Library, K.W. Sldg

NOV 27 1961

ecknlcal riote

\j5oulder laboratories

PB 161603

^e. 102

PERFORMANCE PREDICTIONS

FOR SINGLE TROPOSPHERIC

COMMUNICATION LINKS AND

FOR SEVERAL LINKS IN TANDEM

BY

A. P. BARSIS, K. A. NORTON,

P. L. RICE, AND P. H. ELDER

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950. These include the development and maintenance of the national standards of measurement and the provision of means and methods for making measurements consistent

with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials, devices, and structures; advisory services to government agencies on scientific and technical problems; invention and development of devices to serve special needs of the Government; and the development of standard practices, codes, and specifications. The work includes basic

and applied research, development, engineering, instrumentation, testing, evaluation, calibration services,

and various consultation and information services. Research projects are also performed for other government

agencies when the work relates to and supplements the basic program of the Bureau or when the Bureau's

unique competence is required. The scope of activities is suggested by the listing of divisions and sections on the inside of the back cover.

Publications

The results of the Bureau's research are published either in the Bureau's own series of publications or

in the journals of professional and scientific societies. The Bureau itself publishes three periodicals avail- able from the Government Printing Office: The Journal of Research, published in four separate sections,

presents complete scientific and technical papers; the Technical News Bulletin presents summary and pre- liminary reports on work in progress; and Basic Radio Propagation Predictions provides data for determining

the best frequencies to use for radio communications throughout the world. There are also five series of nonperiodical puolications: Monographs, Applied Mathematics Series, Handbooks, Miscellaneous Publications,

and Technical Notes. A complete listing of the Bureau's publications can be found in National Bureau of Standards Circular 460, Publications of the National Bureau of Standards, 1901 to June 1947 ($1.25), and the Supplement to National Bureau of Standards Circular 460, July 1947 to June 1957 ($1.50), and Miscellaneous Publication 240,

July 1957 to June 1960 (Includes Titles of Papers Published in Outside Journals 1950 to 1959) ($2.25); avail- able from the Superintendent of Documents, Government Printing Office, Washington 25, D. C.
NATIONAL BUREAU OF STANDARDS

102

August 1961

PERFORMANCE PREDICTIONS FOR SINGLE TROPOSPHERIC

COMMUNICATION LINKS AND FOR SEVERAL LINKS IN TANDEM

by

A. P. Barsis, K. A. Norton, P. L. Rice, and P. H. Elder

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. They are for sale by the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C.

DISTRIBUTED BY

UNITED STATES DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D. C.

Price $ 3 . 00

TABLE OF CONTENTS

Page

No.

ABSTRACT

1. INTRODUCTION 1

2. DETERMINATION OF SYSTEM POWER

REQUIREMENTS 3

3. THE CONCEPT OF SERVICE PROBABILITY 23

4. SERVICE PROBABILITY CALCULATIONS 29

5. PERFORMANCE OF A CHAIN OF SYSTEMS

OPERATING IN TANDEM 35

6. PREDICTION IMPROVEMENT BY MEASUREMENTS 48

7. COMPARISON OF PREDICTED AND MEASURED

DATA AND AN EXAMPLE OF PREDICTION

IMPROVEMENT 63

8. ADDITIONAL CONSIDERATIONS AND CONCLUSIONS 76

9. ACKNOWLEDGEMENTS 86

APPENDIX I: ANALYSIS OF VARIANCE OF LONGTERM MEDIAN TROPOSPHERIC TRANSMISSION LOSS DATA

1-1. Introduction 87

1-2. A Statistical Model 89

1-3. Description of Data 99

1-4 Analysis of the Effect of Sample Length on the

Variability of Long-Term Medians 101

2

1-5. Estimation of the Path-to-Path Variance <r_ 113

2

6

1-6. Estimation of cr (p,n), the Variance of |a.(p,n) 118

APPENDIX II: ESTIMATING PREDICTION UNCERTAINTY

FOR ONE PATH USING MEASUREMENTS

MADE OVER ANOTHER PATH 120
Page

No.

APPENDIX III: THE EFFECTIVE NOISE FIGURE OF

A RECEIVING SYSTEM 128

REFERENCES 151
ABSTRACT

Performance of long-distance tropospheric communication

circuits, either singly or in tandem, is predicted in terms of the

probability of obtaining a specified grade of service or better for

various percentages of time. The grade of service is determined by

the minimum acceptable ratio of hourly median predetection-RMSsignal-to-RMS-noise for the type of intelligence to be transmitted.

The standard deviation of prediction errors depends upon

the percentage of hours the specified grade of service is required

and on parameters characterizing the propagation path. The possibility of reducing this standard deviation by making path-loss

measurements is discussed. It is shown that no improvement is

possible unless the test path is very nearly the same as the proposed

operational path; in particular, unless the test path and operational

paths have terminals less than one mile apart it is shown to be doubtful in most cases whether the observations will be useful in improving

the reliability of the predicted performance of the proposed system.

Assuming that the test path and proposed operational paths are

identical, estimates are given of the number of days of observations

required to halve the prediction uncertainty or, alternatively, to

reduce it to 3 db; in some cases several years of observations are

required.

A tutorial discussion is given in Appendix III of the concepts

of the effective noise figure and the effective input noise temperature

of a receiving system.

PERFORMANCE PREDICTIONS FOR SINGLE TROPOSPHERIC

COMMUNICATION LINKS AND FOR SEVERAL LINKS IN TANDEM

by

A. P. Barsis, K. A. Norton, P. L. Rice, and P. H. Elder

1. INTRODUCTION

The utilization of tropospheric radio wave propagation for

long distance communication circuits has become more and more

important in recent years. The underlying physical principles for

this mode of propagation are still a subject of discussion and some

controversy and will not be treated here. Of more importance to

the engineer is that a signal transmitted at very high and ultra-high

frequencies through the troposphere over a long distance is subject

to substantial amplitude variations which may be described statistically. Information contained in the signal will be affected by these

variations, and may be misinterpreted by the receiving system if the

amplitude happens to be too low compared to the inherent noise level

of the receiving system or its surroundings.

Taking into account the characteristics of the propagation

path, the receiving system, the type of information to be transmitted,

and the nature of the signal variations, one may estimate the amount

of transmitter power and the antenna size necessary to provide a given

degree of reliability of the system for a specified percentage of the

time. This degree of reliability is most easily thought of in terms of a maximum permissible error rate in teletype transmission and may

be identified with a "grade of service". Conversely, for a given system

the percentage of time that it is possible to provide a given reliability
-2-

of communication or grade of service may be estimated. Both

results will be functions not only of the percentage of time for which

this given grade of service is desired, but also, because of errors in

the prediction, will be dependent on another statistical variable,

namely the probability that the actual system performance and path

parameters may be expected to conform to the theoretical assumptions made.

It will be shown how the probability of obtaining a predetermined grade of service or better during a given percentage of

time is determined as a function of the system parameters. This

probability will be called "service probability, " and the percentage

of time concerned will be termed "time availability, " and expressed

in percent of all hours. The grade of service will be defined as an

hourly median predetection RMS -signal-to -RMS -noise ratio

necessary to provide the desired maximum permissible teletype error

rate, a corresponding quality of voice transmission, or any other

desired measure of circuit reliability. For any particular communication link the procedure consists in calculating first the distribution

of hourly median basic transmission loss values over the time period

(generally all hours of the year) for which the prediction is desired.

Relations between basic transmission loss and transmitter power

requirements are easily established by the use of equipment parameters and specifications for the desired grade of service. Then,

using reasonable assumptions about the prediction uncertainty, the

service probability of a given system as a function of the time availability may be calculated and presented in graphical form.

Finally, a study is made of the possible reduction of prediction uncertainty by means of transmission loss measurements.
-3-

2. DETERMINATION OF SYSTEM POWER REQUIREMENTS

In the design of a complete communication system, there are

a number of factors determining the optimum frequency, transmitter

power and diversity configuration, as well as the service probability

for a specified grade of service. In general, the transmitter power

is related to equipment parameters, basic transmission loss, and

path antenna gain by a simple formula given by Norton [ 19 56 ]:

P = L, -G + F + L + R + B - 204 (1)

t b p t

where

P = transmitter power in decibels above one watt

L, = basic transmission loss over the path in decibels

b

G = average path antenna gain in decibels

F = effective noise figure of the receiving system in decibels,

including the effects of the receiver itself, the receiving

antenna transmission line losses and the effects of external noise in the receiving antenna. See Appendix III. L = transmitting antenna transmission line and associated

t &

losses, in decibels

R = hourly median RMS-signal-to-RMS-noise ratio in decibels

in the predetection pass band. A given minimum value Rm

characterizes a specified required grade of service. The

RMS signal may consist of a modulated carrier with its

sidebands, or only the modulated sidebands in the case of

suppressed carrier systems.

B = 10 log b, where b is the predetection noise bandwidth in

cycles per second as defined in Appendix III. The band
b should be made sufficiently wide so that it allows for

the effect of drift between transmitter and receiver

oscillators.

204 = a constant equal to (-10 log kt ), where k is Boltzmann's

constant, and the reference temperature t is taken to be

288. 39° Kelvin.

For a long-distance tropospheric propagation path the terms

F, L , R , and B are determined by the choice of the equipment

and the specified required grade of service. They are frequency

dependent, but otherwise may be considered fixed when the frequency

of the system has been selected. The basic transmission loss Ll

and the path antenna gain G , however, are time dependent, and the

resulting transmitter power P from (1) will be the same function of

time as (L, - G ). If (1) is solved for one of the other parameters,

b p

the solution will also be a function of time.

It has been found convenient and practical to distinguish

between short-term and long-term variations of basic transmission

loss. Variability within a single hour is arbitrarily classified as

"short-term", and is allowed for by a minimum hourly median

required signal-to-noise ratio, Rmm. The quantity R is simply

that number of decibels by which the predetection hourly median

RMS signal level exceeds the RMS noise level in order to provide the

specified grade of service during that hour. The study of long-term

variability is then concerned with the variability of these hourly

medians.

Within-the-hour fading characteristics are nearly always

adequately described by the Rayleigh distribution, particularly in the

case of those weak fields which determine the required power for
-5-

long-distance communication links. The required predetection hourly

median RMS -signal -to -RMS -noise ratio R may be directly related

to the binary or teletype error rate for various schemes of demodulation and various diversity configurations [ Florman and Tary, 1961 ],

assuming Rayleigh-distributed within-the-hour fading. For voice

modulation, R may also be related to a voice channel signal-torn noise ratio required for a specified intelligibility under Rayleigh

fading conditions. The grade of service given by R may be

specified by the requirements of the user of the communication circuit.

The designer of the circuit only has to meet the requirement that the

grade of service defined by R , or better, will be available during a m

specified percentage of the hours, or alternatively, he has to determine the percentage of hours during which a specified grade of

service, or better, is available, when certain types of equipment and

various amounts of power are used. Thus the grade of service and its

time availability are independent quantities which both enter into the

final determination of service probability. Statistical variations in

the short-term fading rates and fade durations are accounted for in

determining R ; they do not otherwise affect time availability.

Methods for calculating the hourly median basic transmission loss L and its variability, and the average or hourly bm ° median path antenna gain G as functions of carrier frequency, path

parameters, and antenna size are contained in an NBS Technical Note

[Rice, Longley, and Norton, 1959], and an Air Force Technical

Order, [ 1961 ], which also include references to earlier work by

NBS and other authors. Time variations of the path antenna gain G

are included in the time variations of basic transmission loss, so that

G in (1) may be assumed to be an average value independent of time

IT variations.
-6-

The prediction uncertainty concept which will be discussed

later ties in directly with the transmission loss calculation procedures

discussed in the Air Force Technical Order, [ 1961 ]; therefore, it

is recommended that this work be used for performance predictions.

The expected variability of hourly median transmission loss

values is expressed by an empirical function, V(p), which is combined

with a reference value, L, , to give the value of basic transmission bm

loss, L (p), not exceeded during p percent of all hours within the

time period considered. The percentage of hours, p, is called the

time availability. In terms of basic transmission loss,

L

b

(p) = Lbm - V(p) db (2a)

and in terms of field strength,

E (p) = E + V(p) db (2b)

b bm

where E (p) and E are the field strength values corresponding

to the basic transmission loss values L, (p) and L, , respectively. b bm

For the purpose of analyzing long-term variability, the year has been

divided into eight time blocks, four of which apply to the winter

months, and four to the summer months [Norton, Rice, and Vogler,

1955].* In accordance with the notation used in the Air Force Technical Order, \ 1961 1 , L, is the Time Block 2 (November - April, L J bm

1300 - 1800 hours) median of hourly median basic transmission loss

values. This value corresponds to the average winter afternoon hour

and is identical to the reference value calculated by the methods of

See Table 1-1, page 88.
-7-

Rice, Longley, and Norton, [ 1959 ], or the Air Force Technical

Order, [ 1961 ].

The time variability function, V(p), has also been discussed

in detail in the Air Force Technical Order, [ 1961 ]. For long extradiffraction tropospheric propagation paths V(p) depends on the

angular distance 9. Figs. 1, 2, and 3 are graphs of V(p, 9) for all

hours of the year, for the winter time block group (all hours for

November through April), and for the summer time block group (all

hours for May through October). Figs. 1, 2, or 3 may be used where

the angular distance 9 can be accurately determined from terrain

profiles. V(p, 9) is most reliable for 9 greater than 10 milliradians

and for frequencies near 100 Mc/s. For the most recent prediction

methods, the function V(p, 9) is replaced by a function V(p, d), where

d is the path distance in statute miles. Methods for determining

V(p, d) and pertinent graphs are given in the Air Force Technical

Order, [ 1961 ].

Calculation of the path antenna gain G as a function of the

P

free space antenna gains at the path terminal and other path and

circuit parameters is given in a paper by Hartman and Wilkerson

[ 1959 ], and also in the Air Force Technical Order, [ 1961 ].

Equation (2a) shows the relation between the basic transmission loss, L (p), expected to be exceeded during (100-p)% of

b

all the hours in a given time block, and the winter afternoon reference

value of basic transmission loss L . It is now practical to combine bm

all constant or specified terms of (1) into a constant K and subo

stitute into (2a), in order to obtain either an expression for the

transmitter power required to deliver a fixed predetection hourly

median RMS -signal -to -RMS -noise ratio R as a function of the m
8-

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percentage of hours p and the path and equipment parameters, or an

expression for the predetection hourly median RMS-signal-to-RMSnoise ratio available for p percent of all hours, using a fixed value

of transmitter power P over the same path. Both results may be

obtained for various values of carrier frequency, but for any particular carrier frequency value and a given path the calculated basic

transmission loss value L, and the path antenna gain G are bm r &

p

constants.

The auxiliary constant K is given by:

K = L, - G - 204 + F + B + L. , (3) o bm p t

where the symbols on the right-hand side are the same as in (1) and

have been defined above. Substituting (3) in (1), and changing notation

slightly,

P = K + R , (4) tm o m

where P is the transmitter power required to provide the grade of

service R for the Time Block 2 median hour, as L was used in m bm

(1) and (3). The relation betv/een the transmitter power P (p),

exceeded for pr percent r of all hours and the required power P

trn

for the Time Block 2 median hour is the same as the relation between

the corresponding transmission loss values given in (2a). By a simple

substitution,

P

t< P) = P

tm - V<P> = Ko + Rm " V<P>' (5 >
-12-

is obtained, where P (p) is the transmitter power required to provide a grade of service, R , or better, during p% of all hours.

If, on the other hand, the transmitter power is assumed

fixed or predetermined at a value P , the predetection hourlymedian RMS-signal-to-RMS-noise ratio R will exceed R(p) for the

percentage of hours or time availability which may be determined

from:

R(p) = P

q

- K

q + V(p) . (6)

Thus, (6) determines the grade of service as a function of time

availability. Here the designer is able to determine the grade of

service R(p) expected to be exceeded during a given percentage of

all hours.

The term "expected" has been used here because of the

empirical nature of the prediction formulas, especially of the time

variability V(p). For a given set of path and equipment parameters,

measurement results may vary substantially from path to path.

Average, or statistically expected values are provided by (5) and (6),

whereas prediction errors will be taken into account later on. At this point it is advantageous to illustrate the procedure

outlined so far by means of an example. This will also aid in the

understanding of the remaining discussion and procedures.

The National Bureau of Standards, at the request of U. S.

Air Force agencies, measured transmission loss over a 200-mile

path between North Africa and southern Spain during January and

February, 1957. This path lends itself well as an illustrative example

because of the opportunity to compare the results of the measurements

with the original predictions.
-13-

The terrain profile for this path is shown on Fig. 4, with the

pertinent horizon distances and other important parameters [Rice,

Longley, and Norton, 1959 ] indicated thereon. The communication

requirements and applicable equipment specifications for this path

are as follows:

System: Quadruple Diversity, Frequency Modulation

Carrier Frequency Range: Optimum between 400 and 2000

Mc/ s. Number of Channels and Modulation Requirements: 24-voice

channels, each 4kc/s wide including guard band. It is

assumed that up to 25% of the channels are active

simultaneously at any one time, and that two voice

channels carry 16 teletype channels each, using standard

multiplex methods. The teletype signal utilizes the 5-unit,

start-stop code at the rate of 60 words per minute.

Grade of Service Required: Average short-term teletype

character error rate of 0. 01% or better (not more than 1

in 10, 000 character errors), corresponding to an hourly

median signal-to-noise ratio of 40 db per cycle of noise

bandwidth in the output teletype channel. The corresponding signal-to-noise ratio in the output voice channel is

taken as 70. 5 db per cycle of noise bandwidth. Both

figures apply to a Rayleigh-fading carrier under the

assumption of quadruple diversity.

Transmitter Power: Up to 10 kw.

Antenna Sizes: Up to 60-ft. diameter for parabolic

reflectors.
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Receiver Noise Figure: 2 db over the frequency range

considered based on the use of parametric front-ends.

Transmission Line Losses: Based on the use of low-loss

concentric lines or wave guide of appropriate dimensions

for the frequency considered.

Typical values of line loss and noise figures are given in

Table I, below.

TABLE I

Typical Line Loss and Noise Figure Values as Functions of Frequency

Carrier

Frequency

Mc/ s

Effective Transmitting

System Noise Antenna Line Losses

Figure, F, db L

t' db

3. 5 1. 5

3. 8 1. 8

4. 2.

4. 5 2. 5

4. 0* 2. 0*

400

600

800

1000

2000

^Assumes use of wave guide.

The effective system noise figure is discussed in Appendix

III. For the frequency range considered, external noise contributions

can be roughly approximated by setting f - 1 and if we further

3- assume f = 1 and t = t , the effective system noise figure F

c to

simply exceeds the receiver noise figure F by the receiving

antenna transmission line loss, 10 loe,^ I , if all quantities are s10 t expressed in decibels. For the parametric receiver front-ends

specified here the receiver noise figure F may be assumed to be
-16-

2 db over the entire frequency range and the values of F in Table I were determined on the assumption that the transmission lines at

the transmitting and receiving ends of the path introduce the same

losses.

For the assumed loading conditions and the specified output

signal-to-noise ratios, a functional dependence may be set up between

the average short-term teletype error rate and the predetection hourly

median RMS-signal-to-RMS-noise ratio R, using the methods given

by Florman and Tary [ 1961 ]. The results for the specified system

of 24-voice channels are shown on Fig. 5 for various assumptions of

the deviation ratio. In all cases the width of the base band is

assumed to be 108 kc/ s. As each deviation ratio corresponds to a

definite predetection radio or intermediate frequency channel bandwidth, several curves are shown, each labeled with the total radio

frequency bandwidth B expressed in decibels. The possibility of

trading off signal-to-noise ratio R for bandwidth B is thereby

demonstrated.

The reference value L, of median basic transmission bm

loss and the path antenna gain G have been calculated in this report

in accordance with the methods of Rice, Longley, and Norton [ 1959 ]

t using the applicable path parameters and atmospheric constants.

This reference value, L. , is shown as a function of carrier fre- bm

quency for two given antenna heights on Fig. 6a.

Substitution of all known values including the reference

value of basic transmission loss L, into (1) results in curves of bm

transmitter power versus frequency. These curves are shown on

Fig. 6b for two antenna sizes, and the values of R = 7. 2 db and m

B = 60. 3 db corresponding to a deviation ratio of 2. 6 and the 0. 01%
-17-

TELETYPE ERROR RATE VERSUS PRE-DETECTION CARRIER-TO- NOISE RATIO R FOR

24 VOICE CHANNEL QUADRUPLE DIVERSITY FREQUENCY MODULATION SYSTEMS

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Figure 5
-18-

CALCULATED MEDIAN BASIC TRANSMISSION LOSS VERSUS

FREQUENCY FOR AVERAGE WINTER AFTERNOON HOURS

(TIME BLOCK 2 MEDIANS)

Sample Path Sidi Slimane to Moron

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Figure 6a

2,000
-19-

REQUIRED TRANSMITTER POWER FOR

AVERAGE WINTER AFTERNOON HOURS

(TIME BLOCK 2 MEDIAN) 202.7 Mi. SAMPLE PATH

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CARRIER FREQUENCY IN Mc/s

Figure 6b

2000
-20-

short-term teletype error rate. The resulting transmitter power

values P correspond to the reference basic transmission,loss

tm

value L, , and thus apply to the median of all Time Block 2 hourly bm

medians; i.e. , they are applicable to the average winter afternoon

hour, and are not final results. They serve to demonstrate the frequency dependence of the required transmitter power for the given

path geometry and equipment parameters.

In comparing Figs. 6a and 6b one should note that the frequency dependence of transmitter power is quite different from the

frequency dependence of basic transmission loss alone. This is due

to the effects of path antenna gain, and the frequency -dependent

equipment parameters. The curves of Fig. 6b permit selection of

an optimum frequency, or an optimum frequency range, where P

tm

is a minimum. The optimum frequency is, of course, usually

different for each equipment combination. For the sample path, the

optimum frequency is 1050 Mc/ s for 2 8 -ft dishes and 550 Mc/ s for

60 -ft dishes. The example shows (and it is generally true) that the

optimum frequency decreases with increasing antenna size. If

conventional receiver front-ends are used the optimum frequency

would be lowered, as in this case the receiver noise figure also

increases with frequency.

In the calculations leading to Fig. 6b, no consideration has

been given to the problem of selective fading as a function of carrier

frequency. It may be assumed from available data and theoretical

studies that the bandwidth of the medium is sufficiently large relative

to the highest modulation frequency considered for a path of this type.

From the curves of Fig. 6b, values of transmitter power,

P , may be determined for any desired frequency range and equip-
-21-

ment combination„ For the example, the following are chosen in

order to conform to the design characteristics of available equipment.

TABLE II

Frequency or Range Transmitter Power

of Frequencies M P

tm

Dish Size

28 ft 60 ft 60 ft

1000-1200 Mc/ s 14. 1 dbw

500-600 Mc/ s 4. 4 dbw

1000 Mc/ s 5. 4 dbw

It has already been shown, and should be restated here,

that the transmitter power value P was determined from the tm

median of all Time Block 2 hourly median values; i. e. , the value

required for the average winter afternoon hour. The constant K , and the distribution of all P values over the entire year may be

obtained by (3) and (6a). The angular distance 9 is accurately

known (9 = 40. 2 milliradians) and the V(p, 9) curves of Fig. 1 are

used to determine the all-day, all-year variability of the hourly

medians. The resulting cumulative distributions curves of transmitter power or predetection signal-to-noise ratio are shown on Fig. 7

for the 28-ft antenna size operating in the 1000-1200 Mc/ s frequency

range.

Fig. 7a indicates the expected value of transrritter power

P (p, 9) necessary to provide the specified grade of service (or better)

given by R = 7. 2 db during a given percentage, p, of all hours of

the year. Similarly, Fig. 7b shows the expected distribution of the

predetection hourly RMS-signal-to-RMS-noise ratio R(p) if the
-22-

EXPECTED TIME AVAILABILITY FOR SAMPLE COMMUNICATION CIRCUIT

d = 202.7 mi., 0=40.2 MILLIRADIANS

QUADRUPLE DIVERSITY FM SYSTEMS ON 1,000 -1,200 Mc/s WITH 28ft. DISHES

-15

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Figure 7

99.9 99.99
-23-

transmitter power is fixed at P =30 dbw (1 kw). The ordinate values

o on Fig. 7 are clearly labeled "expected values", because they are

based, in part, on empirical terms in the prediction formula, and

there may be appreciable deviation of predicted from observed values

in many cases. It will be shown in the next section how these prediction errors are accounted for in systems design.

3. THE CONCEPT OF SERVICE PROBABILITY

Although Fig. 7a is a graph of time availability versus transmitter power for a specified grade of service, the design problem is

still not completely solved. The prediction formula for the hourly

median basic transmission loss has empirical terms in it, and its

comparison with data shows appreciable deviations of predicted from

observed values. There are factors influencing the transmission loss

and thus the required transmitter power which are not taken into

account by the prediction formula, because it is impractical to allow

in more detail for the actual characteristics of the terrain and the

atmosphere and because of uncertainty in the estimate of equipment

performance terms, especially the predetection hourly median signalto-noise ratio R , which defines the grade of service. Therefore, m

if a number of tropospheric propagation paths are considered which

have identical geometrical and meteorological parameters entering

the prediction formula, and result in the same predicted all-day,

all-year distribution of hourly median basic transmission loss (or

required transmitter power, similar to Fig. 7a), an extensive longterm measurement program of transmission loss, or performance

factors, for all those similar paths would result in a range of all-day,

all-year distributions as well as in a range for the long-term overall

medians rather than in a single result.
-24-

Fig. 8 illustrates the concept of service probability. The

heavy dashed curve shows the calculated value L, (p) of basic

be

transmission loss expected to be exceeded by (100 - p) per cent of

observed hourly median values Ll . For the same set of values of

prediction parameters such as angular distance and frequency, the

lighter solid curves show "true" infinite-time cumulative distributions L (p) versus p for six randomly different paths. It is

assumed that values L, (p) are normally distributed with a variance

2

b°

0" (p)» which depends upon p, about the calculated value L. (p), as

c be

explained in Appendix I. It is shown, there, for instance, that the

prediction uncertainty for winter afternoon median values may be

characterized by a standard deviation cr = 3. 57 db, independent of

the angular distance, 8.

In order to obtain values of standard deviation which are

applicable to all hours of the year, and for percentage points other

than the median, a substantial number of available field strength

measurements have been analyzed, as described in more detail in

Appendix I. This analysis constitutes a first estimate in the evaluation of a difficult statistical problem and results in standard deviation values or (p) which characterize the prediction uncertainty for

c p% of all hours of the year. Curves of <r (p) versus p are shown

c by the dashed curve on Fig. 9. The estimation of a

- (p) is based

upon predictions using the time availability functions V(p, 0), depending on the angular distance 0; the values of <y (p) so obtained, however, are also considered to be applicable to predictions based upon the

time availability functions V(p, d).

The uncertainty in estimating the equipment performance

parameters F (effective noise figure) and R (predetection hourly
-25-

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-27-

median RMS-signal-to-median-RMS-noise ratio required) is

expressed by a standard deviation cr which has somewhat arbitrarilybeen taken to be equal to 2 db in the absence of sufficient test data.

The total prediction uncertainty for p% of all hours may thus be

characterized by a standard deviation cr (p), which is given by

°" rc

(P) = <Mp) +% ( ? )

under the reasonable assumption that the variables concerned are

uncorrelated. See Fig. 9.

Prediction uncertainty is defined in terms of the probability

of obtaining a specified grade of service or better for a given percentage of time. This is a measure of the chance of success for a particular path, and is termed "service probability". Values of

service probability are obtained by combining the all-day, all-year

distribution of expected transmitter power or carrier-to-noise ratio

values (Fig. 7) with the sta»ndard deviation cr (p) as shown below.

re

The method is based on the assumption that the errors are normally

distributed. Service probability F(t) lies between zero and one;

an "expected value" Ef L (p) 1 = L (p) corresponds to F(t) = 0.5,

o c and a normally satisfactory allowance for errors will be made by

setting F(t) = 0. 95.

For a fixed R , the system power P[F(t),pl required to

rn provide the specified grade of service or better during p% of all

hours with the probability F(t) may now be related to the expected

value of system power P[ 0. 5, p ] = P (p) by
.28-

P[F(t),p] = P[0.5.p] +t • cr rc

(p) (8)

where

-I' "5ZT J - F(t)= —^—

\ -: p(-z

2

/'2) ds (9j

n/~2tt

F(t), as stated above, is the service probability, t is the standard

normal deviate, while <r (p) is defined by (7). The time distribution

of the expected transmitter power P[0. 5, p ] = P (p) is determined

by the V(p, 0) curves of Figs. 1, 2, and 3. As service probability calculations usually are made for a

specific path, the parameter 9 is not necessary any more for

identification, and has been dropped in (8). However, a new parameter, namely F(t), has been added in order to identify clearly the

value of service probability considered. A value of the service

probability F(t) = 0. 5 corresponds to the statistically expected

value, which has formed the basis for calculations up to now. For

F(t) = Q. 5, the standard normal deviate t is zero, and (8) reduces

to an identity, as it should.

Depending on whether Rmoor P is considered fixed, (5)

and (6) may now be generalized to include the prediction uncertainty

term t • cr (p) as follows:

re

P[F(t),p] = R +K - V(p, 0) + t • <r (p) (10a) J m o re

R[F(t),p] = P

q

- K

q + V(p, 0) - t • o- rc

(p) (10b)
-29-

These equations are now complete expressions either for

the required transmitter power with a given R or for the predetection

hourly median RMS-signal-to-RMS-noise ratio with a given P , taking

into account the prediction uncertainty associated with a given percentage of hours, or time availability, p.

4. SERVICE PROBABILITY CALCULATIONS

For purpose of illustration, the sample path data will be

used again. Let it be required to calculate the power P(0.95, 99%)

for the same equipment parameters as were used for drawing Fig. 7a

(28-ft dishes on 1000-1200 Mc/s). This amount of power would

provide the specified grade of service (with R =7.2 db) during

99% of all hours with a 0. 95 probability of success. This number,

0.95, means that out of, say, 100 paths all having the same parameters, 95 would be expected to provide the specified grade of

service or better during 99% of all hours, and 5 would probably fail

to do so. In other words, the designer takes a risk corresponding

to the probability that 5 out of 100 (or 1 out of 20) similar paths

might fail to perform as required.

The power P(0. 95, 99%) is determined from (8) by reading

the value P(0. 5, 99%) = 20. 5 db from the curve of Fig. 7a, and adding to it the term t • o- (99%) which, in this case, is the product

1. 645 X 5. 35= 8.8 db. The value <r (99%) is read from Fig. 9 re while t = 1.645 for F(t) = 0.95 is obtained from probability tables

[Bennett and Franklin, 19 54]. The result, in this case, is 29. 3 dbw

for the transmitter power P(0. 9 5, 99%).

Table III, below, shows additional values of t for various

service probabilities F(t), as taken from probability tables in

Bennett and Franklin [ 1954].
-30-

TABLE III

F(t)

0, 50

0. 70

0. 90

0. 95

0. 98

0. 99

0. 995

0. 998

0. 999

0. 524

1. 282

1. 645

2. 054

2. 326

2, 576

2, 878

3. 090

For service probabilities less than 50%, t is negative and

F(-t) = 1 - F(t) (11)

By choosing several values for F(t), the time availability

graphs of Fig. 7 may be augmented. The curves drawn originally

were for F(t) = 0. 50, or t = 0. New curves for any desired value

of F(t) may be obtained by substitution into (10a) or (10b). Fig. 10

shows the results for the sample path with the values for F(t) = 0.90,

0. 95, and 0. 99, added to the original curves of Fig. 7. It is seen

from the upper graph (a) that more power is needed if values of

service probability F(t)> 0. 5 are taken into account. Similarly,

from the lower graph (b), the hourly median predetection RMS-signalto-RMS-noise ratio R expected to be exceeded with the probability

F(t) for a fixed transmitter power P is substantially reduced when

values of F(t) > 0. 5 are considered.
-31-

TIME AVAILABILITY FOR SAMPLE COMMUNICATION CIRCUIT

AT VARIOUS VALUES OF SERVICE PROBABILITY

d= 202.7 mi., = 40.2 MILLIRADIANS

QUADRUPLE DIVERSITY FM SYSTEMS ON 1,000- 1,200 Mc/s WITH 28 ft. DISHES

5

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12 5 10 20 30 40 50 60 70 80 90 95 PERCENT OF ALL HOURS

FIGURE 10

99.9 99.!
-32-

It is also possible to determine the service probability F(t)

as a function of p for fixed values of both Rmoand P by setting

P[F(t),p] = P in (8) and solving for t:

P - P(0. 5,p)

t = ~ 7Z\ (12) %C

(P)

Note that t is simply a function of p and o- (p) and that F(t) can

re

be determined from t using the appropriate probability tables

[Bennett and Franklin, 1954]. As an example, Table IV gives the

calculations of F(t) versus p for P = 30 dbw with 28-ft dishes at

1000-1200 Mc/s, using the sample path.

TABLE IV

Service Probability Calculations

(1) (2) (3) (4) (5) (6)

Time Availability, p,

in percent P(0.5,p)

P

o-P(0.5,p)

re

of all hours in db in db

4. 3

in db

7. 28

t F(t)

99.99 25.7 0. 590 0.722

99.9 23. 3 6.7 6. 35 1.056 0.854

99 20. 5 9.5 5. 35 1.776 0.962

95 18. 1 11.9 4. 49 2.65 0.9960

90 16. 1 13.4 4. 09 3.28 0.99948

The values in Column (2) of Table IV were determined from

Fig. 7a, while the values in Column (4) were determined from Fig.

9. Column (5) is obtained by dividing (4) into (3), and (6) is obtained
-33-

from (5) by the use of the table of standard normal deviates. The

resulting values of F(t) may now be plotted as abscissa values to

the ordinates p on a double probability display, and the points

connected by a smooth curve, as shown on Fig. 11. This is an

additional way of presenting the service probability-time availability

relation, using a fixed value of transmitter power and a fixed value

of hourly median carrier -to-RMS-noise ratio in addition to the other

fixed parameters of the path and the system. This form of presentation shows how time availability and service probability may be

traded off.

If the term P - P(0.5,p) is negative, t will be negative

and the corresponding values of F(t) will be less than 0. 5. Using

Bennett and Franklin [ 1954], the values in the column headed

1 - F(t) are applicable in this case, as shown by (11).

The smooth curves drawn on Fig. 11 are the final results of

the study and represent the relationship between service probability

and the time availability for the specified grade of service. Besides

the system used for the example, calculations were made for other

system combinations and the resulting service probability curves

are also shown on Fig. 11. The intersection of the curve marked

60 ft dishes - 1000 Mc/s - 1 kw with the 99.9% ordinate, for example,

determines an abscissa service probability value of 0. 990. This is

the probability that the specified grade of service or better will be

obtained during 99. 9% of all hours by the use of the 1000 Mc/ s-60 ft dishes 1 kw transmitter power combination. It may also be seen that

by lowering the carrier frequency to the 600 Mc/s range and for the

same 1 kw value of transmitter power the probability of obtaining

the specified grade of service or better during 99. 9% of all hours is

increased to 0. 9955.
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34-

SERVICE PROBABILITY FOR SAMPLE PATH

QUADRUPLE DIVERSITY FM SYSTEMS WITH

24 VOICE CHANNELS, Rm

= 7.2 db

99.99

99.95

99.9

99.8

99.5

99

98

95

90

80

70

Y

\

28 DISHES

1000-1200 Mc/S

-.10 kw XMTRS

^v

V

^

-28 DISHES

1000-1200 Mc/S

10 kw XMTRS

60 DISHES

kw XMTRS,

500-600 Mc/S

1000 Mc/S

1V

X

0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.98 0.99 0.995 0.998 0.999

SERVICE PROBABILITY, F (t)

Figure
-35-

Each of the curves on Fig. 1 1 may be labeled with a dollar

tag denoting cost of installation, operation, and maintenance; thus an

estimate may be made of the additional cost necessary to obtain a greater probability of success; i. e. , an increase from a 0. 854 to a 0.9955 probability shown for 28-ft dishes at 99.9% time availability

is obtained by increasing the transmitter power from 1 to 10 kw.

This means that the additional installation and operational expenses

"buy" a better chance of obtaining the desired results. Fig. 11 shows

also that for this particular path an increase in antenna size to 60 ft

provides essentially the same probability value as a transmitter power

increase from 1 to 10 kw with 28-ft dishes; thus the costs for the two

possibilities may be compared on the basis of the service probability

values read from the curves.

The entire risk versus cost picture is provided by diagrams

such as Fig, 11 and this makes possible an intelligent determination

of the best system to employ. For example, in some cases it may

be desirable to install a lower powered system with the knowledge

that inadequate performance which might occur by chance with this

lower power can later be improved by the addition of a power amplifier.

5. PERFORMANCE OF A CHAIN OF SYSTEMS

OPERATING IN TANDEM

In this section consideration will be given to the method of

calculation of the performance expected when two or more systems

are operated in tandem. Under these circumstances teletype or other errors tend to add as the signal is transmitted via the successive links in the chain, and, as a consequence, the performance of
-36-

the chain will always be somewhat inferior to the performance over

the poorest of its individual links. It would be possible to determine

the error rate over the chain at a particular time, e.g., for a period of one hour, by an appropriate combination of the error rates of

the individual links for that hour. Then the final performance would

have to be determined by an appropriate statistical combination of

each of these hourly values, making appropriate allowance for the

fact that the fading would occur more or less independently on the

separate links. Fortunately the performance can be evaluated adequately and much more directly in terms of the concept of the time

availabilities on the several links, and this will be the approach

adopted here.

It will be convenient to express the time availability, p-,

on the ith link in terms of a probability p! = (100 - p.)/ 100 of not

obtaining a given grade of service R , or better, over that link for m

a specified long period of time; this is usually taken to be all hours of

the year. The value of p. may be determined for the i link from

the equation:

V(p.,li9.) = K . +R - P (13)

oi m ti

where K . denotes the value of K , as given by (3), for the i l link

and P . denotes the transmitter power for the i link. The value

ti of p. is determined, for the given value of V(p.,9.), by interpolation

from the appropriate graphs on Figs. 1, 2, or 3. In carrying out

this interpolation it will be desirable to plot V(p., 9.) vs. p! for the

angular distance 9. corresponding to the i link; for example, see

Fig. 12.
-37-

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V \

\ \

\ \

\

\

I

I

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-38-

First the case of two links operating in tandem will be considered. Let p' and p' denote the probabilities of not obtaining

X <—i the grade of service, R , or better, over each of these links con- m

sidered separately. The general theory of probabilities of combined

events [Feller, 1950, page 13] may be used for obtaining the

probability p

1 of not obtaining the grade of service R , or better,

over the chain of two links in tandem. Let A and A denote the

JL L* events of not obtaining the grade of service R , or better, over the m

first and second links, respectively. Now the probability P(A or A )

of not obtaining the grade of service R , or better, over the chain m

of two links is given by:

P(A

X or A

2

) = P(A

X

) + P(A

2

) - P^ and A

£

) (14)

P(A or A.,), P(A.) and P(A_) may be readily identified with p

1 , L C L C C

p' , and p' , respectively, ' and it remains to discuss P(A and A ).

It will be convenient to think of P(A and A ) now as the fraction of the

time that the grade of service is worse than R on both links at the m

same time. Without loss of generality, we will assume that

P(A ) < P(A ); now it follows that the maximum value of P(A and A )

Ct x X *— is equal to P(A?

), and the minimum value of P(A and A_) is the

larger of the two values [ P(A ) + P(A ) - 1 ] or zero. If the fading

is well correlated on the two paths, service failures A and A will

tend to occur at the same times, and P(A and A ) will then approach

the value P(A ). On the other hand, if enhanced signals occur on the

*The identification of p^ with P^ or A^) i s only approximate, but,

as will be discussed below, introduces negligible error in practice.
-39-

first link whenever fades occur on the second link, P(A and A ?

) approaches [P(A ) + P(A~) - 1] or zero, whichever is the larger.

Thus, regardless of the nature of the fading, the following inequalityis obtained:

p' < p

1 < The smaller of p' + p' or 1 (where p' < p' ) (15) ^1 ^c 1 2 r

2 1

In terms of time availabilities, the above may be written:

The larger of or (p + p - 100) < p < p (where p < p ) (16)

1 L, C 1 J. <-

The above results may easily be generalized to n links:

n

The larger of or > p. - 100(n - 1) < p < p (17)

i = l

Here p denotes the smallest of the n time availabilities. Note

s that the inequality (17) contains a quantitative formulation of the conclusion, stated without proof at the beginning of this section, that the

performance p of the chain will be at best somewhat worse than the

c performance of its poorest link. As an example of the use of the above

general formula, a chain of four links will be assumed with time

availabilities, as determined by (13), equal to p = 99%, p ? = 98%,

p_ = 97% and p = 98%. It follows from (17) that the expected time

availability, p , of service of grade R , or better, at the end of

the chain of these four links will satisfy the following inequality:

92% <p < 97% (18)
-40-

The above inequality is valid for any possible correlations between

the occurrences of the fading on the four links; if the fading tended

to occur simultaneously on the four links, i. e. , with large positive

correlations between the transmission losses on these links, then

p would be nearer to 97% whereas, if there were no overlap in

the long-term fading on the four links, i.e. , with large negative correlation between the transmission losses on these links, then p

c would be nearer to 92%. In most cases in practice the correlation in

the fading on the several links would be expected to be somewhat

positive. However, in the case of a line-of-sight link and an adjacent

beyond-the -horizon link, the correlation between the variations in

the transmission losses on the two links might well be found to be

negative.

It seems most likely that the correlations in the transmission

losses on the several links would not be very large, either positive

or negative, and thus we are led to examine the case of zero correlation, i.e. , the assumption that the long-term variations occur

independently on the several links* Consider first the case of a chain

of only two links. In this case of independent long-term variations in

the hourly medians:

P(A

X and A2

) = P(Aj) • P(A

2

) (19)

Using the above, the following expression is obtained from (14):

P

c

= Pl + P2

" P

1P2 (20)
-41-

Note that the assumption of independence in the long-term variations

yields an explicit formula for p

1 rather than an inequality. Since the

long-term variations would be expected to occur more or less

independently on the several links of a chain, (20) will be adopted as

the best readily available estimate of the probability, p' of not

c obtaining a given grade of service R , or better, over a chain of m

two links; the corresponding best estimate of the time availability

is given by p = 100 (1 - p' ).

In order to generalize (20) to a chain of n links, the following product will be used which is applicable to the case of independent

fading:

P(A, and A^ and . . . A. . . . and A ) = P( A, )

• P(

A

). . . P( A. ). . . P( A ) Y L i ni^in

(21)

Using (21) in a general formula for combining events given by

Feller [ 1950, page 61 ], the following expression is obtained for a

chain of n links:

p' = S. - S 9 + S_ - S. + - ... ± S (22) *c 1 2 3 4 n

S

1 = Ep!, S

2

= 2p!p!, S

3

= 2p!p!p^ ... (23)

The S. given above refer only to the case of independent events.

Here i < j < k . . . < n, so that in the sums each combination appears

once and only once; hence S has {n(n - 1). . . (n - r + 1) } r

/ {1.2 ,..(r-l) • r} terms. The last sum S reduces to the

n
-42-

single term p' • p' . . . p' which is the probability of the simultaneous

1 Z n

realization of all n independent events. As an example, the above

formula will be applied to the same chain of four links discussed

above. For this chain p' = 0.01, p' = 0.02, p

1 = 0.03, and

JL Ct j

p' = 0. 02; S = 0. 08, S = 0. 0023, S = 0. 000028, and

S = 0. 00000012, and thus:

p' = 0. 08 - 0. 0023 + 0. 000028 - 0. 00000012 = 0. 07772788 (24)

c

From this it follows that the best estimate of the time availability at

the end of this particular chain of four links is expected to be

p = 100(1 - p' ) = 92. 227212%. As discussed above, it is more likely

c c that the correlation of the fading on the several links will be somewhat positive rather than negative, and this would lead to a slightly

larger value of p ; thus the use of (24) normally leads to a slightly

conservative estimate of the time availability for the chain.

If it is found that the time availability for the chain is too

small, then it will be necessary to improve one or more or all of the

individual links in order to increase p to a satisfactory level. In

c general it will be best to improve the poorest of the n links first,

since this will usually yield the largest increase in p for a given

expenditure.

It was indicated that A and A denote the events of not

obtaining the grade of service R , or better, over the first and

second links, respectively, and then P(A or A ) was identified with

X £

the probability p

1 of not obtaining the grade of service R , or

better, over the chain of two links in tandem, i. e. , it was assumed

in effect that the grade of service R for the chain is exactly the

c
-43-

same as the grade of service R achieved on each link independently. m

This is an extremely good approximation, but is not quite exact for

the following reason. Suppose that the required grade of service R

m

corresponds to an 0.01% teletype error rate. As the received

signals fade up and down on the two links, the hourly median signalto-noise ratios R, and R on these two links will also vary. Now

1 2

y the error rate for the chain for any given hour is approximately the

sum of the error rates corresponding to R and R ; since the

short-term fading will be independent, the chain error rate would be

given by 100* = 100(6, + e_ -e, • €_) and, for small error probc 1 2 1 2

abilities e and e~, the term e e is negligible. It is seen that

100 e can deviate from the larger of the two link error rates at most

c

B

by a factor of two, i. e„ , when R = R . Thus for a particular hour

such that R, = R_ = R , the error rate for the chain would be

1 2 m

0. 019999% rather than the design value of 0. 01%. Reference to

Fig. 5 shows that a change of error rate from 0. 01% to 0. 02%

corresponds to a change in R of the order of one decibel so that m

R should be about one decibel greater than R if R, and R_, were

c B

o 1 2

not varying. However, since R will nearly always be substantially

different from R , because of the fading, the net error in identifying

P(A or A ) with p

1 will be a very small fraction of one decibel,

depending on the magnitudes of the long-term fading ranges on the

two links, and may be neglected for all practical purposes. To the

degree that one may neglect the above -described small correction,

it follows that this analysis concerns only the variability of the hourly

transmission loss medians rather than the variability of signal-tonoise ratios. Consequently, to this same degree, it is immaterial in

estimating the time availability for the chain whether the noise is
-44-

removed at a relay station (as in the case of teletype transmissions),

or the entire modulation of signal plus noise is re -transmitted.

The time availability p = 100 (1 - p' ) determined for the

c c chain of n links in tandem from the value of p' given by (22) is c the expected value corresponding to F(t) - 0. 5, and it is still necessary to consider the influence of the errors of estimation as measured

by the variances cr (p.) for the n separate links, and thus to

determine p for various values of the service probability F(t).

c A precise formulation of this problem will be given showing how a

solution may be obtained in practice. However, the solution obtained

is somewhat tedious to use, and the reader may well decide to forego

the luxury of such a solution and to settle instead for the expected

time availability p for the chain of n links in tandem, together

with the graphs of time availability versus service probability for the

n individual links.

It will now be shown how the cumulative distribution of the

time availability, p 9 versus service probability is determined for

c a chain of n links in tandem. In what follows it will be assumed

that the errors of estimation for the individual paths, as measured

2

by the variances cr (p.), are independent from one link to the next. re l We first determine p! versus F(t) for each of the n links separately and then determine for each link m different values of p'

IK

corresponding to F(t ) = (2k - l)/2m, k - 1 to m. These values

p! may be read off the graphs of p! versus F(t) or may be deter1K 1 mined by the inverse solution of the equation:

*k° {v**ix ei\- y<*i'W'Tcl *u>' (25)
-45-

where V(p.,9.) is the expected value of V(p, 9) for path i„ For

this expected value, F(t ) = 0. 5 and t = 0.

Next the p' , k = 1 to m, are reordered at random for

lk

each value of i from i = 2 to n. This reordering can be accomplished by using (n - 1) sets of random numbers [Rand Corporation,

1955], For each value of k, p' may be calculated by means of

CiC

(22),, The corresponding values of p are finally rearranged in

order of decreasing5 magnitude s so that p ^ P > . . . p .. . . > p

*ci c2 cj *cm

and plotted versus

F.(t) = (2j - l)/2m. (26)

This is the desired approximate relation between p and F(t), and

c should be reasonably reliable for j less than m-4 or m-5. For

instance, for m = 100, the time availability p corresponding to a

tandem service probability F (t) = 0.945 is the smallest value of

p for which a good estimate of service probability is provided by

this method.

In order to illustrate the calculation of tandem service

probability for two links, an example is chosen for which values of

0, K , P , and R are listed in Table V.

o o m

TABLE V

Circuit Parameters Link 1 Link 2

Angular distance, 9

Auxiliary constant, K

Transmitter power, PQ

Signal-to-noise ratio, Rm

V(p., 0.) for F(t) = 0. 5

l l

17. 5 mr 53. 8 mr

-5. 7 db 16. 1 db

30 dbw 47 dbw

8.0 db 13. 5 db

-27. 7 db -17. 4 db
-46-

Fig. 12 shows V(p., 0.) for the two links, i= 1 and i= 2,

plotted versus p! = (100 - p.)/ 100. The calculation scheme is set

up in Table VI as follows: Column 2 shows 100 values of F(t, ) evenly spaced between and 1, and Column 3 shows the corresponding values of the standard normal deviate t , determined from

probability tables such as those given by Bennett and Franklin [ 1954].

The curves on Fig. 13 show p = 100 (1 - p

1 ) plotted versus F(t , ) and p = 100 (1 - p' ) plotted versus F(t , ); these curves are

2rC 2lC 2K

simply the service probability curves for the individual links, similar

to those shown for a different path on Fig. 11. Now, columns 4 and

5 of Table VI list values of p' and p' corresponding to the

liC 2rC values of F(t, ) from Column 2 and corresponding to the curves of

Fig. 13.

Next it is necessary for p

1 and p' to be placed in a ranlrC 2K

dom relative order. The order of p' , as given in Column 4 is left

unchanged. The index k from Column 1 is reordered, using a table

of randomly ordered two-digit numbers [Rand Corporation, 1955],

uniformly distributed between 00 and 99 with no repetitions.

Column 6 shows the corresponding randomly reordered values p

12K

from Column 5. The randomly ordered values of p in Column 7

are obtained as p = 100 ( 1 - p

1 ), where p' is given by (20), CK Ci£ CIC the special case of (22) appropriate for n = 2. Arranging these

same values p in descending order, they were relabeled p . in ck cj

Column 8 of Table VI and plotted versus F(t ) in Fig. 13. The

circled dots in Fig. 13 show the tandem service probability corresponding to the random ordering shown in Column 6 of Table VI, and

the x's in Fig. 13 represent a second estimate obtained by the same
47-

ESTIMATE FOR TANDEM SERVICE PROBABILITY

a:o

X

o

o

a:

LjJ 0.

o

Q.

>-

GD<

I

L±J

99.99

99.95

99.9

99.8

99.5

99

95

90

70 t 60

50.

e

G ~ Xv.

\

\

\

^fc^ \

POINTS FOR TANDEM

SERVICE PROBABILITY CURVE

V

6 Q <

. o FIRST ESTIMATE

x SECOND ESTIM ATE

> i, J

IND PR(IVIDUAl

DBABIL

. SERVICE

ITY CURVES y? N

LINK \^

LINK 2^"

0.0001

0.0005

0.001

0.002

0.005

0.01

0.02

0.05

0.1

0.2

03

0.4

o

Q.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 055 0.98 0.99 0.995 0.998 0.999 SERVICE PROBABILITY F(t)

Figure 13
-48-

method with another set of random numbers used to reorder the p'

values. Note that, for all values of F(t), p must be less than

CK

p (i = 1 to n) and, as a consequence, the one x and one o at

IK

F(t) = 0.995 are clearly in error. The p' ordinate scale

CiC

corresponding to the tandem time availability values p is indicated

CK

on the right hand side of Fig. 13.

As noted before, the tandem service probability estimates

of Fig. 13 are reliable only for values of F(t) less than about

0.945. An increase in m would extend the range of reliability.

The example given corresponded to n = 2; the extension to n links

involves the addition of m randomly reordered values of p'., for

each value of i = 3 to n and then the use of (22) to calculate the

m values of p' . . ck

6. PREDICTION IMPROVEMENT BY MEASUREMENTS

In preceding sections a method was presented enabling the

engineer to evaluate his performance predictions for a tropospheric

communication circuit in terms of the service probability concept.

Service probability has been defined as the probability of obtaining a specified grade of service or better during a given percentage of

hours within a specified time period, usually taken to be all the hours

of the year. Basically, the service probability concept takes into

account the uncertainties contained in the prediction process. These

uncertainties are represented by the standard deviation or (p) of

re

a normal distribution of prediction errors. The standard deviation,

o- (p), includes a term cr (p) for errors in the estimate of trans

re c mission loss, as well as a term cr which allows for uncertainties

r in estimating equipment characteristics such as the noise figure F
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oon

OnON

ON

ON

ON

ON

ONON

ON

ON

ON

ON

ONONON

ON ONON

ONON

ON

ON

ONON ON

ON ONON

ON

ON ONON ONON

ON

ON ON

ON

ON

ON

00

vO 00 m ^

<m" vO

00

I

O O CO

I

CO

CO

I

co

fNj

vO nO nO

i ini in

i ini mi in

i mi 1 <*

1 1

vO co vO f-H vO fM * -X) f-H sO f-H vO fM CO in t> f—

1

»—

*

(M CO Tf vO vO i—

(

f-H fM

tf

I COI coi coi

o ^ o vO in

00 fM sO ON co

t- 00 00 00 O ON m oo co o

-h in © m

o © •—' •—

m in in in m m in m in m in in in in m in in

00 on o i—i fM co * in ^o l> 00 on © —

«

fM co ^t< r- r- oo 00 00 CO oo oo 00 00 CO 00 ON ON on ON ON

CO

I fM

I in^o ooincotN-^H oo ^ vO On -• (M CO i>. ^h

1 1 1 i 1 i ^i 1 "*1 1 CO1 co

1 CO

1 CO1 CO

1 CO1 CO

1 CO1 CO

1 CO

i o CO r- fM in o m fM ON [N- sO [N> f-H fM •* IN- o ^ o 00 co ^

CO CO co * ^ in in nO* vO IN- 00 ON f-H f-H f-H r—

1

fM (M co CO

• in ON

o ^ f-H fM o * CO m fM o o nO o in f-H > ^ f-H on on f-H vO r- r*

fM fM CO CO * m m nO 00 ON ?—

i

in

fM fM

in in in in in m vo [N- oo on

ON On On s On o o

ON o l-H fM co rh in o IN* 00 ON o f-H fM CO * in nO r~ 00 o

l> 00 00 00 00 00 00 00 oo 00 00 ON ON fjN ON ON ON ON ON fjN ON o
-54-

and the signal-to-noise ratio R required for a. specific grade of m

service. If o- (p) can be reduced below the values given on Fig. 9,

the estimate of the power required for high service probabilities such

as F(t) = 0. 95 will be reduced accordingly. Also, service probability curves determined by these methods will be less steep, and the

expected value of the time availability p, for F(t) = 0. 5, would

constitute a much better estimate since it would not differ as much

from the estimates for higher and lower values of service probability.

The question is how much the standard deviation of the prediction

error can be decreased by taking into account the results of transmission loss measurements made for various periods of time. The

uncertainty in the assumption of the required predetection medianRMS-signal-to-median-RMS-noise ratio R and of the effective m

noise figure F, which has been expressed by the standard deviation

cr , will not be affected by the availability of transmission loss or

field strength measurements, and only the o" (p) portion of the

standard deviation cr (p) can be reduced.

re

Analogous to the prediction uncertainty cr (p) which is

assigned to calculated field strength or transmission loss values, an

equivalent prediction uncertainty, again expressed by a standard

deviation, may be assigned to observed field strength or transmission

loss values. This standard deviation has been designated cr (p, n),

as it is a function of the percentage of the hours p for which the

improved prediction is desired, a function of the number of days n

in the period for which measurements are available, and it contains

a component cr to take into account calibration and reading errors

inherent in the measurement methods. If now, in conformance with

statistical practice, weights are defined as the reciprocals of
-55-

variances (or squared standard deviations), the improved prediction

may be obtained by calculating weighted averages of predicted and

measured values, usually for a number of percentage values.

The statistical basis for prediction improvement, and the

detailed derivations of the expressions used are given in Appendices

I and II. The results of these derivations are used here to obtain

the proper weights and other parameters for any specific problem.

The procedure will be described in what follows.

The general problem considered here is the improvement of

predictions made for a specific tropospheric propagation path, using

measurements obtained over the same or a slightly different path.

It will often be the case that measurements are made only during a few weeks or months, while the improved prediction should be

applicable to all hours of the year.

As the calculations for tropospheric communication systems

are usually in terms of basic transmission loss, L , the method

b

of prediction improvement will also be treated in these terms. It should be remembered, however, that the two quantities, field

strength and basic transmission loss, are directly related, and an

hourly median value of field strength E(p) exceeded during at least

p percent of all hours is equivalent to a basic transmission loss

value L (p) not exceeded during at least p percent of all hours.

In the following discussion, all L. values are hourly medians of

basic transmission loss and the p-dependence is implied, although

the symbol p and the subscript b may be dropped for convenience.

If an observed transmission loss value L and a calculated

o value L , both in decibels, are available for a path, the improved

c prediction L is given by:
-56-

w (p) L (p) + w (p) L (p)

t / \ ° ° c c l->n\ L

t(P) = w(p) + w(p) (27)

o c

where w and w are weights assigned to the observed and calculated values for the path, and L constitutes a weighted average.

In order to simplify notation, the transmission loss L, (p, T) exceeded

for 100 - p percent of the hourly medians of a T-hour recording, is

referred to in this section as either L (p) or L . In Appendix I, o o however, L (p) is used only to denote a "true " infinite time value

o which does not include the effect of random measurement errors or a finite recording period.

The weight w (p) is the reciprocal of the squared standard

2

C

deviation <r (p) assigned to a calculated transmission loss value

L . It does not contain the component cr due to errors in estimating

c r equipment characteristics of the ultimate communication circuit, as

in this analysis only field strength or transmission loss values are

compared:

w (P) =—^— (28)

c L

The standard deviation cr (p) is the lower (dashed) curve on

c Fig. 9, and the corresponding weight w (p) may be readily calculated from (28).

The weight w (p) assigned to the observed transmission

loss value L (p) is the reciprocal of the sum of the variances

o
-57-

w (p) = -y-^ = —7 y (29) o L L L

°"oe (p>n)

°"o(p ' n)+0

"e

The variance <r due to equipment and reading errors

caused by the measurement method may in most cases be assumed

2 2

to be equal to 2 db . The variance cr (p, n) is derived in Appendix

I. It is given by

o-^(p.n) = 0.0786 g(n, H) f(0) cr^(p) (30)

The significance of the various terms will be explained below.

The weight w (p) assigned to the improved prediction L (p)

is the denominator of (27), namely the sum of the weights assigned

to the calculated and the measured values, respectively. It is evident that this new weight is greater than either of the individual

weights from which it is obtained; thus its reciprocal, the variance

2

or (p) associated with the improved prediction, is smaller than

either of the original component variances representing the prediction

and measurement uncertainties. The following expressions may be

written:

w

t(p) = W

Q

(p) + w (p) (31)

'tW-w

t(p)-wo

(p) + w

c

(pJ -,2,2 oe c
-58-

It is seen that an improved prediction with less uncertainty than the

original prediction is obtained for any value of p, the percentage of

hours.

2

Equation (29) shows that the variance cr (p, n) is a function

of the percentage of hours, p, the angular distance, 9 °, and the

number of days of measurements, n. The parameter H in (30) is

the number of hours per day as explained in Appendix I. For the

purpose of application it is only necessary to select that curve of the

variance ratio g(n, H), which corresponds to the number of consecutive days, n, and the number of hours per day, H, for which

measurements were made. The year-to-year variation f(0) is

shown on Fig. 14, and curves for g(n, H) are shown on Fig. 15 for all

hours of the year and for various time block combinations, depending

on whether there are 5 hours, or 6 or more hours in a time block.

Recapitulating the results obtained so far, the improved

prediction L (p) of the distribution of basic transmission loss as a function of the percentage of hours is given by (27), and the remaining

prediction uncertainty cr (p) assigned to it is given by (32). This

corresponds exactly to the assignment of the original prediction

uncertainty cr (p) to the calculated value L (p). If it is desired to

° ° 2. obtain system power requirements, the term cr has to be added to

2

' r

cr (p) in order to obtain a measure of the total prediction uncertainty

taking into account the uncertainty in estimating the system parameters

In cases where parameters other than the angular distance are used

to describe the time availability V(p), it is permissible to use an

equivalent angular distance value corresponding to the V(p) distribution. This value may be determined from inspection of Fig. 1. For

paths within the radio horizon (0 < 0), the angular distance 9 may be

assumed equal to zero for the purpose of estimating cr (p, n).
-59-

Q>

LUO

~ZL i i < CO

cr

60 ^ 1 O

cr 1 < ^

LU _l

>- • i I LU O II h-1 Q>

cr<

M—

LU

>-

LUX

h- oO

CDO

LO

V)<<

cr o 2

sj-

— <fr Q> (D *» i_

Ld 13 O en o Z Li_ ro O O

C\J

cr<

_i3< o

00 <X> CM 2 aa isiiteu
60

<

UJ

5QI

LUO

>

UJ

<

UJ

>3

<

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>-

<

5!: LU

tr o

< <

U_ LU

2o

q:

— ' ::i: ' 1,: gg

|K :j:U#ii giiM!U ii-M 1 ! !

' -

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: : pi =

•

:

-: /J rr ' : :!

:

: ::' : iH:- : 1 H= T

^~SHI-^lllll^l

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: M-

' _ : • -^ jI4-_x } _

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-

|^^^ M

jf A< ~Aj «? "

: . : : :? :: : ; ; ~ : "" *:- --:!-=

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;

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CD

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-

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yl'zz —IS a nnn -^ >- ^ CD - //' •' IS It "^ cc <r .^ Cr -f :::ffii$.j±[:::::: ..1J2.S

: I

IB

< < Q ^ §

^

/ // ,7 In <\ UJ ~ OJ fO

/if 7 > ljl LT -Jr _ 7^4-^1— cr ^ _, LU CO LU (y) 1111

1 H?y 4__ ^ :rr :/+ui 1 a y ^-> ^t rv c_) i-K o ==== n

f~fjz -:. y- c

iiiiiiil

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UJ*-

. z> Q dq =Wm

> ° m ° m ==== 4/ ^

U_ ' Q . .UJ . ,UJ

, UJ ^ UJ ^ ----

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7

* tf-f

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1— Is- < O 1— O 1— ====

/ / l

IJ Win'Jw lllllllllllll 1

, 'i !

! 1

1

,

l ,/ / /

o

o

o

Q

o O

o

m q:

LU

CL

o~^

(

>

1 — o Q CvJ ct: o

CJ

LU

o cr

(

)

<

2

o

IO ^>L

<t ui Q -

<D LU ^ > - 51 o hCM 3

O

LU

CO

O ~z. o

o

U_O ID or

lu

m^

3

C\J ~Z.

II C

LO CvJ LO

O

CM

(H'u)6
-61-

F and R as discussed in Section 2. Thus, the prediction uncero tainty is characterized by a standard deviation cr (p) as a function

of p, if one deals with transmission loss or field strength values,

or by a standard deviation cr (p), if predictions are to be made for

the performance of a complete system including the uncertainties

inherent in estimating equipment parameters. „ <p) is given by

°\JP) = 0\(P) *<rZ (33)

tr t r

The preceding discussion dealt with prediction improvement

by taking into account measurements over exactly the same path for

which the prediction is made. There might be cases, however,

where an improved prediction is desired for a path "b", whereas

measurements are available only for a different path "a", which

might be in the same region, or have similar parameters and

characteristics. Calculated values of hourly median basic transmission loss, however, may be obtained for both paths. If these are

denoted L and L , , an estimate of the improved prediction L , ca cb tb

for Path "b n may be made as follows:

L =L +(L-L ) (34)

tb ta cb ca'

where L. is the improved prediction for Path "a", obtained in the

ta

Considerations of antenna beam width and lobing effects lead to the

conclusion that the use of two different frequencies over the same

geometrical path may constitute a difference between paths in the

sense used here, especially if very narrow beam antennas are used.
-62-

manner described above. All terms in (34) are of course functions

of the percentage of hours p, as before, and p has been dropped

for convenience. An estimate of the prediction error associated with

L is not only a function of the prediction errors associated with

tb

the terms on the right-hand side of (34), but also depends on the

correlation between the values L and L . . This matter is disca cb

cussed in detail in Appendix H„ The result obtained is an expression

for the prediction uncertainty cr assigned to the improved pretb

diction L , which is a function of the prediction uncertainties cr , tb ^ ca

or , , and cr , as well as the correlation coefficient p . . cb ta ab

o_2 =Q_2 +(r 2_ o_2_ 2o_ o

. (1 - o- 2

/cr 2

) (35)

tb cb ca ta ca cb ab ta ca

If Path "a" is the same as Path "b", L, =L , the

tb ta

variances cr and cr , are equal, and p , = 1; in this case, (35)

ca cb ab

2 2 reduces to cr , = cr as it should

tb ta

It is shown in Appendix II that the correlation coefficient p ab

between estimates L and L . over the two Paths "a" and "b"

ca cb

has to be at least 0. 5 in order to reduce the standard error of estimating L below the standard error of L . In general, it will be

tb cb

quite difficult to make an accurate estimate of the correlation

coefficient p . The results of the same study show that p , has

3-D 3.D to be very close to unity in order to obtain a substantial improvement

in the estimate of L , over the estimate L , obtained without any

tb cb ' measurements. Available data which may be used to estimate p

3-D are contained in papers by Kirby and Capps [ 1956 ] and Kirby [ 1957 ]

It is shown there that for a fixed transmitter location the correlation
-63-

coefficient between field strength values observed at adjacent measuring locations drops to 0. 5 within a distance of one -half to one mile.

On the other hand, Kirby., et al, [ 1956 ] have shown that there is

sometimes relatively high correlation between different frequencies

received over essentially the same paths using antennas of moderately

wide beam widths. All these results were obtained on much shorter

paths than are normally used for tropospheric scatter communication

systems. However, to the extent that such measurements are applicable, it would appear that path-loss measurements made on paths

with one or the other of their terminals separated by as much as one

mile from the terminals of the path for which a prediction is required

will be of little or no direct value, especially in rough terrain.

2

After havinge obtained the variance cr , assigned to L , tb fe tb

from (35) using an appropriate assumption of the correlation coeffi-

' 2 2

cient p , , the variance or has to be added to <r , for the purpose

ab r tb

of system design, just as in (33);

' trb<P> " 'tb<P> + 'I (36 >

Having obtained L, (p) or L (p) and cr (p) or or (p)

as functions of p, service probability values may be calculated in

the same way as was done in Section 4 for the theoretical prediction.

The complete procedure v/ill be demonstrated in the next section by

means of an example.

7. COMPARISON OF PREDICTED AND MEASURED DATA

AND AN EXAMPLE OF PREDICTION IMPROVEMENT.

Although the uncertainty of any percentage point of the predicted transmission loss curve is given by the cr (p) curve of Fig. 9,
-64-

additional allowance has to be made for the departure of observed

values from their true long-term medians. The departures of both

the predicted and the observed values from the true values are

independent random variables and are assumed to be normally distributed with standard deviations o- (p) and cr (p, n), respectively.

c oe The difference between the predicted transmission loss and the

observed system loss for a given time availability p will then be

normally distributed with a standard deviation cr(p) determined by:

2 2 2

o- (p) = cr (p) + cr (p,n) (37)

c oe

An example will now be given to illustrate how this analysis

may be used to determine whether an observed distribution deviates

significantly from the predicted distribution, For this purpose the

observations on the 200-mile (9 = 40.2 milliradians) path described

in Section 2 will be used. Transmission loss measurements were

performed in February of 1957, using a 1046 Mc/s unmodulated

horizontally polarized carrier. The transmitter power was approximately 15 watts, and ten-foot diameter dipole-fed parabolic antennas

were used with their centers 26 feet above ground. The receiver

bandwidth was approximately 1000 cps, and the carrier envelope was

recorded both on strip charts and on time -totalizing equipment. Fig.

16a shows the distribution of the 378 measured hourly medians,

together with the predicted all hours - winter distribution for this

path.

In order to determine whether the observed and the predicted distributions are significantly different, confidence limits for
-65

COMPARISON OF PREDICTED TRANSMISSION LOSS

AND MEASURED SYSTEM LOSS FOR

1046 Mc/S SAMPLE PATH WITH

EVALUATION OF THEIR DIFFERENCES

140

CO

COQ

150

CO

_JUJ

CDo 160 UJQ

170

(a)

,p REC

N

ICTEC TR4kNSMIS sior

(

J LC

TIME

SS

: bi

DIS _ocTRI

K G

3UTI ROU

ON FO

P F)

R AL _ HO JRS , WINTER

Nr^ 1

S

C^\"v x

> ^

*»»_ ^ ^. *< ^

\^ •^

,

^•Lc(p)

\1EASLJRED J

n = l7 I

5YST

DAYS

EM

, FE

LOS

BRL

s c

AR^

)IST

' 19

RIBU

57

TION '- -L (p)

CO

_lUJm

-p' <-> UJ Q

UJ

c-j

CO QJCO -^ CO o_J

_l

i Q -~ UJ 3>oCC

_J UJ

CO

CD

CO o UJ ^ ^sUJ < rruj Q

U_ UJ

30

Q

UJ

rr

a.

20

(b)

, C=(195 CONFI DEN(:e l IMI T FOR l L c (P)-L()(P)|

/

-l L C (P)- L (p)

1

12 5 10 20 30 40 50 60 70 90 95 98 99 99.9 PERCENT OF ALL HOURS

Figure 16
-66-

the difference between the observed and the predicted values are

given on Fig. 16b. Since positive and negative differences are

equally likely, differences less than f cr(p) are expected with a probability C = 2 F(t) - 1. It is evident on Figs. 16a and 16b that the

observed and predicted values in this case do not differ significantly

for any p at the confidence level C = 0.95. In general, if the differences |L (p) - Li (p) j for all values of p are less than t« cr(p) at a confidence level C = 0.9 5, it is desirable to use both the observed and

calculated values for determining an improved prediction. If JL. (p) - L (p) | deviates by more than t * cr(p) at a confidence level C = 0.9 5

for some value of p, it is desirable to carefully re-examine the

measurements to see whether they are reliable and to re-examine the

nature of the path to see whether the calculated values might reasonably be expected to apply. For example, unusual ducting conditions

which are sometimes encountered on paths over sea water might well

be poorly predicted for the smaller values of p and even for values

of p > 50 % in cases where these ducts are almost always present. It should be noted that V(p) and <r (p) are based on a very large amount

of transmission loss data but transmission paths may occasionally be

found with true distributions which depart more than ±2 o~ (p)from

L

c

(p).

The terms "system loss" and "transmission loss" used in

this analysis and appearing on Fig. 16 have been discussed by Norton

[ 1959 ]. One calculates transmission loss as ten times the logarithm

of the ratio of the power delivered to the transmitting antenna terminals

to the power available at the receiving antenna terminals under the

assumption of no circuit losses in the antennas. The measured

system loss is the same power ratio, but includes antenna circuit

losses. In the case of tropospheric propagation the antenna circuit
-67-

losses can usually be neglected, except when inherently lossy

antennas like rhombics are involved. For the example used here,

however, the approximation is applicable and transmission loss and

system loss are considered to be nearly equal. Basic transmission

loss L and transmission loss L differ only by the effective path

antenna gain G . P

Since the calculated and observed losses in this example

were not found to differ significantly, they may be combined to obtain

an improved prediction. Hourly medians of system loss shown on

Fig. 16a are first converted to basic transmission loss by adding the

calculated value of G , and their distribution is shown by the dotted

P

line marked L' (p) on Fig. 17. In conformance with the analysis

of long-time variability by time blocks already discussed in Section 2,

the measurements represent the winter time -block group (designated

by the subscript "F"), as they were performed during February and

include all hours of the day.

The measured values are first adjusted to all-day, all-year

values by subtracting a term AV (p, 9). This term is the difference

between the V(p, 9) values for all-day, all-year (read from Fig. 1 at

the path angular distance value 9 = 49. 2 milliradians), and the V(p, 9)

values for all hours, winter (read from Fig. 2 at the same 9 = 49. 2

milliradians). It is assumed here that errors in L, (59) and any bo

AV(p, 9) are uncorrelated, and that the variance assigned to a AV

2 2

value is identical to a corresponding variance a (p, n) - <r (59,n)

assigned to the difference of observed values. The adjusted values,

L (p) = L' (p) - AV (p, 9) are shown by the broken line on Fig.

17. This curve represents the observed values adjusted to represent

all-day, all year values. The weight assigned to each value L (p) bo
-68-

IAI31SAS 31dlAIVS dOd

Mqp Nl

(d '90)d ddMOd ddllllAlSNVdl

to O ^

CO CD

z. ^r < o

Q

Ld -Z. ^ O X

OURLY N

LOSS

1—s

Ld

_J

n.

-i- O S

Li_ CO <

o to CO co^ LT Z CO o

o -z- u_

BUT! TRA E y

h- co

CO <

Q OQ

CO

en

3)O

X

<

Ll_ o

LUo

crLU

Q.

Q.

<

<

LU

hS138I03C] Nl SSOH NOSSItAISiWdl 3ISV8
-69-

is obtained from (30) using Figs. 14 and 15. For n = 17 days, the

all-day, all-year g(n, H) = 2.8 and for 6 = 40.2 milliradians,

2 2

f(0) = 2.02(db) . Thus for the median (p = 50%), c (50, 17) is

2 2 °

2 2 determined to be 5.66 (db) . To each cr (p, 17) a term or =2. (db) o e

is added in order to allow for path-to-path equipment (reading and

calibration) errors. The resultant values cr (p, 17) represent the

uncertainty assigned to each observed value L (p), as adjusted to

all-day, all-year values. The weight w (p) assigned to each value

2 °

L, (p) is the reciprocal of each cr (p, 17). bo oe For the same 1046 Mc/s frequency, the Time Block 2 median

basic transmission loss value is calculated using the methods of

Rice, et al [ 1959 ]. It should be remembered that the sa.mple calculations of Sections 2 and 4 refer to the same path. The calculated

Time Block 2 median value, L, = 214. 7 db, together with the bm

V(p, 0) values of Fig. 1, produces the calculated distribution of

hourly medians for all -day, all-year:

L (p,0) = L - V(p, 0) (38) be bm

The calculated values L, (p, 0) are shown by the dashbe

dotted line on Fig. 17. They may be redesignated as L, (p), as the

be

angular distance does not enter any more. The prediction uncertainty cr (p) associated with each L (p) is given by the dashed

c be

curve of Fig. 9. Prior to combining predictions and measurements,

the system error cr (representing principally the uncertainties in

estimating the system parameters F and R) does not enter. The

weights w (p) are the reciprocals of the squared cr (p) values

c c read from Fig, 9.
-70-

Now the weighted average of L, (p) and L (p) may be

be bo

obtained in accordance with (27), resulting in the distribution of

hourly medians L (p), the improved prediction. This is the solid

curve on Fig. 17 and the corresponding weights w (p) are obtained

by adding w (p) and w (p). c o The performance of a communication system utilizing this

path may now be predicted on the basis of three methods: first, using

the calculated values along; second, using only observed values; and

third, using the weighted average of the two, which constitutes the

improved prediction. Prediction uncertainties for all three cases

are shown on Fig. 18. It is necessary here to keep in mind that the

desired communication system utilizes a path slightly different from

the one applicable to the measurements, as larger antennas (28 ft. versus 10 ft„ ) at a higher elevation (50 ft. versus 26 ft. ) are involved.

However, in order to permit the use of this example, this difference

is neglected here so that the assumption p = 1 can be made in the

ao

sense of the discussions of the preceding section,

The first case has already been discussed in Section 4. For

this case the distribution L, (p) of Fig. 17 is \ised with the prebc

diction uncertainty given by the dash-dotted cr (p) curve of Fig. 18,

re

which is the same curve as the solid curve of Fig. 9. The system

error cr is included in cr (p), as explained in Section 4. r re

°"rc (P> =°'c (P> +

°"r (?)

For the second case the observed distribution (adjusted to

all-day, all-yeo.r) L (p) of Fig. 17 is used, together with the prebo

diction uncertainty cr (p, n) shown by the dashed curve on Fig. 18.
71

S~138I03Q 'AlNIVld3DNn NOI10IQ3dd
-72-

As the prediction applies to a complete system, cr (p, n) also

contains the system errors cr : r

2 2 2

o- (p f n) = <r (p,n) + o- (39) or oe r

Finally, the improved prediction curve L (p) of Fig. 17

is used together with the prediction uncertainty cr (p) shown by the

solid curve on Fig. 18. Again, cr (p) contains the system error

cr , and is given by

°"tr (p) =°"t (p) +

°"r

(40)

2 where cr (p) is the reciprocal of the sum of the weights w (p) and

w (p) in accordance with (32).

Fig. 18 shows that the prediction uncertainty is appreciably

reduced by the use of these measurements, based, of course, on the

somewhat questionable assumption of perfect correlation between

"paths" represented by p = 1. As the last step, service probability curves will be calculated in accordance with the method first outlined in Section 4. In

this particular case, the effects of higher transmission loss and lower

prediction uncertainty values tend to balance each other, as will be

seen by the final comparison of service probabilities for 99 or 99.9

percent of all hours of the three cases, using both 1 kw and 10 kw

transmitter power.

Briefly repeating the steps already explained in Sections 3

and 4, the relation between the system power P[ F(t), p] required
-73-

to provide the specified grade of service or better during p% of all

hours with the probability F(t) and the expected value of system

power P(0. 5,p) is given by:

P[F(t),p] = P(0. 5,p) +t ' <r(p) (8)

where P(0. 5,p) here is related to the three curves of basic transmission loss values L. (p), L. (p), and L (p), and for cr(p) be bo bt

either a (p), cr (p, n) or cr (p) are substituted, depending on

re or tr

the three cases considered. For the same system as used in

Section 4 (28-ft. parabolic reflectors, parametric receiver frontends, quadruple diversity FM, 1000-1200 Mc/s carrier frequency),

and the same specified grade of service (24-voice channels, several

of which are loaded with standard frequency-shift teletype signals),

which has been assumed equivalent to R = 7. 2 db, it can be shown m

by substitution into (1) that the required transmitter power P in

decibels above one watt is given by the hourly median basic transmission loss L. minus 199. 4 db. Thus a fixed transmitter power

b

P = 1 kw (30 dbw) corresponds to 229. 4 db basic transmission loss

o on the curves of Fig. 17 for this system. A corresponding scale of

power versus transmission loss may now be set up on Fig. 17 for

this system and values of P(0. 5, p) read for substitution into (8).

Now, (8) is solved for t:

P - P(0. 5,p)

«"(P)
-74-

and from t, the service probability F(t) may be found from appropriate probability tables [Bennett and Franklin, 19 54].

For each of the three cases, F(t) may be now determined

as a function of p, with P(0. 5,p) determined from the right hand

ordinate scale and the appropriate L curve of Fig. 17 and o"(p) b

read from the corresponding curve of Fig. 18. P is fixed at 1 kw

and 10 kw. The resultant curves of service probability are shown on

Fig. 19.

On the basis of the observed values alone, and assuming a transmitter power of 1 kw, Fig. 19 would predict a time availability

greater than 99% for only 36 installations out of 100 over this type

of path. On the basis of the calculated values alone, Fig. 19 would

predict a time availability greater than 99% for 96 installations out

of 100. The improved prediction curve, which is based on both the

observed values L (p) and the calculated values L, (p), indicates

bo be

that the time availability would exceed 99% for 71 installations out

of 100. Increasing the transmitter power to 10 kw will provide a time availability greater than 99% for 999 installations out of 1000,

if either the theoretical curve or the one corresponding to the

improved prediction is used.

For any path of system, measurements add to the information gained from theoretical considerations alone. The relative

positions of the "calculated" and the "improved prediction" curve on

Fig. 19 are due to the discrepancy between calculated and measured

values at the high end of the time availability scale (see Fig. 17).

An example of much better agreement between calculated and measured

values is shown in a recent paper by Barghausen and Peterson [ I960 ].

There the "improved prediction" curves are generally above the
-75-

SERVICE PROBABILITY FOR SAMPLE PATH QUADRUPLE

DIVERSITY FM SYSTEMS, IOOO-I200 MC/S,

24 VOICE CHANNELS, 28 ft. DISHES, Rm

= 7.2 db

°§ 99.99

LU

UJ v_ < LlI Q= -r 99.95 ^ h99.9 y o 99 8

Li_ 7T W O cc

UJ ^ 99.b

p- o (/) x 99 T-i -i 98

fefe

CL h- 95

z

>- UJ ^

90

d LU 00 0- < _

_l ^ 80 <LU

TO

UJ i? ^ LU 1- o

> \

\ \

V

V I

i

i k

. 10 kw

\TRANSMITTERS

\

\

v

> // \

N

S, \

1 N

... >,

.

\-

> >. /

> » f

>

>

—

»

V

X ^ X

^^

'"V » .*. ^\;.

r>M ft /I OD DDrnir ATinki Q ikw

" BAjlli \J\\ imrnuvtu rni.ui^Hi wjiv O TRANSMIT i t h'

5

- BA

- BA SED

UN

ON

UBbb

CALC

.KVEC

:ulat

VALUt

ED VAL

sUES

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.98 0.99 0.995 0.9980.999 SERVICE PROBABILITY, F (t)

Figure 19
-76-

"calculated" curves, resulting in greater values of time availability

for the "improved prediction" over the entire range of service probability values shown.

8. ADDITIONAL CONSIDERATIONS AND CONCLUSIONS

It has been shown first how the performance of a communication system may be predicted, and how the probability of its success

may be evaluated by the introduction of the "service probability"

concept. Thus the accuracy of the prediction depends on the magnitude of the standard error values cr (p) as functions of the time

re

availability p in percent of all hours, and of the equipment error

cr . The standard error values cr (p) were obtained from an extenr re

sive and very general measurement program. As an extension of this

method, and based on data from the same measurement program, it has been shown how this prediction method may be modified to take

into account transmission loss measurements over exactly the same

path or a path similar to the one for which the final system is intended.

The "improved prediction" obtained from forming a weighted

average of calculated and observed values will always result in a reduction of the prediction uncertainty because the resultant weight

is greater than each of the individual weights assigned to either the

calculated or the observed values.

In some cases it may be of interest to determine the expected

number of days of measurement necessary to result in a prediction

uncertainty substantially less than the prediction uncertainty associated with the calculated value. In terms of the preceding discussions,

2

this means setting (30) equal to an appropriate fraction of cr (p) as re given on Fig. 9 and solving for n, the number of days of measure-
-77-

ments, as a function of the angular distance 0, and the time availability p in percent of all hours.

Before deriving a solution, one has to consider that a certain minimum number of days is necessary to define a given

percentage value for a distribution of hourly medians. While one

single hourly median may be considered a 50% value, it is necessary

to have at least 50 hourly medians in order to define a value exceeded

for 99% or 1% of all hours, and 500 hours to define a value exceeded

for 99. 9% or 0. 1% of all hours. This is based on the method customarily used in obtaining cumulative distributions of hourly medians.

It is also important to know how many and which hours per day were

measured in order to use the appropriate g(n, H) curve of Fig. 15.

The model which has been used specifies a strong dependence of the variance of a limited number of observed hourly medians

on angular path distance, but no such dependence appears to exist

for the variance associated with calculated hourly medians.

2

Let it be desired that the variance o- (p) characterizing

tr

the uncertainty of an improved transmission loss prediction L (p)

2 2

be reduced to a fraction a of the variance a- (p) characterizing

re

the uncertainty associated with the transmitter power obtained from

a calculated transmission loss value L, (p). As actual systems are

be

considered here, both variances contain the equipment uncertainty

2

term <r , which allows for errors in estimating equipment parameters, especially the required carrier-to-noise ratio R and the

noise figure F. This may be written as

°-tP (p, = a2<r

rc

(p) (41)
-78-

2 2 2 2 2 2 where cr (p) = cr (p) + cr , and cr (p) = cr (p) + cr , as before,

tr t . r re c r Also, (32) may be written in the following form:

111 1 1 + -5— = -1 5 + -3 (42)

2 2 2 2 2 2

o- t (p) °" oe

^ °" C

^ P ^ °"o* P,n* +0

"e °~ c*

P *

2 2

Solving (42) for cr (p) and substituting (30) for cr (p,n),

2 2

2 2 2 °"t (p) °" r (p)

cr (p)= 0.0786 g(n,H)f(0)o- (p) + <r = -= —y (43) oe c e c Ci °" Jp) - *Ap) c t

is obtained.

2 2

The usual assumptions of cr =4 and cr =2 are now made,

r

2 2 e and (43) is solved for g(n, H), using cr (p) = cr (p) + 4,

tr t 2 2

cr (p) = cr (p) + 4: re c

12. 73 {a cr (p) - 6cr (p) [

1 - a ] - 8[ 1 - a ] }

g(n, H) = = £-_ (44)

f(9)cr (p)cr (p) [1-a ]

For any desired value of a < 1, and any desired percentage

point p, g(n, H) may now be calculated, using the cr (p) and the

cr (p) curves of Fig. 9. Having obtained g(n, H), corresponding

values of n may be determined from Fig. 15, using the curve

appropriate to the time block or time block combination, or number

of hours per day for which measurements are available. The result

is n, the number of days of measurements necessary to reduce the
-79-

standard deviation cr (p) associated with an improved prediction

to a fraction "a" times the standard deviation cr (p) associated

re

with the calculated values; all applicable to p% of all hours.

Depending on the desired reduction a and the percentage

p, (44) may break down, as in some instances a value of g(n, H)

less than zero results. This is to be interpreted in this manner:

2 2

due to the fixed component variances cr and cr it is not possible

to reduce cr 2

(p) by any arbitrary amount.

re

In other words, there

re

is a lower limit for the factor a. A relation between p and a may

be established by setting the curly bracket in (44) equal to zero, and

2 2

solving for cr (p) as a function of a. For a = 0. 5, the limit cr (p)

for which g(n, H) equals zero is found to be 19.25, or cr (p) = 4.4.

This means that the prediction uncertainty cannot be reduced by onehalf unless the original cr (p) is at least 4. 4 db. From Fig. 9, this

corresponds to a percentage value p less than 27%, or greater than

97. 5%. For percentages between these two values, the prediction

uncertainty cannot be reduced by one -half, because of the assumed

fixed component calibration and equipment errors.

Results of the preceding analysis are shown on Figs. 20 and

21. The number of days, n, of measurements is shown which is

necessary to obtain an improved prediction uncertainty cr^p) equal

to 0. 5 cr (p) (on Fig. 20), or equal to 0.707 cr (p) (on Fig. 21).

The resulting curves were calculated for all-day, all-year measurements using the solid g(n, H) curve of Fig. 15. They are functions

of the angular path distance G, and assume a shape similar to the

f(0) curve on Fig. 14. This means that more measurements are

necessary for paths with intermediate lengths (10 < 6 < 20) than for

a long path (0 > 50) or for a short path (9 < 2). On Fig. 20, curves
80

LENGTH OF MEASUREMENT PERIOD NECESSARY

TO OBTAIN AN IMPROVED PREDICTION UNCERTAINTY

EQUAL TO 0.5 crrc (p)

10,000

7,000

5,000

3,000

2,000

to

h-

(T

Z>to<

o

to

>-<

o

tr

LlI tn

700

500

300

200

iI1

1//

/

/ /

// a =0.5, p= 9 —r

t~t—y

9 %

\ II f III/

l 1 1

III 1

III I til

Q = 0.5, p = IO%

I

a =0.5, p = 99.9%

= 0.5, p= | < Vo

20 30 40 50 60 70 ANGULAR DISTANCE, Q, IN Ml LLI RADIANS

Figure 20
-81-

LENGTH OF MEASUREMENT PERIOD NECESSARY

TO OBTAIN AN IMPROVED PREDICTION UNCERTAINTY

EQUAL TO 0.707 crrc (p)

7,000

5,000

3,000

c TOO

COH 500 P.

UJ

Ul 300

tr->co ?on <

UJ5

U.o 100 CO

>-<

TO n

.SO u.o

hi 30

QQ•52

zu

10

T

5

3

2

0.707, p = 9 9.9 % -

a =(3.707, p = !50% AND 9 0%--^

707, p = 9 3 %

""*' 1= 0.707, p = 10%

1 \^_a = C 707, p= \°i'o

1 J ? 3 4 5 6() 7 8 9 3 K ANGULAR DISTANCE, 9, IN MILLIRADIANS

Figure 21
-82-

for a = 0. 5, and p = 50 and 90% are missing, as a result of the

limitations explained in the preceding paragraph. Similar reasoning

also explains the seeming paradox: that it required less extensive

measurements to reduce the prediction uncertainty to a fixed fraction for the extreme percentage point values. At 99.9% or 1% the

calculated prediction uncertainty <r (p) is substantially larger than

its component <y ; therefore, cr (p) can easily be reduced to a

r' tr

given fraction of cr (p). For percentage points closer to the minimum between 50 and 90%, cr (p) is already small, and cr (p)

re tr

cannot easily be reduced.

On Fig. 21, the 99.9% curve for a = 0.707 has been

broken off at the limit n = 20. 8 days, which here corresponds to the

minimum number of hourly medians necessary to obtain a 99.9%

value

.

As an example, Fig. 20 shows that for a path characterized

by an angular distance = 40 milliradians, 190 days of measurements (more than half a year) are necessary to reduce the prediction

uncertainty associated with 99. 9% of all hours to one-half of the

value, cr (99.9%) = 6. 35 db. In the study of interference fields, it re would take 300 days of measurement to reduce the prediction uncertainty associated with 10% of all hours to one -half of the value

cr (10%) = 6.0 db.

re

In some applications, it might be desired to reduce the prediction uncertainty to some fixed value, say K decibels. In this

case (41) may be written as

K = a o- (p) , (45)

re
-83-

and "a" determined first, before substituting in (44). In this case,

Ma" will be a function of the desired percentage value p.

Curves for K - 3 db are shown on Fig. 22. There is no

real paradox in this approach, as it obviously takes more hours of

measurements to reduce a large prediction uncertainty to a fixed

value than to a very small one. This graph, however, illustrates the

substantially greater prediction uncertainty associated with lowpercentage fields which are important for jamming or interference

studies. While it takes 145 days of measurements to reduce the

prediction uncertainty associated with 99% of all hours (service

fields) to 3 db, it would take more than 10, 000 days (27 years) of

measurements to reduce the prediction uncertainty of 1%-of-all

hours interference fields to the same 3 db value.

Figs. 20, 21, and 22 should not be interpreted as indicating

where and for how long measurements should be made. This question still depends on the relative cost of measurements versus, say,

the cost of more transmitter power or larger antennas. If, for

instance, the calculations for a particular path result in the requirement of 10 kw transmitter power and 60-ft parabolic reflectors,

which would provide the specified grade of service or better with a probability of 0.95 during 99% of all hours, and if two weeks of

measurements might be expected to show that 28-ft antennas would

provide a similar service probability, the designer will have to

balance the cost of the measurements against the cost of installing

and maintaining the larger antennas with due regard to the fact that

the larger antennas would, in any case, provide a better service.

Of course, there is always a possibility that measurements will

show that more power, or larger antennas are required, as indicated
-84-

10,000

7,000

5,000

3,000

2,000

e 700

CO 500

P"

LlJ

300

tr3

CO<

200 u

s

U-o 100 CO>

< 70 Q

U. 50 o

tr

ijj 30 m

LENGTH OF MEASUREMENT PERIOD NECESSARY

TO REDUCE PREDICTION UNCERTAINTY

cr

tr (p) TO 3 db

- - I /o

\- P

; 99.9%

p_= 10%

p

= 99%

P= E 0%AND 9(>%

20 30 40 50 60 70 ANGULAR DISTANCE, 9, IN MILLIR ADIANS

Figure 22
-85-

by the example in the previous section. In general, it is quite likely

that measurements over a particular path will only serve to confirm

the estimates made solely on the basis of calculations.

The methods outlined in this section may also be used as a basis for specifying acceptance tests for a completed communications

link. This could be done by specifying a range of values of hourly

basic transmission loss medians calculated on the basis of (43) and

graphs similar to Figs. 20 and 21. Measured values may then be

expected to lie within this range, which is a function of angular path

distance and the number of days of testing required. It is intended

to study this matter further and to publish results at a later date.

The basic concept of service probability and the methods

for making predictions and for determining prediction uncertainty

are expected to be valid in spite of possible changes in the detailed

procedures. It has already been pointed out that the estimates of

the prediction uncertainty for percentage values other than the median

are based on an approximate statistical analysis. The fundamental

problems involved are under study now, and their solution will possibly provide a slightly different curve of cr (p), replacing the one

c shown on Fig. 9. Re -evaluation of available data, and the addition

of new data may also affect the curve. Final results in any particular case may also be affected by the estimate of the calibration

error <r , and the equipment error or . The data and constants

used in this paper, however, constitute the best available estimates

to date, and may be used without serious reservation in the prediction of the performance of long-distance tropospheric communication circuits.
-86-

9. ACKNOWLEDGEMENTS

The authors are indebted to Messrs. R. S. Kirby and

J. W. Herbstreit for their careful review of the original material,

and for their suggestions. Drafting work on the figures was done

by Mr. J. C. Harman and his group. Personnel of the PropagationTerrain Effects Section participated in the measurement program

and data evaluation used for the examples. The manuscript was

typed by Mrs. D. J. Hunt, Mrs. R. H. Rotherham, Mrs. L. A.

Charles, Mrs. J. L. Nadgwick, and Miss S. J. Bush. The studies

leading to this paper were in part sponsored by the U. S. Air Force

Ground Electronics Engineering Installation Agency and by U. S.

Air Force Project 480-L in connection with the establishment of

telecommunication performance standards for tropospheric communication circuits.
-87-

APPENDIX I ANALYSIS OF VARIANCE OF LONG-TERM MEDIAN

TROPOSPHERIC TRANSMISSION LOSS DATA

1. 1 Introduction

It will be recalled from Section 6 that it is desired to assign

a standard deviation <r (p, n) to observed field strength or transoe

mission loss values, analogous to the standard deviation cr (p)

assigned to theoretical values. The standard deviation cr (p, n) is a

function of the number of days of measurements n, and the percentage

p of the hours within a given time block for which an improved prediction is desired. Weights are defined as the reciprocals of squared

standard deviations or variances, and the improved prediction is

obtained as a weighted average of predicted and measured values.

Variations in transmission loss are classified arbitrarily as

short-term and long-term, with the hourly median value of observed

transmission loss the basic unit for studying long-term variations.

Seven hundred thousand values of hourly median transmission loss for

300 propagation paths and for frequencies between 40 and 4000 megacycles per second were grouped into eight "time blocks", defined in

Table I-L in order to allow for observed seasonal and diurnal trends.

This appendix assumes then that data recorded within a single time

block are statistically homogeneous. Transmission loss is less in

summer than in winter on the average, and diurnal trends are usually

most pronounced between May and October in the temperate latitudes

of the northern hemisphere, with maximum transmission loss occurring

in the afternoon.
-88-

Table 1-1

Time Blocks

Months

1 . Nov. - Apr.

2 Nov. - Apr.

3 , Nov. - Apr.

4 May- - Oct.

5 May - Oct.

6 May - Oct.

7. May - Oct.

8 Nov. - Apr.

Hours

0600 - 1300

1300 - 1800

1800 - 2400

0600 - 1300

1300 - 1800

1800 - 2400

0000 - 0600

0000 - 0600

The hours in Table 1-1 refer to local standard time at the receiver.

Methods for examining the statistical behavior of data within

these groups or any other time blocks similarly defined are developed

in this appendix. The following sections explain the theory for a

statistical model, describe the data used to obtain variances and

correlation functions, and explain how the data were analyzed.

Estimates of the path-to-path variance have been obtained by

analyzing deviations of observed time block medians from predicted

values obtained by the methods of Rice, et al [1959]. A total of 223

full time blocks representing 53 different propagation paths shows a

mean square deviation of observed from predicted time block medians

2

equal to 19. 35 db and a mean square value for the year-to-year

variance of time block medians equal to 4.62 db . The difference of

2

these variances, 14. 73 db , is taken as the best available estimate of

2

path-to-path variance, and 2 db of this variance is attributed to

equipment and reading errors which vary from path to path. The

2 2 remaining variance, cr = 12. 73 db is the error assigned to the

prediction of an infinite -time median value for a given path. The
-89-

corresponding standard error, cr (p = 50%) = 3.57 db, represents the

standard error of prediction of any time block median.

In order to evaluate the effect of the length of a recording

period on the reliability of measured long-term median values,

data samples recorded over six months to several years are first

broken down into smaller samples, and the variance of the medians

of these shorter periods about their mean is analyzed in Section 1.4.

Section 1.5 of this appendix shows how cr (p) is estimated for

values of time availability other than 50 per cent, and Section I. 6

describes the effect of the length of a recording period on the variance

of percentiles L. (p) other than the long-term median transmission

loss, L (50). o

I. 2 A Statistical Model

Let L denote a single hourly median value of radio transmission loss as observed over a given propagation path during a given

time block:

L=L + b + 6 + £ , (1-1)

where b is the unknown fixed bias of the calculated value L , while

2

C

6 is a random variable with mean zero and variance cr _ associated

o with the particular path being considered, and 5 is a random variable

2

l with mean zero and variance cr. associated both with the hour of

recording and with the path; 6 does not vary with time, but only from

path to path, while £ varies from hour to hour and from path to path.

The random variables 6 and £ are defined for all paths and

for all possible atmospheric conditions that might be associated with

each path, assuming a given set of values of the parameters used by

Rice et al [1959] for predicting transmission loss.
-90-

Since the mean square deviation between the observed and

predicted time block medians for 223 nearly full time blocks was

found to be equal to 19. 35 db , it follows that the mean square bias

2 / 2 . , 2

b < (19. 35/223) = 0. 087 db which is small compared to either <r or (r. • Consequently we may neglect b and write (1-1) as:

L = L + 6 + £ (1-2)

t c t For a given hour, t, the average of any random variable such

as L over the ensemble of all paths is referred to as the conditional

expected value E (L ). The unconditional expected value of L over

the total ensemble of all paths and all atmospheres is E(L ).

In a very complex fashion, L. is determined by propagation

c theory and data including those whose variance is analyzed here. To

a first approximation, each calculated value depends on an average

of all available data, using propagation theory to normalize data to

a single set of values of the prediction parameters.

The path-to-path deviation, 6, is assumed equal to the sum of a

random error, 6 , which allows for random propagation phenomena,

c and a reading and equipment error, 6 . Each of these random

variables is fixed for a given path and varies from path to path, but

not from hour to hour:

6=6+6 . (1-3)

c e

It is assumed that

E(6 ) = E (6 ) = 0, E(6 ) = E (6 ) = 0, (1-4)

v c re e re

and that prediction errors, 6 , are not correlated with reading and

c equipment errors, 6 . Variances of these quantities are

E(6

Z

) = o- 2 = o- 2 + o- 2

, cr 2 = E(6

2

), o- 2 = E(6

2

) (1-5)

6 c e c c e e
-91-

The error £ in (1-2) allows mainly for random propagation

phenomena, as well as for equipment and reading errors not allowed

for by 6 . It is assumed that

e T

EUt

> = E

r

(i

t

) = E

t (gt

) = lim I £ i

t = (1-6)

T—ex. t=l

With these assumptions, (1-2) implies that

E(L ) = E (L ) = L , (1-7)

t r t c

It also follows that

E (L ) = lim ^ ) L = (L - 5 ) i tt ^ T Cj t cc e T—*oo

t^l

Thus an ergodic hypothesis is made that the average of L over

all time equals the conditional expected value E (L ) over the infinite

ensemble of all possible atmospheres that might be associated with a particular path, r. It is also assumed that the process which produces

random values of 4 from hour to hour within a given time block over

path r is stationary, or, in this case, that the expected statistical

distribution and serial correlation of T successive values of £ does

not depend on t. The expected distribution in time of 4 does depend

on the path, r.

The engineer proposing to install a communication link over a

given path is usually interested in the "true value", (L + 6 ), for

this path. The true value is never known; it is a random variable

Z with a variance about the predicted value equal to <r , as defined

2

C

by (1-5) and estimated to be 12. 73 db .

There is assumed to be no correlation between any pair of the

variables 6 , 6 > and %J
-92-

E(6 5 ) = E(6 U = E(6 £ ) = , (1-9)

c e c t e t

so that the variance of all hourly medians, L. , over the total ensemble

of paths and atmospheres is

E[(L - L ) Z

] = o- Z + a

2 + <r Z (1-10)

t c c e s

where the variance of t, over the total ensemble is

t

*\ = E(|2

t

) . (1-11)

The conditional variance of £ for all possible atmospheres and a

single given path, r, is

cr 2

= E.uf), (1-12)

r t t

2 2

another random variable whose expected value E(cr ) is or,. The

r §

conditional variance of L - L over all possible atmospheres

that might be associated with a given path is

E [(L - L ) 2

J = S

2 + o- 2

. (1-13)

t

t c' J r

The sample hour-to-hour variance s . of 4 for a sequence § , £ ,

. . . , £ , . . . , £ differs somewhat from cr , as may be seen in what

t T r follows:

T

s £

(t)= tVi )a~

?)Z

= uZ " i2) (t^t)» (i " 14)

t=i

where sample mean- square and sample mean values of £ for a given

path are

T T

t=l t=l
-93-

Consecutive hourly medians recorded over a given path are

serially correlated in time. The random process which selects

values of £ from all possible atmospheres introduces a covariance

with respect to time of hourly medians separated by |t - t J hours,

It is assumed that

^tiW^rV^- q=lvt2

|, (1-16)

and that the normalized hour-to-hour autocorrelation function

p (q) is not a random variable, but is the same for every path. It n

follows that

^tlW = *g Ph

( *> (1-17)

From (1-14) and (1-16):

W= 7 i

E

t (tt» - i i

E

t

( «ti^»]

(.

V T - 1 t=l V1 V1

(1-18)

The double summation in (1-18), may be expressed as

T-l

2 /rT,, 2

o- z (T) = o- r h r 7

+ 7 2,

(T - q) ph

(<*> q=l

(1-19)

so that

^ / 2 4 2 u ^](tTi) (1-20)

2 2

E( S|) = a

e

1 - z, (T) h JVt-i; (1-21)
-94-

If hourly medians £ were not correlated, z (T) would equal

2 2 2 2

l/T; in such a case E (s.) = cr and Efs.) = cr.. . If hourly medians

were perfectly correlated for all values of q, z (T) would be unity and

2 \ Z

s . would be zero, regardless of the values of & and cr.. e B

r |

If consecutive hourly medians were not correlated, the variance

among estimates of the long-term medians obtained from several

samples of the same size would be inversely proportional to the

number of hourly medians in each sample. With tropospheric transmission loss data, there is a high degree of correlation of observations

obtained from hour to hour, less from day to day, and almost no

measurable correlation from year to year.

As a general rule, p (q) may be set equal to zero for all

q greater than some fixed value, and then T may be chosen sufficiently large so that z (T) is approximately equal to l/T. The

2

sample variance s.(T) is in such a case a fairly good estimator of

cr 2

, and the average

^2

of cr over a large number of paths is a good

r r estimator of cr..

.

Section I. 4 shows how data are used to analyze the effect of

sample length on the variability of long-term medians, obtaining

estimates of the expected variance of the median of a sample

recording of any length. The form of the hour-to-hour autocorrelation

function p (q) required by the statistical model described in this

section is determined indirectly.

Let L, (t) be the mean of the k consecutive T-hour sample in a total period of T hours:
-95-

Tk

L

k

( T) = i I h, (1-22)

t=l+T(k-l)

where k = 1, 2, 3, . . . , and k = T/t.

The index t used in this appendix numbers hours chronologically

from t = 1 to t = T for the year y = 1, from t = T + 1 to 2T for y = 2,

and so on; t goes from 1 + T(y - 1) to Ty in any given year, where

y = 1, 2, 3, . . . , Y. L. is the "expected value" E[L ] of transmission

loss for an ensemble which includes an infinity of paths having

identical prediction parameters, and L + 6 is the time average for an

infinite ordered set of hours, t, including H hours per day, N days per

year, and an infinite number of years.

Besides the hour index t and year index y, a day index, v, is

defined. Each of the indices t, v, and y may range from one to

infinity, or for finite recording periods they may range from one to

T, N, and Y, respectively. For simplicity it is assumed that the

total number of days in a year3 N, always equals 180 for any of the

time blocks of Table 1-1. T = HN equals 900 for Time Blocks 2 and 5

where the number of hours H per day is 5. For Time Blocks 1, 3, 4,

and 6, H = 6 and T = 1080, and for Time Blocks 7 and 8, H = 7 and

T = 1260. Note that as day follows day, in Time Block 2, for instance,

the gap between 1800 hours one day and 1300 hours the next day is

considered no greater than the gap between consecutive hours of a

single day; and similarly, April 30 is followed immediately not by

May 1, but by November 1. Each of the eight time blocks of Table 1-1

is considered to be a homogeneous statistical sample, for which Rice

et al [1959] estimate the expected cumulative distribution, L, (p), of

hourly median transmission loss values.
-96-

Let L represent the mean value for a single day of the H

v hourly median values recorded during that day, and let L represent

a yearly mean of hourly medians. Symbols X and y are introduced

to denote the error of the daily mean and yearly mean, respectively,

assuming that these values are used to estimate L + 5: ° c L = L + 6 + a , u-2 ) t c t

Hv

L = ~ ) L=L+6+X (1-23)

v H Li t c v t=l+H(v-l)

HNy

L = ~ > L = L + 6 + \ , (1-24)

y HN £j t c y

t=l+HN(y-l)

where from (1-6) it follows that mean values of X. and v are zero over

all paths, all atmospheres, or all hours.

If it were known how many independent observations n

1 are

represented by a sample of n days, the theory for assigning weights

to long-term medians obtained with short samples would be quite

simple; the weights would be directly proportional to n

c

. There is so much more correlation from hour to hour during a single day than

there is from day to day that it was decided to use the number of days

in a sample rather than the number of hours as a first estimate of the

number of observations upon which the weight of a sample should be

based.

If each T-hour period in (1-22) includes n = t/H days, then

nk

U (n) = - ) L (1-25) k n Cj v v=l+n(k-l)
-97-

as defined originally by (1-22), is also the mean of the k n-day

sample in a total period of T hours, where T = Kt and t = nH.

The mathematics used to evaluate (1-18) is used here to write

2 2 expressions r for the conditional variances E (X ) and E (y ) of X and

t v t

v y v y in terms of the function z, defined bv (1-19):

y h

o-.2 = E (X

2

) = a

2 z (H), E(o-.2

) = cr 2

z, (H) , (1-26)

X t v rh \ bh

cr 2 = E (y

2

) = cr 2

z (T), E(cr

2 yty ) = <£ z (T) (1-27) rh y £ h

Defining a day-to-day autocorrelation function p(s) as well as

the hour-to-hour autocorrelation function, p (q) defined by (1-16): h

E(X X ) = cr 2

p(s), s=|v -v | (1-28)

t v

% v z X 12

H

P(S)= -^I 1

E

t [l

t, + H(, - 1) K + H(v - 1)1 H \ t =1

112 2

,p (sH) "_1

,

(H) {—a" + 72 1

(H " »> [ph<

SH + '» + Ph(sH

Z

" *>!}

h

(H) l H

H

q=l

(1-29)

Section I. 4 evaluates p (q) for H = 5 and H = 6 and uses these

estimates and (1-29) to derive p(s) for H = 5 and H = 6. A function,

z(n) which depends on p(s) is defined in the same way as z (T), which

depends on p (q):

n-1

z(n) = - +4" ) (n - s) p(s) (1-30) n ^
-98-

In standard texts dealing with continuous functions and with

integrals instead of summations,

n

,(n) = | ydx(l - x/n) p(x)

2

It is seen that the time variance cr of yearly mean values for a fixed

y

path r, given by (1-27), may also be written

cr 2 = erf z(180) (1-31)

y

x

assuming 180 days in every time-block year.

The expected variance in time of observed n-day mean values

L, (n) relative to the true value L + 6 for path r is

k c

cr 2

(n) = E {[L ( n) - (L + 6)] Z

} = cr.2

z(n) = or 2z (t) (1-32) o t k c a. r h

where t = Hn, and H = 5, 6, or 7, as may be seen from Table 1-1.

Section 1.4 uses (1-32) with other relationships to define p (q), defines

p(s) in terms of p (q) for H = 5 and H = 6, using (1-29), and finally

2 ^2 estimates cr (n) = E[cr (n)], an average over all paths of the conditional

.2° ° variance cr (n) defined by (1-32). The desired weight for the mean or

median of an n-day sample is taken to be l/cr (n), where

oe

a- 2

(n) = E{[L (n) - (L + 6 )]

2

} = ^ + a" 2

z (n) (1-33)

oe k c c e 5 h

2 2

and cr is approximated by fitting curves through estimates of cr as

b r obtained from data for 73 time blocks representing 29 different paths.

Each path corresponds to different values of the prediction parameters,

so it is assumed that the statistical model outlined in this section is the

same for each of the 73 double ensembles. Values for some of the
-99-

parameters depend on the time block involved, so Section I. 4 refers

to o-.(H),

2

p (q,H), z (q,H), p(n,H), z(n,H),

^22 cr (h,H), and a (n,H), in

order to reflect this fact.

I. 3 Description of Data

During the period from 1949 to 1954 the Central Radio

Propagation Laboratory of the National Bureau of Standards had

contracts with the following agencies to measure the hourly medians

of field strength of a large number of commercial television stations

and FM radio stations over the United States:

University of Washington

University of Illinois

University of Texas

Pennsylvania State College

United Broadcasting Company

Collins Radio Corporation

Federal Communications Commission
-100-

In addition, CRPL operated transmitters on Cheyenne Mountain and

Pikes Peak and at Fort Carson near Colorado Springs in Colorado.

Radio transmission loss over paths from these transmitters was

measured at sites near Kendrick, Karval, Haswell, and Sheridan

Lake, Colorado, and near Garden City and Anthony, Kansas, and

Fayetteville, Arkansas.

Data collected from these various sources were put on IBM

punch cards and listed in descending order of field strength within

each of the eight time blocks of Table 1-1. A recent NBS Technical

Note [Williamson, Fuller, et al , I960], tabulates some of these data

within summer and winter (all day) time blocks. The total possible

number of hours of data in a full time block ranges from 1288 for a

seven-hour period to 910 for a five -hour period. No time block was

completely filled with data since there were times when either the

transmitter was off the air for some time or the receiver was out of

service. The largest number of hours actually recorded for a time

block was 1199 while the lowest was eight. The average number of

hourly medians per time block was 365. From these two thousand

6-month, 6-hour groups of data, 73 were selected, representing 29

different propagation paths and a wide range of frequency, distance,

angular distance, and antenna height. Of the 73 sets of data, 51 time

blocks were nearly full, including more than 700 of the total possible

number of T observed hourly medians. The major propagation path

characteristics for these data are listed by Williamson et al [i960].

Their selection was based on the following criteria:

1. In general, recordings were selected representing as

nearly full time blocks as possible.

2. The paths selected cover a wide geographical area of

the United States to get sampling of a variety of terrain and climatic

conditions.
-101-

3. A wide range of frequency is represented in the study.

4. Time blocks were selected which had a minimum number

of "greater than" and "less than" values. Sometimes field strength

goes above the highest value that can be measured with the equipment

in use or drops into the noise level. These values were listed as

"greater than" the highest value which could be measured or "less

than" the lowest value above the noise level. They will be referred

to later as cut-off values.

To obtain the weights derived in this report, each of the

73 selected time blocks was divided into smaller and smaller

consecutive non-overlapping groups from the first to the last day

in each time block.

In determining the median of each of the K samples of size

n days, "cut-off values" in a sample were counted as valid pieces of

data if all the "greater thans" were above and the "less thans" were

below the median value. If "cut-off" values affected the median, as

for instance when one or more "less than" was above the median or

when the median itself was a "cut-off" value, these data were

eliminated from the analysis.

I. 4 Analysis of the Effect of Sample Length

on the Variability of Long-Term Medians

Given K consecutive non-overlapping samples of length n days

recorded over a given path within a given time block, the mean, L (n),

of each of the K samples was approximated by its median; then the

variance of these medians about their mean L was calculated for each

y

group of K medians:

2

(n) =^y [L, (n) - L f (1-34) o K (_j k y

k=l

where L and L, (n) are defined by (1-24) and (1-25).

y k
-102-

Corre sponding to the 73 values of s (n) and to ten pre-selected

values n = l/H, 0. 5, 1, 2, 5, 10, 15, 30, 60, and 90 days, there were

730 values of sample variance as defined by (1-23). For n = l/H,

t = 1; for n = 0. 5, t = 3 in the four time blocks where H = 6, and t is

alternately 2 and 3 for H = 5 and alternately 3 and 4 for H = 7.

2

Studying these values of s , it was apparent that they depend upon the

o angular distance, Q, corresponding to each of the 29 propagation paths

represented. The angular distance is the angle between horizon rays

from transmitter and receiver. Methods for its determination and its

importance to various aspects of radio wave propagation by diffraction

and tropospheric forward scatter are discussed in a number of recent

reports, [Rice et al, 1959; Norton, Rice and Vogler, 1955]. The

symbol s (n, 6) will be used now to denote variances defined by (1-34).

2 . Figure 1-1 shows s (n, 9) for the four fixed values n = l/H,

1, 10, and 30 days plotted versus the angular distance, 9. Also shown

on Figure 1-1 is the function

f (0) = [(0 - 0. 3) exp(-0. 003 9 Z

) + 2] (1-35)

which was selected to fit these data as well as any other simple

arbitrary function of 9 having a single maximum between finite values

f (0) and f (co). The selection of a function of 9 which would fit the

data well enough to be usable was simplified by the following process:

overlapping intervals were chosen along the abscissa scale of the

data plot, and for each interval the median ordinate was plotted

against the median abscissa. The curve represented by (1-35) was

chosen as the simplest one that could be made to go through or very

near to these median values and which would at the same time

extrapolate to reasonable values for 9=0 and 9 — oo. The decision

as to what is a reasonable value for 9 —» co was made by studying all

the data compiled by Williamson et al [i960], rather than only the

data used in this study. The set of points for 47 milliradians corresponds
103.

THE VARIANCE OF n -DAY MEDIANS VS

SAMPLE LENGTH AND ANGULAR DISTANCE,

THE CURVES SHOW f, (0)g, (n) FOR EACH VALUE OF

f, (0)= [(0-0.3) exp (-0003 2

) +2]

200

100

_l

1

50

c 20

_l 10

u_o 5

LlJ DC<

o

2

1 <

UJ 0.5

0.2

0.1 10 15 20 25 30 35 40 45 50

ANGULAR DISTANCE, 9, IN MILLIRADIANS

Figure I- I
-104-

to time blocks with fewer than 700 hours of recordi ng available.

Figure I~l shows f (0) normalized to fit the data, as explained in Step 1, below.

Note that Figure 1-1 shows s (n, 6) plotted on a logarithmic

scale; it is assumed that

in[s

2

(n, 0)J = E {in[ S 2

(n, 6)]} + e , (1-36)

o o

where e is a random variable with a variance independent of 0, but

dependent on n; the scatter of in[s (n, 6)] shown on Figure 1-1 seems

consistent with this assumption. It is proposed to determine f(0) byleast squares curve fits of the data infs (n, 0)1 to functions of the form

o in f(G) + ing(n). The steps in this process ares

1. As a first estimate of g(n), define g,(n) for each of the

ten preselected values of n as the median of the 73 values of

s (n, 6)/f (6) available for a given n.

2. As a second estimate of f(Q), use the 730 values of in(s / g )

to determine, by unweighted least squares, coefficients A and B for

the assumed relation

in f (G) = A + !n[(6 - B) exp (-0. 003

2

) + 2] . (1-37)

The results are A = 0. 0560, B = 1. 167, exp A = 1. 0576, so

f (0) = 1. 0576 [(6 - 1. 167) exp (-0. 003 G

2

) + 2] (1-38)

is obtained as a second estimate of f(G).

3. Estimate the variance of in(s ) for each m o

73 73

) {in[s

2

/f (0.)J - ~ ) ir

i_j l 2 i 73 /_j

i=l j=l

ofin(s

2

) = JL ) {in[s

2

/f (G.)J - -L V in [ S 2

/f (G.) J}

2

(I-3« o 1c. (_j i2i 73 /_/ j2j
-105-

4. Denote by the symbol w(n) a weight defined as the reciprocal

of each variance defined by (1-39):

Table 1-2

u n

u

w(n ) u hK] u n

u

w(n ) u Sl<

n

u>

1 1/H 2.92 5.98 6 10 1.19 1.95

2 0.5 2.91 5.84 7 15 1.06 1.40

3 1 2. 75 5.07 8 30 0.92 0.83

4 2 2.43 3.54 9 60 0.53 0.45

5 5 1.82 2.41 10 90 0.27 0. 14

5. Determine a weighted average value y. of in(s /g ) for

i = 1, 2, 3, . . . , 73 and find by weighted least squares coefficients A,

B, C for the assumec relation

ln[£ (0)] = A + in [(0 - B) exp (-0. 003 6

2

) + C] , (1-40)

where the data fitted by least squares to (1-40) are

10 10

y.r. = y(9.) = { Y w(n ) in[s

2

(n , 6 )/g ( n )]}/ ) w(n ) (1-41)

1 i Lj u iuuuIu Lj u

u=l u=l

(See Table 1-2). As explained at the beginni ng of this section, it 2

may be assumed that £n(s ) = in(fg) + e, where e is a zero-mean random

o variable whose variance l/w(n) is independent of 0. The least squares

determinations of the coefficients in (1-40) which define the third estimate

f (6) of f(0) are A = 0. 1155, B = 1. 9, C = 2. 6.

Next, the year-to-year variance cr defined by (1-27) is

estimated. There are available 65 pairs of "full" time blocks for

successive years, and 25 triplets, each comprising three successive

years of observations in the same time block over the same path. For
-106-

this purpose, wherever more than 700 hours of observations are

available, a time block is considered full. Make the following assumption:

E(cr ) = o- (180) = C f,(9) = f(0) (1-42)

"Y o 3

Define C using the pooled variance of year-to-year variances

determined from the 65 pairs and 25 triplets described above, where

A. is the deviation of a time block median from the average of a

pair or triplet of "full "time blocks:

205

I

[A

2/f

3

(e)]

C = — = 0. 76155 (1-43)

205-65-25

From (1-39), (1-42), and (1-43),

E(oZ

) = f(0) = 0. 7 [(9 - 1. 9) exp (-0. 003 9

Z

) + 2.6] (1-44)

The year-to-year variance f(9) is plotted as Figure 14 of the main

body of this paper.

Now the assumption that s (n, 9) = f(G) g(n) is modified as

o

cr 2(n, 6)/f(0) = g ( n, H) = {f(G) + E [s

2

(n, H, 9)]}/f(9) (1-45) o o

2 2

Here, s (n, H, 9) is the same as s (n, 9), but the H is fixed at 5, 6, or

o o

7, and the requirement is added that only time blocks containing more

than 700 hours of observations are to be used in estimating g(n, H).

Among the 73 time blocks used to define f(9), 21 correspond to H = 5,

29 to H = 6, and only one to H = 7, of those representing more than

700 hourly median values of transmission loss. Table 1-3 lists median

values of [f(0) + s 2

(n,H, 9)]/f(9) for H = 5 and H = 6 hours and for n= l/H,
-107-

0.5, 1, 2, 5, 10, 15, 30, 60, 90 days as estimates g of g. Corresponding

values of t are also listed, as well as the final values of g, whose

calculation will shortly be described.

Table 1-3

Median Values of [f(6). + S 2

(n,H, 0)]/f(0) = g (n, H)

.

(r 2

(n) = f(9) g( n ,H)

o

H = 5: H = 6:

n T g g

1/5 1 6.47 6.96

1/2 2.5 6.42 6.06

1 5 5.50 5.40

2 10 4.62 4.60

5 25 3.06 3.57

10 50 2.79 2.87

15 75 2.54 2.50

30 150 1.96 1.96

60 300 1.51 1.51

90 450 1.19 1. 30

180 900 - 1.00

n T g g

1/6 1 11. 79 12.90

1/2 3 10.52 10.26

1 6 9.22 8.44

2 12 6.25 6.80

5 30 4.55 4.85

10 60 3. 73 3.67

15 90 2.54 3.09

30 180 1.95 2.28

60 360 1.62 1.67

90 540 1.11 1.38

180 1080 - 1.00

Since g(n, H) is the ratio of the variance of an n-day mean to

the variance of a yearly mean, g(180, H) = 1 for any H; this explains

the last line of Table 1-3.

Someuseful relations are summarized and combined as

follows:

T-l

1 2

h

(T) =r + ~

Z

(T " q) Ph

(q) (1-19)

q=l
Define

2

-108-

E[oZ

(n)l = <r\ z, (t), t = nH (I-46a) o % h

E(o2

) = <r Z

, z, (T) = <r 2

(180) = f(9) (I-46b)

V 5 n o

g(n,H) = o- 2

(n)/f(9) . z (t,H)/z (T.H) (I-47a) o h h

g(l/H. H) = zh(l, H)/zh(T, H)= l/zh(T, H), (I-47b)

zh

(T,H) = g(v,H)/g(l/H,H). (1-48)

T-l

x(t) = 1- [z(t) - I] = £ (t - q) ph

(q), h(r) a g(n, H) - 1 . (1-49)

q=l

Then,

T-l

x(t) - x(t - 1) = £ ph

(q) (1-50)

q=r

p (t) = x(t + 1) + x(t - 1) - 2x(t)

2 + ( T + 1) Z

h(T + 1) + (T - l) Z

h(T - 1) - 2T2h(T)

= 2g(]/H.H)

(I - 51)

The hour-to-hour serial correlation p

(t) for H = 5 and

h

for H = 6 was somewhat arbitrarily assumed to be of the form
-109-

and values a =3, k = 0.6 for H = 5 and a = 4.2, k= 0.5 for H = 6

were found to give good agreement between the statistical model and

the data listed in Table 1-3. The method of procedure was to select

arbitrary values of g(l/H, H), a, and k, and to compare calculated

values of g(n) = g(l/H, H)z (t) with the data, plotting both g and t on

logarithmic scales: The parameter k determines the slope of g(n, H)

at large t, the parameter M a

s

' moves the curve parallel to the log t

axis, and the parameter g(l/H, H) moves the curve up or down, parallel

to the log g axis. Values of g(l/H, H) listed in Table 1-3 were

determined last, to satisfy the condition g(180, H) = 1. Figure 1-2 is a picture of the curves and data of Table 1-3. Figure 1-3 shows p

(t)

plotted versus t for H = 5 and H = 6. The functional form given by

(1-52) is proportional to t /t for large t and therefore has a Fourier

cosine transform for < k < 1, satisfying the Khintchine theorem

[Takacs, I960], which specifies a necessary and sufficient condition

2 2 2

for a correlation function. For small t, (1-52) approaches 1 - k a t , which has a slope of zero at t = 0.

Next, (1-29) was used to calculate the serial correlation p(n)

of daily mean values separated by n days for H = 5 and H = 6.

Figure 1-4 shows these two autocorrelation functions.

The major difference between g(n) and p (t), or p(n) for

Time Blocks 2 and 5 and the other time blocks arises not from the

fact that H is less, but from the fact that there is little diurnal trend

of the data in the afternoon time blocks. It is assumed that for Time

Blocks 7 and 8 the functions g(n) and p(n) are the same as those

labeled H = 6, and that if one hour a day were recorded in any of these

time blocks, p(n, H) should be set equal to p (tH, H). The dashed

curves of Figure 1-4 represent these functions.

2

With one more assumption, the final estimates of cr (n, H) for

o Time Blocks 2 and 5 (H = 5), Time Blocks 1, 3, 4, 6, 7, 8 (H = 6),
-110 <

Ld

q:1 LU

h» ^ <

>-

U. \ O cr

hi < o LU

>- < <

cr u_ < o > LU

LU o

"Z. h- <

U_ cr o <> o

hi

55

cr

i-o o iLU

l-<

</)

LU /Jo

//• I :>/ 1 °l

h 11 •

in /

ii / ^ /

v/° Y

°l 1 CD lO

II II X X o •

CD

II X *

JoU

oo om 8

in CM ooooomooC\J ooom8

in

CM

O

crLU

0_

CDa

croo

LU

or

CO

(r

Z>OX

l_i_ O

crLU

GO

CVJ O

£o/(H 4 U) °-0 = (H'U)6
111- <Q

UJ

cr co OX

CO3 h 2 O O

CO

"Z. co o ^

h- C/) <

LU

rv <

q:oo

q:

lu

CO

*{?r " :; !iHti " - n'-hiH H-! !

~n : " i i ; ;/. ::!: ::'.:! :::: /

:!.':'. • ' n i ' : : ;:;;;; ::: . ; 1 ' ; ; :. H:! •1; / ' , ;

:::::; r::: :r:: rT: "I

'.:: ' ; : : i 1 : : -- ••I! ::: L

iiii :: rzxf / 1 ;;: '

TKF ' ::.. ' -

;'i"p::; : : ::::

::.: :: CD

11 |

;.

::::::: : — —

J_

J - X7 / IflJBffl 1 Tml w /

.... ... Ill "X-

.... "1 -j-H—1 —I i

':':'. / / : : • : : : : : ; :

;;jl|i; :;•:; / — : . :::: .:::

fn ; :jR "

-~ : " "i"i : .:' :

lb ....

:: : : : ::: : / r JH-)

:.: .:: | .. . .. |. . • • : 1 -H- 1 : 7 T tf tM —r

ttt.:...

:: . .

j: ::::

.'.:'. ._ !!:! iiii ^ /

yj f — ; ::::

i:: : iiii iiii !

!..j ;.:, :: . ,-::

—h-

;.: . '' ." fm i y

'".'.'-, ; : | :.- : i ; : ; ; j jr :':•: : : : : . fr-ii will! y-ii ;

::]:. Y:k :.•: zj=zz=br|-; - :: : :

p:-i

p ::: i +H j : :::: . -

;' i \V I' ; 1 : :: : ...

— ^- / ^f-~ : ;:;

.::

—

J :.k. : ' 1 IIII \fm\

~1-

/' 'fa-*-fir ; "

ffi: : ?: : J

-----

—t~r, ::.: ;::; .::- : : : tjtjfjj

nji j ' [11 1 . ! fj : lj \

II ill ... LOo

CMa o ooooom

>-

if) < 2 o o < rr o O in

CVI LU Q_ ^ oo O >- (Z O _l _>

or o

Z>

X

O

OX

X

m ll Ll. o O

7* *: o o o

CO I— i < CD

C\J

uW

m CVJ o o o c> oo <

Q_

LU

if)

II if)

(Z

LU

ILU

_JO' CO< oX

mi

CD

i—
-112. <

LU— CO < o Q -J o y

CO

~Z. CO O

LU

(Too

cr

LU

CO

CO<

Iu_ O

<\J O

m CO o o a o o

c>

(u)

d
-113-

and all hours (all day and all year) may be plotted. Assuming that

data obtained in any one time block are statistically independent of

2

data obtained in any other, the variance cr (n, H) for all hours is a

Z Z ° Z weighted average: cr (n, Z4) = [Zcr (n, 5) + 6cr (n, 6)J/8. The heavy

o o o solid curve of Fig. 15 of the main body of this paper shows this

Z. estimate. As indicated, two of the curves in Fig. 15 show cr (n, 5) Z Z ° and cr (n, 6) from Fig. I-Z, and two represent cr (n, 1) for Time

o o Blocks Z, 5, and 1, 3, 4, 6, 7, 8, respectively. This latter estimate

Z Z is obtained from (I-3Z), replacing cr by cr (H) and p(s,H) in (1-30)

* 9

by ph(sH,H).

The curves of Fig. 15 may be used to assign weights to

mean or median values of transmission loss observed for n consecutive

days in any of the time blocks of Table 1-1. For other larger time

blocks or where hours or days are not recorded consecutively, the

statistical model of Section I. 3 may still be used to calculate estimates

2

of cr (n, H), with assumptions similar to those of the preceding paragraph. A weight is defined as the reciprocal of the total variance of

Z the quantity being considered. The variance cr (n, H) includes timevarying equipment and reading errors and some of the prediction

errors arising from not taking sufficient account of time trends and

appropriate propagation mechanisms in the prediction methods. As

(1-33) shows, a path-to-path equipment and reading error variance is Z Z added to cr (n, H) to obtain the variance cr (n, H) of an n-day median

o oe value Li, (n) about the "true" path median L + 6 . k c c

2

1.5 The Prediction Uncertainty cr (p) for

Arbitrary Values of Time Availability, p

Let L (p, n) denote the observed transmission loss exceeded

by (100«»p) per cent of the hourly medians recorded during an n-day

period, and let L (p, co) represent an infinite-time value, with L (p) o c
-114-

the corresponding calculated transmission loss. Replace (1-2) with

L

Q

(p,n) = L

c

(p) + 8

c (p,n) + 6e> (1-53)

lim L (p, n) = L (p, co) = L (p) + 6 (p) + 6 (1-54) o o c c e

Both L. (p, n) and L. (p, co) include a measurement error 6 in addition

o o e

to a path-to-path prediction error 6 (p). Assuming that

L (50) = L , 6 (50) = 6 (1-55)

c c c e

and that the mean and median of hourly medians L are the same for

an infinite-time sample,

L (50, co) = L +6 + 6 . (1-56) o c c e

The prediction uncertainty cr (p) is the path-to-path variance

c of the difference between the calculated percentile L. (p) and a "true"

value, L. (p, co) - 6 : o e

o- 2

(p) = E{[L (p) - L (p, co) + 6 ] 2

} = E[6

Z

(p)] (1-57)

c co e c

2

Data used to estimate cr (p) involve L. (p, n) rather than L (p, co), c o o but for this analysis the effect of finite samples is considered negligible

compared to the large path-to-path variance of L. (p,n). Both in the

main body of the paper and in what follows, L (p) is used as an

o abbreviated form of L. (p,n).

o Predicted and observed values of basic transmission loss,

md L (p) were used to estimate cr (p) as follows:

bo c (A) For time blocks 2 and 4 and for 70 propagation paths, the

mare and variance of L,

b<

p - 1%, 10%, 50%, 90%, and 99%.

mean square and variance of Ll (p) - L (p) were determined for

be bo
-115-

(B) For all hours and a total of 80 paths for two frequencies

and two distance ranges (80 and 120 miles), the mean square and

variance of L. (p) - L. (p) were determined for the same values be bo

of p as in (A). (This was also done for time blocks 2 and 4, expected

to represent fairly extreme conditions, and results were almost the

same.

)

2

Ideally, estimates of cr (p) should be based on a comparison of

observed and calculated values L, (p) and L. (p) at each percentage

bo be

level p for a wide diversity of propagation paths for which all-day, allyear cumulative distributions are available. Data set (A) represent a wide diversity of paths, but not all hours, and (B) represents all hours

over a large number of paths which are located in a single region in Ohio.

Long-term recordings over a large number of paths would

be needed to establish accurately cr (p) for each possible combination

of the prediction parameters. The mean error of prediction will

average to zero over the entire range of propagation path parameters

represented by the available data, but this bias is not expected to be

zero in a given region for given values of distance, angular distance,

antenna height, frequency, and surface ref ractivity. Here, the mean

square deviation of observed from predicted values of transmission

2 2

loss, cr (p) + cr , is the sum of (1) a variance about the mean prediction

c e error, (2) the square of this bias, and (3) the equipment and reading

2

error cr . eThe variance of prediction error about its mean value was

assumed independent of all prediction parameters except p and was

estimated from data set (B), observed and predicted values for

p = 1%, 10%, 50%, 90% and 99% of all hours corresponding to two Ohio

FM stations, each recorded for one year at 80 and 120 miles over a

total of 80 propagation paths. The first row of Table 1-4 lists the
-116-

average values for these variances as a function of the percentage

of time availability p.

The mean square error of prediction for Time Blocks 2 and 4

is listed in Row 2 of Table 1-4, averaging values for 70 paths at each

2 2

percentage level [data set (A)]. Tentative estimates of cr (p) + cr appear in Row 3 of Table 1-4, as the sum of variance and mean square

2

bias from Rows 1 and 2. Rows 4 and 5 show the final variance cr adjusted to 12. 73 db at p = 50%, and the corresponding standard

2

deviations are given in Row 6. The variances cr (p) are plotted versus

the time availability p in Fig. 1-5. The standard deviation, cr (p) is

shown as a function of the time availability p by the broken line on

Fig. 9 in the main body of the paper.

Table 1-4

Steps Used in Estimating Standard Deviations a (p)

2

db for per c ent p of all hours

i% 10% T5^ 90% 99%

(1) Average variance

E{[„A(p) - A]

2

} from Ohio data. 69.44 39.07 19.81 19.03 29.31

(2) Mean square bias from

TB 2 and TB 4 data, normalized

to zero at p = 50%.

(3) Sum of (1) and (2).

(4) (3) minus the average of the

year-to-year variances 4. 75 db 65.77 34.54 15.06 15.03 26.94

2

(5) (4) adjusted to 12. 73 db at

p = 50% 63.44 32.21 12.73 12.70 24.61

(6) Standard deviation cr (p) in db. 7.97 5.68 3.57 3.56 4.96

c

1.08 0.22 0.00 0. 75 2.38

70.52 39.29 19.81 19.78 31.69
117

PATH -TO -PATH VARIANCE

THE VARIANCE OF BASIC TRANSMISSION LOSS

L b0 (p) EXCEEDED FOR p PERCENT OF ALL HOURS, RELATIVE

TO THE CALCULATED VALUE L bc (p)

(M

OQQ

CM

LU

o

90

80

70

60

S< 50

CH

2

o

Hi X

40

30

20

10

um ::::._ -^f |fff{ ^|

h

j |||::::::::::I||::::: T — i ±__. p

nTT1 - m ....

1 m4

M

"Y

III

ji'tty-j'.

:

tHFfrT

£:: :

"if

- TO — t :;;;§::::!!i;iiliklb -H—1—1

- H ^ — 1 lllfl i"i ~ tltt J —w /

I "

1 Hill 1+--- _ .ii in -) - 4-

"1 . .. ... ::;:|j:||j;S:g||:::|:::r|r:^:::E

v. : _ ::_ 4 iT^: == ::: gIlllllliilpii £::: —

:

/

"H ^ :

: " -JJ-LLiL - - , ,< "Tr S

jl if

[

j_j

Mill : +

:::

-i—H--

"i --i—

p

— - \--T|-n—

p

'

TTT ^ , l|T' —

TT \1\ TT TrrfT

"

i M UWrommff^PiJH-:-#:::Spi .. :±::: f rrn

:::: — ippiiff-jffrSf ifffij'if!"**S ;±

: ft: —— m liiipi

:E- -: H 4-fJ "5 . ""H i -H-+ -1— 1 --- :f¥ | - - i

|

: :

- ::i: -L|

T

|, | H-+-- — ^-H-' : rflfrl 1 ^;-±Ei - -i—i

0.01 0.1 10 50 90 99 99.9 99.99

PERCENT OF ALL HOURS, p

Fig. 1-5
-118-

I. 6 Estimation of cr (p, n) o

The variance of L (p, n) relative to the infinite-time percentile

o value L (p, co) given by (1-54) is written as

o-^(p, n) = E{[L

o

(p, n) - L^p, co)]} (1-58)

2 2 2

For p = 50 per cent, cr (p, n) = cr (50, n) = cr (n) which was estimated

o o o

in Section I. 4:

o-^(50, n) =f(9) g( n,H) , (1-59)

and for any particular combinations of values of n, H, and 0, the

appropriate curves of Figs. 14 and 15 are used. See the discussion in

connection with (1-45) and (I-47a).

In order to obtain the variance cr (p, n) for values of p other

o than 50%, the assumption is made that a similar relation exists here

as for the variance cr (p) assigned to the calculated values; i. e. ,

cr (p,n) = cr (50, n) h(p) , (1-60) o o

2 2 where h(p) is given by cr (p)/cr (50), and values may be determined

C °

2 from the cr (p) curve of Fig. 9. Noting that cr (50) = 12. 73, (1-59) may

c c be rewritten using (1-60), resulting in

<r Z

(p,n) = 0.0786 g(n,H) f(9) o- 2

(p) (1-61)

This equation is designated as (30) in the main body of the paper. It has already been mentioned that the procedures used to derive cr (p,n),

oe

especially the relation between cr (50, n) and cr (p, n) as well as

o o

cr (50) and cr (p) constitute the best available estimates to date. The

c c
-119-

entire problem is being restudied, and it is expected that these

relations will be formulated later on a statistically more refined

basis. The final results, however, are not expected to differ materially

from the ones obtained using the methods outlined here, so that the

procedures given in this paper will result in reasonably good estimates.
-120-

APPENDIX II ESTIMATING PREDICTION UNCERTAINTY FOR ONE PATH

USING MEASUREMENTS MADE OVER ANOTHER PATH

This problem has been discussed briefly in Section 6 of the

main body of the paper. Here a detailed derivation of (35) in

Section 6 is given, which is a formula for calculating the standard

deviation cr ,

. This standard deviation is assigned to the transmission

tb 5

loss estimate L , for a Path "b" Both L , and cr , depend on

tb tb tb r calculated values for Paths a and b, measurements for Path a, and an

assumed correlation coefficient p , between the prediction errors

ab

6 and 6 . . All concepts have been explained in Section 6 and

ca cb

Appendix I. Equation (35), which is to be derived, is repeated here:

2 2 2 2 2 , 2 . cr = cr + cr -cr -2cr cr p , (1 - cr /cr ) (II-l)

tb cb ca ta ca cb ab ta ca

For convenience, the notation in this appendix will be simplified

compared to the notation in Appendix I. The relationships to be

developed will apply to any value of time availability. The observed

transmission loss exceeded for p per cent of the hours in a T-hour

recording over Path a, for instance, is L : oa

For Path a, L (p, n) = L (H-2)

o oa

The "true" infinite-time mean value L. (p, co) - 6 for Path a is L : o e a

For Path a, L (p, oo) - 6 = L (II-3) o e a

Calculated values L (p) for Paths a and b are L. and L ,

: c ca cb

L = L (p) = L - 6 (p) , (II-4)

ca ca a ca r
-121-

L = L (p) = L - 6 (p) , (II-5) cb cb b cb

Variances of the random prediction errors 6 (p) and 6 (p) are

ca cb

defined by (1-5 7) and are written as o* and c , respectively. Instead

ca c D

of regarding L. and L. , as fixed and L and L as random, it is ca cb a b

convenient here to regard the "true" values L. and L as fixed, and

a b

the calculated values L, and L , as depending on random determinaca cb r- )=> tions of the prediction parameters for Paths a and b.

Similarly, the variance of the observed percentile L relative

? oa

to the true value L is given by (1-58) or (1-60) plus cr :

o- 2 + o- 2

(p, n) = E[(L - L ) 2

1. (II-6) e oa oa a

Reciprocals of these variances are weights:

2 , 2

w = l/o- , w , = l/cr , (n-7)

ca ca cb cb

2 2

w = l/[<r (p,n) + o- ] (II-8) oa oa

It was assumed that measurements are available only for Path a.

The improved prediction, L , for Path a is analogous to (27)

ta

of Section 6 of the main body of the paper, and is written here as: w L. + w L

oa oa ca ca , Q

. Li - Vll-V)

ta w + w

oa ca

2

where the variance <r assigned to L is the reciprocal of the sum of

ta ° ta

the weights w and w : oa ca

2 1 1 (11-10)

0" = =

ta w w + w

ta oa ca

From standard statistical texts, the correlation coefficient

p between the quantities x and y is given by
-122-

= E(xy) - E(x) E(y) , xy cr • o* x y

2 2 where E denotes expected values, as before, and a and cr are

x y

the variances of x and y, respectively. Also, the variance of a sum

of three terms

u = x + y + z (11-12)

is given by

2 2 2 2 cr=cr+cr+cr+2crcrp + 2cr cr p + 2cr cr p (11-13)

u x y z x y xy x z xz y z yz

This may now be applied to (34) of Section 6 which gives the estimated

relation between calculated and improved prediction values of Paths

nan and "b n and is repeated here:

L = L + (L - L ). (11-14)

tb ta cb ca

Using (11-13),

2 2 2 2

cr = cr +0" + cr - 2cr cr p „ + 2cr cr p - 2cr cr p„

tb ta ca cb ta ca

r

12 ta cb HL3 ca cb r 23

(11-14)

where

P12" P L L ' p 13

5 P L L ' and p 23

= Pab

S P L L ' ta ca ta cb ca cb

The first two correlation coefficients are now expressed in

terms of the last one, as for instance,

E[L L ]-E(L )E(L ) E[(L - L )(L -L)J-E[(L - L )]E[(L - L )]

ta ca ta ca ta a ca a ta a ca a

12 cr cr cr cr ta ca ta ca

(H-15)
-123-

Any constant value (L ), subtracted from each of two variables,

does not affect their correlation.

From (II-9),

w (L - L ) + w (L - L ) T T oa oa a ca ca a /TT ,

.

J_i - L, - (11- lb)

ta a w + w

oa ca

E[(L - L )] = — {w E[(L - L )] + w E[(L - L )]} = (11-17)

ta aw oa oa a ca ca a

ta

E(L - L )] = E[6 ] = (11-18) ca a ca

E[(L - L )(L - L )] = —{w E[(L - L )(L - L )] + w E[(L -L ) 2

]} ta a ca a w oa oa a ca a ca ca a

ta

(11-19)

E[(L - L )(L - L )] = (11-20) oa a ca a

as errors of prediction and observation are assumed to be independent.

2 2

As E[(L - L ) J = o- , (11-15) is reduced to ca a ca

2 2

cr w cr o" ta ca ca ca / T -r -i-iv p = = (11-21

12 cr 0" ta ca ca

Similarly,

E[L L J-E[L ]E[L,] E[(L - L )(L , - L )] - E[(L - L )]E[L -L)]

ta cbJ L ta

J L cbJ LV ta a

v cb a

J L ta a

J L cb aJ p = = 13 cr cr cr cr ta cb ta cb

(11-22)

where L 1 -L = L. - L -6 n , and L. - L is a constant, cb a b a cb b a
-124-

E[(L 1 -JL )(L -L )] = — {w E[(L - L )(L - L - 6 J] + w E[(6 -6,)]}

ta a cb a

J w

l oa L oa a b a cb J ca

L ca cb

ta

(11-23)

According to (11-11). E[(6 • 6 , )] = p , <r cr , . The first term on the

ca cb ab ca cb

right hand side of (11-23) and both factors of the product E[(L. - L )]

ta a

times E[(L« « L )] in (11-22) are zero, so that

ca a

cr cr p cr c ca cb ab ta .__ _ .. p = cr —- = p 11-24

13 ta 2 cr cr cr r ab

cr ta cb ca

ca

Finally , substituting the results for p and p into (11-14) and

remembering that p has been denoted p , the expression given as

(II— 1) is obtained:

2 2 2 2 ta , _ ta -. cr =cr +cr + cr - 2cr cr + Zcr cr p - Zcr cr p

tb ta ca cb ta ca cr ta cb cr ab ca cb ab

ca ca

2 2 2 2 t 2 °"cb . = cr +cr + cr - 2cr +Zcr p - Zcr cr p

ta ca cb ta ta cr ab ca cb ab

ca

2

1) 2 2 Z

cr,+cr -cr -2cr7 ^ tSL \ /TT cr p f 1 - -^- (II- cb ca ta ca cb ab \^ c

cr J

ca

This result is identical to (35) given in Section 6:

The special case where cr = cr , , p , = 1, and the paths are

ca cb ab

identical has already been mentioned in Section 6. In this case, (II— 1) 2 2

reduces to cr = cr , as it should,

tb ta

If the two paths, Path a and Path b, are so different that the

correlation coefficient p , between the errors 6 and 6 , assigned

ab ca cb

to the calculated values may be set equal to zero, (II— 1) is reduced to

0- =cr + cr , - cr (11-25)

tb ca cb ta
-125-

In this case no improvement for Path b is obtained on the

2

basis of measurements for Path a, as cr will always be larger than

2

tb

cr . The latter is the prediction uncertainty of the calculated value

cb

for Path b without any measurements, and in this case it will be more

advantageous to use L than L . The improved prediction uncertainty

2

cb tb

z

cr for Path a will, of course, be always smaller than cr , based only

ta ca

on calculated values, and the right-hand side of (11-25) will always be

2

larger than cr ,

. s cb

The case of most interest in practical applications is the one

where the paths are nearly equal (i.e., in the same general area

of similar meteorological or terrain characteristics, with perhaps

one of the terminals identical). In this case the prediction uncertainties

cr 2,2and cr may be set equal to each other, but the correlation

ca cb

coefficient p , is not necessarily unity: ab

2 2 2

cr = a , = cr (11-26)

ca cb c

and (II-l) is transformed to

erf = o-f (2p - 1) + 2cr Z

(1 - p ). (11-27)

tb ta ab c ab

2 2

Dividing by cr , and denoting as r the prediction improvement

2,2 C

n , 2

ratio cr /cr for Path "b", r may be obtained as a function of the

tb c prediction improvement ratio for Path "a" (for which the measurements

, 2 2

are available), cr /cr = k , and the correlation coefficient p , : ta' c ab

r

Z = kZ (2p - 1) + 2(1 - p ) (11-28) ab ab

2

Curves for r have been calculated and are shown on Fig. II-l.

2

Only values of r < 1 represent an improvement in the prediction for
-126-

Path nb" (or < <r ). Fig. (II-l) shows that p . has to be at least 0.5,

tb c ab

and the improvement for Path "b", of course, depends on the degree

2 2,2 of improvement k = cr /o" obtained by measurements for Path "a".

2

ta c

2

For k = 1, p drops out of (11-28) and r = 1 for any value of p .
127

k

1 -4"" ~T^^^§r

--,^-—-" J^^ .///«'

—.—*"JI ^-^ S\ ///ffl

L -i j-4—

"-^"""" ^"^ S S//*//( rTftl ^. — —- u^**^ j,^ ly ////h \0-M ^ — ^"~ ^^^ ,• S\ ////if ^--^^ _ ,--*?! ^

/* /^/ //J)

0.9 ""^ ^^ ^ ^ ////// "~

i* ** ^^ / */ / y ///

*^^ ^

,x' /* /- / / f J. -X**\ ^ i*

00 s S / j / / fj

Cqp" ^ ^ x > /T / j ///

V-fi^^*" ^^ ^ ^ y J f(f

** j-'/' / / / / / // **'*'** <* S / / / '{(

**** 4^ / / / / / f J n Q rf*"^'

J* ^\ "^ /* r y / ' f

<^, ^\ _j^'

r- /* /* y / / 2 / \P^<^^' y / X / / f

^r S ~/ * y > 7 / X y^ </ / '/if S^ ^ / / / f \f / S y/\ >

,'1^ y^ > y if

£. / (b > -iT ^ / fj.

S^ v^^^ ^ y ~f i / f y jc\ / * / f f i

n -j S

.f / / f / / I / y^X ~? / * i j / S s^SY / / f / f ~/ "v^/'r / / i i J > ^T / \f f / / /\- * / f / y x / /^y*/ / / f f

1 / / \d'/s j / i J >^ / ^//* - / / ' / y / y f/ws/ / / J

r\ r / /. ^ /o?/J J f

'/ * 'x?/// y f j / / v\fc\i> * / f / / / /r^y/jL f J

/- J- -/ ^^/vy\V/ f / > * ^wA'if y ^ y / / IS ff^}/

7 f y j r ^If 7 ~/ > / ~fy*Jf n / / / J f \ J r a c * 2 / / ' / f ^ ~y Zj -i -/ f / f~ / ^ / y f / j Z _y LZ L L

y / / / / /- f- * f- y y / / /

-> _y '

' C. /] / / f J

(\ a ^- Z -/ -y £

y ' / J

c./

-j /if7- .j / ~t- J- f / / ' / / / / f ~? ~7_ r j j ,' 7 7

-y y_ / _> 03 -^ j y I PREDICTION IMPROVEMENT RATIO r V 4 4 A9 A FIIMPTinM OF IMPROVFMFMT k

>^ 'f't FOR SIMILAR PATH, AND CORRELATION

7 7 y 2 -t t ~ BETWEEN THE TWO PATHS

°- 2 t 4 o p t 4

r 2 tb k

2 ta ^^ y cr 2 o2 f f_ c c / /

y jj

0.1 j

7

y^

y7

y7

j t _

1.0 0.9 0.8 0.7

/>ab

0.6 0.5 0.4

Figure I-
-128-

APPENDIX III THE EFFECTIVE NOISE FIGURE OF A RECEIVING SYSTEM

For several years now the concepts of an effective noise figure

of a radio receiving system and of an effective input noise temperature

have been found to be quite useful at the Central Radio Propagation

Laboratory of the National Bureau of Standards. An operating noise

factor was originally defined in a paper by North [1942], and characterizes the performance of the entire receiving system as contrasted

to the receiver noise figure which characterizes only the performance

of the receiver itself. Later, Norton [1953] gave a slightly more

general expression for this factor and designated it as an effective

noise figure. Thus the effective noise figure makes appropriate

allowance for the external noise picked up by the receiving antenna as

well as the noise introduced by the receiver itself, together with the

effects of any losses in the antenna circuit and in the transmission line.

The purposes of this appendix are to derive equation (1) in Section 2,

to provide a more general formulation for the effective noise figure of a receiving system, to describe its general properties, and to relate

it to the effective input noise temperature of the receiving system. In

contrast to the approach to unity of the noise figure of an essentially

noise-free network, the effective noise figure of an essentially noisefree receiving system approaches zero.

Essentially all of the basic concepts used in this discussion

originated in the early papers by Burgess [1941], North [1942] and

Friis [1944] and in the discussion by North [1945] of the paper by Friis.

A precise definition of the noise figure of a four-terminal network was

given by Friis [1944], and his definition forms the basis for the

following formulation of the effective noise figure of a receiving system.
-129-

All signal and noise powers will be expressed in watts. It will be

convenient in this appendix to make use of the concept of available

power, i.e. , the power which would be available in the output of a generator if the output circuit were matched to the generator impedance.

It is evidently desirable to calibrate a signal generator in terms of

its open circuit voltage v, and then its available power is given

simplv bv s = v /(4 R ) for a generator with impedance Z = R + j X . g g g g g

Note that the available power from the generator has the desirable

property of being independent of the l oad impedance, and this desirable

property makes this concept especially useful for the definition and

measurement of noise figures.

Let the impedance of the load be Z = R + j X ; now it may be

shown that the ratio of the available signal power s to the signal

power s delivered to the load is given by:

2

s Z + Z

= I > 1 (HI-1)

s

,

4 R R„ m I g I

The mismatch loss factor i will be equal to 1, and the delivered power m

s will be equal to the available power s when the load and generator

impedances V are matched, i.e. , when R. = R and X„ = -X . Since it i g i g

is confusing to use the decibel scale when noise powers are added, the

convention of using lower case letters to denote powers in watts, or

power ratios, will be adopted, and capital letters will be used to denote

their decibel equivalents. For example, d will denote the available

o signal power at the terminals of the receiving antenna expressed in

watts, and P = 10 log p dbw.

The gain, g , of a four-terminal network for a c. w. frequency v

is defined as the ratio of the available signal power, s , at the output

terminals of the network to the available signal power, s , at the output

terminals of the signal generator.
-130-

g = s /s (s /s > 1) (in- 2)

nv o g o g

Alternatively, the loss factor, i ,of a four-terminal network is given

nv

by:

i = s /s (s /s > 1) (III- 3) nv g o go

Note that g and i inherently contain a mismatch factor i determined

n v n v m

from the signal generator and network input impedances, and thus

depend upon the generator impedance as well as upon the characteristics

of the network itself. The four-terminal network will have some kind

of band-pass characteristic, and the gain, g , is often defined as the

maximum gain in this pass band. More precisely, g should be considered to be the gain for the c. w. frequency v at which the noise

figure is to be specified or for the modulated band of frequencies typical

of signals expected to be transmitted through the network. Let s gV

denote the available signal power density per cycle per second in the

modulated band of frequencies intended to be transmitted through the

network. Now the network gain f o r a modulated band of signal

frequencies may be expressed in the following form:

ooc

\ s e dv

J gv to nv

o

s o n oo

\ s dv

J g

v

s g

(III-4)

Similarly the network loss for a modulated band of frequencies may be

expressed in the following form:
-131-

oor

\ s dv

J g

v

s

I = = -& (III- 5) n oo s o

\ (s /I )dv

I [V nv

Since the gain will vary over the pass band of the network, we

will follow North [1945] and define its effective noise bandwidth in cycles

per second as:

v

2 co

b = — c g dv = -L.

\ g dv (in-6)

g J

6nv hg J

& nv

n n

In the above, v and v are chosen so as to include only the principal

response of the network, i.e., a band of frequencies in which the desired

signal power density is a maximum, and which is sufficiently wide so

that g& , and g „ are negligibly small relative to the maximum value of

nvl &nv2 5 6/

g within this band. The factor h in (III-6) is a measure of any spurious

responses which may be present in the receiver. The discussion in this

appendix refers particularly to superheterodyne receivers and might

have to be modified for other types of receivers. Note that h > 1 and

may sometimes be greater than 2 for superheterodyne receivers with

little or no selectivity at the frequency converter input. In such receivers,

spurious responses will be generated by cross modulation of signals or

noise on various frequencies appearing on the frequency converter grid

of the receiver. The frequency converter mixes the signal s , on a desired gvd

frequency v within the principal response band v to v with the

d 12

oscillator frequency v of the receiver to produce a signal with intermediate frequency v. = ±(v - v ). Additional undesired voltages may

i d o
-132-

arise in the intermediate frequency output by virtue of beats between

the m harmonic of the oscillator frequency and the n harmonic of a

spurious signal or noise with radio frequency v not in the band v to

th

u 1 v to produce the w subharmonic of the intermediate frequency. The

spurious frequencies v which may produce such additional responses

in the intermediate frequency output may be determined from the

following relation:

± (n v - mv ) = v./w (III- 7) u o 1

The above may be solved for v by setting n, m, and w equal to various

integers and v. equal to some value within the range J v - v

J to

Jv - v J;the most important spurious response usually corresponds to

n = m = w = 1, and the spurious frequencies v for this response differ

by 2v. from the frequencies v within the principal response band. We will

assume that the noise voltages in the principal and in the spurious

responses are not correlated so that the resulting noise power in the

receiver intermediate frequency output is simply the sum of the output

noise powers arising from the several responses; this assumption

may not be valid in the case of atmospheric or man-made noise, and

in such cases the factor h would require a different evaluation.

Note that it would have been possible to define the noise bandwidth by integrating over the entire band, thus obtaining the larger

bandwidth hb. However, the intermediate frequency band in which the

noise finally appears has the bandwidth b, and this is the motivation

for the definition (III-6). Receivers having spurious responses will

have noise figures based on the definition of b given by (III-6) which

are larger by the factor h than the noise figures of corresponding

receivers not having such responses.
-133-

Note that g and thus b will depend upon the frequency, or band

of frequencies, for which g is determined. When we later consider

several networks in tandem, b is to be determined using g and g as & &n nv

determined from the values of s and s available at the input and output,

g o respectively, of this chain of networks. It will later be convenient to use

the db ratio of b to one cycle per second, B = 10 log b.

The noise figure, f , of a four-terminal network was defined by

n

Friis [1944] as the ratio of the available signal-to-noise ratio at the

signal generator terminals to the available signal-to-noise ratio, r ,

at its output terminals, with the available noise at the signal generator

terminals set equal to the reference available Johnson noise power,

kt b, from a resistance at the absolute reference temperature t . o o

s /(kt b)

f = —

S

— (Friis 1 Original Definition) (III- 8) n ro

In the above, k is Boltzman's constant. If n denotes the available

o noise power at the output, the output available signal-to-noise ratio

r = s /n , and when this, together with (III-2) or (III- 3) are substituted

o 00

in (III- 8) , the following alternative definitions are obtained for the noise

figure of a network in terms of the available output noise power, n , and loss factor I or sain s : n

5 5n

fnono = n I /(kt b) = no°no /(g kt b) (Equivalent Definitions) (III- 9)

The above definitions will now be used to derive an expression

for the noise figure of the simple network of Fig. III-l with loss

i caused by its resistance, R , at an absolute temperature t . Let

n n n

the resistance, R , of the signal generator be at the reference

temperature t . The available signal power from the signal generator

at the network input terminals is given in terms of its open circuit

voltage, v, by:
-134-

^AAAA/ O VWW-

<5>

INPUT

*~ OUTPUTfn

= l+(/n -l)(tn /t )

Figure HI -I
-135-

2

s v7(4R ) (111-10)

and the available signal power at the network output terminals is given

s = v

2

/[4(R + R )] (III-ll) o g n

Thus, from (III-3):

I = s /s = (R + R )/R (111-12)

n g o g n g

is obtained.

The available noise power at the output of the network of

Fig. Ill— 1 is given by the weighted average of the Johnson noises from

the resistances R and R at temperatures t and t : g n on

( R t + R t , n =kbj \° +R

n "

I =^{t +t (l -1)} (111-13) o R+R i i onn

g n n When this is substituted in (III-9), the result is:

f = 1 + (I - l)(t /t ) (111-14)

n n n o Note that the noise figure f depends upon the generator impedance as

well as upon the characteristics of the network itself. Note also that

the noise figure of a passive network at the reference temperature t

is simply equal to its loss factor, i.e. , when t no= t , it follows from

(111-14) that £ = £ . n n

Next, (111-14) will be derived in another way, and it will be

shown that it is applicable in general to any lossy passive network with

loss i and temperature t , i.e. , having arbitrary input and output

impedances. Note that the noise output, n , of the lossy network can

o be expressed as the sum of the two terms:

kb(t - t ) n = kb t + -. — (111-15) o n in
-136-

The first term would represent the available Johnson noise power

from the network if its source resistance were also at the temperature

t , while the second term represents a correction arising from the n

fact that the temperature t of the source resistance is different from

o that of the network, either higher or lower. When n , as given by

(III- 15) , is substituted in the noise figure definition (III- 9), (111-14)

is obtained as before.

Using the above definitions and conventions, the effective noise

figure of the receiving system illustrated on Fig. III-Z may now be

discussed. Let p be the available signal power from the loss-free

receiving antenna, let p be the available signal power from the actual

o receiving antenna, i.e., the available signal power at the antenna

circuit output terminals, and let i be the loss factor of the receiving

antenna circuit; thus p = i p . Let t denote the effective absolute

a c o c temperature of the antenna circuit, and (111-14) may be used to determine

the noise figure of the antenna circuit network;

f = 1 + (i - l)(t /t ) (111-16)

C C CO

Similarly the noise figure of the transmission line network with

absolute temperature t and the line loss factor I is given by;

f = 1+ (i - l)(t /t ) (111-17)

t t to

The noise figure of the receiver itself will be designated f r Friis [1944] gives an expression for the noise figure of three

networks in tandem. This will be used to determine the noise figure

of the three-network system (c), (t), and (r) of Fig. III-2:

f = f + i (£ - 1) + i I (f - 1) (111-18)

ctr c c t c t r
LU

1- >" rv < o F, , 5 z y t

LU ^ ^ £ w rr 5? °

i- ui <

z. 9 9 <i i> or

LU>

LUO

LU

oc 9 (p <1) <b -z. o

en^ U ! ^ 2

CO _l

-z. <

crh- 9 9 6 6

lb2 3

LU

O

I- CC

~Z. 7T < °

<\> (; ) c) i» <2 ! 2 LU LU

LU -J CO 2 < 2

I\ < =i f] -Q

LU <£ < O

f£ or u^

Ll

X LU co t b 8^2

_J

-137- s ** o so o a

s°

CO

II s

s°+*—>» „—. o o -4-»

.«-» \ \ «-»

»-» +-»

-4-» *-, -* N*^-• r -1^ """" ^

;- +

ii N°

-j

M— +

<—»o ,,«*» -t-» o \ -t-» o \ -t-» o **—*

-4-» """^ T

s° + + — «•""» n 1 hP *p"** XI+

II

a;

COm

(D
-138-

It will be convenient to represent the external noise power in

the band dv which is available at the terminals of the loss-free receiving

antenna by kt dv where t is the receiving antenna noise temperature

at the frequency v. Representative values of t are given by Crichlow

[1955] and by the C.C.I. R. [1956 and 1959] for frequencies v < 10 and

r / i 8 10

are given by Blake [1961] for frequencies within the range 10 < v < 10

Several useful additional sources of information relative to t are

av

available in the January 1958 Radio Astronomy Issue of the Proceedings

of the I. R. E. Now the available noise power n at the output of the

complete receiving system may be represented, with the antenna replacing the signal generator having a reference temperature t , as the sum

of two terms;

oo n = f kbtg +k\(t -t)g dv (III- 19 > ctr o

6ctr J av o6 ctrv

In practice, g may sometimes best be determined from its

components * using the relation g = g /(£ i ). The first term in 6 & ctrv & iV c v tv

(III- 19) would represent the available noise power at the output of the

receivingB system y if t = t for all values of v, while the second term

a^ o represents r a correction arising from the difference between t and t . & av o Since the integral in (111-19) has the limits to oo, it allows properly

for all of the spurious responses as well as for the principal response.

It is useful now to define an effective antenna noise temperature

t : a

oo oo f

r

\ t g dv \ t g dv

J av &ctrv j av to ctrv

t = °

= ° _ (111-20) a oo hb g ctr

i

a dv & ctrv
-139-

Using this value of t , (111-19) may be expressed:

a

n = f kbt g +khb(t -t)g (111-21)

ctr o

ectr a o

&ctr

If this value of n is substituted in (III-9) for n .and g for g , the

o ctr n

following expression is obtained for the effective noise figure of the

receiving system:

f = h(t A ) + (f - h) (III- 22) a o ctr

Finally, if we replace (t /t ) by f , substitute (111-18) for f , (111-17) a o a ctr

for f and (111-16) for f , the following general expression for the

effective noise figure of the receiving system described by the network

of Fig. (III-2) is obtained:

f = 1 + h(f - 1) + (i - l)(t A ) + i K - l)(t,A ) + i i (f - 1) (111-23) a c co ct to ctr

For the special case of a receiver with no spurious responses so that

h = 1 and in which the antenna circuits and transmission line are at the

reference temperature t , it follows that f = t , f = i , and (III- 23) o c c t t becomes:

f = f-l + fff=f-l+!If (t = t = t and h = 1) a ctra ctr cto

(111-24)

The above is the expression for the effective noise figure previously

given by Norton [1953].

This effective noise figure concept has been adopted by the

Consultative Committee on International Radio [1956 and 1959].

Examples of its application are given in several recent reports of

the Central Radio Propagation Laboratory [Crichlow et al, 1955;

Norton, 1956; Rice et al, 1959; Florman and Tary, 1961; and Air Force

Technical Order, 1961].
-140-

The effective noise figure is useful for determining the relation

between the signal power, p , available from the loss-free receiving

3, antenna and the corresponding signal-to-noise ratio r at the receiver

output. It is usually more convenient and is essential in the case of FM

receivers to consider r to be the predetection signal-to-noise ratio

available at the output of the intermediate frequency amplifier- Now,

if f is substituted in (III-8) for fnag , p for s , and r for r , the result is: o

p = f r k t b (111-25) a o

Just as Friis defined the noise figure of a four-terminal network

as the ratio of the input available signal-to- reference-noise ratio to

the output available signal-to-noise ratio: f = [s /(kt b)]/r • here

n g o ° the effective noise figure of a receiving system has been defined to be

the ratio of the available signal-to- reference-noise ratio from the lossless receiving antenna to the receiver output available signal-to-noise

ratio: f = [p /(kt b)l/r. As pointed out by Friis, definitions in terms

a o

of available signal and noise powers rather than for necessarily matched

conditions are advantageous since the use of mismatch conditions in

amplifier input circuits often lead to a reduction in the noise figure of

networks containing such amplifiers, [Llewellyn, 1931; Haus and Adler,

1958; Haus et al, I960].

Expressed in decibels, (111-25) may be written

P = F + R + B - 204 dbw (111-26) a where F = 10 log f, R = 10 log r, and 10 log kt = -204. 00 when

t = 288. 39° and k = 1. 38044 x 10" 23 [Cohen et al, 19571; since t must

o o be somewhat arbitrarily chosen in any case, the above value was

assigned so that the constant in (111-26) is equal to 204.
-141-

If P denotes the power radiated from the transmitting antenna,

r then the radio path transmission loss [Norton, 1959]

L = P -P = P - L, - P = L _ G . whe re P = P - L and the

r attabp rtt

remaining terms in this equation are defined on pages 3 and 4. Now

the following simple equation is obtained for determining the transmitter power P required to provide a given predetection signal-tonoise ratio R:

P=L -G +L+F+R+B- 204 dbw (111-27)

t b p

t

Note that the effective noise figure f for a noise-free

receiving system, i.e. , with t = f =0, f = f = 1 and f = h is equal ° a a c t r to zero. We may establish that f = h for a noise-free receiver with

r spurious responses as follows. The noise n available at the output of

such a receiver may be expressed:

co n = kt \ g v dv (111-28)

o

If we substitute this value in (III-9), we obtain:

co

f = -\ \ g dv = h (III- 29)

r g b J °rv

o

In fact, practical receiving systems have been developed with effective

noise figures f substantially less than unity, so that F is actually

negative; for example, DeGrasse et al [i960] have reported a value of

F = -12 db.

Some experimenters prefer to specify the performance of low

noise receiving systems in terms of an effective input noise

temperature t , and this is related to f and t or to h, t and f e o a ctr

by t eoactr =ft =ht +(f -h)t. However, it appears from the above

o discussion that the effective input noise temperature t depends upon

the antenna impedance, the spurious responses of the receiving system
-142-

and the modulation characteristics of the signals intended to be received.

Consequently, it would appear to be better to use f, t and b to provide

o a measure of the expected performance of a receiving system rather

than adding one more to the already large number of different

temperature concepts [Ko, 1961] currently in use in the electronic

engineering literature. In this connection, reference may also be made

to recent discussions by Rhodes [1961] and Talpey [1961]. The reference

temperature t = 288.39 is preferred for the reason given above, but

corresponding noise figures would be less than 0. 0<i4 db larger than

those specified relative to a reference temperature t = 290 ; however,

o the use of a reference temperature t = 1 K would lead to noise figures

o which would be 24.6 db larger.

A temperature in degrees Kelvin may be related to temperatures

in degrees centigrade or in degrees Fahrenheit by the relations:

t = 273.16 + t_ .. , = 255.38 + (5/9) t_ , , .. (111-30) Kelvin Centigrade Fahrenheit

If the signal generator used for receiver noise figure measurements

has its impedance at a temperature t gorather than t , then a term

[l-(t A )] h should be added to the value so measured to determine f . L g ' o

J r

It follows that an error in f of less than ± O.lh will be made ift lies

r g

within the range 260 K to 317 K, i.e., within the range -13 C to 44 C

o o or within the range 8 F to 111 F.

The effective noise figure of the receiving system could be

referred to the receiver input terminals rather than to the terminals

of the loss-free receiving antenna. It would then be smaller by the

factor I t and would thus not represent a proper measure of the noise

c t performance of the entire receiving system. The loss-free receiving

antenna terminals clearly represent the only proper reference point

for the effective noise figure of most communications receiving systems,
-143-

although an argument can be made [Dyke, 1961] by the radio astronomers

for referring the effective noise figure and the corresponding effective

input noise temperature to outer space--the location of their target--by

including tropospheric and ionospheric losses with appropriate

temperatures in determining f.

In radiometry, the desired signal is the noise available from the

loss-free receiving antenna, and such a signal will be present in the

spurious responses as well. In this case g and b, as defined by (III-4) r r ctr

and (III-6) are indeterminate; however, the factor hbg , which is required in (111-10) for the evaluation of t , may be evaluated [See (III-6)] by:

CO

hbg = \ g s dv (III- 31)

ctr J

& ctrv

o

The radiometer measures only an average antenna temperature t , and

it is evident by (111-20) that the presence of spurious responses over a wide band will not yield y a good s estimate of t if this has different

av

values for the different receiver responses. Furthermore, the output

noise voltages from various parts of the response band must be

uncorrelated to the same extent as those from the calibrating white

noise source for a valid determination of t . a Normally the effective noise figure of a receiving system would

not be measured directly; instead each of its components would be

measured, and then (111-23) used to combine these components. The

following systematic procedure may be followed for determining each

of these components under the actual conditions of matching or mismatching of the various elements in the receiving system. Let R

denote the theoretical radiation resistance of the receiving antenna

referred to its input terminals, and let R be its measured input m

resistance; in this case S. = R /R . This method of determining I c m a c may sometimes be used at low radio frequencies where R and R m a

often differ substantially; a more general discussion including the
-144-

effects of insulator losses is given by Crichlow [1955]. At the higher

frequencies, t will usually be negligibly different from unity; however,

in the case of reception with a unidirectional rhombic terminated in its

characteristic impedance, i may be greater than 2 [Harper, 1941],

since nearly half of the received power is dissipated in the terminating

impedance and some is dissipated in the ground. Next the transmission

line loss factor i is measured as the ratio s /s where s denotes the

t g ° g

available power from a signal generator at the input to the line with its

impedance adjusted so as to be equal to the measured impedance of the

receiving antenna, and s denotes the resulting available power at the

output of the line, i.e., the power which would be delivered to an

output impedance which matches the output impedance of the line with

the antenna connected to the input to the line. Finally, the noise figure

f of the receiver is to be determined by using a c.w. signal generator

r at its input having the same impedance as the above-defined output

impedance of the transmission line. Let s denote the available signal

power at the frequency v from the generator with its impedance at

temperature t which is required to double the receiver output power

(signal plus noise) relative to the output noise power with the signal

generator turned off; then f = [s , g /(g kt b)] + [l-(t A )] h. The

r d tv &r o go

above procedures will yield the appropriate values of i , i and f

to be used in (111-17) for the actual conditions of matching of the

transmission line either to the receiving antenna or to the receiver.

It will always be desirable to match the line to the receiving antenna,

since this will result in the smallest value for i and will eliminate

echoes which could be quite objectionable on long lines. However, it

will sometimes be advantageous to mismatch the transmission line to

the receiver and thus minimize f ; this latter point is discussed in the

r paper by Llewellyn [1931] and in the I. R. E . Standards [Haus et al, I960].

Next, g&

is determined over a frequency range which includes all of

ctrv m y s the receiver responses, and g , b, h and t are then determined by

ct r 3, means of their defining equations (III-4), (III-6), and (111-20).
-145-

The use of a dispersed signal source [noise generator] has been

recommended [Haus et al, I960] as a simpler method of measurement

of the receiver noise figure since the bandwidth need not be separately

measured in determining f . However, if this method is used in the

r case of a multiple response receiver, it is essential that the noise

density from the generator be constant over a band which includes all

of the receiver responses, and the apparent noise figure obtained in

this way must be increased by the factor h in order to determine f ; thus it would still be necessary in this case to measure both b and h

in order to determine the noise figure of a multiple response receiver.

Furthermore, the ultimate use of f is for the determination of p , r a

and it is evident by (111-25) that its determination requires an equally

accurate determination of both b and f. Consequently, the specification

of the noise performance of a receiving system (or of a receiver)

requires the determination of b and h as well as f (or f ). Note that

r b should be made somewhat larger than the band ideally required for

the reception of the desired signal, in order to accommodate relative

drifts of the transmitter and receiver oscillators.

The over-all merit of a receiving system is best measured by

p = f r ktb = r ktb where r denotes the minimum value of r m m o m e m

required in the system considered for a specified grade of service.

Although p provides a direct and unambiguous measure of the merit

of a receiving system, it should not be overlooked that a reduction in

pm can sometimes be achieved by the use of very wide band modulation

schemes, and the use of such wide bands is not necessarily desirable

for the efficient use of the spectrum; in many cases the use of higher

power transmitters and narrower modulation bands might well be

advantageous.

In measuring an available loss or gain of a four-terminal network,

it may be inconvenient in some cases to make the signal generator output

impedance the same as the output impedance of the preceding networks

and the load impedance equal to the network output impedance with its
,.146-

input connected to the preceding networks. There are two cases in

which this can be avoided. In case the load impedance is matched to

the network output impedance, we may determine £ or g by (III-2)

or (III- 3) by replacing s by s i J£ where £ denotes the

g g mnp mng mnp

mismatch factor (III-l) between the network input impedance and the

output impedance of the preceding networks while £ denotes the c r c ° mng

mismatch factor between the network input impedance and the signal

generator impedance; when the signal generator impedance is made

equal to the impedance of the preceding networks, £ = £ and

mnp mng

no correction is involved. In case the signal generator impedance is

made equal to the output impedance of the preceding networks, we

may determine £ or g by (III- 2) or (III- 3) by replacing s by si

where s. denotes the power delivered to the load and £ . denotes the

£ * moi

mismatch factor between the network output impedance and the load

impedance.

Many radio relay systems contain a duplexer between the

transmission line and the receiver so as to permit simultaneous transmission and reception from a single antenna. In case this duplexer has

the same absolute temperature t as that of the transmission line, we

t may modify (III- 23) as follows to determine the effective noise figure of a receiving system containing such a duplexer: £ is now measured as

the ratio s /s where, as before, s denotes the available power from

g o g a signal generator at the input to the line with its impedance equal to

that of the receiving antenna, but s is now the resulting power available

at the output of the duplexer. In addition, the following term is added

» . co Vt r

to (111-23): —; — \ p , g dv where p , is the undesired local transg kt b J dv °rv 'dv

r o o mitter power density available at the output of the duplexer; it is assumed

that the receiver output voltages arising from this source from the various

responses are uncorrelated.
-147-

It is important to emphasize that the noise figure of a receiver,

or of any four-terminal network, depends upon a mismatch factor im

between the source and the network, and thus depends upon the source

impedance as well as the characteristics of the network itself. In

particular, the effective noise figure of a receiving system, depends

upon the impedance of the receiving antenna.

In those cases, particularly at the lower frequencies, where

the transmitting and receiving antenna circuit losses are large, it might appear to be advantageous to use the concept of system loss

L = P 1 - P where P 1 is the power input to the transmitting antenna

s t o t r F &

and P is the available power from the actual lossy receiving

o antenna. In this case and assuming a single response receiver with

h = 1, (III- 27) may be written:

P* = L +F +R+B- 204 dbw (111-32)

t s s F = 10 log {(t /t ) + (i - l)(t /t ) + l(f - 1)} (h=l) (111-33)

s 10 as o t to t r

Here F is the effective noise figure of the receiving system referred

now to the terminals of the actual lossy receiving antenna, and kt b

as

represents the external noise available at the terminals of the lossy

receiving antenna. Note, however, that (tasoaccoc /t ) = {f + (i - l)(t /t )}/i , and thus i and t must still be known for a proper determination of

c c the performance of the system. For this reason it appears that (111-32)

is no simpler to use than (III- 2 7) and, of coarse, these two equations

must yield identical results when each of their terms is properly

evaluated.

In many applications, particularly where atmospheric noise

is involved, t and f will be quite variable with time, and in such

a a cases it is useful to consider f to be a random variable and to

describe it in terms of appropriate statistical characteristics. In

still other applications, such as space- satellite communications, f will be found to vary with the receiving antenna orientation since ta

will vary as the antenna is pointed in different directions.
-148-

At very high frequencies or at very low temperatures, the

available noise power from a source at absolute temperature t will

be less than ktb by the factor (hv/kt)/{exp(hv/kt) - l} as was shown

by Nyquist [1928]; here h denotes Planck's constant. Since

(hv/kt) = 0.0479932 v(Gc/s)/t [Cohen et al, 1957], this correction

represents less than 0.1 decibel reduction in the available noise power

when v(Gc/s)/t is less than 0.955895, i.e., when v < 276 Gc/s at the

reference temperature t or when v < 9. 5 Gc/s for a temperature

O r -, t = 10 K. Balazs [195 7

J has shown that the Johnson noise power available

from a conductor also depends on the shape of the conductor at very

high frequencies.

There has been some discussion in the literature [Haus and

Adler, 1957 and Siegman 1961] of difficulties with "negative" resistances

and their associated "negative" temperatures in some types of amplifiers;

such considerations are important, however, only in the design and

description of the various internal components of the receiver. In those

cases where the receiver output impedance has a negative resistance,

r may be re-interpreted to be the signal-to-noise ratio in the load

impedance rather than the available signal-to-noise ratio at the intermediate frequency output. With the use of this convention in such cases,

the noise figure f of any practical stable receiver can still be defined

since such a receiver must have positive source and load resistances

with positive temperatures at its input and output.

It is sometimes possible to reduce the effective noise figure f

of a receiving system and thus to improve its performance by

rearranging the order of its component parts. To see how this may

be accomplished, we may use the formula of Friis for two networks

in tandem. For two networks p and q, with noise figures f and f , p q

the two networks in tandem with p preceding q will have a noise figure

given by:

f = f + (f - l)/g (III- 34)

pq p q p
-149-

Alternatively, if q precedes p, we have:

f = f + (f - l)/g (III- 35)

qp q p q

From the above we obtain the condition which must be satisfied for

f < f : pq qp

f + (f - D/g <f +(f - D/g (Forf <f ) (III- 36)

p q p q p q pq qp

We will first consider the conditions under which it is

advantageous to use a preamplifier p at the antenna terminals

preceding the transmission line represented by network q. In this

case g = l/i and f = 1 + (i - l)(t A ) so that we may write for the sq

' t q t t' o

y reduction in f with the preamplifier preceding the transmission line:

Af=f -f = (i -l){(f - 1) + (t /t )(!- — )} (Ill- 37)

qp pq t p togP

Since Af is inherently positive, it follows that f will always be decreased

by having a preamplifier precede the transmission line, and this

reduction represents an improvement, expressed in decibels, given

by AF = - 10 log. n{(f + Af)/f }. 10 pq pq

Consider next the question of which of two amplifiers in a chain should precede the other. In this case we may subtract 1 from

each side of (111-36) and, provided g > 1 and g > 1, we may re-write

r t. this inequality in the form:

(Forf <f provided (111-38)

pq qp

Ss > 1 and s > 1) P *q

If the above inequality is satisfied, it will be advantageous to have the

amplifier p precede the amplifier q.

f - 1 f - 1 p <r q

1 1 g^ g

p q
-150-

Acknowledgements: This appendix was written after discussions with

E. F. Florman and W. Q. Crichlow, both of whom independentlyderived the expression (111-14) by arguments differing from those given

herein. Discussions with L. V. Blake of the Naval Research

Laboratory also proved to be helpful.
-151-

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Crichlow, W. Q. , D. F. Smith, R. N. Morton, and W. R. Corliss,

Worldwide radio noise levels expected in the frequency band 10 kc

to 100 Mcs, NBS Circular 557 (August 1955); the derivation in this

report for f is correct only in the case where t and t are equal

to t . c t o
-152-

DeGrasse, R. W. , D. C. Hogg, E. A. Ohm, and H. E. D. Scovil,

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Florman, E. F. , and J. J. Tary, Required signal-to-noise ratios,

carrier power and bandwidth to achieve a given performance

of multichannel radio communication systems, to be submitted

for publication as an NBS Technical Note (1961).

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pp. 419-422 (July 1944).

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Haus, H. A., and R. B. Adler, An extension of the noise figure

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~~~

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Rhodes, D. R. , On the definition of noise performance, Proc. IRE 49,

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.

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data and their long-term variability, NBS Technical Note 43,

PB 151402 (Mar. 1960). Order by PB number from the Office

of Technical Services, U. S. Department of Commerce,

Washington 25, D.C. Foreign remittances must be in U.S.

exchange and must include one -fourth of the publication price

to cover mailing costs. $2. 25.
V. S. DEPARTMENT OF COMMERCE

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NATIONAL BUREAU OF STANDARDS

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