MOV 2 2 '967 NBS TEGHNIC

357

Signal Design for Time

Dissemination: Some Aspects

Q

/'

*»T OF

\»« U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards
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NBS TECHNICAL NOTE 357

ISSUED NOVEMBER 2, 1967

SIGNAL DESIGN FOR TIME

DISSEMINATION: SOME ASPECTS

J. L. JESPERSEN

Radio Standards Laboratory

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

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TABLE OF CONTENTS

Abstract 1

1. Introduction 1

2. Information Theory and Time Dissemination 6

3. Bandwidth and the Time of Arrival Problem 10

3. 1 Recognition of the Presence of the

Signal ........... 11

3.2 Measurement of Signal Arrival Time 13

3.3 Measurement of Time Interval . . „ 14

3.4 Simultaneous Measurement of Time

of Arrival and Frequency .............. 16

4. The Multiple CW System . 17

5. The VLF Two-Frequency Timing System . . 21

6. Characterization of the Noise .............. 23

7. Appendix 29

7.1 Effects of Noise on Two-Frequency

Timing Systems . 29

8. Acknowledgments „ 35

9. References 38

in
LIST OF FIGURES

Figure 1 „ Elements of a timing system ........... 2

Figure 2. Variation in the required transmission

time, T, in seconds, as a function of the

number of signals transmitted, N ........ . 20

Figure 3. Variation in the required transmission

time, T, as a function of the ratio of the

frequency actually broadcast, f , to the

optimum frequency, f .............. 24

CO

Figure 4. Schematic representation of scattering

of satellite signal „ 26

Figure 5. Variance of the signal arrival time,

Var t , as a function of frequency

separation, f 36

s

i

IV
SIGNAL DESIGN FOR TIME DISSEMINATION: SOME ASPECTS

J. L. Jespersen

The purpose of this paper is to discuss, in a general way, some problems of time signal dissemination in a noisy environment. Most of the paper applies

to any timing system, but particular emphasis is given to a CW two-frequency system. This is done for two reasons:

first, as will be shown, a two-frequency CW system evolves

naturally from fundamental considerations to meet certain

user requirements; and second, a two-frequency VLF

system is being investigated experimentally at the present

time.

Key Words: VHF, HF, VLF, time and frequency dissemination, synchronization, satellite, noise.

1. Introduction

A timing signal usually consists of some waveform which is repeated at uniform intervals of time, AT, as illustrated in figure 1. To

know the time means knowing how many units and fractional units of AT

have elapsed from some starting point T (see fig. 1). In most timing

systems, as illustrated in figure 1, there is some finite path delay, 6t,

between the time the signal is broadcast and the time it is received.

Although determining the path delay is not a trivial problem, it will be

assumed known in this paper.

In the design of signals for time dissemination, it is important to

know whether the user is primarily interested in obtaining T or AT or both from the timing signal. In addition, these determinations must be

made in the presence of noise, which will, for example, cause the

timing signal arrival time to jitter as illustrated by the double arrow in
oOOaeco

+-cc —

rO- <

in

CsJ- 7<1 o - o c-\ oooo -

-I ro-

-o> -

<D C\J o

0)

crc -

en

Pi

oo

CO

CODOao

a>g

cu i—

W

CDU

UJ —i
figure 1. Although there is a wide variety of noise which may be encountered in actual practice, we will, for the most part, assume noise with

a uniform spectral density NQ (white noise).

Although we will be considering various aspects of the timing

signal design in some detail in later sections, it will be useful at this

point to consider a simple example which will serve to illustrate the kind

of problems which will be encountered.

Consider a timing system which consists of a single pulse of CW

signal at frequency f and duration q, which we know ahead of time is to

designate a certain moment in time with respect to T The standard

deviation in the time of arrival of the leading edge of this signal is

[Skolnik, 1962]

t °(t) = ^-17

(2S/N) >z

where t is the pulse rise time and S/N is the pulse signal-to-noise

power ratio. The bandwidth occupied by the signal is given approximately

by l/t . If we decreased the pulse length q while maintaining the same

energy in the pulse, we could better our measurement of the time of

arrival, since we have increased the pulse S/N. On the other hand,

since the signal consists of a single pulse, we have no way of obtaining

interval information, AT, unless we measure the carrier frequency, f,

of the pulse. The standard deviation of the carrier frequency measurement a (f) is a function of the reciprocal of the duration, q, of the pulse.

(See Section 3. 3) Therefore, we see that for a single pulse the time of

arrival measurement and the frequency measurement are related, and

we can only improve one measurement at the expense of the other.

Suppose now that we have a single pulse of energy which lasts for

q seconds, and we find that we can measure the frequency of the carrier

to sufficient precision for our needs. Is there any way in which we could
improve our time of arrival measurement without sacrificing the frequency measurement? The time of arrival measurement depends upon

the S/N at the leading edge of the pulse and does not depend upon how

long the pulse lasts. This suggests that we could improve our measurement of arrival time if we could have more leading edges. Thus,

suppose we took our pulse of duration q and signal level S and divided it

into m pulses each of duration q/m with the same signal level S. Our

total signal now consists of m pulses but the duration of these m pulses

is still q seconds so that we haven't changed the precision of our frequency

measurement. On the other hand, we now have m leading edges to

measure so that we have decreased the standard deviation of our arrival

time measurement by a factor (m) , if each of the m measurements

are independent. This implies that the time between successive pulses

is at least l/B seconds where B is bandwidth; otherwise, the noise

fluctuations on successive pulses will not be independent. We have now

improved our time of arrival measurement without damaging the frequency

measurement. But in doing so, we have introduced another difficulty.

Unless we have some additional information, we no longer know which

one of the m pulses represents the time. Thus, for the privilege of

improving our time of arrival measurement, while maintaining the same

frequency error, we have introduced ambiguity.

Although we have introduced the idea of several pulses from the

point of view of obtaining precise frequency and time of arrival measurements simultaneously, there is. another important practical reason for

having timing signals available frequently and that is simply that the user

does not wish to wait long periods of time for timing information. Again,

however, the penalty for frequent timing signals is ambiguity, unless

means are taken to keep the signals distinguishable.

However, to keep the signals distinguishable, we must also pay a price. To illustrate how this comes about, consider a timing system
which consists of one per second pulses. In this system no user has to

wait more than one second for a second marker; however, if the user

does not already know the time to the nearest second, the system is

ambiguous for him. Presumably we could make the system unambiguous

for our user by going to a lower pulse rate, but in doing so we would

make the users who already knew the time to the nearest second unhappy;

because, now they have to wait longer for a timing mark. If we wanted

to maintain the time markers at the one per second rate and still satisfy

the user who didn't know the time to the nearest second, we would have

to provide additional information to remove the ambiguity. (WWV uses

such a system where the basic pulse train is supplemented by voice

announcements and omissions of certain pulses, )

If only a certain amount of energy is available, the energy used

for identification (to reduce the signal ambiguity) will result in reduced

knowledge of the time of arrival of the pulses. This is attributable to

the lowered signal-to-noise power ratio, S/N. That is, the timing signal design has traded measurement precision for reduced ambiguity.

This trade-off can be carried too far. The user of a timing pulse occurring each second may wish certain pulses to be omitted (more identification). A second user may be satisfied more quickly by leaving in all

the pulses to improve his measurement precision. Thus, a signal can

be too precise (ambiguous) for one user and overidentified for another.

A timing signal may be used simultaneously for both "time of

arrival" and identification. We could transmit a signal consisting of a group of coded pulses. One of the pulses in the group could be designated

as the synchronization pulse. It would then appear that both synchronization and identification have been satisfied. It will be shown later,

however, that optimum determination of the time of arrival of the signal

requires a priori knowledge of the signal shape. But if we already know

the shape, then the signal does not contain any new identification
information; thus, we cannot optimumly determine the signal arrival

time and simultaneously gain identification information. If, however,

the incoming signal were one of a number of possible signals, then we

could gain information by using a number of parallel decoders optimized

for each of the possible signals.

There does not appear to be any general method for designing signals which will simultaneously satisfy the requirements of ambiguity,

time of arrival, and interval (or frequency) measurement precision. It

is intuitively obvious that the reduction of ambiguity will require a

signal whose frequency components are spread throughout the available

bandwidth. However, for the case of increased precision in time of

arrival measurement, we will see that the frequency components should

be concentrated near the edges of the available bandwidth, whereas for

increased precision in the frequency (or interval) measurements, the

signal should last as long as possible.

2. Information Theory and Time Dissemination

Before considering in detail the problems of ambiguity, time of

arrival and frequency measurement precision, it will be useful, using

information theory, to determine how well the optimum system should

operate for a given bandwidth, B, signal-to-noise power ratio, S/N, and

observing time, t. Ordinarily, the user does not need a signal which completely

specifies the time, but rather he already knows the time down to some

interval, AT, and wishes to know the time down to some smaller interval,

At. For example, he may know the time to the nearest hour (AT = 1 hour)

and wish to know the time to the nearest minute (At = 1 minute). If we

assume that the correct time is equally likely at all points within the

interval AT, then the time has equal probability of being in any one of
k intervals where

k = AT/ At.

The amount of information in bits required to designate one of the k

intervals is

I = log

2 k = iog

2 AT/At. (1)

A system which is optimum with a bandwidth B, and a signal-to-noise

power ratio S/N can transmit information at a maximum rate [Schwartz,

1959] given by

R =Blog2

(l + S/N).

The total amount of information transmitted during a time t will be

I =tBlog

a

(l + S/N). (2)

AT Equating (1) and (2) we see that the improvement factor . , which may

be obtained with the optimum system with a given B, S/N and observing

time t is

AT/ At = (1 +S/N)+tB

. (3)

To illustrate this result, consider a timing system which consists

of a sine wave at frequency f. The standard deviation, a(t), in the time

of arrival of the zero-crossings of the sine wave is

... 1 (see Section 3.2,

if S/N » 1. [Skolnik, 1962; also see Section 2 for further discussion of

the sine wave system. ] In this system it is assumed that the user knows
the time to the period of the sine wave (or less) so that

AT = 1/f.

In addition, the smallest interval which may be distinguished by the user

is a (t); so that, At = a (t). Therefore

(if) = 2 " < S/N' \ / sine wave

In order to compare this result with the optimum system, we will

assume that tB = 1. This condition implies that the sine wave system

bandwidth is just wide enough to accommodate the spectral width of a

sine wave which has a duration of t seconds. With this assumption, and

with the assumption already made that S/N » 1, (3) becomes approximately

£ /*

Thus the optimum system is a factor (S/N) /277 better than the sine

wave system.

If we increased B for the sine wave case and kept the same observing time, we would not improve our result since we would be obtaining

more independent samples per unit time on the one hand and letting in

more noise on the other. This is easily shown from (4) since independent

samples may only be obtained every — seconds. We can, however, B

improve our result for the optimum case by increasing B. To illustrate

this, assume that the noise is white with spectral density NQ

: then

N = NQ B. As we increase B, the term S/NQ B becomes small compared

to 1 and we can approximate (2) by

tBlog (1 + S/N B) = (t) 1.45 S/NQ
which allows us to rewrite (3) as

AT = 2

t (1.45 S/NQ) At

A signal which approached this limit might consist of a long sequence of

pulses almost completely masked by noise. The information could be

recovered by multiplying the incoming signal, pulse by pulse, by a locally- stored replica of the incoming signal. However, for timing we

are generally interested in signals with S/N » 1 which are readily received by simple equipment. Nevertheless, low signal-to-noise signals

with large bandwidths are of interest in specialized cases, as might be

encountered in transmitting time to a distant space craft.

Information theory does not tell us how to design an optimum

system. In most systems there is a high degree of redundancy. But in

systems approaching the optimum, there can be little redundancy. In

fact, systems which approach the optimum transmit only an error signal

[Pierce and Cutler, 1959]. From a timing point of view, this means

that the user estimates the time as well as he can from the preceding

signal and asks that the future signal contain only such information as

will reduce his error. Since no two users of the same timing signal are

likely to have the same errors as a function of time, the optimum signal

for the two users quickly diverges. Such a system would be difficult

and expensive to provide for a large number of users. Nevertheless,

there are some instances in time dissemination where this kind of procedure might be approached. These systems might be classified as "active" systems where the user has some way of communicating with

the signal source. The meteor trail [Gatterer, 1967] and satellitetransponder [Markowitz, et al. , 1966] timing systems are possible

examples. In these systems, a signal may be transmitted from the
master to slave clock and back again, allowing the propagation delay and

the slave clock error to be determined. At this point, the signals from

the master station could be advanced in time so as to arrive on time at

the slave station. Thus, in a sense, a signal is being transmitted which

is tailored to the users' specific needs.

3. Bandwidth and the Time of Arrival Problem

In a recent CCIR document [l966] it is stated:

"There is an increasing number of applications

requiring the use of very precise time reference. In an

effort to achieve greater precision, it is expected that the

system of time distribution will tend to make use of an

increased bandwidth up to the limits imposed by the allocated

band, the instabilities of the propagation path, or by consideration of noise and interference. "

The truth of this statement is well demonstrated by (3). Nevertheless,

as stated previously in instances where the problem is essentially "time

of arrival" rather than identification, what is required for an optimum

time of arrival determination is that the signal be concentrated in a narrow frequency range which implies reduced rather than increased

bandwidth B.

To further amplify this point, consider the similarity between the

problem of using a timing signal for synchronization and the problem of

detecting and using a radar echo. To demonstrate this similarity, consider the steps which are required in the radar operation:

1. Transmit a signal at some known time.

2. Listen for the return echo.

3. If one is present, measure its arrival time so that the

range to the reflecting object may be determined.

4. Determine the carrier frequency of the" echo.

10
Steps 2 and 3 must also be taken by the user of the timing signal. In

addition, if the user needs the time interval as well as the epoch, he

may want to take step 4.

People interested in the radar problem have devoted considerable

effort to the problem of radar signal design so the required steps can be

taken efficiently. Asked simply, for a given energy and noise environment, what signal shape allows the four operations listed above to be

accomplished with the least uncertainty? Approaches have been developed

to answer the signal design problem for radar and have obvious application

to the design of timing signals. We will consider these steps in detail

now.

3. 1 Recognition of the Presence of the Signal

Increasing the amount of energy in the signal for a given noise

environment will certainly enhance the chances of recognizing the presence

of the signal. However, for a given amount of energy, might it be easier

to recognize the signal if it had some distinctive shape ? To make meaningful comparison of various shapes requires that

the means of detection be specified. A device known to maximize the

output S/N is the so-called "matched filter". We will assume in the

signal comparison that a matched filter is always used.

Suppose that the input signal s(t) has a voltage spectrum S(f), that

is, S(f) is the Fourier transform of s(t). Further, let H(f) be the

frequency-response function of the filter (not necessarily a matched

filter), and let NQ be the input noise power per unit bandwidth. Then it may be shown [Skolnik, 1962] that the signal-to-noise power ratio R out

of the filter is given by

| f S(f) H(f) exp (j 27Tft)df| :

^ r iH(f)i 2 df

Ra

N, < 5 >

11
If H(f) = S* (f) exp (-j 2ft) (* means complex conjugate), then the equalitysign in the above equation holds. That is, H(f) is now a matched filter

for s(t). Since

f |S(f)| 2 df = E, the total energy,

we obtain for the optimum case

R -M

No"

The important thing to note about this result is that it is independent of

the signal shape S(f). That is, as long as we use a matched filter, only

the energy in the signal and the noise determine our ability to recognize

the presence of the signal. Note, however, that (5), which gives the

expression for the matched filter, involves the signal spectrum. This

is not surprising since we expect to have a better chance of recognizing

the signal if we know in advance what it looks like.

It is relatively difficult to build matched filters for signals with

complicated spectra. On the other hand, the detection of CW

signals by phase-lock receivers closely approaches a matched filter

under certain conditions. Phase-lock receivers are not usually described

in terms of matched filters, but rather are considered in terms of crosscorrelation where the incoming signal is multiplied by a replica of itself.

It may be shown [Skolnik, 1962], however, that the output of a matched

filter is proportional to the input signal multiplied by a replica, thus

indicating the equivalence between matched filters and cross-correlation

or phase-lock detectors.

12
3. 2 Measurement of Signal Arrival Time

We have seen in the previous discussion that the signal shape is of

no importance in determining its presence when a matched filter is used.

Let us assume now that the signal-to-noise ratio is large and we are

using a matched filter. Then, does our ability to measure the arrival

time depend upon the signal shape ? There are several approaches to

this problem. Using a statistical approach, Berkowitz [1965] shows

that the standard deviation in the time of arrival of the signal is given by

a (t) = 1 -7- . (6)

S(2E N ) '2

Notice that this expression depends upon the total signal energy E, the

noise background density NQ

, and also involves an additional factor 8 , defined by

8

s = -j^ f°° (277f)2 |S(f)| 2 df;

an expression which depends very much upon the signal shape. Whereas

recognizing the presence of the signal was independent of signal shape,

measuring the arrival time is not. 8 is called the effective bandwidth

and differs from the usual definition of bandwidth.

Equation (6) shows that the time-of-arrival error for a given

signal-to-noise ratio may be minimized by maximizing 8. The form of

8 is similar to the moment of inertia of a body about its center of mass:

the moment of inertia of a body confined to some radius r will be maximized by putting all of the material in the body at the radius r. So, if

external factors limit our bandwidth to B, then the spectrum which produces the largest effective spectrum 8 is one with all of the energy

13
crowded at the two ends of the band, i. e. , two sine waves at frequencies

f - B/2 and f + B/2, where fQ

is the carrier frequency and B is the

bandwidth.

The spectrum of such a signal is given by

6(f) = 6 (f - f

Q

- B/2) + 6(f - f + B/2)

where 6 is the delta function. Substituting 6(f) into (5), we obtain

a(t)„ =

27T (2fQ

2 + 2fQ B + Bs

) ,z (2E/NQ)

If only the envelope of the signal is considered, the standard

deviation in time of arrival becomes

G (t)e = V {7)

2 77B (2E/NQ ) /2

which is the same as (4) in Section 2. At the present time [Morgan, 1959;

Watt, et al. , 1961; Fey and Looney, 1966], a two-frequency timing

system is being tested in the VLF radio spectrum which does make use

of both the carrier and the envelope. This system will be discussed in

more detail later.

3. 3 Measurement of Time Interval

If a relatively low frequency timing signal is transmitted as modulation on a higher frequency carrier, time interval may be measured by

demodulating the signal or by direct observation of the carrier itself.

The statistical approach mentioned in the preceding section may also be

14
used to estimate the precision of the interval measurement: either

carrier, modulation, or both. The result is similar to that obtained for

the time of arrival problem [Berkowitz, 1965]; namely,

a (f ) = ±-

a (2E/N p

2

where

f (2 7Tt)2 2

(t) dt

a = %

a is called the effective time duration of the signal. The primary implication of this result is that the longer the signal lasts, the more precisely

its frequency and therefore its interval may be measured. For an

application of this result, suppose a signal consists of a sine wave one

cycle long with frequency f ; i. e. ,

S(t) = A sin 2 7T ft, <t < l/f

= otherwise.

From the two equations above

a(f) = >/6~f

^

V(87T

2

- 3) ^2e"

V

3Nof

2 TT V^E"

15
As long as E/N » 1 [Skolnik, 1962], E, in the expression above, can

be considered the total signal energy resulting from several measurements. Thus, if the sine wave power = P, E = Pt where t is the duration

of the signal. With this result, we can rewrite the expression for a (f)

to obtain

a(f) =

This result shows the 1 A/t dependence in the standard deviation of the

frequency measurement which was pointed out in the introduction.

3. 4 Simultaneous Measurement of Time of Arrival and Frequency

The product of the rms time of arrival and frequency errors is

a(f)a(t) = a 8 (2E/NQ

)

*

This shows that the time of arrival and frequency (or interval) may be

measured simultaneously with as small an error as desired by increasing

the S/N or the a 8 product, or both. As a matter of interest, it may be

shown that the product aB always satisfies the relation

[Skolnik, 1962 J. This result is referred to as the radar uncertainty

relation. Thus,

a (f) ° (t> a ; (2e/n )

• < 8 >

16
For a Gaussian- shaped pulse

8a = 77

so that from (8), this pulse is the poorest possible waveform for the

simultaneous measurement of time of arrival and frequency where time

of arrival refers to the pulse itself and where f means the carrier frequency for the pulse.

4. The Multiple CW System

It was shown in the previous sections that if signal is restricted to

some bandwidth B, a sine wave is optimum for time of arrival measurements. But this system will be ambiguous to all who do not know the

time unambiguously to at least the period of the sine wave. The system

could serve a larger class of users by transmitting a code, as discussed

previously, or by increasing the period of the sine wave with the cost,

however, of making some users who could use the higher frequency wait

longer for a timing mark. To satisfy more users, we can transmit two

sine waves. As an illustration, suppose we have some users who know

the time to the nearest second and another group of users who know the

time to the nearest 10 seconds. We could satisfy both of these groups

by transmitting two sine waves simultaneously with periods of 1 and 10

seconds. As more requirements were added, more CW components

could be transmitted at both higher and lower frequencies. In the limit,

the spectrum of the composite signal would approach that of a pulse, which, of course, is unambiguous but occupies more bandwidth B.

Consider this multiple CW system in somewhat more detail.

Suppose two sine waves are transmitted one after the other with the same

power P at frequencies f and f where f > f . Assume that f is the

lowest frequency which is not ambiguous for users. In addition, assume

17
that the user requires a measurement precision of time of arrival to a seconds. To use the system requires that the position of the zero-crossing

of the lowest frequency f be known well enough to coarsely identify the

correct cycle of the higher frequency f That is, from (7), the phase

jitter on the signal at frequency f ,

/

N° \

l/

> j_

to correctly identify a cycle on the signal with frequency f . . Since we

are transmitting a sine wave of constant amplitude, we may replace E

by PT where P is the power in the signal and T is the observing time.

Solving this equation for T, the minimum time required to identify a cycle at f is

N,

Ti =

8772 P

Similarly, the time required to obtain the desired precision a by observing the signal at f . is

87T

2 P

f

2 °

The total time required is

T = T + T 1 2 ^H) + fe?) \

In a similar fashion, if we have n frequencies, f in. . . f . the time

18
required to obtain a precision a starting at the lowest frequency f is

i. w- fey • N n ,f

T= y i-^^i f ([--—--i

87T

2 P

1 = 1

To determine the optimum spacing of the frequencies, differentiate this

expression with respect to frequencies f through f , and obtain a

set of simultaneous equations. It may be shown that T is a minimum if

\ = ffeY"

1

(f i)

n " 1 + J

I i = 2, 3, ... n+ 1. (10)

Substituting (10) into (9), we obtain

N

n + 1 o min (af

1

) 2/(n+ 1)

87r 3p

Now, to optimize the number of frequencies used, differentiate this ex- pression with respect to n„ We find that if

n = - (1 + 2 In o-f

1

) = - (1 + 4. 65 log^afj

= n

op,

then T is a minimum with respect to both n and the f . , and is given by

N N

T . = (-4.65 log.. CTf.) — = (n + 1) — . mm ° 10 1 „? T^ op ' 2 _

877 2 P

r

8tt

2 P

Figure 2 shows how T varies with n for the case f = 100 Hz,

19
100,000 - I

I I I I I I I I I I I I I I I I

I I I I I I I I I I I H

10,000

|Q I I I I I I

I 10 20 30

N

Figure 2. Variation in the required transmission time, T, in

seconds, as a function of the number of signals

transmitted, N.

20
87T

2 P/N = 1000, G = ljus, and where the f . have the optimum spacings

given by (10),, The figure illustrates that a significant gain in precision

may be made by using more than one frequency, but beyond n = 2 or 3,

the gain diminishes rather rapidly as the value of n, which minimizes T,

is approached. In some situations the value of n, which minimizes T,

may not be realistic in the following sense. As n approaches n , the

transmission time per frequency becomes small. If this transmission

time becomes short compared to the period (i. e. , l/f) of the transmitted

signal, then the foregoing discussion becomes invalid since <j(t) is no

longer accurately given by (6). Although we have transmitted the signals

sequentially in this discussion, the same result is obtained, if all signals are transmitted simultaneously, since in this case each signal-tonoise ratio must be reduced by a factor 1/n to maintain constant energy.

5. The VLF Two-Frequency Timing System

In the VLF two-frequency timing system presently being used, the

low frequency, Af, (called f in the previous section), is obtained by

observing the beat note between two CW VLF signals at frequencies

f = 20 kHz and f = 19. 9 kHz. The next higher frequency can be either

f f or the average carrier frequency

As mentioned, the principle of the system is to observe the low frequency

long enough to resolve the ambiguity at the higher frequency and then observe this unambiguous higher frequency until the desired timing precision

is obtained.

In this system, used over an ionospheric path, the magnitude of the

phase fluctuations on the two carriers are not necessarily equal and may

have some correlation p. For these conditions, it may be shown from

21
(9) and (16) (see Appendix) that the optimum value of carrier frequency

is given by

V ~V?

' 'all- 2yo + y

2

) /2

f = W 77 • (ID

~yp + y

2

)

Here y is the ratio of the rms phase fluctuations on the signals at frequencies f 2 and f , respectively.

As an illustration, assume p=0, y=l, Af = 100 Hz, and the user

would like a measurement precision of a = 10 /is. From (11) the

optimum carrier frequency is

If p= 0. 5, then

and for p = 0. 8

f = 8. 4 kHz.

c

f = 10. kHz

c

f = 22. kHz

c

which is near the presently used carrier frequencies. As p approaches

1. 0, f approaches infinity if y is held at 1. 0. In actual practice y

would probably not approach 1. as p approached 1. 0. The reason for

this is that a highly correlated motion of the carriers at frequencies f

x and f would probably be due to a general movement of the entire reflection

region in the propagation medium (path length fluctuations). This would

result in the delay of the two carriers changing the same amount in time

rather than in phase angle. Since y refers to standard deviation of the

phase, an equal movement of the two carriers in time requires that

y = £ 2 /£ lm For this particular case, the standard deviation of the motion

of the envelope is just the standard deviation of the motion of either one

22
of the carriers. Equation (11) reduces to

in this case.

It is not always possible, because of other considerations, to use

the optimum f given in (11). Figure 3 shows the variation in observation

time T as a function f /f (where f is the optimum carrier frequency)

C C °

C ° 8tt 2 P

for several values of P where y = 1.0, —— = 1000.0, and Af = 100 Hz.

o

6. Characterization of the Noise

Noise affecting a timing system may be placed into two categories.

The first is usually called additive noise, and refers to noise energy

added to the spectral region occupied by the signal. Examples are:

ignition noise, noise generated by the receiver, atmospheric noise, and

cosmic noise. The effects of additive noise can be made as small as

desired b increasing the strength of the signal. There is also multiplicative noise, which cannot be diminished by increasing the signal

strength. This noise is produced, for example, when the propagation

medium distorts or modulates the signal. This distortion is always

present in the signal no matter what its strength. Diversity techniques

have been developed to offset multiplicative noise for communication

channels, but appear to have had little or no application with respect to

time dissemination. The concept of diversity is to obtain a greater

number of independent measurements of the signal for a given amount of

signal energy. Thus, the variance of several independent measurements

23
DTT i—i r i i i i—i—

r

o

I I I 1 I I I

CO

H >. . „3 o

<u rt u

(0O ° U 1) c 2, **—**

rJ d cu CO d cr j-i Q 2 n c Z

TM S a

4-> p O

T3 ^ 4-> <u o ft o •H3 .r- <U UJ ^ ti ^ CD ro *" CO

l

the

i

»f

the £

,

to

c

in 1- .3 u . O

gure

3.

Variatio

a

functic

broadcas

o

o

o

o

o

24
is given by

a

2 a

2 = m

n

where a

2 is the error for each independent measurement, and n is the m

number of independent samples: independence meaning the noise is

uncorrelated from one sample to the next. As an example of multiplicative noise, consider a signal from an artificial earth satellite which

passes through a set of ionospheric irregularities whose average size is

L (see fig. 4). If L is much larger than the signal radio frequency

wavelength \, and if the satellite is far from the irregularities, then

each irregularity will scatter the signal over an angle Q = \/L. To an

observer who is at a distance D from the irregularities, it will appear

to him as though the signal is coming from an area subtended by the

angle at the distance D. Suppose now the satellite transmits a pulse.

Part of the pulse energy will travel directly to the observer along the

path d1# Other portions of the pulse will travel over more distant paths

out to the greatest path d given by

d 2 = ^ D

2 + D

2 92 /4

= D V 1 + \

2 /2L 2

.

Thus, a pulse observed at the ground will be spread over a time interval

d - d

2 1 t = D(l - Vl + \

2 /2L 2

) V V

where V = group velocity of the pulse.

25
o o CD O CD CD <D

Observer

Figure 4. Schematic representation of scattering of satellite

signal.

26
In addition, the irregularities may have motion. This would produce a random phase modulation of the received signal so that its spectrum

would appear to be spread. If the satellite is transmitting a CW signal

of frequency f , the received signal will appear to have spectrum spread

about f . This is true only if there is no average motion of the irregularities either away from or toward the observer. Otherwise, the

spectrum will also have its average value Doppler shifted by some amount

f . If the spectrum is spread about the average frequency over a range

BHz, the received signal will fade with a period of approximately l/B

seconds.

We thus arrive at a fairly simple picture of the fading produced by

this channel. First, the signal will appear to fade at a rate B c/s; and

second, portions of the signal spectrum within 1/t c/s of each other

will appear to fade coherently. The fading rate is governed by the

motion of the irregularities, and the correlation is governed by the

spread in the multi-path.

If we have a signal which is spread over T seconds in the time domain and over F c/s in the frequency domain, then we could think of

the signal as containing N independent diversity elements where

N = H/BHi/t) - < Bt>

<

TF>-

The precision 3

—, obtained over a time T with a signal spread over a TF

spectrum F is

a

3 a

2

p m m

a

2TF

=" —N

- --" —(Bt) (TF)

In principle at least, if the noise in our measurement always adds

incoherently and the signals add coherently, the lack of measurement

27
precision due to the multiplicative propagation noise can be made arbitrarily small if the available energy is divided into enough diversity

elements. Eventually the additive noise will dominate over the multiplicative noise and will represent the fundamental limitation.

In most practical cases there is another type of limitation which is

related to the particular detector being used. It is a characteristic of

most detectors that the output signal to noise deteriorates rapidly when

the input signal to noise reaches some minimum value (threshold effect).

If the available energy is divided into too many elements, the detector

input signal-to-noise ratio will be too small for the detector to operate

efficiently. For example, in a square law detector where S is the input

signal and N is the noise, the output of the detector is

(S + N) 2 = S

2 + 2NS + N2

.

The output signal-to-noise ratio is

S

2 r J. 1 f 1 »

°~2NS + N

2 ^ N

1 N

If S/N « 1

o \n) r =

which deteriorates rapidly for small S/N.

With some kinds of diversity it is not necessary to divide the

available energy into packets thus reducing the signal energy per packet.

An example is space diversity. Here the signal is observed simultaneously at points on the ground sufficiently far apart for the noise fading to

be uncorrelated. Other examples are polarization and angle of arrival

diversity.

28
The primary purpose of the preceding discussion was to point out

the possible utility of using diversity techniques in time dissemination

and not to discuss any particular application in detail. However, there

is evidence in the literature that multiplicative noise rather than additive

noise is a limiting factor in some existing time and frequency systems.

In a study made by Blair and Morgan [ 1965] of the short term frequency

stability of WWVL as recorded at Greenbelt, Maryland, they found the

standard deviation of the frequency fluctuations decreased by a factor of

two during the daytime when the transmitted power was increased by a factor of about 50. If the frequency fluctuations had been produced

entirely by additive noise, the expected decrease would have been by a factor of about seven (see (6)). In this particular instance the multiplicative noise was probably due both to propagation effects and to frequency

deviations in the signal as transmitted. During the night there was no

appreciable change in the standard deviation of the phase fluctuations

from low to high transmitted power. If we assume that the part of the

multiplicative noise due to fluctuations in the transmitted signal was about

the same both day and night, then it appears that for the night time

observations the propagation effects dominate over both additive noise and

transmitted signal fluctuations. In any event, these observations demonstrate that one must be aware of the relative importance of multiplicative

and additive noises as they affect a specific system, before any meaningful steps may be taken to improve the situation.

7. Appendix

7. 1 Effects of Noise on Two-Frequency Timing Systems

If the phase fluctuations on the two carriers are not perfectly

correlated, the position of the envelope will move considerably in excess

of the noise on either one of the carriers. In the event that the noise on

29
the two carriers becomes completely uncorr elated, the fluctuation in the

position of the beat note becomes inversely proportional to the frequency

separation [Watt, et al. , 1961]. Consider the general case where the

correlation between the two carriers phase fluctuations is arbitrary and

where the standard deviation of the fluctuations on each of the two carriers

are not necessarily equal.,

The two carriers may be written in the form

x (t) = A sin 2 7T f (t - T ) x

and

y(t)= Asin2fffJt-akT + UT1

) 2 xx

where t is the total displacement of the zero crossing of x (t) about its x mean position, kr is the portion (0 ^ k <. 1) of the displacement of the

zero crossing of y (t) which is correlated with x (t), r

1 is the uncorrelated part, and a is some constant to take care of the possibility that

the variance of the zero crossing times of x (t) is not the same as that

for y (t). Then*

x(t)y(t) = V£cos [2 7Tf (t-T )- 277f (t-akT + ttT

1 )]

- '/2 cos [2 7Tf (t-T )+ 2 7Tf-(t- akT + ttT

1

)]

.

1 X 2 XX

The first term represents the envelope component and the second is the

carrier frequency.

We want to evaluate the jitter of the envelope, i. e. , we are

interested in the times t when the envelope is zero. Thus

o

x- . ,

Arg(E) = 2 7Tf,(t -T)-2fff,(t -akT + ar1 "ox ) = ± mr, n= 0, 1, 2

^ O XX

*The beat note may also be produced by adding the signals, although it

does not change the result obtained here.

30
or rewriting

Arg(E)=2 7Tt (f - f ) - 27TT (f . - a kf _) - 2 TTOiT

1 f _ = ± nff.

O 12 X 1 2 X 2

Solving for t ,

±n7T + 27TT (f, - akf.) + 2 TT (XT

1 f x 1 2 x 2 *p

" 2ff(f

1 -f

3

)

Then the variance of t is o

Var t Q

= r

f r^f 2i 3 r

f

2 r

where a is the variance of t and a 1 is the variance of t

T xr x

x x

Now

T = kr + T

1 X XX

• a

2 = k

2

ct 2 + a t or

» • T T X

X

a

2 = a2 (1- k

2

) . (13)

T T

31
Substituting (12) -» (13)

Var t •^(Trrvl^)' =

(f 1 2'

X

(f - f ) 2

x 1 2'

[f 2

- 2akfl f 2 + a

2

f

2

2

] . (14)

The total displacement of the phase of y(t) is ar or

Var {ar ) = a

2 = a

2 a

2

. x' T T

y x

Therefore, a

2 represents the ratio of the variance of time variation for

y(t) and x(t)„ Also the phase correlation, p, between x(t) and y(t) is

P =

<T T > x y

ct a

T T

x y

<(T )a(kT + T

1 )) x' X X

a a T T

x y

(akT2

> (ar r

1

> X XX +

oT

a

T

ao

T T

x y x y

32
But the last term equals zero since t and t

1 are not correlated

x x

aka 3 T

.*. P - —^ - k. (15)

CCO TX

Substituting (15) -» (14), we obtain

G

3 T

Var to =

(£ i- C)

2

[ ^ " 2apfi f 2 +a2f^

<p 2

[1 - 2yp + y

2

] (16)

4772 (f

1 - fj a

where

<p 2 y

2 = -x

^x

and (o 3 and en 2 are the variances in phase rather than in time on the ^y ^x

two carriers. Some special cases:

A. p = 1, a = 1

Var t_ = a

3 T ° x

That is, the jitter on the beat note is just the jitter on either one of the

carriers.

B. p = 0, a

2 = 1

Var t = a

3 T ° x U

3

- fa7

33
(2 7Tf

1 ) 2 C. p = 0, a

2 = , i.e., <<p 2 >x = <(p 2

> y

(27Tf

2

) 2

which is the more usual case since ((p 2 )a , „ Then

Var t Q

= x

(V f

s r

r 2 t

f3 + f2

1 2 2 W2 J

(f

t

x

r ff f

2

2 1

F^ [

l2+ "?~J

a' x

<f

l " V N•

Now

2 x

T " 477 2 f

2 x i

Var t

2<cp 2

> x

o 47^(f 1 -f )"

or ^cpx

'O «l - W 2

which is the result given by Watt, et al. [1961].

34
Although the results obtained here apply generally, the system is

presently being investigated at VLF. However, very little information

is available about the correlation bandwidth at VLF. Presumably, it is

considerably greater than 100 c/s at VHF and up; that is, two signals

separated by less than 100 c/s would have their fading highly correlated

while signals separated by frequencies appreciably greater than 100 c/s

would undergo uncorrelated phase fading. For purposes of discussion,

assume that the correlation has a gaussian dependence. Therefore,

<P2 Var t Q

= pr

s (

2 " exp

" w)

where f = f - f . Figure 5 shows Var t assuming that the correlation

s l 3 6 o &

bandwidth is 100 c/s and with the frequency separation varying from to

1000 c/s. It is obvious from this figure that the amount of correlation

between the phase jitter of the two carriers is not nearly as important

a factor as is the frequency separation. This may be easily seen by

looking at (16), which shows that for a given frequency separation, f , that Var t will only be a factor of two larger for the case of p = as compared to the p = 1 case. On the other hand, for a given value of , the variance may be reduced by a factor of two by increasing the frequency

separation by a factor of -dz and by a greater amount for greater frequency separations. However, this cannot go on indefinitely because the

problem of ambiguity will crop up.

8. Acknowledgments

I am indebted to E. L. Crow for suggestions and comments on the

material contained in the Appendix. In addition, I have had many stimulating discussions with L. E. Gatterer, G. Kamas, and A. H. Morgan

35
T 1 III I I I I I 1 Ml

M I I I

(ZH ) SJ o

CVJiO >

roiOio s . ^ o > nJ u ci u a, , eft

CTJ a

UG

<L>

cr

a>u

O

mO

a)O

•r-i > O

•i-i u Io

in 1 1 i I i i I Ll j I I oo io

36
on most of the material covered in this paper. I also thank R. Ruppe

for checking the mathematical results and for developing computer

programs to calculate the material presented in figs. 2, 3, and 5.

37
9. References

Berkowitz, R. S. (1965), Modern Radar, Part III.

Blair, B. E. , and A. H. Morgan (1965), Control of WWV and WWVH

standard frequency broadcasts by VLF and LF signals, Radio Sci. ,

J. Res., NBS 69D, No. 7, 915-928.

CCIR Document VII/1017-E (1966), Freq. spectrum conservation for

high precision time signals.

Fey, L. , and C. H. Looney, Jr. (Dec. , 1966), A dual frequency VLF

timing system, IEEE Trans, on Instr. & Meas. , Vol. IM-15,

No. 4, 190-195.

Gatterer, L. E. (1967), Private communication.

Markowitz, W. , C. A. Lidback, H. Uyeda, and K. Muramatsu (Dec. , 1966), Clock synchronization via relay II satellite, IEEE Trans,

on Instr. & Meas. , Vol. IM-15, No. 4, 177-183.

Morgan, A. H. (1959), Private communication.

Pierce, J. R. , and C. C. Cutler (1959), Advances in space science,

55-108.

Schwartz, M. (1959), Information transmission, modulation, and noise,

Chapter 6.

Skolnik, M. I. (1962), Introduction to radar systems, Chapters 9 and 10.

Watt, A. D. , R. W. Plush, W. W. Brown, and A. H. Morgan (Nov. - Dec. , 1961), Worldwide VLF standard frequency and time signal

broadcasting, (Radio Prop. ), J. Res., NBS 65D, No. 6, 617-627.

38

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