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NBS TECHNICAL NOTE 679

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

and facilitate their effective application for public benefit. To this end, the Bureau conducts

research and provides: (I) a basis for the Nation's physical measurement 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 Institute for Computer Sciences and Technology, and the Office for Information Programs.

THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United

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1 Headquarters and Laboratories at Gaithersburg. Maryland, unless otherwise noted; mailing address Washington. DC. 20234.

- Located at Boulder. Colorado 80302.
Frequency Domain Stability Measurements:

i \2 1976|

\ 'CO

X A Tutorial Introduction

'6

-

David A. Howe

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

Vau of */

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

James A. Baker, III, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued March 1976
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 679

Nat. Bur. Stand. (U.S.), Tech Note 679, 27 pages (Mar. 1976)

CODEN: NBTNAE

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

(Order by SD Catalog No. C13. 46:679) $0.65
FREQUENCY DOMAIN STABILITY MEASUREMENTS: A TUTORIAL INTRODUCTION

David A. Howe

This report introduces the concept of stability measurements of oscillators by

spectral analysis. Development of topics does not rely heavily on mathematics. The

equipment and set-up for stability measurements in the frequency domain are outlined.

Examples and typical results are presented. Physical interpretations of common noise

processes are discussed. The last section provides a table by which typical frequency

domain stability characteristics may be translated to time domain stability characteristics.

Keywords: Fractional frequency fluctuations; frequency stabil i ty; .phase fluctuations; power law noise processes: spectral density: SDectrum analysis.

I. The Sine Function and Noise

A sine wave signal generator produces a voltage that changes in time in a sinusoidal

way as shown in fig. 1. The signal is an oscillating signal because the sine wave repeats

itself. A cycle of the oscillation is produced in one period "T". The phase is the angle "$"

within a cycle corresponding to a particular time "t". Therefore, we have i = j .

+11 -

1 1

2 ii/r

1 -

0° 90° 180

rt\

T/2

270° /360° ' t

T TIMEFIGURE 1

It is convenient for us to express angles in radians rather than in units of degrees, and

positive zero-crossings will occur at even mulitiples of iT-radians. The frequency "v" is the

number of cycles in one second, which is the reciprocal of period (seconds per cycle). The

expression describing the voltage "V" out of a sine wave signal generator is given by V(t) =

V sin [
(t)] where V is the peak voltage amplitude. Equivalent expressions are

V(t) = V

p sin^2TT \\

and V(t) V sin (2Trvt; (1)

Consider figure 2. Let's assume that the maximum value of "V" equals 1, hence "V " =1. We say that the voltage "V(t)" is normalized to unity. If we know the frequency of a signal and

the signal is a sine wave , then we can determine the incremental change in the period "T" (denoted by At) at a particular angle of phase.

FIGURE 2
Note that no matter how big or small At may be, we can determine AV. Let us look at this

from another point of view. Suppose we can measure AV for a particular At. From this there

is a sine wave at a unique minimum frequency corresponding to the given AV and At. For small

At, this frequency is called the instantaneous frequency at time t. The smaller the interval At, the better the approximation of instantaneous frequency at t.

When we speak of oscillators and the signals they produce, we recognize that an oscillator

has some nominal frequency at which it operates. The "frequency stability" of an oscillator

is a term used to characterize the frequency fluctuations of the oscillator signal. There

is no formal definition for "frequency stability". However, one usually refers to frequency

stability when comparing one oscillator with another. As we shall see later, we can define

particular aspects of an oscillator's output then draw conclusions about its relative frestability stability. In general terms,

"Frequency stability is the degree to which an oscillating signal produces the same value of frequency for any interval, At, throughout a specified period of time".

Let's examine the two waveforms shown in figure 3. Frequency stability depends on the amount

of time involved in a measurement. Of the two oscillating signals, it is evident that "2"

is more stable than "1" from time t, to t, assuming the horizontal scales are linear in time.

UNSTABLE FREQUENCY

LINEAR TIME

STABLE FREQUENCY

l --

LINEAR TIME

FIGURE 3

2
From time ' t, to t

?

, there may be some question as to which of the two signals is more stable,

but it's clear that from time t

? to t.,, signal "1" is at a different frequency from that in interval t, to t„.

If we want an oscillator to produce a particular frequency v then we're correct in stating that if the oscillator signal frequency deviates from v over any interval, this is a result of something which is undesirable. In the design of an oscillator, it is important

to consider the sources of mechanisms which degrade the oscillator's frequency stability. All undesirable mechanisms cause noise or a noise process to exist along with the sine wave signal of the oscillator. Simply stated, noise is anything which is undesirable . To account

for the noise components at the output of a sine wave signal generator, we can modify

equation (1) and express the output as

V(t) = [Vq + e(t)] sin [2TMQ

t + ^(t)], (2)

where V„ = nominal peak voltage amplitude,

e(t) = deviation of amplitude from nominal,

v

~ = nominal fundamental frequency,

<j)(t) = deviation of phase from nominal.

Ideally "e" and "<j>" should equal zero for all time. However, in the real world there are no perfect oscillators. To determine the extent of the noise components "e" and "<f>", we shall turn our attention to measurement techniques.

II. Phase Spectral Density

One method of characterizing noise is by means of spectrum analysis. Let's examine the

waveform shown in figure 4.

+ i --

FIGURE 4

Here we have a sine wave which is perturbed for short instances by noise. Some loosely

refer to these types of noises as "glitches". The waveform has a nominal frequency over one cycle which we'll call "^"(^ = j). At times, noise causes the instantaneous frequency to differ markedly from the nominal frequency. If a pure sine wave signal of frequency v is subtracted from this waveform, the remainder is the sum of the noise components. These

components are of a variety of frequencies and the sum of their amplitudes is nearly zero except for the intervals during each glitch when their amplitudes momentarily reinforce each

other. This is shown graphically in figure 5.
+ 1

-1

J,-

NOISE COMPONENT

*

ir r-i ^^r

^ h /

SINE WAVE AND NOISE

FIGURE 5

One can plot a graph showing rms power vs. frequency for a given signal. This kind of

plot is called the power spectrum . For the waveform of figure 5, the power spectrum will have

a high value at v and will have lower values for the signals produced by the glitches.

Closer analysis reveals that there is a recognizable, somewhat constant repetition rate associated with the glitches. In fact, we can deduce that there is a significant amount of

power in another signal whose period is the period of the glitches as shown in figure 5. Let's call the frequency of the glitches v . Since this is the case, we will observe a noticeable amount of power in the spectrum at v with an amplitude which is related to the

characteristics of the glitches. The power spectrum shown in figure 6 has this feature. A predominant fundamental component has been depicted, and other harmonics also exist.

SPECTRAL DENSITY

+ 1 --

POWER

'0 FREQUENCY

FIGURE 6

Some noise will cause the instantaneous frequency to "jitter" around v

fl

, with

probability of being higher or lower than v

n

- We thus usually find a "pedestal" associated

with v as shown in figure 7.
SPECTRAL DENSITY

POWER

FREQUENCYFIGURE 7

The process of breaking down a signal into all of its various components of frequency

is called Fourier expansion . In other words, the addition of all the frequency components,

called Fourier frequency components, produces the original signal. The value of a Fourier

frequency is the difference between the frequency component and the fundamental frequency.

The power spectrum can be normalized to unity such that the total area under the curve equals

one. The power spectrum normalized in this way is the power spectral density . The power spectrum, often called the RF spectrum, of V(t) is very useful in many applications. Unfortunately, if one is given the RF spectrum, it is impossible to determine whether

the power at different Fourier frequencies is a result of amplitude fluctuations "e(t)" or phase fluctuations "cj>(t). The RF spectrum can be separated into two independent spectra, one being the spectral density of "e(t)" often called the AM power spectral density and the other

being the spectral density of "<j>(t)" . For our purposes, the phase-fluctuation components are the ones of interest. The spectral

density of phase fluctuations is denoted by S,(f) where "f" is Fourier frequency. For the

frequently encountered case where the AM power spectral density is negligibly small and the

total modulation of the phase fluctuations is small (mean-square value is much less than one

rad ), the RF spectrum has approximately the same shape as the phase spectral density. However, a main difference in the representation is that the RF spectrum includes the fundamental

signal (carrier), and the phase spectral density does not. Another major difference is that

the RF spectrum is a power spectral density and is measured in units of watts/hertz. The

phase spectral density involves no "power" measurement of the electrical signal. The units

2 are radians /hertz. It is tempting to think of S,(f) as a "power" spectral density because

in practice it is measured by passing V(t) through a phase detector and measuring the detector's output power spectrum. The measurement technique makes use of the relation that for

small deviations ( 6<J> « 1 radian),

2

S,(f) =

V (f) rms

V (3)
where V (f) is the root-mean-square noise voltage per .J Hz at a Fourier frequency "f", and

V is the sensitivity (volts per radian) at the phase quadrature output of a phase detector

which is comparing the two oscillators. In the next section, we will look at a scheme for

directly measuring S .(f). One question we might ask is, "How do frequency changes relate to phase fluctuations?"

After all it's the frequency stability of an oscillator that is a major consideration in many applications. The frequency is equal to a rate of change in the phase of a sine wave. This tells us that fluctuations in an oscillator's output frequency are related to phase

fluctuations since we must change the rate of "<}>(t)" to accomplish a shift in "v(t)", the

frequency at time t. A rate of change of total "$ (t)" is denoted by"$ (t)". We have then T T

2 ttv( t) .(t) (4]

The dot denotes the mathematical operation of differentiation on the function $y with respect

to its independent variable t.* From equation (4) and equation (2) we get

2Trv(t) = $ ft) = 2ttv + <j>(t) T

Rearranging, we have 2irv(t) - 2ttvq

= <()(t)

or

/ . x (fr(t) v(t) " v

= 2V (5)

The quantity v(t) - v„ can be more conveniently denoted as 6v(t), a change in frequency at time t. Equation (5) tells us that if we differentiate the phase fluctuations <j>(t) and divide

by 2tt, we will have calculated the frequency fluctuation 6v(t). Rather than specifying a frequency fluctuation in terms of a shift in frequency, it is useful to denote 6v(t) with respect

to the nominal frequency v~. The quantity 6v(t) v

is called the fractional frequency

fluctuation** at time t and is signified by the variable y(t). We have

y(t) = IvM

v

o

<fr(t] (6]

* As an analogy, the same operation relates the velocity of an object with its acceleration.

The acceleration is equal to the rate of change of velocity.

** Some international recommendations replace "fractional" by "normalized".
The fractional frequency fluctuation y(t) is a dimensionless quantity. When talking about

frequency stability, its appropriateness becomes clearer if we consider the following

example. Suppose in two oscillators fiv(t) is consistently equal to + 1 Hz and we have

sampled this value for many times t. Are the two oscillators equal in their ability to produce their desired output frequencies? Not if one oscillator is operating at 10 Hz and

the other at 10 MHz. In one case, the average value of the fractional frequency fluctuation

is 1/10, and in the second case is 1/10,000,000 or 1 x 10" . The 10 MHz oscillator is then more

precise. If frequencies are multiplied or divided using ideal electronics, the fractional

stability is not changed.

In the frequency domain, we can measure the spectrum of frequency fluctuations y(t).

The spectral density of frequency fluctuations is denoted by S (f) and is obtained by

passing the signal from an oscillator through an ideal FM detector and performing spectral

analysis on the resultant output voltage. S (f) has dimensions of (fractional frequency) /Hz

1 f or Hz" . Differentiation of 4>(t) corresponds to multiplication by — in terms of

spectral densities. With further calculation, one can derive that

S (f) y

We will address ourselves primarily to S,(f), that is, the spectral density of phase fluctuations. For noise-measurement purposes, S,(f) can be measured with a straightforward, easily

duplicated equipment set-up. Whether one measures phase or frequency spectral densities is of

minor importance since they bear a direct relationship. It is important, however, to make the

distinction and to use equation (7) if necessary.

III. The Phase-Locked Loop

Suppose we have a noisy oscillator. We wish to measure the oscillator's phase fluctuations

relative to nominal phase. One can do this by phaselocking another oscillator (called the

reference oscillator) to the test oscillator and mixing the two oscillator signals 90° out of phase

phase quadrature). This is shown schematically in figure 9. The two oscillators are at the same frequency in long term as guaranteed by the phase-lock loop (PLL). A low-pass filter is used after the mixer since the difference signal is the one of interest. By holding the

two signals at a relative phase difference of 90°, short-term phase fluctuations between

the test and reference oscillators will appear as voltage fluctuations out of the mixer.
WE HAVE THIS, BUT...

S IS WHAT WE WANT TO MEASURE

^>w^>H..-t- AfJj**»>s,tftW>**+i\.s<A>"*J*.^

FIGURE 8

VOLTAGE CCNTRC

OF FREQUENCY —

PLL FILTER

FIGURE 9

With a PLL, if we can make the servo time constant very long, then the PLL bandwidth as

a filter will be small. This may be done by lowering the gain A of the loop amplifier. We want to translate the phase modulation spectrum to baseband spectrum so that it is easily

measured on a low frequency spectrum analyzer. With a PLL filter, we must keep in mind that

the reference oscillator should be as good or better than the test oscillator. This is be- cause the output of the PLL represents the noise from both oscillators, and if not properly

chosen, the reference can have noise masking the noise from the test oscillator. Often, the

reference and test oscillators are of the same type and have, therefore, approximately the same noise. We can acquire a meaningful measurement by noting that the noise we measure

is from two oscillators. Many times a good approximation is to assume that the noise power

is twice that which is associated with one oscillator. S (f) is general notation depicting

spectral density on a per hertz basis. A PLL filter output necessarily yields noise from

two oscillators.

The output of the PLL filter at Fourier frequencies above the loop bandwidth is a voltage

representing phase fluctuations between reference and test oscillator. It is necessary to make the time-constant of the loop long compared with the inverse of the lowest Fourier fre1 quency we wish to measure. That is, x

c >

2tt n mf( lowest,

This means that if we want to
measure S (f) down to 1 Hz, the loop time-constant must be greater than y- seconds. One can measure the time-constant by perturbing the loop (momentarily disconnecting the battery is convenient) and noting the time it takes for the control voltage to reach 70% of its final

value. The signal from the mixer can then be inserted into a spectrum analyzer. A preamp

may be necessary before the spectrum analyzer. The analyzer determines the mean square volts

that pass through the analyzer's bandwidth centered around a pre-chosen Fourier frequency f.

It is desirable to normalize results to a 1 Hz bandwidth. Assuming white phase noise (white <|>M),

this can be done by dividing the mean square voltage by the analyzer bandwidth in Hz. One may

have to approximate for other noise processes. (The phase noise sideband levels will usually

be indicated in rms volts-per-root-Hertz on most analyzers.)

IV. Equipment for Frequency Domain Stability Measurements

(1 ) Low-noise mixer

This should be a high quality, double-balanced type, but single-ended types may

be used. The oscillators should have well-buffered outputs to be able to isolate

the coupling between the two input RF ports of the mixer. Results that are too

good may be obtained if the tv.o oscillators couple tightly via signal injection

through the input ports. We want the PLL to control locking. One should

read the specifications in order to prevent exceeding the maximum allowable input

power to the mixer. It is best to operate near the maximum for best signalto-noise out of the IF port of the mixer and, in some cases, it is possible to drive the mixer into saturation without burning out the device.

f

FIGURE 10

[2) Low-noise DC amplifier

The amount of gain A needed in the loop amplifier will depend on the amplitude of

the mixer output and the degree of varactor control in the reference oscillator. We may need only a small amount of gain to acquire lock. On the other hand, it may be necessary to add as much as 80 dB of gain. Good low-noise DC amplifiers are available from a number of sources, and with cascading stages of amplification, each con- tributing noise, it will be the noise of the first stage which will add most significantly to the noise being measured. If a suitable low-noise first-stage amplifier

is not readily available, a schematic of an amplifier with 40 dB of gain is shown
in figure 11 which will serve nicely for the first stage. The response of the

amplifier should be flat from DC to the highest Fourier frequency one wishes to measure. The loop time-constant is inversely related to the gain A and the

determination of A is best made by experimentation knowing that

1 v T < -r- r

OVERLOAD /—-. C 2-rrf (lowest) ' i'<DKAT0R(f!nV327 lamp

2 EA

IN? 70 P

INPUT T . I /-? 0-

DC OUT

ERO AOJ

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2N4250 O.HF 332

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CELLS IN PARALLEL

LCW NOISE AMPLIFIER FIGURE 11

[3) Voltage-controlled reference quartz oscillator

This oscillator should be a good one with specifications available on its frequency

domain stability. The reference should be no worse than the test oscillator. The varactor control should be sufficient to maintain phase-lock of the reference. In general, low quality test oscillators may have varactor control of as much as 1 x 10" fractional frequency change per volt. High quality test oscillators may have only

1 x 10 fractional frequency change per volt. Some provision should be available on the reference oscillator for tuning the mean frequency over a frequency range that will

enable phase-lock. Many factors enter into the choice of the reference oscillator, and

often it is convenient to simply use two test oscillators phase-locked together. In this

way, one can assume that the noise out of the PLL filter is no worse than 3 dB greater

then the noise from each oscillator. If it is uncertain that both oscillators are contributing approximately equal noise, then one should perform measurements on three oscillators taking two at a time. The noisier-than-average oscillator will reveal itself.

|4) Spectrum analyzer

The signal analyzer typically should be capable of measuring the noise in rms volts

in a narrow bandwidth from near 1 Hz to the highest Fourier frequency of interest. This may be 50 kHz for carrier frequencies of 10 MHz or lower. For voltage measuring analyzers,

it is typical to use units of "volts per Hz". The spectrum analyzer and any associated

input amplifier will exhibit high-frequency rolloff. The Fourier frequency at which

the voltage has dropped by 3 dB is the measurement system bandwidth f , or io h = 2^.

This can be measured directly with a variable signal generator.

Rather than measure the spectral density of phase fluctuations between two oscillators,

it is possible to measure the phase fluctuations introduced by a device such as an active

filter or amplifier. Only a slight modification of the existing PLL filter equipment

set up is needed. The scheme is shown in Figure 12.

10
REF OSCILLATOR

DEVICE UNDER

TEST T

L

TAD ~^k 50fi f^ .OOlyF

lOCOfi'

90 u PHASE SHIFTER T

O.OlyF

JUMPER TO MEASURE INTRINSIC NOISE OF SET-UP

FIGURE 12 Figure 12 is a differential pnase noise measurement set-up. The output of the

reference oscillator is split so that part of the signal passes through the device under test. We want the two signals going to the mixer to be 90° out of phase, thus, phase fluctuations

between the two input ports cause voltage fluctuations at the output. The voltage fluctuations

then can be measured at various Fourier frequencies on a spectrum analyzer.

To estimate the noise inherent in the test set-up, one can in principle bypass the device

under test and compensate for any change in amplitude and phase at the mixer. The PLL filter

technique must be converted to a differential phase noise technique in order to measure inherent test equipment noise. It is a good practice to measure the system noise before

proceeding to measurement of device noise.

A frequency domain measurement set-up is shown schematically in figure 13. The component

values for the low-pass filter out of the mixer are suitable for oscillators operating at around 5 MHz.

osc PAD

MIXER

osc PAD

VARACTOR'

son

-W\riooo^ =pn.oivF

FIGURE 13

11
The active gain element (A ) of the loop is a DC amplifier with flat frequency response. One may replace this element by an integrator to achieve high gain near DC and hence, maintain

better lock of the reference oscillator in long term. Otherwise long-term drift between the

reference and test oscillators might require manual re-adjustment of the frequency of one or the other oscillator.

V. Procedure and Example

At the input to the spectrum analyzer, the voltage varies as the phase fluctuations in short-term „

sAf) =

f v (f) rms

v '

V is the phase sensitivity of the mixer in volts per radian. Using the previously described

equipment set-up, V can be measured by disconnecting the feedback loop to the varactor of the

reference oscillator. The peak voltage swing is equal to V in units of volts/rad if the

resultant beat note is a sine wave.*

The value for the measured S,(f) in decibels is given by:

S, (f) = 20 log

V Voltage at f rms ^ V full-scale c|>-detector voltage*

Examp-le: Given a PLL with two oscillators such that,

at the mixer output : V = 1 volt/rad

s V (45 Hz) = 100 nV per root hertz

rms Solve for S, (45 Hz) .

S,(45 Hz) = 100 nV (Hz 5 )

1 V/rad

10 rad Hz 10 14 rad

Hz

In decibels

,

S (45 Hz) = 20 log 10 ° nV

= 20 log ±£-

1 V 10

= 20 (-7) = -140 dB at 45 H:

*This may not be the case for state-of-the-art S (f) measurements where one must drive the mixer very hard to achieve low mixer noise levels. Hence, the output will not be a sine wave.

12
VI . Power-Law Noise Processes

Power-law noise processes are models of precision oscillator noise that produce a particular slope on the spectral density plot. We often classify these noise processes into one of

five categories. For plots of S,(f), they are:

1.

2.

3.

4.

5.

Random walk FM (random walk of frequency), S A plot goes down as 1/f

\ 3 Flicker FM (flicker of frequency), S^ plot goes down as 1/f . White FM (white of frequency), Sa plot goes down as 1/f . Flicker <J>M (flicker of phase), Sa plot goes down as 1/f.

White $M (white of phase), SA plot is flat.

Power law noise processes are characterized by their functional dependence on Fourier frequency.

The spectral density plot of a typical oscillator's output usually is a combination of

different power-law noise processes. It is very useful and meaningful to categorize the noise

processes. The first job in evaluating a spectral density plot is to determine which type of

noise exists for a particular range of Fourier frequencies. It is possible to have all five

noise processes being generated from a single oscillator, but, in general, only two or three

noise processes are dominant. Figure 14 is a graph of S.(f) showing the five noise processes

on a log-log scale. Figure 15 shows the spectral density of phase fluctuations for a typical

high-quality oscillator.

SPECTRAL DENSITY OF PHASE

50

-70 _

S if) dE

90

10 -

130 -

150

-170

10

RANDOM WALK FM

10' 10" io- four; cr frcquucy ( f

)

FLICKER 0M

FIGURE 14

13
SPECTRAL DENSITY OF PHASE

-llOf-

-120

S if) dB

10' 10

1 10' FOURIER FREQUENCY ( f ]

4 10 Hz

FIGURE 15

He can make the following general remarks about power-law noise processes:

1. Random walk FM (1/f ) noise is difficult to measure since it is usually very close to the

carrier. Random walk FM usually relates to the osci liator' s physical environment

If random walk FM is a predominant feature of the spectral density plot then mechanical

shock, vibration , temperature , or other environmental effects may be causing "random"

shifts in the carrier frequency.

3 2. Flicker FM (1/f ) is a noise whose physical cause is usually not fully understood but may

typically be related to the physical resonance mechanism of an active oscillator

or the design or choice of parts used for the electronics, or even environmenta

properties . Flicker FM is common in high-quality oscillators, but may be masked by white

FM (1/f ) or flicker *M (1/f) in lower-quality oscillators.

3. White FM (1/f ) noise is a Common type found in pass ive- resonator frequency standards . These contain a slave oscillator, often quartz, which is locked to a resonance feature of

another device which behaves much like a high-Q filter. Cesium and rubidium standards have

white FM noise characteristics.

4. Flicker *M (1/f) noise may relate to a physical resonance mechanism in an oscillator, but it usually is added by noisy electronics . This type of noise is common, even in the highest

quality oscillators, because in order to bring the signal amplitude up to a usable level,

amplifiers are used after the signal source. Flicker 4>M noise may be introduced in these

stages. It may also be introduced in a frequency multiplier. Flicker 4» M can be reduced

with good low-noise amplifier design (e.g., using rf negative feedback) and hand-selecting

transistors and other electronic components.

5. White CM (f ) noise is broadband phase noise and has little to do with the resonance mechanism. It is probably produced by similar phenomena as flicker tM (1/f) noise.

stages of amp i i f icat ion are usually responsible for white 4>M noise. This noise can

be kept at a very low value with good amplifier design, hand-selected components, the

addition of narrowband filtering at the output, or increasing, if feasible, the power of

the primary frequency source.

14
VII. Translation from Frequency Domain Stability Measurement

to Time Domain Stability Measurement

This section assumes the reader has some knowledge of the measurement of

time domain stability. Our attention will be on how to translate a plot

of S, (f) to a (t), the pair variance*. Elaboration on time domain stability measurements is beyond the intent of this Technical Note. A suggested

reference is NBS Technical Note #669, "The Measurement of Frequency and

Frequency Stability of Precision Oscillators," by D.W. Allan. Other

references are listed in the Bibliography.

From the plot of phase spectral density, it is possible to make general comments about the

time domain stability of an oscillator. Remarks are presented in the previous section. Recall

that inferences of time domain stability are made based on specific kinds of power-law noise

processes over a range of Fourier frequencies. More specifically, when dealing with noise

processes, a power-law in the frequency domain corresponds to a particular power-law in the

time domain. A convenient measure of frequency stability in the time domain is the pair

2 variance, a (x). We have

M-1

2 V(t) = zmi X (7k+1 " 7k) (8)

k=l

where M is the number of data values and y, is the average over time t of the kth data point

(similarly, y. +

, corresponds to kth + 1 point.)

2 Example: Find the pair variance , (T) , of the following sequence of

fractional frequency fluctuation values y , each value

averaged over one second.

~y = 4.36 x 10~ 5~y = 4.47 x 10~ 5 — -5 — -5

y„ = 4.61 x 10 y r = 3.96 x 10 - -5 -6 -5

y = 3.19 x 10 y = 4.10 x 10

~y . = 4.21 x 10~5 ~y n = 3.08 x 10~ 5 4 8 (Assume no dead-time in measurement of averages)

Since each average of the fractional frequency fluctuation values is for one second, then the

first pair variance calculation will be at T = Is. We are given M = 8 (eight values);

*0ften called the Allan variance.

15
therefore , the number of pairs in sequence is M-l = 7 . We have:

Data values First differences First differences squared

Tjk

(x 10~ 5

)

4.36

4.61

3.19

4.21

4.47

3.96

4.10

3.08

(gk+ l - V (x 10~ 5

> (yk+l -y/ (xio- 10

) — __ 0.25 0.06

-1.42 2.02

1.02 1.04

0.26 0.07

-0.51 0.26

0.14 0.02

-1.02 1.04

M-l ZX} - y

k

) 2

= 4.51 x 10" 10

k=l

Therefore

,

O

2

(ls> -

4 - 51 *f

l ° - 3.2 X 10- 11

y 2(7)

J3.2 x --11 and [O (Is)] = V3.2 x 10 = 5.6 x 10

Using the same data, one can calculate the pair variance for T = 2s by averaging pairs of

adjacent values and using these new averages as data values for the same procedure as above. For three second averages (T = 3s) take adjacent threesomes and find their averages and

proceed in a similar manner. More data must be acquired for longer averaging times.

One sees that with large numbers of data values, it is helpful to use a computer or programmable

calculator. The confidence of the estimate of the pair variance improves nominally as the

square root of the number of data values used. In this example, M=8 and the confidence can be expressed as being no better than 2/\/S x 100% = 35%. This then is the allowable error

in our estimate for the T = Is average.

16
Knowing how to measure S (f) for a pair of oscillators, let us see how to translate the power-law

noise process to a plot of a (t). First, consider S (f), the spectral density of frequency

fluctuations. There are two quantities which completely specify S (f) for a particular power-law

noise process: (1) the slope on a log-log plot for a given range of f and (2) the amplitude.

The slope we shall denote by "a"; therefore f is the straight line (on log-log scale) which

relates S (f) to f. The amplitude will be denoted "h "; it is simply the coefficient of f for a range of f. When we examine a plot of spectral density of frequency fluctuations, we are looking at a representation of the addition of all the power-law processes. We have

S (f) y E h f

L a

In the preceeding section, five power-law noise processes were outlined with' respect to S (f) These five are the common ones encountered with precision oscillators. Equation (7) relates

these noise processes to S (f). One obtains

1. Random Walk FM

2. Flicker FM

3. White FM

4. Flicker <j>M

5. White <J)M

with resoect to S

(f 2

: if 1

: in

(f)

(f

2

) «J

a

= -2 a

= -1 a

=

a

= 1 a

= 2

(f) L slope

Table 1 is a list of coefficients for translation from a

2 (t) to S (f) and from SAf)

to a

2 (t). The left column is the designator for the power-law process. Using the middle

column, we can solve for the value of S (f) by computing the coefficient "a" and using the

measured time domain data a

2 (x). The rightmost column yields a solution for a2 (t) given

frequency domain data S (f) and a calculation of the appropriate "b" coefficient.

17
S (f) = ha

fa S y( f ) = *°y< T > y

a = b =

b

<P

3 f

2 (2tt) h

2 t2 f

2 (white phase)

3 £ (2ti)2 t2 y2

o

1

(2-rr)

2 t2 f [3.81 + 3 ln(ui, x)]f

(flicker phase) , 01 , , , -. v ^ ' r -, ^i z 2 3.81 + 3 ln(a),T) ( v 2tt ) x v

n o

- f

2 Z x (white frequency) 2 * o

-1 1 2 ln(2) • f

: (flicker frequency)

2 ln(2 )

• f v

o

-2 6 (2^) 2 x f

(random walk frequency) 2 (2tt) 2 x f

2 6 v

2

TABLE 1. Conversion table from time-domain to frequency-domain and from

frequency-domain to time-domain for common kinds of integer power

law spectral densities; f, {= u),/2tt) is the measurement system bandwidth. Measurement response should be within 3 dB from

D.C. to f. (3dB down high-frequency cutoff is at f

h

). Vf) =-^r Vf)

18
SPECTRAL DENSITY OF PHASE

v„ = 1MHz

S*(f)

10"

10 -9

10 10

10

-1

1

10

12

10

10

-13

-14

10'

MEASUREMENT BW f. r

A ,2 10' 10" 10 -

FOURIER FREQUENCY (f)

FIGURE 16

In the phase spectral density plot of figure 16, there are two power-law noise processes. For region 1, we see than when f increases by one decade (that is, from 10 Hz to 100 Hz), S (f) goes down

-11 -14 3-3 ^ by three decades (that is, from 10 to 10 ). Thus, S (f) goes as 1/f = f . For region 1, we

<P identify this noise process as flicker FM. The rightmost column of Table 1 relates a

2 (x) to S (f). y <p The row designating flicker frequency noise yields:

oiM • a$SL^ s if)

One can now pick (arbitrarily) a convenient Fourier frequency f and determine the corresponding

values of S (f) given by the plot of figurel8. Say, f = 10, thus SJ10) = 10" . Solving for

2 (t), given v~ = 1 MHz, we obtain:

a

2

( T ) = 1.39 x 10 -20

10

therefore, a (t) = 1.18 x 10~ ,u

. For region 2, we have white c|>M. The relationship between

Oy(x) and SJf) for white <|>M is:

a*(x]

(2tt) 2 t

2 v: S*( f)

Again, we choose a Fourier frequency, say f = 100, and see that S (100) = 10 -14

. Assuming f^ = 10 Hz,

we thus obtain: a

2

(-r) = 7.59 x 10" 24 y

. \t

therefore,

a v (x) = 2.76 x 10" 12 1

y t The resultant time domain characterization is shown in figure 17.

19
TIME DOMAIN STABILITY

lO"7

-8 . 10

-9 10 o

v (x) y 10" T0

10 -n

10 -12

1.18 x 10 10

10 -3

j-Z.76 x 10* 12 1

10" 10

'

T(S)

10

v 10'

FIGURE 17. The translation of S^f) of figure 16 yields

this o (t) plot.

Figures 18 and 19 show a plot of time-domain stability and a translation to frequency domain.

The procedure is left as an exercise for the reader.

VIII. Conclusion

Starting with the basic concepts associated with the sine-wave signal generator, this

writing attempts to present fundamental aspects of spectral analysis with only light mathematical treatment. The RF spectrum is not a useful means of inferring the frequency stability

of an oscillator, so a discussion of S (f) and S (f) is put forth as well as the technique

for measurement. It was felt that there was a need for a tutorial writing that explains frequency-domain measurements without elaborating on existing time-domain techniques. A table

for translating data from one domain to the other has been included, and the use of this table

requires knowledge of the Allan Variance. The author has tried to be general in the treatment

of topics, and a bibliography is attached for readers who would like details about specific

i terns

20
10

10

-10

10

10

12

13

*"*»_ WH I TE NOISE FH V* u = -1

'>» FLICKER NOISE FM

U

=

RANDOM WALK FM x ——m**£Z&*

_L J_

10 10 10

SAMPLE TIME, t (s)

10'

FIGURE 18. A a (t) versus t plot of some actual oscillator data. y

P./2 Notice that if a (t) = T

M ,then a (t) - x

M/ hence

-1/2 y y

the t slope for y = -1 , etc.

10

10"

10 -20

S (f)

10

21

10

RANDOM WALK FM

3xl0" 30 Hz

23

10" 6 10" 5

FLICKER NOISE FM

-25 2.9x10 "

WHITE NOISE FM

h 9.6xlO" 23 /Hz

10" - 3 - 2 10 J 10 10" 10

f (Hz) FIGURE 19. A plot of S (f) as transformed from the time-domain data

plotted in Figure , 18.

21
BIBLIOGRAPHY

General References

A. November or December of even-numbered years, IEEE Transactions on Instrumentation and

Measurement (Conference on Precision Electromagnetic Measurements, held e\/ery 2 years).

B. Proceedings of the IEEE . Special issue on frequency stability, vol. 54, February 1966.

C. Proceedings of the IEEE . Special issue on time and frequency, vol. 60, no. 5, May 1972.

D. 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.

E. Proceedings of the Annual Symposium on Frequency Control . Prepared by the U.S. Army

Electronics Command, Fort Monmouth, New Jersey. Copies available from Electronics

Industry Association, 2001 Eye Street, N.W., Washington, D.C., 20006, starting with

23rd Annual. Price per copy, $8.00.

F. J. A. Barnes, A.R. Chi, L.S. Cutler, et. al., "Characterization of Frequency Stability,"

IEEE Trans, on Instr. and Meas. IM-20 , No. 2, pp. 105-120 (May 1971). Prepared by the

Subcommittee on Frequency Stability of the Institute of Electrical and Electronics

Engineers.

G. Time and Frequency: Theory and Fundamentals , Byron E. Blair, Editor, NBS Monograph 140,

May 1974.

Additional Specific References (selected)

1. D.W. Allan, "Statistics of Atomic Frequency Standards," Proc. IEEE, vol. 54,

pp. 221-230, February 1966.

2. D.W. Allan, "The Measurement of Frequency and Frequency Stability of Precision

Oscillators," NBS Tech. Note 669, July 1975.

3. 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,

pp. 753-760, September 1968.

4. E.J. Baghdady, R.D. Lincoln, and B.D. Nelin, "Short-Term Frequency Stability: Theory,

Measurement, and Status," Proc. IEEE, vol. 53, pp. 704-722, 2110-2111, 1965.

5. J. A. Barnes, "Atomic Timekeeping and the Statistics of Precision Signal Generators,"

Proc. IEEE, vol. 54, pp. 207-220, February 1966.

6. J. A. Barnes, "Tables of Bias Functions, B-, and B-, for Variances Based on Finite Samples

of Processes with Power Law Spectral Densities," NBS Tech. Note 375, January 1969.

7. J. A. Barnes and D.W. Allan, "A Statistical Model of Flicker Noise," Proc. IEEE, vol. 43,

pp. 176-178, February 1966.

22
8. 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

1960.

9. W. R. Bennett, Electrical Noise. New York: McGraw-Hill, 1960.

10. 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.

11. R. B. Blackman and J. W. Tukey, The Measurement of Power Spectra . New York:

Dover, 1958.

12. E. 0. Brigham and R. E. Morrow, "The Fast Fourier Transform," IEEE Spectrum,

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

13. R. A. Campbell, "Stability Measurement Techniques in the Frequency Domain," Proc. IEEE-NASA Symposium on the Definition and Measurement of Short-Term Frequency Stability,

NASA SP-80, pp. 231-235, 1965.

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 1966.

15. W. A. Edson, "Noise in Oscillators," Proc. IRE, vol. 48, pp. 1454-1466, August 1960.

16. C. H. Grauling, Jr. and D. J. Healey, III, "Instrumentation for Measurement of the

Short-Term Frequency Stability of Microwave Sources," Proc. IEEE, vol. 54,

pp. 249-257, February 1966.

17. H. Hellwig, "Atomic Frequency Standards: A Survey," Proc. IEEE, vol. 63, pp. 212-229,

February 1975.

18. H. Hellwig, "A Review of Precision Oscillators," NBS Tech Note 662, February 1975.

19. H. Hellwig, "Frequency Standards and Clocks: A Tutorial Introduction,"

NBS Tech Note 616-R, March 1974.

20. Hewlett-Packard, "Precision Frequency Measurements," Application Note 116, July 1969.

21. G. M. Jenkins and D. G. Watts, Spectral Analysis and its Applications . San Francisco: Holden-Day, 1968.

22. 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.

23. P. Kartaschoff and J. Barnes, "Standard Time and Frequency Generation," Proc. IEEE,

vol. 60, pp. 493-501, May 1972.

24. D. B. Leeson, "A Simple Model of Feedback Oscillator Noise Spectrum," Proc. IEEE L.

,

vol. 54, pp. 329-330, February 1966.

25. P. Lesage and C. Audoin, "Characterization of Frequency Stability: Uncertainty due to the Finite Number of Measurements," IEEE Trans, on IM, vol. IM-22, pp. 157-161, June 1973.

26. 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. 215-227, November

1970.

23
27.' C. D. Motchenbacher and F. C. Fitchen, Low-Noise Electronic Design . New York:

John Wiley & Sons, 1973.

28. J. Shoaf, "Specification and Measurement of Frequency Stability," NBS Internal Report

74-396, November 1974.

29. J. L. Stewart, "Frequency Modulation Noise in Oscillators," Proc. IRE,

vol. 44, pp. 372-376, March 1956.

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

Characteristics from Measurements in the Frequency Domain," Proc. IEEE,

pp. 199-206, February 1966.

31. F. Walls and A. Wainwright, "Measurement of the Short-Term Stability of Quartz

Crystal Resonators and the Implications for Quartz Oscillator Design and

Applications," IEEE Trans, on IM, pp. 15-20, March 1975.

24
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NBS TN-679

2. Gov't Accession

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4. TITLE AND SUBTITLE

"Frequency Domain Stability Measurements

A Tutorial Introduction"

3. Recipient's Accession No.

5. Publication Date March 1976

6. Performing Organization Code

277.04

7. AUTHOR(S) David A. Howe

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WASHINGTON, D.C. 20234

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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 introduces the concept of stability measurements of oscillators by

spectral analysis. Development of topics does not rely heavily on mathematics.

The equipment and set-up for stability measurements in the frequency domain are outlined. Examples and typical results are presented. Physical interpretations

of common noise processes are discussed. The last section provides a table by

which typical frequency domain stability characteristics may be translated to

time domain stability characteristics.

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)

Fractional frequency fluctuations; frequency stability; phase fluctuations;

power law noise processes; spectral density; spectrum analysis.

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