• TABLE OF CONTENTS
HIDE
 Title Page
 Copyright
 Dedication
 Acknowledgement
 Table of Contents
 List of Figures
 Abstract
 Introduction
 Mathematical prelimenaries
 Synthesis of single-sideband...
 Examples of single-sideband signal...
 Analysis of single-sideband...
 Examples of single-sideband signal...
 Comparison of some systems
 Summary
 Appendix
 Reference
 Biographical sketch
 Copyright














Title: Synthesis and analysis of real single-sideband signals for communication systems
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00082485/00001
 Material Information
Title: Synthesis and analysis of real single-sideband signals for communication systems
Physical Description: xiv, 125 leaves. : illus. ; 28 cm.
Language: English
Creator: Couch, Leon W
Publication Date: 1968
 Subjects
Subject: Communication research   ( lcsh )
System analysis   ( lcsh )
Signal theory (Telecommunication)   ( lcsh )
Electrical Engineering thesis Ph. D
Dissertations, Academic -- Electrical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida, 1968.
Bibliography: Bibliography: leaves 121-123.
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082485
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000955703
oclc - 16961426
notis - AER8332

Table of Contents
    Title Page
        Page i
    Copyright
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
    Table of Contents
        Page v
        Page vi
        Page vii
    List of Figures
        Page viii
        Page ix
        Page x
        Page xi
        Page xii
    Abstract
        Page xiii
        Page xiv
    Introduction
        Page 1
        Page 2
        Page 3
    Mathematical prelimenaries
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
    Synthesis of single-sideband signals
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
    Examples of single-sideband signal design
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
    Analysis of single-sideband signals
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
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        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
    Examples of single-sideband signal analysis
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
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        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
    Comparison of some systems
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
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        Page 91
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        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
    Summary
        Page 102
        Page 103
        Page 104
    Appendix
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
    Reference
        Page 121
        Page 122
        Page 123
    Biographical sketch
        Page 124
        Page 125
    Copyright
        Copyright
Full Text










SYNTHESIS AND ANALYSIS OF REAL

SINGLE-SIDEBAND SIGNALS

FOR COMMUNICATION SYSTEMS









By

Leon Worthington Couch, II













A Dissertation Presented to the Graduate Council of

The University of Florida

in Partial Fulfillment of the Reauirements for the

Degree of Doctor of Philosophy








UNIVERSITY OF FLORIDA

1968
































Copyright by

Leon Worthington Couch, II

1968















DEDICATION



The author proudly dedicates this dissertation to his parents,

Mrs, Leon Couch and the late Rev. Leon Couch, and to his wife, Margaret

Wheland Couch,














ACKNOWLEDGMENTS


The author wishes to express sincere thanks to some of the many

people who have contributed to his Ph.D program. In particular,

acknowledgment is made to his chairman, Professor T. S. George, for his

stimulating courses, sincere discussions, and his professional example.

The author also appreciates the help of the other members of his super-

visory committee. Thanks are expressed to Professor R. C. Johnson and

the other members of the staff of the Electronics Research Section,

Department of Electrical Engineering for their comments and suggestions.

The author is also grateful for the help of Miss Betty Jane Morgan who

typed the final draft and the final manuscript.

Special thanks are given to his wife, Margaret, for her inspi-

ration and encouragement.

The author is indebted to the Department of Electrical Engi-

neering for the teaching assistantship which enabled him to carry out

this study and also to Harry Diamond Laboratories which supported this

work in part under Contract DAAG39-67-C-0077, U. S. Army Materiel Com-

mand.
















TABLE OF CONTENTS


ACKNOWLEDGMENTS . . .

LIST OF FIGURES . .

KEY TO SYMBOLS . .

ABSTRACT . . .

CHAPTER


I


I. INTRODUCTION . . . .

.1, MATHEMATICAL PRELIMINARIES . .

I. SYNTHESIS OF SINGLE-SIDEBAND SIGNALS *

V. EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN

4.1. Example 1: Single-Sideband AM with
Suppressed-Carrier ....... *

4,2 Example 2: Single-Sideband PM *

4.3. Example 3: Single-Sideband FM *

4,4. Example 4: Single-Sideband a *

V. ANALYSIS OF SINGLE-SIDEBAND SIGNALS *

5. 1 Three Additional Equivalent Realiza

5.2, Suppressed-Carrier Signals *

5.3. Autocorrelation Functions * *.

5.4. Bandwidth Considerations .. *.

5.4-1. Mean-type bandwidth *

5.4-2. RMS-type bandwidth . .

5.4-3, Equivalent-noise bandwidth *

5 5, Efficiency . . .

5.6, Peak-to-Average Power Ratio .* *

v


Page
* iv

* viii

* x

* xiii



* 1


4

* 9

* 18


tions

* *


. . 18

19








. 28

* 30



. 3 42

* .* 43

*. 44

* 45

* .* 45

. 46


* *


* *








Page

VI EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS . .. .48

6o1. Example 1: Single-Sideband AM With
Suppressed Carrier. . ................ 48

62. Example 2: Single-Sideband PM . . . 51

6.3. Example 3: Single-Sideband FM * * * 68

6.4. Example 4: Single-Sideband a . . . 71

VII, COMPARISON OF SOME SYSTEMS . . . .. 75

7.1. Output Signal-to-Noise Ratios . . . 76

7 1-1 AM system . . . . 76

7.1-2. SSB-AM-SC system . . . 77

7.1-3. SSB-FM system . . . .. 78

7.1-4. FM system . . . . .. 84

7.1-5. Comparison of signal-to-noise ratios. * 85

7.2, Energy-Per-Bit of Information .......... *89

7.2-1. AM system . . . . .. 93

7.2-2. SSB-AM-SC system . . . . 93

7.2-3. SSB-FM system . . . .. 94

7.2-4. FM system . . . . 94

7.2-5. Comparison of energy-per-bit for
various systems *. ..............** 95

7.3. System Efficiencies ................. 97

7,3-1. AM system . . . . .. 98

7.3-2c SSB-AM-SC system . . . . 98

7.3-3. SSB-FM system . . . .. 98

7.3-4. FM system . . . . .. 99

7.3-5. Comparison of system efficiencies * 100








Page

VIII. SUMMARY . . . . . .. 102

APPENDIX

I. PROOFS OF SEVERAL THEOREMS . . . .. .105


II. EVALUATION OF e j(x + jy) . . . . 119

REFERENCES . . . . . . .. 121

BIOGRAPHICAL SKETCH . . . . . . 124
















LIST OF FIGURES


Figure

1. Voltage Spectrum of a Typical m(t) Waveform * *.

2. Voltage Spectrum of the Analytic Signal Z(t) .*

3. Voltage Spectrum of an Entire Function of an Analytic
Signal * .......................

4. Voltage Spectrum of the Positive Frequency-Shifted
Entire Function of the Analytic Signal * *.

5. Voltage Spectrum of the Synthesized Upper Single-
Sideband Signal . . . . .

6. Voltage Spectrum of the Negative Frequency-Shifted
Entire Function of an Analytic Signal . .

7. Voltage Spectrum of the Synthesized Lower Single-
Sideband Signal . . . .

8. Phasing Method for Generating USSB-AM-SC Signals -

9. USSB-PM Signal Exciter--Method I . . .

10. USSB-FM Signal Exciter . . .

11. Envelope-Detectable USSB Signal Exciter ...... .

12. Square-Law Detectable USSB Signal Exciter ......

13. USSB-PM Signal Exciter--Method 11 . . .

14. USSB-PM Signal Exciter--Method III . . .

15. Power Spectrum of a(t) . *

16. AM Coherent Receiver . . . .

17. SSB-AM-SC Receiver . . . . .

18. SSB-FM Receiver . . . . .

19. Output to Input Signal-to-Noise Power Ratios for
Several Systems .* . * .


Page

9

11


* 16

* 20

. 22

. 24

* 26

. 27

* 53

. 54

* 67

* 76

* 78

* 78


* 86


vi11


* *


* *








Figure Page

20 Output Signal-to-Noise to Input Carrier-to-Noise Ratio
for Several Systems * .* ..................... 87

21. Output Signal-to-Noise to input Signal-to-Normalized-
Noise Power Ratio for Various Systems .......... * 90

22, Output Signal-to-Noise to Input Carrier-to-Normalized-
Noise Power Ratio for Various Systems . . . 91

23. Comparison of Energy-per-Bit for Various Systems . .. .96

24. Efficiencies of Various Systems . . . 101

25. Contour of Integration . . . . .. 107

26. Contour of Integration * .................. 115














KEY TO SYMBOLS


A0 = Amplitude Constant

AM = Amplitude-Modulation

b = Baseband Bandwidth (rad/s)

B = RF Signal Bandwidth (rad/s)

Cb = Baseband Channel Capacity

CB = RF Channel Capacity

Ci = Input Carrier Power

(C/N)i = Input Carrier-to-Noise Ratio

(C/N)I = Input Carrier-to-Normalized-Noise Ratio

D = Modulator Transducer Constant

FM = Frequency-Modulation

F(w) = Voltage Spectrum

F(-) = The Fourier Transform of (*)

g(W) = U(W) + jV(W) = An Entire Function

GN = Gaussian Noise

LSSB = Lower Single-Sideband

m(t) = Modulating Signal or a Real Function of the Modulating Signal
(see e(t) below)

M = Either Multiplex or Figure of Merit

Ni = Input Noise Power

NI = Normalized Input Noise Power

P()) = Power Spectral Density








PM = Phase-Modulation

R(,) ; Autocorrelation Function

Re(-) = Real Part of (*)

RF = Radio Freouency

Si = Input Signal Power

So: = Output Signal Power

(S/N)i = Input Signal-to-Noise Ratio

(S/N)I = Input Signal-to-Normalized-Noise Ratio

(S/N)o = Output Signal-to-Noise Ratio

SC = Suppressed-Carrier

USSB = Upper Single-Sideband

-(W) = The "Suppressed-Carrier" Function of U(W)

V(W) = The "Suppressed-Carrier" Function of V(W)

X(t) = A Real Modulated Signal

XL = Lower Single-Sideband Modulated Signal

XU = Upper Single-Sideband Modulated Signal

Z(t) = m(t) + jm(t) = The Analytic Signal of m(t)

a = Modulation (as defined in the text)

S= System Efficiency

6 = Modulation Index

n = Efficiency

e(t) = Modulating Signal (when m(t) is not the Modulating Signal)
o0 = Variance

om Average Power of m(t)
W = Angular Frequency

wrms = RMS-Type Bandwidth








AO = Eouivalent-Noise Bandwidth

S = Mean-Type Bandwidth

* = The Convolution Operator

(.)* = The Conjugate of (*)

(*) = The Hilbert Transform of (*)

(*) = The Averaging Operator








Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy



SYNTHESIS AND ANALYSIS OF REAL
SINGLE-SIDEBAND SIGNALS
FOR COMMUNICATION SYSTEMS






By

Leon Worthington Couch, II
June, 1968


Chairman: Professor T. S. George
Major Department: Electrical Engineering


A new approach to single-sideband (SSB) signal design and ana-

lysis for communications systems is developed. It is shown that SSB

signals may be synthesized by use of the conjugate functions of any

entire function where the arguments are the real modulating signal and

its Hilbert transform. Entire functions are displayed which give the

SSB amplitude-modulated (SSB-AM), SSB frequency-modulated (SSB-FM),

SSB envelope-detectable, and SSB square-law detectable signals. Both

upper and lower SSB signals are obtained by a simple sign change.

This entire generating function concept, along with analytic

signal theory, is used to obtain generalized formulae for the properties

of SSB signals, Formulae are obtained for (1) equivalent realizations

for a given SSB signal, (2) the condition for a suppressed-carrier SSB

signal, (3) autocorrelation function, (4) bandwidth (using various-de-

finitions), (5) efficiency of the SSB signal, and (6) peak-to-average

power ratio. The amplitude of the discrete carrier term is found to be


xiii








equal to the absolute value of the entire generating function evaluated

at the origin provided the modulating signal is AC coupled. Examples

of the use of these formulae are displayed where these properties are

evaluated for stochastic modulation.

The usefulness of a SSB signal depends not only on the pro-

perties of the signal but on the properties of the overall system as well.

Consequently, a comparison of AM, SSB-AM, SSB-FM, and FM systems is

made from the overall viewpoint of generation, transmission with additive

Gaussian noise, and detection. Three figures of merit are used in these

comparisons: (1) Output signal-to-noise ratios, (2) Energy-per-bit of

information, and (3) System efficiency.

In summary, the entire generating function concept is a new tool

for synthesis and analysis of single-sideband signals.


xiv











CHAPTER I

INTRODUCTION

In recent years the use of single-sideband modulation has become

more and more popular in communication systems. This is due to certain

advantages such as conservation of the frequency spectrum and larger post-

detection signal-to-noise ratios in suppressed carrier single-sideband

systems when comparison is made in terms of total transmitted power.

A single-sideband communication system is a system which generates

a real signal waveform from a real modulating signal such that the Fourier

transform, or voltage spectrum, of the generated signal is one-sided about

the carrier frequency of the transmitter. In conventional amplitude-modu-

lated systems the relationship between the real modulating waveform and

the real transmitted signal is given by the well-known formula:


XAM(t) = Ao [1 + m(t)] cos Wot m(t)|j 1 (1.1)


where Ao is the amplitude constant of the transmitter

m(t) is the modulating (real) waveform

Wu is the carrier frequency of the transmitter.

Likewise, frequency-modulated systems generate the transmitted waveform:


t
XFM(t) = Ao cos [Wot + D f m(t')dt'] (12)







where A0 is the amplitude constant of the transmitter

m(t) is the modulating (real) waveform

w0 is the frequency of the transmitter
D is the transducer constant of the modulator.

Now, what is the corresponding relationship for a single-sideband system?

Oswald, and Kuo and Freeny have given the relationship:


XSSB-AM(t) = Ao [m(t) cos "ot m(t) sin owt] (1.3)


where A0 is the amplitude constant of the transmitter

m(t) is the modulating signal

m(t) is the Hilbert transform of the modulating signal

(o is the frequency of the transmitter [1, 2].

This equation represents the conventional upper single-sideband suppressed-

carrier signal, which is now known as a single-sideband amplitude-modulated

suppressed-carrier signal (SSB-AM-SC). It will be shown here that this is
only one of an infinitely denumerable set of single-sideband signals. In-

deed, it will be shown that any member of the set can be represented by


XSSB(t) = Ao [U(m(t), m(t))cos wot T V(m(t), m(t)) sin wot] (1.4)


where Ao is the amplitude constant of the transmitter
U(x,y) and V(x,y) are the conjugate functions of any entire
function

m(t) is the modulating (real) waveform

m(t) is the Hilbert transform of m(t)

Wo is the transmitter frequency.

Various properties of these single-sideband signals will be analyzed in





3

general for the whole set, and some outstanding members of the set will

be chosen for examples to be examined in detail.

It should be noted that Bedrosian has classified various types of

modulation in a similar manner; however, he does not give a general repre-

sentation for single-sideband signals [3].










CHAPTER II

MATHEMATICAL PRELIMINARIES


Some properties of the Hilbert transform and the corresponding

analytic signal will be examined in this chapter. None of the material

presented in this chapter is new; in fact, it is essentially the same

as that given by Papoulis except for some changes in notation [4]. How-

ever, this background material will be very helpful in derivations pre-

sented in Chapter III and Chapter V

The Hilbert transform of m(t) is given by


1. m(x)dx 1
m(t) =- P m(t) -- (2,1)
St-x Ttt


where (*) is read "the Hilbert transforms of (*)"

P denotes the Cauchy principal value

indicates the convolution operation.

The inverse Hilbert transform is also defined by Ea (2.1) except that a

minus sign is placed in front of the right-hand side of the equation. It

is noted that these definitions differ from those used by the mathema-

ticians by a trivial minus sign. It can be shown, for example, that the

Hilbert transform of cos wot is sin mot when 0o > 0 and that the Hilbert

transform of a constant is zero. A list of Hilbert transforms has been

compiled and published under work done at the California Institute of Tech-

nology on the Bateman Manuscript Project [5].







The Fourier transform of m(t) is given by


Fm(w) = [-j sgn (w)] Fm(a) (2.2)


where

+ 1 W > 0

sgn (w) = 0 0 = (2.3)

1 W < 0

and Fm(w) is the Fourier transform of m(t). In other words, the Hilbert

transform operation is identical to that performed by a -900 all-pass

linear (ideally non-realizable) network.

From Eq. (2.2), it follows that


F^(w) = [-j sgn (w)]2 Fm(w) = -Fm(W) (2.4)

or

M(t) = -m(t). (2.5)

The (complex) analytic signal associated with the real signal

m(t) is defined by


Z(t) = m(t) + jm(t). (2.6)

The Fourier transform of Z(t) follows by the use of Ea. (2.2),

and it is


FZ(w) = Fm(w) + j[-j sgn (w)] Fm(w)







or


2Fm(w) >

FZ(w)= Fm(w) W = 0 (2.7)

L 0 W < j

Now suppose that m(t) is a stationary random process with auto-

correlation Rmm(t) and power spectrum Pmm(w)o Then the power spectrum

of m(t) is


Pim(w) = Pmm(w) 1-J sgn () = Pmm(w). (2.8)


This is readily seen by use of the transfer function of the Hilbert trans-

form operator given by Eq. (2.2). Then, by taking the inverse Fourier

transform of Eq. (2.8), it follows that


Rmf(-) = Rmm(r). (2.9)


The cross-correlation function is obtained as follows:


Rim(T) = m(t + T)m(t)


S-m(t + T A)m(t)dx

where () is the averaging operator. Thus



where (.) is the averaging operator. Thus


Rmm(i) = Rmm().


(2.10)







It follows that the spectrum of the cross-correlation function is given by


fmmif) [-j sgn (w)] Pmalnw)


(2,11)


it. is i..'fd that Pmm(() is a purely imaginary function since Pmm(w) is a

real function. Then


1
Rmm(L ) = I


[-j sgn (,)] Pmm(w) ejwT


which, for Pmm(w) a real even function, reduces to


" '0


Pinfri L ) sin t d.


Thus the cross-correlation function is an odd function of L:

Rnm(i) = -Rmm(-)

= -m(t [)m(t) = -m(t + 1)mr(t)


or


Rmm(' ) :. -Rm(-') = -Rmm(i)o


(2.12)


(2.13)


(2.14)


The autocui elation for the analytic signal is found as follows:

RZZ(i) = Z(tt+)Z*(t)

= [m(t+) + jm(t+r)] [m(t) jm(t)]

m(tr,)m(t) + m(t+T)m(t) + jm(t+. )m(t) jm(t+r)m(t)


= Rmm(') + RWm(T) + jRim(T) jRmm(i).










Using Eqs. (2.9), (2.10) and (2.14) we obtain


RZZ(T) = 2[Rmm(i) + jRm(t)] = 2[Rmmn() + 3Rmm(s)]. (2.15)


Thus (1/2)Rzz(T) is an analytic signal associated with Rmm(v). By use of
Eq. (2.7) it follows that


"4Pmm() W 0

PZZ(W) = 2Pmm(w) a 0. (2 16)

L 0 < O_












CHAPTER I ;

SYNTHESIS OF SINGLE: -SiUDEAIND GN'LS


Eq. (1.4), which specifies thc set of single-sideband signals

that can be generated from a given modulating waveform or process, will

be derived in this chapter. The equation must be a real function of a

real input waveform, m(t), since it represents the generating function

of a physically realizable system--the single-1sideband transmitter--and,

in general, it is non-linear. Analytic signal techniques will be used

in the derivation. It will be shown that if we have a complex function

k(z) where k(z) is analytic for z = x + jy in the upper half-plane (UHP),

then the voltage spectrum of k(x,O) k(t) is zero for < 0. In order

to synthesize real SSB signals from a real modulating waveform, an UHP

analytic generating function of the complex time real modulating process

must be found regardless of the particular (physically realizable) wave-

form that the process assumes.

Let m(t) be either the real baseband modulating signal or a reaL

function of the baseband modulating signal e(t), Then the amplitude of

the voltage spectrum of m(t) is double sided about the origin, for ex-

ample, as shown by Figure 1.


Fm( i)






Figure 1. Voltage Spectrum of a Typical m(t) Waveform






10

Since m(t) is generated by a physically realizable process, it con-

tains finite power for a finite time interval. This, of course, is equiva-

lent to saying that m(t) is a finite energy signal or, in mathematical

terms, it is a L2 function. By Theorem 91 of Titchmarsh, m(t) is also a

member of the L2 class of functions almost everywhere [6]. Now the complex

signal Z(t) is formed by


Z(t) = m(t) + jm(t). (3,1)


It is recalled that Z(t) is commonly called an analytic signal in the

literature. By Theorem 95 of Titchmarsh there exists an analytic func-

tion (regular for y > 0), Z (z), such that as y 0


Z1(x + jy) Z(t) = m(t) + jm(t) x t


for almost all t and, furthermore, Z(t) is a Ll (-, function [6]

It follows by Theorem 48 of Titchmarsh that the Fourier transform of Z(t)

exists [6].

Theorem I: If k(z) is analytic in the UHP then the
spectrum of k(t,O), denoted by Fk(w), is zero for
all w < 0, assuming that k(t,0) is Fourier trans-
formable,

For a proof of this theorem the reader is referred to Appendix I.

Thus the voltage spectrum of Z(t) is zero for w < 0 by Theorem I

since Z(t) takes on values of the analytic function Z7(z) almost every-

where along the x axis. Furthermore, since Z(t) is an analytic signal--

that is, it is defined by Eq. (3.1)--its voltage spectrum is given by





Eq. (2.7), which is


Fz(,) F j


where Fm(w) is the voltage spectrum of the signal m(t)
Figure 2 for our example used in Figure 1.
,> j F )


This is shown in


Figure 2 Voltage Spectrum of the Analytic Signal Z(t)
Figure 2. Voltage Spectrum of the Analytic Signal Z(t)


Now let a function g(W) be given such that


g(W) = U(ReW,ImW) + jV(ReW,ImW)


(3.3)


where g(W) is an entire function of the complex variable W.

Theorem. II: If Z(z) is an analytic function of z in
the UHP and if g(W) is an entire function of W, then
g[Z(z)] is an analytic function of z in the UH z-plane,

A proof of this theorem may be found in Appendix I.
Thus g[Z1(z)] is an analytic function of z in the UH z-plane, and
by Theorem I g[Z(t)] has a single-sided spectrum: the spectrum being
zero for w < 0. This is illustrated by Figure 3, where Fg(w) = F[g(Z(t))].


r1


(3 2)











SFga ()


Figure 3. Voltage Spectrum of an Entire Function
of an Analytic Signal


Now multiply the complex baseband signal g[Z(t)] by eJ"ut to

translate the signal up to the transmitting frequency, o,. It is noted

that g[Z,(z)] and ejWoz for mo > 0 are both analytic functions in the UH

z-plane, By the Lemma to Theorem I in Appendix I, g[Z,(z)]eJawz is ana-

lytic in the UH z-plane. Therefore, by Theorem I the voltage spectrum

of g[Z(t)]e JOt is one sided about the origin. Furthermore,


F[g(Z(t))eJoWt 1 TeF[g(Z(t))] F[eJ t]


= Fg(w) 6(w-Lo)


or


F[g(Z(t))ej ot] Fg(w-wo) 'o 0 (3.4)


This spectrum is illustrated in Figure 4.



















Figure 4. Voltage Spectrum of the Positive Frequency-
Shifted Entire Function of the Analytic Signal

The real upper single-sideband signal can now be obtained from

the complex single-sideband signal, g[Z(t)]eJot, by taking the real

part, This is seen from Theorem III,

Theorem III. If h(z) is analytic for all z in the
UHP and F[h(x,O)] = Fh(w), then for wc > 0,



Fh(w-w)o a wo

F{Re[h(x,0)eJwox]} = 0 i W < mo

-Fi(46-w.) w s-w


This theorem is proved in Appendix I.
Thus the upper single-sideband signal for a given entire function is


XUSSB(t) = Re{g[Z(t)l]ejot}

= Re{[U(ReZ(t),ImZ(t)) + jV(ReZ(t),ImZ(t))]ej 't}

= Re{[U(m(t),m(t)) + jV(m(t),m(t))]ejot}


)ejwut i i











I XUSSB(t) = U(m(t),m(t)) cos t- V(m(t),m(t)) sin wmt I


where U(ReW,ImW) is the real part of the entire function g(W)

V(ReW,ImW) is the imaginary part of g(W)

m(t) is either the modulating signal or a real function of the
modulating signal e(t)

m(t) is the Hilbert transform of m(t),

Using Theorem III the voltage spectrum of XUSSB(t) is


FUX(w) = F[XUSSB(t)] =


Fg(- )

0

F*(-w-0)
g


, O -< coo

, ii < O 0


< --


(3.6)


This spectrum is illustrated by Figure 5.


IFUX( )i


-WO 0
Figure 5. Voltage Spectrum of the Synthesized
Upper Single-Sideband Signal


The lower single-sideband signal can be synthesized in a similar

manner from the complex baseband signal. Now we need to translate the

complex baseband signal down to the transmitting frequency instead of up,


(3.5)





15

as in the upper single-sideband synrthc:.. Then the Fourier transform of

the down-shifted complex baseband signal is


F[g(Z(t))e-Lot] [g ]
2;r


F[e-Jut]


Fg(v) I(ulJo)


F[g(Z(t))e-Jmot] Fg(w+u4)


, WU


(3,7)


This spectrum is illustrated in Figure 6.


IF[g(Z(t))e-jot]


Figure 6.
Shifted


W0 W4
Voltage Spectrum of the Negative Frequency-
Entire Function of an Analytic Signal


Theorem IV: If h(z) is analytic for all z in UHP and
F[h(x,O)] = Fh(w) where Fh(Q) = 0 for all > wo, then
for wo > 0


F{Re[h(x,O)e-JmWx]


0

Fh(w+wo)


, O < WO

, i > Wo

,0 > W ~-W


This theorem is proved in Appendix I.







Thus the real lower single-sideband signal for a given entire

function is



XLSSB(t) = Reg(Z(t))e-j 't


= Re{[U(m(t),m(t)) t jV(m(t),fi(t))]e-j o t


XLSSB(t) = U(m(t),m(t)) cos wot + V(m(t),m(t)) sin wt.


Using Theorem IV the voltage spectrum of XLSSB(t) is


(3.8)


FLX(M) = F[XLSSB(t)] =


F*(-wtwo)

0

Fg(W+Wo)


, 0 < < WO

, Ki > Wo

, O m u -W


It is noted that the requirements that Fg(w) be zero for w > wo is to

prevent spectral overlap at the origin. This requirement is satisfied

(for all practical purposes) for wo at radio frequencies.

The spectrum of FLX(M) is illustrated by Figure 7.


FLX(w)


(3.9)


-WO I O w
Figure 7. Voltage Spectrum of the Synthesized
Lower Single-Sideband Signal








To summarize, it has been shown that once an entire function 9gW)

is chosen, then an upper or lower single-sideband signal can be obtained

from the signal m(t). The signal m(t) is either the modulating signal

or a real function of the modulating signal b(t). The generalized ex-

pressions, which represent SSB signals, are given by Eq, (3 5) for the

USSB signal and by Eq. (3.8) for the LSSB signal. These expressions are

obviously the transfer functions that are implemented by the upper and

lower single-sideband transmitters respectively. Since there are dn in-

finitely denumerable number of entire functions, there are an inrinuiely

denumerable number of upper and lower single-sideband signals that can be

generated from any one modulation process, In Chapter IV some specific

entire functions will be chosen to illustrate some well-known single-

sideband signals.













CHAPTER IV

EXAMPLES OF SINGLE-SIDEBAND SIGNAL DESIGN

Specific examples of upper single-sideband signal design will now

be presented. Entire functions will be chosen to give signals which have

various distinct properties. In Chapter VI these properties will be ex-

amined in detail. Only upper sideband examples are presented here since

the corresponding lower sideband signals are given by the same equation

except for a sign change (Eq. (3.5) and Eqo (3.8)).


4.1. Example 1: Single-Sideband AM With Suppressed-Carrier

This is the conventional type of single-sideband signal that is

now widely used by the military, telephone companies, and amateur radio

operators. It will be denoted here by SSB-AM-SC.

Let the entire function be


g!(W) = W (4.1)

and let m(t) be the modulating signal. Then substituting the corresponding

analytic signal for W


g (Z(t)) = m(t) + jm(t)

or


UM(m(t),6(t)) = m(t) and Vj(m(t),m(t)) = m(t). (4.2 a,b)








Substituting Eqs. (4.2a) and (4o2b) into Eq, (3.5) we obtain the

upper single-sideband signal:


XUSSB-AM-SC(t) = m(t) cos wot m(t) sin wot (4.3)


where m(t) is the modulating audio or video signal and m(t) is the Hil-

bert transform of m(t). It is assumed that m(t) is AC coupled so that

it will have a zero mean.

The upper single-sideband transmitter corresponding to the gene-

rating function given by Eq. (4.3) is illustrated by the block diagram

in Figure 8. It is recalled that this is the well-known phasing method

for generating SSB-AM-SC signals [7, 8],,


4.2. Example 2: Single-Sideband PM

Single-sideband phase-modulation was described by Bedrosian in

1962 [3].
To synthesize this type of signal, denoted by SSB-PM, use the

entire function:


g,(W) = eJ (44)


Let m(t) be the modulating audio or video signal. Then


g2(Z(t)) = e(m(t) + j(t)) = e-(t) em(t)

or

U2(m(t),m(t)) = e-(t) cos m(t) (4.5a)












m(t) cos


Radio Frequency
Oscillator, wo


XUSSB-AM-SC(t)
+"^ -Modulated RF Output


Phasing Method for Generating USSB-AM-SC Signals


m(t)


Input


Hilbert Transform
{-90 Phase Shift
over Spectrum of
m(t)}


Figure 8.








and


V2(m(t),m(t)) = e-m(t) sin m(t). (4.5b)


Substituting Eqs. (4.5a) and (4.5b) into Eq. (3.5) we obtain the upper

single-sideband signal:


XUSSB-PM(t) = e-(t) cos m(t) cos wot e-m(t) sin m(t) sin wot


or


XUSSB-PM(t) = e-m(t) cos (cot + m(t)). (4.6)


It is again assumed that the modulation m(t) is AC coupled so that its

mean value is zero. The single-sideband exciter described by Eq. (4.6)

is shown in Figure 9.

4.3. Example 3: Single-Sideband FM

Single-sideband frequency-modulation is very similar to SSB-PM

in that they are both angle modulated single-sideband signals. In fact

the equations for SSB-FM are identical to those given in Section 4.2 ex-

cept that

t
m(t) = D f e(t)dt (47)
-00


where e(t) is now the modulating signal (instead of m(t)) and D is the

transducer constant.

Experiments with SSB-FM signals have been conducted by a number

of persons and are reported in the literature [9, 10].













m(t)
Modulating Input


XUSSB-PM(
Modulated RF Output


cos (mot+m(


Figure 9. USSB-PM Signal Exciter--Method I


Phase Modulator
at
Radio Frequency mo


Hilbert Transform
{-90 Phase Shift
over Spectrum of
m(t)}








The SSB-FM exciter as described by Eas. (4.6) and (4.7) is given
in Figure 10.


4.4. Example 4: Single-Sideband a
The term single-sideband a (SSB-a) will be used to denote a sub-
class of single-sideband signals which may be generated from a particular
entire function with a real parameter a. This notation was first used by
Bedrosian [3].
Let the entire function be


g3(W) = e"W (4.8)


where a is a real parameter, and let


m(t) = ln[l + e(t)] (4.9)


where e(t) is the video or audio modulation signal which is amplitude
limited such that le(t)| < 1. It is assumed that m(t) is AC coupled
(that is, it has a zero mean). Note that these assumptions are
usually met by communications systems since they are identical to the
restrictions in conventional AM modulations systems. Then


g3(Z(t)) = e[1m(t)+jm(t)]= e m(t) emja(t)


or


U3(m(t),6(t) = eam(t) cos (am(t)) (4.10a)










































Figure 10.


USSB-FM Signal Exciter











V3 (m(t),m(t)) = em(.t) sin (am(t)). (410b)


Substituting Eqs. (4.10a) and (4.10b) into Eq. (3.5), the SSB-a signal is


XUSSB-a(t) = em"(t) cos (am(t)) cos wot

eam(t) sin (am(t)) sin wot


or


XUSSB-a(t) = eam(t) cos (mot + am(t)). (4.11)



In terms of the input audio waveform, Eq. (4.11) becomes


XUSSB(t) = ealn[l+e(t)] cos (wot + aln[l+e(t)])


or


XUSSB_-(t) = [1+e(t)] cos (Wot + aln[l+e(t)]). (4.12)


For a = 1 we have an envelope-detectable SSB signal, as is readily
seen from Ea. (4.12). Voelcker has recently published a paper demon-
strating the merits of the envelope-detectable SSB signal [11]. The real-
ization of Eq. (4.12) is shown in Figure 11.
For a = 1/2 we have a square-law detectable SSB signal. This type
of signal has been studied in detail by Powers [121. Figure 12 gives the
block-diagram realization for the square-law detectable SSB exciter.











e(t)
Modulating Input


DC Level
of +1


XUSSB-o = 1(t)


[1+e(t)]


Phase Modulator
at
Radio Frequency w0


Envelope-Detectable USSB Signal Exciter


Hilbert Transform
{-90 Phase Shift
over Spectrum of
In[1+e(t)]}


cos (w~t+i'[l +e(t) I


Figure 11.









e(t)
Modulating
Input


Positive Square I [l+e(t)]1
Root Circuit


Square-Law Detectable USSB Signal Exciter


Figure 12.












CHAPTER V

ANALYSIS OF SINGLE-SIDEBAND SIGNALS

The generalized SSB signal, that was developed in Chapter III,

will now be analyzed to determine such properties as equivalent gener-

alized SSB signals, presence or absence of a discrete carrier term,

autocorrelation functions, bandwidths, efficiency, and peak-to-averaige

power ratio. Some of these properties will depend only on the entire

function associated with the SSB signal, but most of the properties will

be a function of the statistics of the modulating signal as well.


5.1. Three Additional Equivalent Realizations

Three equivalent ways (in general) for generating an upper SSB

signal will now be found in addition to the realization given by Eq. (3.5).

Similar expressions will also be given for lower SSB signals which are

equivalent to Eq. (3.8). It is very desirable to know as many equivalent

realizations as possible since any ore of them might be the most econom-

ical to implement for particular SSB signal.

Theorem V: If h(x,y) = U(x,y) + jV(x,y) is
analytic in the UHP (including UH-) then

h(t,O) = U(t,O) + j[O(t,O)+k1] (5.1)

or
h(t,O) = [-(t,O)+k2] + jV(t,0) (5.2)








or
h(t,O) = [-V(t,O)+k2] + j[O(t,O)+k1] (5.3)
where
T-
k, = lim f V(R cos e,R sin e)de a real constant (5.4)
R** 0

k2 = lim U(R cos e,R sin e)de a real constant (5.5)
R-- o

A proof of this theorem is given in Appendix I.

Theorem V may be applied to ,the generalized SSB signal by letting
h(z) = g(Z1(z)) where g(.) is an entire function of (.), Z1(z) is analytic
in the UHP, and lim Z (z) = lim Z (t + jy) = m(t) + j6(t). Thus Theorem V
yO y+O
gives three additional equivalent expressions for g(Z(t)) in addition to


g(z(t)) = U(m(t),m(t)) + jV(m(t),M(t)) (5.6)


which was used in the derivation in Chapter III. Therefore, following
the same procedure as in Chapter III, equivalent upper SSB signals may be

found. Using Eq. (5.1) we have for the first equivalent representation
of Eq. (3.5):

XUSSB(t) = Re{g(Z(t))ejwOt}


= Re{g(m(t),m(t))eJwot}


= Re{[U(m(t),m(t) + jU(m(t),m(t)) + jkl]eJWOt}

or

XUSSB(t) = U(m(t),m(t)) cos wot [U(m(t),m(t))+ k,] sin o0t. (5.7)







Using Eq. (5.2) the second equivalent representation is

XUSSB(t) = [-V(m(t),m(t))+k2 cos mot V(m(t),m(t)) sin mot. (5.8)

Using Eq. (5.3) the third equivalent representation is

XUSSB(t) = [-V(m(t),m(t))+k2] cos mot [U(m(t),m(t))+k1],sin wot. (5.9)

Likewise the three lower SSB signals, which are equivalent to
Eq. (3.8), are

XLSSB(t) = U(m(t),m(t)) cos awt + [U(m(t),m(t))+k1] sin mot (5.10)


XLSSB(t) = [-V(m(t),m(t))+k2] cos mot + V(m(t),6(t)) sin wot (5.11)


and

XLSSB(t) = [-9(m(t),m(t))+k2] cos w0t + [U(m(t),i(t))+kj] sin wot.(5.12)


It should be noted, however, that if for a given entire function
k, and k2 are both zero, then all four representations for the USSB or
the LSSB signals are identical since by Theorem V, U = -V and V = U under
these conditions.


5.2. Suppressed-Carrier Signals
The presence of a discrete carrier term appears as impulses in
the (two-sided) spectrum of transmitted signal at frequencies wo and -woo
The impulses may have real, purely imaginary, or complex-valued weights
depending on whether the carrier term is cos wot, sin mot, or a com-
bination of the two. Thus the composite voltage spectrum of the modulated








signal consists of a continuous part due to the modulation plus impulse

functions at w0 and -mo if there is a discrete carrier term. As defined

here, the "continuous" part may contain impulse functions for some types

of modulation, but not at the carrier frequency. Taking the inverse

Fourier transform of the composite voltage spectrum it is seen that if

there is a discrete carrier term, the time waveform must be expressible

in the form:

X(t) = [f1(t)+c1] cos Wot [f2(t)+c2] sin mot (5.13)

where cI and c2 are due to the discrete carrier

f1(t) and f2(t) are due to the continuous part of the spectrum
and have zero mean values.

Thus Eq. (5.13) gives the condition that c2 and cl are not both zero if

there is a discrete carrier term.

To determine the condition for a discrete carrier in an upper

SSB signal, Eq. (5.13) will be identified with Eq. (5o9), which represents

the whole class of upper SSB signals. It is now argued that both U and V

have a zero mean value if the modulating process is stationary. This is

seen as follows:





U(m(t),m(t))= P U(m(t'),m(t')) dt' .
-00


But U[m(t'),m(t')] = c, a constant, since m(t) is wide-sense stationary.

Thus


U(m(t),m(t)) = IP c--dt' = 0.
-00








Likewise V has a zero mean value. Then, identifying Eq. (5.13) with
Eq. (5.9), it is seen that


fi(t) 4 -V(m(t),m(t)) (5.14a)


f2(t) = U(m(t),m(t)) (5.14b)

c, = kg and c2 k (5.14c,d)


Similarily, for lower SSB signals Eq. (5.13) can be identified

with Eq. (5.12).
Thus the SSB signal has a discrete carrier provided that k1 and k2
are not both zero.
As an aside, it is noted that the criterion for a discrete car-
rier, given by Eq. (5.13), is not limited to SSB signals; it holds for

alt modulated signals. For AM Eq. (5.13) can be identified with Eq. (1.1).
Here

f (t) Aom(t) (5.15a)

f2(t) = 0 (5.15b)

c, = Ao and c2 = 0 (5.15c,d)

because m(t) has a zero mean due to AC coupling in the modulator of the
transmitter. Thus for AM it is seen that there is a discrete carrier
term of amplitude c, = A which does not depend on the modulation. For FM
Eq. (1.2) can be expanded. Then, assuming sinusoidal modulation of fre-
quency wa, we obtain








XFM(t) = [Ao cos (- cos wat)] cos ot
wa


[A0 sin (-- cos wat)] sin wot. (5.16)


To identify Eq. (5.16) with Eq. (5.13) we have to find the DC terms of

f (t) + c A cos (- cos mat)
1 0 a

and


f (t) + c2 Ao sin (-cos wat).

These are

C, = A0 COS (L COS Wat)
wa


SAo T cos ( cos at)dt


= A ( D) (5.17a)
wa

and

c2 = A sin ( D cos wat)
Sa
T
= Ao0 sin (-k- cos wat)dt
T 0 wa

= 0 (5.17b)


Then for sinusoidal frequency-modulation it is seen that the discrete
carrier term has an amplitude of AoJo(D/ta) which may or may not be zero

depending on the modulation index D/wa. Consequently, for FM it is seen

that the discrete carrier term may or may not exist depending on the

modulation. Prof. T. S. George has given the discrete carrier condition








for the case of FM Gaussian noise [13].

Continuing with our SSB signals, it will now be shown that k, and

k2 depend only on the entire function associated with the SSB signal and

not on the modulation. From Theorem IV we have


k = I lim
R-*oo





k2 = lim
SRT


f V[ml(R cos e,R sin e) m (R cos e,R sin e)]de
0





U[m1(R coS e,R sin e) m(R cos e,R sin e)]de
0


where U and V are the real and imaginary parts of the entire function

Z1(z) = m1(z) + jm1(z) is the analytic function associated
with the analytic signal Z(t) of m(t).

It is seen that if


(5.18a)


lim m (R cos e,R sin e) = 0 0 e s
R+o1


lim m,(R cos e,R sin e) = 0
R-+o


(5.18b)


, 0 O


then k and k2 depend only on U and V of the entire function and not on

m. Thus we need to show that Ea. (5.18a) and (5.18b) are valid. By the

theory of Chapter III there exists a function Z1(z) = m (z) + j0(z)

which is analytic in the UHP such that (almost everywhere) lim Z1(t + jy)
y*
= Z(t) = m(t) + jm(t) where the Fourier transform of Z(t), F(o), is

L.2(-o, o). Then we have










F(w)ejzwdw.


It follows that


lim IZ (Rej')I2
R+.-


= lim (-)2
R -*O "


[F(w)][e-(R sin O)wej(R cos o)]dJ2


By use of Schwarz's inequality this becomes


lim 1Z1(Rej3) |2
R-o


12


f


IF(w)12d } {lim
R-xo


e-2(R sin O)adw,}


But F(M) e L ("', -) so that


f
0


IF(w) 12dw K.


Vim
R ~


e-(2R sin e)wdw = 0
e di=


, 0 < 0 < T.


Therefore we have


lim jZ(ReJ )I (-)2 K 0 = 0


Also


Z, (z) =


, 0 < e < n.








For e = 0 or e =



2(+) e = 0

lim Z I(ReJ)I = 0
R-o





since


Z(t) e L2(-, 2).


Then


lim jZi(ReJe)I = lim IZ (R cos e,R sin O)| = 0 0 s e s
R-co R-


which shows that Eqs. (5.18a) and (5.18b) are valid statements. Thus,

the presence (kI and k2 not both zero) or the absence (k, = k = 0) of

a discrete carrier depends only on the entire function associated with

the SSB signal and not on the modulation. Furthermore, it is seen that

the amplitude of the discrete carrier is given by the magnitude of the

entire function evaluated at the origin (of the W plane), and the power

in the discrete carrier is one-half the square of the magnitude.

For every generalized USSB signal represented by Eq. (3.5),

there exists a corresponding suppressed-carrier USSB signal:


XUSSB-SC(t) = '-(m(t),(t)) cos Wot *(m(t),m(t)) sin Wot


(5.19)








where the notation SC and denote the suppressed-carrier functions.
But what are these functions U and W? The condition for a suppressed
carrier is that ki = k2 = 0. By comparing Eq. (5.19) with Eq. (5.9) it

follows that th -V and V Furthermore by Theorem V of Section 5.1,
U = -V + kg and V = U + ki, Thus

U -V = U k2 (5,20)

and

V- U = V ki. (5,21)

It is also noted that 4 and + are a unique Hilbert transform pair. That
is, V- is the Hilbert transform of i, and U- is the inverse Hilbert trans-

form of -. This is readily shown by taking the Hilbert transform of

Eq. (5.20), comparing the result with Eq. (5.21), and by taking the in-

verse Hilbert transform of Eq. (5.21), comparing the result with Eq. (5.20).
Thus Eq. (5,19) may be re-written as



XUSSB-SC(t) = tt(m(t),'(t)) cos wot -(m(t),m(t)) sin o0t (5.22)

or


XUSSB-SC(t) = -V(m(t),m(t)) cos wot V(m(t),m(t)) sin wot (5.23)


where U and V-are given by Eq. (5.20) and Eq. (5.21).
It is interesting to note that the form of the USSB signal given

above checks with the expression given by Haber [14]. He indicates that

if a process n(t) has spectral components only for I)w > wo then n(t)

can be represented by










n(t) = s(t) cos wot 9(t) sin mot. (5.24)

Thus Eq. (5.22) checks with Eq. (5.24) where U = s(t), and Eq. (5.23)
checks also where -V E s(t).
The corresponding representations for LSSB suppressed-carrier

signals are given by



XLSSB-SC(t) = tJ(m(t),m(t)) cos wot + 4(m(t),m{t)) sin mot (5.25)

and

XLSSB-SC(t) = --(m(t),m(t)) cos wot + *(m(t),m(t)) sin mot (5.26)


where 4- and V are given by Eq. (5.20) and Eq. (5.21).
This representation also checks with that given by Haber for pro-

cesses with spectral components only for Iwl < wo which is

n(t) = s(t) cos wot + s(t) sin mot. (5o27)


5.3. Autocorrelation Functions
The autocorrelation function for the generalized SSB signal and

the corresponding suppressed-carrier SSB signal will now be derived.
Using the result of Chapter III, it is known that the generalized
upper SSB signal can be represented by


XUSSB(t) = Re{g(m(t),m(t))ej(ot+)} (5.28)








where a uniformly distributed phase angle > has been included to account
for the random start-up phase of the RF oscillator in the SSB exciter.
Then, using Middleton's result [15], the autocorrelation of the USSB sig-
nal is


RXU(t) = XUSSB(t+T)XUSSB(t) = Re{eJwoTRg(-)}


(5.29)


where


Rg(T) = g(M(t+T)mi(t+T))g*(m(t),rm(t))


(5.30)


and


g(m(t),m(t)) = U(m(t),m(t)) + jV(m(t),m(t)).


(5.31)


The subscript XU indicates the USSB signal. For the generalized LSSB
signal the corresponding formulae are


XLSSB(t) = Re{g(m(t),rn(t))eJj(bOt+f) }


and


IRXL(T) = Re{e-JWOTRg(T)}.


These equations can be simplified if we consider the autocorre-
lation for the continuous part of the spectrum of the SSB signal. The
suppressed DC carrier version of g, denoted by gSC, will first be found
in terms of g, and then the corresponding autocorrelation function Rg-sc()


(5.32)





(5.33)









will be determined in terms of Rg(T).
By examining Eqo (5o19) and comparing this equation to Eq. (35),
with the aid of Eq. (3.3) it is seen that the suppressed DC carrier
version of g is given by


gsc(m(t),m(t)) = f(m(t),m(t)) + jW(m(t),m(t)) (5.34)


where 4 and V are the suppressed-carrier functions defined by Eq. (5.20)
and Eq. (5.21). Then it follows that


g(m(t),m(t)) = gsc(m(t),m(t)) + [k2+jk,]o (5.35)


It is noted that the mean value of gSC is zero. This is readily seen via
Eqs. (5.34), (5.20), and (5.21) since it is recalled that the mean value
of U and V was shown to be zero in Section 5,2. Then, using Eq. (5.35),
the autocorrelation of g is obtained in terms of the autocorrelation of

9SC:


Rg(T) = Rg-SC(T) + (k 2+k ). (5.36)


Therefore the autocorrelation functions for the USSB signal,
Eq. (5.29), and the LSSB signal, Eq. (5.33), become


RXU(T) = Re{eJwo( 22+k22) + Rg-SC(r)]} (5o37)

and


RXL(') = Re{&WoT[ k2+k22) + Rg-SC(T)]}


(5.38)








It may be easier to calculate the autocorrelation for the USSB or
LSSB signal using this representation rather than that of Eq. (5.29) and
Eq. (5,33) since RgSC(T) may be easier to calculate than Rg(r). This
is shown below.
A simplified expression for Rg-SC(T) will now be derived. First,
it is recalled from Section 5.2 that and V- are a unique Hilbert trans-
form pair. Thus gSC, given by Eq. (5.34), can be expressed in terms of
two analytic signals:


gsc(m(t),m(t)) = t(m(t),m(t)) + j4(m(t),m(t)) (5.39)


and


gSC(m(t),m(t)) = -V(m(t),m(t)) + j-(m(t),m(t)) (5.40)


where Eq. (5.39) is the analytic signal associated with and Eq. (5.40)
is the analytic signal associated with -V-. Using Eq. (5.39) and Eq. (2.15),
the autocorrelation of gSC is given by


Rg-SC(T) = 2[R (T) + JR4.(T)] (5.41)


or by using Eqs. (5.40), (2.15),and (2.9)


Rg-SC(T) = 2[R.(T) + jR (T)]o (5.42)


Thus Rg-SC(T) may be easier to calculate than R (T) since only Ri,(T) or
R .(T) is needed. This, of course, is assuming that the Hilbert trans-








form is relatively easy to obtain. On the other hand Rg(T) may be calcu-
lated directly from g(m(t),m(t)) or indirectly by use of RUU.(T), Rvv(),
RUV(T), and RVU(T).
The autocorrelation functions for the generalized USSB and LSSB
signals having a suppressed-carrier are readily given by Eq. (5.37) and
Eq. (5.38) with k, = k2 = 0:


RXU-SC(i) = Re{eJmwTRg-SC(T)}


= CR. o) cos W"o Rtt() Sin wfo


= RV(7T) cos wo' RW.(T) sin moT






RXL-SC(t) = -Re{ eeoRg-SC()}

= R 1.(,) COs WOT + RiJ.(T) sin wor


= R ,(T) cos W0T + Rv.(T) sin wor.


(5.43a)


(5.43b)


(5.43c)






(5.44a)


(5.44b)


(5.44c)


It follows that the power spectral density of any of these SSB

signals may be obtained by taking the Fourier transform of the appro-
priate autocorrelation function presented above.


5.4. Bandwidth Considerations
The suppressed-carrier autocorrelation formulae developed above
will now be used to calculate bandwidths of SSB signals. It is noted








that the suppressed-carrier formulae are needed instead of the "total sig-
nal" formulae since, from the engineering point of view, the presence or
absence of a discrete carrier should not change the bandwidth of the sig-
nal. Various definitions of bandwidth will be used [16, 17]1


5o4-1. Mean-type bandwidth
Since the spectrum of a SSB signal is one-sided about the carrier
frequency, the average frequency as measured from the carrier frequency
is a measure of the bandwidth of the signal:

f WPg.SC(w)d Rg-Sc(O)
0 = ------- (5.45)
f Pg-SC(w)dw Rg-SC(O)


where Pg_SC(w) is the power spectral density of gSC(m(t),m(t))and the
prime indicates the derivative with respect to r. The relationship is
valid whenever R'_SC (0) and Rgsc(O) exist. Substituting Eg. (5.41)
into Eq. (5.45) we have

2[R (O) + jR (O)]
I 2[R*(0) + jRi(0O)]


But it recalled that Rju.(T) is an even function of T and, from Chapter II,
RUS.(T) is an odd function of T. Then R%,(0) = R(O) = 0 and it follows
that


R44u(O) R(0)
P"Ry(0) Rft(O)


(5,46)







I
It is noted that this formula is applicable whenever Ru.(0) and R. (O) or
R.(0O) and R,,(0) exist. That is, R,44(0), R 4(0), R (O), and R^_(0) may

or may not exist since Rg-SC(T) is analytic almost everywhere (Theorem 103

of Titchmarsh [6]).


5o4-2, RMS-type bandwidth

The rms bandwidth, wrms, may also be obtained.


CO
2 2 PgSC()d -Rg-SC(O)
(wrms)2 2 -0o
f Pg-SC(w)dw Rg-SC(O)
-Substituting Eq (5.41) once again, we have00


Substituting Eq. (5.41) once again, we have


(rms)2


(5.47)


-2[R~(0) + jR (0)]

2[R i (0) + JRi (O)]


Since RUU(T) is an odd function of T, R (O) = u(0) = 0, and we have


2 -R (0)

R-BB(0)


-RW(0)

R4.(0)


(5.48)


It is noted that this formula is applicable whenever R 1(0) and Ry,.(0)

or R,4(0) and RW (0) exist.








5,4-3. Equivalent-noise bandwidth

The equivalent-noise bandwidth, Aw, for the continuous part of

the power spectrum is defined by


(2Aw) Pg-SC(O)] 2= Pg-SC(w)d = Rg-SC(O)
-00


(5.49)


But


PgSC(O) = f Rg-SC(t)dT
COO


Thus


(AoW) =


1
Rg-SC(O)


f Rg-SC(r)dT
-00


Substituting for Rg-SC(T) by using Eq. (5.41) or Eq. (5.42) we

obtain (noting once again that R (T) is even and R (r) is odd)



TT i7
(Aw) (5.50)
1 f- R,,(T)dT R1 f Rv(T)dr
R(0) Rw() )



5.5. Efficiency

A commonly.Used definition of efficiency for modulated signals

is [18]


n = Sideband Power/Total Power.


I -


(5o51)








This definition will be used to obtain a formula expressing the efficiency
for the generalized SSB signal. Using Eq. (5.43) and Eq. (5.44) the side-
band power in either the USSB or LSSB signal is


RXU-SC(O) = RXL-SC(O) = RWo(O) = RW(O) (552)


It is also noted that Rg-SC(O) is not equal to the total power in the
real-signal sidebands since gSC is a complex (analytic) baseband signal;
instead, (l/2)Re[Rg_sc(O)] : Rwu(O) = Rwv(O) is the total real-signal
power. This is readily seen from Eq. (5.43a) and Eq. (5.44a).
Similarily the total power in either the USSB or LSSB signal is
obtained from Eq. (5.37) or Eq. (5.38):


Rxu(O) = RXL(O) = [kl2 + k2 + 2R4(0)]

= 1[k 2 + k22 + 2R (O)] (5.53)


Thus the efficiency of a SSB signal is


2R 4U(0) 2RV(0)
S= (5.54)
k 2 + k2 + 2R (0) k2 + k2 + 2R4.(0)
1 2 +12R2



5.6. Peak-to-Average Power Ratio
The ratio of the peak-average (over one cycle of the carrier-
frequency) to the average power for the SSB signal may also be obtained.
The expression for the peak-average power over one carrier-
frequency cycle of a SSB signal is easily obtained with the aid of
Eq. (3.5) and Eq. (3.8). Recalling that U and V are relatively slow








time-varying functions compared to cos mot and sin mot, we have for the
peak-average power:


P pv {[U(m(t),m(t))] + [V(m(t),m(t))]l}
pt i tpeak


(5.55)


where tpeak is the value of t which gives the maximum value for Eq. (5.55).
Using Eq. (5.20) and Eq. (5.21), Pp-Av can also be written as


Pp-Av -= {[ + k 2 + [+ k i]2 t = tpeak
2 2
= {[U-+ k2] + [*-+ kj] t tpeak
peak

= {[-V-+ k21 + [2 + kI t
peak .

The average power of the SSB signal was given previously by Eq, (5.53).
Thus the expression for peak-to-average power ratio for the generalized
SSB signal is

Pp-Av {[U(m(t),m(t))] + [V(m(t),m(t))] It = t (556a)
-- peak
PAv k 2 + k 2 + 2R (0)
2 o6 (2
{[U(m(t)m(t))] + [V(m(t),m(t))] t = tpeak (556b)
k2 + k2 + 2Rv (0)

{[U(m(t),m(t))+k2]2 + [4(m(t),m(t))+k,]2} it tpeak (5 56c)
k 2 + k22 + 2R.4(0)
{[-V-(m(t),m(t))+k2]2 + [V-(m(t),m(t))+k]21}
= ____________________t tpeak. (5.56d)
k, 2 + k22 + 2R .(0)

Several equivalent representations have been given for peak-to-average
power since one representation may be easier to use than another for a
particular SS1B signal.












CHAPTER VI

EXAMPLES OF SINGLE-SIDEBAND SIGNAL ANALYSIS


The examples of SSB signals that were presented in Chapter IV

will now be analyzed using the techniques which were developed in

Chapter V.


6.1. Example 1: Single-Sideband AM With Suppressed Carrier

The constants kI and k2 will first be determined to show that

indeed we have a suppressed carrier SSB signal. By substituting

Eq. (4.2b) into Eq. (5.4) we have

Tr
k = lim m(R cos e,R sin e)de
0

But from Eq. (5.18b) it follows that


lim m1(R cos e,R sin e) = 0 0 < e < .
R+->


Thus

k, = 0 (6.1)

Similarily substituting Eq. (4.2a) into Eq. (5.5) we have




k = lim P m(R cos e,R sin e)de = 0 (6.2)
2 R-wc J










since lim m(R cos e,R sin e) = 0 for 0 < e < x from Eq. (5.18a). Further-
R-x
more, since both k and k are zero, the equivalent realizations for the

SSB signals, as given by the equations in Section 5.1, reduce identically

to the phasing method of generating SSB-AM-SC signals (which was given

previously in Figure 8).

The autocorrelation for the SSB-AM-SC signal is readily given by

use of Eq. (4.2a) and Eq. (5.20). Thus


U-(m(t),m(t)) = m(t). (6.3)


Then the autocorrelation of the suppressed-carrier USSB-AM signal is

given via Eq. (5.43b), and it is


RXU-SC-SSB-AM() = Rmm () cos Wmo Rmm(T) sin wmo. (6.4)



Likewise, by use of Eq. (5o44b) the autocorrelation for the suppressed-

carrier LSSB-AM signal is


RL-SC-SSB-AM() = Rmm(T) cosw + mm(t) sin wor. (6.5)



From Eq. (6.4) it follows that the spectrum of the USSB-AM-SC

signal is just the positive-frequency spectrum of the modulation shifted

up to a0 and the negative-frequency spectrum of the modulation shifted

down to -wo. That is, there is a one-to-one correspondence between the

spectrum of this SSB signal and that of the modulation. This is due to

the fact that the corresponding entire function for the signal, g(W) = W,

is a linear function of W. Consequently, the bandwidths for this SSB









signal are identical to those for the modulation. This is readily shown

below.


The mean-type bandwidth

is given by use of Eq. (6.3) in




MSSB-AM


where 'm = Rmm(0), the power in

the rms bandwidth is


(when the numerator and denominator exist)

Eq. (5.46):


Rmm(O) Rmm(O)
(6.6)
Rmm(O) m


the modulating signal. By using Eq. (5.48)




M = (0) (6.7)


whenever Rmm(O) and 'm exist. By using Eq. (5.50) the equivalent-noise

bandwidth is



(Am)SSBAM = (6.8)
f- Rmm(r)dT



Thus the bandwidths of the SSB-AM-SC signal are identical to those of the

modulating process m(t).
The efficiency of the SSB-AM-SC signal is obtained by using

Eq. (5.54):


2Rmm(0)
nSC-SSB-AM -
2Rmm(0)


(6.9)










The peak-to-average power ratio for the SSB-AM-SC signal follows

from Eq. (5.56c), and it is


Pp-Av {[m(t)]2 + [m(t)]2}t = tpeak

PAV /SC-SSB-AM 2*m


(6.10)


6.2. Example 2: Single-Sideband PM

The SSB-PM signal has a discrete carrier term. This is shown by

calculating the constants k, and k2. Substituting Eq. (4.5b) into

Eq. (5.4) we have


k I lim e-m,(R cos e,R sin e)sin [m,(R cos e,R sin e)]de.
0


But from Eqs. (5.18a) and (5.18b) lim m1(R cos e,R sin e) = 0 for
R-o
0 _s e < T and lim m (R cos e, R sin e) = 0 for 0 e rr. Thus
R--o


ki = 0. (6.11)


Likewise, substituting Eq. (4,5a) into Eq. (5.5) we have


k = e-0 cos 0 de = 1. (6.12)
IT J
0


Thus the SSB-PM signal has a discrete carrier term since k2 = 1 f 0.

There are equivalent representations for the SSB-PM signal since

k and k2 are not both zero. For example, for the upper sideband signal,

equivalent representations are given by Egs. (5.7) and (5.8). It is

noted that Eq. (5.8) is identical to Eq. (5.9) for the USSB-PM signal









since k, = 0. Thus the two equivalent representations are:



XUSSB-PM(t) e= e(t)cos m(t)] cos Wot e acos m(t)] sin wot (6.13)


and


XUSSB-PM(t) =-(e-(t)sin m(t))+1]cos wot [e-m(t)sin m(t)]sin ot. (6.14)



The USSB-PM exciters corresponding to these equations are shown in Figure
13 and Figure 14. They may be compared to the first realization method
given in Figure 9.

The autocorrelation function for the SSB-PM signal will now be

examined. In Section 6.1 the autocorrelation for the SSB-AM-SC signal

was obtained in terms of the autocorrelation function of the modulation.

This was easy to obtain since 4 = m(t). However, for the SSB-PM case 4

and Vare non-linear functions of the modulation m(t). Consequently, the

density function for the modulation process will be needed in order to

obtain the autocorrelation of the SSB-FM signal in terms of Rmm(T).
To calculate the autocorrelation function for the SSB-PM signal,

first RV_(r) will be obtained in term of Rmm(T). Using kL = 0, Eq.(5.21),

and Eq. (4.5b) we have


V(m(t),m(t)) = V(m(t),m(t)) = e-m(t) sin m(t). (6Jo5)

Then

Rm jm(t) -jm(t) -Jm(t-)- jm(t-r)
R (1) = e ep-m(t-T) e-2j
W e 2, e-2/_














m(t)
Modulating Input


jcos -.t


XUSSB-PM(t)


OutDut


USSB-PM Signal Exciter--Method II


e4(t) cos m(t)


Figure 13.











m(t)
Modulating Input


e-m(t) sin m(t)


(e sin m(t)) Balanced [-(e tsin m(t})+1]cos wot
Modulator
+ -
DC Level
of +1


RF Oscillator
at mro


i sin wot
-90 Phase _
~ shift at wO


XUSSB-PM(t)

Modulated
RF Output


[e-&(t) sin m(t)]sin


Figure 14. USSB-PM Signal Exciter--Method III









or


R(r) ej[xi(tr)+jy(t)+jy )] ej[X2(tT)+jy(tT)]


eJ[x3(t,.)+jy(t,T)] + ej[x4(t,T)+jy(t,T)] (6.16)


where x1(t,r) = m(t) m(t-T.)
x2(t,T) m(t) + m(t-r)
x3(t,t) -m(t) m(t-T) = -x2(t,T)
x4(t,T) E -m(t) + m(t-t) = -x (t,T)
y(t,r) H i(t) + i(t-T).
Now Zet the modulation m(t) be a stationary Gaussian process with zero mean.
Then x1(t,T), X2(t,T), X3(t,T), x11(t,r), and y(t,T) are Gaussian processes
since they are obtained by linear operations on m(t). They are also stat-
ionary and have a zero mean value. It follows that x (t,T), y(t,r);

x2(t,i), y(t,7); x3(t,T), y(t,T); and x4(t,r), y(t,r) are jointly Gaussian
since the probability density of the input and output of a linear system
is jointly Gaussian when the input is Gaussian [15]. For example, to show
that x,(t,T) and y(t,r) are jointly Gaussian, a linear system with inputs
m(t) and m(t-r) can readily be found such that the output is y(t,r). Now
the averaging operation in Eq. (6.16) can be carried out by using the fol-
lowing formula which is derived in Appendix II:


eJ{x(t)+jy(t)} = e-{x +j2Pxy-ay2} (6.17)

where x(t) and y(t) are jointly Gaussian processes with zero mean,

x2 = x2(t)

y2 = 2(t)











uxy = x(t)y(t) .


Thus


2 2---- -7
O' [m(t)-m(t-t)] = 2[am -Rmm(T)]


ax r[m(t)+m(t-r)]2 2[om2+Rmm(T)]


2
OX3


[-m(t)-m(t-T)]= 2[ om2+Rmm( )]


x 2 = [-m(t)+m(t-t)]2= 2[am2-Rmm()]


and


y 2 [m(t)+m(t-1i)] = 2[cm2+Rmm([)] .


From Chapter II
-Rmm() so that


it is recalled that Rm(0) = 0 and Rm(i) = -Rlm() =


the p averages are


= [m(t)-m(t-r)][m(t)+m(rt-T)]


= -2Rmm()


[m(t-(tT( =


Sx3Y


and


X y = -[m(t)-m(t-T)][m(t)+m(t-r)] 2Rmm(T) .


Ix y
I


Ix2 =









Therefore, using Eq. (6.17), Eq. (6.16) becomes


R (T) = e-{2[am -Rmm(T)] + j2[-2Rmm(T)] 2[om2+Rmm(T)]}


e-2{2[m2 +Rmm(i)] + j2.0 2[am2+Rmm(T)]}


.- e-{2[m2+Rmm(,)] + j2 2[a m2+Rmm(T)]}


+ 1 e-{2[am2 -Rmm(T)] + j2[2Rmm(l)] 2[Om2+Rmm(T)]}

which reduces to


RVW-SSB-PM-GN(r) = {e2Rmm(T) cos (2Rmm(r)) 1} (6.18)

where W -SSB-PM-GN denotes the autocorrelation of the imaginary part of
the entire function which is associated with the suppressed-carrier SSB-
PM signal with Gaussian noise modulation.
It is noted that Eq. (6.18) is an even function of i, as it should
be, since it is the autocorrelation of the real function V(m(t),6(t)).
Furthermore R (0) is zero when Rmm(O) = 0, as it should be, since the
power in any suppressed-carrier signal should be zero when the modulating
power is zero.
The autocorrelation of the USSB-PM signal is now readily obtained
for the case of Gaussian noise modulation by substituting Eq. (6.18) into
Eq. (5.42) and using Eq. (5.37):

RXU-SSB-PM-GN(T) = Re eJ0 T{[e2Rmm() cos (2Rmm(())]


+ j[e Rmm T)cos (2Rmm(T)]}] (6.19)









Likewise, the autocorrelation of the LSSB-PM signal may be obtained by
using Eq. (5.38).
The autocorrelation of the suppressed-carrier USSB-PM signal with
Gaussian modulation is given by using Eq. (5.43a):


RXU-SC-SSB-PM-GN(t) = Re [eJ m([e2Rmm(t) cos (2Rmm(r)) 1]


+ j[e2Rmm( cos (2Rmm(r)]} (6.20)

Similarly, the autocorrelation of the suppressed-carrier LSSB-FM signal
may be obtained by using Eq. (5,44a).
The mean-type bandwidth will now be evaluated for the SSB-PM

signal assuming Gaussian noise modulation. From Eq. (6.18) we obtain

001 e2Rmm( ) cos (2Rmm(x)dA
P(tA)2


Then

1 e2Rmm() cos [2Rmm(x)]dx
R V_(O) P 2 (6o21)


and from Eq. (6.18)

R *(O) = [e2%m 1] (6.22)

where m = m2 is the average power of m(t). Substituting Eqs. (6.21)
and (6.22) into Eq,, (5.46) we have the mean-type bandwidth for the
Gaussian noise modulated SSB-PM signal:

17 P e2Rmm(A) cos[2Rmm(A)]dA
(W)SSB-PM-GN (6-23)
e29m -1










where m is the noise power of m(t), It is seen that Eq. (6.23) may or
may not exist depending on the autocorrelation of m(t).

The mis-type bandwidth can be obtained with the help of the second

derivative of Eq. (6.18):


R () {-e2Rmm()

+ {-e2Rmm(r)

+ {-e2Rmm ()

+ { e2Rmm ( )


+ {-e2Rmm(,)

+ { e2Rmm(t)


sin [2Rmm(T)]} 2[Rmm(T)]2

cos [2Rmm(T)]J 2[Rmm(T)l


sin [2Rmm(T)]}


Rmm ()


cos [2Rmm(I)]} 2[Rmm(i)]2


sin [2Rmm(T)]}

cos [2Rmm(T)]}


Imm
Rmm (T)


RW(O) = e2m {Rmm(0) 2[Rmm(0)]2}


Substituting Eq. (6.24) and Eq. (6.22) into Eq. (5.48) we have for the
rms-type bandwidth of the SSB-PM signal with Gaussian noise modulation:


/2(2[Rnm(O)12 Rm(O)}


6(; ?q)


v'rmsJssB-PM-GN 1 e-2m


This expression for the rms bandwidth may or may not exist depending on

the autocorrelation of m(t). It is interesting to note that Mazo and Salz

have obtained a formula for the rms bandwidth in terms of different para-
meters [19]. However both of these formulae give the same numerical re-
sults, as we shall demonstrate by Eqo (6.29).


Thus


(6.24)


2RMM(T)MM(T)


t \1 A
ll ?










The equivalent-noise bandwidth is obtained by substituting

Eq. (6.18) into Eq. (5.50):


(AW) =

1 e2Rmm() cos [2Rmm(T)] 1} dT
[e22m-l].

or


'T(e2m 1)
(Am)SSB-PM-GN = (6.26)

S{e2Rmm(T) cos [2Rmm(T)] l}dT


It is noted that the equivalent-noise bandwidth may exist when the formu-

lae for the other types of bandwidth are not valid because of the non-

existence of derivatives of Rmm(t) at T = 0.

It is obvious that the actual numerical values for the bandwidths

depend on the specific autocorrelation function of the Gaussian noise.

For example, the rms bandwidth of the SSB-PM signal will now be calculated

for the particular case of Gaussian modulation which also has a Gaussian

spectrum. Let
2
-w)

Pm(w) = e 232


where Pm(w) is the spectrum of m(t)
no = m is the total noise power in m(t)

2a is the "variance" of the spectrum.

The autocorrelation corresponding to this spectrum is


Rmm(1) = e 0, (6.27)








The Hilbert transform of Rmm(T) is also needed and is obtained by the

frequency domain approach. It is recalled from Chapter II that


P () =
mm


-j Pmm()

0

j Pmm


S> 0


Then


SRmm(T) ~ = T)



P .1
2i


00

f Pim(w) ejwt dw

0 -,2 0 -W
S e2 ej de J e2-' e2a ejTd]
f f
-0 0


which reduces to


w 2




integral is evaluated by using the
of the Bateman Manuscript Project,
1 [5]:
S 2 -

2V"
o


sin wT dm .


formula obtained from page 73,

Tables of Integral Transforms,



Erf Re a > 0
\2v/a /


Erf (x) e't dt.
O


This

#18,

vol.


where


a < 0







Thus


Rmm J) (- 0 e e-22T2) Erf (-4 2)




Rmm(T) = -j Rmm() Erf or .
\/j2 /


(6.28)


From Eao. (6.27) it follows that

Rmm(O) = -oo2

and from Ea. (6.28) we have

Rim(O) = 2
/27

Substituting these two equations into Ea. (6.25) we get



(Grms) 2, -e-2o--

Thus if m(t) has a Gaussian spectrum and if the modulation has a Gaussian
density function, the SSB-FM signal has the rms bandwidth:


20o2 [4( oo/n) + ]
("rms)SSB-PM-GN = l -2' -- (6.29)


where *o is the total noise power in m(t)
a'is the "variance" in the spectrum of m(t).










This has the same numerical value as that obtained from the result given

by Mazo and Salz [19]. The result may also be compared to that given by
Kahn and Thomas for the SSB-PM signal with sinusoidal modulation [20].
From Eq. (19) of their work


(wrms)ssB-PM-S = wa6 (6.30)


where ma is the frequency of the sinusoidal modulation and 6 is the modu-
lation index. For comparison purposes, equal power will be used for m(t)

in Eq. (6.30) as was used for m(t) in Eq. (6.29). Then Eq. (6,30) becomes


(wrms)SSB-PM-S = /2 wa 'u (6.31)


Thus it is seen that for Gaussian modulation the rms bandwidth is propor-

tional to the power in m(t) when the power is large (%o > > T/4), and for
sinusoidal modulation the rms bandwidth is proportional to the square root
of the power m(t).
The efficiency for the SSB-PM signal with Gaussian modulation

will now be obtained. Substituting En. (6.22) into Eq. (5.54) we have


e2m-_l
tSSB-PM-GN = + (e2m-1l)


or

SSB-PM-N e2m (6.32)


where 1m is the noise power of m(t).
The peak-average to average power ratio for Gaussian m(t) is given








by use of Eas. (4.5a), (4.5b), and (6.22) in Ea. (5.56b):


Pv
VAv


{[e-m(t) cos m(t)] [el(t) sin m(t)]2 t tpeak
1 + (e-2mm-l)


(6,33)


Note that m(t) may take on large negative values because it has a Gaussian
density function (since it was assumed at the outset that the modulation
was Gaussian), However, it is reasoned that for all practical purposes,
m(t) takes on maximum and minimum values of +3am and -3am volts where am
is the standard deviation of m(t). This approximation is useful only for
small values of am since e+2(3am) approximates the peak power only when the
exponential function does not increase too rapidly for larger values of am.
Thus the peak-to-average power ratio for the SSB-PM signal with Gaussian
noise modulation is


-Av
( v )SSB-PM-GN


Se6am e6/im-2 m
e -


when 'm is small.
It is noted that the efficiency and the peak-to-average power
ratio depend on the total power in the Gaussian modulation process and not
on the shape of the modulation spectrum. On the other hand the autocorre-
lation function and bandwidth for the SSB signal depend on the spectral


(6.34)










shape of the modulation as well.
The dependence of bandwidth on the spectrum of the Gaussian noise
modulation will be illustrated by another example. Consider the narrow-

band modulation process:


m(t) = a(t) cos (wat + 4) (6.35)


where a(t) is the (double-sideband) suppressed-subcarrier amplitude
modulation
ma is the frequency of the subcarrier
( is a uniformly distributed independent random phase due to
the subcarrier oscillator.
That is, we are considering a SSB signal which is phase modulated by the

m(t) given above. Then


Rmm(T) = Raa(f) cos wa{ (6.36)

where Raa(T) is the autocorrelation of the subcarrier modulation a(t).

Rmm(T) can be obtained from Eq. (6.36) by use of the product theorem [21].
Thus, assuming that the highest frequency in the power spectrum of a(t)
is less than wa,


Rmm(T) = Raa() sin aT (6.37)


Furthermore let a(t) be a Gaussian process; then m(t) is a
narrow-band Gaussian process. This is readily seen since Eq. (6.35) may
be expanded as follows:

m(t) = [a(t) cos (wat+q) a(t) sin (wat+f)]

+ [a(t) cos (a t+p) + a(t) sin (wat+4)] (6.38)








The terms in the brackets are the USSB and LSSB parts of the suppressed-

subcarrier signal m(t). But these USSB and LSSB parts are recognized

as the well-known representation for a narrow-band Gaussian process.

Thus m(t) is a narrow-band Gaussian process.

Now the previous expressions for bandwidth, which assume that

m(t) is Gaussian, may be used. The mean-type bandwidth for the multi-

plexed SSB-PM signal is then readily given via Eq. (6.23), and it is
00
eRaa(X) cos wax cos[Raa(x) sin wax]dA
-________ ______________ (6.39)
(u)M-SSB-PM-GN ~ e8a 1


where ag is the average power of the Gaussian distributed subcarrier

modulation a(t). Obtained in a similar manner, the rms bandwidth is


Wa2 ( a+1) Raa(0)
(wrms)M-SSB-PM-GN = e*a (6.40)



and the equivalent-noise bandwidth is



f[e2a-1]
(Aw)MSSB-PMN = 0 (6.41)
f eRaa(T) cos waT cos[Raa(r) sin waT]
-00


Thus, it is seen once again that the bandwidth depends on the spectrum of
the modulation, actually the subcarrier modulation a(t).

To obtain a numerical value for the rms bandwidth of the multi-

plexed SSB-PM signal assume that the spectrum of a(t) is flat over

a Wo < w"a









67


Pa()M



0 W W-+
Figure 15. Power Spectrum of a(t)


From Figure 15 we have



-Wo

or

NoWo sin Wo.
Raa(r) (6.42)


and

0a 0 (6.43)
IT

Then

Raa(O) (6.44)
31
Substituting the last two equations into Ea. (6.40) we obtain the rms

bandwidth for the SSB-PM multiplexed signal:


2 NOWO( N0W0 N oWo_
a -- -- +1 + -

(mrms)M-SSB-PM-GN = NoWo/f (6.45)
1 l -e-W/

where ma is the subcarrier frequency

No is the amplitude of the spectrum of the subcarrier Gaussian
noise modulation

Wo is the bandwidth of the subcarrier noise modulation.








Thus the rms bandwidth is proportional to the power in the subcarrier

modulation as No becomes large.


6.3. Example 3: Single-Sideband FM

As was indicated in Section 4.3. the representation for the SSB-FM

signal is very similar to that for the SSB-PM signal. In fact it will be

shown below that all the formulae for the properties of the SSB-PM signal

(which were obtained in the previous section) are directly applicable to

the SSB-FM signal.

The SSB-FM signal has a discrete carrier term since the entire

function for generating the SSB-FM signal is identical to that for the

SSB-PM signal, which has a discrete carrier term.

The other properties of the SSB-FM signal follow directly from

those of the SSB-PM signal if the autocorrelation of m(t) can be obtained

in terms of the spectrum for the frequency modulating signal e(t). It is

recalled from Eq. (4.7) that

t
m(t) = D ef (t')dt', (6.46)
-00

First, the question arises: Is m(t) stationary if e(t) is stationary?

The answer to this question has been given by Rowe; however, it is not

very satisfactory since he says that m(t) may or may not be stationary [22].

However, it will be shown that m(t), as given by Eq. (6.46), is stationary

in the strict sense if e(t) is stationary in the strict sense; and,

furthermore, m(t) is wide-sense stationary if e(t) is wide-sense stationary.

It is recalled that if


y(t) = L[x(t)]







where L is a linear time-invariant operator, then y(t) is strict-sense
stationary if x(t) is strict-sense stationary and that y(t) is wide-sense
stationary if x(t) is wide-sense stationary [4]. Since the integral is a
linear operator, we need to show only that it is time-invariant, that is
to show that


y(t+e) = L[x(t+E]

or


I (tl)dt f e(t2+e)dt2
e(tl)dt1


This is readily seen to be true by making a change in the variable,
letting tI = t2 + e. Thus, if 8(t) is stationary, then m(t) is stationary.
Moreover, in the same way it is seen that if m(t) had been defined by

t
m,(t) = D e,(t')dt' (6.47)
to

then m (t) is not necessarily stationary for e,(t) stationary since the
system is time-varying (i.e. it was turned on at to). But this should
not worry us because, as Middleton points out, aZZll physically realizable
systems have non-stationary outputs since no physical process could
have started out at t = -m and continued without some time variation in
the parameters 05]. However, after the "time-invariant" physical systems
have reached steady-state we may consider them to be stationary processes--
provided there is a steady state. Thus by letting t -m we are con-
sidering the steady-state process m(t) which we have shown to be stationary.
Now the autocorrelation of m(t) can be obtained by using power-spectrum
techniques since m(t) has been shown to be stationary. From Eq. (6.46)










we have


e(t) dm(t) (6.48)


Then in terms of power-spectrum densities


P ee() = W2Pmm(a) (6.49)

As Rowe points out, Pmm(w) must eventually fall off faster than k/w2,
where k is a constant, if e(t) is to contain finite power; and if Pmm(w) =
k/w2, Pee(w) will be flat and, consequently, white noise. Thus we have a
condition for the physical realizability of m(t): Pmm(w) falls off faster
than -6 db/octave at the high end. This condition is satisfied by physi-
cal systems since they do not have infinite frequency response. From

Eq. (6.49) we have


1
Pmm(W) PeO() (6650)


Immediately we see that if POe() takes on a constant value as IwI + 0
and at w = 0, m(t) will contain a large amount of power with spectral
components concentrated about the origin. In other words, m(t) has a
large block of power, located infinitely close to the origin which is
infinitely large. Thus m(t) contains a slowly varying "DC" term with a
period T and m2(t) -- -. By examining Eqo (6.46) we obtain the same
result from the time domain. That is, for Pee(w) equal to a constant,
e(t) contains a finite amount of power located infinitely close to the
origin which appears as a slowly varying finite "DC" term in e(t) such
that T -. Then by Eq. (6.46), m(t) has a infinite amplitude and,








consequently, infinite power, In other words, the system does not have a

steady-state output condition if the input has a power around w = 0. Thus,

this system is actually conditionally stable, the output being bounded

only if the input power spectrum has a slope greater than or equal to +6

db/octave near the origin (and, consequently, zero at the origin) as seen

from Eq. (6.50). It is interesting to note that for the case of FM, eJm(t)

is stationary regardless of the shape of the spectral density Pee(w). This

is due to the fact that ejm(t) is bounded regardless of whether m(t) is

bounded or not.

From Eq. (6.50) we can readily obtain Rmm(r) for any input process

e(t) which has a bounded output process m(t). Thus

00
Rmm()) : 2 eJWT dw (6.51)
-00

Furthermore, R"m(0), Rmm(M), and Rmm(O) may be obtained in terms of

Pee(w). By substituting for these quantities in the equations of Section

6.2, the properties of a SSB-FM signal can be obtained in terms of the

spectrum of the modulating process.



6.4. Example 4: Single-Sideband a

The SSB-a signal has a discrete carrier term. This is readily

shown by calculating the constants k, and k2. Substituting Eq. (4.10b)

into Eq. (5.4) we have

iT
ki = 'I lim em1(R cos eR sin e) sin am,(R cos e,R sin e)de .
SR_00 f
0

But lim m,(R cos e, R sin e) = 0, for 0 s e s 7 and lim m (R cos e,
R--o R4-
R sin e) = 0 for 0 s e To, Thus

k, = 0. (6.52)










Likewise, substituting Eq. (4.10a) into Eq. (5.5), we have

k2 = 1. (6.53)

Thus the SSB-a signal has a discrete carrier term.
It follows that equivalent representations for the SSB-a signal

are possible since k2 0. This is analogous to the discussion on equiva-

lent representations for SSB-PM signals (Section 6.2) so this subject will
not be pursued further.
The autocorrelation function for the SSB-a signal will now be ob-

tained in terms of Rmm(T)o Using Eq. (5.21) and Eq. (4.10b) we have



RW(T) = [eem(t) sin am(t)][eam(t-[) sin am(t-T)]


or


R (,) = k{ea[m(t)+m(t-T)]} eja[m(t)-m(t-T)] -eja[m(t)+m(t-T)]


+ {eam(t)+m(t-)]} {eJ[-m(t)-m(t-)] + ej[-m(t)+(t-T)}.

(6.54)

The density function of m(t) has to be specified in order to carry out
this average. It is recalled that m(t) is related to the modulating
signal e(t) by the equation:


m(t) = In [l+e(t)] o


Now assume that the density function of the modulation is chosen such
that m(t) is a Gaussian random process of all orders. Eq. (6.54) can
then be evaluated by the procedure that was used to evaluate Eq. (6.16).








Assuming a Gaussian m(t), Eq. (6.54) becomes


RW(T)SSB-a-GN = {e2a2Rmm(T) cos [2a2mm(T)] 1} o (6,55)

But this is identical to Eq. (6.18) except for the scale factor a2.
Thus for envelope detectable SSB (a = 1) with m(t) Gaussian, the auto-
correlation and spectral density functions are identical to those for
the SSB-PM signal with Gaussian m(t). Moreover, the properties are
identical for SSB-a and SSB-PM signals having Gaussian m(t) processes
such that (*m)SSB-PM = a2(m)SSB-a_
It is also seen that if |e(t)l < < 1 most of the time then


m(t) : e(t).

Thus, when e(t) is Gaussian with a small variance, m(t) is approximately
Gaussian most of the time. Then Eq. (6.55) becomes



RW(t)SSB-a-GN {e22Ree () cos [2a2Ree(f)] 1} (6.56)

when le(t)l < < 1 most of the time. Consequently, formulae for the auto-
correlation functions analogous to Eqs. (6.19) and (6.20), may be further
simplified to a function of Ree(i) instead of Rmm(t). Then the auto-
correlation functions for USSB-a and LSSB-a signals, assuming Gaussian
modulation e(t) with a small variance, are



RXU-SSB-a-GN(r) Re ejo'T {[e2a2Re(T) cos (2otee(T))]


+ j RLe2a2Ree() cos (2a2Reo())]} (6)57)










and


RXL-SSB-a-GN(,) Re e-jwo([e2a2Ree(T) cos (2c2(ee(T))]


+ j [e22Ree) cos (2W2 ee(r))]} (6.58)


The efficiency is readily obtained by substituting Eq. (6.56)

into Eq. (5.54):



nSSB-a-GN = 1 e"22m (659)

where Pm is the power in the Gaussian m(t) and le(t)| < < 1. This result
may be compared for a = 1 to that given by Voelcker for envelope-detectable

SSB(a = 1) [11]. That is, if e(t) has a small variance, then m(t) = e(t);
and Eq. (6.59) becomes


nSSB-a-GN 1. 1 e-2e2 z 20e2, (6o60)

This agrees with Voelcker's result (his Eq. (38)) when the variance of the
modulation is small--the condition for Eqo (6o60) to be valid.
The expressions for the other properties of the SSB-a signal, such
as bandwidths and peak-to-average power ratio, will not be examined further
here since it was shown above that these properties are the same as those
obtained for the SSB-PM signal when (Wm)SSB-PM '= 2(m)SSB-a as long as
m(t) is Gaussian.















CHAPTER VII

COMPARISON OF SOME SYSTEMS



In the two preceding chapters properties of single-sideband sig-

nals have been studied. However, the choice of a particular modulation

scheme also depends on the properties of the receiver. For example,

the entire function g(W) = W2 can be used to generate a SSB signal, but

there is no easy way to detect this type of signal.

In this chapter a comparison of various types of modulated sig-

nals will be undertaken from the overall system viewpoint (i.e. generation,

transmission and detection). Systems will be compared in terms of the

degradation of the modulating signal which appears at the receiver out-

put when the modulated RF signal plus Gaussian noise is present at the

input. This degradation will be measured in terms of three figures of

merit:

1. The signal-to-noise ratio at the receiver output

2o The signal energy required at the receiver input for

a bit of information at the receiver output when com-

arison is made with the ideal system (Here the ideal

system is defined as a system which requires a minimum

amount of energy to transmit a bit of information as

predicted by Shannon's formula.)

3. The efficiency of the system as defined by the ratio

of the RF power required by an ideal system to the RF

power required by an actual system.(Here the ideal sys-

75










tem is taken to be a system which has optimum trade-off

between predetection signal bandwidth and postdetection

signal-to-noise ratio.)
Comparison of AM, FM, SSB-AM-SC, and SSB-FM systems will be made using

these three figures of merit. It is clear that these comparisons are

known to be valid only for the conditions specified; that is, for the

given modulation density function, and detection schemes which are used

in these comparisons.


7.1. Output Signal-to-Noise Ratios

7.1-1. AM system

Consider the coherent receiver as shown in Figure 16 where the

input AM signal plus narrow-band Gaussian noise is given by


X(t) + ni(t) = {Ao[l+6 sin wmt] cos wot}

+ {xc(t) cos )ot xs(t) sin wmot (7.1)

where X(t) is the input signal, ni(t) is the input noise with a flat spec-

trum over the bandwidth 2wm, and 6 is the modulation index.

AC Couple
X(t)+ni(t) Low Pass ___ Output


2k cos cot

Figure 16. AM Coherent Receiver

Then the output signal-to-noise power ratio, where Aok6 sin wmt is the
output signal, is given by

(S/N)o (S/N)i (7.2)
1 + _26









or

(S/N)o 62(C/N)i, (7.3)

where (S/N)i = The input signal-to-noise power ratio

(C/N)i = The input carrier-to-noise power ratio

and the spectrum of the noise is taken to be flat over the IF bandpass

which is 2wm(rad/s).


7.1-2, SSB-AM-SC system

Consider the coherent receiver (Figure 16) once again, where

the input is a SSB-AM-SC signal plus narrow-band Gaussian noise. Then

the input signal plus noise is


X(t) + ni(t) = {Ao[m(t) cos wot m(t) sin wot]}

+ [xc(t) cos wot Xs(t) sin mot] (7.4)

where

m(t) = 6 sin mt

and xs(t) = xc(t) if the IF passes only upper sideband components. The

input noise is assumed to have a flat spectrum over the bandwidth m.o

Then the output signal-to-noise power ratio, where Ak6 sin wmt is the

output signal, is given by [23]


(S/N)o = (S/N)i (7.5)

where the spectrum of the noise is taken to be flat over the IF bandpass

which is wm(rad/s)o

It is interesting to note that the same result is obtained from a









more complicated receiver as given in Figure 17. However, in some practi-

cal applications the receiver in Figure 17 may give much better perform-

ance due to better lower sideband noise rejection. That is, in Figure 17

the lower sideband noise is eliminated as the result of the approximate

Hilbert transform filter realized about w = 0; whereas, in Figure 16 the

lower sideband noise is rejected by the IF filter realized about w = mo.

Thus, in order to obtain equal lower sideband noise rejection in both

receivers, the IF bandpass for the receiver in Figure 16 would have to

have a very steep db/octave roll-off characteristic at w = wo.




Low Pass
Filter

X(t)+ni(t) + Output
2k cos wot
2k sin mot +

Low Pass Hilbert
Filter Filter


Figure 17. SSB-AM-SC Receiver


7.1-3. SSB-FM system

Now consider a FM receiver which is used to detect a SSB-FM sig-

nal plus narrow-band Gaussian noise as shown in Figure 18.



X(t)+ni(t) FM Output
Receiver


Figure 18. SSB-FM Receiver









The input signal plus noise is given by


X(t) + ni(t) = Aoe-(t) cos [wot + m(t)] + ni(t)


(7.6)


where Ao = The amplitude of carrier

o =s The radian frequency of the carrier
m(t) = D _t v(t) dt

m(t) = m(t) = The Hilbert transform of m(t)
ni(t) = Narrow-band Gaussian noise with power spectral density Fo
over the (one-sided spectral) IF band
and v(t) is the modulation on the upper SSB-FM signal. The independent
narrow-band Gaussian noise process may be represented by


ni(t) = R(t) cos [mot + (t)j = xc(t) cos wot xs(t) sin wot


where xs(t) = xc(t) since the IF passes only the frequencies on the upper
sideband of the carrier frequency.

Then the phase of the detector output is obtained from Eq. (7.6)
and is


p(t) = k tan-1


(7.7)


which reduces to


i(t) = km(t)


+ k tan- K


R(t) sin [p(t) m(t)]


(7,8)


where k is a constant due to the detector. The detector output voltage
is given by dt)- o Eq. (7.8) is identical to the phase output when the
input is conventional FM plus noise except for the factor e -(t)









For large input signal-to-noise ratios (i.e. Aoe-m(t) > > R(t)
most of the time), Eq. (7.8) becomes

kR(t)
p(t) = km(t) + ---- sin [p(t) m(t)] (7.9)
Aoe-m(t)
dno(t)
Then the noise output voltage is -d--- where

kem(t)
n (t) = R(t) sin [<(t) m(t)]. (7.10)
Ao

Now the phase p(t) is uniformly distributed over 0 to 2r since the input
noise is a narrow-band Gaussian process. Then for m(t) deterministic,

[<(t) m(t)] is distributed uniformly also. Furthermore, R(t) has a

Rayleigh density function. Then it follows that R(t) sin [q(t) m(t)]
is Gaussian (at least to the first order density) and, using Rice's
formulation [24, 25],


R(t) sin [p(t) m(t)] = xs(t) = l 2F(n) a sin [(Un- )t + en]
n=l 2i-T

where F(w) = F0 is the input noise spectrum and {en} are independent
random variables uniformly distributed over 0 to 2i. Actually it is
known that the presence of modulation produces some clicks in the out-

put [26], but this effect is not considered here. Eq. (7.10) then be-
comes
kem(t) _____
ke r
no(t) = 2F(wn) -w sin [(Pn-wo)t + On]
Ao n=l 2'







or

dno(t) kem ---
dt Ao nt 1 2F(tn) (wn-wo) cos [(n-mo)t+ On]
dt A0 n f n 2n


ke(t) dm(t) i F(n) Am n]
+ t-- 1 Fn) sin [(n-)t + n
Ao Ldt-] n=1 2Tn

Noting that {on} are independent as well as uniformly distributed and
that the noise spectrum is zero below the carrier frequency, the output
noise power is

dno(t) 2

Wm Wm
k k e2m(t) F + 2 e2m(t) 2dO(t)] id
^-dT2
Ao2 t F 2dw + e 2- Fd

k 2 (t F m3 k2 2m(t) dm(t) F0
2 [ -- 2 m (7.11)
Ao2 2 3 A0 dt 2n

where ,-) is the averaging operator and wm is the baseband bandwidth
(rad/s) o Now let v(t) = -Am cos Wmt then, averaging over t, we have

2e / 2
S2m(tj- wm e26 cos Wmt dt = Io(26) (7.12)

and

e2m(t) ^l(t)2 1 (m6)2 [io(26) 12(26)]

= m 26 I (26) (7.13)
2 m 1








where 6 = DAm/wm, the modulation index


Eq. (7.13) into Eq, (7,11) we obtain for the output noise power


k2Fo 0m3
N 0
0


1 (2


1a'


(7.14)


Referring to Eq. (7.9), the output signal power is


k2 2
= -T (DAm) .
2


dkm(t)
0 dt


Then the output signal-to-noise ratio is


k2
(DAm)2


S Ak2L
2TWn A 02 3


AO2 2


Fo
2--
21T


m I-o(26)


1
+ 61
2


1
+ -
2


i(26)





(7.16)


611(26)


Referring to Eq. (7.6), the signal power into the detector is



Si = Ao2 e2(t) cos2 [.ot + m(t)] = 1 A 2 e2(t)
=A2 e~~~(2


1 2
= Ao
2


Io(26).


Kahn and Thomas have given the ratio of the rms bandwidths (taken about


(7.15)


(S/N), =







(S/N)o =


(7.17)


Substituting Eq. (7.12) and






83

the mean of the one-sided spectrum) for a SSB-FM signal to a conventional

FM signal [20], and it is


BSSB-FM


BFM


(7.18)


I 1 2(26)
102(26)


It is known that the bandwidth (in rad/s) of a


FM signal is approxi-


mately


BFM = 2(6+1)wm.


(7,19)


Thus, to the first approximation, the SSB-FM bandwidth is


BSSB-FM 2


I 2(2 )
2 1 2=


(7.20)


(6+1)w .
m


Then, taking the IF bandwidth to be that of the SSB-FM signal, the input

noise power is


V 0
N B SSB-FM'


(7 21)


Consequently, the input signal-to-noise ratio is


Io(26)


(S/N)i =


(7,22)


mm (6+1) /2


I 2(26)
1- 2)
Io (26)


Fo
4 -
2-u








Using Eq. (7.16) and Eq. (7.22), we have


(7.23)


for the case of SSB-FM plus Gaussian noise into a FM detector.
The signal-to-noise output can also be obtained in terms of the
unmodulated-signal-to-noise ratio (i.e. the carrier-to-noise power at
the input). From Ea, (7.6) we obtain


(S/N)i lo(26) (C/N)i


and Eq. (7.23) becomes

iO2(26)
6 62(6+1) /2 1 ----
(S/N)o = 102(26) (C/N)i
Io(26) + 61,(26)


(7.24)


(7.25)


where (C/N)i is the carrier-to-noise power ratio.


7.1-4. FM system
The signal-to-noise ratio at the output of a FM receiver for a
FM signal plus narrow-band Gaussian noise at the input can be obtained
by the same procedure as used above for SSB-FM. The factor e (t)of
Eq. (7.6) is replaced by unity, and the bandwidth of the input noise
is given by Eqo (7.19). Then the output signal-to-noise ratio becomes


(S/N)o = 3 62(6+1) (S/N)i


6 62(6+1) /2 21 12(6)
(S/N)o = 2 (S/N)i
102(26) + T 61o(26)11(26)


(7.26)








when the input signal-to-noise ratio is large. It is also noted that


(S/N)i = (C/N)i. (7.27)




7.1-5. Comparison of signal-to-noise ratios

A comparison of the various modulation systems is now given by

plotting (S/N)o/(S/N)i as a function of the modulation index by use of

Eqs. (7.2), (7.5), (7.23), and (7.26). This plot is shown in Figure 19.

Likewise (S/N)o/(C/N)i as a function of the modulation index

is shown in Figure 20, where Eqs. (7.3), (7.25), (7.26), and (7.27) are

used. It is noted that in both of these figures the noise power band-

width was determined by the signal bandwidth.

When systems are compared in terms of signal-to-noise ratios, a

useful criterion is the output signal-to-noise ratio from the system

for a given RF signal power in the channel--that is, (S/N)o/Si. This

result can be obtained from (S/N)o/(S/N)i, which was obtained previously

for each system, if the input noise, Ni, is normalized to some convenient
constant. This is done, for example, by taking only the noise power in

the band 2mm (rad/s) for measurement purposes. (The actual input noise

power of each system is not changed, just the measurement of it.) Then

the normalized input noise power for all the systems is

F0
N = 2wm
2n

where the subscript I denotes the normalized power. Then the ratio

(S/N)o/(S/N), is identical to Ni[(S/N)o/Si] where NJ is the constant de-

fined above. Thus, to within the multiplicative constant NJ, comparison

of (S/N)o/(S/N)I for the various systems is a comparison of the output










3.5 ---- I
I


3.0 -
I

FM


(S/N)o f
(S/N)i

I


/ SSB-FM--FM Detection

1.5 5 --

SSB-AM-SC

1 ____ _




0.5 /

/ AM--Coherent
/ Detection


0 0.5 1.0 1.5 2.0 2.5
Modulation Index (6)


Figure 19. Output to Input Signal-to-Noise
Power Ratios for Several Systems




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