MODULATION AND DEMODULATION OF RF

SIGNALS BY BASEBAND PROCESSING

By

JORGE A. CRUZ-EMERIC

A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL

OF THE UNIVERSITY OF FLORIDA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1976

DEDICATION

The author proudly dedicates this dissertation to his

wife, Sara Hilda Quifones de Cruz.

ACKNOWLEDGEMENTS

The author wishes to acknowledge his chairman, Professor

L. W. Couch, for his advice, sincere cooperation, and encour-

agement. Thanks are given to other members of his supervisory

committee and of the staff of the Department of Electrical En-

gineering for their comments and suggestions.

Thanks are given to his wife, Sara, for her patience and

inspiration. She also had the difficult task of typing the

final manuscript.

The author is indebted to the University of Puerto Rico,

Mayaguez Campus, and the University of Florida for their fi-

nancial support during his graduate studies.

iii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ........................................

LIST OF TABLES................. .. *. *. *. .****

LIST OF FIGURES .................. ........... ....

ABSTRACT ................... *...................... .....

CHAPTER

I. INTRODUCTION ...................................

1.1 Problem Statement..........................

1.2 Historical Background......................

1.3 Research Areas.............................

II. MATHEMATICAL PRELIMINARIES ....................

2.1 Introduction...............................

2.2 The Complex Envelope............... .........

2.3 Analytic Functions.........................

2.4 Modulation Techniques........................

2.5 Signal Parameters and Properties...........

2.5.1 Bandwidth ..........................

2.5.2 Dynamic Range.......................

2.5.3 Measure of Error....................

2.6 Equivalence Between Continuous and Discrete

Signals ...................................

2.7 Summary......... ....................... ....

III THE COMPLEX ENVELOPE ............................

3.1 Introduction .......... ....................

Page

iii

viii

ix

xii

1

1

2

5

6

6

7

8

10

12

13

19

19

21

22

23

23

Page

CD

3.2 The Complex Envelope for Different Types

of Modulation............... .. ..... ...

3.2.1 Amplitude Modulation (AM).........

3.2.2 Linear Modulation (LM)............

3.2.3 Angle Modulation (XM).............

3.2.4 Compatible Single Sideband

Modulation (CSSB) ................

3.3 Autocorrelation Function of the Complex

Envelope Components .....................

3.3.1 General Expressions for the

Autocorrelation Function.........

3.3.2 Special Case: Analytic Complex

Envelope .........................

3.3.3

3.3.4

3.3.5

3.3.6

3.3.7

3.3.8

3.3.9

3.3.10

3.4 Second

3.4.1

3.4.2

3.4.3

3.4.4

3.4.5

3.4. 6

3.4.7

AM ...............

DSB-LM...........

SSB-LM...........

PM ...............

FM*...,...,.o..,*

CSSB-AM. .........

CSSB-PM...........

CSSB-FM.........

Moment Bandwidth

AM ...............

DSB-LM...........

SSB-L ..........

PM ...............

FM ..............

CSSB-AM ... .......

CSSB-PM..........

v

..oooo

..o.o.

.O...O

oi***o

o**oo*

of the

ooooo.

o.....

o.oooo

oooooo

****o*

o.oo.o

.o..oe

***********

.....o..coo

ooooooo.o..

**oo*****o*

*o**o*o***o

oeoo..o*oo.

....log.gee

Components

...........

...........

*oo********

*o*oooooooo

...........

ooooo*oo***

oooeooeooo

3.4.8 CSSB-FM ...........................

3.5 Summary................ ............. .....

IV. THE UNIVERSAL TRANSMITTER....................

4.1 Introduction.............. ... ...... ......

4.1.1 The Complex Envelope as a Vector..

4.1.2 Physical Modulators...............

4.1.3 The AM/PM Modulator ..............

4.1.4 The Quadrature Modulator...........

4.1.5 The PM/PM Modulator...............

4.1.6 Other Configurations ..............

4.1.7 Criteria for Comparison and

Evaluation of the Modulators.....

4.2 The AM/PM Modulator........ .............

4.2.1 Bandwidth Requirements............

4.2.2 Dynamic Range......................

4.3 The Quadrature Modulator.................

4.3.1 Bandwidth Requirements............

4.3.2 Dynamic Range......................

4.4 The PM/PM Modulator......................

4.4.1 Bandwidth Requirements............

4.4.2 Dynamic Range ....................

4.5 Discussion of Results...................

4.5.1 The AM/PM Modulator...............

4.5.2 The Quadrature Modulator...........

4.5.3 The PM/PM Modulator...............

4.5.4 Which One is Better?...............

4.6 Physical Realization of the Easeband

Preprocessors ...... .....................

vi

Page

54

54

56

56

57

59

62

65

65

67

72

77

77.

85

90

90

101

105

110

113

117

119

120

122

122

123

4.6.1 Continuous-Time...... ...............

4.6.2 Discrete-Time Continuous-Amplitude..

4.6.3 Discrete-Time Discrete-Amplitude...

4.7 Summary ......... ............... ..........

V. THE UNIVERSAL RECEIVER ......................

5.1 Introduction..............................

5.1.1 Demodulation as the Inverse of

Modulation ...................... ..

5.1.2 The Synchronization Problem.........

5.1.3 Types of Physical Demodulators......

5.1.4 Coherent Universal Demodulators....

5.1.5 Incoherent Universal Demodulators..

5.1.6 Criteria for Comparison and

Evaluation of Demodulators.........

5.2 The AM/PM Demodulator.........................

5.3 The Quadrature Demodulator................

5.4 The AM/FM Demodulator......................

5.5 Practical Considerations and Comparison of

Demodulators ..............................

5.6 Summary .................................

VI. CONCLUSIONS ................................. ...

APPENCICES

A. AUTOCORRELATION FUNCTION OF x(t)/x(t)..........

B. COMPUTER PROGRAMS.. ..... ........... ........... .

REFERENCES .... ................ ....................... .

BIOGRAPHICAL SKETCH .............. .....................

vii

Page

123

124

124

129

131

131

132

134

136

140

144

144

145

152

156

159

163

164

167

172

178

183

LIST OF TABLES

Table Page

III-1. Complex Envelopes.. ............................ 31

IV-1. Magnitude and Phase Functions for the AM/PM

Modulator........... ..... ........... ........ 78

IV-2. Equivalent-Filter Bandwidth Requirements for

the a(t) and p(t) Signals in the AM/PM

Modulator ............................. ....... 84

IV-3. Quadrature Component Functions for the

Quadrature Modulator.......................... 91

IV-4. Equivalent-Filter Bandwidth Requirements for

the i(t) and q(t) Signals in the Quadrature

Modulator........ ......... .................. 102

IV-5. The X(t) and p(t) Functions for the PM/PM

Modulator................................... 109

IV-6. Equivalent-Filter Bandwidth Requirements for

the X(t) and p(t) Signals in the PM/PM

Modulator . . . ........ .... ......... ..... 115

viii

LIST OF FIGURES

Figure Page

1.1. The proposed universal transmitter............. 3

1.2. The proposed universal receiver................ 3

2.1. Procedure used to calculate the equivalent-filter

bandwidth......... ........ ................... 16

4.1. Block diagram of the universal transmitter..... 57

4.2. Possible vector representations of the complex

envelope, (a) the complex envelope vector,

(b) polar components, (c) quadrature components,

(d) two amplitude-modulated vectors, and (e) two

phase-modulated vectors....................... 58

4.3. The balanced modulator......................... 62

4.4. The AM/PM modulator............................ 64

4.5. The improved AM/PM modulator................... 64

4.6. The quadrature modulator....................... 66

4.7. The PM/PM modulator............................ 66

4.8. Meewezen's independent sideband modulation in

terms of the quadrature components............ 71

4.9. Method used to calculate the equivalent-filter

bandwidth ........... ..... ............ .... 74

4.10. Magnitude of the spectrum of the test message

signal............ ... ....................... 75

4.11. IM distortion of the AM/PM modulator a(t) and

p(t) functions for DSB-LM and SSB-LM as a

function of Bf................................ 80

4.12. Sideband suppression factor (S(%)) of the AM/PM

modulator a(t) and p(t) functions for SSB-LM

as a function of Bf........................... 82

4.13. Relation between S(%), Bf, and m for the CSSB-AM

p(t) function for the AM/PM modulator......... 83

ix

4.14. Relation between S(%), Bf, and Dp for the

CSSB-PM a(t) function for the AM/PM modulator. 83

4.15. Relation between Bf, D, and S(%) for CSSB-FM

a(t) function for the AM/PM modulator......... 84

4.16. Relation between Bf, Dp, and IM distortion for

the PM quadrature components, (a) only one of

the components is filtered, and (b) when both

components are filtered....................... 92

4.17. Relation between Bf, D, and IM distortion for

the FM quadrature components, (a) when only

one component is filtered, and (b) when both

components are filtered....................... 93

4.18. Relation between Bf, m, and IM distortion for

the CSSB-AM quadrature components (all cases). 96

4.19. Relation between Bf, D, and IM distortion for

the CSSB-PM quadrature components (a) when only

one component is filtered, and (b) when both

components are filtered....................... 97

4.20. Relation between Bf, D, and IM distortion for

the CSSB-FM quadrature components, (a) when

only one component is filtered and (b) when

both components are filtered.................. 98

4.21. Relation between S(%), m, and Bf for the CSSB-AM

case where only one of the quadrature components

is filtered ....... ... ... ............ .. ..... 100

4.22. Relation between S(%), D and Bf for the CSSB-PM

case where only one of the quadrature

components is filtered........................ 100

4.23. Relation between S(%), D, and Bf for the CSSB-FM

case where one of the quadrature components is

filtered ....... .. ...... .................... 101

4.24. Another PM/PM modulator........................ 108

4.25. The best PM/PM modulator....................... 108

4.26. Relation between Bf, m, and IM distortion for

the AM and CSSB-AM $(t) function.............. 111

4.27. IM distortion for the DSB-LM and SSB-LM '(t)

component as a function of Bf ................ 112

Figure

Page

Figure Page

4.28. The function cos-1(y) and a linear

approximation .... .......... ......... ..... 114

4.29. Relation between S(%), Dp, and Bf for the CSSB-PM

X(t) function....... .... ................. .. 114

4.30. Relation between S(%), D, and Bf for

CSSB-FM f(t) function. ....................... 114

4.31. Block diagram of a digital baseband

preprocessor ........... ...... ... ..... ... .... 126

4.32. Block diagram of the digital computing system

for a general baseband preprocessor........... 128

5.1. The universal receiver......................... 132

5.2. The homodyne detector......................... 136

5.3. Effect of nonzero phase error in the output of

the phase detector............................ 138

5.4. The AM/PM demodulator.......................... 142

5.5. The quadrature demodulator..................... 142

5.6. The AM/FM demodulator ......... ............... 144

5.7. Block diagram for the general baseband

postprocessor #1 for the noiseless case with

perfect synchronization. ...... ........... ... 148

5.8. Block diagram for the general baseband

postprocessor #2 for the noiseless case with

perfect synchronization....................... 154

5.9. Improved baseband postprocessor with programmable

lowpass filters....................... .... . 160

5.10. Baseband postprocessor #4 for the AM/PM

demodulator .......... .. ..................... 162

ABSTRACT OF DISSERTATION PRESENTED TO

THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

MODULATION AND DEMODULATION OF RF

SIGNALS BY BASEBAND PROCESSING

By

Jorge A. Cruz-Emeric

August 1976

Chairman: Leon W. Couch, Ph.D.

Major Department: Electrical Engineering

A technique for modulating a carrier by a variety of mo-

dulation laws is analyzed. The technique is based on the

separate generation of the complex envelope function by a base-

band preprocessor. The complex envelope is then used to mo-

dulate the carrier. Since the complex envelope function de-

pends exclusively on the modulation law and on the message

signal, the baseband preprocessor can be programmed to obtain

the desired modulation law. Direct generation of complex

valued signals is not physically possible so various alter-

natives exist to realize the modulation process, but it was

found that only three are of practical interest.

All systems considered were a combination of two modu-

lators and two baseband preprocessors. The modulator combi-

nations were an amplitude modulator with a phase modulator,

a pair of balanced modulators, and a pair of phase modulators.

These configurations were based on the possible representa-

xii

tions of a complex function in terms of real-valued compo-

nents. Equations were obtained for the complex envelope com-

ponents and their corresponding autocorrelation functions.

The bandwidth required by these components were analyzed to

determine what specifications the modulators must meet.

Using the digital computer, graphical results were obtained

which give the bandwidth for the component as a function of

the modulation law parameters.

The modulation laws considered were amplitude modula-

tion, double-sideband linear modulation, single-sideband li-

near modulation, phase modulation, frequency modulation, com-

patible single-sideband amplitude modulation, compatible

single-sideband phase modulation, and compatible single-side-

band frequency modulation.

Also studied was the possibility of demodulating carriers

by recovering the complex envelope of the modulated carrier

and then extracting the message signal with a baseband post-

processor. It was found that only two systems are practical.

These were a combination of an envelope detector and a phase

detector; and the combination of two homodyne detectors. The

latter was found to be better.

harman

Chairman

xiii

CHAPTER I

INTRODUCTION

1.1 Problem Statement

Modulation is defined as the systematic alteration of a

carrier wave in accordance to the message to be transmitted

[1]. The modulation law is the mathematical rule used to

alter the carrier properties. The carrier and the modulation

law are selected to make better use of the transmission medium

[2]. An optimum choice of a carrier waveform and a modulation

law for every possible situation does not exist since there

are many alternatives.

Each modulation law requires an specific modulator cir-

cuit. This is a problem if it is necessary to have a trans-

mitter capable of operating with various modulation laws with-

out drastic modifications. A similar problem exists for the

receiver. This dissertation proposes a solution to these pro-

blems with the concepts of universal modulation and universal

demodulation.

Universal modulation and universal demodulation are de-

fined as techniques used to modulate and demodulate carriers

following different modulation laws with a two-stage process

that separates the modulation process into a part that depends

exclusively on the modulation law and a part that depends only

on the carrier waveform. The block diagrams of the proposed

universal transmitter and the universal receiver are shown in

2

Figures 1.1 and 1.2 respectively.

The baseband preprocessor in the transmitter modifies

the input message to produce a signal that if applied to the

carrier modulator produces a modulated carrier that follows

a prescribed modulation law. The carrier modulator is inde-

pendent of the modulation law. The carrier demodulator and

the baseband postprocessor perform the inverse operations at

the receiver.

1.2 Historical Background

This section presents a literature review of previous

work done by other researchers that is relevant to the re-

search problem.

The concept of complex envelope was popularized by

Dugundji [3] and Bedrosian [4] as a technique to analyze modu-

lated carriers. The relationship between the complex envelope

and the complex variable theory was studied in detail by

Bedrosian [4], Voelcker [5, 6], and Lockhart [7]. The mathe-

matical form of the complex envelope is well known for most

types of analog modulation. The idea of combining amplitude

and phase modulation was used by some authors [4, 8, 9, 10, 11]

to study new forms of modulation.

The properties of the most common types of modulation

laws are well known and are available from most communications

theory textbooks [12, 13, 14, 15]. The newer types of modu-

lation, like compatible single sideband modulation (CSSB),

have been studied in detail by Glorioso and Brazeal [16], Mazo

and Salz [17], Kahn and Thomas [18], Couch [19, 20], and others.

Figure 1.1. The proposed universal transmitter.

Modulated

Carrier and

Noise

Estimate

of the

Message

Figure 1.2. The proposed universal receiver.

There is still some controversy among researchers about

the virtues and flaws of CSSB. The transmission bandwidths

required for distortionless transmission (or with tolerable

distortion) for the modulated carrier are well known proper-

ties. It is claimed that CSSB modulated carriers can occupy

more bandwidth than their double-sided counterparts and that

any bandwidth economy depends on the modulation parameters.

Also known is the performance of these modulated carriers in

the presence of noise. The CSSB modulated carriers have a

poorer signal-to-noise ratio when detected with the receiver

for which they were designed to be compatible [203.

The idea of designing a modulator capable of following

various modulation laws have been proposed by Voelcker [5],

Lockhart [7], Couch [19], and Thomas [14]. Voelcker and

Lockhart proposed a combination of a phase modulator with an

amplitude modulator. Couch proposed a combination of two ba-

lanced modulators to obtain different modulated carriers with

the same general structure. Meewezen [211 proposed the in-

dependent modulation of the sidebands. Cain [22] proposed

modifications to a SSB transmitter to obtain CSSB modulated

carriers. Leuthold and Thoeny [233 proposed the use of a two

dimensional semirecursive filter to obtain some forms of mo-

dulation.

The idea of a universal receiver has been suggested by

those researchers working on the transmitter problem. Most

effort has been channeled toward the development of optimum

receivers for specific modulated carriers under specific

5

restrictions [24-37] rather than to obtain generalized re-

ceivers. Nonoptimum receivers are well known and are dis-

cussed in many communications theory textbooks.

1.3 Research Areas

Although the idea of universal modulation and demodula-

tion is not new, little attention has been paid to the fea-

sibility of such structures. This dissertation studies the

possible structures for the universal transmitter and the

physical limitations imposed on the realizations.

To achieve this goal, it is necessary to study the pro-

perties of the complex envelope components as related to the

different modulation laws, which to the author's knowledge,

have not been studied before. This is the subject of Chapter

III.

The universal transmitter is covered by Chapter IV.

Here various system block-diagrams are proposed as universal

modulators and their bandwidth and parameter constraints are

studied to determine their feasibility as practical systems.

The receiver case is the subject of Chapter V. The re-

ceiver is studied as an extension of the transmitter analysis

and general requirements are stated for the receiving problem.

Finally, Chapter VI states the conclusions and highlights

the important results obtained throughout the dissertation.

CHAPTER II

MATHEMATICAL PRELIMINARIES

2.1 Introduction

Any cosinusoidal carrier wave is completely described by

its amplitude and the absolute phase angle Consequently,

there are only three distinct ways to modulate a carrier:

(1) modulate only the amplitude, (2) modulate only the phase

angle, and (3) modulate the amplitude and the phase angle si-

multaneously. The modulated carrier is described by the gene-

ral expression

y(t) = a(t)cos[e(t)], (2-1)

where a(t) represents the carrier amplitude, 8(t) represents

the carrier absolute phase angle, and y(t) is the modulated

carrier. The function a(t) is also known as the real envelope

of y(t) or simply the envelope [4]. The absolute phase angle

is

8(t)'= wot + p(t), (2-2)

where the first term depends exclusively on the carrier fre-

quency in radians/seconds and the time, and the last term

kPolarization modulation by modulating the transversal

components of the radiated RF field will not be considered

here.

7

represents relative phase angle with respect to cos(w t) and

includes any other phase-angle variations.

The modulated carrier is a minimum-phase signal if

--

and it is defined as nonminimum-phase otherwise.

2.2 The Complex Envelope

It is a well known fact that a cosinusoidal waveform can

be described in terms of a complex exponential using Euler's

formula:

exp(jw t) = cos(w t) + jsin(wot), (2-4)

or

cos(wot) = Re -exp(jwot)} (2-5)

Define the complex modulated carrier, z(t), as

z(t) = a(t)exp[jwot + jp(t)], (2-6)

where a(t) and p(t) are real-valued functions. Comparing

Equations (2-4) and (2-5) reveals that the complex modulated

carrier is related to the physical modulated carrier by

y(t) = Re{ z(t)} (2-7)

Equation (2-6) can be rewritten as

z(t) = v(t)exp(jwot),

(2-8)

where

v(t) = a(t)exp(jp(t)). (2-9)

The function v(t) is defined as the complex envelope of y(t)

[4] and depends exclusively on the message signal and the

modulation law.

If v(t) and z(t) are square-integrable functions for

-oo

00

Z(w) = f z(t)exp(-jwt)dt, (2-10)

-00

and

00

V(w) = f v(t)exp(-jwt)dt. (2-11)

-co

Direct substitution of Equation (2-8) into Equation (2-10)

yields

Z(w) = V(w wo). (2-12)

This means that the spectrum of the complex modulated carrier

is the result of a linear translation of the spectrum of the

complex envelope to the carrier center frequency. The spec-

trum of z(t) is usually concentrated in the vicinity of w ;

therefore, the spectrum of v(t) is a baseband function be-

cause it is concentrated in the vicinity of zero frequency.

2.3 Analytic Functions

A complex function is classified as Analytick if the real

AThe upper case A is used to denote that the function is

analytic in the upper-half t plane as opposed to other regions.

and imaginary parts form a Hilbert pair [4]. This means that

the complex function s(t) is an Analytic function if

Im[s(t)]= H{Re[s(t)]l (2-13)

where

H{r(t)} = r(t) r(u) du (2-14)

-00o

is the Hilbert transform of the argument [12], and Im{.i} and

Re-C} are the imaginary and real part operators.

If s(t) is an Analytic function, the following properties

are true [4]:

1. The Fourier transform of s(t) vanishes for all nega-

tive frequencies.

2. The real and imaginary parts have identical autocor-

relation functions.

3. The complex function is completely described by either

the real part or the imaginary part.

Any complex modulated carrier, z(t), is not necessarily

Analytic. The spectrum of most types of modulated carriers

is concentrated near the carrier center frequency and gradually

drops to zero as the frequency approaches zero, and remains

zero for negative frequencies. Under this condition, z(t) can

be considered to be approximately Analytic [4].

The complex envelope is Analytic if and only if V(w) is

identical to zero for all negative frequencies. This is sa-

tisfied whenever the complex modulated carrier is Analytic

and if at the same time Z(w) is zero for all the frequencies

in the interval [0, wo]. This type of modulated carrier spec-

trum is defined as a single-sided spectrum.

There are single-sided spectrums that do not satisfy

Bedrosian's definition of Analytic. A simple example is when

s(t) in Equation (2-13) has a nonzero mean value; however,

this case can be analyzed because the effect of the nonzero

mean value can be considered separately.

2.4 Modulation Techniques

Modulated carriers can be classified in terms of:

(1) how the modulation law acts on the spectrum of the message,

(2) the distribution of the spectrum of the modulated carrier,

and (3) how the effect of modulation is observed in the time

domain waveform.

A modulation law is defined as linear modulation if the

spectrum of the modulated carrier is a translation of the

spectrum of the message and if superposition applies. The mo-

dulation law is defined as nonlinear otherwise.

The spectrum of the modulated carrier is classified as

double-sided if it is nonzero over at least a finite frequency

range on both sides of the carrier frequency. The spectrum

is classified as single-sided when all the energy (or power)

is concentrated in only one side of the carrier frequency.

In the time domain waveform, modulation of the carrier

can be accomplished by the systematic alteration of the car-

rier amplitude, the carrier phase angle, or both the amplitude

and phase simultaneously.

The following are the types of modulation to be considered

in this dissertation. In all cases the message signal is

assumed to be a real analog signal.

Amplitude modulation (AM). In this case the message

signal is carried exclusively by the real envelope of the car-

rier. It is a translation of the spectrum of the message and

the spectrum of a constant to the carrier center frequency.

Due to the constant involved in the modulation process, the

modulation law is nonlinear. The AM modulated carrier has a

double-sided spectrum.

Linear modulation (LM). It is a translation of the spec-

trum of the modulating signal to the carrier center frequency

1]) This can be a mixture of amplitude and phase modulation

and it can be single-sided or double-sided since both side-

bands carry the same information. The single-sided version

is known as single-sideband linear modulation (SSB-LM) and

the double-sided version is called double-sideband linear mo-

dulation (DSB-LM). If the DSB-LI. modulated carrier is li-

nearly filtered by an asymmetrical bandpass filter centered

around the carrier frequency, the output is classified as

vestigial sideband linear modulation (VSB-LM).

Angle modulation (dM). The information is carried by

the phase angle of the carrier (15) and it can be recovered

from the zero crossings of the carrier. It is nonlinear.

There are two main types, frequency modulation (FM) and phase

modulation (PM). In FM the modulating signal is proportional

to the derivative of the phase angle, while in PM the modulat-

ing signal is proportional to the phase angle itself.

12

Compatible single sideband (CSSB). The objective of CSSB

modulation is to have a modulated carrier with a single-sided

spectrum while at the same time retain compatibility at the

receiving end with the common types of double-sided demodula-

tors. It is a mixture of amplitude and angle modulation and

is nonlinear. There are various alternatives to achieve CSSB

[4, 10, 11, 22, 32, 33]. This dissertation restricts its

attention to CSSB types of modulations whose complex envelope

is an Analytic signal [4]. The three types to be considered

are CSSB-AM, CSSB-PM, and CSSB-FM.

2.5 Signal Parameters and Properties

This section defines and discusses the properties and

parameters that are of interest in this dissertation.

A deterministic signal is completely described by the

time-domain waveform or by its Fourier transform if it exists.

If the signal is produced by a random process, there are many

possible time-domain waveforms. This situation requires a

more general description based on the statistical properties

of the signal source. The autocorrelation function of the

random process is a satisfactory description in most cases,

but it may not be appropriate for nonlinear problems.

The autocorrelation function of the random process s(t)

is defined as

Rs(t,t +r ) = E{s(t)s*(t +z ) (2-15)

where E{*} is the expectation operator and s (t) is the

complex conjugate of s(t). The expectation operator is de-

finedk as

Ess2 If ls2 P(s1, s2)ds1ds2, (2-16)

-00

where the variables s1 and s2 represent s(t) and s(t + -r) res-

pectively. The function p(sI, s2) is the joint probability

density function of the random variables s1 and s2.

The autocorrelation function is related to the power spec-

tral density function by the Fourier transform relationship

00oo

Gx(w) = J Rs(r)exp(jwt)dt, (2-17)

-c00

where

Rs(r) = Rs(t,t + ) (2-18)

The right hand side of Equation (2-18) denotes the time ave-

rage of Rs(t,t + r). Equation (2-17) is known as the Wiener

and Kinchine theorem.

2.5.1 Bandwidth

Most signal processing problems involve filters. In some

cases, the filter is required by the system realization, in

others it is present due to the circuit limitations; therefore,

it is necessary to determine the bandwidth required by the

signal before the system components are specified. There are

various definitions for bandwidth. The most frequently used

Assuming a stationary process.

definitions will follow. All definitions are in terms of Hz.

3 dB down bandwidth (Bd). It is defined for a real

valued signal as the frequency where the power spectral den-

sity function decreases by 3 dB relative to the maximum level.

This is the same as

G(B) = max[G(f)] (2-19)

m% power bandwidth (Bp). It is defined for a real-valued

process as the frequency band that contains m% of the total

power in the signal. The mathematical definition is

Bp

JG(f)df = JG(f)df. (2-20)

o o

Equivalent noise bandwidth (Bn). The equivalent noise

is the width of a rectangle with a height of G(O) that con-

tains the same area as the power spectral density curve.

This can be written as

oo

Bn 2 f G(f)df. (2-21)

This equation can be expressed in terms of the autocorrelation

function

G(O)

Second moment bandwidth (Bs). It is defined as

2 jf2G(f)df

Bs = -. (2-23)

s G(f)df

-00o

or, in terms of the autocorrelation function

15

2 -R (0) (2-2

(2r) R(0)

where Ro (T) denotes the second derivative of R(r) with res-

pect to r Bs is also known as the root mean squared (RMS)

bandwidth.

Equivalent-filter bandwidth (Bf). The equivalent filter

bandwidth is best defined by the block diagram of the proce-

dure used to calculate it rather than by a mathematical ex-

pression. This procedure is illustrated by Figure 2.1. There

are two signal paths, one through an ideal lowpass filter and

the other through a delay line that compensates the delay in-

troduced by the filter. The two signals are applied to the

same kind of signal processor and the two outputs are compared

to determine the error introduced by the filtering action.

The signal processor includes all processing done between the

point where the bandwidth is measured to the point where the

error is measured. Define the equivalent-filter bandwidth as

the width of the passband of the ideal lowpass filter that

produces a determinate amount of error. The equation is

k = E f[d[z(t)]] f[z(t)eh(t)]-, (2-25)

where f[C- represents the signal processor algorithm, E-',*}

is the error calculation operation, d[*] is the time delay ope-

ration, h(t) is the impulse response of the ideal lowpass

filter, and k is the value of the measure of error. The ideal

lowpass filter impulse response is

i sin(2lTBft)

h(t) = 1- (2-26)

1T t

Control

Figure 2.1. Procedure used to calculate the equivalent-filter

bandwidth.

Signal

The 3 dB bandwidth is often used to specify the useful

frequency range of amplifiers and filters. It should not be

used to specify the bandwidth of a signal because it does not

take into account the signal properties.

The power bandwidth is useful in situations where the

evaluation of the power distribution is the prime considera-

tion. It has the disadvantage that Equation (2-20) cannot be

solved in terms of Bp except in a few very simple cases; there-

fore it must be calculated with a numerical iterative process.

It does not specify the actual bandwidth required by a signal

unless the power is the only consideration.

The noise equivalent bandwidth and the second moment band-

width have the advantage of being directly related to the sta-

tistical properties of the signal. The disadvantage is that

if the signal is subsequently processed by a nonlinear system,

the distortion introduced may not be simply related to these

bandwidth definitions, although in most cases it is possible

to relate the input bandwidth to the output bandwidth. Abramson

[343 found that the second moment bandwidth of an arbitrary

zero-memory nonlinear transformation of a stationary random

process is given by

2 E g'(x).x'}

Ss (2-r)2E{g2(x) (2-27)

where x(t) is the input to the nonlinear device, s = g(x) is

the nonlinear transformation of the input, x'(t) is the deri-

vative of x(t) with respect to time, and g'(x) is the deriva-

tive of g(x) with respect to x(t). For the case where x(t)

and x'(t) are statistically independent, Abramson showed that

2 E[g.'(x)]2 Ex2(t)} 2

B B (2-28)

ss E{g2(x) sx

where all expectations are with respect to x(t). This rela-

tionship shows that Bss and Bsx may not be simply related.

It should be pointed out that the calculation of some of these

expectations may be very difficult.

The equivalent-filter bandwidth has the advantage of re-

lating the actual bandwidth requirements of the signal at a

point in the system to the criteria used to measure the per-

formance of the system. Therefore, the equivalent-filter band-

width is the bandwidth required for the actual design at the

point where it is measured. It has two disadvantages. The

first one is that a closed-form expression is almost impos-

sible to obtain; therefore, it is necessary to simulate the

system in order to calculate the bandwidth. This is not a

great disadvantage if the reader realizes that all the other

definitions may require numerical solutions in complicated

cases. The second disadvantage is that the definition is

based on a subjective consideration of what is a good perfor-

mance criterion for a system; therefore, universal acceptance

of this definition cannot be expected. However, distortion

is precisely what the design engineer will consider as a de-

sign criterion; therefore, this definition is acceptable. A

similar technique is often used to define the bandwidth of FM

signals as a function of distortion [35-37] Even in the case

of the second moment bandwidth, where an analytical expres-

sion may be found, the design engineer cannot use it directly

as the design bandwidth without subjective considerations.

2.5.2 Dynamic Range

The voltage (current) dynamic range of an analog signal

is defined as the maximum positive voltage (or current) and

the maximum negative voltage (or current) that can be found

in the signal under consideration. This is important because

physical devices and circuits have definite maximum voltage

(or current) levels that, if exceeded, may result in improper

operation or physical damage to the components.

The numerical dynamic range is defined for digital signals

as the maximum (positive or negative) number that can be found

in the signal sequence and the minimum number that must be re-

solved by the computing circuit. These numbers may determine

if overflow or underflow will occur and how many bits are nec-

essary to represent the sequence.

2.5.3 Measure of Error

There are two common measures of error. These are the

mean-squared error and distortion.

-2

The mean-squared error 2 is defined as [14]

E 2(t)J = E{[x(t) x(t)]2}, (2-29)

where x(t) is the reference against which x(t) is compared.

It involves knowledge of the statistics of the signal so it

is a widely accepted theoretical criteria of goodness of a

system. Equation (2-29) can include the effect of noise and

other disturbances so the minimization of the mean-squared

error is often the objective of optimization problems. The

main disadvantage is that it is difficult to measure because

any procedure used requires time delay equalization of x(t)

and x(t) to guarantee that the measured E-[ 2 }comes from x(t)

and not from the measuring procedure.

The other measure of error is distortion. It is based

on the selection of a deterministic test message that is used

as the reference against which the distortion is calculated.

Distortion is usually specified as harmonic distortion or in-

termodulation distortion. The total harmonic distortion (THD)

is defined for a single tone test signal as

T P(nf)

THD(o) = -2 x 100%, (2-30)

P(f )

o

where P(nfo) is the power of the n-th harmonic of the test

signal. The intermodulation distortion (IM) is defined as

/ ZP(nfl + mf2) if n=0, m/1,

IM(fo) = \ n x 100%, (2-31)

P(f) + P(f2) if m, n .

The intermodulation distortion definition takes into account

the mixing of the test signal frequency components due to the

nonlinearity. It includes the harmonic distortion so it is

usually higher than the THD. Since two tones are involved,

there are many possible combinations of relative amplitudes

and of relative frequency separation. There is no universally

21

accepted standard. The typical arrangement is a 4:1 amplitude

ratio and frequency ratio [38], but there is no technical rea-

son for using it over any other combination. It is advisable

to state clearly what is the test signal together with the

distortion readings or calculations.

From the engineer point of view, the distortion figures

are easier to measure and to understand than the mean-squared

error because they are self-normalizing and no phase equaliza-

tion is required. The disadvantage of the distortion defini-

tions is that they do not take into account the effect of noise

and other disturbances.

2.6 Equivalence Between Continuous and Discrete Signals

In some situations it is desirable to replace a conti-

nuous-time signal with the sampled or discrete-time version

of it. If certain conditions are met, both representations

contain the same information.

Let XA(w) be the Fourier transform of the continuous time

signal and XD(exp(jwT))k the discrete Fourier transform (DFT)

of the sampled signal. The sampled signal and its DFT are

related by

T/T

T

x(nT) = XD(exp(jwT))'exp(jnwT)dw, (2-32)

xTr/T

where T is the spacing between samples in the time domain.

AThe notation XD(exp(jwT)) is a common practice in digi-

tal signal processing literature. It is the result of evalu-

ating the z-transform of x(nT) with z = exp(jwT).

It can be shown [39] that the Fourier transform of the conti-

nuous-time signal is related to the DFT of the sampled sig-

nal by

1 2tT

XD(exp(jwT)) = XA(w + m), (2-33)

m=-oo

which is a periodic function in the frequency domain. If the

spectrum of the continuous-time signal is absolutely bandli-

mited to the range Iwlis/T, then

XD(exp(jwT)) = XA(w) (2-34)

provided that Iwl

lost by sampling the continuous-time signal. If the function

XA(w) is not bandlimited to Iw i$ r/T, XD(exp(jwT)) will con-

sist of overlapping frequency-shifted spectrums of the conti-

nuous signal. This is known as aliasing. The signal can

still be sampled, but the sampling rate should be high enough

to reduce the effect of aliasing to a tolerable level.

2.7 Summary

This chapter presented the definitions and the concepts

that will be used throughout the dissertation. It also dis-

cussed the various alternatives to the definition of band-

width in order to help select the best one for the particular

application.

CHAPTER III

THE COMPLEX ENVELOPE

3.1 Introduction

The complex envelope is known to completely describe

the modulation process since it contains all the message in-

formation. It is reasonable to assume that the generation of

the complex envelope is all that is needed to produce a modu-

lated carrier that follows any specified modulation law. This

chapter presents a compilation of the complex envelope func-

tions for well known types of modulated carriers and analyzes

some properties that have not been considered before in the

literature.

Since physical systems cannot handle complex voltages or

complex currents, it is necessary to study the representation

of the complex envelope in terms of real-valued functions.

These components will be used in Chapters IV and V to study the

possible structures of the universal transmitter and receiver.

The general properties of the complex envelope are well

known and were already presented in Section 2.2. This discus-

sion concentrates on the properties of the components that

have received little attention in the literature.

Any complex number can be represented geometrically as

a point or a vector in the two-dimensional complex plane de-

fined by the real and imaginary axis. The two most common

descriptions of a complex function are in terms of absolute

23

value and the argument, and in terms of the two quadrature

components. These representations are

v(t) = a(t)exp[jp(t)], (3-1)

where a(t) is the absolute value and p(t) is the phase angle

v(t), and

v(t) = i(t) + jq(t), (3-2)

where i(t) and q(t) are the quadrature components. In terms

of a vector, a(t) is the magnitude or the length of the vec-

tor v(t) and p(t) is the phase angle with respect to the real

axis. The quadrature components are equivalent to two ortho-

gonal vectors that are the projections of the complex enve-

lope vector along the real and the imaginary axes.

3.2 The Complex Envelope for Different Types of Modulation

The general approach to obtain the equation of the com-

plex envelope for a modulated carrier is to specify certain

conditions that the modulated carrier must satisfy. Using

available mathematical techniques plus some ingenuity, equa-

tions may be found that satisfy all or some of the conditions.

These conditions usually require an specific relationship be-

tween the message signal and one of the complex envelope com-

ponents and an specific characteristic of the modulated car-

rier. For practical purposes, one may like to specify three

parameters, namely the magnitude, the phase, and the bandwidth

occupancy of the modulated carrier. Unfortunately, there are

only two components; therefore, the arbitrary specification of

the three requirements is not possible.

Voelcker [5, 6] developed and Lockhart [7] expanded a

theory that serves to study the complex envelope of modulated

carriers where the message is a periodic waveform. It is based

on the zero-singularity patterns of the z-transform of the com-

plex envelope. By examining and manipulating the location of

the zeroes or singularities it is possible to determine if the

desired properties can be obtained or if they are not compatible.

This section considers the types of modulation defined

in Section 2.4.

3.2.1 Amplitude Modulation (AM)

By definition, amplitude modulation requires the modulated

carrier to have a spectrum identical to the translation of the

spectrum of the message (except for a carrier line) and the

complex envelope to have a magnitude proportional to the mes-

sage. A complex envelope that satisfies both conditions is

v(t) = C[1 + mx(t)], (3-3)

provided that

mx(t),-1, (3-4)

where m is the AM modulation index, x(t) is the message, and

C is the unmodulated carrier peak amplitude.

3.2.2 Linear Modulation (LM)

Linear modulation is the linear translation of the spec-

trum of the message to the carrier center frequency. This re-

quires a complex envelope

26

v(t) = Cx(t). (3-5)

The spectrum is double-sided so this is called double-sideband

linear modulation (DSB-LM). Comparison of Equation (3-5) with

Equations (3-1) and (3-2) reveals that

a(t) = CIx(t)l (3-6)

'0, if x(t) > 0

p(t) =- (3-7)

.IT, if x(t) < 0,

i(t) = Cx(t), (3-8)

and

q(t) = 0. (3-9)

Equation (3-7) can be rewritten as

p(t) = sgnx(t)] (3-10)

where

1, if u < 0

sgn(u) = (3-11)

-1, if u > 0,

The single-sided version is obtained by replacing the

message with an equivalent analytic signal that has a spectrum

identical to one of the sides of the spectrum of the message.

This is the same as requiring the complex envelope to be

v(t) = C[x(t) + jx(t)] ,

(3-12)

where x(t) is the Hilbert transform defined by Equation (2-14).

This is known as single-sideband linear modulation (SSB-LM).

The required components are

a(t) = C x2(t) + 2(t) (3-13)

p(t) = tan (t) (3-14)

Lx(t).

i(t) = Cx(t), (3-15)

and

q(t) = Cx(t). (3-16)

3.2.3 Angle Modulation (XM)

Angle modulation requires that the relative phase angle

of the modulated carrier be a linear function of the message.

Phase modulation (PM) requires that the relative phase angle,

p(t), be directly proportional to the message, while frequency

modulation (FM) requires that the message be proportional to

the derivative of the phase angle. The required complex enve-

lope is

v(t) = C exp[jr(t)], (3-17)

where

r(t) = Dpx(t) for PM, (3-18)

and

dr(t)

d(t = Dfx(t) for FM. (3-19)

dt

28

The constant D and Df are the PM modulation index and the FM

frequency deviation constant, respectively

The required complex envelope components are

a(t) = C, (3-20)

p(t) = r(t), (3-21)

i(t) = C cos[r(t)] (3-22)

and

q(t) = C sin[r(t)]. (3-23)

3.2.4 Compatible Single Sideband Modulation (CSSB)

The idea behind compatible single sideband modulation

(CSSB) is to obtain a modulated carrier that is compatible

with receivers for conventional AM, PM, and FM, while at the

same time obtain the bandwidth savings of a single-sided spec-

trum. This requires the selection of a(t) in the CSSB-AM case,

and p(t) in CSSB-PM and CSSB-FM cases. This leaves only one

function to specify. This function should result in a single

sided spectrum that at the same time is narrower than the spec-

trum of the conventional case. This may not be possible in

all cases.

Several solutions have been proposed for the CSSB-AM

signal by Villard [9], Kahn [111, Bedrosian [4], Powers [10],

and others. The Villard's system has been shown to be non-

bandlimited and not completely single-sided. The Powers' sys-

tem is absolutely bandlimited and single-sided but it is not

exactly compatible with conventional AM receivers. Kahn's

and Bedrosian's systems are very similar, they have single

sided spectrums but they have been shown not to be absolutely

bandlimited. These systems do show some "effective" band-

width reduction compared with the double-sided cases but the

reduction depends on the message level [7].

Solutions to the CSSB-FM and CSSB-PM cases have been pro-

posed by Bedrosian [4]. Glorioso and Brazeal [16], Mazo and

Salz [17], Kahn and Thomas [18], Dubois and Aagaard [40], and

Barnard [41] have studied the CSSB-PM and CSSB-FM carrier spec-

tral properties. It is known that CSSB-PM and CSSB-FM require

less bandwidth than conventional PM and FM for low modulation

levels but that the bandwidth reduction depends on the modula-

tion level.

Bedrosian also showed that if an Analytic periodic signal,

s(t), is used to form the complex envelope

v(t) = C exp[s(t)], (3-24)

a single-sided complex envelope results. This signal, depends

on what characteristics are desired. For CSSB-AM, s(t) is

s(t) = In[l + mx(t)]+ jHln[l + mx(t)] (3-25)

provided that Equation (3-4) is satisfied. This results in

the following complex envelope components

a(t) = C[1 + mx(t)], (3-26)

p(t) = H{ln[l + mx(t)]}, (3-27)

i(t) = C[1 + mx(t)]cos[p(t)],

(3-28)

and

q(t) = C[1 + mx(t) sin[p(t)]. (3-29)

The CSSB-PM and CSSB-FM cases require that

s(t) = r(t) + jr(t), (3-30)

where

r(t) = Dpx(t) (3-31)

for CSSB-PM, and

r(t) = Dffx(u)du (3-32)

-00

for CSSB-FM. Couch [19] showed that other SSB signals could

be obtained by the use of entire functions to obtain analytic

complex envelope functions.

The complex envelopes are summarized in Table III-1.

3.3 Autocorrelation Function of the Complex Envelope Compo-

nents

This section presents the autocorrelation function of the

complex envelope and derives the autocorrelation function of

the components for different types of modulation laws.

If the autocorrelation function is available, the second

moment bandwidth can be calculated using Equation (2-24).

33.31 General Expressions for the Autocorrelation Function

The autocorrelation function of the complex envelope can

be derived in terms of the a(t) and p(t) functions:

R(t,t + t) = Ea(t)a(t + r)exp[jp(t) jp(t +r )] (3-3)

Complex Envelopes

Types of

Modulation Complex Envelope

AM C[i + mx(t)],

mx(t)- -1 v t

LM Cx(t)

SSB-LM C[x(t) + jx(t)]

PM C exp[jDpx(t)]

t

FM C exp[jDfJ x(u)du]

-00

CSSB-AM C exp[r(t) + jr(t)],

r(t) = ln[l + mx(t)]

CSSB-PM C exp jDp(x(t) + jx(t))]

CSSB-FM C exp jDf fix(u) + jx(udu]

t-o

Table III-1.

which can be written as

Rv(t,t +T ) = E{a(t)a(t + ) cos[p(t) p(t +T )1+

+ jE{a(t)a(t +- ) sin[p(t) p(t +- )j]. (3-34)

No further simplification is possible unless more is known

about the nature of a(t) and p(t). This equation illustrates

that it is not possible to separate the effects of the ampli-

tude and phase functions on the modulated carrier complex

envelope.

It is possible to define the second moment bandwidth of

v(t) in terms of a(t) and p(t). For this purpose, define

d(t) as

d(t) = v'(t), (3-35)

where v'(t) is the first time-derivative of v(t). Observe

that the Fourier transform of d(t) is given by

D(w) = jwV(w); (3-36)

therefore, the spectrum of d(t) can be obtained as the output

of a filter whose transfer function is

H(w) = jw (3-37)

and whose input is v(t). The power spectral density of d(t)

can be expressed in terms of the power spectral density of

v(t) by

Gd(w) = lwI2Gv(w) (3-38)

where Gd(w) and Gv(w) are the power spectral density functions

of d(t) and v(t) respectively. Apply the Fourier transform

differentiation theorem to obtain the autocorrelation func-

tion of d(t) in terms of the autocorrelation function of v(t)

Rd(-r) = -R;'(-c)

(3-39)

Using this equation, Equation (2-24) can be written as

B2 E-Iv'(t)l 2}

sv (2-v)2E{Iv(t)I 2}

Using Equation (3-1), observe that

v'(t) = a'(t) exp[jp(t)] +

+ j[a(t)p'(t)] exp[jp(t)]

iv'(t)I 2 = [a'(t)]2 + la(t)]2p (t)]2,

and the second moment bandwidth of v(t) is given by

2

B

sv

1

= 1 .

(2-r)2

E{[a'(t)] 2}+ E{[a(t) 2 p' (t)] 2}

E{[a(t)32]

B = B + 1

sv Sa (2w)2

(3-44)

E-a[at p'(t)] )]2

E[a(t)]2}

(3-40)

(3-41)

(3-42)

(3-43)

where Bv is the second moment bandwidth of the complex enve-

lope and Bsa is the second moment bandwidth of the amplitude

function. This is similar to a result obtained by Kahn and

Thomas [18] and shows that the complex envelope bandwidth is

at least as wide as the bandwidth of the a(t) function, and

that it depends on both a(t) and p(t).

If the a(t) and the p(t) functions are statistically in-

dependent, Equation (3-44) becomes

B2 = B2 + 1 Et(p'(t)]2 (3-45)

sv sa (21T)2

Under the same condition, Equation (3-33) becomes

Rv(t,t + T ) = Ra(t,t + T) E{exp[jp(t) jp(t + )]}.

(3-46)

The autocorrelation function of v(t) can be derived in

terms of the quadrature functions,

Rv(t,t + t) = E-i(t)i(t + -)} + E{q(t)q(t + T )} +

+ jE{q(t)i(t + -c)} jE{i(t)q(t +r )},

(3-47)

or

Rv(t,t + -) = Ri(t,t + t) + Rq(t,t + t)

+ j[Rqi(tt + t) Riq(t,t + )] (3-48)

where Ri(t,t +T') and Rq(t,t + r) are the autocorrelation

functions of i(t) and q(t), respectively,

Rqi(t,t + ) = E{q(t)i(t + )} (3-49)

and

Riq(t,t + ) = E[i(t)q(t + )}. (3-50)

Using Equation (3-39), the second moment bandwidth can be

written as

B2 +E{[i'(t)]2 +[q'(t)]2} (3-51

sv (2-)2E{i2(t) + q2(t)

The autocorrelation functions of i(t) and q(t) are easily

related to that of v(t). Observe that i(t) can be written as

i(t) = a(t)cos[p(t)] (3-52)

so

i(t) = [v(t) + v*(t)] ; (3-53)

therefore,

Ri(t,t +T ) = Re{E.v(t)v*(t + )}}+ ReE,

(3-54)

Ri(t,t + ) = 2 Re{Rv(t,t + ) + R v(t,t +x ) A

(3-55)

Since q(t) can be written as

q(t) = [v(t) v*(t)], (3-56)

2j

so following the same procedure,

Rq(t,t + -) = Re-Rv(t,t +T) Rvv*(t,t +T )}.

(3-57)

All these equations point out that, in general', knowledge of

the autocorrelation function of the complex envelope does not

necessarily imply knowledge of the autocorrelation functions

of a(t), p(t), i(t), and q(t). The converse is also true.

These statements are also valid for the second moment band-

width.

If a(t) and p(t) are statistically independent, Equations

(3-55) and (3-57) can be simplified. Using Equation (3-46),

Equation (3-55) simplifies to

R.(t,t +T) = Re a(t,t +-c) EEexp[jp(t) jp(t +c)] -+

1 2

+ exp[jp(t) + jp(t +r )] -, (3-58)

and Equation (3-57) simplifies to

ARvv(t,t + T) = Rv(t,t + )

Ra(t,t + )

Rq(t,t + ) = Rea 2- E exp[jp(t) jp(t + )]+

2

exp[jp(t) + jp(t + T )111. (3-59)

33.32 Special Case: Analytic Complex Envelope

If the complex envelope is an Analytic signal, by

definition

A

q(t) = i(t), (3-60)

Rq(t,t +T ) = Ri(t,t + ), (3-61)

Riq(t,t + ) = Ri(t,t + ), (3-62)

and Equation (3-48) becomes

Rv(t,t +-) = 2Ri(t,t + -) + 2jRi(t,t +S ) (3-63)

where Ri(t,t + T) is the Hilbert transform of Ri(t,t +r).

Observe that if Equation (3-60) is valid, then

E [i'(t)12) = E[q' (t)]2}; (3-64)

therefore, Equation (3-52) becomes

B2 E-[i'(t)] 2 (3-65)

Sv (21) 2Ei2(t)}

Remember that i(t) and q(t) have the same autocorrelation

functions, so

B = B = B (3-66)

v "i -q

A similar simplification of Equation (3-34) is not

possible under this condition; however, when v(t) is Analy-

tic,

q(t) = a(t) sinp(t)

= H{a(t) cos[p(t)]} (3-67)

33.33 AM

Assume that the message, x(t), is a zero-mean, unit

power, real process with normalized autocorrelation function,

p(T). For the AM case the following autocorrelation func-

tions were obtained using the equations listed on Table

III-1k

Rv(T) = C2[1 + m2p(t)], (3-68)

Ri(-) = Ra(r) = C2[1 + m2?(T)] (3-69)

and

Rp(-) = Rq(t) = 0. (3-70)

where it is assumed that m is low enough to prevent frequent

violations of Equation (3-4).

AIt should be remembered that these equations assume

I1 + mx(t)l= (1 + mx(t)).

3.3.4 DSB-LM

The autocorrelation function of the DSB-LM complex enve-

lope is

Ry(T) = C2 (); (3-71)

therefore,

Ri(T) = C2p(T), (3-72)

and

Rq(T) = 0. (3-73)

The autocorrelation function for a(t) can be calculated

by observing that Equation (3-6) is similar to the output of

a full-wave linear rectifier. The autocorrelation function

of this system with zero mean gaussian noise (ZMGN) at the

input was calculated by Middleton [42]; therefore, adapting

that result to this situation

Ra() = C2r2 )arcsin[c()1 + 1 2 (3-74)

The autocorrelation function for p(t) is obtained by

substituting Equation (3-10) in the definition of autocor-

relation function

R (-) = [ 2 Et[1 sgn(x(t))] [1 sgn(x(t +1 ))]-

(3-75)

If x(t) is a zero mean stationary random process

E{sgn x(t)} = 0, (3-76)

so

Rp(r) = [2E1 + sgn[x(t)] sgn[x(t + T)]}. (3-77)

The second term is similar to the autocorrelation function of

odd-symmetry limiter. Using published results for this device

and ZMGN [42] the autocorrelation of the phase function is

Rp(T) = [ T2+ arcsin[p(T) (3-78)

33.35 SSB-LM

The autocorrelation function of the complex envelope is

easily found by comparing Equation (3-12) with Equation (3-65):

Ry(T) = 2C2 p(T) +j(-) (3-79)

and by the definition of an Analytic signal,

Ri(T) = Rq(T) = C2p(T). (3-80)

The situation for the a(t) and p(t) functions is not as

simple because Equations (3-13) and (3-14) involve nonlineari-

ties and memory. Rewrite Equation (3-13) as

a(t) = C FbtT), (3-81)

where b(t) is

b(t) = x2(t) + x2(t).

(3-82)

The new process b(t) is not gaussian. The autocorrelation

function of b(t) is

Rb(r) = E[b(t)b(t +T)}

= EBx2(t)x2(t + T ) + E{x2(t)A2(t + ) +

+ E{x2(t)x2(t + T ) + E {2(t)X2(t + T )j.

(3-83)

If x(t) is ZMGN, x(t) is also ZMGN and

E{x2(t)x2(t + T)} = 1 + 292(-), (3-84)

E{2(t)x2(t + r)} = 1 + 2p2(T). (3-85)

Ex2(t)2(t + T)} = 1 + 292(-), (3-86)

and

E{ 2(t)x2(t +- )} = 1 2 2(T). (3-87)

Equation (3-73) reduces to

Rb(T) = 4 [1 +92(r (3-88)

The power spectral density of b(t) is bandlimited for band-

limited x(t), so it is well known that the power spectral

density of b(t' is not absolutely bandlimited [7]. A closed

solution for the autocorrelation function of a(t) does not

seem possible. Bowen [43] presented a method to calculate the

autocorrelation of instantaneous nonlinear devices with non-

gaussian noise that yields an infinite series expansion in

terms of the input autocorrelation function. This method can

be used if the exact autocorrelation function is known.

An approximation can be obtained for the autocorrelation

A

function of a(t) by observing that x(t) and x(t + ) are un-

correlated [13] for small This means that a(t) is approxi-

mately a two-dimensional chi process. A chi process is de-

fined by

r = u (3-89)

r k=1 k]

where uk represent independent gaussian variables. Miller,

et al [44] obtained an expression for the autocorrelation

function of a two-dimensional chi process

2 2 r(3/2)

R -(T) = 2R (0) 1 9 ()]2. ] 2( )

S2F1[3/2, 3/2; 1; (()] (3-90)

where Ru(0) is the variance of ui, pu(t) is the normalized

autocorrelation function of u, r(n) is the Gamma function

[45] given by

r(n) = fexp(-t)t (-ndt (3-91)

0

o

and 2F1(., .; .; .) is a gaussian hypergeometric function

given by

r(c) 1a-1 c-a-1

2Fl(a, b; c; z) = r t (1 -t) c-a

(a)r (c a) o

-b

* (1 zt) dt.

(3-92)

43

For the case under consideration

r(3/2) = (3-93)

r(1) = 1, (3-94)

and

Ru(0) = C2 (3-95)

so Equation (3-90) becomes

RaT)=: c2I -p2(T)]2 2 F [3/2, 3/2; 1;p (r)] (3-96)

The derivation of the autocorrelation function of p(t)

is very complicated. The autocorrelation function of the in-

termediate variable w(t), defined as

w(t) = x(t)/x(t), (3-97)

is calculated in the Appendix A and is listed below:

R (T)= 2() +

[i -92() -* [ (-.

1 -9 2() (3-98)

For a baseband gaussian process bandlimited to W with the auto-

correlation function of Equation (3-33) yields

R (0) =c0. (3-99)

This means that the spectrum is not absolutely bandlimited and

requires infinite power. Since w(t) must be applied to the

nonlinear operation tan- (*) it is not possible to predict

the effect on p(t); however, it is reasonable to expect a

very wideband signal.

3.3.6 PM

The autocorrelation function of the PM complex envelope

is [13]

Rv(r) = C2exp[-D2[1 9()]] (3-100)

provided that x(t) is a stationary ZMGN. Also

Rvv*() = C2exp[ -D2[1 + (r) (3-101)

The autocorrelation functions of the amplitude and phase

functions are

Rv(T) = C2 (3-102)

and

R p() = D? p(). (3-103)

The autocorrelation function of the quadrature components,

i(t) and q(t), can be found by substituting Equations (3-100)

and (3-101) into Equation (3-55). The result is

Ri(r) = exp -Dp (1 xp+

(3-104)

which can be simplified to

Ri(r) = C2exp(-D2)cosh[D2 9()] (3-105)

Similarly, substituting Equations (3-100) and (3-101) into

(3-57) yields

Rq(T) = C2exp(-D2)sinhLD?(u)J (3-106)

33.37 FM

If the random process resulting from the integral of x(t)

given by Equation (3-19) has zero mean and is stationary, the

results derived for PM can be extended to FM and it is only

necessary to replace x(t) by X(t), where

X(t) = Dffx(u)du (3-107)

-0o

with an autocorrelation function of [13]

2 00

R(rT) = 2JGx(w exp(jwr)dw, (3-108)

where

oo

Gx(w) = f9(T)exp(-jwr)dr. (3-109)

If $(t) is ZMGN, the required equations are

Rv(r) = C2exp[-Rp(0) + Rp(T)], (3-110)

RA(T) = C2, (3-111)

Rp(t) = RH(r), (3-112)

Ri(T) = C2exp[-RX(0)]cosh(R(rc)), (3-113)

and

R q() = C2exp[-RX(0)]sinh(R(t-)). (3-114)

3.3.8 CSSB-AM

A closed form expression for the autocorrelation func-

tion of the complex envelope cannot be found in the literature.

Preliminary work done by the author reveals that it is neces-

sary to use series expansions that are very difficult to com-

pute and virtually impossible to work with when put into use.

That is, no satisfactory result could be obtained for the auto-

correlation function of the complex envelope.

The autocorrelation function of the conjugate functions

depends on the availability of the result for the complex en-

velope as shown by Equation (3-65).

The situation for the autocorrelation of the amplitude,

a(t), is different. If

(1 + mx(t))> 0 Vt (3-115)

the autocorrelation function of a(t) is given by Equation

(3-69).

The autocorrelation of the phase function, p(t), is

Rp(r) = E[H{ln[1 + mx(t)]} Hf ln[1 + mx(t + T )]}1 (3-116)

It is well known that the autocorrelation function of the

argument of the Hilbert transform is the same as the autocor-

relation function of the original function [13]; therefore,

R p() = El ln[l + mx(t)] ln[l + mx(t + )] (3-117)

A closed-form solution of this equation is not possible. If

mx(t) <<1, Equation (3-111-) can be approximately solved by

47

using the first two terms of the series expansion of ln(l + x):

2 2

Rp(-) EL.[mx(t) 2- x2(t)] [mx(t + ) - x2(t + )

(3-118)

which for a ZMGN message signal reduces to

R (-r) m2 p() + m4(l + 2 2(()) (3-119)

3.3.9 CSSB-PM

The autocorrelation functions of the CSSB-PM components

can be derived in terms of the autocorrelations functions ob-

tained for PM. The required modification is to replace the

real-valued process x(t) with the complex valued process s(t)

indicated by Equations (3-30) and (3-31). Since x(t) is a

gaussian process, x(t) is also a gaussian process and the sum

of the two variables is also a gaussian process; therefore,

s(t) is a complex gaussian process. Miller [46] showed that

the same formula used to calculate the moments of a real mul-

tivariate gaussian process may be used to calculate the moments

of a complex gaussian process. The autocorrelation function

of the complex process is

Rs(r) = 2D [?(t) + j A(C)] (3-120)

and substituting this expression in Equation (3-100)

Rv() = C2exp[- 2D2(1 p() j p(T))]. (3-121)

The autocorrelation function of a(t) is obtained as

Ra(T) = E{C2exp[-Dpx(t) DpX(t +r )] (3-122)

but since [13]

E exp[jclxi+ jc2xd = exp 1[ R(0O)(cf + c2) Rx(r)cic2]

(3-123)

where ci and c2 are constants, x1 and x2 are ZMGN variables,

and Rx(r) is the autocorrelation function between xI and x2,

Equation (3-122) becomes

Ra () = C2exp[ D [1 + p(T)]j. (3-124)

The autocorrelation function of p(t) is

Rp(T) = D p(T). (3-125)

The autocorrelation functions of i(t) and q(t) are found

by expanding Equation (3-123) in terms of its conjugate compo-

nents and comparing the expansion with Equation (3-57). Note

that if v(t) is Analytic, i(t) and q(t) have the same auto-

correlation function so

R.(t) = R (-) = 2 exp -2Dp l (T)] cos 2DP(T) ,

(3-126)

3.3.10 CSSB-FM

If the conditions set for in Section 3-3.9 are met, then

the equations derived for CSSB-PM are modified to yield:

Rv(r) = C2exp[-2R(0O) + 2RO(r) + j2R(t)] (3-127)

49

Ra(x) = C2exp[2Rp(0) + 2R(rT)J, (3-128)

Rp(T) = R(T-) (3-129)

and

2

R (r) = R (T) = exp[-2R (0) + 2R(Tr)] cos[2RX(T)].

q 2

(3-130)

3.4 Second Moment Bandwidth of the Components

Using Equation (2-24), the second moment bandwidth of the

components can be calculated.

3.4.1 AM

Using Equations (3-68), (3-69), and (3-70) together with

the definition of second moment bandwidth yields

2 2

2 -m2 2 2

B? (0) = B_ (3-131)

sa 1 + m2(0) 1 + m2 sx

Bs = B (3-132)

i! Sa

and

B = B = not defined, (3-133)

S s

q p

where Bs is the second moment bandwidth of the message. In

practice the carrier term can be neglected and Bsa = Bsx

a X

3.4.2 DSB-LM

qhe second moment bandwidth of a(t) can be found as fol-

lows. First find Ra (T) using Equation (3-74)

Ra'( 2 2 "(r) (T)arcsirp(T)]. (3-134)

a ( I) (.t)

Then

R (0) = C2; (3-135)

a

therefore,

B =B (3-136)

Sa sx

The bandwidth of the phase function requires the calcula-

tion of the second derivative of R (T):

P

,, p(T)[' (2(t) + 2 T)P )]2L()

R (3-137)

p 2 [ 2()] 3/2

which can be rewritten as

"T P(') + P(- )[Rp(' )]2

R ()= 1 2 ) (3-138)

P 2 1 -p ()

where

R () = (3-139)

P 1 2(.)

Let T.0, and observe that Rp(0) = 0, so

Rp (0) = lim R (r) = oo (3-140)

This implies that

B = oo (3-141)

Sp

The second moment bandwidths ofi(t) and q(t) are

51

Bsi = Bx (3-142)

and

Bq = not defined. (3-143)

3.4.3 SSB-LM

Expressions for the second moment bandwidths of a(t) and

p(t) cannot be obtained because the autocorrelation functions

are not available. Equation (2-27) can be used, but unfortu-

nately the evaluation of the required expectations is not

physically tractable.

The second moment bandwidths of the quadrature components

are easily found by inspection

B = Bs (3-144)

and

Bs = BSA = B (3-145)

q x x

3.4.4 PM

The second moment bandwidths of the amplitude and phase

function are easily found

B = 0 (3-146)

and

B = B (3-147)

p x

The bandwidth expressions for the quadrature components

are obtained as follows. For the real quadrature component,

-Dp (0)sinh(Dp)

B2 P (3-148)

si (21)2cosh(Dp)

or

2 = D2 BS tanh(D2). (3-149)

si p x p

Similarly,

B2 = D2 Bx coth(D ). (3-150)

Sq p sx p

This illustrate that the i(t) and q(t) components require

different bandwidths.

3.4.5 FM

For the FM case, use the same Equations as for PM, except

that D2 is replaced with R(0),

D2 f G(w)

R(0) =- w dw (3-151)

-00

where Gx(w) is given by Equation (3-109).

3.4.6 CSSB-AM

Since the expressions for the autocorrelation function of

i(t) is not available, the calculation of the second moment

bandwidth of i(t) is not possible. Since q(t) also depends

on i(t), the bandwidth for this signal cannot be found.

The expression for the bandwidth of a(t) is the same as

Equation (3-131) found for AM. The phase function has an

autocorrelation function given by Equation (3-119) from which

the second moment bandwidth is obtained

2 B2 [m2 + m4

B = B 2 2 + m if mx(t)cl. (3-152)

sp sx m2 + m4

which illustrates a dependence of the phase function band-

width on the modulation index.

3.4.7 CSSB-PM

The second moment bandwidth of the amplitude function is

found by substituting Equation (3-124) into Equation (2-24).

This yields

Bsa = DpBsx (3-153)

The phase function has a bandwidth of

B B (3-154)

sp x

The bandwidth of the quadrature components is found using

Equation (3-126) and the fact that both components have the

same autocorrelation function. It is easily shown that

Ri (0) = 2DP (0)Ri(0) L2Dp?(0)]2Ri(0); (3-155)

therefore,

B2 = B2 = 2D 2 B + 2D2 (0)]2. (3-156)

For the case of a bandlimited ZMGN message with flat spectra,

bandlimited to W radians/sec.,

sin(Wr)

p(r) = S -W (3-157)

SO

-B

A(0) = -W (3-158)

2 2 \F_

where W is the absolute bandwidth of the message. Substitu-

tion of this equation into Equation (3-156) yields

2

2 2 i2 2+ j

B = D B2 2 + D (3-159)

si p x

which shows a faster increase in bandwidth as Dp increases

compared to the FM case.

3.4.8 CSSB-FM

The equations for CSSB-FM are easily obtained by re-

placing R((O), given by Equation (3-151), in place of Dp in

Equations (3-153) thru (3-159).

3.5 Summary

This chapter presented additional complex envelope pro-

perties that are not available in the literature. Mathema-

tical expressions were derived for the second moment band-

width and the autocorrelation function of the complex enve-

lope components for most of the modulation laws under consi-

deration. The bandwidth equations show how the complex enve-

lope bandwidth is related to the modulation law parameters

and the message statistics.

The equations obtained for the bandwidth of CSSB-PM qua-

drature functions reveals that these functions may require

55

larger bandwidth as compared to their double-sided counter-

parts, but any comparison must be done on the basis of a value

of Dp and the message statistics.

CHAPTER IV

THE UNIVERSAL TRANSMITTER

4.1 Introduction

This chapter presents the idea that a transmitter can be

designed so that it is capable of producing almost any type

of modulated carrier. The general structure of the transmit-

ter was already shown in Figure 1.1. First, the complex en-

velope is generated from the message and the modulation law.

A complex carrier is subsequently modulated with this complex

envelope, the real part is extracted and amplified. This is

an universal transmitter since the baseband preprocessor can

be programmed to implement the desired modulation law. The

block diagram of this system is shovn in Figure 4.1.

The system proposed in the previous paragraph is unreal-

izable because it requires the generation of complex-valued

voltages. A complex function can be described by real-valued

functions, so by manipulation of the different representations

of the complex envelope specified by the operations illustrated

in Figure 4.1, a physically realizable system can be obtained.

In a practical transmitter it is always possible to sepa-

rate the power amplifier from the actual modulator circuit if

a linear amplifier is used. Since amplification is a linear

operation, nothing is lost in the analysis if the power ampli-

fier is disregarded. This system without power amplification

will be defined as the universal modulator.

56

BASEBAND COMPLEX REAL PART

.PREPROCESSOR MULTIPLIER OPERATOR

COMPLEX

MESSAGE CARRIER POWER

SOURCE GENERATOR AMPLIFIER

Physical Modulated

Carrier

Figure 4.1. Block diagram of the universal transmitter.

4.1.1 The Complex Envelope as a Vector

It is well known that a complex-valued function can be

represented as the trajectory of the tip of a vector in the

two-dimensional plane described by the real and the imaginary

axes. A vector in a two-dimensional plane can be described

in terms of its phase angle and its magnitude or in terms of

a linear combination of two nonparallel vectors. Four of

these possibilities are illustrated in Figure 4.2. The polar

representation consists of the magnitude, a(t), and the phase

angle, p(t). The quadrature components are two orthogonal

vectors that represent the quadrature functions, i(t) and q(t).

These are the conventional representations for a complex-valued

,(t)

(a)

(t)

V(t)

v(t)

(c)

61

/\ Re

1(t) If 1

i---- -2 3(t)

2 2(t) R

L

"2 Re (d)2

6-1 L _r2(t)

(e)

Figure 4.2. Possible vectbr representations of the complex

envelope, (a) the complex envelope vector, (b) polar compo-

nents, (c) quadrature components, (d) two amplitude-modulated

vectors, and (e) two phase-modulated vectors.

function and correspond to the sketches shown in Figure

4.2(b) and 4.2(c).

There are two more representations that are a direct

consequence of the vector representation. The linear combi-

nation of two nonorthogonal vectors is shown in Figure 4.2(d).

Here the magnitudes of the two vectors are adjusted to repre-

sent the complex envelope vector while their phase angles re-

main fixed. In the last case, illustrated by Figure 4.2(e),

the length of the vectors remain constant and the phase angle

is adjusted. These equivalent representations will generate

the same complex envelope, but a certain representations may

have economic advantages in terms of circuit realization

energy consumption.

4.1.2 Physical Modulators

This section is concerned with the models for the physi-

cal modulators. It is necessary to understand the modulator

limitations in order to take them into account in the design.

The function of the modulator is to alter the properties

of the carrier in accordance to the modulating signal. Ideal

modulators do just that, but practical modulator circuits have

definite maximum input levels and a limited frequency response.

If the maximum input level is exceeded, the circuit may not

operate as intended or the circuit is damaged. The effect on

the modulated carrier is distortion or no carrier at all. The

limited frequency response must be taken into account because

if the modulator circuit cannot follow the modulating signal,

the signal modulated into the carrier is different from what

it is supposed to be.

The limitations on the maximum input level and on the

frequency response can be modelled as a saturating amplifier

and an ideal lowpass filter respectively. The modulator can

be approximated as the cascade combination of these two blocks

followed by an ideal modulator. There are two possible arrange-

ments that are not equivalent. The final selection depends

on which model describes better the modulator under considera-

tion. It is possible that the maximum input level might be

frequency-dependent or that the frequency response might be

input-level-dependent. In these cases a new model is necessary.

Signal levels can be attenuated, so the maximum input

level problem can be solved in cases where the message does

not have such a large dynamic range that circuit noise becomes

a problem. The solution to the frequency response problem is

not that simple because there is no way of reducing the band-

width occupied by a signal without changing the signal itself.

This means that the modulator model can be simplified to the

cascade combination of an ideal lowpass filter and an ideal

modulator.

There are two basic modulators. These are the amplitude

modulator and the phase modulator. The ideal modulator is de-

fined by the following equation

Ka[l + Kbe(t)]cos(wot), if Kbe(t)- -1

y(t) =

0, if Kbe(t)< -1, (4-1)

where e(t) is the input modulating signal, Ka is the carrier

peak amplitude in the absence of a modulating signal, Kb is

a modulator constant, wo is the carrier frequency, and y(t)

is the modulated carrier.

The ideal phase modulator is defined by following equa-

tion:

Kacos[wot + Kce(t)] iflKce(t)l,<-

y(t) = -

y(-Kacos(wot), if IKce(t)l> (4-2)

where e(t), Ka, and y(t) are the same as above; and K is a

c

modulator constant. This equation assumes that the modulator

is not capable of producing a phase shift of more than iT ra-

dians. This is in agreement with concept of minimum-phase

signal defined in Section 2.1. If this restriction is lifted,

the inequalities in Equation (4-2) have to be modified. This

kind of modulator is defined as an ideal nonminimum-phase

modulator.

Two ideal amplitude modulators can be combined as shown

in Figure 4.3 to form an ideal balanced modulator. This is

described by the following equation:

y(t) = Kae(t)cos(wot). (4-3)

If the ideal phase modulator is proceeded by an ideal in-

tegrator, an ideal frequency modulator is obtained. It is

described by the following equation:

t

y(t) = Kccos[wot + Kc Je(u)du]. (4-4)

Note that for most frequency modulators the phase is not res-

tricted to +V.

e(t)

-1 AMPLITUDE

MODULATOR

OSCILLATOR

cos(wot)

Figure 4.3. The balanced modulator.

A balanced modulator can be used in place of an ampli-

tude modulator if a constant is added to the input. If eb(t)

is the input to the balanced modulator and ea(t) is the input

to the amplitude modulator, they are related by

eb(t) = [1 + Kbea(t)]. (4-5)

The phase modulator can operate as a frequency modulator,

if the minimum-phase condition is lifted. If the modulating

signal has a zero mean value, the frequency modulator can re-

place a phase modulator with a differentiator at the input.

If the modulating signal has a nonzero mean value the differ-

entiator will destroy the mean value. Rowe [13] considers in

detail the relation between the PM and FM functions and their

power spectral density limitations.

4.1.3 The AM/PM Modulator

The first structure to be considered to serve as an uni-

versal modulator is the AM/PM modulator [6, 7]. It consists

of a phase modulator followed by an amplitude modulator to-

gether with a pair of baseband preprocessors. The baseband

preprocessor #1 generates a(t) while the baseband preproces-

sor #2 generates p(t). To relate a(t) to the input of the

amplitude modulator, compare Equation (4-1) with Equation (2-1)

and observe that

a(t) = Ka[1 + Kbea(t) (4-6)

so the input to the amplitude modulator, ea(t), is given by

e (t) [a(t) -1 (4-7)

a K I Ka

To determine the input to the phase modulator, compare Equa-

tions (4-2) and (2-1). These equations require that

p(t) = Kcep(t), (4-8)

where ep(t) is the input to the phase modulator. The required

arrangement is shown as a block diagram in Figure 4.4.

Observe that if Equation (4-5) is used, a simpler block

diagram results because the amplitude modulator can be re-

placed by a balanced modulator. If eb(t) is the input to the

balanced modulator, it is easy to see that

eb(t) a(t), (4-9)

Ka

This arrangement is shown in Figure 4.5 and from now on will

be referred to as the AM/PM modulator.

a(t)

x(t)

Figure 4.4. The AMi/PM modulator.

(t)

Figure 4.5. The improved AM/PM modulator.

4.1.4 The Quadrature Modulator

The arrangement of two balanced modulators in phase qua-

drature is a well known method of generation SSB-LM [47]. The

idea has been extended to other types of modulation by Thomas

[14] and others. The block diagram for the quadrature modu-

lator is shown in Figure 4.6. It is desired to generate Equa-

tion (2-7) with the complex envelope quadrature components,

so

y(t) = Re{[i(t) + jq(t)]exp(jwot)}. (4-10)

This can be simplified to

y(t) = i(t)cos(wot) q(t)sin(wot), (4-11)

so the baseband preprocessor #3 has to generate i(t) while

the baseband preprocessor #4 generates q(t). The i(t) and

q(t) functions will be defined as the quadrature components.

These quadrature components are related to the inputs of the

balanced modulators by

ebi( i(t = (4-12)

ebi(t) Ka

and

e (t) = (t) (4-13)

q K

4.1.5 The PM/PM Modulator

The idea of combining two phase modulators to generate

an AM carrier was proposed by Chireix [48]. This system is

i(t)

x(t)

q(t)

BASEBAND BALANCED

PREPROCESSOR MODULATOR

900

OSCILLATOR go

PHAASE

cos(wot) SHIFTER

Figure 4.6. The quadrature modulator.

x(t)

The PM/PM modulator.

Figure 4.7.

marketed by RCA under the trademark "Ampliphase". The same

idea can be extended to generate other types of modulation.

If Figure 4.2(e) is examined, it is seen that the complex en-

velope can be obtained by adding two constant amplitude vectors

with an specific phase arrangement. This is equivalent to the

addition of two phase modulated carriers with constant ampli-

tude. The block diagram of the proposed system is shown in

Figure 4.7. The details will be worked out later.

4.1.6 Other Configurations

It is possible to obtain the complex envelope by com-

bining two vectors that are not orthogonal as shown in Figure

4.2(d). The required block diagram of a system like this is

similar to the block diagram for the quadrature universal mo-

dulator, except that the carriers going into the balanced mo-

dulators are not in phase quadrature. Observe that in general

'(t) = u (t) + 2(t), (4-14)

but each vector 'l(t) and u2(t) can be written as a linear

combination of 1(t) and q(t). Let ki and kq be unit vectors

in the real and imaginary axis directions, then

ul(t) = alki + a2kq (4-15)

and

=2(t) = blki + b2kq. (4-16)

Observe that

al = Iul (t) cos(b1),

a2 = ulU(t)l sin(61),

bl = 1'2(t) cos(62),

b2 = u2(t)l sin(b2);

where u-l(t)l

sines,

and |i2(t) are obtained using the law of the

a(t)sin[2 p(t)

sin( 1 2 + Y)

and

a(t)sin[p(t) ]

sin( 1 2 + )

which simplify to

q. (t)| =

and

-i(t)sin(62) + q(t)cos(62)

sin(61 62)

-q(t)cos(1b) + i(t)sin(6l)

sin( 6 6 2)

Substitute Equations (4-23) and (4-24) into Equations

thru (4-20) and substitute these into Equations (4-15) and

(4-16) to obtain:

-i(t)sin(S2) + q(t)cos(62) c

si* cos( )k

sin(61 b2)

+ sin(62)k

(4-25)

and

(4-17)

(4-18)

(4-19)

(4-20)

(4-21)

(4-22)

(4-23)

(4-24)

(4-17)

ri (t) =

I (t)

[U2 (t)

and

1(t) --q(t)cos(61) + i(t)sin(61)

sin(61- 62)

S[sin(sin()i+ in(+2)gq], (4-26)

so ul(t) and i2(t) can be expressed in terms of a linear com-

bination of i(t) and q(t); therefore, the analysis is very

similar to the quadrature modulator case. This nonorthogonal

representation has a serious disadvantage. The magnitude of

the vectors i(t) and q(t) is never greater than the magnitude

of v(t). The magnitude of 1(t) and u2(t) can be several times

larger than the magnitude of v(t) as shown by Equations (4-21)

and (4-22). This can be avoided with a slight modification

of&1 and 62, for example, 00 and 900. This results in the

quadrature modulator which has been considered before.

Meewezen [21] proposed a system where the modulation pro-

cess is done by independent modulation of the sidebands. This

will be shown to be similar to the quadrature universal modu-

lator. The complex envelope is obtained as the combination

of two functions,

v(t) = ru(t) + rl(t) (4-27)

where ru(t) represents upper sideband modulation and rl(t)

represents lower sideband modulation. These two signals are

single-sided; therefore, they are Analytic, so each one can

be described as

70

r (t) = u(t) + ju(t) (4-28)

and

rl(t) = 1(t) jl(t) (4-29)

where u(t) and l(t) are real-valued functions not yet deter-

mined. Compare these equations with Equation (3-2) and ob-

serve that

i(t) = u(t) + 1(t) (4-30)

and

q(t) = u(t) 1(t), (4-31)

so

i(t) q(t)

u(t) = (4-32)

2

and

i(t) + q(t)

l(t) = (4-33)

2

This means that the analysis done for the quadrature modu-

lator can be applied to Meewezen's approach to modulation.

The block diagram of this system is shown in Figure 4.8.

Observe that only i(t) and q(t) have to be obtained, that the

system requires two SSB-LM modulators and that a Hilbert trans-

former is necessary. The complexity of such system is not

compensated by any advantage over the other systems already

under consideration. For this reason this will not be con-

sidered in detail.

ru(t)

Figure 4.8. Meewezen's independent sideband modulation

in terms of the quadrature components.

y(t)

72

Cain [22] studied the generation of CSSB modulations with

a SSB-LM transmitter. This is similar to the Meewezen's ap-

proach if only one sideband is modulated.

4.1.7 Criteria for Comparison and Evaluation of the Modulators

The two main criteria for comparison are the bandwidth

required by the signals and their dynamic range. These are

the two most important factors to consider in the design and

it is important to study their effect for different types of

modulation laws.

The point where the signal will be studied is at the out-

put of the baseband preprocessor. These are the reasons to

justify that selection: (1) this is the point where all mo-

dulation-type-dependent signal-processing ends and where the

signals go to the modulation-law-invariant part of the system;

(2) this is the point where digital-to-analog (D/A) conversion

is most likely to take place; and (3) at this point the signal

bandwidths are independent of the properties of the actual

circuits that will be used to build the baseband postproces-

sor or the modulators.

The modulation parameters and the message signal have in-

fluence on the bandwidth of the complex envelope component

functions and their dynamic range. In addition, the mathema-

tical definition of these components may impose additional

constraints that must be considered.

Bandwidth was defined in five different ways in Section

2.5.1. The definition to be used in this discussion is the

equivalent-filter bandwidth because it is directly related to

the actual bandwidth required for the system. The measure of

error is the intermodulation distortion (IM) at the output of

an ideal receiver in the absence of noise. The analysis and

calculations will relate the bandwidth of the complex envelope

components at the output of the baseband preprocessor to the

distortion of the message at the receiver.

The method to be used to calculate the IM distortion is

best illustrated by Figure 4.9. There is a control signal

that is obtained from the output of the baseband preprocessor,

modulated, and demodulated by the receiver. The spectrum of

the demodulated control signal is calculated to detect any

nonlinearities acting on the message signal. Ideally, the IM

distortion should be zero. In practice it is nonzero due to

round-off errors and aliasing of sampled signals. The control

signal is considered acceptable if the IM distortion is less

than 0.0005%. The signal whose bandwidth is to be measured

is filtered by an ideal lowpass filter and from there on it

follows the same path as the control signal.

It is necessary to perform three different tests because

two complex envelope components are involved. For example,

the first test consists of filtering a(t) and combining it

with the control component, p(t). The second test reverses

the situation, that is, a(t) is the control component. The

third test involves filtering both a(t) and p(t). The pur-

pose of the test is to determine if a(t) or p(t) requires

more bandwidth than the other, and to check what will happen

if both components are filtered simultaneously.

Control Signal

Distortion

Filtered Signal

Distortion

Figure 4.9. Method used to calculate the equivalent-filter

bandwidth.

The test signal consists of two tones with equal ampli-

tudes, but at two frequencies, fl and f2, that are not harmo-

nically related. The intermodulation distortion is calculated

following Equation (2-31), but instead of including all the

frequencies to infinity, it is modified to include all compo-

nents up to the third harmonic of the first order intermodu-

lation product (fl + f2). The test message signal is shown

in Figure 4.10. The frequency scale is easily changed to re-

present other tones that have the same frequency relation-

ship. Since the signal is deterministic, the peak amplitude

is normalized to unity so the modulation parameters take care

of the denormalization. For gaussian random signals the

theoretical peak amplitude is infinite so it is necessary to

normalize the variance and ignore those amplitudes that exceed

a given level, for example, five times the standard deviation.

Another consideration in measuring bandwidth is that

there are cases, CSSB-AM is an example, where one of the com-

plex envelope components (p(t) in this case) is ignored by

the ideal receiver in the noiseless case. The purpose of the

Ix(f)I

4 f

-5 -2 0 2 5

Figure 4.10. Magnitude of the spectrum of the test message

signal.

neglected component is to produce a single-sided modulated

carrier. The effect of filtering this component is reflected

as incomplete sideband suppression and can be measured as the

ratio of the power in the unwanted sideband to the total power,

expressed as a percentage. This is given by the following

equation:

P

S() = x 100% (4-34)

Pt

where P is the power present in the suppressed sideband, Pt

is the total power, and S(%) is defined as the sideband sup-

pression factor expressed as a percentage.

The calculations of the intermodulation distortion and

the suppression factor were obtained with a computer simula-

tion of the systems shown in Figures 4.6, 4.7, and 4.8. This

requires the use of discrete signals to represent continuous

signals. This was discussed in Section 2.6. To avoid errors

due to aliasing of nonbandlimited signals, the control signal

is monitored and the sampling rate is adjusted to reduce the

distortion to acceptable levels.

The results obtained in Chapter III for the complex enve-

lope components, their autocorrelation functions, and the se-

cond moment bandwidth are used to check the reliability of

the calculations. They are not directly applicable because

these autocorrelation functions were derived for the special

case of a zero mean gaussian random process. However, the

same general properties have to apply.

4.2 The AM/PM Modulator

This section analyses the AM/PM modulator whose block

diagram is shown in Figure 4.5. This system was described

in Section 4.1.3. The mathematical expressions for the trans-

fer characteristics of the baseband preprocessors #1 and #2

are the same as the equations obtained in Section 3.2 for

the magnitude and phase of the complex envelope. These func-

tions are summarized on Table IV-1.

4.2.1 Bandwidth Requirements

The bandwidths required for the signals a(t) and p(t)

are defined in terms of the equivalent-filter bandwidth.

These are calculated following the procedure outlined in Sec-

tion 4.1.5.

The equivalent-filter bandwidths of the a(t) and p(t)

functions for AM, PM, and FM are easily calculated. The only

nonlinearity is the addition of a constant in the AM a(t) func-

tion, In these cases, the equivalent-filter bandwidth can be

defined for zero distortion at the receiver. This is achieved

whenever the ideal lowpass filter shown in Figure 4.9 is at

least as wide as the message absolute bandwidth B. For the

cases under consideration, the equivalent filter bandwidth is

B. In practice all signals are considered to be bandlimited

[49].

In the DSB-LM and SSB-LM cases, a computer simulation of

the system is necessary due to the complexity of the analysis.

The computer program used in the simulation is listed in

Appendix B. The results of the simulation are presented as

Table IV-1.

Magnitude and Phase Functions for the AM/PM

Modulator.

Baseband Preprocessor Transfer Characteris-

tics.

Type of #1 #2

Modulation a(t) p(t)

AM C[i + mx(t)] 0

LM C x(t) sgn[x(t)]]

SSB-LM C x2(t) + 2(t) arctan[x(t)/x(t)]

PM C D x(t)

t

FM C Df j x(u)du

CSSB-AM C[I + mx(t)] H-ln[l + mx(t)]}

CSSB-PM C exp[-Dpx(t)] Dpx(t)

t t

CSSB-FM C exp [-Df I (u)duj Df x(u)du

-o>

a graph that indicates the relationship between the calcu-

lated intermodulation (IM) distortion and the equivalent

filter bandwidth. The plots for the a(t) and p(t) components

for DSB-LM and SSB-LM are shown in Figure 4.11. The band-

width scale is normalized with respect to the message equiva-

lent-filter bandwidth. These plots were obtained for the

case where one of the two components is filtered and the other

is taken from the experimental control signal. The cases where

both a(t) and p(t) are filtered are very close to the plots

for the filtered phase alone. These plots indicate that the

SSB-LM phase function requires the widest bandwidth of all

the modulation types under consideration. The reason for such

a large bandwidth is that p(t) exhibits phase jumps in the

SSB-LM and DSB-LM cases.

In the CSSB cases, there is usually a component that is

ignored by the receiver. For example, CSSB-FM is detected by

an FM receiver that ignores any amplitude modulation present

in the carrier. This component is necessary to obtain a

single-sided modulated carrier; therefore, its bandwidth must

be determined. The p(t) function for CSSB-PM and CSSB-FM,

and the a(t) function for CSSB-AM are identical to the PM, FM,

and AM cases already considered. Instead of trying to mea-

sure the distortion in the "unused" component it is better to

measure how this component affects the undesirable sideband

suppression. The procedure was outlined in Section 4.1. The

sideband suppression factors,S(%),for all the single-sided

carriers under consideration are presented as a function of

10.

5.

2.1

1.0-

II

0.5

!I /DSB-LM

DSB-LMl

a(t)

0.051

H, a(t)

0.02

0 20 40 60 0 100 120 140 160 180 2b0 220

Normalized Equivalent-Filter Bandwidth (Bf/B)

Figure 4.11. IM distortion of the AM/PM modulator a(t) and

p(t) functions for DSB-LM and SSB-LM as a function of Bf.

the modulation parameters in Figures 4.12 thru 4.15.

Figure 4.12 presents the S(%) curve for the SSB-LM case.

This curve shows that a(t) and p(t) also affect the distri-

bution of power between sidebands. Again, the SSB-LM phase

function demands more bandwidth than all the other cases under

consideration. This is expected because a large distortion

figure was obtained at the output of the receiver indicating

that the real part of the complex envelope is distorted; there-

fore, it is reasonable to expect that the imaginary part is

also distorted and the resulting complex envelope is far from

being Analytic.

The plots for FM and CSSB-FM are in terms of D, the de-

viation ratio, defined as [1 ]

fd Df

D (4-35)

B 21TB

where fd is the frequency deviation in Hz, B is the message

bandwidth, and Dfis the frequency deviation in radians/sec.

The CSSB plots for the sideband suppression factor are

presented in Figures 4.13 thru 4.15. These curves show how

the equivalent-filter bandwidth normalized with respect to

the message bandwidth is related to the modulation constants

when the sideband suppression factor is used as a parameter.

These modulation constants are the AM modulation index in the

CSSB-AM case, the PM modulation index in the CSSB-PM case,

and the FM deviation ratio in the CSSB-FM case. These modu-

lation constants were defined in Equation (3-3), (3-31), and

(4-35) respectively. All the curves exhibit the same general

10.

5.

2.0(t)

1.0-

0.5-

0.2-

0.1-

0.05-1 a(t)

0.02-

0.01 "'" i

0 20 40 0 80 100 120 140 160

Normalized Equivalent-Filter Bandwidth (Bf/B)

180 200

Figure 4.12. Sideband suppression factor (S(%)) of the

AM/PM modulator a(t) and p(t) functions for SSB-LM as a

function of Bf.

Normalized

Equivalent

Filter

Bandwidth

Bf/B

4-

I I I yI i -I I Ii- i

0.2 0.4 0.6 0.8

AM Modulation Index (m)

1.0

Figure 4.13. Relation between S(%), Bf, and m for the

CSSB-AM p(t) function for the AM/PM modulator.

Normalized 4

Equivalent

Filter

Bandwidth

Bf/B

1.0

Figure 4.14. Relation between S(%), Bf, and Dp for the

CSSB-PM a(t) function for the AM/PM modulator.

s() = o.5%

3 s(M) = 0.1%

s(M) = 0.01%

Bf/B

4- S(W) = 0.01o

S() = 0.1%70

2--

1 0.5 1.0 1.5 2.0

FM Deviation Ratio (D)

Figure 4.15. Relation between Bf, D, and S(o) for the

CSSB-FM a(t) function.for the AM/PM modulator.

Table IV-2.

Equivalent-Filter Bandwidth Requirements for

the a(t) and p(t) Signals in the AM/PM Modu-

lator.

Equivalent Filter Bandwidth (Hz.)

Type of

Modulation a(t) p(t)

AM B 0

DSB-LM Figure 4.11 Figure 4.11

SSB-LM Figure 4.11 Figure 4.11

PM 0 B

FM 0 B

CSSB-AM B Figure 4.13

CSSB-PM Figure 4.14 B

CSSB-FM Figure 4.15 B

85

trend. The bandwidth requirements increase with the modula-

tion parameters. In the worst case, an equivalent-filter

bandwidth of 2 or 3 times the original message bandwidth is

satisfactory.

The equivalent-filter bandwidth requirements of all the

cases considered in this section are summarized on Table IV-2.

4.2.2 Dynamic Range

Since physical modulators have definite maximum input

levels, it is clear that the modulation parameters may be

constrained to a particular range. Let Eb and E be the max-

imum input allowed to the balanced modulator and the phase mo-

dulator respectively. Using Equations (4-8) and (4-9) it is

clear that the maximum values allowed for a(t) and p(t) are

max a(t)] KaEb (4-36)

and

max[p(t)]< KcEp, (4-37)

where KcEp is the maximum phase shift produced by the phase

modulator.

Assume that the message is normalized to unit peak ampli-

tude. The constraints on the unmodulated carrier amplitude,

C, and the modulation parameters can be found by substituting

the expressions listed in Table IV-1 for the a(t) and p(t)

functions.

Consider the AM a(t), Equation (4-36) requires that

max[C[1 + mx(t)]] KEb (4-38)

but since

max[x(t)] = 1, (4-39)

this requires that

C 'aEb (4-40)

1 + max(m)

if m is specified,or

KaEb C

max(m) K (4-41)

when C is specified. Since p(t) is zero, no constraint is

placed by the phase modulator.

Take the DSB-LM case, Equation (4-36) requires that

max[C Ix(t)I]< KcEb, (4-42)

so using Equation (4-39), the carrier peak amplitude, C, is

constrained to

max(C) KcEb. (4-43)

The phase function has only two values, 0 or-a; therefore,

Equation (4-37) requires that

-T:KcEp (4-44)

so the phase modulator should be capable of producing at least

a +1800 phase shift.

The SSB-LM case has a new mathematical constraint. The

peak value of x(t) depends on the waveform of x(t). Squires

and Bedrosian [50] studied the problem of the peak to average

power ratio for a deterministic signal and found that it can

vary from unity for a sine wave to infinite for a square wave.

It is only possible to define

Xp = max[x(t)], (4-45)

where xp is the Hilbert transform peak amplitude. The cons-

traint on the carrier constant, C, can be found in terms of

xp. Substitute the expression for a(t) for the SSB-LM case

in Equation (4-36)

max[C[x2(t) + 2(t)] KcEb (4-46)

using Equations (4-39) and (4-45), Equation (4-46) simplifies

to

KaEb (4-47)

max(C)< a 2 (447)

[1 +x p

Since C is related to the average power of the SSB-LM carrier,

Equation (4-47) puts a limit on how high this power can be.

The phase function is constrained to the 1800 because tan-l(*)

is constrained to that range; therefore, the phase modulator

must satisfy Equation (4-44).

The PM and FM magnitude functions are equal to the con-

stant C. The only restriction placed on C is that

max(C) < KaEb* (4-48)

To obtain the restrictions on the PM modulation index, Dp,