SPECTRALLY EFFICIENT ANTI-JAMMING SYSTEM DESIGN IN WIRELESS
NETWORKS
by
Lei Zhang

A DISSERTATION
Submitted
to Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Electrical Engineering
2011

ABSTRACT
SPECTRALLY EFFICIENT ANTI-JAMMING SYSTEM DESIGN IN WIRELESS
NETWORKS
by
Lei Zhang
In wireless networks, one of the most commonly used techniques for limiting the effectiveness of an opponent’s communication is referred to as jamming, in which the legitimate
user’s signal is deliberately interfered by the adversary. Along with the wide spread of various wireless devices, especially with the advent of user reconfigurable intelligent devices,
jamming attack is no longer limited to military applications, but has become an urgent and
serious threat to civilian communications as well. Motivated by this observation, in this
dissertation, we consider hostile jamming modeling, classification, and spectrally efficient
anti-jamming system design and analysis.
First, we investigate the cognitive jamming modeling and classification in wireless communications. Instead of using existing jamming models that assume the jamming remains
invariant during the signal transmission period, we focus on time-varying jamming and its
classification based on time-frequency analysis and the relative correlation between the signal
and the jamming interference: (i) We introduce the general concepts of time-varying jamming coherence time and time-frequency jamming coherence bandwidth, and propose a new
jamming classification scheme based on these parameters; (ii) We introduce the concept of
disguised jamming, where the jamming is highly correlated with the signal, and has a power
level close or equal to the signal power; (iii) Based on time-frequency analysis and approximation theory, we propose algorithms to estimate the time-varying coherence time and the
time-frequency coherence bandwidth for both stationary and locally stationary jamming.
Next, we break new ground on anti-jamming system design in wireless networks based
on message-driven frequency hopping (MDFH). MDFH is a highly efficient spread spectrum

technique that is particularly robust under strong jamming. However, it experiences considerable performance losses under disguised jamming. To overcome this drawback, we propose
an anti-jamming MDFH (AJ-MDFH) system. The main idea is to transmit a secure ID
sequence along with the information stream. The ID sequence is generated through a cryptographic algorithm using the shared secret between the transmitter and receiver, it is then
exploited by the receiver for signal extraction. It is shown that AJ-MDFH can effectively reduce the performance degradation caused by disguised jamming while remains robust under
strong jamming. We investigate ID constellation design and its impact on the performance
of AJ-MDFH under both noise jamming and disguised jamming. In addition, we extend
AJ-MDFH to a multi-carrier scheme, which can increase the system efficiency and jamming
resistance significantly through jamming randomization and frequency diversity, and can
readily be used as a collision-free multiple access system. Our analysis indicates that: while
AJ-MDFH has strong anti-jamming property, its spectral efficiency is very close to that of
MDFH, which is several times higher than that of the conventional FH.
Finally, we analyze the capacity of MDFH and AJ-MDFH under disguised jamming using
the arbitrarily varying channel (AVC) model. We show that under the worst case disguised
jamming, as long as the secure ID sequence is unavailable to jammer (which is ensured by
AES), the AVC corresponding to AJ-MDFH is nonsymmetrizable. This implies that the
deterministic code capacity of AJ-MDFH with respect to the average probability of error
is positive. On the other hand, due to lack of shared randomness, the AVC corresponding
to MDFH is symmetric, resulting in zero deterministic code capacity. We further calculate
the capacity of AJ-MDFH and show that it converges as the ID constellation size goes to
infinity, which echoes with convergence result for the probability of error of AJ-MDFH. We
also extend the capacity analysis to multiuser AJ-MDFH system (MC-AJ-MDFH) and show
that it outperforms the multiple access scheme for conventional FH (FHMA).
Future research will be conducted on adaptive transceiver design under cognitive jamming
scenario.

Dedicated to my family and friends

iv

ACKNOWLEDGMENTS

I would like to take this opportunity to express my deep appreciation to my advisor, Dr.
Tongtong Li, for her constant support, guidance and encouragement throughout my Ph.D.
years. She makes great effort to help me through many difficulties in academic research and
personal development and growth. She herself sets a great example on these areas for me.
I want to thank Dr. Jonathan Hall from Department of Mathematics, Dr. Subir Biswas
and Dr. Selin Aviyente from Department of Electrical and Computer Engineering for serving
on my committee. I am deeply indebted to them for their kind support, either in the
classroom or in all thoughtful correspondences. I would also like to thank Dr. Jian Ren
from Department of Electrical and Computer Engineering, who introduces me to the area of
network security and provides many insights on the security related issues in my research.
I am deeply indebted to my labmates including Dr. Huahui Wang, Dr. Qi Ling, Dr.
Weiguo Liang, Dr. Leonard Lightfoot, Ms. Abdelhakim Mai and Mr. Xiaochen Tang, for
their valuable discussions on the research issues, as well as their helpful advices on the daily
life on and off the campus. I am also grateful to all my friends who have made my life
at Michigan State University an enjoyable experience. I would like to extend my heartfelt
thanks to Yun Li, Di Tang, Leron Lightfoot, Wenbo Qiao, Jian Li, Jiankun Liu, Mingwu
Gao, Guanqun Zhang, Ming Gu, Wei Wang, Ya Mo and Dave Conger for being my great
friends and for all the fun we have together.
I would like to thank my parents for their unyielding love and continuous support through
all the ups and downs, without which nothing could ever be possible. A special thanks goes
to my roommate, Mr. Hao Wen, for providing valuable suggestions and encouragement
during last two years of my Ph.D. program.

v

TABLE OF CONTENTS

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

ix

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

x

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Jamming Interference in Wireless Communications . . . . . . . . . . . . . .
1.1.0.1 System Inherent Jamming . . . . . . . . . . . . . . . . . . .
1.1.0.2 Hostile Jamming . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Problem Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Limitations of Existing Models . . . . . . . . . . . . . . . . . . . . .
1.2.2 Limitations of Existing Jamming Resistant Systems . . . . . . . . . .
1.2.2.1 Existing Spread Spectrum Systems . . . . . . . . . . . . . .
1.2.2.2 Limitations of Existing Spread Spectrum Systems . . . . . .
1.3 Proposed Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Cognitive Jamming Modeling and Classification . . . . . . . . . . . .
1.3.2 Anti-jamming System Design Based on Message-Driven Frequency
Hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Capacity Analysis of MDFH Based Systems under Disguised Jamming
1.4 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 2 COGNITIVE JAMMING MODELING AND CLASSIFICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Cognitive Jamming Modeling and Classification . . . . . . . . . . . . . . . . 17
2.2.1 Jamming Modeling using Time-Varying Power Spectral Density . . . 17
2.2.2 Jamming Classification Based on Time-frequency Analysis . . . . . . 18
2.2.3 Strong Jamming and Disguised Jamming . . . . . . . . . . . . . . . . 20
2.3 Estimation of Time-Varying Jamming Coherence Time and Time-Frequency
Jamming Coherence Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.1 Stationary Jamming . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3.2 Locally Stationary Jamming . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2.2 Best Basis Search and Spectrum Estimation . . . . . . . . . 24
2.3.3 Binary Tree Based Basis Search Algorithm . . . . . . . . . . . . . . . 27
2.3.3.1 Dictionary Construction . . . . . . . . . . . . . . . . . . . . 27
2.3.3.2 Best Basis Search Using Dynamic Programming . . . . . . . 29
2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
CHAPTER 3 ANTI-JAMMING MESSAGE-DRIVEN FREQUENCY HOPPING SYSTEM DESIGN . . . . . . . . . . . . . . . . . . . . . 36
vi

3.1
3.2

3.3

3.4

3.5

3.6
3.7
3.8
3.9

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Brief Review of Message-Driven Frequency Hopping . . . . .
3.2.1 System Description . . . . . . . . . . . . . . . . . . . . .
3.2.2 Performance of MDFH under Hostile Jamming . . . . . .
Anti-jamming MDFH (AJ-MDFH) System . . . . . . . . . . . .
3.3.1 Transmitter Design . . . . . . . . . . . . . . . . . . . . .
3.3.2 Receiver Design . . . . . . . . . . . . . . . . . . . . . . .
3.3.2.1 Demodulation . . . . . . . . . . . . . . . . . . .
3.3.2.2 Signal Detection and Extraction . . . . . . . .
ID Constellation Design and its Impact on System Performance
3.4.1 Design Criterion and Jamming Classification . . . . . . .
3.4.2 Constellation Design under Noise Jamming . . . . . . . .
3.4.3 Constellation Design under ID Jamming . . . . . . . . .
Multi-carrier AJ-MDFH . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Secure Group Generation . . . . . . . . . . . . . . . . . .
3.5.2 Multi-Carrier AJ-MDFH without Diversity . . . . . . . .
3.5.3 Multi-carrier AJ-MDFH with Diversity . . . . . . . . . .
Spectral Efficiency Analysis . . . . . . . . . . . . . . . . . . . .
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proofs of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . .
3.9.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . .
3.9.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . .

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CHAPTER 4 CAPACITY ANALYSIS OF MDFH BASED SYSTEMS
UNDER DISGUISED JAMMING . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 MDFH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 AJ-MDFH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Capacity of MDFH under Disguised Jamming . . . . . . . . . . . . . . . . .
4.4 Capacity of AJ-MDFH under Disguised Jamming . . . . . . . . . . . . . . .
4.4.1 AVC Symmetricity Analysis . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Capacity Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Capacity of Multiuser AJ-MDFH under Disguised Jamming . . . . . . . . .
4.6 Capacity of MDFH under Noise Jamming . . . . . . . . . . . . . . . . . . .
4.6.1 Capacity Derivation for Carrier Information Transmission Channel . .
4.6.2 Capacity Derivation for Ordinary Information Transmission Channel
4.7 Capacity of AJ-MDFH under Noise Jamming . . . . . . . . . . . . . . . . .
4.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Proofs of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10.1 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10.2 Calculation of the Probability Matrix W1 . . . . . . . . . . . . . . .

vii

36
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111

CHAPTER 5 CONCLUSIONS AND FUTURE WORKS . . . . . . . . . . 115
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.2.0.1 Disguised Jamming Analysis under Different Wireless Systems118
5.2.0.2 Adaptive Transceiver Design under Cognitive Jamming . . . 118
APPENDIX A LIST OF ABBREVIATIONS AND ACRONYMS . . . . . 121
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

viii

LIST OF TABLES

2.1

The best basis search algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.1

The bit rate and spectral efficiency comparison in the single-user case . . . . . .

59

ix

LIST OF FIGURES

2.1

An example of window functions. . . . . . . . . . . . . . . . . . . . . . . . . . .

23

2.2

An example of full admissible binary tree with depth ND = 2 and corresponding
window functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

2.3

Example 1: The true and estimated jamming PSD SJ (f ). . . . . . . . . . . . .

31

2.4

Example 2: The true jamming autocorrelation function RJ (t, τ ). For interpretation of the references to color in this and all other figures, the reader is
referred to the electronic version of this dissertation. . . . . . . . . . . . . . . .

32

2.5

Example 2: The estimated RJ (t, τ ) using time-frequency analysis. . . . . . . . .

33

2.6

Example 2: The magnitude of the true time-varying jamming PSD |SJ (t, f )|.

.

34

2.7

Example 2: The magnitude of the estimated |SJ (t, f )| using time-frequency
analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.1

The nth block of the information. . . . . . . . . . . . . . . . . . . . . . . . . . .

39

3.2

Transmitter and receiver structure of MDFH, here ABS means taking the absolute value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Performance comparison under single band jamming, Eb /N0 = 10dB, Nc =
64, Nh = 3. MDFH uses QPSK modulation and conventional FH uses 4-FSK
modulation. In this case, the spectral efficiency of MDFH is roughly 3.3 times
that of conventional FH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3.4

AJ-MDFH transmitter structure. . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.5

AJ-MDFH receiver structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.6

Transmitter and receiver structure of MC-AJ-MDFH. . . . . . . . . . . . . . . .

54

3.7

Example of the Secure Permutation Algorithm for Nc = 8 channels and Ng = 2
subcarriers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.8

Performance of MC-AJ-MDFH and E-MDFH in multiuser case. . . . . . . . . .

60

3.9

Performance of FHMA in multiuser case. . . . . . . . . . . . . . . . . . . . . . .

61

3.10 Example 1: The performance of AJ-MDFH with different constellation size,
under single-band ID jamming. . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

3.3

x

3.11 Example 2: Performance comparison under single band jamming. . . . . . . . .

63

3.12 Example 3: Performance comparison under 2-band disguised jamming. . . . . .

64

4.1

MDFH transmitter structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.2

Transmitter and receiver structure of AJ-MDFH. . . . . . . . . . . . . . . . . .

76

4.3

AJ-MDFH capacity with different PSK constellation size, under the worst case
single band disguised jamming (ID jamming). Nc = 64. . . . . . . . . . . . . . . 103

4.4

Capacity of MC-AJ-MDFH and FHMA under the worst case single band disguised jamming. Nc = 64, SN R = 10dB. Here, per channel use means the
total bandwidth of all used channels over one hopping period. . . . . . . . . . . 104

4.5

Capacity of AJ-MDFH and MDFH under the worst case single band noise
jamming. For MDFH, upper bounds for capacity of the ordinary information
transmission channel as well as for the overall channel capacity are provided.
The number of hops per symbol period is Nh = 3. . . . . . . . . . . . . . . . . . 106

xi

Chapter 1
INTRODUCTION

In the past few decades, wireless communication has fundamentally changed people’s way of
lives. Accelerated by the breakthroughs in wireless technologies such as OFDM and MIMO,
wireless communication nowadays is able to provide high speed connections for broadband
multimedia applications including Video on Demand (VoD) and videoconferencing [1, 2, 3].
Comparing with its wireline counterpart, wireless communication does not only provide
comparable data rate but also possesses other attractive features such as ubiquitous network
coverage and mobility support. Due to these notable features, wireless communication sees
an explosive growth in the recent years [4].
As more and more critical information is transmitted wirelessly for applications such
as e-commerce and e-banking services, the security issues in wireless communication pose
serious challenges for the development of next generation wireless networks, and wireless
communication security becomes an active research field [5, 6, 7]. Although most of the
higher layer security threats against wireline network can also be applied to wireless network, the latter suffers from additional vulnerabilities. Due to lack of the protective physical
boundary, wireless communication systems are susceptible to PHY layer vulnerabilities such
as unauthorized detection, interception and jamming interference [8, 9]. Jamming interference can introduce several forms of distortion to the transmitted signal and disrupt the legal
user’s communication. Therefore, it becomes a major challenge for secure and reliable PHY
layer design in wireless networks. Motivated by this challenge, in this dissertation, we mainly
focus on the jamming issue in the wireless networks.

1

1.1

Jamming Interference in Wireless Communications

Depending on the sources of the interference, the jamming can be classified into two categories with distinctive characteristics: when the interference is introduced by the sources
within the wireless system, it is called system inherent jamming; when the interference is
introduced by a malicious jammer, it is called hostile jamming.

1.1.0.1

System Inherent Jamming

In the single user environment, the wireless system includes the transmitter, the receiver
and the wireless channel as the transmission medium. To maximize the usage of precious
spectrum resource, wireless systems usually employ multiple access techniques to support
multiple users transmitting simultaneously within the same frequency range. In the multiuser environment, these active transmitting users are also part of the wireless system. Both
wireless channels and multiuser environment impose fundamental limitations on the performance of wireless systems [10, 11, 12]. (i) Due to the physical factors such as multipath
propagation, wireless channel introduces inter-symbol interference (ISI) where the delayed
replicas of previous symbols interfere with the current transmitted symbol. (ii) When orthogonality among different users are broken either by wireless channel or by transmitter
and/or receiver, the multiuser interference (MUI) will be introduced by the signals from
other active transmitting users.
Since the sources of system inherent jamming are located within the system, their characteristics can be either known to or controlled by the system. Therefore, the system inherent
jamming can be effectively mitigated by system design techniques and system management
protocols.
• The wireless channel can be well characterized by some physical parameters such as
coherence time and coherence bandwidth, which can be obtained from appropriate
channel models [13, 14]. With the knowledge of these channel characteristics, ISI can

2

be suppressed using appropriate design techniques. For example, in frequency selective
fading channels, the ISI of PAM signals can be suppressed by adding a channel equalizer
at the receiver [15, 16, 17]; for OFDM signals, ISI can be eliminated by inserting guard
intervals between adjacent symbols [18, 19].
• Since signal characteristics of the users can be controlled by the system, MUI can
be canceled by employing system management protocols. For example, in OFDMA
systems, the MUI can be eliminated by scheduling the users to transmit on nonoverlapping orthogonal subcarriers [20]; in multiuser MIMO systems, MUI can be
mitigated by precoding techniques such that the signal transmits only towards the
direction of the intended user without causing interference to users at other directions
[21, 22].
Many of these techniques and protocols have been incorporated to the system design for
3G/4G wireless networks [23, 24].

1.1.0.2

Hostile Jamming

The malicious jammer with appropriate transmitter can intentionally interrupt the legitimate user’s communication by saturating its receiver with noise or false information through
deliberate radiation of radio signals [25, 26]. The jammer differs from active users in two
ways: (i) The jamming can vary arbitrarily and is unpredictable. Its characteristics are difficult to obtain and are often not available to the system. (ii) The malicious jammer does not
obey the system management protocols, thus the conventional MUI mitigation techniques
no longer work under hostile jamming.
Due to these harmful characteristics, hostile jamming becomes an effective way to carry
out denial-of-service (DoS) attack and is often used in military fields. However, with the
advent of reconfigurable cognitive radios widely available, hostile jamming attack is much

3

easier to be launched and has become an urgent and serious threat to civilian communications
as well.

1.2

Problem Descriptions

In this dissertation, we focus on hostile jamming modeling, classification and spectrally
efficient anti-jamming system design and analysis.

1.2.1

Limitations of Existing Models

Traditionally, hostile jamming has been characterized in either frequency domain or time
domain:
• Tone-jamming [27], where the jamming power is concentrated around carrier frequencies.
• Band-jamming [28, 29, 30, 31, 32], in which the jamming signal is modeled as a zeromean wide sense stationary Gaussian random process. In general, band-jamming is
further classified into full-band [28, 29], partial-band [30, 31, 32]. The power of fullband jamming is uniformly distributed over the bandwidth of interest with PSD NJ .
Partial-band jamming is characterized by the additive Gaussian noise interference with
N
PSD J over a fraction ρ of the total bandwidth and negligible interference over the
ρ
remaining fraction (1 − ρ) of the band.
• Partial-time jamming [33, 34, 35], where the jamming occurs at certain time periods
during the signal transmission. It is basically a two-state Markov process. When the
jammer is in state 0, it is off; when it is in state 1, the jammer emits the interfering
signal. State 1 occurs with probability of ρ, along with the variance of jamming signals
NJ
. (1 − ρ) is the probability of occurrence of state 0, for which the variance of
ρ
jamming signals is 0.

4

In reality, a smart jammer can be used for more effective jamming attacks. The smart
jammer is often equipped with receiver that captures the transmitted signal of the legitimate
user. With sufficient intelligence, the smart jammer can determine the transmission scheme
used by the legitimate user and adjust the jamming strategy accordingly to maximize the
adverse effect. The jamming generated by the smart jammer is called cognitive jamming,
also known as adaptive jamming or time-varying jamming. Apparently, cognitive jamming
cannot be accurately characterized by existing jamming models, and cognitive jamming
modeling and classification technique needs to be developed.

1.2.2

Limitations of Existing Jamming Resistant Systems

In general, wireless communication systems do not possess jamming resistant features except
the spread spectrum systems. Spread spectrum techniques have been considered for secure
communication since mid 1950 [36, 28, 29, 25, 37]. In this section, we provide a brief overview
of existing spread spectrum systems and their characteristics [28, 25, 38].
By definition, spread spectrum systems occupy a larger bandwidth than the minimum
necessary to transmit the information. In other words, the bandwidth of spread spectrum
signal W is much greater than the bit rate of information messages R, and the expansion
W
is usually referred to as the processing gain. The spreading is accomplished
factor Gp =
R
using a code which is independent of the data. The receiver is synchronized with the transmitter and uses the identical code for despreading and subsequent data recovery. To prevent
interception or jamming by malicious user, the code should be shared only between the corresponding transmitter and receiver of legitimate user and kept secret from other undesirable
users.

1.2.2.1

Existing Spread Spectrum Systems

Two techniques are often employed for spread spectrum systems: Direct-Sequence Spread
Spectrum (DSSS) and Frequency Hopping Spread Spectrum (FHSS). The DSSS systems have
5

been extensively studied in the literature [39, 40, 41, 42, 43] and successfully incorporated
into the 3G wireless communication standards. On the other hand, FHSS systems have
been widely adopted in military communication systems. Other spread spectrum systems
based on the hybrids of these two spreading techniques have also been proposed, but their
performances do not significantly differ from those of these two basic ones.
Direct Sequence Spread Spectrum System Direct sequence spread spectrum is a
technology that is more suitable for integration with bandwidth-efficient linear modulation
such as QAM or PSK. In DSSS, the key operation of spectrum spreading is achieved by
a PN sequence, also known as PN code ore PN chip. The PN sequence is usually binary,
consisting of 0’s and 1’s, which are polar signaling of +1 and −1. To minimize interference
and to facilitate chip synchronization, the PN sequence has certain desirable autocorrelation
and cross-correlation properties. By spreading the overall signal energy over a broader
bandwidth, DSSS systems provide nice security features against potential hostile jamming
and interception:
• It is difficult for the unauthorized receiver to recover the information signal without
the knowledge of PN code used at the transmitter.
• By applying sufficient processing gain, the power spectrum of the DSSS signal distributes over a broad bandwidth, resulting in a low flat power spectral density which
has the similar characteristic as that of noise. It is then difficult for malicious user to
separate spread spectrum signal from background noise. In this way, the spread spectrum signal can “hide” within the noise floor and prevent itself from being detected by
malicious user.
• DSSS is especially robust under narrowband jamming by reducing the jamming power
through the despreading process. Since the narrowband jamming interfere with only
a small portion of signal energy and does not block out the entire signal spectrum,
narrowband jamming is not effective against DSSS signals.
6

Frequency Hopping Spread Spectrum System In Frequency Hopping Spread Spectrum (FHSS) system, each user can vary its carrier frequency according to the predetermined,
pseudorandom pattern, thereby effectively occupies a broader spectrum. By using PN sequence to control the frequency synthesizer, the transmitted signal hops to a new carrier
frequency at the beginning of each hopping period. At the receiver, the received signal is
shifted back to original signal band by using the identical PN sequence. Based on the duration of the hopping period, FHSS systems can be further divided into two categories: fast
hopping (FFH) scheme and slow hopping (SFH) scheme. In an FFH system, the carrier hops
several times during one symbol period, while in an SFH system, each hop lasts at least one
symbol period. There are several unique features of FHSS system:
• Since FHSS can produce signals of much wider bandwidth compared to DSSS, it can
achieve much higher processing gain.
• As the carrier hops randomly over a wide range of frequencies, it is hard for the
adversary to track or jam the active transmission without knowledge of the its hopping
pattern.
• FHSS system is more robust to wideband jamming, since the signal power can be
concentrated on a narrower frequency band during each hopping period. The receiver
can filter out the jamming locates outside the current signal transmission band.

1.2.2.2

Limitations of Existing Spread Spectrum Systems

There are two major drawbacks in existing spread spectrum systems:

Inadequate Security In spread spectrum system, the PN sequence should only be shared
between the transmitter and receiver of the legitimate user and kept secret from the malicious
user. In practice, the PN sequence is usually generated by linear feedback shift register
(LFSR) and can be uniquely determined by the initial state of LFSR (i.e., seed). Therefore,

7

it is more convenient for legitimate user to share the seed instead of sharing the PN sequence
directly. Again, the seed should be kept secret from the malicious user. The inherent security
of spread spectrum systems is based on the assumption that the malicious user without the
knowledge of the seed is unable to predict the PN sequence. However, the seed can be
consistently estimated based on the noisy observations of the generated PN sequence, which
can be easily extracted from received spread spectrum signal [44, 45]. To overcome this
drawback, a secure PN sequence generation algorithm needs to be designed.

Low Spectral Efficiency In multiuser environment, due to the system limitations, MUI
can become a severe problem for spread spectrum systems:
• In conventional DSSS system, the channel multipath and asynchronous transmission
activity can introduce nonzero cross-correlation among the users’ signals. This leads
to self-jamming to the DSSS system. Although some MUI suppression techniques such
as multiuser detection algorithms have been proposed to tackle this problem [46, 47],
their complexity may be too high to be affordable in many applications.
• In conventional FHSS system, each user hops independently based on its own PN
sequence. A collision occurs whenever there are more than one users transmitting over
the same frequency band. When there is a collision, it is reasonable to assume that
the probability of error is 0.5. Therefore, the overall information bit rate of spread
spectrum can be significantly limited, resulting in low spectral efficiency.
Existing spread spectrum systems work reasonably well for voice centric communications
which only requires relatively narrow bandwidth. However, their low spectral efficiency
provides insufficient information capacity for today’s high speed multimedia wireless services.
This turns out to be the most significant obstacle in developing anti-jamming features for
high speed wireless wireless communication systems, for which spectrum is one of the most

8

precious resources. In this dissertation, we will develop spectrally efficient anti-jamming
systems as a potential solution to this problem.

1.3

Proposed Research Directions

This dissertation focuses on the study of hostile jamming modeling, classification, and spectrally efficient anti-jamming wireless network design by integrating advanced signal processing techniques and cryptographic techniques into the PHY layer transceiver structure.
More specifically, the proposed research directions are briefly summarized in the following
subsections.

1.3.1

Cognitive Jamming Modeling and Classification

In literature, jamming signals are generally categorized into band jamming, tone jamming
and partial-time jamming. Existing work on jamming detection and jamming prevention was
generally targeted at a particular jamming model at a time. That is, the jamming pattern
is assumed to be known and invariant during the signal transmission period. In practice,
however, the jammer may very likely switch frequently from one pattern to another, with
each jamming pattern only lasting a very short period of time. In other words, the jammer
may launch smart, cognitive jamming, also known as adaptive jamming or time-varying
jamming. Note that no particular anti-jamming system is effective under all jamming attacks,
for optimum jamming mitigation, the transmitter has to be cognitive as well. Effective
cognitive anti-jamming system design depends on successful jamming detection, modeling
and classification.
In this part of dissertation, we focus on cognitive jamming modeling and classification
based on time-frequency analysis and the relative correlation between the signal and the
jamming interference: (i) We introduce the general concepts of time-varying jamming coherence time and time-frequency jamming coherence bandwidth, and propose a new jamming

9

classification scheme based on these parameters; (ii) We introduce the concept of disguised
jamming, where the jamming is highly correlated with the signal, and has a power level close
or equal to the signal power. It is complementary to the traditional strong jamming; (iii)
Based on time-frequency analysis and approximation theory, we develop algorithms to estimate the time-varying jamming coherence time and the time-frequency jamming coherence
bandwidth for both stationary and locally stationary jamming.

1.3.2

Anti-jamming System Design Based on Message-Driven Frequency Hopping

Mainly limited by multiple access interference, the spectral efficiency of existing jamming
resistant systems are very low due to inefficient use of the large bandwidth. Recently, a threedimensional modulation scheme, known as message-driven frequency hopping (MDFH) was
proposed in [48, 49]. The most significant property of MDFH is that: by embedding a large
portion of information into the hopping frequency selection process, additional information
transmission is achieved with no extra cost on either bandwidth or power [50]. In fact,
transmission through hopping frequency control essentially adds another dimension to the
signal space, thus increases the system spectral efficiency by multiple times. MDFH is
particularly robust under strong jamming. However, it experiences considerable performance
losses under disguised jamming from sources that mimic the true signal.
To improve the performance of MDFH under disguised jamming, in this part of dissertation, we propose an anti-jamming MDFH (AJ-MDFH) scheme. The main idea is to insert
some signal identification (ID) information during the transmission process. This ID information is generated through a cryptographic algorithm using the shared secret between the
transmitter and the receiver. Therefore, it can be used by the receiver to locate the true
carrier frequency or the desired channel. At the same time, it is computationally infeasible to
be recovered by malicious users. Comparing with MDFH, AJ-MDFH can effectively reduce
the performance degradation caused by disguised jamming. At the same time, it is robust

10

under strong jamming just as MDFH. The spectral efficiency of AJ-MDFH is very close
to that of MDFH, which is several times higher than conventional FH. We investigate the
ID constellation design and its impact on the performance of AJ-MDFH under both noise
jamming and disguised jamming. It is shown that when noise is present, the detection error
probability of AJ-MDFH under the worst case disguised jamming converges as the size of
the constellation increases. This result justifies the use of practical finite size constellations
in AJ-MDFH. It is observed that multi-carrier AJ-MDFH (MC-AJ-MDFH) can increase the
system efficiency and jamming resistance significantly through jamming randomization and
enriched frequency diversity.

1.3.3

Capacity Analysis of MDFH Based Systems under Disguised Jamming

In this part of dissertation, we analyze the capacity of MDFH and AJ-MDFH under disguised
jamming. Both MDFH and AJ-MDFH can be modeled as arbitrarily varying channels, which
is characterized as W : X × J → S, where X is the transmitted signal space, J is the
jamming space and S is the estimated information space. For any x ∈ X , J ∈ J and s ∈ S,
W (s|x, J) denotes the conditional probability that s is detected at the receiver, given that
x is the transmitted signal and J is the jamming. If J = X and W (s|x, J) = W (s|J, x) for
any x, J ∈ X , s ∈ S, the AVC is said to have a symmetric kernel [51]. More generally, if
the jammer can choose a stochastic method to generate J, such that W becomes symmetric,
then the AVC is said to be symmetrizable. The deterministic code capacity of the AVC for
the average probability of error is positive iff the AVC is nonsymmetrizable. [52, 53]
Based on the results in AVC capacity analysis, we show that under the worst case disguised jamming, the AVC corresponding to MDFH has symmetric kernel, resulting in zero
deterministic code capacity. On the other hand, under the worst case disguised jamming - ID
jamming, as long as the ID sequence is unavailable to the jammer, the AVC corresponding to
AJ-MDFH is nonsymmetrizable, resulting in positive deterministic code capacity. Note that
the secure ID in AJ-MDFH is generated using AES, to symmetrize AJ-MDFH is equivalent

11

to break AES, which is computationally infeasible in practical systems. We derive the capacity of AJ-MDFH under ID jamming, for which the mutual information is maximized for
all possible input probability distributions and minimized for all possible jamming distributions. We show that the capacity converges as the constellation size M goes to infinity and
extend the capacity analysis to multiuser AJ-MDFH system (MC-AJ-MDFH). It is observed
that: under reasonable SNR levels (≥ 10dB), the capacity of AJ-MDFH under ID jamming
is close to the jamming-free case, and it outperforms the conventional FH systems by big
margins. These results confirm the superior performance of the AJ-MDFH under disguised
jamming.

1.4

Overview of the Dissertation

In the dissertation, we aim to address the following problems:
• How to model the cognitive jamming phenomenons and classify jamming patterns
effectively?
• How to improve the anti-jamming feature of MDFH, while maintaining its high spectral
efficiency?
• How do the MDFH and AJ-MDFH perform under various jamming scenarios?
The dissertation is structured as follows.
Chapter 2 explores the cognitive jamming modeling and classification in wireless communications. First, we introduce the time-varying autocorrelation function and time-varying
power spectral density to characterize time-varying jamming phenomenons and to include
all the existing jamming models as special cases. Second, we introduce the concepts of timevarying jamming coherence time and time-frequency jamming coherence bandwidth based
on the characteristics of these parameters. We also introduce two types of jamming classification criterions: (i) Based on time-frequency analysis of jamming statistics, we can classify
12

the jamming into fast jamming versus slow jamming and flat jamming versus frequency selective jamming; (ii) Based on relative power and correlation between signal and jamming,
we can classify the jamming into strong jamming and disguised jamming. Finally, we develop algorithms based on time-frequency analysis and approximation theory to estimate
the coherence time and coherence bandwidth for stationary jamming and locally stationary
jamming.

Chapter 3 presents spectrally efficient anti-jamming system design based on messagedriven frequency hopping (MDFH). First, we provide a brief review of MDFH, which improves spectral efficiency considerably by embedding part of information into the process
of hopping frequency selection. Despite being robust under strong jamming, MDFH experiences considerable performance losses under disguised jamming from sources that mimic
the true signal. Anti-jamming MDFH is developed to mitigate this security vulnerability by
inserting a secure ID sequence in the transmission process. Second, we investigate ID constellation design and its impact on the performance of AJ-MDFH under both noise jamming
and disguised jamming. Third, we extend the single carrier AJ-MDFH to multi-carrier AJMDFH (MC-AJ-MDFH), which can increase the system efficiency and jamming resistance
significantly through jamming randomization and enriched frequency diversity. Finally, we
analyze the performance of the proposed system.

Chapter 4 analyzes the capacity of MDFH and AJ-MDFH under disguised jamming using
the AVC model. First, we show that under the worst case disguised jamming, the AVC kernel
corresponding to MDFH is symmetric. That is, the deterministic code capacity of MDFH
under worst case disguised jamming is zero. Second, we prove that under the worst case
disguised jamming - ID jamming, as long as the ID sequence is unavailable to the jammer,
the AVC corresponding to AJ-MDFH is nonsymmetrizable. Note that the secure ID in AJMDFH is generated using AES, to symmetrize AJ-MDFH is equivalent to break AES, which
is computationally infeasible in practical systems. This result implies that AJ-MDFH has
13

positive capacity under ID jamming. Third, we derive the deterministic capacity of AJMDFH under ID jamming. We show that the capacity converges as the constellation size M
goes to infinity. Finally, we extend the capacity analysis to the multiuser AJ-MDFH system
(MC-AJ-MDFH) and show that it outperforms the multiple access scheme for conventional
FH (FHMA).

Chapter 5 summarizes the contributions and conclusions of the dissertation. An outline
of future work is also provided.

14

Chapter 2
COGNITIVE JAMMING MODELING AND CLASSIFICATION

In this chapter, we consider cognitive jamming modeling and classification based on timefrequency analysis and the relative correlation between the signal and the jamming interference. We introduce the general concepts of time-varying jamming coherence time and
time-frequency jamming coherence bandwidth, and propose a new jamming classification
scheme based on these parameters. We also introduce the concept of disguised jamming,
for which the jamming is highly correlated with the signal, and has a power level close or
equal to the signal power. We point out that very often, disguised jamming can be much
more harmful than the traditional strong jamming. Algorithms based on time-frequency
analysis and approximation theory are developed to estimate the time-varying jamming coherence time and the time-frequency jamming coherence bandwidth for both stationary and
locally stationary jamming. Simulation examples are provided to demonstrate the proposed
approaches.

2.1

Introduction

One of the most commonly used techniques for limiting the effectiveness of an opponent’s
communications is referred to as jamming. Intentional jamming, also known as hostile jamming, intends to disable the legitimate transmission by saturating the receiver with noise
or false information through deliberate radiation of radio signals, and thus significantly decreasing the signal-to-interference-plus-noise ratio (SINR). In literature, jamming signals are
generally categorized into three classes: (i) Band jamming, generally modeled as a zero-mean
wide sense stationary Gaussian random process with a flat power spectral density (PSD) over
the bandwidth of interest. Band jamming is further classified into full-band jamming [28] and
partial-band jamming [31]; (ii) Tone jamming, typically a sinusoid waveform whose power

15

is concentrated on the carrier frequency. Tone jamming includes single-tone jamming and
multi-tone jamming [27]. (iii) Partial-time jamming, modeled as a two-state Markov process,
for which the jammer is on in state 1, and is off in state 0. State 1 occurs with probability
of ρ, and state 0 occurs with probability (1 − ρ) [33].
Existing work on jamming detection and jamming prevention was generally targeted at
a particular jamming model at a time. That is, the jamming pattern is assumed to be
known and invariant during the signal transmission period, see [54, 55, 56], for example.
In practice, however, the jammer may very likely switch frequently from one pattern to
another, with each jamming pattern only lasting a very short period of time. In other
words, the jammer may launch smart, cognitive jamming, also known as adaptive jamming
or time-varying jamming. Note that no particular anti-jamming system is effective under all
jamming attacks. For optimum jamming mitigation, the transmitter has to be cognitive as
well. Cognitive jamming modeling and classification are critical for dynamic anti-jamming
system design.
This chapter focuses on cognitive jamming modeling and classification based on timefrequency analysis. We introduce the concepts of time-varying jamming coherence time and
time-frequency jamming coherence bandwidth. By comparing the time-varying jamming
coherence time with the signal symbol period, we classify the jamming into fast jamming
and slow jamming; By comparing the time-frequency jamming coherence bandwidth with
the signal bandwidth, we classify the jamming into flat jamming and frequency selective
jamming. Note that at one time instant, the jamming coherence bandwidth may vary from
one frequency to another. Therefore, a multi-band signal may experience flat jamming and
frequency selective jamming simultaneously at different frequency bands. We also introduce
the concept of disguised jamming, where the jamming is highly correlated with the signal, and
has a power level close or equal to the signal power. It is complementary to the traditional
strong jamming. Algorithms based on time-frequency analysis and approximation theory
are provided to estimate the time-varying coherence time and time-frequency coherence

16

bandwidth for stationary jamming and locally stationary jamming. Simulation results are
provided to demonstrate the proposed approaches.

2.2

Cognitive Jamming Modeling and Classification

Jamming interference can be divided into two classes: system inherent jamming and hostile
jamming. The system inherent jamming, such as the intersymbol interference caused by
multipath propagation, or multiuser interference due to simultaneous multiple access to the
same frequency bands, can generally be reduced or minimized using various interference
cancelation methods. In this section, we will focus on the characterization of random hostile
jamming through two perspectives: (i) The time-frequency analysis of the jamming statistics;
(ii) The correlation between the jamming and the signal.

2.2.1

Jamming Modeling using Time-Varying Power Spectral Density

We start with a single-input single-output AWGN channel. Let s(t) be the transmitted
signal, then the received signal can be written as:
r(t) = s(t) + n(t) + J(t),

(2.1)

where n(t) is the white Gaussian noise, J(t) represents the hostile jamming signal. J(t) can
either be stationary or nonstationary. Let RJ (t, τ ) = E{J(t + τ )J(t)} be the autocorrelation
function of J(t). The time-varying PSD is defined as
∞

SJ (t, f ) = F{RJ (t, τ )} =

−∞

RJ (t, τ )e−j2πf τ dτ.

(2.2)

∞

The time-varying jamming power is given by PJ (t) =

−∞

SJ (t, f )df. We assume that PJ (t)

is finite and PJ (t) ≤ PJ,max , where PJ,max is the maximum jamming power. The reason that
we allow the jamming power to be time-varying, rather than always using the total available
jamming power, is because that strong jamming that uses the whole jamming power may

17

not always be the worst jamming [57]. If J(t) is wide-sense stationary, then RJ (t, τ ), SJ (t, f )
and PJ (t) all become time-invariant.
It turns out that all the existing jamming models can be characterized using the timevarying power spectral density, SJ (t, f ). In fact, let f0 and f1 be the start and ending
frequency of the available frequency band, respectively; [t0 , t1 ] the time duration period of
the message signal, and PJ the constant jamming power.
PJ
NJ , ∀f ∈ [f0 , f1 ], ∀t ∈ [t0 , t1 ], then J(t) with PSD SJ (t, f ) is
• If SJ (t, f ) =
f1 − f0
reduced to the traditional full-band jamming. Partial-band jamming can be defined in
a similar manner.
• If SJ (t, f ) = PJ δ(f − fk ), ∀t ∈ [t0 , t1 ], where fk ∈ [f0 , f1 ] and δ is the Dirac delta
function, then we obtain the single-tone jamming. For multi-tone jamming,
K

K

PJ (t, k)δ(f − fk ), s.t.

SJ (t, f ) =
k=1

PJ (t, k) = PJ ,

(2.3)

k=1

where K is the number of jamming tones, and PJ (t, k) stands for the jamming power
allocated for the kth tone fk .
• If ∀f ∈ [f0 , f1 ],
SJ (t, f ) =


 0,




if t ∈ [t0 , tm ) or t ∈ (tn , t1 ],

PJ
, if t ∈ [tm , tn ],
f1 − f0

(2.4)

where tm and tn (tm ≤ tn ) are certain intermediate time instants within [t0 , t1 ], then
we obtain the partial-time jamming.
Motivated by the observations above, we propose to model J(t) through 2D analysis of
RJ (t, τ ) and its Fourier transform SJ (t, f ).

2.2.2

Jamming Classification Based on Time-frequency Analysis

In this section, we introduce the concepts of time-varying jamming coherence time and timefrequency jamming coherence bandwidth for J(t).
18

• Time-varying jamming coherence time: For a given time instant t, let [0, Tc (t)] be the
period over which RJ (t, τ ) is essentially non-zero and flat. This implies that J(t + τ )
and J(t) are highly correlated when τ ≤ Tc (t). We define Tc (t) as the time-varying
jamming coherence time of J(t). Tc (t) is used to characterize the time-varying nature
of J(t) in the time domain. In other words, Tc (t) is a statistical measure of the time
duration over which J(t) is essentially invariant.
• Time-frequency jamming coherence bandwidth: For a given time instant t and frequency
Bc (t, υ)
Bc (t, υ)
, υ+
] be the frequency range over which the magnitude of the
υ, let [υ−
2
2
time-varying jamming PSD, |SJ (t, f )|, is essentially nonzero and can be considered to
be flat. That is, at time instant t, two frequency components around υ with separation
greater than Bc (t, υ) are affected differently by the jamming. We define Bc (t, υ) as the
time-frequency jamming coherence bandwidth at time t and frequency υ.
Let Ts and Bs be the symbol period and bandwidth of the information signal, respectively,
where Ts and Bs can be time-varying as well, such as in adaptive transmitters. We further
introduce the following jamming classification criterions:
• For any given time instant t, if Ts > Tc (t), then it means that the jamming changes
rapidly within the symbol duration of the signal, we say that the signal is experiencing
fast jamming at time t. Otherwise, we say that the signal is experiencing slow jamming.
• For any given time instant t, let fc (t) be the center frequency of the signal.

If

Bc (t, fc (t)) > Bs , that is, the jamming coherence bandwidth at υ = fc (t) is larger
than the signal bandwidth, we say that the signal is experiencing flat jamming at time
t. Otherwise, we say that the signal is experiencing frequency selective jamming. For
multi-carrier signals, Bc (t, υ) needs to be evaluated at each carrier frequency. Therefore, a multi-band signal may experience flat jamming and frequency selective jamming
simultaneously at different frequency bands.

19

Remark 1 Comparing with the traditional time-varying channel modeling, it should be
pointed out that our jamming model is based on the time-frequency statistics of J(t), while
the channel model is based on the statistics of the time-varying channel impulse response
h(t, τ ), which is the system response at t to an impulse applied at t − τ .

2.2.3

Strong Jamming and Disguised Jamming

Let Ps (t) and PJ (t) denote the time-varying signal power and jamming power, respectively. If
PJ (t) >> Ps (t), we say that the signal is experiencing strong jamming. Most communication
systems become paralyzed under strong jamming attacks. However, with recent advances
in anti-jamming system design [57], we can see that strong jamming may not be the worst
jamming. On the other hand, disguised jamming, where the jamming is highly correlated
with the signal and has a power level close or equal to the signal power, can be more harmful.
Consider the cross-correlation of the signal s(t) and jamming interference J(t) defined as
Rs,J (t1 , t2 ) = E{s(t1 )J(t2 )}. For band jamming or tone jamming, we are more interested
in the case t1 = t2 = t. During the (k + 1)th symbol interval [kTs , (k + 1)Ts ], we can
approximate the cross-correlation between s(t) and J(t) with the time average
ρk =

Ts

(k+1)Ts
1
s(t)J(t)dt,
Ps,k PJ,k kTs

(2.5)

(k+1)Ts
(k+1)Ts
1
1
s2 (t)dt and PJ,k =
J 2 (t)dt.
where Ps,k =
Ts kTs
Ts kTs
Note that 0 ≤ |ρk | ≤ 1. We say that J(t) is a disguised jamming over [kTs , (k + 1)Ts ] if

1. s(t) and J(t) are highly correlated. More specifically, |ρk | > ρ0 , where ρ0 is an
application-oriented, predefined threshold.
P
2. The jamming-to-signal ratio (JSR) is close to 0dB. More specifically, | J − 1| < P ,
Ps
where P is an application-oriented, predefined jamming-to-signal ratio threshold.

20

Disguised partial-time jamming is generally targeted at partial-time transmissions, for which
the signal is transmitted on certain time slots within a frame. To locate disguised jamming
in this scenario, (2.5) should be extended to the case t1 = t2 .
Intuitively, disguised jamming makes it difficult for the receiver to identify the true signal
from disguised interference. To combat with disguised jamming, the transmitter and the
receiver need to have some preshared secret information, which can be used as the secure
ID for the true signal so that it can be effectively extracted [57]. On the other hand,
study on disguised jamming can also be used in electronic interference of an opponent’s
communication.

2.3

Estimation of Time-Varying Jamming Coherence Time and
Time-Frequency Jamming Coherence Bandwidth

For jamming classification, we consider two scenarios: (i) J(t) is stationary; (ii) J(t) is
locally stationary [58, 59, 60], which means that for any t, there exists a time interval of size
l(t)
l(t)
,t +
], such that J(t) can be approximated by a stationary process within
l(t), [t −
2
2
this interval. The size of the intervals, l(t) may change with time t. This assumption is
reasonable in the sense that a particular jamming pattern may last for a short time and then
the jammer switches to another pattern.

2.3.1

Stationary Jamming

When J(t) is stationary, the autocorrelation function RJ (t, τ ) = E{J(t + τ )J(t)} and its
Fourier transform SJ (t, f ) are both independent of t. We have RJ (τ ) = E{J(t + τ )J(t)}
∞

and SJ (f ) = F{RJ (τ )} =

−∞

RJ (τ )e−j2πf τ dτ. In this case, Tc (t) and Bc (t, υ) become

time-invariant, and are denoted as Tc and Bc (υ), respectively.
By definition, [0, Tc ] is the period over which RJ (τ ) is flat and essentially nonzero. That
is,
|RJ (τ )| > α0 RJ (0),
21

(2.6)

where α0 ∈ [0.5, 0.95] is a predefined constant for coherence time estimation.
Bc (υ)
Bc (υ)
For any given frequency υ, let [υ −
,υ +
] be the frequency range over which
2
2
SJ (f ) is essentially nonzero and can be considered to be flat. Consider the case that SJ (f )
has only one main lobe in the positive frequency range, centered at fc . (i) When fc = 0,
J(t) is a baseband stationary process. For υ = 0, Bc (0) can be estimated through
Bc (0)
2
Bc (0)
− 2

|SJ (f )|2 df ≥ α1 SJ 2 ,
2

(2.7)

where α1 ∈ [0.5, 0.95] is a predefined constant for coherence bandwidth estimation. For any
Bc (0)
υ = 0, Bc (υ) can be estimated from Bc (0) as: Bc (υ) = Bc (0) − 2|υ|, if 0 < |υ| <
;
2
Bc (0)
Bc (υ) = 0, if |υ| ≥
. (ii) When fc > 0, J(t) is a passband stationary process. For
2
|υ| = fc , Bc (υ) can be estimated through
B (υ)
fc + c2
B (υ)
fc − c2

|SJ (f )|2 df ≥

α1
S 2.
2 J 2

(2.8)

For |υ| = fc , Bc (υ) can be estimated from Bc (fc ) as: Bc (υ) = Bc (fc ) − 2||υ| − fc |, if
Bc (fc )
Bc (fc )
0 < ||υ| − fc | <
; Bc (f ) = 0, if ||υ| − fc | ≥
.
2
2
2.3.2
2.3.2.1

Locally Stationary Jamming
Definition

First, partition the active time axis into intervals [tp , tp+1 ] of size lp = tp+1 − tp , with
lim tp = ∞ and

p→∞

lim tp = −∞. We cover each interval [tp , tp+1 ] with a window function

p→−∞

gp (t). We construct the window function gp (t) such that it satisfies the following conditions:
(i) For ∀p ∈ Z, the support of gp (t) only intersects with the support of gp−1 (t) and gp+1 (t).
The supports of gp (t) and gp−1 (t) intersects in [tp − ηp , tp + ηp ]. (ii) gp (t) and gp−1 (t) are
symmetric with respect to tp , i.e., gp (t) = gp−1 (2tp − t) over [tp − ηp , tp + ηp ]. (iii) For
∞

|gp (t)|2 = 1. The window functions are illustrated in Figure 2.1. It can be

∀t ∈ R,
p=−∞

22

shown that under these three conditions, the local cosine family defined by
φp,k (t) = gp (t)

2
π(k + 1/2)
cos
(t − tp )
lp
lp

(2.9)
k∈N,p∈Z

formulates an orthonormal basis of L2 (R) [61]. From (2.9), we can see that the support of
π(k + 1/2)
φp,k (t) is [tp − ηp , tp+1 + ηp+1 ], and its center frequency is ξp,k =
.
lp

g p −1 (t )

tp −ηp t p t p + η p

t p −1

g p +1 (t )

g p (t )

t p +1

Figure 2.1: An example of window functions.
For RJ (t, τ ) = E{J(t+τ )J(t)}, let s = t+τ , then RJ (t, τ ) can be rewritten as RJ (t, s) =
E{J(t)J(s)}. For ∀f ∈ L2 (R), define the covariance operator by
∞

T f (t) =
−∞

RJ (t, s)f (s)ds.

(2.10)

Following [61], locally stationary jamming processes can be defined as follows.
Definition 1 A jamming J(t) is said to be locally stationary if there exists a local cosine
basis {φp,k (t)}k∈N,p∈Z such that for some constant µ < 1 and A > 0, we have that for all
p = q,
max(lp , lq )
≤ A|p − q|µ ,
min(lp , lq )

(2.11)

and for all n > 1, we can find a constant Qn such that for all (p, q, j, k) ∈ Z2 × N2 ,
T φp,k , φq,j

≤

Qn
n )(1 + | max(l , l )(ξ
(1 + |p − q|
p q
p,k

− ξq,j )|n )

.

(2.12)

Here {lp } specify the supports of {φp,k (t)} and indicate the intervals over which J(t) is
approximately stationary. Condition (2.12) implies that | T φp,k , φq,j | decays rapidly as we
23

increase |p − q| and |ξp,k − ξq,j |. This means that the operator T is “almost” diagonalized
by {φp,k (t)}. In other words, each φp,k (t) is “almost” an eigenvector of T .
In fact, define u = t + τ /2, then
RJ (t, τ ) = E{J(t + τ )J(t)}
τ
τ
= E{J(u + )J(u − )}
2
2
∞

Define WJ (u, f ) =

−∞

CJ (u, τ ).

(2.13)

CJ (u, τ )e−j2πf τ dτ. Note that for a locally stationary process, if

l(x)
l(x)
,x +
], then
u ∈ [x −
2
2
CJ (u, τ ) ≈ 0, if τ > d(x),

(2.14)

l(x)
. By definition,
2
the time-varying jamming coherence time at t = x should satisfy Tc (x) ≤ d(x). For any
l(x)
ξ
1
ξ
1
l(x)
,x +
]×[
−
,
+
], WJ (u, f ) can be
ξ ∈ R and ∀(u, f ) ∈ [x −
2
2
2π
2d(x) 2π
2d(x)
approximated by WJ (u, f ) ≈ WJ (x, ξ). Let gx (t) be a smooth window function with support
l(x)
l(x)
[x −
,x +
]. The local cosine function corresponds to the time-frequency rectangle
2
2
l(x)
l(x)
ξ
1
ξ
1
[x−
, x+
]×[ −
, +
] can be represented as φx,ξ (t) = gx (t) cos(ξt+θ).1
2
2
2π 2d(x) 2π 2d(x)
Following the argument in [61],
where d(x) is the so-called decorrelation length, and generally d(x) <

T φx,ξ (t) ≈ WJ (x, ξ)φx,ξ (t).

(2.15)

That is, φx,ξ (t) is almost an eigenvector of T .

2.3.2.2

Best Basis Search and Spectrum Estimation

Let {φn (t)}n∈N be an orthonormal basis of L2 (R), which implies that for any f (t) ∈ L2 (R),
f (t) can be decomposed as f (t) =

f, φn φn (t). It follows that T f (t) =
n∈N

1 The

f, φn T φn (t).
n∈N

local cosine function φp,k (t) defined in (2.9) can be represented as φxp ,ξ (t) =
p,k

gxp (t) cos(ξp,k t + θp,k ) with xp =

1
(tp + tp+1 ), gxp (t) =
2

θp,k = −ξp,k tp .
24

π(k + 1/2)
2
gp (t), ξp,k =
and
lp
lp

That is, the operator T is completely determined by T φn (t). For any m ∈ N,
T φm (t) =

T φm , φn φn (t).

(2.16)

n∈N

Note that in practice, T f (t) can only be approximated by a finite sum. A natural question
is: for a fixed number of items in the sum, which basis will result in the best approximation?
γ

Let D = {B γ }γ∈Γ be a dictionary of orthonormal basis B γ = {φn }n∈N of L2 (R), indexed by
γ

γ ∈ Γ where Γ denotes the collection of all the index sets in the dictionary. Define BK as


 T φγ , φγ , if |n − m| ≤ K,
m n
γ γ
γ
BK φm , φn =
(2.17)
 0,

Otherwise.
γ

For a fixed K, the best basis is the one which minimizes the norm T − BK s , defined as
γ

γ

T − BK s = sup

(T − BK )f 2 .

f 2 =1

(2.18)

Since local cosine functions are approximate eigenvectors of the corresponding covariance
operator T , we search the best basis in a dictionary of local cosine bases, with K = 0 in
γ

(2.17) such that the operator T is diagonalized. Once the best basis {φn0 }N is selected,
n=1
we have
RJ (t, s) = E{J(t)J(s)}


N

 N
γ0 γ0
γ0 γ0
J, φn φn (s)
≈ E
J, φm φm (t)


n=1

m=1

N

N

=

E

γ

γ

0
J, φm J, φn0

γ

γ

0
φm (t)φn0 (s),

(2.19)

m=1 n=1

where E

γ

γ

0
J, φm J, φn0

E

γ0
J, φm

can be rewritten as

γ
J, φn0

∞

= E
∞

γ0
J(t)φm (t)dt ·

−∞
∞

=
−∞ −∞
∞
∞

=
=

−∞ −∞
γ
γ0
T φn0 , φm

25

∞

γ

J(s)φn0 (s)ds

−∞
γ0
γ
E{J(t)J(s)}φm (t)φn0 (s)dtds
γ

γ

0
R(t, s)φn0 (s)ds · φm (t)dt

.

(2.20)

Following from (2.19) and (2.20), we have
N

N

γ

γ

γ

N

γ

0
0
T φn0 , φm φm (t)φn0 (s) ≈

RJ (t, s) =
m=1 n=1

γ

γ

γ

γ

T φn0 , φn0 φn0 (t)φn0 (s).

(2.21)

n=1

γ
γ
γ
Let dn = T φn0 , φn0 = E{ | J, φn0 |2 }, then the time-varying autocorrelation function can

be estimated as

N

γ

γ

dn φn0 (t)φn0 (t + τ ).

RJ (t, τ ) ≈

(2.22)

n=1
γ

Let xn and ξn denote the center time and the center frequency corresponding to φn0 , respectively. That is,
γ

φn0 (t) = gxn (t) cos(ξn t + θn ).

(2.23)

In the time domain, the smooth function gxn (t) is approximately nonzero and flat over the
l(xn )
l(xn )
, xn +
]. Assuming J(t) has a finite duration, and can be covered
time support [xn −
2
2
γ
by P window functions {gp (t)}P associated with the best basis functions {φn0 }N . Let
p=1
n=1
Np denote the set of indexes for the best basis functions corresponding to gp (t). Then,
2
gp (t) for any n ∈ Np , and ∪P Np = {1, · · · , N }. We have
gxn (t) =
p=1
lp
P

RJ (t, τ ) ≈
p=1

2
gp (t)gp (t + τ )
lp

dn cos(ξn t + θn ) cos(ξn t + ξn τ + θn ).

(2.24)

n∈Np

For a given time t ∈ (tp , tp+1 ), RJ (t, τ ) is mainly determined by |Np | cosine terms in (2.24)
γ

corresponding to a set of basis functions {φn0 (t)}n∈Np (See Figure 2.1). Note that dn can
be estimated from Q realizations of J(t) using the best basis as
1
ˆ
dn =
Q

Q

γ
| J q , φn0 |2 .

(2.25)

q=1

ˆ
Therefore, RJ (t, τ ) can be estimated by replacing dn with dn in (2.24). For any t, Tc (t) can
then be estimated from RJ (t, τ ) using the method described in (2.6). Note that at the border
time instant t = tp (p = 2, · · · , P ), RJ (tp , τ ) is determined by cosine terms associated with
both Np−1 and Np . For conservative estimation, we have Tc (tp )
τ )} where τ is a small time difference.
26

min{Tc (tp − τ ), Tc (tp +

Following from (2.22), the time-varying PSD can be obtained as
N

γ

γ

dn ej2πf t φn0 (t)Φn0 (f ),

SJ (t, f ) ≈

(2.26)

n=1
∞

ξn
) and Gxn (f −
2π
−∞
ξn
ξn
1
ξn
1
) are approximately nonzero and flat over the frequency ranges [ −
,
+
]
2π
2π 2l(xn ) 2π 2l(xn )
ξn
1
ξn
1
and [−
−
,−
+
], respectively. Note that
2π 2l(xn ) 2π 2l(xn )
γ

where Φn0 (f ) =

γ

φn0 (τ )e−j2πf τ dτ . In the frequency domain, Gxn (f +

γ

Φn0 (f ) =
∞

where Gxn (f ) =
P

SJ (t, f ) ≈
p=1

−∞

1
ξn jθn
ξn −jθn
,
Gxn (f −
)e
+ Gxn (f +
)e
2
2π
2π

(2.27)

gxn (τ )e−j2πf τ dτ . It then follows from (2.24) - (2.27) that

1
gp (t)ej2πf t
lp

dn cos(ξn t + θn ) Gp (f −
n∈Np

ξn jθn
ξn −jθn
)e
+ Gp (f +
)e
.
2π
2π
(2.28)

For any t ∈ (tp , tp+1 ), |SJ (t, f )| is also mainly determined by |Np | terms in (2.28) correspondξn
1 ξn
1
γ
,
]
ing to {φn0 (t)}n∈Np . More specifically, over the two frequency range [
−
+
2π
2lp 2π 2lp
ξn
1
ξn
1
dn
and [−
−
,−
+
], |SJ (t, f )| can be approximated by | cos(ξn t + θn )|. Hence,
2π 2lp 2π 2lp
lp
Bc (t, υ) can be estimated from |SJ (t, f )| using the method described in (2.7)-(2.8). Similarly,
min{Bc (tp − τ , υ), Bc (tp + τ , υ)}.

at each border time instant t = tp , Bc (tp , υ)

2.3.3

Binary Tree Based Basis Search Algorithm

We now discuss a practical binary tree based algorithm for fast “best basis” search.

2.3.3.1

Dictionary Construction

To reduce the complexity of the best basis search, a dictionary with admissible binary tree
structure is preferred [62, 63]. The admissible binary tree is defined by any binary tree whose
nodes have either 0 or 2 branches. Each tree node corresponds to a time interval. Assume the

27

jamming J(t) is observed over 0 ≤ t ≤ M . The root of the tree corresponds to the whole time
interval [0, M ]. The left and right branch nodes correspond to the time interval [0, M/2] and
[M/2, M ], respectively. Each node is further split until the tree depth ND is reached. The
node (p, j) at depth j and position p corresponds to the time interval [pM 2−j , (p + 1)M 2−j ].
j

The window function gp (t) is used to cover the time interval corresponding to each node
(p, j). Given a smooth function β(t) which satisfies: β(t) = 0 if t < −η, β(t) = 1 if t > η,
and β 2 (t) + β 2 (−t) = 1. The window function can be constructed as


 β(t − pM 2−j ),
if t < pM 2−j + η,



j
gp (t) =
1,
if pM 2−j + η ≤ t ≤ (p + 1)M 2−j − η,




 β((p + 1)M 2−j − t), if t > (p + 1)M 2−j − η,

(2.29)

where M 2−j ≥ 2η. A full admissible binary tree with depth ND = 2 and corresponding
window functions are illustrated in Figure 2.2.
0
g 0 (t )

j =0

j =1

1
g1 (t )

1
g 0 (t )

j=2
2
g12 (t ) g 2 (t )

2
g 0 (t )

2
g3 (t )

Figure 2.2: An example of full admissible binary tree with depth ND = 2 and
corresponding window functions.
j

Note that the window function gp (t) is associated with the local cosine family through
j

j

φp,k (t) = gp (t)

1
2
cos π k +
−j
2
M2

t − M 2−j p
M 2−j

.

(2.30)

0≤k<M 2−j

For leaf nodes from the admissible binary tree indexed by γ, i.e., (p, j) ∈ γ, the local cosine
j
family B γ = {φp,k (t)}(p,j)∈γ,0≤k<M 2−j formulates an orthogonal basis [63]. The dictionary

28

D = {B γ }γ∈Γ is constructed with all admissible binary trees of depth no more than ND .
Given an orthogonal basis B γ and Q realizations of J(t), following (2.25), the corresponding
j
j
j
j
diagonal coefficients dp,k = T φp,k (t), φp,k (t) = E{ | J, φp,k (t) |2 } can be estimated as

ˆj
dp,k

1
=
Q

2.3.3.2

Q

j
| J q , φp,k |2 .

q=1

Best Basis Search Using Dynamic Programming
M 2−j −1

j

γ

ˆ
|dp,k |2 . The Hilbert-Schmidt norm of the diagonal operator B0 defined

Let C(p, j) =
k=0

γ
in (2.17) can be estimated as B0 2 =
h

C(p, j). To obtain the best basis, we need to
(j,p)∈γ

γ
p
search in the dictionary for best basis that maximizes B0 2 . Let Bj denote the set of basis
h
j

p

functions {φp,k (t)}0≤k<M 2−j and Oj the set of best basis functions associated with node
γ
p
(p, j). It can be shown that B0 2 is maximized if we construct the Oj using the following
h

rule [63]:

 2p
2p+1
 O
p
j+1 ∪ Oj+1 , if C(p, j) < C(2p, j + 1) + C(2p + 1, j + 1),
Oj =
 Bp ,

otherwise.
j

(2.31)

By making best basis selection at each node recursively using bottom-up progression, the
0
overall best basis can be determined at the root node B γ0 = O0 . The computational

complexity of the best basis search is reduced to O(M [log2 M ]2 ). The best basis search
algorithm is summarized in Table 2.1.

2.4

Simulation Results

In this section, we illustrate the estimation of the time-varying jamming coherence time and
the time-frequency jamming coherence bandwidth through simulation examples. M = 4096
samples are obtained from the random jamming process, which is assumed to be real-valued
zero-mean Gaussian. The time duration of the jamming is normalized to 1, and the frequency

29

Table 2.1: The best basis search algorithm.
for j = (J − 1) : −1 : 0 and p = 0 : (2j − 1) do
M 2−j −1

j

ˆ
|dp,k |2

Compute C(p, j) =
k=0

if j == J − 1 then
Mark node (p, j) as leaf
else
if C(p, j) ≥ C(2p, j + 1) + C(2p + 1, j + 1) then
Mark node (p, j) as leaf
else
Leave node (p, j) unmarked and C(p, j) = C(2p, j+1)+C(2p+
1, j + 1)
end if
end if
end for

is normalized by the sampling rate. The jamming-to-noise ratio (JNR) is defined as the ratio
of the jamming power to the noise power.
Example 1: Stationary jamming The jamming here has coherence time Tc = 7/4096,
and coherence bandwidth Bc (0) = 0.1056. The JNR is set to be 10dB. We use the WelchBartlett method [64] to estimate the jamming PSD SJ (f ), and choose α0 = 0.5, α1 = 0.95.
The true and estimated SJ (f ) are illustrated in Figure 2.3. The estimated jamming coherence
ˆ
ˆ
time and bandwidth are Tc = 6/4096 and Bc (0) = 0.1060, respectively. We can see that the
estimation results are quite accurate.
Example 2: Locally stationary jamming In this example, the jamming consists
of 3 stationary intervals with equal length. The JNRs in the three intervals are set to
be (10, 15, 15)dB, and the normalized center frequencies of SJ (t, f ) in the three intervals
are (0, 0.15, 0.3), respectively. The estimated best basis consists of local cosine functions
associated with 5 different window supports, corresponding to intervals l1 ∼ l5 . The true and
estimated RJ (t, τ ) and |SJ (t, f )| are illustrated in Figure 2.4 - Figure 2.7, respectively. It can
ˆ
be observed that |SJ (t, f )| concentrates within the same time-frequency areas as |SJ (t, f )|.

30

20
Bc /2

SJ(f)

0

PSD (dB)

−20

Estimated SJ(f)

ˆ
Bc /2

−40
−60
−80
−100
0

0.1

0.2

0.3

0.4

0.5

f
Figure 2.3: Example 1: The true and estimated jamming PSD SJ (f ).
For example, for t = 0.4375 and υ = 0.15, Tc (0.4375) = 2/4096 and Bc (0.4375, 0.15) =
ˆ
0.0801. With α0 = 0.5, α1 = 0.7, the estimated results are: Tc (0.4375) = 1/4096 and
ˆ
Bc (0.4375, 0.15) = 0.0762.

2.5

Summary

This chapter provides a 2D framework for cognitive jamming modeling and classification. For
the first time in literature, all the existing jamming models are summarized and extended to
the time-varying case under one general scheme. We further introduce the concepts of timevarying jamming coherence time and time-frequency jamming coherence bandwidth, and
classify jamming as fast versus slow jamming, and flat versus frequency selective jamming.
We also introduced the concept of disguised jamming, as it can be much more harmful
than the traditional strong jamming. Time-frequency analysis and approximation theory

31

Figure 2.4: Example 2: The true jamming autocorrelation function RJ (t, τ ). For
interpretation of the references to color in this and all other figures, the reader is referred
to the electronic version of this dissertation.
based algorithms are developed to estimate these statistics for both stationary and locally
stationary jamming.

32

Figure 2.5: Example 2: The estimated RJ (t, τ ) using time-frequency analysis.

33

Figure 2.6: Example 2: The magnitude of the true time-varying jamming PSD |SJ (t, f )|.

34

Figure 2.7: Example 2: The magnitude of the estimated |SJ (t, f )| using time-frequency
analysis.

35

Chapter 3
ANTI-JAMMING MESSAGE-DRIVEN FREQUENCY HOPPING SYSTEM
DESIGN

In this chapter, we consider robust and spectrally efficient anti-jamming system design in
wireless networks. We start with message-driven frequency hopping (MDFH), which is recently proposed as a highly efficient spread spectrum technique. It is observed that while
MDFH is robust under strong jamming, it experiences considerable performance losses under
disguised jamming from sources that mimic the true signal. To overcome this limitation,
we propose an anti-jamming MDFH (AJ-MDFH) system. The main idea is to transmit
a secure ID sequence along with the information stream. The ID sequence is generated
through a cryptographic algorithm using the shared secret between the transmitter and the
receiver, it is then exploited by the receiver for effective signal extraction. It is shown that
AJ-MDFH can effectively reduce the performance degradation caused by disguised jamming,
and is also robust under strong jamming. In addition, we extend AJ-MDFH to multi-carrier
case, which can increase the system efficiency and jamming resistance significantly through
jamming randomization and frequency diversity, and can readily be used as a collision-free
multiple access system.

3.1

Introduction

As a widely used spread spectrum technique, frequency hopping (FH) was originally designed
for secure communication under hostile environments [65, 66, 67, 68, 69]. In conventional
FH, each user hops independently based on its own PN sequence, a collision occurs whenever
there are two or more users transmitting over the same frequency band. Mainly limited by
the collision effect, the spectral efficiency of conventional FH is very low [70, 50]. To improve
the spectral efficiency, FH systems that exploit high-dimensional modulation scheme have

36

been studied in the literature [71, 72, 73, 74, 75]. However, the performance of these systems
are still limited by the collision effect, also known as self-jamming.
Recently, a three-dimensional modulation scheme, known as message-driven frequency
hopping (MDFH), was proposed in [49]. The basic idea of MDFH is that part of the message
acts as the PN sequence for carrier frequency selection at the transmitter. More specifically,
selection of carrier frequencies is directly controlled by the encrypted information stream
rather than by a pre-selected pseudo-random sequence as in conventional FH. The most
significant property of MDFH is that: by embedding a large portion of information into the
hopping selection process, additional information transmission is achieved with no extra cost
on either bandwidth or power. In fact, transmission through hopping frequency control adds
another dimension to the signal space, and the resulted coding gain can increase the system
spectral efficiency by multiple times [49].
In this chapter, we analyze the performance of MDFH under hostile jamming. It is
observed that: MDFH is particularly powerful under strong jamming scenarios, and outperforms the conventional FH by big margins. The underlying argument is that: for MDFH,
even if the signal is jammed, strong jamming can enhance the power of the jammed signal
and hence increases the probability of correct detection. When the system experiences disguised jamming, that is, when the jamming is highly correlated with the signal, and has a
power level close or equal to the signal power, it is then difficult for the MDFH receiver to
distinguish jamming from the true signal, resulting in performance losses.
To improve the performance of MDFH under disguised jamming, in this chapter, first,
we propose an anti-jamming MDFH (AJ-MDFH) scheme. The main idea is to insert some
signal identification (ID) information during the transmission process. This secure ID information is generated through the Advanced Encryption Standard (AES) [76] using the shared
secret between the transmitter and the receiver. The ID information can be exploited by the
receiver to locate the true carrier frequency. Moreover, protected by advanced encryption
techniques, it is computationally infeasible for malicious users to recover the ID sequence.

37

In other words, in AJ-MDFH, secure ID signals are introduced to distinguish the true information channel from the disguised channels invoked by jamming interference. Our analysis
indicates that: comparing with MDFH, AJ-MDFH can effectively reduce the performance
degradation caused by disguised jamming. At the same time, its spectral efficiency is very
close to that of MDFH, which is several times higher than that of conventional FH.
Second, we investigate ID constellation design and its impact on the performance of
AJ-MDFH under both noise jamming and disguised jamming. Noise jamming, where the
jamming is modeled as Gaussian noise, has been widely adopted in literature [77, 78, 79].
But disguised jamming can be much more harmful for most communication systems. For
AJ-MDFH, the worst case disguised jamming is ID jamming, for which the jammer tries to
mimic the ID signal, and sends symbols from the same constellation as that of the ID signal.
We show that under noise jamming, the detection error probability is mainly determined by
the signal to jamming and noise ratio. In this case, for a given power constraint, constant
modulus constellation delivers the best results in terms of detection error probability. Under
ID jamming, the situation is more complex. In the ideal case when the system is noisefree, increasing the ID constellation size can increase the ID uncertainty, hence reduce the
probability of error. In this case, the ideal constellation size is M = ∞. However, when noise
is present, we prove that the detection error probability converges as M goes to infinity. In
other words, there exists a threshold Mt , increasing the constellation size over Mt will result
in little improvement in error probability. This result justifies the use of practical, finite size
constellations in AJ-MDFH.
Finally, we extend the single carrier AJ-MDFH to multi-carrier AJ-MDFH (MC-AJMDFH). It is observed that by exploiting secure group generation mechanism, MC-AJMDFH can increase the system efficiency and jamming resistance significantly through jamming randomization and enriched frequency diversity. Moreover, by assigning different carrier groups to different users, MC-AJ-MDFH can also be used as a collision-free multiple
access system. We analyze the proposed system through both theoretical derivation as well

38

as numerical results. Our analysis indicates that: while AJ-MDFH is much more robust
than MDFH under various jamming attacks, its spectral efficiency is very close to that of
MDFH, which is several times higher than that of conventional FH.
This chapter is organized as follows. In Section 3.2, the anti-jamming performance of
MDFH is analyzed under both strong jamming and disguised jamming. In Section 3.3, the
proposed AJ-MDFH scheme is introduced, together with the carrier detection metrics. ID
constellation design is investigated in Section 3.4. The multi-carrier extension of AJ-MDFH
is introduced in Section 3.5. Spectral efficiency is analyzed in Section 3.6. Simulation
examples are provided in Section 3.7 and we conclude in Section 3.8.

3.2

A Brief Review of Message-Driven Frequency Hopping

In this section, we briefly review the message-driven frequency hopping (MDFH) system,
and evaluate its performance under different jamming scenarios.

Ordinary
Bit Vector

Carrier Bit Vectors
X

n,1

X

n,2

X
Xn

n, Nh

Yn

Figure 3.1: The nth block of the information.

3.2.1

System Description

The basic idea of MDFH is that a major part of the information is transmitted through
carrier frequency selection in the hopping process. In other words, the hopping pattern is
determined by the encrypted message information itself.
Let Nc be the total number of available channels, with {f1 , f2 , · · · , fNc } being the set of
all available carrier frequencies. The number of bits used to specify an individual channel
39

Carrier Frequency
Carrier Bits
Selection

S/P
Encrypted
Partitioning
Information
Baseband Signal
Sequence
Generation
Ordinary Bits

Modulation

(a) Transmitter structure

BPF 1

ABS

BPF 2

ABS

BPF Nh

ABS

Carrier
Local
Detection Oscillator

Table
Look-up

P/S
Merger

Baseband
Demodulation
(b) Receiver structure

Figure 3.2: Transmitter and receiver structure of MDFH, here ABS means taking the
absolute value.
here is Bc = log2 Nc , where x denotes the largest integer less than or equal to x. If
Nc is a power of 2, then there exists a 1-1 map between the Bc -bit strings and the total
available channels; otherwise, when Nc is not a power of 2, we will allow some Bc -bit strings
to be mapped to more than one channel. Without loss of generality, here we assume that
Nc = 2Bc .
Let Ω be the selected constellation that contains M symbols, each symbol in the constellation represents Bs = log2 M bits. Let Ts and Th denote the symbol period and the hop
Ts
. We
duration, respectively, then the number of hops per symbol period is given by Nh =
Th
assume that Nh is an integer larger than or equal to one.
The transmitter and the receiver structure of MDFH is shown in Figure 3.2. We start
by dividing the encrypted information stream into blocks of length L

Nh Bc + Bs . Each

block is parsed into Nh Bc carrier bits and Bs ordinary bits. The carrier bits are used to
determine the hopping frequencies, and the ordinary bits are mapped to a symbol which
is transmitted through the selected channels successively. Denote the nth block by Xn , as

40

illustrated in Figure 3.1. Note that in MDFH, the whole block Xn is transmitted within one
symbol period.
At the receiver, the transmitting frequency is captured using a filter bank as in the FSK
receiver rather than using the frequency synthesizer. Recall that {f1 , f2 , · · · , fNc } is the
set of all available carrier frequencies. To detect the active frequency band, a bank of Nc
bandpass filters (BPF), each centered at fi (i = 1, 2, · · · , Nc ), is deployed at the receiver front
end, followed by a demodulator equipped with matched filter and sampler. At each hopping
period, the carrier frequency (hence the information embedded in frequency selection) can
be blindly detected at each hop. The ordinary bits can then be extracted from the selected
channels through the regular demodulation process.
In a jamming-free environment, carrier frequency detection can be performed by selecting
the channel with maximum received power. However, when jamming is present, the Jamming
detection algorithm can be incorporated at the receiver to improve the system performance.
One way to do this is to use a threshold based detector as follows [80]:
1. Classify each channel according to the power detection threshold η. Let Pi denote the
received signal power over the ith channel. If Pi < η, then there is only noise over
channel i, we say that the channel is inactive; otherwise, if Pi ≥ η, we say that the
channel is active.
2. Let Ia = {i|Pi > η, i = 1, 2, · · · , Nc } be the set of the index of all the active channels,
ˆ
then the estimated hopping frequency index, denoted by k, is determined by
ˆ
k = arg min {Pi }.
i∈Ia

(3.1)

ˆ
The carrier bits can be recovered based on k; the ordinary bits are extracted following
the regular demodulation process.

41

3.2.2

Performance of MDFH under Hostile Jamming

First, we introduce the concept of disguised jamming. Disguised jamming denotes the case
where the jamming is highly correlated with the signal, and has a power level close or equal
to the signal power. More specifically, let s(t) and J(t) be the user’s signal and jamming
interference, respectively. Define
ρ=

t2
1
√
s(t)J ∗ (t)dt
T0 Ps PJ t1

(3.2)

as the normalized cross-correlation coefficient of s(t) and J(t) over the time period [t1 , t2 ],
where T0 = t2 − t1 ,
Ps =

t2
1
|s(t)|2 dt
T0 t1

PJ =

t2
1
|J(t)|2 dt.
T0 t1

and

We say that J(t) is a disguised jamming to signal s(t) over [t1 , t2 ] if
1. J(t) and s(t) are highly correlated.

More specifically, |ρ| > ρ0 , where ρ0 is an

application-oriented, predefined correlation threshold.
P
2. The jamming to signal ratio (JSR) is close to 0dB. More specifically, | J − 1| < P ,
Ps
where P is an application-oriented, predefined jamming-signal ratio threshold.
In this dissertation, we consider disguised jamming over each hopping period, that is,
[t1 , t2 ] = [mTh , (m + 1)Th ] for some integer m. In the worst case, the constellation Ω and the
pulse shaping filter of the information signal are known to the jammer, the jammer can then
disguise itself by transmitting symbols from Ω over a fake channel using the same power
level. That is, J(t) = eiθ s(t) for some phase θ.
We compare the performance of MDFH with that of conventional FH in AWGN channels,
under both noise jamming and disguised jamming. The result (with no channel coding) is
P
shown in Figure 3.3. The jamming-to-signal ratio is defined as JSR = J , where PJ and
Ps
42

2

10

conventional FH: Disguised Jamming
conventional FH: Noise Jamming
MDFH: Disguised Jamming
MDFH: Noise Jamming

0

BER

10

−2

10

−4

10

−20

−10

0
JSR(dB)

10

20

Figure 3.3: Performance comparison under single band jamming, Eb /N0 = 10dB,
Nc = 64, Nh = 3. MDFH uses QPSK modulation and conventional FH uses 4-FSK
modulation. In this case, the spectral efficiency of MDFH is roughly 3.3 times that of
conventional FH.
Ps denote the jamming power and signal power per hop, respectively. As can be seen,
MDFH delivers excellent performance under strong jamming (i.e., JSR

1) scenarios, and

outperforms conventional FH by big margins. Note that in this case, the spectral efficiency
of MDFH is 3.3 times that of conventional FH. The underlying argument is that: when the
jamming power is much stronger than the signal power, jamming can be easily distinguished
from the true signal when they are in different bands; even if jamming collides with the
signal, the true carrier frequency can still be detected as jamming can even enhance the
power of the jammed channel and hence increases the correct detection probability. For
conventional FH, on the other hand, once the jamming power reaches a certain level, the
system performance is mainly limited by the probability that the signal is jammed.

43

However, we also notice that under disguised jamming, the system experiences considerable performance losses, since it is difficult for the MDFH receiver to distinguish jamming
from the true signal. The sensitivity of MDFH to disguised jamming is influenced by the
SNR. To enhance the jamming resistance of MDFH under disguised jamming, we introduce
the anti-jamming MDFH (AJ-MDFH) system.

3.3
3.3.1

Anti-jamming MDFH (AJ-MDFH) System
Transmitter Design

The main idea here is to insert some signal identification (ID) information during the transmission process. This secure ID information is generated through a cryptographic algorithm
using the shared secret between the transmitter and the receiver, and can be used by the receiver to locate the true carrier frequency. Our design goal is to reinforce jamming resistance
without sacrificing too much on spectral efficiency.

Encrypted
Carrier Frequency
Information Channel
Interleaving
Coding
Selection
Initial
Modulation
Vector, Key PN Sequence
Symbol
Baseband Signal
Encryption
Generation
Mapper
Generation
Secure ID Generation
Figure 3.4: AJ-MDFH transmitter structure.

The transmitter structure of AJ-MDFH is illustrated in Figure 3.4. Each user is assigned
a secure ID sequence. We propose to replace the ordinary bits in MDFH with the ID bits.
In order to prevent impersonate attack, each user’s ID sequence needs to be kept secret from
the malicious jammer. The ID sequence can be generated using two steps as in [81]:
1. Generate a pseudo-random binary sequence using a 42-bit linear feedback shift register

44

(LFSR) specified by the characteristic polynomial
x42 + x35 + x33 + x31 + x27 + x26 + x25 + x22 + x21 + x19
+x18 + x17 + x16 + x10 + x7 + x6 + x5 + x3 + x2 + x + 1.
2. Take the output of LFSR as the plaintext, group it into blocks of length KL bits
(KL = 128, 192 or 256), and feed it into the Advanced Encryption Standard (AES)
[76] encrypter of key size KL . The AES output is then used as our ID sequence.
Recall that Bc = log2 Nc and Bs = log2 M , where Nc is the number of channels, and M
is the constellation size. We divide the source information into blocks of size Bc and divide
the ID sequence into blocks of size Bs . Denote the nth source information block and ID bits
block as Xn and Yn , respectively. Let fXn be the carrier frequency corresponding to Xn
and sn the symbol corresponding to ID bit-vector Yn . The transmitted signal can then be
represented as
s(t) =

√

∞

sn g(t − nTh )e

2Re

j2πfXn t

n=−∞

=

√

2Re


 ∞


Nc

αi,n sn g(t − nTh )ej2πfi t

n=−∞ i=1




,

(3.3)



where Th is the duration of each hop, g(t) is the pulse shaping filter,


 1 if f
Xn = fi ,
αi,n =
 0 otherwise.


3.3.2

Receiver Design

The receiver structure for AJ-MDFH is shown in Figure 3.5. The receiver regenerates the
secure ID through the shared secret (including the initial vector, the LFSR information
and the key). For each hop, the received signal is first fed into the bandpass filter bank.
The output of the filter bank is first demodulated, and then used for carrier bits (i.e., the
information bits) detection.
45

BPF 1
BPF 2

Demodulation

BPF Nc
Initial
Vector, Key

Recovered
Signal
Detection Information
&
Extraction

Secure ID
Generation

Symbol
Mapper

Figure 3.5: AJ-MDFH receiver structure.
3.3.2.1

Demodulation

Let s(t), J(t) and n(t) denote the ID signal, the jamming interference and the noise, respectively. For AWGN channels, the received signal can be represented as
r(t) = s(t) + J(t) + n(t).

(3.4)

We assume that s(t), J(t) and n(t) are independent of each other.
For i = 1, 2, · · · , Nc , the output of the ith ideal bandpass filter fi (t) is ri (t) = fi (t) ∗ r(t).
For demodulation, ri (t) is first shifted back to the baseband, and then passed through a
matched filter. At the nth hopping period, for i = 1, · · · , Nc , the sampled matched filter
output corresponds to channel i can be expressed as
ri,n = αi,n sn + βi,n Ji,n + ni,n ,

(3.5)

where sn , Ji,n and ni,n correspond to the ID symbol, the jamming interference and the
noise, respectively; αi,n , βi,n ∈ {0, 1} are binary indicators for the presence of ID signal and
jamming, respectively. Note that the true information is carried in αi,n .

3.3.2.2

Signal Detection and Extraction

Signal detection and extraction is performed for each hopping period. For notation simplicity, without loss of generality, we omit the subscript n in (3.5). That is, for a particular
46

hopping period, (3.5) is reduced to:
ri = αi s + βi Ji + ni ,

for i = 1, · · · , Nc .

(3.6)

Define r = (r1 , . . . , rNc ), α = (α1 , . . . , αNc ), β = (β1 , . . . , βNc ), J = (J1 , . . . , JNc ) and
n = (n1 , . . . , nNc ), then (3.6) can be rewritten in vector form as:
r = sα + β · J + n.

(3.7)

For single-carrier AJ-MDFH, at each hopping period, one and only one item in α is
nonzero. That is, there are Nc possible information vectors: α1 = (1, 0, . . . , 0), α2 =
(0, 1, . . . , 0), · · · , αNc = (0, 0, . . . , 1). If αk is selected, and the binary expression of k is
b0 b1 · · · bBc −1 , with Bc = log2 Nc , then the estimated information sequence is b0 b1 · · · bBc −1 .
At each hopping period, the information symbol α, or equivalently, the hopping frequency
index k, needs to be estimated based on the received signal and the ID information which
is shared between the transmitter and the receiver. When the input information vectors are
1
for i = 1, 2, . . . , Nc , the MAP (maximum a posteriori
equiprobable, that is, P (αi ) =
Nc
probability) detector is reduced to the ML (maximum likelihood) detector. For the ML
ˆ
detector, the hopping frequency index k can be estimated as:
ˆ
k = arg max P r{r|αi }.
1≤i≤Nc

(3.8)

Recall that the information signal, the ID signal, the jamming interference and the noise
are independent to each other. When n1 , . . . , nNc , J1 , . . . , JNc are all statistically independent, r1 , . . . , rNc are also independent. In this case, the joint ML detector in (3.8) can be
decomposed as:
ˆ
k = arg max

Nc

1≤i≤Nc
j=1

P r{rj |αi }

Nc

P r{rj |αj = 0} · P r{ri |αi = 1}

= arg max

1≤i≤Nc

j=1,j=i
Nc

= arg max

1≤i≤Nc
j=1

P r{rj |αj = 0} ·

47

P r{ri |αi = 1}
.
P r{ri |αi = 0}

(3.9)

Nc

P r{rj |αj = 0} is independent of i, (3.9) can be further reduced to the likelihood

Since
j=1

ratio test
P r{ri |αi = 1}
ˆ
k = arg max
,
1≤i≤Nc P r{ri |αi = 0}
where P r{ri |αi = 1} =

(3.10)

P r{ri |αi = 1, βi }P (βi ) and P r{ri |αi = 0} =
βi

P r{ri |αi =
βi

0, βi }P (βi ), with βi ∈ {0, 1}. In the ideal case when βi is known for i = 1, · · · , Nc , the ML
detector above can be further simplified. If we assume that n1 , · · · , nNc are i.i.d. circularly
2
symmetric Gaussian random variables of zero-mean and variance σn , and J1 , · · · , JNc are
2
i.i.d. circularly symmetric Gaussian random variables of zero-mean and variance σJ , then
i
it follows from (3.6) and (3.10) that
2

r −s
exp{− i 2 }
σ

i
ri 2
exp{− 2 }
σi
ri 2 − ri − s 2
arg max
,
2
1≤i≤Nc
σi

ˆ
k = arg max

1≤i≤Nc

=

1
2
πσi

1
2
πσi

(3.11)

2
2
2
where σi = βi σJ + σn .
i
2
2
Note that σi is generally unknown. If we replace the overall interference power σi with

the instantaneous power of the received signal ri 2 , then it follows from (3.11) that:
ri 2 − ri − s 2
ri 2
1≤i≤Nc
ri − s 2
= arg min
.
ri 2
1≤i≤Nc

ˆ
k = arg max

(3.12)

For more tractable theoretical analysis, we can replace ri 2 with average signal power
ri − s
√
observed in channel i, Pi = E{ ri 2 }. Define Zi
, then we have
Pi
ˆ
k = arg min Zi .
1≤i≤Nc

48

(3.13)

3.4

ID Constellation Design and its Impact on System Performance

For AJ-MDFH, ID signals are introduced to distinguish the true information channel from
disguised channels invoked by jamming interference. In this section, we investigate ID constellation design and its impact on the performance of AJ-MDFH under various jamming
scenarios.

3.4.1

Design Criterion and Jamming Classification

The general design criterion of the ID constellation is to minimize the probability of error
under a given signal power. Under this criterion, the following questions need to be answered:
(1) How does the size of the constellation impact the system performance? (2) How does
the type or shape of the constellation influence the detection error? Which type should we
use for optimal performance? In this section, we will try to address these questions under
different jamming scenarios.
In literature, jamming has generally been modeled as Gaussian noise [77, 78, 79], referred
as noise jamming. Recall that disguised jamming denotes the jamming interference which
has similar power and spectral characteristics as that of the true signal. For AJ-MDFH,
when the ID constellation is known to, or can be guessed by the jammer, the jammer can
then disguise itself by sending symbols taken from the same constellation over a different or
fake channel. In this case, it could be difficult for the receiver to distinguish the true channel
from the disguised channel, leading to high detection error probability. We refer to this kind
of jamming as ID jamming or ID attack. That is, ID jamming is the worst case disguised
jamming for AJ-MDFH.

49

3.4.2

Constellation Design under Noise Jamming

Without loss of generality, we consider the case where the ID symbol is transmitted through
channel 1, i.e.,
α = (α1 , · · · , αNc ) = (1, 0, · · · , 0).

(3.14)

Recall that for i = 1, · · · , Nc , ri = αi s + βi Ji + ni . Let ni = βi Ji + ni , and denote its variance
˜
2
2
2
2
as σi = βi σJ + σn . σi may vary from channel to channel. Following the definition in (3.13),
i
n −s
˜
n1
˜
for 2 ≤ i ≤ Nc . It can be seen that Z1 is a
, and Zi = i
we have Z1 =
σi
2
s 2 + σ1
Rayleigh random variable with probability density function (PDF)
z2

z − 1
pZ1 (z1 ) = 1 e 2σ2 ,
σ2
where

σ2

z1 > 0,

(3.15)

2
σ1
=
. For 2 ≤ i ≤ Nc , Zi is a Rician random variable with PDF
2
2( s 2 + σ1 )
2

2

z +ν
zi − i 2
zν
pZi (zi ) = 2 e 2σ I0 i2 ,
σ
σ

where ν =

zi > 0,

(3.16)

1
s
, σ = √ and I0 (x) is the modified Bessel function of the first kind with
σi
2

order zero.
According to (3.13), the carrier can be correctly detected if and only if Z1 < Zi for all
2 ≤ i ≤ Nc . Assuming that the symbols in constellation Ω are equally probable, then the
carrier detection error probability is given by
Pe = 1 −

P r{Z1 < Z2 , . . . , Z1 < ZNc |s}pS (s)
s∈Ω

1
= 1−
|Ω|

∞ Nc
s∈Ω 0

i=2

P r{Zi > z1 |s, Z1 = z1 }pZ1 (z1 )dz1 .

Note that Z2 , · · · , ZNc are i.i.d. Rician random variables, then it follows from (3.15) and
(3.16) that

Pe = 1 −

1
|Ω|

√

∞ Nc
s∈Ω 0

Q1
i=2

2

σi

√
s , 2z1

50

2( s

2
+ σ1 ) −
z1 e
2
σ1
2

2
s 2 +σ1 2
z1
2
σ1
dz1 ,

(3.17)

where Q1 is the Marcum Q-function [82]. We have the following result:
Proposition 1 Assuming the true channel index is k. Under noise jamming, an upper
bound of the carrier detection error probability Pe can be obtained as:

Nc −1 

2
s 2 ( s 2 +σk )
− 2


2

σk
1


σ ( s 2 +2σ 2 ) 
U
k 
e m
Pe =
1 − 1 −
,



2 + 2σ 2
|Ω|


s

(3.18)

k

s∈Ω

2
where m = arg max{σl } for 1 ≤ l ≤ Nc , l = k.

Proof : See Section 3.9.1.

2

s 2
1 −ζ x +x Nc −1
and a(x) = 1−(1−
e x+2 )
.
2
x+2
σk
U
U
The upper bound of the detection error probability, Pe can be written as Pe = a(x) with
2 +x
(Nc − 1) −ζ x
2 2
e x+2
a(x). We now show that
˜
ζ = σk /σm . Note that when x
1, a(x) ≈
x+2
when x
1, a(x) is a convex function. The first and second derivative of a(x) can be
˜
˜
Assuming the true channel index is k, let x =

obtained as
2

ζx2 + (4ζ + 1)x + (2ζ + 2) −ζ x +x
a (x) = −(Nc − 1)
˜
e x+2
(x + 2)3

(3.19)

2

ζx2 + (4ζ + 2)x + (4 − 2ζ) −ζ x +x
e x+2
a (x) = (Nc − 1)
˜
(x + 2)4
2

ζ(x2 + 4x + 2)[ζx2 + (4ζ + 1)x + (2ζ + 2)] −ζ x +x
+(Nc − 1)
e x+2
(x + 2)5

(3.20)

Note that the second term of a (x) is always positive when x > 0. Note that the quadratic
˜
equation ζx2 + (4ζ + 2)x + (4 − 2ζ) = 0 with discriminant ∆ = 6ζ 2 + 1 > 0 has two
1/ζ 2 + 6 − (1/ζ + 2) denote the larger real root. Define
y
c(y)
y 2 + 6 − (y + 2) such that x0 = c(1/ζ). Note that c (y) =
−1 < 0
y2 + 6
√
√
and c(0) = 6 − 2. Since 1/ζ > 0, x0 = c(1/ζ) < c(0). Hence for x > 6 − 2 > x0 ,
distinct real roots. Let x0 =

ζx2 + (4ζ + 2)x + (4 − 2ζ) > 0 and a (x) is positive. Thus we proved that a(x) is convex
˜
˜
√
when x > 6 − 2.

51

By Jensen’s inequality [83], we have
U
Pe ≈

1
|Ω|

a
˜
s∈Ω

s
2
σk





2

≥ a
˜

1
2
|Ω|σk

˜
s 2 = a
s∈Ω

Ps
2
σk

.

(3.21)

The equality is achieved if and only if s 2 = Ps for all s ∈ Ω. This implies that: under the
s 2
U
condition that the signal to jamming and noise ratio over channel k satisfies 2
1, Pe
σk
is approximately minimized when the constellation is constant modulus, that is, s 2 = Ps
for all s ∈ Ω.
An intuitive explanation for this result is that the signal power in constant modulus
constellations is always equal to the maximal signal power available. Furthermore, it can be
U
2
seen that Pe is independent of the constellation size |Ω|, but is only a function of Ps /σk .

Next, we will investigate how the constellation size affects the system performance under ID
jamming.

3.4.3

Constellation Design under ID Jamming

Theorem 1 For a given SNR and assuming PSK constellation is utilized, under ID jamming, the carrier detection error probability Pe is a function of constellation size M and
¯
lim Pe (M ) = Pe .

M →∞

In other words, for any given

(3.22)

> 0, there always exists an Mt such that for all M > Mt ,

¯
|Pe (M ) − Pe | < .
¯
The expression of Pe and the proof of the theorem can be found in Section 3.9.2. This
theorem essentially says that: for a given SNR, due to the noise effect, increasing the constellation size over a threshold Mt will result in little improvement in detection error probability.
This result justifies the use of finite constellation in AJ-MDFH.

52

3.5

Multi-carrier AJ-MDFH

For more efficient spectrum usage and more robust jamming resistance, in this section, we
extend the concept of MDFH to multi-carrier AJ-MDFH (MC-AJ-MDFH). The transmitter
and the receiver structure of the MC-AJ-MDFH system are illustrated in Figure 3.6. The
idea is to split all the Nc channels into Ng non-overlapping groups, and each subcarrier hops
within the assigned group based on the AJ-MDFH scheme. To ensure hopping randomness
of all the subcarriers, the groups need to be reorganized or regenerated securely after a
pre-specified period, named group period. In the following, we will first describe the secure
group generation algorithm, and then discuss the design of MC-AJ-MDFH with and without
additional frequency diversity.

3.5.1

Secure Group Generation

In this section, we propose a secure group generation algorithm to ensure that: (i) Each
subcarrier hops over a new group of channels during each group period, so that it eventually
hops over all the available channels in a pseudo-random manner; (ii) Only the legitimate
receiver can recover the transmitted information correctly. Secure group generation is synchronized at the transmitter and the receiver. At the receiver, the received signal is fed to a
bank of single-carrier AJ-MDFH receivers for signal extraction and recovery.
Recall that we assume there are a total of Nc available channels and there are Ng subcarriers in the system. For l = 0, · · · , Ng − 1, the number of channels assigned to subcarrier
i
i is denoted as Ng . As different subcarriers transmit over non-overlapping set of channels
Ng
i
Ng = Nc .

and we have
i=0

For secure group generation, first, generate a pseudo-random binary sequence using a
32-bit linear feedback shift register (LFSR) as in Section 3.3, which is initialized by a secret
sequence shared between the transmitter and the receiver. Encrypt the generated sequence
Nc
into a ciphertext using the AES algorithm and a secure key. Pick an integer L ∈ [ , Nc ]
2
53

Channel Coding
and Interleaving

Subcarrier Selection
Over Group 1

S/P

Encrypted
Information

Channel Coding Subcarrier Selection
and Interleaving Over Group Ng
Initial
Vector, Key
Secure Group Generation
Initial
Vector, Key Secure ID
Generation

Modulation

Symbol Mapping &
Baseband Signal Generation
(a) Transmitter structure

BPF 1
Secure
Group Deassignment

BPF 2

BPF Nc
Initial
Vector, Key Secure ID
Initial
Generation
Vector, Key

Group 1
Subcarrier Detection
Group Ng
Subcarrier Detection

Recovered
Message Re- Information
construction

Symbol Mapper
(b) Receiver structure

Figure 3.6: Transmitter and receiver structure of MC-AJ-MDFH.
and let q = L log2 Nc = LBc . Take q bits from the ciphertext and put them as a q-bit vector
e = [e1 , e2 , · · · , eq ].
Second, partition the ciphertext sequence e into L groups, such that each group contains
Bc bits. For k = 1, 2, · · · , L, the partition of the ciphertext is as follows
pk = [e(k−1)∗Bc +1 , e(k−1)∗Bc +2 , · · · , e(k−1)∗Bc +Bc ],

(3.23)

where pk corresponds to the kth Bc -bit vector.
For k = 1, 2, · · · , L, denote Pk as the decimal number corresponding to pk . And denote
P = [P1 , P2 , · · · , PL ] as the permutation index vector. For k = 0, 1, 2, · · · , L, denote Ik =
54

[Ik (0), Ik (1), · · · , Ik (Nc − 1)] as the index vector at the kth step. The secure permutation
scheme of the index vector is achieved through the following steps:
0. Initially, the index vector is I0 = [I0 (0), I0 (1), · · · , I0 (Nc − 1)] and the permutation
index is P = [P1 , P2 , · · · , PL ]. We start with I0 = [0, 1, · · · , Nc − 1].
1. For k = 1, switch I0 (0) and I0 (P1 ) in index vector I0 to obtain I1 . In other words,
I1 = [I1 (0), I1 (1), · · · , I1 (Nc − 1)], where I1 (0) = I0 (P1 ), I1 (P1 ) = I0 (0), and I1 (m) =
I0 (m) for m = 0, P1 .
2. Repeat the previous step for k = 2, 3, · · · , L. In general, if we already have Ik−1 =
[Ik−1 (0), Ik−1 (1), · · · , Ik−1 (Nc −1)], then we can obtain Ik = [Ik (0), Ik (1),· · · , Ik (Nc −
1)] through the permutation defined as Ik (k − 1) = Ik−1 (Pk ), Ik (Pk ) = Ik−1 (k − 1),
and Ik (m) = Ik−1 (m) for m = k − 1, Pk .
3. After L steps, we obtain the channel center frequency vector as FL = [fI (0) , fI (1) , · · · ,
L
L
fI (Nc −1) ].
L
4. Vector FL is used to assign the channels to Ng groups. We assign channels {fI (0)
L
, fI (1) , · · · , fI (N 0 −1) } to the first group; Assign {fI (N 0 ) , fI (N 0 +1) , · · · , fI (N 0 +N 1 −1) }
L
g
L g
L g
L g
L g
to the second group, and so on.
Because each frequency index appears in FL once and only once, the proposed algorithm
ensures that all the subcarriers are transmitting on non-overlapping sets of channels. In the
following, we illustrate the secure group generation algorithm though a simple example.
Example:

Assume the total number of available channels is Nc = 8, to be equally

divided among Ng = 2 subcarriers; the permutation index vector P = [4, 7, 4, 0], and the
initial index vector I0 = [0, 1, 2, 3, 4, 5, 6, 7], as shown in Figure 3.7. Note that, the initial
index vector I0 can contain any random permutation of the sequence {0, 1, · · · , Nc − 1}, and
Nc
Nc
.
L ∈ [ , Nc ]. In this example, we choose L =
2
2

55

Figure 3.7: Example of the Secure Permutation Algorithm for Nc = 8 channels and Ng = 2
subcarriers.
At Step 1, k = 1, and Pk = 4, thus we switch I0 (Pk ) and I0 (k − 1) of the index vector
I0 . After the switching, we obtain a new index vector I1 = [4, 1, 2, 3, 0, 5, 6, 7].
At Step 2, k = 2, and Pk = 7, thus we switch I1 (Pk ) and I1 (k − 1) of the index vector
I1 . We obtain the new index vector I2 = [4, 7, 2, 3, 0, 5, 6, 1]. Below are the remaining index
vectors for k = 3, 4:
I3 = [4, 7, 0, 3, 2, 5, 6, 1], I4 = [3, 7, 0, 4, 2, 5, 6, 1].
The channel frequency vector is F4 = [fI (0) , fI (1) , · · · , fI (Nc −1) ]. Channels {f3 , f7 , f0 , f4 }
4
4
4
are assigned to subcarrier 0 and channels {f2 , f5 , f6 , f1 } are assigned to subcarrier 1.

3.5.2

Multi-Carrier AJ-MDFH without Diversity

In this case, each subcarrier transmits independent bit stream. The spectral efficiency of
the AJ-MDFH system can be increased significantly. Let Bc = log2 Nc and Bg = log2 Ng ,
56

then the number of bits transmitted by the MC-AJ-MDFH within each hopping period is
BM C = (Bc − Bg )Ng = (Bc − log2 Ng )Ng . BM C is maximized when Bg = Bc − 1 or
Bg = Bc − 2, which results in BM C = 2Bc −1 . Note that the number of bits transmitted by
the AJ-MDFH within each hopping period is Bc , it can be seen that BM C > Bc as long as
Bc > 2. Take Nc = 256 for example, then the transmission efficiency of AJ-MDFH can be
2Bc −1
B
= 16 times.
increased by M C =
Bc
Bc
We assume that jamming is random and equally distributed among all the groups. Then
the overall carrier detection error probability Pe of MC-AJ-MDFH is equal to that corresponding to each subcarrier fk for k = 1, · · · , Ng . Let Pe,k denote the carrier detection error
probability corresponding to the kth subcarrier or the kth group, then we have Pe = Pe,k ,
and
Pe,k = P0,k · P {incorrect carrier detection|signal not jammed}
+ P1,k · P {incorrect carrier detection|signal jammed},

(3.24)

where P0,k , P1,k denote the probability that the kth subcarrier is jamming-free or jammed,
respectively.

3.5.3

Multi-carrier AJ-MDFH with Diversity

Under the multi-band jamming, diversity needs to be introduced to the AJ-MDFH system
for robust jamming resistance especially. A natural solution to achieve frequency diversity
is to transmit the same or correlated information through multiple subcarriers. The number
of subcarriers needed to convey the same information differs in different jamming scenarios.
Ideally, the number of correlated signal subcarriers should not be less than the number of
jammed bands.
At the receiver, the received signals from different diversity branches are often combined
for joint signal detection. To achieve better performance, appropriate diversity combination
schemes need to be selected for different metrics used. In [84], the product combination

57

scheme was used for the square-law metrics. In [85, 86], the equal gain combination scheme
was used for the normalized square-law metrics. If metric Zi (see eq. (3.13)) is used for
MC-AJ-MDFH, we propose to use the equal gain combination scheme, since Zi can also be
regarded as a normalized square-law metric.
Assume that the same information is transmitted through the hopping frequency index
of Nd subcarriers over Nd groups {Gn1 , Gn2 , . . . , GnN } simultaneously, each group has the
d

same number of channels, denoted as Ngc . Note that the secure group generation algorithm
ensures that the channel index in each group is random and does not necessarily come in
n

ascending or descending order. Let Zi l denote the likelihood ratio of ith channel in group
Gnl , then the active hopping frequency index can be estimated as
ˆ
k = arg

Nd

max

1≤i≤Ngc

n

Zi l ,

(3.25)

l=1

The diversity order Nd can be dynamic in different jamming scenarios to achieve tradeoff
between performance and efficiency.
Overall, as demonstrated in Section 3.7, MC-AJ-MDFH can increase the system efficiency and jamming resistance significantly through jamming randomization and frequency
diversity. Moreover, by assigning different carrier groups to different users, MC-AJ-MDFH
can also be used as a collision-free multiple access system.

3.6

Spectral Efficiency Analysis

The spectral efficiency ν is defined as the ratio of the information bit rate Rb to the transmisR
sion bandwidth Wt , i.e., ν = b . In this section, we will analyze and compare the spectral
Wt
efficiency of the existing and proposed frequency hopping schemes, including conventional
FH, MDFH and (MC-)AJ-MDFH.
We start with the single-user case. That is, no multiple access interference needs to be
considered. Recall that Ts and Th denotes the symbol period and the hopping duration,
respectively; Nh = Ts /Th is the number of hops per symbol period. For fair comparison,
58

we assume that all systems have: (i) The same number of available channels Nc ; (ii) The
same hopping period Th to ensure the hopping channels have the same bandwidth Wc =
2/Th ; (iii) The same frequency spacing ∆f between two adjacent subcarriers, where ∆f ≥
1/Th is chosen to avoid inter-carrier interference. Note that under these assumptions, all
systems have the same total bandwidth Wt = (Nc − 1)∆f + Wc . For conventional FH using
MFSK modulation, log2 M bits are transmitted during each symbol period. The bit rate
log2 M
log2 M
=
, and the corresponding
of conventional FH can be calculated as Rb =
Ts
Th Nh
log2 M
R
. The bit rate and spectral
spectral efficiency can be obtained as ν = b =
Wt
Th Nh Wt
efficiency of other frequency hopping schemes can be obtained similarly. The results are
listed in table 3.1. An illustration of the spectral efficiency comparison in the single-user
case can be found in Section 3.7.
Table 3.1: The bit rate and spectral efficiency comparison in the single-user case

Rb (b/s)
ν (b/s/Hz)

conventional FH
log2 M
Th Nh
log2 M
Th Nh Wt

MDFH
Nh Bc + Bs
Nh Th
Nh Bc + Bs
Th Nh Wt

AJ-MDFH
Bc
Th
Bc
Th Wt

MC-AJ-MDFH
(Bc − log2 Ng )Ng
Th
(Bc − log2 Ng )Ng
Th Wt

Next, we consider the more general multiuser case. The multiple access scheme for
conventional FH, namely FHMA, was proposed in [66]. The multiple access extension of
MDFH, denoted as E-MDFH, has been analyzed in [49]. Due to the variability in multiple
access system design, a closed-form expression of the spectral efficiency is hard to obtain.
Here we compare the total information bits allowed to be transmitted by each system under
the same BER and bandwidth requirements, and illustrate the spectral efficiency comparison
through the following example.
Let Nu denote the number of users and we choose the number of channels be Nc = 64
(i.e.,Bc = 6). For MC-AJ-MDFH, we choose Ng = Nu = 4; For E-MDFH, we choose 8-PSK
to modulate Bs = 3 ordinary bits and Ng = Nu = 4, Nh = 3. For FHMA, we choose
64-FSK modulation, Nh = 3 and consider Nu = 2, · · · , 4, respectively. The required BER is
59

10−4 . Figure 3.8 and Figure 3.9 depict the performance of these multiuser systems. From
Figure 3.8, it can be seen that MC-AJ-MDFH and E-MDFH achieve the desired BER at
E
Eb
≈ 6.5dB and b ≈ 6.4dB, respectively. From Firgure 3.9, it can be observed that: due
N0
N0
to severe collision effect among different users, FHMA can only accommodate up to 2 users
E
at b ≈ 6.5dB for the desired BER. In this particular example, the spectral efficiency of
N0
both MC-AJ-MDFH and E-MDFH are approximately 4 times and 5 times that of FHMA.
−1

10

MC−AJ−MDFH
E−MDFH

−2

BER

10

−3

10

−4

10

5

6

7
8
Eb/N0 (dB)

9

10

Figure 3.8: Performance of MC-AJ-MDFH and E-MDFH in multiuser case.

Based on our analysis above, as well as the performance analysis of AJ-MDFH under
various jamming attacks in Section 3.7, it will be shown that: while AJ-MDFH is much
more robust than MDFH under various jamming attacks, it can achieve a spectral efficiency
that is very close to MDFH, which is several times higher than that of conventional FH.
A comprehensive capacity analysis for MDFH and AJ-MDFH under disguised jamming is
provided in Chapter 4.

60

−1

10

Nu=2
N =3
u

−2

N =4

BER

10

u

−3

10

−4

10

5

6

7
8
Eb/N0 (dB)

9

10

Figure 3.9: Performance of FHMA in multiuser case.

3.7

Simulation Results

In this section, simulation examples are provided to illustrate the performances of the proposed AJ-MDFH and MC-AJ-MDFH schemes under various jamming scenarios. For all
the systems considered in the following examples, we assume the total number of available
channels is Nc = 64, that is, Bc = 6.
Example 1: Impact of the ID constellation size In this example, we consider the
impact of the ID constellation size on the BER performance of AJ-MDFH under single-band
ID jamming. From Figure 3.10, it can be seen that in the ideal case where the system is
noise free, the BER performance of AJ-MDFH improves continuously as the constellation
size increases. However, when noise is present, the BER converges once the constellation size
reaches a certain threshold Mt . For example, for Eb /N0 = 15dB, we can choose Mt = 36.
This example demonstrates the theoretical result in Theorem 1. In the following examples,

61

we choose Eb /N0 = 10dB and use 32-PSK to modulate ID signal for AJ-MDFH and MCAJ-MDFH.
0

10

Eb/N0=5dB
Eb/N0=10dB
E /N =15dB
b

−1

0

Noise−free

BER

10

−2

10

−3

10

10

20

30
40
50
60
Constellation size |Ω|

70

80

Figure 3.10: Example 1: The performance of AJ-MDFH with different constellation size,
under single-band ID jamming.

Example 2: Performance comparison under single band jamming In this example, we consider both noise jamming and disguised jamming. SNR is taken as Eb /N0 = 10dB
and Jamming-to-Signal Ratio (JSR) is defined as the ratio of the jamming power to signal
powers during one hop period. For conventional FH, 4-FSK modulation scheme is used and
we assume that the adjacent frequency tones in 4-FSK correspond to the center frequencies
of the adjacent MDFH channels. For MDFH, QPSK is used to modulate ordinary bits and
the number of hops per symbol period is Nh = 3. From Figure 3.11, it can be observed that
AJ-MDFH can effectively reduce the performance degradation caused by disguised jamming,
while remaining robust under noise jamming. Note that JSR=0dB under disguised jamming
corresponds to the ID jamming for AJ-MDFH. It can also be seen that the performance of
62

AJ-MDFH improves significantly when the jamming power differs from the signal power.
This implies that uncertainty in the signal power is another dimension in combating ID
jamming.

2

10

MDFH: Disguised Jamming
MDFH: Noise Jamming
conventional FH: Disguised Jamming
conventional FH: Noise Jamming
AJ−MDFH: Disguised Jamming
AJ−MDFH: Noise Jamming

1

10

0

10

−1

BER

10

−2

10

−3

10

−4

10

−20

−10

0
JSR(dB)

10

20

Figure 3.11: Example 2: Performance comparison under single band jamming.

Example 3: Performance comparison under multi-band disguised jamming In
this example, Eb /N0 = 10dB. For multi-band disguised jamming, the jammed bands are
independently and randomly selected, and the jammer takes symbols randomly from the
same constellation as the ID signal. For MC-AJ-MDFH without diversity, the channels are
divided into 32 groups to maximize the spectral efficiency; for MC-AJ-MDFH with diversity,
each symbol is transmitted simultaneously over 4 subcarriers to achieve frequency diversity.
The equal gain combination scheme is adopted for the joint detection metric at the receiver.

63

From Figure 3.12, it can be seen that MC-AJ-MDFH delivers much better performance than
AJ-MDFH under multi-band disguised jamming.

2

10

MDFH
conventional FH
AJ−MDFH
MC−AJ−MDFH: without diversity
MC−AJ−MDFH: with diversity

0

BER

10

−2

10

−4

10

−6

10
−20

−10

0
JSR(dB)

10

20

Figure 3.12: Example 3: Performance comparison under 2-band disguised jamming.

3.8

Summary

In this chapter, we proposed a highly efficient anti-jamming scheme, AJ-MDFH, based on
the message-driven frequency hopping technique. It was shown that by inserting a secure
ID sequence in transmission, AJ-MDFH can effectively reduce the performance degradation
caused by disguised jamming, while being robust under strong jamming. The impact of ID
constellation design on system performance was investigated under both noise jamming and
ID jamming. It was proved that for a given power constraint, constant modulus constellation

64

delivers the best results under noise jamming in terms of detection error probability; While
under ID jamming, when noise is present, the detection error probability converges as the
constellation size goes to infinity. Moreover, AJ-MDFH can be extended to MC-AJ-MDFH
by allowing simultaneous multi-carrier transmission. With jamming randomization and enriched frequency diversity, MC-AJ-MDFH can increase the system efficiency and jamming
resistance significantly, and can readily be used as a collision-free multiple access scheme.

3.9

Proofs of Chapter 3

3.9.1

Proof of Proposition 1

It follows from (3.17) that
1
Pe = 1 −
|Ω|
≤ 1−

1
|Ω|

√

∞ Nc

Q1

s∈Ω 0

l=2
∞

s∈Ω 0

QNc −1
1

√
s , 2z1 pZ1 (z1 )dz1
σl
√
√
2
s , 2z1 pZ1 (z1 )dz1
σm
2

(3.26)

2
where m = arg max {σl }. The inequality follows from the fact that for fixed s and z1 ,
2≤l≤Nc
√
√
2
s , 2z1 is a monotonically decreasing function with respect to σl . The equality
Q1
σl
can be achieved when σ2 = · · · = σNc .
U
Assume Nc ≥ 2. For Nc = 2, it is easy to show that Pe = Pe with σm = σ2 . Note that
√
√
2
s , 2z1 = P r{Zm > z1 |s, z1 = Z1 } is a function of z1 . And
for fixed s and σm , Q1
σm

for Nc > 2, f (x) = xNc −1 is convex when x > 0. By Jensen’s inequality, we obtain
√
Nc −1
∞
∞
√
2
Nc −1
Q1
P r{Zm > z1 |s, Z1 = z1 }pZ1 (z1 )dz1
s , 2z1 pZ1 (z1 )dz1 ≥
σm
0
0
= [P r{Z1 < Zm |s}]Nc −1 .

(3.27)

According to [87], P r{Z1 < Zm |s} can be calculated as
P r{Z1 < Zm |s} = 1 −

s

2
s 2 ( s 2 +σ1 )
− 2
2
2
2
σ1
e σm ( s +2σ1 ) .
2 + 2σ 2
1

65

(3.28)

Following (3.26)-(3.28),


Pe ≤

1
|Ω|

s∈Ω

Nc −1
2
s 2 ( s 2 +σ1 )
− 2

2 
2

e σm ( s +2σ1 ) 





2

σ1

1 − 1 −

2

s 2 + 2σ1







U
= Pe for Nc > 2. (3.29)

U
Overall, Pe ≤ Pe for Nc ≥ 2.

3.9.2

Proof of Theorem 1

Note that the system is under ID attack. If the power of the M -ary PSK constellation is Ps ,
then the signal and jamming symbol can be written as s =

Ps e

j 2πms
M

, Jj =

2πm
j MJ

Ps e

respectively, where M = |Ω| and 0 ≤ ms , mJ ≤ M − 1. Without loss of generality, we
assume that: (i) Both the ID and jamming take the symbols in Ω with equal probability
1/M ; (ii) The signal is transmitted in channel 1 (α1 = 1) and channel j is jammed (βj = 1).
We consider the following two scenarios:
(i) When j = 1, jamming collides with the ID signal. In this case, r1 = s + J1 + n1 and
n −s
J1 + n1
2
, where σn is
and Zl = l
rl = nl for l = 2, . . . , Nc . We have Z1 =
2 + σ2
σn
s + J1
n
the noise variance. The detection error probability in this case can be calculated as
Pe1 = 1 −

1
M2

∞
s∈Ω J1 ∈Ω 0

[P r{z1 < Z2 |s, J1 , z1 }]Nc −1 pZ1 (z1 )dz1 .
2

(3.30)

2

z +ν
z1 − 1 2
z ν
Note that Z1 is a Rician random variable with PDF pZ1 (z1 ) = 2 e 2σ I0 12 , where
σ
σ
√
2
Ps
σn
2 =
ν=
,σ
and Zl ’s are i.i.d. Rician random variables
2
2
2( s + J1 2 + σn )
s √ J1 2 + σn
+
Ps 2 1
with ν =
, σ = . Then (3.30) can be further written as
σn
2
√
∞
2
2Ps √
1
s + J1 2 + σn
Nc −1
Pe1 = 1 −
Q1
, 2z1 2z1
2
σn
|Ω|2
σn
0
s∈Ω J1 ∈Ω
2 2
( s+J1 2 +σn )z1 +Ps
−
2
σn
·e
I0

2z1
2
σn
66

2
Ps ( s + J1 2 + σn ) dz1 .

(3.31)

For M-PSK constellation with power Ps , s + J1 2 can be rewritten as
2πms
2πmJ
2πms
2πmJ
s + J1 2 = Ps cos
+ cos
+ j(sin
+ sin
)
M
M
M
M

2

π(ms + mJ )
π(ms − mJ )
π(ms + mJ )
π(ms − mJ ) 2
cos
+ j sin
cos
M
M
M
M
π(ms − mJ )
= 4Ps cos2
M
2π(ms − mJ )
= 2Ps 1 + cos
(3.32)
M
= 4Ps cos

(ms − mJ ) mod M , which is uniformly distributed over [0, M − 1], then s +
2πκ
) and
J1 2 = 2Ps (1 + cos
M
Define κ

1
Pe1 = 1 −
M

M −1
κ=0

∞
0

√
QNc −1
1

2Ps
2Ps √
, 2z1 2z1
2
σn
σn

2
s
− 2Ps 1+cos 2πκ +1 z1 − P2
2
M
σn
σn I
·e
0

2z1
2
σn

1 + cos

2
2Ps 1 + cos

2πκ
M

2πκ
M

+1

2
+ Ps σn

dz1 ,
(3.33)

(ii) When j = 2, . . . , Nc , jamming does not collide with ID signal. In this case, r1 = s+n1 ,
Jj − s + nj
n1
n −s
rj = Jj + nj and rl = nl . We have Z1 =
, Zj =
and Zl = l
2
2
σn
Ps + σn
Ps + σ n
for l = 2, . . . , Nc , l = j. The detection error probability in this case can be calculated as
Pe2 = 1 −

1
|Ω|2

∞
s∈Ω Jj ∈Ω 0

P r{Zj > z1 |s, Jj , z1 }[P r{Zl > z1 |s, z1 }]Nc −2 pZ1 (z1 )dz1 .
(3.34)

z2
z1 − 12
Note that Z1 is a Rayleigh random variable with PDF pZ1 (z1 ) = 2 e 2σ , where σ 2 =
σ
2
2
Jj − s
σn
σn
, Zj is a Rician random variable with ν =
, σ2 =
and Zl ’s
2
2
2
2(Ps + σn )
2(Ps + σn )
Ps + σn
√
Ps 2
1
are i.i.d. Rician random variables with ν =
, σ = . Following similar derivation as
σn
2

67

2πκ
in (3.32), we have Jj − s 2 = 2Ps (1 − cos
) and (3.34) can be further written as
M
√
∞
2
2
2(Ps + σn )
1
Jj − s ,
z1
Pe2 = 1 −
Q1
σn
σn
|Ω|2
0
s∈Ω J1 ∈Ω

2

√
·QNc −2
1
1
= 1−
M

Ps +σn 2
2
2(Ps + σn ) − σ2 z1
n
dz1
z1 e
2
σn

2Ps √
, 2z1
σn

M −1
κ=0

∞
0

2
σn

Q1

2πκ
1
,
M
σn

2
2(Ps + σn )z1

2

√
·QNc −2
1

Ps 1 − cos

Ps +σn 2
2
2(Ps + σn ) − σ2 z1
n
z1 e
dz1 .
2
σn

2Ps √
, 2z1
σn

(3.35)

The overall detection error probability in noisy environment is given as
1
Nc − 1
Pe1 +
Pe2 .
Nc
Nc

Pe = P r{j = 1}Pe1 + P r{2 ≤ j ≤ Nc }Pe2 =
When

(3.36)

Ps
is fixed, it follows from (3.33) and (3.35) that Pe is a function of M given as
2
σn
Pe =

1
M

M −1

b
κ=0

2πκ
M

,

(3.37)

where
∞
1
QNc −1
b(x) = 1 −
1
Nc 0

√

2Ps √
2Ps
(1 + cos x) + 1
, 2z1 2z1
2
σn
σn

2
s
− 2Ps (1+cos x)+1 z1 − P2
2
σn
σn I
·e
0

Nc − 1 ∞
2
Q1
Nc
σn
0
√
2Ps √
Nc −2
, 2z1
·Q1
σn
−

2z1
2
σn

2
2
2Ps (1 + cos x) + Ps σn dz1

Ps (1 − cos x),

1
σn

2
2(Ps + σn )z1

P +σ 2 2
2)
− s 2 n z1
2(Ps + σn
σn
z1 e
dz1 .
2
σn

(3.38)

As M approaches infinity, Pe converges. In fact, we have,
b 2πκ · 2π
M
M
¯
Pe = lim Pe = lim
2π
M →∞
M →∞
M·M
κ=0
M −1

=
=

1
lim
2π M →∞
1
2π 0

M −1

b
κ=0

2πκ
M

·

2π
M

2π

b(x)dx.

68

(3.39)

Note that b(x) is the detection error probability with 0 ≤ b(x) ≤ 1. We have
¯
0 ≤ Pe =

2π
1
b(x)dx ≤ 1.
2π 0

(3.40)

¯
That is: ∀ > 0, there always exists an integer Mt such that ∀M > Mt , |Pe − Pe | < .

69

Chapter 4
CAPACITY ANALYSIS OF MDFH BASED SYSTEMS UNDER DISGUISED
JAMMING

In Chapter 3, we point out that under disguised jamming, where the jammer mimics the
signal of the legal user, MDFH experiences considerable performance losses like other wireless
systems. To overcome this, we propose an anti-jamming MDFH scheme, which enhances the
jamming resistance of MDFH by enabling shared randomness between the transmitter and
receiver using an AES generated ID sequence transmitted along the information stream. In
this chapter, we analyze the capacity of MDFH and AJ-MDFH under disguised jamming.
We show that under the worst case disguised jamming, as long as the secure ID sequence
is unavailable to the jammer (which is ensured by AES), the AVC corresponding to AJMDFH is nonsymmetrizable. This implies that the deterministic capacity of AJ-MDFH
with respect to the average probability of error is positive. On the other hand, due to lack
of shared randomness, the AVC corresponding to MDFH is symmetric, resulting in zero
deterministic capacity. We further calculate the capacity of AJ-MDFH under the worst case
disguised jamming, and show that it converges as the ID constellation size goes to infinity.

4.1

Introduction

The basic idea of MDFH is that part of the information message acts as the PN sequence
for carrier frequency selection at the transmitter. Transmission through hopping frequency
control adds another dimension to the signal space, and the resulted coding gain can increase the system spectral efficiency significantly. In Chapter 3, we pointed out that MDFH
is sensitive to disguised jamming, where the jammer mimics the signal of the legal user.
Performance losses occur since it is difficult for the MDFH receiver to distinguish the disguised jamming from the signal. To overcome this limitation, we proposed the anti-jamming

70

MDFH (AJ-MDFH) scheme. The idea is to insert some secure signal identification (ID)
information during the transmission process. The ID information can be regenerated at the
receiver based on the pre-shared secret, and then be used for signal detection and extraction.
We further explored ID constellation design and its impact on the performance of AJMDFH. It was observed that constant modulus constellation would result in minimum probability of error under noise jamming, as the signal power is always equal to the maximal
signal power available. Under the worst case disguised jamming, in which the jamming tries
to mimic the ID symbols of the legal user (referred to as ID jamming [88]), we showed that
under the ideal case when the system is noise-free, increasing the ID constellation size can
increase the ID uncertainty, and hence reduce the probability of error. In this case, the ideal
constellation size is M = ∞. However, when noise is present, we proved that the detection
error probability under ID jamming converges as M goes to infinity. This result justifies the
use of practical, finite size constellations in AJ-MDFH.
In this chapter, we analyze the capacity of MDFH and AJ-MDFH under disguised jamming. Both MDFH and AJ-MDFH are modeled as arbitrarily varying channels (AVCs)
[89, 90, 91, 53, 92, 93], which is characterized as
W : X × J → S,

(4.1)

where X is the transmitted signal space, J is the jamming space and S is the estimated
information space. For any x ∈ X , J ∈ J and s ∈ S, W (s|x, J) denotes the conditional
probability that s is detected at the receiver, given that x is the transmitted signal and J is
the jamming. If J = X and W (s|x, J) = W (s|J, x) for any x, J ∈ X , s ∈ S, the AVC is said
ˆ
to have a symmetric kernel [51]. Define W : X × X → S by
ˆ
W (s|x, J)

π(y|J)W (s|x, y),

(4.2)

y∈Y

where π : X → Y is a probability matrix, and Y ⊆ J . If there exists a π such that
ˆ
ˆ
W (s|x, J) = W (s|J, x), ∀x, J ∈ X , ∀s ∈ S,
71

(4.3)

then W is said to be symmetrizable. The deterministic code1 capacity of the AVC for the
average probability of error is positive iff the AVC is nonsymmetrizable [52, 51, 53, 92].
The main contributions of this chapter can be summarized as follows:
• Under the worst case disguised jamming, the AVC corresponding to MDFH has symmetric kernel. That is, the deterministic code capacity of MDFH under the worst case
disguised jamming is zero.
• For AJ-MDFH, under the worst case disguised jamming - ID jamming, we prove that:
as long as the ID sequence is unavailable to the jammer, the AVC corresponding to
AJ-MDFH is nonsymmetrizable. Note that the secure ID in AJ-MDFH is generated
using AES [76, 94], to symmetrize AJ-MDFH is equivalent to break AES, which is
computationally infeasible in practical systems. That is, the AVC corresponding to
AJ-MDFH is computationally infeasible to be symmetrized. This result ensures that
AJ-MDFH has positive capacity under ID jamming.
• We derive the capacity of AJ-MDFH under ID jamming, for which the mutual information is maximized for all possible input probability distributions and minimized for
all possible jamming distributions. We show that the capacity converges as the constellation size M goes to infinity. It is observed that: (i) Under reasonable SNR levels
(≥ 10dB), the capacity of AJ-MDFH under ID jamming is close to the jamming-free
case, and it outperforms the conventional FH system by big margins; (ii) For AJMDFH, since the information bits are transmitted through hopping frequency control,
it is very robust to additive noise.
This chapter is organized as follows. A brief system description for MDFH and AJMDFH is provided in Section 4.2. The capacity of MDFH under disguised jamming is
analyzed in Section 4.3. In Section 4.4, the capacity of AJ-MDFH is analyzed and derived
1A

deterministic (n, k) code means that each k-bit data word is mapped to a unique n-bit
codeword.

72

under ID jamming. In Section 4.5, we extend the capacity analysis to the multiuser AJMDFH system (MC-AJ-MDFH). The numerical examples are provided in Section 4.8 and
we conclude in Section 4.9.

4.2
4.2.1

System Description
MDFH

The basic idea of MDFH is that part of the message acts as the PN sequence for carrier
frequency selection at the transmitter. More specifically, selection of carrier frequencies
is directly determined by the encrypted information stream rather than by a pre-selected
pseudo-random sequence as in the conventional FH.
Let Nc be the total number of available channels, with {f1 , f2 , · · · , fNc } being the set of
all available carrier frequencies. The number of bits required to specify an individual channel
is Bc = log2 Nc , where x denotes the largest integer less than or equal to x. Without loss
of generality, we assume that Nc = 2Bc . Let Ω be the selected constellation of size M , and
denote Bs = log2 M bits. Let Ts and Th denote the symbol period and the hop duration,
Ts
.
respectively, then the number of hops per symbol period is given by Nh =
Th
The transmitter structure of MDFH is shown in Figure4.1. The encrypted information
stream is divided into blocks of length L

Nh Bc + Bs . Each block is parsed into Nh Bc

carrier bits and Bs ordinary bits. The carrier bits determine the hopping frequencies, and the
ordinary bits are mapped to a symbol in Ω and transmitted through the channels identified
by the carrier bits. The structure of the nth block, Xn = [Xn,1 , Xn,2 , · · · , Xn,N , Yn ], is
h

shown in Figure 4.1a.
Let fX

n,l

be the carrier frequency corresponding to Xn,l , l ∈ {1, · · · , Nh }, sn the symbol

corresponding to ordinary bit vector Yn , and g(t) the pulse shaping filter. For the mth
hopping period [mTh , (m+1)Th ] with m = nNh +l, the transmitted signal can be represented

73

Ordinary
Bit Vector

Carrier Bit Vectors
X

n,1

X

X

n,2

Xn

Yn

n, Nh

(a) Information block structure

Carrier Frequency
Carrier Bits
Selection

S/P
Encrypted
Partitioning
Information
Baseband Signal
Sequence
Generation
Ordinary Bits

Modulation

(b) Transmitter
Figure 4.1: MDFH transmitter structure.

as
s(t) =

√

2Re


 Nc


αi,m sn g(t − mTh )ej2πfi t




,

(4.4)



i=1

where αi,m = 1 if fX = fi , and αi,m = 0 otherwise.
n,l
Let s(t), J(t) and n(t) denote the signal, the jamming interference and the noise, respectively. For AWGN channels, the received signal can be represented as
r(t) = s(t) + J(t) + n(t).

(4.5)

We assume that s(t), J(t) and n(t) are independent of each other. Feeding r(t) into a bank
of Nc bandpass filters, each centered at fi (i = 1, 2, · · · , Nc ), the output of the ith ideal
bandpass filter fi (t) is ri (t) = fi (t) ∗ r(t). For demodulation, ri (t) is first shifted back to
the baseband, and then passed through a matched filter. At the mth hopping period, for
i = 1, · · · , Nc , the sampled matched filter output corresponds to channel i can be expressed
as
ri,m = αi,m sn + βi,m Ji,m + ni,m ,

74

(4.6)

where sn , Ji,m and ni,m correspond to the signal symbol, the jamming interference and
the noise, respectively; αi,m , βi,m ∈ {0, 1} are binary indicators for the presence of signal
and jamming over channel i at the mth hopping period, respectively. Note that the user’s
information is carried in both αi,m and sn . For notation simplicity, without loss of generality,
we omit the subscript m and n in (4.6). That is, for a particular hopping period, (4.6) is
reduced to:
ri = αi s + βi Ji + ni , i = 1, · · · , Nc .

(4.7)

The carrier bits and the ordinary bits can then be estimated from ri [88].

4.2.2

AJ-MDFH

AJ-MDFH was proposed to improve the capacity of MDFH under disguised jamming by
adding shared randomness between the transmitter and receiver. This is achieved by replacing the ordinary bits in MDFH with a secure identification (ID) information during the
transmission process.
The system structure of AJ-MDFH is illustrated in Figure 4.2. Each user is assigned a
secure ID sequence. For each hopping period, AJ-MDFH can also be characterized as
ri = αi s + βi Ji + ni , i = 1, · · · , Nc ,

(4.8)

except that s is now the ID symbol instead of the signal symbol. To prevent impersonate
ID attack, the ID symbol is refreshed at each hopping period. For AJ-MDFH, the user’s
information is only carried in αi . Recall that for AJ-MDFH, the ML receiver reduces to a
normalized minimum distance receiver. Define Pi = E{ ri 2 }, and
Zi =

ri − s
√
.
Pi

(4.9)

Let Ic = {1, · · · , Nc }. Assuming αi = δ(k −i) for some k ∈ Ic , at the receiver, k is estimated
as
ˆ
k = arg min Zi .
i∈Ic

75

(4.10)

Encrypted
Carrier Frequency
Information Channel
Interleaving
Coding
Selection
Initial
Modulation
Vector, Key PN Sequence
Symbol
Baseband Signal
Encryption
Generation
Mapper
Generation
Secure ID Generation
(a) Transmitter structure

BPF 1
BPF 2
BPF Nc
Initial
Vector, Key

Demodulation

Secure ID
Generation

Recovered
Signal
Detection Information
&
Extraction
Symbol
Mapper

(b) Receiver structure
Figure 4.2: Transmitter and receiver structure of AJ-MDFH.

For more efficient spectrum usage, the system can be extended to multi-carrier AJ-MDFH
(MC-AJ-MDFH). The idea is to split all the Nc channels into Ng non-overlapping subgroups,
and each carrier hops within the assigned subgroup based on the AJ-MDFH scheme [88].

4.3

Capacity of MDFH under Disguised Jamming

In this section, we will show that in MDFH, due to the fact that there is no shared secret
between the transmitter and receiver, when the constellation Ω and the pulse shaping filter
g(t) are known to the jammer, the capacity of MDFH under worst case disguised jamming
is actually zero.
Following (4.7), let r = (r1 , . . . , rNc ), α = (α1 , . . . , αNc ), β = (β1 , . . . , βNc ), J =
(J1 , · · · , JNc ) and n = (n1 , . . . , nNc ), the MDFH system under hostile jamming can be

76

modeled as:
r = αs + β · J + n.

(4.11)

Note that the information is contained in both α and s. Define x = αs, J = β · J , then we
have
r = x + J + n.

(4.12)

Nc

Let A = {α = (α1 , . . . , αNc )|αi is 0 or 1, and

αi = 1}. Define X = {αs|α ∈ A, s ∈
i=1

Ω} be the set of all possible information signal x, and J = {J = (β1 J1 , · · · , βNc JNc )|Ji ∈
ˆ
ΩJ , βi = 0, 1, i = 1, · · · , Nc } where ΩJ is the set of all possible jamming symbols. Let x
be the estimated version of x at the receiver, and W0 (ˆ |x, J) the conditional probability
x
ˆ
that x is estimated at the receiver given that the signal is x, and the jamming is J. The
jammed MDFH system can be modeled as an arbitrarily varying channel (AVC) which is
characterized by the probability matrix
W0 : X × J → X ,

(4.13)

with
W0 (ˆ |x, J) ≥ 0,
x

ˆ
x, x ∈ X , J ∈ J ,

(4.14)

W0 (ˆ |x, J) = 1, ∀x ∈ X , ∀J ∈ J .
x

(4.15)

ˆ
x∈X

W0 is called the kernel of the AVC.
Under the worst case single band disguised jamming,
J = {βb|β ∈ A, b ∈ Ω} = X .

(4.16)

That is, the jamming and the information signal are fully symmetric. Note that in MDFH,
no shared randomness is exploited for signal detection at the receiver, the recovery of x is
fully based on r. Hence, we further have
W0 (ˆ |x, J) = W0 (ˆ |J, x).
x
x
77

(4.17)

This implies that the kernel of the AVC corresponding to MDFH, W0 , is symmetric.
In [92], it has been proved that the deterministic capacity (i.e., the largest rate achieved
with deterministic codes) of an AVC with symmetric kernel is zero. Therefore, we have the
result below.
Proposition 2 The deterministic capacity of MDFH under the worst case single band disguised jamming is zero.

4.4

Capacity of AJ-MDFH under Disguised Jamming

In this section, first, we will show that due to the shared randomness introduced by the secure
ID sequence, the resulted kernel of the AVC corresponding to AJ-MDFH is nonsymmetrizable
even under the worst case disguised jamming - ID jamming. We will further calculate the
capacity of AJ-MDFH under ID jamming.

4.4.1

AVC Symmetricity Analysis

Recall that for AJ-MDFH,
r = αs + J + n.

(4.18)

Under the worse case single band disguised jamming, J ∈ X , and can be represented as
J = βb for some β ∈ A and b ∈ Ω. Note that the information is only transmitted through
α, the AVC corresponding to AJ-MDFH can be characterized using the probability matrix
W : X × X → A,

(4.19)

with
ˆ
W (α|x, J) ≥ 0,

ˆ
x = αs ∈ X , J = βb ∈ X , α, α, β ∈ A,

ˆ
W (α|x, J) = 1, ∀ (x, J) ∈ X 2 .
ˆ
α∈A

78

(4.20)

ˆ
Here α is the estimated version of α. In this section, we will first prove that: under reasonable
SNR levels, the kernel W defined in (4.19)-(4.20) is nonsymmetric. Then prove a stronger
result: W is actually nonsymmetrizable.
Recall that W is symmetric if and only if
ˆ
ˆ
ˆ
W (α|x, J) = W (α|J, x), ∀x, J ∈ X , ∀α ∈ A.

(4.21)

To prove that W is nonsymmetric, we need to show that there always exist some x, J and
ˆ
α, such that the equality above does not hold. Following the discussion on ID constellation
design in Chapter 3, we assume that Ω is a PSK constellation with power Ps , and define a
mapping v : Ic → A as
v(k) = α if αi = δ(k − i), ∀i ∈ Ic .

(4.22)

Lemma 1 Suppose X, Y are independent continuous random variables. If Z1 , · · · , ZN are
i.i.d. continuous random variables, which are also independent of X, Y , then
P r{X < Y and X < Zi , ∀1 ≤ i ≤ N } ≥ P r{X < Y } − N P r{X ≥ Zi0 },

(4.23)

for any fixed 1 ≤ i0 ≤ N .
Proof : Let A = {X < Y } and B = {X < Zi , ∀1 ≤ i ≤ N } =
1≤i≤N

¯
corresponding complement, B, can be written as
¯
B=

{X < Zi }. The

{X ≥ Zi }.

(4.24)

1≤i≤N

Note that P r{A} = P r{A

B} + P r{A

¯
B},

P r{X < Y and X < Zi , ∀1 ≤ i ≤ N } = P r{A

B}

= P r{A} − P r{A

¯
B}

¯
≥ P r{A} − P r{B}
N

≥ P r{A} −

P r{X ≥ Zi }
i=1

= P r{X < Y } − N P r{X ≥ Zi0 },
79

(4.25)

for any fixed 1 ≤ i0 ≤ N .
Proposition 3 Assuming Ω is a PSK constellation with power Ps . Let x = αs, J = βb,
where α, β ∈ A, α = β, and s, b ∈ Ω, then
b−s 2

where

1 −
2
W (α|x, J) ≥ 1 − e 2σn − ,
2
Nc − 2
γ(γ + 1)
Ps
=
exp{−
} with γ = 2 denoting the SNR.
γ+2
γ+2
σn

(4.26)

Proof : Let α = v(k) and β = v(j). When β = α, we have j = k and
W (α|x, J) = W (α|αs, βb)
= P r{Zk < Zj and Zk < Zi , ∀i ∈ Ic , i = j, k|x, J}.

(4.27)

It then follows from Lemma 1 that
W (α|x, J) ≥ P r{Zk < Zj |x, J} − (Nc − 2)P r{Zk ≥ Zi0 |x, J},

(4.28)

for any fixed i0 ∈ Ic , i0 = k, j. For any i ∈ Ic , it follows from (4.8) and (4.9) that signal ri
and the corresponding metric Zi can be written as









 s + nk , i = k,




Zi =
ri =
b + nj , i = j,









 n,
i = j, k,

i


nk

,
2
Ps + σn
b − s + nj

i = k,

(4.29)
, i = j,
2
Ps + σ n
ni − s
,
i = j, k.
σn
Then, P r{Zk < Zj |x, J} = P r{ nk < b − s + nj |x, J}. For any s, b ∈ Ω, both nk and
2
b − s + nj are circularly symmetric complex Gaussian random variables with nk ∼ CN (0, σn )
2
and b − s + nj ∼ CN (b − s, σn ). Then P r{Zk < Zj |x, J} can be calculated as [87, page 49]
−

b−s 2
2
2σn .

1
P r{Zk < Zj |x, J} = 1 − e
(4.30)
2
2
ni − s
nk
σn
∼ CN (0,
), 0
∼
Similarly, for any fixed i0 ∈ Ic , i0 = k, j, we have
2
2
σn
Ps + σ n
Ps + σ n
s
CN (− , 1) and
σn
P r{Zk ≥ Zi0 |x, J} = P r{

nk
2
Ps + σn

≥

80

ni0 − s
σn

1 − γ(γ+1)
}=
e γ+2 ,
γ+2

(4.31)

Ps
where γ = 2 . It then follows from (4.28) that
σn
b−s 2

1 −
2
W (α|x, J) ≥ 1 − e 2σn
2
Note that

− .

(4.32)

is determined by the SNR γ as well as the number of channels Nc . When

SN R ≥ 10dB and Nc = 512, for example, ≤ 0.004.
Theorem 2 Assuming Ω is a PSK constellation with power Ps . Let x = αs, J = βb, where
Ps
Nc − 2
γ(γ + 1)
α, β ∈ A, α = β, and s, b ∈ Ω, s = b. Let γ = 2 and =
exp{−
}, then
γ+2
γ+2
σn
−

W (α|x, J) − W (α|J, x) ≥ 1 − e

b−s 2
2
2σn

−2 .

(4.33)

Proof : Following Proposition 3, we have
b−s 2

1 −
2
W (β|J, x) ≥ 1 − e 2σn
2

− .

(4.34)

An upper bound for W (α|J, x) can be derived as
W (α|J, x) = 1 − W (β|J, x) −

ˆ
W (α|J, x)
α=α,β
ˆ

≤ 1 − W (β|J, x)
≤

1
e
2

−

b−s 2
2
2σn

+ .

(4.35)

It then follows form (4.26) and (4.35) that
−

W (α|x, J) − W (α|J, x) ≥ 1 − e

b−s 2
2
2σn

−2 .

(4.36)

Proposition 4 Assuming Ω is a PSK constellation with power Ps . Let x = αs, J = βb,
where α, β ∈ A, α = β, and s, b ∈ Ω, s = b, then
W (α|x, J) > W (α|J, x),
whenever

b−s 2
1
> 2 ln
.
2
1−2
σn
81

(4.37)

This result follows directly from Theorem 2. It implies that as long as s and b are
“distinguishable” under the additive noise, the channel symmetricity between the jammer
and the legal user is broken, and this increases the probability of correct decision.
ˆ
Define W : X × X → A by
ˆ ˆ
W (α|x, J)

ˆ
π(y|J)W (α|x, y),

(4.38)

y∈Y

where π : X → Y is a probability matrix, and Y ⊆ X . If there exists a π such that
ˆ ˆ
ˆ ˆ
ˆ
W (α|x, J) = W (α|J, x), ∀x, J ∈ X , ∀α ∈ A,

(4.39)

then W is said to be symmetrizable. Next, we will show that under ID jamming, as long as
the ID sequence is unavailable to the jammer, the AVC corresponding to AJ-MDFH is not
only nonsymmetric, but also nonsymmetrizable.
Ps
,
=
2
σn
Nc − 2
γ(γ + 1)
1
x(x + 1)
exp{−
} and dmin =
min
s1 −s2 . Let f (x) =
exp{−
}.
γ+2
γ+2
x+2
x+2
s1 ,s2 ∈Ω,s1 =s2
For Nc > 2 and M > 2, under the condition2 that
√
2
1
−1 ( 1 ) and dmin > max( √ 2 ln Nc
γ>f
)
(4.40)
, 2 ln
√
2
2Nc
1−2
σn
2γ − ln Nc
Theorem 3 Assuming Ω is a M-PSK constellation with power Ps . Let γ =

the kernel W for the AVC corresponding to AJ-MDFH is nonsymmetrizable.
ˆ
We are going to show that for any probability matrix π, there exist some α0 ∈ A, and
x0 , J0 ∈ X , such that
ˆ ˆ
ˆ ˆ
W (α0 |x0 , J0 ) = W (α0 |J0 , x0 ).

(4.41)

To prove this result, we need the following two Lemmas:
Lemma 2 For any given π : X → X , there exists a pair x0 = αs and J0 = βb, α, β ∈ A,
s, b ∈ Ω, such that β = α, b = s and π(−x0 |J0 ) + π(βs|J0 ) < 1.
2 When

M is fixed, the two conditions in (4.40) can be reduced to one condition on SNR.
As an example, for M = 32 and Nc = 64, the kernel W is nonsymmetrizable when γ > 8.3dB.
82

˜b
˜
˜s
˜ b ˜
Proof: Suppose for all x = α˜ and J = β˜ with β = α, ˜ = s, the equality π(−x|J) +
˜s
π(β˜|J) = 1 holds. For Nc > 2 and M > 2, consider x0 = αs, J0 = βb with β =
α, b = s and any x1 = λc, λ ∈ A, c ∈ Ω with λ = α, β and c = b, s. On one hand,
π(−x0 |J0 ) + π(βs|J0 ) = 1, which implies that J0 can only be mapped to −x0 and βs. On
the other hand, we also have π(−x1 |J0 ) + π(βc|J0 ) = 1, which implies that J0 can only be
mapped to −x1 and βc. Since −x1 = −x0 and βc = βs, this is a contradiction. Hence, we
can always find a pair x0 and J0 such that π(−x0 |J0 ) + π(βs|J0 ) < 1.
Lemma 3 X can be partitioned into six subsets with respect to x0 as X = ∪6 Xi , where
i=1
X1

{α(−s)}, X2

{αs0 |s0 ∈ Ω, s0 = −s}, X3

X5

{α0 s|α0 = α, β}, X6

Under the condition that γ >

{βs}, X4

{βs0 |s0 ∈ Ω, s0 = s}

{α0 s0 |α0 = α, β, s0 = s}.

f −1 (

(4.42)

√
d2
1
1
min > max( √ 2 ln Nc
) and 2
, 2 ln
),
√
2Nc
1−2
σn
2γ − ln Nc

W (α|x0 , y) = W (β|x0 , y), ∀y ∈ Xi , i = 1, 3.

(4.43)

W (α|x0 , y) − W (β|x0 , y) > 0, ∀y ∈ Xi , i = 2, 4, 5, 6.

(4.44)

Proof : See Section 4.10.1.
For any x ∈ X , y ∈ Y, we assume 0 ≤ π(y|x) ≤ 1. Here the value 1 corresponds to the
case that Y is a single item subset; the value 0 excludes certain points in X , and results in
the case that Y is a proper subset of X . Without loss of generality, we can assume that
Y = X and prove that (4.39) is not true for any probability matrix π : X → X .
Proof of Theorem 3: Following Lemma 2, we pick x0 , J0 such that β = α, b = s
ˆ
ˆ
and π(−x0 |J0 ) + π(βs|J0 ) < 1. We will prove that W (α|x0 , J0 ) = W (α|J0 , x0 ) and
ˆ
ˆ
W (β|x0 , J0 ) = W (β|J0 , x0 ) cannot hold simultaneously, by showing that
ˆ
ˆ
ˆ
ˆ
W (α|x0 , J0 ) − W (β|x0 , J0 ) > W (α|J0 , x0 ) − W (β|J0 , x0 ).

(4.45)

ˆ
By Lemma 3, X = ∪6 Xi . For any α0 ∈ A, we have
i=1
6

ˆ ˆ
W (α0 |x0 , J0 ) =

ˆ
π(y|J0 )W (α0 |x0 , y).
i=1 y∈Xi

83

(4.46)

It then follows from (4.43) - (4.44) that
6

ˆ
ˆ
W (α|x0 , J0 ) − W (β|x0 , J0 ) =

[W (α|x0 , y) − W (β|x0 , y)]π(y|J0 ) ≥ 0, (4.47)
i=2, y∈Xi
i=3
6

π(y|J0 ) = 0, i.e., π(−x0 |J0 ) + π(βs|J0 ) = 1.

with the equality holds if and only if
i=2, y∈Xi
i=3

Recall that we pick x0 , J0 such that β = α, b = s and π(−x0 |J0 ) + π(βs|J0 ) < 1. Therefore,
ˆ
ˆ
W (α|x0 , J0 ) − W (β|x0 , J0 ) > 0.

(4.48)

Similarly, X can be partitioned into six subsets with respect to J0 = βb, defined as
J1

{β(−b)}, J2

{βb0 |b0 ∈ Ω, b0 = −b}, J3

J5

{β 0 b|β 0 = α, β}, J6

{αb}, J4

{αb0 |b0 ∈ Ω, b0 = b}

{β 0 b0 |β 0 = α, β, b0 = b},

and

(4.49)

6

ˆ ˆ
W (α0 |J0 , x0 ) =

ˆ
π(y|x0 )W (α0 |J0 , y).

(4.50)

i=1 y∈Ji

Then we have
6

ˆ
ˆ
W (α|J0 , x0 ) − W (β|J0 , x0 ) =

[W (α|J0 , y) − W (β|J0 , y)]π(y|x0 ).

(4.51)

i=2, y∈Ji
i=3

Moreover, under the same condition as in previous case,
ˆ
ˆ
W (α|J0 , x0 ) − W (β|J0 , x0 ) ≤ 0.

(4.52)

ˆ
From (4.48) and (4.52), we can see that (4.45) holds, which implies that W (α|x0 , J0 ) =
ˆ
ˆ
ˆ
W (α|J0 , x0 ) and W (β|x0 , J0 ) = W (β|J0 , x0 ) cannot hold simultaneously.
Note that the secure ID in AJ-MDFH is generated using AES, to symmetrize AJ-MDFH is
thus equivalent to break AES, which is computationally infeasible in practical systems. That
is, the AVC corresponding to AJ-MDFH is computationally infeasible to be symmetrized.
This result ensures that when the ID sequence is unknown to the jammer, the deterministic
capacity of AJ-MDFH is positive, and equal to the random code capacity [51, 92].
84

4.4.2

Capacity Calculation

Note that in AJ-MDFH, the message information is only transmitted through the carrier
bits. Consider x = αs where s ∈ Ω and α = (α1 , · · · , αNc ) ∈ A. Let iS and iJ be the signal
channel index and jamming channel index, respectively, and ˆS the detected signal channel
i
index at the receiver. For capacity analysis, define
ˆ
W1 (k|k, j)

ˆ
P r{ˆS = k|iS = k, iJ = j}.
i

(4.53)

ˆ
ˆ
Let x = αs, J = βb with α = v(k), β = v(j). Let α = v(k), and assuming s and b are
uniformly distributed over Ω, then the relationship between W1 and W can be characterized
as
1
ˆ
W1 (k|k, j) =
|Ω|2

ˆ
W (α|x = αs, J = βb).

(4.54)

s∈Ω b∈Ω

The detailed representation of W1 is provided in Section 4.10.2, where we prove that W1 has
the following properties:
(P1): W1 (k|k, k) and W1 (i|k, k) are fixed values for any i, k ∈ Ic , i = k.
(P2): W1 (k|k, j), W1 (j|k, j) and W1 (i|k, j) are fixed values for any i, j, k ∈ Ic , j = k, i = j, k.
Denote the set of all probability distributions on Ic as P(Ic ). Let P and ζ denote the
probability distribution associated with iS and iJ , respectively. P, ζ ∈ P(Ic ). Let Wζ denote
the averaged probability matrix for a given ζ
ˆ
ˆ
Wζ (k|k) = Wζ (ˆS = k|iS = k)
i
ˆ
W1 (k|k, j)ζ(iJ = j).

=

(4.55)

j∈Ic

Let I(P, Wζ ) denote the mutual information [92] between the input and the output for the
AJ-MDFH channel, defined as
ˆ
P (iS = k)Wζ (k|k) log

I(P, Wζ )
ˆ
k∈Ic k∈Ic

85

ˆ
Wζ (k|k)
,
ˆ
(P W )ζ (k)

(4.56)

ˆ
where (P W )ζ (k) =

ˆ
Wζ (k|k )P (k ). Following Theorem 3, the AVC corresponding to
k ∈Ic

AJ-MDFH is nonsymmetrizable. Its channel capacity for the average error probability is
positive and can be calculated as [52, 53]
C=

max

min I(P, Wζ ) =

P ∈P(Ic ) ζ∈P(Ic )

min

max I(P, Wζ ).

ζ∈P(Ic ) P ∈P(Ic )

(4.57)

It can be observed from (4.57) that the legal user tries to choose P to maximize the mutual
information, while the jammer tries to minimize it by choosing an appropriate ζ [95, 96]. Let
(P, ζ) ∈ P(Ic ) × P(Ic ) be a pair of mixed strategy chosen by the user and the jammer. The
capacity can be achieved when a pair of saddle point strategy (P ∗ , ζ ∗ ) are chosen, which can
be characterized by the following two inequalities for all (P, ζ) ∈ P(Ic ) × P(Ic ) [97, 98]:
I(P, Wζ ∗ ) ≤ I(P ∗ , Wζ ∗ ) ≤ I(P ∗ , Wζ ).

(4.58)

Following the same arguments in [99], we have the following Lemma:
Lemma 4 In an AJ-MDFH channel, the saddle point strategy pair can be reached when
both P and ζ are uniform distributions over Ic . That is,


 1 , j∈I ,
 1 , k∈I ,


c
c
Nc
Nc
ζ ∗ (j) =
P ∗ (k) =


 0,
 0,
otherwise,
otherwise.

(4.59)

Proof: We need to show that both inequalities in (4.58) are satisfied if (P ∗ , ζ ∗ ) is given
in (4.59). First, assuming ζ = ζ ∗ , it can be shown that Wζ ∗ is a symmetric matrix given in
(4.80). It is shown in [100] that I(P, Wζ ∗ ) is maximized when P = P ∗ . Hence
I(P, Wζ ∗ ) ≤ I(P ∗ , Wζ ∗ ).

(4.60)

Next, assuming P = P ∗ , we will show that I(P ∗ , Wζ ) is minimized when ζ = ζ ∗ . For
notation simplicity, we will consider the case Nc = 2, as the proof can be extended to Nc > 2.
For Nc = 2, Ic = {1, 2},
ˆ
Wζ (k|k) =

ˆ
ˆ
W1 (k|k, j)ζ(iJ = j), ∀k, k ∈ Ic .
j∈Ic

86

(4.61)

Define





 Wζ (1|1) Wζ (2|1) 
Wζ = 
.
Wζ (1|2) Wζ (2|2)

(4.62)

For any ζ ∈ P(Ic ), let q1 = ζ(iJ = 1), q2 = ζ(iJ = 2) = 1 − q1 ,


 W1 (1|1, 1)q1 + W1 (1|1, 2)q2 W1 (2|1, 1)q1 + W1 (2|1, 2)q2 
Wζ = 
.
W1 (1|2, 1)q1 + W1 (1|2, 2)q2 W1 (2|2, 1)q1 + W1 (2|2, 2)q2

(4.63)

Let p1 = W1 (1|1, 1) and p2 = W1 (1|1, 2). Note that W1 (k|k, k) is a fixed value for any k ∈ Ic
and W1 (k|k, j) is also a fixed value for j, k ∈ Ic , j = k. We have
W1 (2|2, 2) = W1 (1|1, 1) = p1 ,
W1 (2|2, 1) = W1 (1|1, 2) = p2 ,

(4.64)

and
W1 (1|2, 2) = W1 (2|1, 1) = 1 − p1 ,
W1 (1|2, 1) = W1 (2|1, 2) = 1 − p2 .

(4.65)

It follows from (4.63)-(4.65) that


p1 q1 + p2 q2
(1 − p1 )q1 + (1 − p2 )q2 

Wζ = 
.
(1 − p2 )q1 + (1 − p1 )q2
p2 q1 + p 1 q2

(4.66)

Consider ζ ∈ P(Ic ) with ζ (iJ = 1) = q2 = 1 − q1 and ζ (iJ = 2) = q1 , the corresponding
probability matrix


Wζ = 


(1 − p2 )q1 + (1 − p1 )q2 
.
(1 − p1 )q1 + (1 − p2 )q2
p1 q1 + p2 q2
p2 q1 + p1 q2

1
For the uniform distribution ζ ∗ , i.e., ζ ∗ (iJ = 1) = ζ ∗ (iJ = 2) = ,
2


p1 + p2
p + p2
1− 1


2
2
Wζ ∗ = 
p 1 + p2
p1 + p2  .
1−
2
2
87

(4.67)

(4.68)

1
1
ˆ
ˆ
ˆ
ˆ
It can be observed that Wζ ∗ (k|k) = Wζ (k|k) + Wζ (k|k) for all k, k ∈ Ic . That is,
2
2
1
1
ˆ
Wζ ∗ = Wζ + Wζ . Since I(P ∗ , Wζ ) is a convex function of Wζ (k|k) [100, Theorem 2.7.4],
2
2
according to Jensen’s inequality,
1
1
1
I(P ∗ , Wζ ∗ ) = I(P ∗ , Wζ + Wζ ) ≤ [I(P ∗ , Wζ ) + I(P ∗ , Wζ )].
2
2
2

(4.69)

Recall that
ˆ
P (iS = k)Wζ (k|k) log

I(P, Wζ )
ˆ
k∈Ic k∈Ic

ˆ
Wζ (k|k)
,
ˆ
(P W )ζ (k)

(4.70)

ˆ
Wζ (k|k )P (k ). Note that

ˆ
where (P W )ζ (k) =
k ∈Ic

Wζ (1|1) = Wζ (2|2), Wζ (1|2) = Wζ (2|1)
Wζ (2|1) = Wζ (1|2), Wζ (2|2) = Wζ (1|1).

(4.71)

1
When the input is equiprobable, i.e., P ∗ (1) = P ∗ (2) = , it can be observed that
2
1
(W (1|1) + Wζ (1|2))
2 ζ
1
=
(W (2|2) + Wζ (2|1)) = (P W )ζ (2),
2 ζ

(P W )ζ (1) =

(4.72)

and similarly (P W )ζ (2) = (P W )ζ (1). It then follows from (4.70) and (4.72) that
I(P ∗ , Wζ ) =
k ∈Ic k∈Ic

ˆ
Wζ (k|k)
1
ˆ
Wζ (k|k) log
ˆ
2
(P W )ζ (k)

ˆ
Wζ (k|k)
1
ˆ log
=
W (k|k)
ˆ
2 ζ
(P W )ζ (k)
k∈Ic
k ∈Ic
= I(P ∗ , Wζ ).

(4.73)

Then
I(P ∗ , Wζ ∗ ) ≤ I(P ∗ , Wζ ).

(4.74)

Combining (4.60) and (4.74), we proved that I(P ∗ , Wζ ∗ ) is a pair of saddle point strategy.

88

In AJ-MDFH, when the jammer chooses the strategy ζ ∗ as in (4.59), the averaged probability matrix can be calculated as
ˆ
Wζ ∗ (k|k) =

Nc

ˆ
W1 (k|k, j)ζ ∗ (j).

(4.75)

j=1

ˆ
(i) When k = k, (4.75) can be expanded as
ˆ
Wζ ∗ (k|k) = W1 (k|k, k)ζ ∗ (k) +

W1 (k|k, j)ζ ∗ (j).

(4.76)

j∈Ic ,j=k

Following property (P1) and (P2) of W1 , we have
ˆ
Wζ ∗ (k|k) =

1
Nc − 1
W1 (k0 |k0 , k0 ) +
W1 (k0 |k0 , j0 ),
Nc
Nc

(4.77)

for any fixed j0 , k0 ∈ Ic , j0 = k0 .
ˆ
(ii) When k = k, (4.75) can be expanded as
ˆ
ˆ
ˆ ˆ
ˆ
Wζ ∗ (k|k) = W1 (k|k, k)ζ ∗ (k) + W1 (k|k, k)ζ ∗ (k) +

ˆ
W1 (k|k, j)ζ ∗ (j).

(4.78)

ˆ
j∈Ic ,j=k,k

Following property (P1) and (P2) of W1 , we have
1
Nc − 2
1
ˆ
ˆ
ˆ
ˆ
ˆ
W1 (k0 |k0 , k0 ) +
W1 (k0 |k0 , k0 ) +
W1 (k0 |k0 , j0 ),
Wζ ∗ (k|k) =
Nc
Nc
Nc
ˆ
ˆ
ˆ
for any fixed k0 , k0 , j0 ∈ Ic , k0 = k0 , j0 = k0 , k0 . Define w1
ˆ
ˆ
Wζ ∗ (k|k), k = k, then Wζ ∗ can be obtained as

Wζ ∗ (2|1) · · · Wζ ∗ (Nc |1)
 Wζ ∗ (1|1)

 W ∗ (1|2)
Wζ ∗ (2|2) · · · Wζ ∗ (Nc |2)
ζ

Wζ ∗ = 
.
.
.
..

.
.
.
.
.
.
.


Wζ ∗ (1|Nc ) Wζ ∗ (2|Nc ) · · · Wζ ∗ (Nc |Nc )





 
 
 
 
=
 
 
 

(4.79)

Wζ ∗ (k|k) and w2



w1 w2 · · · w2 

w2 w1 · · · w2 

.
. ..
. 
.
.
. . 
.
.
. 

w2 w2 · · · w1

.

Nc ×Nc

(4.80)

ˆ
Due to the structure of the matrix Wζ ∗ , it then follows that for any k, k ∈ Ic ,

ˆ
Wζ ∗ (k|k ) =
k ∈Ic

ˆ
Wζ ∗ (k|k ) = 1, and
ˆ
k∈Ic

ˆ
(P ∗ W )ζ ∗ (k) =
k ∈Ic

1
ˆ
.
Wζ ∗ (k|k )P ∗ (k ) =
Nc
89

(4.81)

It then follows from (4.56) and (4.57) that
C = I(P ∗ , Wζ ∗ )
=
ˆ
k∈Ic k∈Ic

=
ˆ
k∈Ic k∈Ic

ˆ
W ∗ (k|k)
1
ˆ log ζ
W ∗ (k|k)
1
Nc ζ
Nc

1
ˆ
W ∗ (k|k) log Nc +
Nc ζ
Nc

= log Nc +

ˆ
k∈Ic k∈Ic

1
ˆ
ˆ
W ∗ (k|k) log Wζ ∗ (k|k)
Nc ζ

ˆ
ˆ
Wζ ∗ (k|1) log Wζ ∗ (k|1)

ˆ
k=1

= log Nc + w1 log w1 + (Nc − 1)w2 log w2 .

(4.82)

Following the discussions above and similar argument as in the proof of Proposition 1 in
Chapter 3, we have:
Ps
. The
2
σn
deterministic capacity of AJ-MDFH under ID jamming is a function of M, Nc and γ of the
Theorem 4 Assuming Ω is an M-PSK constellation with power Ps . Let γ =

form C = C (M, Nc , γ) . As M approaches infinity, C converges to
¯
C = log Nc + w1 log w1 + (Nc − 1)w2 log w2 ,
¯
¯
¯
¯

(4.83)

where w1 = lim w1 and w2 = lim w2 .
¯
¯
M →∞

M →∞

Proof : It can be observed from Section 4.10.2 that when Ω is an M-PSK constellation
ˆ
ˆ
with power Ps , for any j, k, k ∈ Ic , W1 (k|k, j) can be written as a function of the form
ˆ
W1 (k|k, j) =

1
M

M −1
κ=0

fj,k,k (
ˆ

2πκ
, Nc , γ).
M

(4.84)

It then follows from (4.76) that
1
w1 =
M

M −1

fw1 (
κ=0

2πκ
, Nc , γ) ,
M

(4.85)

where
fw1 (

2πκ
1
2πκ
Nc − 1
2πκ
, Nc , γ) =
fk0 ,k0 ,k0 (
, Nc , γ) +
fj0 ,k0 ,k0 (
, Nc , γ),
M
Nc
M
Nc
M
90

(4.86)

for any fixed j0 , k0 ∈ Ic , j0 = k0 . Similarly, from (4.78), we have
w2 =

1
M

M −1

fw2 (
κ=0

2πκ
, Nc , γ) ,
M

(4.87)

where
fw2 (

2πκ
1
2πκ
1
2πκ
, Nc , γ) =
fk ,k ,k (
, Nc , γ) +
fk ,k ,k (
, Nc , γ)
ˆ
ˆ
ˆ
M
Nc 0 0 0 M
Nc 0 0 0 M
Nc − 2
2πκ
+
fj ,k ,k (
, Nc , γ),
ˆ
Nc
0 0 0 M

(4.88)

ˆ
ˆ
ˆ
for any fixed k0 , k0 , j0 ∈ Ic , k0 = k0 , j0 = k0 , k0 . Since both w1 and w2 are functions of
M, Nc and γ, it then follows from (4.82) that the channel capacity C is a function of M, Nc
and γ of the form C = C(M, Nc , γ).
As M approaches infinity, w1 converges to w1 as
¯
M −1

w1 =
¯

lim w1 = lim

M →∞

1
lim
=
2π M →∞

M →∞

κ=0

M −1

fw1 (
κ=0

fw1 ( 2πκ , Nc , γ) · 2π
M
M
M · 2π
M

2πκ
2π
, Nc , γ) ·
M
M

2π
1
fw1 (x, Nc , γ)dx.
=
2π 0

(4.89)

Similarly, we have
w2 = lim w2 =
¯
M →∞

2π
1
fw2 (x, Nc , γ)dx.
2π 0

(4.90)

Note that w1 , w2 ∈ (0, 1) and x log x is continuous within this range. As M approaches
¯
infinity, C converges to C as
¯
C = lim C = log Nc + lim w1 log w1 + (Nc − 1) lim w2 log w2
M →∞

M →∞

M →∞

= log Nc + w1 log w1 + (Nc − 1)w2 log w2 .
¯
¯
¯
¯

4.5

(4.91)

Capacity of Multiuser AJ-MDFH under Disguised Jamming

In this section, we will analyze the deterministic capacity of MC-AJ-MDFH system under
the single band worst case disguised jamming - ID jamming.
91

Recall that in MC-AJ-MDFH, Nc channels are divided into Ng non-overlapping groups,
and each subcarrier hops within the assigned group based on AJ-MDFH scheme. For each
hopping period, let sm and αm be the symbol and the indicator vector for mth subcarrier,
ˆ
respectively. Let xm = αm sm denote the signal over mth subcarrier, and αm the estimated
version of αm . Let Gm denote the indexes of the subcarrier group for mth subcarrier and
¯
Gm = Ic \Gm the complement of Gm in Ic . The set of indicator vector used by mth subcarrier,
denoted by Am , can be obtained by Am = {v(i)|∀i ∈ Gm }. Define Xm = {αm sm |αm ∈
Am , sm ∈ Ω} be the set of all possible xm . Note that the ID jamming is denoted by
J = βb ∈ X . The AVC corresponding to the mth subcarrier can be characterized by the
probability matrix
˜
W m : Xm × X → A m

(4.92)

with
˜ ˆ
Wm (αm |xm , J) ≥ 0,

˜ ˆ
Wm (αm |xm , J) = 1,

(4.93)

ˆ
αm ∈Am

ˆ
where xm = αm sm ∈ Xm , J = βb ∈ X , αm , αm ∈ Am , β ∈ A.
Note that in MC-AJ-MDFH, both secure group information and ID sequence are protected by AES. Without the knowledge of the shared secret, the jammer is unable to deter˜
mine the subcarrier group Gm , which implies that Xm = X thus kernel Wm is nonsymmetric.
˜
Following the similar arguments as in Section 4.4, it can be further shown that kernel Wm is
also nonsymmetrizable and the AVC corresponding to mth subcarrier has positive capacity
under ID jamming.
Let iS,m and ˆS,m be the signal channel index and its estimated version at the receiver,
i
respectively. Recall that iJ denote the jamming channel index. The probability matrix for
carrier bits information of mth subcarrier can be defined as
ˆ
ˆ
W1,m (km |km , j) = P r{ˆS,m = km |iS,m = km , iJ = j}.
i

(4.94)

Let αm = v(km ), km ∈ Gm , sm ∈ Ω and J = βb with β = v(j), j ∈ Ic and b ∈ Ω. Assuming

92

˜
sm and b are uniformly distributed over Ω, W1,m can be obtained from Wm by
ˆ
W1,m (km |km , j) =

1
|Ω|2

ˆ
˜
Wm [v(km )|v(km )sm , v(j)b],

(4.95)

sm ∈Ω b∈Ω

ˆ
˜
where Wm [v(km )|v(km )sm , v(j)b] can be calculated as
ˆ
˜
Wm [v(km )|v(km )sm , v(j)b] = P r{Zk
ˆ

m

ˆ
< Zi , ∀i ∈ Gm , i = km |xm = αm sm , J = βb}.
(4.96)

Assuming Ω is an M-PSK constellation with power Ps .
(i) When j ∈ Gm , W1,m can be calculated following the similar derivations in Section
4.10.2, except that the detection is performed on subcarrier group Gm instead of on Ic .
¯
(ii) When j ∈ Gm , the group is jamming-free. Zi can be obtained as

nkm


, i = km ,

2
Ps + σ n
Zi =
 ni − s m


, i ∈ Gm , i = km .
σn

(4.97)

2
zk
zkm − m
e 2σ2 ,
σ2

Zkm is a a Rayleigh random variable with PDF pZ (zkm ) =
where σ =
km
σn
. For any sm ∈ Ω, when i = km , Zi ’s are i.i.d. Rician random variables with
2
2(Ps + σn )
√
z 2 +ν 2
zi − i 2
1
zi ν
Ps
PDF pZi (zi ) = 2 e 2σ I0
and σ = √ . It can be observed
, where ν =
2
σn
σ
σ
2
that the jamming does not affect the signal detection performance in this case. For any
¯
xm = v(km )sm , J = v(j)b with j ∈ Gm , we have
W1,m (km |km , j)
=

1
|Ω|2

=

1
|Ω|2

P r{Zkm < Zi , ∀i ∈ Gm , i = km |xm , J}
sm ∈Ω b∈Ω
sm ∈Ω b∈Ω 0

√

∞

=
0

Q1

∞
i∈Gm ,i=km

2Ps √
, 2zkm
σn

P r{Zi > zkm |xm , J, zkm }pZ (zkm )dzkm
km

|Gm |−1

93

2
2zkm (Ps + σn ) −
e
2
σn

2 2
(Ps +σn )zk
m
2
σn
dzk

m,

(4.98)

and for any i, km ∈ Gm , i = km ,
W1,m (i|km , j) =

1
[1 − W1,m (km |km , j)].
|Gm | − 1

(4.99)

Let Wζ,m denote the averaged probability matrix of mth user under ID jamming with
distribution ζ on Ic given as
ˆ
Wζ,m (km |km ) =

ˆ
W1,m (km |km , j)ζ(iJ = j).

(4.100)

j∈Ic

Due to the pseudorandom property of the secure group generation algorithm, iJ is uniformly
distributed over Nc channels after the secure groups are recovered at the receiver, which is
the same as the saddle point strategy given in (4.59). Following the similar derivations for
AJ-MDFH in (4.75)-(4.78), we have the following results:
ˆ
(i) When km = km ∈ Gm ,
ˆ
Wζ ∗ ,m (km |km ) = W1,m (km |km , km )ζ ∗ (km ) +

W1,m (km |km , j)ζ ∗ (j)
j∈Gm ,j=km

W1 (km |km

+

, j)ζ ∗ (j)

¯
j∈Gm

=

|Gm | − 1
1
W1,m (k0,m |k0,m , k0,m ) +
W1,m (k0,m |k0,m , j0 )
Nc
Nc
Nc − |Gm |
+
W1,m (k0,m |k0,m , u0,m ),
Nc

¯
for any fixed j0 , k0,m ∈ Gm , j0 = k0,m and any fixed u0,m ∈ Gm .
ˆ
ˆ
(ii) When km , km ∈ Gm , km = km ,
ˆ
ˆ
ˆ
ˆ
ˆ
Wζ ∗ ,m (km |km ) = W1,m (km |km , km )ζ ∗ (km ) + W1,m (km |km , km )ζ ∗ (km )
ˆ
W1,m (km |km , j)ζ ∗ (j) +

+

¯
j∈Gm

j∈Gm
ˆ
j=km ,km

=

W1,m (km |km , j)ζ ∗ (j)

1
1
ˆ
ˆ
ˆ
W1,m (k0,m |k0,m , k0,m ) +
W (k |k , k )
Nc
Nc 1,m 0,m 0,m 0,m
|Gm | − 2
Nc − |Gm |
ˆ
ˆ
+
W1,m (k0,m |k0,m , j0 ) +
W1,m (k0,m |k0,m , u0,m ),
Nc
Nc

ˆ
ˆ
ˆ
¯
for any fixed k0,m , k0,m , j0 ∈ Gm , k0,m = k0,m , j0 = k0,m , k0,m and any fixed u0,m ∈ Gm .
94

ˆ
ˆ
Wζ ∗ ,m (km |km ), km = km . Wζ ∗ ,m can be

Wζ ∗ ,m (km |km ) and w2,m

Define w1,m

written into a symmetric matrix of size |Gm | × |Gm | similar as in (4.80). The deterministic
capacity for mth subcarrier group under ID jamming, Cm , can be obtained following (4.82)
that
Cm = log |Gm | + w1,m log w1,m + (|Gm | − 1)w2,m log w2,m ,

(4.101)

which is achieved when iS,m is uniformly distributed over Gm . Hence, the capacity of MCAJ-MDFH can be obtained as

Ng

CM C =

Cm .

(4.102)

m=1

4.6

Capacity of MDFH under Noise Jamming

Let x = αs ∈ X and J ∈ J denote the transmitted signal and the jamming, respectively,
2
ˆ
ˆs
and x = αˆ ∈ X the estimated version of x = αs at the receiver. Let σJ denote the jamming

power and C the set of all complex numbers. Under the single band noise jamming, we have
J = βb where β ∈ A and b ∈ C is a circularly symmetric complex Gaussian random variable,
2
i.e., b ∼ CN (0, σJ ). The corresponding noise jamming space is

J = {βb|β ∈ A, b ∈ C}.

(4.103)

Therefore, MDFH under noise jamming can be modeled as an AVC characterized by the
probability matrix
W0 : X × J → X ,

(4.104)

with
W0 (ˆ |x, J) ≥ 0,
x

ˆ
x, x ∈ X , J ∈ J ,

W0 (ˆ |x, J) = 1, x ∈ X , J ∈ J .
x

(4.105)
(4.106)

ˆ
x∈X

For the kernel to be symmetric, it is required that J = X . Since J ⊃ X for noise
jamming, the kernel corresponding to MDFH under noise jamming, W0 , is nonsymmetric.
95

Recall that to symmetrize W0 , we need to find a probability matrix π(y|J), where J ∈ X
and y ∈ Y, Y ⊆ J . This implies that to symmetrize X also requires J ∈ X . However, under
the noise jamming, the jammer simply transmits random Gaussian noise and J ∈ J . This
implies that the jammer has no intention to disguise itself as the true signal. Hence, the
MDFH channel under noise jamming is nonsymmetrizable and has a positive capacity.
Recall that in MDFH, the information bits are divided into carrier bits and ordinary bits.
At the receiver, the carrier bits are first detected by identifying the carrier channel, then
the ordinary bits are extracted from the estimated carrier using conventional demodulation
scheme. Following the MDFH receiver design, we have
ˆ
W0 (ˆ = αˆ|x, J) = W0,c (α|x, J)W0,o (ˆ|α, x, J).
x ˆs
s ˆ

(4.107)

Here, W0,c : X × J → A is the AVC kernel corresponding to the carrier bits transmission
channel, and W0,o : A × X × J → Ω is the AVC kernel corresponding to the ordinary bits
transmission channel. Note that W0,o depends on the output of W0,c . We now derive the
capacity of the carrier bits transmission channel and that of the ordinary bits transmission
channel, respectively.

4.6.1

Capacity Derivation for Carrier Information Transmission Channel

Let iS and iJ be the signal channel index and jamming channel index, respectively, and ˆS
i
the detected signal channel index at the receiver. For capacity analysis, define
ˆ
ˆ
W1,c (k|k, j) = P r{ˆS = k|iS = k, iJ = j}.
i

(4.108)

Note that the signal symbol s is selected from Ω by the encrypted information, which is
2
assumed to be random, and the jamming noise b ∼ CN (0, σJ ) is independent of s. Let

ˆ
ˆ
x = αs, J = βb with α = v(k), β = v(j), and α = v(k). Then, the relationship between
W1,c and W0,c can be characterized as
1
ˆ
W1,c (k|k, j) =
|Ω|

s∈Ω C

ˆ
W0,c [v(k)|v(k)s, v(j)b]pB (b)db,

96

(4.109)

1
b 2
exp{− 2 } is the PDF for Gaussian noise b.
2
πσJ
σJ
Assuming Ω is a PSK constellation with power Ps . Let Yi = ri denote the square root

where pB (b) =

of the received signal power in ith channel. When the threshold based (with threshold η)
ˆ
carrier detector is used, channel k is detected if Yk is the smallest above the threshold η, and
ˆ
ˆ
W1,c (k|k, j) can be obtained as follows:
(i) When j = k, Yi can be obtained as


 s + b + n , i = k,
k
Yi =
 n ,

i = k,
i

(4.110)

2
2
2
2
Since b ∼ CN (0, σJ ), we have rk ∼ CN (s, σJ + σn ), ri ∼ CN (0, σn ) and

W1,c (k|k, k)
=

=

1
|Ω|
Q1

s∈Ω C

P r{Yk > η and Yi < η, ∀i ∈ Ic , i = k|x, J}pB (b)db

2Ps
,
2
2
σJ + σn

η2

− 2
2
η (1 − e σn )Nc −1 .
2
2
σJ + σn

(4.111)

For i = k, W1,c (i|k, k) can be obtained following (4.165) in Section 4.10.2.
(ii) When j = k, Yi can be calculated as


 s + nk , i = k,



Yi =
b + nj , i = j,




 n ,
i = j, k.
i

97

(4.112)

2
2
2
2
Note that rk ∼ CN (s, σn ), rj ∼ CN (0, σJ + σn ) and ri ∼ CN (0, σn ). We then have

W1,c (k|k, j)
1
|Ω|

=

s∈Ω C

P r{Yk > η and Yj < η and Yi < η, ∀i ∈ Ic , i = k|x, J}

+ P r{Yk > η and Yk < Yj and Yi < η, ∀i ∈ Ic , i = k|x, J} pB (b)db


2
yk
η2
√
√
− 2
∞ − 2


2
2Ps
2
σJ +σn
σ +σ 2
Q1
,
η (1 − e
)+
e J n pY (yk )dyk 


k
σn σn
η

=

η2
− 2
·(1 − e σn )Nc −2 ,
2

(4.113)

2

y +ν
yk − k 2
y ν
where pY (yk ) = 2 e 2σ I0 k2
k
σ
σ

with ν =

σn
Ps and σ = √ , and
2

W1,c (j|k, j)
=

=

1
|Ω|

s∈Ω C

P r{Yk < η and Yj > η and Yi < η, ∀i ∈ Ic , i = k|x, J}

+ P r{Yj > η and Yj < Yk and Yi < η, ∀i ∈ Ic , i = k|x, J} pB (b)db


η2
√
√


√
√
− 2


∞
2
2Ps
2
2Ps
2
σJ +σn
Q1
+
1 − Q1
,
η
e
,
yj pYj (yj )dyj


σn σn
σn σn
η


η2
− 2
·(1 − e σn )Nc −2 ,

(4.114)

2
yj
yj − 2
e 2σ
σ2

σ
with σ = √J . For i = k, j, W1,c (i|k, j) can be obtained following
2
(4.169) in Section 4.10.2. The capacity of the carrier information transmission channel,

where pYj (yj ) =

denoted as Cc , can be obtained similarly following the capacity result in (4.82).

4.6.2

Capacity Derivation for Ordinary Information Transmission Channel

ˆ
Let S be the signal symbol selected by the ordinary bits, B the Gaussian noise, and S the
estimated version of S. For capacity analysis, define
ˆ ˆ
W1,o (ˆ|s, b) = P r{S = s|S = s, B = b}.
s
98

(4.115)

Note that iS is selected by the encrypted carrier bits information, which is assumed to be
uniformly distributed over Ic , and iJ is also assumed to be uniformly distributed over Ic .
Then, the relationship between W1,o and W0,o can be characterized as
W1,o (ˆ|s, b) =
s

1
2
Nc

ˆ
W0,o [ˆ|v(k), v(k)s, v(j)b]W1,c (k|k, j).
s ˆ

(4.116)

ˆ
k∈Ic k∈Ic j∈Ic

2
For noise jamming, b ∼ CN (0, σJ ), and the averaged probability matrix for W1,o can be

obtained as
Wo,pB (ˆ|s) =
s
=

C

W1,o (ˆ|s, b)pB (b)db
s

1
2
Nc

ˆ
¯
W0,o [ˆ|v(k), v(k)s, v(j)]W1,c (k|k, j),
s ˆ

(4.117)

ˆ
k∈Ic k∈Ic j∈Ic

where
¯
W0,o [ˆ|v(k), v(k)s, v(j)] =
s ˆ

C

W0,o [ˆ|v(k), v(k)s, v(j)b]pB (b)db.
s ˆ

(4.118)

When the minimum distance detector s = arg min rk − s is used for symbol s detection,
ˆ
ˆ ˆ
s∈Ω
ˆ

¯
W0,o [ˆ|v(k), v(k)s, v(j)] can be calculated as follows:
s ˆ
ˆ
(i) When k = j = k, rk = rk = s + b + nk . Let Ds denote the decision region for symbol
ˆ
ˆ
2
2
2
2
s. Since b ∼ CN (0, σJ ) and nk ∼ CN (0, σn ), we have rk ∼ CN (s, σJ + σn ) and
ˆ

¯
W0,o [ˆ|v(k), v(k)s, v(k)] =
s

Ds
ˆ

prk (rk )drk ,

(4.119)

rk − s 2
1
exp{− 2
} is the PDF of rk .
where prk (rk ) =
2
2
2
π(σJ + σn )
σJ + σn
2
ˆ
(ii) When j = k and k = k, rk = rk = s + nk . We then have rk ∼ CN (s, σn ) and
ˆ
¯
W0,o [ˆ|v(k), v(k)s, v(j)] =
s

Ds
ˆ

prk (rk )drk ,

(4.120)

1
r −s 2
where prk (rk ) =
exp{− k 2
}.
2
πσn
σn
ˆ
(iii) When j = k and k = j, rk = rj = b + nk . That is, the detected channel contains
ˆ
only noise. In this case, the detected symbol is randomly distributed over Ω
1
¯
W0,o [ˆ|v(j), v(k)s, v(j)] =
s
.
|Ω|
99

(4.121)

ˆ
ˆ
Similarly for j = k, k = k and j = k, k = j, k, we have rk = nk and
ˆ
ˆ
1
¯
¯
.
W0,o [ˆ|v(k), v(k)s, v(k)] = W0,o [ˆ|v(k), v(k)s, v(j)] =
s ˆ
s ˆ
|Ω|

(4.122)

¯
After obtaining W0,o and W1,c from carrier information transmission channel, Wo,pB (ˆ|s)
s
can be calculated as (4.117). Let Q denote the probability distribution associated with S.
Then the capacity of the ordinary information transmission channel can be obtained as
Co = max I(Q, Wo,pB ).
Q

(4.123)

It can be shown that Wo,pB is a symmetric matrix. Hence, the capacity is achieved when
Q = Q∗ which is the uniform distribution over Ω. Overall, the channel capacity for MDFH
under noise jamming can be calculated as
CM D = Cc + Co .

(4.124)

Note that I(Q∗ , Wo,pB ) is a convex function of Wo,pB , according to Jensen’s inequality,
the capacity can be upper bounded by
Co = I(Q∗ , Wo,pB ) ≤

1
2
Nc

ˆ
¯
W1,c (k|k, j)I(Q∗ , W0,o [ˆ|v(k), v(k)s, v(j)]),
s ˆ
ˆ
k∈Ic k∈Ic j∈Ic

(4.125)
ˆ
¯
where I(Q∗ , W0,o [ˆ|v(k), v(k)s, v(j)]) is the capacity of the detected carrier channel k when
s ˆ
signal and jamming are in kth and jth channel respectively. The capacity of the ordinary
information transmission channel is upper bounded by the averaged capacity over all carrier
detection scenarios.

4.7

Capacity of AJ-MDFH under Noise Jamming

2
2
Under the worst case single band noise jamming, J = βb, b ∼ CN (0, σJ ) where σJ = Ps ,

and
J = {βb|β ∈ A, b ∈ C},

100

(4.126)

where C denotes the set of all complex numbers. The AVC corresponding to the AJ-MDFH
can be characterized by the probability matrix
W :X ×J →A

(4.127)

with
ˆ
W (α|x, J) ≥ 0,

ˆ
W (α|x, J) = 1,

(4.128)

ˆ
α∈A

ˆ
where x = αs ∈ X , J = βb ∈ J , α, α, β ∈ A. Following the similar argument in Section
4.6, it can be seen that the AVC corresponding to AJ-MDFH under noise jamming is also
nonsymmetrizable and has a positive capacity. Recall that for capacity derivation, we define
the information transmission channel in AJ-MDFH as
ˆ
W1 (k|k, j)

ˆ
P r{ˆS = k|iS = k, iJ = j},
i

(4.129)

which corresponds to the carrier information transmission channel in MDFH case. As the
ID symbol s is randomly selected from Ω, the relationship between W1 and W can be
characterized as
1
ˆ
W1 (k|k, j) =
|Ω|

s∈Ω C

ˆ
W [v(k)|v(k)s, v(j)b]pB (b)db.

(4.130)

Note that b ∼ CN (0, Ps ) in this case. Assuming Ω is the PSK constellation with power
Ps . Following from similar calculation scheme of W1 under disguised jamming in Section
4.10.2, we have
W1 (k|k, k)
=
=

1
|Ω|
1
|Ω|

s∈Ω C
s∈Ω 0

P r{Zk < Zi , ∀i ∈ Ic , i = k|x = αs, J = βb}pB (b)db
∞
1≤i≤Nc ,i=k

P r{Zi > zk |x, J, zk }pZ (zk )dzk .
k

(4.131)
z2

z − k
For any s ∈ Ω, Zk is a Rayleigh random variable with PDF pZ (zk ) = k e 2σ2 for 0 ≤
k
σ2
2
Ps + σn
zk < ∞, where σ =
; for i = k, Zi ’s are i.i.d. Rician random variables with
2
2(2Ps + σn )
101

z 2 +ν 2

√
Ps
1
for 0 ≤ zi < ∞, where ν =
and σ = √ . Then
σn
2

zν
z − i
PDF pZi (zi ) = i e 2σ2 I0 i2
2
σ
σ
(4.131) can be simplified as
∞

W1 (k|k, k) =

0

√
QNc −1
1

2Ps √
, 2zk pZ (zk )dzk .
k
σn

(4.132)

For i = k, W1 (i|k, k) can be obtained following (4.165) in Section 4.10.2.

z2

z − k
When j = k, Zk is Rayleigh distributed with PDF pZ (zk ) = k e 2σ2 where σ =
k
σ2
2

2

z +ν
zj ν
zj − j 2
σn
; Zj is Rician distributed with PDF pZj (zj ) = 2 e 2σ I0
2
σ
σ2
2(Ps + σn )
Ps
1
ν=
and σ = √ . Hence, we can obtain
2
Ps + σ n
2

where

W1 (k|k, j)
=

1
|Ω|

∞
s∈Ω 0

∞

2Ps √
, 2zk
2
Ps + σn

Q1

=
0

Pr{Zj > zk |x, J, zk } [P r{Zi > zk |x, J, zk }]Nc −2 p(zk )dzk
√
QNc −2
1

2Ps √
, 2zk pZ (zk )dzk ,
k
σn

(4.133)

and
W1 (j|k, j)
=

1
|Ω|

∞
s∈Ω 0

∞ −

=

e
0

P r{Zk > zj |x, J, zj } P r{Zi > zj |x, J, zj }

2 2
(Ps +σn )zj
2
σn
QNc −2
1

√
2Ps √
, 2zk pZ (zj )dzj .
k
σn

Nc −2

pZj (zj )dzj

(4.134)

For i = k, j, W1 (i|k, j) can be obtained following (4.169) in Section 4.10.2. The capacity of
AJ-MDFH, denoted as CAJ ,can be calculated following (4.82).

4.8

Numerical Results

In this section, simulation examples are provided to illustrate the capacity of the proposed
AJ-MDFH and MC-AJ-MDFH schemes under the worst case disguised jamming. For all the
102

systems considered, we assume the total number of available channels is Nc = 64, that is,
Bc = 6.
Example 1: Impact of the ID constellation size on capacity In this example,
we consider the impact of the ID constellation size on the capacity of AJ-MDFH under
the single band worst case disguised jamming (ID jamming). In Figure 4.3, it can be seen
that the capacity under the ID jamming converges as the constellation size increases. Under
reasonable SNR levels (for example, we require γ > 8.3dB when the constellation size M = 32
¯
to ensure the AVC corresponding to AJ-MDFH is nonsymmetrizable), the capacity limit C
is close to the corresponding jamming-free case indicated by the dashed line.

6.5

Capacity(bits/symbol)

6
5.5
5
SNR=10dB: ID jamming
SNR=10dB: jamming−free
SNR=15dB: ID jamming
SNR=15dB: jamming−free

4.5
4
3.5

5

10

15
20
25
30
Constellation size |Ω|

35

40

Figure 4.3: AJ-MDFH capacity with different PSK constellation size, under the worst case
single band disguised jamming (ID jamming). Nc = 64.

Example 2: Capacity of the multiuser systems under ID jamming In this
example, we choose MC-AJ-MDFH and FHMA as the multiuser systems for AJ-MDFH
103

and conventional FH, respectively, and consider the capacity of the two systems under the
single band worst case disguised jamming [66, 101]. The SNR is taken as Eb /N0 = 10dB and
Th = 1s. For FHMA, we choose Nh = 3; for MC-AJ-MDFH, we choose 32-PSK constellation.
From Figure 3.10, it can be observed that due to the collision-free design and the use of ID
sequence, under disguised jamming, MC-AJ-MDFH can effectively support much more users
than FHMA, which is mainly limited by the collision effect among users.

35

Capacity (bits/channel use)

30
25
MC−AJ−MDFH
FHMA

20
15
10
5
0

5

10

15
20
Number of users

25

30

Figure 4.4: Capacity of MC-AJ-MDFH and FHMA under the worst case single band
disguised jamming. Nc = 64, SN R = 10dB. Here, per channel use means the total
bandwidth of all used channels over one hopping period.

Example 3: Capacity of AJ-MDFH and MDFH under worst case single band
noise jamming Recall that in Section 3.2.2, it can be observed that the worst case noise
2
jamming occurs when the noise and the signal has the same power, i.e., σJ = Ps . For

104

MDFH, we choose the uniform ring constellation (i.e., M -PSK when M → ∞) [102]. for the
upper bound analysis; for AJ-MDFH, we choose 32-PSK constellation for ID constellation.
Figure 4.5 illustrates the capacity of AJ-MDFH and MDFH in this case. The capacity
of AJ-MDFH under the worst case single band disguised jamming - ID jamming is also
provided as comparison. It can be observed that AJ-MDFH outperforms the carrier/ordinary
bits transmission channel of MDFH significantly under noise jamming. Recall that due
to a symmetric kernel, MDFH has zero capacity under the worst case disguised jamming.
However, it has positive capacity under noise jamming. It can also be observed that AJMDFH has a lager capacity under noise jamming comparing with that under ID jamming.
Therefore, noise jamming is not as effective as disguised jamming in terms of using the given
jamming power.

4.9

Summary

In this chapter, we analyzed that capacity of MDFH and AJ-MDFH under the worst case
disguised jamming. We proved that: (i) For MDFH, the corresponding AVC is symmetric,
which implies that the deterministic capacity of MDFH is zero; (ii) For AJ-MDFH, due to
shared randomness between the transmitter and receiver provided by the secure ID sequence,
the corresponding AVC is nonsymmetrizable, which implies that the deterministic capacity
of AJ-MDFH is positive, and equal to the random code capacity. We calculated the capacity
of AJ-MDFH and showed that it converges as the ID constellation size goes to infinity. This
echoes our result in previous chapter, where we showed that the probability of error of AJMDFH converges as the ID constellation size goes to infinity. It can be observed that shared
secure randomness between the transmitter and receiver plays a critical role in anti-jamming
system design.

105

9

Capacity(bits/channel use)

8
7

AJ−MDFH: ID Jamming
AJ−MDFH: Noise Jamming
MDFH Carrier: Noise Jamming
MDFH Ordinary: Upper bound, Noise Jamming
MDFH Overall: Upper bound, Noise Jamming

6
5
4
3
2
1
0
5

10
SNR(dB)

15

Figure 4.5: Capacity of AJ-MDFH and MDFH under the worst case single band noise
jamming. For MDFH, upper bounds for capacity of the ordinary information transmission
channel as well as for the overall channel capacity are provided. The number of hops per
symbol period is Nh = 3.

4.10

Proofs of Chapter 4

4.10.1

Proof of Lemma 3

Proof of Lemma 3: Note that X can be partitioned into six subsets with respect to x0 = αs.
Define B1

{α˜|˜ ∈ Ω}, B2
ss

{β˜|˜ ∈ Ω} and B3
ss

{α0 s|˜ ∈ Ω, α0 = α, β}. We have
˜s

X = ∪3 Bi . It then follows from the definition of subset Xi in (4.42) that B1 = X1 ∪ X2 ,
i=1
B2 = X3 ∪ X4 , B3 = X5 ∪ X6 , and X = ∪6 Xi .
i=1
(i) We consider the cases where y ∈ Xi , i = 1, 3. When y ∈ X1 (y = −x0 ), the
jamming cancels the true signal and the received signal only contains noise, resulting in

106

1
ˆ
, ∀α0 ∈ A. When y ∈ X3 (y = βs), the jamming has the same ID
Nc
symbol as the true signal and the receiver cannot distinguish between the two, resulting in
ˆ
W (α0 |x0 , −x0 ) =

W (α|x0 , βs) = W (β|x0 , βs). Hence, W (α|x0 , y) = W (β|x0 , y) holds in both cases.
(ii) When y ∈ X2 , we have y = αs0 where s0 ∈ Ω, s0 = −s. Assuming α = v(k), the
received signal in the ith channel ri and corresponding Zi defined in (4.9) can be calculated
as


 s + s0 + n , i = k,
k
ri =
 n,
 i
i = k,

Zi =









s 0 + nk
2
s + s0 2 + σ n
ni − s
,
σn

, i = k,
(4.135)
i = k.

Then we have
W (α|x0 , y) = P r{Zk < Zi , ∀i ∈ Ic , i = k|x0 , y}
≥ 1−

P r{Zk ≥ Zi |x0 , y}
i=k

= 1 − (Nc − 1)P r{Zk ≥ Zi0 |x0 , y},

s
s 0 + nk
∼
, 1) and
2
σn
σn
s + s0 2 + σ n
1
s 2
γ
1
). Define S1 =
−
= , N1 =
and
2 + σ2
2
σn
2
2
n
2
Ps
σn
. It then follows
, N2 =
2
2
s + s0 2 + σn
2( s + s0 2 + σn )

for any fixed i0 = k. Since s0 = −s, we have
CN (

s0

,

2
σn

2
s + s0 2 + σ n
s + s0
1
s0
2 =
S2 =
2 + σ2
2
s + s0
n
from the results in [87] that

ni0 − s

(4.136)

∼ CN (−

√
√ √
√ √
C+D
N1
P r{Zk ≥ Zi0 |x0 , y} = Q1 ( C, D)−
e− 2 I0 ( CD) < Q1 ( C, D), (4.137)
N1 + N2
2
2S2
2Ps
2Ps ( s + s0 2 + σn )
2S1
where C =
=
and D =
= 2
. Since
2
2
N1 + N2
N1 + N2
s + s0 2 + 2σn
σn ( s + s0 2 + 2σn )

D > C, P r{Zk ≥ Zi0 |x0 , y} ≤ e
dmin that

−

(D−C)2
2

[103]. It then follows from (4.136) and s + s0 ≥

γd2
W (α|x0 , y) > 1 − (Nc − 1) exp[−2( 2 min 2 )2 ],
dmin + 2σn

(4.138)

and
W (β|x0 , y) =

γd2
1
[1 − W (α|x0 , y)] < exp[−2( 2 min 2 )2 ].
Nc − 1
dmin + 2σn
107

(4.139)

Hence, we have
γd2
W (α|x0 , y) − W (β|x0 , y) > 1 − Nc exp[−2( 2 min 2 )2 ].
dmin + 2σn

(4.140)

Under the condition
γ>

√
d2
1
min > √ 2 ln Nc
ln Nc and 2
,
√
2
σn
2γ − ln Nc

(4.141)

W (α|x0 , y) − W (β|x0 , y) > 0.
(iii) When y ∈ X4 , we have y = βs0 where s0 ∈ Ω, s0 = s. Note that W (α|y, x0 ) =
W (β|x0 , y). Since s0 − s ≥ dmin , it follows from Proposition 2 that

W (α|x0 , y) − W (β|x0 , y) ≥

d2
− min
2
1 − e 2σn

−2 .

(4.142)

Under the condition
<

d2
1
1
and min > 2 ln
,
2
2
1−2
σn

(4.143)

W (α|x0 , y) − W (β|x0 , y) > 0.
(iv) When y ∈ X5 , we have y = α0 s where α0 ∈ A, α0 = α, β. It follows from Lemma
1
2 that W (α|x0 , y) ≥ − . Note that W (α|x0 , y) = W (α0 |x0 , y), we have
2
W (β|x0 , y) =

2
1
[1 − 2W (α|x0 , y)] ≤
.
Nc − 2
Nc − 2

(4.144)

Hence, we have
W (α|x0 , y) − W (β|x0 , y) ≥

1
Nc
−
.
2 Nc − 2

(4.145)

Under the condition
<

Nc − 2
,
2Nc

(4.146)

W (α|x0 , y) − W (β|x0 , y) > 0.
(v) When y ∈ X6 , we have y = α0 s0 where α0 ∈ A, α0 = α, β and s0 ∈ Ω, s0 = s. It
follows from Lemma 2 that
1
W (α|x0 , y) ≥ 1 − e
2
108

−

s0 −s 2
2
2σn

− .

(4.147)

Assuming α = v(k) and α0 = v(k0 ), it follows from Lemma 1 that
W (α0 |x0 , y) ≥ P r{Zk0 < Zk |x0 , y} − (Nc − 2)P r{Zk0 ≥ Zi0 |x0 , y}
s0 −s 2

1 − 2σ2
n
e
=
2

− .

(4.148)

Then we have
W (β|x0 , y) =

2
1
[1 − W (α|x0 , y) − W (α0 |x0 , y)] ≤
.
Nc − 2
Nc − 2

(4.149)

Hence, we have

W (α|x0 , y) − W (β|x0 , y) ≥ 1 −

d2
− min
1
2
e 2σn

2

−

Nc
.
Nc − 2

(4.150)

Under the condition
<

d2
Nc − 2
Nc
and min > −2 ln 2(1 −
),
2
Nc
Nc − 2
σn

(4.151)

W (α|x0 , y) − W (β|x0 , y) > 0.
Next, we will show that the conditions in (4.141), (4.143), (4.146) and (4.151) can be
reduced into the two conditions
√
d2
1
2 ln Nc
1
),
γ > f −1 (
) and min > max( √
, 2 ln
√
2
2Nc
1−2
σn
2γ − ln Nc

(4.152)

as presented in Lemma 3.
Step 1: Recall that f (x) =

1
x(x + 1)
exp{−
}, then
x+2
x+2

f (x) = −

x2 + 5x + 4 − x(x+1)
e x+2 .
(x + 2)3

(4.153)

Note that the SNR γ > 0, hence f (γ) < 0, which implies that f (γ) is a monotonically
1
1
decreasing function. Therefore, when γ > f −1 (
), we have f (γ) <
, which implies
2Nc
2Nc
that
Nc − 2
γ(γ + 1)
Nc − 2
=
exp{−
} = (Nc − 2)f (γ) <
.
(4.154)
γ+2
γ+2
2Nc
109

Nc − 2
1
implies the following four conditions: < ,
2Nc
2
d2
1
Nc − 2
Nc − 2
Nc
<
, γ >
ln Nc and min > −2 ln 2(1 −
). First, when <
, it
2
Nc
2
Nc − 2
2Nc
σn
1
Nc − 2
Nc − 2
is obvious that < and <
. Next, we will show that when <
, we also
2
Nc
2Nc
1
have γ >
ln Nc . Consider Nc > 2 (i.e., Nc ≥ 3). Note that f (γ) is a monotonically
2
1
decreasing function. Hence, γ ≥ f −1 ( ) ≈ 1.02, which implies that γ > 1. Define h(γ)
6
2γ 2 −γ − γ + 2 , then
e
2
Step 2: We now show that

<

2
1
h (γ) = (4γ − 1)e2γ −γ − ,
2

(4.155)

2
h (γ) = [(4γ − 1)2 + 4]e2γ −γ .

(4.156)

Since h (γ) > 0, then h (γ) is a monotonically increasing function. When γ > 1, h (γ) >
1
h (1) = 3 · e − > 0, which implies that h(γ) is also a monotonically increasing function.
2
2
3
1 −γ
1
Hence, h(γ) > h(1) = e − > 0, which implies that
e > e−2γ . Then we have
2
γ+2
2
2
1 −γ
1 − γ(γ+1)
1
1
>
e γ+2 >
e > e−2γ ,
2Nc
γ+2
γ+2
2

(4.157)

Nc − 2
1
ln Nc . Therefore, we proved that when Nc > 2, <
2
2Nc
1
Nc − 2
Nc
implies γ >
ln Nc . Finally, when <
) < 0. Hence,
, we have −2 ln 2(1 −
2
2Nc
Nc − 2
d2
min > −2 ln 2(1 − Nc ) holds automatically.
2
Nc − 2
σn
√
2
1
−1 ( 1 ) and dmin > max( √ 2 ln Nc
Hence, when γ > f
, 2 ln
), all the condi√
2
2Nc
1−2
σn
2γ − ln Nc
tions in (4.141), (4.143), (4.146) and (4.151) are satisfied and W (α|x0 , y) − W (β|x0 , y) > 0
which implies that γ >

for any y ∈ Xi , i = 2, 4, 5, 6.
For M-PSK constellation with power Ps , we have s =

j 2πms
M

Ps e

and b =

2πm
j MJ

Ps e

where ms , mJ ∈ [0, M − 1]. It is shown in Section 4.10.2 that b − s 2 = 2Ps [1 −
d2
2π(ms − mJ )
2π
2π
1 2π
2π 2
cos
] and min = 2γ(1 − cos ). For large M , cos
≈ 1 − ( )2 = 1 − 2
2
M
M
M
2 M
σn
M
√
2
2
2
d
d
4π
2 ln Nc
and min ≈ 2 γ. Using this approximation, the condition min > √
can be
√
2
2
σn
M
σn
2γ − ln Nc
110

rewritten as
1
M2
ln Nc γ − 2
2
4π

γ2 −

2 ln Nc > 0,

(4.158)

which holds when
γ>

1
2

1
M2
ln Nc + 2
2
π

1
1
ln Nc +
2
2

2 ln Nc .

(4.159)

d2
1
Similarly, the condition min > 2 ln
can be rewritten as
2
1−2
σn
2
− 2π2 γ
e M
2
− 2π2 x
e M

+

γ(γ + 1)
2(Nc − 2)
exp{−
} < 1.
γ+2
γ+2

(4.160)

x(x + 1)
2(Nc − 2)
exp{−
}, then the condition in (4.160) holds
x+2
x+2
−1
when γ > f1 (1). Therefore, for a given PSK constellation of size M , the condition γ >
√
2
1
−1 ( 1 ) and dmin > max( √ 2 ln Nc
, 2 ln
f
) can be reduced to one condition on
√
2
2Nc
1−2
σn
2γ − ln Nc
SNR as
Define f1 (x)

γ > max[f −1 (

4.10.2

+

1
1
),
2Nc 2

1
1
ln Nc +
2
2

1
M2
ln Nc + 2
2
π

−1
2 ln Nc , f1 (1)].

(4.161)

Calculation of the Probability Matrix W1

Let x = αs, J = βb with α = v(k), β = v(j), and k, j ∈ Ic , s, b ∈ Ω. Assume Ω is an
M-PSK constellation with power Ps .
(i) When j = k, the received signal in the ith channel ri and corresponding Zi defined
in (4.9) can be calculated as


 s + b + n , i = k,
k
ri =

 ni ,
i = k,

Zi =









b + nk
2
s + b 2 + σn
ni − s
,
σn

, i = k,
(4.162)
i = k.

Note that n1 , . . . , nNc are i.i.d. circularly symmetric Gaussian random variables of zero
2
mean and variance σn . For any s, b ∈ Ω, Zk is a Rician random variable with PDF
√
z 2 +ν 2
zk − k 2
zk ν
Ps
pZ (zk ) = 2 e 2σ I0
for 0 ≤ zk < ∞, where ν =
and σ =
2
k
2
σ
σ
s + b 2 + σn

111

σn

; for i = k, Zi ’s are i.i.d. Rician random variables with PDF pZi (zi ) =
√
z 2 +ν 2
zi ν
1
Ps
zi − i 2
and σ = √ . We have
e 2σ I0
for 0 ≤ zi < ∞, where ν =
2
2
σn
σ
σ
2
2
2( s + b 2 + σn )

W1 (k|k, k)
=

1
|Ω|2

=

1
|Ω|2

=

P r{Zk < Zi , ∀i ∈ Ic , i = k|x = αs, J = βb}
s∈Ω b∈Ω

∞

s∈Ω b∈Ω 0

1≤i≤Nc ,i=k
∞

1
|Ω|2

Q1

s∈Ω b∈Ω 0
2 2
( s+b 2 +σn )zk +Ps
−
2
σn
I0
·e

P r{Zi > zk |x, J, zk }pZ (zk )dzk
k

√
2Ps √
, 2zk
σn
2zk
2
σn

Nc −1

2
2( s + b 2 + σn )zk
2
σn

2
Ps ( s + b 2 + σn ) dzk ,

(4.163)

where Q1 is the Marcum-Q function and I0 is the modified Bessel function of the first kind
with order zero. For M-PSK constellation with power Ps , we have s =
b=

2πmJ
j
Ps e M

j 2πms
M

and

where ms , mJ ∈ [0, M − 1], then (4.163) can be simplified as

1
W1 (k|k, k) =
M

M −1
κ=0

√

∞
0

Q1

2Ps √
, 2zk
σn

2
2[2Ps (1 + cos 2πκ ) + σn ]zk −
M
·
e
2
σn

·I0
where κ

Ps e

2zk
2
σn

Ps [2Ps (1 + cos

Nc −1

2 2
[2Ps (1+cos 2πκ )+σn ]zk +Ps
M
2
σn

2πκ
2
) + σn ] dzk ,
M

(4.164)

(ms − mJ ) mod M is uniformly distributed over [0, M − 1]. Since Zi ’s are i.i.d.

∀i ∈ Ic , i = k, then
W1 (i|k, k) =

1
[1 − W1 (k|k, k)], ∀i ∈ Ic , i = k.
Nc − 1

We have (P1): W1 (k|k, k) and W1 (i|k, k) are fixed values for any i, k ∈ Ic , i = k.

112

(4.165)

(ii) When j = k, the received signal ri and corresponding Zi can be calculated as


nk


,
i = k,


 s + nk , i = k,

2


Ps + σn



 b−s+n
j
(4.166)
ri =
Zi =
, i = j,
b + nj , i = j,


2
Ps + σ n





 n −s
 n,

i = j, k,

i
i

,
i = j, k.
σn
z2

z − k
For any s, b ∈ Ω, Zk is a Rayleigh random variable with PDF pZ (zk ) = k e 2σ2 , where σ =
k
σ2
2

2

z +ν
zj ν
zj − j 2
; Zj is a Rician random variable with PDF pZj (zj ) = 2 e 2σ I0
,
2
σ
σ2
2(Ps + σn )
b−s
σn
where ν =
and σ =
; for i = j, k, Zi ’s are i.i.d. Rician random
2
2
2(Ps + σn )
Ps + σ n
√
z 2 +ν 2
zi − i 2
Ps
1
zi ν
variables with PDF pZi (zi ) = 2 e 2σ I0
, where ν =
and σ = √ . Then,
σn
σ
σ2
2
W1 (k|k, j) can be calculated as

σn

W1 (k|k, j)
=

1
|Ω|2

=

1
|Ω|2

=

1
|Ω|2

P r{Zk < Zj and Zk < Zi , ∀i ∈ Ic , i = k, j|x, J}
s∈Ω b∈Ω

∞

s∈Ω b∈Ω 0

P r{Zj > zk |x, J, zk } [P r{Zi > zk |x, J, zk }]Nc −2 pZ (zk )dzk
k

∞
s∈Ω b∈Ω 0

Q1

√

2Ps √
, 2zk
σn

·QNc −2
1
=

1
M

M −1
κ=0

∞

Q1

0

√
·QNc −2
1

2
σn

2Ps √
, 2zk
σn

√
2
z
b−s , k
σn

2
2(Ps + σn )
σn

2
2zk (Ps + σn ) −
e
2
σn

Ps 1 − cos

2 2
(Ps +σn )zk
2
σn
dzk ,

2πκ zk
,
M

2
2zk (Ps + σn ) −
e
2
σn

113

2
2(Ps + σn )
σn

2 2
(Ps +σn )zk
2
σn
dzk ,

(4.167)

and
W1 (j|k, j)
=
=

=

1
|Ω|2
1
|Ω|2

P r{Zj < Zk and Zj < Zi , ∀i ∈ Ic , i = j, k|x, J}
s∈Ω b∈Ω

∞

P r{Zk > zj |x, J, zj } P r{Zi > zj |x, J, zj }

s∈Ω b∈Ω 0

2 2
(Ps +σn )zj
∞ −
2
σn
e
QNc −2
1

1
|Ω|2

s∈Ω b∈Ω 0
P +σ 2 2 b−s 2
−( s 2 n zj +
2 )
σn
σn
·e
I0

=

1
M

M −1

∞ −

e
κ=0

0

2zj
b−s
2
σn

2 2
(Ps +σn )zj
2
σn
QNc −2
1

P +σ 2 2
−[ s 2 n zj + 2Ps (1−cos 2πκ )]
2
M
σn
σn
·e
I0

√

pZj (zj )dzj

2
2zj (Ps + σn )
2
σn

2Ps √
, 2zj
σn

2
Ps + σn dzj

√
2Ps √
, 2zj
σn
2zj
2
σn

Nc −2

2
2zj (Ps + σn )
2
σn

2Ps (1 − cos

2πκ
2
)(Ps + σn ) dzj . (4.168)
M

Since Zi ’s are i.i.d. for any i ∈ Ic , i = j, k, then
W1 (i|k, j) =

1
[1 − W1 (k|k, j) − W1 (j|k, j)], i ∈ Ic , i = j, k.
Nc − 2

(4.169)

We have (P2): W1 (k|k, j), W1 (j|k, j) and W1 (i|k, j) are fixed values for any i, j, k ∈ Ic , j =
k, i = j, k.

114

Chapter 5
CONCLUSIONS AND FUTURE WORKS

5.1

Conclusions

This dissertation considered secure and effective communication in wireless networks under
hostile jamming attacks. First, we established a framework for the modeling and classification of cognitive jamming. Second, we proposed to enhance the spectral efficiency and
security of spread spectrum system by by integrating advanced signal processing techniques
and cryptographic techniques into the transceiver design. Finally, we analyzed the capacity
of the proposed systems under an effective jamming scenario. More specifically, based on
theoretical analysis and simulation results, we had the following conclusions:
On cognitive jamming modeling and classification:
• An innovative two-dimensional time-varying jamming model was proposed to characterize jamming signals in both time and frequency domain. It includes all the existing
jamming models as special cases, and can be used to characterize and track time-variant
jamming attacks.
• By examining the time-varying autocorrelation function and power spectral density, we
introduced the concepts of time-varying jamming coherence time and time-frequency
jamming coherence bandwidth. By comparing the jamming coherence time with the
signal symbol period, we classified the jamming into fast jamming and slow jamming;
By comparing the jamming coherence bandwidth with the signal bandwidth, we classified the jamming into flat jamming and frequency selective jamming.
• Based on relative power and correlation between the signal and the jamming, we introduced the concept of disguised jamming, which is complementary to the traditional
strong jamming. It was observed that strong jamming may not always be the worst
115

case. Disguised jamming, which is highly correlated with the signal and had a similar
power level, can be more harmful to the system, as it is more difficult to be detected
and eliminated.
• Based on time-frequency analysis and approximation theory, we proposed algorithms
for time-varying coherence time and time-frequency coherence bandwidth estimation
for both stationary jamming and locally stationary jamming. It enables dynamic jamming pattern detection and is of great significance for adaptive transmission design.
On spectrally efficient anti-jamming system design:
• By embedding part of information into the process of hopping frequency selection,
additional information transmission is achieved by MDFH with no extra cost on either
bandwidth or power, which increases the system spectral efficiency by multiple times.
• MDFH has distinctive performance under strong jamming and disguised jamming:
Under strong jamming, even if the signal is jammed, the power of the jammed signal is
enhanced and hence increased the probability of correct detection. As a result, MDFH
is particularly robust under strong jamming scenarios. When the system experiences
disguised jamming, it is difficult for the MDFH receiver to distinguish jamming from
the true signal, resulting in performance losses.
• To overcome the drawback of MDFH, we proposed the anti-jamming MDFH system, in
which a secure ID sequence is transmitted along with the information stream. The ID
sequence is generated through a cryptographic algorithm using the shared secret between the transmitter and the receiver, it is then exploited by the receiver for effective
signal extraction. It was shown that AJ-MDFH can effectively reduce the performance
degradation caused by disguised jamming, while remaining robust under strong jamming. AJ-MDFH also retained the high spectral efficiency of MDFH.

116

• We investigated ID constellation design and its impact on the performance of AJMDFH under both noise jamming and ID jamming. For ID jamming, it was shown that
under the ideal case when the system is noise-free, increasing the ID constellation size
can increase the ID uncertainty, and hence reduces the probability of error. When noise
is present, we proved that the detection error probability converges as the constellation
size goes to infinity. In other words, there exists a threshold, increasing the constellation
size over the threshold would result in little improvement in error probability. This
result justifies the use of practical, finite size constellations in AJ-MDFH.
• Single carrier AJ-MDFH was extended to multi-carrier AJ-MDFH (MC-AJ-MDFH).
By exploiting secure group generation algorithm, MC-AJ-MDFH can increase the system efficiency and jamming resistance significantly through jamming randomization
and enriched frequency diversity. Moreover, by assigning different carrier groups to
different users, MC-AJ-MDFH can also be used as a collision-free multiple access system.
• By incorporating the cryptographic techniques and secure permutation scheme, we
proposed a secure ID generation algorithm and a secure group generation algorithm to
maximize the security of the proposed AJ-MDFH and MC-AJ-MDFH systems.
On capacity analysis of MDFH based systems under disguised jamming
• Using the AVC model, we showed that the AVC corresponding to MDFH is symmetric,
which implies that the deterministic capacity of MDFH is zero.
• For AJ-MDFH, due to the shared randomness between the transmitter and receiver
provided by the secure ID sequence, we proved that the corresponding AVC is nonsymmetrizable, which implies that the deterministic capacity of AJ-MDFH is positive.
• We calculated the capacity of AJ-MDFH under ID jamming and showed that it converges as the ID constellation size goes to infinity. This echoed our result in AJ-MDFH
117

system design, where we showed that the probability of error of AJ-MDFH converges
as the ID constellation size goes to infinity.
• We extended the capacity analysis to the multiuser AJ-MDFH system (MC-AJ-MDFH)
and showed that it outperforms the multiple access scheme for conventional FH (FHMA).
• We observed that shared secure randomness between the transmitter and receiver plays
a critical role in anti-jamming system design.

5.2

Future Work

We plan to extend our research in the following two directions.

5.2.0.1

Disguised Jamming Analysis under Different Wireless Systems

In this dissertation, it can be observed that disguised jamming can degrade the performance
of MDFH significantly by mimicking the signal of legal user. To prevent the disguised
jamming attack to other popular wireless systems such as OFDM and MIMO, it is essential
to extend the disguised jamming research to these systems as well. The future research in
disguised jamming can be extended to
• Identify and quantitatively characterize disguised jamming for different wireless systems.
• Analyze the performance of these systems under disguised jamming.
• Propose effective methods to combat disguised jamming for these systems.

5.2.0.2

Adaptive Transceiver Design under Cognitive Jamming

Based on the proposed cognitive jamming modeling and classification scheme, we will consider adaptive transceiver design under cognitive jamming. For this purpose, we will conduct

118

comprehensive performance analysis on DSSS system, collision-free FH system, MDFH system, OFDMA based collision-free FH system. Each system will be tested and analyzed under
various jamming attacks. The transmitter can then be adjusted accordingly to achieve optimal jamming mitigation under cognitive jamming. Adaptive transceiver design will include
• Parameter adjustment or reconfiguration of a particular selected anti-jamming system.
• Cognitive switching among different types of anti-jamming systems.

119

APPENDIX

120

Appendix A
LIST OF ABBREVIATIONS AND ACRONYMS

3G

Third Generation

AES

Advanced Encryption Standard

AJ-MDFH

Anti-Jamming Message-Driven Frequency Hopping

AWGN

Additive White Gaussian Noise

AVC

Arbitrarily Varying Channel

BER

Bit Error Rate

BPF

BandPass Filter

BPSK

Binary Phase-Shift Keying

CDMA

Code Division Multiple Access

DPSK

Differential Phase-Shift Keying

DS-CDMA

Direct-Sequence Code Division Multiple Access

DSSS

Direct-Sequence Spread Spectrum

E-MDFH

Enhanced Message-Driven Frequency Hopping

FFH

Fast Frequency Hopping

FH

Frequency Hopping

FHMA

Frequency Hopping Multiple Access

FHSS

Frequency Hopping Spread Spectrum

FSK

Frequency-Shift Keying

IFFT

Inverse Fast Fourier Transform

ISI

Inter-Symbol Interference

JSR

Jamming-to-Signal Ratio

LFSR

Linear Feedback Shift Register

MAC

Medium Access Control

121

MAI

Multiple-Access Interference

MAP

Maximum A Posteriori

MC-AJ-MDFH

Multi-Carrier Anti-Jamming Message-Driven Frequency Hopping

MDFH

Message-Driven Frequency Hopping

MFSK

Multiple Frequency-Shift Keying

ML

Maximum Likelihood

MMSE

Minimum Mean Square Error

MUI

Multiuser Interference

PAM

Pulse Amplitude Modulation

PHY

Physical

PN

Pseudo-Random

PSD

Power Spectral Density

PSK

Phase-Shift Keying

OFDM

Orthogonal Frequency Division Multiplexing

OFDMA

Orthogonal Frequency Division Multiple Access

QAM

Quadrature Amplitude Modulation

QPSK

Quadrature Phase-Shift Keying

SFH

Slow Frequency Hopping

SINR

Single-to-Interference-Noise Ratio

SNR

Signal-to-Noise Ratio

STFT

Short-Time Fourier Transform

WSS

Wide Sense Stationary

122

BIBLIOGRAPHY

123

BIBLIOGRAPHY

[1]

G. Stuber, J. Barry, S. Mclaughlin, Y. Li, M. Ingram, and T. Pratt, “Broadband
MIMO-OFDM wireless communications,” Proceedings of the IEEE, vol. 92, no. 2, pp.
271–294, 2004.

[2]

H. Sampath, S. Talwar, J. Tellado, V. Erceg, and A. Paulraj, “A fourth-generation
MIMO-OFDM broadband wireless system: design, performance, and field trial results,” IEEE Communications Magazine, vol. 40, no. 9, pp. 143 – 149, Sep. 2002.

[3]

A. Ghosh, D. Wolter, J. Andrews, and R. Chen, “Broadband wireless access with
WiMax/802.16: current performance benchmarks and future potential,” IEEE Communications Magazine, vol. 43, no. 2, pp. 129–136, 2005.

[4]

T. Teng, F. Sideco, and J. Rebello, “Mobile Handset Market Tracer,” iSuppli,
Tech. Rep., 2010. [Online]. Available: http://www.isuppli.com/Mobile-and-WirelessCommunications

[5]

J. Allen and J. Wilson, “Securing a wireless network,” in Proceedings of the 30th annual
ACM SIGUCCS conference on User services. ACM, 2002, p. 215.

[6]

M. Maxim and D. Pollino, Wireless security. McGraw-Hill Osborne Media, 2002.

[7]

B. Potter, “Wireless security’s future,” IEEE Security and Privacy, vol. 1, no. 4, pp.
68–72, 2003.

[8]

P. Ashley, H. Hinton, and M. Vandenwauver, “Wired versus wireless security: the
Internet, WAP and iMode for E-Commerce.” IEEE Computer Society, 2001, p. 0296.

[9]

M. Frater, M. Ryan, and M. Ryan, Electronic warfare for the digitized battlefield.
Artech House Publishers, 2001.

[10] A. Goldsmith and P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Transactions on Information Theory, vol. 43, no. 6, pp. 1986 –1992, Nov.
1997.
[11] R. Cheng and S. Verdu, “Gaussian multiaccess channels with ISI: capacity region and
multiuser water-filling,” IEEE Transactions on Information Theory, vol. 39, no. 3, pp.
773 –785, May 1993.
[12] R. Knopp and P. Humblet, “Information capacity and power control in single-cell multiuser communications,” in Proceedings of IEEE International Conference on Communications, Jun. 1995.
[13] K. Witrisal, Y.-H. Kim, and R. Prasad, “A new method to measure parameters of
frequency-selective radio channels using power measurements,” IEEE Transactions on
Communications, vol. 49, no. 10, pp. 1788 –1800, Oct. 2001.
124

[14] W. Jakes, “Microwave mobile communications,” 1975.
[15] D. Godard, “Channel equalization using a kalman filter for fast data transmission,”
IBM Journal of Research and Development, vol. 18, no. 3, pp. 267–273, 1974.
[16] S. Qureshi, “Adaptive equalization,” Proceedings of the IEEE, vol. 73, no. 9, pp. 1349–
1387, 1985.
[17] L. Tong, G. Xu, and T. Kailath, “Blind identification and equalization based on secondorder statistics: A time domain approach,” IEEE Transactions on Information Theory,
vol. 40, no. 2, pp. 340–349, 1994.
[18] D. Falconer, S. Ariyavisitakul, A. Benyamin-Seeyar, and B. Eidson, “Frequency domain
equalization for single-carrier broadband wireless systems,” IEEE Communications
Magazine, vol. 40, no. 4, pp. 58–66, 2002.
[19] P. Schniter, “Low-complexity equalization of ofdm in doubly selective channels,” IEEE
Transactions on Signal Processing, vol. 52, no. 4, pp. 1002–1011, 2004.
[20] D. Kivanc, G. Li, and H. Liu, “Computationally efficient bandwidth allocation and
power control for OFDMA,” IEEE Transactions on Wireless Communications, vol. 2,
no. 6, pp. 1150–1158, 2003.
[21] Q. Spencer, C. Peel, A. Swindlehurst, and M. Haardt, “An introduction to the multiuser MIMO downlink,” IEEE Communications Magazine, vol. 42, no. 10, pp. 60–67,
2004.
[22] H. Sampath, P. Stoica, and A. Paulraj, “Generalized linear precoder and decoder
design for MIMO channels using the weighted MMSE criterion,” IEEE Transactions
on Communications, vol. 49, no. 12, pp. 2198–2206, 2001.
[23] E. Dahlman, 3G evolution: HSPA and LTE for mobile broadband.
2008.

Academic Press,

[24] J. Zyren and W. McCoy, “Overview of the 3GPP long term evolution physical layer,”
Freescale Semiconductor, Inc., white paper, 2007.
[25] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications Handbook. McGraw-Hill, 1994.
[26] D. Adamy, Introduction to electronic warfare modeling and simulation. Artech House
Publishers, 2003.
[27] B. Levitt, “FH/MFSK performance in multitone jamming,” IEEE Journal on Selected
Areas in Communications, vol. 3, no. 5, pp. 627–643, 1985.
[28] R. Pickholtz, D. Schilling, and L.B.Milstein, “Theory of spread spectrum communications - a tutorial,” IEEE Transactions on Communications, vol. 30, pp. 855–884, May
1982.

125

[29] C. Cook and H. Marsh, “An introduction to spread spectrum,” IEEE Communications
Magazine, vol. 21, pp. 8–16, Mar. 1983.
[30] P. Crepeau, “Performance of FH/BFSK with generalized fading in worst case partialband gaussian interference,” IEEE Journal on Selected Areas in Communications,
vol. 8, pp. 884–886, Jun. 1980.
[31] M. Pursley and W. Stark, “Performance of reed-solomon coded frequency-hop spreadspectrum communications in partial-band interference,” IEEE Transactions on Communications, vol. 33, pp. 767–774, Aug. 1985.
[32] W. Stark, “Coding for frequency-hopped spread-spectrum communication with partialband interference-part ii,” IEEE Transactions on Communications, vol. 33, pp. 1045–
1057, Oct. 1985.
[33] J.-W. Moon, J. Shea, and T. Wong, “Jamming estimation on block-fading channels,”
in Proceedings of IEEE Military Communications Conference, vol. 3, Oct. 31- Nov. 3
2004, pp. 1310–1316.
[34] J. Tan and G. Stuber, “Multicarrier spread spectrum system with constant envelope:
antijamming, jamming estimation, multiuser access,” IEEE Transactions on Wireless
Communications, vol. 4, pp. 1527– 1538, Jul. 2005.
[35] J.-W. Moon, J. Shea, and T. Wong, “Collaborative mitigation of partial-time jamming
on nonfading channels,” IEEE Transactions on Wireless Communications, vol. 5, pp.
1371–1381, Jun. 2006.
[36] A. Viterbi, “Spread spectrum communications: myths and realities,” IEEE Communications Magazine, vol. 17, no. 3, pp. 11–18, 1979.
[37] R. Peterson, R. Ziemer, and D. Borth, Introduction to spread-spectrum communications. Prentice Hall, 1995.
[38] B. Lathi, Modern digital and analog communication systems, 3rd ed. Oxford University
Press, 1995.
[39] L. Milstein, S. Davidovici, and D. Schilling, “The effect of multiple-tone interfering signals on a direct sequence spread spectrum communication system,” IEEE Transactions
on Communications, vol. 30, no. 3, pp. 436–446, 1982.
[40] L. Milstein, “Interference rejection techniques in spread spectrum communications,”
Proceedings of the IEEE, vol. 76, no. 6, pp. 657–671, 1988.
[41] S. Zhou, G. Giannakis, and A. Swami, “Digital multi-carrier spread spectrum versus
direct sequence spread spectrum for resistance to jamming and multipath,” IEEE
Transactions on Communications, vol. 50, no. 4, pp. 643–655, 2002.
[42] J. Laster and J. Reed, “Interference rejection in digital wireless communications,”
IEEE Signal Processing Magazine, vol. 14, no. 3, pp. 37–62, 1997.

126

[43] S. Buzzi, M. Lops, and H. Poor, “Code-aided interference suppression for DS/CDMA
overlay systems,” Proceedings of the IEEE, vol. 90, no. 3, pp. 394–435, 2002.
[44] M. Mihaljevi´ and J. Goli´, “A comparison of cryptanalytic principles based on iterac
c
tive error-correction,” Advances in Cryptology, vol. 547, pp. 527–531, 1991.
[45] ——, “Convergence of a Bayesian iterative error-correction procedure on a noisy shift
register sequence,” Advances in Cryptology, vol. 658, pp. 124–137, 1993.
[46] S. Verdu, Multiuser detection. Cambridge University Press, 1998.
[47] A. Duel-Hallen, J. Holtzman, and Z. Zvonar, “Multiuser detection for CDMA systems,”
IEEE Personal Communications, vol. 2, no. 2, pp. 46–58, 1995.
[48] T. Li, Q. Ling, and J. Ren, “A spectrally efficient frequency hopping system,” in
Proceedings of IEEE Global Telecommunications Conference, Nov. 2007, pp. 2997–
3001.
[49] Q. Ling and T. Li, “Message-driven frequency hopping: Design and analysis,” IEEE
Transactions on Wireless Communications, vol. 8, no. 4, pp. 1773–1782, Apr. 2009.
[50] Q. Ling, J. Ren, and T. Li, “Spectrally efficient spread spectrum system design:
message-driven frequency hopping,” Proceedings of IEEE International Conference on
Communications, pp. 4775–4779, May 2008.
[51] T. Ericson, “Exponential error bounds for random codes in the arbitrarily varying
channel,” IEEE Transactions on Information Theory, vol. 31, no. 1, pp. 42 – 48, Jan.
1985.
[52] R. Ahlswede, “Elimination of correlation in random codes for arbitrarily varying channels,” Probability Theory and Related Fields, vol. 44, no. 2, pp. 159–175, 1978.
[53] I. Csiszar and P. Narayan, “The capacity of the arbitrarily varying channel revisited:
Positivity, constraints,” IEEE Transactions on Information Theory, vol. 34, no. 2, pp.
181–193, 1988.
[54] N. Pronios and A. Polydoros, “Jamming optimization in fully-connected, spreadspectrum networks,” in Proc. IEEE Military Commun. Conf. IEEE, pp. 65–70.
[55] M. Pursley and J. Skinner, “Turbo product coding in frequency-hop wireless communications with partial-band interference,” in Proceedings of IEEE Military Communications Conference, vol. 2, Oct. 2002, pp. 774–779.
[56] R. Nikjah and N. Beaulieu, “On jamming capacity of general multiuser CDMA systems,” in Proc. IEEE Wireless Commun. Networking Conf. IEEE, 2007, pp. 191–196.
[57] L. Zhang, H. Wang, and T. Li, “Jamming resistance reinforcement of message-driven
frequency hopping,” in Proceedings of IEEE International Conference on Acoustics,
Speech and Signal Processing, Mar. 2010, pp. 3974 –3977.

127

[58] M. Priestley, “Evolutionary spectra and non-stationary processes,” Journal of the
Royal Statistical Society. Series B, pp. 204–237, 1965.
[59] ——, Non-linear and non-stationary time series analysis. Academic Press, 1988.
[60] R. Dahlhaus, “On the kullback-leibler information divergence of locally stationary
processes,” Stochastic processes and their applications, vol. 62, no. 1, pp. 139–168,
1996.
[61] S. Mallat, G. Papanicolaou, and Z. Zhang, “Adaptive covariance estimation of locally
stationary processes,” Annals of Statistics, vol. 26, no. 1, pp. 1–47, 1998.
[62] N. Saito and R. Coifman, “Local discriminant bases and their applications,” Journal
of Mathematical Imaging and Vision, vol. 5, no. 4, pp. 337–358, 1995.
[63] R. Coifman and M. Wickerhauser, “Entropy-based algorithms for best basis selection,”
IEEE Transactions on Information Theory, vol. 38, no. 2, pp. 713 –718, Mar. 1992.
[64] D. Manolakis, V. Ingle, and S. Kogon, Statistical and adaptive signal processing.
Artech House, 2005, vol. 1.
[65] G. Cooper and R. Nettleton, “A spread spectrum technique for high capacity mobile
communuzation,” IEEE Transactions on Vehicular Technology, vol. 27, pp. 264–275,
Nov. 1978.
[66] A. Viterbi, “A processing-satellite transponder for multlple access by low rate mobile
users,” IEEE Journal on Selected Areas in Communications, Oct. 1978.
[67] M. Simon and A. Polydoros, “Coherent detection of frequency-hopped quadrature
modulations in the presence of jamming–part I: QPSK and QASK modulations,”
IEEE Transactions on Communications, vol. 29, pp. 1644–1660, Nov. 1981.
[68] R. Pickholtz, D. Schilling, and L. Milstein, “Theory of spread-spectrum communications: A tutorial,” IEEE Transactions on Communications, vol. 30, no. 5, pp. 855–884,
May 1982.
[69] M. Pursley and W. Stark, “Performance of reed-solomon coded frequency-hop spreadspectrum communications in partial-band interference,” IEEE Transactions on Communications, vol. 33, pp. 767–774, Aug. 1985.
[70] K. Choi and K. Cheun, “Performance of asynchronous slow frequency-hop multipleaccess networks with MFSK modulation,” IEEE Transactions on Communications,
vol. 48, no. 2, pp. 298–307, Feb. 2000.
[71] L.-L. Yang and L. Hanzo, “Overlapping M-ary frequency shift keying spread-spectrum
multiple-access systems using random signal sequences,” IEEE Transactions on Vehicular Technology, vol. 48, no. 6, pp. 1984–1995, Nov. 1999.

128

[72] S. Glisic, Z. Nikolic, N. Milosevic, and A. Pouttu, “Advanced frequency hopping modulation for spread spectrum WLAN,” IEEE Journal on Selected Areas in Communications, vol. 18, no. 1, pp. 16–29, Jan. 2000.
[73] J. Cho, Y. Kim, and K. Cheun, “A novel frequency-hopping spread-spectrum multipleaccess network using M-ary orthogonal Walsh sequence keying,” IEEE Transactions
on Communications, vol. 51, no. 11, pp. 1885–1896, Nov. 2003.
[74] K. Choi and K. Cheun, “Maximum throughput of FHSS multiple-access networks
using MFSK modulation,” IEEE Transactions on Communications, vol. 52, pp. 426–
434, Mar. 2004.
[75] Y. Kim, K. Cheun, and K. Yang, “A bandwidth-power efficient modulation scheme
based on quaternary quasi-orthogonal sequences,” IEEE Communications Letters,
vol. 7, no. 7, Jul. 2003.
[76] Advanced Encryption Standard, FIPS-197, National Institute of Standards and Technology Std., Nov. 2001.
[77] J. Lee and L. Miller, “Error performance analyses of differential phase-shiftkeyed/frequency-hopping spread-spectrum communication system in the partial-band
jamming environments,” IEEE Transactions on Communications, vol. 30, no. 5, pp.
943–952, May 1982.
[78] J. Kang and K. Teh, “Performance of coherent fast frequency-hopped spread-spectrum
receivers with partial-band noise jamming and AWGN,” IEE Proc. Commun., vol. 152,
no. 5, pp. 679–685, Oct. 2005.
[79] C. Esli and H. Delic, “Antijamming performance of space-frequency coding in partialband noise,” IEEE Transactions on Vehicular Technology, vol. 55, no. 2, pp. 466–476,
Mar. 2006.
[80] L. Zhang, J. Ren, and T. Li, “Spectrally efficient anti-jamming system design using
message-driven frequency hopping,” in Proceedings of IEEE International Conference
on Communications, Jun. 2009.
[81] T. Li, Q. Ling, and J. Ren, “Physical layer built-in security analysis and enhancement
algorithms for CDMA systems,” EURASIP J. Wireless Commun. Networking, vol.
2007, pp. Article ID 83 589, 7 pages, 2007.
[82] J. G. Proakis and M. Salehi, Digital Communications, 5th ed.
Hill, 2008.

New York: McGraw-

[83] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities:
Cauchy’s Equation and Jensen’s Inequality, 2nd ed. Springer, 2009.
[84] R. Viswanathan and K. Taghizadeh, “Diversity combining in FH/BFSK systems to
combat partial band jamming,” IEEE Transactions on Communications, vol. 36, no. 9,
pp. 1062–1069, Sep. 1988.
129

[85] J. Lee, L. Miller, and Y. Kim, “Probability of error analyses of a BFSK frequencyhopping system with diversity under partial-band jamming interference–part II: Performance of square-law nonlinear combining soft decision receivers,” IEEE Transactions
on Communications, vol. 32, no. 12, pp. 1243–1250, Dec. 1984.
[86] L. Miller, J. Lee, and A. Kadrichu, “Probability of error analyses of a BFSK frequencyhopping system with diversity under partial-band jamming interference–part III: Performance of a square-law self-normalizing soft decision receiver,” IEEE Transactions
on Communications, vol. 34, no. 7, pp. 669–675, Jul. 1986.
[87] S. Stein, “Unified analysis of certain coherent and noncoherent binary communications
systems,” IEEE Transactions on Information Theory, vol. 10, no. 1, pp. 43–51, Jan.
1964.
[88] L. Zhang, H. Wang, and T. Li, “Anti-jamming message-driven frequency hopping:
Part I – system design,” IEEE Transactions on Wireless Communications, 2011, under
review.
[89] D. Blackwell, L. Breiman, and A. Thomasian, “The capacities of certain channel classes
under random coding,” The Annals of Mathematical Statistics, pp. 558–567, 1960.
[90] I. Csiszar and J. K¨rner, Information theory: coding theorems for discrete memoryless
o
systems. Academic press, 1981, vol. 244.
[91] I. Csisz´r and P. Narayan, “Arbitrarily varying channels with constrained inputs and
a
states,” IEEE Transactions on Information Theory, vol. 34, no. 1, pp. 27–34, 1988.
[92] A. Lapidoth and P. Narayan, “Reliable communication under channel uncertainty,”
IEEE Transactions on Information Theory, vol. 44, no. 6, pp. 2148 –2177, Oct. 1998.
[93] A. Sarwate, “Robust and adaptive communication under uncertain interference,” Technical Report No. UCB/EECS-2008-86, University of California at Berkeley, Tech. Rep.,
2008.
[94] J. Daemen and V. Rijmen, The Design of Rijndael: AES–the Advanced Encryption
Standard. Springer, 2002.
[95] J. Borden, D. Mason, and R. McEliece, “Some information theoretic saddlepoints,”
SIAM journal on control and optimization, vol. 23, p. 129, 1985.
[96] T. Basar and Y. W. Wu, “Solutions to a class of minimax decision problems arising in
communications systems,” J. Optim. Theory Appl., vol. 51, pp. 375–404, Dec. 1986.
[97] T. Ba¸ar, “The gaussian test channel with an intelligent jammer,” IEEE Transactions
s
on Information Theory, vol. 29, no. 1, pp. 152–157, 1983.
[98] T. Ba¸ar and G. Olsder, Dynamic noncooperative game theory. Society for Industrial
s
Mathematics, 1999, vol. 23.

130

[99] I. Stiglitz, “Coding for a class of unknown channels,” IEEE Transactions on Information Theory, vol. 12, no. 2, pp. 189 – 195, Apr. 1966.
[100] T. Cover and J. Thomas, Elements of information theory. Wiley-Interscience, 2006.
[101] J. Goh and S. Maric, “The capacities of frequency-hopped code-division multiple-access
channels,” IEEE Transactions on Information Theory, vol. 44, no. 3, pp. 1204–1211,
1998.
[102] A. D. Wyner, “Bounds on communication with polyphase coding,” Bell Labs Technical
Journal, pp. 523 –559, Apr. 1966.
[103] M. Simon and M. Alouini, “Exponential-type bounds on the generalized marcum qfunction with application to error probability analysis over fading channels,” IEEE
Transactions on Communications, vol. 48, no. 3, pp. 359 –366, Mar. 2000.

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