National Bureau of Standards

library, N.W. Bldg

OCTt -1962

^ecknlcai vjote 158

EFFICIENT USE OF THE

RADIO SPECTRUM

KENNETH A. NORTON

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

technical Nccte

158

APRIL 1962

EFFICIENT USE OF THE

RADIO SPECTRUM

Kenneth A. Norton

NBS Boulder Laboratories

Boulder, Colorado

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.

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 45 cents
ABSTRACT

Methods are given for determining the transmitter power

required for satisfactory operation in the presence of noise on anygiven telecommunications link. It is then suggested that sufficiently

high power should be used so that a satisfactory grade of service

will be achieved at all times. The above conclusion has reference

only to the problem of reception in the absence of interference from

other telecommunications systems. It is then shown that optimum

use of the spectrum can be achieved only when interference from

other signals rather than from noise provides the ineluctable limit

to satisfactory reception. The fact that interference, rather than noise,

should provide the ineluctable limit to satisfactory reception indicates

that greater stress should be placed on the use of various techniques

for making systems free of mutual interference rather than designing

them simply with the objective of overcoming noise.

Most of the report deals with statistical methods of using the

concept of transmission loss on the propagation paths in order to

achieve the above described optimum use of the spectrum.

11
CONTENTS

Page

No.

ABSTRACT "

1. INTRODUCTION 1

2. CRITERIA FOR SATISFACTORY SIMULTANEOUS

USE OF TWO OR MORE COMMUNICATION LINKS 5

3. THE PRINCIPLE OF A HORIZONTAL INCREASE

IN POWER 17

4. THE SERVICE PROBABILITY APPROACH TO FACILITY

ASSIGNMENT DECISIONS 20

5. THE EFFICIENT ASSIGNMENT OF FACILITIES

TO BROADCASTING SERVICES 21

6. SOURCES OF INFORMATION ON THE PREDICTION

OF TRANSMISSION LOSS AND ITS VARIABILITY 27

REFERENCES 34

APPENDIX I: On Figures of Merit for Radio Receiving

Systems in the Presence of Noise or

Unwanted Signal Interference and the

Concepts of Operating Noise Factor and

Operating Noise Temperature

1. Introduction I1

2. Definitions of wanted and

unwanted available signal powers 1-2

3. The operating noise factor of a receiving system 1-4

4. The operating signal-to-noise ratio 1-18

5. The required value P of wanted

signal power 1-21

in
CONTENTS

(continued)

Page

No.

6. Reference point for the operating

noise factor 1-23

7. Other considerations 1-26

8. Radiometry 1-28

9. Measurement of the operating

noise factor 1-29

10. The measurement of R 1-32

u

References to Appendix I 1-35

List of Symbols 1-38

IV
EFFICIENT USE OF THE RADIO SPECTRUM

by

Kenneth A. Norton

1. INTRODUCTION

The radio spectrum is a prime natural resource which is available

for use by the citizens of all nations throughout the world and, indeed,

by the inhabitants of all planets throughout the universe. The extent

to which the spectrum is actually used in various countries is currently limited primarily by the extent of the economic development of

their resources. However the use of certain parts of the spectrum is

currently limited in many regions of the world by mutual interference

between two or more radio transmissions simultaneously occupying

the same frequency bands. This simultaneous occupancy of a given

band of frequencies will be called frequency sharing.

In the United States the responsibility for the efficient use of the

spectrum is shared by the Director of Telecommunications Management, who reports directly to the President, and the independent

governmental agency, the Federal Communications Commission.

Internationally this responsibility rests with the International Telecommunications Union. Up to the present time the attitude towards

efficient use of the spectrum may be described somewhat as follows.

First, various uses for the spectrum have been considered socially

desirable, and then frequency bands have been allocated to these uses.

Currently allocations have been made throughout the range from

10 kc/s to 40 Gc/s, i.e. to all of bands 5 to 10 and including parts

1
*

- 2 - of bands 4 and 11 . Internationally, the approved uses for the

spectrum are given in the Radio Regulations of the I. T.U. [ 1 959 j

.

These allocations may be different in the different regions of the world.

Within the United States frequency allocations conform to those in the

Radio Regulations but are further divided between government and

non-government users by joint agreement between the Director of

Telecommunications Management and the F.C.C. Currently approved

allocations are given in the Rules and Regulations of the Federal

Communication Commission [Pike and Fischer], The D. T.M. and

F.C.C. also have the responsibility for assigning frequencies, transmitter powers, antenna locations and heights, and modulation characteristics to individual users of the spectrum. Recently the F. C.C.

has also requested authority to control television receiver characteristics as well.

Opportunities for making more efficient use of the spectrum arise

either in the method of transmission or in the assignment of a specific

band of radio frequencies for use by a particular radio transmitter for

communication from location A to locations B^ C, D, etc. A fixed

service is characterized by a single communications link from an

antenna at a fixed geographical location A to an antenna at a fixed

geographical location B. A broadcasting service is characterized by

a single transmitting antenna and multiple receiving locations. A

mobile service is characterized by transmitting and, or, receiving

locations which may change with time. However, all of the above

types of service may be characterized as consisting simply of one or

more point-to-point communication links with a transmitter at one

terminal and a receiver at the other terminal. The purpose of the

communication link is to reproduce at the receiver, with a given

n . „n *Band n extends from 0.3 x 10 to 3 x 10 cycles per second.
-3-

degree of fidelity, the information available at the source. The difficulty of achieving this reproduction depends on (a) the nature of the

information to be transmitted, (b) the rate at which the information is

to be transmitted, and (c) the fidelity of the received signal. Precise

definitions of these three aspects of a transmitted signal are available

in the literature on information theory. Information theory provides a very fundamental and formally satisfactory basis for describing the

efficiency of use of a single communications link. However, this

rather sophisticated theory will not be used in the following elementary

discussion since this would unnecessarily complicate the exposition of

other fundamental technical principles which must be applied in order

to achieve efficient use of the spectrum. The application of information theory indicates that presently used methods of transmitting

various types of information over a particular link are often quite

inefficient. However, it is also generally recognized that a substantial

improvement in the efficiency of transmission of a given type of information will usually involve the development of more complex or more

powerful equipment. Consequently the extensive use of these more

efficient methods of transmission will largely be limited by economic

considerations. The technical principles described below are

concerned with the efficient^ oint, simultaneous use of all of the

individual communication links which occupy the same or adjacent

frequency bands. Thus the application of these technical principles

can improve the efficiency of use of the spectrum to a large extent

independently of the efficiency of use of the individual links and may

thus be used to advantage without resort to more complex or more

powerful equipment. The achievement of this more efficient use

depends only upon good spectrum management, i.e. upon the proper

assignment of currently available facilities to the individual communication links.
-4-

Thus optimum use of the spectrum involves not only the efficient

use of the individual communication links but also the efficient joint

use of many such links either by time sharing or by frequency sharing

of the same frequency bands. This report proposes as figures of merit

for a receiving system the protection ratios r (between the hourly

median wanted and hourly median unwanted signal powers available

from a loss-free receiving antenna which is otherwise equivalent to

the actual receiving antenna) which are required for the satisfactory

reception of the information carried by a specified wanted signal in

the presence of specified kinds of unwanted signals. The use of

receiving systems having the smallest values of r will permit the

same portions of the spectrum to be used simultaneously by the maximum

number of users and thus the minimization of these protection ratios

represent an important means for better spectrum utilization. On the

other hand, economic considerations, as contrasted to spectrum

conservation considerations, indicate that radio receiving systems

should be designed so that the minimum possible transmitter power is

required for the satisfactory reception of the wanted signals in the

presence of noise. This report proposes as an additional figure of

merit for a receiving system the hourly median wanted signal power p

which is required at the terminals of the equivalent loss-free receiving

antenna for the satisfactory reception of the information carried by the

wanted signals in the presence of noise but in the absence of any other

unwanted signals.

For optimum use of the spectrum by the maximum number of

simultaneous users, the transmitting and receiving systems of the

individual links should be designed with the primary objective of

minimizing the various values of r involved and then sufficiently high

transmitter powers should be used so that the hourly median wanted
-5-

signal power pm exceeds p for a sufficiently large percentage of the

n ;ox- o hours during the intended period of reception at each receiving location.

This approach to frequency assignment problems will be unrealistic in

a few cases such as the cleared channels required for radio astronomy,

but these rare exceptions merely serve to prove the otherwise general

rule that optimum use of the radio spectrum can be achieved only when

interference from other signals rather than noise provides the

ineluctable limit to satisfactory reception.

2. CRITERIA FOR SATISFACTORY SIMULTANEOUS

USE OF TWO OR MORE COMMUNICATION LINKS

If two or more communication links are operated anywhere in the

spectrum they can, in principle, mutually interfere. Naturally this

interference is usually greatest when these links are operated in the

same or in nearly the same frequency bands but may be significant

even for operations in widely separated frequency bands by virtue,

for example, of harmonic radiation. By virtue of this mutual interference the fidelity of the information received on each of the links will

be deteriorated not only by noise but also by the cross talk arising

from the operation of the other links. For example, consider the joint

operation of n links with transmitters having radiated powers P.,

i= 1 to n, expressed in decibels above one watt. Let R.. denote the

n

wanted signal-to-operating noise ratio at the input to the receiving

system which is required for the satisfactory reception of the wanted

signal in the absence of all other kinds of interference. A precise

definition of R.. is given in Appendix I where it is designated R . ii & n

Note that R.. will depend not only on the nature of the noise and
-6-

the nature of the information transmitted over the i link, together

with its rate of transmission, but also depends on the grade of service,

i.e. the degree of fidelity of the received signal which is considered to

be satisfactory. Let N.(t) denote the hourly median operating noise

power level at the time t referred to the loss-less receiving antenna

terminals of the i link:

N.(t) = F.(t) + B - 204 dbw (1)

In the above F.(t) denotes the hourly median operating noise factor at

the time t of the receiving system expressed in decibels, B = 10 1og b

10 where b is the effective noise bandwidth of the receiving system

expressed in cycles per second and 204 = -10 log (kT ) where k is

10 ° Boltzman's constant and the reference temperature T =288.39°

o Kelvin. Precise definitions for the operating noise factor and effective

noise bandwidth of a receiving system are given in Appendix I. Let

L..(t) denote the hourly median transmission loss [Norton, 1953, 1959

XJ th

and CCIR 1959] between the receiving antenna of the i link and the

th

transmitting antenna of the j link at the time t. Let P..(t) denote

the hourly median signal power in dbw which is available at the lossless receiving antenna terminals of link i at the time t from the

transmitter of link j. A definition of signal power is given by (1) in

Appendix I. P..(t) denotes a wanted signal power when j=i and an

unwanted signal power when j^i. The following equation defines the

transmission loss L..(t):

L..(t) ep. - P..(t) . (2)
-7-

Finally let R. . denote the ratio of the i

L wanted hourly median signal

1J th

power to the j unwanted hourly median signal power, expressed in

decibels, which is required for the satisfactory reception of the i wanted signal in the presence of the j unwanted signal alone. A

precise definition of R. . is given in Appendix 1 where it is designated

R . Note that R.. will depend not only upon the nature and rate of

transmission of information over the links i and j and the degree of

the spectrum overlap between the channels i and j but also upon the

nature of the within-the-hour fading of both the wanted and unwanted

th

signals and the degree of fidelity of the i wanted signal which is

considered to be satisfactory. It follows from the above definitions

that it is necessary that the following inequalities be satisfied if the

th

i received signal is to be considered a satisfactory replica of the i transmitted signal over a period of one hour at the time t:

P..(t) - N.(t) = P. - L..(t) - N.(t) > R.. (3)

ii i l li i li

P..(t) - P..(t) = P. - L..(t) - P. + L..(t) > R . (4)

ii ij i ii j ij ij

The second inequality above must be satisfied for j=l to n excluding

the case j = i which is covered by the first inequality. The above

inequalities would also be sufficient conditions for the satisfactory

reception of the i wanted signal at the time t if it were correct to

assume that several sources of interference present simultaneously do

not deteriorate a wanted signal more than it would be deteriorated by

each of the individual sources of interference acting alone. This

assumption is clearly not precisely correct, however, and a practical
-8-

method is given by Norton, Staras and Blum [1952] for making allowance for interference present from several sources simultaneously.

For the purpose of the present report however, we will consider the

effects of the interference from each source independently and the

reader should remember that this assumption will always be at least

to some extent unrealistic and in some cases completely inappropriate

The use of this assumption will, however, greatly simplify our subsequent analysis and it will not vitiate the important fundamental

principles of frequency assignment which we expect to establish.

We will now show how (3) and (4) may be used to determine the

percentages of time p..(j=l to n) that the i link will provide a satis1J th

factory signal in the presence of the j source of interference alone;

th when j=i, p.. is the percentage of time that the i link will provide a

satisfactory signal in the presence of noise. The percentage of time

th

p.. is called the time availability of satisfactory service on the i 1J th

link in the presence of the j source of interference. Consider the

cumulative distribution of the random variable L..(t). Let L..(p)

denote the hourly median transmission loss which is not exceeded

during p percent of all hours within a given time period, e.g. all of

the hours in the winter or all of the hours in a year. Now we maydefine y..(p) as follows:

iJ

yii (p) S L

ii (50) " Lii (p) * (5)

Note that y..(p) is positive for p < 50% and negative for p > 50%.

Similarly for the random variable N.(t) we may define z.(p) as follows

z.(p) =N.(50) - N.(p) . (6)

i i i
-9-

In the above N.(p) is the hourly median noise level exceeded for p

percent of the hours in the same time period used for defining y..(p).

The random variables L..(t) and N.(t) will tend to be normally

ij i distributed and we may let p.. denote the correlation coefficient

ij

between L. . (t) and L. . (t) and p.. the correlation coefficient between

11 ij ii L..(t) and N.(t).

n l Using the above definitions we may define factors S.. and S , expressed in decibels, which provide measures of the extent to which

the inequalities (3) and (4) are satisfied for specified time availabilities

p.. and p. .« respectively:

ii ij

S s P - L... (50) - N. (50) - R..

ii i n i ii

- {Ai <Pii> + z

i (p

ii» + 2p

ii yu

tp

ii) z

i (pii» I

(7)

S =P - L.. (50) - P. + L.. (50) - R..

ij i ij J iJ iJ %

~ {^

2

ii^ij

)+ ^ij

< 100 - *i? ~ 2p

ij yii (p

ij )y

ij

(10 °

- Pij>}

The above definitions would be exact if the hourly medians L (t) and

ij

N (t) were normally distributed random variables. For the types of

i distributions actually encountered in practice (7) and (8) provide

entirely adequate approximations.
-10-

If we set S and S . equal to zero then (7) and (8) may be solved

11 1J

i_ - th

for the percentage of the hours p.. that the wanted signal on the l li link is "expected" to be received with satisfactory fidelity in the

presence of noise and the percentage of the hours p that the wanted

signal on the ith

link is "expected" to be received with satisfactory

fidelity in the presence of the j interfering signal.

The values of S and S.. defined by (7) and (8) may be calculated

by means of prediction formulas for the hourly median transmission

loss, for the hourly median noise level, and for the long term

variabilities of these quantities. Section 6 of this report contains a resume of the formulas available in the literature. Rather comprehensive formulas suitable for the prediction at frequencies above

40 Mc/s of the transmission loss and its long term variability for

propagation paths involving only tropospheric modes of propagation

have recently been published by Rice, Longley and Norton [1962]. It

is convenient to assume that the errors of predicting S are normally

distributed with a standard deviation o" which may be estimated by the

methods given in the above reference. Now we may determine the

service probability q.. that the wanted signal on the i link will be

received with satisfactory fidelity in the presence of noise for p.n

percent of the hours by setting (S../cr ) equal to the standard normal

ii o .

.

n

deviate Mq..) which is tabulated in Table I. The inverse function

ii q(X) is defined by the following equation:

dxexp(-x2 /2). (9)
-11-

Alternatively the service probability q{ M may be defined in terms of

erf (x) , the error function of x:

q(M = 0.5 -I 1 - erf Mq) / *JT 1;

For q(\)<0. 5, Mq) <0 (10)

q(M = 0.5 -I 1 + erf Mq) /\TT For q(\)>0.5 f \(q) >0 (11)

Table I q(M Mq) = (S/crs

)

0.001 -3.090

0.002 -2.878

0.005 -2.576

0.01 -2.326

0.02 -2.054

0.05 -1.645

0. 1 -1.282

0.3 -0.524

0.5

0.7 +0.524

0.9 +1.282

0.95 +1.645

0.98 +2.054

0.99 +2.326

0.995 +2.576

0.998 +2.878

0.999 +3.090
-12-

Sirailarly we may determine the service probability q. . that the wanted

th 1J signal on the i link will be received with satisfactory fidelity in the

th

presence of the j unwanted signal for p.. percent of the hours by

setting ( S../o_ ) equal to the standard normal deviate Mq..).

v ij s..y ij

The above analysis provides quantitative criteria for determining

the merit of an assignment plan for the simultaneous operation of n

links. These criteria may also be used to determine the merit of

adding one more link to those already operating. As an example of how

these criteria might be used, consider the problem of adding one more

link to a line-of-sight microwave relay system. The first step in the

procedure would be to tabulate the characteristics of all of the other

communication links which might be expected to cause interference to

the proposed additional link. Next terrain profiles should be obtained

between the receiving terminal of the proposed link (i = n) and the

terminals (j = 1 to n) of all of the potentially interfering links. Using

these profiles predictions are next made of the cumulative distributions

of the transmission loss and the noise for each of these paths; the noise

on a microwave link may be considered constant so that z.(p )S0 for

1 n

all the values of p . Finally, setting S and S = in (7) and (8), n n ij

these equations may be solved for the expected values of the time

availabilities p . (j = 1 to n) of the desired signal on the proposed new

link in the presence of (n - 1) interfering signals (j = 1 to n - 1) and noise

(j=n).
-13-

It is convenient to express these n time availabilities, p ., in

terms of n probabilities p

1

. = (100 -p .)/100 that a satisfactory

nj nj

signal will not be available on the new link in the presence of the n

sources of interference considered one at a time. Using these n

probabilities we will now derive upper and lower bounds for the

probability p* that a satisfactory signal will not be available on the

n link; this probability p

1 will be derived from the n values of p

1

. n nj

and is thus also based on the simplifying assumption made earlier

that several sources of interference present at a particular time do

not further deteriorate the wanted signal more than it would be

deteriorated by the worst of these n sources Of interference acting

alone. Thus (1-p 1 ) is simply the probability that no one of the n

sources of interference, acting independently, will deteriorate the

wanted signal on the n link so that it is no longer satisfactory.

The general theory of probabilities of combined events [Feller,

1957] may be used for estimating the probability p

1 that a satisfactory

signal will not be available on the n link. Let A. (j = 1 to n) denote the

events of not having a satisfactory signal available in the presence of

each of the n sources of interference; thus the probability P(A.) that

A. will occur is equal top 1 .. Now the probability p' = P(A or A or

J nj e r n j 2

• • • • A.* • • • or A ) that a satisfactory signal will not be available on the

th i- 1 • K n link is given by:

P(A or A or- • -A.- • • or A ) = S -S + S -S + - • 12 • -± S (12)

J n 1 2 3 4 n

where S = SP(A ), S = SP(A and A.), S = SP(A. and A. and A ), etc.

1 12 1 J 3

1 J k
-14-

In the above i<j < k < • • • * <n so that in the sums each combination

appears once and only once; hence S has

r {n(n-l)' • -(n - r + 1) } / { 1 • 2 (r - 1) • r}

terms. The last sum S is the probability of the simultaneous realization of all n events. Feller ( 1957] has shown that S is an upper

l bound to P(A or A or- • • • A. • • • • or A ) and that this upper bound

l a J n

^

will be realized only when the n events A. are mutually exclusive;

thus the smaller of S or 1 is the upper bound top 1

. Let p

1 denote

l n

r ni

the largest of the n probabilities p' . and it is obvious that this is the

minimum value which p

1 can have. Thus we have established the

following bounds for p

1 : n

n

p' < p

1 < The smaller of ) p

1

. or 1 (13)

For example, if the fading is well correlated for all of the unwanted

signals, the wanted signal will tend to be deteriorated only during

that small fraction of the time represented by p

1 F y

. and p* will then

^ni *n

approach this lower bound; for uncorrelated fading or for negatively

correlated fading p

1 will be larger and may approach the upper bound

given by (13)» It follows from (13), regardless of the nature of the

fading, that the time availability p = I00(l-p f ) of satisfactory reception

on the n link in the presence of n sources of interference is bounded

by:

n

The larger of zero or \ 100 - ) (100 - p .) [ <p <p (14)

I l_i nj j n ns

3=1
-15-

In the above p denotes the smallest of the n time availabilities p .. *ns nj

As an example of the use of the above general formula, we will assume

that three sources of unwanted signal interference and noise are present at the receiver of a fourth link and that the wanted signal on the

fourth link will have separate time availabilities in the presence of

these three interfering signals given by P41 = 99%, P4?

= 98%, p =97%

and a time availability as regards noise given by p =98%. It follows

from (14) that the wanted signal on link 4, regardless of the correlation

of the fading of the unwanted signals, is expected to be satisfactory for

a percentage of time bounded by:

92%^p4

<97% (15)

The above bounds for the time availability in the presence of n

sources of interference should be quite useful in those situations,

which often occur in practice, for which it is not known whether the

fading of the unwanted signals is completely independent or is

positively or negatively correlated. If it is reasonable to assume that

the long term variations in the unwanted signals occur completely

independently, then specific estimates can be obtained for p

1 and p

n n

as follows. For independent events P(A and A and A ) = p' p

1 p1 l

j k ni nj nk

so that

S =)?' ., S =)p' .p» ., S =)p' .p» .p» , etc. (16)

1 Li ni a Li nir

nj 3 Li nir nj^ nk

When the above values are substituted in (12) we obtain the following

expression for p' which is applicable when the long term fading/inay

be considered to be independent for the n sources of interference.
-16-

P

' =yP

' . -Yp' .p* . + •••-... ± P

« p' ••• P

- (17) *n ^^ni /j ni nj nj na nn

As an example the above formula will be applied to the same special

case treated above. For this case p

1 = 0. 01, p' = 0. 02, p' = 0. 03

41 42 43 andp 1 =0.02; S =0.08, S =0.0023, S = 0. 000028 and

44 1 2 3 S =0.00000012, and thus

4

p

1 = 0.08 - 0.0023 + 0.000028 - 0.00000012 = 0.07772788.

n

Note, however, that the formula (17) for p

1 for independent fading can

n

also be written in the simpler form:

d-p' ) = d-p- )(i- P

» )..... d-p 1 ) (is)

n ni n2 nn

The above formula could have been derived more simply from the fact

that the probability (1-p 1

) of receiving a satisfactory signal in the

presence of n sources of interference is simply equal to the product

of the n probabilities of receiving a satisfactory signal in the presence

of each interfering source considered separately when these interfering

sources have independent probabilities of causing interference.

For the above example we find that the time availability in the case

of these four sources of independently fading interference is given by

P ~ 92. 22 7212%. It is interesting to note that this value of p for

independent fading lies quite near the lower bound given by (15). In

most applications, however, the fading of the n interfering sources

will tend to be positively correlated, rather than independent, and

under these circumstances p will be somewhat larger. Thus, for

n

most practical purposes, we may consider that the value of p

determined from (18) will be a more realistic lower bound for p than

n

the lower bound determined by (14).
-17-

In order to decide whether the fourth link should be added at a

particular geographical location it is desirable to determine not only

the value of p but also the values of the time availabilities p , p , and p of satisfactory reception on the other three links both with

3 and without the addition of the fourth link. An examination of these

seven values of time availability should provide an objective quantitative basis for deciding (a) whether to add the fourth link at the location

initially under consideration, (b) whether to choose some alternate

location or (c) whether to abandon the fourth link altogether.

The above development of criteria for the satisfactory use of two

or more communication links, although applied in the above example

to links in a radio relay system, is actually quite general and provides

a quantitative and objective basis for deciding on the desirability of

proposed assignment procedures for any kinds of service operating

anywhere in the radio spectrum. However, because of the somewhat

unrealistic assumption made earlier in this section, the analysis will

yield somewhat optimistic values for the time availabilities and thus

it may sometimes be desirable to use the substantially more complicated but more accurate method of Norton, Staras and Blum [1952].

3. THE PRINCIPLE OF A HORIZONTAL INCREASE IN POWER

We will now show that the increase in the powers of n communication links by the same number of decibels will improve the operation

of each of these n links by the reduction of the effects of noise but will

not, contrary to popular belief, increase their mutual interference at

all. This is true for every instant of time t as is evident from (3) and

(4) if we interpret the values of L..(t), L..(t), N.(t), R.. and R. . in

n ij l n ij

these equations as being instantaneous values. Thus, if P. and P.

1

J
-18-

are increased by the same number of decibels, the wanted signal-tonoise ratio P..(t) - N.(t) will increase on all n links i = 1 to n by this

11 1

J same number of decibels. However none of the wanted-to-unwanted

signal ratios P..(t) - P..(t) -< i=l to n and j=l ton ^ will change at all

since P. - P. will not change: this follows since P. and P. are both

1

J i J increased by the same amount for all combinations of i and j. If we

set S.. and S.. equal to zero in (7) and (8) we will find that a horizontal

increase in power will increase the time availability p.. but will not

cause any change in p... It follows from this and equations (12) or (18)

that the time availabilities p. (i=l to n) of satisfactory reception on the

la

n links after the horizontal increase in power will necessarily be larger

than the time availabilities p. (i=l to n) before the horizontal increase

in power. In the important special case of independent fading (18) may

be expressed:

p. = P- ..p.. P- /(100) n_1

. (19)

i 1I1Z x in

It follows from (19) that:

P. =p. -[p.. /p..} . (20)

la l i na nJ

Thus we conclude that the use of higher transmitter power on all

communication links will usually lead to a more efficient use of the

spectrum for any given assignment of stations. One usually minor

technical qualification must be made to this otherwise general rule.

The equations (3) and (4) are strictly applicable only to linear systems

and non-linearities in the system may, in fact, lead to more harmonic

radiation and to this extent, more mutual interference when higher

power is used.
-19-

A decision relative to the desirability for making a horizontal increase in the powers of several mutually interfering links may be reached simply by deciding whether the

larger time availabilities p. (i=l to n) represent a

13. sufficient improvement in performance to warrant the additional cost.

If the values of p. are determined largely by interference then p. will

be much less than p.. and there will be little advantage in increasing

11 the power. However, if p. is determined largely by noise, then p.

will be only slightly less than p.. and the outage time can be reduced

by a large factor by increasing the power. This point can be illustrated

by an example. Suppose p.= 90%, p.. = 99% and p.. =99.9%: such a

i n na

change in p.. would require a large increase in power but would only

increase p. from 90% to 90.82%, i.e. a reduction outage time from

10% to 9. 18%. On the other hand if p. = 98. 9%, p.. = 99% and p.. =99.9%,

i n na

then the same increase in power would result in an increase of p.

l from 98. 9% to 99. 8% and a reduction in outage time from 1. 1% to 0. 2%,

i.e. by a factor greater than 5.

In most cases it would appear to be better to increase the power

by different amounts on each link with the objective of making all

values of p. (i=l to n) the same and equal to a sufficiently large value

so that all links achieve their objectives. In this case all of the

probabilities p.. will change and the simple formula (20) is no longer

available

.
-20-

4. THE SERVICE PROBABILITY APPROACH

TO FACILITY ASSIGNMENT DECISIONS

Frequently it will be found that a single source of interference

will be of dominant importance. In such cases it is convenient to

use the concept of service probability for making a facility assignment

decision. This concept has already been described briefly in Section 2.

To use this concept a particular value is chosen for the desired time

availability p.. and S.. is then evaluated by (8). This value of S. . can

then be used, together with an estimate a" of its standard error, to

IJ determine Mq..)-S.. /<*"_, . This value may then be used in (9), ( 10) or

ij ij S.

ij

(11) to determine the service probability q.. that the signal received

th 1J on the i link will be of satisfactory fidelity or better for the desired

percentage of the hours p... Such probabilities have the advantage of

making appropriate quantitative allowance for the effects of any

errors in the prediction process and thus form a better basis for

making a decision about the desirability of some new assignment.

We may also use the following procedure to obtain an estimate of

the service probability q. of obtaining a satisfactory signal on the

th

lg

i link for a given percentage of time p. in the presence of n sources

of interference. We first set S.. = in (8) and determine the

ij

'expected" value of p. from:

i

p. = 100(l-p\) = 100(l-p». ) U-p. ) (I-P 1

. ) < 2i ) fJ r ! ^ ii 12 in
-21-

In the above the individual values of pf . =(100 -p. .) /100 are obtained from

the n solutions of (7) and (8) with j= 1 to n. If the "expected" value of

p. determined by (20) is either larger or smaller than the given value

p. , then it follows that q. will be either larger or smaller, respectig ig

ively, than 0.5. If p > p ,then (8) should be solved for the n values

i ig

of p obtained by setting all of the S.. equal to some positive value

ij ij

which is to be chosen so that the resulting value of p. as determined

i by (20) will be equal to p . The proper choice for S . would have to

ig ij

be made by a process of successive approximation. Using this value

of S.., together with the largest of the o\.(j=l to n), an estimate of

q can be obtained.

ig

If such estimates of q are obtained for the n links (i= 1 to n)

ig

and these values are all found to be greater than say 0.95, then the n

proposed facility assignments may be considered to be technically

feasible.

5. THE EFFICIENT ASSIGNMENT OF FACILITIES

TO BROADCASTING SERVICES

It was mentioned in the introduction that all kinds of communications services, including broadcasting, can be considered to consist

of a number of point-to-point communication links and thus all of the

analysis in the preceding sections is also applicable to broadcasting.

However, in the case of broadcasting, particularly in bands 8 and 9,

it is more convenient to describe the service in a somewhat different

way. Thus, suppose we are at a fixed distance d over land from a

100 Mc broadcasting station. Now consider all of the possible

receiving locations at this distance as we change through 360° the

azimuth of these locations relative to the broadcast transmitting
-22-

antenna location. For each azimuth there will be a different terrain

profile between the transmitting and receiving antennas and thus, for

a given time availability p.., a different value of S.. , may be obtained

11 nd

from (7) for each azimuth. Now S.. , for the given distance d may be nd J considered to be a random variable which is approximately normally

distributed with a standard deviation cr . By setting b

(S.. , /cr ) =Mq.. -,) w® may determine the probability q . , that a randomly nd b lid nd J chosen receiving location at the distance d will be provided with

satisfactory service for at least p . per cent of the time. The

n

magnitude of cr will vary with the nature of the profiles between the

transmitting and receiving antennas, with the radio frequency, and,

to some extent, with the receiving antenna height.

As the distance is increased for a given transmitter power,

frequency and antenna heights, the percentage of receiving locations,

100 q, receiving satisfactory service will decrease from a value

near 100% at short distances to a value near zero at very large

distances.

The prediction of q.. , requires a prediction of both cr, and the nd -ix k

expected transmission loss as a function of distance for a given

frequency and antenna height. Note that cr will be systematically

larger than cr.. since cr, includes an allowance for the variance of the B

ii b

transmission loss at a fixed distance for the varying angular distances

and other terrain profile features at the various azimuth angles.

The above analysis applies only to a clear channel broadcasting

problem and the problem of mutual interference between two or more

stations is substantially more complicated. As an introduction to this

problem we will consider a typical pair of co-channel VHF television

broadcasting stations. In the example given, the stations have a separation of 220 miles; this separation is typical since about half of
-23-

the separations between neighboring co-channel stations in the United

States are greater than 220 miles. On figure 1 the two irregular dotted

curves show the decreasing field intensity of station A when using either

100 KW or 1000 KW of radiated power, and the two irregular solid

curves show the decreasing field intensity of station B when using

either 100 KW or 1000 KW of radiated power. The fields are shown

along a line joining the two stations. Note that, although the fields

decrease irregularly with distance because of the influence of terrain

irregularities, at every location the fields are exactly 10 db stronger

when the radiated power is 1000 KW rather than 100 KW. The fields

shown are typical of a particular time of day and, in practice, the

fields are known to vary from minute -to -minute and from hour-to-hour

somewhat more at the larger than at the smaller distances; however,

for any particular time, the 1000-KW fields will always be exactly

10 db stronger than the 100-KW fields.

There are two conditions which must be satisfied by the fields at

a particular receiving location in order to provide a satisfactory

picture to a television receiving installation: (1) the field of the

desired station must be sufficiently strong so that it can overcome the

noise level in the receiver, i.e., eliminate the obscuring effects of

the video noise which appears on the face of the tube and has the

appearance of snow, and (2) the differences in the fields from the

desired and any undesired stations must be sufficiently great so that

these undesired stations do not produce objectionable bars or other

effects on the picture tube which interfere with the proper reception

of the desired station. Both of these requirements on the desired and

the undesired fields depend upon the excellence of the receiving

installation. For example, when an indoor antenna and an inexpensive

receiver are used, the field required to override the noise may be

about 50 db above one microvolt per meter, and the desired field must
24

FIELDS EXPECTED ALONG A LINE BETWEEN TWO TYPICAL

VHF (100 MC) TELEVISION BROADCASTING STATIONS

ANTENNA HEIGHTS: 1000 FEET AND 30 FEET

IdIUJ

uj

Q.o

>

o

<ro

Ld>O

CD<

CO

GOO

UJQ

CO

-z. UJ

UJ

u.

220 200 180

120 140 DISTANCE IN MILES FROM STATION A

160 140 120 100 80 60 DISTANCE IN MILES FROM STATION B40 20

Figure 1
-25-

be 25 db* greater than the field from an undesired co-channel offset

carrier in order to override the interference. We will call such an

installation a grade-B receiving installation. On the other hand, for a grade-A receiving installation where a typical outdoor receiving

antenna is used in conjunction with a fairly good (but not necessarily

very expensive) receiver, the field required to override the noise

will be about 40 db above one microvolt per meter, and the desired

field need be only 15 db greater than the field from an undesired

co-channel offset carrier in order to override the interference. The

typical antenna characteristics used in arriving at the above figures

were obtained from a private communication by A. C. Wilson. A 6-db

gain and a 10-db front-to-back ratio were assumed; some of the

antennas tested indicated more than 12 db gain and more than 20 db

front-to-back ratio, so that the values assumed for our grade-A

installation are far from being the best available.

The following table shows the minimum and maximum distances

at which the fields satisfy the above two conditions for 100 KW and

1000 KW radiated power, and for grade-A and grade-B receiving

installations. The minimum distances represent the nearest locations, as one leaves the transmitter, at which the fields first fail to satisfy

one or the other or both of the above conditions, and the maximum

distances represent the most remote locations from the transmitter

at which the fields still satisfy those conditions. It is clear that the

"effective" range of the station lies somewhere between the minimum

* At this field intensity difference and with a zero front-to-back ratio

(typical of an indoor antenna), half of the TASO observers rated the

received picture to be somewhere between fine and excellent; see

Fig. 41 of the TASO report [1959].
-26-

and maximum distances, at which both conditions are satisfied.

Appropriate statistical methods have been developed to give a more

precise meaning to the term "effective "range, but we need not go into

that much detail in order to illustrate the technical principle involved

in a horizontal increase in power.

Table I Service Ranges Along the Line Between

the Two Stations of the Accompanying Figure

100 KW 1000 KW

Minimum Maximum Minimum Maximum

Distance Distance Distance Distance

Miles Miles Miles Miles

53 71 78 86

73 84 73 84

53 71 73 84

66 86 88 97

87 87 87 87

66 86 87 87

Grade B - Noise Condition (1)

Interference Condition (2)

Both Conditions Satisfied

Grade A - Noise Condition (1)

Interference Condition (2)

Both Conditions Satisfied

It is easy to verify the distances in the above table by reference to

figure i. For example, the minimum distance at which the noise

condition is satisfied for a grade-B receiving installation when 100 KW

is used is 53 miles; this is the first distance at which the 100-KW field

falls below 50 db above one microvolt per meter. As another example,

73 miles is the shortest distance at which the difference between the

two 100-KW fields is less than 25 db.

A large number of general conclusions can be reached by an

examination of the figures in the above table:
-27-

(1) In the first place it is most important to notice that the distance

at which the interference condition is satisfied does not depend upon the

power level, assuming that the power is changed by the same factor,

i.e., by the same number of decibels, for all of the interfering stations.

(Z) When higher power is used at all stations, the effective

service ranges are increased and the fields at all receiving locations

are improved.

(3) It is possible to increase the power to the point where

interference, rather than noise, limits the reception. For the

example given, the noise condition limits the effective range when

100 KW is used for both grade-A and grade-B receiving installations,

while the interference condition limits the effective range when 1000 KW

is used for both grade-A and grade-B receiving installations.

(4) The increase in power is more important for the less

expensive grade-B receiving installation. For any broadcasting

service, it is clearly more economical to use an expensive, highpower transmitting installation than many thousands of expensive

receiving installations.

The above conclusions are generally applicable to all broadcasting

services. Thus it appears that broadcasting stations should always

be encouraged to use the maximum power which is consistent with

economic feasibility up to the point at which interference, rather than

noise, limits the service. In fact one can only claim to have made

really efficient use of the spectrum for a broadcasting service when

sufficient power is used for a given geographical configuration of

stations so that it is not possible to find a single receiving location

at which noise, rather than station interference, limits the reception

for any percentage of the time. For typical VHF television broadcasting stations separated by 220 miles, we see by the above example that

the technically desirable power is of the order of 1000 KW.
-28-

The above example illustrates some aspects of the nature of a broadcasting service. With a fixed separation between stations it appears that efficient use of the spectrum can be obtained only when

sufficient power is used so that interference, rather than noise,

provides the ineluctable limit to satisfactory reception. On the other

hand, if the power is fixed, it has been shown [Norton and Fine 1949,

Reference E and Norton 1950, Addendum to Reference E] that there is

an optimum separation between the stations for efficient use of the

spectrum. Recently Decker [1959 and 1962] has applied these concepts

to determine an efficient allocation scheme for an airborne television

network in band 9.

6. SOURCES OF INFORMATION ON THE PREDICTION OF

TRANSMISSION LOSS AND ITS VARIABILITY

A. Tropospheric Propagation

Air Force Technical Order, Ground Telecommunications Performance Standards, Part 5 of 6, Tropospheric Systems prepared

by the Central Radio Propagation Laboratory of the National

Bureau of Standards under sponsorship of the Ground

Electronics Engineering and Installation Agency (Directorate

of Engineering, ROZM), United States Air Force, T. O.

31-Z-10-1, published under authority of the Secretary of the

Air Force (June 15, 1961).

Barghausen, A. F. , and C. F. Peterson, Path loss measurements

versus predictions for long distance tropospheric scatter

circuits, IRE Transactions on Communications Systems,

PGCS, 9, No. 4, 439-445 (December 1961).
-29-

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

Performance predictions for single tropospheric communication links and for several links in tandem, NBS Technical

Note 102 (PB161603) (August 1961). $3.00*

Bullington, K. , Characteristics of beyond-the-horizon radio

transmission, Proc. IRE 43, No. 10, p. 1175 (October 1955).

Dickson, F. H.,J. J. Egli , J. W. Herbstreit, and G. S. Wickizer,

Large reductions of VHF transmission loss and fading by the

presence of a mountain obstacle in beyond- line -of- sight paths,

Proc. IRE 4.1, p. 967 (1953). See also subsequent correspondence by Crysdale and rebuttal by Dickson, et al, in Proc.

IRE 43, p. 627 (1955).

Dutton, E. J., and G. D. Thayer, Techniques for computing

refraction of radio waves in the troposphere, NBS Technical

Note 97 (PB 161598) (October 17, 1961). $1.50*

Dutton, E. J., On the climatology of ground-based radio ducts and

associated fading regions, NBS Technical Note 96 (PB 161597)

(June 15, 1961). $1.75*

Florman, E. F. , and J. J. Tary, Required signal-to-noise ratios,

RF signal power, and bandwidth to achieve a given performance

for multichannel radio communications systems. To be

published as NBS Technical Note 100 (1962).

Hartman, W. J., and R. E. Wilkerson, Path antenna gain in an

exponential atmosphere, NBS Journal of Research 63D, p. 273

(1959).

Herbstreit, J. W. , and P. L. Rice, Survey of Central Radio

Propagation Laboratory research in tropospheric propagation

1948-1956, NBS Technical Note 26 (PB 151386) (January 1957),

$4.00*

Joint Technical Advisory Committee of IRE-EIA, Frequency

allocations for space communications (March 1961).

*Order by PB number from the Office of Technical Services, U. S.Dept.

of Commerce, Washington, D.C. Foreign remittances must be in U.S.

exchange and must include one-fourth of the publication price to cover

mailing costs.
-30-

Kirby, R. S. , J. W. Herbstreit, and K. A. Norton, Service range

for air-to-ground and air-to-air communications at

frequencies above 50 Mc, Proc. IRE 40, p. 525 (1952).

Kirby, R. S. , H. T. Dougherty, and P. L. McQuate, Obstacle gain

measurements over Pikes Peak at 60 to 1046 Mc, Proc. IRE 43,

No. 10, p. 1467 (October 1955).

Kirby, R. S. , Measurement of service area for television broadcasting, IRE Transactions on Broadcast Transmission Systems

BTS-7, p. 23 (1957).

Kirby, R.S., P. L. Rice and L. J. Maloney, Characteristics of

point-to-point tropospheric propagation and siting considerations, NBS Technical Note 95 (PB 161596) (October 19, 1961).

$2.50*

Kurihara, Y. , Trans-horizon microwave propagation over hilly

terrain, Proc. IRE 43, No. 10, p. 1 362 (October 1955)

Mellen, G. L. , W. E. Morrow, A. J. Pote, W. H. Radford, and

J. B. Wiesner, UHF long-range communication systems,

Proc. IRE 43 , No. 10, p. 1269 (October 1955).

Muchmore, R. B. and A. D. Wheelon, Line-of- sight propagation

phenomena, -- I. Ray treatment, -- II. Scattered Components,

Proc. IRE 43, No. 10, p. 1437 (October 1955).

Norton, K. A., and A. C. Omberg, The maximum range of a radar

set, Proc. IRE 35_, p. 4(1947).

Norton, K. A., and P. L. Rice, Gapless coverage in air-to-ground

communications at frequencies above 50 Mc, Proc. IRE 40,

p. 470 (1952).

Norton, K. A., P. L. Rice, H. B. Janes, and A. P. Barsis, The

rate of fading in propagation through a turbulent atmosphere,

Proc. IRE 43, No. 10, p. 1341 (October 1955).

*Order by PB number from the Office of Technical Services, U.S.Dept,

of Commerce, Washington, D.C. Foreign remittances must be in U.S.

exchange and must include one-fourth of the publication price to cover

mailing costs.
-31-

Norton, K. A., P. L. Rice, and L. E. Vogler, The use of angular

distance in estimating transmission loss and fading range for

propagation through a turbulent atmosphere over irregular

terrain, Proc. IRE 43, No. 10, p. 1488 (October 1955).

Norton, K. A., L. E. Vogler, W. V. Mansfield, and P . J. Short,

The probability distribution of the amplitude of a constant

vector plus a Rayleigh-distributed vector, Proc. IRE 43,

No. 10, p. 1354 (October 1955).

Norton, K. A. , Point-to-point radio relaying via the scatter mode

of tropospheric propagation, IRE Transactions on Communication Systems CS-4, No. 1, p. 39 (1956).

Norton, K.A., Transmission loss in radio propagation - II, NBS

Technical Note 12 (PB 151371) (June 1959) $3.00*

Rice, P. L. , A. G. Longley, and K. A. Norton, Prediction of the

cumulative distribution of ground wave and tropospheric wave

transmission loss (Part I), NBS Technical Note 15 (July 1959).

Staras, H. , Antenna-to-medium coupling loss, IRE Transactions

on Antennas and Propagation, AP-5, p. 228 (1957).

Trolese, L. G. , Characteristics of tropospheric scattered fields,

Proc. IRE 43, No. 10, p. 1300 (October 1955).

B. Ionospheric Propagation

Basic Radio Propagation Predictions, CRPL D Series, issued

monthly by the National Bureau of Standards.

Ionospheric Radio Propagation, U. S. Department of Commerce,

National Bureau of Standards Circular 462, (U. S. Government

Printing Office, June 25, 1948).

Jones, W. B., and R. M. Gallet, Ionospheric mapping by

numerical methods, Telecommunication Journal (December

I960).

*Order by PB number from the Office of Technical Services, U.S.Dept.

of Commerce, Washington, D.C. Foreign remittances must be in U.S.

exchange and must include one-fourth of the publication price to cover

mailing costs.
-32-

Kift, F. , The propagation of high frequency radio waves to

long distances, Proc. I.E.E. 107 , Pt. B, No. 32, 127-140

(March i960).

Laitinen, P. O. , and G. W. Haydon, Analysis and prediction of

sky-wave field intensities in the high-frequency band, Technical Report No. 9, RPU 203, U. S. Army Signal Radio

Propagation Agency, Ft. Monmouth, New Jersey ( 1950)

Piggott, W. R. , The calculation of the median sky wave field

strength in tropical regions, Special Report No. 27, Department of Scientific and Industrial Research, Slough, England

(1959).

Rawer, K. , Calculation of sky-wave field strength, Wireless

Engineer, 287-301 (November 1952).

Zacharisen, D. H. , World Maps of F2 critical frequencies and

maximum usable frequency factors, NBS Technical Note 2

(1959 and I960).

C. Ground- Wave Propagation.

Norton, K. A., The calculation of ground-wave field intensity over

a finitely conducting spherical earth, Proc. IRE 29, p. 623

(1941).

Norton, K. A. , Low and medium radio propagation from the book,

Electromagnetic Wave Propagation, edited by M. Desirant

and J. L. Michiels, Academic Press (London and New York

1960).

Rice, S. O. , Diffraction of plane radio waves by a parabolic

cylinder, Bell System Techn. Journal 33, p. 417 (1954).

Vogler, L. E. , Smooth earth diffraction calculations for horizontal

polarization, NBS Journal of Research 65D, No. 4, p. 397

(1961).
-33-

Wait, J. R. , and H. H. Howe, Amplitude and phase curves for

ground- wave propagation in the band ZOO cycles per second to

500 kilocycles, NBS Circular 574 (May 21, 1956). Available

through Superintendent of Documents, U. S. Government

Printing Office, Washington 25, D.C., $.20.

Wait, J. R. , and A. M. Conda, Diffraction of electromagnetic

waves by smooth obstacles for grazing angles, NBS Journal

of Research 63D, p. 181 (1959).

Wait, J. R. , The electromagnetic fields of a horizontal dipole

in the presence of a conducting half- space, Canadian Journal

of Physics 39 (July 1961).

Wait, J. R. , The propagation of electromagnetic waves along the

earth's surface, Diffraction Theory edited by R. E. Langer,

(University of Wisconsin Press, Madison, Wisconsin, 1961).

D. Radio Noise

Cottony, H. V., and J. R. Johler, Cosmic radio noise intensities

in the VHF band, Proc. IRE 40, 1053-1060 (September 1952).

Crichlow, W. Q. , R. T. Disney, and M. A. Jenkins, Quarterly

radio noise data, NBS Technical Note 18 (PB 151377)

(published quarterly).

Crichlow, W. Q. , D. F. Smith, R. N. Morton, and W. R. Corliss,

Worldwide radio noise levels expected in the frequency band

10 kilocycles to 100 megacycles, NBS Circular 557 (August 25,

1955).

Crichlow, W. Q. , Noise investigation of VLF by the National

Bureau of Standards, Proc. IRE 45, No. 6, p. 778(1957).

Crichlow, W. Q. , J. J. Roubique, A.D. Spaulding, and W. M.

Beery, Determination of the amplitude -probability distribution

of atmospheric radio noise from statistical moments, NBS

Journal of Research 64D, No. 1, p. 49(1960).
-34-

Dinger, J. E. , Report on URSI Commission IV - Radio noise of

terrestrial origin, Proc. IRE 46, No. 7, p. 1366(1958).

Fulton, F. F. , Jr. , The effect of receiver bandwidth on amplitude

distribution of VLF atmospheric noise, National Bureau of

Standards, VLF symposium paper 37, Boulder, Colorado (1957).

Menzel, D. H. , Cosmic Noise Survey. (A set of cosmic noise

grids and overlays prepared in part under DOD contract

DA 49-170-sc-2386).

Obayashi, T. , Measured frequency spectra of very-low-frequency

atmospherics, NBS Journal of Research 64D, No. 1, p. 41

(1960).

Report on revision of atmospheric radio noise data, CCIR Report

No. 65, VHIth Plenary Assembly, Warsaw, 1956 (International

Radio Consultative Committee, Secretariat, Geneva, Switzerland) .

Spaulding, A. D. , C. J. Roubique, and W. Q. Crichlow, Bandwidth

conversion of the amplitude - probability distribution function

and its first two moments for atmospheric radio noise, to be

published in NBS Journal of Research Part D, Radio Propagation.

Taylor, W. L. , and A. G. Jean, Very-low-frequency radiation

spectra of lightning discharges, NBS Journal of Research 63D,

No. 2, p. 199 (1959).

Watt, A. D. , and E. L. Maxwell, Measured statistical characteristics of VLF atmospheric radio noise, Proc. IRE 45, No. 1,

p. 55 (1957).

Watt, A. D. , and E. L. Maxwell, Characteristics of atmospheric

noise from 1 to 100 kc, Proc. IRE 45, No. 6, p. 787 (1957).

Watt, A. D. , R. M. Coon, E. L. Maxwell, and R. W. Plush,

Performance of some radio systems in the presence of

thermal and atmospheric noise, Proc. IRE 46, No. 12,

p. 1914 (1958).
-35-

REFERENCES

C.C.I. R. , Documents of the IXth Plenary Assembly I, 128,

Los Angeles (1959).

Decker, M. T. , Service area of an airborne television station,

NBS Technical Note 35 (PB 151394) (October 1959). $.75*

Decker, M. T. , Airborne television coverage in the presence of

co-channel interference, NBS Technical Note 134 (PB 161635)

(January 1962). $2.00*

Feller, W. , An introduction to probability theory and its applications,

(John Wiley and Sons, Inc. , New York, N. Y. , Second

Edition 1957).

I.T.U. 1959, Final Acts of the International Telecommunication and

Radio Conferences, Geneva (1959).

Norton, K. A. , and H. Fine, A study of methods for the efficient

allocation of radio frequencies to broadcasting services

operating in the range above 50 Mc. , Reference "E" to the

report of the ad hoc committee of the FCC for the evaluation

of the radio propagation factors concerning the television and

frequency modulation broadcasting services in the frequency

range between 50 and 250 Mc (August 1, 1949). See also

Addendum to Reference "E" by K. A. Norton (November 16,

1950).

Norton, K. A. , H. Staras and M. Blum, A statistical approach to the

problem of multiple radio interference to FM and television

service, IRE Transactions on Antennas and Propagation,

PGAP-1, 43-49 (February 1952).

Norton, K. A., Transmission loss in radio propagation, Proc. IRE 41,

p. 146 (1953).

*Order by PB number from the Office of Technical Services, U.S. Dept.

of Commerce, Washington, D.C. Foreign remittances must be in U.S.

exchange and must include one-fourth of the publication price to cover

mailing costs.
-36 = Norton, K. A., System loss in radio wave propagat ion, NBS Journal

of Research 63D, No. 1, p. 53(1959).

Rice, P. L. , A. G. Longley and K. A. Norton, Tropospheric Wave

Transmission Loss Prediction, C.C.I.R. Study Group V

Document V123-E (March 1, 1962). To be published as NBS Technical Note 101.

Rules and Regulations of the Federal Communications Commission,

Frequency Allocations and Radio Treaty Matters, General

Rules and Regulations, Pike and Fischer Radio Regulation,

Vol. I, Par 52, (Pike and Fischer, Inc., Washington, D.C.)
1-1

APPENDIX I

ON FIGURES OF MERIT FOR RADIO RECEIVING

SYSTEMS IN THE PRESENCE OF NOISE OR UNWANTED SIGNAL

INTERFERENCE AND THE CONCEPTS OF OPERATING

NOISE FACTOR AND OPERATING NOISE TEMPERATURE

1. Introduction

The optimum use of the radio spectrum requires that radio

receiving systems be so designed that the reception of the wanted signals is immune to the greatest degree possible from interference byunwanted radio signals occupying the same or other radio frequency

channels. This report proposes as figures of merit for a receiving

system the protection ratios r (between the hourly median wanted

u

and the hourly median unwanted signal powers available from a lossfree receiving antenna, which is otherwise equivalent to the actual

receiving antenna) which are required for the satisfactory reception of

the information carried by a specified wanted signal in the presence of

specified kinds of unwanted signals. The use of receiving systems

having the smallest values of ru

will permit the same portions of the

spectrum to be used simultaneously by the maximum number of users.

On the other hand, economic considerations, as contrasted to

spectrum conservation considerations, indicate that radio receiving

systems should be designed so that the minimum possible transmitter
1-2

power is required for satisfactory reception of the wanted signals in the

presence of noise. This report proposes as an additional figure of

merit for a receiving system the hourly median wanted signal power

p (available at the terminals of the equivalent loss -free receiving

antenna) which is required for the satisfactory reception of the wanted

signals in the presence of noise, but in the absence of any other unwanted

signals.

The figure of merit ruinvolves only the ratio between the

hourly median values of the wanted and unwanted signal powers, while

p is a measure of the required magnitude of the hourly median wanted

signal power. For optimum use of the spectrum by the maximum number

of simultaneous users, the transmitting and receiving systems of the

individual links should be designed with the primary objective of

minimizing the various values of r„ involved; and then sufficiently high

transmitter powers should be used so that the hourly median wanted

signal power p m exceeds p for a sufficiently large percentage of the

hours during the intended period of reception at each receiving location.

This approach to frequency assignment problems will be unrealistic in a few cases, such as the cleared channels required for radio astronomy,

but these rare exceptions merely serve to prove the otherwise general

rule that optimum use of the spectrum can be achieved only when interference from other signals rather than noise provides the ineluctable

limit to satisfactory reception.

2. Definitions of "Wanted and Unwanted Available Signal Powers

All signal and noise powers will be expressed in watts. Since

it is confusing to use the decibel scale when noise powers are added, the

convention will be adopted of using lower case letters to denote power

in watts, or power ratios, and capital letters will be used to denote
1-3

their decibel equivalents. For example, p will denote the wanted signal power available at the terminals of the equivalent loss -free receiving

antenna expressed in watts, and P = 10 log,

Q p dbw.

Let p = (dp /dv) denote the wanted signal power density in

av a watts per cycle per second at the radio frequency v which is available

at the terminals of the actual lossy receiving antenna, and p is defined

by:

p=\i pdv=i p . (1) J rev rav re a

o

where i is a factor greater than one which provides a measure of

rev s *

the antenna circuit loss at the frequency v while p is the wanted signal

cL power available at the terminals of the actual lossy receiving antenna.

Similarly, P » P and p denote corresponding quantities for the

u uav rua

unwanted signal. The available power from a generator, in this case

the receiving antenna, is the power which would be available at the

output of the generator if the output circuit were conjugately 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 simply by s = v /(4 R ) for a generator with

impedance Z = R + j X . Note that the available power from the

g g g

generator has the desirable property of being independent of the load

impedance, and this desirable property makes this concept especially

useful for the definition and measurement of noise factors.

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 power s g

F

I delivered to the load is given by:

s |Z + Z J

2 -*=-! Z— S JL 2:1. (2)

S£

4 R

g

R

i -
1-4

This mismatch loss factor JL will be equal to 1, and the delivered

power s will be equal to the available power s when the load and

generator impedances are conjugately matched, i. e. , when R„ = R and X = -X . 1 § * g

The necessity for referring the wanted signal power to the

terminals of an equivalent loss-free receiving antenna rather than to

the actual antenna terminals or to the receiver input terminals will be

established later but the use of this reference point clearly has the

advantage of making the wanted signal powers independent of

antenna circuit losses and this provides an added reason for defining

the wanted signal powers in this way.

3. The Operating Noise Factor of a Receiving System

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 factor which characterizes

only the performance of the receiver itself. Later, Norton [1953 and

1961] gave slightly more general expressions for this factor and

designated it as an effective noise figure. This generalized operating

noise factor 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 purpose of this section is to

give this more general formulation for an operating noise factor f of a receiving system, to describe its general properties, and to show

how the operating noise temperature T of the receiving system may

be determined from f. In contrast to the approach to unity of the noise

factor f of an essentially noise-free network, the operating noise

factor f of an essentially noise-free receiving system approaches zero.
1-5

Essentially all of the basic concepts used in this discussion

originated in the early papers by Burgess [1941], North [194Z] and

Friis [1944] and in the discussion by North [1945] of the paper by Friis.

A precise definition of the noise factor f of a four-terminal network

n

was given by Friis [1944], and his definition forms the basis for the

following formulation of the operating noise factor of a receiving

system. Friis designated his noise factor f by the term noise figure,

but North's terminology seems preferable since f is simply a dimensionless factor.

The available gain, g , of a four-terminal network for a c. w. & &nv

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.

g = s /s (s /s >1). (3) °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) . (4)

nv g o go

Note that g and I inherently contain a mismatch factor i determined

nv nv 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 its effective signal gain or loss

will depend upon the spectral distribution of the wanted signal within

the pass-band. Two examples will be given to demonstrate the nature

of the effective signal gain, g . Consider first a single side band suppressed carrier signal

with input available power spectral density s . In this case the

g

v effective signal gain for the four terminal network is given by:
1-6

OO r

\ s g dv

J gv °nv

o

. (5) OO

o

s dv

g

v Note that g will be less than or equal to the maximum gain, g , within

the pass band. In particular, if the receiver pass band is sufficiently

detuned relative to the input signal the effective signal gain may even

decrease to a value less than unity and its reciprocal would, in such a case, be considered to be the effective signal loss, I . Consider next a carrier, amplitude modulated with a single tone, so that the signal power is all contained in the two side bands with radio

frequencies v and v and input r.m. s. voltages v and y which will a b a b

be assumed to be in phase. Since the side band voltages will add

coherently after detection the following general expression may be

obtained for the effective signal power gain in this case:

v Vg + v, \Tg " )

. a

5nva b6 nvb /

For the usual case of equal sideband input voltages (v = v ) the above

becomes :

gs

= 0.25 g + 0. 25 g , + 0. 5 \Tg g , s 6 nva 5 nvb 6nva 5nvb .

If the network is tuned so that g = g , , then g = g and this &nva 6nvb 6

s 6nva

will usually be somewhat less than the maximum gain, g , within the

pass band.

The effective signal gain for other types of modulation may be

handled in a similar manner and the desirability for having a definition

of g will be discussed later. However it is more convenient to define

s
1-7

the noise factor and the effective noise bandwidth simply in terms of

the maximum value, g , of g within the pass band or the minimum

n nv

value, £ . of £ within the pass band and then to allow separatelv for

n nv r f i the effect on the required signal power, P , of the difference between

n

g and g or the difference between £ and £ . 6

s &n s n

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 »

b=-z— \ g dv = -tr—- \ g dv . (7)

gn J

Snv Eg J

gnv

v o

When we later consider several networks in tandem, b is to be

determined using g and g as determined from the values of s and & &n

6nv g

s available at the input and output, respectively, of this chain of o networks:

co

gnv

dV

h = ^

• (8)

2

\ g dv

V

l

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

wanted 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 g within this band. The factor h in (7) and (8) is a measure

nv

of any spurious responses which may be present in the receiver. The

following discussion refers particularly to superheterodyne receivers

and might have to be modified for other types of receivers. Note that
1-8

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 frequency v within the principal response

band v to v with the oscillator frequency v of the receiver to produce

a signal with intermediate frequency v. = ± (v - v ). Additional

l d o undesired voltages may arise in the intermediate frequency output by

virtue of beats between the m harmonic of the oscillator frequency

th

and the n harmonic of a spurious signal or noise with radio frequency

, , , , , th . , , v not in the band v . to v, to produce the w subharmonic of the

u 1 2 r 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 - m v ) = v. / w (9)

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 Jv - v | to

v o ~ v J the most important spurious response usually corresponds

2 o

to n = m = w = 1, and the spurious frequencies v for this response

differ by 2 v. 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.
I - 9

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 (7). Receivers having spurious responses will have noise

factors based on the definition of b given by (7) which are larger by

the factor h than the noise factors of corresponding receivers not

having~such responses. Consequently receiving systems having small

values of H = 10 log._h will be superior to otherwise equivalent systems

having larger values of H. H is a numerical measure of the response

of a receiving system to spurious signals or noise which may be

present outside of its principal response band v to v , and thus

provides a conditional figure of merit for a receiving system.

The noise factor, f , of a linear four-terminal network was

n

defined by Friis [1944] as the ratio of the available c. w. signal-to-noise

power ratio at the signal generator terminals to the available c.w.

signal-to-noise power ratio, [s /n ], at its output terminals, with the

available noise at the signal generator terminals set equal to the

reference available Johnson noise power, k T b, from a resistance at o the absolute reference temperature T and with the c.w. signal tuned

to the maximum response of the network band-pass characteristic:

[s /(kT b)]

f = ^—

n [s /n ] o o

o (Friis' Original Definition^ (10)

In the above, k is Boltzman's constant. When £ or g is substituted

n n

for the ratio of s goand s in (10), the following& alternative definitions

are obtained for the noise factor of a network in terms of the available

output noise power, n , and loss factor SL or gain g : o n n

fnono =n f /(kT b) = nono /(g kT b) (Equivalent Definitions). (11)
I - 10

Note that the noise factor f depends upon the generator

n

impedance as well as upon the characteristics of the network itself, since g depends upon the generator impedance. Thus one cannot

usefully describe the noise performance of a network in terms of its

noise factor alone without also specifying the impedance of the

generator used in determining this noise factor. On the other hand the

operating noise factor does provide a complete description of the noise

performance of a receiving system since the generator impedance is

the receiving antenna impedance in this case and this is an integral

part of the receiving system.

The above definitions will now be used to derive an expression

for the noise factor of the simple network of figure 1 with loss £ caused

by its resistance, R , at an absolute temperature T . Let the resistance,

1 n n. R , of the signal generator be at the reference temperature T . The

g ° available signal power from the signal generator at the network input

terminals is given in terms of its open circuit voltage, v, by:

s = v

Z /(4R ) (12)

g g

and the available signal power at the network output terminals is given

by:

s = v

2

/[4(R + R )] . (13) o g n

Thus, from (6):

i = s /s = (R + R )/R (14) n g o g n g

is obtained.

The available noise power at the output of the network of figure 1

is given by the weighted average of the Johnson noises from the

resistances R and R at temperatures T and T : p n on
1-11

Rg»T ^ri'Tn

AAAAA/ O VWW-

<5>

INPUT — OUTPUTfn

= l+(/n -l)(T n /T ) ^T^n + Rg) /Rg

Figure I
I- 12

, R T + R T (go n n ; -uu "o

= kb { R +R }^- {T

c +Tn Wn1,) - (15)

u g

T n

J n

When this is substituted in (11), the result is:

f = 1 + (I - 1)(T /T ). (16) n n n o Note that the noise factor of a passive network at the reference temperature T is simply equal to its loss factor, i.e., when T = T , it o no

follows from (16) that f = I . n n

Next, (16) will be derived in another way, and it will be shown

that it is applicable in general to any lossy passive network with loss

t and temperature T , i.e., having arbitrary input and output

n n

impedances. Note that the noise output, n , of a linear passive

network can be expressed as the sum of the two terms:

kb(T - T ) o n

n = kbT + 1

. (17) ° n

n

The first term would represent the available Johnson noise power from

the network if its source resistance were also at the temperature T , n

while the second term represents a correction arising from the fact

that the temperature T of the source resistance is different from that o

of the network, either higher or lower. For example, suppose that

Ton> T ; in this case the second term in (17) is the excess noise power

kb(Ton- T ) available at the input reduced by the factor I in passing

n

through the network. When n , as given by (17), is substituted in the

o noise factor definition (11), (16) is obtained as before.

Using the above definitions and conventions, the operating noise

factor f of the linear portion of the receiving system illustrated on

Fig. 2 may now be discussed. Let p be the available signal power from

an equivalent loss-free receiving antenna, let p be the available signal

a
1-13 >

LU

h- >- < CJ> Q 2

LU LU

0- UJ

LU a;

q:

LU I13 6

a:>

UJo

LU

(T

Q

O

CO

^2 uj

co _j <

ui o

z: ui uj

ui -J co H 2 o

CO t t s ^ ^

-Q oco o

s° I

UI 1 CO

>- o -—

CO ""^- ^< I + >

i^* \ O UI

-4-3 .»_»

U

k_ + o

N" h- S"

H UI

or + T

i-±~ < C\J

ii .+_

— u.

-4-1 H—

w. 3 i O 0)3 o ^ :z D>

i^ + + o iZ \ o 1-° o

h^

H CL \ —

^^

|-° 1 cr o o 'o o <

S" 1- ^ ^

UJ2 +

II i o+ o

II +

JZ

II h-°

M—

III !-*

_i

UIX

1—
1-14

power from the actual receiving antenna, i.e., the available signal

power at the antenna circuit output terminals, and let i be the loss

re factor of the receiving antenna circuit; thus p = I p . Let T denote

re a c the effective absolute temperature of the antenna circuit resistance

exclusive of its radiation resistance, and (16) may be used to determine

the noise factor of this portion of the antenna circuit network:

f = 1 + (l - 1) (T /T ). (18)

c re c o

Similarly the noise factor of the transmission line network with

absolute temperature T and the line loss factor Jt is given by:

f = 1 + (I - i)(T /T ). (19)

t rt t o The noise factor of the receiver itself considered as a four-terminal

network will be designated f . 5

r Friis [1944] gives an expression for the noise factor of three

networks in tandem. This will be used to determine the noise factor

of the three-network system (c), (t), and (r) of figure 2:

f = f + I (f - 1) + I i (f - 1). (20)

ctr c re t re rt r

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 bv; kT dv where T is the receiving antenna radiation

av av to resistance noise temperature at the frequency v. The concept and

method of calculation of an effective temperature T of the receiving

antenna have been described by Slater [1942] and by Lawson and

Uhlenbeck [1948]. Representative values of T are given by Crichlow

[1955] and by the C.C.I. R. [1956 -and 1959] for frequencies v < 10 ,and

O 4 Q

are given by Blake [1961] for frequencies within the range 10 < v < 10
I - 15

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. and in papers by Hansen [1959] and Hogg and Mumford [I960],

Now the available noise power n at the predetection 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:

n =f kbT g + k \ (T -T)g dv . (21) o ctr o

6 ctr J av o6 ctrv

o

In practice, c g may sometimes best be determined from its to ctrv

components using the relation g * s s = g / (£ i ). The first term

ctrv 5 rv rev rtv

in (21) would represent the available noise power at the output of the

receiving5 system y if T = T for all values of v, while the second term

av o represents F a correction arising& from the difference between T and T , av o Since the integral in (21) has the limits to °°, 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 T g dv \ T g dv av°ctrv J av to ctrv

o o

= —Ebi

• (22)

5 ctr

a I

g& dv

ctrv

Using this value of T , (21) may be expressed:

el

n = f kbT g + khb(T - T )g . (23) o ctr o

s ctr a o& ctr
I - 16

If this value of n is substituted in (11), and g for g , the following

o ctr n

6

expression is obtained for the operating noise factor of the receiving

system:

f = h(T /T ) + (f - h). (24) a o ctr

Finally, if we follow C. C. I. R. Report No. 65 [1956] and replace (T /T ) a o by f , substitute (20) for f , (19) for f and (18) for f , the following

a ctr t c &

general expression for the operating noise factor of the receiving

system described by the network of figure 2 is obtained:

f = hf + (i -i)(T/T)+I (i -l)(T/T) + {! i (f -l)-(h-l)} .(25) a re co re rt to re rt r 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 = I , f = S. , and (25) o c re t rt

becomes:

f = f -1 + f f f = f -1 +i i f (T=T=T andh=l). (26) a ctra rcrtr cto

The above is the expression for the operating noise factor derived by

Norton [1953] and given in C. C. I. R. Report No. 65.

Note that the operating noise factor f for a noise-free receiving

system, i. e. , with T = f =0, f = f = 1 and f = h is equal to zero.

a a c t r We may establish that f = h for a noise-free receiver with spurious

r responses as follows. The noise n available at the output of such a

o receiver may be expressed:

oo n = kT \ g dv . (27) o o J

s rv

o
I - 17

If \^e substitute this value in (11), we obtain:

f =—- f g dv = h • 28

r g b J

& rv

r o Some experimenters prefer to specify the performance of low

noise receiving systems in terms of an operating noise temperature

T , and this is related to f and T or to h, T and f by:

e o a ctr

T =fT =hT + (I -1) T +! (i -1)T + {1 I (f -l)-(h-l)} T . e o a re c re rt t re rt r o

(29)

It appears from the above that the operating noise temperature T

depends not only upon the effective antenna temperature T and the

excess noise temperature (f -1)T of the receiver, but also depends

r o upon the losses, mismatch conditions and spurious responses of the

receiving system. Thus T can be identified with an actual temperature

only by virtue of the fact that it has the dimensions of a temperature.

This follows from the fact that the operating noise factor f is a dimensionless positive factor which is usually much greater than one, but

which may be very much less than one for microwave receiving systems

employing low-noise receiving antennas and masers. In fact, practical

receiving systems have been developed with operating noise factors f

substantially less than unity, so that F is actually negative and

T <<T ; for example, DeGrasse et al [I960] have reported a value of

T = 18°K which corresponds to F = -12 db.

In summary, just as Friis defined the noise factor of a fourterminal network as the ratio of the input available c.w. signal-toreference-noise ratio to the predetection output available c.w. signalto-noise ratio: f = [s /(kT b)]/[s /n 1, here the operating noise

n g o

J L o o

J

' * 5
1-18

factor of a receiving system has been defined to be the ratio of the

available c.w. signal-to-reference noise ratio from the equivalent

loss-less receiving antenna to the predetection receiver output available c.w. signal-to-noise ratio: f = [p/(kT b)]/[s /n 1. In this

o o o expression p denotes the unmodulated c.w. signal power available from

the equivalent loss -free receiving antenna and the frequency v of this

unmodulated signal is adjusted so that g

is equal to its maximum

value g . Note now that [s /n 1 = p/[fk T b] = p/[k T b] so that

ctr o o o ^ e the c.w. signal-to-noise ratio [s /n ] at the predetection output is

equal to the ratio of the c.w. signal power p to the operating noise

power fkT b or kT b available at the terminals of the equivalent

loss-free receiving antenna. As pointed out by Friis [1944], definitions

in terms of available signal and noise powers rather than for necessarily

matched conditions are advantageous not only because of their greater

generality, but also because the use of mismatch conditions in amplifier

input circuits often leads to a reduction in the noise factor of networks

containing such amplifiers, [Llewellyn, 1931; Haus and Adler, 1958;

Haus et al, I960].

The noise factor concept is useful primarily for receiving

systems with adequate predetection gain so that their maximum usable

sensitivities are "noise limited" as discussed in C.C.I. R. Recommendation No. 234 [C.C.I. R. 1959]; this report is primarily concerned with

such receiving systems but is also applicable to "gain limited"

receiving systems as discussed in Section 5.

4. The Operating Signal-To-Noise Ratio

The operating signal gain, g , of the linear predetection portion

of the receiving system may be defined as the ratio of the wanted

signal power available at the predetection output of the receiver to the

wanted signal power available at the terminals of the equivalent loss-
I - 19

free receiving antenna:

J

i p q dv

rev r av °ctrv

\ i p dv

J rev r av

(30)

Note that this operating signal gain is exactly the same as the receiving

system effective signal gain defined by (5) in the case of a suppressed

carrier single side-band system* but will usually be somewhat different

for other systems since it includes all of the radio frequency energy

and not just that portion carrying the signal information. Now the

operating signal-to-noise ratio, r, at the predetection output of the

receiving system will be smaller than the tuned c.w. signal-to-noise

ratio [s /n 1 by the factor [g /g 1 so that r = [s /n 1 [g /g 1 = o o o ctr o o o ctr

[p/(f k T b)][g /g ]. Expressed in decibels, the relation between the

o o ctr

wanted signal power, P, available at the terminals of the equivalent

loss-free receiving antenna and the operating signal-to-noise ratio R

at the predetection output of the receiving system may be written:

P = F + B+R + G -G - 204 dbw . (31) ctr o

where

F = 10 log

1() f, R = 10 log

1Q

r, B = 10 log

1() b, and 10 log^kl^ = -204. 00

when T = 288. 39 ° Kelvin and k = 1 . 38044 x 10" 23 [Cohen et al, 1957].

o Since T must be somewhat arbitrarily chosen in any case, the above

o

*This follows from (5) when it is noted that the generator input power

spectral density available at the terminals of the equivalent loss-free

receiving antenna is equal ^ to s = S. p gv rev rav
1-20

value was assigned so that the constant in (31) is equal to 204 for the

currently best estimate of k; this choice of T is consistent with

o C.C.I. R. Report No. 65 [1956], In engineering practice it is more

convenient to use this easily remembered even decibel noise level

reference -204 dbw or -174 dbm than to adopt the previously proposed

values T = 300°, 290°, or 1° Kelvin as a reference. Note that the

o use of T = 288. 39° results in noise factors less than 0. 024 db larger

than those specified relative to a reference temperature T = 290° : o however, the use of a reference temperature T = 1° K would lead to o noise factors which would be 24.6 db larger.

The difference [GL - G ] between the tuned c.w. signal gain

ctr o

J 5 5

and the operating signal gain will usually be negligibly small since the

receiver pass band will normally be designed to have a width somewhat

larger than that ideally required for the reception of the wanted signal.

This small difference could be absorbed in the definition of the effective

noise bandwidth B but then B would depend upon the wanted signal

characteristics as well as the characteristics of the receiving system

and this seems undesirable. For this reason this small correction is

given explicitly.

A temperature in degrees Kelvin is related to temperatures in

degrees Celsius (centigrade) or in degrees Fahrenheit by:

T = 273.16 + T . = 255.38 + (5/9)T_ , ... (32) Kelvin Celsius Fahrenheit

If the signal generator used for receiver noise factor measurements

has its impedance at a temperature T rather than T , then a term

g o

("1 - (T /T )lh should be added to the value so measured to determine f . g o r

It follows that an error in f of less than ± 0. Ih will be made if T lies

r g

within the range 260°K to 3l7°K, i.e., within the range -13°C to 44°C

or within the range 8°F to 111°F. Thus the use of the precise value

T =288.39° K = 15. 23°C = 59.42° F will be required only in connection

g

with very precise noise factor measurements.
1-21

5. The Required Value P of Wanted Signal Power

The sensitivity of a receiving system and its over-all merit as regards its ability to overcome noise may conveniently be measured by

p = r fkT b ; r kTb watts where r denotes the value of the r n n o n e n

product r(g /g ) required for the system under consideration to

provide the specified grade of service. Note that r will depend upon

the degree and the nature of the modulation of the wanted signal, as well as the degree to which the receiver pass-band matches and is

aligned with the spectral characteristics of the wanted signal. In

particular any drift of the pass-band will cause g to decrease and

then r and thus p will increase correspondingly. In practice it

n ^n c to } c

is not necessary to measure g since r is most easily determined

s n

from directly measured values of p and b together with a measured

value of either f or T by using the relations: r = p /(fkT b) = e n n o

p /(kTb). The value of p is measured by changing the wanted

signal power p, as defined by (1), until the grade of service actually

provided is equal to the grade of service specified. For example a curve might be plotted of the error rate for a teletype receiving system

versus the wanted signal power p and then p will be equal to that

n

value of p corresponding to the value of error rate associated with the

specified grade of service.

In many applications, particularly where atmospheric noise is

involved, T and f will be quite variable with time, and in such cases

a a

it is useful to consider f and n to be random variables and to describe

o them in terms of appropriate statistical characteristics. In still other

applications, such as space -satellite communications, f and n will be

found to vary with the receiving antenna orientation since T , and thus

f and T will vary as the antenna is pointed in different directions.
1-22

Since both the wanted signal and noise power may vary from

minute to minute in a random unpredictable fashion, it is convenient to

include the effects of these short-term variations of s os, g and n in p

o n

and thus in r . Thus p should be considered to be the measured median

n r n

value of p over a short period of time, say one hour, for which the grade

of service provided under typical signal and noise fading conditions is just

equal to the grade of service specified. To allow for possible effects of

receiver drift on p , its value measured with the receiver tuned may be

increased by the factor (g /g ) where g and g are the effective

signal gains with the receiver respectively tuned and then detuned by the

amount expected to be exceeded under operating conditions for half of

the time.

The figure of merit p may be expressed in decibels above one

watt as: , P = R + F + B - 204 dbw . (33)

n n rn

In the above F denotes the median value of the operating noise factor F. m

Suppose now that the value R of operating signal-to-noise ratio required

to provide the specified grade of service is determined directly at the

predetection output of the receiver. In that case it follows from (31) that

P is given by:

P = R + G -G+F +B- 204 dbw (33a)

n r ctr o m

The above expression is applicable only to "noise limited" receiving

systems with adequate predetection gain. If the directly determined

required value P of wanted signal power is greater than the value

determined by (33a) then the receiving system is said to be "gain limited".

For such systems the concept of operating noise factor is not useful and

the value of R determined by (33) will be greater than (R + G J - G ). n

' & r ctr o

In the special rase of an amplitude modulated signal it is usually

more convenient to determine the hourly median value of the wanted signal

carrier power P required to provide the specified grade of service. If R denotes the median value of the operating carrier-to-noise ratio re r- &

required to provide the specified grade of service at the predetection output

then: _P =R +F +B- 204 dbw . 33b

n c re m

The above is applicable only for a "noise limited" receiving system tuned

to the carrier.
1-23

6. Reference Point For the Operating Noise Factor

It has been suggested that the operating noise factor should be

referred to the accessible input terminals of the receiver, i.e. , at the

output of the transmission line, rather than to the input to the equivalent

loss-less receiving antenna terminals. The receiver input terminals

might appear to have some advantage as a reference point since the

signal-to-noise ratio can be directly measured at this point. However,

it is easy to show by means of an example that the use of this point of

reference will not yield an operating noise factor which provides an

appropriate measure of the performance of the entire receiving system

and this, after all, was the only purpose for defining this factor. For

simplicity in the following example, we may let h = 1 and assume that

all of the ambient temperatures are equal to the reference temperature.

In this case the operating noise factor f referred to the equivalent

loss-free receiving antenna terminals is given by (26). If an operating

noise factor f were defined to be the ratio of the available signal to o

e reference noise ratio at the output of the transmission line to the

predetection receiver output signal to noise ratio, then

f - 1 f orcrtjgjg = ill JL = -^ + f (T = T = T and h = 1) . (34) re rt

r cto

Now consider two systems with f' = 3, I . = 2, I , = 3, f , = 3, and

al rcl rtl rl

f = 5, I - 4, S. ._ = 3, and f _ = 3; for these systems, f, = 20 and a2 rc2 rt2 r2 ' 1

f = 40 when referred to the equivalent loss-free antenna terminals,

while f . = f = 10/3 when referred to the output of the transmission

line. Although the first system is twice as good as the second, since

the value of the power p required to provide the same signal-to-noise

ratio at the output is half as large, one would conclude that the two
I - 24

systems have the same performance if the operating noise factor were

referred to the output of the transmission line. It is concluded therefore

that the only proper reference point for the operating noise factor and

for the corresponding operating noise temperature of a receiving system

is at the terminals of the equivalent loss -free receiving antenna.

Using the definitions given in C.CI.R. Recommendation No. 241

and in C.CI.R. Report No. 112 [1959] for the transmission loss L, the

basic transmission loss L and the path antenna directivity gain G , the b

p

following expressions relate the power P available from the equivalent

loss-less receiving antenna to the power P 1 radiated from the transmitting antenna:

p = p;- L =p; + G

p -V <35)

The radiated power p' is less than the power p

fT delivered by the

transmitter to the transmission line by a loss factor £ which includes

tr

transmission line, mismatch and transmitting antenna circuit losses.

Note that a concept of available power from a transmitter is not a

useful one and thus L = 10 log, JL is better defined as above and

thus simply as the difference in decibels between the power delivered

from the transmitter and the power radiated from the transmitting

antenna

L EP'i.p', (36)

tr t t Now (31), (35), and (36) may be combined to give the following general

expression for the transmitter power required to provide a given

predetection signal-to-noise ratio R:

P" = L +L -G +F + R + G -G +B-204dbw. (37)

t tfr b p ctr o
1-25

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

s defined in C.C. I. R. Recommendation No. 241 [1959]:

L = P - P (38)

s t a where P is the power input to the transmitting antenna and P is the

t a available power from the actual lossy receiving antenna. The power

input to the transmitting antenna p is less than the transmitter power

p " delivered to the transmission line by a factor £ which may include

transmission line and mismatch losses. Thus L, = 10 log, A is

tt 1 (J tt simply the difference in decibels between the transmitter power P "

delivered to the transmission line and the power P delivered to the

transmitting antenna:

L = P" - P . (39)

tt " t t Now (37) may be expressed:

P "= L + L +F +R + G -G +B- 204 dbw (40)

t tt s s ctr o

F =10 1og..{(T /T )+(!.- i)(T./T ) +i(f -l)}(h= 1) . (41)

s 10 aso rt to rtr

Here F is a noise factor for the receiving system referred now to the

terminals of the actual lossy receiving antenna, and kT b represents

ct S the external noise available at these terminals. Note, however, that

(T /T ) = {f + (£ -1)(T /T )}/£ , and thus £ and T must still be

as o a re c o re re c known for a proper determination of the performance of the system.

For this reason it appears that (40) is no simpler to use than (37) and,

of course, these two equations must yield identical results when each

of their terms is properly evaluated.
I - 26

7. Other Considerations

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(h v/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

T = 10°K. Balazs [1957] 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 factor 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.
I - 27

It is sometimes possible to reduce the operating noise factor 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 factors f and f , the

p q

two networks in tandem with p preceding q will have a noise factor

given by:

f =f + (f -l)/g (42)

pq p q p

Alternatively, if q precedes p, we have:

f = f +(f -i)/g (43)

qp q p q

From the above we obtain the condition which must be satisfied for

f < f : pq qp

f + (f -l)/g <f + (f -l)/g (For f <f ) (44)

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 = 1/j0 J and f = 1 + (I -1)(T/T ) so that we may write for the

q rt q rt t o reduction in f with the preamplifier preceding the transmission line:

Afsf -f = (1 -i){(£ -1) + (T /T )(l--h} (45)

qp pq rt p to g

p

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 1n {(f + Af)/f }. 10 pq pq
1-28

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 (44) and, provided g > 1 and g > 1, we may re-write this

P 3

inequality in the form:

f„

~ i f "

1 < -3 — (For f <f provided g > 1 and g > 1) . (46)

i _ J_ i _ J_ pq qp p q

p q

If the above inequality is satisfied, it will be advantageous to have the

amplifier p precede the amplifier q.

8. Radiometry

In radiometry, the desired signal is the noise available from the

equivalent 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 (5) and (7) are indeterminate; however, the factor hb g , which is

ctr

required in (22) for the evaluation of T , may be evaluated [See (7)] by: a

h b g = 5 \ g dv . (47)

ctr J

5 ctrv

o The radiometer measures only an average antenna temperature T , and s— a

it is evident by (22) that the presence of spurious responses over a wide band will not yield a good estimate of T if this has different

values for 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 .
I - 29

9 . Measurement of the Operating Noise Factor

Normally the operating noise factor of a receiving system would

not be measured directly; instead each of its components would be

measured, and then (25) 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

cL denote the theoretical radiation resistance of the receiving antenna

referred to its input terminals, and let R. be its actual measured input

resistance; in this case i = R./R • A more general discussion of the

re l a

&

method of determining S. , including the effects of insulator losses is

given by Crichlow [1955]. At the higher frequencies, S. will usually

be negligibly different from unity; however, in the case of reception

with a unidirectional rhombic terminated in its characteristic impedance,

£ may be greater than 2 [Harper, 1941], since nearly half of the

re received power is dissipated in the terminating impedance and some

is dissipated in the ground. Next the transmission line loss factor £

rt

is measured as the ratio s /s where s denotes the available power

g o g

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

conjugately matched to the output impedance of the line with the

antenna connected to the input to the line. Finally, the noise

factor f of the receiver is to be determined by using a c.w.

signal generator at its input having the same impedance as the

above-defined output impedance of the transmission line. Let

the generator impedance have a temperature T and let s . denote

g d

the available signal power at the frequency v which is required to

double the receiver output power (signal plus noise) relative
1-30

to the output noise power with the signal generator turned off; then

f rdrvro = [s ,g /(g k T b)] + [1 - (T /T )] h. The generator need not be g°

tuned to the receiver in making this noise factor measurement; it is

only necessary to use the appropriate values of g and of the product

g b. The above procedures will yield the appropriate values of i ,

£ and f to be used in (25) for the actual conditions of matching of the

rt r transmission line either to the receiving antenna or to the receiver.

It will usually be desirable to use non- reflective matching of the line

to the receiving antenna, since this will eliminate echoes which could

be quite objectionable on long lines; to the extent that the characteristic

impedance of the line is real, non-reflective matching will also result

in the smallest value for S. . However, it will sometimes be advantrt

ageous to mismatch the transmission line to the receiver in such a way as to minimize f ; this latter point is discussed in the 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 the

receiver responses, g is the maximum value of g and b, h and

ctr ctrv

T are then determined by means of their defining equations (7), (8), a and (22).

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 factor 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 factor obtained in
I - 31

this way must be increased by the factor h in order to determine f ; r thus it would still be necessary in this case to measure both b and h

in order to determine the noise factor of a multiple response receiver.

Furthermore, the ultimate use of f is for the determination of P for

r a given value of R, and it is evident by (31) that its determination re- quires 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 ). r 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 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 i or g n n by replacing s in (3) or (4) by s i I & where I denotes the

g g mnp mng mnp

mismatch factor (2) between the network input impedance and the output

impedance * of the preceding v B networks while i denotes the mismatch

mng

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, i -i and no correction

mnp mng

is involved. In case the signal generator impedance is made equal to

the output impedance of the preceding networks, we may determine

i or g by replacing s in (3) or (4) by s. S. „ where s. denotes the

n n o S. mojE X. power delivered to the load and I . denotes the mismatch factor moi

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
1-32

duplexer has the same absolute temperature T as that of the transmission line, we may modify (25) as follows to determine the operating

noise factor of a receiving system containing such a duplexer: i is

now measured as the ratio s /s where, as before, s denotes the g° g

available power from a signal generator at the input to the line with

its impedance equal to that of the receiving antenna, but s is now the

o resulting power available at the output of the duplexer. In addition,

the following term is added to (25):

oo

S. i —r c

, c, r

, \ p , g dv, where p , is the undesired local transmitter skTb \ *dvs rv ' Fdv B

r o J

o 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 unc or related

10. The Measurement of R

u The value r of the wanted-to-unwanted signal power ratio

required for a given quality or grade of service will depend upon the

nature of the wanted and unwanted signals. Since both the wanted

signal power p and the unwanted signal power p available at the

terminals of the equivalent loss-free receiving antenna will usually

vary from minute to minute in a random unpredictable fashion, it is

convenient to include the effects of these short-term variations of

p and p in determining r , and thus to consider R ^10 log^^ r F F

u

e u u

5 10 u

to be the value of the ratio of the median wanted signal power p to the

median unwanted signal power p required to provide a specified

grade of service over a specified period. For example, R for

Rayleigh fading wanted and unwanted teletype signals could be

specified as the value of 10 log, Jp /p ) which will provide a teletype °10 m um
1-33

character error rate E equal to 0. 001, i. e. , 0. 1% , over a period of

one hour. In case the rate of fading is sufficiently small so that

fading durations at signal levels, say 20 db, below the median signal

level are large on the average with respect to the duration of a teletype character, then it is possible to determine R for Rayleigh

fading wanted and unwanted teletype signals and a character error

rate E as follows. When p is Rayleigh distributed about a median

value p while p is independently distributed in a Rayleigh distribution m u

about a median value p , the probability Q that the instantaneous

um

ratio (p /p) will be less than x is given by the simple expression:

x(p /p ) Q[(P /p) < xj = TM ,Um

. (48)

u 1 + x(p /p ) m um

Now let E (x) denote the empirically determined probability of error

distribution for steady non-fading unwanted and wanted signals as a function of their ratio x. By combining the E (x) distribution with the

Q distribution of (p /p), it is possible to determine the error rate

probability distribution E as a function of (p /p ). rn um

OO E (Pm/p ) = V E U) |^- dx • (49) um J s 3 x

' o

The function (49) may be used to determine R = 10 log. (p /p ) u

s 10^m Fum'E

corresponding to a teletype error rate E. In determining R , the power

P must be much larger than P so that noise does not contribute to m n

the e::ror rate.

Telecommunication systems should be designed to have the

smallest values of R which can be obtained so as to permit the same

u

portions of the spectrum to be used simultaneously by the maximum

number of simultaneous users. An example of large reductions in the
I - 34

required values of R obtainable by careful engineering is the use of

precise offset carrier frequencies in television. It has been shown

[Behrend, 1956 and Television Allocation Study Organization, 1959]

that the interference between geographically adjacent television

broadcasting stations occupying the same channel can be substantially

reduced by the use of precisely offset carrier frequencies. Fine [I960]

has made a detailed study of the Television Allocation Study Organization

results and found that the ratio of the wanted-to-unwanted carrier levels

required to provide a "passable " picture may be decreased from 23. 3 db

to 18. 5 db by using carriers precisely offset by 10, 010 ± 5 c/s as

compared to the use of a nominal offset of 9985 ± 1000 c/s. Similarly,

he found that this required ratio is reduced from 29. 3 db to 20. 2 db by

using a precise offset of 20, 020 ±5 c/s as compared to a nominal offset

of 19, 995 ± 1000 c/s.
1-35

REFERENCES FOR APPENDIX I

Balazs, Nandor L. , Thermal fluctuations in conductors, Phys. Rev.

105 , No. 3, 896-899 (Feb. 1957).

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

Performance predictions for single tropospheric communication

links and for several links in tandem, NBS Tech. Note 102,

(PB161603) (August 1961). $3.00*

Behrend, W. L.. , Reduction of co-channel television interference byprecise frequency control of television picture carriers,

IRE Transactions on Broadcast Transmission Systems, BTS-7 , p. 6 (February 1957).

Blake, L. V., Recent advancements in basic radar range calculation

technique, IRE Transactions on Military Electronics, MIL-5 , No. 2, 154-164 (April 1961).

Burgess, R. E. , Noise in receiving aerial systems, Proc. Phys.

Soc. 5_3, 293-304 (May 1941).

C.C.I.R., Report No. 65, Report on revision of atmospheric radio

noise data (Warsaw, 1956, available as a separate document

from the International Telecommunications Union, Geneva, or

in Vol. I of the Documents of the VUIth Plenary Assembly of

the C.C.I.R., Warsaw (1956). Also in Documents of the IX th

Plenary Assembly, Vol. Ill, pp. 223-256, Los Angeles (1959).

See also Vol. I, pp. 56-74 and pp. 128- 1 30 and Vol. Ill, pp. 94-99

Cohen, E. R. , K. M. Crowe, and J. W. M. Dumond, The fundamental

constants of physics, Interscience Publishers, New York, N. Y. , (1957).

Crichlow, W. Q. , D. F. Smith, R. N. Morton, and W. R. Corliss,

Worldwide radio noise levels expected in the frequency band 1 kc

to 100 Mc, 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

^Order by PB number from the Office of Technical Services, U. S.

Department of Commerce, Washington, D.C. Foreign remittances

must be in U.S. exchange and must include one-fourth of the publication

price to cover mailing costs.
I - 36

DeGrasse, R. W. , D. C. Hogg, E. A. Ohm, and H. E. D. Scovil,

Ultra-low noise receiving system for satellite or space

communication, Proc. Nat. Elect. Conf. _15, 370-379 (1959);

also published as Bell Telephone System Monograph 3624

(August I960).

Dyke, E. , Corrections and coordination of some papers on noise

temperature, Proc. IRE, 49, No. 4, 814-815 (April 1961).

Fine, Harry, A further analysis of TASO Panel 6 data on signal to

interference ratios and their application to description of

television service, Federal Communications Commission, T.R.R.

Report No. 5.1.2 (April I960); also published in IRE Transactions on Broadcasting, BC-7, No. 1, 22-38 (January 1961).

Friis, H. T. , Noise figures of radio receivers, Proc. IRE 32, No. 7,

419-422 (July 1944).

Hansen, R. C. , Low noise antennas, Microwave Journal 2, No. 6,

19-24 (June 1959).

Harper, A. E. , Rhombic antenna design, D. VanNostrand Co., (1941).

Haus, H. A. , W. R. Atkinson, G. M. Branch, W. B. Davenport, Jr. , W. H. Fonger, W. A. Harris, S. W. Harrison, W. W. McLeod,

E. K. Stodola and T. E. Talpey, Representation of noise in

linear twoports, Proc. IRE 48, No. 1, 60-74 (January I960).

Haus, H. A. , and R. B. Adler, An extension of the noise figure

definition, Proc. IRE 45, No. 5, 690-691 (May 1957).

Haus, H. A., and R. B. Adler, Optimum noise performance of linear

amplifiers, Proc. IRE 46, No. 8, 1517-1533 (August 1958).

Hogg, D. C, and W. W. Mumford, The effective noise temperature of

the sky, Microwave Journal 2> No. 2, 80-84 (March I960).

Jelley, J. V., The 21 -CM maser installation at Harvard College

Observatory, Microwave Journal, 5, No. 2, 77-83 (February 1962),

Ko, H. C, Temperature concepts in modern radio, Microwave

Journal 4, No. 6, 60-65 (June 1961).
I - 37

Lawson, James L. , and George E. Uhlenbeck, Threshold signals 24, of the Radiation Laboratory Series, 103-108, McGraw-Hill

Book Co., New York, N. Y. (1950).

Llewellyn, F. B., A rapid method of estimating the signal-to-noise

ratio of a high gain receiver, Proc. IRE 1_9, No. 3, 416-421

(March 1931).

North, D. O. , The absolute sensitivity of radio receivers, RCA

Review 6, 332-343 (January 1942).

North, D. O. , Discussion on noise figures of radio receivers by

H. T. Friis, Proc. IRE 3^, No. 2, 125-127 (February 1945).

Norton, K. A., Transmission loss in radio propagation, Proc. IRE 41 , No. 1, 146-152 (January 1953).

Norton, K. A., System loss in radio-wave propagation, Proc. IRE 47,

No. 9, p. 1661 (September 1959).

Nyquist, H. , Thermal agitation of electric charge in conductors,

Phys. Rev. 32_, No. 1, 110-112 (July 1928).

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No. 1, p. 376 (January 1961).

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June 13, 1960.
1-38

List of Symbols

f = (T /T ) operating noise factor for the receiving system

f hourly median value of the operating noise factor f

f = (T /T ) effective receiving antenna noise factor

a a o

f noise factor of the receiver

r

f noise factor of the receiving antenna circuit

c 5

f noise factor of the transmission line to the receiver

f noise factor of a four terminal network

n

T reference temperature equal to 288.39° Kelvin

T equivalent loss-less receiving antenna noise temperature

in degrees Kelvin at the radio frequency v

T effective antenna noise temperature in degrees Kelvin

for the particular receiving system with effective noise

bandwidth b and spurious response factor h

T = f T operating noise temperature in degrees Kelvin for the

particular receiving system

T ambient temperature in degrees Kelvin of the receiving

antenna circuit

T ambient temperature in degrees Kelvin of the transmission

line to the receiver

p or (p ) wanted or (unwanted) radio frequency signal power

available at the terminals of the equivalent loss-free

receiving antenna

p or (p ) wanted or (unwanted) radio frequency signal power available at the terminals of the actual receiving antenna

p or (p ) wanted or (unwanted) radio frequency signal power density

(watts per cycle per second) available at the terminals of

the actual receiving antenna
1-39

p hourly median value of the wanted signal power p which

is required to provide the specified grade of service in

the presence of noise alone

p hourly median value of wanted signal carrier power

nc available at the terminals of the equivalent loss -free

receiving antenna which is required to provide the

specified grade of service in the presence of noise alone

p power delivered to the transmitting antenna

p

' total power radiated from the transmitting antenna

p

" power delivered from the transmitter to the transmission

line

g gain factor for the radio frequency v; ratio of the power

available at the output of a four terminal network to the

power available at its input

g maximum value of g within the pass band of the 6n , 6nv ^ network

g receiving system gain factor for the radio frequency v; ratio of the power available at the predetection output of

the receiver to the power available at the terminals of

the equivalent loss -free receiving antenna

g maximum value of g within the pass band of the B ctr . . 6ctrv r receiving system

g effective signal gain factor for a four terminal network;

this is the gain factor for the signal bearing components

of the wanted radio frequency signal power

g operating signal gain factor for the receiving system;

ratio of the total wanted radio frequency signal power

available at the predetection receiver output to that

available at the terminals of the equivalent loss-free

receiving antenna

£ loss factor for the radio frequency v; ratio of the powers

nv

available at the input and output, respectively, of a four

terminal network
1-40

S. receiving antenna circuit loss factor

re

s

i loss factor for the transmission line to the receiver

rt

ft = (p "/p ') ratio of the transmitter power delivered to the transTT" L t mission line to the total power radiated from the transmitting antenna

i = (p "/p ) ratio of the transmitter power delivered to the transmission line to the power delivered to the transmitting

antenna

b effective noise bandwidth of a four terminal network;

also used to denote the effective noise bandwidth of the

linear portion of the receiving system with input terminals at the equivalent loss -free receiving antenna and

output terminals at the predetection receiver output.

h spurious response factor of the receiving system; also

used to denote Planck's constant.

k Boltzman's constant

fkT b = kT b operating noise power available at the terminals of the

equivalent loss-free receiving antenna; a fictitious power.

r operating signal-to-noise ratio; ratio of the wanted signal

power to the noise power available at the predetection

output terminals of the receiver

r median value of r required to provide the specified

grade of service

r nnmo = p /(f kT b) ratio of the hourly median wanted signal power to

. ,. . . ...

hourly median operating noise power required at

the terminals of the equivalent loss -free receiving

antenna to provide the specified grade of service

in the absence of all other sources of interference

r ratio of the hourly median wanted signal power p to

u , \m

hourly median unwanted signal power p required at

um

the terminals of the equivalent loss-free receiving

antenna to provide the specified grade of service in the

absence of all other sources of interference
1-41

204. 00 = - 10 log (kT ) the signal power at a radio frequency v in

decibels below one watt which is required at

the terminals of the equivalent loss -free

receiving antenna to produce an operating

signal-to-noise ratio r = 1 at the predetection output of a receiving system with an

effective noise bandwidth b = 1 cycle per

second and an operating noise factor f = 1 or operating noise temperature T = T

L = 1 log,5 „ (p '/p) = L, - G the transmission loss over the propa10 *V F/ b p . , , F * gation path between loss -free antennas

which are otherwise equivalent to the

actual transmitting and receiving

antennas

L the basic transmission loss; the transmission loss expected

over the propagation path between loss-free isotropic

transmitting and receiving antennas

G = L - L path antenna gain between loss-free transmitting and

receiving antennas which are otherwise equivalent to the

antennas used on the propagation path

L =10 log (p /p ) system loss between the actual transmitting and

s 10ta .. receiving antennas used on the propagation path

6 U. S. GOVERNMENT PRINTING OFFICE : 1962 O - 656392

U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

In general, each section carries cut specialized research, development, and engineering in the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

front cover. WASHINGTON, D. C.

Electricity. Resistance and Reactance. Electrochemistry. Electrical Instruments. Magnetic Measurements.

Dielectrics. High Voltage.

Metrology. Photometry andColorimetry. Refractometry. Photographic Research. Length. Engineering Metrology.

Mass and Scale. Volumetry and Densimetry.

Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics.

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment.

Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Pure Substances. Spectrochemistry. Solution Chemistry. Standard Refer- ence Materials. Applied Analytical Research. Crystal Chemistry.

Mechanics. Sound. Pressure and Vacuum. Fluid Mechanics. Engineering Mechanics. Rheology. Combustion

Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry. Polymer Physics. Polymer Characterization. Polymer Evaluation and Testing. Applied Polymer Standards and Research. Dental Research.

Metallurgy. Engineering Metallurgy. Microscopy and Diffraction. Metal Reactions. Metal Physics. Electrolysis and Metal Deposition.

Inorganic Solids. Engineering Ceramics. Glass. Solid State Chemistry. Crystal Growth. Physical Properties.

Crystallography.

Building Research. Structural Engineering. Fire Research. Mechanical Systems. Organic Building Materials. Codes and Safety Standards. Heat Transfer. Inorganic Building Materials. Metallic Bunding Materials. Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering. Mathematical Physics. Operations Research. Data Processing Systems. Components and Techniques. Computer Technology. Measurements Automation.

Engineering Applications. Systems Analysis.

Atomic Physics. Spectroscopy. Infrared Spectroscopy. Solid State Physics. Electron Physics. Atomic Physics.

Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- ments. Basic Instrumentation. Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chemistry. Molecular Spectroscopy. Molecular Kinetics. Mass Spectrometry. Office of Weights and Measures. BOULDER, COLO.

Cryogenic Engineering Laboratory. Cryogenic Equipment. Cryogenic Processes. Properties of Materials. Cryogenic Technical Services. CENTRAL RADIO PROPAGATION LABORATORY

Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Research. Prediction Services. Sun-Earth Relationships. Field Engineering. Radio Warning Services. Vertical Soundings Research.

Radio Propagation Engineering. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements.

Tropospheric Analysis. Propagation-Terrain Effects. Radio-Meteorology. Lower Atmosphere Physics.

Radio Systems. Applied Electromagnetic Theory. High Frequency and Very High Frequency Research. Modulalation Research. Antenna Research. Navigation Systems.

Upper Atmosphere and Space Physics. Upper Atmosphere and Plasma Physics. Ionosphere and Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

RADIO STANDARDS LABORATORY

Radio Physics. Radio Broadcast Service. Radio and Microwave Materials. Atomic Frequency and Time-Interval

Standards. Millimeter-Wave Research.

Circuit Standards. High Frequency Electrical Standards. Microwave Circuit Standards. Electronic Calibration

Center.
NBS