National Bureau of Standards

Library, N.W. Bldg

AUG 8 1962

^ecltnical ^lote 100

REQUIRED SIGNAL-TO-NOISE RATIOS,

RF SIGNAL POWER, AND BANDWIDTH FOR

MULTICHANNEL RADIO COMMUNICATIONS SYSTEMS

E. F. FLORMAN AND J. J. TARY

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

technical ^Kiote

100

JANUARY, 1962

ORIGINAL: JULY, 1960

REVISED: NOV., 1960; JUNE, 1961; AND NOV. 1961

REQUIRED SIGNAL-TO-NOISE RATIOS,

RF SIGNAL POWER, AND BANDWIDTH FOR

MULTICHANNEL RADIO COMMUNICATIONS SYSTEMS

E.F. Florman and J.J. Tary

NBS Boulder Laboratories

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 $1.00

CONTENTS

Page

No.

1. INTRODUCTION 1

2. RADIO COMMUNICATION SYSTEM 4

2. 1 Performance of System 5

2.2 Design and Testing of Equipment and Systems ' 7

3. MODULATOR -TRANSMITTER PERFORMANCE 9

3. 1 Required Radio Frequency Signal-Spectrum 10

Bandwidth for Radio Communication Systems

3. 1. 1 Radio Frequency Bandwidth Required for 10

SSBSC-AM Systems

3. 1.2 Radio Frequency Bandwidth Required for 10

FM Systems

3. 1.2. 1 Required FM Radio Frequency 11

Bandwidth for a Single Sinsoidal

Modulating Signal

3.1.2.2 Required FM Radio Frequency 15

Bandwidth for Various Types of

Modulating Signals

3.1.2.3 Required FM Radio Frequency 16

Bandwidth for a "White Noise

Modulating Signal

3.2 Radio -Frequency Signal Time -Amplitude 23

Characteristics for Radio Communication Systems

3.2.1 Radio-Frequency Signal Time -Amplitude 23

Characteristics for SSBSC-AM Systems

3.2.1.1 Radio-Frequency Signal Time- 23

Amplitude Characteristics of

SSBSC-AM Systems for a Single

Sinusoidal Modulating Signal

111
Page

No.

3.2.1.2 Radio -Frequency Signal Time- 24

Amplitude Characteristics of

SSBSC-AM Systems for a WhiteNoise Type of Modulating Signal

3. 2. 2 Radio-Frequency Signal Time -Amplitude 27

Characteristics for FM Systems

3. 3 Peak Power Requirements for SSBSC-AM Systems 27

RADIO RECEIVER -COMBINER PERFORMANCE 29

4. 1 Radio Frequency Signal Characteristics and 30

Power Levels in the Receiving System

4.2 Pre-Detection Noise-Power Levels in the Re- 33

ceiving System

4. 3 Post-Detection Noise-Power Levels in tne Re- 41

ceiving System

4. 3. 1 Post-Detection Noise Due to Pre-Detec- 43

tion Amplitude Noise

4. 3. 2 . Post-Detection Noise Due to Noise at the 43

Modulator Input

4.3.3 Equipment -Intermodulation Post-Detection 44

Noise

4.3.4 Post-Detection Path-Modulation Noise 47

4.3.5 Total Post-Detection Noise 49

4. 3. 6 Effect of Post-Detection Noise on 50

Receiver Characteristics and Methods of

Correcting System -Design Curves

4.4 System Parameters which Affect the Message- 53

Channel Signal -to-Noise Ratio at the Radio

Receiver Output

4.4.1 Dependence of Receiver-Output Signal-to- 54

Noise Ratio on the Modulating -Signal

Power Level, Pm

IV
Page

No.

4.4.2 Dependence of Receiver -Output 56

Message-Channel Signal-to-Noise Ratio,

S /N , on the Message Channel Signaloc oc / Power Loading Ratio, P /p , in the

Modulating Baseband Signal.

4.5 Performance Characteristics of SSBSC-AM Radio 59

Receivers

4. 5. 1 Receiver -Input Radio Frequency Signal 6l

Power Requirements for SSBSC-AM

Systems Using Steady Received

Signals

4. 5.2 Receiver -Input Radio Frequency Signal 66

Power Requirement for SSBSC-AM Systems

Using a Rayleigh-Fading Type of Received

Signal

4.6 Performance Characteristics of FM Radio Re- 67

ceiver s

4. 6. 1 FM Radio Receiver Characteristic Curves 68

4. 6. 1. 1 Linear Region of FM Receiver 69

Characteristic Curves

4.6.1.2 Threshold Region of FM Receiver 71

Characteristic Curves

4. 6. 2 FM Radio Receiver Input -Power and 73

Bandwidth Requirements for Steady

Received Signals

4.6. 3 FM Radio Receiver Input-Power and Band- 77

width Requirements for a Rayleigh-Fading

Type of Received Signal

4.6.4 Pre-Emphasis - - De-Emphasis Techniques 81

4. 6. 5 Bandwidth Compression in FM Receivers 83
Page

No.

PERFORMANCE CHARACTERISTICS OF MESSAGE- 88

SIGNAL DECODER UNIT

RADIO COMMUNICATION SYSTEM DESIGN AND TEST 95

PROCEDURES

6. 1 Determination of the Basic System Parameters: 97

S /N , B , P /p , f , TA and T for oc oc c m m m A er Specified Average Message Error Rates, Number

of Available Message Channels, and Specified

Receiving -System Parameters.

6.2 Equipment Design Procedures, SSBSC-AM 101

Systems, for a Rayleigh-Fading Type of Received

Signal, Dual Diversity, and Maximal -Ratio

Combining

6. 3 Equipment Design Procedures, FM Systems, 103

for a Rayleigh-Fading Type of Received Signal,

Dual Diversity, and Maximal -Ratio Combining

6.4 Test Procedures for SSBSC-AM and FM Systems 108

6. 5 System Performance Estimates for SSBSC-AM 108

and FM Systems

REQUIRED TRANSMITTER-OUTPUT POWER 114

CONCLUSIONS 119

ACKNOWLEDGEMENTS 121

REFERENCES 122

LIST OF SYMBOLS 126

LIST OF FIGURES 132

VI
Page

No.

APPENDICES

A. PRE-DETECTION NOISE-POWER LEVELS IN

THE RECEIVING SYSTEM 135

B. DERIVATION OF THE MESSAGE-CHANNEL

SIGNAL -POWER LOADING RATIO P^/p , IN

THE MODULATING BASEBAND SIGNAL 163

C. DERIVATION OF FM RECEIVER PERFORMANCE

CHARACTERISTICS 167

D. METHOD OF ESTIMATING THE PERFORMANCE

CHARACTERISTICS OF RADIO RECEIVERS AND

MESSAGE-SIGNAL DECODING UNITS, FOR A

TIME -VARYING SIGNAL 174

D. 1. Estimated Performance Characteristics of

Radio Receivers 174

D. 1. 1 Estimated Performance of Radio

Receivers with Diversity j 78

D. 2 Estimated Performance Characteristics

of the Decoder Unit, for a Time -Varying

Input Signal-to-Noise Ratio 1?9

vn
ABSTRACT

A method is outlined for determining the relationships between

the grade of performance, or message error rate, of a radio

communication system and the system parameters. Results are

presented in the form of design equations and system-design curves

The system parameters and variables considered are the following

definable and measurable factors: signal-to-noise ratios, carriersignal power level, message load, receiver noise figure, etc.

These factors are used with the design curve scales as normalizing

factors in order to yield quantitative results.

vm
REQUIRED SIGNAL-TO-NOISE RATIOS, RF SIGNAL POWER,

AND BANDWIDTH FOR MULTICHANNEL

RADIO COMMUNICATION SYSTEMS

by

E. F. Florman and J. J. Tary

1. INTRODUCTION

The objective of this work is to obtain the basic relationships between the grade of service, or the message error-rate, and the

various system parameters, for radio communication systems. The

results are presented in the form of design equations and sets of

system-design curves, in which the system parameters are used as

variables and as scale -normalizing factors. The design curves

are basic and are general in form; however, they may easily be converted to families of curves which apply directly to specific cases,

involving specified system parameters.

This paper outlines a method for obtaining design equations and

sets of design curves for radio communication systems, based primarily on the combined performance characteristics of the radio

receiver, the diversity combiner, and the message-signal decoder

unit. Results are obtained in terms of the important specified, definable, and measurable system parameters, such as message error

rates, modulating -signal power levels, modulating baseband signal

bandwidth, signal-to-noise ratio at the radio receiver output, radio

receiver noise figure, order of diversity, number of available message channels, etc. Through the application of these results it is

possible to determine directly the optimum values for the important

factors, such as the required radio-frequency signal power and the

optimum bandwidth.

-1-
-2-

It is assumed that the grade of service for the system is directlyassociated with the radio receiver output signal-to-noise ratio; therefore, this ratio is used as a primary factor in determining the system

performance

.

In this work the radio receiver output message-channel signal-tonoise ratio, S /N , is determined in terms of: oc oc

(1) the radio- receiver input total RF- signal power and the total

noise power levels,

(2) the pre-detection signal-to-noise ratio,

(3) the receiver noise figure,

(4) the relative power levels of the modulating signals at the

radio transmitter,

(5) the power- spectrum bandwidth of the composite modulating

signal,

(6) the power-spectrum bandwidth of the modulated radio-frequency signal,

(7) the modulation index, and

(8) position of the signal in the receiver-output baseband signal

spectrum.

The effect of other factors on the receiver output signal-to-noise

ratio may also be incorporated; these additional factors include:

(9) effects of bandwidth compression in the radio receiver IF,

or at the pre-detection point,

(10) radio frequency path bandwidth capability, or effects of frequency selective fading on the radio-frequency signal, and

(11) intermodulation noise generated within the equipment.

The statistical characteristics of the signal-to-noise ratio at the

output of the radio receiver depend upon the time-varying characteristics of (a) the received radio-frequency signal and (b) the total noise

power, at the receiver input. The combined radio receiver output

signal-to-noise characteristics also depend upon the order of diversity,

type of diversity, type of combining, and the point within the receiver
-3-

at which combining takes place, such as pre-detection and/or postdetection. For example, the distribution of the receiver-output

signal-to-noise ratio, for a time-varying received RF signal, may be

determined by combining the steady- signal characteristic of the radio

receiver, with the combiner characteristics, for a particular comulative distribution of the time -varying received radio-frequency signal. This procedure is used here to determine the cumulative distribution of the combined receiver- output signal-to-noise ratio, for

various orders of diversity; assuming that the received radio-frequency carrier signal is Rayleigh distributed. Other types of received

RF signals may also be considered, provided that the statistical characteristics of these signals are known.

The cumulative distribution of the short-term message error

rate is obtained by combining the steady- signal performance char -

acteristic of the message-signal decoder unit with the cumulative

distribution of the time-varying signal-to-noise ratio at the receiver

output, or at the decoder-unit input. From the resultant cumulative

distribution of the message error-rate we obtain the performance

characteristics of the decoder unit for a time-varying signal-to-noise

ratio at the decoder unit input. The above procedures are used for

non, dual and quadruple diversity.

It should be noted in the above analysis that the performance of

the radio receiver is considered in terms of the total RF-signal and

total noise powers at the receiver input versus the receiver output

signal-to-noise ratio. The performance of the decoder unit is taken

to be in terms of the input signal-to-noise ratio versus the outputmessage error rate. The reason for the difference between the selected input -terminal factors for these equipment units is that the RFsignal power at the receiver input is a separable factor, which may
-4-

be adjusted independently (within limits) while the signal-to-noise

ratio at the receiver output or the decoder unit input, must be treated

as a ratio because of the characteristics of the radio receiver.

2. RADIO COMMUNICATION SYSTEMS

From a technical viewpoint, a radio communication link should

transfer information between two or more points at a specified rate,

for a given amount of transmitted power, a required RF spectrum

bandwidth, and with an acceptably small received-information error

rate. An exchange between these major requirements can be effected;

however, their mutual interdependence usually cannot be eliminated.

Decisions involving the choice of the type of communication system to

be used involve such factors as total initial cost, operational cost,

time of delivery of equipment, etc. ; most of these factors are obtainable from a technical evaluation of the various types of systems.

This paper deals with the comparative performance of point-topoint radio communication systems employing either SSBSC-AM

(single sideband suppressed carrier, amplitude modulated) or FM

(frequency modulated) types of modulation. Frequency division type

of multiplexing is considered and the number of voice channels ranges

from 1 to 120, or more. However, the subject is treated on the basis

of message-error rates, required baseband frequency bandwidth,

available received radio carrier power levels, signal-to-noise ratios,

etc. Hence, the results are not restricted to f requency-division

type of multiplexing nor to broadband systems, and are of general use

in the field of electronic communications.

Results are presented in a form which should be useful to the

system designer who has the responsibility of selecting the optimum

combination of parameters; the results can also be used as a guide in

setting up system performance tests.
-5-

The method of analyzing and evaluating a radio communication

system, as outlined below, is to consider the performance characteristics of the major sections of the system and then optimize the performance of each of these sections. By this procedure, it is possible

to compare directly the overall performance and relative merits of

different types of equipment selected to perform the same function

within a communication system. Referring to figure 2-1, the major

sections comprising a radio communication system are seen to be:

(A) The encoding units which accept and encode the information

into the form .of an electronic message -signal, and a multiplexing unit which forms a composite baseband signal from

the set of encoded information-bearing message signals.

(B) The modulator -radio transmitter transmitting- antenna section which generates, modulates and then directs a radiowave signal to the radio receiver. This baseband- signal

modulated radio-wave signal acts as a carrier to convey the

baseband signal to the receiving end of the communication

system.

(C) The radio-wave transmission paths having characteristics

which are dependent upon such factors as antenna beam width,

radio-carrier frequency, radio frequency spectrum, path

length, etc.

(D) The receiving antenna, radio receiver^and diversity-combiner

unit, considered as a section which accepts the modulated

radio-frequency signals and delivers a replica of the transmitted composite baseband signal.

(E) The receiving multiplexing unit and decoding unit combined,

which converts the received baseband signal to information in

its original form.

2. 1 Performance of System

Optimum performance of a radio communication system can be

achieved only by proper considerations of the design parameters involved in the functioning of the various sections of the system. Referring to figure 2-1, the factors to be considered in this paper are:
BLOCK DIAGRAM OF BASIC UNITS

IN A MULTICHANNEL TROPOSPHERIC RADIO COMMUNICATION SYSTEM

TRANSMITTER TRANSMISSION

LINE

RADIO RECEIVERS

-RADIO WAVE

TRANSMISSION PATHS

.MODULATING OR BASEBAND SIGNAL

RADIO RECEIVER OUTPUTS

RECEIVED COMBINED" BASEBAND SIGNAL

MULTIPLEX

(TRANSMIT)

UNIT

MULTIPLEX

(RECEIVE)

UNIT

-CODED MESSAGE- SIGNAL CHANNELS

ENCODING

UNITS

INFORMATION BEARING CHANNELS

(TO BE TRANSMITTED)

INFORMATON BEARING CHANNELS

(RECEIVED)

Figure 2-1
-7-

(1) Spectrum bandwidth required for the composite baseband

signal, which contains the information to be transmitted.

(2) Cumulative amplitude distribution of the composite baseband

or modulating signal; that is, the modulating- signal power

levels exceeded for various percentages of the time.

(3) Required transmitted RF spectrum bandwidth.

(4) Signal-to-noise ratio in the radio receiver IF circuits at the

demodulator input, and the statistical characteristics of this

(received) signal-to-noise ratio.

(5) Received power level, P , at the radio receiver input and the

characteristics of this received signal such as the cumulative

amplitude distributions, fade rate, fading range and fadeduration distribution.

(6) Radio receiver noise level, combined with the receivingantenna system noise power, the radio transmitter -output

noise and the intermodulation noise, all referred to the radio

receiver input terminals.

(7) Diversity gain, both theoretical and measured.

(8) Message-channel signal-to-noise power ratio, S /N , at

, ,. . oc oc the radio receiver output.

(9) The relationship between S /N at the radio receiver out- oc oc

put and the message error rate of the received information.

From a study of the above definable and measurable parameters

of a radio frequency communication system it is possible to determine

the performance of various types of systems. Direct comparisons can

then be made between the various systems at common points within

the systems, in order to obtain relative figures of merit.

2. 2 Design and Testing of Equipment and Systems

The principal requirement, which may be used as a guide when

designing and testing radio communication systems, is that the average (received) message error rate should not exceed a specified value

for specified message-load conditions on the system. This allowable

average message error rate determines a required time-average
-8-

signal-to-noise ratio at the message-signal decoder-unit input. This

required time-average signal-to-noise ratio at the message-signal

decoder-unit input may be used as a starting point in the design of the

radio frequency section of a communication system. It is understood

that the signal level, at the input to the message -signal decoding unit,

is required to be above a particular minimum level for proper oper -

ation of the decoding unit; proper amplification will maintain this

signal level. However, the signal-to-noise ratio cannot be enhanced

by amplification.

The design of each type of radio communication system can

be carried out for various message-error rates and message loading;

the different types of communication systems can then be compared

for similar grades of service in terms of factors such as radio frequency power required at the transmitter output, required radio frequency spectrum, total cost of system, operational complexity, etc.

The performance and the cost of a radio communication system are

governed largely by the radio-wave transmission loss, fading range,

fading rate and the amount of frequency selective fading of the transmitted radio frequency signal. However, it is convenient to design

the system for a non-fading or steady radio-frequency signal and then

estimate the system performance with a fading type of radio frequency

carrier signal; this is the procedure to be followed in this work. This

procedure is feasible under conditions where the fade rate of the re- ceived carrier signal is low compared with either the duration of the

binary pulse or the lowest frequency in the message signal. Methods

of estimating the performance of radio communication systems, based

on the above principles, are outlined in Sections 6 and 7 and Appendices B. C, and D.
-9-

3. MODULATOR-TRANSMITTER PERFORMANCE

The transmitter -output RF-signal characteristics, such as re- quired spectrum bandwidth, B , average power level, P , and timeri t amplitude distribution of the envelope, depend upon the type of modulation and the type of modulating signal. In order to obtain useful

results it is necessary to derive the relationships between the average

power level of the modulating signal, P , and the characteristics of

the RF signal. For convenience, a sinusoidal modulating signal, with

a power level P at the Modulator input, is used as a "standard" of ms

comparison. Proper or equivalent modulating- signal power levels,

P , for noise -type modulating signals are then obtainable relative to

P by means of the measurable or the specified characteristics of

ms

the Modulator. For amplitude modulation, the Modulator characteristics are in terms of P versus the amplitude -modulation index, ms

m ; for frequency modulation, P versus the carrier-frequency devia

n 3 ms n ' ation, AF, would be required. The amplitude -modulation index, m , may vary from to 100% without serious intermodulation problems;

modulating- signal power levels greatly in excess of those required for

100% amplitude modulation will yield excessive intermodulation noise,

and should not be used. For frequency modulation, the modulating

signal power level, P , determines the RF-signal spectrum bandwidth, B r , and hence the chosen level for P depends upon the choice

rf m

of B and the FM Modulator characteristics,

rf

In this section quantitative relationships are derived between: (a)

the modulating -signal average power level, P , (b) the RF signal

bandwidth, B r

, and (c) the distribution of the time -varying envelope of

rf

the transmitter -output RF signal. The results apply to both a sinusoidal modulating signal and a white -noise type of modulating signal.

SSBSC-AM and FM systems are considered in this analysis.
-10-

3. 1 Required Radio Frequency Signal Spectrum

Bandwidth for Radio Communication Systems.

The required radio-frequency signal spectrum bandwidth depends

upon the bandwidth of the modulating baseband signal and the type of

modulation employed. In turn, the baseband- signal bandwidth depends

upon the number of information or message channels, the bandwidth of

each channel, and the type of multiplexing used to form the basebandsignal spectrum. The problem can be simplified somewhat by considering the radio-frequency signal bandwidth required in terms of a given baseband- signal bandwidth. This procedure makes possible a direct comparison of the radio-frequency signal bandwidth requirements for various types of radio communication systems, with the

same baseband signal, or message load. The separate problem of

minimizing the baseband signal bandwidth required, for a given

number of information channels or message load, involves a consideration of the principles of encoding [ Nyquist, 1924, 1928; Hartley, R.

V. L. , 1928] and multiplexing techniques [ Landon, 1948] and is not

considered in this paper.

3. 1. 1 Radio Frequency Bandwidth Required for SSBSC-AM Systems

In an SSBSC-AM (single sideband suppressed carrier, amplitude

modulated) system the required radio frequency signal bandwidth is

approximately equal to the bandwidth of the composite baseband or modulating signal.

3.1.2 Radio Frequency Bandwidth Required for FM Systems.

In an FM system the required radio frequency bandwidth depends

upon the baseband or modulating- signal bandwidth, the modulation

index, and also the amplitude distribution of the modulating signal, as

will be shown later. The required radio-frequency signal bandwidth
-11-

for FM systems is usually greater than the modulating- signal bandwidth, by a factor ranging from 2. 5 to 25, or more.

3.1.2. 1 Required FM Radio Frequency Signal Bandwidth

for a Single Sinusoidal Modulating Signal.

When a single sinusoidal modulating signal, of frequency f , is

employed to frequency-modulate a (radio-frequency) carrier signal,

the resultant modulated-carrier power spectrum (theoretically) consists of an infinite number of components [ Carson, 1922, 1929; Van

der Pol, 1930; Hund, 1942] . For our purpose we restrict the

required or transmitted radio-frequency signal spectrum bandwidth so

as to include only the "significant" spectral components; that is, we

arbitrarily include only those components having amplitudes equal to

or greater than one percent of the amplitude of the unmodulated carrier-signal amplitude. Under these conditions the "required" radiofrequency signal bandwidth B is given by

rf

B . = f x<j> (AF/f ) (3. 1)

rf c c

where, f = Frequency of the sinusoidal modulating signal,

in c/s.

AF = Peak deviation of the radio-frequency carrier

signal from its un-modulated frequency, in c/s.

This is the frequency deviation above and below

the carrier-signal frequency.

cj> (AF/f ) = Function of AF/f , which gives the number of

significant sidebands in the radio-frequency

spectrum.

For a given value of AF, corresponding to a particular value of

modulating- signal average -power level, P , the radio-frequency

bandwidth increases with increasing values of f . Hence, the maximum ° c value of B P is determined for the case where the frequency

rf
-12-

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-13-

of the sinusoidal modulating signal is equal to f , the highest frequency in the modulating baseband signal. Therefore, throughout this

report the required (maximum) radio-frequency signal spectrum bandwidth, B ., is derived from,

rf

B , = f x <j> (AF/f ) (3.2)

rf m m

where, f = Highest frequency in the modulating baseband

signal, in c/s.

The function (j> (AF/f ) is shown plotted in figure 3-1 for both a sinusoidal modulating signal and for a "noise-type" modulating signal;

the latter case is discussed in Subsection 3.2.3. Note that a smoothed

curve is drawn for the case of a sinusoidal modulating signal.

The decision to include only the RF spectral components having

amplitudes greater than one percent of the unmodulated radio-frequency carrier signal, is based on measured results such as shown

in figure 3-2. The curves in figure 3-2 indicate the effects of bandwidth clipping of the radio-frequency spectrum. These curves also

show the performance of an FM receiver, in terms of the receiveroutput signal-to-noise ratio, S /N , for a received radio-frequency

oc oc

signal at and above the "threshold level" of the receiver. At threshold (or below), the receiver IF bandwidth, B , should be somewhat

greater than the radio-frequency spectrum bandwidth, B .; as given

by (3.2), or the curve in figure 3-1. Above threshold, the receiver

IF bandwidth may be somewhat less than B , without seriously degrading the receiver output signal-to-noise ratio. Hence the use of

(3.2) and figure 3-1 results in system performance estimates which

are conservative above the receiver threshold level and are slightly

optimistic below the receiver threshold level.
14-

RECEIVER OUTPUT Soc /N c RATIO VERSUS B r f/Bjf FOR

VARIOUS RECEIVER- INPUT TOTAL RF SIGNAL POWER LEVELS

STEADY RECEIVER -INPUT SIGNAL

SINUSODIAL MODULATING SIGNAL

-5

-10

.a

o

o

-15

o

-20

-25

-30

-40

1^-2l

RECEIVER INPUT- SIGNAL POWER

RELATIVE TO THRESHOLD LEVEL, db

0.5 1.0 1.5 2.0 25 10

B r f _ RF SPECTRUM BANDWIDTH

B if BANDWIDTH OF RECEIVER IF

Figure 3-2
-15-

3.1.2.2 Required FM Radio Frequency Signal Bandwidth

for Various Types of Modulating Signals.

In the preceding analysis the maximum value of the required

radio-frequency signal bandwidth, B , was derived on the assumption

that the modulating signal was sinusoidal, with a frequency f , equal m

to the top frequency in the baseband, and that all of the modulatingsignal power was concentrated in this one signal. Under these conditions the radio-frequency signal spectrum and the required bandwidth, B f

, will be a maximum for a given set of system parameters

and for a single sinusoidal modulating signal of frequency f We wish to determine the relationship between the required radiofrequency signal bandwidth, B , for a single sinusoidal modulating ns signal at the top baseband frequency, f , and the required radiofrequency signal bandwidth, B . , for a composite modulating signal.

rfn

The spectral components of the composite modulating signal are

assumed to be distributed uniformly throughout the baseband modulating-signal spectrum and the cumulative amplitude distribution of this

signal may be of various types. Radio-frequency signal bandwidth

comparisons are made on the assumption that f is the highest f rein

quency in the composite modulating signal.

The radio-frequency carrier- signal frequency deviation, AF,

varies directly with the amplitude of the (voltage) envelope of the composite modulating signal, and hence the cumulative distribution of the

envelope of the modulating signal is used to determine A F on a statistical basis. That is, the "short-term" values of A F can be determined for various percentages of time, including the maximum or

peak value of A F which occurs during voltage or power peaks of the

modulating signal. It should be noted that the short-term period is

assumed to be short compared with the time variations of the envelope

of the modulating signal.
-16-

The short-term value of the required radio-frequency signal

spectrum bandwidth B , is determined by both the deviation ratio

rf

A F/f and f ; where f is the short-term frequency of the modulatc c c ing signal. The time included in the short-term period is assumed to

be large compared with one cycle of the modulating frequency; during

this period, AF and f are assumed to be constant. However, the

c values of A F and f are assumed to vary independently, with time.

The radio-frequency signal bandwidth, B „ , required for a comrfn

posite-type of modulating signal, is defined here as the radio-frequency signal spectrum bandwidth containing 99. 99 percent of the

spectral energy of the modulated RF carrier signal. This definition

of B r is equivalent to the above definition of B - , for the case

rfn ris

where the modulating signal is a single sinusoid. However, this

bandwidth is exceeded for a small percentage of the time, during

which the (power) peaks in the composite modulating signal exceed

the level required to generate a radio-frequency signal spectrum

greater than B . . Furthermore, these modulating -signal powerlevel peaks, which are required to generate radio-frequency signal

spectrum bandwidths exceeding B r

, will also depend upon the simulrfn

taneous short-term frequency of the modulating signal.

From the above considerations it is evident that both AF and

s AF are the peak values of the short-term AF; while AF is seen to

n

be proportional to the (voltage) envelope of the modulating signal.

3. 1.2.3 Required FM Radio Frequency Signal Bandwidth for a White -Noise Modulating Signal.

Reliable information is scarce on the relationship between the

required radio-frequency signal spectrum bandwidth, B f

, and

A F/f , for modulating signals composed of groups of sinusoidal
-17-

signals arranged in various frequency combinations in the modulating

signal baseband spectrum. However, the relationship between B

rfn

and A F /f has been calculated and measured bv Medhurst [ 19561

n m 7 l / j for the case where 2tt (f -f ) equal- amplitude, randomly-phased sinus- m o oidal signals were assumed to simulate white-noise across the baseband; this artificial white-noise signal was used in his calculations

as the composite modulating signal, where f was the lowest frequency

and f was the highest frequency in the modulating signal. The results

m

of his work are shown plotted in figure 3-1, see "noise modulation"

curve; see also [ Middleton, 1951; Stewart, 1954] .

In figure 3-1, AF is defined as the peak value of the frequency

deviation of the carrier signal. When a sinusoidal modulating signal

is used, AF is constant, and is proportional to the peak amplitude

s of the modulating- signal voltage. For noise-signal modulation, AF

is proportional to the "peak amplitudes" of the voltage envelope of

the noise -modulating signal; these peak amplitudes are exceeded for

only . 04 percent of the time, and are estimated to be 11 db above the

average power value of the noise modulating signal. [Gladwin, 1947;

Alversheim and Schafter, 1 952; Medhurst, 1956]. AF should be

n

considered as a peak deviation which is 11 db greater than the "rms

AF " frequency deviation. The rms AF frequency deviation detern n

mines or generates the RF signal spectrum having a bandwidth, B . Both AF and rms AF are quantitatively defined (and measured) so

n n

as to make proper allowances, for the fact that the short-term deviation, AF, and the short-term modulating- signal frequency, f , vary

with time for noise-signal frequency modulation.

The following derivations are for the purpose of obtaining the

relationships between the frequency deviations, AF and AF , and

the modulating -signal average-power levels, P and P ; for the 6 6 e r ms mn

conditions:
-18-

B , = B , = B (3.3)

rf rfs rfn

and,

f (for sinusoidal-signal mod. ) = f (for noise-signal mod. )(3.4)

The above factors are defined as follows:

AF = Peak deviation of the RF carrier signal for a sinusoidal

modulating signal, in c/s.

AF = Peak deviation of the RF carrier signal exceeded for

n

only . 04 percent of the time; for a white-noise modulating signal, in c/s.

P = Average long-term power level of the sinusoidal modulating signal; measured at the Modulator -input terminals,

with a true rms reading meter.

P = Average long-term power level of the noise modulating mn . , 5 5

, *\ • . , . . 6

signal; measured at the Modulator-input terminals,

with a true rms reading meter.

The advantage of referring our results to sinusoidal- signal modulation conditions is due to the fact that B . , A F , and P may

rfs s ms

either be calculated or may be measured conveniently with standard

types of test equipment; these measured factors may then be used as reference levels. It is inconvenient to calculate or to measure B - , rfn

AF , and P , because AF can only be measured indirectly in

n mn n

terms of B , . rfn

Figure 3-3 shows the "relative levels", in db units, associated

with B ., AF, and P , for sinusoidal-signal and for white-noiserf m ° signal frequency modulation. From figure 3-3,

10 Log (AF /AF ) = 10 Log. _ (P /p ) + 8 (3.5) 10 n s

to l mn ms v

Values of the factor 10 Log, . (iiF /A F ), for particular values of

10 n s B Ji , were obtained from figure 3-1 and substituted in (3. 5) to obtain rf m ° v ' corresponding values for 10 Logn .(P /P ); the results of .these

10 ms mn

calculations are shown in figure 3-4.
19'

RELATIVE LEVELS OF AF AND Pm, SINUSOIDAL AND NOISE

MODULATING SIGNALS AND FOR EQUAL RF SPECTRUM BANDWIDTHS

AFn , db t

NOTE , Brfs

= Br fn

= Brf

f

= {(rmsAFn

) + ll}-{(rms

r„-AFt ),db^ j

x _, , ROM Fig. 3-D I. s \

pmn + "/ \

pms + 3

J

»

(AF,

(FROM Fi

AFS

) +3}

,

db

db

3 db

AFS

, db

rms AFS , db

OR

rms AFn , db

OR

pmn. dbm

II db

SINUSOIDAL SIGNAL

MODULATION

NOISE SIGNAL

MODULATION

Figure 3-3
20-

30

RF SPECTRUM BANDWIDTH, Br f,

VERSUS RELATIVE MODULATING-SIGNAL POWER LEVELS,

FOR FREQUENCY MODULATION

i—

I

1—i—

i

1—i—

i

r~

25

<

CD3

crh-oUJ

o.COu_

ccQ

UJ

CC3

o

UJ

cc

Pms = AVERAGE POWER LEVEL OF THE TOTAL

SINUSOIDAL MODULATING SIGNAL.

WHITE-NOISE MODULATING SIGNAL

NOTE , Brfs = Brfn = B r f

CO

C3 20

3QO

2 15

>-o

z

UJ3

oui

cc

u.

tj E cor4

10 LOG ms

io\P,mn>

Figure 3-4

db
-21-

It should be noted that the RF spectrum bandwidth, B ., is either

rf

given or it may be chosen by the system designer. Also the highest

modulating frequency in the modulating signal, f depends upon the m

number and the type of message channels and also the type of multiplexing equipment being used. Hence the value of B Ji is available

rf m

for use in figures 3- 1 and 3-4.

Figure 3-1 may be used to determine values for A F /f and

s m

AF /f ; for the conditions given by (3.3) and (3.4). The AF vs n m ° ' * s n/P characteristic of the (frequency) modulator should be linear

ms

for values of A F equal to A F , for sinusoidal- signal modulation; or equal to AF , for a white -noise modulating signal. If the modulator

characteristic is non-linear for values of A F , obtained from figure

3-1, intermodulation "noise" will be added to the system noise, and

the signal-to-noise ratio at the receiver output will be degraded.

Figure 3-4 may be used to obtain correct values for P /P ms mn

which correspond to particular values of B /f ; assuming the conditions given by (3.3) and (3.4). The sinusoidal-signal modulatingpower level, P , which is required to produce a frequency deviation

ms

of the RF carrier signal, AF , and an RF spectrum bandwidth, B, , ° s frs

may be obtained from the measured AF vs n/P characteristic

s ms

of the frequency modulator. The required noise -signal modulating - power level, P , may then be obtained from figure 3-4. If pre- mn

emphasis is used, the pre-emphasis circuit must be inserted at a

point in the system which follows the modulator-input terminals;

note that P and P are measured at the modulator -input terminals

ms mn

It is common practice to estimate B . from, r rf

B = f x z(

^ + 1 rf m V fm
-22-

The required bandwidth obtained from (3.6) will include only the

RF spectral components having amplitudes which are approximately

10 percent of the unmodulated RF carrier- signal amplitude, or greater.

More precisely, 99 percent of the RF- spectrum energy will be within

the bandwidth determined from (3.6). Hence the use of (3.2) and

figures 3-1 and 3-4 yield required-bandwidth values, B , (99.99%)

rf

which are somewhat greater that the B A 99 %) bandwidths, which would

rt

be obtained from (3.6). The following relationship between B f (99. 99%

bandwidth) and B A 99 % bandwidth) has been determined, using informarf

tion similar to that used to obtain the curves in figure 3-1 (see Medhurst,

1956):

<t> AF

m

99. 99% bandwidth

J « 1. 4

m ' 99 % bandwidth

or,

B ( 99. 99% bandwidth

J » 1. 4 B J 99% bandwidthJ (3.7)

The use of the smaller RF spectrum bandwidth, obtained from (3.6),

will result in a degradation of the receiver-output signal-to-noise ratio.

The amount of this degradation is shown in figure 3-2.
-23-

3.2 Radio-Frequency Signal Time -Amplitude Characteristics

for Radio Communication Systems

The time -amplitude characteristics, or the distribution of the

envelope of the transmitter-output RF signal, depend upon the statistical characteristics of the modulating baseband signal and the type of

modulation. Other considerations relating to the baseband signal are

the same as discussed in section 3.1.

3.2. 1 Radio-Frequency Signal Time -Amplitude Characteristics

for SSBSC-AM Systems

An SSBSC-AM Modulator translates the baseband- signal spectrum

to an RF- signal spectrum. In an ideal SSBSC-AM system, wherein

there is no modulator distortion, the distribution of the RF signal is

identical to the distribution of the baseband signal. Furthermore, the

average transmitter -output power level, P , is related to the average

power level of the modulating signal, P , in terms of the combined

characteristics of the modulator and the transmitter. For maximum

modulation efficiency and to limit over -modulation effects, the average power level of the modulating signal, P , should be adjusted so as not to exceed a short-term amplitude -modulation index, m , of

100% for more than a given percentage of the time.

3.2.1.1 Radio-Frequency Signal Time-Amplitude Characteristics

of SSBSC-AM Systems for a Single Sinusoidal Modulating Signal

When a single sinusoidal modulating signal of frequency, f , and

power level, P , is employed in an SSBSC-AM system to amplitude

ms

modulate a (radio-frequency) carrier signal, the resultant transmitteroutput RF signal consists of a single (sinusoidal) RF signal.
-24-

The average power level of the sinusoidal modulating signal re- quired for 100 percent amplitude modulation, P (100%), maybe

obtained from the modulator characteristics, in terms of P versus

ms

percent amplitude -modulation, m ; the modulator characteristics may

either be specified or measured. The above value of P (100%), for

ms

100 percent amplitude modulation, may be used as a "limiting reference

level" for the average power of the modulating signal- -for various types

of modulating signals. From a practical viewpoint, it is convenient

to estimate or to measure the performance of an amplitude modulator

using a sinusoidal modulating signal.

3.2. 1.2 Radio-Frequency Signal Time-Amplitude Characteristics

of SSBSC-AM Systems for a White-Noise Type of Modulating Signal

In Section 3.2.1.1 it was noted that the maximum value of (sinusoidal) modulating- signal average power, P , (100%), was obtainable

ms

from the SSBSC-AM modulator characteristics, on the assumption

that the modulating signal was sinusoidal and that all of the modulatingsignal power, P , was concentrated in this one signal, ms ° We wish to determine the average power level, P , of a commn

posite (noise-type) modulating signal which will insure that 100 percent

amplitude modulation of the RF signal is exceeded for no more than a specified percentage of the time. The noise-type modulating-signal

average power level, P , is to be referred to the previously-deter- mn

mined sinusoidal modulating-signal power level, P (100%). mn
-25-

The amplitude of the radio-frequency signal varies directly with

the amplitude of the (voltage) envelope of the composite modulating

noise-type signal, for values of modulating- signal power less than the

"100 percent modulation" level, P (100%). Hence, for noise-signal

modulation the " 1 00 percent modulation peaks", P , correspond to

mnp

particular "peak amplitudes" of the voltage envelope of the noise-modulating signal; these peak amplitude values, P , may be defined in

mnp

terms of P (100%) and also in terms of the average power level of

mn

the noise -type modulating signal, P

For a white-noise signal, the envelope of the signal will be 11 db

or more above the average power level, P , for . 04 percent of the

time [Gladwin, 1947; Alversheim and Schafter, 1952; Medhurst, 1956] . Since the peak value of the sinusoidal modulating signal for 100

percent amplitude modulation, P (100%) is 3 db above its average msp

value, P (100%), and P (100%) has the same level as P (100%), ms msp mnp

it follows that the average sinusoidal modulating- signal power level,

P (100%), is 8 db higher than the average power level of the noisemodulating signal, P --for the condition that the noise -modulating mn

signal will exceed 100 percent amplitude for . 04 percent of the time.

Following the above line of reasoning, combined with the distribution of a noise-type signal [ Plush, R. W. , et al, 19&0] , the ratio,

P (100%)/P , was calculated for a range of percentage-of-time ms mn

during which amplitude modulation exceeded 100 percent. The results of

these calculations are shown in figure 3-5.
-26-

AMPLITUDE MODULATION EXCEEDING 100 PERCENT VERSUS

RELATIVE MODULATING -SIGNAL AVERAGE POWER LEVELS,

FOR SSBSC-AM

99

98

95

90 Q

3

j z 80

:* u

< oc /0 UJ

•- O. z 60 Soa: UJ 50

uj R 0. UJ

UJ 40 o y o x

— UJ 30 o —

i z ?0 $ o

15tr 3 10 3 a

Q o

uj 2

5

0.5

0.2

0.1 005

0.01

Pms ( 100%) = AVERAGE POWER LEVEL OF TH

SINUSOIDAL MODULATING SIGNAL F

E TOTAL

EQUIRED_

FOR 100% AMPLITUDE MODULATION

Pmn = AVERAGE POWER LEVEL OF THE

TOTAL WHITE-NOISE MODULATING SIGNAL

6 -< -\! ( ) i i i ; t i 3 12 iolog io dss =mmr, l0 [ pms( |00%)

] t db

' mn

Figure 3-5
-27-

Figure 3-5 may be used as a guide in determining the (approximate)

proper average power level, P , for the noise-type modulating signal,

in terms of the measurable average power level, P , of a sinusoidal

ms

modulating signal, and the percentage of time that 100% amplitude

modulation conditions are exceeded. However, figure 3-5 does not

yield accurate estimates of the intermodulation power generated within

the Amplitude -Modulator, due to over-modulation; this work should be

extended.

The factor, D = P (100%)/P , in the abscissa scale of ss ms mn

figure 3-5, is required in the design or the performance-testing of

SSBSC-AM systems, for proper adjustment of the modulating- signal

power level, at the modulator input terminals.

3.2.2 Radio-Frequency Signal Time -Amplitude Characteristics

for FM Systems.

In an FM system the amplitude of the transmitter -output RF

signal is independent of the characteristics of the baseband modulating

signal.

3. 3 Peak Power Requirements for SSBSC-AM Systems

In AM (amplitude -modulated) systems, the transmitter-output

short-term peak power levels, P , should be related to the transmit - — — tp

ter-output average power level, P , in precisely the same manner

that the short-term peak power levels, P , of the composite modu-

— — mnp

lating signal, are related to the average total power level of the modulating signal, P . These relationships are required in the transmit - mn

ter output in order to avoid excessive intermodulation distortion in the

transmitter -output RF signal during periods of peak modulating power.
-28-

The above peak-power capability requirements for the transmitter

depend directly upon the amplitude -time distribution of the envelope of

the composite baseband modulating signal. A composite modulating

signal which is composed of a number of independent voice-message

signals combined with teletype -message and data-phone message

signals, may be assumed to have a distribution which approximates

the distribution of a white-noise signal. For a white-noise signal

having a normal distribution, the peak power exceeded for one percent

of the time is 7.3 db above the average power level of this signal.

Other combinations of modulating signals, such as 8 or more equalamplitude randomly-phased sinusoids, have distributions [Slack, 1946] ,

[ Plush, R. W. , et al, I960] in which the composite-signal peakpower-level amplitude, P , exceeds the average power level, P , mnp m

by only 8 db for one percent of the time. Estimates of the percentage

of time during which the transmitter -output short-term power exceeds

its average power level, P , by 1 1 db, may be obtained from figure 3-5.

One method of qualifying the transmitter characteristics is to

specify that the transmitter (power amplifier) must be linear to within

a particular degree, usually one percent, for transmitter-output-power

peaks, P , approximately 1 db above the average power -output level,

tp

P , of the transmitter.
-29-

4. RADIO RECEIVER- COMBINER PERFORMANCE

This section deals with the performance of the radio receiver considered as a unit in the radio communication system. The method of

analysis is general and may be applied to any type of receiver, provided that the steady received- signal performance characteristics of

the receiver are available, either measured or calculated.

In the following work, the receiver performance characteristics

are first determined for a steady received signal. The receiver performance, that is, the receiver output signal-to-noise ratio, is then

calculated for a time -varying received signal by combining the steadysignal receiver characteristics with the cumulative distribution of the

received radio -frequency signal. The effects of various orders of

diversity on the resultant signal-to-noise ratio in the combined baseband signal are included in the analysis by combining the steady- signal

characteristics of the radio receiver with the performance characteristics of the combiner; at this point, there may be a choice between

predetection and post detection combining.

Radio receiver performance is defined here in terms of the relationship between the receiver-output signal-to-noise ratio, S /N , and the following factors:

(1) receiver-input radio-frequency signal total available power,

P ; see Appendix A for definition of available power,

r

(2) available receiver-input noise power, N ; which is composed

of the noise power received by the antenna plus the noise

power which originates in the receiving-antenna system,

(3) noise power contributed by the radio receiver, (F - l)kT B

= kT B. r and referred to the receiver input,

er if

(4) the power level of the modulating signal, P ,

(5) the signal-power loading ratio, P /p ., in the modulating

baseband signal message channel.
-30-

(6) message -signal bandwidth, B , and

(7) the highest frequency in the modulating baseband signal, f The performance of a radio receiver, as defined above, depends

upon the type of modulation being used and also depends upon the particular individual characteristics of the receiver. The required radiofrequency total signal power, P , at the receiver input, is directly

related to the total noise power at the same point, (see Appendix A).

The nature of the radio receiver output signal-to-noise ratio is

dependent upon the statistical characteristics and the median power

level of the radio-frequency signal at the receiver input; for example,

the characteristics of the output noise of a FM receiver, when operating in its threshold region, differ considerably from the output noisepower characteristics above the threshold region. The message error

rate, for a given receiver output signal-to-noise ratio, will depend

upon the characteristics of the noise at the receiver output, hence this

factor is important when considering the overall performance of a radio communication system.

The above-listed factors (1) to (7) inclusive, are treated in Section

4. Detailed methods of applying the above analysis, to determine the

radio receiver output signal-to-noise ratios, are given in Subsections

4.5 and 4.6; in which are developed quantitative performance characteristics in graphical form, for SSBSC-AM and FM receivers, respectively.

This work includes steady received-signal and Rayleigh-fading re- ceived-signal performance curves for non-diversity, dual diversity,

and quadruple diversity. Maximal- ratio type of combining is assumed.

4. 1 Radio-Frequency Signal Characteristics and Power Levels in the

Receiving System.

The radio frequency signal power level at the receiver input, P ,

is a key factor in the performance of a radio communication system;

this condition applies for either a steady or a time-varying received
-31-

signal. The power which the radio transmitter must provide is directlyrelated to the power P required at the radio receiver input through

the total transmission path loss. Furthermore, the performances of

various types of radio receivers are best compared in terms of the

receiver-input RF signal power levels required for the same value of

receiver -output signal-to-noise ratio; this method of comparing

receivers determines the receiver "sensitivity" and makes the proper

allowance for the receiver noise figure and the receiver IF noisepower bandwidth. In practice, it is also found to be more convenient

to measure the receiver-input RF signal power than to measure the

pre-detection signal -to -noise ratio. This situation is particularly

apparent to the system-testing engineer when measuring a time-varying received signal, and for cases where automatic gain control (age)

and/or limiting is used in the receiver.

The following characteristics of a time-varying received radiofrequency signal are of importance, insofar as they affect the receiver

output signal-to-noise ratio:

(a) Median power level, for the sampling-time or sampling-period

(b) Cumulative distribution of the envelope of the received radiofrequency signal

(c) Fade rate at the median power level

(d) Time variability of the amplitude and phase cross-correlation,

between the spectral components of the received radio frequency signal- -due to "selective fading" effects and multipath radio-wave propagation conditions.

The effects of the above listed received- signal characteristics (a),

(b), and (c) are considered later in this paper, to estimate radio receiver performance for a time-varying radio-frequency signal. For

tropospheric-propagated radio-frequency signals, (a) may be calculated [Rice, Longley and Norton, 1959] , (b) has been extensively

measured and has been found to closely approximate a Rayleigh distribution, and (c) has been found to vary between approximately 0. 1 cps

to 1 cps.
-32-

Additional experimental work is required to determine an

accurate measure of the relationship between characteristic (d ) and

the radio receiver output signal-to-noise ratio; at present, experimental measurements of this type are very meager. In cases where

relatively wide radio -frequency spectra are used, it is necessary to

consider the non-correlated amplitude and phase variations between

the signal components across the radio frequency and the baseband

spectrums. These factors influence the receiver output signal-tonoise ratio and the message error rate and are usually involved in the

transmission characteristics of the radio-wave path. The path-bandwidth cross -correlation or covariance factor is probably a function of

the beam width of the antenna -pattern, length of path,' and carrier frequency. Measurements of the tropospheric radio-path bandwidth capability have been made by Clutts, Kennedy and Trecker [ 19&0] on a 185 mile path between Florida and Cuba.

From theoretical work by Staras [1955] we have the following

relationship:

P = ^3— Mc/s (4.1)

where d = radio wave transmission path length in hundreds

of miles

(3 = correlation (radio frequency) bandwidth in Mc.

By definition, the cross -correlation coefficient

of the radio-frequency spectral components

spaced (3 Mc/s is 0.5.

Equation 4. 1 is probably conservative and hence can be used to

obtain an estimate of the usable radio frequency bandwidth in cases

where antenna beam widths are equal to or greater than the angle subtended by the effective "scatter volume". The usable radio frequency

bandwidth is much greater for cases where the antenna beam width is
-33-

considerably smaller than the angle subtended by the scatter volume.

Where the antenna beams are of the order of 1 degree or less a somewhat larger value of usable radio frequency bandwidth could be assumed.

Narrow-beam antennas reduce the multi-path effects and thereby result

in an increase of usable radio frequency bandwidth; see section 4. 3.

4. 2 Pre-Detection Noise-Power Levels in the Receiving System

The noise performance of the receiving system depends upon the

amount of noise power which is contributed by the various units in the

receiving system, and which subsequently interferes with the pre-detection received signal, P. f

. In this analysis an estimate is made of

the total noise power contributed by the entire receiving system, from

and including the receiving antenna to the pre -detection point in the

radio receiver.

In order to make the results of this analysis convenient for use in

system design work and performance tests, the total of the noise power

contributed by the receiving system is. referred to the radio receiver

input, in terms of the receiving-system parameters. Using this procedure, the system designer can accurately estimate the level of the

noise power at the receiver input; this same noise-power level can be

conveniently measured, when performance tests are made, by using a calibrated (linear) receiver—where the receiver is calibrated in terms

of receiver -input RF signal power versus receiver IF-output signal

power or, the receiver age voltage. Only amplitude -type of noise

power is considered, as distinguished from phase or frequency-modulation noise; the latter is measurable only in the output of an FM

receiver, and does not appear as amplitude noise in the predetection

circuits of the receiver. Post-detection noise and its effects on the

receiver-output signal-to-noise ratio is considered in section 4. 3.
-34-

In this work all noise powers in the receiving system are added

algebraically on an average power-level basis, as measured by a true

rms-reading meter with an appropriate time constant. The noise

power developed within the receiving-system units is assumed to be

"thermal" type noise and to have a flat power spectrum over the bandwidth accepted by the receiver. Interference from other radio transmissions, which will introduce an effective noise component designated

as N .in Appendix A, are excluded from the present analysis, since

their inclusion would require a consideration of the statistical characteristics of the interfering signal. This task is beyond the scope of

this report.

The transmitter -output "noise" power N , associated with the

received RF signal, is composed of "transmitter amplitude noise" and

"transmitter intermodulation noise. " Transmitter amplitude noise usually originates in the transmitter

output stages; this noise is present even though the transmitter is not

being modulated, and may be measured, relative to the un-modulated

RF carrier signal, at the transmitter output. It may be measured by

connecting a transmitter and a receiver back-to-back through an attenuator, and measuring the receiver baseband -output noise power,

under conditions of no modulation on the transmitter carrier signal.

The receiver-output total noise, N , (see figure 4-A) is measured

for a range of receiver-input carrier-signal powers and the results

plotted. The relative value of transmitter noise is indicated at that

portion of the curve where the receiver-output noise does not decrease

with further increase in the received carrier-signal power. The transmitter-output signal-to-noise ratio should be high enough to insure that

the "leveling-off" portion of the curve will be above the operating or

required RF signal-power levels of the receiver performance characteristic.
-35-

The net effect of transmitter -output noise is to place an upper

limit on the receiver -output signal-to-noise ratio, and is similar to

the effects of (a) the signal-to-noise ratio at the modulator input

terminals, (b) equipment intermodulation noise, and (c) path-modulation

effects. The latter subjects are discussed in section 4.3.

The conclusions to be drawn from the above discussion of the

transmitter-output amplitude noise, N , are that proper modifications

at

for the effects of this noise power can be made on the system design

curves and also on system-performance test results, and that N

at

need not be considered when calculating the noise levels or the performance of the receiving -antenna system. Consequently, N does

not appear in the expression for the receiving-system available noisepower level, N , or effective noise temperature, T , at the receiverJ- J\. input terminals; see figure 4-1 and Appendix A.

Following is an outline of the method used to estimate the total

(average) available noise-power level, N , at the (radio frequency)

receiver input. Referring to figure 4-1, the noise power, N , contributed by each unit in the receiving system, and subsequently referred to the receiver-input terminals, is estimated in terms of the parameters of each unit, such as its thermal temperature T , transmission

x

loss Lxx, noise figure F , etc. In the following derivations it is

assumed that non- reflecting impedance -matched conditions exist at the

junctions of the various units in the receiving system, except (possibly) <

at the receiver-input terminals, and that each unit is constructed with

input and output positive -resistance terminations; hence each unit,

preceding the receiver, delivers the "available" noise power, N , at

its output terminals. See Appendix A for methods of estimating noise

power levels in the receiving system, including a discussion of the

effects of impedance -mismatching at the receiver -input terminals.
36.

UJ

h-V)

>-CO

UJoUJ

a:<

ccCD<OO

_J

CD o oo

10 ZK

o: P > < i _i

UJ 3 o Q ui o

cc 2

UJa

«•- •^ £ z

,_ CD - ' 1* ^

ff,

m * m"

.> - - -

p. !_. -* UJ* i* ll ii

CM i ,. Z

or3 + „ o 3 q!" or N k> ^ * "1

rl 5" z J

—

\ 1d| ^-6 ' -J S °Lj" 1 »- z Xm < t O to

0? 4*1

ht 3am 1 X

u. -1 z<

p:

u.

F

T ttT

4- 1 o!p.i z z 1 2 «. J 1 uj * zi 1 ?

UI

J-O i

uT 1 z

i- • r1 oJP! z° aa <Z t h?

CD.o z 3 l~ UJ U "

-3£ 1- o: -i — z< o u° 1 CJ _lT o 1 Q-, <z

toUI

UJ <

UJ

to ._ E 1 UJ 1 _J

a

_l<

o: 2 z a . 1 CD > z

u. z1 UJ

w Z O 5 h- 1 < a

CO

a

a. (O H Li- _l > UI ' < o

</> z ui Z. \- I 5 UJ u_ o < o °

UI z

tr UlH°

tooz < ct cc
-37-

Referring to figure 4-1, the total receiving-antenna system output

noise power, N' , available at the receiver-input terminals and within

the bandwidth B (=B ) "accepted" by the receiver, is given by (A. 25)

of Appendix A and is re-written here as (4.2)

N' = kT. B. r (4.2) T A if v '

-23 where, k = Boltzman's constant = 1.3804x 10 joules

per degree Kelvin

T = Effective output noise temperature of the receiving-antenna system, degrees Kelvin

B = Effective receiver IF bandwidth, in c/s, see

1 (A. 18)

The effective output noise temperature of the receiving- antenna

system, T , is defined by (4.2). T is a function of the receivingantenna system parameters and is given by (A. 30) of Appendix A, and

reproduced here as (4. 3),

*

T

a+( V1)T

c +Lc (Lt1) VLc Lt (Ldu- 1)Tdu PH

t

a= rrr + S53i < 4 - 3) c t du if D

The value of T , in degrees Kelvin, may be calculated by means

of (4. 3), by substituting values for the various receiving-system parameters; these parameters are completely described and defined in Appendix A.

If the thermal temperatures of the various units in the receivingantenna system are equal to T , and if impedance -matching conditions

exist throughout the receiving-antenna system, (4.3) becomes,

T -T P..

a o n , a -, \ + T + ——=

(4.3a) A L L Is, o kB.,Ln c t du if D

The portion of the noise power, N' , which is "delivered" to or

absorbed by the receiver input circuit is,
-38-

N

dT =Mir

N

T

' 4 - 4)

where, M. = Impedance mismatch factor, at the receiver

ir

input terminals. This factor is a function of

the impedance at the antenna- system output

terminals, Z , and the receiver-input impedance, Z. , and is defined by (A. 16) as,

ir

4R R.

M. = ^_il (A . l6 ) " l Z

H +Z -

I du ir

where, R , R = Resistive or "real" component of the complex

ir du

impedances Z and Z , , respectively,

ir du

The noise power contributed by the linear section of the receiver,

and delivered to the receiver-input terminals, is closely approximated by,

M. {(F -l)kT B.J = M. kT B., (4.5)

ir r oif ir erif

where, F = Receiver noise figure [ IRE Comm. on noise,

I960] , as a ratio

T = Standard reference temperature, 290 degrees

Kelvin [ IRE Comm. on noise, I960]

T _ = Effective receiver input noise temperature [ IRE

Comm. on noise, i960] , degrees Kelvin

From (4. 5) we obtain:

T = (F - 1)T (4.6)

er r o

The total noise power delivered to the receiver-input terminals

is equal to the sum of the noise powers given by (4.4) and (4.5) or,

N , + M. {(F -l)kT B.,} = M. (T A +T )kB.. (4.7) dT ir r o if ir A er if
-39-

The dependence of the total noise power at the receiver input,

relative to the noise temperatures T . and T , is clearly evident from A er

(4.7).

The total predetection noise power level is equal to the total noise

power' ^ at the receiver input, ^ M. (T . + T )kB. r , multiplied by the

lr A er if r y receiver gain, G . The load-reflected [ Siegman, 19&1] noise power

in the predetection circuit of the receiver is assumed to be negligible

for communication-type receivers, and is not considered here; however, for extremely low-noise negative-resistance and/or bilateral

types of receivers this noise power may be important. Hence, the

total average noise power at the receiver-detector input point, N. .,

is given by:

N. r = G M. [T A + T ] kB. r (4.8)

if r lr A er if

where, G = Receiver power gain, as a ratio. This factor is

defined as the ratio of the signal power delivered

to the receiver input terminals to the available

signal power in the receiver IF circuit.

When using (4.8) to estimate the predetection total noise power

level, N.,, the first step is to determine the numerical value of the

factor T . From (4.3) it is seen that the numerical value of T

depends upon the receiving-system parameters. All of these parameters except T may be either measured or estimated with a high order

of accuracy. T varies with time and antenna orientation and hence ' a may (in some instances) be estimated only to a low order of accuracy.

The antenna noise temperature, T , or noise power, N , depends

a a upon the RF carrier- signal frequency [ CCIR Report 65, 1957] , [Crichlow, Smith, Morton and Corliss, 1955] , [Hogg and Mumford,

I960] . However, in the microwave frequency range it may be assumed [ Grimm, 1959] that T varies between 50 to 300 degrees Kelvin,

a
-40-

provided that the main lobe of the antenna pattern does not include the

sun or an intense radio star.

The remaining° factors in (4.8), G , T , and B. r , are obtainable

r er if

from the receiver specifications; the substitution of these factors in

(4.6) and (4.8) together with the previously calculated Value of T , gives the predetection noise power level, N. f

, and the receiver-input

total delivered noise power level, M. (T . + T )kB.., for the

lr A er if estimated and the specified receiving- system parameters.
-41-

4. 3 Post-Detection Noise Power Levels

in the Receiving System

Post- detection noise, which appears at the receiver output, is

generated at various points within the system and is due to various

causes as listed below:

(1) Pre -detection (amplitude) noise, N , at the detector input.

(2) Noise power at the modulator input, N , relative to the m

total modulating- signal power, Pm

(3) Inter modulation noise, generated within the equipment.

(4) Radio-wave path-modulation noise, imposed on the RF signal

by transmission over the radio-wave path.

A complete analysis of post-detection noise requires a consideration of the statistical properties of each of the above noise powers in

order to obtain the characteristics of the resultant total noise power

per message -signal channel, N , at the receiver output. The effects

of N on system performance may then be determined in terms of

oc

the resultant average signal power -to -average noise power ratio,

S /N , at the receiver output,

oc oc

Throughout this analysis it is understood that the post-detection

noise is "available" at the point where it is being measured; usually,

noise -power measurements are made for impedance -matched conditions and within a particular and carefully defined effective bandwidth.

In FM systems, the post- detection noise level will depend upon its

position in the baseband- signal spectrum, and will be a maximum in

the top channels of the baseband signal, when the receiver -input

signal is above the threshold point. If noise power levels are meassured at a point beyond the receiver output, such as the output terminals of a multiplex unit, due allowances must be made for the noise

properties of the intervening units and the power gains or losses in

these units
42-

FM RECEIVER-OUTPUT SIGNAL AND NOISE IN

THE TOP CHANNELS OF THE BASEBAND SIGNAL

3

Q.I-3O

i

>

UJ .J a

o < a: o z

Z O H

w d ^ z or o < UJ

-

• i

UJ I Seco

to

Ul

o

o

CO

I o

COo

z

<

Pr

RECEIVER -INPUT RF-SIGNAL AVERAGE POWER

Figure 4 -A
-43-

4.3. 1 Post-Detection Noise Due to Pre-Detection

Amplitude Noise

The effect of pre -detection noise, item (1) above, on the postdetection noise level, shown by the N curve in figure 4- A, is considered in detail in sections 4. 5 and 4. 6. The design equations and

the design curves developed in this report are based primarily on the

effects of the pre -detection noise with the understanding that the

transmitter -output (amplitude) noise, N , was not included in the

at

pre -detection noise. The reason for this omission may be seen from

an inspection of figure 4-A, where the N curve gives the receiveroutput noise, which is due to the pre -detection noise, N , excluding

the transmitter -output noise. The receiver -output noise which is

due to the transmitter -output noise is dependent upon the transmitter

output signal-to-noise ratio and is shown by the N curve. It is

evident that the receiver -output noise, N , effectively limits the

maximum value of the receiver -output r signal- b to-noise ratio, S /N

oc oc Modifications of the receiver-performance characteristics for

the effects of that portion of the post-detection noise which is due to

items (2), (3), and (4), are considered below.

4.3.2 Post-Detection Noise Due to Noise

at the Modulator Input

The design equations and curves in this report were based on the

assumption that the signal-to-noise ratio in the modulating (baseband)

signal, at the modulator input, was very much greater than the required

(baseband) signal-to-noise ratio at the radio receiver output terminals.

The net effect of noise at the modulator input is to place an upper limit

to the receiver -output signal-to-noise ratio. In general, we cannot

expect to obtain a signal-to-noise ratio at the receiver output higher

than the signal-to-noise ratio at the modulator input. The post-detection
-44-

or receiver-output noise -power level, due to noise N , at the modulator input, is shown by the N curve in figure 4-A.

Estimated and measured system-performance characteristics must

involve the baseband signal-to-noise ratio at the modulator input, as a

factor which limits the radio receiver -output signal-to-noise ratio.

The actual value of the receiver -output baseband signal-to-noise ratio

will be lower than the modulator -input baseband signal-to-noise ratio,

and will depend upon the signal-to-noise "degradation factor" of the

radio -frequency section of the communication system.

4.3.3 Equipment-Intermodulation Post-Detection Noise

Intermodulation noise is generated within the equipment whenever

the limits of linear modulation or demodulation are exceeded and when

the receiver IF effective bandwidth, B._, is insufficient. For FM

if

systems, the post-detection noise power level, due to equipment

intermodulation noise, is shown in figure 4-A, by the N curve.

Hence, the post-detection (equipment) intermodulation noise-power

levels may be estimated from the characteristics of the modulator,

the demodulator, and the passband characteristics of the receiver IF

circuits, using the material in Section 3.

For amplitude modulation, the required modulator characteristic

is in terms of the sinusoidal modulating- signal power level, P , versus the percent modulation. The percent of time during which

100% amplitude modulation is exceeded, for particular power-levels

of a noise -type modulating signal, P , may be obtained from figure

3-5. The percent of time during which 100% amplitude modulation is

exceeded may be used as an estimate of the percent of the modulating

signal power which is converted to noise power. By arbitrarily assigning a limit to the percent of time during which 100% amplitude modulation
-45-

may be exceeded, the value for D __ may be obtained from figure 3-5.

At the same time, the corresponding value for the average power level

of the noise -type modulating signal, P , may be obtained from

mn

figure 3-5. It is evident from figure 3-5 that intermodulation noise

in the modulator is dependent upon the factor D ; low values of inter

ss modulation noise require high values for D , and vice versa.

ss

If the amplitude demodulator is linear to 1 00% amplitude modulation of the RF signal, it will not add appreciable intermodulation noise

power to the system.

For frequency modulation, the required modulator characteristic

is in terms of the sinusoidal modulating- signal power level, P , & B ^ ms

versus the peak frequency deviation, AF . The modulator characters istic should be linear for values of A F somewhat greater ° than AFs in order to prevent excessive "distortion" of the RF spectrum during

the time when the peaks of the noise -modulating signal occur. This

distortion is comparable to that produced in an amplitude modulator

when the modulating- signal power level exceeds P (100%); the net

° ° ms

effect results in distortion of the baseband signal at the demodulator

output, and the production of receiver -output noise, shown by the N . curve

.

The relationship between P /P , for FM systems, and the

ms mn

required RF spectrum bandwidth, B f

, is shown in figure 3-4. The

effect of "bandwidth clipping" in the receiver, on the receiver -output

signal-to-noise ratio, S /N , is shown in figure 3-2. By combining

oc oc

the modulator characteristic with the data in figures 3-2 and 3-4, it is

possible to estimate the relationship between the increase of the postdetection noise level or the degradation of the receiver-output
-46-

signal-to-noise o

ratio, and the modulating- signal ° power level, P

mn

The results will be relative to the FM receiver threshold point and

the bandwidth ratio, B r/B...

rf if

If the FM demodulator characteristic is linear to ± A F it will n

not add an appreciable amount of intermodulation noise; however, if

this condition does not hold the demodulator characteristics

should be included in the above analysis of intermodulation noise

effects

.

The portion of the receiver-output noise power, N , due to

modulator-transmitter -receiver intermodulation effects, may also be

measured by means of a transmitter-receiver back-to-back test. For

this test, the receiver performance characteristic is first measured

using a modulated RF signal having a negligible amount of intermodulation noise; this condition is attained by using a comparatively low

value for the modulating -signal power level, P . For multichannel

systems, the modulating signal should be white noise which is transmitted through a band-rejection slot filter; noise and signal measurements are made at the receiver output within the clear channels or spectrum "slots." Meaningful results will only be obtained if the

entire system is properly adjusted for normal operation. The modulating- signal power level, P , is then increased in steps, and either

or both the receiver -output increase in noise or the change in the

signal-to-noise ratio are measured for each modulating-signal power

level, P . , and for a range of receiver -input RF signal power levels,

P . From the results of these tests it is possible to determine the

r r relationship between the modulating-signal power level, P , and the

amount of intermodulation-noise power; in terms of the received RF

signal power, P , required to produce a given receiver -output signalto-noise ratio--for particular values of the modulating-signal power,
-47-

P . For FM systems the results would be similar to those shown in m

figure 4-A.

Since the design curves in this report were measured and calculated

under conditions of negligible intermodulation noise power; the above

information may be used to make the proper corrections for the effects

of equipment intermodulation noise in the design curves; see section

4.3.6. Alternatively, departures of system-performance measured

results from design-curve values may be attributed in part to equipment intermodulation-noise effects, and corrections may be made

from the above data, after due allowances have been made for all other

factors.

The results of the above intermodulation-noise tests will include

intermodulation effects in both the modulator -transmitter unit and the

receiver. Furthermore, the relationship between the modulatingsignal power level, P , and the system intermodulation noise power

will be obtained; these equipment-performance data should be used as

guides for setting up system-performance tests and for operation of

the particular system under test. The above types of tests may also

be conducted on a complete system, utilizing an over -the -path signal.

In fact, this method of testing is necessary in order to determine the

performance characteristics of the combiner and to obtain measurements of the path -modulation noise; see section 4.3.4. A completesystem test is not as convenient to conduct as an equipment back-toback test, primarily because of the lack of control of the received RF

signal, in the former case.

4.3.4 Post-Detection Path- Modulation Noise

Radio wave path -modulation effects on the transmitted RF signal

are due to "frequency selective fading" of the RF-signal spectrum
-48-

components, which results in distortion of the received-signal spectrum

and consequent distortion of the baseband- signal components. Path

modulation arises primarily because of the differences in the relative

delays of the (received) RF- signal spectrum components due to multipath conditions. In a multichannel FM system, the effects of path

modulation appear at the receiver output as interchannel modulation

noise. In an AM system the effect of path modulation on the receiveroutput noise is somewhat different from that in an FM system; however,

the overall effects on system performance are similar for both types

of systems.

The severity of selective fading increases with the RF-spectrum

bandwidth and hence determines the "transmission bandwidth capability"

of a radio-wave path. Selective fading, and therefore, transmission

bandwidth capability, depend also on the type of RF signal, such as HF via ionospheric reflection, tropospheric, line of sight, etc.

A measure of the degree or the amount of selective fading may be

obtained from the cross -correlation of the amplitudes of the spectral

components of the received RF-signal. However, a more direct and

useful measure of the transmission-bandwidth of a radio-wave path

may be obtained by measuring the path -modulation noise in the baseband signal at the receiver output [Clutts, C. E. , et al, I960] . The

power level of the path -modulation noise will depend upon the RFsignal spectrum bandwidth, B , and the various system-path parameters, such as path length, antenna beamwidth, and to some extent,

the order of diversity and the RF-signal frequency. This noise is

time -varying and hence its study requires statistical treatment.

Path -modulation acts the same as signal modulation and sets a

lov/er limit to the receiver -output noise and hence, determines an

upper limit to the receiver -output signal-to-noise ratio, regardless
-49-

of the power transmitted or received. For a given system, RF bandwidth, and type of RF signal, any apparent dependency of the median

receiver -output path -modulation noise level on the median received

RF signal level is due primarily to variations of the type of received

RF signal. However, the path -modulation noise level at the receiveroutput will also appear to be dependent upon the median received RF

signal, for RF- signal levels below the receiver threshold point; see

figure 4-A. Hence, the net effect of path -modulation noise is similar

to the effects of either noise at the modulator input, or equipment

intermodulation noise. Receiver -output noise power, due to pathmodulation effects, is shown by the N curve in figure 4-A.

4. 3. 5 Total Post-Detection Noise

The total receiver -output noise -power per message -signal channel,

relative to the received RF- signal, P , is shown by the N curve in

r oc

figure 4-A. The absolute levels of the noise-power curves, N to

N , inclusive, are dependent upon the various factors described

05

above. These curves are positioned as shown in order to indicate

their relative effect on the level of the noise curve, N , and the

oc

resultant effect on the receiver -output signal -to -noise ratio curve,

S /N . The shapes of the curves are dependent upon the type of

oc oc

modulation-demodulation employed and include the effects of age

(automatic gain control) and limiting, in the receiver. For the case

where system back-to-back and over-the-path tests are made, the

curves in figure 4-A may be measured accurately. However, it

should be understood that the absolute levels of the post-detection or

receiver-output noise powers, N to N , inclusive, are dependent

upon power gains (or power losses) within the receiver, therefore,

these noise-power levels are meaningless unless they are referred
-50-

to the post-detection or receiver -output signal - power level, S

Hence, estimates and measurements of receiver -output noise power

levels should only be made for conditions where signal-power levels

are properly adjusted throughout the system.

4. 3. 6 Effect of Post-Detection Noise on Receiver

Characteristics and Method of

Correcting System-Design Curves

The actual performance characteristic of the receiver will be as

indicated by the "corrected" receiver -output signal-to-noise ratio

curve, (S /N )', in figure 4-A. The corrected curve is obtained

oc oc from the "ideal" curve byy reducingB the ideal-curve values of S /N

oc' oc an amount equal to the noise -power-level increase (in db units) between

the 1ST . curve and the N curve; for particular values of P . These

01 oc r corrections also apply to the design curves in this report, which are

identical to the ideal curve shown in figure 4-A. In practice, corrections to the design curves would be in terms of the additional required

receiver -input RF-signal power, AP . This additional power would

be required to attain a particular value of receiver-output signal-tonoise ratio, (S /N ) . , or a particular grade of service. The actual

oc' oc i r 6

value of AP would be approximately equal to the noise -power -level

difference between the N . curve and the N curve. Note that this

01 oc

type of correction is limited by the "levelling -off " point of the corrected

receiver-characteristic curve; beyond this point, no improvement in

the receiver -output signal-to-noise ratio will be obtained by increasing

the receiver-input RF signal power, P . r The net effect of post-detection noise on the performance of a radio

communication system may be determined from the corrected receiver

characteristic curve. The use of the corrected receiver characteristic

curve results in an accurate determination of the distribution of the
-51-

time-varying post-detection (baseband) signal, noise and signal-tonoise ratio; these distributions are very important factors in the performance of either single-link or tandem-link radio systems. The

performance of single-link and tandem-link radio systems should be

determined from the distribution of the time-varying post-detection

(combined) baseband signal-to-noise ratio. Any attempt to estimate

the performance of radio systems solely in terms of the pre-detection

signal-to-noise ratio, is likely to lead to the erroneous conclusion that

this type of solution is complete and general.

For single-link systems, the characteristics of the post-detection

or receiver-output signal, noise, and the signal-to-noise ratio would

be obtained by the methods outlined above.

In tandem-link systems, the signal and the noise at the intermediate points in the system may be "transferred" either at the pre-detection point or at the post-detection point. If the signal and the noise are

transferred at the pre-detection point, signal de-modulation is not involved at the intermediate points in the system and the receivers are

used as linear repeaters at these points. Hence, in this type of system

the characteristics of the intermediate -point pre-detection combined

signal-to-noise ratio will influence the transmitted and the received

(RF) signal-to-noise ratio. However, in a linear-repeater type of

system, the system-intermodulation noise which is developed within

each of the tandem links, is transferred through the system and

arrives at the output of the terminal demodulator. Therefore, in a tandem-link system which employs linear repeaters at the intermediate

points, the total system-intermodulation noise at the terminal-demodulator output may be approximated by the sum of the system-intermodulation post-detection noise in each of the tandem links.
-52-

In tandem-link systems employing demodulation at the intermediate points in the system, the characteristics of the post-detection or receiver-output signal and noise at each intermediate point would be

determined by the methods outlined above for single-link systems.

The total receiver-output noise, N , at an intermediate point in the oc system would be applied to the input terminals of a modulator, and

hence this noise power (N , see section 4. 3. 2) would influence the m

system-intermodulation noise developed in the succeeding link of the

system. Under these conditions, the performance of each succeeding

link in a tandem-link system becomes progressively worse because of

the progressive deterioration of the signal-to -noise ratio at the modulator input terminals.

Therefore, the characteristics of the post-detection noise at the

output terminals of a tandem- link system will depend somewhat upon

whether the intermediate links are operated as

(a) linear repeaters, which transfer the received signal and

noise at each intermediate point in the system to another

portion of the RF spectrum; or

(b) as receiver-demodulator units which produce the baseband

noise (and signal) at each receiving-point in the system.

In either case, the characteristics of the post detection noise may be

determined by the methods outlined in section 4. 3.

The case of re-construction of the message signal at each intermediate point in a tandem-link system is equivalent to single-link

operation, and the total message errors would be equal to the sum of

the message errors in the individual links.
-53-

4.4 System Parameters Which Affect the Message-Channel

Signal-to-Noise Ratio at the Radio Receiver Output

In the previous Subsections we discussed the effects of predetection and post-detection noise on the receiver -output signal-to-noise

ratio, S /N . In this Subsection we will consider the relationship

oc oc r between the message -channel signal-to-noise ratio, S /N , at the oc oc

radio receiver output, and the following factors: (1) the average total

power of the composite modulating signal, P , (2) the individual message-channel average signal-power level, p , in the modulating baseband signal, relative to the total average power level of the modulating baseband signal, P , and (3) the message-signal spectrum bandwidth, B . It is shown later that these parameters determine the per

c message -channel signal-to-noise ratio at the radio receiver output,

in terms of the total RF signal power-to-total noise power ratio and

the RF spectrum bandwidth, B ., at the receiver input.

It should be noted that the message-channel signal-to-noise

ratio, S /N , at the message-signal decoder-unit input, is related

to the radio receiver output total signal-to-noise ratio, S /N , through

the characteristics of the receive-multiplex equipment. The net result

of this situation is that for some types of message -signal multiplexing

equipment [ Landon, 1948] , the message-channel signal-to-noise ratio,

S /N , cannot be measured directly in the radio receiver output baseoc oc

band signal. For SSB frequency-division multiplex, being considered

in this paper, S /N may be measured directly in the receiver output oc oc

baseband signal, in terms of the message-signal spectrum bandwidth

and its position in the baseband signal. For example, in FM multiplex,

S /N cannot be measured directly in the radio receiver-output baseoc oc

band signal, and must be determined at the output terminals of the

receive-multiplex equipment.
-54-

Hence the multiplex receiver characteristics must be included

in the analysis, in terms of the total baseband signal-to-noise ratio,

S /N , at the multiplex receiver input, and the message -channel

o o signal-to-noise ratio, S /N , at the multiplex receiver output. The

oc oc above considerations make proper allowances for any change in the

message -channel signal-to-noise ratio which might be attributed to

the message- signal multiplexing system.

4.4.1 Dependence of Receiver Output Signal-to-Noise Ratio

on the Modulating -Signal Power Level, Pm

In SSBSC-AM systems the power level of the modulating signal,

P , affects the receiver output signal-to-noise ratio only in terms

of the intermodulation noise generated within the modulator. The

dependence of the receiver output signal-to-noise ratio on the average

power level, P , of the noise-type modulating signal is referred to

the power level, P ^ (100%) of a sinusoidal modulating signal; where

P ^ (100%) is the modulating- signal power level required to provide

100% a.mplitude- modulation of the RF carrier signal. The factor,

P /P (100%), may be obtained from figure 3-5.

mn' ms * ' ' B

Note that for a sinusoidal modulating signal, P or, P (100%) m ms

for SSBSC-AM systems and P for FM systems, would be obtained

ms ' from the modulator characteristic. For a noise-type modulating

signal, P or, P would be obtained from figure 3-4 for FM systems, m mn ° ' and from figure 3-5 for SSBSC-AM systems.

In FM systems, the modulating-signal power level, P , determines the required RF- signal spectrum bandwidth, B . The relationrf

ship between the factor, P /p , and B ,. may be obtained from

mn ms rf

figure 3-4. Note that for the FM case, the reference power level is

P and is not a li ms

SSBSC-AM system.

P and is not a limiting factor as is P (1 00%), which is used in the ms ms
-55-

A value for B . is required in order to establish the noise-power

rf c level at the radio receiver input, see Section 4. 2. The radio receiver

IF bandwidth, B.., is associated with B „, and hence these factors

if rf

influence the radio receiver output signal-to-noise ratio. The relationship between the receiver output signal-to-noise ratio, S /N , o o and B , for FM systems, is derived fully in Sections 3.2, 4.6, and

Appendix C. In SSBSC-AM systems, the radio-frequency signal spectrum bandwidth is independent of the modulation index, and is approximately equal to the highest frequency in the modulating baseband signal.

The portion of the radio-receiver output noise power which is related to the modulating signal power, P , is known as "intermodulation

noise". This intermodulation noise is generated in a radio communication system whenever the limits of linear modulation or demodulation are exceeded, in either or both the radio transmitter and the

radio receiver. It is difficult to estimate intermodulation noise power

levels and accurate results must be based on the equipment performance; however, it is usually assumed that the intermodulation noise

power level is equal to the thermal noise power level, for practical

system design. See section 4. 3 on equipment intermodulation effects.
-56-

4.4.2 Dependence of Receiver Output Message-Channel Signal-to-Noise

Ratio, S /N , on the Message -Channel Signal- Power Loading

oc oc

Ratio, P /p , in the Modulating Baseband Signal.

The message-signal power, p ., which must be allocated to each

individual message signal, in the modulating baseband signal, depends

upon the required signal-to-noise ratio at the radio receiver output.

This required radio-receiver output signal-to-noise ratio depends upon

the specified grade of service or message error rate, for a particular

message. The relationship between the radio receiver output messagechannel signal-to-noise ratio and the message error rate is discussed

in Section 5.

In this work it is assumed that the individual message -signal

average power level in the modulating baseband signal, must be adjusted so as to give a specified or required average message-channel

signal-to-noise ratio at the receiver output. Under these conditions

(see Appendix B), the message-channel signal-power loading ratio

in the modulating baseband signal is:

M

m

oc

cV N

oc

+ . . + M

oc

cV N

+ . . +M

oc/J

I N

B

oc

cV N

oc mi

B

ci

S

oc N^ N~~ J. oc A

X

(4.9)
-57-

where, p . = Average power level of a particular message mi

signal in the modulating baseband signal.

P = Average total power level of the composite

modulating baseband signal, under conditions

when a specified number of simultaneouslyactive message channels are operating. For

sinusoidal- signal modulation, P = P , and

j- • -, . _ m _ ms

for noise -signal modulation, P = P m mn

B . = A particular message -signal channel bandwidth

in c/s, at the receiver output. This channel

bandwidth is assumed to be equal to the spectrum

bandwidth of the message signal.

S = A particular message-channel signal average

power level, at the radio receiver output, or a related message-channel signal average power

level at the decoder -unit input.

N __ = Average noise-power level of a particular

mess age -channel having a bandwidth B , at the

radio receiver output or, a related noise-signal

average power level at the decoder -unit input.

S

oc N\ —— ) = Required radio receiver output signal-to-noise

oc/ i ratio, or required decoder-unit input signal-tonoise ratio plus a degradation factor for the

multiplex receiver, in a particular messagesignal channel, for a specified allowable average

message error rate.

M = Number of simultaneously-active message signals each of which requires the same value for

the product {B (S /N )} A. c oc oc A

MT = Ditto, for the product {B (S /N )} t I c oc oc I M„T = Ditto, for the product {B (S /N )>. N

^ c oc

7 oc N

As noted in Subsection 4. 3, for some types of message-multiplexing systems, it may be necessary to determine the message-channel

signal-to-noise ratio at the output of the multiplex receiver because of

the impossibility of measuring this ratio directly in the receiver-output baseband signal. Conversely, at the transmitting end of the system
-58-

for some types of message-signal multiplexing, it may not be possible

to measure the modulating signal power levels directly in the modulating baseband signal: under these conditions, p and P must be

measured at the multiplex transmitter input. This situation will not

arise where SSB frequency-division multiplex equipment is employed.

In order to use (4.9) it is necessary to determine the required

message-channel 6 signal-to-noise & ratio, S /N , and the messageoc/ oc B

signal bandwidth B , at the radio receiver output. This required information would be obtained from measured or estimated data such as shown in figure 5-3 of Section 5.

The specified message error rate depends upon the type of message. Consequently, the required receiver -output signal-to-noise

ratio will be different for each type of message. However, if different

types of messages require the same value for the product, \B (S /N )} , then these messages may be grouped in the same class, M . Proper methods for adjusting the power levels of the individual message signals, at the terminals of the multiplexing equipment, are outlined in the manuals for this equipment; the instructions in these manuals should be followed carefully in order to avoid overloading of the

multiplex circuits, and thereby prevent consequent intermodulation

distortion. The power levels p . and P must be considered in terms mi m

of their effect at the radio-transmitter modulator input; if the multiplex

baseband power level is inadequate, a power amplifier or an attenuator

is required at this point in the system.
-59-

4. 5 Performance Characteristics of SSBSC-AM Radio Receivers

In this section the performance of SSBSC-AM (single sideband

suppressed carrier amplitude modulation) radio receivers is defined

in terms of the signal-to-noise ratio per message channel at the

receiver output, S /N , versus: (a) the total received RF signal

oc oc &

power and (b) the total available noise power at the radio receiver

"front-end" input. In order to obtain accurate estimates of the radio

receiver performance it is necessary to consider the statistical

characteristics of the received RF signal power and the noise power,

at both the receiver input and at the receiver output. Methods for

estimating the noise level and the characteristics of the noise at the

receiver input are outlined in Section 4. 2 and Appendix A. It is

assumed here that the total noise -power at the receiver input approximates "white " noise and that the power level of this total noise is proportional to the RF spectrum bandwidth accepted by the receiver.

The effects of system-intermodulation noise are not included in

the analysis of receiver performance, at this point, and hence, the

results will be in terms of "ideal" receiver performance. Modifications required for the effects of system-intermodulation noise on

the ideal receiver-performance characteristics, are outlined in

section 4. 3. These remarks apply also to FM-receiver performance,

as developed in section 4.6.
-60-

Since SSBSC-AM receivers are linear, it follows that the characteristics of the receiver -output noise are similar to the characteristics

of the noise at the receiver input, within the limits of receiver linearity.

The ideal SSBSC-AM receiver does not exhibit a "threshold effect",

that is, there is a linear relationship between the receiver-input signal

power level and the receiver -output signal-to-noise ratio. The following procedures for determining SSBSC-AM receiver performance are

also applicable in case the receiver characteristic curve is nonlinear, and is known.

In this work the relationship between the receiver output signalto-noise ratio and the receiver input-signal power level was calculated

for SSBSC receivers, assuming that the receiver-input RF signal was

steady, that is non-varying with time. To obtain the receiver-performance characteristic for a time-varying receiver-input RF signal, the

above steady receiver-input signal receiver -output signal-to-noise

ratio characteristic was combined with the distribution of the timevarying receiver-input signal; see Appendix D.
-61-

4.5. 1 Receiver -Input Radio-Frequency Signal Power Requirements

for SSBSC-AM Systems Using Steady Received Signals.

For an ideal SSBSC-AM receiver we have, — = ——X (4.10)

o if

where, P = Total of the average IF signal power at the

receiver demodulator input, in watts.

N. f = Total of the average noise power at the receiverdemodulator input, for IF bandwidth B. r

, in watts

if

S = Total of the average baseband signal power at

the demodulator output, in watts

N = Total of the average noise power at the demodulator output for baseband bandwidth ~B...

On a per message-channel basis, with a message-channel signaler loading5 ratio of P /p in the modulatin

iri *m

radio transmitter (see Section 4.4. 2) we have,

power loading ratio of P /p in the modulating baseband signal at the

S S p B p -2£ = -°

x

-^

x _Jt!

( 4.ii) N N P BV oc o m c

or, from (4. 10) and (4. 11)

S p B - f P. f oc m if it x N P B N

oc m c x tt1- (4.12)

if

where, S = Signal power in the receiver output, per message-signal channel, in watts.

N = Noise power in the receiver output in a messageoc signal bandwidth, B , in watts.

p = Average power level of a message-signal in the

m modulating baseband signal, in watts,
-62-

P = Total average power level of the composite

modulating baseband signal; determined under

conditions when a specified number of simultaneously-active message signals are operating,

in watts.

B = Message- signal bandwidth, in c/s.

B. = B = Effective bandwidth at the radio receiver IF

if rf

output, or demodulator input.

In an SSBSC-AM system the radio-frequency spectrum bandwidth

B , and also the receiver IF bandwidth, B , are approximately equal

to the highest frequency, f , in the modulating baseband signal; hence,

B = B. * f (4.13)

rf if m

Combining (4. 12) and (4.13) and rearranging terms, we obtain,

S B P P.,

oc c m it . . , .

.

ST"

x - x — = n~ (4 - 14)

oc m m if

The term, P /N , in (4.14) is the required predetection signalto-noise ratio, associated with a specified value of S /N at the oc oc

receiver output, for particular values of the parameters: f , B , m c and Pmm/p . However, in radio communication system design and

test work it is much more convenient to deal with the radio-frequency

signal power and the noise power at the radio receiver input terminals

than it is to measure P._/N._ in the IF section of the receiver. Hence

if if the form of (4.14) was modified in the following derivations to include

the total received signal power, P , and total received noise power.

r First,

P.. = G M. P (4.15)

if r lr r P P

*The symbol R is sometimes used to designate —-—or 10 Log

N. r 6

10 N. f if if
and (see 4. 8),

-63-

N = G M. {T A + T }kB. r

it r lr A er if

or, since B. . « f for the SSBSC-AM case,

if m

N. f = G M. {T A +T }kf (4.16)

if rirAerm

therefore,

P., P

if r N., k(T A + T ) f

if A er m

(4.17)

where, G = Power gain of the cascaded RF and IF linear

r sections of the SSBSC radio receiver

P = Total average radio-frequency signal power

available at the receiver input, in watts

-23

k = Boltzman's constant = 1.38 04 x 10 joules

per degree Kelvin

M. = Defined by (A.16); see Appendix A.

f = Highest frequency in the modulating signal

T = Effective receiving-antenna system output noise

temperature, degrees Kelvin. See Section 4. 2

and Appendix A on noise at receiver input.

T = Effective receiver-input noise temperature,

degrees Kelvin [ IRE Comm. on Noise, I960] . Substituting (4. 17) in (4. 14) we obtain,

S B P P

oc c m r IA — ... X T— X — = k(T A+ T )f

(4 ' 18) oc m m A er m

The factor f was retained in both sides of (4. 18) in order to m

obtain power-level values of P on the abscissa scale of the design

r curves shown in figure 4-2, relative to the total noise power,

k (T + T )f , at the receiver input. A er m
64

SSBSC-AM RADIO RECEIVER PERFORMANCE CURVES

STEADY AND RAYLEIGH- FADING TYPES OF RECEIVED SIGNAL

o_

eL

E

- 35

Q

i

1

!

! !

'

'

, ' ' '

' 1 /y y s

. "h_.J_!_Ll

-f MAXIMAL-RATIO COMBINING -

1 I

.

\

1 S s r X

•

' III XX XX

X^ i/^ rS f s ± sS

jf

r r r r

, |

QUADRUPLE C -iJ i XX

. II III

i DUAL on/bHSIIY-^ >

1 1 ^> Lf

JT 'S jr * rJr-—r NON DIVERS TY ± * * Jr n I III ! ^1 i ~^-STEADY S IA1 III M

I

! ! ! Mi:

/ i 1

1 r Jr J*

J

! :

i

.

.

r r nil

f y !/T *

& j»

1 '

>T r f

....:.... ^

!

'

'

1

1 / 1

' Jr '

'

, . ^_

1II

-5

-15 -240 -235 -230 -225 -220 -215 -210 -205 -200 -195 -190 -185 -180

10 L06 o

P. -10 LOGc

fm

-10 UX3C <W>

AVERAGE OF THE TOTAL SIGNAL POWER , WATTS

AJ receiver wput

J: -?:

/STEADYN P, VSIGNAL^ WW AVERAGE OF THE TOTAL RELATIVE NOISE POWER

/TIME-VARYING

\ SIGNAL

SAMPLING -PERIOD MEDIAN OF THE NON- DIVERSITY AVERAGE TOTAL SIGNAL POWER, WATTS

")" SAMPLING -PERIOD AVERAGE OF THE TOTAL RELATIVE NOSE POWER

Figure 4-2

a *£:£.£= N«n
-65-

Equation (4.18) may be used to calculate the required radio-frequency signal power level, P , at the (SSBSC-AM) radio receiver input,

to yield a specified receiver-output signal-to-noise ratio, S /N

oc oc

-for particular values of the parameters: f , P /p , B , T , and m m m c A

T . However, it is more convenient to use the information coner

tained in this equation in a graphical form. Hence, (4.18) was plotted

as shown in figure 4-2 for a time -invariant or steady RF signal at the

receiver input. Scale values in figure 4-2 are normalized in terms of

the parameters in (4.18).

To obtain quantitative values for P from figure 4-2, correct

values must be used for S /N , f , and p /P . The required

oc oc m mm

value of S /N is obtained from data similar to those shown in

oc oc

figure 5-3. f is taken to be equal to the highest frequency in the modulating baseband signal, in c/s. The message-channel signal-power

loading ratio, P /p , may be calculated by means of (4.9). The

factor T may be calculated from (4. 3). The radio-receiver input

noise temperature, T , depends largely upon the carrier (radio)

frequency and the type of receiver; T may be calculated by means of

(4.6). Procedures for using the above information to estimate the

performance of SSBSC-AM systems are outlined in Section 6.

P

if The predetection signal-to-noise ratio, —— , which corresponds

if

to a particular value of P , may be obtained from (4.17). It should be

noted that (4.18) involves the modulation index, m , which is tacitly

assumed to be unity, for SSBSC-AM systems, that is, P (100%) is

7 J ms

used as the reference level of the modulating- signal power. If the

modulation index exceeds unity the resultant (intermodulation) distortion will appear as additional noise at the radio receiver input, and at

the receiver output and will degrade the signal-to-noise ratio at the

receiver output, for all values of receiver input power, P .
-66-

4.5 .2 Receiver -Input Radio-Frequency Signal Power Requirements for

SSBSC-AM Systems Using a Rayleigh-Fading Type of Received Signal.

The value of the required receiver-output signal-to-noise ratio,

S /N , will not be the same for time-varying received RF-signal

oc oc

amplitudes as for steady received signals for the same message error

rate. Also, S /N will depend upon the order of diversity. The

oc oc value of the ratio, S /N , to be inserted in (4. 9) may be obtained

oc oc

from either steady-signal or varying -signal data similar to those

shown in figure 5-3. However, each different type of message will

require a different set of curves similar to those shown in figure 5-3.

The curves in figure 4-2 for non-diversity, dual diversity and

quadruple diversity were obtained by combining the SSBSC-AM steadysignal characteristic curve (in figure 4-2) with the (maximal- ratio)

combiner characteristics given in figure D-lb, Appendix D. Details

of the method used to combine these characteristic curves are given

in Appendix D. It should be noted that the abscissa-scale values in

figure 4-2 for the fading-type signals refer to the median of the nondiversity received total RF signal average power, P , while the

ordinate- scale values yield the average of the time-varying signal-tonoise ratio, S /N , at the receiver output; all other parameters are OC OC f > f

the same as for the "steady-signal" case. The 1.59 db displacement

between the "steady-signal" curve and the "non-diversity" curve in

figure 4-2 is due to the fact that the average value of the Rayleighdistributed received RF signal power, P , is (1.59 + 10 Log n)db

r 10

greater than the median value of P ; where n is the order of diversity.

r

' Dual and quadruple-diversity curves are similarly displaced. Note

that maximal-ratio combiner performance is assumed.
-67-

In the above derivations no allowance was made for degradation of

the signal-to-noise ratio in the Multiplex unit; the effect of this degradation is to increase the required signal-to-noise ratio at the

receiver output. In the absence of information on this degradation

factor, it may be assumed to be approximately 1 to 2 db for SSB-SC

frequency division multiplex equipment.

Procedures for using the information in figure 4-2, to estimate

the performance of SSB systems, are outlined in Section 6 for a Rayleigh-fading type of received signal and for non-diversity, dualdiversity and quadruple -diversity.

4.6 Performance Characteristics of FM Radio Receivers

The performance of an FM (frequency modulated) radio receiver

may be defined in terms of the signal-to-noise ratio per message channel at the receiver output versus: (a) the total available RF-signal

power, (b) the total available noise power at the receiver front-end

input, and (c) the radio-frequency signal spectrum bandwidth; other

system parameters may be considered as normalizing factors.

A method for determining the above relationships for FM systems

* follows; this method is based on a combined mathematical and graphical analysis of typical characteristic curves of FM radio receivers.

These curves consist of an upper linear portion, and a lower nonlinear portion called the "threshold" region. In the work that follows,

the slope and the position of the upper linear portion of the receiver

curve were calculated and the curvature of the threshold region was

obtained from measured results on several different types of FM

radio receivers; however, the threshold region portion of the curve

may als"o be calculated [Stumpers, 1948] . The position of the threshold-level point was assumed to correspond to a signal-to-noise power

ratio (P /N ) of 1 db, at the input to the receiver limiter-discriminator.
-68-

Accurate estimates of the FM radio receiver performance requires

quantitative data on the statistical characteristics of the received RF

signal and the noise power, at the receiver input, and corresponding

data on the signal power and the noise power at the receiver output.

4.6.1 FM Radio Receiver Characteristic Curves

Throughout this work, covering the development and use of the FM

radio receiver characteristic curves, the following ideas and assumptions are used as guides:

(a) The modulating signal is sinusoidal and its frequency is equal

to the highest frequency in the modulating baseband signal.

Furthermore, all of the power in the modulating baseband

signal is contained in this single sinusoidal modulating signal.

The work is then extended to cover receiver performance for

modulating signals at lower frequencies. Receiver performance is shown to be improved, above the threshold region,

directly proportional to the square of the ratio of the highest

baseband- signal frequency to the frequency of the modulating

signal being considered.

(b) The radio-frequency spectrum bandwidth required for a noisetype modulating signal is then calculated in terms of the results

obtained for the conditions in (a).

(c) Receiver-input noise is unweighted, that is, its power spectrum

is uniform or flat.

(d) Limiting in the radio receiver is used, preceding the frequency

discriminator.

(e) The radio receiver IF bandwidth, B. r , is assumed to be only

if

wide enough to pass the radio frequency spectrum, B /, that

rf

is, B = B . If "bandwidth-compression1

' type of receivers

are used, then B < B . The effects of bandwidth-compression

techniques on the system design curves are considered.
-69-

(f) The effects of pre-emphasis and de- emphasis techniques are

considered.

(g) Companders and limiters (for speech messages) are not considered.

4.6. 1. 1 Linear Region of FM Receiver Characteristic Curves

For the condition where the pre-detection signal-to-noise ratio,

P.

f/N , exceeds 10 db, it can be shown (see Appendix C, (C-15)) that,

S B P 2 P

oc c m /AF \ r x 7— x — = -T- x ,,,/^

^w (4.19)** N f p V f J 2k(T A + T )f

oc m ^m v m ' A er m

where,

S = Average message-signal power level per message-channel, at the receiver output, in watts.

N = Receiver output average noise power level, per

message-signal spectrum bandwidth, in watts.

B = Message -signal spectrum bandwidth, in c/s.

f = Frequency of the modulating sinusoidal signal, in

c/s. This is also the highest frequency in modulating baseband signal.

p = Average power level of an individual modulating

signal in the baseband signal.

P = Total average power in the composite modulating

baseband signal. This is the power level which

exists at a time when a specified number of

simultaneously-active message signals are being

transmitted.

-^Acceptable system performance may be obtained in the "threshold

region" where pre-emphasis and de-emphasis do not improve system

performance, see Section 6.

##For a discussion of the term (T + T ) and the effects of impedance

mismatching in the receiving system, see Appendix A.
-70-

AF = Peak radio-frequency carrier deviation, in c/s,

for a single sinusoidal modulating signal, having

an average power level P equal to the total

average power level of the composite modulating

baseband signal.

P = Total available carrier- signal power at the input

terminals or "front-end" of the FM receiver, in

watts

.

-23

k = Boltzman's constant = 1.3804 x 10 Joules per

degrees, Kelvin.

T . = Receiving antenna-system output noise temperature, degrees Kelvin; see Appendix A.

T = Radio receiver effective input noise temperature

[IRE Comm. on Noise, 19&0J , degrees Kelvin.

AF _ . . = Deviation ratio.

fm

P and S /N are to be considered as the "variables" in (4. 19);

r oc oc

A F/f is the "variable-parameter" and the remaining terms in this m

equation are considered as "scale-normalizing factors".

Since (4. 19) was developed for the purpose of calculating the

performance of FM radio receivers, a discussion of the limitations

and application of this equation is appropriate. It should be noted that

S /N __ is the receiver output signal-to-noise ratio for a particular

message signal having a power spectrum centered at a frequency of f m

cycles per second in the baseband. Furthermore, this equation is valid

only for values of the predetection signal-to-noise ratio, P /N , greater than approximately 10 db. In other words, there is a linear relationship between the predetection signal-to-noise ratio in the FM receiver

IF, and the signal-to-noise ratio at the receiver output, only for values

of the predetection signal-to-noise ratio exceeding approximately 10 db.

In this respect (4. 19) can be deceptive because it does not include the
-71-

non-linear threshold region. Hence, (4. 19) should be used only to

locate the position and the slope of the FM radio receiver characteristic curve above the threshold region.

The position of the "threshold-level" point of the FM radio receiver

characteristic performance curve is determined from the relationship

P., P P

if r r N

if k(T.+T )B.- k(T A +T )f x<j/

where,

A er if A er m \ fm

= 10 (4.20)

<j> (AF/f ) = Function of A F/f , calculated [ Hund, 1942] and

plotted in figure 3-1 for "sinusoidal- signal" FM

modulation. This function gives the number of

significant sidebands in the radio frequencycarrier spectrum; assuming that the modulating

signal is a single sinusoidal voltage wave at the

highest frequency f , in the modulating baseband

, m

signal.

Equation 4.20 was used to determine the "threshold level" points

on the curves in figure 4-3; these points are the limits at which the

curves cease to be linear for decreasing value of P . The slope of the

r upper linear portion of the receiver characteristic curves, shown in

figure 4-3, was determined from (4.19 )•

4.6.1.2 Threshold Region of FM Receiver Characteristic Curves

The shape or curvature of the non-linear "threshold region" of

the radio receiver characteristic curve was obtained from averaged

measured data on several different types of conventional FM radio

receivers. Calculated threshold-level points are shown in figure 4-3

as the intersections of the threshold-level curve and the receivercharacteristic curves.
-72-

FM RADIO RECEIVER CHARACTERISTIC CURVES

STEADY RECEIVED SIGNAL

: 230 -225 -220 -215 -210 -205 -200 -195 -190 H85 -180 -175 -170 -165 H60 H55

10 Log|0 Pr - 10 Log |0 fm-IO Log (TA +Ter)

ivV'e

Figure 4-3

S_ AVERAGE OF THE TOTAL SIGNAL POWER.WATTS AT RFnriwre , NPLrr

fm(TA

+Ter) "AVERAGE OF THE TOTAL RELATIVE NOISE POWER' Al TM

lulllv,in MNru '
-73-

From the above outline it is possible to determine the shape and

the position of an FM radio-receiver characteristic curve, in terms of

the receiver -input RF-signal power to the receiver-input relative noise

P

r power ratio, r—p= —

—— ; and the receiver output (message)

A er m

channel- signal power to channel-noise power ratio, S /N ; for a oc oc

given set of system parameters.

4.6.2 FM Radio Receiver Input-Power and Bandwidth Requirements for Steady Received Signals.

Following is a description of the method used to construct the FM

radio receiver curves shown in figure 4-3, using the scale normalizing

parameters as noted in the abscissa and in the ordinate titles. The

shape of these curves and their locations, relative to the absicssa and the

ordinate scales and relative to each other, are important if they are

to be used to obtain accurate determinations of the required bandwidth

B A = B.

f

) and the receiver input-power, P , required for particular

grades of service, or for particular values of receiver -output signalto-noise ratios, S /N

oc oc

The non-linear threshold-region portions of the curves were obtained from averaged measurements on a number of FM radio receivers

of different types, as indicated by the measured points shown in figure

4-3. It was assumed that the threshold- region curvature of the curves

did not vary appreciably with change in the deviation ratio AF/f , hence all curves shown in figure 4-3 are identical in shape.

The positions and the slopes of the upper (linear) portions of the

FM radio receiver curves in figure 4-3 were calculated, using (4.19)

and (4. 20); each of the scale-normalizing factors, B , f , and x ' °

c m

P /p were set equal to unity. T was set equal to the standard
-74-

temperature, T (= 290 degrees Kelvin); this condition is equivalent to o the assumption that the modulated RF signal was obtained from a signal

generator with an output impedance temperature of 290 degrees Kelvin.

T was assumed to be degrees Kelvin; this assumption corresponds

er

to a receiver noise figure of unity, or db. Note that A F/f is considered here as the "variable" parameter and not as a scale-normalizing factor. The upper linear portions of the curves are displaced

relative to each other on the ordinate scale an amount which is depend-

/ 2

ent upon the relative values of ( A F/f ) . m

The threshold-region section of the receiver curves are drawn

tangent to the upper linear portions of the curves, at the points given

by (4. 20); these points correspond to an IF signal-to-noise ratio

p /N =10. This value of 1 db for P. r/N. r , applies only to the case

if if if if where the received radio-frequency signal is non-varying or steady.

The curves in figure 4-3 were drawn for a value of f assumed to 6 m

be equal to the highest frequency in the modulating composite baseband

signal; therefore, the receiver output signal-to-noise ratio values, for

the linear portion of these curves, would be increased by a factor,

(f /f ) , or 20 Log —— db, for modulating signals of frequency f - c -where f cm< f . However, when minimum receiver-input power P is r the criterion, the system parameters should be adjusted so that the

receiver will be operated in its threshold region for a small percentage

of the time; under these conditions, when P is above the thresholdr level value, the receiver output signal-to-noise ratio exceeds the

required S /N value and pre-emphasis techniques are of doubtful

oc oc ^ value. In the threshold region, the receiver output signal-to-noise

ratio, S /N , approaches the same value for all frequencies in the oc oc ^ modulating baseband signal; see figure 4-8 and Section 4.6.4. Hence,

the curves in figure 4-3 will yield accurate results in the threshold

region and conservative results above the threshold level.
-75-

The radio-frequency spectrum bandwidth, B , associated with

rf

particular values of AF/f and f (see (C. 8)) is given by,

B = f x ^AF/f ) (4.2 1) rf m m

where,

B = Radio-frequency spectrum bandwidth which

includes all components or sidebands having

amplitudes equal to or greater than one percent

of the unmodulated carrier amplitude.

The relationship between AF/fmand 4> (AF/f ) was calculated m

[Hund, 1942] and is shown in figure 3-1, for sinusoidal- signal modulation. Values for B /f , associated with particular values of AF/f , ri m m

were obtained by combining (4.2 1) and figure 3-1; these associated

values of AF/f , B Ji and d> (AF/f ) are shown in figure 4-3.

' m rf m v ' m 6

The receiver characteristic curves in figure 4-3 were re-plotted

in a more convenient and useful form, as shown in figure 4-4.

The interrelationships between three important factors in an FM

radio communication system are shown in figures 4-3 and 4-4. These

factors are: ( 1 ) required bandwidth or radio frequency spectrum, B ., ri

(2) grade of service or minimum permissible receiver output signalto-noise ratio, S /N , and (3) required radio frequency carrier

oc oc signal power, P , relative to the noise-power level, k (T . + T )f , & *

r ^ A er m

at the receiver input.

The designer of a radio communication system is usually given

the specified number of voice channels, that is, the baseband spectrum

bandwidth, from which he can determine f ; the grade of service or the

m

maximum tolerable average message error rate is also specified, from

which the system designer can establish the required receiver output

signal-to-noise ratio, S /N . The initial problem, is to choose

oc oc

the required radio-frequency carrier spectrum bandwidth B (where
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-77-

FM systems are involved), and the required radio frequency carriersignal power level, P , at the receiver input. The curves in figures

4-3 and 4-4 make it possible to select required sets of values for B

and P , for a particular grade of service, or S /N , for specified

r oc oc

types of messages. The curves in figure 4-4 can also be used as guides

when it is desired to trade or interchange required P , for required

r bandwidth, B r

. rf

The required deviation ratio, A F/f , associated with the required m

bandwidth ratio, B /f , and for a sinusoidal modulating signal, may

be obtained from the right-hand scale in figure 4-4. Hence, the required deviation AF can be obtained for a given & baseband bandwidth B, or

b

a particular value of f . It should be understood that the data in m

figures 4-3 and 4-4 applies only to the case of a steady or non- varying

radio frequency received signal. Procedures for using the design

curves in figure 4-4 are outlined in Section 6.

4. 6. 3 FM Radio Receiver Input-Power and Bandwidth Requirements for a Rayleigh-Fading Type of Received Signal.

FM receiver characteristic curves, were obtained for a Rayleighfading type of received signal and for non-diversity, dual diversity,

and quadruple diversity by graphically combining the curves in figure

4-3 (for steady received signal) with the appropriate maximal-ratio

Combiner -output distributions [ Brennan, 1958] shown in figure D-lb;

details of the method used to combine the FM receiver performance

characteristic for steady received signal with the Combiner characteristics are given in Appendix D. The three sets of curves obtained by

this procedure were then scaled and replotted as shown in figures 4-5,

4-6 and 4-7; for non-diversity, dual diversity, and quadruple diversity,

respectively.
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The major differences between the design curves in figures 4-4,

4-5, 4-6 and 4-7 are that the P values for the steady received signals

are average values while the P values for the fading type of received

signals are median values of the non-diversity average-power levels of

the received signal. The positions of the curves, relative to the abscissa and the ordinate scales, are of course different for the different

orders of diversity; furthermore, their relative positions are also

dependent upon the type of combining and the actual performance of the

Combiner. This latter point greatly influences the accuracy of systemperformance estimates based on the design curves which are developed

in this report, compared to actual system-performance measurement

results. Procedures for using the design curves in figures 4-4, 4-5,

4-6, and 4-7 are outlined in Section 6.

4.6.4 Pre-emphasis-De-emphasis Techniques

In FM systems, the rms noise voltage in the receiver -output baseband signal is proportional to the frequency in the baseband- signal

spectrum. If equal power levels are assigned to each spectral component in the modulating baseband signal at the modulator input, regardless of the positions of the modulating signals in the baseband signal,

it will be found that the receiver-output signal-to-noise ratio will vary

as shown in figure 4-8.

Pre-emphasis of the spectral components in the modulating baseband signal, proportional to the frequency of the component in the baseband, and a corresponding de-emphasis of the baseband components in

the radio receiver, result in an equilization of the radio receiver output signal-to-noise ratio for all components in the baseband signal.

However, the above conditions apply only when the radio receiver is

operating above its "threshold level"; below the receiver threshold
-82-

FM RECEIVER CHARACTERISTIC CURVES, WITHOUT PRE-EMPHASIS

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-83-

level the use of pre-emphasis-de-emphasis techniques results in a degradation of receiver-output signal-to-noise ratio. This condition is

evident from figure 4-9. Hence, pre-emphasis-de-emphasis techniques

will only be effective when the FM receiver is operated above its threshold level.

FM system performance may be estimated without pre-emphasisde-emphasis techniques as outlined in this paper; the results will be

accurate in the receiver threshold region and will be conservative

above the threshold level. Final design estimates may be modified, in

terms of the baseband- signal frequency, by the use of data similar to

that shown in figure 4-8; however, there may be some question as to

the practicability of this procedure.

4. 6. 5 Bandwidth Compression in FM Receivers.

All of the evidence to date shows that the various proposed receiver bandwidth-compression schemes have the effect of lowering the

"threshold-level" of the FM receiver, relative to the received carriersignal power and the total noise power "accepted" by the receiver. The

accepted noise power is that portion of the total receiver-input noise

power which appears at the input to the FM receiver limiter-discriminator, or demodulator unit. It should be noted that the position of the

threshold level also depends upon the characteristics of the limiter and

hence the threshold level may, in some cases, possibly be lowered by

improvements in the limiter characteristics.

The overall effect attained by compressing the receiver bandwidth

is shown in figure 4-10. It is evident that there is no enhancement of

the bandwidth-compressed receiver output signal-to-noise ratio above

the threshold level point; this situation is due to the fact that the predetection frequency deviation, AF, is reduced when the bandwidth is

compressed, and hence the net effect is that there is no improvement

in the receiver performance above the threshold level.
-8i+-

EFFECT OF PRE-EMPHASIS ON FM RECEIVER CHARACTERISTIC CURVES

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-85

-

EFFECT OF BANDWIDTH COMPRESSION ON

FM RECEIVER CHARACTERISTICS

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-86-

The FM- receiver design-curve data presented in this report may

be modified to include receiver bandwidth- compression or modifiedlimiter effects by incorporating the data from figure 4-10. The data

shown in figure 4-10 may be obtained by either (1 ) performing the

indicated comparative measurements on the FM receivers, for the

conditions shown in the figure, and for identical frequency-modulated

RF signals at the input terminals of the radio receivers; or (Z) by

estimating the value of the bandwidth compression factor, K , and

modifying the design curves as shown in figure 4-10 or figure 4-11.

The factor K is defined in terms of the compressed-bandwidth

c receiver threshold-point power level, P

1 relative to the similar

power level, P", for an un-compressed-bandwidth receiver. The

value for K may be obtained from the measured characteristics of

c the receivers, as shown in figure 4-10.

The modified design curves, well below threshold and for high

values of B r/f , will parallel the original curves and will be shifted

to lower values on the relative P scale; this decrease in scale values

r will be equal to K , in db. The modified curves will converge with

and will join the original design curves at the original threshold-level

points. Above the threshold-level points, the modified design curves

are identical to the original design curves- -which apply to FM receivers

without bandwidth compression. The effect of receiver-bandwidth compression on the shape of the FM system-design curves, is shown in

figure 4-1 1.
-87-

EFFECT OF BANDWIDTH COMPRESSION ON

FM SYSTEM DESIGN CURVES

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Fig. 4-11
-88-

5. PERFORMANCE CHARACTERISTICS OF MESSAGE-SIGNAL

DECODER UNIT

The performance of the message-signal decoder unit is defined

here in terms of the relationship between the message-channel signalto-noise ratio, S /N . at the decoder-unit input, and the error rate

oc oc

in the message at the decoder-unit output. This definition of the message-signal decoder unit performance is general and may be applied to

any type of decoder, provided that proper consideration is given to the

statistical characteristics of both the message-channel signal and the

message-channel noise at the decoder-unit input, and the resultant

message error rate.

At the decoder-unit input, both the short-term message-channel

signal power, S , and the short-term message-channel noise power,

N , will vary independently with time; furthermore, the distributions oc

of S and N are dissimilar. Hence, in order to obtain measurable

oc oc and definable results it is necessary to analyze the performance of the

message- signal decoder unit in terms of the average signal power-to

average noise power ratio, S /N , at the decoder input. The 2— oc oc problem is further complicated by the fact that the ratio S /N varies

oc oc with time and hence its distribution must be considered; finally, the

distribution of the noise power affects the message error rate, for

particular values of S /N , as shown in figures 5- 1 and 5-2. oc oc

In this paper the decoder-unit performance is determined, for

the above conditions at the decoder input, as follows:

(1) The decoder-unit performance is either estimated or measured for a range of values of the average signal power-to-average

noise power ratio, S /N , at the decoder input. S /N

oc oc oc oc

is assumed to be non-time varying or steady, and the decoder-input noise, N , is assumed to be "white". The

oc
-89-

CUMULATIVE AMPLITUDE DISTRIBUTION OF NOISE AT RECEIVER OUTPUT

FOR RECEIVED CARRIER POWER LEVELS ABOVE AND BELOW THE

FM RADIO RECEIVER THRESHOLD LEVEL

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Figure 5- I
-90-

MEASURED RELATIONSHIP BETWEEN THE STEADY -SIGNAL POWER TO

MEAN NOISE POWER RATIO, AT THE TELETYPE RECEIVER INPUT TERMINALS,

AND THE AVERAGE TELETYPE CHARACTER ERROR RATE

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10 12 14 16 II 20

io log 10 N

»0£. OC TELETYPE MESSAGE- SIGNAL AVERAGE POWER LEVEL PER CHANNEL

AVERAGE NOISE POWER LEVEL PER TELETYPE MESSAGE CHANNEL B c MEASURED AT TELETYPE RECEIVER (DECODER -UNIT) INPUT

db

Fig. 5-2
-91-

"measured steady signal" curve shown in figure 5-3 is typical

for narrow-band FSK teletype decoder-unit performance -

(2) The performance of the decoder unit, for time-varying

(decoder-input) signal-to-noise ratios, is then estimated by

combining the above steady- signal decoder-unit characteristic with the distribution of the time-varying decoder-unit

input signal-to-noise ratio. Details of this procedure are

given in Appendix D. 2; results are shown in figure 5-3 for

non, dual, and quadruple diversity, assuming maximal- ratio

combining.

(3) Under conditions where "impulse -type" of noise appears at

the decoder-unit input, the procedures outlined in (1) and (2)

above would he followed; the resultant decoder-performance

curves would be similar to those shown in figure 5-3 (for

white noise at the decoder input) but they would be shifted to

higher values of S /N , for given values of message error

rates. See figures 5-1 and 5-2.

The shapes and the positions of the curves in figure 5-3 are

determined by the distribution of the signal-to-noise ratio, S /N , y B oc' oc

at the decoder-unit input; in turn, this distribution (for our purpose)

is dependent upon: (a) the distribution of the received RF signal at

the radio receiver input, (b) the radio-frequency receiver characteristics, (c) the order of diversity, and (d) the combiner characteristics.

Combiner performance was assumed to be as shown in figure D-lb, for

a Rayleigh-fading type of radio-frequency carrier signal, using maximal-ratio combining, and for SSBSC and FM radio receiver characteristics as given in Sections 4.5 and 4.6.

The principal requirement, which may be used as a guide when

designing and testing radio communication systems, is that the (received) average message error rate should not exceed a specified value for
92-

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-93-

particular message-load conditions on the system. Message errors

usually occur in bursts or groups, and are associated either with

signal fades or noise peaks, or both. Hence, the "short-term message

error rate" will vary from zero to some comparatively high value

during the sampling period; therefore, the term "average message

error rate" is used throughout this paper, in order to differentiate

between the short-term message error rate and the message error

rate which is obtained by averaging the short-term message error

rates over the entire sampling period. For example, the average

teletype character error rate is defined here as:

Total number of teletype character errors occurring during

sampling period

Total number of teletype characters transmitted during the

sampling period

The interval of time included in the sampling period is subject

to the mutual agreement of all parties involved in the design and

acceptance-tests of the radio communication system.

Narrow band FSK (frequency- shift-keyed) teletype equipment

employing a frequency shift of approximately ± 40 cps is generally

available in multichannel tropospheric radio communications systems,

and considerable data are available on the correlation between the

teletype character error rate and the signal-to-noise ratio at the

teletype-receiver input or in the received baseband signal. Hence,

teletype character error rates may be used conveniently as the criteron in designing and testing radio communication systems. Other

types of messages, such as voice, digital data, or facsimile, could

be used of course, provided that the relationship between the message

error rate, i.e., grade of service, and the signal-to-noise ratio at the

dec oder -unit input is known.
-94-

For a specified allowable teletype -message error rate, from the

data shown in figure 5-3, we can determine the required messagechannel signal-to-noise ratio, S /N , at the radio receiver output; ° oc oc

proper corrections are required for the effect of the message-signal

multiplexing equipment on this signal-to-noise ratio. Similar information is required for the other types of messages being transmitted.

These required radio-receiver output signal-to-noise ratios are then

used in (4.9), Section 4, and with the proper set of system-design

curves in figures 4-2, 4-4, 4-5, 4-6 or 4-7, to determine correct

values for P , B , AF, etc. The method of applying the above procedures is outlined in Sections 4.5 and 4.6. A typical design problem

for SSBSC-AM and FM radio communications systems is solved in

detail in Section 6.

From figures 5-1 and 5-2, it is evident that the performance of

the FSK Teletype Decoder is not influenced, to a large degree, by the

statistical characteristics of the noise at the decoder input terminals.

This situation is due primarily to the effectiveness of the decoder

limiter in removing the "amplitude" noise, which is associated with

the signal. FM decoders for other types of messages would be expected to give results similar to the results obtained above for FSK teletype messages.

There is a need for up-dating voice-message performance data,

similar to that given in figure 5-3 for teletype message errors vs. signal-to-noise ratios. In the system-design example worked out in

section 6, the required voice message signal-to-noise ratio was obtained from the work of French and Steinberg, 1947; Eagan, 1944;

and Beranek, 1947. An alternative solution to this problem is to

accept the required voice message signal-to-noise ratio specified by

the user of the system.
-95-

6. RADIO COMMUNICATION SYSTEM DESIGN AND TEST

PROCEDURES

A typical system-performance problem is solved in this Section,

in order to indicate the procedures to be followed when using the system

design curves to estimate the following major factors: (A) averagepower level of the total modulating signal, P , (B) required peak

frequency deviation, AF (for FM systems), (C) required radiofrequency spectrum bandwidth, B , and (D) required receiver-input

ri

RF-signal power level, P . The "required" values of these factors

are associated with a specified maximum message error rate or

grade of service. For the condition where certain parameters and

factors such as B ., transmitter power, P , antenna gain, G , message load, etc. are given, the design curves may also be used to

estimate the grade of service which will be provided by the system.

Calculations are carried out for SSBSC-AM and for FM systems using

dual diversity and maximal-ratio combining, and assuming a Rayleighfading type of received RF signal; for other orders of diversity the

analysis is identical using the appropriate design curves.

It is assumed that the following system specifications are given:

(1) Number of available message (voice) channels.

(2) Specified number of simultaneously-active message channels.

(3) Number of messages of each type, M , and bandwidth, B ,

J. *— required for each message signal.

(4) Maximum allowable average message error rate.

(5) Type of Multiplex equipment to be used, and the baseband

spectrum bandwidth required to accomodate the specified

number of available message channels.

(6) Transmission losses in the transmitting and receiving

antenna systems.
-96-

(7) Antenna noise-power temperature, T .

(8) Receiver noise figure, F , or receiver -input temperature, T

The procedures for using the design curves in this report are as

follows:

(a) Determine the required minimum value of the receiver output

average signal power-to-average noise power ratio, S /N , associated with the specified maximum allowable average

message-error rate, for a specified order of diversity. For

teletype messages this information may be obtained from

figure 5-3. For other types of messages, data similar to

that given by figure 5-3 would be required, and could be

developed using methods outlined in Appendix D.

(b) Determine P /p , the message-signal power-loading ratio

in the modulating baseband signal, using (4. 9) and associated

information. Where p is the individual message-signal

average modulating power and P is the total average message-signal modulating power in the modulating baseband signal, measured at a time when the specified number of simultaneously-active channels are operating. The maximum

number of simultaneously-active channels should be specified, otherwise the designer is compelled to choose a value

for this important factor.

(c) Determine the highest modulating frequency, f , from the

specified number of available voice channels, or from the

modulating66baseband- signal bandwidth. The value of f will m

depend upon the type of multiplex system being used; f is

always equal to the highest frequency in the modulating baseband signal.
-97-

(d) Calculate the antenna- system temperature, T , and the effective receiver-input temperature, T , from specified

er

receiving-system parameters: LLL ,T,T,T,T , c t du act du

and F . See Appedix A.

r The above information on the system parameters is combined with

the data in the appropriate equipment-performance design curves, in

order to determine the required RF bandwidth, B , and the required

receiver-input RF signal power level, P ; for the specified grade of

service or permissible message error rate.

6.1 Determination of the Basic System Parameters:

S /N , B , P /p , f , T . , and T

oc oc c m m m A er

for Specified Average Message Error Rates,

Number of Available Message Channels, and

Specified Receiving-System Parameters.

This Subsection outlines the method of calculating the information

required in items (a), (b), (c), and (d) in Section 6, from the system

specifications which are outlined in Items (1 ) through (5) in Section 6.

The parameters S /N , B , P /p , and f obtained from these

oc oc c m m m

calculations apply to both SSBSC-AM systems and FM systems. A

specific example is considered in order to indicate more clearly the

design procedures, and the calculated values of the above parameters

are then used in Subsections 6. 2 and 6.3.

Radio Communication System Specifications

Number of available voice channels 36

Maximum (specified) number of simultaneously- active

channels 9

M = Maximum number of teletype message channels

simultaneously-active (16 teletype messages per

voice channel) 32
-98-

M = Maximum number of simultaneously-active

voice-message channels (continuous talkers)

Bandwidth B required for each teletype signal

Bandwidth required for each voice signal

7

110 c/s

3200 c/s

Maximum allowable average teletype character

error rate

Type of Multiplex

Multiplex baseband bandwidth, 4 x 36 = Type and order of Diversity

Transmission-line and receiving-antenna

loss factor, L L L

,

c t du

Antenna noise temperature, T

Receiver noise figure, F

0. 01%

SSBSC, Frequency div.

144 kc

Space, Dual

1.26(=ldb)

300 deg. K

5 db

Calculations

From figure 5-3, the required signal-to-noise ratio at the teletype-message decoder unit input, for dual diversity operation, is 28 db;

and allowing 2 db degradation of the signal-to-noise ratio through the

Multiplex unit, we have

lOLogfS /N /

telet VPe ^ 10 oc oc ^messageJ

= 3 db

and,

S /N f

teletYPe V 10 00. oc oc y messagey

From the system specifications listed above we have,

B (teletype) = 110. c/s

B (voice) = 3200 c/s

c
-99-

The message-signal loading ratio in the modulating baseband signal, P /p ., is determined from (4. 9), see p. 56 . This equation m mi

.th shows that P /p . for the i— message signal depends upon the required receiver-output signal-to-noise ratios, S /N , for each of the

oc oc message signals in the modulating baseband signal. The required

decoder-input signal-to-noise ratio for teletype message signals was

obtained from figure 5-3, and was found to be 28 db. The required

decoder-input signal-to-noise ratio for voice message signals (dual

diversity) is assumed to be 33 db [French and Steinberg, 1947; Eagan,

1944; Beranek, 1947] . The estimated message intelligibility for

voice messages, corresponding to a signal-to-noise ratio of 33 db, is

90 percent. Therefore, in the receiver baseband-signal at the receiver

output, we have,

S

oc / voice

N

oc

message

= 33 + 2 = 35 db = 3160,

/

Substituting the above information in (4.9) p. 56, we obtain,

Pm /per teletype \ _ (32 x 110 x 1000) + (7 x 3200 x 3160)

p . I message )' 110x1000 mi K ° /

and,

10 Log

10

m

m

per teletype

message )

= 10 Log

1()

(677.) = 28.3 db

In systems which require the transmission of pilot tones, it is

necessary to include the pilot-tone modulating signal power levels, pm, and the number of pilot tone signals, M, in the above calculations. The

net effect of the inclusion of these pilot tone- signals will be to require

an increase in the transmitter -output power; in order to transmit the

pilot-tone signals without altering the grade of service of the "useful"

message load.
-100-

The value of f is equal to the highest frequency in the modulating m

baseband signal and is,

f = 12 + (36 x 4) = 156. kc m

or,

10 Log, rt f = 52. db, s 10 m

Note that the lowest-frequency in the multiplex is 12 kc and that

each voice channel occupies a bandwidth of 4 kc in the modulating

baseband signal including the guard bands; this simple relationship

applies only to SSB frequency division type of Multiplex systems.

The value for T is obtained from either (4. 3) or (4. 3a), p. 37.

Using (4.3a), and assuming P /k B L to be 200 degrees Kelvin

and T = 300 degrees Kelvin

a

6

T - T

T a = ~r—% T- A L L L.

2- + T

o

+ P,

it'

/kB.

ifr LD

c t du

300 — - 290

6

7 + 290 + 200 = 498 deg K

The receiver noise figure F is assumed to be 5 db or, F = 3. 16; r r therefore, from (4.6), p. 38.

T = (3. 16 -1) 290 = 625 degrees Kelvin

er

*Note that the required value of the Duplexer isolation factor , L , depends upon P. , k, B.. and the allowable level of the Local-Transit ii mitter "noise temperature", at the Duplexer output.
-101-

6.2 Equipment Design Procedures, SSBSC-AM Systems,

For a Rayleigh-Fading Type of Received Signal,

Dual Diversity, and Maximal-Ratio Combining

It is required to determine P from the SSBSC-AM radio-receiver

r performance curves in figure 4-2; using the above listed specification

items (6), (7), and (8), and the values of the parameters which were

derived in Subsection 6. 1.

For SSBSC-AM systems, the modulator-demodulator conversion

factor, D , may be obtained from figure 3-5. Assuming D = 4 db,

o o S S the (noise-type) modulating- signal power level, P , (= P ) required mn m

so as to insure that 100% amplitude -modulation conditions will be

exceeded for only 1. 2 percent of the time, will be 4 db below the power

level, P (100%). The value for P (100%), corresponding to 100%

ms ms

amplitude modulation, may be obtained from the modulator characteristic, which is in terms of sinusoidal moi

versus percent amplitude modulation, m

istic, which is in terms of sinusoidal modulating- signal power, P ,

a The power level, p . , of the individual message-signal, at the mi

modulator input, may now be obtained from the previously calculated

value for P /p . (= P /p .). Note that this calculated value for m mi mn mi

p . applies only to the case where the maximum number of simultan- mi

eously-active message channels is one-fourth of the available channels,

as specified in this example. Other specified message-loading conditions will require ^ different values for pmi

From the above derivations we have

10 Log lrt D = 4 db

10 ss 10Log in P =10Logl „P (100%)- 4 db & 10 mn 6 10 ms

10Log 1A p . =10Log in P -28.3db & 10 *mi 6 10 mn
-102-

Note that both P and p . are dependent upon the measurable or mn mi

estimated value of P (100%), obtainable from the modulator -per- ms

formance characteristics.

The next step in the design procedure is to determine the operating

value, on the normalized ordinate scale in figure 4-2. This scale value

is obtained by substituting the above information on S /N , B , P /p ., and f in the "title" of the ordinate scale, which is, m mi m

10LogS 1/x ( t^- J + 10Log lrt B - lOLog, f +10Log lrt (^^ 10VN J

5

10 c 5 10 m 10Vp • oc mr

or,

lOLog^dOOO)^1^) 10Log

10 (110) - 10Log10 (156000)

+ lOLog (677. ) = 30 + 20.4 - 52. + 28. 3 = 26. 7 db

From figure 4-2, using the above calculated ordinate -scale value

of 26 . 7 db and the dual -diversity curve, we obtain a value of -207- 5 on

the normalized abscissa-scale. Therefore,

10 Log

10

P

r

- 10 Log

1() £m

- 10 Log

10

(TA !„! = -207. 5

Substituting& the values of T . and T (and f =52 db) in the above A er m

expression for the abscissa- scale value, we obtain for the SSBSC-AM

system,

10 Log.. P =52+10 Log,„ (498 + 625) - 207. 5 = -125.1 dbw

10 r °10

However, this value of P (-125.1 dbw) must be corrected for the

r effects of system intermodulation noise; see section 4. 3. 6. Assuming

that the receiver -output system intermodulation noise is equal to the

receiver -output noise due to the pre-detection amplitude noise, the

required correction to the above calculated value of P is 3 db. Hence,

r
-103-

10 Log. A P = -125. 1 + 3 = -122. 1 dbw (SSBSC-AM)

1 U r

The value of P (-122. 1 dbw) is the median non-diversity received

r

J (total) RF signal-power level at the SSB receiver input, required to

transmit the specified message load, with an average teletype -message

error rate of . 01 percent and a predicted voice-message intelligibility

of 90 percent. This value of P is also the value to be substituted in

r

(7. 1) of Section 7, together with appropriate values of the other factors

in (7. 1), to obtain the required transmitter -output average power level,

p

f

In a SSBSC-AM system the radio -frequency spectrum bandwidth is

approximately equal to the highest frequency in the modulating baseband signal; therefore,

B., = B , = f =156 kc.

if rf m

6.3 Equipment Design Procedures, FM Systems, for a

Rayleigh- Fading Type of Received Signal, Dual

Diversity, and Maximal -Ratio Combining

It is required to determine B and P from the performance

rf r curves for FM systems using the system specifications and the values

of the derived parameters in Subsection 6. 1. Note that for the case

of an FM receiver which does not use bandwidth compression, the

"bandwidth compression factor" K (figure 4-10) is equal to 1 and,

B

rf

= B

if

• The first step in the design procedure is to ascertain which one of

the design-performance curves in figure 4-6 p. 79 (dual diversity), is to

be used. The particular design-performance curve to be used is found

by substituting the above calculated values for S /N , B , f , and ' ° oc oc c m

P /p . in: m mi

10 Log&

(S /N )+ lOLog, B - 10 Log f + lOLog, (P /p ) 10 oc oc 10 c s 10 m 6 10 m *m
-104-

This procedure is identical to that used in the case of SSBSC-AM

design calculations, except that the factor D (= P (100%)/P ) is & c ss ms mn

not involved for the FM case. A corresponding factor, P /P , is

ms mn

derived at the point in the FM- system design where the bandwidth ratio,

B Ji , is chosen. Therefore, making the proper substitutions in the

rf m

above expression we obtain,

10Log

10(1000^fg^)+ 10Log

10(110)-10Log10(152000) + 10Log10

(677.,

= 30 + 20. 4 - 52 + 28. 3 = 26. 7 db

The particular design-performance curve in figure 4-6, which has

a "scale-value" of 26. 7 db, is now used to determine B . and P . Any

rf r - point on this curve yields a combination of values of B Ji and

ri m

P /f (T . + T ) such that the average teletype character error rate

r' m A er 6 yt^

is . 01 percent, for the above values of f and P /p . The system m m m

designer may now decide to what extent he should interchange or trade

bandwidth, B - or B.., for required receiver-input power, P , that is,

ri if r he may choose any point on the "26. 7 db" design curve in figure 4-6

as the operating point.

For operation at the receiver threshold-level point on the "26. 7 db"

design curve in figure 4-6, we have,

B Ji =10.5

rf m

or

B . = 10.5 x 156 = 1638 kc

rf
-105-

and

A F/f =2.8 m

or

AF = AF = 2.8 x 156 = 436.8 kc

s

The required total modulating- signal average power level, P , ms

may be determined from the above required value of A F (= 436.8 kc)

s and the measured or the estimated modulator characteristics, AF

s versus P , where the modulating- signal is a single sinusoidal signal.

The required modulating- signal power level, P , for a white-noise

mn

modulating signal, may be obtained from figure 3-4, relative to P , using B Ji - 10.5; that is,

rf m

ms

10 Log P =10Log. P -5.5db

10 mn 10 ms

The correct power level, p ., of the individual message-signal,

at the modulator input may now be obtained from the previously- calculated value of P /p . (= 28. 3 db) or, m mi

10 Log,5 ^p . = 10Log in P -28.3db

10 Vi s 10 mn

Note that this calculated value for p . applies to this specific example, mi

where it was specified that the maximum number of simultaneouslyactive channels is one-fourth of the available channels. Other message'

loading ratios will require different values for p .. Also, P and mi mn

p . are in terms of the measurable sinusoidal modulating- signal mi

power level, P

ms
-106-

The above calculated value of the radio-frequency signal bandwidth,

B r , is the required bandwidth which includes 99. 99 percent of the

rf

spectral energy in the FM radio-frequency spectrum. B , is the same

for either a sinusoidal modulating signal or a noise -modulating signal,

provided that the proper values are used for P and P

ms mn

The selected value for B Ji (teletype-voice) = 10.5, may now

rf m

be used in figure 4-6 to determine the required median of the nondiversity receiver-input RF signal-power level, P , as follows: at

the intersection of the B Ji - 10.5 level on the ordinate scale, and

rt m

the "design curve" for 26. 7 db, we have,

10 Log

10

P

r

- 10 Log

1() fm

- 10 Log

1()

(T A+ T ] = -210. 5

or, for the FM system

10 Log P = 52. + 30.4 - 210. 5 = -128. 1 dbw.

This value of P (-128. 1 dbw) must be corrected for the effects

r of system intermodulation noise; see section 4. 3. 6 and figure 4-A.

Assuming that the receiver-output system intermodulation noise is equal

to the receiver-output noise due to the pre-detection amplitude noise,

the required correction to the above calculated value of P is 3 db.

r Hence,

10 Log lrt P =-128.1+3. =-125. 1 dbw (FM)

1 r

The above calculated value of P (-125. 1 dbw) is the median nonr diversity received (total) RF signal-power level at the FM receiver

input, required to transmit the specified message load, with an average

teletype -message error rate of . 01 percent and a predicted voicemessage intelligibility of 90 percent. This value of P is also the
-107-

value to be substituted in (7.1) of section 7, together with appropriate

values of the other factors in (7. 1), to obtain the required transmitteroutput average power level, P . Note- -The assumed value of 3 db for FM system intermodulation

noise may be too low in a system which employs an RF spectrum bandwidth of several Mc/s. A more realistic value might be 3 db for equipment intermodulation noise and 2 db (B = 1.64 Mc/s) for path-modurf

lation noise. With the additional correction of 2 db for path modulation

effects, the required value for median P (FM) = - 123. 1 dbw.

The above design procedures apply to the condition where particular system specifications and a grade of service are given and it is

required to determine the total average modulating- signal power level,

P , the individual message-channel average modulating-signal power mn

level, p ., the RF signal spectrum bandwidth, B r , or B. r , and the rmi rf if required receiver -input RF signal-power level, P . The system

design curves may also be used to estimate system performance when

either or both B and P are specified. Furthermore, system perrf r formance may be estimated for a time - varying median- value of P ;

this time -varying median P is obtainable by methods outlined in

section 7.
-108-

6.4 Test Procedures for SSBSC-AM and FM Systems

Acceptance tests on radio communication systems involve the

measurement of the system performance, in terms of the message

error rate for particular message loads and transmitter output power.

Test programs should be based on acceptable message-error rates,

for a specified number of simultaneously-active message channels,

with a given radio-frequency spectrum bandwidth, and for a particular

value of transmitter -output power. The information contained in the

system design curves of this report may be used as a guide for determining proper test procedures, and also for estimating equipment testlevel values for the system variables and parameters, as applied to

SSBSC-AM and FM multi-channel radio communication systems.

Acceptance-test programs may be arranged conveniently by following

the system design procedures outlined in Sections 6.1, 6. Z, and 6.3.

The major differences between design procedures and test procedures

are due to the fact that in the former case certain parameters are to

be estimated while in the latter case these same parameters are

specified and the system performance is to be measured. In either

case the design equations and the design curves in this report may be

used to obtain quantitative estimates of the performance of the radio

communication system.

6.5 System Performance Estimates for

SSBSC-AM and FM Systems.

Using the procedures outlined in Sections 6.1, 6. Z, 6. 3, and the

appropriate design curves and equations given in this report, the performance of a specific radio communication system may be estimated

in terms of the received RF-signal power, P , and the corresponding

r message error rate. One form of presentation of these results is
-109-

shown in figure 6-1. Note that the curves in this figure are only presented as being typical and are not accurate, hence they do not apply

to a particular system; the same is true of figures 6-2 and 6-3.

A typical measured or estimated distribution [ Rice, Longley,

Norton; 19^9] of a received tropospheric radio signal is shown in

figure 6-2. The "sampling periods" should be restricted to intervals

during which the distribution of the received RF signal approximates

a Rayleigh distribution. The sampling period may also be defined in

terms of an interval of time which is very large compared with the

duration of an information bit; this definition permits the transmission

of a "usable" sample of the message during the sampling period. The

sampling-period median received-power level, P , is sometimes referred to as the "hourly median", with the (implied) understanding that it

is not necessarily the median for an hour but that the period of time

might only be a fraction of an hour, or it might be several hours; the

above definitions of the sampling period may be used to clarify the

meaning of the hourly median, of the received RF signal-power level.

The total time involved in the percentage-of- sampling periods scale

must be very much greater than the sampling-period time.

Combining the data in figure 6-1 with that in figure 6-2, using the

method outlined in Appendix D, we obtain the distribution of the message error rate as shown in figure 6-3. Note that the positions of

these curves, relative to the message error-rate scale, depend

directly upon the median power level, P , of the received RF- signal,

as given in figure 6-1. The total time to be considered in figure 6-3 is

identical to the total time indicated in figure 6-2; that is, if figure 6-2

represents the distribution of the received RF-signal power levels

for one day, then figure 6-3 will show the distribution of the message

error rate for the same period of time, that is, for the same day;

similar results may be obtained for a portion of a day, for one week,

one month or for any specified total time.
-110-

MESSAGE ERROR RATE VERSUS RECEIVED RF-SIGNAL

-105 Pr

, NON -DIVERSITY SAMPLING-PERIOD MEDIAN

OF RECEIVED RF-SIGNAL POWER, dbw

Figure 6-1
-Ill

<

Q

UJ

o-d

a: . UJ _i

Q_ <

I

11 t>— lil

UJ a;

i c

DISTRIBUTION OF MEDIANS OF RECEIVED

RF-SIGNAL POWER

-115

-120

-125

-130

-135 I 0.5 5 10 20 30 40 50 60 70 80 85 90 95 98 99 99.5 PERCENTAGE OF SAMPLING PERIODS ORDINATE VALUES ARE EXCEEDED

Figure 6-2
112

DISTRIBUTION OF MESSAGE ERROR RATE

IU

7

5

3

2

i

NON-DIVEF*SITY

i

0.7

" DUAl^^*"

0.5

0.3

0.2

0.1

QUADRUPLE

0.07

0.05

0.03

0.02

001

0.01 I 5 10 20 304050 60 70 80 90 95 98 99 99.5 99.9 99.95 PERCENTAGE OF SAMPLING PERIODS DURING WHICH THE MESSAGE

ERROR RATE IS LESS THAN THE ORDINATE VALUE

99.99

Figure 6-3
-113-

The information in figure 6-3 is presented in a form which may be

easily and unambiguously interpreted. The statistical nature of the

message error rate is clearly indicated by these curves. From data

of the type shown in figure 6-3 it is possible to predict the system

performance for various periods of time such as, all-year, worst-day,

etc.

The data in figure 6-1 would apply to a particular message load on

the specified system. The material in this report may also be used

[Barsis, et al. , 196l] to estimate radio communication system

performance for various assumed system-parameter values, and the

results may then be presented in terms of the percent message error

and the various parameters such as, the RF- signal spectrum bandwidth,

B , number of message channels, N, percent message load, etc. This

ri

type of presentation of system performance estimates may be applied

to specific cases, and it is sometimes more convenient to use this form

of presentation than to use the general form of design curves shown in

this report. However, these design curves are basic and of course are

not restricted to specific cases.

Test procedures involve the calibration of the receiver so that it

may be used to indicate the power level of the received RF signal. The

receiver-input signal power, P , is available at the receiver -input

terminals. Hence, when calibrating a receiver with a signal generator,

using impedance mismatch conditions at the receiver input, in order to

simulate operating conditions, care must be exercised to make certain

that the values of P obtained from the signal generator output-power

r scale are properly corrected. These corrections would be in terms of

the output impedance of the signal generator and the characteristics of

the impedance -changing unit, which may be required between the

generator output and the receiver input. Proper corrections (see Appendix A) will yield the available RF signal power at the receiver input

terminal.
-114-

7. REQUIRED TRANSMITTER-OUTPUT POWER

Under conditions where the signal power is transmitted over a

radio path, the required transmitter -output RF- signal power, P , may be calculated from the following (equivalent) equations:

P = (P LT L, L L. L, )/G (7.1)

t r T bm c t du p

or,

10Log

1()

P

t =10Log1()

P

r + 10Log10

L

T

+ 10Log

1() L^

(7.1a)

- 10Log,„ 5 G + 10Log lrt (L L L n ), dbw

10 p

5 10 c t du'

where,

P = Transmitter output average power, watts.

P = Required available receiver -input median of the

non-diversity signal power level, watts. Median

values of P are used primarily because of the

fact that calculations of the path transmission

loss, L, , yield results in terms of median

values for L» . Furthermore, it is easier to

obtain measured values of median P (or L. ) from their (measured) distributions than it is to

obtain average values for these factors.

L = Total antenna- system transmission loss factor

at the transmitter, as a ratio.

L, = Median of the basic transmission loss. This bm

factor is defined as the ratio of the power input

to the terminals of a loss-free isotropic transmitting antenna to the power available from the

terminals of a loss-free isotropic receiving

antenna. The median basic transmission loss,

L , and the effective path gain, GD

, may be

calculated by methods outlined in NBS Technical

Note #15. [Rice, Norton, Longley, 19 59] and NBS

Technical Note #101 [Rice, et al. , 1962] .
-115-

*

' , L , = Transmission loss factors for the (receiving) antenna

c t du . , , , , circuit, transmission line, and the duplexer network,

respectively, as a ratio. These factors are defined

as the ratio of the power delivered to the input of a network to that portion of the input power which is

available at the network output. These factors are

conveniently measured for non- reflecting impedancematched conditions at the input and at the output of

the network.

G = Effective path gain [ Hartman, Wilkenson, 1959;

JTAC, I960] of the transmitting and receiving

antennas combined, relative to an isotropic antenna,

as a ratio.

Eq. (7. 1) may also be written [ Norton, 1956] as:

Pt = (R B. r F L L kT )/G (7.2)

t if T bm o" p

x

or,

10Log

1()

P

t = 10Log

1()

R + 10Log

i0

B.

f + 10Log

1()

F + 10Log

1()

L

T

(7.2a)

+ 10Log in L. - lOLog, _ G - 2 04, dbw °10 bm °10 p

The factor F in (7.2) is the "effective noise figure," [ Barsis,

Norton, Rice, Elder, 19&1, Appendix III] . For a receiver with no spurious responses,

T + (L - 1) T + L (L - 1) T + L LJF - 1) a

x c c c t t c t r , ? 3

>

o

The factors in (7. 3) are identical to the factors used in (4. 3) and

(4.3a), provided that impedance -matched conditions exist in the receiving antenna system; these factors are described in Appendix A.

* These factors apply only to the case where non- reflecting impedancematching conditions prevail in the transmitting and receiving antenna

systems, except at the receiver input terminals.
-116-

The factor R in (7.2) is the predetection signal-to-noise ratio,

P.yN.,.. The relationship between R and P maybe obtained by

if if r combining (7.1) and (7. 2), and is

RB.,FkT

P = . 1

. ° (7.4)

r L L. L n c t du

Multiplying both sides of (7.4) by l/f (T + T ), we obtain

P RB.JkT

lf ° (7.5)

f (T A + T ) L L LJ mA f (T A + T ) er ctdumA er

From (4.3) and (7.3), and neglecting the term P /k B L , we have

L L. L, (T. + T ) F= C t du^ A ^-

(7.6) T

o

Combining (7. 5) and (7. 6) we obtain

P R B k R B k

r if rf

f (T A + T ) f f m A er m m

(7.7)

Equation (7.7) is useful when it is desired to obtain values of the

predetection signal-to-noise ratio, R, from the system design curves;

this equation gives the appropriate substitution of abscissa-title factors

in the design curves in figures 4-2, 4-4, 4-5, 4-6, and 4-7.

The required transmitter -output power, P , may be determined in

terms of either:
-117-

(1) the required receiver -input power level, P , which is obtainable

r directly from the system design curves and then substituted in

(7. 1) or (7. la), or

(2) the required predetection signal-to-noise ratio, R and the required

bandwidth, B (= B.J; values for these factors are inserted in

rf if

(7.2) or (7. 2a).

The choice between the use of either of the above procedures,

(1) or (2), depends entirely upon the designer's preference; however,

where performance -test measurements are made it is very much

more convenient to measure P than to measure R. Furthermore,

r (7.1) and (7.1a) contain only power levels, loss factors and gain

factors and hence these equations are easily understood. Equations

(7.2) and (7. 2a) involve the additional factors, F and R, each as a

ratio; this situation apparently is confusing to many engineers.

The following relationships between P /N. f and the various

system parameters have been found to be useful, and are included

here for convenient reference:

P.

f P

R = —

= (7.8)

N. r k(T A + T ) B..

if A er if

P

a L L L k(T + T )B. r c t du A er if

(7.9)

P G

t p

L L, L L L, k(T A + T ) B.

,

T bm c t du v A er' if

(7.10)
where,

-118-

-23

k = Boltzman's constant = 1.3804 x 10

T = Effective receiving-antenna system output

noise-power temperature, in degrees Kelvin;

see Appendix A, pp. 157-161.

T = Effective receiver-input temperature, in

er degrees Kelvin. This factor is given by

T = (F - 1) x 290.

er r P = Received RF-signal, available at the terminals

of the "loss-free" antenna.

From (7.8), (7.9), and (7. 1 0) we have,

P

(7.11)

r L L L,

c t du

P

r

' LT U "I L L

<7 ->2 > T bm c t du
-119-

CONCLUSIONS

The equipment-performance design curves presented in this report

are based on the performance of "ideal" radio receivers and maximalratio type of combiners and hence these curves will yield optimistic

results. However, for FM systems, placing the single sinusoidal

modulating signal in the highest (baseband) frequency partially compensates for the optimistic assumptions regarding receiver (ideal) performance and combiner operation. Furthermore, proper allowance

may be made for the effect of system intermodulation noise using the

methods outlined in subsection 4. 3. 6 and following the examples worked

out in section 6.

In practice, the degree of conformity between calculated and measured system performance will depend largely upon the performance

of the combiner, compared to the (ideal) performance of the maximalratio combiner. In fact, a comparison between the calculated and the

measured performance of a system constitutes a reliable check on the

combiner performance, after due allowance is made for all other factors

The method used to obtain the equipment-performance curves

implies that pre -detection diversity combining is used; the use of these

results, where post-detection combining is employed, might be

questioned. However, if the combiner operates so as to improve the

distribution of the signal-to-noise ratio of the (combiner) input signals,

then our methods are justified; since the overall results should be similar whether the combiner precedes or follows the receiver demodulator.

The results of this work are presented in a form such as to be

directly usable in determining system performance when combined with

effective antenna gain and radio -path transmission-loss estimates.
-120-

The design curves shown are basic and may be used to determine

sets of curves showing the system-performance characteristics for

various combinations of the system parameters. These design curves

involve the fundamental parameters of the system and hence may be

used to establish equipment-performance standards on a quantitative

basis.

It should be understood that the technical analysis of FM and

SSBSC radio communication systems carried out in this paper does

not involve economic factors or operational complexities. However,

this type of technical analysis is required as a starting point for the

determination of a complete analysis and comparison of the different

types of systems.

In conclusion, a method has been outlined in this report, and a

set of system-design curves have been developed based on the combined performance characteristics of the receiver, diversity combiner,

modulator, and the message -signal decoder unit. The net results of

this work are a set of curves in which all of the important system

parameters are used as scale factors. The technique employed in

this work essentially provides a link between information theory results

and the application of these results.
-121

ACKNOWLEDGE ME NTS

The material for this report was obtained through the assistance of

a large number of individuals. Of these, the more important are:

M. J. Ogas, who assisted in obtaining financial support from the USAF;

Colonel Wm. E. Geyser, USAF, for permitting us to make use of the

"Pole -Vault" tropospheric radio link between Gander, N. F. , and

Pepperrell AFB, St. Johns, N. F. ; and Mr. Carlton J. Modlin, who

made a major contribution to the experimental work carried out on the

tropo system at Verona, N. Y. We are indebted to Mr. R. W. Brauer

of the Comm. Sys. Eng. Div. , USASEA, for providing the tropo link

between the East Coast Relay Station, Frederick, Md. , and La Plata,

Md. , to carry out system-performance tests; and also the funding of

the project. The cooperation of the Commanding Officers at the ECRS

and the La Plata Station, Major Alfred K. Granschow and Major Howard

L. Hall, and the assistance of the personnel at these Radio Comm.

Stations, is gratefully acknowledged. Other persons who were of

assistance in this program are: Sgt. George H. Farmer, R. F.

Kaltenbach, and Sen. Sgt. R. L. Prather, all of Pepperrell AFB;

Richard V. Locke, Burt E. Nichols, and Dave Karp of Lincoln Laboratories; Joe S. Turner, Wm. Long, L. R. Goodell, and Keith Garlets

of Collins Radio Co.; Don Glen, B. F. Quereau, J. A. Clark, B. D.

Samsel, and H. R. Dahms of the National Bureau of Standards, Boulder

Laboratories. Mr. John C. Harman and his assistants in the Drafting

Department of the Boulder Laboratories are to be commended for the

accuracy of the drawings. Mrs. Marylyn Olson, Mrs. Doris Hunt,

and Mrs. Ruth H. Rotherham deserve credit for the tedious job of

typing this report.

For constructive criticism of this report, we are grateful to:

K.A.Norton, R.C.Kirby, C. Gordon Little, A.P.Barsis, W.O.

Crichlow, R.W.Beatty, W. Beery, P. Hudson, C. Allred, and many

others.
-122-

REFERENCES

Albersheim, W. J., and Schafter, J. P., Echo distortion in the FM

transmission of frequency-division multiplex, Proc. IRE 40, 316-328 (March 1952). ~~

Barsis, A. P., Norton, K. A., Rice, P. L. , and Elder, P.H., Performance predictions for single tropospheric communications

links and for several links in tandem, Tech. Note #102, National

Bureau of Standards, Boulder Laboratories (Aug. 1961).

Becking, A. G. Th. , Groendijk, H. , and Knol, K.S., The noise

factor of four -terminal networks, Philips Res. Rept. 10, 349-

357 (1955).

Bell, R. L. , Induced grid noise and noise factor, Proc. IRE (Sept. 1951).

Beranek, L. L. , The design of speech communications systems, Proc.

IRE, 35, 880 (1947)

Black, Harold S. , Modulation theory, p. 224 (D. Van Nostrand and

Co., 1953).

Black, HaroldS., Modulation theory, p. 221 (D. Van Nostrand and

Co., 1953)

Brennan, D. G. , Noise methods II: Linear diversity techniques,

Lincoln Lab. Group Report 36-29, p. 56, p. 59 (Mar. 1, 1958).

Carson, J. R. , Notes on the theory of modulation, Proc. IRE, 10, 57

(1922); Proc. IRE 7, 187(1929).

Clutts, C. E., R. N. Kennedy, and J. M. Trecker, Results of bandwidth tests on the 185-mile Florida-Cuba tropospheric scatter

system (I960).

CCIR Report No. 65, Revision of atmospheric radio noise data, International Telecommunication Union, Geneva (1957).

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 (Aug. 25, 1955).

Eagan, J. P., Articulation testing methods II, U.S. Dept. of Commerce

Report No. PB22848 (1944).
-123-

Ewen, Harold I., A thermodynamic analysis of Maser systems, Microwave Journal, 41-46 (March 1959).

French, N. R. , and J.C. Steinberg, Intelligibility of speech sounds,

J. Acous. Soc. of Amer. _19, 90 (1947).

Friis, H.T., Noise figure of radio receivers, Proc. IRE, 419-422

(July 1944).

Gladwin, A.S. , Energy distribution in the spectrum of a frequency

modulated wave, Part II, Phil. Mag., 38^ 229-251 (April 1947).

Grimm, H.H., IRE Trans, on Instru. , 97-103 (Dec. 1959).

Grimm, H.H., Noise temperature in passive circuits, Microwave

Jl., 52-53 (Feb. I960).

Hansen, R.C., Low noise antennas, Microwave Jl. , 19-24 (June 1959).

Harris, Donald P. , An expanded theory for signal to noise performance

for FM systems carrying frequency division multiplex, IRE Convention Record, Part 8, 298 (1958).

Hartley, R. V. L, Transmission of information, B.S.T.J., vol. 7,

July 1928, 535-563.

Hartman, W.J. and R.E. Wilkerson, Path antenna gain in an exponential atmosphere, Jour, of Res. of the Nat. Bur. of Stda. -D,

Nov. -Dec, 1959.

Haus, A.H. , and R. B. Adler, Circuit theory of linear noisy networks,

Tech. Press of MIT and John Wiley & Sons, 1959.

Haus, A.H. , and R. B. Adler, Optimum noise performance of linear

amplifiers, Proc. IRE, Aug. 1958.

Hogg, D.C. and W.W. Mumford, The effective noise temperature of

the sky, Microwave Jl. , 80-84 (March I960).

Hund, August, Frequency modulation, first ed. , 33 (McGraw-Hill

Book Co. , Inc. , 1942).

IRE Committee on Noise, Standards on electronic devices: Methods of

measuring noise, Proc. IRE, 891-896 (July 1953); 60-68 (Jan. I960),

IRE Committee 7. 9 on Noise, Representation of noise in linear twoports,

Proc. IRE, 69-74 (Jan. I960).

ITT Handbook, Reference data for radio engineers, Fourth Edition

(1956).

Joint Technical Advisory Comm. , Radio transmission by ionospheric

and tropospheric scatter, Proc. IRE, (Jan. I960).
-124-

Landon, V.D., Theoretical analysis of multiplex transmission, RCA

Review (June, Sept. 1948).

Medhurst, R. G. , RF bandwidth of frequency-division multiplex using

frequency modulation, Proc. IRE, 189-199 (Feb. 1956).

Middleton, D. , The distribution of energy in randomly modulated waves,

Philosophical Mag. , 42, 689-707 (July, 1951).

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

tropospheric propagation, IRE Trans, on Communications Systems

CS-4, No. 1, 39-49 (1956).

Nyquist, H. , Certain factors affecting telegraph speed, B.S.T.J., vol.

3, Apr. 1924, pp. 324-326.

Nyquist, H. , Certain topics in telegraph transmission theory, AIEE

Trans., vol. 47, Apr. 1928, pp. 617-644.

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Rev. 32, 110 (July 1928).

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(I960).

Rice, P. L. , A.G. Longley and K. A. Norton, Prediction of the cumulative distribution with time of ground wave and tropospheric wave

transmission loss, NBS Tech. Note #15 (July 1959). Available

at a cost of $1.50 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.

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predicition for tropospheric communication circuits, N.B.S.

Technical Note #101.

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(June 1956).

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Siegman, A.E. , Thermal noise in microwave systems, Part II, Microwave Journal, 66-73 (April 1961).

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combined in random phase, J. IEE 93, Part III, 76-78 (March

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-125-

Stewart, J. L. , The power spectrum of a carrier frequency-modulated by Gaussian noise, Proc. IRE, 1539-1542 (Oct. 1954).

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IRE, 36, 1081-1992 (Sept. 1948).

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(July 1956).

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-126-

LIST OF SYMBOLS

B Composite baseband signal spectrum bandwidth, c/s.

b

B Message -signal spectrum bandwidth in c/s. In a single - sideband frequency-division multiplex system B is also

the bandwidth occupied by the message signal in the composite modulating baseband signal.

B.. Effective (noise-power) bandwidth of the receiver IF

circuit, at the receiver demodulator input.

B Effective bandwidth of a network, in c/s.

n

B f Radio-frequency signal power -spectrum bandwidth of

the total radio-frequency signal, in c/s.

B - Radio-frequency signal FM spectrum bandwidth for a white-noise type of modulating signal.

B Radio-frequency signal FM spectrum bandwidth for a

sinusoidal modulating signal.

D Modulation-demodulation conversion factor. This

s s factor is defined as, P (100%)/P , and applies

only to amplitude modulation.

F Effective noise figure (Barsis, et al, Appendix III, I960).

f , f , f Frequency in c/s of particular signals in the modulating

composite baseband signals.

f Maximum value of the modulating- signal frequency in

c/s. Also, the highest frequency in a composite baseband modulating signal.

F Noise figure of a network, as a ratio; see figure A-l.

F Noise figure of (radio) receiver, or amplifier, as a

r ratio, measured according to IRE Standards (IRE Comm.

1953, I960).
-127-

G

r Power gain of the linear section of the (radio) receiver,

or amplifier. This factor is defined as the ratio of the

available output power to the power delivered to the

receiver input. This is not the available gain.

-L 3 k Boltzman's constant = 1.38042 x 10 joules per degree

Kelvin.

K Bandwidth- compression factor, applied to the receiver.

This factor indicates the degree of IF bandwidth compression in the receiver; note that < K < 1. c L Transmission loss factor for the antenna network, due

c to the lossy elements in the receiving antenna. This

factor is equal to the ratio of the non- reflecting impedance-matched delivered power at the network input to

the non-reflecting impedance-matched power at the

output of the network.

L Ditto, for the duplexer.

L Ditto, for the antenna- system transmission line.

L Transmission loss factor of a network; defined as the

ratio of the non- reflecting impedance-matched input

power to the non- reflecting matched-impedance output

power.

L. Isolation factor for the duplexer. This factor is equal

to the ratio P« f /N. , see figure 4-1.

L Total transmitting antenna system transmission loss,

as a ratio.

m Amplitude -modulation index, in percentage.

M. Impedance mismatch factor, at network input,

in

M Impedance mismatch factor at the receiver input termlr \ mals.

M Number of simultaneously- active message signals each

of which requires the same value for the product

< B

c<So</

N

oc> >AMT Ditto, for the product (B (S /N ) ) T

. I c oc oc I
-128-

M Ditto, for the product (B (S /N ) ) . N c oc oc N

N Receiving-antenna environmental noise power, in watts.

3

N Ditto, for interfering signals.

3.1 N Available noise power at the output terminals of a netao , r- a i work, in watts; see iigure A-l.

N Receiving antenna noise power, transmitted with the

RF signal, in watts.

N Available noise power at receiving-antenna system output, due to antenna temperature, T , plus noise gener3. ated within the antenna- system units.

N Noise power generated within and contributed by a network, in watts. This noise power is referred to the

input terminals of the network, see figure A-l.

N Total of the average noise power at the IF output, or at

the demodulator input, in watts.

N Receiver output average total noise power, in watts.

N Average noise-power level of a message-signal channel having an effective bandwidth of B , at the receiver

output; or, a relative average noise -power level per

bandwidth, B , at decoder-unit input terminals,

c N Total available noise power from the receiving-antenna system

p Average power level of a message-signal in the modu- m

mi

lating baseband signal, in watts

1"U p Average power level of the i— message signal in the

modulating baseband signal, in watts.

P Receiver IF-output total average signal power, or,

average signal power at the demodulator input, in watts

P Power output of the local transmitter, in watts.
-129-

P Total average power level of the baseband modulating

signal. It is understood that this power level is determined under conditions when a specified number of

simultaneously-active message signals are in operation.

P

Message loading ratio, being the ratio of the total

m average power level of the modulating signal to the

average power level of a message signal.

P Average power level of a "white -noise" modulating

signal, in watts; where the modulating signal is distributed uniformly across a specified portion of the modulating baseband.

P Peak-power level in the composite modulating signal,

' exceeded for only a specified percentage of the time.

P Average power level of a sinusoidal modulating (system

test) signal, in watts; under conditions where only one

modulating signal is used. For FM systems the frequency of this modulating signal is f --the highest

frequency in the baseband signal.

P (100%) Sinusoidal- signal power level required to produce 100%

amplitude modulation.

P Total average radio-frequency signal power available at

the receiver -input terminals, in watts. For a time-varying RF signal, the median value of P is used in this

report.

P Transmitter average total power output, in watts.

P Peak-power level at the transmitter output, exceeded for

tp

P

N

only a specified percentage of the time.

R= T7^— Predetection total signal-to-total noise ratio. For the

if case of a varying received signal, R will be in terms of

its median value.

S Receiver output (baseband) average total signal power,

o

in watts.
-130-

S Average message-signal power in the receiver output, or

a relative message-channel signal average power level

at the decoder -unit input terminals.

T Receiving-antenna noise (power) temperature, associated

with the sky noise and the environmental noise, in degrees

Kelvin. This noise temperature, or its equivalent noise

power N , is available at the output terminals of the lossfree antenna.

T Ditto, for interfering- signal power.

T Ditto, for the transmitter -output noise power transmitted

with the carrier signal.

T Thermal temperature of the lossy elements in the (receiver ing) antenna-circuit network; see figures A-l, 4-1, and

Appendix A.

T Ditto, for the duplexer in receiving-antenna system.

T Effective input noise (power) temperature of a network,

ei

er

n

o

degrees Kelvin; see figure A-l

T Effective output noise-power temperature of a network,

degrees Kelvin; see figure A-l.

T Effective receiver-input temperature, degrees Kelvin.

(IRE Comm. on Noise, I960).

T Output noise-power temperature of source or signal

° generator, degrees Kelvin; see figure A-l.

T Thermal temperature of the receiving-antenna transmission line, degrees Kelvin.

T Thermal temperature of the lossy elements in a passive

network, degrees Kelvin; see figure A-l

T Standard reference temperature, assumed to be 290.

degrees Kelvin. (Ire Comm. on Noise, I960).

T Effective output noise (power) temperature of the

receiving-antenna system, degrees Kelvin. T is

defined by: N = kT B , see Appendix A.

1 J\ IX
-131-

z

du Characteristic output impedance of the duplexer. In

general this is the characteristic output impedance of

the network which precedes the receiver.

z

^ n

Input impedance of network or amplifier, in ohms.

z - Input impedance of the radio receiver.

z Output impedance of source or signal generator, in ohms

z

on

Output impedance of network or amplifier, in ohms.

z j. Image or characteristic input impedance of a network

or transducer.

ZJri Image or characteristic output impedance of a network

or transducer.

Io

AF Peak radio-frequency carrier deviation above and below

the un-modulated carrier frequency, in c/s; for a single sinusoidal modulating signal having an average

power level P , equal to the total average power level

of the composite modulating signal.

AF^ Peak radio-frequency carrier deviation in c/s; due to

the message-signal modulating power, p

c m —AF

Deviation ratio, in radians. This ratio is equal to the

"phase deviation" of the radio-frequency carrier, that

is, the extreme angular displacement of the carrier

from its "average" position.

P Correlation (radiofrequency) bandwidth in Mc/s. This

bandwidth is defined as the frequency separation in c/s

of spectral components whose cross-correlation coefficient has an average value of 0. 5.

c|>(AF/f) Calculated ratio of the radio-frequency spectrum bandwidth B -, to the deviating frequency f ; applicable only

to the case where a single sinusoidal modulating signal

is used--see figure 3-1.
-132-

Figure No. Page No.

2-1 Block Diagram of Basic Units in a Multichannel Tropospheric Radio Communication

System 6

3-1 Relationship Between the Deviation Ratio

AF/fm and RF Spectrum Bandwidth, B f

, which Includes 99. 99 Percent of the RF

Spectral Power 12

3-2 Receiver Output S /N Ratio Versus B e/V>- c oc oc rf if for Various Receiver -Input Total RF Signal

Power Levels Steady Receiver-Input Signal

Sinusoidal Modulating Signal 14

3-3 Relative Levels of A F and P^,, m Sinusoidal

and Noise Modulating Signals, and for Equal

RF Spectrum Bandwidths 19

3-4 RF Spectrum Bandwidth, B f

, Versus Relative Modulating-Signal Power Levels, for

Frequency Modulation 2

3-5 Amplitude Modulation Exceeding 100 Percent

Versus Relative Modulating-Signal Average

Power Levels, For SSBSC-AM 26

4-1 Block Diagram of Receiving System 136

4-A FM Receiver-Output Signal and Noise in the

Top Channels of the Baseband Signal 42

4-2 SSBSC-AM Radio Receiver Performance

Curves Steady and Rayleigh-Fading Types

of Received Signal 64

4-3 FM Radio Receiver Characteristic Curves

Steady Received Signal 72

4-4 FM Radio Receiver Performance Curves

Steady Signal 76
Figure No,

4-5

4-6

4-7

4-8

4-9

-133-

FM Radio Receiver Performance Curves

Rayleigh-Fading Signal, Non-Diversity

FM Radio Receiver Performance Curves

Rayleigh-Fading Signal, Dual Diversity

FM Radio Receiver Performance Curves

Rayleigh-Fading Signal, Quadruple

Diversity

FM Receiver Characteristic Curves,

Without Pre-Emphasis

Effect of Pre-Emphasis on FM Receiver

Characteristics Curves

4-10 Effect of Bandwidth Compression on FM

Receiver Characteristics

4-11 Effect of Bandwidth Compression on FM

System Design Curves

5-1 Cumulative Amplitude Distribution of Noise

at Receiver Output for Received Carrier

Power Levels Above and Below the FM

Radio Receiver Threshold Level

Page No,

78

79

80

82

84

85

87

89

5-2 Measured Relationship Between the SteadySignal Power to Mean Noise Power Ratio,

at the Teletype Receiver Input Terminals,

and the Average Teletype Character Error

Rate

5-3 Measured and Calculated Relationship

Between the Signal Power to Mean Noise

Power Ratio, at the Teletype Receiver

Input Terminals, and the Average Character

Error Rate at the Teletype Printer Output

90

92
-134-

Figure No. Page No,

6-1 Message Error Rate Versus Received RF

Signal 110

6-2 Distribution of Medians of Received RF

Signal Power 111

6-3 Distribution of Message Error Rate 112

A-l Network Noise Powers and Noise Temperatures 145

D-l Graphical Method for Determining FM

Receiver Performance 177

D-la FM Receiver Performance CharacteristicSteady Received Signal 177

D-lb Cumulative Distribution of Maximal-Ratio

Combiner Output, for Dual and Quadruple

Diversity, for Rayleigh-Distributed Input Signal 177

D-lc Distribution of FM-Receiver Output Signal-toNoise Ratio-Time Varying Received Signal 177

D-ld FM Receiver Performance CharacteristicRayleigh-Fading Signal 177

D-2 Graphical Method for Determining MessageDecoder Performance 181

D-2a Performance Characteristic of Teletype

Decoder Unit-Steady Signal-to-Noise Ratio 181

D-2b Distribution of FM-Receiver Output Signal-toNoise Ratio-Time Varying Received Signal 181

D-2c Distribution of Teletype Character Error

Rate for Time-Varying Received Signal 181

D-2d Performance Characteristic of Teletype

Decoder-Unit for Time-Varying Received

Signal 181
-135-

APPENDIX A

Pre-Detection Noise-Power Levels in the Receiving System.

The objective is to obtain the functional relationship between the

predetection noise-power level, N in figure 4-1, and the various

receiving-system parameters: F , L , G , T , etc., as indicated

x x x x

in this figure. The method used to determine the total pre-detection

noise-power level in the receiver consists of properly adding the noisepower outputs of the receiving-system (linear ) twoport network

units in cascade [ Ewen, 1959; Hansen, 1959] . Basically, the noise

performance of each network is considered in terms of the amount

of noise contributed by the network, and the resultant effect of this

noise on the network- output signal-to-noise ratio.

In this analysis, the noise performance of each twoport network

unit in the receiving system is determined in terms of the total noise

power delivered* to and absorbed at the input of the unit (including the

noise power originating in the unit) and the resultant available* noise

power at the output of the unit. The use of delivered noise powers at

the input of the units results in noise-performance factors which depend primarily upon the internal characteristics of the network

units, such as power gain, effective bandwidth, image or characteristic impedance, and transmission loss . Furthermore, network

noise figures, and noise performance characteristics, may be defined

in terms of delivered (signal and noise) powers instead of available

powers at the network input, without any loss in generality. The

use of the delivered-power concept has the additional advantage of

*If impedance mismatch conditions exist at the input terminals of the

network, the delivered power is dependent upon the input impedance of

the network and the output impedance of the preceding network. The

available power is the maximum power which may be obtained from the

network, at its output terminals; see following subsection.
136-

LUco

>•

coCD2>

q:<o<QoO

_iCD i •
-137-

making it possible to define and to calculate the impedance mismatch

factors in terms of the various system impedances, independently of

the power gain or the dissipative transmission loss, and the effective

bandwidth of the networks.

The available -power concept, as applied to the noise power (and

also the signal power) at the network output terminals, is valid

because we are restricting this analysis to receiving antenna- system

units having positive input and output resistances. In other words, the

"exchangeable-power" concept is not required in this case [Haus,

Adler, 1959] . The various factors discussed above such as delivered power,

available power, transmission loss, etc. are defined and discussed

in the following subsections.

Effects of Impedance Matching on

Delivered, Matched-Impedance, and Available Powers.

In this appendix the effects of impedance mismatching is treated

in a general manner, for the purpose of indicating how the impedance

mismatch factors affect the noise-power level, or the noise temperature, at the receiving-antenna system output terminals. No attempt

is made here to derive the mismatch factors as explicit functions of

the antenna system impedances.

The noise performance properties of a linear lossy passive bilateral twoport network, of the type generally used in a receivingantenna system, are dependent upon impedance -matching conditions

at the network input and output terminals. The effects of impedance

matching on network-input and output powers are determined from

the following considerations, based on information in the Radio Engineers Handbook [ Terman, 1943] and the ITT Handbook [pp. 549-616] .
-138-

The power delivered to and absorbed by a load having an input

impedance Z ( = R + j X ), when connected to the output terminals of a source or generator having an internal impedance Z ( = R + jX ), g g g

and an internal (generated) voltage, E , is

2

l E I P

del

=

I * I R

i =

', ^TT X R

i

(A - l)

Z + Z I *

1 g i

'

where, I = Current flowing in the load.

If the generator internal (or output) impedance is non-reflecting

matched to the load impedance, that is, Z = Z , the power in the load

becomes

i i 2

I E I P . = L- x R (A. 2) mat . l „ | c. a 4 Z

K

g

Note that the impedance-matched power, P , is absorbed in the

mat

load under non- reflecting impedance-matched conditions at the input

terminals of the load.

The maximum power will be obtained from the generator when the

generator impedance is conjugate to the load impedance; that is,

Z = Z„ , and hence R = R. , and X = -X„ . The available power is

g

* g* gi F

defined as the maximum power which may be obtained from a given

source, and is

|E | Z |E | Z

P s ^- x R. = S— (A. 3) aV 4RZ l 4R

g g

A study of (A. 1), (A.Z), and (A. 3) shows that:

(a) the available power closely approximates the non- reflecting

matched-impedance power only when the ratio X /R is very much

g g

less than unity; that is, X /r < <i } anc}
-139-

(b) conjugate -impedance matching is a special case of impedance

(mis)matching and hence its use results in power reflections at the

conjugate -impedance -matched point.

The impedance mismatch factor is defined as the ratio of the

power delivered to the load without matching to the power delivered

with either type of impedance -matched conditions. Thus, the impedance mismatch factor may be defined in terms of either the non-reflecting matched-impedance power or the conjugate -impedance matched

(available) power. In terms of the non- reflecting matched-impedance

power, the impedance -mismatch factor is defined as

P , _ 4 | Z |V 4(R

2 + X

2

) R

M s

del_

= g_i = g gi {A4)

mat P

mat |Z +Z.| R | Z + Z J

2

R

g

| g gI g

In terms of conjugate -impedance matched or available power, the

impedance -mismatch factor is defined as

P, , 4 R R„

M =—- del g * ik c\ = s (A. 5)

av P i „ „ i c av Z + Z J

g ^ From (A. 4) and (A. 5) it is evident that the impedance-mismatch

factors for matched-impedance power and for available power are

identical only when the ratio X /R is equal to zero. Hence, these ~

g g

powers are approximately equal when the ratio X /R is very much

less than 1. In other words, maximum power transfer and non-reflecting impedance -matching can be attained simultaneously only when

X /R < < 1

.

g g

When a network is inserted between the generator and the load,

the impedance-mismatch factors at the network input and output terminals may be determined in terms of the net or effective input and output

impedances, Z and Z . In turn, Z. and Z may be calculated ^ in on in on
-140-

from the network input and output image impedances, Z and Z , and the generator and load impedances, Z and Z [Terman, 1943] . The input image impedance of a network is defined as the impedance

looking into the input terminals of the network with the output terminals terminated by the output image impedance; note that the input

and output terminals may be interchanged in this definition. The

image impedance of a network may be determined from

Z T = \T~Z Z =RT + iXT (A. 6)

I oc sc I

J I

v

where Z = Network input impedance with the network output

oc : , M , terminals open-circuited

Z ^ = Network input impedance with the network output

terminals short-circuited.

In a symmetrical network such as a transmission line, the input

and output image impedances are equal to each other and also equal to

the characteristic impedance of the line.

The image impedances of a network may also be calculated in

terms of the equivalent T or it network impedances [Terman, 1943,

pp. 208-210] . Conversely, the equivalent T or it network impedances

may be calculated for known or assumed values of network image impedances. These latter relationships are useful where it is desired to

calculate the net or effective input and output impedances, Z. and Z , in on

of a network in a tandem-network system in terms of network impedances

In a system of tandem networks, the impedance mismatch factors,

M. , at the input terminals of the networks are functions of the various

in

network image impedances, or the equivalent T or tt network impedances, and the generator and the load impedances. Each point or junction involving impedance mismatching may be treated as being

equivalent to the simple case of impedance mismatch between a source

(or generator) and a load; Z being replaced by Z and Z being
-141-

replaced by Z. . The network-input impedance mismatch factors may

in

be calculated by means of either (A. 4) or (A. 5); however, as stated

previously, no attempt is made in this Appendix to derive the networkinput mismatch factors explicitly in terms of the various impedances

of the system.

The results of the above derivations are to be applied to the analysis of the noise performance of linear lossy passive bilateral networks

in the antenna system; see pp. 147-151.

Note that if non-reflecting (image or characteristic) impedance

matching conditions are assumed to exist at all points in the system

except at the receiver input terminals, and if the ratios, X / R , are

assumed to be very much less than 1, throughout the antenna system,

under these conditions the available power will be approximately

equal to the matched-impedance power. Furthermore, at all of the

impedance -matched impedance points the available power will be

approximately equal to the delivered power. However, in the general

case, where impedance mismatched conditions exist at more than one

point in the system, and the ratios, X /R , are not each very much

less than 1, it is possible to define and use mismatch factors (to

estimate delivered power) in terms of either the non- reflecting matched-impedance power or the available power.

Noise Performance of a Transducer Network

Each lossy (bilateral) transducer unit, or network, in a receivingantenna system contributes (average) available noise power which is

equal to (Fnon -l)kT B [ Friis, 1944] ; where F is the noise figure of

n

the transducer unit, k is Boltzman's constant, and T is the "standard

o reference temperature" of the generator impedance used to measure

F and is assumed to be 290 degrees Kelvin [ IRE Comm. on Noise,

n

I960] . If the generator or source-impedance temperature, T , is
-142-

not T degrees, the correct of "standard temperature" value of F is ° T T

F = (1 - —°- ) + -=-°- F' ; where F' is the measured value of the noise

n T T n n

o o figure * for the condition that T is not equal to T [ IRE Committee on

g

H

o Noise, I960, p. 64, eq. (6. a)] . The available noise power contributed by the network, N ., (see

ei

figure A-l) is referred to the input of the unit, and is added to the

noise power available from the preceding unit, N , to give a total

available noise power, N . + N . The "available -power transmission

•ei ag

loss factor" is defined here as N . + N

_ ei ag . . , . ao

where, N is the available noise power at the network output terminao

als.

In terms of delivered noise power at the network input and available noise power at the network output, the "delivered power transmission loss factor" is defined here as,

M. ) (N . + N ) ^' ln)

n "

ag

< A ^>

ao

where f M. ) is the network-input impedance -mismatch factor, in

\ my av

terms of available powers, and may be obtained from (A. 5).

From (A. 6a) and (A. 6b) we obtain,

l =/Sk < a- 6c > an /M. \

av

in N

(

in

>

For Impedance-matched conditions at the network input, M. = 1 in

and X =0 we have,

g

L, =L (A.6d)

an n
-143-

Hence, the transmission- loss factor, L , may be considered as a n

special value of L or L. , and is seen to be an internal characteran dn

istic of the network.

Note that in practice L, may be measured conveniently by using

non- reflecting impedance -matched conditions at the input and output

terminals of the network with the (test) signal-power-level, P , much greater than the transducer noise -power level, (F - l)kT B . Note also, that L is the "insertion loss" factor as defined by IRE

n

Standards [59 IRE 2. Si] . The transmission loss factor, L , is dependent upon the "real"

part of the complex network "image transfer constant", Q, and is a measure of the dissipative loss of power in the network [ Terman,

1943] . The image transfer constant, Q is given by

tanh e = -x / ^(A. 7)

Where transmission lines are involved, L , in decibels is pron

portional to the product of the length of the transmission line, S. , and

the "real" part (q; ) of the complex propagation constant, y; where

Y = a + j£ . That is,

L cc e 2Qi

(A. 7a)

n

Network Bandwidth and Temperature Effects on Noise Power

The noise power level at the output of a network depends upon

the effective bandwidth of the network, B . The power accepted by

n

and processed in the network is proportional to B . Hence in a cascaded system of twoport networks, each of which may have

different effective bandwidths, the network with the narrowest

bandwidth will be the power-level and bandwidth-controlling unit.
-144-

It can be shown [ Siegman, 196la] that the exact expression for the

(maximum or available) noise power originating in a lossy passive

network having a thermal temperature of T degrees and an effective

bandwidth, B , is given as,

n

h f B

N

n= exp(hf/kT)-l (A - ?b)

where, h = Planck's constant

f = Frequency

hf

since < < 1 even at microwave frequencies and normal temperatures, exp (hf /kT) ~ 1 + —r^ and (A. 7b) becomes

iC JL N =kTB (A. 7c)

n n

and is valid for the radio frequencies and ambient temperatures used in

radio communication systems.

The effective bandwidth of the network, in c/s, is defined as:

B

n

5L;IiV d£ (A - 7e)

where L' is the effective network transmission-loss factor; see disn

cussion of the various types of L' above, and L(f) is the network transn

mission-loss factor, as a function of the frequency, f

.

Noise-Power Levels in a Network

Noise power levels, and "effective noise temperature" [Strum,

1956, 1958] , [Grimm, 1959, I960] , [Siegman, 196la, 196lb] , [ IRE

Committee on Noise, I960] , at the input and output terminals of a linear passive lossy bilateral twoport network unit are shown diagramatically in figure A-l. Figure A-l applies to the case of impedance-mismatching conditions at the network input and output
-145-

terminals. The interrelationships between: noise figures, F , the

respective effective input noise powers of the network and receiver,

N . and N , and the corresponding effective input noise temperatures,

T and T , are derived below,

ei er

The objective is to determine the available noise -power output

from the network, N , in figure A-l, in terms of the various system

ao

parameters.

The (fictitious) noise power N ., is contributed by the network and

is referred to the network input terminals. This noise power is applied

through the mismatch plane, M. , [ Rothe and Dalke, 1956; Haus and

Adler, 1958; IRE Subcommittee 7.9 on Noise, 19&0; Bell, 1951;

Becking, et al. , 1955] ; this condition is shown diagramatically in

figure A-l. This concept of network noise power or current flowing

through the generator impedance and hence, through the mismatch

plane at the network input, is derived from the form of the equivalent

"noise-source circuit" which is used in the above references.

The level of the noise power, N ., depends upon the network noise

ei

figure, F [Friis, 1944] , and is given by

N . = (F - l)kT B (A. 8)

ei n on

The available noise power from the source or generator is

N =kT B (A. 9) ag g g

-23

where, k = Boltzman's constant = 1.3804 x 10

T = Generator output-impedance (noise-power) temperature

B = Generator effective bandwidth

g

For the general case, involving impedance mismatching conditions,

and when B < B , the available noise power at the network output,

n g

N , is given by

ao & y

-145a -

NETWORK NOISE POWERS AND NOISE TEMPERATURES

M in

Tg , B

g %i

Na g=kTg Bg

•in

'din

LINEAR. PASSIVE

LOSSY BILATERIAL

TWOPORT NETWORK

Bn. Fn » L n i Tn

Mjn ^ogBn /B

g tNei

)' L dn

'ei <

-(Fn -l)kT Bn M;

n ( Tq -Hej)kBn

.= (Ln -l)kTn Bn (Mj n =l)

w= k Te j Bn * TeoBnV

>N,

Mij = M ir y

•on

Tei=(Fn-OT

= (Ln -l)Tn (Mj n

= |)

( Tg+Tei )M.

n

'eo - i . •dn

LOAD

OR

Zj» RECEIVER

(=Z ir ) Ndir

-N

'dir

J=(Fr -l) k T Bjf

er JjkTer B if

(Nao Bjf/Bn+Ner)Mjr MTeo+Ter )Bif M ir

Ter =(Fr -l) T

NOTE, B

g >Bn >Bif

Bx

= EFFECTIVE PASSBAND, c/s

i -

N din ao

L dn = L n (FOR Mip = I) F = NOISE FACTOR

k = BOLTZ MAN'S CONSTANT

Ln

= TRANSMISSION LOSS FACTOR, AS A RATIO

Na

= AVAILABLE NOISE POWER

Nd

= DELIVERED NOISE POWER

Tei Ter

= EFFECTIVE INPUT NOISE TEMPERATURE , DEGREES KELVIN T

g Jeo

= EFFECTIVE OUTPUT NOISE TEMPERATURE , DEGREES KELVIN

Tn

= THERMAL TEMPERATURE, DEGREES KELVIN

T = STANDARD REFERENCE TEMPERATURE

Z in = INPUT IMPEDANCE AT NETWORK INPUT TERMINALS

Zon s OUTPUT IMPEDANCE AT NETWORK OUTPUT TERMINALS

M; = IMPEDANCE MISMATCH FACTOR

Fig.A-l
-146-

N = -r^ (A. 9a) a° L

dn

( (N B /B ) + N . ^ V ag n' g ex J

L

an

(A. 9b)

(N B /B ) + (F - l)kT B ] ag n g n o n/

L

an

(A. 9c)

For the case of a transmission line or lossy passive twoport

network, impedance -matched at its input and output. [ Siegman, 1961a,

p. 86] , see also [Strum, 19^6, 19^8; Grimm, 1959, I960] ; we have

where,

N . = (L - l)kT B (A. 10)

ei n n n

T = Thermal temperature of the lossy elements of the

network.

L = Dissipative transmission loss factor, defined above,

n

From (A. 8) and (A. 10) it is evident that

F = 1 + (L - 1)(T /T ) (A. 10a)

n n n o

Effects of Impedance Mismatching

In the Receiving System

Non-reflecting impedance -matched conditions are necessary

throughout a receiving-antenna system, in order to avoid multiple

reflections and consequent delayed or ghost signals at the receiver
-14 7-

input terminals. However, an impedance mismatch at the receiver

input is sometimes beneficial, provided that proper impedance mismatching at this point results in an improvement of the receiver noise

figure. When the receiver -input impedance is not matched to the output impedance of a (matched-impedance) antenna system, the power

which is reflected from the receiver-input terminals will be transmitted back through the antenna system where it will be absorbed in the

transmission line and also re-radiated from the antenna, and multiple

reflections will not exist in the receiving system. Consequently, if

impedance-mismatch conditions exist only at the receiver input, interfering or ghost signals will not be developed at the receiver-input

terminals.

From the above discussion it is evident that impedance mismatch

conditions, at more than one point in the receiving-antenna system,

cannot be tolerated in a multichannel or broadband system. Hence, in

the following analysis it will be assumed that non-reflecting impedancematched conditions exist throughout the receiving-antenna system, except

that impedance -mismatch conditions may (or may not) exist at the

receiver input. Under these conditions, the "load" in figure A-l is

replaced by the radio receiver, and the impedance mismatch factor,

M. (= M ), will be equal to or less than unity.

Furthermore, in practically all radio receiving systems, the characteristic impedance of each network unit is essentially "real", hence

the following conditions apply:

X /R < < 1 (A. 11)

n n

From (A. 11), (A.6d), and the fact that the system is always

matched at every point looking toward the antenna

L =L (A.12)

an n
-148-

From the previous subsection and for the conditions given by

(A. 11), and (A. 12), and also referring to figure A-l, we have

((N B /B ) + N . ) V ag v! g eij N = N "5 " 5

^

(A. 14)

ao L

n

or (see (A. 10) ),

N B /B + (L - l)kT B

ag n g n n n, N = A - s-r: *- (A. 15)

ao L,n

Equation (A. 15) may be used to determine the available noise

power output from a system of linear lossy twoport networks in tandem, under conditions of impedance matching at the input terminals

of the networks, looking toward the antenna.

Noise Levels in the Receiving-Antenna System Networks

Referring now to figure 4-1, the "loss-free" antenna unit by definition does not contribute any noise; however, this unit receives and

has available at its output ^ the noise power ^ N A = N + N . + N from A at ai a the following sources:

(a) radio-transmitter output amplitude noise, N , associated

with the received radio-frequency carrier signal;
-149-

(b) interfering radio-frequency signals, N .; 3-1 (c) the antenna- received environmental noise power plus atmospheric and galactic noise, all of which add up to give N . el The effects of the received noise power, N , are discussed in

at

Sections 4.2 and 4.3; where it is shown that proper allowances may be

made for the effects of this noise power by using the results of equipment back-to-back tests to modify the (ideal) system-design curves.

Therefore, N , need not be included at this point in the analysis of

noise powers in the receiving system.

Also, the case of interference (noise) signals is not considered

in this report. The effects of interfering- signals are dependent upon

a variety of factors, including the "capture -effect" in FM systems;

with the result that the interfering- signal noise power, N ., is not

3,1 simply additive. Hence, for our purpose the available noise power at

the output terminals of the loss -free antenna is

N A = kT B (A. 16) A a a

v '

where B is the effective bandwidth of the receiving antenna, and T

a

& a

is the antenna received-noise temperature.

Each of the passive lossy units, in the receiving-antenna system,

contributes noise power, and a portion of each of these noise powers

is subsequently available at the duplexer output (receiver input) terminals. The total available noise power at the duplexer output, due to

the noise contributed by each lossy antenna unit plus the antenna-temperature noise, is designated & here as N and is assumed to be uniformly

au ' distributed over the effective bandwidth of the duplexer, B n

. du

Interfering- signal noise power is also added to the receiving system

from the local transmitter through the duplexer. The level of this

noise power at the duplexer output (receiver input) terminals, is
-150-

dependent upon the power output from the local transmitter, P , and

the "isolation factor" of the duplexer, Lp. The available local-transmitter noise power level at the duplexer output (receiver input) terminals is _p

<t N

it

=—

(A. 17)

where, P = Power output of the local transmitter, in watts.

L = Transfer isolation factor of the duplexer. This

factor is defined as the ratio of P to that portion

of P. which is available at the duplexer output

(receiver input) terminals, and is within the receiver

RF spectrum effective bandwidth B -( = B. r ). rf if The position r of the main portion r of the RF spectrum ^ of P„ must

it

be well outside of the RF spectrum included by B and the receiver

"image-frequency" spectrum. Note that in the above definitions the

available noise power, N , is distributed uniformly over the effective

au

bandwidth of the duplexer, B, , while the available noise power N. is r du ^ it

(assumed) to be distributed uniformly over the effective bandwidth of

the receiver, B . or B... The selectivity characteristics of the duplexrf if

er and the receiver-input circuits are relied upon to exclude the localtransmitter power (outside the bandwidth B .) from the receiver IF

rf

circuits.

The sum of the noise powers in the receiving-antenna system at

the duplexer output terminals is N = N + N . This noise power,

N > is available at the duplexer output, that is, at the receiver input

terminals.

Because of the manner in which L. has been defined, N, is the D it

noise power from the local transmitter as measured by the receiver

having an effective bandwidth B . The value of N. will depend upon
-151-

the position of the effective bandwidth of the power-measuring device,

(in this case the receiver) relative to the position and the shape of

the power spectrum of P. . Effective bandwidths of the cascaded units in the receiving-antenna

system are assumed to be greater than the effective bandwidth of the

receiver, B . Hence, as noted previously, the noise power "accepted" at the receiver-input terminals will be proportional to the receiver

effective bandwidth, B.,., and to the noise-power densities, N /B , , if au du

and N. /B . Furthermore, if impedance-mismatch techniques are

employed at the receiver -input terminals, in order to minimize the

receiver noise figure F , then the portion of N which is delivered

to and absorbed at the receiver input terminals is dependent upon the

value of the impedance-mismatch factor, M. . Therefore, the total

lr

receiving-antenna system noise power delivered to the receiverinput terminals, N , and subsequently processed in the receiver and

transferred to the receiver IF circuits, is

/

N

au

N

it\ N^ = M. (

-^ + -ii )B. £ (A. 18) Td irV B n B.,. i if

v du if /

In (A. 18), M. is the impedance mismatch factor at the receiver

input

.

In terms of available power at the duplexer output, M is

lr

4 R R.

(M. ) = ^-ir

, (A. 19) irav |z, +z. | 2

du lr

In terms of non- reflecting matched-impedance power at the duplexer

output, M. is

du du ir

(M. ) =

? (A. 20)

ir mat i „ . „ i £-n Z, +Z. ! R, du ir du
-152-

where,

R, = Resistive component of the output impedance of the

du j i duplexer.

Z = Output impedance of the duplexer. In general, this

is the output impedance of the network which precedes the receiver.

Z = Input impedance of the receiver,

ir

In order to carry out the above procedures, and obtain an estimate

of N , the noise power output of a linear passive lossy bilateral network, N , is determined in terms of the noise power input to the unit,

ao

N , the thermal temperature, T , transmission loss factor, L , and

ag n a

the noise figure F of the unit; see figure A-l and (A.1'5).

Referring to figure 4- 1, the output noise powers from each of the

tandem units may be obtained by substituting the proper factors in

(A. 15). This procedure will be used in a later section of this Appendix

to calculate the value for N , the total noise power at the duplexer

output terminals, preceding the receiver input terminals.

Receiver Noise

The noise power contributed by the receiver [ Bell, R. L. , 195l]

[Becking, et al. , 1955] , [ Rothe and Dalke, 1956] , [ IRE Subcommittee

7.9 on Noise, 19&0] and delivered to the receiver-input terminals is

very closely approximated by,

M. N = M. (F - l)kT B. r (A. 21)

ir er ir r o if

where,

M. = Impedance mismatch factor at receiver-input terminals, as given by (A. 16)

F = Noise figure of the receiver [ IRE Standards, Proc.

IRE, Jan., 19&0] , as a ratio.
-153-

The receiver IF effective bandwidth, B.., is defined by:

if

> if = "^- Cg (f)df (A. 22)

r

where,

G = Effective receiver power gain, as a ratio; this

factor is defined as the ratio of the signal power

delivered to the receiver input terminals to the

available output signal power in the receiver predetection or IF circuit at an arbitrarily specified

frequency. This frequency may correspond to

that of maximum gain.

G(f) = Receiver power gain as a function of the frequency, f.

Since our results apply to radio communication type of receivers,

B , is assumed to be the effective single-response IF channel bandwidth.

The receiver noise figure, F , as used above must be measured

r using a "source" or generator output impedance which is equal to

Z. . In practice, the ratios X n /R n and X. /R. are each very du du du lr lr

much less than unity, and hence conjugate -impedance matching and

non- reflecting impedance matching are equivalent. Furthermore,

the signal generator or source impedance, Z , may be varied in

order to obtain the optimum or minimum value for the receiver noise

figure, F [Bell, 1951] , [Becking, et al. 1955] , [ Rothe and Dalke,

1956] , and [ IRE Subcommittee 7.9 on Noise, Jan. , 19&0] ; the particular value of Z which gives F would establish the correct value

for Z required to insure optimum impedance-mismatch operating

conditions at the receiver input.
-154-

Pre-Detection Noise Power, N.-

if

The receiver-contributed noise power, M. N , is added to the

lr er

delivered noise power, N (see (A. 18)), from the receiving system

which precedes the receiver, and this total noise power level is

multiplied by the receiver power gain.G , to obtain the predetection

r noise power, N , that is,

N

if= < NTd+M

ir

N

er

)G

r

< A ' 23 >

The pre-detection noise power, N. f

, obtained from (A. 23) applies

only to a linear receiver having sufficient selectivity, in the input

section, to eliminate "image-frequency" or "spurious-response"

(signal and noise) power in the receiver IF circuits. The total IF

noise power in multiple -response receivers depends upon the number

of channels accepted by the receiver; this total IF noise power is

equal to the sum of the noise power contributed by all of the channels.

In radio communication systems only the "principal-response" channel is used. In receivers employing either or both age and limiting,

the receiver power gain, G , will be a function of the power level of

the receiver-input RF signal, P . The above ideas are now used as a guide in determining the functional relationship between the pre-detection noise power level, N , and the receiving-system parameters. The predetection load-reflected noise power [ Siegman, 19&lb] referred to in Section 4.2 is assumed

to be negligible, and is not included in the following analysis. However, this factor would be important for very-low-noise bilateral

receivers with an impedance-mismatch load connected to the receiver

IF output terminals. Referring to figure 4-1, and substituting (A. 18)

and (A. 21) in (A. 23) we obtain the following relationship for N , the

available pre-detection noise power,
N e = G

if r

-155-

N i

;

.

(A. 24)

it\ M + —-]B , + M. (F - 1) kT B., lA B

du B

if^

lf ir r °

lf

Each of the two terms inside the overall brackets on the righthand side of (A. 24) represents components of the total noise power

delivered to the receiver-input terminals. The noise power contributed by the receiver, and delivered to the receiver input [ IRE Subcommittee 7.9 on Noise, i960] is given by , M. (F - l)kT B. r ; ir r o if

see (A. 1 21 ). The noise power, N' = (N /B , + N„ /B.,)B.-, is com- >. ^ T au' du it' if if

posed of the noise power received by the antenna, N , plus the noise

power generated within the lossy passive networks in the receivingantenna system; including the noise power N. , which is contributed

by the local transmitter; N' is available within the RF spectrum bandwidth B „ (or B..) at the receiver input terminals. This noise power,

rf if

N' , is available at the output of the network which precedes the

receiver, within the RF spectrum bandwidth B ,(or B.J accepted by

rf if

the receiver, and is given by

N' = T

N N, -1 au it

^ B

du " B

i£ - B., (A. 25)

II

Note that N' - N B /B only when the various noise powers in

the receiving -antenna system are uniformly distributed over the effective bandwidth of the duplexer, B, . du

The factors within the brackets of (A. 25) may now be determined

in terms of the receiving antenna- system parameters, using the ideas

outlined above to estimate the total noise power at the output of the

tandem units in the antenna system, as follows:
-156-

From (A. 15), (A. 17), and figure 4-1 we obtain

N N

au it

B

du

N B n /B (L -l)kT B, L du t du du du

du du

it

B

du if D

N B /B +(L -l)kT B BA (L -l)kT,

c t ac t t t \ du du du du

B

L L n t du J-\ + du B

du

it

B

if

LD

(N A B /B ) + (L -l)kT B B (L - l)kT B A ac' a c c ac t t t t

c t du ac L L

t du

du

(L -l)kT B

du du du it

B

du if D

(A. 26)

Since N. = kT B , see (A. 16), (A. 26) may be written as A a a N N

au x-t du if

T + (L - 1)T (L - 1)T (L, - 1)T,

a c c _t t_ du du

L L L

c t du t du du

it

if D

(A. 27)

where,

L , L , L

c t du

Transmission (dissipative) loss factors for the

antenna circuit, transmission line, and the

duplexer network, respectively; as a ratio^ see p. 143

^Antenna-earth "proximity effects" on the antenna circuit losses

[Wait, J. R. , 1959; Vogler, L. E. , 19&2] , are included in the

transmission loss factor, L . c
-157-

The noise power available to the receiver within the receiver

IF bandwidth, B. r , is given above by (A. 25) as:

if

N' = T

r N N.

au , it

- B

du if J

B. (A. 25)

Re-arranging terms in (A. 27) and substituting in (A. 25), we

obtain

T

rrT + (L - 1 ) T + L (L-l)T + L L (L, -1)T, P^

a c c c

x t ' t c tx du ' du it

c t du B

if

L

D ^ kB

Equation (A. 28) may be written as

(A. 28)

N' = kT B. r T A if

(A. 29)

From (A. 28) and (A. 29) we obtain:

r T + (L - 1)T + L (L - 1)T + L L (LJ

- 1)T J a c c c t t c t du du

c t du

it

kB

i£

L

D

(A. 30)

The factor, T , is the "effective receiving antenna- system (output) noise-power temperature", and is completely defined by (A. 29).

Because of its fundamental nature and simplicity of application, the

concept of effective noise temperature is in general use in the fields

of radio communication and radio astronomy, and is referred to

throughout the literature in these fields.

From (A. 29) and (A. 30) it is seen that T is a measure of the

total noise power received by the antenna, plus the noise power originating within the receiving antenna system. The noise power, NL, is
-158-

" available" at the duplexer output terminals. The portion of N' that

lies within the bandwidth B is processed by the receiver and is

available at the "pre-detection" point within the receiver.

The factors * L , L , L, , and L^ in (A. 30) have been defined

c t du D

previously. T will be in degrees Kelvin if T , T , and T are in

J\ c t du

degrees Kelvin, and also if P. is in watts and B. r is in c/s of bandit if width.

The effective antenna-system output noise temperature, T , as determined from (A. 30) is seen to depend upon: (1) received noisepower temperature of the antenna, T , (2) thermal temperatures of

the units in the receiving-antenna system, T , T , and T , (3)

receiving antenna- system transmission loss factors, L , L , and L, , c t du

and (4) other factors such as, the receiver IF bandwidth B , the

power output of the local transmitter, P. , and the isolation factor of

the duplexer, L„.

It should be noted that (A. 30) determines the value of T . for un- A —

restricted values of the thermal temperatures of the antenna- system

networks, T , T , and T . Hence, this equation is applicable to the

c t du

case where the networks in the receiving-antenna system are at a low

thermal temperature; such as might be achieved with appropriate refrigeration, or perhaps in a "space" receiving system.

It should also be noted that (A. 30) is restricted to the case where

non- reflecting impedance matched conditions exist throughout the

receiving-antenna system, when looking towards the antenna. The

receiver input, may or may not be matched; (A. 30) is further restricted to the case where, X /R < < 1.

n n

See discussion of L on p. 142.

n
-159-

The effect of the loss factors, L and L. , on the antenna- system

c t ... v temperature, T , may be seen by applying (A. 30) to a typical example.

The receiving antenna system is assumed to consist of only an antenna

and a transmission line, with the following paramaters:

T = 300 degrees Kelvin

T

a

= T = 290 degrees Kelvin

L

c =1.12 (= 0.5 db)

L

C

= 1.58 (= 2. db)

From the above specified parameters and (A. 30), we obtain

300 + (1.12 - 1) 2 90 + 1.12 (1.58 - 1) 2 90

A

" 1.12 x 1.58

= 170 +19.6 + 106 = 295.6 deg. Kelvin

The noise-temperature contribution from each unit in the antenna

system is evident from these calculations. The calculated noise temperature, T (= 295.6 K) is seen to be only slightly different from the

assumed antenna temperature, T (= 300 K); hence, the loss factors,

L and L. , have the dual role of effectively reducing the antenna temperature, T , and also contributing to the resultant noise temperature,

cL T , at the receiver input terminals.

However, the receiver-input signal, P , will also be reduced by the

r product of the loss factors, L L , (= l/l.l2 x 1 58 = 0.56), and the net

result will be a degradation of the predetection signal-to-noise ratio, W

Equation (A. 30) may be simplified if the thermal temperatures of

the receiving-antenna passive-network units are each at the same

(standard) temperature, T , and if impedance -matching conditions looking toward the antenna, exist throughout the receiving-antenna system;

that is: T =T =T, =T (A. 31)

c t du o

* In this example a duplexer was not used.
-160-

Substituting conditions (A. 31) in (A. 30) and combining terms, we

obtain:

T - T P

T . = T~^—

T

2— +T + , „ T (A. 32)

c t du if D

If (a ) the thermal temperature of each of the receiving-antenna

elements is T , (b) X /R < < 1, and (c) non- reflecting impedancematching conditions exist throughout the receiving-antenna system

looking toward the antenna, the effective antenna- system output temperature, T , maybe calculated from (A. 32).

Equation (A. 2 3) may now be written as,

N. r = G M. [T + (F - 1)T ] kB m (A. 33)

if r lr A r o if

where F is the receiver noise figure, (see p. 153), and M. is der i r fined by (A. 16), for "available" noise power at the duplexer output.

The factor, (F - 1)T , in (A. 33) is the effective receiver (input)

r o noise temperature, T , [ IRE Comm. on Noise, 19&0] . Hence (A. 33)

er

may be written as,

N. r = G M. [T A +T ] kB.. (A. 34)

if r ir A er if

Equation (A. 34) is used throughout this paper to estimate the

receiver IF noise power level, N . The performance of a radio receiver depends upon the receiver

IF total signal power-to-total noise power ratio, P /N . An inspection of (A. 34) shows that the receiver IF noise power level, N , depends

upon both the receiving antenna-system noise power level, kT ^.-,

and the receiver-contributed noise power level, kT B... c er if
-161-

The noise power, kT B.., may be assumed to be thermal noise

er if

with a steady average power level. In the frequency range above

(approximately) 100 Mc, the received noise power which is associated

with the antenna noise temperature, T , may also be assumed to be

thermal noise. Therefore, for radio communication systems operating with RF signals above 100 Mc, the receiver IF noise, N , may

be assumed to be "white", and its average total power level may be

estimated from (A. 34). In the frequency range well below 100 Mc,

the noise signal, kT B , may have the characteristics of atmos3- IX pheric noise. Under these conditions, N , may be estimated from

(A. 34) provided that kT B is assumed to be an "average" value;

obtained by averaging over a period of time which is long compared

with the average period of time between noise peaks.

-163-

APPENDIX B

Derivation of the Message-Channel Signal-Power Loading

Ratio P /p , in the Modulating Baseband Signal

Assuming that the message -channel signal power S , at the radio

receiver output, is proportional to the modulating signal-power loading ratio, p /P , we have: 6 *m m

Pm

S = K, x G x-^=- (B. 1)

oc 1 r P

x m

where,

S = a particular message-channel average signal power

level, at the radio receiver output, or decoder input.

K - constant of proportionality, dependent upon the radio

system parameters.

G = gain of the linear section of the radio receiver.

r

° p = average power level of a particular message signal m

m

in the composite modulating baseband signal.

average of the total power level of the composite modulating baseband signal.

Note that for SSBSC time -division message -signal multiplexing

systems which are being considered in this paper, p will be confined

within a definite bandwidth in the baseband- signal spectrum and may

be measured directly in the baseband. For other types of multiplexing

systems (see Subsection 4. 4) it may not be possible to measure or

adjust p directly in the modulating baseband; hence, modifying factors

might be required in (B. 1).

The message-channel average noise-power level at the radio receiver output is,

N = K x G x B (B.2)

oc Z r c
-164-

where, K = constant of proportionality, dependent upon the radiosystem parameters.

B = bandwidth of a particular message-signal channel, at

the receiver output, in c/s. This bandwidth is assur

to be equal to the spectrum of the message signal.

Since the radio receiver gain, G , is dependent upon the receiverinput radio-frequency signal power level, it is inconvenient to use

either (B. 1 ) or (B. Z) separately. Hence, by combining (B. 1 ) and (B. Z)

we obtain the message-channel signal-to-noise ratio at the radio receiver output, S /N , and also eliminate the variable radio- receiver

oc oc

gain factor, G . Combining (B. 1) and (B.Z), we obtain,

r

oc m

The "constant" K in (B. 3) is a function of the radio system parameters; its functional form and its value depend upon the type of radio

system--see Sections 4.5 and 4.6.

From (B. 3) we obtain,

p B (S /N ). ml cl oc oc 1 p " B (S /N ).

(B * 4)

mZ cZ oc oc Z

or, in general;

p . B (S /N ). ~ml cl oc oc 1 " B . (S /N T mi ci oc oc l (B.5)

and,

p . B .(S /N ). rni ci oc oc l ._ , . r- = B..(S. ,/N ) (B - 6)

mn en oc oc n
-165-

From (B. 4) to (B. 6) we obtain:

B AS /N ). cl oc oc 1 Pml" PmnB (S 7n T (B.7)

en oc oc n

or, in general:

B _(S /N ).

cZ oc oc 2 . . m2 = Pmn B (S /N ) {B ' 8)

en oc oc n

B .(S /N ). Cl OC OC 1 .„ .. "mi^mn B (S /N ) (B - 9)

cn oc oc n

[B (S /N )] a c

v oc' oc

/J A

PmA = Pmn B (S /n ) (B ' 10)

cn oc oc n

The subscript A denotes a "class" of message signals, each class

ires the same value of the product [ receiver output. Continuing, we obtain:requires the same value of the product [ B (S /N )] , at the radio

c oc oc

[ B (S /N >] „ c

x o& oc'

J B mm

mB = Pmn B (S /n ) (B - H)

cn oc oc n

[ B (S /N )] ^T c

v oc/ oc

/J N ._. PmN = Pmn B (S /N ) (B ' 12)

cn oc oc n

The total power level, P , in the modulating baseband signal is, m

P = M A p . + M^ p „ + ...+ MnvT p (B.13) m A *mA B

ymB N ^mn v
-166-

where, M = number of simultaneously- active message signals

each of which requires the same value for the product

[B (S /N )] c oc oc A

M„ = Ditto, for the product [ B (S /N )] . B

^

c oc' oc

,J B

M„ = Ditto, for the product [ B (S /N )] . N c oc oc N

Combining (B. 10) to (B. 12) and (B. 13) we obtain,

m

P / /S mn / r f oc

B (S /N ) (

MA[B

c( 5T

cn oc oc n oc K +mJb/-^\ + ... B

L c^N

oc

x oc' x oc (B.14)

Combining (B. 6) and (B. 14) we obtain:

S % .S

m

mi

x OC' x OC' x oc/ N

B .(S /N T

Cl OC OC 1 (B.15)

Equation (B. 15) is reproduced as (4. 9) in Section 4. 3. 2,
-167-

APPENDIX C

Derivation of FM Receiver Performance Characteristics

For the condition where the total signal power-to-total noise

power ratio, P. f/N. f

, at the IF output, that is, at the input to the limiter of a "conventional" type FM radio receiver, exceeds (approximately) 10 db, it is possible to calculate the relationship between the signalto-noise ratio, S /N , at the receiver output, and the ratio of the radioo o frequency signal power level, P , to the total noise power at the receiver input; k (T + T )B . These calculated results apply only to the

upper linear portion of the FM receiver characteristic curve, above

the threshold level. The following deriviations indicate the method

used to calculate the upper linear portion of the FM radio receiver

characteristic curve, and the parameters to be considered.

The average total noise power in a one-ohm circuit at the output

of an FM discriminator [Black, 19^3] through a low pass filter having

a cutoff frequency of f c/s is:

4 * d r 3 lf N = —-— x —

x fn x —

o 3 P.. k B. r if if

(C.l)

where,

D n = demodulator conversion factor, d

P = total average IF signal power at the receiver limiter

input, in watts.

N = total average IF noise power at the receiver limiter

input, in watts.

B. = effective bandwidth of the receiver IF circuits,

if Equation (C.l) indicates that the discriminator-output noise voltage spectrum is triangular, increasing with frequency up to the cutoff

frequency f . This equation also shows that the total discriminator-

\
-168-

output noise power is inversely proportional to the IF signal power P.,»

and directly proportional to the noise power density N./B., in the IF

passband.

From (C. 1) the total average noise power out of an FM discriminator in a message-channel passband from f to f is: 6 * a b

N

oc

= — X P- X(f

b ' V X IT (C - 2 » if if

Equation (C.2) gives the difference in discriminator -output noise

power levels between the triangular-shaped output-noise voltage

spectra having cutoff frequencies f and £ . The average signal power in a one-ohm load at an FM discriminator output, associated with a sinusoidal modulating signal of frequency

f is [Black, 1953] : c

S = 2 it 2D^ AF2

(C.3)

oc d c

where, AF = peak frequency deviation of the radio frequency carrier, in cycles per second, due to the sinusoidal modulating signal of frequency f .

Combining (C.2) and (C.3) we obtain,

S _ AF2

B. f P.. — = - x , C * x -^ (C.4) N 2 3 3 w

oc (f - f ) if

b a

The value of AF is determined by the power level, p , of the

c m

modulating sinusoidal signal at the radio-transmitter frequency-modulator input and the modulator characteristics. This modulating power

level may be adjusted, relative to the total baseband- signal power level so that:

\
-169-

Vz AF = (p /P )' AF (C.5)

c mm

where,

p = average power level of a particular sinusoidal modulating signal in the baseband; the frequency of the

modulating signal being f c/s.

P = total average modulating- signal power level in the

baseband signal.

AF = peak radio-frequency carrier deviation in c/s for a

single sinusoidal modulating baseband signal, having

an average power level equal to the total average

power level of the total baseband signal, Pm

The message-channel signal-power loading ratio, pm/P » is

determined by methods outlined in Section 4.4.2 and Appendix B.

We are concerned with a comparatively narrow portion of the

baseband spectrum and hence f ~f, . For this condition it can be

a b

shown that,

The portion of the baseband- signal spectrum which is given by

(f Da- f ), is related to the message -signal bandwidth B , as follows,

c

f - f = K B (C.6a)

b a M c

where,

K = factor which depends upon the type of multiplex M

system.

B = message- signal channel bandwidth, in c/s.
-170-

For SSBSC frequency-division multiplex, being considered in this

paper, K = 1 therefore, M

fK" f = B . (C.6b) b a c

The term (f + f,)/^ in (C.6) is the mid-frequency, i t between

c

f and f_ , hence

a b

= f (C.6c)

2 c

Substituting (C.6b) and (C.6c) in (C.6) we obtain,

where,

f

3

- f 3 = 3 B f 2

(C.6d) b a c c

f = Center frequency of the message channel in the

baseband signal spectrum, in c/s.

The radio-frequency spectrum bandwidth (B .) , associated with

rf c particular values of A F /f and f , is given by:

(B ) =f x <^(AF /f ) (C.7)

rf c c c c

However, the value of B ., is determined under conditions where

rf

all of the modulating- signal power is in a single sinusoidal modulating

signal having a frequency = f ; tha

AF = AF; hence, B . is given by

c rf

signal having a frequency = f ; that is, p /P = 1, f = f , and m m m c m

B r =f x <)> (AF/f ) (C.8)

rf m m
-171-

where,

B . - Radio-frequency spectrum bandwidth which includes

3TX all components or sidebands having amplitudes equal

to or greater than one percent of the unmodulated

carrier amplitude.

cf) (A F/f ) = function of A F/f . This function gives the number of

significant sidebands in the radio-frequency spectrum,

assuming that the modulating signal is a single sinusoidal voltage wave having a frequency f , in c/s.

The function <j> (A F/f ) in (C. 8) has been calculated [Hund, 1942] m

and is shown in figure 3-1.

Assuming that the FM receiver IF bandwidth, B , is adjusted to

be only wide enough to pass the significant sideband components in the

radio-frequency spectrum, we have,

B = B (C.9)

rf if

Therefore,

B. r = f x <j> (AF/f ) (CIO)

if m m

Substituting (C.5), (C.6d) and (C.10) in (C.4) and re-arranging

terms we obtain:

S

oc

B

c Pm 1 ,/AF\ /AF \

2 P

if . X J— X = - X <j)(- X ( -r— X —

(C.ll) N f p 2

T Vf J V f J N

oc m m x m /v c /

The term P /N in (C.ll) is the required predetection total signal

power-to-total noise power ratio associated with a specified value of

S /N at the receiver output, and for particular values of the paramoc oc

eters: f , B , p /P , cj>(AF/f ) and (A F/f ). However, in radio m c m m m c
-172-

communication system design work and system performance testing it is

usually more convenient to deal with the total radio-frequency signal

power and the total noise power at the receiver input. Hence, the form

of (C. 11) was modified as follows:

P. r = G M. P (C.12)

if r lr r

From (A. 30) in Appendix A and (C. 10) above, we have,

N. r = G M. [T.+T ] kf x $(^-\ (C.13)

if r lr A er m If 7 m

therefore,

where,

P., P

if r NT " [T A + T ]~kf x <(> (A F/f ) (C14)

if A er m m

G = power gain of the cascaded RF and IF linear sections

of the FM receiver.

P = total average radio-frequency signal power at the

receiver input, in watts.

-23

k = Boltzman's constant = 1.3804 x 10 joules per degree

Kelvin.

f = highest frequency in the modulating baseband signal.

T = receiving antenna- system output noise temperature,

degrees Kelvin; see Appendix A.

T = radio receiver-input effective noise temperature [ Proc.

IRE Comm. on Noise, I960] , degrees Kelvin.

er

Substituting (C.14) in (C.ll) we obtain,

s B P , A ^ x2 P

oc cm/AF \ r . <-. X X = ( — \ X —

;

t— —

rrz (C.13) N f p \ £ J 2k(T A +T)f

oc m rn v c ' A er m
-173-

Note that the left-hand side of (C.15) coincides with that of (C.ll);

this similarity was preserved in order to compare the predetection

total signal-to-noise ratio, P /N , with the RF signal-to-relative

noise power ratio, P /k(T . + T )f , at the receiver-input terminals.

r A er m

It is evident from (C.15) that the receiver-output signal-to-noise

ratio, S /N , above the threshold region, is inversely proportional

oc oc

to the square of the modulating frequency f , for a given set of systemparameter values. This situation does not apply to the receiver performance in the threshold region; in this region the receiver -output

sisnal-to-noise ratio, S /N , approaches the same value for all 5 oc oc *^

frequencies in the modulating baseband signal. Hence f in (C. 15) was

made equal to f , the maximum frequency in the modulating signal,

in order to obtain a design equation which is conservative above the

threshold region. Equation (C. 15) is reproduced as (4.19) in Section

4.6.1.1, withf = f . c m
-174-

APPENDIX D

Method of Estimating the Performance Characteristics of Radio

Receivers and Message-Signal Decoding Units,

for a Time-Varying Signal.

Estimated equipment performance, for a time-varying signal, is

outlined by a method of graphically combining the steady- signal performance characteristic of the equipment with either the distribution

of the time-varying signal power or the distribution of the time-varying signal-to-noise ratio at the equipment input.

The analysis of radio receiver performance in this Appendix does

not include system intermodulation noise; see Section 4.3. Hence, the

estimated performance will apply to the "ideal" receiver characteristic, as shown in figure 4-A. Proper modifications of the estimatedperformance results may be made for the effects of system-intermodulation noise, using the procedures outlined in Section 4. 3.6.

D. 1 Estimated Performance Characteristics of Radio Receivers.

The total post-detection ^ signal-to-noise Bratio, S /N , at the outo' o put of an "ideal" radio receiver depends upon the total received radiofrequency signal power, P , and the total noise power, N' + N , available at the receiver input. Where N' is the antenna- system noise

power which is available at the receiver input terminals, and lies within the RF signal spectrum bandwidth B A=B. r ). N is the noise con- 5 ^ rf if er

tributed by the receiver, referred to the receiver input. See Appendix

A. N' + N may be estimated by adding the antenna- system noise

and the receiver noise on a power basis and the resultant (for the case

of radio systems operating at and above VHF) is assumed to have the

characteristics of "white noise", with a steady average value. In the

following derivations the total signal-to-noise ratio, S /N , is converted to the message -channel signal-to-noise ratio, S /N (see Section 6 6 oc oc

4), in order to conform to symbols used in Sections D. 1 and D. 2.
-175-

In general, the statistical characteristics of both P and N must

r T

be considered. The characteristics of P to be considered are (1) the

r cumulative (time) distribution and (2) the fade rate. The characteristics of N to be considered are (1) the cumulative (time) distribution,

including average and median power levels, and (2) the spectral density

in the frequency range of the spectrum included in B f.

In this work the received RF signal, P , is assumed to be of the

r Rayleigh-fading type. Other types of received signals may be considered, provided that their distributions are known and also provided

that the performance characteristics of the combiner are known for

these other distributions. The fade rate of P at its median level is

r assumed to be lower than the lowest binary-bit rate in the modulating

message signal. In other words, it is assumed that there is no appreciable change- in the value of P during the duration of a message -signal

x binary pulse, or during the period of time required for one cycle of

the lowest frequency in the modulating (message) signal.

At this point it should be noted that the characteristics of the

receiver -output noise, N , for an FM receiver, will depend upon the

oc

level of operation of the receiver with reference to the "threshold region". Hence, this situation must be considered when determining the

performance characteristic of FM receivers.

The method of determining the receiver performance characteristic for a time-varying received RF signal, P , is based on the idea

of combining the receiver-input steady RF signal performance characteristic with the cumulative power-level distribution of the received RF

signal. Following is the detailed procedure for applying this method

to determine the performance characteristic of an FM receiver for

a Rayleigh-fading type of received RF signal.
-176-

Figure D-la shows the measured steady-signal performance characteristic for an FM receiver, covering a range of receiver-input

values of P /f (T . + T ) and receiver output message -channel

r' m A er' r 6

signal-to-noise ratios, S /N . Normalized abscissa and ordinate

oc oc

scales are used to conform to other work in this report; see Section

4.6, figure 4-3. The normalizing factors are: f , T A , T , B , and m A er c Pmm/p . Note that A F/f = 1 for this curve and hence curves derived m

from this characteristic, for a time-varying receiver -input relative

z AF

signal-to-noise ratio, P /f (T . + T ), apply only to —— = 1. P is 6 r' m A er ff y y ^ r m

the power level of the total RF signal, available at the receiver-input

terminals. The total receiver-input noise consists of thermal noise

from the signal generator and from the receiver; hence T. = T (gen- A g

erator output ^ noise-power ^ temperature) ^ ' and T = (F - 1) T . N' + N

er

v r o T er may be obtained from: N' + M - k (T A + T )B. r

. Note that the 7 T er A er if

relative value of N' + N (= k(T A + T )f ) is used in the abscissa

T er

v A er m

titles in figure D-l.

Curve A-A in figure D-lb gives the cumulative distribution of the

Rayleigh-fading received RF signal, P . Any median value may be

selected for P by proper adjustment of the ordinate scale values.

The method of obtaining the estimated FM receiver performance

characteristic for a time-varying received RF signal is as follows: In

figure D-lb, an abscissa value of percentage of time t is selected, and

3. the corresponding normalized ordinate- scale value P' , is noted. The

value of P

1 from figure D-lb was used in figure D-la to determine an

ordinate- scale value on the normalized scale; this normalized signalto-noise ratio is denoted here as (S /N )'. This value of (S /N )' oc oc oc oc

was then plotted versus t as shown in figure D-lc. The above procedure was repeated to obtain additional points for the curve in figure

D-lc.

The average value of the distribution of the relative signal-to-noise

ratios was obtained by numerical integration of the curve and is shown

in figure D-lc as (S /N ) . Note that the integration must be & v oc' oc'ave
-177-

GRAPHICAL METHOD

FOR DETERMINING FM RECEIVER PERFORMANCE

F

111 <-> a

H

o a m HI Vo

H

o 14

Jfe?

STEAD f RECI3VE0 iSIGNAL

AF 1

fei

b

?t?

-210 -205 -200 -195 -ISO -1(5 -180

10 LOGiq Pr - 10 LOGio fm - 10 LOGio (TA+ \r

)

Flj. O-lo

50

55

50

«5

40

55

JO

25

20

15

10

RAYLEIGH- FADING NON-DIVERSITY RECEIVED 9GNAL

lNoc> ove\ 'm —

7

1

/

-5 /

/

/

(Pr)med

-225 -220 -215 -210 -205 -200 -195 -190 -*5 -ISO -IT5 -ITO -IB lOUOGm Pr -lOLOGKjfm -K)LjOG io (Ta +Tgr

) Fl» D-ld

-160

j

h* -ITO +

og -IM

w\XI VIAL -RATIOCOMBINING

(P| )medV 4--t\ c

i J -190 o1

° -200

...

•^B

1 OT

a.- X. 3 -210 O \a

-220 to

1

a§

2

a?

Q

0*ooTM*** «. Jkrt

to \. ! &r

i \

'o

001 I N 30 50 TO 90 95 99 995 99.9 99S5 99 99 PERCENT OF TIME ORDINATE IS EXCEEDED

Flfl. D-lb

aa i to so so n so 15 nsu susms mm

PERCENT OF TIME ORDINATE IS EXCEEDED

Fl»0-lc
-178-

carried out using ordinate -scale values as numerical ratios instead of

decibel units. This value of (S /N ) , together with the value of oc oc ave. P (med.)from figure D-lb, were plotted as shown in figure D-ld. Addr itional points for the FM receiver performance characteristic curve in

figure D-ld were obtained from similar distributions (curve A-A) in

figure D-lb but with different P (med. ) values. The resultant FM

receiver-performance characteristic curve in figure D-ld is in terms

of the median of the time -varying received RF signal, P , and the

r average of the distribution of the time-varying receiver-output relative

signal-to-noise ratio, S /N . The shape and the position of the B oc oc r ^ curve in figure D-ld is related to the shape and the position of the

curve in figure D-la in terms of their respective abscissa and ordinate

scales; hence, the accuracy of the estimated receiver-performance

curve in figure D-ld depends directly upon the accuracy of the measured receiver-performance curve in figure D-la and the accuracy of the

combiner-performance curves in figure D-lb.

D. 1. 1 Estimated Performance Characteristics of

Radio Receivers-With Diversity

The statistical characteristics of the "combined" receiver-output

signal-to-noise ratio, S /N , depend upon both the receiver char- & oc' oc

^ ^ acteristics and the combiner characteristics.

The curves in figure D-lb are essentially the combiner-performance characteristics and apply to maximal-ratio type of combining;

these curves were obtained from Brennan's work [19^8] . Curve A-A

is the distribution of the signal at the combiner input, curves B-B and

C-C, are the corresponding distributions at the combiner output, for

dual and quadruple diversity, respectively. Curve A-A may be considered as the distribution of the signal-to-noise ratio at the combiner
-179-

input, instead of the distribution of the signal at this point. Proper

phasing of the combiner -input signals is required in order to obtain

accurate measurements of the radio receiver output signal-to-noise

ratio.

The system design curves presented in this report were developed,

for non, dual, and quadruple diversity, using the curves A-A, B-B,

and C-C in figure D- lb, by the method outlined in Section D. 1 .

D.2 Estimated Performance Characteristics of the Decoder Unit,

for a Time-Varying Input Signal-to-Noise Ratio.

The number of message errors, and hence the average message

error rate occurring during a sampling period, depends upon the message-channel signal-to-noise ratio, S /N , at the decoder-unit in- 5 5 oc' oc

put. More in detail, we need to know the relationship between the

average message-error rate and,

(1) the value of the average signal-to-average noise power ratio,

S /N , and

oc oc

(2) the statistical characteristics of:

(a) the message-channel signal power, S ,

(b) the message-channel noise power, N ,

(c) x the message-channel signal-to-noise ratio, S /N ' 6 & oc' oc

In this work it is assumed that the performance characteristic of

the decoder unit for a time-varying (input) average signal-to-average

noise power ratio, S /N , may be determined bv combining the deoc oc

coder-unit performance characteristic for the non-time -varying

S /N condition, with the cumulative distribution of the time -varying

oc oc

S /N . It is assumed that the time interval required for an apprecioc oc

able change6 in S /N is large compared with the duration of a mesoc' oc B * sage- signal binary pulse or the period of time required for one cycle
-180-

of the lowest frequency in the modulating message signal. The results

obtained by the above procedure will also depend upon the statistical

properties of the noise, N , at the decoder-unit input; each "type" of

noise will have a different effect on the slope and the position of the

decoder-unit performance-characteristic curve.

Measurements of the average mess age -channel signal-power level,

S , and the average message-channel noise-power level, N , may oc oc easily be made with the proper type of power-measuring meters having

appropriate time - constants . However, if either or both of the average

power levels, S or N , vary (independently) with time, their separate measurements and also the measurement of the ratio, S /N , oc oc

require special equipment capable of measuring the cumulative distribution of these variables.

Following is an outline of the method used to calculate the performance of a decoder unit for a time-varying signal-to-noise ratio, S /N , oc oc

at the decoder-unit input. In this case, the decoder unit consisted of a teletype receiver and a teletype printer, combined.

Figure D-2a shows the measured performance characteristic for

the teletype-message decoder, for a range of values of steady signalto-noise ratios, S /N , and assuming that N was thermal noise.

oc oc oc

Figure D-2b shows the estimated distribution of the time-varying

signal-to-noise at the output of an FM radio receiver, for the non-diversity case, and assuming a Rayleigh-fading type of signal at the

radio receiver input. The method of obtaining this distribution, and its

/

Soc\

average value, ( —— j

, is shown in Section D. 1. Note that figure

^ octave.

D-2b is identical to figure D-lc, except that the ordinate scale values

in figure D-2b are decreased over those in figure D- lc by the factor

10 Log B (= 20.4 db). This scale modification is required so that
-181-

GRAPHICAL METHOD

FOR DETERMINING MESSAGE- DECODER PERFORMANCt

S3 *

i 03

i 0.2

! »

0.07

! 0.05

I

j 0.03

1 0.02

0.01

0.007

0.005

0.003

M—

°N V

r-'_ \

y

• = E- —«-

i

. Y- \ -o

—Ii(Noc

'

10 log.. ,db

4 6 I 10 12 H 16 II 20 STEADY- SIGNAL MESSAGE-CHANNEL AVERAGE SIGNAL POWER

MESSAGE-CHANNEL AVERAGE NOISE POWER MEASURED AT TELETYPE RECEIVER INPUT

Fig. D-2o

3u

-20on

N 3TE (i) Bc

= 108 C^

6») R£ NON DIVERSITY YLEIGH-FADING

-ANoc/ave

1

1

RF Q4RRIER SIGNAL

Woe)

N

y- 10 .J # 5 K 3 u 2

1 -E~,« 1 1 1 — c 1 1 o «•' * 05 Z

° 3

a- Oi j 01

1 S^ION DIVERSITY

fc! 007 -1 e, 005 ? \ 0.03 -DUAL

SIT < ao2 > DIVER Y

•- 0.007

- \

- £ 0005

;\ —

o 0j0O2 S 01 £ 00007

QUADRUPLEA_

DIVERSITY

3 00005 nm-. \

~UJ \1«

£

\

0.0001 c

/ave I ^ "DI52025J03540455055 AVERAGE OF THE T1ME-\«RYING Sxmx AT THE TELETYPE RECEIVER INPUT, db

Fifl. D-Zd

0.7 05

0.3

0.2

0.I

0j07

0.05

0.03

002

0.0I

0.007

0:005

0.003

0.002

.._. ._: L .

_—

,

1

, ,_,—

|

4 .

1 _ mm i m

1

Sjve 1 I

—

" " "I

_ L^i_

E"j

|

i ill 4^4

-j*-

.

i TT ta

l / 1

10 50 50 70 9095 99 995 9935 PERCENT OF TIME ORDINATE IS EXCEEDED

Fig. D-2b

00 I D 30 50 70 90 95 39 945 9995 99 PERCENT OF TIME ORDINATE IS NOT EXCEEDED

Fig. D-2c
-182-

the ordinate -scale values in figure D-2b correspond to the abscissascale values in figure D-2a. In other words, the procedures outlined

in figure D-2 apply to a particular type of message-signal decoder

unit.

The distribution of the teletype-character error rate, shown in

figure D-2c, was obtained as follows: In figure D-2b, an abscissa

value of the percentage of time, t , was selected and the corresponding

3, normalized value of (S /N ) was noted. The value of (S /N )', oc oc oc oc from figure D-2b, was used in figure D-2a to determine E'. This

value of E' was then plotted versus t as shown in figure D-2c. The

3. above procedure was repeated to obtain other points, E", E" 1 , etc. , for the curve shown in figure D-2c.

The averageB value of the distribution shown in figure 6 D-2c, E , ave. was obtained byy numerical integration 6

of the curve. This value of E

ave. together with the value of (S /N ) . from figure D-2b, were plotted 5 v oc' oc'ave B ^ as shown in figure D-2d. Additional points for the non-diversity curve

in figure D-2d were obtained using the above procedures and the nonS oc diversity distribution in figure D-2b, but with different

values for this distribution. Various values may be obtained for

(S /N ) in figure D-2b by proper adjustment of the ordinate

oc oc ave

.

scale values.

Teletype decoder-unit performance curves for dual and quadruple

diversity, shown in figure D-2d, were obtained using the same procedure as outlined above and using S /N distributions, in figure D-2b,

oc oc

for dual and quadruple diversity, from figure D-lc.

U.S. GOVERNMENT PRINTING OFFICE 19« —*47SM
. 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.

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Standards. Millimeter-Wave Research.

Circuit Standards. High Frequency Electrical Standards. Microwave Circuit Standards. Electronic Calibration Center.






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