measurement setup as shown in the block diagram of figure 9 can also be used at microwave frequencies utilizing a computing counter

for time domain measurements. Extensive measurements of frequency

stability have been made on stabilized X-band sources [12]. Time

domain data obtained via the computing counter have been compared

with frequency domain data obtained via several methods.

An example shown in Appendix C translates the frequency domain

data of figure 14 into estimated time domain performance shown in

figure 18. In the same figure we have plotted time domain data taken

directly via a computing counter (see fig. 9).

5 . SUMMARY

Terminology and concise definitions for the specification of

frequency stability have been given. Recommended techniques and

measurement systems were described in detail for both high frequency

and X-band signals. Experimental results are compared using various

systems. Examples of computations are also included.

A. E. Wainwright, Howard E. Bell, and David W. Allan have

assisted in the development of the measurement systems reported here.

Their contributions are gratefully acknowledged. The authors appreciate the assistance of Mrs. E. Helfrich in the preparation of the manuscript.

28
ST a:

ID LU

QCM

1— >-

C_J _l

LU <C

Q- Z

co <: i i ( » , s IM in

-i! oo

loV3 / —' \ CQ / a. \ .' / s:

LU / <ci \ CD<C

h-_lO>^o1—1

1—o Q

LU ^ CC

DC <C LU

OC CQ

X

O 1 i—i O M x s:

N

nz n:

CD CD

LO LO

en CT> Q LU O LU

a: c_) Z C_> ca id 1 o

CQ ZD I o X CO X CO

go>-

go

LU Q_

SI

O

LU

Qi I GO O <c >

lu cc:

>-

<C

I—

OO

oo

^:

CDI LU

OO

c_>

C3

LU

cc:

LO

Cd

CD

29
s: ac

!=> LU

cc rvi I— >- O —1

UJ >=C O- Z

IS) <c

LU CD

1— CD

CO 1 >-

CO CD>

h- Qi

'^^ LU

LU CO

LU ^L

cc c_> J CD

CO 1 <c 1 LU LU

;^~ co

<c >- t : f— Q_ 1 >- i—i

CD

CQ t^*1 <C LU

1— ZZ) CO CD

LU

2^ Qi

CD LL_

^z. 1

cC UJ

CQX

x s:

Q LU Z (_) <C CC

ca rD i o X oo i LU CC

crzr. lu UD

rsl lu 1— r^i <-H

31 cc z: >—

co Ll_ >- CO i_U oo CH

LO :=) oo CD

CD CO

C£ LL. U_ CD

Q LU

z: o

<: a:

ca :z> 1 o X 00

30
s: q:

rD uj

o; rvi i— >- O —1

uj <C

Q- Z.

00 "=C nt > zO UJ M O /_i\

1— <c / Q. \ O 1— / ^ \

UJ _l / "* \ i o a OO 1 ' 1 M

UJ Qi o : Oi

31 UJ - s: UJ Z i—• X1—4 >- oo £

OO

_i • <C (_> 1— 00 x o O

N o :n

O UJ

CO ZD I o x oo

UJx

ON

3Z -— s: I— 00 x o

Q_O

CD

UJ1

CD

GO > >- c£

CO UJ

oo \—^ ^ UJ CD ^"'' CD

LU 1 C£ 1 —)

UJ

CO CO <c <c UJ

SI Q_ >- 1

1— UJ

t—1

oo _J u_ 1— u_

PQ CD

<C 1 1

>-

oo cd~7^~ >- UJ

<_> rD ^: CD

UJ UJ

T> ctr C3 1 1.. UJ

cc: UJ

u_ CD

Cd

r^

(JD

Q UJ z o

<c Qi

CO ZD 1 o X 00

31
dll 1 1 1 1 1 [III II 1 1 1 |l II 1 1 1 1 1 -

-

-

o — —

— — — - - -

en =C

1—

UJ "=£

ZD O — <L3 Z

i—i

|—" X °l —

_ u. <C i-, _ — s: a r-^-l _

— ^ O z oo — - i—i Q UJ

Q. - O o <

o

o

1— LU Q

UJ

oo

!0M M

FREQ 14A

6

V

z~ u_ s: uj ~

— O o£ ~

— Q "^ => ~

UJ u_ to "

— ^ i—i — - —. Q u. <C UJ - " h- h- uj

cq <: uj O —1 00

-

ZD < o

1— —1 —

| o | —

z o z _

1 -

- - _

o A

_

r - ^9 = X 3U3HM 'z/^h°l -

Ill i i I l i In i i i i i 1 llll 1 1 1 ' '

oo

(15 COaco

ooo

LUO

£3

I X

__l

O

CO

0T>

32
6. REFERENCES

[1] Shoaf, John H. , Specification and measurement of frequencystability, NBS Report 9794, 7 September 1971 (supersedes

report dated 30 June 1971), sponsored by Army/Navy/Air Force

Joint Technical Coordinating Group for Metrology and Calibration.

[2] Barnes, J. A. , Chi, A. R. , Cutler, L. S. , et al. , Characterization of frequency stability, NBS Technical Note 394 (October

1970); also published in IEEE Trans, on Instr. and Meas. IM-20 , No. 2, pp. 105-120 (May 1971).

[3] Allan, D. W. , Statistics of atomic frequency standards, Proc.

IEEE 54, No. 2, pp. 221-230 (February 1966).

[4] Van Duzer, V. , Short-term stability measurements, Proc. of the

IEEE- NASA Symposium on the Definition and Measurement of

Short-Term Frequency Stability, NASASP-80, pp. 269-272(1965).

[5 ] Meyer, D. G. , A test set for measurement of phase noise on high

quality signal sources, IEEE Trans, on Instr. and Meas. IM- 19 , No. 4, pp. 215-227 (November 1970).

[6] Shields, R. B. , Review of the specification and measurement of

short-term stability, Microwave Journal, pp. 49-55 (June 1969).

[7] Cutler, L. S. , and Searle, C. L. , Some aspects of the theory and

measurement of frequency fluctuations in frequency standards,

Proc. IEEE 54, No. 2, pp. 136-154 (February 1966).

[8] Hewlett-Packard, Computing counter applications library, Nos. 7,

22, 27, 29 (1970).

[9] Ashley, J. R. , Searles, C. B. , and Palka, F. M. , The measurement of oscillator noise at microwave frequencies, IEEE Trans,

on Microwave Theory and Techniques MTT-16, No. 9, pp. 753-760

(September 1968).

[10] Ondria, John, A microwave system for measurements of AM and

FM noise spectra, IEEE Trans, on Microwave Theory and

Techniques MTT-16, No. 9, pp. 767-781 (September 1968).

33
[11] Reeve, G. R. , Signal detection systems, NBS Report 9767

(October 1967), sponsored by Army/Navy/ Air Force

JTCG-METCAL/CCG on Contract 69-25b.

[12] Risley, A. S. , et al. , Frequency stabilization of X-band sources

for use in frequency synthesis into the infrared, to be published.

34
APPENDIX A

Glossary of Symbols

A Peak-to-peak voltage of a beat frequency at output of mixer

ptp

B High frequency cutoff f (bandwidth)

B Analysis bandwidth (frequency window) of the spectrum

analyzer

f Fourier frequency of fluctuations

f Defined as B, high frequency cutoff (bandwidth)

h Coefficient of f in spectral density representation

a. K Calibration factor used in the single oscillator stability

measurement system for microwave frequencies,

K = (Av) /V

rms rms

k, n Integers (used as index of summation)

s£(f) Frequency domain measure of phase fluctuations;

Script =£(f) is defined as the ratio of

Power density (one phase modulation sideband)

Power (total signal)

For small 60, S „ (f) m 2=?(f)

60

M Total number of data values available (usually M »N)

N Number of data values used in obtaining a sample variance

P , Total power of signal

total F e

r Parameter related to dead time; r = T/T

S (f) Spectral density of frequency fluctuations

S (f) Spectral density of voltage fluctuations

S _ _ (f

)

Spectral density of phase fluctuations; S _ .(f) = — = S (f)

—

60 60

f2 y

f 2

35
APPENDIX A cont.

S /

(l/1 ) Spectral density of the (square root of the) radio frequency

V RFP

power

5 (f) Spectral density of y (Spectral density of fractional frequency fluctuations)

T Time interval between the beginnings of two successive

measurements

t Time variable

v Root-mean- square (noise) voltage at output of mixer as measured by a spectrum analyzer

V Root-mean- square voltage of the output of an FM

r ms demodulator due to intentional modulation

W The 3-dB resonance linewidth of the loaded discriminator

c cavity

dx x Time interval fluctuations; —— = y, hence x = 6T

dt

y Fractional frequency fluctuations, y = —

o y Average of y over a specified time interval T

\ / Time average operator (usually over a large but finite

time interval T ) A Difference operator

6 Fluctuation operator

§V Frequency fluctuations

60 Phase fluctuations

6V, v Voltage fluctuations

V Signal frequency (carrier frequency) variable

V Average frequency of source (nominal frequency)

36
APPENDIX A cont.

cr Square root of a variance

2

CT (t) Specific Allan variance -where N = 2, T = T y

2

cr (N, T,T, f ) Sample variance of N averages of y(t), each of duration T

and repeated every T units of time (Allan variance)

measured in a post- detection noise bandwidth of £,h

T Sampling time interval

T Post-detection averaging time of the spectrum analyzer

3,

Cl Signal angular frequency (carrier angular frequency),

CI s 2-rrv

CO Fourier angular frequency of fluctuations, CO = 2tt£

37
APPENDIX B

CO 0" oo o

.5 A

A

A

* - _ - -|

AJ C IH

jlltM V

-' Z ^

(>J ^

* '?

CM

1-1

.-i o

£. M—

i _^ ' m h CM M ^

~J3 « u r£

CM CM* H ' +

t-~ i CM

-3 O

" O ^ S

*" a

J- —

i

c 1

2

E-i*

1= - 1 + *—

2 2

z III >

C+"IN

E2

+

-.|fM -|^ O O

Swl |

t-

(NJ

J3" N 5wj -

" a 2

M_

2 + a

z -| £. A

X + x +

, 1 z\ 7

s ri. -i| z

2

^- C?3 J3 fo

tN | *- ^.B 2 ^

h- c ^ CM T 2

2 r4

E-i h + CM +

r^lfM "_

'rj z

i_i 2 z

o

j-

' X

' M >. t=

t> J3 z (M

t- z

" M

.c

r\Ja

P

IT

ja

t-

(= M

\r- B

vO P >H fl CM "5"

ifc t= N >. CM -<|<\]

t. (M + o

"-

rvj

2^ .B

, ^l^ -C J3

1 J3

' K

t-

,B f

II u

., ^ (N] ^~~iTM~* *

"x" (M o

J3

rj tM a o> 2 a

g 2 y q m /y

/y c TM A

(= M

J3

' !» J3 E"

/ 0> 1 — A s ^ •A

.3 "

8 •

o Q- Q

"

a""

0) Lh

X CT ^ X

ii h _ J3 a _ >. A

< CT «

>

// a s

// H

A

A H B

rtH

is2

Q

rt

s §/

2

a

1- U

« A

rt>w

E-i o

u :\-

NlIN

J3 1 -< 3 0) Jl t 2 —

^1 ^t ^ 3; ii ii r^k W

^ >. >. hTM CO in U3 w

a -2 2

o rt u h2h

w rt "

>. u >. u h nl a 5

g.W

1) W (M £ a ^

o a 2

S -a •

rt u o>

° c -2 £ £ a

0)

< 2

38
APPENDIX C

Translation of Data from Frequency Domain

Into Time Domain Performance Using the Conversion Chart

Referring to the frequency domain plot in figure 14 it is

-3

determined that S , (f ) indicates approximate f behavior over the

50 _!

total range plotted. Therefore S (f) is nearly proportional to f

(i.e. , flicker frequency noise). At f = 1000 Hz, S (f) is equal to -0.3 dB relative to 1 Hz (see Table 1). The carrier frequency, v , is

9.5 GHz.

s W . fffi£ = "°"°-° 3 "*>

= ,.04 X lO"

20 ^- 1

. (CI)

y

v (9.5 X 10 Hz)

o

for f = 1000 Hz.

(f) y

S —— (see conversion chart) (C2)

h = S (f) Xf = (1.04 X 10

_20Hz"

1

) X (10

3 Hz) = 10. 4 X 10 -1 ' -1 y

(C3)

CT.2

(T) = h • 21n2 = 10.4 X 10" 18

X 1. 39 = 14.5 X

10" 18

y (C4)

O (T) = 3. 8 X 10" 9

. (C5)

For a flicker frequency noise there is no T dependence. A dashed line

at this calculated value is plotted on the same graph as data taken directly

in the time domain (fig. 18).

39
APPENDIX D

Spectral Densities: Frequency Domain Measures of Stability

Donald Halford

Stabilities in the frequency domain are commonly specified as spectral densities. We have used the concept of spectral density ex- tensively in the preparation of this report. The spectral density

concept is simple and very useful, but care must be exercised in its use. There are at least four different, but related, types of spectral

densities which are used in this paper. In this Appendix, -we state and

explain some of the simple, often-needed relations among these four

often-used types of spectral densities.

The four types 'which we have used, and -which are most relevant

to frequency and phase fluctuations, are

S (f) Spectral density of fractional frequency fluctuations

(noise, instability, modulation). The dimensionality

is Hz . The range of f is from zero to infinity.

S (f) Spectral density of frequency fluctuations (noise,

instability, modulation). The dimensionality is Hz

per Hz. The range of f is from zero to infinity.

S ,(f) Spectral density of phase fluctuations (noise, insta60

2 bility, modulation). The dimensionality is rad

per Hz. The range of f is from zero to infinity.

a?(f) Script =£(f) is a frequency domain measure of phase

fluctuations (noise, instability, modulation). Script

s£(f) is defined as the ratio of the power in one phase

noise sideband, referred to the input carrier frequency, on a per hertz of bandwidth spectral density

basis, to the total signal power, at Fourier frequency

f from the signal's average frequency v , per one

40
APPENDIX D cont.

device. The dimensionality is Hz . The range

of f is from minus v to plus infinity.

Each of these spectral densities is one-sided and is on a. per

hertz of bandwidth density basis. This means that the total meansquare fluctuation (the total variance) of frequency, for example, is given mathematically by

co /S (f ) df , o

and, as another example, since Script a?(f) is a normalized density,

that +co /

^(f) df V

o

is equal to unity.

Two-sided spectral densities are defined such that the range of

integration is from minus infinity to plus infinity. For specification of

noise as treated in this paper, our one-sided spectral density is twice

as large as the corresponding two-sided spectral density. That is,

+co +00 +00

y*[s

Two-Sided

] df = zj[s

Two-Sided] ^ /^One- Sided

] ^

-co o o (Dl)

Two-sided spectral densities are useful mainly in pure mathematical

analysis. We recommend and use one-sided spectral densities for

experimental 'work. References [1] and [2] also use one-sided spectral densities.

41
APPENDIX D cont.

We use the definition

r-¥-.V m>

o

and it follows that

V

o (tj

S (f) m S £

(f) = / —) S c

(f) . (D3)

To relate frequency, angular frequency, and phase -we use

2ir[v(t)3 = 0(t) = ^& . (D4)

This may be regarded as a definition of instantaneous frequency v{t),

From equation (D4), a direct result of transform theory is

„2 . .2

(i) s 6n (£) (?)

s 6,

S <£) - (D5: 6/» '"

Script s£(f) can be related in a simple way to S (f), but only for

the condition that the phase fluctuations occurring at rates f and faster are small compared to one radian. Otherwise Bessel function algebra

must be used to relate Script a?(f) to S (f). Fortunately, the "small

6©

angle condition" is often met in random noise problems. Specifically

we use

\2 rad

2 / — ]S

60

(f) , (D6)

42
APPENDIX D cont.

provided that + 00 / S ff')df « 1 rad

2

. (D7) J 60

For the types of signals under discussion and for |f | < V , we

use as a good approximation

=£(-f) » =£(f) . (D8)

Script 3?(f) is the normalized version of S / (V) , with its frequency

parameter f referenced to the signal's average frequency v as the

origin such that f equals v - v .

Some Mathematics of Phase Sideband Power as Related to Phase

Fluctuations: A simple derivation of equation (D6) is possible. We

combine the derivation with an example which illustrates the operation

of a double-balanced mixer as a phase detector. Consider two

sinusoidal 5 -MHz signals (having negligible amplitude modulation)

feeding the two input ports of a double-balanced mixer. When the two

signals are slightly out of zero beat, a slow sinusoidal beat with a period of several seconds at the output of the mixer is measured to have a peak-to-peak swing8 of A

Ptp

43
APPENDIX D cont.

Without changing their amplitudes, the two signals are retuned

to be at zero beat and in phase quadrature (that is, 7T/2 out of phase

with each other), and the output of the mixer is a small fluctuating voltage centered on zero volts. Provided this fluctuating voltage is small

compared to A , /2, the phase quadrature condition is being closelymaintained, and the "small angle condition" is being met.

Phase fluctuations

, respectively, -where

Phase 60 bet-ween the two signals of phases and

T

60 = 6(0

2

- 0j) , (D9)

will give rise to voltage fluctuations §V at the output of the mixer

A

6V « -$£ 60, (D10)

where we have used radian measure for phase angles, and we have

used

sin 60 m 60 (Dll)

for small 60 (60 << 1 rad). We solve equation (D10) for 60. square

both sides and take a time average

< AptP

)

If we interpret the mean- square fluctuations of 60 and of 6V",

respectively, in equation (D1Z) in a spectral density fashion, we may

write

44
APPENDIX D cont. S

6 V

(f) 60

2 (A )*

rms

where we have used

(A ) Z= 8 (A ) 2

, (D14)

ptp rms

which is valid for the sinusoidal beat signal.

For the types of signals under consideration, by definition the

two phase noise sidebands (lower sideband and upper sideband, at -f and +f from v , respectively) of a signal are coherent with each other.

As already expressed in equation (D8), they are of equal intensity also.

The operation of the mixer when it is driven at quadrature is such that

the amplitudes of the two phase sidebands add linearly in the output of

the mixer, resulting in four times as much power in the output as would be present if only one of the phase sidebands -were allowed to contribute to the output of the mixer. Hence for f I < V we obtain

s

6y<l f

l> 5

v^f1^o + £) — r ** 4

p

' (D15)

(A ) total

rms

and, using the definition of Script sf(f),

Vm^'o + £) *« " ^p— ~ ^S (|£|) , (D16)

total

provided the phase quadrature condition is approximately valid.

45
APPENDIX D cont.

The phase quadrature condition -will be met for a time interval at least

T long, provided

oo / S (f ')df << 1 rad2

, (D17)

(Znr)' 1

and hence equation (D16) is useful for values of f at least as low as

(2ttT) . Equatior

(D7) respectively.

Equations (D16) and (D17) correspond to equations (D6) and

46
APPENDIX E

A Sample Calculation of Script =£

Script =£(f) is easily measured using a double-balanced mixer

as a phase sensitive detector, together with a spectrum analyzer 'which can measure at the frequency f. Equation (El) is valid for the case

•where the reference signal has negligible phase noise compared to the

test signal.

_ 2 / one unit \ . (El) n \ ptp /

However, eq (E2) is the valid equation when we have two equally noisy

signals (test and reference) driving the mixer.

(E2)

In case the device being measured has amplification of times n

(i.e. , frequency multiplication or frequency synthesis), the definition

2 of Script :£(f) requires that the factor n appear in eqs (El) and (E2).

For example, a frequency synthesizer with output at 45.55 MHz, and

with its input driven at 5 MHz, is characterized by n equal 9.11

corresponding to a phase fluctuations increase expressed in decibels of

19. 2 dB.

2 10 log n = 20 log n = 20 log 9.11 = 19. 2 dB. (E3)

Both (El) and (E2) require that the beat signal out of the mixer be

sinusoidal when the two input signals are slightly out of zero beat. The

peak-to-peak amplitude of the beat is A

ptp

47
APPENDIX E cont. For convenience of computation and plotting it often is advantageous to set the beat frequency voltage (before locking) to some special

voltage such as —, volts (0. 316 V) peak-to-peak at the mixer output.

Then (after lock) with the output of the phase detector expressed in rms

nanovolts per root hertz, direct plotting is facilitated for Script ;£(f) in

decibels versus frequency in hertz. In this case 1000, 100, and 10

nanovolts per root hertz correspond to -110, -130, and -150 dB

respectively. A sample calculation demonstrating this convenience is

shown below.

Given:

A, = 0.316V (i.e. , 1_ V), (E4)

ptp ,/TcT

v= 100 nV • Hz"

1 ' 2 rms (E5)

at f equal to 20 Hz for a pair of equally noisy devices having no frequency

multiplication (n = 1):

*(f) = f

Vtw

Aounits

j V A

Ptp

/ 100 nVHz /2 \ 0. 316 V yT^1 (E7)

lO'^Hz"1 = -130 dB (E8)

at f equal to 20 Hz. Or, using logarithms;

48
APPENDIX E cont.

„, r , _„ , / two units \ 4{i) = 20 log

1() — (E9)

\ Ptp /

(10 V- Hz /

2

) = 20 log * rjT = 20(-7 + 0.5) = -130 dB (E10) 10 (10 /a V)

at f equal to 20 Hz.

If the phase noise were to follow flicker law (1/f), at f equal to

1 Hz the mean square noise would be 20 times worse (13 dB greater).

That is

=£(1 Hz) = -130 dB + 13 dB = -117 dB. (Ell)

49
APPENDIX F

2 A—Sample Calculation of Allan Variance, (j (T) y

k=l

(Fl)

in the example below:

Number of data values available, M = 9 Number of differences averaged, M - 1 = Sampling time interval T = Is

Data values First differences

(y) ( ^k+ l ~\]

892

809 - 83

823 14

798 - 25

671 -127

644 - 27

883 239

903 20

677 -226

M-l

I

k=l

Fir st differences squared

6889

196

625

16129

729

57121

400

51076

(yk+1 -y/ = 133165

Based on these data:

a

2

{T) = ^T^ = 8322.81, (F2)

y 2(8)

[0; (T)] = x/8322. 81 = 91. 23, N = 2, T = T = 1 s. (F3)

In this example, the data values may be understood to be expressed in

12

parts in 10

50
APPENDIX F cont.

Using the same data as in the above example it is possible to

calculate the Allan variance for T = 2 s by averaging pairs of adjacent

data values and using these averaged values as new data values to proceed with the calculation as before. Allan variance values may be

obtained for T = 3 s by averaging three adjacent data values in a similar

manner, etc. , for larger values of T.

Ideally the calculation is done via a computer and a large

number, M, of data values should be used. (Typically M = 25 6 data

values are used in the NBS computer program. ) The statistical confidence of the calculated Allan variance improves nominally as the square

root of the number, M, of data values used. For M = 256, the confidence of the Allan variance is not expected to be better than approximately , X 100% ~ 7% in the rms sense. The use of M » 1 v/256

is logically similar to the use of B • T » 1 in spectrum analysis

a a measurements, where B is the analysis bandwidth (frequency window)

of the spectrum analyzer, and T is the post-detection averaging time

of the spectrum analyzer.

51
APPENDIX G

COMPUTING COUNTER a (T) PROGRAM

y

USING AN EFFICIENT OVERLAPPING ESTIMATOR

David W. Allan

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(ID

(12)

(13)

(14)

(15)

(16) bx'y

(17) + (add)

MANUAL (19) REPEAT

Enter carrier or basic frequency (20) X FER PROGRAM

(21) Nx y [skip to (33) if program is already in]

(22 N x y LEARN

(23 + (add)

CLEAR x b^x *

(24

(25)

--* a x «_• bx y MODULE or PLUG-IN

a x

(26

(27

axy

-r (divide)

X FER PROGRAM

(28

^

MODULE or PLUG-IN

(29 c x y

axy

(30

(31

i -=- (divide)

DISPLAY x

- (subtract)

(32 PAUSE

xy

(33 RUN

X (multiply)

(34 START

(18) Vx

Program will automatically re- peat unless righthand PAUSE

switch is in HALT position

T = Sample time (computing counter "measurement time")

T - r - 0.003 seconds (compute + cycle time)

N = 2

Number set on repeat loop corresponds to the number of estimates of the

variance. For good confidence levels 100 or more estimates usually are

required.

52
APPENDIX H

Some Important References

for

Measurement and Specification of Frequency Stability

General References

1. November or December of even-numbered years IEEE

Transactions on Instrumentation and Measurement (Conference on Precision Electromagnetic Measurements, held every two years).

2. February 1966 Proceedings of the IEEE, Special Issue on Frequency Stability (IEEE-NASA Symposium). Very useful.

3. Proceedings of the IEEE-NASA Symposium on the Definition and

Measurement of Short-Term Frequency Stability at Goddard Space

Flight Center, Greenbelt, Maryland, November 23-24, 1964.

Prepared by Goddard Space Flight Center (Scientific & Technical

Information Division, National Aeronautics and Space Administration, Washington, D. C. , 1965). Copies available for $1.75 from

the U. S. Government Printing Office, Washington, D. C. 20402.

4. The annual Proceedings of the Symposium on Frequency Control

(Fort Monmouth). The Proceedings are not edited nor reviewed.

5. J. A. Barnes, A. R. Chi, L. S. Cutler, et al. , "Characterization

of Frequency Stability, " NBS Technical Note 394 (October 1970);

also published in IEEE Trans, on Instr. and Meas. IM-20 , No. 2,

pp. 105-120 (May 1971). This is the most definitive discussion to date of the characterization and measurement of frequency stability.

It was prepared by the Subcommittee on Frequency Stability of the

Institute of Electrical and Electronic Engineers.

6. September 1968 IEEE Transactions on Microwave Theory and

Techniques , Special Issue on Noise.

7. Frequency and Time Standards , Application Note 52, HewlettPackard Company (November 19 65).

53
APPENDIX H cont.

Some Specific Papers

8. D. W. Allan, "Statistics of Atomic Frequency Standards, " Proc. IEEE, vol. 54, pp. 221-230, February 1966.

A thorough understanding of this paper is important for

everyone -who •wishes to measure and quote performance of

frequency standards in the time domain, e.g., a versus T

plots. The data analysis must take into account the number

of samples taken and how they are used.

9. David W. Allan, B. E. Blair, D. D. Davis, and H. E.

Machlan, "Precision and Accuracy of Remote Synchronization via Portable Clocks, Loran C, and Network Television

Broadcasts, " Proc. 25th Annual Symposium on Frequency

Control, Fort Monmouth, N. J. , pp. 195-208, April 1971.

10. J. R. Ashley, C. B. Searles, and F. M. Palka, "The

Measurement of Oscillator Noise at Microwave Frequencies, " IEEE Trans, on Microwave Theory and Techniques MTT- 1 6 , No. 9, pp. 753-760, September 1968.

11. E. J. Baghdady, R. N. Lincoln, and B. D. Nelin, "ShortTerm Frequency Stability: Characterization, Theory, and

Measurement," Proc. IEEE, vol. 53, pp. 704-722, July 1965.

Among many other topics, the possible problem of AM noise

is discussed.

12. Helmut Brandenberger, Frederic Hadorn, Donald Halford,

and John H. Shoaf, "High Quality Quartz Crystal Oscillators:

Frequency Domain and Time Domain Stability, " Proc. 25th

Annual Symposium on Frequency Control, Fort Monmouth,

N.J., pp. 226-230 April 1971. An example of 1970 "stateof-the-art" measurements.

13. L. S. Cutler, "Present Status in Short Term Frequency

Stability," Frequency, vol. 5, pp. 13-15, September-October

1967. This is a well-written, concise progress report with

indications for future effort. Caution: His examples I and II were not in fact state-of-the-art, while example IV has some

factor-of-ten typographical errors as given.

54
APPENDIX H cont.

14. L. S. Cutler and C. L. Searle, "Some Aspects of the Theory

and Measurement of Frequency Fluctuations in Frequency

Standards, " Proc. IEEE, vol. 54, pp. 136-154, February

19 66. This is a useful treatment of some of the theory, mathematics, and measurement techniques- -with physical insight

into the noise processes of practical concern.

15. W. A. Edson, "Noise in Oscillators, " Proc. IRE, vol. 48,

pp. 1452-1466, August I960. This paper is an "old classic."

16. C Finnie, R. Sydnor, and A. Sward, "Hydrogen Maser

Frequency Standard, " Proc. 25th Annual Symposium on Frequency Control, Fort Monmouth, N.J. , pp. 348-351,

April 1971. State-of-the-art H maser stability.

17. D. J. Glaze, "Improvements in Atomic Beam FrequencyStandards at the National Bureau of Standards, " IEEE Trans,

on Instr. andMeas. IM-19 , No. 3, pp. 156-160, August 1970.

18. D. B. Leeson, "A Simple Model of Feedback Oscillator

Noise Spectrum, " Proc. IEEE (Letters), vol. 54, pp. 329-

330, February 1966. Descriptive, in a simple, practical manner.

19. J. A. Mullen, "Background Noise in Nonlinear Oscillators, " Proc. IRE, vol. 48, pp. 1467-1473, August I960. An "old

classic.

20. J. G. Ondria, "A Microwave System for Measurements of AM

and FM Noise Spectra, " IEEE Trans, on Microwave Theory

and Techniques MTT-16 , No. 9, pp. 767-781, September 1968.

21. H. P. Stratemeyer, "The Stability of Standard-Frequency

Oscillators," General Radio Experimenter, vol. 38, pp. 1-16,

June 1964. Easy reading.

22. V. Van Duzer, "Short-Term Stability Measurements,"

IEEE-NASA Symposium on Short-Term Frequency Stability,

Washington, D.C.: U. S. Government Printing Office, pp. 269-

272, NASA-SP80. The basic method described by Van Duzer is simple, elegant, easily instrumented, easily modified, extremely

versatile, and capable of the best resolution which is attained at today's state-of-the-art.
BIBLIOGRAPHY

1. D. W. Allan, "Statistics of Atomic Frequency Standards, " Proc.

IEEE, vol. 54, pp. 221-230, February 1966.

2. David W. Allan, B. E. Blair, D. D. Davis, and H. E. Machlan,

"Precision and Accuracy of Remote Synchronization via Portable

Clocks, Loran C, and Network Television Broadcasts, " Proc.

25th Annual Symposium on Frequency Control, Fort Monmouth,

N. J., pp. 195-208, April 1971.

3. David W. Allan, James E. Gray, and H. E. Machlan, " The

National Bureau of Standards Atomic Time Scales: Generation,

Dissemination, Stability, and Accuracy, " IEEE Trans, on Instrumentation and Measurement IM-2 1 , No. 4, pp. 388-391,

November 1972.

4. David W. Allan, H. E. Machlan, and James Marshall, "Time

Transfer Using Nearly Simultaneous Reception Times of a Common Transmission, " Proc. 26th Annual Symposium on Frequency Control, Fort Monmouth, N. J. , pp. 309-316,

June 1972.

5. A. E. Anderson and H. P. Brower, "A Tunable Phase Detector

for Short-Term Frequency Measurement, " Proc. IEEE-NASA

Symposium on the Definition and Measurement of Short-Term

Frequency Stability, NASASP-80, pp. 295-302, 1965.

6. J. R. Ashley and C. B. Searles, "Microwave Oscillator Noise

Reduction by a Transmission Stabilizing Cavity, " IEEE Trans, on Microwave Theory and Techniques MTT-16, No. 9, pp. 743-748,

September 1968.

7. J. R. Ashley, C. B. Searles, and F. M. Palka, "The Measurement

of Oscillator Noise at Microwave Frequencies, " IEEE Trans, on Microwave Theory and Techniques MTT-16, No. 9, pp. 753-760,

September 1968.

8. W. Atkinson and L. Fey, "Statistical Aspects of Clock Errors, " NBS Report 6085, January 5, I960.

9. W. R. Atkinson, L. Fey, and J. Newman, "Spectrum Analysis of

Extremely Low Frequency Variations of Quartz Oscillators, " Proc. IEEE (Correspondence), vol. 51, p. 379, February 1963.

56
BIBLIOGRAPHY cont.

10. E. J. Baghdady, R. D. Lincoln, and B. D. Nelin, "Short-Term

Frequency Stability: Theory, Measurement, and Status," Proc.

IEEE-NASA Symposium on Short-Term Frequency Stability,

NASASP-80, pp. 65-87, 1965; also Proc. IEEE, vol. 53,

pp. 704-722, 2110-2111, 1965.

11. J. J. Bagnall, Jr. , "The Effect of Noise on an Oscillator

Controlled by a Primary Reference, " NEREM 1959 Record,

pp. 84-86.

12. J. A. Barnes, "Studies of Frequency Stability, " NBS Report 9739,

August 15, 19 69.

13. J. A. Barnes, "Frequency Measurement Errors of Passive

Resonators Caused by Frequency-Modulated Exciting Signals, " IEEE Trans, on Instrumentation and Measurement IM-19, No. 3,

pp. 147-152, August 1970.

14. J. A. Barnes, "Atomic Timekeeping and the Statistics of Precision

Signal Generators," Proc. IEEE, vol. 54, pp. 207-220, February

1966.

15. J. A. Barnes, "Tables of Bias Functions, B i and B?, for

Variances Based on Finite Samples of Processes with Power Law

Spectral Densities, " NBS Technical Note 375, January 1969.

16. J. A. Barnes and D. W. Allan, "A Statistical Model of Flicker

Noise, " Proc. IEEE, vol. 54, pp. 176-178, February 1966.

17. J. A. Barnes and D. W. Allan, "Effects of Long-Term Stability on the Definition and Measurement of Short-Term Stability, " Proc. IEEE-NASA Symposium on Short-Term Frequency Stability,

NASASP-80, pp. 119-123, 1965.

18. J. A. Barnes, A. R. Chi, L. S. Cutler, et al. , "Characterization

of Frequency Stability, " NBS Technical Note 394, October 1970;

also published in IEEE Trans, on Instrumentation and Measurement IM-20, No. 2, pp. 105-120, May 1971.

19. J. A. Barnes and L. E. Heim, "A High- Re solution Ammonia

Maser Spectrum Analyzer, " IRE Trans, on Instrumentation I- 10. No. 1, pp. 4-8, June 1961; also NBS Report 6098, April 29, I960.

57
BIBLIOGRAPHY cont.

20. J. A. Barnes and R. C. Mockler, "The Power Spectrum and Its Importance in Precise Frequency Measurements, " IRE Trans, on Instrumentation 1-9, pp. 149-155, September I960.

21. Richard A. Baugh, "Low Noise Frequency Multiplication, " Proc.

26th Annual Symposium on Frequency Control, Fort Monmouth,

N.J., pp. 50-54, June 1972.

22. J. S. Bendat, Principles and Applications of Random Noise Theory. New York: John Wiley and Sons, p. 79, 1958.

23. W. R. Bennett, Electrical Noise . New York: McGraw-Hill, I960.

24. w. R. Bennett, "Methods of Solving Noise Problems," Proc. IRE,

vol. 44, pp. 609-638 (Eq. 223 ff. ), May 1956.

25. I. Berstein and G. Gorelik, "Frequency Modulation Noise in

Oscillators, " Proc. IRE (Correspondence), vol. 45, p. 94,

January 1957.

26. C. Bingham, M. D. Godfrey, and J. W. Tukey, "Modern

Techniques of Power Spectrum Estimation, " IEEE Trans. AU,

vol. 15, pp. 56-66, June 1967.

27. R. B. Blackman and J. W. Tukey, The Measurement of Power

Spectra . New York; Dover, 1958.

28. Helmut Brandenberger, Frederic Hadorn, Donald Halford, and

John H. Shoaf, "High Quality Quartz Crystal Oscillators:

Frequency Domain and Time Domain Stability, " Proc. 25th

Annual Symposium on Frequency Control, Fort Monmouth, N. J. , pp. 226-230, April 1971.

29. E. O. Brigham and R. E. Morrow, "The Fast Fourier Transform,"

IEEE Spectrum, vol. 4, pp. 63-70, December 1967.

30. J. R. Buck and D. J. Healey, III, "Calibration of Short-Term

Frequency Stability Measuring Apparatus, " Proc. IEEE L. , vol. 54, p. 305, February 1966.
BIBLIOGRAPHY cont.

31. J. R. Buck and D. J. Healey, III, "Short Term Frequency Stability

Measurement of Crystal Controlled X-Band Source, " Proc.

IEEE-NASA Symposium on Short-Term Frequency Stability,

NASASP-80, pp. 201-209, 1965.

32. J. R. Buck, D. J. Healey, III, and M. Meiseles, "Measurement

of Phase Stability of Quartz Crystal Oscillators for Airborne Radar

Applications," IEEE International Conv. Rec. , pt. 8, pp. 34-42,

March 1964.

33. G. A. Campbell and R. M. Foster, Fourier Integrals for

Practical Applications . Princeton, N. J. : D. Van Nostrand, 1948.

34. R. A. Campbell, "Stability Measurement Techniques in the

Frequency Domain, " Proc. IEEE-NASA Symposium on the

Definition and Measurement of Short-Term Frequency Stability,

NASASP-80, pp. 231-235, 1965.

35. E. R. Chenette, "Low Noise Transistor Amplifiers, " Solid State

Design, vol. 5, pp. 27-30, February 1964.

36. J. Chramiec, "Noise Properties of a Superregenerative

Parametric Amplifier, " Proc IEEE, vol. 54, p. 704, April 19 66.

37. E. L. Crow, "The Statistical Construction of a Single Standard

from Several Available Standards, " IEEE Trans, on Instrumentation and Measurement IM- 13, pp. 180-185, December 1964.

38. L. S. Cutler, "Some Aspects of the Theory and Measurement of

Frequency Fluctuations in Frequency Standards, " Proc.

IEEE-NASA Symposium on the Definition and Measurement of

Short-Term Frequency Stability, NASASP-80, pp. 89-100, 1965.

39- L. S. Cutler, "Present Status in Short Term Frequency Stability, " Frequency, vol. 5, pp. 13-15, September -October 1967.

40. L. S. Cutler and C. L. Searle, "Some Aspects of the Theory and

Measurement of Frequency Fluctuations in Frequency Standards, " Proc. IEEE, vol. 54, pp. 136-154, February 1966.

41. W. B. Davenport and W. L. Root, Random Signals and Noise . New York: McGraw Hill, Ch. 6, 1958.

59
BIBLIOGRAPHY cont.

42. M. M. Driscoll, "Two-Stage Self- Limiting Series Mode Type

Quartz Crystal Oscillator Exhibiting Improved Short-Term

Frequency Stability, " Proc. 26th Annual Symposium on Frequency Control, Fort Monmouth, N. J. , pp. 43-49, June 1972.

43. E. V. Dvornikov, E. A. Mavrin, and N. V. Morkovin,. "A HighStability Transistorized Crystal Oscillator, " Pribory i Tekhn.

Eksperim. (USSR), No. 4, July-August 1963. For English

translation see Instr. Exper. Tech. (USSR), pp. 672-674, JulyAugust 19 63.

44. W. A. Edson, "Noise in Oscillators, " Proc. IRE, vol. 48,

pp. 1454-1466, August I960.

45. W. A. Edson, "Progress and Problems in Short Term Stability, " Proc. 19th Annual Symposium on Frequency Control, Fort

Monmouth, N. J. , pp. 43-48, 20-22 April 1965.

46. R. Esposito and J. A. Mullen, "Noise in Oscillators with General

Tank Circuits, " IRE International Conf. Rec. , pt. 4, pp. 202-208,

March 1961.

47. C. Finnie, R. Sydnor, and A. Sward, "Hydrogen Maser Frequency

Standard, " Proc. 25th Annual Symposium on Frequency Control,

Fort Monmouth, N. J. , pp. 348-351, April 1971.

48. L. Fey, W. R. Atkinson, and J. Newman, "Obscurities of

Oscillator Noise, " Proc. IEEE (Correspondence), vol. 52,

pp. 104-105, January 1964.

49. E. A. Gerber and R. A. Sykes, "State of the Art- -Quartz Crystal

Units and Oscillators, " Proc. IEEE, vol. 54, pp. 103-116,

February 1966.

50. E. A. Gerber and R. A. Sykes, "Quartz Frequency Standards,"

Proc. IEEE, vol. 55, pp. 783-791, June 1967.

5 1. D. J. Glaze, "Improvements in Atomic Beam Frequency

Standards at the National Bureau of Standards, " IEEE Trans, on Instrumentation and Measurement IM- 19, No. 3, pp. 156-160,

August 1970.

60
BIBLIOGRAPHY cont.

52. M. J. E. Golay, "Monochromaticity and Noise in a Regenerative

Electrical Oscillator, " Proc. IRE, vol. 48, pp. 1473-1477,

August I9 60.

53. C. H. Grauling, Jr. and D. J. Healey, III, "Instrumentation for

Measurement of the Short-Term Frequency Stability of Micro-wave

Sources," Proc. IEEE, vol. 54, pp. 249-257, February 1966.

54. P. Grivet and A. Blaquiere, "Nonlinear Effects of Noise in

Electronic Clocks, " Proc. IEEE, vol. 51, pp. 1606-1614,

November 1963.

55. J. D. Hadad, "Basic Relation Between the Frequency Stability

Specification and the Application, " Proc. IEEE-NASA Symposium

on the Definition and Measurement of Short-Term Frequency

Stability, NASA SP- 80, pp. 11-18, 19 65.

5 6. E. Hafner, "The Effects of Noise on Crystal Oscillators, " Proc.

IEEE, p. 179, February 1966.

57. Donald Halfor d, "A General Mechanical Model for |f|a

Spectral

Density Random Noise with Special Reference to Flicker Noise

l/|f|, " Proc. IEEE, vol. 56, No. 2, pp. 251-258, March 1968.

58. Donald Halford, A. E. Wainwright, and James A. Barnes,

"Flicker Noise of Phase in RF Amplifiers and Frequency

Multipliers: Characterization, Cause, and Cure, " (Summary)

Proc. 22nd Annual Symposium on Frequency Control, Fort

Monmouth, N. J. , pp. 340-341, April 1968.

59. D. J. Healey, III, "Flicker of Frequency and Phase and White

Frequency and Phase Fluctuations in Frequency Sources, " Proc.

26th Annual Symposium on Frequency Control, Fort Monmouth,

N. J., pp. 43-49, June 1972.

60. Helmut Hellwig, "Frequency Standards and Clocks: A Tutorial

Introduction, " NBS Technical Note 616, April 1972.

61. Hewlett-Packard, "Frequency and Time Standards, " Application

Note 52, 1965.

62. Hewlett-Packard, "Precision Frequency Measurements , " Application Note 116, July 19 69.

61
BIBLIOGRAPHY cont.

63. Hewlett-Packard, "Computing Counter Applications Library, " Nos. 7, 22, 27, 29; 1970.

64. R. H. Holman and L. J. Paciorek, "Short-Term StabilityMeasurement Techniques and Results, " Proc. IEEE-NASA

Symposium on the Definition and Measurement of Short-Term

Frequency Stability, NASASP-80, pp. 237-252, 1965.

65. S. L. Johnson, B. H. Smith, and D. A. Calder, "Noise Spectrum

Characteristics of Low-Noise Microwave Tubes and Solid-State

Devices," Proc. IEEE, vol. 54, pp. 258-265, February 1966.

66. P. Kartaschoff, "Shot- Effect Influence on the Frequency of an

Oscillator Locked to an Atomic Beam Resonator, " Proc.

IEEE-NASA Symposium on the Definition and Measurement of

Short-Term Frequency Stability, NASASP-80, pp. 303-308, 1965.

67. R. F. Lacey, A. L. Helgesson, and J. H. Holloway,

"Short-Term Stability of Passive Atomic Frequency Standards, " Proc. IEEE, vol. 54, pp. 170-176, February 1966.

68. R. "W". Larson, "Power Spectrum of Oscillator and Frequency

Multiplier Phase Jitter," Proc. IEEE, vol. 54, p. 796, May 1966.

69. D. B. Leeson, "A Simple Model of Feedback Oscillator Noise

Spectrum, " Proc. IEEE L. , vol. 54, pp. 329-330, February 1966.

70. d. B. Leeson and G. F. Johnson, "Short-Term Stability for a Doppler Radar: Requirements, Measurements, and Techniques, " Proc. IEEE, vol. 54, pp. 244-248, February 1966.

71. R. M. Lerner, "The Effects of Noise on the Frequency Stability

of a Linear Oscillator, " Proc. Natl. Electr. Conf. , vol. 7,

pp. 275-280, 1951.

72- l# Mailing, "Phase-Stable Oscillators for Space Communications,

including the Relationship between the Phase Noise, the Spectrum,

the Short-Term Stability, and the Q of the Oscillator, " Proc. IRE,

vol. 50, pp. 165 6-1664, July 19 62.

73. J. C. McDade, "Measurement of Additive Phase Noise Contributed

by the Step-Recovery Diode in a Frequency Multiplier, " Proc.

IEEE L. , vol. 54, pp. 292-294, February 1966.

62
BIBLIOGRAPHY cont.

74. D. G. Meyer, "A Test Set for Measurement of Phase Noise on High Quality Signal Sources, " IEEE Trans, on Instrumentation

and Measurement IM- 19, No. 4, pp. 215-Z27, November 1970.

75. A. H. Morgan and J. A. Barnes, "Short-Time Stability of a Quartz-Crystal Oscillator as Measured with an Ammonia Maser, " Proc. IRE (Letter), vol. 47, No. 10, p. 1782, October 1959.

7 6. J. A. Mullen, "Background Noise in Nonlinear Oscillators, " Proc. IRE, vol. 48, pp. 1467-1473, August I960.

77. J. A. Mullen, "Background Noise in Oscillators with R. F. Pushing,

J. Electronics and Control, vol. 10, pp. 127-138, February 1961.

78. Takanori Okoshi, Sumihisa Hashiguchi, and Michio Kotani, "A

New Method of Measuring Microwave Oscillator Noises (Long-Line

Method)," Electronics and Communications in Japan, vol. 53-B,

No. 8, pp. 80-85, 1970.

79. J. G. Ondria, "A Microwave System for Measurements of AM

and FM Noise Spectra, " IEEE Trans, on Microwave Theory and

Techniques MTT-16, pp. 767-781, September 1968.

80. R. E. Paradysz and W. L. Smith, "Measurement of FM Noise

Spectra of Low-Noise VHF Crystal Controlled Oscillators, " IEEE Trans, on Instrumentation and Measurement IM- 15 , pp. 202-

211, December 1966.

81. Max E. Peterson, "The Design and Performance of an Ultra LowNoise Digital Frequency Synthesizer for Use in VLF Receivers, " Proc. 26th Annual Symposium on Frequency Control, Fort

Monmouth, N.J., pp. 55-70, June 1972.

82. v. Prabhu, "Noise Performance of Abrupt- Junction Varactor

Frequency Multipliers, " Proc. IEEE L. , vol. 54, pp. 285-287,

February 1966.

63
BIBLIOGRAPHY cont.

8.3. J. Rarity, L. Saporta, and G. Weiss, "Study of Short Term

Stability of Crystal Oscillators, " New York University, New York,

N. Y. , Tech. Rpt. Cont. DA36-039 -SE- 87450, DA Project

3A-99-15-02, 02-02.

84. G. R. Reeve, "Signal Detection Systems, " NBS Report .9767,

October 1967, sponsored by Army/Navy/ Air Force

JTCG-METCAL/CCG on Contract 69-25b.

85. K. H. Sann, "The Measurement of Near-Carrier Noise in

Microwave Amplifiers, " IEEE Trans, on Microwave Theory and

Techniques MTT- 16, No. 9, pp. 761-766, September 1968.

86. William K. Saunders, "Short Term Frequency Stability in

Coherent Radar Applications, " Proc. 2 6th Annual Symposium on Frequency Control, Fort Monmouth, N.J., pp. 21-28, June 1972.

87. C. L. Searle, R. D. Posner, R. S. Badessa, and V. J. Bates,

"Computer- Aided Calculation of Frequency, " Proc. IEEE-NASA

Symposium on the Definition and Measurement of Short-Term

Frequency Stability, NASASP-80, pp. 273-277, November 1964.

88. R. B. Shields, "Review of the Specification and Measurement of

Short-Term Stability, " Microwave Journal, pp. 49-55, June 1969.

89. W. L. Smith and W. J. Spencer, "Quartz Crystal Controlled

Oscillators, " B. T. L. , Final Report, Cont. DA 36-039 sc- 85373,

Dept. of the Army, Proj. 3G-26-05-00 1 -01 , March 15, 1963.

89. W. J. Spencer and W. L. Smith, "Precision Crystal Frequency

Standards, " Proc. 15th Annual Symposium on Frequency Control,

Fort Monmouth, N. J. , pp. 139-155, 1961.

90. S. R. Stein and J. P. Turneaure, "A Superconducting-CavityStabilised Oscillator of High Stability, " Electronics Letters, vol. 8 No. 13, pp. 321-323, June 1972.

91. J. L. Stewart, "Frequency Modulation Noise in Oscillators, " Proc. IRE, vol. 44, pp. 372-376, March 1956.

64
BIBLIOGRAPHY cont.

92» H. L. Stover, "Theoretical Explanation for the Output Spectra of

Unlocked Driven Oscillators, " Proc. IEEE L. , vol. 54, pp. 310-

311, February 1966.

9 3- H. P. Stratemeyer, "The Stability of Standard- Frequency

Oscillators," General Radio Experimenter, vol. 38, pp.. 1-16,

June 1964.

94, R. Sydnor, J. J. Caldwell, and B. E. Rose, "Frequency Stability

Requirements for Space Communications and Tracking Systems, " Proc. IEEE, vol. 54, pp. 231-236, February 1966.

95, R. A. Sykes, W. L. Smith, and W. J. Spencer, "Performance of

Precision Quartz -Crystal Controlled Frequency Generators, " IRE Trans, on Instrumentation I- 1 1, pp. 243-247, December 1962.

9 6. V. Troitsky, "Certain Problems of the Theory of Fluctuations in

Oscillators. The Influence of Flicker Noise, " Izv. Vysshikh

Uchebn. Zavedenii, Radiofiz. , vol. 1, pp. 20-33, 1958.

97. A. Tykulsky, "Spectral Measurements of Oscillators, " Proc.

IEEE L. , vol. 54, p. 306, February 1966.

98. V. Van Duzer, "Short-Term Stability Measurements, " Proc.

IEEE-NASA Symposium on the Definition and Measurement of

Short-Term Frequency Stability, NASASP-80, pp. 269-272, 1965.

99. R.F.C. Vessot, L. F. Mueller, and J. Vanier, "A CrossCorrelation Technique for Measuring the Short-Term Properties

of Stable Oscillators, " Proc. IEEE-NASA Symposium on the

Definition and Measurement of Short-Term Frequency Stability,

NASASP-80, pp. 111-118, 1965.

100, R. F. C. Vessot, L. Mueller, and J. Vanier, "The Specification of

Oscillator Characteristics from Measurements made in the

Frequency Domain," Proc. IEEE, p. 199, February 1966.

101, A. L. Whitwell and A. N. Williams, "A Microwave Technique for

Determining Noise Spectra at Frequencies Close to the Carrier, " Microwave J. , vol. 2, pp. 27-32, November 1959.

65
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBS-TN-632

2. Gov't Accession No.3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Frequency Stability Specification and Measurement:

High Frequency and Microwave Signals

5. Publication Date January 1973

6. Performing Organization Code

7. AUTHOR(S)

John H. Shoaf, Donald Halford, A. S. Risley

8. Performing Organization

10. Project/Task/ Work Unit No. 2730444

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS , Boulder Labs,

DEPARTMENT OF COMMERCE

Boulder, CO 80302

11. Contract/Grant No. 71-56

12. Sponsoring Organization Name and Address Army/Navy/ Air Force

Joint Technical Coordinating Group for Metrology and

Metrology Engineering Center Calibration

Pomona, CA Attn: Peter Strucker

13. Type of Report & Period Covered Final (6/70 - 12/72)

14. Sponsoring Agency Code

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.) This report gives concise definitions for specifying frequency stability for

measurements in the frequency domain and time domain. Standards of terminology

and of measurement techniques are recommended. Measurement systems in the

high frequency and microwave regions are described in adequate detail so that the

systems may be duplicated.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Allan variance; frequency stability measurements; measurement system description;

phase noise, spectral density; stability definitions; terminology standards,

18. AVAILABILITY STATEMENT

[X] UNLIMITED.

7J FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

70

22. Price

65

USCOMM-DC 66244-P'i
NBS TECHNICAL PUBLICATIONS

PERIODICALS JOURNAL OF RESEARCH reports National Bureau of Standards research and development in

physics, mathematics, and chemistry. Comprehensive

scientific papers give complete details of the work,

including laboratory data, experimental procedures, and theoretical and mathematical analyses. Illustrated with photographs, drawings, and charts. Includes

listings of other NBS papers as issued.

Published in two sections, available separately:

• Physics and Chemistry

Papers of interest primarily to scientists working in these fields. This section covers a broad range of physical and chemical research, with major emphasis on standards of physical measurement, fundamental

constants, and properties of matter. Issued six times a year. Annual subscription: Domestic, $9.50; $2.25

additional for foreign mailing.

• Mathematical Sciences

Studies and compilations designed mainly for the mathematician and theoretical physicist. Topics in mathematical statistics, theory of experiment design, numerical analysis, theoretical physics and chemistry, logical design and programming of computers

and computer systems. Short numerical tables. Issued

quarterly. Annual subscription: Domestic, $5.00;

$1.25 additional for foreign mailing.

TECHNICAL NEWS BULLETIN

The best single source of information concerning the Bureau's measurement, research, developmental, co- operative, and publication activities, this monthly

publication is designed for the industry-oriented

individual whose daily work involves intimate contact with science and technology

for engineers, chemists,

physicists, research managers, product-development

managers, and company executives. Includes listing of

all NBS papers as issued. Annual subscription: Domestic, $3.00; $1.00 additional for foreign mailing.

Bibliographic Subscription Services

The following current-awareness and literature- survey bibliographies are issued periodically by

the Bureau: Cryogenic Data Center Current Awareness Service (weekly), Liquefied Natural Gas (quarterly), Superconducting Devices and

Materials (quarterly), and Electromagnetic

Metrology Current Awareness Service (monthly) . Available only from NBS Boulder Laboratories. Ordering and cost information may be

obtained from the Program Information Office, National Bureau of Standards, Boulder, Colo- rado 80302.

NONPERIODICALS Applied Mathematics Series. Mathematical tables, manuals, and studies. Building Science Series. Research results, test methods, and performance criteria of building materials, components, systems, and structures. Handbooks. Recommended codes of engineering and industrial practice (including safety codes) de- veloped in cooperation with interested industries, professional organizations, and regulatory bodies. Special Publications. Proceedings of NBS conferences, bibliographies, annual reports, wall charts, pamphlets, etc. Monographs. Major contributions to the technical

literature on various subjects related to the Bureau's

scientific and technical activities. National Standard Reference Data Series. NSRDS provides quantitative data on the physical

and chemical properties of materials, compiled from

the world's literature and critically evaluated. Product Standards. Provide requirements for sizes, types, quality, and methods for testing various industrial products. These standards are developed co- operatively with interested Government and industry groups and provide the basis for common understanding of product characteristics for both buyers and

sellers. Their use is voluntary.

Technical Notes. This series consists of communications and reports (covering both other-agency and

NBS-sponsored work) of limited or transitory interest. Federal Information Processing Standards

Publications. This series is the official publication within the Federal Government for information on

standards adopted and promulgated under the Public Law 89-306, and Bureau of the Budget Circular A—86

entitled, Standardization of Data Elements and Codes

in Data Systems. Consumer Information Series. Practical information, based on NBS research and experience, cover- ing areas of interest to the consumer. Easily understandable language and illustrations provide useful background knowledge for shopping in today's tech- nological marketplace.

CATALOGS OF NBS PUBLICATIONS NBS Special Publication 305, Publications of the NBS. 1966-1967. When ordering, include Catalog No. CI 3. 10: 305. Price $2.00; 50 cents additional for foreign mailing.

NBS Special Publication 305, Supplement 1, Publications of the NBS, 1968-1969. When ordering, include Catalog No. C13.10:305/Suppl. 1. Price $4.50; $1.25 additional for foreign mailing.

NBS Special Publication 305, Supplement 2, Publications of the NBS, 1970. When ordering, include Catalog No. C13.10:305/Suppl. 2. Price $3.25; 85 cents additional for foreign mailing.

Order NBS publications (except Bibliographic Subscription Services) from: Superintendent of Documents, Government Printing Office, Washington, D.C. 20402.
U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington. D.C. 20234

OFFICIAL BUSINESS

Penalty for Private Use. $300

POSTAGE AND FEES PAID

U.S. OEPAP.TMENT OF COMMERCE

215