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dissertation entitled

A SIMULATION ANALYSIS OF INTERFERENCE
INDUCED BY A FREQUENCY HOPPING SIGNAL
TO AN FM SIGNAL

presented by

Ghulam Haider Raz

has been accepted towards fulfillment
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Ph.D. degree in Electrical
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A SIMULATION ANALYSIS OF INTERFERENCE
INDUCED BY A FREQUENCY HOPPING SIGNAL
TO AN FM SIGNAL
By

Ghulam H. Raz

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1992

ABSTRACT
A SIMULATION ANALYSIS OF INTERFERENCE
INDUCED BY A FREQUENCY HOPPING SIGNAL
TO AN FM SIGNAL
By

Ghulam H. Raz

There is a growing interest in utilizing Spread Spectrum communication techniques
for civilian applications. As we enter to the twenty first century more bandwidth will
be required for growing communication industry. Unfortunately the existing bands are
limited and most of them are already occupied by different licensees. Most of these
users are utilizing conventional communication systems. One possible way to utilize the
spectrum more efficiently is to overlay spread spectrum systems on top of conventional
system. Two important implementations of spread spectrum are popular at the present
time, namely direct sequence (DS) and frequency hopping (FH) spread spectrum. These
two techniques will produce some interference to amplitude modulation (AM) and
frequency modulation (FM) systems. Therefore one can consider four interference cases:
i) DS to AM, ii) DS to FM, iii) FH to AM, iv) FH to FM. Frequency hopping
interference to PM systems is the most severe case. Consequently this study consider
FH interference to FM systems. The FM receiver presented adapts a zero-crossing
scheme for demodulation.

A mathematical analysis of an FM zero-crossing is presented. It has been shown

theoretically that the FM signal can be detected from its zero-crossings by a low pass

filter. A mathematical model for this detection has been developed, and expressions for
every component of the FM zero-crossing detection is presented.

In the ideal zero-crossing technique for FM demodulation which is considered, there
is no need to differentiate the modulated signal. Samples of the received signal are
stored in memory, and are immediately processed by an appropriate microprocessor.

To demonstrate the potential compatibility of spread spectrum and conventional PM,
a detection simulation study was conducted and algorithms developed for the detection.
The results of the simulation confirm that as one gets closer to ideal zero-crossing, the
performance of the detector improves remarkably. The algorithms developed were used
for simulating FM detection in the presence of white Gaussian noise and FH signals. It
is shown for the simple case simulated that FH as an interfering signal induces less
distortion to PM system than the background noise normally present. The conclusion is
that for many applications conventional FM systems could coexists with appropriately

designed frequency hopping spread spectrum systems.

DEDICATION

I dedicate this dissertation to:
my parents
my teachers

and

my family

TABLE OF CONTENTS

LIST OF TABLES vii

LIST OF FIGURES viii

CHAPTER I: INTRODUCTION 1

1.1 Spread Spectrum 1

1.2 Historical Background 4

1.3 Spectrum Management 13

1.4 Problem Identification 13

1.4.1 Literature Review 14

1.4.2 Problem Statement 16

CHAPTER II: INTERLEAVING 18

2.1 Introduction 18

2.2 interference Effects of DS on AM system ’ 19

2.2.1 Receiver Model 19

2.2.2 Signal Model 19

2.2.3 System Analysis 21

2.3 Interference effects of DS on FM system 25

2.3.1 Signal Model 25

2.3.2 System Analysis 27

2.4 Interference Effects of PH on AM and FM systems 28

2.5 Conclusion 32
CHAPTER III: MODELING AND SIMULATION OF FM

RECEIVER BY ZERO-CROSSING 33

3.1 Introduction 33

3.2 The Zero-crossing Demodulation of an FM signal 34

3.3 The Simulation Model 39

3.4 Zero-crossing Detection Algorithms 43

3.5 The Multi—tone FM Signal 50

3.6 The Frequency Hopping and Noise signals 53
3.6.1 The Noise Signal 54
3.6.2 Frequency Hopping signal 54
3.6.3 Digital Data and PN Sequence 57
3.7 PM Detection in the Presence of Noise and FH 59
3.7.1 FM Signal in the Presence of Noise 59
3.7.2 FM Signal in the Presence of PH 72
3.8 System’s Performance Measure 85
3.8.1 Signal to Noise Ratio 85
3.8.2 Calculation of SNR and SIR 86
3.8.3 Simultaneous Performance 94
3.9 Conclusion 99

CHAPTER IV: THE SPECTRUM OF FM ZERO-CROSSING 101

4.1 Introduction 101

4.2 A Mathematical Discussion of FM Zero-Crossing 102

4.3 The Spectral Components z(t) 109

4.3.1 A thee term Approximation of x(t) 109

4.3.2 Decomposition of z(t) in its Components 112

4.4 A Practical Aspect of FM Zero-Crossings 115

4.5 Summary and Conclusion ‘ 117

CHAPTER V: SUMMARY AND CONCLUSIONS 118

5.1 Interleaving Overlay 118

5.2 Zero-Crossings and Simulation 119

5.3 Areas of Further Study and Recommendations 121

5.4 Final Remarks 123
APPENDICES

APPENDIX A: SIGNAL TO NOISE AND INTERFERENCE RATIO FOR AM 124

APPENDIX B: INTERLEAVING OVERLAY 130

APPENDIX C: SIGNAL TO NOISE AND INTERFERENCE RATIO FOR FM 138

APPENDIX D: STATISTICS OF INTERFERING SIGNALS 143

APPENDIX E: SIGNALING SCHEMES FOR DS AND PH 145

BIBLIOGRAPHY 149

vi

LIST OF TABLES

Table
3.1 MSE (in vz/sec) for single-tone modulation
3.2 MSE results for multi-tone modulation

3.3 MSE results for a single-tone modulation in the
presence of noise

3.4 MSE results for a single-tone modulation in the
presence of frequency hopping interference

vii

page
48

51

59

72

LIST OF FIGURES

figure Page
1.1 Spread spectrum transmitter receiver 2
2.1 An AM receiver model 20

2.2 A overlay concept demonstrated by the spectra ofthree adjacent channels 22

2.3 FM receiver model 26
2.4 Hopping pattern for three users 30
3.1 An FM signal and its zero crossings instant 35
3.2 A plot of 9(t) versus t 35
3. 3 The message signal is recovered by a moving average scheme

m(t) lS plotted for reference 38
3.4 A block diagram of the simulation process 40
3.5a The received FM signal s(t) 41
3.5b The zero-crossings function z(t) 41
3.6 2(1), the zero-crossing of z(t) 42
3.7 A plot of m(t) and m’(t) 42
3.8 The message is recovered by Algorithml 45
3.9 The message is recovered by Algorithm II 46
3.10 The message is recovered by Algorithm III 47

3.11 The message is recovered by Algorithm IV 49

viii

3.12

3.13

3.14

3.15
3.16a
3.16b

3.17

3.18
3.1%
3.1%
3.19c
3.20a
3.20b
3.20c
3.21a
3.21b
3.21c
3.22a
3.22b
3.22c
3.23a

3.23b

The multi-tone is recovered by Algorithm 1

The multi-tone is recovered by Algorithm II

The multi-tone is recovered by Algorithm Ill

The multi-tone is recovered by Algorithm IV ‘

The noise signal n(t)

The amplitude spectrum of noise N(t)

Frequency hopping with M-ary scheme of signaling
Frequency hopping signal and its spectrum

The message signal is recovered by ALG. l
(The noise level = 10)

The message signal is recovered by ALG. I
(The noise level = 15)

The message signal is recovered by ALG. l
(The noise level = 20)

The message signal is recovered by ALG. II
(The noise level = 10)

The message signal is recovered by ALG. II
(The noise level = 15)

The message signal is recovered by ALG. II
(The noise level = 20)

The message signal is recovered by ALG. III
(The noise level = 10)

The message signal is recovered by ALG. III
(The noise level = 15)

The message signal is recovered by ALG. III
(The noise level = 20)

The message signal is recovered by ALG. IV
(The noise level = 10)

The message signal is recovered by ALG. IV
(The noise level = 15)

The message signal is recovered by ALG. IV
(The noise level = 20)

The message signal is recovered by ALG. I

(The F H interference level = 10)
The message signal is recovered by ALG. l
(The FH interference level = 15)

ix

51

52

52

53

54

54

56

58

61
62

63

65

66

67

68

69

70

71

73

74

3.23c
3.24a
3.24b
3.240
3.25a
3.25b
3.25c
3.26a
3.26b
3.26c
3.27
3.28
3.29
3.30
3.31
3.32
3.33
3.34
4.1
4.2
4.3
4.4

4.5

The message signal is recovered by ALG. I
(The FH interference level = 20)

The message signal is recovered by ALG. II
(The FH interference level = 10)

The message signal is recovered by ALG. II
(The FH interference level = 15)

The message signal is recovered by ALG. II .
(The FH interference level = 20)

The message signal is recovered by ALG. III
(The F H interference level = 10)

The message signal is recovered by ALG. III
(The FH interference level = 15)

The message signal is recovered by ALG. III
(The FH interference level = 20)

The message signal is recovered by ALG. IV
(The FH interference level = 10)

The message signal is recovered by ALG. IV
(The FH interference level = 15)

The message signal is recovered by ALG. IV
(The FH interference level = 20)

SDR versus input SNR and SIR for Algorithm I

SDR versus input SNR and SIR for Algorithm II

SDR versus input SNR and SIR for Algorithm III

SDR versus input SNR and SIR for Algorithm IV

The performance of the message under Algorithm 1

The performance of the message under Algorithm II
The performance of the message under Algorithm III
The performance of the message under Algorithm IV
The FM signal s(t) which is superimposed on m(t)

The function 20 x(t) marks the zero-crossings of s(t)
The function y(t) marks the bipolar zero-crossings of s(t)
The function z(t) marks the zero-crossings of s(t) unipolarly

The amplitude spectrum of z(t)

75
76
77
78
79
8O
8 l
82
83

84

91

92

93

95

96

97

98

103

103

104

107

111

4.6

4.7a

4.7b

4.7c

4.8a

4.8b

4.80

A.1

B.1

E. 1a

E.1b

E.1c

E.1d

m’(t) the recovered version of m(t)

The signal z(t)

Z(t), the amplitude spectrum of z(t)

The message signal m(t) and its recovered version
The signal z,(t)

21(1), the amplitude spectrum of z,(t)

The message signal m(t) and its recovered version
AM receiver Model

FM receiver Model

Digital data d(t)

PN sequence PN (t)

PN(t)d(t)

(1{[1’1‘1(t)d(t)]}/<it

xi

111

114

114

114

116

116

116

125

135

146

146

146

146

CHAPTERI'

INTRODUCTION

1.1 Spread Spectrum:

A new class of communication system named spread spectrum has been emerging in
recent years. ”Spread spectrum is a means of transmission in which the signal occupies
a bandwidth in excess of the minimum necessary to send the information; the band
spread accomplished by means of a code which is independent of data, and a
synchronized reception with the code at the receiver is used for despreading and
subsequent data recovery. " [1]

A block diagram of a general transmitter-receiver spread spectrum system is
shown in figure 1.1. From this diagram one can notice the following two important

observations:

1) Modulation and demodulation are independent of the type of
spread spectrum used. This will become more clear as one

discusses different types of spread spectrum modulation.

MESSAGE

.....................
..................

   
 
  
 
     

MODULATION

 

SPREADING
MODULATOR

 

 

 

 

 

 

 

JAMMING
NOI SE ----- spgggmc
INTERFERENCE ~
FROM OTHER -5
USERS ~.-.:i?:3: .......

 

DESPREADING
MODULATOR ' .

 

MES SAGE
DEMODULATOR

 

 

MESSAGE

Figure 1.1 Spread spectrum transmitter receiver

3
2) The spreading modulator and demodulator both are using the same

code. Thereforethe code in transmitter and receiver should be in
synchrony, otherwise one has to use an acquisition scheme until
synchrony is achieved. A discussion and generation of these codes see

[2] and [3].

Using spread spectrum techniques one can spread the power across a wide band
of frequencies. As a result the power density (watt/Hz) is very low, and make it
impossible to detect the signal without the knowledge of the spreading code. Thus the
most important application of this technique has been in the military. The military
system designers combined these techniques with the rapid evolution in integrated circuit
technology and made it possible for the development of extremely wide band direct
sequence and frequency hopping spread spectrum systems.

Bandwidth spreading by direct modulation Of a data modulated carrier by a
wideband spreading signal or code is called direct sequence (DS) spread spectrum [4].

In a Frequency Hopping (FH) signal, the frequency is constant in each time chip,
but changes from chip to chip. If the hopping rate is greater than the bit message rate,
it is called fast hopping. The number of frequencies over which the signal may hop is
usually a power of 2 [5].

There are many reasons for spreading the spectrum. Some of these are:

1) Anti Jamming
2) Anti Interference

3) Low probability of Intercept

4

4) Multiple user random access communications
with selective addressing capability
5) High resolution ranging

6) Accurate Universal Timing

The above features of spread spectrum are used primarily by military
communication systems. However, there has been a growing interest in utilizing spread
spectrum techniques for civilian systems for example mobile radio communications,
packet radio, radio telephony, amateur radio, timing and some specialized applications
in satellites. The suggested application which received considerable attention in

published literature is that of cellular mobile radio [6].

1.2 Historical Background:

Military usage of spread spectrum signal is well established. In many military

applications there is no alternative to spread spectrum.
The civil applications of spread spectrum are different in character, and represent
relatively new areas of investigation. The details of some of these cases are examined
in this chapter. The purpose of this section is to present some of the historical
background and the growing importance of this area of communication.

Preliminary discussions on a possible study of non-government uses of spread
spectrum techniques was first held between Mr. Jack Robinson of the FCC (Federal
Communication Commission) and Dr. Feisal Keblawi of the MITRE Corporation. After

their meeting on the subject, a proposal was prepared, and after coordination with the

F CC’s Chief Scientist, Dr. Stephen Lukasik, a contract was awarded. Dr. Michael

 

5

Marcus of the FCC’s office of science and technology was named as the coordinator.
The MITRE corporation prepared the following document.

”Potential Use of Spread Spectrum Techniques in Non-Govemment Applications” ,By
Walter C. Scale, December 1980, MTR - 80W335CONTRACT SPONSORzFederal
Communications Commission CONTACT:FCC-O320 Project 14570 [7].

The document which is 215 pages long consists of 7 sections and 3 appendices. A list
of individuals who had contributed to the study is given in Footnote '.

As a result of the MITRE Report, the FCC issued a notice of inquiry, Docket 81-
413. In the matter of "authorization of spread spectrum and other wideband emissions
not presently provided for in the FCC Rules and Regulations.”

This inquiry notice raised many questions in communication circles. The most important
questions which are related and, in reality motivated this study will be repeated with their
answers later.

At the April 1982 Communication Theory Workshop (Wi‘ckenberg, Arizona) the
communication theory committee of the IEEE Communication Society decided to

generate an IEEE submission to the FCC Docket NO. 81-413. A drafting committee had

 

Footnote 1. A list of contributors to the MITRE report

Yarsolav Kaminsky MITRE Joe Fee MITRE

Jim Morrell MITRE George Steelman MITRE

Art Smith MITRE Mary Ann Davis MITRE

Bill Zeiner MITRE Bob Noyer MITRE

Richard Horn MITRE Paul Ebert COMSAT

Aaron Weinberg Stanford Telecom Richard Tell EPA/Las Vegas
Ray W. Nettleton Michigan State University George Cooper Purdue University
Jerry Silverman Magnavox Charles Cahn Magnavox

Paul Goodman Bell Lab/Holmdel Thijs de Haas DOC, NT IA

Carl S. Mathews DOC, Maritime John Juroshek DOC, NTIA

Samuel Musa DOD Don I. Torrieri DOD

6

been formed which was led by Ray Nettleton of Michigan State University. At the June
ICC’82 ( in Philadelphia, PA ) the communication theory committee received the draft
which was prepared by Ray Nettleton [8], and subsequently established a "Review
Committee," led by Don Schilling of the City College of New York. After two iterations
between the Nettleton and Schilling committees a final report was generated, entitled
"Before the Federal Communications Commission, Washington, DC 20554 General
Docket No. 81-143. In the matter of: Authorization of spread spectrum and other
wideband emission not presently provided for in the FCC Rules and Regulations." The
report met the deadline set by FCC and later on was published in the IEEE
communication magazine entitled "FCC Testimony.“

In the remainder of this section the questions which are related to this study and raised
by FCC are repeated along with a summary of reply comments which were provided by
IEEE Communication Society. For details the reader is referred to The FCC Testimony
[9].

Question: What services can be accommodated by the use of spread spectrum system?
Can they be implemented by band overlay or will they require dedicated frequency

allocations?

Answer: The answer to the above question is provided in two parts. Part 1 lists the
recommended potential applications and Part 11 lists the applications which were not

recommended.

Potential Applications

a. Low Power Density Application:

This category which could cover a distance of less than one kilometer in
Open terrain includes almost all transmissiOn applications that do not
require an FCC license at that time. Some of the applications are as
follows: Biomedical and other telemetry; remote control of domestic and
other appliances and machinery; wireless telephone and microphone;
walkie-talkie; wireless intercom tracking; homing and trailing; intrusion
alarms; wireless data terminals, and radio location in urban areas. If the
total power spectral density of all simultaneous users within a given
interference zone is limited to levels below the ambient noise level these

systems could be overlaid onto existing services.

b. Multiple Access Systems:

Multi users each with low duty cycles at ranges up to 10 Km.
Representative applications are: mobile telephone and data networks,
pagers telemetry, a ”citizen’s band,” radio location and automatic status
reporting devices such as proposed for law enforcement and tracking
fleets. If the power spectral densities are higher than ambient noise then
the use of overlay would be undesirable. With appropriate constraints
overlay is always technically feasible, the question remains whether or not

it would be desirable.

8

Another area that overlay may be seriously considered is in wide industrial
scientific, medical (ISM) bands where the other uses of the spectrum are not for
communication purposes, and only a minimal set of rules relating to interference would

be required.

11. Applications Which are not Recommended:
Any spread spectrum overlay on an existing service that generates a high level of
spectral interference power.
Services that would require a dedicated band for spread spectrum use, for

example most multiple channel links that numerous transmitters are collocated.

Question: In the case of band overlay, should a spread spectrum system be required to

operate on a noninterference basis with conventional system?

Answer: Overlay will always cause perceptible interference to a narrowband service.
The interference would be smaller if an overlay is to be used to exploit an underutilized

existing service.

Question: What narrowband receiver characteristic should be considered in determining
the interference potential of spread spectrum systems to conventional narrowband
emissions? Do these characteristics affect the possibility of having spread spectrum

systems and conventional systems coexist in the same frequency band?

Answer: The response of a narrowband receiver to a spread spectrum interference

depends on both the narrowband receiver characteristics and the nature of spread

9

spectrum signal. Two important characteristics of the narrowband are: bandwidth image

rejection, capture ratio.

Question: Should each wideband modulation technique be considered on its own merits

as to the spectrum use and efficiency?
Answer: Yes.

Question: Will special test equipment be necessary to evaluate spread spectrum

emission?
Answer: Yes.

Question: Can systems which use different wideband modulation method be evaluated

by the same measurement techniques?

Answer: No.

Question: Will transmitter duty cycle affect the measurement technique and results?
Answer: Yes.

Question: If the commission chooses to authorize spread spectrum systems, will detailed

technical standards be needed?

Answer: Yes.

l0

There were some general questions about the cost of the new system, such as
cordless telephone, and questions with lengthy answers which were not reported here,
however the above questions and answers clearly show that both the industry and FCC
were interested in utilizing the spread spectrum in civilian applications at that time.

In early 1984 FCC proposed to allow spread spectrum utilization to operate on any
range of frequencies above 70 MHZ without any restriction on their occupied bandwidth
but with limited radiation. Dr. Michael Marcus of the FCC was assigned to receive
replies and comments about the further notice of inquiry.

On May 9, 1985 the FCC adopted new regulations regarding authorization of spread
spectrum and other wideband emission which did not previously exist in the Rules and
Regulation of FCC.

The new Rules and Regulation appeared in the FCC 85-245, 35747 document which
is 20 pages long. Some of the new Rules and Regulations which supports the relevance

of this study regarding frequency hopping spread spectrum are as follows:

1) Spread spectrum system may be operated in the 902-928 MHZ, 2400 -

2483.5 MHZ and 5725 - 5850 MHZ frequency bands subject to:

(a) Maximum peak output power of 1 watt.

(b) RF output power outside these bands over any 100 KHZ must be 20
dB bellow that in any 100 KHZ bandwidth within the band of operation.
(c) They will be operated on a noninterference basis to any other
operations.

((1) For frequency hopping systems at least 75 hopping frequencies,

ll

separated by at least 25 KHZ shall be used. The average occupancy time
on any frequency shall not be greater than .4 second within a 30 second

period. The maximum bandwidth of the hopping channel is 25 KHZ.

2) Spread spectrum transmitters may be operated on public safety frequencies

between 37 and 952 MHZ, subject to:

(a) Output power of 2 watts.
(b) At least 20 hopping frequencies shall be used.
(c) The average occupancy time on any frequency shall not be greater

than .1 sec. in every 2 sec.

The new Rules and Regulations clearly show that FCC acknowledges the
utilization of spread spectrum in civilian applications.

Recently FCC released another document GE docket No. 89-354 in the matter of
amending the Rules and Regulations of 1985. This document, which was adopted on
June 14, 1990 and released on July 9 and became effective August 24, 1990, relaxed
some of the conditions which were previously imposed in 1985. The conditions which

affect frequency hopping are as follows:

i) Frequency hopping systems operating in 902-928 MHZ band shall use at least
50 hopping (previously 75) frequencies. The maximum allowed 20 dB bandwidth
is set to 500 KHZ (previously 100 KHZ). The average time of occupancy can
not be greater than .4 seconds within a 20 second period.

ii) Frequency hopping systems operating in the 2400-2483.5 MHz and 5725-5850

l2

MHz bands shall use at least 75 hopping frequencies. The maximum 20 dB
bandwidth of the hopping channel is set to 1 MHz. The average time of
occupancy on any frequency can be as much as .4 seconds within a 30 second

period.

This new amendment has further promoted the use of spread spectrum in public
domain applications. At the present time someof the proposed applications have already
been realized, such as cordless telephone and microphone etc, and active research is
conducted toward implementation or improvement of new spread spectrum systems for
civilian applications [10]-[ l 7] .

Recently three prominent investigators in spread spectrum technology, namely
Donald Schilling, Raymond Pickholtz, and Laurence Milstein, reexamined spread
spectrum applications in public domain in an article " Spread Spectrum Goes
Commercial" [18]. The article reflects the great potential of spread spectrum in peace
time after so many years of cold war. The frequency spectrum is extremely congested,
and spread spectrum is going to provide more utilization of the available spectrum and
permit sharing the existing spectrum. The article also discusses many aspects of spread
spectrum, including overlay and spectrum sharing with conventional systems. The article
addresses many points that are being considered in this study.

In summary the recent regulation of FCC, the research activities in this field, and
most important of all the recent article by three famous professors in this field reveals
that this study is worthwhile. Hopefully this study will enhance spectrum sharing and

overlaying, as well as additional investigation of spread spectrum technology.

1.3 Spectrum Management:

Although there are several justifications for using spread spectrum communication
systems, there are some draw- backs also. The main problem which will be caused by
spread spectrum to narrow band system is the problem of? interference. Spread spectrum
produces some interference to existing conventional narrowband AM and FM
communication systems. Therefore there will be a spectrum efficiency dilemma. To
manage this problem there are two basic methods for implementing spread spectrum.
The first method involves locating and allocating the existing spectrums by reserving
some band of frequencies to be used by spread spectrum users. In this approach
narrowband systems would be prohibited from operating in the same band. The second
approach would allow spread spectrum systems to operate in the same band as
conventional narrowband systems. This approach will allow the spread spectrum signals
to overlay the signal on top of the existing narrowband signals.

The first method, reserving a band for spread spectrum, is not feasible or at least is
not an easy task to accomplish. If spread spectrum proved to be more efficient than the
existing narrow band system, it would gradually phase out the narrow band; and this is
an effort toward that task. Therefore a concise study of the coexistence of these two

systems is crucial.

1.4 Problem Identification:
The second method, which is overlaying spread spectrum signals on top of current
existing systems, will also cause complications. In lower frequencies where the band

is crowded, spread spectrum signals produce interference and, as a result, distortion.

14

The only frequencies for which spread spectrum overlay seems feasible are above 800
MHz; since in this portion of the band block allocation is wide, spread spectrum signals
would overlay fewer services. In any circumstance in this application, one should deal

with interferences between narrowband and spread spectrum signals.

1.4.1 Literature Review:

Besides the growing demand for spectrum resources, which was discussed in section
1.2, there is still enough room for more users. An FCC report [6] indicates that as few
as 48% of the 25-47 MHz land mobile channels monitored in Los Angeles had a very
"high occupancy” (defined as at least 60 percent peak hour message occupancy).

By overlaying spread spectrum on existing narrowband systems, it might be possible
to improve the overall efficiency of spectrum management. In any kind of band sharing
or overlay between spread spectrum and narrowband systems, three distinct types of

interferences will arise.

l) Interference to spread spectrum signals from other spread spectrum signals:
This type of interference was the topic of investigation in many papers, for example [6]
and [19]-[35]. This kind of interference mainly occurs in multiple-access applications
of spread spectrum. To study this type of interference one needs to specify the type of

application and spread spectrum technique which will be utilized.

2) Interference to spread spectrum systems from conventional broadcasting
systems: The survivability of spread spectrum signals in bands of conventional signals

operations depends on the following parameters:

15

I) Number of users in the conventional band.
11) The maximum power which is utilized by different users.
III) The frequency spacing of the conventional signals.

IV) The specific spread spectrum technique.

In direct sequence systems for example the interfering (conventional) signals are at-
tenuated by an amount equal to the receiver’s processing gain. This has been studied in
[l], [2],[36] and [37]. Slow frequency hopping systems can be distorted severely by a
number of conventional interferers at widely separated frequencies than by a single strong

narrow band interferer [38].

3) Interference to conventional services from spread spectrum signals:

This kind of interference was studied experimentally for some applications. The
elementary studies on band-sharing between spread spectrum and television signals were
published in 1978 by Ormandroyd [39] and Juroshek [40].

Ormandroyd used a black and white TV for this experiment. He found that spread
spectrum signals generated by modulating an AM signal with a direct sequence
pseudonoise code caused much less degradation than a conventional narrowband AM
signal as an interfering signal.

Juroshek found that direct sequence spread spectrum signals produce about the same
amount Of interference to color television as narrowband FM signals of the same energy
level, provided that spread spectrum bandwidth is about 2 MHz or less. For spread
spectrum bandwidth greater than 6 MHz, spread spectrum should cause little interference

relative to narrow band FM signals simply because of the TV receiver‘s ability to

l6

attenuate interference outside of its nominal 6 MHz pass band.

A frequency hopping signal can produce short bursts of relatively unattenuated
interference in a narrowband receiver. A concise comprehensive summary of methods
for analyzing spread spectrum interference to conventional receiver is given in [41].

None of the above works addressed the third kind of interference mathematically.
Therefore mathematical modeling and simulation of these interferences is the purpose of

this study.

1.4.2 Problem Statement:

There has been a more receptive attitude by the regulating agencies toward the
possibility of examining the coexistence of spread spectrum system with the narrowband
system. This has been reported by the Federal Communication Commission (FCC),
(General Docket No. 81-413 Sept. 15, 1981). In response to the FCC inquiry a reply
has been provided by the IEEE communication theory committee [42].

Most spread spectrum communication systems utilize the entire band for there
applications. An interesting application, which is also spectrally efficient, is when spread
spectrum overlays on top of narrowband signals. This process could be for various

reasons. The most important one are as follows:

I) Utilization of more frequency.
2) Employing for emergency services.

3) Using jamming in a hostile environment.

17

These kind of applications are not considered seriously. Any kind of spread
spectrum overlay or coexistence must be studied very carefully, to the extent that it either
phases out the narrowband completely, or such that it coexists with the narrowband
system on a permanent basis. Therefore the topic of this study is to determine whether
the coexistence of spread spectrum with the existing narrowband is possible; and, if it
is, what would be the tolerance of the narrowband system in the presence of spread

spectrum?

CHAPTERII

INTERLEAVING

2.1 Introduction:

In this chapter the desired AM and FM signals, together with their receiver models
are presented . The signal models of direct sequence and frequency hopping systems are

also presented. Considering these signal sets one gets four cases of interference which

are:

l) Interferences of DS to AM
2) Interferences of DS to FM
3) Interferences of PH to AM

4) Interferences of Hi to FM

From the study of the input and output signals and interferences in the presence of
additive noise one can derive expressions for input signal to noise and interference ratio
(SNIRI), and output signal to noise and interference ratio (SNIRQ). Finally from the
study of their signal to noise and interference ratios, one can determine the worst
interference case among the four cases, and in the next chapter a simulation study of the

worst case will be performed.

18

19
2.2 Interference Effects of DS System on AM System:

The system models in this case are composed of the following subsystems.

2.2.1 Receiver Model:

The AM receiver model consists of a predetection band pass filter, an envelope
detector, and a post detection low-pass filter. The band-pass filter has a bandwidth of
2wm and the low-pass filter has a band width of wm, where wm is the band width of the

message signal. A block diagram of the model is given in Figure 2.1.

2.2.2 Signal Model
The received signal is composed of three signals; the desired AM signal s(t), the
white Gaussian noise n(t) and a set of direct sequence spread spectrum signals i(t).

The desired signal s(t) can be represented by
s(t) = Am(t)cos(wct+O) . g (2.1)

where A“,(t) is the amplitude and O (which is a constant) represents the phase angle,
w,,=2‘irfc is the angular frequency of the carrier, of the AM signal.
The noise n(t) is assumed to be additive white Gaussian noise (WGN). The direct

sequence signal i(t) can be described in the form of

20

s(t)

4444444444

n(t) i(t)

 

 

 

B A N D P A 8 8

F I L T B R

 

x(t)

 

 

E N V E L O P E

D B T B C T O R

 

Yttl

 

 

P I L T B R

 

L O I P A 8 8 "

z(t)

 

Figure 2.1 AM receiver Model

21

15;; dejttmtht) sintwjgej) (2-2)

where u is the number of users, I,- is the signal amplitude, wj is the angular frequency and
O,- is the phase angle for the jth user, dj(t) is the digital data (message) carried by the jth
user and PNj(t)e{l,-1} is its pseudo-noise code sequence. Furthermore dj(t)e{l,—1} over
the code signal duration Tc for all j, j = 1,2,3, ...u. The statistical properties of interfering

signal i(t) are derived in Appendix A, and are as follows:

 

ETEFO (2.3a)
i;(t)=Ij (2.313)

where the overbar denotes the ensemble average. These expressions play an important

role in the derivation of the signal to noise and interference ratios.

2.2.3 System Analysis:

Consider the worst interference case. In this case wj=wc for all j, j= l,...,u., thus
the interfering power at the output of the predetection band-pass filter is maximum. This
kind of communication will only happen in a hostile situation i.e. in deliberate jamming
circumstance. If the spread spectrum system designer uses the guard band of AM for
spread spectrum users, or in other words interleaving the spread spectrum between two
conventional AM signals, only the side lobes of the neighboring signal will contribute
some distortion. This idea which could also be applied to FM signals is shown in Figure
2.2. Thus the output power of the band-pass filter will be reduced. Since this power

reduction is due to interfering signals, meaningful communication is possible.

22

 

 

 

 

 

User 1 Guard Band User 2 Guard Band User3

Figure 2.2 The overlay concept demonstrated by the
spectra of three adjacent channel

Now consider the AM receiver step by step. The predetection band—pass filter has
a bandwidth just sufficient for passing the desired signal, therefore it suppresses the out
of band components of noise and interfering signal. As a result one gets the desired
signal, narrowband noise and narrowband interference at the output of the band-pass
filter. Using in phase and quadrature representation of each of these narrowband signals
one can write x(t) the output of band pass filter as follows:

u
x( t) = [Am( t) +nc( t) +§ icj(t) ] cos (wct+6)

u )1 a) (2'4)
-[n,,(t)+ i, (t sin(wt+
J); .

where nc(t) and n,(t) are the in phase and quadrature component of noise and iq-(t) and
i,,-(t) are the in phase and quadrature components of interference signal for j=1,...,u.

Assuming that n(t), ij(t) and s(t) are independent random variables one can write the

23

following expression for signal to noise and interference ratio (SNIR) at the input of the

 

 

detector.
gr:
SNRI= 1 _2 7 1 u (2.5)
—2- (He-+118) +3; (1c12+'1812)
or
SNRI=SNR§ 1
(2.6)

u — —

-'r— -v— 2 2

1+; (chz+1812)/(nc+ns)
'1

where SNIR,’ is the input signal to noise ratio, and SNR,’ is the usual signal to noise
ratio at the input of the AM detector. If one denotes the interference to noise ratio by

INR, then (see Appendix A for details and an expression for INR)

SNIRI=SNR§fiIR . (2 . 7)

where SNR,’ is the ordinary input signal to noise ratio of the AM system. The key
parameter in this expression is INR. In Appendix A an expression for INR has been
derived INR=ITcu/No. The parameters in this expression are: the noise power No,
number of users It, interfering signal’s power I, and code duration Tc. From Appendix

A one gets the following expression.

I I
SNR; SNR,
SNIR = = (2 .8)
I 1+ITcU/No 1+INR

 

as INR > 0, SNIR. > SNR.’.

 

 

Similarly an expression of signal to noise and interference ratio for output of AM

24
system (SNIRQ) have been derived, the result of which is

_ SNRa (2 9)
SNR". 1+INR
where SNR; is the ordinary signal to noise ratio of the AM system.

The detection gain G is given as follows:

SNIRO
G:
SNIRI

 

01'

G: (31mg) (1+INR) = SNR$=G,.

I I (2.10)
(1+INR) (SNR!) SNRI

 

For instance the detection gain (G) in the presence of interference can be regarded as the
detection gain (6’) in the presence of Gaussian noise.

In the worst possible case which is all direct sequence spread spectrum signals
transmit on top of an AM system, that is w,- = w,2 for all j, then INR will be very large.
As a result one gets a very small SNIRO, and possibly loss of the signal.

In conclusion, for one user the performance is like noise [43]; for a large number
of users the signal will be lost if wj = wc for all j. However if one uses interleaving
overlay properly the coexistence of direct sequence spread spectrum and AM system is

feasible. For a mathematical discussion see Appendix B.

25
2.3 Interference Effects of DS on FM System:

The receiver in this case is an FM receiver, and is composed of a predetection band-
pass filter, a limiter discriminator, and a post detection low pass filter. The band-pass
and low pass filters have a bandwidth of 2wm(1+B) and wm respectively, where 13 is the
modulation index of the FM signal. The discriminator output is proportional to the time
derivative of the phase angle of the input to the discriminator. A block diagram of the

FM receiver is given in Figure 2.3.

2.3.1 Signal Model:
The received signal is composed of three signals, the desired FM signal s(t), white
Gaussian noise n(t) and a set of direct sequence spread spectrum signals i(t).

The desired FM signal can be represented by

s(t)= Acos[wct+¢m(t)] (2.11)

where A, wc and m(t) are the amplitude, the carrier frequency, and the phase angle of

the FM signal. For FM modulation system ¢m(t) has usually the following form

t
¢,(c>= wmfm(t)dr (2.12)

where wm is the maximum instantaneous frequency deviation, m(t) is the base band
modulating waveform, or the message signal. The direct sequence signal is modeled in

the previous section. The system analysis is now considered.

26

s(t)

4444444444

n(t) i(t)

 

x(t)

 

 

B A N D P A 8 8

P I L T B R

 

x(t)

 

 

L I N I T E R

D I 8 C R I N I N A T O R

 

Yltt)

 

B A 8 B B A N D

F I L T B R

 

Yzltl

A 8 8
B R

l z(t)

 

 

I."
'10
PII
1."
6'6

 

Figure 2.3 FM receiver Model

27
2.3.2 System Analysis:

Like the AM case the worst interfering case occurs when wj = wc for all j’s,
j=1,. . . ,u, because the interfering power at the output of the predetection band-pass filter
is maximum.

Again the solution is interleaving overlay. Since the interfering signal’s full power
will be filtered as described in the AM case, only the side-lobes of the neighboring signal
will contribute some distortion. Since FM operate in the MHz range, in this case one
will have enough space in the guard bands for interleaving spread spectrum signals. The
above technique is shown in Figure 2.2.

The predetection band-pass filter of FM has a bandwidth just sufficient for passing
the desired signal, therefore it suppresses the out of band component of noise and
interference. As a result one gets the desired signal, narrowband noise and narrowband
interferences. Using the in phase and quadrature representation of each of the
narrowband signals one can obtain the following expression at the output of predetection

filter [38].

x(t)=[A+nc(t)+ ic (t)]cos[wct+ m(t)]
Z3. 1 4’

 

u (2.13a)
- [n.( t) +2: i,j(t) 1 sintwct+d>mt t) J
= R( t) COS-IWct+¢m( t) +<I> ( t)]
with
R(t) =\[[A+nc(t) +§ ic,(t) ] 2+ [118(13) +2; 191(t) ] 2 (2.13b)

and

28

n,(t)+ isjtt)
O (t) =tan'1 ‘1 . (2 . 13c)

11

A+nc(t) +; icj(t)
-1

 

Using this representation one can determined an expression for SNIRl and SNIRO. These

expressions have been obtained in Appendix C. The results are:

/

 

SNIRf-ibfl- (2 . 14)
1+INR
and
I
SNIRO= SNR° (2 - 15)
1+INR

where SNR,’ and SNR,’ are the ordinary SNR for input and output of an FM system and
INR is the same as AM system except that there should be some more interference due
to digital data and discriminator of FM.

In conclusion one can observe that the form for SNIR of FM is similar to SNIR of
AM, however the interference which will be produced by discriminator will reduce the

output SNIRo of FM.

2.4 Interference Effects of FH on AM and FM Systems:

In this section a study of the interference effects of frequency hopping spread
spectrum on AM and FM systems is considered. The AM and FM signals have been
modeled in the previous sections, therefore one can start the analysis by modeling the
frequency hopping signal.

The frequency hOpping signal i(t) can be described in the following form:

29

i(t) =§ dejtt)c031wj(t) +61] (2.16)
tst<kr

where wj(t) in this case is given by the following expression [44] and [45].

wj(t) = wc + hj(t)°6w (2.17)

i = 1.....U; (k-1)r$t<kr

where 6w is the minimum frequency shift, and hj(t) is an integer for jth user such that
no other user can have that same integer in a given interval of time tE [(k-1)r,kr] with
r is the hopping duration or hOpping width, k€[l,K] with K=L=T/r, and T is the
period of one complete hopping pattern. Thus h(t) =h(t+T) and L is the number of slots
for hopping.

Consider Figure 2.4 as an example. In this example only three users are considered,
namely (i-1)th, jth, and (j+l)th users. The following values are‘chosen for different

parameters:
wc =50MHz,r= 1 ms, wi = lOkHz,T= 8ms

thus L=8. Therefore, the function h(t) will be repeated each 8 ms. The function hj(t)

is determined pseudorandomly by a PN sequence. Figure 2.4 shows the periodic nature

0f hid“), hj“), and hit-1“):

w

M

f) 2 rd C K3

30

- (j—1)th user, 44 jth user, R (j+1)th user

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

we+8wi :
. - S
4444 4444 :
4444 4444 :
0 :
4444 4444 4444 :
4444 4444 4444 :
0 :
4444 4444 444:
4444 4444 444:
0 :
we+3wi :
we+2wi :
4444 :
4444 :
we
0 11 Zr 3T 41 Sr 6T 7T 81 91
T I M E
r = 1 ms, we = 50 MHz, wi = 10 KHz

kr<t<(k+1)r k=0 k=1 k=2 k=3 k=4 k=5 k=6 K=L=8

hj_,(t) 1 2 1 7 3 l 2 1

hj(t) 4 6 5 5 0 4 6 5

hj+,(t) 3 1 2 1 7 3 1 2

 

 

 

 

 

Figure 2.4 Hopping pattern for three users

 

 

 

31

The statistical properties of i(t) for the frequency hopping signal are similar to the

direct sequence statistics and are derived in Appendix D, the results of which are given

 

as follows:
iItI=0 (2.18a)
12(t)=I. (2.1813)

Using the above statistical results one can write the following expressions for input and

output signal to noise and interferences ratios,

SNR;

SNIRI= a
1+; (re—3+3? / (n§+n§)
-1

 

(2.19)

SNR;

°- u ~___r_ .7 .3 (2.20)
1+3-1 (1e12+183)/(ne+n8)

 

where SNR,’ and SNRe’ are the ordinary SNR. If the system is an AM system, the SNR
terms are for AM; and if the system is an FM system, the SNR terms represent the SNR
for the FM system. Thus as far as mathematical analysis is concerned one gets the same
expression in both cases, but one should remember that these results are for average
SNIR’s, the instantaneous SNIR is different in each case, and will be studied by
simulation. The details of a mathematical consideration is given in Appendix B. In
Appendix E a signaling scheme for DS and FH is considered. Based on this inference
and observations of Appendix B one may come to a conclusion which is given in the next

section.

32

2.5 Conclusion:

It is shown in Appendix E that the time interval for a signal set is important for
frequency hopping only, and direct sequence is independent of these intervals. Thus the
main difference between DS and FH is the hopping function h(t), which causes an
unavoidable interference.

As a conclusion one can make the following statement: "The worst kind of spread
spectrum interference to conventional systems is the interference of FH systems to FM
systems". To see the behavior of this type of interference, a simulation study of this

case is considered in the next chapter.

CHAPTER III
MODELING AND SMULATION OF

FM DETECTION BY ZERO-CROSSING

3.1 Introduction:

In this chapter a signal model of frequency hopping spread spectrum system and FM
system will be considered. The demodulation of FM signals in the presence of
interfering FH and noise, have been presented also. The FM receiver will adapt a zero
crossing scheme for demodulation. This technique, which has existed in the
communication system literature for quite some time [49]-[53], has emerged as a major
technique for frequency discrimination and spectrum analysis [54]e[58].

The zero crossing technique will be a convenient method of demodulation that will
be used with the new generation of micro-processors, especially with those customized
digital signal processing chips which have fast Fourier transform (FFT) capability.

In the zero crossing technique of FM demodulation there is no need to differentiate
the modulated carrier signal. The entire blocks of limiter and discriminator will be
replaced by a simple zero crossing sensor; however, samples of this information need
to be stored in a block of memory. The data which is stored in the memory can be

processed with appropriate digital signal processors.

33

34

This technique can be applied to satellite communication systems as well as many
other data communication systems, such as packet switching and any time division
multiple access systems.

To demonstrate the zero crossing technique, several algorithms have been examined.
When compared with the ideal zero crossing algorithm, the result of the simulation
confirmed that as one gets closer to ideal zero-crossing, the performance of the detector

improves greatly.

3.2 The Zero - Crossing Demodulation of an FM Signal:

A discussion of the demodulation of an FM signal by zero-crossing techniques is
given in some communication texts, for example [5], [59] and [60]. In the following
discussion a unified representation of the signal, demodulation and a practical algorithm
for demodulation is summarized.

The PM signal is usually expressed as follows:

s(t)= Acosrweupfmam] (3.1)
0

Where m(t) is the derivative of the message signal, and B is a measure of the

modulation index of the FM signal. Thus one can write s(t) as follows:

s(t)= Acos[0(t)] (3.2)

with

0(1)= wct+fl f m(r)d1: (3.3)
0

35

The message m(t) can be recovered from a knowledge of the zero crossing as
follows: Let tl and t; be the times of the two adjacent zero-crossing of s(t) as shown in

Figure 3.1.

ll/\ Alllllflllllli NW.
_. UWUWWVI’VWWUV WV

Figure 3.1 An FM signal and its zero-crossings instant

 

 

H

 

 

 

 

From the above diagram one can write

900-901) = 1
900-902) = 1
900-903) = T (3.4)

The above observation can be graphed as shown in Figure 3.2.

9(t)

 

Figure 3.2 A plot of O(t) versus t

36

Let us concentrate at the two adjacent arbitrary zero crossing at t1 and t2 of Figure 3.2.

9(12)-O(t,)=(m+ 1)1r-m1r=1r (3.5)
or

‘2 ‘t
n =wct2+ [i f m(r)dt -w¢tl - [3 f m(r)dt (3'6)

0 0

Thus
‘2

n=we(t2-t,)+fl f med: (3.7)

‘1
Assuming that the FM carrier frequency we is much higher than the message’s largest
frequency we, (which is usually the case). Then in the interval of [t,, t2], m(t) can be

assumed as a constant and can be factored out of integration, thus

1r=w.(t¢-ti) +6 m(t)(t2-tt) (3.8)
t, < t < t,
01'
Iva-H9 m(t)](trtt)=w (3.8)
but the term
we+fl m(t) =d(s(t))/dt=w,(t) (3.9)

where w, is the instantaneous frequency of the FM signal. Therefore one gets:

37

 

 

w,=wc+[3m(z)=_”_ (3.10)
‘2":
or
f.(t)=f +—"-m(r)= 1 (3.11)
c 21: 2(t2-tl)
If the number of zero crossing in an interval of T is nT, then one gets
T
n a (3.12)
T t2"tr
or
T
t -t -°-— 3.13
2 I "r ( )
with
1 l
_<T<_ (3.14)
f. .. .
thus
9 "r (3 15)
= +-’" t u— e
f‘ f‘ 21: 0 2T
or
(3.16)

.21 2:-
m(t) 5 (2T f.)

This shows that the message m(t) can be recovered from the number of zero crossing

me. Other terms in this expression include a dc bias and possibly amplitude distortion

38
which ean be adjusted electronically.

In conclusion, one can recover m(t) from averaging The nT’s as shown in Figure
3.3.

Cm ’(t) m(t)

l/I\ I’ll /I\ /I\ /I\ /I\ /I\
-. \I/ \I/ \I/ \I/ \I/ \I/ \I/ \I

Figure 3.3 The message signal m’(t) is recovered by a moving average
scheme. m(t) is plotted for reference

In the above discussion one can actually recover the FM signal from its zero-
crossings by moving average method. Averaging is in reality a low-pass filtering
process; therefore, one can recover the FM signal from its zero-crossing by a low pass
filter.

0 A rigorous mathematical analysis of why one can recover the FM signal from its
zero—crossings by a low pass filter is presented in chapter 4. However, in the following
sections the algorithm and methods which are presented are based on the results of
chapter 4.

The zero crossing demodulation technique is also considered in [5] , [59] and [60] and
the realization is proposed by a four step algorithm. This algorithm together with other

algorithms are discussed in section 3.4.

39
3.3 The Simulation Model:

In chapter two it was stated that the worst type of interference which is considered
in this study is the interference of a frequency hopping spread spectrum signal with an
FM signal. It has also been established that the FM Signal can be recovered from the
zero crossings of the received FM signal, thus an FM demodulator in the presence of a
frequency hopping interfering signal and noise is considered. Since there is a growing
interest in zero crossing detection of signals and their spectra, this scheme is also adapted
for the simulation model of this study. Thus the demodulation of an FM signal by zero
crossing in the presence of frequency hopping interference and noise is modelled.

A block diagram of the simulation model is given in Figure 3.4. After the FM signal
s(t) and the frequency hopping interfering signal i(t), and the noise signal n(t) are
received in the antenna, they are added together and are applied to the first stage of the
model. The first stage is composed of a zero crossing scheme which will be discussed
in more detail in the next section. This stage simply estimates .
the location of the zero’s of incoming signal. To illustrate the technique let us consider
a tone-modulated FM signal alone and process this signal point by point in the model.

Consider an FM signal s(t) as follows:

s(t)= Acostweus f m(r)d1:) (3.17)
0

for A=20, and m1(t)=5cos101rt one gets a signal set at point 1 and point 2 of the

simulation model as shown in Figures 3.5a and 3.5b respectively.

samples ofs(t)
l44444444444444444444l

to point: 1

ZERO

 

 

CROSSING

 

 

SENSOR

F point 2

BUFFER II

If

( in frequency domain)

 

 

LPP

 

 

 

INVERSE rr'r
Iraurraa I]

n(t)

 

 

 

 

 

Figure 3.4 Block diagram of the simulation process

41

willlllllllfmltlllltl "“lltlll MW“ 1
t t lllllttll llllllllllll ll

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.5a The received signal s(t)

2: llllllll e l

Figure 3.5b The zero-crossing function z(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The next stage of the model consists of a Buffer, where an array of 512 points is
stored, and then it is used to calculate the FFT in the next stage. Stage III finds the fast
Fourier transform of the corresponding zero crossing array. This stage involves low pass
filtering in frequency domain, and that is the reason for calculating FFT of the zero-
crossing of the signal. After FFI‘ have been calculated, the lower frequency components
could be used for low pass filtering, and the high frequency components could be ignored
or simply would not be calculated. Hence, the above scheme is very suitable for VLSI
digital signal processing applications. For example one can program the chip for
calculating the FFT for lower frequencies, and that will serve as a low pass filter in the
frequency domain. Consequently this stage could be considered as two steps. In step
one the FFT is calculated, and in the second step it is low pass filtered. The result of
FFT stage gives the spectrum of the appropriate signal. This spectrum is given in Figure

3.6.

42

At

 

 

 

 

 

 

|.. 1]. ..l.Il lrluiliililliiiiill illlllr Ii llllll‘l‘llllsl l1l|||lllllll|| I I'll. Illlillliiullllllrl 11.1.. .11 .. I

0 256 Hz
Figure 3.6 2(1) The spectrum of z(t)

The last stage simply calculates the inverse Fourier Transform (IFFI‘) of the filtered
spectra or the partial FFI‘. As one expects it should return the message m(t). To _

compare the original signal m(t) with m(t) a plot of these signals is given in Figure 3.7.

\ m(t f'\ m’(t) A /'\

I

v" v v v’

 

 

 

 

Figure 3.7 A plot of m(t) and m’(t)

In summary the simulation model determines the zero crossing of an FM signal using
an array of length 512, and then it calculates the FFT of the zero crossing signal and
passes it through a low pass filter. Finally it calculates its inverse FFT (IFFI‘) which is
the approximation to the desired signal. In the next section, different algorithms have
been developed to determine the zero crossing of the incoming FM signal, and the rest
of the simulation process will not be changed. There will be some distortion and
amplitude scaling in the output, since this type of distortion is very common in practice

and have been examined. The simulation model is also adjusted to include these type of

43

distortions. The simulation in the presence of noise and interference is considered in
section 3.5 . In the next section different algorithms for zero crossings detection are

examined.

3.4 Zero Crossings Detection Algorithms:

In this section some algorithms for zero crossing detections are presented. Each
algorithm approximates the location of the zero crossings of the FM signal in the interval
of (0,21) , however none of the algorithm determines the exact locations of the zero
crossings. The first algorithm has existed in the literature for quite some time; the
second and the third algorithms have been proposed by the author; and the last algorithm,
which is called an ideal zero crossing detector, is also considered. Each of these
algorithms have been tested by the simulation model. The results are reported through
some pictorial diagrams of the original signal, and the corresponding simulation outputs.
A comparison of the results shows that the third and fourth algorithms are the ones that
shOuld be considered for the future. If the zero crossing is going to be handled by hard-
ware the third algorithm is suitable. However, if some application permits a software
handling of the data, the fourth algorithm is the one which is most suitable. Now these
four algorithms are presented according to their order of improvement toward ideal zero

crossing detection.

Algorithm I:
This algorithm is based on the discussion of section 3.2 and it is proposed in [5] and
[59]. One can refer to this as a classical algorithm. This algorithm can be realized by

a four step procedure as follows:

44

5

Half wave rectify the received FM signal s(t).

II) Differentiate to accentuate the zero crossing points.

III) Rectify once again to eliminate the negative pulses due to the negative
going zero crossings and generates z(n).

IV) Pass the signal through a low pass filter.

The above algorithm is applied to an FM signal s,(t), which is a single tone

modulation.
s,(r)= 20eos[2n 128000t+25nftcos(2n50001:dt) (3.18)
0

Notice that if one consider t in seconds, the FM signal s,(t) will have a carrier
frequency of 64 Hz. However, if one consider t in microseconds, then the FM signal will
have a carrier frequency of 128 MHz. For simplicity of discussion (without loss Of
generality) a period of 1 ms have been considered. The above signal s,(t) is sampled
with a sampling frequench of 512 Hz, and s,(n) is obtained. The signal s,(n), 512
samples of s,(t) is processed by all algorithms.

The result of the simulation when it applied to s,(n) is given in Figure 3.8.
Obviously the above classieal algorithm is working, but it can not recover the message
completely. For example there is some distortion. The MSE (mean square error) for
this algorithm is 8.71298 V’lsec.

In the following algorithm the above method is modified to improve the zero-crossing
technique. This algorithm determines the zero crossings more closely to the theoretieal

OI'ICS.

45

2(n)

2(1)

O 256 Hz

m(t)

\ /\ /\ /\ [“x /:
0\_7 \_/ \c/ \./‘\_/1s

 

Figure 3.8 The message signal is recovered by Algorithm I

Algorithm II:

In this algorithm the first stage which is a half wave rectifier is preceded by a
limiter, but this limiter is not similar to the classic limiter which was in use for FM
demodulators. This limiter uses a comparator to increase the level of a signal which falls

above some predetermined level aL as follows:

A
L(n)= Lsgntstn» Istn) | 20,, (3.19)
30‘) L904) 4“],

where AL is in the order of 1 to 5 volts and aL is a fraction of a volt, and sgn is the sign

function.

46
This algorithm is also applied to the FM signal s,(n) in order to detect m(t). The

result of the simulation shows an improvement compared to the first algorithm. The

MSE for this algorithm is 6.9826 Vzlsec. The output signal which is recovered by

algorithm 11 is shown in Figure 3.9.

 

 

l l l l I
0 1 s
2(011
7 .. -I..‘. i !ll 51 l . ..
0 256 Hz
I)
m(t)

\ /\ /\ /\ /\ /,
0V V V \/ V”

 

Figure 3.9 The message signal is recovered by Algorithm 11

Comparing Figure 3.9 with Figure 3.8 and their MSE, one can conclude that

Algorithm II demonstrates an improvement compared to algorithm 1. However,

algorithm 111 will bring further improvement.

47
Algorithm III:
In this algorithm the limiter which existed in algorithm 11 is preserved, but the first
half wave rectifier is removed. Algorithm III has the following 4 steps:
1) Limiter (As described in algorithm 11) '
2) Differentiator
3) Half Wave Rectifier, generates z(n)

4) Low Pass Filter

The result of the simulation when applied to the same signal set s,(n) is given in

Figure 3.10. The MSE in this case is 3.0626 VZ/sec.

 

 

 

l I l H I

0 ' .
2(0‘

-.---._-..11i.i. .. . ; t. .. .. .e. h . t. . w. w. ..

o w 25on
met

\ /\ /\ /\ /\ C
°V V V V V13

 

 

Figure 3.10 The message signal is recovered by Algorithm III

48

By a careful study of Figures 3.8 - 3.10 and their MSE, one can conclude that
algorithm 111 is superior to algorithms I and II. The final algorithm is a simple zero -
crossing scheme. Due to the nature of the algorithm it is referred as the ideal zero

crossing algorithm.

Algorithm IV: (Ideal Zero-Crossing)

In this algorithm, two adjacent signal samples are multiplied. If the result is
negative, this indicates a zero crossing has occurred. Thus we have the following 2 step
algorithm:

I) detect the zero-crossing by examining the product of two adjacent samples as

follows:

0 s(r)s(t+l)20
s(r)= { (3.19)

A L s(t)s(t+l)<0
2) low pass filter the result.

A simulation result of this algorithm is shown in Figure 3.11. The MSE for this
algorithm is 0.02161 V2/seO, a drastic improvement. In order to compare these

algorithms, the quantitative results or MSE results are summarized in Table 3.1.

Table 3.1 MSE (in V2/sec) results for single tone modulation

 

Algorithm I Algorithm 2 Algorithm 3 Algorithm 4

 

8.71298 6.9826 3.0626 0.02161

 

 

 

 

 

 

49

z(n)

 

2(1)

0 256 HZ

"“°\ /\ /\ /\ /\ /.
v v v v V”

Figure 3.11 The message signal is recovered by Algorithm IV

 

In all of these simulations, the signal samples which were obtained by specific
algorithms were processed in the frequency domain; for example the spectra of the
signals were obtained by taking FFI‘ of the zero crossing samples, and then low passed
filtered in frequency domain. Finally by taking inverse FFI‘ (IFFT) the message signal
is detected.

Since the low pass filtering is performed in frequency domain, it is more appropriate
to find the FFT of the signal up to the band width of the low pass filter. This will save

a great amount of time in the realization process of the system.

50
3.5 Multi-Tone FM Signal:

In the previous section the FM signal was generated by a single tone message signal.
In this section the FM signal sz(t) is generated by a multi-tone message signal m(t) as

follows:

s,(r)= 20eos(wet+ pfm(r)dr) (3.20)
0

where

3 (3.21)
m(t) =2 a,sin(21tfet)

i-l

with a, and fi being random numbers uniformly distributed in their range of variation.

For illustration purposes the following values have been considered.
a,=2 V; a2=3 V; a3=5 V; f,=4 kHz; f2=6 kHz and f3=8 kHz

The amplitudes are measured in volts while the frequencies are in kilo Hertz. Although
the choice may be random, the above values has been chosen to illustrate the multi-tone
modulation and demodulation of the FM signal s2(t). These values give a reasonable
amplitude and frequencies to compare this case with the single-tone situation.

The four algorithms which were introduced in the previous section have been applied
to the multi-tonc signal set sz(n), (512 samples of s2(t)) and the results of the simulation
are given in Figures 3.12-3.15. The MSE for these cases have been calculated and

summarized in Table 3.2.

51

Table 3.2 MSE results for multi-tone modulation

 

 

 

 

 

 

 

Algorithm 1 Algorithm 2 Algorithm 3 Algorithm 4 I
11.7085 10.8202 4.9352 0.0267 I
M r —
2(I!)
i I ..lil l
0 .1 s
2(0
0 256 Hz
A
m(t)

oVV

 

”V

\AAA /\ AAA /
\/

V v ’
Vls

Figure 3.12 The multi-tone signal is recovered by Algorithm I

 

1

Is

2(1)

0 256 Hz

WlflAAAAAA/
VVVV VVVV“

 

Figure 3.13 The multi-tone signal 18 recovered by Algorithm II

    

“U

0

z(t)“
0 256Hz

“O‘BAAAA/LAA/
VVVVV VV”

Figure 3.14 The message s1 "gnal 1s recovered by Algorithm III

 

 

53

z(n) . ,
l1!»

2(1)

 

0 256Hz

mm._\ /\/\/\ AAAA/T.
VVVVVVVV

Figure 3.15 The multi tone signal is recovered by Algorithm IV

 

These results confirm the previous inferences about these algorithms, and
furthermore these exhibit the demodulation of the FM signal in a more realistic manner.
In the following section the demodulation of the FM signal in the presence of noise and

frequency hopping interfering signal is considered.

3.6 The Frequency Hopping and Noise Signals:
Although the frequency hopping signal set was given in previous chapter, in this
section the noise and the frequency hopping signals are considered for the simulation

purpose.

54
3.6.1 The Noise Signal:

The noise signal is considered to be Gaussian, where each seed is obtained from a
uniform distribution in the interval of [-1,1], and then by a Box-Muller transformation
the Gaussian or Normal deviates are obtained, [62] and [63]. The resulting noise has a
mean of zero and a variance of unity.

The noise signal which is generated by the above scheme is given in Figure 3.16a.
The spectra of the above noise is given in Figure 3.16b. As one expects the noise has

components at every frequency.

Figure 3.16a The noise signal n(t)

   

Figure 3.16b The amplitude spectrum of noise N(f)

3.6.2 Frequency Hopping Signal:

Frequency Hopping unlike AM and PM can be represented in many ways. The most
popular representation which is M-ary Frequency shift keying MFSK has been adapted
for the simulation model, [4] and [61]. In the M-ary frequency shift keying
representation of the frequency hopping signal (FH/MFSK), the data bits determine the

increment frequency to be added to the carrier frequency. Thus in a FH/MFSK the

55
increment of frequency is added to the carrier frequency which is determined by a PN

code. Thus in this system it is very diffith for an intentional jammer to jam the
communication system. One can represent the above modulation system by the following

mathematical expression for interfering frequency hOpping signal i(t).

i(t) = Acos{[w(t)+d(t)dw]t} (3.22)

where w(t) =w,+h(t)0wi with h(t) is the hopping function which could be generated by
a PN sequence, and w, is the hopping increment to be multiplied by h(t) and then added
to the carrier frequency w,. The frequency wc in this case is the carrier frequency of the
FM system. d(t)e{0,l} is the digital data which regulates the M frequencies that each
group of bits determines. Thus dw is the increment of frequency which separates two
adjacent frequencies in the M-ary frequency shift keying. This scheme of signalling ean
be demonstrated by a frequency versus time diagram as shown in Figure 3.17. Notice
that as the time proceeds the frequency will change in each half period of the hopping
chirp due to the pattern of the data, and at the end of each hop increment the PN
sequence determines the next hopping slot by h(t), and again in every hopping slot there
are two frequency increments which are determined by d(t). A common practice is to
choose the symbol rate R, for the MSK signal as an integer multiple of the hopping rate
R.I or vice versa. Thus if R, is an integer multiple of Rh, several bits could be
transmitted in each frequency hopping. This option is adapted for the simulation model
and in each hopping slot two of the M-ary frequencies are transmitted. Since each M-ary
frequency is determined by two bits (4 frequencies), a total of four bits is transmitted in

each frequency hopping slot.

0< 02 t1] C210 MN ’11

“k + 8wi

wc+3wi

wc-t-ZWi

wc+w-

56

- (i-1)th user, . . jth user, m (j+l)th user

 

....
....

 

....
....

 

 

....
....

....
....

....
....

 

....
....

....
....

..
..

 

 

 

 

....
....

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r= 1 ms,

W

e

128 KHZ,

w.=

4 KHz,

61

71

81

kr<t<(k+1)r|k=0|k=1|k=2|k=3|k=4|k=5lk=6|k=L=8

 

h,-<t>

|1|2lo|7|3|1|3|

Figure 3.17 Frequency hopping with M-ary scheme of signaling

dw=1KHz

57

One can notice that all the slots of the frequency hopping are not used, thus by using
PN sequences of large length one can utilize coding scheme to achieve multiple access.
This technique is known as code division multiple access (CDMA) which has applications

in military and satellite communication systems.

3.6.3 Digital Data and PN Sequence:

The digital data are random in nature, since they usually come from the output of a
pulse code modulation (PCM) system or simply from an analog to digital (AID)
converter. Thus the digital data are considered to be random bits of zero’s and
ones. The pseudonoise sequences are generated by specific polynomials via shift
registers. These polynomials are known as primitive polynomials [64]. The following
recursive relation, which is obtained from the primitive polynomial 1+X3 +X‘, is adapted

to generate the corresponding pseudonoise sequence for the simulation model.

. . (3.23)
PN(t) = PM: -3) ° PN(1-4)
with the initial loading of the register as follows:
PN(0)=-1; PN(1)=-1; PN(2)=1; PN(3)=-1 and PN(4)=1 (3.24)

Based on the above digital data and PN sequence the following digital data and PN

sequence is obtained

digitaldata:101010010111101110101

PNsequence: 001010000111011001010000111011001

58
Now each pair of the digital binary data is grouped as 00, 01,10, and 11 to

determine the four decimal numbers 0,1,2, and 3. One of these numbers would be
multiplied by dw, and dw is considered to be 1 KHz in this model. Thus the frequency
within the M-ary changes from 0 to 4 KHz. Since'Rs is an integer multiple of Rh, each
three bits of the PN sequence is grouped to determine 23:8 different hopping slots out
of 16 possible slots. Thus the hopping starts with the binary coded decimal of the four
bits, which could be any value from 000B=OD to 111b=7D where B denotes the binary
and D stands for decimal in this case. From the total of 8 possible hopping slots only
5 of them have been used by the simulation model. The hopping increment is considered

to be 4 KHz. A frequency hopping signal and its spectrum are shown in Figure 3.18.

l H w l E‘ ‘

ls

 

 

 

 

 

 

 

 

O

 

l
1

__ ____________ I- ..1'11“ ‘11!) (it

0 25 6 Hz
Figure 3.18 The Frequency Hopping signal and its spectrum
The frequency separations of the hopping scheme are obvious in the spectrum of the
frequency hopping signal. In the next section the interfering FH signal has been added

to the FM signal, and the results have been compared.

59
3.7 FM Detection in the Presence of Noise and FH:

In this section the noise and frequency hopping signals which have been modeled in
the previous section will be added to the FM signal. Three cases have been considered.
In order to compare the various results indisputably, in all of these cases the FM signal

is considered to be a tone modulated signal.

I. A single tone FM in the presence of noise only.

11. A single tone FM in the presence of FH only.
In all of these cases an amplitude of 20 is considered for the FM signal.

3.7.1 FM signal in the presence of Noise:

In this case while the amplitude of the FM signal is fixed, the amplitude of the noise
is changed to different levels, namely 10, 15,and 20. The corresponding simulation
result for the four algorithms of zero crossings are shown in Figures 3.19A - 3.220. A
simulation study of MSE is also conducted, and the result of simulation is reported in

Table 3.3.

Table 3.3 MSE results for a single tone modulation in the presence of noise

 

 

 

 

 

 

 

 

 

 

 

ALG 1 ALG 11 ALG 111 - ALG IV 1
5 v 9.153 6.988 3.071 0.004
10 v 10.120 7.954 4.255 0.417 I
15 v 10.977 9.826 7.348 3.572 I
20 v 11.329 11.349 10.153 8.142 J

 

60

One can observe from these figures or Table 3.2 that as the noise level becomes
larger more distortion results. From these Figures and Table 3.2 algorithm III an IV

appears to perform noticeably better for this example

0 4 time x ‘ l s

a) The zero-crossing function z(t)

1.J.Ll.l.h.ll..lluIIII:ullllmhlI1.111.ll..luilllulllullLullaflullnthlll.mjlllLllml”111111111”thilllullllllhllllllfl-

0Hz frequency 256 Hz

b) 2(1) the spectrum of z(t)

m(t)

K

| 0
|
l

 

 

c) The message m(t) and m’(t)

Figure 3.1% The message signal is recovered by ALG. 1

(the noise level = 10)

61

I . " I

0 time 1. s

 

a) The zero-crossing function z(t)

1111.11111111111111 1111.11.11111111111. l..11111111111111111111.111111.1111111111111111.11111111111111111.

frequency 256 Hz

b) 2(1), the spectrum of z(t)

 

 

 

c) The message m(t) and m’(t)

Figure 3.1% The message signal is recovered by ALG. l

(the noise level = 15)

62

11111111111111 _  l 1 1111111

0 time 1 s

 

 

 

 

 

 

 

a) The zero-crossing function z(t)

 

1.11 .11111111111111111111|1.1.11111111111111111111111111111.111.11111111111111.11111111111111111111111111111111111

0 Hz frequency 256 Hz

b) 2(1), the spectrum of z(t)

 

 

 

c) The message m(t) and m’(t)

Figure 3.19c The message signal is recovered by ALG. 1

(the noise leve1=20)

63

 

 

11 11 1

0 t1me 1 s
a) The zero-crossing function z(n)

 

 

O111.1111111111uhd11111111.1.1111.11111111111111111111111110111101101111111111111111111111111111

 

 

 

frequ ucen y 256 Hz
b) z(t)
N” ‘\ A f
6 x/ b\/ \‘r V1?
c) The message m(t) and m’(t)

Figure 3.2081 The message signal is recovered by ALG. 2

(the noise level = 10)

111 111 , 1 111 11

me
a) The zero-crossing function z(n)

 

:fi
{fl

 

11111.1.1.1.111111111.111.11.111.11111111111111.1111.....1111111111111111111111.11111111111111111!z

bZ)(0

11:11-41

c)Them Nessa: m(t)and m’(t)

 

 

 

Figure 3.20b The message signal is recovered by ALG. 2

(the noise level = 15)

65

111 1 111

0 time 1 s
a) The zero-crossing function z(n)

..1111.1.1111.11.1...1..1.1111.111..11111.1111111111111111111111111111.111111.111111111111111

0 Hz frequency 256 Hz

b) z(t)

 

 

c) The message m(t) and m’(t)

Figure 3.20c The message signal is recovered by ALG. 2

(the noise level =20)

111 111

a) The ezero—crossing func

11111111111.11.1....11111.1.111.....11..111.1..11111.11.l11.I.1.111.1111111.11111111111.1111111111111111111

0 Hz frequency 256 Hz
b) 2(1)

1118. c A A /,
“V‘V V V V“

c) The message m(t) and m’(t)

111 111

1s

 

 

 

 

 

 

Figure 3.21a The message signal is recovered by ALG. 3

(the noise level = 10)

67

111 1 11

a) The zero—crossing function z(n)

1mm.mumblmmmnhmm11111111111111.1111;

1 11 11 11

time 11 s

 

fl

 

 

 

0 Hz frequency , 256 Hz
b) z(t)
11“” A A A /
6V1 V V V VI:
c) The message m(t) and m’(t)

Figure 3.21b The message signal is recovered by ALG. 3

(the noise level = 15)

68

 

1 111 111 11

time 1

 

ri_

a) The zero-crossing function z(n)

Mummmmmmnnmwm

frequency 256 Hz

b) 2(1)

 

 

 

c) The message m(t) and m’(t)

Figure 3.21 The message signal is recovered by ALG. 3

(the noise level =20)

69

 

a) The zero—crossing function z(n)

ll1IilmllldllmllmlduflllluthaIJuJIIIuIILIMMWIMMMWMLUMMMMIIIHInlllldluhlllallllluh
0 Hz frequency 256 Hz

b) Z(t)

 

 

c) The message m(t) and m’(t)

Figure 3.223 The message signal is recovered by ALG. 4

(the noise level = 10)

7O

  

11 1

Is

a) The zero—crossing function z(n)

WWmemummmmm

0 Hz frequency 256 Hz

b) Z(t)

 

 

c) The message m(t) and m’(t)

Figure 3.22b The message signal is recovered by ALG. 4

(the noise level=15)

71

11111

 

0 time

a) The zero—crossing function z(n)

lJWJlnlulahalibut-I11lulu.ulluI1.1.;lulwlumlnmuhnlwnmmmuhmwJuullnulflulu
0 Hz frequency 256 Hz

b) Z(f)
0

Figure 3.22c The message signal is recovered by ALG. 4

 

 

 

c) The message m(t) and m’(t)

(the noise level =20)

72

3.7.2 FM Signal in the Presence of FR:

In this case while the amplitude of FM is fixed at 20, the frequency hopping signal’s

amplitude is changed to 10, 15, and 20. The corresponding simulation results are shown

in Figures 3.23a - 3.26c, and the quantitave results are tabulated in Table 3.4.

Table 3.3 MSE results for a single tone modulation in the presence of PH interference

 

 

 

 

 

 

 

 

 

 

fl.‘
ALG 1 ALG n ALG m ALG 1v
5 v 8.418 6.962 3.043 0.0016
10 v 8.677 7.113 3.135 0.1047 I
15 v 9.182 8.112 4.445 0.8610 I
9.670 9.424 7.098 3.4313 I

I 20V

 

 

 

From a study of the above figures and tables, one can see that the distortion which

is eaused by the frequency hopping is less than the noise for the same level of frequency

hopping and noise level. One can notice that the simulation results are applicable to the

multi-tone modulation as well. The single tone has been considered to demonstrate the

different algorithms geometrically. The simulation model thus has been used to test the

four algorithms. It showed that the FM signal can be detected in the presence of noise

and FH signal.

73

1 1 time 11 s

 

a) The zero-crossing function z(n)

MMWIMMMWWJMMIMLLWIWM1.1km“

Hz frequency 256 Hz

b) 2(1)

 

 

c) The message m(t) and m’(t)

Figure 3.23a The message signal is recovered by ALG. 1

(the FH interference level = 10)

74

1 1 1111 .

0 time 1 s

a) The zero-crossing function z(n)

11111111111111mi..ul.11111.1111111.11111111111.111.111.11111111111111.1111111111111111.11111111111.1111111.111.111.1111111I1111111111

0 Hz frequency 256 Hz

TD) 2(1)

 

 

 

 

c) The message m(t) and m’(t)

Figure 3.23b The message signal is recovered by ALG. l

(the PH interference 1evel=15)

75

. 1 .

a) The zero-crossing function z(n)

 

 

 

01111111 ..111111 11.1111I1II.I1|III...1III111.111.11.111I111111111.11 111111111 1111 11111111111111111.1111 11 11111111 11111

frequency 256 Hz

b) 2(1)

A. A___A_.A _,
‘ V V V V ‘

c) The message m(t) and m’(t)

A

 

Figure 3.23c The message signal is recovered by ALG. l

(the FH interference level =20)

76

1 1 ;
1 1

0 1 time 11 s

 

a) The zero-crossing function z(n)

 

 

II."IIIII1IIIIuIII|.III.I1III..II.I.I1I1.1-.1111111111111111111Ill-III1111111111111111I IIII111111I|111I1111111IIII1111111111111111111111111

0 Hz frequency 256 Hz
13) 2(1)

W” /\ A A /
c) The message m(t) and m’(t)

Figure 3.24a The message signal is recovered by ALG. 2

(the PH interference level = 10)

77

 

 

 

 

 

 

 

 

 

0 time 1 s

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a) The zero-crossing function z(n)

 

.II 1.1.1111 IIIII1.111.111IIIIIIIIIIIII1.11.1111 11111111.1111111111111111111111111111111111111111 II1II11111I1I11|111111111111

0 Hz frequency 256 Hz
b) Z(t)

A //\ f
V V V”

c) The message m(t) and m’(t)

 

 

Figure 3.24b The message signal is recovered by ALG. 2

(the FR interference level=15)

78

 

W HI-MlL-HIEL.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

0 time

a) The zero-crossing function z(n)

 

 

 

 

 

 

 

.111.

 

 

 

 

 

 

 

 

Ls

1111.111.IIIII1IIIIIIIIII11IIIIII1IIIIILIII1.1111111:11111.111IIIIIIIII1111111111111l111111|III11111111111111111111111.1111111

0 Hz frequency
b) 2(1)

 

 

c) The message m(t) and m’(t)

Figure 3.24c The message signal is recovered by ALG. 2

(the PH interference level =20)

256 Hz

 

79

1 11‘ I 1 '
11. , 1 1
1' 11 :1: 11 1 1

0 time 1 l s

 

a) The zero-crossing function z(n)

 

IIIiIlIIIIIIIIIIIIIIIII.IIIIIILIIIIJHIII11Llnllluullul111111.111In11III]IIIIII1I|1|I1111111111111111.111111I1111II111111II11I111111111III1111111111111111111111
0 Hz frequency 256 Hz

1)) 2(1)

”1% /\ A /\ /
v V v V v“

c) The message m(t) and m’(t)

 

 

Figure 3.25a The message signal is recovered by ALG. 3

(the PH interference level = 10)

80

0 time S

a) The zero-crossing function z(n)

IaIlLIIIJJIIILLIIIlllllllll1“ll“mullllltll.“llllJulIllllllllb'l'llil'lhmlilllllhlfillli‘IIIJ‘1IIIJII1H1M.lll-I“Ilium|1II".lI111Ih||ll1.1h"III'IJIJIIIIMI'

0 Hz frequency 256 Hz

b) Z(t)

1311A A A A /
v v v v

Figure 3.25b The message signal is recovered by ALG. 3

(the PH interference level = 15)

81

11111 1

(11 time s

 

a) The zero—crossing function z(n)

u.||.l.1111l11111.|..1rl111ll

 

.thmnn11.4.1111““lullluunmn.Iu1.111111n.1lmlu..|hnll.111111111ll|111111111111I1111l11|lhlui1111I|1I

0 Hz frequency 256 Hz

b) Z(t)

 

 

c) The message m(t) and m’(t)

Figure 3.25c The message signal is recovered by ALG. 3

(the PH interference level =20)

82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O I - time

 

 

a) The zero—crossing function z(n)

 

III-Illumlnaluminum-Inuit.ll..l|.|h.|.al1ml1|.II.II..I...II.II.I.I.lIlllhdmdlmlilllilldilhlnlilIla.JudiIIILII'IJI..I'IL.lullllllsilln.IlulmihlllIJIMILII
0 Hz frequency 256 Hz

11) z(t)

Y”“’/\ /\ /\  /”\ /
\/ V \/ \/ V”

c) The message m(t) and m’(t)

 

 

Figure 3.26a The message signal is recovered by ALG. 4

(the PH interference level = 10)

83

 

a) The zero—crossing function z(n)

 

u: allldjllu“llll.Illnflnlflllmtlufllllaluldllll.Ilulilllhtlllllllllllll‘Illn'.lnllluallufldlilul'lllJJMuM“llll”.I"I'lllislulnlllblllllillllllll'llllhllhlhllll
0 Hz frequency 256 Hz

b) Z(f)

A AA /,
\r V V V”

c) The message m(t) and m’(t)

  

Figure 3.26b The message signal is recovered by ALG. l

(the FH interference level=15)

84

1111111

a) The zero-crossing function z(n)

 

II. «lullInhl..muumml|.11.:1.l|.|.u1.1.l.nliu.l|.l.llhulul."mum1Il:l..l|.|l.|.lrlullhmu11111.1..I111.Ihllll.mlJ.|l.ml1||..lu.llll.nllunalullm.
0 Hz frequency 256 Hz

MA A
«V «

Figure 3.26c The message signal is recovered by ALG. 4

b) Z(f)

 

  

c) The message m(t) and m’(t)

(the PH interference level =20)

85

From the above simulation figures, tables and from the discussions in the beginning
of this chapter, one can conclude that the FM signal can be detected in the presence of
noise and interfering frequency hopping signal i(t) by a zero crossing scheme. The zero
crossing schemes that perform best under the test conditions are algorithm III and IV
which were developed in this study. If FM detection is going to be performed by
utilizing a hard-ware system, then algorithm 111 would be the proper choice, and if the
FM detection is considered by employing a software system, then algorithm IV would
be a suitable candidate. Furthermore from the detection of signal in the presence of
noise and frequency hopping interfering signal one can conclude that the FM system
performs better in the presence of frequency hopping than that of the noise. In the next
section different performance measures such as signal to noise ratio (SNR), signal to

interference ration (SIR), and signal to distortion ratio (SDR) are considered.

3.8 System’s Performance Measure:

A standard performance measure for Analog Communication systems such as FM
systems are signal to noise ratio (SNR), signal to interference ratio (SIR), signal to noise
and interference ratio (SNIR), and signal to distortion ratio (SDR), [65]. The above

mentioned measures are adapted for this model also.

3.8.1 Signal to Noise Ratio:

Signal to noise ratio is the ratio of the signal power to the noise power. The signal
to noise ratio must be calculated at the input of the detector as well as at the output of
the detector, then a plot of the output signal to noise ratio (SNRO) versus the input signal

to noise ratio (SNRI) will determine the performance of the system. In this simulation

86

study the signal to noise ratios are calculated for all different algorithms. Since the
signal power is usually very large compared to the noise power the signal to noise ratio
is measured in dB. Before plotting different signal to noise or interference ratio, a brief

discussion of the calculation procedures is given. '

3.8.2 Calculation of SNR and SIR:

Considering the FM signal s,(t), the signal’s power in the input of the receiver can
be determined by the amplitude of the signal. In this case A2/2 = (2O)2/2 = 200 watts,
where A is the amplitude of the FM signal. This value is the theoretical value of the
power for the FM signal. But in simulation the input power of the signal 31 can be

calculated as follows:

N

ng s,(i)2/N (3.25)

where Si represents the input average power, and s,(i) is the same as s,(t) with t is
replaced by UN for simulation. This power has been calculated for an amplitude of 20
and the result is 200.78475 watts which is very close to the theoretical value. This value
is more realistic, since A2/2 is actually the power of a pure sinusoidal signal, and it is
only an approximation to the real power of the FM signal. It has been also observed that
the power of an FM signal depends on the carrier and the message frequencies. Thus
this method of calculation is also utilized for calculation of the noise power and the
interfering frequency hopping signals power. The input noise power Ni and the input

interfering frequency hopping power I, have been calculated as follows:

87

N
N,=£n(k)2/N (3.26)
k-l
and
N ( )
I= i(kV/N 3-27

where n(k) and i(k) are the simulation expression for n(t) and i(t) respectively, with t
replaced by k/N in their corresponding expressions.

The output powers are calculated as follows:

First, the FM signal is detected in the absence of noise and the interfering frequency
hopping signal. This signal is actually the message signal m,(t) and it is denoted by
m,(t), the detected message signal at the out put of the demodulator. Thus the power of

the FM signal at the output of the detector 8, is

N
S,= m (1)2/N (3.28)
1);: d

where m,(i) is the simulation expression of m,(t) with t is replaced by i/N in the
simulation process.

The noise is then added to the FM signal at the input of the detector and the
corresponding output is the output signal plus noise at the output of the detector SN,,

giving the following expression for SN,:
SN, = S, + N, (3.29)

where N, is the output noise at the output of the zero crossing detector in the absence of

FM signal. Thus the output noise N, is calculated as follows:

88

N
1170‘; [n,(i)-m,<i)12/N (3.30)

where n,(i) and m,(i) are the simulation expressions for the detected message and noise
at the output of the detector.
Similarly if frequency hopping interfering signal is added to the FM signal, then the

output interfering power I, is

N
IO=2 [si,(j)-md(t)]2/N (3.31)
j-1

where si,(t) is the output detected FM signal in the presence of the frequency hopping
interfering signal. Finally if noise and interference are present, the corresponding output
signal is denoted by sni and the corresponding output power of noise plus interference

is NI, which is given by the following sum

N
NI0=; [sni,(j) —m,(j)]2/N . (3.32)

Having these parameters calculated, the different signal to noise ratios, signal to
interference ratios and signal to noise and interference ratios at the input and output of

the detector are obtained as follows:

SNRI The input signal to noise ratio

SNRO The output signal to noise ratio

SIRI The input signal to interference ratio

SIRO The output signal to interference ratio

SNIRI The signal to noise and interference ratio at the input

SNIRO The signal to noise and interference ratio at the output

89
each of the above parameters is obtained in dB’s, for example SNRI is 1010g,o(S,/N9.

The output parameters such as SNRO, SIRO and SNIRO are categorically referred as
signal to distortion ratio SDR [65]. The parameter SDR can be used to determine and
compare the performance of the system under different disturbances. Thus this
parameter is also adapted for this study, to compare the distortion due to noise and
frequency hopping interfering signal. The plots of SDR are given in Figures. 3.27-3.30.
Notice that these parameters are plotted for different algorithms. Once again, one can
come to the same conclusion that algorithm 3 and 4 are performing better than other
algorithms. By comparing the SDR’s one can also conclude that the distortion which is
caused by frequency hopping interference is less than the distortion which caused by the
noise. Therefore the distortion effects of frequency hopping signal is less than the
distortion which is caused by the noise. To see the performance of the FM signal in the

presence of FH signal and noise, a simultaneous performance is considered.

9O

OUTPUT SDR 1‘1. dB

m6c$ mm: mom <chm ins mZm 85 mi 3: >633: _

 

mo

mo.

5.

 

 

 

mTle. mom $85 mi 3: >633: _
Q ...... o mom <35 mzm 3: >603: _

 

 

 

b D F F D r b

 

 

-6

m a
ZUE. mzm >20 m5 _2 am

no

91

OUTPUT SDR |N_ dB

36:6 mmm mi 565 .85 mi m3 mi 3: >633: __

 

mo

>0.

mo.

mo.

 

£11.13. IIIIIIII a
\<\\mflhtmlu1ullmll ....... m
\4VD\
\i
\ ,
\Q
X .\
\ a
a... \
a .8.
N m
n a.

 

31114 mi <35 mi 2: >633: __
m-|im_ mom <2mcm mi *2 >633: __

 

 

 

 

 

m Am
22:. mZi >20 mi _2 am

mm

92

OUTPUT SDR IN dB

36:8 mum m0i <35 59: mzi m3 mi 3 >633:: ___

 

 

 

 

 

 

 

 

mo
\hwhflhflfihflhflhlla
so . \flxaxxm
\a\ X
N .
at &
“WC ' \ \\
\4 E.
\
mo . mint .3
a
a: a.
a . am as:
fillld m0i <2mcm mi 3 >633: ___
mmummmm m-l.im_ m0i <2mcm mi 3 >633: E
o . .
-m m a

25: mZi >20 mi _2 60

mm

93

OUTPUT SDR 1N dB

m6c$ mac moi <83 39: mi m3 mi 3 >633: _<

 

mo

>0.

mo.

moj

 

 

 

fillld m0i <29; mi 3 >633: _<
m-.lim m0i <2mcm mzi 3 >633:: _<

 

 

 

 

m 2m
23: mZi >20 mi _2 60

mm

94
3.8.3 Simultaneous Performance:

In the previous section the performance of the system through averaging signal to
noise and interference have been examined. By a plot of these parameters it was
concluded that best algorithms are algorithm 111 "and IV, and the noise induces more
distortion than frequency hopping signal to the FM system which was considered.
Although those calculations and plots determine the overall performance of a system,
they fail to reveal other system behavior, such as group delay or phase shifts. With the
help of advancements in minicomputers and their graphics capability one can determine
the simultaneous behavior of the signal in the presence of different level of noise or
interference. This is referred as the simultaneous performance measure.

In average performance measure one determines the behavior of the system in one
complete period of the signal for example over a length of 512 points. In the
simultaneous performance one can see the behavior of the system point by point
pictorially. The simultaneous performance is actually a sUmmary or combination of
several figures, in one figure. One can think of it as a multi channel oscilloscope, which
shows the effects of different levels of additive noise or interference to a signal. The
simultaneous performance of the algorithms in the presence of different levels of noise
or FH interference is shown in Figures 3.31-3.34. These Figures demonstrate the
amplitude and phase behaviors of the recovered signal in the presence of noise and/or

interference.

95

m(t)

 

 

 

 

Fig. 3.31b A Simultaneous performance of m(t) in the presence of PH

Figure 3.31 The performance of the message under Algorithm I

96

 

 

 

 

Fig. 3.32a Simultaneous performance of m(t) in the presence of noise

m(t)

 

 

 

Fig. 3.32b Simultaneous performance of m(t) in the presence of FH

Figure 3.32 The performance of the message under Algorithm II

97

 

 

 

 

m(t)

 

 

 

Figu. 3.33b Simultaneous performance of m(t) in the presence of PH

Figure 3.33 The performance of the message under Algorithm HI

98

m(t)

 

 

 

 

Fig. 3.34a Simultaneous performance of m(t) in the presence of noise

4
m(t)

 

 

 

Fig. 3.34b Simultaneous performance of m(t) in the presence of PH

Figure 3.34 The performance of the message under Algorithm IV

99

Accordingly these figures confirm the result which were obtained by the averaged
performance measure. Furthermore they reveal the phase distortions which were not
obvious in the preceding figures.

In summary, the simultaneous performance shows the amplitude and phase
distortions, together with group delays that exists in the output of the detector. All of
these graphs and studies show that the FM signal could coexist with a frequency hopping
interfering signal. This is possible by a careful system design which manages the bands

of each system carefully.

3.9 Conclusion:

The demodulation of an FM signal by zero crossing scheme has been considered.
The proposed algorithm which exist in the literature has been examined. Recovering the
signal by frequency domain methods showed a better result. The frequency domain
detection consists of an ideal low pass filter, followed by taking the inverse Fourier
transform (IFFI‘) of the filtered signal. The output is the recovered version of the
message signal which was employed for generation of the FM signal. The frequency
domain detection procedure is applied to all algorithms. The result of the simulation
shows that these algorithm bring more improvement as they get closer to the ideal zero-
crossings.

The result of the simulation also showed that algorithm III and IV are the best
schemes for applications. Furthermore, it has been observed that algorithm III is suitable
for hardware applications, and algorithm IV which is ideal zero-crossing could be

adapted for software applications. The above observations have been confirmed by a

100

study of performance measure such as SNR, SIR and SDR. Finally the simultaneous
performance of the system has been plotted point by point. The result of the study
showed that the distortion that resulted from frequency hopping was less than that of
noise of the same amplitude. The conclusion is that the zero-crossing method of
demodulation should emerge as a major alternative method of detection for the decades
to come. Since FH causes less distortion to FM system compared to the noise, FM

could coexist with a frequency hopping spread spectrum system.

CHAPTERIV

THE SPECTRUM OF FM ZERO-CROSSING

4.1 Introduction:

The idea of FM detection by zero-crossing has existed in the literature for quite
sometime [59], and it has been used in a number of applications. In this part of the
study it is verified that the FM signal can be detected by low pass filtering its zero-
crossings.

In this analysis an FM signal s(t) is considered. The signal x(t) is constructed from
the ratio of the signal s(t) and its absolute value |s(t)|. The function x(t) and s(t) have
the same zero-crossings. However, the signal x(t) is a non-uniform square wave, which
is similar to a hard limited version of s(t). The zero-crossing instants are determined by
taking the derivative of the signal x(t). Since the function x(t) is a non-uniform square
wave, its derivative y(t) is a non-uniform sequence of 6-functions. consequently every
zero-crossing instant t, is marked by positive amplitude or negative amplitude 6(tk) or -
(i(tk). The low-pass filtering of the signal y(t) is considered next. Low-pass filtering in
frequency domain approximates integration or averaging in time domain. Since y(t) is
an alternating sequence of positive and negative 6 functions the result of low-pass
filtering would be zero. Accordingly it is necessary to mark the zero-crossing instants

by positive amplitude 6-functions only. Thus z(t) which is the square of y(t) is

101

102

considered. z(t) has a variety of different spectral components. It will be shown in the
next section that one of the components is the amplitude scaled version of the message
signal m(t). This signal could easily be recovered by a low~pass filter.

In the following sections a mathematical analysis of the FM zero-crossing has been

presented, followed by an example.

4.2 A Mathematical Discussion of FM Zero-Crossings:

In the following discussion an FM signal s(t) is considered, the zero-crossings of this
signal are investigated using a trigonometric power series expansion. Subsequently an
example is presented to demonstrate the theory.

Let the FM signal s(t) be defined as follows:

s(t) = Asin<wct+pfo‘m(r)dr) (4.1)
or simply

s(t)=Asin[6(t)] (4.2)
with

6(t) = wct+flfotm(r)dr (4.3a)
and

6’(t) = w,+Bm(t) (4.3b)

Where B is the frequency deviation, and m(t) is the message signal. The FM signal

s(t) which is generated by modulating signal m(t) is shown in figure 4.1.

103

 

s(t)

m(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Osec lsec

Figure 4.1 The FM signal s(t) is superimposed on m(t)

Consider the function x(t) which is constructed from s(t) as follows:

s(t)

This function changes the FM signal s(t) to a normalized square wave type FM
signal. However, the zero-crossings of s(t) and x(t) are the same, as shown in figures

4.1 and 4.2.

20v

 

s(t) 1

 

-20v

 

Osec lsec

Figure 4.2 The function 20 x(t) marks the zero-crossings of s(t)

Now one can consider x(t) in the following form.

Asin0(t) - sin0(t:)

x”) ' |Asin9(t)l Isin0(t)l

 

(4.5)

since |sin9(t)| = [l-cos29(t)]‘”; thus x(t) can be written as follows:

Expanding [1 - cosze(t)]"’2 in a series approximation [66] one gets the following

104

 

 

x(t)= Sine”) , (4.6)
[1-cos26 ( t) ] '5
expression for x(t)
N
= . (2k-1)O! 2 4.7)
x(t) Sln6(t)1+1; (2k)e! cos m(t) (

where (2k-l)o! = 1-3-5- -(2k-1); product of odd numbers up-to 2k—1.

(Thus 0!, or odd factorial can be defined as the product of odd numbers, and similarly
e! , or even factorial is the product of even numbers could be expressed as follows:
(2k)e! = 2 -4 ~6- - (2k); product of even numbers up-to 2k.)

(note: Zero factorial is equal to one by definition.)

Since x(t) is the square wave version of the FM signal s(t), by taking the derivative
of x(t) one gets nonuniform (in period) samples of 6-functions. Accordingly y(t) =x’(t)
can locate the zero-crossings of x(t), and consequently it also designates the zero-
crossings of the FM signal s(t). Figure 4.3 shows the function y(t) which is constructed

from x(t), and it clearly marks the zero-crossings of x(t) and s(t).

 

20V

1111 11111111 11111111 11111111 1111
-...111 111111111111111111111111 111

Osec lsec

 

 

 

 

Figure 4.3 The function y(t) marks the bipolar zero—crossings of s(t)

105
By taking the derivative of x(t), one gets y(t) as follows:

1

N
y( 1:) =0’( t) cose ( c>11+£ ”(12:12,“ cos21‘0 ( t)
k-l - ,

 

(4.8)

 

 

N . d
-e'( c) sin=e ( c)[ (21"1101 cosZk’16(t)
1;; (2k-2) 9! J

After substituting l-cos’6(t) for sin29(t), multiplying the cosine terms, regrouping the

terms, and finally factoring 9’(t), y(t) can be expressed in equation (4.9).

 

1 N
y( c) =e'( c) cosB(t) i: 13:319.“ cosz"*16 < c)
L k=1 -

.N .
- I (2k-1)o! 2H (4.9)
6 (t) p1 (2k-2)e!C°S 6(t)d

N

+e’( c) 12 1:11:31: cosz"*16( c)
101 '

 

 

 

From the equation (4.9) it is obvious that each term is composed of an odd power
of cos 9(t). The coefficient of each cosine term can be calculated separately. For
instance the coefficient of cos 9 itself is (1-1)=O; and the coefficient of any other
arbitrary odd power of cosine is zero except the last term. Consider an arbitrary term,
for example the rth term. In this case one must find Cr the coefficient of the term cosme
when k=r< N. The coefficient Cr can be found from equation (4.9), and it is given in

equation (4.10).

106

C = [2(1'-1)-1]o!_(2r-1)o!+ [2(r-1) -1] o!

 

 

 

 

 

‘ (2r-2)e! (Zr-2)e! [2(r-1)-2]e!
= (2r-3)0!'(2r-1)O!+[(2I-3)0!] (ZI‘Z) (4.10)
(2r-2)e!
= (2r-3)o! [1-(2r-1) +2r-2] =0
(2r-2)el

Thus the coefficients of cos”"9(t) is zero for all k< N. Now consider the Nth term,
the case for which k=N. In this case the power of cosine is 2k+l, and it has only
positive coefficients. Thus the coefficient of the last term CN or the coefficient of

cosm+16 is given by equation (4.11)

 

(2?.N-1)Ol+(2N-l)01= (2N-l)0|+(2N-1)O! (2N)

 

CN= _
(2N)e! (2N 2)el (2N)e! (4.11)
= (2N-l)o! (1+2N) = (2N+1)O!
(2N) e! (2N) 9!
Therefore the coefficient of the last term survives. Thus
y(t) = x’(t) = 0’(t) (2N+1)°'cosz"*16(t) (4.12)

 

(2N)e!

One can notice that y(t) converges slowly to its ideal version, which is a sequence
of 5-function non-uniformly spaced in time. The value of CN shows that the convergence
is slow. Nonetheless, theoretically (by applying the ratio test) the coefficient CN of the
cosine term in the above equation approaches infinity as N approaches infinity.

Since y(t) becomes very large as N becomes very large, the above function y(t)
becomes a non-uniformly spaced distribution of 6—functions, and the locations of these
6—functions mark the zero-crossings of the FM signal s(t). The function y(t) is composed
of positive and negative amplitudes (6—functions). Furthermore, the positive and negative

5—functions alternate. Consequently, one can not extract the message signal m(t) from

107

this signal by using a low-pass filter. This is due to characteristic of the low pass filter.
Low—pass filtering in the frequency domain is equivalent to integration or averaging in
the time domain. However, it is possible to take the absolute value of y(t) or the square
of y(t) to mark the unipolar 5—functions at the zero-crossing of the FM signal s(t), and
then low pass the resulting signal. Since squaring y(t) is easier to deal with
mathematically, it is considered next. Accordingly squaring y(t) one gets z(t) as follows:

z(t)= y2(t)=[e'(t)]211[2(12V;n—11;]1212cosm26(t) (4.13a)
which after substituting the value of 9’(t) becomes

z( t) = [w§+2|3wcm( t) +9sz (t) ] MCOS‘N‘ZG (t) (4 . 13b)

[(2N)e!]2

Figure 4.4 shows the function z(t) which is generated from y(t).

1211111 11111111111111

Figure 4.4 The function z(t) marks the zero-crossings of s(t) unipolarly

 

 

 

lsec

Since the power of cosine becomes even, the series has a dc value and the cosines
are all of even powers. To find the exact value of the dc component, and the coefficients
of different components involving m(t) and m2(t), the term cos‘"”6(t) is expanded as

follows: [66]

108
cos‘"*20=—i— 2%? 2 4N+2 cosZ(2N+1+k)6 4N+2 (4-14)
24N+2 k-o 1 k T12N+11
After utilizing equations (4.14), one can identifies a number of different signal sets

in equation (4.13). The signals that one can identify are:

I. A large dc value
H. The message signal m(t).
III. DSB AM type signals, with the message signals m(t) and m2(t).

IV. FM signals with a carrier frequency of at least 2w,.

Using the above relations, one can extract the message by using a dc blocking

capacitor and a low-pass filter. The recovered signal will be

 

1 (2N+1)o! 2 4N+2
2w,p 2mm) ( (2N)e! ) (2N+1)m(t) (4.15)

Thus the recovered message is an amplitude scaled version of m(t), and could be
adjusted to an appropriate level electronically. Notice that one also gets a large dc
component which involves w}. However, the dc component could be blocked by a
capacitor, or simply ignored in software applications. To compare the coefficients of
different signals which are expressed in equation (4.13), in the following section different

spectral components of z(t) are considered.

109
4.3 The Spectral Components of z(t):

The above signal z(t) which marks the zero-crossings of the FM signal has been
considered to show mathematically the extraction of the message signal from a
complicated FM modulated wave. However, as it was discussed in the simulation
process one need not proceed as above. It is possible to obtain the zero-crossings by a
simple comparison of two adjacent samples as described in the simulation process. In
summary the above mathematical discussion showed that the spectrum of the message
signal is located in the lower portion of the frequency band of the corresponding FM
zero—crossing waveform, and as a result one can recover the message signal from its
zero-crossings waveform by using a low-pass filter. Beside the message signal and the
d.c. components which were mentioned one also gets a variety of FM signals with
different carrier frequencies, and double sideband (DSB) AM types of waveforms. Using
equation (4.14) one can determine the coefficients of each of the signal components. To
illustrate this method of demodulation and compare the coefficients of different
components, an approximation for N=3 will be presented next. In this approximation

the least value of N is chosen to make the mathematical discussion simple.

4.3.1 A Three Term Approximation of x(t):
In this section an expansion of x(t) for N =3 is considered, and then the function y(t)
and z(t) are obtained. In the next section different components of z(t) are discussed.

Let 9(t)=9 (for convenience) in equation 4.7, and expand it for N=3 as follows:

110

1 1 3

x(t)= sin0(1+-2—cosz0+-2—— 4cos"0+1cos‘50) (4.16)

3 5
2 4' 6
Once again notice that x(t) is the square wave version of the FM signal. by taking the
derivative of x(t) one can mark the zero—crossings 'of x(t), and therefore the zero-
crossings of the corresponding signal s(t). Thus y=x’(t) becomes
y( t) = 0’c030 (1+lc0520+—3-cos‘0+1—5cos"0) (4 .17)
2 8 48
+sin0 [0+c030 ( -sin0) 0’+%cos30 ( -sin0) 0’+l8-5-c0550 ( -sin0) 0’]
factoring 0’, replacing -sin20 with c0326 - l, and multiplying the cosine terms one gets
the following value for y(t)
= I _1_ 3 3 S _!~_5_ 7
y( t) 0 (c030+ 2 cos 0+ 8cos 0+ 48COS 0)

-0’( ~cos0-gcos30-% coss0+cos30+ '3 00350+ 355- c0370)

=0’[0 (cos0+cos30+coss0) + L40: c0670] ' (4 .18)
or simply
y(t): x’(t) = —%—0’(t)cos70(t) (4.19)
squaring y(t) one gets z(t) as follows:
z(t)=y2(t)= (L405) [0’(t)]2cos“0(t) (4.20)

substituting the value of 9’(t) and expanding cos”9 in terms of multiple angles [66], one

obtains:

111

z—(t) =(-14— (5)093 +20wcm(t)+[32m 2(:))— 2—.l“[(14)+2 1:)c032(7 406] (4.21)

The above signal z(t) is composed of a variety of different signals, such as a large d.c. ,
the message signal,the square of the message signal, FM signals with high frequency
carrier, and DSB AM type signal with the FM signals being its carrier. The spectrum
of z(t), and the recovered version of m(t) from z(t) by low pass filtering are shown in

Figures 4.5 and 4.6

 

 

 

 

 

 

 

1:..1111L11JI11111111111 J1.1.11. 1111111111111.111111

0 Hz 256 Hz
2(1)

1 l

L

 

 

 

 

Figure 4.5 The amplitude spectrum of z(t)

 

20V

m’(t)

 

 

-20V
0 sec time 1 sec

 

Figure 4.6 m’(t) the recovered version of m(t)

112

To comprehend this signal, an expression for each component has been calculated,

and presented in the following section.

4.3.2 Decomposition of z(t) in its Components:

The signal z(t) is a very complicated signal. In this section a classification of
different components of the function z(t) is presented. considering the expression for z(t)

in equation (4.21), One can identify the following components from z(t).
1) a dc component with the following magnitude

w 2(1_os)1[_1_114) (4.22)
‘ 43 214 7

the above dc value is easily blocked by a capacitor in hardware implementation or simply

ignored in software applications.
2) the detected message signal with amplitude scale as follows:

213w (_1__05 21:11:11“) (4.23)

One can easily recover this signal by a low-pass filter, and then magnitude scale it in
hardware or software. The filter must exclude the zero frequency in order to block the

dc component. This scheme is also used in simulation study of chapter 3.

3) a term involving the square of the message signal given as follows:

2 PE _1_.14 2 4.24
9 1431214171“) ( )

the above term which involve the square of the message signal can induce some

113
interference in the band of the message signal, specially if the message signal is

composed of frequencies with some harmonics and their multiples. Nonetheless, by
comparing the coefficient of m(t) with that of m2(t) one can notice that the ratio of their
coefficients is 2w,/B. Since this ratio is very large the interference introduced by m2(t)

is negligible.

4) seven term FM signals, containing higher carrier frequencies given as follows:

2 6
2w,2 E51 i 14 14 [2(7 -k)6] (4.25)
48 214 7 M k
The above signals are centered around 2wc up—to 7w,. Therefore, all of the above signal

will be in the upper portion of the spectrum. However if the value of B is large it could

cause some interference very small in magnitude.

5) seven terms involving components, at multiple of w,, multiplied by m(t). The

mathematical expression for these terms are given as follows:

10511 1 l4 6 l4
4ch ( 48 ) 2141 7 ) m(t); ( k )ws[2('7 k)0]

The above equation represents a set of double sideband AM type of the message signal
m(t). However, it is not an ordinary double sideband AM modulation, since the carrier
portion consist of frequency modulated signal rather than cosw,t. The last components
which are given by equation (4.27) are similar to the above scenario, except for the
message signal which is replaced by m2(t).

The critical point in the above deliberation is that the spectrum of each of those

signals is centered around 2wc or higher. Consequently, they are not inducing any

114

21371105]1[2i 1174) m(t)2£(l:)cos[2(7 -k)6] (4.27)

interference to the lower portion of the band which the message signal is located. In the
case of m2(t) it has been noticed that the amplitude of the m(t) component is much higher
than that ofm m(.t) Thus one can deduce that, the message signal which 18 modulated by
an FM scheme can be recovered from the zero-crossings of the corresponding FM signal
by low-pass filtering the zero-crossing signal z(t). The above argument is illustrated in
Figure 4.7, where the multi-tone signal m’(t) is recovered from the zero—crossing

function z(t) by low-pass filtering.

 

 

           

 

z(t)
1

.11111.11,11);.11..1i.1.1111 111.111111'1111111111111

0 ls
a)

2(0
11

IllJL.J. A I; llllljljlllllllil.|lll. , l|1l|l|||llllr:i’
0 b) 256Hz

m(t)

11 11W 1,

ovMVV VMVV VAAVVV MVV >1.
C)
Figure 4.7 a) The function z(t)

b) 2(1), the amplitude spectrum of z(t)
c) The message signal m(t) and its recovered version

115

In the next section a modified version of z(t) is considered to show that the
interference induced by m2(t) is caused by squaring the signal. Therefore, the

interference expected from the spectral component involving m2(t) could be avoided.

4.4 A Practical Aspect of FM Zero-Crossings:

In the above discussion of zero-crossings, z(t) is considered to designates the unipolar
zero-crossing instants of the function x(t). z(t)=y2(t) as given in equation (13). z(t) is
mainly composed of multiplication of two functions, [9’(t)]’, and
cos‘"*29(t)=[cosz"“9(t)]2. Since 6’(t)=w,+6m(t) and wc> >Bm(t) for almost all
practical purpose, thus, one can consider z,(t) the absolute value of y(t) as follows:
zl(t)=[wc+ pm(:)]—(—2(%;)cl—);’l 1cos2N+1e(:)| (4.28)
As far as zero-crossings are concerned the function |cosz”“9|
and cos‘"*29 mark the same unipolar zero—crossing instants, but the absolute value of the
cosine function is not tabulated. Thus it was easy to consider the square of y(t) rather
than its absolute value. Nevertheless, the absolute value of the cosine functions has dc
value. Furthermore, the two approaches has the same zero-crossings, and the only
difference is the amplitude of the corresponding zero-crossings. In this last approach the
function z,(t) is marking the zero-crossing instants without amplitude distortion which is
induced by the squaring process. This last observation shows that the FM zero-crossing
signal z,(t) contains the following signals:

1) A large dc component which involves we

2) The message signal with a spectrum which is located in the lower portion of the

116

zero-crossing spectrum. Thus it could be recovered by a low-pass filter

3) A variety of FM signals, with amplitude wc and their spectrum centered at integer
multiples of w,. Thus they could be filtered out by a low-pass filter.

4) A variety of AM type signals with FM signal serving as their carrier, thus these
signals also have their spectrum centered at integer multiples of we. Thus it could

also be filtered out by a low-pass filter as shown in Figure 4.8.

 

 

 

 

It“)
11 - 11 1111 1 1111
1 1111. 2111111111111111111111 2111 .11 11111111 1111111 11111 11 11
0 S
(2!)
21(1)
1111. a. .-.11IJlI1..11.l.Jlil.IJIIJ 11 ll ”llljln.’
0 (b) 256Hz

 

Figure 4.8 a) The signal z,(t)
b) Z,(t), the amplitude spectrum of z,(t)
c) The message signal m(t) and its recovered version

An important observation that one can make from the above consideration is that z,(t)
guarantees that the lower portion of the spectrum does not have any spectral component
involving m’(t). In conclusion one can recover the message signal m(t) by a low-pass

filter from the zero-crossing of the FM signal which is modulated by the corresponding

117

message signal m(t). Furthermore, there is no interference due to the square of the

message signal m2(t).

4.5 Summary and Conclusion:

A mathematical study of FM zero-crossing has been considered in this chapter. A
square wave type frequency modulation signal x(t) was desired from the corresponding
FM signal s(t). It has been observed that the FM signal and its corresponding square
wave signal have the same zero-crossings. Subsequently, a power series expansion of
the x(t) was obtained. To mark the zero-crossings of the non-uniform square wave, its
derivative y(t) was considered. The function y(t) marked the zero-crossings of the FM
signal with either positive or negative large values (positive or negative 5-functions in the
limit). Although y(t) marked the zero-crossing instants, the value of the function
fluctuated between positive values and negative values. As a result one can not recover
the message signal from y(t). Thus the function z(t), the square of y(t) was considered.

The function z(t) marked the zero—crossings of the FM signal in a unipolar manner.
This function is composed of a variety of components. The spectrum of the message
signal is located in the lower portion of the spectrum z(t). Therefore, it is possible to
separate it from the rest of the spectrum. It has been shown that there would be a
negligible interference due to the squaring process. Finally a practical aspect of the FM
was considered and the function z,(t) has been constructed. This function showed that
in practice the interference which was induced by the squaring process could be
eliminated. In conclusion: It has been shown mathematically that the FM signal can be

recovered using a low-pass filter from the spectrum of its zero-crossings.

CHAPTERV

SUMMARY AND CONCLUSIONS

5.1 Interleaving Overlay:

The interference effects of spread spectrum modulation on the performance of
conventional AM and FM system has been investigated. A mathematical description of
all signals has been presented. A study of signal to noise ratio and signal to interference
ratio for different signal sets has been considered. An expression for signal to nose ratio
at the input and output of AM receiver has been obtained. It has been shown analytically
that when the interference to noise ratio approaches zero, the signal to noise ratio
approaches the signal to noise ratio in the absence of interference. It has also been shown
analytically that the interference to noise ratio is a function of the bit duration 7
[Appendix B].

In studying DS-FM system interference, the tolerance of an FM system over an AM
system has been shown analytically, and an expression for signal to noise ratio has been
obtained. It has been observed that the degradation of signal to noise ratio in this FM
case would be larger than for AM, but due to the wide bandwidth of FM and interleaving
overlay it could be tolerated. It was noticed that the average signal to noise ratio for FH-
AM would be the same as the signal to noise ratio for DS-AM. However there is be an
important difference in their instantaneous value, because in this case the instantaneous

118

119

value would be deterrninistically non stationary, due to F H signalling pattern.

The problem of intersymbol interferences has been touched upon [Appendix E], and

the solutions proposed as follows:

1) To reduce intersymbol interference, the data bit duration time should be an
integer multiple of PN bits duration time.
2) One should use MSK signaling instead of non return to zero to prevent generation

of impulse type interference.

In the final stage of the signal to noise ratio analysis it has been observed that direct
sequence produces click type interference to FM system, which is induced by digital data
and PN code. Since in frequency hopping signaling scheme, one can not combine the
digital data and PN code, the click type interference induced by frequency hopping to
PM system would be the worst type of interference.

Thus in the first portion of this study it has been shown that the worst type of
interference is the interference of PH spread spectrum to FM system. Therefore this

type of interference analysis was chosen for a simulation study.

5.2 Zero-crossing and Simulation:

The demodulation of an FM signal by zero crossing scheme has been considered in
chapter 3. The proposed algorithm which exists in the literature has been examined.
The detection of signals by frequency domain methods appear to show a better result.
The frequency domain detection consists of an ideal low pass filter, followed by an

inverse Fourier transformation (IFFT) of the filtered signal. Three new algorithms have

120
been developed in chapter 3. The simulation procedure was applied to all four

algorithms. The result of the simulation shows that these algorithms brings further
improvement as it gets closer to the ideal zero-crossings.

The result of the simulation also showed that algorithms III and IV show the best
promise. Since algorithm 111 is composed of a limiter, differentiator, and half wave
rectifier, algorithm III is suitable for hardware applications. Similarly algorithm IV
which is the ideal zero-crossing is suitable for software application. This algorithm is
composed of a simple comparator, comparing of two adjacent samples. Thus it is simple
to compare the samples which are stored in the memory or exist in a port by a software
technique. Therefore algorithm IV would be used for software applications in years to
come. The above observations have been confirmed by a study of performance measure
such as SNR, SIR and SNIR. A new type of performance measure has been introduced.
This new measure, which is called the instantaneous signal to noise and/or interference
ratio, demonstrates the performance of the system point by point. The result of these
performance measures showed that the distortions which were induced by frequency
hopping is less than that of the noise for the same amplitude. The conclusion is that the
zero-crossing method of demodulation will emerge as an alternative method of detection.
Furthermore, it has been shown for the simple case simulated that FH as an interfering
signal induces less distortion to the FM system compared with noise. Therefore
frequency modulation system could coexist with the frequency hopping spread spectrum
system. Finally in chapter 4 a mathematical analysis of the FM zero-crossings has been
considered. It has been shown theoretically that the FM signal can be detected from its

zero crossings by a low pass filter. This analysis also revealed that there is some

121

spectrum overlap if the message frequency is composed of a frequency and its multiple.
However, it was shown mathematically that the amplitude of that interference is very

small.

5.3 Areas of Further Study and Recommendations:

The research which was reported in this study consists of two parts. The first part
of this study was devoted to the problem of interleaving overlay, and the coexistence of
different spread spectrum signals with AM and FM systems. Since this concept is
considerably new in the field of communications, it raises a number of questions. Some

of the questions are:

1. In this study only one interfering frequency hopping signal has been considered.
What if one consider more than one, and possibly heavy loads (many users)?

2. There might be some other types of frequency sharing, and coexistence. A first
step is to analyze and study the corresponding system similar to this study.

3. The analysis which is considered in this study was based on just two systems at
one time. It would be desirable if a more general system with more variables
variable could be considered for interference and overlay studies. For example
a study of many different types of modulation occurring at the same time.

4. The FCC has expressed its concern about overlaid systems. The FCC has also
expressed its concern about the use of spread spectrum in the public domain. As
the modern technology advances, and expands, adequate research will be
conducted in this area. The FCC will consider the results of these type of

studies, and certainly field test the corresponding situation before issuing any

122

license.

The second portion of this research was the simulation of the FM modulation in the
presence of frequency hopping interference and noise. The FM zero-crossing which was
adapted for this study has existed in the literature and application. However, questions

to be answered or explored are as follows:

1. The different algorithms that have been developed and examined by simulation,
need to be verified experimentally as well. The experimental outcomes together
with the simulation results will determine the adaption of a particular technique
for a particular application.

2. In this study only one user of frequency hopping has been used. A study of
several user and perhaps considerable load would be of interest.

3. The zero-crossing demodulation could be adapted for microprocessor based
application, such as video cassette recorder and other FM detection systems which
operate in the time (clock) range of a microprocessor. For high frequency
applications one needs to translate the FM signal to the IF band by using a mixer
and then detection could be performed by a zero-crossing algorithm.

4. The concerns and questions which were expressed regarding interleaving overlay

in items 3 and 4 of the previous segment also holds for this segment.

123
5.4 Final Remarks:

The concept of interleaving overlay, which is a new idea in modern
communication technology has been examined. For the purpose of this study there was
no concern about industrial limitations such as cost, components, geophysical limitation,
and finally FCC rules and regulations. Consequently the real credit will belong to those
individuals who actually implement these ideas in the field. However, this study attempts

to demonstrate that the idea has potential benefit.

APPENDICES

APPENDIX A'

SIGNAL TO NOISE AND INTERFERENCE RATIO FOR AM

In this appendix an expression for signal to noise and interference ratio for an AM
signal in the presence of direct sequence spread spectrum interference signal and WGN
is obtained. Furthermore, an expression for interference to noise ratio is also derived.

The study shows that the gain of the AM detector is not changed due to interference.

Consider an AM signal r(t) which is received at the input of the receiver as shown

in figure A. l.
r(t) = s(t) + n(t) + i(t) (A.l)

where s(t) is the transmitted AM signal, n(t) is the additive white Gaussian noise (WGN),

and i(t) is the interference due to direct sequence spread spectrum signal.
The desired AM signal s(t) can be represented as follows:
s(t) = Am(t)cos(w,t+9) (A.2)
Where A,, is the amplitude of the AM signal, wc is the carrier frequency, and 9 is the

phase shift of the AM signal s(t).

124

125

 
 

s(t)
1411444114
n(t) iéllllllllllll i(t)

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

ooooooooooooooooooooooooooooooooooooooooo
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
-----------------------------------------

r(t)

 

 

B A N D P A 8 8

P I L T B R

 

x(t)

 

 

E N V E L O P B

D E T E C T O R

 

Yul)

 

 

P I L T B R

 

L O W’ P A 8 8 H

 

z(t)

 

Figure A.1 AM receiver Model

126
The DS spread spectrum signal can be described in the following form: [4]

iDS(t)= f2dej(t)PNj(t)sin(wjt+6j) (A-3)

where u is the number of users, I,- is the signal amplitude, W5 is the angular frequency and
9,- is the phase angle for the jth user, d,-(t) is the digital data (message) carried by the jth
user and PNj(t)e{l,-l} is its pseudo-noise code sequence. Furthermore dj(t)e{l,-1} over
the code (chip) duration T, for all j, j=1,2,3,...u.

Assuming that the noise is additive white Gaussian noise with zero mean and two
sided power spectral density of N,/2, one gets the following expressions for noise at the

output of band-pass filter [67].

 

nc(t)= nslt)= o (A.4a)

 

nc2 ( t) = n,2(t) = Zme0 (A-4b)
where n,(t) is the in-phase and n,(t) is quadrature component ‘of the noise n(t) at the
output of band-pass filter, and 2w,,, is the bandwidth of the band-pass filter. On the other
hand using statistics of the random data, one gets the following statistical properties for

the interfering signal i(t) [4].

 

1a,“) =isj(c) =0 (11.5)

 

1311 1:) =13” 2:) =fsj1f) IHBPFIde=2ImeC (11.6)

for allj, j=1,...,u

where icj(t) and i,j(t) representing the in-phase and quadrature components of the

127
interfering signal ij(t), 2w,III is the bandwidth of the band pass filter, and u is the number

of spread spectrum users. In worst case that all users of direct sequence spread spectrum
are transmitting in we, the following expression can be written for the input signal to

noise plus interference ratio (SNR,).

 

 

 

 

gr; g3:
311712,: _ _ _ __ = (11.7)
%(n§+n§) +% (i§j+i§j) ZWMN°+2wmIT°U
or
_1—711—2/ (ZWUN) I I
SNR_2N °_SNRI_SNRI
1 1+ ITcu 1+ ITcu 1+INR ”"81
NO NO

where SNR,’ is the SNR of the ordinary AM system in the absence of interference and
INR is the interference to noise ratio thus INR=ITcu/No.
Using the in-phase and quadrature components for noise and interference signals, the

input to the envelope detector of Figure A.1, x(t) can be written as follows:

x( t) = [Am( t) +nc( t) +2 ic1(t) ] cos (cache)
j-1

n (11.9)
- [ns( t) +; 181(t) ] sin(wt+6)
-1

The next stage of the AM demodulation is the envelope detector as shown in Figure
A. 1, it simply detects the envelope of the AM signal. If the SNRl is high enough one

can approximate y(t) as follows:

11

y(t)=Am(t)+nc(t)+; icjm (A-IO)
=1

128
y(t) will pass through a low pass filter. This filter further suppresses all high frequency

components which are higher than the desired message bandwidth. Thus the output of

this filter which is shown in Figure A.1 is simply given by the following expression.

z(t)=A,,,(t)+nc(t)+ 1,. (t) (A-n)

Using the above expression, the signal to noise ratio at the output of post detection

(SNR) is

213-74?

[I
—2 — .—
(nc-ncz) +;: (icz-lc 2)
:1 J J

 

SNR=

O

which after a simple manipulation becomes

311112,, _ SNRg

° u __ -1+INR
(1 2-1 ) (A.12)

 

where SNRO’ is the signal to noise ratio at the output of the AM receiver in the absence
of interferences, and INR is the interference to noise ratio as before. The detection gain
is given by the following expression

smeér I
_ SNRO= 1+INR = 5117120:

SNR: SNR; 311in
1+INR

G G’ (A.13)

 

 

where G’ is the detection gain in the absence of interference. Thus G = G’ , for example

the detection gain in the presence of interference is the same is the same as the detection

129

gain in the absence of interference. Furthermore, from the above relationships the

interference to noise ratio (INR) is defined as follows:

I1VR= c (A.14)

 

APPENDIX B

INTERLEAVING OVERLAY

In this appendix a mathematical discussion of interleaving overlay is presented. The
results exhibit that by using the spectrum of the spread spectrum signal one can design
a system to reduce the interference induced by spread spectrum signals.

To show the interleaving overlay mathematically, consider a general waveform at
the front of the receiver. This waveform includes amplitude and frequency modulations

as its special cases. The waveform could be given by the following expression
s(t) = A(t)cos[wot+90(t)] (8.1)

in the above expression when 9(t)=8°=constant, s(t) represents an AM signal, and
when A(t)=A=constant s(t) represents an FM signal. A general form of the spread

spectrum interference signal could be written as follows

I(t)=;dj(t)cos[wj(t)t+0j] (B.2a)
-1

by using trigonometry it becomes
I(t)=B(t)cos[wj(t)t+6] (B.2b)

The second line of the above equation lumps all interferences into a single complex

130

131

interference.

Consider AM and FM demodulations. For AM coherent demodulation 9(t) =9, a
constant, then A(t) will be recovered by multiplying x(t)=s(t)+l(t) by 2cos(w,,t+9°) and
then low-pass filtering the result. In this case the double frequency terms will be

removed and the following expression will be obtained at the output of the LPF.
A(t)+I(t)cos(w2(t)+62) (3.3)

where w2=w,-wo and 62=G,-80. Now if w2=0 then w,=w0 it becomes similar to
jamming, however by using interleaving overlay this situation will be avoided, and in this
case IWO-WII > B/2, where B is the bandwidth of the filter. Therefore for a reliable
communication one has to overlay the interfering spread spectrum signal in the guard
band portions of the AM system, otherwise there will be distortion and even jamming.

To understand the system behavior in this situation, consider I(t) which is given by
equation (B.2a). By considering only one of the interferences for example
ij(t)=dj(t)cos(wj(t)t+9j) {assuming that 9,- is uniformly distributed in [0,211] and
Pr(dj(t)=l)=Pr(d,-=-1)=1/2 and the random variable dj(t) and 9,- are independent events}

the mean and variance of ij(t) can be evaluated as follows:
E[i,-(t)] = E[dj(t)]E{cos[wj(t)t+9,-]} =0 (B.4)

where E is the expectation operator. Since E[dj(t)]=0 and E{cos[wj(t)t+9,-]}=0,
therefore E[i,-(t)] =0.

To find E[i,-2(t)], first find the autocorrelation function of ij(t)

Rj(‘r) = E[ij(t+1')ij(t)]

132
= E[dj(t+r)cos(wjt+wj-r+6j) dj(t)cos(wjt+9,-)]

= E[d,-(t + r)dj(t)]E[cos(w,-t + wjr+ 9i)cos(wjt + 65)]

‘ARdj(r)coswjr » (13.5)

where Rdj(r) is the autocorrelation of dj(t), if dJ-(t) represents a semirandom binary
transmission RdJ-(r): 1; if dj(t) represents a random binary transmission R,,-(r) is given as

follows [48].

1--111 tsT
Rdj(t)= T (B-6)
0 t>T

where T in this case is the bit duration time.

In any case the following expression is obtained

E[i,-2(t)] = 'AR,,-(O)cosO = l/2R,,,-(O) (3.7)
In case of semirandom binary transmission one gets [48]

E[i,-2(t)] = 1/2 (B8)

and in the case of random binary transmission the result will not be changed, since
r=0 < T in this application. Therefore the following expression for signal to interference
ratio (SIR) is obtained.

A(t)5

_ (13.9)
12(t)

SIR=

 

where

133

1211:) =E[I2(t)]=E[; ij(t)2] (13.10)
-1

Since each ij(t) is independent and is identically distributed one gets the following

2 = u . =2 13.11
I (t) £120,110) 2 ( 1
and

SIR=3A—2u$£)_ (13.12)

Therefore the reduction in SNR will be from 0 dB (overlaying) to 1010g(u/2), in cases
that each user transmits with different amplitudes for example A); j=1,...,u. Then the
reduction is lOlog(‘/$EA,.) ;j=1,2,3,...u. For example for 10 users with unit power the
SNR will be reduced by lOlogS
or about 7dB.

An important conclusion will be obtained by finding the spectral density of the
interference. Since the autocorrelation functions are known from equation (B5), and the
spectral density is the Fourier transform of autocorrelation [48], the following important

relationship is obtained

s(t) = 1/4 [s,,(f-f,)+s,,(f+ 1)] (B. 13)

where 85(1) is the Fourier transform of the autocorrelation of the jth user Rj(r), and de(f)
is the Fourier transform of the autocorrelation of the digital data dj(1). From the above

relationship one can clearly see the importance of overlay. For instance, one can overlay

134

by choosing fj’s properly, or the relationship simply suggests that the system designer can
assign the carrier frequency of the jth user f,- to the guardbands of the narrowband
systems. In the case of a complicated form of interference, for example, the sum of
several interfering signals, the following system of inequalities are necessary for reliable

communication

|w°- j| >‘/2B ;j=l,...,u (B.14)

Now consider the FM case, in this case A(t)=A (a constant). A block diagram
of this system is given in Figure B. 1. One can transform the output of the predetection
filter x(t) to a form suitable for FM detection. Using the in phase and quadrature for the

signals, x(t) can be written as follows

x(t) = {A + B(t)cos[w2t+ 92(1)] }cos[w°t + 90(t)]
- {B(t)sin[w2t+92(t)]}sin[w°t+9°(t)] (B.15)
where w2 = w,-wo and 62 = 61-90.

Using more trigonometry one can write

x(t) = R(t)cos[w,t+e,(t)+<1>(t)] (B. 16)

where

R(t) = {A2+B2(t)+2AB(t)cos[w2t+92(t)]}m (B. 17)

135

s(t)

4444444441

 

 

 

B A N D P A 8 8

P I L T B R

 

x(t)

 

 

L I N I T E R

D I 8 C R I N I N A T O R

Y1“)

 

 

B A 8 B B A N D

F I L T B R

 

YNt)

 

 

 

L O W A
F I B

 

 

8
R

New

| I.

1 z(t)

Figure B.1 FM receiver Model

136

and

B(t) sin1w2t+82(t)]
A+B(t) cos [w2 t+62(t)]

 

¢(t)= tan‘1 (3.18)

In an FM system the message is recovered from the derivative of the phase function,

therefore the output y(t) is given by the following expressions

y(t)=63(t)+¢’(t) (3.19)

(wag) 12131:) cos(w2t+62(t) ) +3210]

1: =3’ 1:
Y() 0( )+ A2+32(t)+2AB(t)cos[w2(t)+9209]

 

where f’(t)=df/dt. Expanding this expression in a Taylor series and assuming that

A/2 > B(t) one gets the following two term approximation

y( c) =eg+ (w,+e;) cos [w,t+e; ( t)]

 

B(t)
A

(3.20)

“317131403 [214213262 (t) ]

From the above expression it follows that with increase of W2 the interference term
increases. For example the best performance will be achieved when one overlays the
interfering signal in the top of FM signal (w2=0, w,=w0), this situation shows the
tolerance of the FM systems that the AM system is lacking. Now consider the terms
92(t) and B(t) which involve discontinuous functions and their derivatives, for example
data dj(t) and code PNj(t). On the one hand one should have w2=0 in order to overlay
on top of FM signal. On the other hand if w2 increase the interfering term is also
increase. To get out of this dilemma one should consider one more stage of FM receiver

which is the post detection filter. In this case one will realize that if overlay has been

137

utilized and w2 is large enough, for instance IWrWol =w2=B’/2 where B’ is the
bandwidth of the post detection filter, then in this case almost all of the interference will
be filtered out and reliable communication is possible.

In the case that W2 is not large enough, for instance w < B’/2 but 2w2 > B’l2, the last

term of equation (3.20) will be filtered, and one gets the following expression

y(t)=63+(w2+6;[3(t)/A]cos(w2t+62) (3.21)
since A > 23. Assuming O to be a constant the determining factor of interference would
be w2 itself. Therefore the reduction in SNR will be from 0 dB (w2 is small) to
approximately 10 logw2 dB. For large value of w2 it could be as large as 40 dB which
could happen in wideband FM.

In summary if w2 is small one can overlay over an FM signal, and the reduction in
SNR would be small. For large values of w SNR will be reduced further. If w2 is large

enough the filter will take care of the loss in SNR [68].

APPENDIX C

SIGNAL TO NOISE AND INTERFERENCE RATIO OF FM

In this appendix an expression for signal to noise and interference for FM is derived.
The receiver in this case is an FM receiver, and is composed of a predetection band-pass
filter, a limiter discriminator, and a post detection low pass filter. The band-pass and
low pass filters have a bandwidth of 2w,,,( 1+6) and w“, respectively, where B is the
modulation index of the FM signal. The discriminator output is proportional to the time
derivative of the phase angle of the input to the discriminator. A block diagram of the
FM receiver is given in Figure 8.].

Similar to the AM case the received signal is composed of three signals, the desired
FM signal s(t), white Gaussian noise n(t) and a set of direct sequence spread spectrum

signals i(t). The desired FM signal can be represented by
s(t) = Acos[wct + ¢m(t)] (c.l)

where A, we and 4),,(t) are the amplitude, the carrier frequency, and the phase angle of
the FM signal. For the FM modulation system ¢m(t) usually has the following form
C
4),..1t)= wmfmumr «2.2)

where wm is the maximum instantaneous frequency deviation, m(t) is the base band

138

139

modulating waveform, or the message signal. The direct sequence signal is modeled in
Appendix A. Assuming that s(t), n(t), and i(t) are independent random variables, the

input signal to noise ratio can be written as before (Appendix A equation (A.8)) as

 

follows:
SNRI= SNR; :51. _
1+icj+iszj (C°3)
Him—3
or
/ 1
SM? = SNR — C.4
I 11+INR ( 1

where SNR'I is the predetection SNR in the absence of interference, and INR is the

interference to noise ratio as in the case of AM system. Thus

SNR} P SNR} as INR —' 0 (C.5)

 

All inferences that have been made for the AM case are also valid for FM predetec-
tion, thus that development is not repeated here. Instead, the discriminator is considered,
which is quite different from the AM detection stage.

Consider the signal x(t), the output of the predetection in Figure B. 1. Using the in
phase and quadrature representation of each of the narrowband signals one can obtain the

following expression at the output of predetection filter [38].

140

U

x( t) = [A+nc( t) +; icy ( t) ] cos [wct+¢m( t)]
+1

-[n,(t)+; i,j(t)]sin[wc(t)+¢m(t)] ((3:53)
=1

= B(t) cos [wct+d>m( t) +1|r(t)]

 

with
u 2 u 2 (0 6b)
R(t)= [A+nc(t)+ .ic (13)] +[ns(t)+ 18113)] °
1 2 j 2:. j
and

n,(t)+ i, (t)
12:; 1

A+nc( t) +; icj ( t)
-1

(C.6c)

 

1|1(t) = arctan

Using this representation the discriminator output can be expressed as follows:
y(t)= lint) (t)+111(t)] ((2.7)
21: d t: m

The lowpass filter suppresses all components higher than the desired message bandwidth.

Thus the output will be
Z(t) = Zm(t) + 2,,(t) + Z,(t) (C.8)

with Zn,(t) is the desired message

141

c. 1 d .
Zm(t)- —21t—t¢m(t) (c 9)
or
d c
= _ = C.1O
Zm(t) t[m(1:)d1: m(t) ( )

where m(t) is the message, and Z,,(t) and Z,(t) are the unavoidable contribution of noise
and direct sequence signals. Z,(t) will contain a collection of 5—functions at points where
dj(t)PN,-(t) changes its polarity, this is in the case of nonretum to zero data format, this
could be improved by using minimum shift keying or raised cosine signaling format [46].

Although one can improve the system performance in this case, however in the worst
case two kinds of distortion will occur, one due to phase and another due to code and
digital data. Thus in this type of interference two kinds of distortions will be noticed.

Similar to AM case, the output signal to noise ratio is given by the following expression

 

 

 

‘—2 —2
Z ‘2
5111120: f: "‘
.131
I 1 I 1
SNRO= SNRO U =SNROW
2 —2
; Zij-zij (Co 11)
1+ '1
—2 4
z,,-zn

where SNR,’ is the signal to noise ratio at the output of the receiver in the absence of
interferences. The statistics of noise and interference at the output of discriminator is

given as follows: [4] and [65]

142

U

2 _ — 2 ITcw;
2,, - 2
3A

. ¥=
' 2‘ 3A2

a?

(c.12)

 

 

Using equations (C.3) and (C. 11) the following value for the gain G and G’ is obtained,

where G’ is the gain without interference.

SNRg
G: SNRO: 1+INR = SNRé:
SNR: SNR} SNR}
1+INR

G’ (c.13)

 

Thus similar to AM case, G = G’ and INR=IT,u/N°.

Appendix D

STATISTICS OF THE INTERFERING SIGNALS

In this Appendix the mean and variance of the direct sequence signal and frequency
hopping signals have been obtained. The derivations involves the in-phase and
quadrature components of the pertinent signals.

Consider ic(t) and i,(t) the in-phase and the quadrature components in the two systems
(direct sequence and frequency hopping). For simplicity consider only one user, for
example the jth user; the generalization to multiple users is straightforward. Any

interfering signal could be written in the following general form.

ij(t)=,/2a(t)cos[wjt+6(t)] (0.1)

where for direct sequence a(t)=d(t)PN(t) and O is a random variable uniform in [0,21].
Furthermore d(t) is random sequence of _1_-1 with E[d(t)]=0, where E denotes the

expectation operator. Consider i(t) in terms of in-phase and quadrature components as

follows:
i(t)=f2-x(t)coswjt-\/§y(t)sinwjt (0.2)
with
x(t) = d(t)PN(t)cose ; y(t) = d(t)PN(t)sin6 (D.3)

In the above expression d(t) and G are stochastic processes and random variables

143

144

respectively. Furthermore assume they are independent of each other, thus
1500 = E[d(t)] EIPN(t)] E(0089) = 0
E(Y) = E[d(t)] E[PN(t)] B(Sme) = 0
since E[d(t)] = 0; therefore E[i(t)] = o, and the variance of i(t) is
Var[i(t)] = E{[i(t)-E(i(t))]2}
= Ewan-Elia)?
= E[i2(t)]
thus
Var[i(t)] = E12a’(t)cosz(wt+9)l
= E[2°10cos2(wt+9)]
since
a2(t) = d2(t)'PN’(t) = 1
Using uniform distribution for 3 one can write the

following expression

Var [.i ( t) ] = alcosz (wjt+0) aB=1
1‘ 0

(D.4)

(D.5)

(13.6)

(D.7)

(3.8)

Now consider the frequency hopping signal. In this case one can have a(t) =d(t) and

wj=wo+h(t)5W, where h(t) is the hopping function, and (SW is the hopping increment.

Since E(d(t))=0 the expectation of the interference signal is also zero or E(i(t))=0 and

the variance becomes unity, as shown:
Var[i(t)] = E{2a2°cosz[wjt+9]} = l

as expected by inspection.

(3.9)

APPENDIX E

SIGNALING SCHEMES FOR DS AND FH

In this appendix a study of the mathematical differences between the different types
of signaling are considered. The discussion shows the difference between direct sequence
and frequency hopping interference. As a result, one can compare the interference case
among the four different types that have been proposed.

Consider DS and FH systems for a single user, the generalization to multi user is
straight forward.

The Direct sequence signal can be written as follows:

i(t),” = f2d(t)PN(t)cos(wot+6) (la)
or
i(t)” = fisin[wo,+e+d(t)PN(t)n/2] (E. lb)

(k-1)1' S t < k1

and for frequency hopping one can have

1(1),, = J25in[wot+h(t)Awt+6 +d(t)1rl2] (5.2)
(k-l)1' S t < kw

In order to prevent intersymbol interference the period of the PN sequence should be

145

146

some integer multiple of the data bit duration. Therefore d(t)PN(t) is a random sequence
of {i1} rectangular waves and the derivative of these type of signals is not continuous

as shown in figure E.1.

 

a)

b)

d)

 

 

 

 

 

a) Digital data d(t) b) PN sequence PN(t)
c) d(t)PN(t) e) d{[D(t)PN(t)]}/dt

Figure E.1 Digital data D(t) and PN sequence PN(t)

As shown in figure E. l—d the derivatives are not defined when the random sequence
changes its signs. Denoting these random points by an impulse function, one can write

an expression for this sequence as follows:

%[d(t)PN(t)] = f: a,15 (Hz) (133)

k. -.

where akE {11} if there is a sign change and zero otherwise.
In the multi-user case the situation becomes worse since one gets the sum of several

sequences of delta functions. Mathematically one gets the following expression.

 

147

if: 4(r)PNJ(r)=)j 2 0,1,50'15) (13.4)
1-1

j-l It»-

with 35,6{11} if there is a sign change for jth user, and zero otherwise.

The difficulty is associated with the rectangular signaling of the channel bits. If one
uses a minimum shift keying (MSK) signal for the channel bits, the discontinuity which
was caused by rectangular signaling could be removed. An expression for MSK is as

follows [46].

s(:)= costwor+b.(t);‘—;1.1 (13.5)

where bK(t)E {—1,1} and ¢KE[O,1r].

This signalling scheme has many important properties [46] and [47]. The property
which is significant in this study is the fact that there is phase continuity in the RF carrier
at the bit transition instant. Thus one can prevent phase discontinuity which was caused
by rectangular signaling. For instance the function is not only continuous, but its
derivative is also continuous. Therefore this kind of signaling can suppress the delta
functions, or at least it smoothers the process.

The above solution is also true for frequency hopping, but in this case an additional
function h(t) has to be included. This is a discontinuous function; these discontinuities
can not be removed. The derivative of this function is a sequence of delta functions
which is deterministically periodic in nature. Thus for FM systems there are two kinds
of interference due to frequency hopping. One due to random data d(t), and the other
due to the hopping function h(t). This frequency hopping interference appears to have

a larger impact than direct sequence interference. Consequently, one can conclude that

 

148

for the cases considered, frequency hopping interference to FM presents the worst
interference case. As a result the interference induced by FH spread spectrum to an FM

signal becomes the subject of this study.

BIBLIOGRAPHY

[1]

BIBLIOGRAPHY

Donald L. Schilling, et al., ”Spread Spectrum Communications A Tutorial" IEEE
mm Vol. COM-30, pp 855-884, 1982.

[2] Dixon, SW, Wiley. 1976.

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

Ray H- Pettit. WWW
Lifelong Publications, Belmont, California, 1982.

Rodger E. Ziemer and Roger L. Peterson, W
W, Macmillan, New York, 1985.

George R. Cooper and Clare D. Mc Gillim, WW5;
5mm, McGraw-Hill New York, 1986.

G.R. Cooper, eta1., "Cellular Land Mobile Radio: Why Spread Spectrum," m
QammunicationsMagazine Vol. 17. N0. 2. 1979.

Walter Scales, Betential Use ef Spread Smetmm Teehniques in fievernmem
AM FCC-0320 Project 14570 the MITRE Corporation, Dec. 1980.

IEEE Ad Hoc Subcommittee on Spread Spectrum; (Ray Nettleton ”Chairman”),
Comments on General Docket No.-8l-413 of the FCC "Authorization of Spread
Spectrum and Other Wideband Emissions not Presently Provided for in the FCC
Rules and Regulations,” IEEE Communication Society, First draft; June 1982.

Sherman Karp Chairman, Communication Theory Committee, Communication
Society, "FCC Testimony,” IEEE Communication Magazine, March 1983.

Didier Verhulst, Michel Mouly, and Jacques Szpirglas "Slow Frequency Hopping
Multiple Access for Digital Cellular Radiotelephone," IEEE Journal on Selected
Areas in Communications, Vol. SAC-2, No. 4, July 1984.

Jay E. Padgett, "Channel Requirements for a Cordless Telephone Spectrum
Allocation,” IEEE Journal on Selected Areas in Communications, Vol. SAC-5,
No. 5, June 1987.

149

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

150

Mohsen Kavehrad and George E. Bodeep, ”Design and Experimental Results for
a Direct-Sequence Spread-Spectrum Radio Using Differential Phase-Shift Keying
Modulation for Indoor, Wireless Communications,” IEEE Journal on Selected
Areas in Communications, Vol. SAC-5, No. 5, June 1987.

Jean-Louis Domstetter and Didier Verhulst, ”Cellular Efficiency with Slow
Frequency Hopping: Analysis of the Digital SFH900 Mobile System,” IEEE
Journal on Selected Areas in Communications, Vol. SAC-5, No. 5, June 1987.

Adel .A.M. Saleh and Leonard J. Cimini, Jr., "Indoor Radio Communications
Using Time-Division Multiple Access with Cyclical Slow Frequency Hopping and
Coding,” IEEE Journal on Selected Areas in Communications, Vol. 7, No. 1,
January 1989.

Sami A. El-Dolil, Wai-Choong Wong, and Raymond Steele, "Teletraffic
Performance of Highway Microcells with Overlay Macrocell, " IEEE Journal on
Selected Areas in Communications, Vol. 7, No. 1, January 1989.

Jeffrey W. Gluck and Evaggelos Geraniotis, "Throughput and Packet Error
Probability of Cellular Frequency—Hopped Spread—Spectrum Radio Networks,”
IEEE Journal on Selected Areas in Communications, Vol. 7, No. 1, January
1989.

Ahmed H. Abdelmonen and Tarek N. Saadawi, " Performance Analysis of Spread
Spectrum Packet Radio Network with Channel Load Sensing,” IEEE Journal on
Selected Areas in Communications, Vol. 7, No. 1, January 1989.

Donald L. Schilling, Raymond L. Pickholtz, and Laurence B. Milstien, ”Spread
Spectrum Goes Commercial", IEEE Spectrum, Aug 1990.

FinalReport, 1' hM l'n frh ki 1
Satellite System contract No. NASS-10744, The Magnavox Company,
Government and Industrial Systems Division.

D.H. Hamsher, Qemmgnieetion System Enginxring HQQMK, Chapter 18,

McGraw-Hill, 1967.

Andrew R. Cohen, et. al. , "A New Coding Technique for Asynchronous Multiple
Access Communication,” EEE Teens, Qommgnication Tech., Oct. 1971.

J.P. Costas, ”Poisson, Shannon, and the Radio Amateur," Proc. IRE, Dec. 1959.

George R. Cooper and Ray W. Nettleton, ffi i ' l l -

WWW P8293264. Purdue
University, 1978.

[24]

[25]

[26]

[27]

[23]

[29]

[30]

[31]

[32]

[33]

[34]

135]

151

G. R. prer and R. W. Nettleton, ”Spread Spectrum in a Fading, Interference

Limited Environment " W June 10-13
1979.

DP. Grybos, et al., ”Probability of Error Performance of the Spread Spectrum
Mobile Communications Receiver in an Non-Rayleigh Fading Environment, 22th

hi 1 hn nf , March 27-30, 1979.

G.R. Cooper and R.W. Nettleton, ”A Spread Spectrum Technique for High-

Capacity Mobile Communications,” IEEE flfrepe. Vehieplg Technolegy, VT-27,
No. 4, Nov. 1978.

D. P. Grybos and G. R. Cooper, "A Receiver Feasibility Study for the Spread
Spectrum High Capacity Mobile Radio Systems," W
fl'ghnplpgy gen fereng, March 22-24,1978.

W., Vol. COM-25, No. 8, August 1977 (special issue on
Spread Spectrum).

Samuel A. Musa and Wasyl Wasylkiwskyj, "Co-Channel Interference of Spread
Spectrum Systems in a Multiple User Environment,“ IEEE Team, 29mm, Vol.
COM-26, No. 10, Oct. 1978.

Mepile Mplp'ple Agsg Study, Final Regn, TRW Defense and Space System
Group, NASA Contract No. NASS-23454, August 1977.

Michael B. Pursley, ”Performance Evaluation for Phase-Coded Spread Spectrum
Multiple-Access Communication--Part 1: System Analysis” , IEEE Teape. Cpmm, ,
Vol. COM-25, No. 8, August 1977.

Leslie A. Berry and E. J. Haakinson, Spgtgtm Effieieney fer Multipl e
. - - ,NTIA Report 78/ ll,

 

153291539, November 1978.

George P. Cooper and Raymond W. Nettleton, ”A Spread Spectrum Technique
for High Capacity Mobile Communications," 7 h IEEE V hi 1 T l
Qpnjemg, March 16-18, 1977.

Raymond W. Nettleton and George R. Cooper, "Error Performance of a Spread
Spectrum Mobile Communication System in a Rapidly-Fading Environment,"

1339'an Telggzmmpnigtipns Cpnferenee, 1977.

Raymond W. Nettleton and George R. Cooper, "Mutual Interference in Cellular
LMR Systems,” Midgn 1277 prggdings, November 8-10, 1977.

[36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

152

A. J. Viterbi, 'Bandspreading Combats Multipath and RFI in Tactical Satellite
Net " W December 1969

AGARD, Advisory Group for Aero-Space Research and Development, Paris
(France), WWW. Lecture Series, July 1973.

Don L Torrieri. WWW AD A067299.
December 1978.

R.F. Ormondroyd, ”Spread Spectrum Communications in the Land Mobile

Service,” MW,
Birmingham, England, April 4- 7, 1978.

John R. Juroshek, A limin E 'm h Eff
W, NTIA Report 76-8, PB—286623, June 1978.
Paul Newhouse, Prggpres fer Anflyzing Ipterfereng Cepsg py Sprgd-

W315, I'I‘I‘ Research Institute Report No. ESD-77-033, AD A0569] 1 ,
Feb. 1978.

Ray W. Nettleton, "Comments on General Docket No—81-413 of the FCC" IEEE
Ad-Hoc Subcommittee on Spread Spectrum, June 1982.

Leonard Farber, ”Spread Spectrum Interference to Voice Communication

SystemS”. WW PP- 282-287. 1978.

Ray W. Nettleton, Effi i n in l l - i1 m ni ' 1) °
W, Ph.D. Thesis, Purdue University, Oct. 1978.

William C.Y. Lee, Mppile Cpmmpnigp’ens Engingring, McGraw Hill, 1983.

Subbarayan Pasupathy, "Minimum Shift Keying" A Spectrally Efficient
Modulation, IEEE Qemmpnicag'ons Maggine, July 1979.

Amoroso, "Pulse and Spectrum Manipulation in the Minimum (Frequency) Shift
Keying (MSK) Format”, IEEE Tram, Cemm,, Vol. COM-24, pp. 381-384,
March 1976

Athanasios Papoolis, Prebability mdom variables and Stghgtic Processee,
McGraw-Hill, New York, 1965.

Herbert B. Voelcker. ”Toward A Unified Theory of Modulation Part 1: Phase
EnveIOPe RelationshiPS”.IEEE_2m1ngs_VQL_fl. No. 3. pp- 340353-

[50]

[51]

[52]

[53]

[54]

[55]

[5 6]

[57]

[5 3]

[59]

[60]

[61]

[62]

[63]

153

Herbert B. Voelcker ”Toward A Unified Theory of Modulation Part H: Zero
Manipulation," IEEE Preegdings; Val, 54 No. 5 pp. 735-755, May 1966.

Andrew Sekey ”A Computer Simulation Study of Real- Zero Interpolation," M

WM 1 pp. 43- 54, March
1970.

Herbert B. Voelcker ”Zero-Crossing Properties of Angle-Modulated Signals, "
WWW N0. 3 119307-315, June 1972.

R.G. Wiley, H. Schwarzlander and DD. Weinter ”Demodulation Procedure For

Very Wide-Band FM,” EEE Teapgetion Cpmmpnigtjon, Vol, COM-25, pp.
318-327, March 1977.

M.Dechambre and J. Lavergbnate, "Frequency Determination of Noisy Signal,

by Zero Crossings Counting," Signal Prgegsing, V91, 3, pp. 93-105, February
1986.

SM. Kay and R. Sudhaker, "A Zero Crossing-Based Spectrum Analyzer,"
Agustie, Smh Signal Processing Vel= ASSP-3 No. 1, February 1986.

Benjamin Kedem, ”Search For Periodicities by Axis- -Crossings of Filtered Time

Series, ”SgpaLELgesstpgloLfl, pp. 129- 144, March 1986.

_, ”On Frequency Detection by Zero Crossings,” i n Pr in l 1 ,
pp. 303-306, April 1986.

_, ”Spectral Analysis and Discrimination by Zero-Crossings,” IEEE Prg, V91,
:75, No. 11, pp. 1477-1493, November 1986.

Mischa Schwartz, Trapgmiseipn, Medulap’on, and Noise, Third Edition, McGraw-
Hill, pp. 286-288, 1980.

Henry Stark and Franz B. Tuteur, Mgem Elgtrieal Cemmpnigtjpne Thgry md
Syater_n§, Second Edition, pp. 344-345, Prentice Hall, 1988.

Simon Haykin, Di 1 mmuni 'ons, Jhon Wiley, New york 1988

Willim Press et 211. W
Cambridge Publication, london 1986.

Donald E. Knuth, imin m ' l rihm 2n v
gemppter Emgpamming, Addison-Wesley Massachusetts, 1981.

[64]

[65]

[66]

[67]

[63]

154

Shu Lin and Daniel Costello, _E_r:er gen eel Qging; Funeamentgs aed
Applieatieps, Printice-Hall, New Jersey 1983.

Robert M. Gagliardi, Inteedeee'en t_o Qemmenigtiens Engineering, Second
Edition, John Wiley & Sons, New York, 1988.

I..S Gradshteyn and LM. Ryzhik, Taele ef Integgls, Series, aed Prug uets,
magnum, Academic Press, New York, 1980.

John M. Wozencraft and Irwin M. Jacobs, bjeeiples ef Cemmenigtjen
Engingrjeg, John Wiley & Sons, New York, 1965.

Ghulam H. Raz, "Coexistance of Frequency Hopping and PM: An Interleaving
Overlay Approach", GLOBECOM 87, Tokyo Japan. Nov. 1987 .

       

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