PERFORMANCE OF MULTITONE DIRECT SEQUENCE SPREAD

SPECTRUM IN THE PRESENCE OF NARROWBAND AND

PARTIALBAND INTERFERENCE

A thesis presented to

the faculty of

School of Electrical Engineering and Computer Science

Russ College of Engineering and Technology of

Ohio University

In partial fulfillment

of the requirement for the degree

Master of Science

Virat Deepak

November 2002

This thesis entitled

PERFORMANCE OF MULTITONE DIRECT SEQUENCE SPREAD SPECTRUM IN

THE PRESENCE OF NARROWBAND AND PARTIALBAND INTERFERENCE

BY

VIRAT DEEPAK

has been approved for

the School of Electrical Engineering and Computer Science

and the Russ College of Engineering and Technology by

David W. Matolak

Assistant Professor, School of Electrical Engineering and Computer Science

Dennis Irwin

Dean, Russ College of Engineering and Technology

DEEPAK, VIRAT. M.S. November 2002. Electrical Engineering

Performance of Multitone Direct Sequence Spread Spectrum in the Presence of

Narrowband and Partialband Interference (103 pp.)

Director of Thesis: David W. Matolak

In this work, we provide new analytical and computer simulation results for the

performance of multitone (MT) DS-SS signaling in the presence of narrowband and

partialband interference. We look at two transformations on the input data, serial to

parallel conversion and replication. We investigate the tradeoff between the number of

subcarriers and the per-subcarrier processing gain, for a fixed data rate and fixed

bandwidth, and also compare with the conventional single-carrier (SC) system. Our

results show that in the presence of single and multiple tone interference, MT DS-SS and

conventional SC system have similar performance.

Approved: David W. Matolak

Assistant Professor,

School of Electrical Engineering and Computer Science

Acknowledgements

First of all, I would like to thank my thesis advisor Dr. David Matolak for his

invaluable support and guidance during the duration of this research. His creative

insights and scientific visions constantly inspire me. I would also like to thank my

committee members, Dr. Jeffrey Dill and Dr. Chris Bartone for reviewing my thesis and

for their instruction in the classes I took with them.

I am also thankful to Frank Alder for his unfailing assistance in my dealing with

Matlab. I would also like to thank the numerous friends I made in the last eighteen

months in Athens who inspired me. Finally, most of all, I thank my parents for their

unconditional love and support.

5

TABLE OF CONTENTS

Abstract ............................................................................................................................... 3

Acknowledgements ............................................................................................................. 4

Table of Contents................................................................................................................ 5

List of Tables ...................................................................................................................... 8

List of Figures ..................................................................................................................... 9

CHAPTER 1 .................................................................................................................... 12

INTRODUCTION........................................................................................................... 12

1.1 Background ........................................................................................................... 12

1.2 Multiple Access Techniques ................................................................................. 14

1.3 Multicarrier CDMA............................................................................................... 18

1.4 Thesis Objective .................................................................................................... 19

1.5 Outline of Thesis ................................................................................................... 20

CHAPTER 2 .................................................................................................................... 22

SYSTEM DESCRIPTION ............................................................................................. 22

2.1 Literature Review.................................................................................................. 22

2.2 System Model........................................................................................................ 24

2.2.1 Transmitter Description................................................................................. 25

2.2.2 Receiver Description..................................................................................... 30

2.3 Description of Impairments................................................................................... 33

CHAPTER 3 .................................................................................................................... 36

MT PERFORMANCE ANALYSIS .............................................................................. 36

3.1 Receiver Statistics and Definitions ....................................................................... 36

6

3.2 Tone Jamming Decision Statistics ........................................................................ 41

3.3 Partial Band Jamming Statistics ............................................................................ 42

3.4 Bit Error Rate Expressions .................................................................................... 46

3.4.1 T=S:P, Single Tone Jammer.......................................................................... 46

3.4.2 T=S:P Multiple Tone Jamming ..................................................................... 48

3.4.3 T= Replication, Single Tone Jamming .......................................................... 48

3.4.4 Single Carrier Partial Band Jamming ............................................................ 51

3.5 Simulation and Analysis Comparison Equations .................................................. 52

CHAPTER 4 .................................................................................................................... 56

SIMULATION DESCRIPTION.................................................................................... 56

4.1 Simulation Transmitter Description...................................................................... 57

4.2 Simulation Channel Description ........................................................................... 59

4.3 Simulation Receiver Description .......................................................................... 60

CHAPTER 5 .................................................................................................................... 62

SIMULATION RESULTS ............................................................................................. 62

5.1 Inter Subcarrier Same User Interference (IS-SUI) ................................................ 62

5.2 Tone Jamming ....................................................................................................... 64

5.2.1 Single Tone Jamming .................................................................................... 64

5.2.2 Multiple Tone Jamming ................................................................................ 70

5.3 Partial Band Jamming ........................................................................................... 75

5.3.1 Rectangular Spectrum Interferer ................................................................... 75

5.3.2 M-ary (“Sinc-Squared” Spectrum) Interferer ................................................ 79

CHAPTER 6 .................................................................................................................... 84

CONCLUSIONS AND FUTURE WORK.................................................................... 84

6.1 Summary of Research ........................................................................................... 84

6.2 Suggestions for Future Work ................................................................................ 85

REFERENCES ................................................................................................................ 87

7

APPENDICES ................................................................................................................. 90

Appendix A: MT Replication Variances..................................................................... 90

Appendix B: DS-SS Cross-Correlations ..................................................................... 93

Appendix C: Matlab Programs.................................................................................... 96

8

LIST OF TABLES

Table 2.1: Relationships between energies, powers, symbol times in MT-DS-SS......... 30

9

LIST OF FIGURES

Figure 1.1: Simplified digital communication system. ................................................... 13

Figure 1.2: Frequency division multiple access.............................................................. 15

Figure 1.3: Time division multiple access ...................................................................... 16

Figure 1.4: Code division multiple access ...................................................................... 17

Figure 2.1: Power spectrum of MT-DS-SS .................................................................... 24

Figure 2.2: DS-SS transmitter for description of SC, MC, & MT waveforms. .............. 27

Figure 2.3: Channel and Receiver (ith subcarrier) ........................................................... 32

Figure 2.4: PSD of MT-DS-SS with single and multiple tone jammers......................... 34

Figure 2.5: PSD of MT-DS-SS with partial band jammer.............................................. 35

Figure 3.1: MT-SS transmitter diagram for description of MT waveforms ................... 37

Figure 4 1: Schematic depiction of MATLAB simulation functional operation

for MT (S:P) ................................................................................................ 58

Figure 4.2: Power density spectrum of partial band jammer with J/S=10dB, and

bandwidth = 0.9Rc ....................................................................................... 60

Figure 5.1: Aggregate Pb vs. Eb/N0 for MT(S:P) with M=3, long different codes,

showing the effect of IS-SUI for various values of N. ................................ 63

Figure 5.2: Aggregate Pb vs. Eb/N0 for MT(S:P) with single tone interference, several

values of the number of subcarriers M and per-subcarrier processing gain N,

for two J/S ratios, using same long codes.................................................... 65

Figure 5.3: Aggregate Pb vs. Eb/N0 for MT(Rep) with single tone interference, several

values of the number of subcarriers M and per-subcarrier processing gain N,

for two J/S ratios, using same long codes. Jammer with random phase. .... 66

10

Figure 5.4: Aggregate Pb vs. Eb/N0 for MT-DS-SS with single-tone interference,

M=3, NMTS=100, NMTR=32, for two J/S ratios, using long codes, showing

MT(Rep) variation with jammer phase. ...................................................... 67

Figure 5.5: Aggregate Pb vs J/S for Eb/N0 of 6 dB. MT(S:P ) with different long codes

and single tone jammer................................................................................ 68

Figure 5.6: Pb vs J/S for Eb/N0 of 6 dB. MT(Rep) with different long codes and single

tone jammer. ................................................................................................ 69

Figure 5.7: Aggregate Pb vs. Eb/N0 for MT-DS-SS with single-tone interference, M=3,

for two J/S ratios, using (same) long codes, showing agreement with

analytical results. For S:P, M=3, NMTS=304; for Rep, M=3, NMTR=100. .... 70

Figure 5.8: Aggregate Pb vs. Eb/N0 for MT(S:P) with M-tone jammer, for two J/S ratios,

using different long codes. SC results for 3 tones. ..................................... 71

Figure 5.9: Aggregate Pb vs. Eb/N0 for MT(Rep) with M-tone jammer, for two J/S ratios,

using different long codes. Jammer with random phase. SC results for 3

tones............................................................................................................. 72

Figure 5.10: Aggregate Pb vs J/S for Eb/N0 of 6 dB. MT(S:P) with same long codes and

M tone jammer. SC results for 3 tones ....................................................... 73

Figure 5.11: Aggregate Pb vs J/S for Eb/N0 of 6 dB. MT(Rep) with same long codes and

M tone jammer. SC results for 3 tones ....................................................... 74

Figure 5.12: Aggregate Pb vs. Eb/N0 for MT-DS-SS with M-tone jammer, M=3, for two

J/S ratios, using (same) long codes, showing agreement with analytical

results. For S:P, M=3, NMTS=304; for Rep, M=3, NMTR=100...................... 75

Figure 5.13 Pb vs. Eb/N0 for MT (S:P) and SC in the presence of a rectangular-spectrum

interferer, for J/S=10dB, and two values of BJ and N/M. Same long codes

on the subcarriers......................................................................................... 76

11

Figure 5.14: Pb vs. Eb/N0 for MT(Rep) and SC in the presence of a rectangular-

spectrum interfe rer, for J/S=10dB, and two values of BJ and N/M. Same

long codes on the subcarriers. ..................................................................... 77

Figure 5.15: Pb vs BJ for MT(S:P) in the presence of a rectangular-spectrum interferer,

for J/S=10dB, and two values of Eb/N0 and N/M, showing variation of BER

with BJ Same long codes on the subcarriers. ............................................. 78

Figure 5.16: Pb vs BJ for MT(Rep) in the presence of a rectangular-spectrum interferer,

for J/S=10dB, and two values of Eb/N0 and N/M, showing variation of BER

with BJ Same long codes on the subcarriers. .............................................. 79

Figure 5.17: Pb vs. Eb/N0 for MT(S:P) and SC in the presence of a M-ary (“sinc

squared” spectrum) interferer, for J/S=10dB, and two values of BJ and N/M.

Same long codes on the subcarriers............................................................. 80

Figure 5.18: Pb vs. Eb/N0 for MT(Rep) and SC in the presence of a M-ary (“sinc

squared” spectrum) interferer, for J/S=10dB, and two values of BJ and N/M.

Same long codes on the subcarriers............................................................. 81

Figure 5.19: Pb vs fJ for MT(S:P) and SC in the presence of a rectangular-spectrum

interferer, for J/S=10dB, BJ=Rc/10 and two values of Eb/N0 and N/M,

showing variation of BER with fJ. Same long codes on the subcarriers..... 82

Figure A.1: Expectations vs. jammer phase for MT(Rep) with M=3 NMTR=100 with

single tone jammer with J/S=10dB, and jammer fJ=1.6f1 showing the

magnitude difference for different cross-product terms. ............................. 91

Figure A.2: Cross term expectations vs. jammer phase for MT(Rep) with M=3

NMTR=100 with single tone jammer with J/S=10dB, and fJ=1.6f1, 2f1 and

2.2f1 showing the variation in the cross-term amplitude with the jammer

center frequency fJ. ...................................................................................... 92

12

Chapter 1

Introduction

1.1 Background

In its basic electrical sense, the term communication refers to sending, receiving and

processing of information by electric means. As such, it started with wire telegraphy in

the eighteen forties, developed with telephony some decades later, and continued with

radio at the beginning of the twentieth century. Radio communication, made possible by

the invention of the triode tube, was generally stimulated by the work done during World

War II. It subsequently became more widely used and refined through the invention and

use of transistor, integrated circuits and other semiconductor devices. More recently, the

use of satellite and fiber optics has made communications even more widespread, with an

increasing emphasis on computer and data communications.

Digital communication is a branch of communications which utilizes discontinuous

signals, i.e., signals which appear in discrete “steps” (0 & 1 in binary) rather than having

the continuous variation characteristic of analog signals. The principle feature of a digital

communication system (DCS) is that during a finite interval of time, it sends a waveform

from a finite set of possible waveforms in contrast to an analog communication system,

which sends an infinite variety of waveform shapes. It is important to note, that though

the waveform transmitted by a DCS has an analog appearance, it is called a digital

13

waveform because it is encoded with digital information. DCS are becoming

increasingly attractive because of the ever growing demand for data communication, and

because digital transmission offers data processing options and flexibilities not available

with analog transmission. In a DCS, the objective at the receiver is not to reproduce a

transmitted waveform with precision; instead, the objective is to determine from a noise-

perturbed signal which waveform from the finite set of waveform was sent by the

transmitter. A simplified DCS block diagram is shown in Figure 1.1. More detailed

structure and functionality of each block can be referred in [1].

Figure 1.1: Simplified digital communication system.

Over the last two decades in the commercial marketplace, a new technique has been

emerging called Spread Spectrum. This field covers the art of secure digital

communications that is now being exploited for commercial and industrial purposes.

Spread spectrum is a means of transmission in which the signal occupies a bandwidth in

excess of the minimum necessary to send the information. Spread spectrum can be

implemented in a number of ways; two prominent among them are Frequency Hopping

Information Source

Source Encode

Spread

Transmitter Receiver

Despread Demodulation

Channel Decode

SourceDecode

Information Sink

Modulation

Channel

Channel Encode

14

and Direct Sequence. Frequency hopping works very much like its name implies. It

takes the data signal and modulates it with a carrier signal whose center frequency hops

from frequency to frequency as a function of time over a wide band of frequencies. With

frequency hopping spread spectrum, the carrier frequency changes periodically. Direct

sequence spread spectrum combines a data signal at the sending station with a higher data

rate bit sequence, which many refer to as a chipping code (also known as spreading

code). The band spreading in this method is accomplished by means of a code that is

independent of the data, and a synchronized reception with the code at the receiver is

used for the de-spreading and the subsequent data recovery. The ratio of chips to

information bits is defined as the processing gain. In the next several years hardly

anyone will escape being involved, in some way, with spread spectrum communications

due to its advantages, namely multiple access capability, robustness against fading, and

anti- interference characteristics. Applications for commercial spread spectrum range

from wireless communication, to wireless LAN's, to integrated bar code scanner/palmtop

computer/radio modem devices for warehousing, to digital dispatch, to "information

society" city or country wide networks for passing faxes, computer data, email, or

multimedia.

1.2 Multiple Access Techniques

In any wireless communication system, there are many users who need to

communicate simultaneously. Therefore, the available radio frequency (RF) resources

must be distributed among these users in a way that allows them to access the

15

communication system. In a coordinated system, such as a cellular network, the

allocation of these resources requires extensive planning.

Perhaps the most natural and fundamental way for multiple users to communicate

simultaneously is to allocate a different subband of the RF spectrum to each user. A

simple bandpass filter at the receiver would then select the bandwidth of interest. This

method, Frequency Division Multiple Access (FDMA) is the oldest method for multiple

access, dating back to the invention of broadcast radio. Different channels in an FDMA

system are simply assigned different frequency bands that do not overlap, as illustrated in

Figure 1.2. One of the main features of FDMA is that each channel is narrowband,

allowing either an analog or digital modulation scheme.[2]

Figure 1.2: Frequency Division Multiple Access.

Instead of splitting the RF spectrum into subbands for each user, multiple non-

overlapping time slots can be created and assigned to each user. The receiver

Tim

e

Use

r 1

Use

r 2

Use

r k

• • •

Frequency

16

synchronizes to the correct time slot to recover the user's information. Figure 1.3

shows resource allocation in a Time Division Multiple Access (TDMA) system, which is a

somewhat more complex technology. Since all users occupy the entire RF bandwidth,

TDMA channels have much wider bandwidths compared with FDMA channels, usually

necessitating equalization to overcome degradation due to multipath.

Figure 1.3: Time Division Multiple Access.

Another system called Code Division Multiple Access (CDMA), allows the

communicators the entire spectrum all of the time (Figure 1.4). CDMA is a multiple

access technique that differentiates between users by assigning unique spreading codes to

them. Although the users sharing the spectrum overlap in time and frequency, the

receiver is able to differentiate each user’s information from that of other users by

correlating the received signal with the desired user’s spreading code. “Encoding” the

user information with its unique code usually leads to the spreading of the user signal

Tim

e

Frequency

User 1

User 2

User k

•

17

bandwidth, which is why it is termed a direct sequence spread spectrum technique. In

a sense, the spreading codes can be viewed as another dimension the signals can occupy

[3].

Figure 1.4: Code Division Multiple Access.

In the early 1990s, CDMA was adapted to civilian applications, most notably

wireless applications. Qua lcomm, a U.S. company, was the driving force in those days

and had its system standardized as US interim standard IS-95 [4]. In that standard, the

information is spread into a 1.25 MHz wide spectrum by multiplication of each bit with a

whole sequence of chips, where each chip is 0.814µs long. The CDMA technology was

later also adapted by Japanese and European standardization bodies for the definition of

Codes

Tim

e

Frequency

User 1

User 2

User k

18

third generation wireless systems called W-CDMA. This system has a bandwidth of 5

MHz and thus will also be wideband in many indoor applications, as opposed to IS-95.

1.3 Multicarrier CDMA

A recent development in this field is the emergence of a new modulation technique,

namely Orthogonal Frequency Division Multiplexing (OFDM). Essentially OFDM

distributes, or spreads, the information to be transmitted onto many orthogonal sinusoidal

subcarriers so that the bits on each subcarrier are much longer, drastically reducing the

effect of any channel dispersion, which causes Intersymbol Interference, or ISI. This

technique can improve system capacity by making transmission more robust to frequency

selective fading; this system capacity increase can be be viewed as enhancing system

spectral efficiency. Frequency division multiplexing can be thought of as one type of

multi-carrier/OFDM scheme [6].

Multicarrier DS-CDMA (MC-DS-CDMA) is a modulation technique that combines

OFDM and DS-CDMA [7]. An MC-CDMA transmitter combines the use of multiple

carriers and spectrum spreading; it spreads the original data stream in the frequency

domain over different subcarriers using a given spreading code, or set of codes. This

approach is in fact an option in the 3rd-generation cellular standard cdma2000. The MC-

CDMA offers better frequency diversity to combat frequency selective fading. Other

versions allow different data on subcarriers for higher data rates. Another technique

called Multitone CDMA (MT-CDMA) [8] is similar to MC-CDMA in the sense that the

incoming bit stream is divided into a number of different bit streams. After that, the

19

spreading of each stream is done with a much longer spreading sequence relative to the

MC case. This results in substantial subcarrier spectral overlap, in contrast to the MC

case where subcarrier spectra are essentially non-overlapping. As with the MC schemes,

typically the MT approach also uses a constant bandwidth for each of the subcarriers.

The key distinction between these techniques is the amount of spectral overlap

among subcarriers—MC has little overlap, with subcarrier orthogonality over chip time

Tc, whereas MT has much overlap, with subcarrier orthogonality over symbol time Ts.

The MC approach can provide explicit frequency diversity, but the MT approach may

have potential to provide higher spectral efficiency, because of its larger per-subcarrier

processing gain [9].

1.4 Thesis Objective

The objective of this research is to apply mathematical analysis and computer

simulations to evaluate the effects of narrowband and partial band interference (jamming)

on the Multitone-DS SS system. We consider transmission schemes that apply two

different transformations to the input data: serial-to-parallel conversion and replication.

This research also addresses the issue of using same and different spreading codes on the

subcarriers. We will investigate the tradeoff between the number of subcarriers and the

per-subcarrier processing gain, for a fixed data rate and fixed bandwidth. We will present

simulation and analytical results in the presence of these impairments on MT-DS SS

system and compare performance to that obtained on a traditional single carrier CDMA

(SC-DS-SS) system. To our knowledge, this type of study has not been done before. We

20

will also consider a generalization of inter-subcarrier crosscorrelations that take into

account the subcarrier sinusoids; these generalized crosscorrelations are of interest in the

code selection process, and can be used to predict degradations due to Inter Subcarrier-

Same User Interference (IS-SUI) when different codes are used on subcarriers for

enhanced security.

The following conditions are assumed for the study:

• For fair comparison between the systems of interest [SC, MT(S:P) and

MT(Rep)], we consider equal data rates and equal bandwidths

• The received powers of the three systems are equal, and

• We use long, random (Bernoulli chip) spreading codes for all the three

systems.

1.5 Outline of Thesis

The remainder of the thesis is organized as follows. In Chapter 2 we precisely

specify the system of interest and define terminology which is helpful in clear

understanding of the subject. We also present a description of the system model used for

the MT-DS-SS system. Chapter 3 provides analytical results derived for the system in

the presence of narrowband and partial band interference. Chapter 4 describes the

simulation method employed for the performance evaluation. The implementation of

transmitter, receiver channel and impairments in software is also explained. Chapter 5

covers the results obtained from the simulation and the comparison of these to the

21

analytical results obtained in Chapter 3. Chapter 6 concludes the thesis and

summarizes the results of the work. Areas of future work are also detailed and briefly

discussed.

22

Chapter 2

System Description

In this chapter we present a brief literature review [11] of the work done in the field

of MT-DS-SS. This literature review identifies the gap in previous work done and helps

us to choose the topics which bridge this gap. This is followed by the description of the

common system model and frame work developed to facilitate the study of three DS-SS

waveforms—the wideband SC-DS-SS, the MC-DS-SS, and the MT-DS-SS.

2.1 Literature Review

The founding paper on MT-DS CDMA is the one by Vandendorpe [7]. As noted in

the title of [7], this paper studies performance on a dispersive channel. It assumes perfect

power control (no near- far problem), and Gaussian MUI statistics. Much of our analysis

parallels this development because of its generality. In [7], the system uses the same

spreading code and different data (our transformation T = S:P) on all subcarriers. In this

paper, detection is coherent and modulation is PSK.

One of the first (recent) papers on MC-DS schemes is the one by Kondo & Milstein

[10]. This paper analyzes performance on a dispersive channel where the fading is

frequency non-selective (flat) over each subcarrier, and a single spreading code, and the

same data (our transformation T = replication) is used on all subcarriers. A comparison is

made to SC DS CDMA performance, and for the channel studied, these systems exhibit

23

identical performance. This paper is described because it has initiated much additional

work in the area, on systems both identical and similar to this one. It is also worth noting

that in this paper, and in others, one of the primary motivations for using multicarrier

schemes is for frequency diversity on frequency-selective fading channels.

Two recent overview papers that compare both the structure and performance of the

MT & MC DS schemes are [11] and [12]. These papers also consider terrestrial cellular

applications. Reference [11] discusses some implementation issues for flexible

modulations, and [12] uses simulation results to compare performance among several

schemes on a chip-synchronous dispersive channel. Receiver structures and algorithms

are also discussed in [12] with its virtue being primarily illustration of the similarities of

the various DS-CDMA approaches.

In [13], the authors study MC CDMA where interleaving is used on the subcarrier

data streams. They provide a well-organized description of the impairments encountered

in MC transmission over a dispersive channel when multiple user signals are present, and

the subcarriers generally overlap. Hence, this scheme is distinct from the MC scheme in

[10], and can be thought of as lying between the non-spectrally-overlapping scheme of

[10] and the fully-spectrally-overlapping MT scheme in [7]. Reference [14] also studies

the effect of subcarrier frequency spectral overlap, but is focused on a particular

dispersive channel; this paper can be considered an extension to the work in [13]. All its

results (analytical only) rely on a Gaussian MUI approximation, useful for large numbers

of simultaneous users, large processing gain, and relatively large error probability.

24

It can be noted from the literature review above, that MT-DS-CDMA has seen

much less attention as compared to MC-DS-CDMA and no work has been done in

evaluating the performance of MT-DS-CDMA in jamming environment.

2.2 System Model

In brief, MT-DS CDMA system is implemented with the subcarriers separated in

frequency by multiples of 1/(NT), with N the processing gain and T is the bit duration.

The data streams on each subcarrier are direct-sequence spread with either a subcarrier-

unique or common spreading code; in the subcarrier-unique-code case the waveforms on

the subcarriers are no longer orthogonal. This results in IS-SUI. In the frequency

domain, the spectrum of the MT-CDMA signal consists of M subcarriers each spread by

a factor N. The bandwidth of each subcarrier after spreading the signal is usually

designed to be larger than the coherence bandwidth of the channel. Figure 2.1 shows a

diagram of the transmitted spectrum with M subcarriers [8].

f

BTBSC

f1 f2 fM...Figure 2.1: Power spectrum of MT-DS-SS.

25

We next develop a common notational framework and set of parameters for the

receiver and transmitter. This will facilitate communication to readers, and more easily

allow for comparisons between the three systems.

This description is a mathematical representation of time-domain waveforms, and

includes a large number of parameters.[9] Analysis of one form or the other can then be

done by proper choice of parameters. For all the three cases long spreading codes are

used, and we assume coherent detection with MPSK modulation, as this modulation

format is most common for the DS-SS schemes. The channel is assumed to be non-

dispersive AWGN.

Spreading codes are called “long” if the period of the code is greater than the symbol

period and “short” if the period of the code is equal to the symbol period [15]. In other

words, if we have the same spreading code on each symbol in a sequence, it is called a

short code, and if we have a different code on each symbol then it is called a long code.

We note here that this description is primarily for the development of the analysis.

The simulation model is naturally slightly different from this and is discussed in the

chapter on Simulation Description (Chapter 4).

2.2.1 Transmitter Description

The waveform description is facilitated by a diagram of the transmission scheme.

This is shown in Figure 2.2. Referring to Figure 2.2, we represent the kth user’s data

waveform as dk(t). Although our notation allows for arbitrary signal sets, our focus will

26

be on the most typical binary case. In the general case the data waveform for the kth

user can be represented as

∑∞

=

−=0

)()(n

Tnb

b nTtpdT

aEtd (2.1)

where the nth data symbol ∈nd {±1}, the bit energy is Eb, the bit duration is Tb, and the

constant a=2 for BPSK, and a=1 for QPSK. The pulse shape pT(t) is rectangular:

≤≤

=else ,0

Tt0 ,1)(tpT (2.2)

The data waveform enters the transformation block denoted T to form 2M parallel

streams for quadrature modulation, and M parallel streams for binary modulation, where

M is the number of subcarriers. This transformation can take several forms, depending

upon the desired waveform characteristics. We consider two transformations; serial-to-

parallel (S:P) and replication (abbreviated “Rep”). In the S:P transformation, the symbol

rate (Rs) is equal to Rb/M, where Rb is the bit rate. In the rep transformation, Rs is equal to

Rb. We denote Iki(m) and Qki(m), as modulation symbols for the data symbols dkn where

Iki(m) is the mth in-phase symbol on the kth user’s ith subcarrier, and Qki(m) is likewise for

the ith quadrature subcarrier, where i ranges from 1 to M.

The modulation waveform for the kth user ith subcarrier I-channel can be represented

as

∑∞

=

−=0

)()()(m

sTb

b mTtpmIT

aEtI

s (2.3)

27

and a similar expression can be obtained for for Qki(t).

Figure 2.2: DS-SS transmitter for description of SC, MC, & MT waveforms.

Each quadrature channel is DS-spread by a spreading code, which may or may not be

unique. The ith subcarrier spreading code for the I-channel is denoted )()( tc Iki , and

similarly for the Q-channel code. These codes are assumed binary. The code waveform

is of the same form as the data symbol waveform:

MifornTtpnctcn

cTI

kiI

ki c.....2,1 )()()(

0

)()( =−= ∑∞

= (2.4)

T ... dk(t)

c(I)k1(t)

Ik1(t)

Σ

vk(t)

Qk1(t)

c(Q)k1(t)

cos(ωc1t+θ1)

c(I)k2(t)

Ik2(t)

Qk2(t)

c(Q)k2(t)

c(I)kM(t)

IkM(t)

QkM(t)

c(Q)kM(t)

-sin(ωc1t+θ1)

cos(ωc2t+θ2)

-sin(ωc2t+θ2)

cos(ωcMt+θM)

-sin(ωcMt+θM)

× hT1(t)

×

×

× ×

×

× ×

×

× ×

×

hT1(t)

hT2(t)

hT2(t)

hTM(t)

hTM(t)

28

where the DS spreading factor, or processing gain is N=Ts/Tc, with Tc=1/Rc the chip

time, and Rc is the chip rate. After spreading, lowpass pulse shape filtering is

implemented on the ith subcarrier with the ith subcarrier filter, with unit-energy impulse

response hTi(t).

The spread data waveforms then modulate the subcarriers. The I-channel ith

subcarrier signal is cos(ωcit+θi), where ωci=2πfci=2π(fc+fi), with fc being a common

carrier frequency, and fi the offset of the ith subcarrier from this common frequency. The

phase shift θi is often set equal to zero for all i (we note that judicious choice of the set of

phases can reduce the peak-to-average power ratio (PAR) of the final RF signal). Finally,

the modulated and spread waveforms are scaled, and then summed.

The system shown in Figure 2.1 can be used to represent any of the three DS-CDMA

waveforms, if the parameters are set as follows:

• SC-DS-CDMA: set M=1 subcarrier

• MC-DS-CDMA: set fi=iRc to yield orthogonal (mostly spectrally non-

overlapping) subcarrier signals, and

• MT-DS-CDMA: set fi=i/Ts to yield spectrally overlapping subcarrier signals.

The final carrier waveform for user k is denoted vk(t):

29

[ ]

])(2sin[)()()/(

])(2cos[)()()/(

)sin()()()cos()()(

)()(

)(,

1

)(,

1

)()(,

1

iicn

cTiQ

kikis

kis

iic

M

i ncTi

Ikiki

s

kis

M

iici

Qkikiici

Ikiki

s

kis

M

ikik

fftnTthncNnQT

E

fftnTthncNnIT

E

ttctQttctIT

E

tvtv

θπ

θπ

θωθω

++

−−

++

−=

+−+=

=

∑

∑ ∑

∑

∑

=

=

=

(2.5)

where x = integer part of x. The quantity Es,ki in (2.5) is the symbol energy of the 2-

dimensional I-Q symbol on the ith subcarrier for user k. The total power transmitted by

user k we denote by ∑=

=M

ikiTk PP

1

, where Pki is the power (W) on the ith subcarrier (both I

and Q channels), and Pki=Es,ki/Ts. For all the transformations we consider, Pki=Ebk/Tb

when the input data waveform is binary with bit energy Ebk.

Table 2.1 [16] describes the relationships between various quantities of interest for

the two different transformations. The constant c in Table (2.1) is equal to 1 for BPSK

and 2 for QPSK.

30

Table 2.1: Relationships between energies, powers, symbol times in MT-DS-SS for equal data rates and equal bandwidth.

Transformation T Rate Rs Es,ki Ts Sk=PTk=Σ i(Es,ki/Ts)

Serial-parallel (S:P) conversion Rb/(cM) cEb cMTb Eb/Tb

Replication (Rep) Rb Eb/M Tb Eb/Tb

As seen from the table, the total power of user k’s waveform is Eb/Tb, the

conventional definition. We can also define a total symbol energy ∑ ==

M

i kissk EE1 , that

represents the energy of the total output of the transformation block during the symbol

interval Ts. The formula for PTk in the column heading of Table 2.1 simplifies

to skisTk TMEP ,= when the power is equal on each subcarrier.

2.2.2 Receiver Description

The conventional receiver diagram for user k’s ith subcarrier is shown in Figure 2.3.

The signal rk(t) is assumed to have already passed through an antenna and any RF

dividing networks, plus any wideband noise limiting filters, plus the low noise amplifier

(LNA). The received signal for user k is denoted as rk(t):

)()()()( tJtntvtr kk ++= (2.6)

where n(t) is AWGN with zero mean and J(t) is an interfering signal. Actual noise

variance will depend upon the receiver noise figure as well as front-end filtering; the two-

sided power spectral density of n(t) is N0/2 W/Hz.

31

The outputs of the carrier acquisition and tracking blocks for the ith subcarrier are

the two sinusoids: cos(ωcit-φki), and )sin( kici t φω −− , where the phases φki (for coherent

detection) are estimates of the received phases 2πfciτk. The pulse shape filters hTi(t)

remove the double, or high-frequency term resulting from this multiplication. The

despreading is accomplished by multiplication by )( kkiI tc τ− and )( kki

Q tc τ− . The

decision circuits next collect the symbol samples for making decisions on the subcarrier

symbols {Iki(m), Qki(m)}, which are translated to the data symbols as appropriate for the

chosen transformation T.

The notation {Ts} denotes the symbol duration. The integrator outputs are sampled at

integer multiples of the symbol time, i.e., at mTs, offset by the appropriate delay τk. The

decision statistic is denoted zki. This correlation is performed for all M subcarriers (hence

requiring M channels of the form of Figure 2.3.

32

Figure 2.3: Channel and Receiver (ith subcarrier)

×

cos(ωcit-φki) c(I)

k i(t-τk)

)( Ikiz

hTi(t)

-sin(ωcit -φki)

ith Carrier & Code Acq &

Track c(Q)ki(t-τk)

)(Qkiz

mTs+τk

hTi(t) × ×

× rk(t)

2Re{hk(τ,t)ejωct}

Jk(t) nk(t)

2Re{hκ(τ,t)ejωct}

2Re{h1(τ,t)ejωct}

Σ ∫ } { s T

∫ } { s T

33

2.3 Description of Impairments

The deliberate radiation, reradiation or reflection of electromagnetic energy, which

disrupts the ability of the receiver to decode the transmitted desired signal information, is

called jamming. The source of jamming could be a power generators, radar sets, high

power RF radio sets, or intentional enemy jammer. In spectral overlay schemes [17], the

jammer can be a narrowband (e.g., TDMA or FDMA) signal. The jamming to signal

power ratio (J/S) is the ratio, usually expressed in dB, of the power of a jamming signal

to that of a desired signal at a given point such as the antenna terminals of a receiver. In

this thesis we investigate performance with two types of jamming/ interference signals.

A single-tone jammer transmits an unmodulated carrier with power J somewhere in

the spread-spectrum signal band. The one-sided power spectrum of this jamming signal

is shown in Figure 2.4. Maximum jamming effect is achieved if the tone is placed on one

of the subcarrier center frequencies [18]. We can also introduce jamming with a multiple

(KJ) number of tones

The model for this jammer is

)](cos[2)(1

ttJtJ jj

K

jj

j

θω += ∑=

(2.7)

where in Wattspower jammer total=J , tonesofnumber =jK , tonein power thj jJ =

tone offrequency ,2 thjjj jff == πω and )2,0( πθ ∈j

34

Figure 2.4: PSD of MT-DS-SS with single and multiple tone jammers.

A partial band pulsed jammer transmits a signal over a band of frequencies for a

certain fraction of time and no signal the rest of the time. This is of interest for several

applications, two of which are in military anti-jam systems, and in the spectral overlay of

narrowband pulsed (e.g., TDMA cellular) signals over DS-SS transmissions [8].

The model for this jammer is given as

[ ]ttJttJJ

tJ JQJI ωωρ

sin)(cos)()( 0 −= (2.8)

... f Hz f 1 f 2 f M

J

...

J/M

f 1 f 2 fM

...

W/H

z W

/Hz

f Hz

35

where the average jammer power is J0 W, the duty cycle is ρ, and ωJ=2πfJ is the

jammer center frequency. Figure 2.5 shows the spectra of MT-DS-SS with a partial band

jammer of bandwidth BJ.

Figure 2.5: PSD of MT-DS-SS with partial band jammer.

In Chapter 3 we provide the results of the derivation for the receiver decision

statistics, and approximations to error probability. For the derivation, we assume that the

jammer decision statistic has a Gaussian probability density function and is independent

of AWGN channel. The simulation system model developed for the MT-DS-SS is

discussed in Chapter 4 and the results obtained from simulation are corroborated with the

analysis in Chapter 5.

f 1 f 2 f M ...

Jo/?

BJ

W/H

z

f Hz

36

Chapter 3

MT Performance Analysis

This chapter presents the derivation of the analytical results for performance in the

presence of the impairments discussed in Chapter 2. We first define the receiver

statistics, followed by the bit error rate (BER) approximations for MT-DS-SS in the

presence of AWGN, single tone jamming, multiple tone jamming and partial band

jamming.

3.1 Receiver Statistics and Definitions

For an easier understanding of the terms, we again present the diagram of the

transmission scheme. Referring to Figure 3.1 The final carrier waveform for user k is

denoted vk(t) [9]:

[ ]

])(2sin[)()()/(

])(2cos[)()()/(

)sin()()()cos()()(

)()(

)(,

1

)(,

1

)()(,

1

iicn

cTiQ

kikis

kis

iic

M

i ncTi

Ikiki

s

kis

M

iici

Qkikiici

Ikiki

s

kis

M

ikik

fftnTthncNnQT

E

fftnTthncNnIT

E

ttctQttctIT

E

tvtv

θπ

θπ

θωθω

++

−−

++

−=

+−+=

=

∑

∑ ∑

∑

∑

=

=

=

(3.1)

37

Figure 3.1: MT-DS-SS transmitter diagram for description of MT waveforms

where

=)(tdk user k’s binary data stream

=)(tIki users k’s “in-phase (I)” binary data stream on the ith subcarrier, i=1,2,3……M

)(tQki = users k’s “quadrature (Q)” binary data stream on the ith subcarrier, i=1,2,3……M

=)(tc Iki users k’s “I” spreading code waveform on the ith subcarrier

=)(tc Iki users k’s “Q” spreading code waveform on the ith subcarrier

=tciωcos ith cosine subcarrier

=tciωsin ith sine subcarrier

T ... dk(t)

c(I)k1(t)

Ik1(t)

Σ

vk(t)

Qk1(t)

c(Q)k1(t)

cos(ω c1t+θ1)

c(I)k2(t)

Ik2(t)

Qk2(t)

c(Q)k2(t)

c(I)kM(t)

IkM(t)

QkM(t)

c(Q)kM(t)

-sin(ω c1t+θ1)

cos(ω c2t+θ2)

-sin(ω c2t+θ2)

cos(ω cMt+θM)

-sin(ω cMt+θM)

× hT1(t)

×

×

× ×

×

× ×

×

× ×

×

hT1(t)

hT2(t)

hT2(t)

hTM(t)

hTM(t)

38

T=Transformation on the input data which can take the following two forms: Serial-to-

Parallel (S:P) and Replication (Rep) “splitting”.

We assume a single user AWGN channel with coherent detection for our study and

also we assume BPSK modulation. To represent equation 3.1 for MT-DS-CDMA with

BPSK modulation, we set

• Qki(m)=0, Es,ki= 2Eb,ki, and Ts=MTb for T=S:P

• Qki(m)=0, Es,ki= 2Eb,ki/M, Ts=Tb for T=replication

• Iki(m) ∈{ 1± }

• fi=i/Ts to yield spectrally overlapping subcarrier signals

The received signal for user k is denoted rk(t):

)()()()( tJtntvtr kk ++= (3.2)

where n(t) is additive white Gaussian noise and J(t) is the jammer or interferer, which can

be a single tone, multiple tones, or partial band interference.

At the receiver (Figure 2.3.), on the ith subcarrier, we demodulate by multiplying the

received signal r(t) by cki(t)g(ωit), and integrating over the symbol period (Ts), where g is

cosine for the I-channel and cki(t) is the kth user’s ith spreading code. We assume phase

coherence and perfect symbol timing [8].

The decision statistic for the ith subcarrier is given by

39

Signal. Desired SUI-IS Jammer AWGN

)(cos)(

)(cos)(cos)()(cos)(

)(cos)(

0

0 0 0 1

0

+++=

+

++=

=

∫

∫ ∫ ∫∑

∫

≠=

S

S S S

S

T

kiciki

kici

T T T M

ill

klkicikici

T

kiciki

dtttctv

dtttcvdtttctJdtttctn

dtttctrz

ω

ωωω

ω

(3.3)

Note: Throughout we assume long random spreading codes on each subcarrier, and

rectangular pulse shapes. Depending on the transformation, we may or may not make

individual bit decisions per subcarrier. Here we now consider the per-subcarrier decision

statistics and consequent error probabilities

We first analyze the AWGN term.

2

0

varianceandmean zero of AWGN

)(cos)(

i

T

kiciki

s

dtttctnnS

=

= ∫ ω (3.4)

[ ]

tionauocorrela noise white theis )(2

N since and

)()(coscos

0

0 0

2

xt

dtdxxctxc?t?n(t)n(x)E kikicici

T T

i

S s

−

= ∫ ∫

δ

σ (3.5)

402 s

iTN

=σ (3.6)

The desired signal term is

40

[ ] tdttctIT

E

dtttctvI

ci

T

kikis

ksi

T

kicikiki

s

S

ω

ω

2

0

2,

0

cos)()(

)(cos)(ˆ

∫

∫

=

=

(3.7)

Conditioned upon sending the 0th symbol Iki(0)=Iki, we obtain the signal mean value of

skibkiki TEII ,21ˆ = (3.8)

for BPSK with perfect carrier and code synchronization.

For AWGN, the error probability of the correlator output sequence is the “tail”

integral of the Gaussian pdf, i.e., a Q-function, with argument equal to the square root of

the following: the square of the mean value of the decision variable, divided by the

decision variable variance. We assume equiprobable data, and via symmetry conclude

that Pb is the same for either data bit (±1) sent, hence we condition on transmission

of 1=kiI , and obtain trivially from (3.8).

skibki TE ,21

1, =µ (3.9)

as the conditional decision variable mean, noting that the AWGN, IS-SUI, and jammer

terms are zero mean. If we use the square of (3.9), and divide by (3.6) we can write the

argument of the Q- function as NWS /

004121 2

/NE

TNTE

WS b

s

sbN == (3.10)

which is the well-known analytical result for coherent BPSK on the AWGN channel.

41

3.2 Tone Jamming Decision Statistics

We approximate the sinusoidal tone jammer’s decision statistics as Gaussian,

independent of the AWGN, which can be added to the variance of the AWGN in equation

3.10 [19]

The model for this jammer is

)](cos[2)(1

ttJtJ jj

K

jj

j

θω += ∑=

(3.11)

where in Wattspower jammer total=J , tonesofnumber =jK , tonein power thj jJ = ,

tone offrequency ,2 thjjj jff == πω and )2,0( πθ Uj ∈

The decision statistic kiχ for the jammer term on user k’s ith subcarrier is given by

∑ ∫∑

∑ ∫

∫

=

+−

=

=

−−=

+=

=

J c

c

J S

S

K

jj

Tm

mTjci

N

mki

j

K

j

T

kijjcij

T

kiciki

dtfftmcJ

dttcttJ

dtttctJ

1

)1(1

0

1 0

0

))(2cos()(2

)()cos(cos2

)(cos)(

θπ

θωω

ωχ

(3.12)

which is for rectangular chip pulses and with the dropping of sum (double-frequency)

term ( jci ff + ) as we have assumed fc>>Rc.

Solving the above integral, we get the decision statistic as

∑ ∑=

−

=

−+∆∆

∆=

JK

j

N

mj

JijckiJ

ijc

Jijccj

ki mfTmcfT

fTTJ

1

1

0

])12(cos[)()sin(

2θπ

ππ

χ (3.13)

42

where jciJ

ij fff −=∆

The variance of the jammer statistic is easily obtained for random codes as

var ])12([cos)(

)(sin

2)(

1

0

2

12

2

jJ

ij

N

mc

K

jJ

ijc

Jijcj

ki mfTfT

fTJJ

θππ

πχ −+∆

∆

∆= ∑∑

−

==

(3.14)

which can be upper bounded by

var ∑=

≤∆

∆≤

JK

j

cJ

ijc

Jijc

jki

NJTfT

fTJ

N

1

2

2

2

2)(

)(sin

2)(

π

πχ (3.15)

and when averaged over the jammer phase we have

var ∑=

≤∆

∆≤

JK

j

cJ

ijc

Jijc

jki

NJTfT

fTJ

N

1

2

2

2

4)(

)(sin

4)(

π

πχ (3.16)

The variance of the jammer decision statistic of equation 3.15 and 3.16 does not

depend on the transformation T, nor does it depend on the use of same and different

spreading codes on subcarriers.

3.3 Partial Band Jamming Statistics

The model for the narrowband, bandpass partial band jamming signal is [20]

[ ]ttJttJJ

tJ JQJI ωωρ

sin)(cos)()( 0 −= (3.17)

where the average jammer power is J0, the duty cycle is ρ, and ωJ=2πfJ is the jammer

center frequency. In this bandpass quadrature form, the quadrature components JI and JQ

are lowpass random processes.

43

For the interferer we investigate two cases:

(1) JI and JQ are Gaussian random processes, with rectangular power spectrum SJ(f) and

autocorrelation RJ(τ) as follows:

≤

=else ,0 ),2/(1

)( JJJ

B|f|BfS (3.18)

τπτπ

τJ

JJ B

BR

2)2sin(

)( = (3.19)

(2) JI and JQ are unfiltered random modulating (PSK or QAM) waves, with power

spectrum SJ(f) and autocorrelation RJ(τ) as follows:

2

2

)()(sin

)(J

JJJ fT

fTTfS

ππ

= (3.20)

≤−

=else 0,

| ,/||1)( JJ

JT|tT

Rτ

τ (3.21)

where the jammer main lobe bandwidth in (3.20) is equal to 2/TJ. The time TJ is the M-

ary interferer’s symbol duration. We also let BJ=1/TJ for case (2). We use (3.17) as the

input to the conventional receiver to obtain the jammer decision statistic on the I-channel,

denoted Iχ , assuming fJ=fc:

∫

∫

=

=

s

s

T

I

T

IciI

dttJJ

dttcttJ

0

0

0

)(21

)()cos()(

ρ

ωχ

(3.22)

where in the last line we have dropped the double frequency term. Assuming rectangular

chip pulses we obtain

44

∑ ∫−

=

+

=1

0

)1(0 )()(

21 N

m

Tm

mTIII

c

c

dttJmcJρ

χ .

3.23)

On the Q-channel an analogous expression is obtained:

∑ ∫−

=

+

=1

0

)1(0 )()(

21 N

m

Tm

mTQQQ

c

c

dttJmcJρ

χ . 3.24)

For random spreading codes and zero-mean jammer lowpass processes, the jammer

statistics are zero mean. The jammer decision variables on the I and Q channels are also

uncorrelated, by virtue of the assumption of independent jammer quadrature components.

(Even if we impose an arbitrary phase rotation of the jammer signal, which yields

correlation between the I and Q components, the final variance expression is the same.)

The variances of the jammer terms are then

[ ]

[ ]∫ ∫

∑∑+ +

×

−=

c

c

cTm

mT

Tn

nTXX

m nXX

dtdyyJtJE

nmRJ

E

)1( )1(

02

)()(

)(4ρ

χ

(3.25)

where X is either I or Q. In (3.25), the random code cross-correlation

mnXXXX ncmcEnmR δ==− ))()(()( , (3.26)

but (3.25) allows for more general cases of correlated codes.

We first address case (1), when the jammer lowpass processes are modeled as

Gaussian, with rectangular power spectral density and autocorrelation given by (3.18).

For this case, the integral in (3.24) becomes

45

dtdyytB

ytBc

c c

Tm

mT

Tn

nT J

J∫ ∫+ +

−−

=)1( )1(

)(2))(2sin(

ππ

XX?

(3.27)

which can be decomposed into an “inner” and “outer” pair of integrations. The inner

integral is given by

c

c

Tn

nTJ

J

ytBSiB

)1(

)](2[2

1+

−ππ

(3.28)

where the sine integral, denoted Si(x), is given by

∫=x

dtt

txSi

0

)sin()( (3.29)

Using (3.28) in (3.27), the final integration yields the following expression for the

jammer variance, after some algebra, and invoking the m=n condition of (3.26) for the

random codes:

( )( )

),(4

//sin2

4)(

1

20

2

2202

cJc

cJ

cJ

c

J

J

cc

RBfNTJ

RBRB

RB

SiBRNTJ

E

ρ

πππ

πρχ

=

−

=

(3.30)

The analysis in [21] arrived at this result assuming the interferer bandwidth BJ is less

than the bandwidth of a front-end ideal bandpass filter, an assumption we do not use.

Also, we need not invoke the assumption of negligible inter-chip interference due to this

bandpass filter since our receiver uses only the correlator.

46

For case (2), when the interferer lowpass process is a random PSK or QAM

modulating wave with autocorrelation given by (3.21), the integral in (3.25) is

[ ]dtdyBytc

c c

Tm

mT

Tn

nTJ∫ ∫

+ +

−−=)1( )1(

||1XX? (3.31)

which easily yields the following result for the interferer variance, when m=n,

),(4

31

4)(

2

20

202

cJc

c

Jc

RBfNTJ

RBNTJ

E

ρ

ρχ

=

−=

(3.32)

We note that (3.31) applies when |t-y|≤BJ, which translates to an interferer symbol rate

RsJ≤Rc (or equivalently, TJ≥Tc).

3.4 Bit Error Rate Expressions

We now derive the BER expressions for the systems of interest. To ensure that the

parameters are set correctly, we refer to Table 2.1, which lists the relationships between

various quantities for the two transformations of interest for BPSK modulation[22].

3.4.1 T=S:P, Single Tone Jammer

To estimate error probabilities in tone jamming, we use the approach that culminates

in (3.10) and combine this with the jammer variance and appropriate parameter settings.

We assume in the following that the jammer tone frequencies are such that the jammer

decision statistics are not zero; for example, if the frequency of a jammer tone is such

47

that ,...}2,1{ , ±±∈=∆ nnf Jij π , the sin(f)/f term in both (3.15) and (3.16) is zero.

Typically, we assume that the jammer tone is at the subcarrier center frequency, which is

a worst-case situation for the communicator 18]. In this case, our Q-function argument

will be

+=

+≥

2/42

)(/ 2

0041

,21

cMTSs

sbIkis

skibNi JTNTN

TEVarTN

TEWS

χ (3.33)

where this relation applies since we are upper bounding the jammer variance and hence

will be upper bounding the error probability. Applying the relation Ts=NMTSTc and also

dividing throughout by N0

+≥

0

0,

/21/2

/NJT

NEWS

c

kibNi (3.34)

We want this expression to be a function of the jammer to signal power ratio

J/S=JTb/Eb. The second term in the denominator of equation (3.34) can be written as

0

00000

)/(2

)/(2)/(2)/(2//22

NE

NMSJ

TE

NNTSJ

TE

NTSJ

SN

TSJSNSJT

NJT

b

MTS

b

b

MTS

s

b

bcccc

=

==== (3.35)

because Ts=NMTSTc and Ts=MTb from Table 2.1. Substituting equa tion (3.35) in the

denominator of equation (3.34) yields

)/)(/)(/(21/2

/0

0

MTSb

bNi NMNESJ

NEWS

+≥ (3.36)

Finally the upper bound on the error probability is

48

+≤

MTS

b

bbib

NNESJM

NEQNESJP

)/)(/(21

/2)/,/(

0

00, (3.37)

The aggregate error probability in the S:P case is the average over the M subcarriers, i.e.,

∑=

≤M

iibbb P

MNESJP

1,0

1)/,/( (3.38)

which for a single tone jammer (KJ=1) at the center frequency of one of the subcarriers,

is

)/,/()/,/( 0,0 NESJPNESJP bibbb ≤ (3.39)

since we can upper bound each subcarrier’s Pb by Pb,i.

3.4.2 T=S:P, Multiple Tone Jamming

In this case, we have M jamming tones, one at each sub-carrier center frequency.

For a fair comparison, each of the M jamming tones must have power J/M, which ensures

that the total jamming power is equal in both the single-tone and M-tone jamming cases.

Since the per-subcarrier jammer variance is still upper bounded by (3.16), the

derivation used for the single tone jammer applies exactly. Equation (3.38) can be used

to describe the performance on each sub-carrier, substituting J with J/M. Since the

equation for each sub-carrier is identical, the aggregate equation for the bound is the

same as the individual sub-carrier equation, i.e., (3.38).

3.4.3 T= Replication, Single Tone Jamming

49

For T= replication, bit decisions are not made on each sub-carrier independently;

rather, the correlator outputs from each sub-carrier are summed, and the final bit

decisions are made using this composite output. The decision statistic for a tone jammer

is given by equation (3.13) and the bound on the variance is given by equation (3.16),

when the decision variable at the receiver is averaged over all possible jammer phases.

The jammer decision statistic at the output of the summing device, after the

correlation is then given by

∑=

=M

iikk

1

χχ (3.40)

where kχ is users k’s composite jammer decision statistic. The variance of (3.40) is

= ∑

=

2

1

varM

iikk ?E)(? (3.41)

As this is a square of a sum term, the variance will have square and cross-product terms,

e.g., for M=3, the variance will be

)222()var( 32312123

22

21 kkkkkkkkkk E χχχχχχχχχχ +++++= (3.42)

The product terms cannot be ignored in the MT (replication) system because the spectral

overlap among subcarriers means that kiχ are correlated.

For finding the expectations of the product terms, we use equation (3.13), and for a

single tone jammer we obtain for user k

50

])12(cos[)])12(cos[)(

)sin()sin(

2)(

1

0

1

0j

Jpjc

N

m

N

nj

Jijcpnim

Jpjc

Jpjcc

Jijc

Jijcc

pi

nfTmfTccE

fT

fTT

fT

fTTJE

θπθπ

ππ

ππ

χχ

−+∆−+∆×

∆∆

∆∆

=

∑∑−

=

−

=

(3.43)

For random codes, only the m=n term expectation is nonzero, and if we upper bound the

trigonometric terms by one, we obtain

4)(

2c

piNJT

E ≤χχ (3.44)

When each expectation )( piE χχ is the same and each variance is the same, then (3.41) is

)(2

)1()()(

2)()var( 22

pipi EMM

MEEM

ME χχχχχχχ−

+=

+= (3.45)

If the cross terms and variance terms are identical, this can be simplified to

]2/)1()[()var( 2 += MME χχ (3.46)

Using the above (3.46) and (3.44) we can upper bound the variance of user k (for

jammer with random phase) as

8)1(

var2

ck

NJTMM)(?

+≤ (3.47)

We now calculate the Q- function argument for this system. As the energy per

subcarrier is Eb/M,

2)/(

21

)/(2

1

1

sbsb

sb

M

i

TMETMEM

TMES

==

= ∑=

(3.48)

51

and

8)1(

4

8)1(

4

20

2

1

0

cMTRs

cMTRM

i

sN

JTNMMTMN

JTNMM

TNW

++=

++

= ∑

= (3.49)

where NMTR is the processing gain of a sub-carrier in the MT system with replication

transformation, the noise variance on each sub-carrier is defined by equation (3.6), and

the jammer variance is upper bounded by equation (3.47).

Using substitutions similar to those in equation (3.35), and using Tc = Ts/N=Tb/N for

BPSK, we have the error probability upper bound as

MTR

b

b

cMTRs

sb

N

NNESJM

NEJTNMM

TMN

TMEWS

2)/)(/)(1(

1

)/(2

8)1(

41

21

/0

02

0

++

=+

+≥ (3.50)

++

≤

MTR

b

bsjMTRb

NNESJM

NEQP

2)/)(/)(1(

1

)/(2

0

0,, (3.51)

Similar to the S:P case, in the Rep case the upper bound in M-tone jamming is

identical to the upper bound with single tone jamming. We investigate the variances for

MT(Rep) in Appendix A.

3.4.4 Single Carrier Partial Band Jamming

52

As noted, modeling the jammer terms in (3.23) and (3.24) as Gaussian allows

expression of the error probabilities as Q- functions [23]. In case (1), the interferer terms

are Gaussian by assumption; in case (2), we invoke the Central Limit Theorem, whose

validity improves as processing gain N increases. The result is as follows:

+=

),()/)(/(

1

/2

00

0

cJb

bb

RBfN

NESJNE

QP

ρ

(3.52)

where f(BJ,Rc) is either f1 or f2, implicitly defined in (3.30) and (3.32), respectively. To

account for the pulsing of the interferer, (3.52) is multiplied by duty cycle ρ, and added to

(1- ρ)Pb(AWGN) for the final error probability expression, where Pb(AWGN) is the error

probability of the modulation in AWGN alone, and we employ the usual assumption that

if any given DS-SS symbol is jammed, it is jammed for the entire DS-SS symbol

duration. The expression derived in 3.52 is for a single carrier system only, for MT

systems, the derivation is left for future work.

3.5 Simulation and Analysis Comparison Equations

To fairly compare the different type of DS-SS we are studying, we must ensure

parameters are set properly, in both simulations and analysis. We present the expressions

for the processing gain of the three systems of interest MT (S:P), MT(Rep) and SC, for

equal bandwidth and equal data rates[24].

For the MT system we have

53

)1()1(

)1(

,,,

,,

−+=−+=

−+=

MTMTMTsMTsMTMTsMT

MTsMTMTcT

MNRRMRN

RMRB (3.53)

where BT is the bandwidth of the system, Rc,MT is the chip rate, RS,MT is the symbol rate,

MMT is the number of subcarriers ,and NMT is the processing gain of the MT system.

For the SC system

SCsSC

SCcT

RN

RB

,

,

=

= (3.54)

where NSC is the processing gain of the SC system.

Now, let us consider the S:P transformation, and for simplicity, binary modulation.

Then from Table 3.1 we have that Rs=Rb/M and Ts=MTb for all cases. We equate (3.53)-

(3.54) and use the relationship between Rs and Rb to obtain

bSCMTMTMTb

SCsSCMTMTMTs

RNMMNR

RNMNR

=−+

=−+

/)1(or

)1( ,, (3.55)

Dividing out Rb we obtain

SCMTMT

MT NMM

N=−+

11 (3.56)

Keeping in mind the definition of N as a ratio

SCcMTMTMTc

SCcMTMTMTcMTc

RNMR

RNMRR

,,

,,,

]/)1(1[or

/)1(

=−+

=−+ (3.57)

Using the above equations, we obtain the expression for the three systems, with equal

data rate and equal bandwidth as

SC:

54

NSC = Rc,SC /Rs = Rc,SC /Rb (3.58)

BT,SC = Rc,SC = NSC Rb (3.59)

MT, serial-parallel transformation:

NMTS = Rc,MTS /Rs = Rc,MTS /(Rb/M) (3.60)

BT,MTS = Rc,MTS + (M − 1)Rb/M (3.61)

= NMTS (Rb/M) + (M − 1)Rb/M (3.62)

= (Rb/M)(NMTS + M − 1) (3.63 )

MT, replication transformation:

NMTR = Rc,MTR /Rs = Rc,MTR /Rb (3.64)

BT,MTR = Rc,MTR + (M − 1)Rb (3.65)

= NMTR Rb + (M − 1)Rb (3.66)

= Rb(NMTR+ M − 1) (3.67)

Given a single-carrier system with processing gain NSC , we can compute the

processing gains of the three “equivalent” systems using equations (3.58)-(3.67):

NMTS = M NSC − M + 1 (3.68)

NMTR = NSC − M + 1 (3.69)

Or, in a single-line equation, we have

NSC = (NMTS + M – 1)/M = NMTR + M – 1 (3.70)

55

We will be presenting the comparison of these analytical results with those

obtained by computer simulation in Chapter 5, which will validate the correctness of the

expressions derived in this chapter. In the next chapter, we present the description of the

simulation model developed for this research.

56

Chapter 4

Simulation Description

A MT-DS-SS simulation model was developed for this research. This model was

developed primarily to study the MT-DS-SS and evaluate its performance in a jamming

environment. This simulation also serves as a platform for future study of system

characteristics, and as an aid in the design of practical applications.

One of the methods used for the performance evaluation of digital systems is

estimation of bit error probability. We employ the Monte Carlo method for this

estimation. The Monte Carlo method is a numerical method for statistical simulation

which utilizes sequences of random numbers to perform the simulation. The simulation

computes an estimate for the bit error probability on each of the individual subcarrier data

streams, and also an aggregate average bit error probability.

We now describe the simulation design and implementation, and begin by listing the

input parameters to the program developed for the simulation. These parameters are

selectable by the program user.

• N : Number of DS spreading code chips per subcarrier symbol (processing

gain)

• M : Number of subcarriers

• T : Transformation on the input data serial-to-parallel (S:P) or replication

(Rep)

• Type of spreading code (e.g., Walsh-Hadamard, random)

57

• Same or different codes on each subcarrier

• Short codes or Long codes on the subcarriers

• Jammer type: Single tone/Multiple tone/Partial band

• Jammer center frequency and bandwidth (for partial band jammer)

• Eb/N0 range

• J/S = “Jammer to Signal” power ratio = power of jamming signal relative to

the desired MT signal, in dB.

Figure 4.1 shows a schematic depiction of this simulation. For clarity, the above

listed input parameters are not shown in the figure. This description is for a single user;

Multiuser Interference (MUI) can easily be generated by duplicating the transmitter block

for as many users as desired.

4.1 Simulation Transmitter Description

The random binary data source is generated using the rand function of Matlab. This

function generates uniform random numbers between (0, 1) with equal probability, and as

we employ antipodal transmission, the numbers between the range (0, 0.5) are output as 1

and between (0.5, 1) as -1. The T block in the figure governs the transformation on the

input source data. This transformation is user selectable, and can be either serial-to-

parallel (different bits on each of the M subcarriers with energy equal to Eb) or replication

(same bits on each subcarrier with energy equal to Eb/M). The spreading code generator

is also a random binary generator similar to the source generator (for the case of random

spreading codes—our dominant model). It generates random number vectors of length N

58

for each source bit and depending on the input settings, the codes can be short (same

code on each bit of the subcarrier) or long (different code on each bit of the subcarrier).

(For long codes, code vector length is N*Nb, with Nb=#bits/subcarrier.)

Figure 4 1: Schematic depiction of MATLAB simulation functional operations for

MT(S:P)

(Note: For Rep case, threshold decisions in receiver are after T-1)

The sinusoidal generators generate sinusoidal vectors for up-converting each of the

M subcarriers. The frequency separation between the subcarriers in MT is Ts, therefore in

the simulation the sinusoidal vectors are separated by 1/N (because we have 1 sample per

chip and for spreading, each bit is oversampled N times).

Random Binary Data Source

Sinusoid Generators

d

Oversample by P

s

AWGN Generator

n +

Jamming Generator

J +

v

Receiver Spreading + Sinusoid Generators

c r s r

Spreading Code Generators ×

c

T

×

Sum over Subcarriers

r

Split into M Subcarriers

×

Accumulators

Threshold Decisions

T-1 Compare

d ^

Transmitter

Receiver

Channel

Pb Estimate

59

The sinusoidal signal multiplied by the spreading signal is called the composite

code signal. This composite code is multiplied by the source bits for each subcarrier and

all the M subcarrier samples are then summed for transmission.

4.2 Simulation Channel Description

The channel is assumed to be AWGN. For the generation of AWGN, the randn

function is used. The variance of AWGN is changed according to the desired Eb/N0

vector.

The single tone and multiple tone jammers are generated using a method similar to

that of the generation of the sinusoids for up-conversion. Amplitude scaling of these

signals is performed to maintain the desired J/S.

Two methods are employed for the simulation of the partial band jammer. For the

partial band jammer with rectangular spectrum, we use a low pass filter whose spectrum

is shown in Figure 4.2. The filter employed for the simulations is a 7th order elliptical

filter with 3dB of ripple in the passband, and 50dB of attenuation in the stopband. For

the M-ary jammer, a circular shifted vector of another random M-ary wave is generated

and its energy is scaled as per the J/S value. The impairments and the transmitted signal

are then added and this combined signal acts as the input to the receiver.

60

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 104

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Lowpass Filter Power Spectral Density, J/S=10dB, BJ=0.9Rc

Frequency

Mag

nitu

de d

B

Figure 4.2: Power density spectrum of partial band jammer with J/S=10dB, and

bandwidth = 0.9Rc

4.3 Simulation Receiver Description

We use a conventional receiver for the detection. This receiver consists of a

correlator, integrator and comparator. The combined signal which enters the receiver is

down-converted and despread by multiplication with the composite code signal by the

correlator. The signal is then integrated (accumulated) over the symbol time, and

threshold decisions are made on the bits in the S:P case; in the T=replication case, we

combine (sum) the outputs of the integrator and make a threshold decision on the

61

composite output of all subcarriers. The bit error calculations are performed by

comparing these received bit estimates with the transmitted bits. This process is different

for the two transformations. In the serial-to-parallel transformations, the bits of each

transmitted subcarrier are compared with the corresponding received subcarrier bit

decisions, and a bit error is recorded if the transmitted and received bits don’t match. The

aggregate bit error of the system is the average of these subcarrier bit errors. In the

replication transformation, the subcarrie r signals are added and then compared to the

transmitted signal bit stream for the bit error calculations.

The computer program for this simulation was written in MATLAB® and the program

files are attached in Appendix C for reference. In the next chapter we present the

simulation results.

62

Chapter 5

Simulation Results

In this chapter, we present the results of the computer simulations. The system

described in Chapter 2 is simulated using the simulation concept and models presented in

Chapter 4.

The simulation provides the results for performance of the MT-DS-SS in the

presence of single-tone, multiple-tone and partial band jamming. The results of MT-DS-

SS with serial-to-parallel and replication transformation on the input data are compared to

the traditional SC-DS-CDMA. The analytical expressions obtained for the bit error rate

(BER) in Chapter 3 are also corroborated with the simulation results

Certain assumptions are used throughout to simplify the simulations. These include

perfect time and frequency synchronization. Also we simulate a single user system. For

the fair comparison of the three system; MT-DS-SS Rep, MT-DS-SS S:P and SC-DS-SS,

we ensure that the parameters are set according to the equations derived in Chapter 4 for

equal bandwidth and equal data rate. Modulation for all results is BPSK.

5.1 Inter Subcarrier Same User Interference (IS-SUI)

In this section, we present the results for IS-SUI. Due to the large spectral overlap in

the MT-DS-SS, when different long codes are used for the spreading of the subcarriers,

63

there is interference among the subcarriers. Figure 5.1 shows the BER curves for

several values of processing gain (N) and number of subcarriers (M) for the S:P case.

0 1 2 3 4 5 6 7 8 9 10

10-4

10-3

10-2

10-1

T=S:P, M=3, Different Long Random Codes

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

N=10N=20N=50N=100N=500AWGN

Figure 5.1: Aggregate Pb vs. Eb/N0 for MT(S:P) with M=3, long different codes, showing the effect of IS-SUI for various values of N.

The only impairment in the channel for the IS-SUI simulation is AWGN. For same

spreading codes, the BER curve is same as BPSK in AWGN, i.e., the IS-SUI is zero, by

virtue of the orthogonality of the composite spreading codes (spreading code times the

subcarrier sinusoids). We note that the use of different spreading codes on the M

subcarriers, while attractive from a security perspective, does induce IS-SUI. This

64

interference is very prominent at lower values of N (4dB loss for N=20 and M=3 at Pb

of 0.01). For large values of N/M, this is mostly negligible at error probabilities of

interest. A value of N/M of approximately 30 ensures insignificant performance losses

for error probabilities down to ≅ 10-3. (An analytical computation of the effect of IS-SUI

appears in Appendix B.)

5.2 Tone Jamming

5.2.1 Single Tone Jamming

In Figure 5.2, we show BER curves for MT(S:P) for several values of N and M, with

two values of J/S. These curves are with the same long random spreading codes on each

subcarrier and for a single tone jammer at center frequency of subcarrier 2 (f2) for M=3

and at f5 for M=5 with random jammer phase. Phase averaging is obtained in the

simulation by the jammer taking a random phase value between 0-2p for each transmitted

symbol. The jammer is placed at the center of the spectrum to achieve maximum effect

[18]. (Note: In all the results presented in this chapter, the jammer is placed at the center

of the spectrum, unless otherwise mentioned.)

For constant data rate and bandwidth, in the MT(S:P) case we keep the ratio of N/M

constant. The equivalent SC results are also shown, and it can be seen that both the

system perform equally. It can also be seen that if N/M is kept constant, then the

performance of MT is not dependent on any one of these parameters (N or M).

65

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

Eb /N0 (dB)

Pro

babi

lity

of B

it E

rror

T=SP, Same long Codes, Single Tone Jamming, fJ=f

C

SC N=102 J/S=10dBM=3 N=304 J/S=10dBM=9 N=910 J/S=10dBM=3 N=304 J/S=15dBM=9 N=910 J/S=15dBAWGN

Figure 5.2: Aggregate Pb vs. Eb/N0 for MT(S:P) with single tone interference, several

values of the number of subcarriers M and per-subcarrier processing gain N, for two J/S ratios, using same long codes.

Figure 5.3 shows analogous results for the equivalent MT (Rep) systems. The

jammer in this case has a random phase also and we see that there is marginal difference

in the performance of MT(S:P) and MT(Rep). This is also predicted from the analysis,

when S:P and Rep are compared for equal data rates and bandwidth with a random phase

jammer.

66

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

T=Repl, Same Long Codes, Single Tone Jammer (random phase), fJ=fC

SC N=102 J/S=10dBM=3 N=100 J/S=10dBM=9 N=94 J/S=10dBM=3 N=100 J/S=15dBM=9 N=94 J/S=15dBAWGN

Figure 5.3: Aggregate Pb vs. Eb/N0 for MT(Rep) with single tone interference, several

values of the number of subcarriers M and per-subcarrier processing gain two J/S ratios, using same long codes. Jammer with random phase.

Figure 5.4 shows the large potential effect of jammer phase in the MT(Rep) system.

It can be seen that the jammer phase has negligible effect in the MT(S:P) case, while the

MT(Rep) could see a wide range of performance depending upon the phase realization at

the MT(Rep) receiver. The jammer is centered at 1.6 f1 where f1 is the center frequency

of subcarrier 1. In Appendix A, we provide further details on this topic.

67

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

Multitone, Same Long Codes, Single Tone Jammer, fJ=1.6f1

S:P,Worst,J/S=9dBRepl,Worst,J/S=9dBS:P,Phase=Best,J/S=9dBRepl,Best,J/S=13dBS:P,Worst,J/S=13dBRepl,Worst,J/S=13dBS:P,Best,J/S=13dBRepl,Best,J/S=13dBAWGN

Figure 5.4: Aggregate Pb vs. Eb/N0 for MT-DS-SS with single-tone interference, M=3,

NMTS=100, NMTR=32, for two J/S ratios, using long codes, showing MT(Rep) variation with jammer phase.

Figure 5.5 shows the BER variation of MT(S:P) with J/S for Eb/N0 value of 6 dB.

The SC curve is also plotted for comparison. It can be clearly seen that at all values of

J/S, if N/M ratio is kept constant then increasing the number of subcarriers M does not

degrade performance, as previously noted.

68

-3 -1 1 3 5 7 9 11 13 15 1710

-3

10-2

10-1

J/S (dB)

Pro

babi

lity

of B

it E

rror

T=S:P,Single Tone Eb/No=6dB, Different Long Codes, fJ=fC,

M=1 N=102M=3 N=304M=9 N=910

Figure 5.5: Aggregate Pb vs J/S for Eb/N0 of 6 dB. MT(S:P ) with different long codes

and single tone jammer

Figure 5.6 show the analogous results for MT (Rep) system, where it can be seen that

at all values of J/S, the system shows no improvement, and performance is same as SC,

when the jammer is not averaged over the phase.

69

-3 -1 1 3 5 7 9 11 13 15 1710

-3

10-2

10-1

J/S (dB)

Pro

babi

lity

of B

it E

rror

T=Repl,Multiple Tone Eb/No=6dB, Same Long Codes, fJ=fC,

M=1 N=102M=3 N=100M=9 N=94

Figure 5.6: Pb vs J/S for Eb/N0 of 6 dB. MT(Rep) with different long codes and single

tone jammer.

Figure 5.7 shows the comparison of analytical and simulation for both the systems

with a single tone jammer for two values of J/S. Very good agreement is obtained for the

MT(S:P) system while the bounds are looser for the MT(Rep) system, because of over

bounding of cross term expectations. We present more on this in Appendix A.

70

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

MT, Single Tone Jamming, NSC=102

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

S:P,Simulation 10dBS:P, Analytical 10dBREP,Simulation 10dBREP,Upper bound 10dBS:P,Simulation 15dBS:P, Analytical 15dBREP,Simulation 15dBREP,Upper bound 15dBAWGN

Figure 5.7: Aggregate Pb vs. Eb/N0 for MT-DS-SS with single-tone interference, M=3,

for two J/S ratios, using (same) long codes, showing agreement with analytical results. For S:P, M=3, NMTS=304; for Rep, M=3, NMTR=100.

5.2.2 Multiple Tone Jamming

Figure 5.8 shows MT(S:P) performance for several sets of N and M and two values

of J/S with M tone jamming. Each tone jammer is centered at the subcarrier center

frequency (fJ= fc ) and has a power of J/M. The SC curve is also plotted for comparison.

The results with M-tone jamming are similar to those obtained with single tone; the

performance of MT(S:P) system is similar to the SC system.

71

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

T=SP, Multiple Tone Jamming, fJ=fC, Different Long Codes

SC N=102 J/S=10dBM=3 N=304 J/S=10dBM=9 N=910 J/S=10dBM=3 N=304 J/S=15dBM=9 N=910 J/S=15dBAWGN

Figure 5.8: Aggregate Pb vs. Eb/N0 for MT(S:P) with M-tone jammer, for two J/S ratios, using different long codes. SC results for 3 tones.

Figure 5.9 shows the analogous results for MT(Rep) system. As seen in the single

tone jammer case, MT(Rep) with M tones also shows performance similar to its

equivalent S:P case when averaged over jammer phase.

72

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

Eb /N0 (dB)

Pro

babi

lity

of B

it E

rror

T=SP, Multiple Tone Jamming, fJ=fC, Different Long Codes

SC N=102 J/S=10dBM=3 N=100 J/S=10dBM=9 N=94 J/S=10dBM=3 N=100 J/S=15dBM=9 N=94 J/S=15dBAWGN

Figure 5.9: Aggregate Pb vs. Eb/N0 for MT(Rep) with M-tone jammer, for two J/S ratios, using different long codes. Jammer with random phase. SC results for 3 tones

Figure 5.10 and Figure 5.11 show the BER variation of MT(S:P) and MT(Rep) with

J/S for Eb/N0 value of 6 dB. The SC curve is also plotted for comparison. The

performance of S:P is better compared to its equivalent Rep case for a constant jammer

phase. Comparing Figure 5.10 and Figure 5.11 to their analogous single tone plots, it can

be observed that for all values of J/S, systems with single or M tones jammer show

similar performance

73

-3 -1 1 3 5 7 9 11 13 15 1710

-3

10-2

10-1

J/S (dB)

Pro

babi

lity

of B

it E

rror

T=S:P,Multiple Tone Eb/No=6dB, Same Long Codes, fJ=fC,

M=3 N=304M=9 N=910

Figure 5.10: Aggregate Pb vs J/S for Eb/N0 of 6 dB. MT(S:P) with same long codes and

M tone jammer. SC results for 3 tones

74

-6 -4 -2 0 2 4 6 8 10 12 1410

-3

10-2

10-1

J/S (dB)

Pro

babi

lity

of B

it E

rror

T=Repl,Multiple Tone Jammer Eb/No=6dB, Same Long Codes, fJ=fC,

M=1 N=102M=3 N=100M=9 N=94

Figure 5.11: Aggregate Pb vs J/S for Eb/N0 of 6 dB. MT(Rep) with same long codes and M tone jammer. SC results for 3 tones

As seen in Chapter 3, the analytical expression for jammer variance in single tone

and M tone interference is same. This is corroborated by Figure 5.12, which shows the

comparison of analytical with the simulation results for both the MT systems. The curves

are plotted for two values of J/S for both the systems, with equivalent N and M.

Excellent agreement is obtained, which corroborates simulation with analysis. The

MT(Rep) bounds are looser, and as noted earlier, explanation is provided in Appendix A.

75

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

Multiple Tone Jamming, fJ=f

C, N

SC=102

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

S:P,Simulation 10dBS:P, Analytical 10dBREP,Simulation 10dBREP,Upper bound 10dBS:P,Simulation 15dBS:P, Analytical 15dBREP,Simulation 15dBREP,Upper bound 15dBAWGN

Figure 5.12: Aggregate Pb vs. Eb/N0 for MT-DS-SS with M-tone jammer, M=3, for two

J/S ratios, using (same) long codes, showing agreement with analytical results. For S:P, M=3, NMTS=304; for Rep, M=3, NMTR=100.

5.3 Partial Band Jamming

5.3.1 Rectangular Spectrum Interferer

Figure 5.13 shows the BER curves for SC and MT (S:P) with a partial band

interferer, for two values of N/M and for a J/S of 10dB. The rectangular spectrum

interferer is centered at f1 (subcarrier 1’s center frequency) and we show the curves for

two interferer bandwidth values. It can be observed from the plot that, with a rectangular

76

spectrum interferer, SC and MT(S:P) perform the same. Also comparing this plot to

the analogous tone jamming plot, we note that the performance is better in the partial

band interferer than with tone interferer. As the bandwidth is reduced, (0.9 Rc to 0.1 Rc)

the performance degrades for all cases, thus it can be inferred that for the SC and MT

(S:P) system, performance is worst with a tone jammer at the center frequency.

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

T=S:P, Partial Band (Rectangular Spectrum) Interferer, J/S=10dB, Same Long Codes

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

SC N=102 BJ=R

c/10

M=3 N=304 BJ=Rc/10M=9 N=910 B

J=R

c/10

SC N=102 BJ=0.9R

cM=3 N=304 B

J=0.9R

cM=9 N=910 BJ=0.9Rc

AWGN

Figure 5.13: Pb vs. Eb/N0 for MT (S:P) and SC in the presence of a rectangular-spectrum interferer, for J/S=10dB, and two values of BJ and N/M. Same long codes on the subcarriers.

Figure 5.14 shows the analogous performance for the MT(Rep) system, and

observations similar to those in the MT(S:P) case can be made.

77

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1T=Repl, Partial Band (Rectangular Spectrum) Interferer, J/S=10dB, Same Long Codes

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

SC N=102 BJ=Rc/10 M=3 N=100 BJ=Rc/10M=9 N=94 BJ=Rc/10SC N=102 BJ=0.9RcM=3 N=100 BJ=0.9RcM=9 N=94 BJ=0.9Rc

AWGN

Figure 5.14: Pb vs. Eb/N0 for MT(Rep) and SC in the presence of a rectangular-spectrum interferer, for J/S=10dB, and two values of BJ and N/M. Same long codes on the subcarriers.

Figure 5.15 and Figure 5.16 show the variation of BER with the partial band

interferer’s bandwidth for both MT(S:P) and MT(Rep) systems respectively. The curves

are for a J/S=10dB, two values of Eb/N0 and for various values of N/M. SC curves are

also plotted for comparison. As previously noted, the performance of all the three

systems is worst for a very narrow band partial jammer and it improves as the bandwidth

of the jammer is increased. The variation of BER as a function of BJ is more prominent

at higher values of Eb/N0.

78

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910

-4

10-3

10-2

BJ as a fraction of RC

Pro

babi

lity

of B

it E

rror

T=S:P, Partial Band (Rectangular Spectrum) Inteferer, J/S=10dB, Same Long Codes,

SC N=102 Eb/No=6dBM=3 N=304 Eb/No=6dBM=9 N=910 Eb/No=6dB

SC N=102 Eb/No=10dBM=3 N=304 Eb/No=10dBM=9 N=910 Eb/No=10dB

Figure 5.15: Pb vs BJ for MT(S:P) in the presence of a rectangular-spectrum interferer,

for J/S=10dB, and two values of Eb/N0 and N/M, showing variation of BER with BJ Same long codes on the subcarriers.

79

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910

-4

10-3

10-2

BJ as a fraction of RC

Pro

babi

lity

of B

it E

rror

T=Repl, Partial Band (Rectangular Spectrum) Inteferer, J/S=10dB, Same Long Codes,

SC N=102 Eb/No=6dBM=3 N=304 Eb/No=6dBM=9 N=910 Eb/No=6dBSC N=102 Eb/No=10dBM=3 N=304 Eb/No=10dB

M=9 N=910 Eb/No=10dB

Figure 5.16: Pb vs BJ for MT(Rep) in the presence of a rectangular-spectrum interferer,

for J/S=10dB, and two values of Eb/N0 and N/M, showing variation of BER with BJ. Same long codes on the subcarriers.

5.3.2 M-ary (“Sinc-Squared” Spectrum) Interferer

Figure 5.17 shows the BER curves for SC and MT(S:P) with a partial band

interferer, for two values of N/M and for a J/S of 10dB. The M-ary interferer is centered

at fc (subcarrier center frequency), and we show the curves for two bandwidth values.

The jammer phase for each transmitted symbol is different and the curves shown are

averaged over phase. Comparing these to the rectangular spectrum interferer, it can be

noted that MT(S:P) performs worse with an M-ary interferer than with a rectangular-

80

spectrum interferer. This performance difference is more prominent for higher values

of BJ. Also the performance of SC and MT(S:P) is same, as also previously noted with

rectangular spectrum interferer. The performance is better than the MT(S:P) system with

tone jamming. Also as the bandwidth of the interferer is increased the performance

improves.

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

T=S:P, Partial Band (sin(f)/f)2 Spectrum Inteferer, J/S=10dB, Same Long Codes

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

SC N=102 BJ=Rc/10 M=3 N=304 BJ=Rc/10M=9 N=910 BJ=Rc/10SC N=102 BJ=0.9RcM=3 N=304 BJ=0.9RcM=9 N=910 BJ=0.9Rc

AWGN

Figure 5.17: Pb vs. Eb/N0 for MT(S:P) and SC in the presence of a M-ary (“sinc squared” spectrum) interferer, for J/S=10dB, and two values of BJ and N/M. Same long codes on the subcarriers.

81

Figure 5.18 shows the analogous MT(Rep) system BER curves. The performance

of MT(S:P) and MT(Rep) is similar, for equal data rates and bandwidths, and all the

observations noted for the MT(S:P) system apply to MT(Rep) system as well.

0 1 2 3 4 5 6 7 8 9 1010

-3

10-2

10-1

T=Repl, Partial Band (sin(f)/f)2 Inteferer, J/S=10dB, Same Long Codes

Eb/N0 (dB)

Pro

babi

lity

of B

it E

rror

SC N=102 BJ=Rc/10 M=3 N=100 BJ=Rc/10M=9 N=94 BJ=Rc/10SC N=102 BJ=0.9RcM=3 N=100 BJ=0.9RcM=9 N=94 BJ=0.9Rc

AWGN

Figure 5.18: Pb vs. Eb/N0 for MT(Rep) and SC in the presence of a M-ary (“sinc squared” spectrum) interferer, for J/S=10dB, and two values of BJ and N/M. Same long codes on the subcarriers.

Figure 5.19 shows the variation of Pb with the jammer center frequency (fJ) for

MT(S:P) with M=3 and N=304 . The M-ary jammer has J/S=10dB and a constant

bandwidth of Rc/10. We plot the curves for two values of Eb/N0. It can be observed from

the plots, that the performance is worst when the jammer is centered on a subcarrier

82

frequency (fc). The SC system is also plotted for comparison, and a similar

observation can be made: Performance is worst when the jammer is placed at the center

frequency. Similar results are obtained for MT (Rep) system (Not included here).

0.5 1 1.5 2 2.5 310

-3

10-2

10-1

fJ Jammer Center Frequency

Pro

babi

lity

of B

it E

rror

T=S:P, Partial Band (sin(f)/f)2 Spectrum Inteferer, J/S=10dB, Same Long Codes

SC N=102 Eb /No=6dBM=3 N=304 Eb /No=6dBSC N=102 Eb /No=10dB

M=3 N=304 Eb /No=10dB

Figure 5.19: Pb vs fJ for MT(S:P) and SC in the presence of a M-ary (“sinc squared”

spectrum) interferer, for J/S=10dB, BJ=Rc/10 and two values of Eb/N0 and N/M, showing variation of BER with fJ. Same long codes on the subcarriers.

In this Chapter, we presented the simulation results and corroborated those with the

analytical expressions obtained in Chapter 3. Agreement between simulation and

analysis is obtained for the systems with all the types of narrowband interference. In the

83

next chapter, we summarize the thesis, and present the conclusions and suggestions for

future work.

84

Chapter 6

Conclusions and Future Work

In this chapter, we summarize the research performed for this thesis. Topics for

future areas of research are also suggested.

6.1 Summary of Research

In this research, we have compared two MT-DS-SS systems with the conventional

SC DS-SS system in the presence of narrowband and partial band interference. This fills

an important literature gap, as this has been never done before.

We developed a common framework and notation for the study of the three systems

MT, MC and SC (MC was not presented here, [8] ). A simulation tool was developed in

Matlab, which calculates the BER for user defined parameters. We looked at two

transformations on the input data, serial-to-parallel conversion and replication. For the

narrowband interference, we investigated both single tone and multiple tone interference.

For partial band interference, interference with rectangular spectra and interference with

sinc spectra were studied.

For the analysis, we developed expressions for the decision statistics in the presence

of the above mentioned interferences. Modeling the jamming statistics as Gaussian

enabled development of closed form analytical, error probability expressions.

85

Excellent agreement between the simulation results and the analytical expressions

were found in the MT(S:P), the simulation performed better in the MT(Rep) system than

the analysis because of overbounding. For the partial band interference, simulation and

analysis were only corroborated in the SC case with sinc spectrum partial band interferer.

The main findings of our work are:

• In the presence of multiple tone jamming, SC and MT-DS-SS system perform

equally.

• The MT Replication case is very sensitive to the jammer phase.

• IS-SUI is negligible for N/M greater than 30, for most BERs of interest.

6.2 Suggestions for Future Work

The computer simulation developed for this work provides a flexible vehicle for

future extensions of related work. Tightening the bounds for the MT (Rep) analytical

expressions and corroboration of these new bounds with simulation results is one of the

suggestions for future work. Deriving analytical expressions for MT system performance

in partial band jamming is another. Implementing different combining techniques for the

MT(Rep) system is another area in both analysis and simulations, which can be of

interest, e.g., maximal ratio combining.

In our work we have assumed perfect time and frequency synchronization between

the transmitter and the receiver. One related work looks at frequency offset or phase

noise [25] but simulation and analysis of the MT system in asynchronous environments

86

will be a very insightful piece of work. Increasing the number of users, and comparing

capacity and performance of an MT system with SC system in a flat fading non

dispersive channel is also suggested.

87

References

[1] B. Sklar, Digital Communications Fundamentals and Applications, 2nd edition,

Prentice-Hall, Upper Saddle River, New Jersey, 1995.

[2] E.A Lee, David G. Messerschmitt, Digital Communications, 2nd edition Kluwer

Academic Publishers, 1993.

[3] V. Deepak, D. W. Matolak, “MT-DS-SS Simulation Report” Version 2,

September 2001.

[4] V. K. Garg, IS-95 CDMA and cdma 2000: Cellular/PCS Systems

Implementation, 1st edition, Prentice-Hall, Upper Saddle River, New, 1999.

[5] Simon, Hinedi and Lindsey, Digital Communication Techniques, Signal Design

and Detection, Prentice Hall, 1995.

[6] S. Hara, R. Prasad, “Overview of Multicarrier CDMA,” IEEE Communications

Magazine, vol. 35, no. 12, pp. 126-133, December 1997.

[7] L. Vandendorpe, “Multitone Spread Spectrum Multiple Access Communications

System in a Multipath Rician Fading Channel,” IEEE Trans. Vehicular Tech.,

vol. 44, no. 2, pp. 327-337, May 1995.

[8] D. W. Matolak, V. Deepak, F. Alder, “Performance of Multitone & Multicarrier

DS-SS in the Presence of Narrowband Interference,” to appear in Proceedings of

12th MPRG/Virginia Tech Symposium, Wireless Personal Communications June

5-7, 2002

[9] D. W. Matolak, J. C. Dill, “CDMA Waveform, Multiuser Detection,

Compatibility, and Network Strategy Evaluation for TTNT,” Ohio University

Quarterly Technical Report TTNT#1-31.08.2001, DARPA contract F33615-01-

C-1834, 31 August 2001.

[10] S. Kondo, L. B. Milstein, “Performance of Multicarrier DS CDMA Systems,”

IEEE Trans. Comm., vol. 44, no. 2, pp. 238-246, February 1996.

88

[11] K-C. Chen, S-T. Wu, “A Programmable Architecture for OFDM-CDMA,”

IEEE Communications Magazine, vol. 37, no. 11, pp. 76-82, November 1999.

[12] S. Hara, R. Prasad, “Overview of Multicarrier CDMA,” IEEE Communications

Magazine, vol. 35, no. 12, pp. 126-133, December 1997.

[13] E. A. Sourour, M. Nakagawa, “Performance of Orthogonal Multicarrier CDMA

in a Multipath Fading Channel,” IEEE Trans. Comm., vol. 44, no. 3, pp. 356-367,

March 1996.

[14] S. M. Elnoubi, A. El-Beheiry, “Effect of Overlapping Between Successive

Carriers of Multicarrier CDMA on the Performance in a Multipath Fading

Channel,” IEEE Trans. Comm., vol. 49, no. 5, pp. 769-773, May 2001.

[15] S. Verdu, Multiuser Detection. New York: Cambridge Univ. Press, 1998.

[16] D. W. Matolak, J. C. Dill, “CDMA Waveform, Multiuser Detection,

Compatibility, and Network Strategy Evaluation for TTNT,” Ohio University

Quarterly Technical Report TTNT#1-30.11.2001, DARPA contract F33615-01-

C-1834, 31 November 2001.

[17] L. B. Milstein, D. L. Schilling, R. L. Pickholtz, M. Kullback, E. G. Kanterakis,

D. S. Fishman, W. H. Biederman, and D. C. Salerno, “On the Feasibility of a

CDMA Overlay for Personal Communications Networks,” IEEE Journ. Select.

Areas in Comm., vol. 10, pp. 655-668, May 1992.

[18] R. L. Peterson, R. E. Ziemer, D. E. Borth, Introduction to Spread Spectrum

Communications, Prentice-Hall, Upper Saddle River, New Jersey, 1995.

[19] M. K. Simon, J. K. Omura, R. A. Scholtz, B. K. Levitt, The Spread Spectrum

Communications Handbook, revised edition, McGraw-Hill, 1994.

[20] D. W. Matolak, V. Deepak, F. A. Alder, “Performance of Multitone and

Multicarrier Direct Sequence Spread Spectrum in the Presence of Partial-Band

Pulse Jamming/Interference,” to appear in Proceedings of IEEE Vehicular

89

Technology Conference (VTC Fall 2002), Vancouver, Canada, 24-29

September 2002.

[21] R. A. Iltis, L. B. Milstein, “Performance Analysis of Narrow-Band Interference

Rejection Techniques in DS Spread-Spectrum Systems,” IEEE Trans. Comm.,

vol. COM-32, no. 11, pp. 1169-1177, November 1984.

[22] D. W. Matolak, V. Deepak, F. A. Alder, “Comparing MT DS-SS Simulations

and Analysis”, Version 4, April 2002.

[23] David W. Matolak, Frank A. Alder, Virat Deepak, “New Results on the

Performance of Direct Sequence Spread Spectrum in the Presence of Partial-

Band Pulse Jamming/Interference,” to appear in Proceedings of Wireless 2002

Conference, Calgary, Alberta, Canada, 8-10 July 2002.

[24] D. W. Matolak, F. A. Alder “Comparing MT, MC, and SC DS-SS System

Parameters for Equal Bandwidth and Equal Data Rate” Version 2, January 2002.

[25] D. W. Matolak, V. Deepak, F. Alder, “Performance of Multitone & Multicarrier

DS-SS in the Presence of Imperfect Phase Synchronization,” submitted to

MILCOM 2002, Anaheim, CA, 7-10 October 2002.

90

Appendices

Appendix A: MT Replication Variances

Referring to Chapter 3, the decision statistic for the MT(Rep) system is given by

equation (3.40) and the variance of the decision statistic is given by equation (3.41). As

the variance has square terms and cross product terms (which cannot be ignored, due to

spectral overlap), equation (3.43) gives the expectation of these cross-product terms

which we have upper bounded by equation (3.44).

For deriving the upper bound BER expression given by equation (3.51), we have

assumed that all the variances and cross term are identical and are given by equation

(3.44). This leads to over bounding of the expression, and thus we see that the analytical

curves and the simulated curves in Figures 5.8 (for single tone jammer) and 5.13 (M-tone

jammer) do not show good agreement--the analytical bounds are loose.

We investigated this further, and note that if the jammer is placed not on a subcarrier

center frequency but is placed between two subcarriers, e.g., 1.6f1 where f1 is the center

frequency of subcarrier 1, then the cross product terms between subcarrier 1 and

subcarrier 2 are large compared to the other cross product terms. Depending upon the

phase realization at the MT(Rep) receiver, the sum of these variances will vary and the

effect of this can be seen in Figure 5.5. Figure A.1 shows a plot of expectation vs.

jammer phase for NMTR=100, M=3 and with a single tone jammer placed at 1.6f1 for J/S

of 10dB. The cross-product term between subcarrier 1and subcarrier 2 (Svar(1,2)) is

91

shown by the solid line with “x”, between subcarrier 2 and subcarrier 3 (Svar(2,3)) by

“o” and between subcarrier 1 and subcarrier 3 (Svar(1,3))“+”. It can be observed that

there is a large difference in the magnitude of these terms, and the total jammer statistic

variance is maximum at phases of approximately 2.9 and 6, and minimum at phases of

1.3 and 4.4 radians. The square terms represented by the dashed lines have the same

magnitude.

202.404

93.541

Svar 1 2, 100, 1.6, qv,( )

Svar 1 3, 100, 1.6, qv,( )

Svar 2 3, 100, 1.6, qv,( )

Sv 1 100, 1.6, qv,( )

Sv 2 100, 1.6, qv,( )

Sv 3 100, 1.6, qv,( )

VarX 100 1.6, qv,( )

6.20

1.3 2.9

qv

0 1 2 3 4 5 6 7100

50

0

50

100

150

200

250Phase vs Magnitude for MT(Rep)

Phase

Mag

nitu

de

Figure A.1: Expectations vs. jammer phase for MT(Rep) with M=3 NMTR=100 with

single tone jammer with J/S=10dB, and jammer fJ=1.6f1 showing the magnitude difference for different cross-product terms.

Figure A.2 shows a plot of cross-term expectation vs. phase for different values of fJ,

for MT(Rep) system with the same parameters as those of Figure A.1. It can be observed

92

that the magnitude of the cross variances is maximum when the jammer is between two

subcarriers.

93.409

93.549

Sc 1 2, 100, qv, 1.6,( )

Sc 2 3, 100, qv, 2,( )

Sc 2 3, 100, qv, 2.2,( )

6.20 qv

0 1 2 3 4 5 6 7100

50

0

50

100Phase Vs Cross-Variance

Phase

Cro

ss-V

aria

nce

Figure A.2: Cross term expectations vs. jammer phase for MT(Rep) with M=3

NMTR=100 with single tone jammer with J/S=10dB, and fJ=1.6f1, 2f1 and 2.2f1 showing the variation in the cross-term amplitude with the jammer center frequency fJ.

Thus we conclude that for achieving tighter analytical bounds, we must come up

with expressions, which are not based on the assumption that all the cross-product

variances are identical. To get a closed form expression which takes into account these

variances individually is a challenge and is included in suggestions for future work.

Although loose, the closed form analytical expression given in equation (3.51) is a valid

upper bound for the MT(Repl) system.

93

Appendix B: DS-SS Cross-Correlations

In this section we present the analytical expressions for the cross-correlations, when

different long codes are used on the subcarriers. This correlation between the subcarriers

causes the IS-SUI.

Cross correlation between user k’s ith and jth subcarrier is given by

dttStST

T

jiij ∫=0

)()(1

ρ (B1)

where Si(t)= )2cos()( tftc cii π which is the composite code for user’s k’s ith subcarrier

Sj(t)= )2cos()( tftc cjj π which is the composite code for user’s k’s j th subcarrier.

We assume rectangular chip pulse shape, so that we can subdivide the integral as

∫∑

∫∑+−

=

+−

=

∆=

=

Tcm

mTij

N

mji

s

Tcm

mTcjci

N

mji

sij

c

c

dttfmcmcT

dttftfmcmcT

)1(1

0

)1(1

0

)2cos()()(21

)2cos()2cos()()(1

π

ππρ

(B2)

where cjciij fff −=∆ , and we have dropped the sum term fci + fcj. Completing the

integration we obtain

∆∆+∆

= ∑−

= )()sin()]12)(cos[(

)()(21 1

0 ij

cijcijN

mji

sij f

TfmTfmcmc

T πππ

ρ (B3)

which can be represented as

94

)]12)(cos[()()()(

)sin(21 1

0

+∆∆∆

= ∑−

=

mTfmcmcf

TfT cij

N

mji

ij

cij

sij π

ππ

ρ (B4)

If same codes are used then ci(m)cj(m)=1 and as dtmttnT

)2cos()2cos(0∫ =0 for

nm ≠ ,and m=0,1…N, and n=0,1…N, the cross correlations are zero. (Also in (B4), the

summation over the cosine is zero when ci(m)=cj(m).)

In the case of different codes on the subcarriers, we are interested in the variance of

the cross correlations. This is

∑−

=

+∆∆

∆=

1

0

22

2

2 )]12)([(cos)(

)(sin

41

)var(N

mcij

ij

cij

sij mTf

f

Tf

Tπ

π

πρ , (B5)

and using NT

T

s

c 1= and

sij T

jif

−=∆ , equation (B5) can be written as

∑

∑

−

=

−

=

+

−

−

−

=

+

−

−

−

=

1

0

22

2

2

1

0

22

2

2

2

)12()(

cos

)(sin

41

)12()(

cos

)(sin

4)var(

N

m

N

m sc

sc

sc

s

cij

mN

ji

NjiN

ji

N

mT

jiT

TjiT

Tji

T

TT

π

π

π

π

π

πρ

(B6)

For large values of N, equation (B6) is given by

Nij 81

)var( =ρ (B7)

95

Thus, as seen in Figure (5.1), for large values of N, the performance of MT, with

different codes and same codes on subcarriers is equivalent.

96

Appendix C: Matlab Programs

%============================================================= % Program for computing BER for various values of N and M. clear all; PMAT=[100,200,500,0;3,3,3,3;0,0,0,0;3e6,3e6,3e6,3e6]; number=3; for pp=1:number pp P=PMAT(1,pp); % Set spreading length M= PMAT(2,pp); % Set # sub carriers jsc=PMAT(3,pp); % SC frequency on which jammer is centered N =PMAT(4,pp); Jdb =10 Sp=1; SPber=SPSJ(P,M,jsc,Jdb,N,Sp); Repber=RESJ(P,M,jsc,Jdb,N,Sp); end Eb_No=[0:1:10]; EBp=0:0.1:max(Eb_No); EBpn=10.^(EBp/10); % Numeric value of Eb/N0 BER=0.5*erfc(sqrt(2*EBpn)/sqrt(2)); figure(2) semilogy(Eb_No,data1,'bo-',Eb_No,data2,'r^-',Eb_No,data3,'g*-',EBp,BER,'k-') axis([0 max(Eb_No) 1E-3 1]) grid; xlabel('E_b/N_0 (dB)'); ylabel('Probability of Bit Error') %============================================================= %=========================================================== % Function SPSJ calculates BER for MT(S:P) with a single tone jammer % This function is the FOR loop version % 4 April 2002 10:58 pm, Virat Deepak, Athens. % the Inputs to the function are %-P=Processing gain of S:P system %-M=Number of subcarriers %-jsc=Tone Jammers center frequency

97

% -Jdb= Desired J/S ratio %-N=number of bits to run; % Sp=samplesper chip function ber=SPSJ(P,M,jsc,Jdb,N,Sp) chip_samp=Sp; Eb_No=[0:1:10]; % Set range of Eb/Nots LE=length(Eb_No); % length of the Eb_No vector N =N-rem(N,M); % Total # bits; E_desired =chip_samp*P; %serial to parallel case desired energy E_total =M*E_desired; J_over_S_dB =Jdb; J_1 = 10^(J_over_S_dB/10); J = E_total*(10^(J_over_S_dB/10)); % Binary antipodal random generator for the input data bits d0=rand(1,N); % Generate random vector of uniform (0,1) variates of length N d0(find(d0>=0.5))=1; d0(find(d0<0.5))=-1; % Conversion to antipodal signal elements Ch_bits=N/M; % Total number of data bits used per subcarrier Chip= Ch_bits*P; % Total number of chips per subcarrier= N/M*P d=reshape(d0,M,Ch_bits); % Data bits serial to parallel converted os_chips = P*(chip_samp); berob=zeros([M length(Eb_No)]); for vv=1:Ch_bits %Same code on each SC cran=rand(1,P); % Generation of length-M*P uniform (0,1) random vector cran(find(cran>=0.5))=1; cran(find(cran<0.5))=-1; % Conversion to antipodal signal elements cranovsamp=ones(1,chip_samp)'*cran; % Conversion to antipodal signal elements same =reshape(cranovsamp,1,chip_samp*P); for ch=1:M ch; dosp=ones(os_chips,1)*d(ch,vv); % Oversample data bit vector by P for each subchannel dosp1=reshape(dosp,1,os_chips); % Reshape input bits and make them equal to total # chips codemod(ch,:)=same; sinmod(ch,:)=sqrt(2)*cos(2*pi*(ch/os_chips)*(0:1:(os_chips-1))); % Generate sinusoid vector for modulation E = sum(sinmod(ch,:).^2); %energy of the signal sinmod(ch,:) = sqrt(E_desired/E)*sinmod(ch,:); % scale so E=E_desired

98

txc(ch,:)=dosp1.*codemod(ch,:).*sinmod(ch,:); end Jv= sqrt(2*J)*cos(2*pi*(jsc/(os_chips))*(0:1:(os_chips-1))); Ejv=sum(Jv.^2); Jv= sqrt(J/Ejv)*Jv; Ejva=sum(Jv.^2); tx=(sum(txc,1))+Jv; Ltx=length(tx); % Generation of AWGN chip_Eb_No= Eb_No-10*log10(chip_samp*P); No= 1./(10.^(chip_Eb_No/10)); % Noise density for different values of Eb_No for i=1:LE noise=sqrt(No(i)*.5)*randn(1,Ltx); rx=tx+noise; % Generate received vector as noise + signal for ch= 1:M despread=rx.*codemod(ch,:).*sinmod(ch,:) ; % Multiplication of received bits by reshaped spreading code dreshape=reshape(despread,os_chips,1); % Reshape received bits for detection dintegrating(ch,:)=sum(dreshape,1); % Integration (accumlation) over symbol detection=dintegrating(ch,:); detection(find(detection>=0))=1; % Make binary decision on received bits detection(detection<0)=-1; out(ch,:)=detection ; count=0; test=d(ch,vv)+out(ch,:); if test==0 count=count+1; end ber_channel(ch,i)=count; end end berob=berob+ber_channel; ber_channel=berob./Ch_bits ; % Calculation of BER per subchannel if rem(vv,1000)==0

99

clc; vv end end ber=sum(ber_channel,1)/M; %============================================================= %============================================================= % Function RESJ calcultaes the BER for MT(Rep) with single tone Jamming %MT Replication transformation, Single tone Jamming, with phase=0, %the chips are oversampled by2 % This function is the FOR loop version, therefore takes lot of time % 4 April 2002 12:00 pm, Virat Deepak, Athens. % the Inputs to the function are %-P=Processing gain of Repl system %-M=Number of subcarriers %-jsc=Tone Jammers center frequency % -Jdb= Desired J/S ratio %-N=number of bits to run; % Sp=samplesper chip function berobsys=RESJ(P,M,jsc,Jdb,N,Sp) chip_samp=Sp; Eb_No=[0:1:10]; % Set range of Eb/Nots LE=length(Eb_No); % length of the Eb_No vector N =N-rem(N,M); E_desired =chip_samp*P/M; %serial to parallel case desired energy E_total =M*E_desired; J_over_S_dB =Jdb ; J_1 = 10^(J_over_S_dB/10); J = E_total*(10^(J_over_S_dB/10)); % Binary antipodal random generator for the input data bits d0=rand(1,N); % Generate random vector of uniform (0,1) variates of length N d0(find(d0>=0.5))=1; d0(find(d0<0.5))=-1; % Conversion to antipodal signal elements Ch_bits=N; % Total number of data bits used per subcarrier=N=Ch_bits Chip=Ch_bits*P; % Total number of chips per subcarrier=N*P d=repmat(d0,M,1);; % Data bits for replication ( bits repeated for replication ) berob=zeros([M length(Eb_No)]); berobsys=zeros([1 length(Eb_No)]); os_chips = P*(chip_samp); %SAME Long CODE ON EACH BIT

100

for vv=1:Ch_bits cran=rand(1,P); % Generation of length-M*P uniform (0,1) random vector cran(find(cran>=0.5))=1; cran(find(cran<0.5))=-1; % Conversion to antipodal signal elements cranovsamp=ones(1,chip_samp)'*cran; % Conversion to antipodal signal elements same =reshape(cranovsamp,1,chip_samp*P); for ch=1:M; % Beginning of transmit vector generation loop- dosp=ones(os_chips,1)*d(ch,vv); % Oversample data bit vector by P for each subchannel dosp1=reshape(dosp,1,os_chips); codemod(ch,:)=same; sinmod(ch,:)=sqrt(2)*cos(2*pi*(ch/os_chips)*(0:1:(os_chips-1))); % Sinusoid V E = sum(sinmod(ch,:).^2); %energy of the signal sinmod(ch,:) = sqrt(E_desired/E)*sinmod(ch,:); % scale so E=E_desired txc(ch,:)=dosp1.*codemod(ch,:).*sinmod(ch,:); end % END of transmit vector generation loop Jv= sqrt(2*J)*cos(2*pi*(jsc/(os_chips))*(0:1:(os_chips-1))+(2*pi*rand(1))); Ejv=sum(Jv.^2); Jv= sqrt(J/Ejv)*Jv; Ejva=sum(Jv.^2); tx=(sum(txc,1))+Jv; Ltx=length(tx); % Generation of AWGN chip_Eb_No= Eb_No-10*log10(P*chip_samp); No= 1./(10.^(chip_Eb_No/10)); % Noise density for different values of Eb_No for i=1:LE i; noise=sqrt(No(i)*.5)*randn(1,Ltx); rx=tx+noise; % Generate received vector as noise + signal for ch= 1:M despread=rx.*codemod(ch,:).*sinmod(ch,:) ; % Multiplication of received bits by reshaped spreading code dreshape=reshape(despread,os_chips,1); % Reshape received despread vector for detection dintegrating(ch,:)=sum(dreshape,1); % Accumulate decision statistic for each bit, per subcarrier

101

detection=dintegrating(ch,:); % Assign subcarrier(ch) correlator outputs to variable detection end clear count clear test % Integration (accumulation) over symbol detection=sum(dintegrating); detection(find(detection>=0))=1; % Make binary decision on received bits detection(detection<0)=-1; out2=detection; count=0; test=d(1,vv)+out2; if test==0 count=count+1; end ber_system(1,i)=count; % out2 is hard decision on combined subcarrier correlator outputs end berobsys=berobsys+ber_system; if rem(vv,200)==0 clc; vv end end berobsys=berobsys./Ch_bits; %============================================================= %============================================================= % Function MTJAM generates M-tone jamming signal. % Virat Deepak % 20 April 2002 Athens %Input parameters %-os_chips= samples*P %-M= Number of tone Jammer %-J=Absolute value of the Jamer (Not J/S dB) function JVM=MTJAM(os_chips,M,J); Jm=J/M; for jj=1:M Jv(jj,:)= sqrt(2*J)*cos(2*pi*(jj/(os_chips))*(0:1:(os_chips-1))+(2*pi*rand(1)));; Ejv(jj)=sum(Jv(jj,:).^2);

102

Jv(jj,:)= sqrt(Jm/Ejv(jj))*Jv(jj,:); Ejva(jj)=sum(Jv(jj,:).^2); end JVM=sum(Jv,1); Ejvm=sum(JVM.^2); %============================================================= %============================================================= % Function LPPBjammer generates Partial band jamming signal % -N=Number of samples %-B=Desired Bandwidth of the jammer %-J=Jammer power function out = LPPBjammer(N,B,D,J) rJ=16; fover=2; % Oversampling rate for plotting with random binary wave [bb,aa]=ellip(7,0.5,60,B); % Generate Elliptical bandpass filter coefficients g=randn(1,N+3000); % Generate Gaussian source vector, zero padded for filter transient x1=filter(bb,aa,g); % Filter WG process clear g; x1=x1/sqrt(mean(x1.*x1))*sqrt(J); % Normalize filtered GN process to have variance=J x=x1(3001:N+3000); % Take DFT of filtered Gaussian vector sw=B01(ceil(N/rJ),D); % Generate binary 0/1 switching process with duty cycle D sw=overN(sw,rJ); % Oversample binary switching process sw=sw(1:N); % Select N samples of random switching wave for pulse waveform jam=x.*sw; % Create the pulsed jammer signal out = jam; %============================================================= %============================================================= %V= The input vector %S= The shift amount(shifts right) % Produces right shifted vector % 28 April 2002

103

% Virat Deepak %Circular shift function [shifted] = circshift(V,S) len = length(V); for i=1:S for k=len:-1:1 V(k+1)=V(k); end V(1)=V(len+1); end shifted = V(1:len); %============================================================= % 27 April 2002 function out = RBWjammer(N, samp, BW, shift) % Generates a random binary waveform to be used as a jammer. % N = number of symbols (chips) in the jammer waveform % samp = number of samples per symbol in the jammer waveform % BW = jammer bandwidth expressed as a fraction of chip rate of signal % shift = samples to circular shift the result RBWsamp = samp*(1/BW); % period of RBW jammer is 1/BW times period of signal RBWsym = N*(samp/RBWsamp); % number of total symbols in jammer waveform x = 2*round(rand(1,RBWsym)) - 1; % random binary vector with elements 1 or -1 x_oversamp = x'*ones(1,RBWsamp); % oversample the vector clear x; % delete the original x vector x = reshape(x_oversamp',1,RBWsym*RBWsamp); % reshape the oversampled result to a row vector out = circshift(x,shift); % return the result %=============================================================