Covert Wireless Communications with Artificial Noise and ...

Covert Wireless Communications

with Artificial Noise and Channel

Uncertainties

Khurram Shahzad

June 2020

A thesis submitted for the degree of

Doctor of Philosophy

Research School of Electrical, Energy and Materials Engineering

College of Engineering and Computer Science

The Australian National University

c© Copyright by Khurram Shahzad (2020)

All Rights Reserved

Dedication

To my parents.

For the inspiration, drive and support they give me !

All I am is because of them.

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Declaration

The contents of this thesis are the results of original research and have not been

submitted for a higher degree to any other university or institution.

The research presented in this thesis has been performed jointly with Assoc. Prof.

Xiangyun Zhou (The Australian National University), Dr. Shihao Yan (Macquarie

University, Australia), Dr. Jinsong Hu (Fuzhou University, China), Prof. Feng Shu

and Dr. Jun Li (Nanjing University of Science and Technology, China).

The substantial majority of this work was my own.

Khurram Shahzad

Research School of Electrical, Energy and Materials Engineering,

College of Engineering and Computer Science,

The Australian National University,

Canberra, ACT 2601,

Australia.

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Acknowledgements

Alhamdulillah !!!

I express my utmost praise and gratitude to Allah, The All-Knowing and The

Most Merciful, for reaching this stage in my career.

I gratefully thank my supervisor Assoc. Prof. Xiangyun (Sean) Zhou for his

continuous support and understanding over the course of my PhD program. I

appreciate his always welcoming attitude for technical discussions and making time

for our weekly meetings. His ability to critically analyse and reaching the crux

of technical issues, always keeping in view the bigger picture is remarkable and

highly admirable. He has taught me analytical thinking, enhanced my writing and

presentation skills and I have learned a lot from his intuitive approach to modeling

and solving problems. I will always be proud to call myself his student and thank

him for giving me the opportunity to work with him.

I would like to thank Assoc. Prof. Salman Durrani and Dr. Nan (Jonas)

Yang for agreeing to be a part of my thesis committee. I am grateful to Salman

for always being supportive and understanding, sharing his sound advice in both

personal and professional matters. More than often, he not only lent a listening

ear, but also offered strong moral and emotional support. I am also thankful to

Salman and Sean for providing me with tutoring opportunities, which not only

helped me greatly in financial matters but also provided with valuable experiences

in my academic career.

Special thanks are due to Dr. Shihao Yan for his help over the course of my

research. Although he was not formally on my thesis committee, majority of the

work carried out during my PhD was in close collaboration with him. He not only

listened to my often absurd ideas with patience but also guided me in the right

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direction. I wish him all the best for his future.

It has been a great pleasure to work in such a supportive and friendly environ-

ment, and I would like to thank all the members of the Communications and Signal

Processing research groups for making it a memorable experience. In particular, I

would like to mention Usama Elahi, Sheeraz Alvi, Noman Akbar and Hassan Iqbal

(Research School of Physics) for their friendship and comradery, and will always

cherish the fun and laughter we had together. I am especially thankful to Usama

for always being there for me and listening to me whenever I felt under the weather.

I am thankful to my parents for their unconditional love and support. I greatly

appreciate all the hardships they have endured and sacrifices they have made during

their whole life for letting me and my siblings pursue our dreams. My father took

a great decision to leave the village for the sake of our education and it will always

resonate with me to provide inspiration and encouragement. My parents have

always been my biggest supporters, providing endless inspirations, and this journey

would not have been possible for me without their continuous encouragement. I

also express my gratefulness to my sisters for their kind love, prayers and assurance.

I would like to express my deepest love, gratitude and appreciation to my wife,

Memoona, and my son, Arham, for their support, understanding and prayers. I am

thankful to my wife for bearing the hardships of student life with me and providing

me with a wonderful companionship. She has been very supportive throughout this

journey, providing me strength and support at hard times. I am thankful to my

son, for bringing a lot of joy in my life since the day he was born and whose smile

always takes away the tiredness of long days at work.

Abstract

Traditional approaches to wireless communication security (e.g., encryption) focus

on maintaining the message integrity so that the contents are only accessible to

the intended recipient. However, detection of the mere presence of a transmission

can have a negative impact, violating the privacy of the communicating parties.

In contrast, Covert Communications (also known as Low Probability of Detection

Communications) hide the transmission of a message from a watchful adversary

while ensuring a certain decoding performance at the receiver. In this thesis, we

focus on exploiting any existing or induced uncertainties at the adversary, develop-

ing novel methods to achieve covertness in wireless scenarios. The insights gained

from this thesis aim to help alleviate the ever-increasing security and privacy con-

cerns in future wireless networks.

The first half of the thesis examines the use of artificial noise (AN) to cause

sufficient confusion at the adversary such that message transmissions cannot be

detected. We first consider a full-duplex information receiver, who generates AN

of varying power causing uncertainty in the adversary’s received signal statistics.

Although the transmission of this AN causes self-interference, it provides the op-

portunity of achieving covertness under carefully managed transmit power levels.

Here, we provide design guidelines for the choice of AN transmission power range.

Furthermore, we demonstrate that if the transmission probability and AN power

can be jointly optimized, the prior transmission probability of 0.5, which amounts

to a random guess by the adversary, is not always the best choice for achieving

maximum covertness. Rather, increasing the transmission probability beyond 0.5

allows an increase in the AN transmit power for satisfying a given covert rate

requirement and can be the difference between strong covertness and almost no

covertness at all.

Relying on the use of AN, we next consider achieving covertness in the domain

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of backscatter radio systems. We assume that the tag (containing the information)

is passive and the reader (transceiver) controls the transmit power to keep the

tag’s response hidden. A non-conventional transmission scheme is proposed where

the reader emits a noise-like signal with transmit power varying across different

communication slots. We analyse the conditions on the transmit power to achieve

a target level of covertness and illustrate the price a backscatter system has to pay

for achieving covert communication.

In the second half of the thesis, we focus on scenarios where users suffer from

uncertainty in their channel knowledge under quasi-static fading conditions. We

first focus on the case where the adversary can make an infinite number of obser-

vations in a time slot, and a public action is used to provide cover for a secret

action. It has been demonstrated that although channel uncertainty adversely

affects the information at the intended receiver, it also provides the opportunity

to hide any transmissions. Secondly, under a finite blocklength assumption, we

investigate Willie’s optimal detection performance in two extreme cases, i.e., the

case of perfect channel state information (CSI) and the case of channel distribu-

tion information (CDI) only. In the large detection error regime, Willie’s detection

performances in these two extreme cases are essentially indistinguishable, implying

that the quality of CSI does not help Willie in improving his detection performance.

We, thus, reveal fundamental differences in the design of covert transmissions for

quasi-static fading channels in comparison to non-fading AWGN channels.

List of Publications

The work in this thesis has been published or has been submitted for publication

as journal articles or conference proceedings. These papers are:

Journal Articles

J1. K. Shahzad, X. Zhou, S. Yan, J. Hu, F. Shu, and J. Li, “Achieving Covert

Wireless Communications Using a Full-Duplex Receiver,” IEEE Trans. Wire-

less Commun., vol. 17, no. 12, pp. 8517-8530, Dec. 2018.

J2. K. Shahzad, and X. Zhou, “Covert Wireless Communications under Quasi-

Static Fading with Channel Uncertainty,” submitted to IEEE Trans. Inf.

Forensics Security, Oct. 2019.

Conference Proceedings

C1. K. Shahzad, and X. Zhou, “Covert Communication in Backscatter Radio,”

in Proc. IEEE Int. Conf. on Communications, ICC’2019, Shanghai, China,

pp. 1-6, May. 2019.

C2. K. Shahzad, X. Zhou, and S. Yan, “Covert communication in Fading Chan-

nels under Channel Uncertainty,” in Proc. IEEE Vehicular Technology Con-

ference, VTC’2017, Sydney, Australia, pp. 1-5, Jun. 2017.

The following publications are also the result of my Ph.D. study but not included

in this thesis:

Journal Article

J3. K. Shahzad, X. Zhou, and S. Yan, “Covert Wireless Communication in Pres-

ence of a Multi-Antenna Adversary and Delay Constraints,” IEEE Trans. Veh.

Technol., vol. 68, no. 12, pp. 12432-12436, Dec. 2019.

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Conference Proceeding

C3. K. Shahzad, “Relaying via Cooperative Jamming in Covert Wireless Com-

munications,” in Proc. Int. Conf. Signal Processing and Communication

Systems ICSPCS’2018, Cairns, Australia, pp. 1-6 Dec. 2018.

List of Abbreviations

AN artificial noise

AWGN additive white Gaussian noise

BER bit error rate

BSC binary symmetric channel

BPSK binary phase shift keying

CAS centralized antenna system

CDI channel distribution information

CSCG circularly symmetric complex Gaussian

CSI channel state information

CW continuous wave

DAS distributed antenna system

DMC discrete memoryless channel

DSSS direct sequence spread spectrum

FD full-duplex

FHSS frequency hopping spread spectrum

i.i.d. independent and identically distributed

IoT internet of things

LPD low probability of detection

MAC multiple access channel

MIMO multiple-input multiple-output

MMSE minimum mean square error

PDF probability density function

PLS physical layer security

RF radio frequency

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RFID radio frequency identification

RV random variable

SNR signal to noise ratio

SINR signal to interference plus noise ratio

UAV unmanned aerial vehicle

UHF ultra high frequency

List of Notations

(·)∗ Conjugate operation

(·)† Complex conjugate operation

EX(·) Expectation operator with respect to X

D(P0||P1) Kullback-Leibler divergence between distributions P0 and P1

P(·) Probability measure

PFA Probability of false alarm

PMD Probability of missed detection

χ2n Chi-squared rv with n degrees of freedom

N (µ, σ2) Real normal distribution with mean µ and variance σ2

CN (µ, σ2) Complex normal distribution with mean µ and variance σ2

exp(·) Exponential function

ln(·) Natural logarithm

max(·, ·) The maximum value

| · | Magnitude of a complex number

U(a, b) Continuous Uniform distribution over a and b

Γ(·) Complete Gamma function

γ(·, ·) Lower incomplete Gamma function

Γ(·, ·) Upper incomplete Gamma function

ψ(·) Digamma function

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Contents

Dedication i

Declaration ii

Acknowledgements iv

Abstract vii

List of Publications ix

List of Abbreviations xi

List of Notations xiii

List of Figures xviii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Overview and Literature Survey of Covert Communications . . . . . 3

1.2.1 Basics of Covert Communications . . . . . . . . . . . . . . . 4

1.2.2 Spread Spectrum Techniques . . . . . . . . . . . . . . . . . . 6

1.2.3 Square Root Law for Covert Communications . . . . . . . . 8

1.2.4 Positive-Rate Covert Communications . . . . . . . . . . . . 9

1.3 Steganography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Thesis Overview and Contributions . . . . . . . . . . . . . . . . . . 12

2 Covert Communications Using a Full-Duplex Receiver with Arti-

ficial Noise 21

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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2.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Communication Scenario . . . . . . . . . . . . . . . . . . . . 22

2.2.2 Proposed Transmission Scheme . . . . . . . . . . . . . . . . 24

2.2.3 Willie’s Detection, Priors and Performance Metrics . . . . . 26

2.3 Detection Scheme at Willie . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Performance of Covert Communication . . . . . . . . . . . . . . . . 29

2.4.1 Transmission Outage Probability from Alice to Bob . . . . . 30

2.4.2 Expected Detection Error Probability at Willie . . . . . . . 31

2.5 Covert Communication Design . . . . . . . . . . . . . . . . . . . . . 31

2.5.1 Optimal Minimum AN Power . . . . . . . . . . . . . . . . . 32

2.5.2 Optimal Priors for Alice’s Transmission . . . . . . . . . . . . 33

2.5.3 Optimal Maximum AN Power . . . . . . . . . . . . . . . . . 35

2.6 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . 36

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Covert Communications in Backscatter Radio using Artificial

Noise 43

3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.1 Proposed Reader Transmission Scheme . . . . . . . . . . . . 45

3.2.2 Tag’s Operation and Willie’s Detection . . . . . . . . . . . . 46


3.4 Reader’s Strategy for Covertness . . . . . . . . . . . . . . . . . . . 52

3.5 Reader’s BER Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 53


3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Covert Communications within a Public Link under Channel Un-

certainty 59

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . 61


4.3.1 Detection using a Radiometer . . . . . . . . . . . . . . . . . 63

Contents xvii

4.3.2 Optimal Threshold for Willie’s Radiometer . . . . . . . . . . 63

4.4 Performance of Covert Communication . . . . . . . . . . . . . . . . 66

4.4.1 Average Detection Error Probability . . . . . . . . . . . . . 66

4.4.2 Outage Probabilities at Carol and Bob . . . . . . . . . . . . 68


4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Covert Communications with Channel Training and Finite Block-

length 73

5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2.2 Training-Based Transmission and Channel Estimation . . . . 76

5.2.3 Performance Metrics . . . . . . . . . . . . . . . . . . . . . . 77

5.3 Detection Analysis at Willie . . . . . . . . . . . . . . . . . . . . . . 78

5.3.1 Detection under Perfect CSI Knowledge . . . . . . . . . . . 78

5.3.2 Detection under Knowledge of CDI only . . . . . . . . . . . 80

5.3.3 Performance Comparison between CSI and CDI Cases . . . 80

5.4 Covertness under Channel Uncertainty . . . . . . . . . . . . . . . . 81

5.4.1 Covert Connection Probability . . . . . . . . . . . . . . . . . 82

5.4.2 Optimization of Transmit Power and Number Of Transmit

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4.3 Suboptimal Solution . . . . . . . . . . . . . . . . . . . . . . 85


5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6 Conclusions and Future Research Directions 93

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . 95

Appendix A 99

A.1 Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . 99

A.2 Proof of Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 103


xviii Contents


Appendix B 107

B.1 Proof of Lemma 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B.2 Proof of Lemma 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Bibliography 113

List of Figures

1.1 Basic covert communication model under consideration. Bob has to

decode information from Alice’s signal while Willie has to decide on

Alice’s transmission state. . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Illustration of a basic spread spectrum communication system. . . . 7

1.3 Different scenarios considered in the thesis for achieving covert com-

munication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Covert communications model under consideration with a FD receiver. 23

2.2 Willie’s observation model, presented as a slotted channel use by

Alice. Each slot contains n symbol periods and there is a certain a

priori probability, π1, of Alice’s communication to Bob in each slot. 25

2.3 Optimal maximum transmit power of Bob’s AN, P ∗max, versus the

covert rate requirement from Alice to Bob, τ , for varying values of

Alice’s transmit power, Pa. . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Optimal maximum transmit power of Bob’s AN, P ∗max, versus the


Bob’s self-interference cancellation coefficient, φ. . . . . . . . . . . . 38

2.5 Optimal choice of transmission probability, π∗1, versus the covert

rate requirement from Alice to Bob, τ , for varying values of Alice’s

transmit power, Pa. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Optimal choice of transmission probability, π∗1, versus the covert

rate requirement from Alice to Bob, τ , for varying values of Bob’s

self-interference cancellation coefficient, φ. . . . . . . . . . . . . . . 39

2.7 The expected detection error probability at Willie, P ∗E, versus the


Bob’s self-interference cancellation coefficient, φ. . . . . . . . . . . 39

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2.8 The expected minimum detection error probability at Willie, P ∗E,

versus Bob’s self-interference coefficient φ, for varying values of covert

rate requirement τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.9 The expected detection error probability at Willie, P ∗E, versus the

covert rate requirement from Alice to Bob, τ , under the proposed

scheme and under the approach where π1 = 1/2. . . . . . . . . . . 41

3.1 System model for covert communication in a backscatter system. . . 46

3.2 Ratio of Pmax and Pmin required for a target covertness. . . . . . . 55

3.3 BER Comparison of non-covert and covert communication schemes.

The tag’s reflection coefficient |Γ| = 0.8. . . . . . . . . . . . . . . . 56

4.1 Illustration of the Covert Communication Scenario . . . . . . . . . 60

4.2 The achievable rate region for Carol and Bob under the effect of

varying channel uncertainty, β. Other parameters are ε = 0.2, α = 3

and daw = dac = dab = 5. . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 The achievable rate region for Carol and Bob under the effect of

varying covertness requirement, ε. Other parameters are β = 0.2,

α = 3 and daw = dac = dab = 5. . . . . . . . . . . . . . . . . . . . . 71

5.1 Covert communications model under consideration. . . . . . . . . . 75

5.2 Willie’s minimum detection error probability, ζ∗w, vs. Alice’s data

transmit power, PD, under perfect CSI and CDI only cases for vary-

ing ND. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Comparison of the optimal data transmit power at Alice, P ∗D, under

the optimal and suboptimal solution vs. the covertness requirement, ε. 89

5.4 The optimal number of data symbols used by Alice, N∗D, under the

optimal and suboptimal solution vs. the covertness requirement, ε.

Note that all four curves in this figure overlap completely. . . . . . 90

5.5 The optimal throughput from Alice to Bob, N∗DRPcc, under the op-

timal approach, suboptimal approach and case of using ND,max vs.

the covertness requirement, ε. . . . . . . . . . . . . . . . . . . . . . 91

A.1 Case-I : |hbw|2Pmax + σ2w < |hbw|2Pmin + |haw|2Pa + σ2

w . . . . . . . . 100

A.2 Case-II : |hbw|2Pmax + σ2w ≥ |hbw|2Pmin + |haw|2Pa + σ2

w . . . . . . . 102

Chapter 1

Introduction

1.1 Motivation

Communication systems using electrical and electronic means have a huge impact

on the modern society. An inspection of the history of communication reveals that,

apart from physical transport, only a few methods appear for exchange of infor-

mation including smoke signals and signal flags. Early attempts to communicate

visual signals by means of the semaphore, a pole with movable arms, were made

in the 1830’s in France. The demonstration of electrical telegraphy by Henry and

Morse in 1832 followed shortly after the discovery of electromagnetism by Christian

Oersted and A-Marie Ampere early in the 1820’s. At the very beginning of the

20th century, developments in radio transmissions by Marconi paved the way for

modern wireless communication systems. Since then, wireless communication has

developed into an integral part of the society and wireless devices have attained an

important role in our day to day life due to their size, mobility and flexibility.

According to Statista Research [1], the number of wirelessly connected devices in

2019 exceeds the world population by more than three times, and the total installed

base of IoT-connected devices is projected to amount up to 75.44 billion worldwide

by 2025. This multi-fold increase in the use of wireless technology translates into

a higher reliance on modern wireless systems for exchange of sensitive and private

information. The results of Australian Cyberawareness Index 2019 [2] indicate

that nearly all (94%) of the survey respondents conducted financial transactions

online including online banking, paying bills, and online shopping. According to

1

2 Introduction

the Fifth Generation Public Private Partnership (5G-PPP), 5G will connect about

7 trillion wireless devices, shrink the average service creation time from 90 hours

to 90 minutes, and enable advanced user controlled privacy [3]. Consequently,

wireless communication systems are already of critical importance and their se-

curity and privacy takes an unmatched precedence in our society. The 5G and

beyond wireless communications will support massive transmission of important

and private information, including personal and financial data, military secrets

and mission-critical industrial control messages. However, the security solutions

and architectures used in previous mobile generations will apparently not suffice

for 5G and future architectures since the dynamics of new services and technolo-

gies such as Cloud Computing [4], Software Defined Networking [5] and Network

Function Virtualization [6] call for new security measures and architectures.

Security of wirelessly transmitted data has never been easy as the open air

interface of wireless transmission is accessible to both legitimate and illegitimate

users. This creates reasonable concerns over the security and privacy of information

transmitted through this medium. The recent remarkable increase in the amount

of information conveyed using the wireless medium has spurred an interest in both

research and academic communities regarding the development of new mechanisms,

enhancing the privacy and integrity of wirelessly transmitted data. On the other

hand, governments and private corporations are also determined to ensure that

their digital assets are properly protected, so that consumers can access information

in confidence. The security and privacy of information transmitted over the air has

always been a concern for wireless design engineers, with a recently renewed interest

owing to the advances and innovations in wireless technologies and their widespread

use in everyday activities.

Traditionally, cryptography has been used by system designers to provide se-

curity for wireless transmissions [7–11], where the message is encoded in such a

way that the eavesdropper or the unintended receiver is unable to decode the

message, at least not without significant computation. In fact, all cryptographic

measures are based on the premise that it is computationally infeasible for them to

be deciphered without knowledge of the secret key, which remains mathematically

unproven. In recent years, physical layer security (PLS) [12–19] has emerged as

an alternative to traditional cryptographic ways of securing wireless information,

1.2 Overview and Literature Survey of Covert Communications 3

where the mechanisms of key exchange and distribution in cryptography impose

varied challenges, especially in dynamic network environments. Furthermore, re-

cent advancements in computing technologies, especially through breakthroughs in

quantum computing [20, 21], have drastically improved the computational resources

available to potential adversaries and hence, have exposed encryption techniques

to certain vulnerabilities.

PLS techniques, on the other hand, introduce a level of information-theoretic

security through exploiting the uncertainties such as fading, interference, noise and

lack of predictability of the wireless channel, minimizing the information obtained

by an unauthorized eavesdropper. This line of work was pioneered by Wyner [22],

who introduced the wire-tap channel and established the possibility of creating

perfectly secure communication links without relying on secret keys, as needed

in common cryptography practices. Based on information theory, the design of

channel security capacity in PLS can be used to support wireless communication

security without encryption and decryption. PLS offers the advantage of being

unconstrained by adversary’s computational capability. Furthermore, it can be

used in conjunction with existing cryptographic approaches, improving the overall

wireless communication security. We note that PLS protects the communication at

the physical layer, while cryptography, being a higher layer approach, protects the

communication after the communication phase, thus they work in different domains

and provide different layers of protection. However, circumstances exist where not

only the privacy and integrity of the information are important, but the users

may also wish to avert any invigilation, looking to hide the very existence of their

communication. Such scenarios, although commonplace in military applications,

are now also arising in non-military situations, relating to civil unrest and even

monitoring of peoples daily activities.

1.2 Overview and Literature Survey of Covert

Communications

Keeping in view the recent concerns on wireless transmission security, and to

augment the existing approaches, a new viewpoint has recently been proposed

termed as Covert Communications or Low Probability of Detection Communica-

4 Introduction

tions [23, 24]. Covert communications advocate to offer a stronger security by

hiding the very existence of the wireless transmission itself. In contemporary so-

cial and political backdrops, there are situations where in addition to protecting

the information content of the transmission, it is imperative to hide the transmis-

sion. For example, hiding communications in a sensitive or hostile environment

is of paramount importance to military and law enforcement agencies. On the

other hand, landing of sensitive information, e.g., pertaining to health issues or

financial transactions of an individual, in the wrong hands can be exploited and

is highly undesirable. Detecting any covert transmissions is also highly desired by

law enforcement and cyber task forces since even the presence of such malicious

activities offers sufficient incentive for them to take action [24]. In above mentioned

and many other potential scenarios, covert communications offer a viable pathway

which can be used in conjunction with existing security approaches to enhance user

privacy.

1.2.1 Basics of Covert Communications

Covert communications intend to obscure the existence of any wireless transmission

from a watchful adversary, referred to as Willie in recent literature of covert com-

munications, while guaranteeing a certain decoding performance at the intended

receiver. These low probability of detection (LPD) communications have drawn

significant research attention and are materializing as a promising prospect for

shielding the future wireless communication networks from unapproved probing

and access.

The basic scenario of covert communication can be explained by considering

the situation shown in Fig. 1.1, where Alice looks to send a message to Bob over a

noisy channel in the presence of warden Willie. Alice has to transmit her message

in such a way that a certain decoding performance is guaranteed at Bob while

the transmission remains hidden from Willie. Willie, on the other hand, is not

interested in the content of the message and only wants to determine whether Alice

transmitted any message to Bob or not. We note that this is in strong contrast to

the role of an eavesdropper is traditional physical layer security schemes where the

eavesdropper is aware of the presence of a message transmission and looks to decode

the information transmitted from Alice to Bob. Thus while the problem at Bob is


Figure 1.1: Basic covert communication model under consideration. Bob has todecode information from Alice’s signal while Willie has to decide on Alice’s trans-mission state.

decoding the information send by Alice, the problem at Willie becomes a detection

one, looking to decide between the two possible states of Alice’s transmission.

From Willie’s detection perspective, the problem reduces to a binary hypothesis

testing problem, where the null hypothesis, H0, states that Alice did not transmit

while the alternative hypothesis, H1, states that Alice did transmit, hence convey-

ing information to Bob. We denote by PFA the probability of a false alarm, i.e.,

Willie declares that Alice did transmit while actually she did not, whereas PMD

denotes the probability of a missed detection, i.e., Alice actually transmits while

Willie fails to raise an alarm. Denoting by π0 and π1 the prior probabilities of

hypotheses H0 and H1, respectively, the total detection error at Willie is given

by π0PFA + π1PMD. Then from Alice’s point of view, the problem of covert com-

munication becomes of maximizing this detection error probability at Willie while

satisfying a certain decoding performance at Bob. Under the assumption of equal

priors of transmission by Alice, we consider Alice achieving covert communication

if, for any ε > 0, a communication scheme exists so that PFA + PMD ≥ 1 − ε, as

n → ∞ [25]. We also note that the detection error probability is no only affected

by Alice’s choices of transmission but also by other factors which may or may not

be beyond Alice and Willie’s control, and include but are not limited to channel

noise and fading, Willie’s receiver noise and presence of jammers and interferers.

Hence a robust design to achieve covertness consists of also exploiting any factors

or impairments that directly affect Willie’s detection capability.

Recent research efforts in the domain of covert communications have explored

different problems in this field, from establishing the achievable fundamental lim-

6 Introduction

its to exploiting any uncertainties at the potential adversary, including channel

noise, fading, network interference and jamming signals introducing uncertainties

in Willie’s observations. Current works in the domain of covert communications

are focused at the potential integration of these techniques into current and future

wireless networks, e.g., 5G and beyond, to protect the privacy and security of wire-

less system users. As pointed out earlier, hiding communications in sensitive or

hostile environments is of paramount importance to military and law enforcement

agencies. Research in this area thus enables the government agencies to enhance

national security and understand how to regulate the use of this new technology

in future wireless communications.

Covert communications technology has drawn significant research interests since

2013, owing to the seminal work presented in [25]. Existing research in this field can

be broadly categorized into three different directions with some significant overlaps.

The first direction works towards characterizing the fundamental limits (e.g., [26]),

which aim to identify the number of information bits that can be transmitted with

a negligible probability of detection under a given communication scenario. Next

are the works that consider encoding schemes to achieve covert communications,

focusing on practical encoding and characterizing the required size of shared secret

among the communicating parties in order to achieve the communication limits

(e.g., [27]). The final category targets enhancements of existing techniques and de-

veloping new schemes to improve covert communications performance in practical

scenarios. This thesis falls into this last category, where we explore strategies to

cause and exploit uncertainties, looking to maximize the detection errors at the

adversary, while augmenting the achievable covert information exchanged between

the covert communication pair.

1.2.2 Spread Spectrum Techniques

One of the closely related technologies to covert communications are spread spec-

trum techniques which aims at providing users with secrecy and privacy. Since its

emergence and early adoption in the 20th century, spread spectrum techniques have

been widely used to protect wireless RF signals against jamming and interference

while providing users with privacy and enabling multiple access of a shared radio

resource. In its most basic form, spreading is achieved by means of a spreading


Figure 1.2: Illustration of a basic spread spectrum communication system.

signal, generally called the spreading code or sequence, which is independent of the

transmitted data. At the receiver side, despreading is accomplished by correlating

the received signal with a synchronized replica of the spreading code. Fig. 1.2 pro-

vides an illustration of this general concept. Using these techniques, a baseband

signal bandwidth is intentionally spread over a large bandwidth, essentially sup-

pressing the power spectral density of the transmitted signal below the noise floor.

A detailed review of these techniques is available in [28–30]. Traditional spread

spectrum techniques include direct-sequence spread spectrum (DSSS), frequency-

hopping spread spectrum (FHSS) and their combinations.

When DSSS is used, the transmitter simply multiplies the signal waveform by

a spreading sequence - a randomly generated binary waveform with a bandwidth

significantly higher than the original signal. This spreading sequence is assumed

to be either shared between the information transmitter and receiver before the

start of the communication, or generated by a synchronized random sequence gen-

erator when needed. The resulting waveform thus possesses a higher bandwidth,

effectively having a reduced power spectral density and reducing the probability

of being detected by an adversary. The ratio (in dB) between the spread base-

band and the original signal is generally referred to as the processing gain, and

values of 10dB to 60dB of processing gains are typical in spread spectrum sys-

tems. We note here that spreading does not spare the limited frequency resource,

however the overuse is well compensated by the possibility that many users will

share the enlarged frequency band through orthogonal spreading sequences. The

receiver uses the same spreading sequence to de-spread the received waveform and

8 Introduction

obtain the original signal. Message privacy is provided by a direct-sequence system

since the transmitted message signal cannot be recovered without knowledge of the

spreading sequence used by the transmitter.

Different from DSSS, FHSS systems re-adjust the transmit frequency for ev-

ery symbol, where the available frequency band is divided into smaller sub-bands.

Similar to the spreading sequence discussed earlier, the frequency-hopping pattern

is generated randomly and is already shared between the communicating parties.

FHSS is a more robust technology with only very little influence from reflections,

noise and other environmental factors. The active system numbers in same ge-

ographical areas is higher than an equivalent number for direct sequence spread

spectrum systems. Thus it is suited well for installations designed to cover large

areas where numerous co-located systems are needed.

1.2.3 Square Root Law for Covert Communications

For a large part of the century, hiding wireless transmissions was addressed by

spread spectrum as the only existing solution in practical scenarios. However, the

fundamental performance limits of covertness under spread spectrum have not been

fully analyzed, which can be partly attributed to the fact that there was no clear

understanding on when or how often spread spectrum fails to hide wireless commu-

nications. As a result, the level of potential covertness achieved by spread spectrum

has not been fully realized. Due to these factors, spread spectrum usage deviated

from hiding wireless transmissions to obtaining high reliability and improved data

rate in the last two decades.

On the other hand, the fundamental limits of covert communications were estab-

lished in [25], presenting a square root law on the amount of information transmit-

ted reliably and with low probability of detection over additive white Gaussian noise

(AWGN) channels. This square root law states that no more than O (√n) bits can

be sent reliably from the transmitter, Alice, to the receiver, Bob, in n channel uses

while lower bounding Willie’s detection error probability of this transmission being

no less than a specific value ε. Under this square root law, we have a zero-rate chan-

nel since limn→0O(√n)/n = 0 bits/symbol. This result has been further extended

to discrete memoryless channels (DMCs), binary symmetric channels (BSCs) and

multiple access channels (MACs) in [27], [31] and [32], respectively. Expanding on


this asymptotic result, the scaling constant associated with the amount of covert

information with respect to√n was specified for AWGN channels and discrete

memoryless channels (DMCs) in [27]. Under the square root law, achieving the

covert information limits generally requires a pre-shared secret key between Alice

and Bob, prior to Alice’s transmission. Regardless of the quality of channels, this

key size was shown to be on the order of√n for DMCs in [26]. Furthermore, [26]

also showed that this secret is unnecessary if the channel quality from Alice to Bob

is better than the channel quality from Alice to Willie.

1.2.4 Positive-Rate Covert Communications

Under the square root law discussed above, the average number of covert bits

per channel use asymptotically reaches zero. However, the square root law holds

under the circumstances when Willie has no uncertainty about his channel statistics

from Alice. To improve upon this result, uncertainties in the probabilistic models

that are either forced upon Willie or are already present haven been shown to

help improve upon this, resulting in the achievement of a positive covert rate.

These uncertainties can be in the form of Willie’s receiver noise power, imperfect

channel knowledge from the transmitter of covert information, interference from

other network users or through a jammer transmitting artificial noise to confuse

Willie. In the following, we present a review of those works that have been proposed

to exploit these uncertainties for achieving a positive covert rate.

Exploiting uncertainty in the knowledge of receiver’s noise power has been con-

sidered in [33–36], where the work in [33] was the first to show that positive rate can

be achieved based on the concept of SNR walls [37], while the work in [35] analyzed

the worst case scenario under the consideration of bounded and unbounded noise

uncertainty models at Willie. In [38], the scenario where an uninformed jammer is

present in the communication environment was considered and it was shown that a

continuously transmitting jammer can help in achieving a positive rate even with-

out any close collaboration with the transmitter. The case when additional friendly

nodes generating artificial noise are present in the environment, causing confusion

at Willie regarding the received signal statistics, is presented in [39], while the use

of an FD receiver that continuously transmits AN to improve covert performance

has been considered in [40, 41]. A study on covert communications in the presence

10 Introduction

of a Poisson distributed field of interferers has been presented in [42], where lever-

aging the total received interference, the effect of interferer’s transmit power and

density on the covert throughput is analyzed. In all the above mentioned studies,

it has been shown that utilization of informed / uninformed jammers or sources of

additional noise can greatly help in achieving positive-rate covert communications.

In addition to receiver’s noise or jamming signals, a positive rate can also be

achieved when Willie has uncertainty on the time instant of the communication

as analyzed in [43, 44]. More recently, [45] and [46] considered the performance

of multi-antenna covert communications in AWGN channels and fading channels

under random wireless networks, respectively. Here [45], evaluating the codebook

scaling rates in the limiting regimes for the number of channel uses (asymptotic

block length) and the number of antennas (massive MIMO), showed that MIMO

has the potential to provide a substantial increase in the file sizes that can be

covertly communicated subject to a reasonably low delay. On the other hand, [46]

considered both centralized and distributed antenna systems (CAS/DAS) in the

presence of randomly located wardens and interferers, and it is demonstrated that

the CAS outperforms the DAS in terms of the covert throughput for the random

network. The throughput gap between the two systems increases dramatically

when the number of transmit antennas becomes higher.

The analysis of covert transmissions in one-way relay networks is presented

in [47, 48], respectively, where in [47], the relay is greedy and opportunistically

transmits its own information to the destination covertly on top of forwarding

the source’s message whereas [48] considers the case of a greedy relay under the

additional constraint of energy harvested from the source’s message under time

switching and power splitting schemes. The work in [49] showed achieving a positive

covert rate in queuing channels where a sufficiently high rate secret key is available,

while multi-hop routing in LPD communications has been considered in [50], which

has been shown to improve the performance of LPD communication relative to

single-hop transmissions. [51] offers a first study in considering a UAV as the

transmitter in the context of covert communications.

Although most of the works in covert communications assume Gaussian sig-

nalling employed at the transmitter, the optimality of Gaussian signalling for covert

communications under the asymmetry of Kulback-Leibler divergence was discussed

1.3 Steganography 11

in [52]. It was shown that while Gaussian signalling is optimal in terms of maxi-

mizing the mutual information of transmitted and received signals for covert com-

munications with an upper bound on D(P0||P1) as the constraint, this is not the

case when considering an upper bound on D(P1||P0) as the constraint. Rather, it

was shown that a skew-normal signalling can outperform the Gaussian signalling

in terms of achieving higher mutual information.

The above-mentioned works consider covert communications under the assump-

tion of an infinite number of channel uses. However, limited storage resources and

requirements of quick updates in modern systems often require a finite, sometimes

small, number of channel uses, and hence the results in the infinite blocklength

regime do not hold anymore. Covert communications under finite blocklength

have also been previously considered in the literature. The authors in [53] and [54]

consider achieving covertness under AWGN channels where the maximum number

of allowed channel uses is constrained. Furthermore, [55] has considered achieving

covertness under strict delay requirements using a full-duplex receiver, where it has

been shown that in contrast to asymptotically infinite channel uses, a fixed power

artificial noise transmission helps improve covert communications. The authors in

[56] have analyzed covert communications under finite blocklength in the presence

of a multi-antenna Willie, while covert communications over slow fading channels

under finite blocklength has been considered in [57], providing an upper bound on

the total power satisfying a desired probability of detection by the adversary.

1.3 Steganography

Steganography is the art and science of communicating using everyday objects in

such a way that the presence of a message cannot be detected. Simple stegano-

graphic techniques have been in use for hundreds of years, and we find their ex-

amples abundantly throughout human history. From Herodotus [58], telling the

story of a slave’s head shaved and tattooed for communicating a hidden message,

to the use of invisible ink during the American Revolution [59], that would glow

when exposed to a flame, history is brimming with examples of stegonagraphy uses.

More recent examples of this art include using special inks to write secret messages

on bank notes and also the multimedia industry using digital watermarking and

12 Introduction

fingerprinting of audio and video for copyright protection [60–63].

Modern-day steganography [64–66] uses finite-length, finite-alphabet covertext

objects, such as images and software codes to hide messages. Embedding these

messages changes the properties of the covertext producing stegotext, where ste-

ganalysis, the study of detecting hidden messages, looks for these changes. The

original covertext is generally unavailable otherwise the task would be a trivial

comparison between the covertext and stegotext. While steganalysts are assumed

to have a complete statistical model of the covertext, the amount of information

that can be hidden depends upon how much knowledge of this model is available

to parties producing the stegotext. The square root law previously mentioned

for covert communications has close parallels in steganography. In finite-alphabet

steganographic systems, at most O (√n) symbols in the original covertext of length

n may safely be modified to hide a message of length O (√n log n) [66]. This result

was extended to Markov covertext [67] and was shown to either require a key linear

in the size of the message [68] or encryption of the message prior to embedding [69].

The similarity of square root law in covert communications to this stegonagraphic

square root law can be attributed to the mathematics of statistical hypothesis test-

ing. The additional log n factor in the stegonagraphic square root law comes from

the fact that the steganographic channel to Bob is noiseless.

While steganography is an active research field in computer sciences, its ap-

plications in covert wireless communications from physical layer perspective are

limited. This can be attributed to the fact that it is fundamentally an application

layer technique. Analysis of steganographic systems generally assumes that the

sent messages are not effected by any channel noise. Furthermore, it essentially

requires the transmission of stegotext or images between the covert transmission

pair when all communication is essentially prohibited. On the other hand, phys-

ical layer covert communication systems exploit channel artifacts and adversary’s

receiver noise variation to hide any transmissions.

1.4 Thesis Overview and Contributions

The main focus of this thesis is on achieving positive rate covert communications in

wireless environments through exploiting any uncertainties that might be present

1.4 Thesis Overview and Contributions 13

or can be induced at Willie. We consider communication scenarios where we first

identify the possibilities of achieving a positive-rate covert communication and then

further design the system parameters in our control to exploit these uncertainties in

such a way that one of the following objectives is achieved: (i) In a given communi-

cation setup, how can we maximize the covert information that can be transferred

from Alice to Bob while Willie’s detection error probability is lower bounded un-

der a given constraint, or (ii) In a given communication setup, while the covert

information transfer from Alice to Bob satisfies a given requirement, how can we

maximize Willie’s detection error probability. In this regard, uncertainty in the

received signal power and lack of channel knowledge from the covert transmitter

to Willie constitute very important aspects that have not been considered before

in the literature. This thesis explores these two possibilities in detail for achieving

the subject purpose. Firstly, we introduce the use of AN for achieving covertness,

causing confusion at Willie in determining the transmission state of Alice. The

scenarios of AN use are considered under the worst case assumption that Willie

is fully aware of the channel information from the covert transmitter, which then

motivates us to consider the case where Willie is uncertain of his channel infor-

mation and only possesses part of the channel knowledge. Both of these scenarios

also consider the impact of these imperfections on the covert communication pair

as well, helping quantify the covert performance from a realistic perspective. A

block-diagram showing the scenarios considered in this thesis is shown in Fig. 1.3.

Figure 1.3: Different scenarios considered in the thesis for achieving covert com-munication.

In the first half of the thesis, we focus on the use of AN by the information

transmitter or receiver that transmits the AN with a varying power. This creates

14 Introduction

confusion at Willie regarding the received signal statistics, forcing errors in his

detection of any covert transmissions. This approach is different to the use of

jamming signals as considered in the previous literature, since it offers superior

control over the power of the AN transmit power level, helping in improving the

decoding ability of covertly transmitted information. In the second half, the focus is

on situations where users suffer from uncertainty about their channel knowledge and

we consider communication scenarios where this lack of knowledge can be exploited

to achieve covertness. Under the channel uncertainty scenarios, we distinguish

between the two important cases of finite and infinite blocklengths i.e., how many

symbols are transmitted by Alice in a given time slot, or alternatively, how many

samples does Willie take before he makes a decision on the transmission state of

Alice.

In the following, we detail the specific contributions of each chapter presented

in this thesis.

Chapter 2 - Covert Communications Using a Full-Duplex receiver with

Artificial Noise

Chapter 2 considers a wireless communication scenario where covertness is achieved

by using a full-duplex receiver. More precisely, the receiver of covert information

generates artificial noise with a varying power causing uncertainty at the adver-

sary, Willie, regarding the statistics of the received signals. Given that Willie’s

optimal detector is a threshold test on the received power, we derive a closed-form

expression for the optimal detection performance of Willie averaged over the fading

channel realizations. Furthermore, we provide guidelines for the optimal choice of

artificial noise power range, and the optimal transmission probability of covert in-

formation to maximize the detection errors at Willie. Our analysis shows that the

transmission of artificial noise, although causes self-interference, provides the op-

portunity of achieving covertness but its transmit power levels need to be managed

carefully. We also demonstrate that the prior transmission probability of 0.5 is not

always the best choice for achieving the maximum possible covertness, especially

when the covert transmission probability and artificial noise power can be jointly

optimized.

The work presented in Chapter 2 is closely related to [38], where a jammer is


assumed to be present in the environment. Although the jammer does not closely

coordinate with the covert transmitter, it is allowed to transmit continuously and

the received power at Willie due to the jammer changes randomly from slot to

slot. In this case, the covert communication pair has no control over the jam-

mer’s transmit power level. In contrast, although we also consider randomizing the

AN power in each slot, our focus is on optimizing the AN transmit power range,

since this choice affects the information decoding at the intended receiver through

self-interference. This important optimization is made possible because the AN is

transmitted by the FD receiver, and hence, controllable by the covert communica-

tion pair. Moreover, instead of satisfying a given covertness constraint, we present

our analysis on the choice of AN transmit power range to achieve the maximum

possible covertness while meeting a given rate requirement.

Performance of communication systems with randomly distributed interferers

has been studied extensively in the literature [70–72]. More recently, a study on

covert communications in the presence of a Poisson distributed field of interferers

has been presented in [42], where leveraging the total received interference, the

effect of density and transmit powers of the interferers on the covert throughput is

analyzed. Our work differs from [42] in that we consider AN generated by the FD

receiver, hence allowing design and optimization of AN power with other design

parameters. Thus, while the authors in [42] study the covert performance for a

given interference scenario, we take a design approach and provide guidelines on

the optimal choice of parameters for achieving covertness. The novel contributions

of the chapter can be summarized as follows:

• We show that the use of an FD receiver is an effective way of achieving

covert communication over fading wireless channels, where the FD receiver

is designed to transmit AN with varying power to cause confusion at Willie.

• Under the assumption of a radiometer (power-detector) at Willie, we analyti-

cally derive the optimal detection threshold of Willie’s radiometer and obtain

its optimal detection performance in terms of the minimum detection error

probability.

• For a given covert rate requirement, we provide design guidelines on the

optimal choices for the range of AN transmit power at the FD receiver and

16 Introduction

the optimal a priori probability of covert transmission in order to maximize

the expected detection error probability at Willie.

• Our analysis reveals that an a priori transmission probability of 0.5 is not

always the best choice. Increasing this transmission probability beyond 0.5

gives more room to increase the AN transmit power for maintaining the same

rate requirement. Thus whether to allow such a change in the transmission

probability can be the difference between achieving strong covertness and

achieving almost no covertness at all.

The results of this chapter have appeared in the following publication [41]:

J1. K. Shahzad, X. Zhou, S. Yan, J. Hu, F. Shu, and J. Li, “Achieving Covert

Wireless Communications Using a Full-Duplex Receiver,” IEEE Trans. Wire-

less Commun., vol. 17, no. 12, pp. 8517-8530, Dec. 2018.

Chapter 3 - Covert Communications in Backscatter Radio using Artifi-

cial Noise

Chapter 3 considers covert communication in backscatter radio systems, where

the transmitter controls its transmit power to keep the transponder’s response

hidden, while a warden tries to detect this covert communication. Backscatter

communication [73, 74] offers the unique advantage of eliminating the need of any

active radio frequency (RF) components, resulting in a prolonged life-span of the

wireless devices and continued network functionality. These wireless devices can

not only harvest energy from the transmitters signal, but can also modulate the

same signal to convey information. Although backscatter communication has been

largely deployed in radio frequency identification (RFID) systems for consumer-

based applications e.g., supply-chain management, RFID cards have also made

their way into more sensitive arenas, e.g., access control, payment systems and

asset tracking. However, the application of backscatter systems in such sensitive

scenarios is limited, owing to their broadcast nature and the ease of snooping

information through eavesdropping. One option to alleviate this issue may be to

use stronger encryption protocols, but the size, cost and power constraints of most

backscatter transponders do not warrant such luxuries [75].


To achieve covertness, we propose a non-conventional transmission scheme where

the transmitter emits noise-like signal with transmit power varying across different

communication slots. Under the assumption of a radiometer as the detector at the

warden, we first derive the optimal detection threshold for this detector. Next,

building upon the detection performance of warden, we analyze the condition on

the transmit power to achieve a target level of covertness. Our numerical results

illustrate the price a backscatter system has to pay, in terms of bit error rate, for

achieving covert communication. The novel contributions of the chapter can be

summarized as follows:

• To achieve covert backscatter communication, we propose to use a noise-like

signal with variable power at the reader when sending its transmitted signal.

This transmission scheme achieves a desired level of covertness by controlling

the variation in readers transmit power.

• Under the proposed scheme, we derive a closed-form expression for the opti-

mal detection threshold for a radiometer at Willie.

• We analytically characterize the condition on the reader’s transmit power to

achieve a target level of covertness and numerically investigate the bit error

rate (BER) performance of the backscatter communication. The tradeoff

between covertness against Willies detection and BER performance at the

reader is presented.


C1. K. Shahzad, and X. Zhou, “Covert Communication in Backscatter Radio,”

in Proc. IEEE Int. Conf. on Communications, ICC’2019, Shanghai, China,

pp. 1-6, May. 2019.

Chapter 4 - Covert Communications within a Public Link under Channel

Uncertainty

Chapter 4 considers a covert communications system under block fading channels,

where users experience uncertainty about their channel knowledge. The transmitter

18 Introduction

seeks to hide the covert communication to a private user by exploiting a legitimate

public communication link, while the warden tries to detect this covert communi-

cation by using a radiometer. We derive the exact expression for the radiometer’s

optimal threshold, which determines the performance limit of the warden’s detec-

tor. Furthermore, for given transmission outage constraints, the achievable rates

for legitimate and covert users are analyzed, while maintaining a specific level of

covertness. Our numerical results illustrate how the achievable performance is

affected by the channel uncertainty and required level of covertness. The novel

contributions of the chapter can be summarized as follows:

• We exploit the uncertainty in channel knowledge under block fading channels

to achieve covertness.

• We derive the exact expression for the optimal threshold of wardens detector

(radiometer).

• Under the constraints required for gaining covertness, we analyze the feasible

rates for given transmission outage constraints of the legitimate and the covert

user.


C2. K. Shahzad, X. Zhou, and S. Yan, “Covert communication in Fading Chan-

nels under Channel Uncertainty,” in Proc. IEEE Vehicular Technology Con-

ference, VTC’2017, Sydney, Australia, pp. 1-5, Jun. 2017.

Chapter 5 - Covert Communications with Channel Training and Finite

Blocklength

Chapter 5 considers covert communications over quasi-static block fading channels,

where users suffer from channel uncertainty. Under the assumption of a radiometer

as the detector of choice at the adversary, Willie, we first investigate Willie’s op-

timal detection performance in two extreme cases, i.e., the case of perfect channel

state information (CSI) and the case of channel distribution information (CDI)

only. It is shown that in the large detection error regime, Willie’s detection per-

formances of these two cases are essentially indistinguishable, which implies that


the quality of CSI does not help Willie in improving his detection performance.

This result enables us to study the covert transmission design without the need to

factor in the exact amount of channel uncertainty at Willie. We then obtain both

the optimal and suboptimal closed-form solution to the covert transmission design.

Our result reveals some fundamental difference in the design between the case of

quasi-static fading channel and the previously studied case of non-fading AWGN

channel. The novel contributions of the chapter can be summarized as follows:

• Under the assumption of a radiometer, we analytically derive Willie’s opti-

mal detection performance. Focusing on large detection errors, we show that

Willie’s detection performance is extremely insensitive to the accuracy of his

channel knowledge. Thus, as long as Willie is forced to stay in the large de-

tection error regime by an appropriate transmission strategy, the accuracy of

Willie’s channel knowledge has almost no impact on his detection capability.

• In order to maximize the communication throughput under a given covertness

constraint, we provide the optimal choice of the number of data symbols

and data transmission power to be used by Alice. Our results reveal some

fundamental differences in the covert transmission design between the case

of AWGN channels and that of quasi-static fading channels.

• We also provide a suboptimal closed-form solution to this problem, which

offers a trade-off between obtaining a closed-form solution and a moderate

reduction in the achievable performance.

The results of this chapter have appeared in the following publication:

J2. K. Shahzad, and X. Zhou, “Covert Wireless Communications under Quasi-

Static Fading with Channel Uncertainty,” submitted to IEEE Trans. Inf.

Forensics Security, Oct. 2019.

Finally, Chapter 6 draws conclusions from this thesis and provides some directions

for future research work.

Chapter 2

Covert Communications Using a

Full-Duplex Receiver with

Artificial Noise

2.1 Background

In this chapter, we consider the use of an FD receiver to achieve covert commu-

nication. Specifically, the FD receiver generates AN with a randomized transmit

power, causing a deliberate confusion and affecting the decisions at Willie regard-

ing the presence of any covert transmissions. Although, not studied before in the

context of covert communications, the use of AN and jamming signals for enhanc-

ing physical layer security has been widely advocated in the literature [78–82, and

references therein]. The use of an FD receiver generating AN provides a cover

for the covert transmission, and offers a multitude of benefits as compared to the

use of a separate, independent jammer. Being equipped with an FD receiver, we

can exercise a better control over the power used for transmitting AN, hence a

better management of system resources to achieve the said purpose of security

is achievable. Furthermore, while Willie will face a strong interference, the self-

interference at the FD receiver can be greatly suppressed by the well-developed

self-interference cancellation techniques [83, 84], providing a significant advantage

to the covert communication pair.

In the considered scenario, covert transmissions can occur in multiple blocks

21

22 Covert Communications Using a Full-Duplex Receiver with Artificial Noise

of time, and Willie is performing the detection on a block-to-block basis. In this

case, the a priori transmission probability becomes an interesting and important

parameter affecting Willie’s detection performance as well as the overall through-

put of covert communications. A general assumption in the literature regarding

the a priori probability of covert transmission is that there is a 50% chance that

transmission occurs in a block of interest. This assumption is understood as a good

choice for covertness, since it renders Willie’s knowledge of Alice’s transmission un-

informative and is equivalent to assuming that Willie has no prior knowledge on

whether Alice transmits or not [25, 44]. We show that an a priori probability of

0.5 is not always the best choice in our considered scenario; rather a joint adjust-

ment of this probability with other system parameters can offer a better covert

performance.

The rest of this chapter is organized as follows: Section 2.1 details our com-

munication scenario, proposed scheme and the assumptions used in this chapter.

Section 2.2 explains Willie’s approach for detection of any covert transmissions,

deriving the conditions for possibility of any covert communications and the op-

timal settings at Willie. Using the knowledge of Willie’s approach, Section 2.3

discusses the parameters that affect the achievable performance of the proposed

communication scheme while Section 2.4 addresses the optimal design of all the

system parameters that we have in our control to achieve the best possible perfor-

mance in covertness. Section 2.5 provides numerical results validating our analysis

and provides further insights on the impact of AN and a priori probabilities, and

Section 2.6 draws some concluding remarks.

2.2 System Model

2.2.1 Communication Scenario

A covert wireless communication system is considered, as shown in Fig. 2.1, where

a transmitter (Alice) possesses sensitive information that needs to be sent to an

information receiver (Bob). Bob operates in FD mode, and Alice seeks to transmit

covertly to Bob with the aid of AN generated by Bob. Under these circumstances,

an adversary (Willie) silently listens to the communication environment and tries

to detect any covert transmission from Alice to Bob. We use the subscripts a, b

2.2 System Model 23

ALICE

Tx

BOB

Rx Tx

WILLIE

Rxhaw hbw

hab hbb

Figure 2.1: Covert communications model under consideration with aFD receiver.

and w to represent the terms associated with Alice, Bob and Willie, respectively.

It is assumed that Willie has complete knowledge of the carrier frequency of any

transmissions, associated antenna gains and the distances between all the nodes.

A communication slot is defined as a block of time over which the transmission of

a message from Alice to Bob is completed. Each slot contains n symbol periods and

we assume that n is large enough, i.e., n → ∞. The slot boundaries are perfectly

synchronized among all the users, and we consider fading wireless channels where

the channel coefficients remain constant in one slot, changing independently from

one slot to another, i.e., quasi-static Rayleigh fading channels are considered. The

channel between any two users i and j is represented by hij, where the channel gain

is assumed to encompass the combined antenna gain of transmit/receive antennas

and the distance between the two users as well. The mean of |hij|2 over different

communication slots is denoted by 1/λij, where subscript ij can be ab, aw, bw or

bb. Hence, the Alice-Bob, Alice-Willie and Bob-Willie channels are denoted by hab,

haw, and hbw, respectively, while the self-interference channel of Bob is denoted by

hbb. We note here that hbb is the loop interference channel at Bob and is modelled

via the Rayleigh fading distribution under the assumption that any line-of-sight

component is efficiently reduced by antenna isolation and the major effect comes

from scattering [85]. Regarding the channel knowledge, it is assumed that Bob

knows hab, while Willie possesses complete knowledge of haw and hbw. Here, the

availability of knowledge regarding haw and hbw at Willie represents the worst case

scenario from the perspective of covert communication design.

The complex additive Gaussian noise at Bob and Willie’s receiver is denoted

by nb ∼ CN (0, σ2b ) and nw ∼ CN (0, σ2

w), respectively. Each of Alice and Willie is


equipped with a single antenna, while apart from a receiving antenna, Bob also has

an additional antenna for the transmission of AN. Due to its full-duplex nature,

Bob suffers from self-interference, causing a degradation in the signal-to-noise ratio

(SNR) of the message signal received from Alice [86, 87]. Since the generated AN

signal is known to Bob, the self-interfering signal, acting as noise for Bob’s receiver,

can be rebuilt and eliminated up to a certain extent by using efficient techniques of

self-interference cancellation [83, 88]. However, owing to computational limitations

and practical non-idealities, we assume that perfect cancellation of self-interference

is not achievable [89]. The self-interference cancellation coefficient is denoted by

φ, where 0 < φ ≤ 1 corresponds to different cancellation levels of interfering AN

signals. The residual interfering link is also modelled as Rayleigh fading channel,

following a common assumption in the literature [85, 90].

The transmit power of Alice and Bob is denoted by Pa and Pb, respectively.

When Alice transmits, the signal received at Bob for each symbol period is given

by

yb(i) =√Pahabxa(i) +

√φPbhbbxb(i) + nb(i), (2.1)

where i = 1, . . . , n represents the symbol index. Here, xa and xb represent the

signals transmitted by Alice and Bob, respectively, satisfying E[xa(i)x†a(i)] = 1

and E[xb(i)x†b(i)] = 1. We also consider an average power constraint on Bob’s

transmit power, denoted by Pavg. We follow the common assumption that a secret

of sufficient length is shared between Alice and Bob [25, 38], which is unknown to

Willie, enables Bob to know Alice’s strategy. Employing random coding arguments,

Alice generates codewords of length n, by independently drawing symbols from a

zero-mean complex Gaussian distribution with unit variance. Here, each codebook

is known to Alice and Bob and is used only once. When Alice transmits in a slot,

she selects the codeword corresponding to her message and transmits the resulting

sequence.

2.2.2 Proposed Transmission Scheme

We propose a communication scheme that allows Bob to receive Alice’s transmission

covertly, exploiting an AN signal generated by Bob, where the transmit power of

2.2 System Model 25

· · · · · ·n

slot k − 1

n

slot k

n

slot k + 1 · · · · · ·

Figure 2.2: Willie’s observation model, presented as a slotted channel use by Alice.Each slot contains n symbol periods and there is a certain a priori probability, π1,of Alice’s communication to Bob in each slot.

this AN changes from one slot to the next. Alice’s transmit power, Pa, is fixed and

publicly known by both Willie and Bob. On the other hand, Pb, defined as the

average power used by Bob for AN transmission in a given slot, changes from slot

to slot, following a continuous uniform distribution over the interval [Pmin, Pmax],

having a probability density function (pdf) given by

fPb(p) =

1

Pmax−Pmin, if Pmin ≤ p ≤ Pmax

0, otherwise.(2.2)

It should be noted here that in the proposed scheme, Bob transmits the AN

signal continuously, regardless of whether or not Alice transmits in a given slot.

In this work, we address the covertness regarding Alice’s existence and message

transmission to Bob, whereas we are not trying to hide the existence of Bob. Apart

from being the information receiver, Bob also plays the role of a cooperative jammer

and hence his presence is known to Willie. Willie has complete knowledge of

Bob’s AN power distribution, but the exact power used by Bob in a given slot is

unknown to Willie. Due to the independent Gaussian nature of Alice and Bob’s

transmitted signals and Willie’s receiver noise, the signal received at Willie is always

Gaussian in any slot, regardless of whether Alice transmits or not, thus Willie can

not make use of any difference in the distribution type of the received signal for

detection purposes. Since Willie knows the channels haw and hbw in any slot under

consideration, for a constant transmit power from Bob, it is straightforward for him

to raise an alarm when an additional power from Alice is received. By introducing

randomness in Bob’s transmit power, we create an uncertainty at Willie, causing

confusion as to whether the increase in received signal power is due to Alice’s covert

transmission or merely a change in Bob’s AN power. This effectively creates an

artificial fading for Willie [91], despite the fact that he has the perfect channel

knowledge for both Alice and Bob.


2.2.3 Willie’s Detection, Priors and Performance Metrics

As described earlier, it is assumed that Willie is unaware of the exact transmit

power used by Bob in each slot, although the distribution of Bob’s AN transmit

power is known to Willie. Also, Willie has full knowledge of the associated antenna

gains, distances among all the nodes and his own receiver’s noise variance. We

consider a covert communications scenario that spans a large number of slots, and

there is a possibility of transmission by Alice in each slot. Due to the shared secret

between Alice and Bob, this constitutes a form of a time-hopping system. Here,

Willie looks to make a decision regarding Alice’s transmission in each slot as he

is interested in knowing for each individual slot that whether Alice transmitted or

not. This means that Willie is not only interested in “whether” a transmission

happens but also “when” it happens, i.e., in which slot. Note that if Willie is only

interested in whether transmission happens but not in when it happens, he needs

to make only a single decision after observing all slots. Such a scenario has been

considered in [44], where the slot selection is kept secret from Willie and he looks

to make a single decision regarding Alice’s transmission over all possible slots.

The knowledge of “when” a transmission happens not only improves upon

Willie’s effectiveness in detecting covert transmission, but also gives him the ca-

pability of taking an action at the required time rather than waiting for the end

of observation interval before intervening (although the corresponding action by

Willie is beyond the scope of this work). Consider a scenario where Alice and Bob

agree upon a certain “pattern” in choosing the slots over which covert messages are

sent. Once Willie is able to detect the pattern based on his per slot decisions, it be-

comes easier for him to efficiently predict the slots over which future transmissions

will happen1.

Willie faces a decision as to whether or not Alice sent any covert information

to Bob. As a result, Willie faces a binary hypothesis testing problem. The null

hypothesis, H0, states that Alice did not transmit while the alternative hypothesis,

H1, states that Alice did transmit, sending covert information to Bob. We define

the probability of false alarm (or Type I error) as the probability that Willie makes a

decision in favor of H1, while H0 is true, denoted by PFA. Similarly, the probability

1Although the proposed scheme will help Willie in being able to predict any such pattern, thisprediction is beyond the scope of this work and is thus not considered here.

2.3 Detection Scheme at Willie 27

of missed detection (or Type II error) is defined as the probability of Willie making

a decision in favor of H0, while H1 is true, and is denoted by PMD. We denote by

π0 and π1 the a priori probabilities of hypothesis H0 and H1, respectively. The

detection error probability at Willie is given by

PE = π0PFA + π1PMD, (2.3)

which serves as a measure of covertness. In the recent literature, the assumption of

both hypotheses being presented with an equal a priori probability has been widely

adopted [33, 44]. The knowledge of a priori probabilities helps Willie improve his

detection performance [25, Fact 4], as his assumption of π0 = π1 = 12

implies that

his observations are of little use to him and his decisions are akin to a random

guess about the transmission state of Alice. Here, we instead consider general, i.e.,

not necessarily equal priors, and assume that Willie happens to know them. Since

PE ≤ min (π0, π1), achieving covert communication guarantees that PE is in close

proximity of min (π0, π1).

2.3 Detection Scheme at Willie

The signals received at Willie under the two possible hypotheses for each symbol

period are given by

yw(i) =

√Pahawxa(i) +

√Pbhbwxb(i) + nw(i), If H1 is true

√Pbhbwxb(i) + nw(i), If H0 is true.

(2.4)

From the independent and identically distributed (i.i.d.) nature of Willie’s received

vector, yw, each element of yw, i.e., yw(i) has a distribution given byCN (0, |haw|2Pa + |hbw|2Pb + σ2w), If H1 is true

CN (0, |hbw|2Pb + σ2w), If H0 is true.

(2.5)

We note that while the distribution of Pb is known to Willie, its value in a given slot

is not known. Based on his observation vector yw = [yw(1), . . . , yw(n)], Willie has to

make a decision regarding Alice’s actions in each communication slot. We assume


that Willie uses a radiometer as his detector [33, 38] due to its low complexity

and ease of implementation. When Willie has the statistical knowledge of his

observations, this assumption is justified and the optimality of radiometer can

be proved along the same lines as the proof of Lemma 3 in [38] using Fisher-

Neyman Factorization Theorem [92] and Likelihood Ratio Ordering concepts [93].

While adopting a radiometer, the total received power at Willie,∑n

i=1 |yw(i)|2 is

a sufficient statistic for Willie’s test. Since any one-to-one transformation of a

sufficient statistic is also sufficient, the term 1/n∑n

i=1 |yw(i)|2 is also a sufficient

statistic. Thus Willie conducts a threshold test on the average power received in a

slot, given by

PwD1

≷D0

γ, (2.6)

where Pw = 1/n∑n

i=1 |yw(i)|2 is the average power received at Willie in a slot, D0

and D1 are defined as the events that Willie makes a decision in the favor of H0

and H1, respectively, and γ is Willie’s detector threshold, which can be optimized

to minimize the detection error probability. The average power at Willie in a slot

under hypothesis H0 is given by

Pw(H0) = limn→∞

(|hbw|2Pb + σ2

w

)χ2

2n/n

= |hbw|2Pb + σ2w,

(2.7)

where χ22n represents a chi-squared random variable with 2n degrees of freedom and

from the Strong Law of Large Numbers, we know that χ22n/n → 1 almost surely.

Similarly, the average power at Willie in a slot under hypothesis H1 is

Pw(H1) = |hbw|2Pb + |haw|2Pa + σ2w. (2.8)

We first analyze the condition under which Willie has non-zero probability of

making detection errors and based on that, we find the optimal setting for Willie’s

detector threshold. We note here that the analysis of Willie’s detection error prob-

ability presented in the following proposition is for given channel realizations as

Willie possesses the full knowledge of his channel from Alice and Bob.

2.4 Performance of Covert Communication 29

Proposition 2.1 Willie has a non-zero probability of making detection errors when:

|haw|2|hbw|2

≤ Pmax − Pmin

Pa. (2.9)

When (2.9) holds, the optimal choice for Willie’s detector’s threshold is

γ∗ =

|hbw|2Pmin + |haw|2Pa + σ2

w, if π1 ≥ π0

|hbw|2Pmax + σ2w, otherwise,

(2.10)

and the corresponding minimum detection error probability at Willie is given by

P∗E =

π0

[1− |haw|2Pa

|hbw|2(Pmax−Pmin)

], if π1 ≥ π0

π1

[1− |haw|2Pa

|hbw|2(Pmax−Pmin)

], otherwise.

(2.11)

Proof

See Appendix A.1.

Remark 2.1 From Proposition 1, when (2.9) does not hold, Willie will have zero

probability of making a detection error by setting the threshold γ in the interval

|hbw|2Pmax + σ2w < γ ≤ |hbw|2Pmin + |haw|2Pa + σ2

w. We also note here that although

Willie’s receiver noise variance, σ2w, is required for the calculation of the optimal

threshold for Willie’s detector, its value does not affect the minimum detection error

probability at Willie. This can be attributed to the fact that as n→∞, there is no

uncertainty at Willie regarding the noise statistics and hence it does not contribute

to an increase or decrease in the detection error probability at Willie.

2.4 Performance of Covert Communication

In this section, we present those system metrics which affect the performance of our

proposed covert transmission scheme. We note that the square root law presented

by Bash et al. [25] holds given Willie has perfect statistical knowledge of the

test statistics. It has been shown in prior works [33, 35, 38] that uncertainties

present (or intentionally introduced) in the test statistics under both the null and

alternative hypotheses at Willie result in a positive rate. Here, the randomness in

Bob’s transmit power introduces the required uncertainty at Willie, and hence we


are able to achieve a positive covert rate. We first calculate the outage probability

for the transmission from Alice to Bob, and then present a measure that helps in

quantifying the performance of our presented covert scheme.

2.4.1 Transmission Outage Probability from Alice to Bob

The signal-to-interference-plus-noise ratio (SINR) at Bob, in case Alice transmits,

is given by

SINRb =|hab|2Pa

φ|hbb|2Pb + σ2b

. (2.12)

We assume a pre-determined rate from Alice to Bob, and denote it by Rab. Due

to the random nature of hab, hbb and Pb, a transmission outage from Alice to Bob

occurs when Cab < Rab, where Cab is the channel capacity from Alice to Bob.

Lemma 2.1 The transmission outage probability from Alice to Bob is given by

δab = 1− λbb exp(−λabµσ2b )

(Pmax − Pmin)λabφµln

[λbb + λabφµPmax

λbb + λabφµPmin

], (2.13)

where µ ,(2Rab − 1

)/Pa.

Proof

From the definition of transmission outage probability, we have

δab = P [Cab < Rab]

= P[ |hab|2Paφ|hbb|2Pb + σ2

b

< 2Rab − 1

]=

∫ Pmax

Pmin

∫ ∞0

∫ µ(φ|hbb|2Pb+σ2b )

0

f|hab|2(x)f|hbb|2(y)fPb(z)dx dy dz

=

∫ Pmax

Pmin

∫ ∞0

[1− exp

(−λabµ(φ|hbb|2Pb + σ2

b )) ]f|hbb|2(y)fPb(z) dy dz

=

∫ Pmax

Pmin

[1− λbb exp (−λabµσ2

b )

λbb + λabµφPb

]fPb(z)dz

= 1− 1

Pmax − Pmin

∫ Pmax

Pmin

[λbb exp (−λabµσ2

b )

λbb + λabµφz

]dz,

(2.14)

and using the solution from [94] for the general form of integral∫

AB+Cx

dx =A log(B+Cx)

Cfor the second term gives the desired result.

2.5 Covert Communication Design 31

2.4.2 Expected Detection Error Probability at Willie

Since Alice and Bob are unaware of their instantaneous channel to Willie, we

consider the expected value of detection error probability at Willie, P∗E, over all

possible realizations of haw and hbw as the measure of covertness from the viewpoint

of Alice and Bob, and this expected detection error probability at Willie is denoted

by P∗E.

Lemma 2.2 Under the optimal detection threshold setting, the expected detection

error probability at Willie is given by

P∗E =

π0 [1 + t ln t− t2] , if π1 ≥ π0

π1 [1 + t ln t− t2] , otherwise,(2.15)

where

t ,λbwPa

λbwPa + λaw (Pmax − Pmin). (2.16)

Proof

See Appendix A.2.

Remark 2.2 We make a few observations regarding the effect of Pmax and Pa on

Willie’s detection performance. Firstly, as Pmax → ∞, the probability of Willie

making detection errors approaches π0 or π1, in respective cases, which represents

the maximum of P∗E. Secondly, if Alice’s transmit power Pa →∞, then t→ 1 and

P∗E → 0. Thus for a given set of {Pmin, Pmax}, Alice can be “loud” enough to be

heard by Willie.

2.5 Covert Communication Design

In majority of the recent literature in covert communications, the detection error

probability is used to measure the level of covertness under the assumption of

equal priors. However, in this work, we propose a different framework and instead

of putting a constraint on the error probability to achieve a said covertness, we

look to maximize it under the given system model. Hence, from Alice and Bob’s

perspective, the objective is to achieve the best possible covertness in transmission,


while being subject to an average power constraint and satisfying a given effective

covert rate requirement which we denote by τ . In this section, we consider optimal

choices for the parameters in our control to achieve the said purpose.

Although Alice’s transmit power, Pa, is assumed to be fixed in this work, to

make the problem feasible, we assume that the value of Pa at least satisfies the rate

requirement from Alice to Bob when no AN is transmitted by Bob. The rest of

the design parameters that affect the performance of covert communication in our

system model are the distribution parameters of Bob’s AN power, {Pmin, Pmax},and the a priori probabilities of Alice’s transmission, {π0, π1}, with π0 = 1− π1.

We state our main problem as following:

P2.1 maximizeπ1,Pmin,Pmax

P∗E

subject to π1Rab(1− δab) ≥ τ,

Pmin + Pmax ≤ 2Pavg,

(2.17)

while

Pa ≥λabσ

2b

(2Rab − 1

)ln (Rab/τ)

(2.18)

is assumed for feasibility. Here, the expression for P∗E is given in (2.15) un-

der Lemma 2.2, δab is the transmission outage probability and is a function of

{Pmin, Pmax}, τ is the minimum required covert rate, and Pavg is the average trans-

mit power for Bob’s AN. We solve P2.1 in a step-by-step manner, as this approach

not only provides the globally optimal solution, but also provides further insights

in the role of different parameters in achieving the said purpose of covertness.

2.5.1 Optimal Minimum AN Power

For a given average transmit power at Bob, we look to minimize the value of

transmission outage probability, in order to satisfy the covert rate requirement,

corresponding to the first constraint in (2.17), while maximizing P∗E. Under these

conditions, in this subsection, we consider finding the optimal minimum AN power

at Bob, Pmin, for any given maximum AN power, Pmax, and prior probabilities of

Alice’s transmission, {π0, π1}.


Proposition 2.2 The optimal choice of Pmin to maximize the expected detection

error probability at Willie, P∗E, while satisfying the effective covert rate requirement

from Alice to Bob is given by P ∗min = 0.

Proof

See Appendix A.3.

As a result of Proposition 1, we can simplify the transmission outage probability

at Bob and the expected detection error probability at Willie as

δab = 1− λbb exp(−λabµσ2b )

Pmaxλabφµln

[λbb + λabφµPmax

λbb

], (2.19)

and

P ∗E =

π0 [1 + s ln s− s2] , if π1 ≥ π0

π1 [1 + s ln s− s2] , otherwise,(2.20)

respectively, where

s ,λbwPa

λbwPa + λawPmax

. (2.21)

2.5.2 Optimal Priors for Alice’s Transmission

Once the optimal value of Pmin has been found, the task from Alice and Bob’s

perspective is to find the optimal a priori probabilities of Alice’s transmission and

Bob’s maximum possible transmit power, Pmax. In this subsection, we consider

finding the optimal choice of Alice’s a priori transmission probabilities for a given

Pmax. We state this problem as:

P2.1a maximizeπ1

P∗E

subject to π1Rab(1− δab) ≥ τ,(2.22)

where the expression for P∗E is now given by (2.20), and the feasibility condition of

(2.18) is still held. The solution to problem P2.1a is presented in the following:


Proposition 2.3 The optimal choice of a priori probabilities for Alice’s transmis-

sion, as a function of maximum AN power, Pmax, is given by

π∗1(Pmax) = max

(1

2,

τ

Rab(1− δab(Pmax))

), (2.23)

and π∗0 = 1− π∗1.

Proof

We consider the two cases for P∗E individually. Note here that δab is now a function

of Pmax only.

Case-I: π1 < π0

In this case, π1 < 1/2, and using the constraint in P2.1a, we have τRab(1−δab(Pmax))

≤π1 < 1/2, which can only happen when τ

Rab(1−δab(Pmax))≤ 1/2. Also in this case,

∂P∗E∂π1

= 1 + s ln s− s2 ≥ 0 for s ∈ [0, 1).

Case-II: π1 ≥ π0

Here, π1 ≥ 1/2, and due to the constraint in P2.1a, π1 ≥ max(

12, τRab(1−δab(Pmax))

).

Also, in this case,∂P∗E∂π1

= −(1 + s ln s− s2) ≤ 0 for s ∈ [0, 1).

Combining these two cases gives the desired result.

From Proposition 2.3, it is evident that the optimal value of π1 depends upon the

choice of Pmax. Thus to satisfy a given rate requirement, any choice of Pmax at Bob,

directly affecting the transmission outage probability through self-interference, will

determine whether π∗1 is equal to 0.5 or not. Since the purpose of our covert scheme

is to maximize the detection error at Willie while satisfying the rate requirement,

it presents an interesting interplay of our choice of these parameters.


2.5.3 Optimal Maximum AN Power

Once the optimal priors for Alice’s transmission i.e., {π∗0, π∗1} have been found in

terms of Pmax, the expected detection error probability at Willie becomes

P∗E(π∗1) =

12κ(s), if τ

Rab(1−δab) ≤12(

1− τRab(1−δab)

)κ(s), else,

(2.24)

where again, κ(s) = (1 + s ln s− s2), and s is as defined earlier in (2.21). We now

consider finding the optimal value for Bob’s maximum transmit power, Pmax, under

the average power constraint. This problem is stated as

P2.1b maximizePmax

P∗E(π∗1)

subject to π∗1Rab(1− δab) ≥ τ,

Pmax ≤ 2Pavg.

(2.25)

We note here that in the statement of P2.1b above, P∗E from (2.20) has now been

replaced by P∗E(π∗1) in (2.24) and the feasibility condition of (2.18) is still held.

Following the step-by-step approach, and due to the monotonicity of P∗E w.r.t Pmin

and π1, P2.1 is now reduced to P2.1b. The solution to this problem is presented

in the following proposition.

Proposition 2.4 The optimal value for Bob’s maximum transmit power under an

average power constraint, Pavg, is given by

P ∗max =

2Pavg, if 2Pavg ≤ P †max

P ‡max, otherwise,(2.26)

where P †max is the solution of δab(Pmax) = 1− 2τRab

for Pmax and P ‡max is the solution

to

maximizePmax

(1− τ

Rab(1− δab(Pmax))

)(1 + s ln s− s2

)subject to P †max ≤ Pmax ≤ 2Pavg.

(2.27)

Proof


See Appendix A.4.

Remark 2.3 The approach we have taken in solving P2.1 guarantees that the ob-

tained solution is globally optimal. Specifically, we first solved for the optimal Pmin

for any value of π1 and Pmax. We next solved for the optimal π1 as a function of any

given Pmax. Finally the optimal Pmax is obtained. The globally optimal solution to

P2.1 can thus be summarized as P ∗min = 0, π∗1 = max (1/2, τ/[Rab(1− δab(P ∗max))])

and P ∗max as given in (2.26)− (2.27).

As discussed in Remark 2.3, increasing the value of Pmax helps improve our

covert performance, but on the other hand, Pmax also affects the covertly conveyed

information through self-interference. Proposition 4 tells us that while satisfying

the average power constraint, the optimal choice of Pmax satisfies the covert rate

requirement with a bare equality, while maximizes the expected detection error

probability at Willie. There might exist values of τ for which the choice of Pmax

satisfying the rate constraint under π = 12

are not the optimal choice of maximizing

P ∗E. Under such a scenario, the freedom of adjusting π1 helps us in meeting the

rate requirement while keeping P ∗E as high as possible.

2.6 Numerical Results and Discussions

In this section, we present the numerical results and study the performance of

our proposed scheme in achieving covertness while satisfying a given covert rate

requirement. Unless otherwise stated, we set the transmit power at Alice Pa =

10dB, a pre-determined rate for Alice to Bob transmissionRab = 1, Bob and Willie’s

receiver noise power σ2b = σ2

w = −10 dB and Bob’s self-interference cancellation

coefficient2 φ = 0.01.The average power constraint on Bob’s AN power is 40 dB,

while for simplicity, the means of all fading channels are considered as 1/λab =

1/λaw = 1/λbw = 1/λbb = 1.

We first show the effect of Pa and φ on the optimal maximum transmit power

for Bob’s AN for varying covert transmission rate requirements, as demonstrated

in Fig. 2.3 and Fig. 2.4, respectively. In Fig. 2.3 with a fixed value of φ, a higher

2Self-interference passive suppression of roughly 34− 44 dB for FD systems has been reportedin the literature [87], while a combination of passive suppression and active cancellation resultingin a total self-interference suppression of 90 dB has also been demonstrated [83].

2.6 Numerical Results and Discussions 37

0.0 0.1 0.2 0.3 0.4 0.515

20

25

30

35

40

45

τ

P∗ max(dB)

Pa = 10 dB

Pa = 20 dB

Pa = 30 dB

Figure 2.3: Optimal maximum transmit power of Bob’s AN, P ∗max, versus the covertrate requirement from Alice to Bob, τ , for varying values of Alice’s transmit power,Pa.

value of Pa allows a higher value of P ∗max to maximize the detection error probability

at Willie, whilst satisfying the given rate requirement. In Fig. 2.4, with a fixed

value of Pa in the feasible range, a higher value of φ (i.e., poorer self-interference

cancellation) requires a lower value of P ∗max (i.e., less self-interference) to satisfy

the same rate requirement. We note here that in such circumstances, a reduced

P ∗max for a given Pa will adversely affect the achievable covertness.

We next consider the effect of Pa and φ on the optimal transmission probability

of Alice’s covert transmissions for varying covert transmission rate requirements,

as demonstrated in Fig. 2.5 and Fig. 2.6, respectively. From Fig. 2.5, we see

that for a given Pa, a choice of π1 = 1/2 is optimal up to a certain value of τ ,

but a further increase in τ results in an increase in optimal π1. For a given Pa, a

rate requirement can be met by decreasing the value of P ∗max, but it will in return

decrease the achievable covertness. Keeping in view the results shown in Fig. 2.3

and Fig. 2.4, the optimal solution dictates that instead of making a drastic change

in P ∗max, a better choice is to decrease P ∗max a little while π1 can be increased to

meet the rate requirement.


0.0 0.1 0.2 0.3 0.4 0.515

20

25

30

35

40

45

τ

P∗ max(dB)

φ = 0.01

φ = 0.05

φ = 0.1

Figure 2.4: Optimal maximum transmit power of Bob’s AN, P ∗max, versus the covertrate requirement from Alice to Bob, τ , for varying values of Bob’s self-interferencecancellation coefficient, φ.

0 0.10 0.44 0.46 0.48 0.50

0.50

0.52

0.54

0.56

τ

π∗1

Pa = 10 dB

Pa = 20 dB

Pa = 30 dB

Figure 2.5: Optimal choice of transmission probability, π∗1, versus the covert raterequirement from Alice to Bob, τ , for varying values of Alice’s transmit power, Pa.


0 0.38 0.41 0.44 0.47 0.5

0.50

0.52

0.54

0.56

0.58

0.60

0.62

0.1τ

π∗1

φ = 0.01

φ = 0.05

φ = 0.1

Figure 2.6: Optimal choice of transmission probability, π∗1, versus the covert raterequirement from Alice to Bob, τ , for varying values of Bob’s self-interferencecancellation coefficient, φ.

0.0 0.1 0.2 0.3 0.4 0.50.25

0.30

0.35

0.40

0.45

0.50

τ

P ∗E

φ = 0.01

φ = 0.05

φ = 0.1

Figure 2.7: The expected detection error probability at Willie, P ∗E, versus the covertrate requirement from Alice to Bob, τ , for varying values of Bob’s self-interferencecancellation coefficient, φ.


0.0 0.2 0.4 0.6 0.8 1.00.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

φ

P ∗E

τ = 0.2

τ = 0.3

τ = 0.4

Figure 2.8: The expected minimum detection error probability at Willie, P ∗E, versusBob’s self-interference coefficient φ, for varying values of covert rate requirementτ .

As the value of Pa is increased, the same effect appears for a little higher value

of τ . Fig. 2.6 shows the effect of increasing φ on the optimal π1 for a given value

of Pa, which is inverse of what is observed for increasing Pa. Since an increase in

φ will have a detrimental effect on the transfer of covert information, thus to keep

the covertness high and to satisfy the rate requirement, an increase in π1 is desired

for an even lower value of τ . To further demonstrate the effect of φ, we show

the expected detection error probability at Willie for different values of φ in Fig.

2.7. For a fixed Pa and a given φ, as τ increases, Pmax at Bob has to decrease in

order to reduce Bob’s self-interference. A lower value of Pmax will result in a lower

P∗E, since it decreases the confusion in received signal statistics at Willie. This

effect is shown more explicitly in Fig. 2.8, where we show the effect of φ on the

performance of proposed covert scheme through the expected minimum detection

error probability at Willie P ∗E. It should be noted here that a value of φ = 0

corresponds to a perfect cancellation of the self-interference while φ = 1 refers to

no cancellation or suppression at all, representing the worst case scenario for the

FD receiver. For a higher value of φ, Bob has to reduce P ∗max to satisfy a certain


0 0.1 0.38 0.42 0.46 0.50.0

0.1

0.2

0.3

0.4

0.5

P ∗E ≈ 0.385

P ∗E ≈ 0.005

τ

P ∗E

Under Proposed Solution

Under π1 = 1/2

Figure 2.9: The expected detection error probability at Willie, P ∗E, versus the covertrate requirement from Alice to Bob, τ , under the proposed scheme and under theapproach where π1 = 1/2.

rate requirement, which in effect, reduces the achievable covertness.

Last but not least, we investigate the advantage of our proposed scheme of

jointly optimizing π1 and Pmax over a benchmark scheme of only optimizing Pmax

while keeping π1 = 0.5. Fig. 2.9 shows the overall performance of our proposed

scheme in terms of the expected detection error probability at Willie versus the

covert rate requirement from Alice to Bob. For τ ∈ [0, 0.44], the proposed joint

optimization scheme performs the same as the benchmark scheme and there is

no discernable difference in P ∗E for the two schemes. However, for τ ≥ 0.44, the

optimal π1 starts to deviate from 0.5, as shown in Fig. 2.5 and Fig. 2.6. Here,

the P ∗E achieved by the joint optimization scheme reduces gradually as the rate

requirement increases, but the P ∗E for the benchmark scheme drops sharply, and

at τ = 0.5, the benchmark scheme offers P ∗E ≈ 0.005, which means almost no

covertness at all. Thus for τ ≥ 0.44, the proposed joint optimization scheme

provides a significant gain in the achievable covertness.


2.7 Conclusion

In this chapter, we have considered the potential of achieving covert communication

using a full-duplex receiver that generates artificial noise to cause detection errors

at a watchful adversary Willie. Considering a radiometer as the detector of choice

at Willie, we have analyzed the conditions under which Willie makes detection

errors, and characterized Willie’s optimal detection performance conditioned over

the fading channel realizations. From the perspective of covert communication

pair, we have provided design guidelines for the optimal choice of transmit power

of full-duplex receiver’s artificial noise. Owing to the self-interference of the full-

duplex receiver, these power levels need to be controlled carefully, otherwise they

affect the transfer of any covert information. We have also shown that contrary to

a commonly adopted assumption, the a priori transmission probabilities of 0.5 are

not always the optimal choice to achieve the best possible covertness.

Chapter 3

Covert Communications in

Backscatter Radio using Artificial

Noise

In Chapter 2, we considered the use of AN by the receiver of covert information

using an Fd mode. In this chapter, we consider the use of AN in backscatter

systems, where the transceiver employs AN of varying power to achieve the said

purpose of covertness.

3.1 Background

The Internet of Things (IoT) foresees integration of every object for interaction

via embedded systems. This will lead to a highly distributed network of devices

communicating with human beings as well as other devices. The IoT devices are

expected to be equipped with millions of sensors and communication capabilities,

making them an intrinsic part of the existing communication systems. It can

be an arduous task to keep these energy-hungry sensors alive, since majority of

these sensors are not easily accessible, due to their deployment in toxic and unsafe

environments, or at places hard to reach.

Backscatter communication [73, 74] offers unique advantages, eliminating the

need of any active RF components, resulting in a prolonged life-span of the wireless

devices and continued network functionality. These wireless devices can not only

43

44Covert Communications in Backscatter Radio using Artificial

Noise

harvest energy from the transmitter’s signal, but can also modulate the same sig-

nal to convey information. Although backscatter communication has been largely

deployed in radio frequency identification (RFID) systems for consumer-based ap-

plications e.g., supply-chain management, RFID cards have also made their way

into more sensitive arenas, e.g., access control, payment systems and asset track-

ing. However, the application of backscatter systems in such sensitive scenarios

is limited, owing to their broadcast nature and the ease of snooping information

through eavesdropping. One option to alleviate this issue may be to use stronger

encryption protocols, but the size, cost and power constraints of most backscatter

transponders do not warrant such luxuries [75].

Security of backscatter systems and specifically RFIDs has been considered

widely in the recent literature. The physical layer security of backscatter systems

has been considered in detail in [95–97], and references therein. In [98], a frequency

hopping RFID system in the presence of an adversarial reader is considered and a

theoretical analysis of decoding error probability is provided. In this chapter, we

present a study on a backscatter system where the reader (i.e., the transmitter)

tries to obtain information from a tag (i.e., the transponder) in such a way that

the transmission from the tag remains covert from a warden, Willie, who is looking

to detect the tag’s transmission to the reader.1 In our considered system, the

reader’s transmitted signal is not intended to be hidden, rather the reader looks to

manipulate its signal such that Willie remains unaware of tag’s response state.

The remainder of this chapter is organized as follows: Section 3.1 details our

system model, stating the communication scenario, and explaining the reader’s

proposed transmission scheme and the tag’s operation. Section 3.2 explains the

detection scheme at Willie, deriving the conditions under which Willie makes detec-

tion errors, and Willie’s optimal choice of detection threshold. Section 3.3 and 3.4

address the covertness strategy adopted by the reader satisfying a given covertness

requirement and provide the reader’s performance analysis. Section 3.5 provides

numerical results and details the effects of proposed scheme on the covert system

performance while concluding remarks are given in Section 3.6.

1We adopt the terms “reader” and “tag” as is commonly used in RFID literature, althoughthe analysis is applicable to a variety of systems employing backscatter communication.

3.2 System Model 45

3.2 System Model

A backscatter communication system with a passive tag is considered, as shown

in Fig. 3.1, where the tag possesses sensitive information that needs to be sent

to the reader. Being passive, the tag has no power supply, thus it cannot initiate

communication on its own and fully relies on the reader’s signal for its operation.

A monostatic reader is considered, whose transmitted signal is not only used by

the tag to harvest energy, but is also modulated by the tag to send information

to the reader. The tag utilizes Binary Phase Shift Keying (BPSK) [73] to send

information to the reader, thus the intentional reflection from the tag has two

possible states in each symbol, depending on the data the tag has to transmit.

We define a communication slot as a block of time over which the transmission of

a message from the tag to the reader is complete. Each slot contains n symbol

periods and we assume that n is large enough, i.e., n → ∞. Under this setting, a

warden Willie is also present as a silent observer, trying to detect whether or not

the tag transmits to the reader in a given slot.

We use the subscripts r, t and w to represent the terms associated with reader,

tag and Willie, respectively. The distances from reader-tag, tag-Willie and reader-

Willie are represented by drt, dtw and drw, respectively. For simplicity, we consider

the time delay among the signals arriving at a node to be negligible. The channel

coefficient between any two users a and b is denoted by hab, and is dependent

upon the combined antenna gain and distance between the two users. The additive

Gaussian noise at the reader’s receiver and Willie is denoted by nr ∼ N (0, σ2r) and

nw ∼ N (0, σ2w), respectively.

3.2.1 Proposed Reader Transmission Scheme

In conventional backscatter communication, the reader transmits a continuous wave

(CW) with a constant amplitude. This approach does not lend itself well to covert

communication, since under the assumption of Willie knowing the reader’s constant

transmit power, it is straightforward for Willie to raise an alarm when an additional

reflection from the tag is received at Willie alongside the reader’s signal.

To achieve covertness, we propose the following transmission scheme: instead

of transmitting a simple unmodulated CW, the reader transmits a noise-like sig-


Noise

READER

Receiver

CW Transmitter

TAG

EnergyHarvesting

ZL2ZL1

WILLIE

Radiometer

Figure 3.1: System model for covert communication in a backscatter system.

nal following Gaussian distribution. This creates confusion at Willie and makes

it impossible for Willie to cancel such a signal. More importantly, the trans-

mit power of the noise-like signal is randomized such that the reader’s transmit

power in each slot, PR, is a random variable, following a uniform distribution,

i.e., PR ∼ U(Pmin, Pmax). The introduction of randomness in the reader’s transmit

power creates uncertainty in Willie’s received power, effectively creating an artifi-

cial fading [99], such that Willie is unsure whether an increase in the received power

is due to the tag’s backscatter or simply a variation in the power of the reader’s

transmitted signal. We note here that the transmitted noise-like signal following

a Gaussian distribution might degrade the energy harvesting efficiency of the tag,

which can be considered as the cost of achieving covertness under the backscatter

scenario.

3.2.2 Tag’s Operation and Willie’s Detection

If the tag has information to send in a slot, it modulates the incident signal by

changing its load impedance. It reflects back a certain portion of the power con-

tained in the signal and absorbs the rest of the power for utilization, including

energy consumption by the tag’s chip, modulation circuitry and antenna [100, 101].

Assuming complex impedances, the wave reflection coefficient at the tag is given

by [73]

Γ =ZL − Z∗AZL + Z∗A

, (3.1)


where ZL and ZA represent the tag’s load and antenna impedance, respectively, and

(·)∗ denotes the conjugate operation. To convey any information to the reader, the

tag chooses an appropriate load impedance,

ZL =Z∗A + ΓxZ

∗A

1− Γx, (3.2)

where, under BPSK, Γx can be Γ−1 or Γ+1, depending upon the information symbol

x ∈ {−1,+1}. In this work, we assume that |Γ−1| = |Γ+1| = |Γ|.Based on the signals received in a slot, Willie has to decide whether the tag

transmitted any information by modulating the reader’s signal. Here, Willie faces

a binary hypothesis testing problem. The null hypothesis, H0, says that the tag

did not send any information to the reader, while the alternative hypothesis, H1,

says that the tag did modulate the reader’s signal, hence sending information to

the reader. It is assumed that Willie is unaware of the exact transmit power used

by the reader in each slot, although the transmission model and distribution of

reader’s transmit power is known to Willie. Also, Willie has full knowledge of the

associated antenna gains, reflection coefficients utilized by the tag under BPSK

and his receiver’s noise variance.


Due to the independent and identically distributed (i.i.d.) nature of Willie’s ob-

servation vector yw = [yw(1), yw(2), . . . , yw(n)], the optimal approach for Willie to

minimize his detection error, according to Neyman-Pearson criterion, is to use the

likelihood ratio test [102],

Λ(yw) =fyw|H1(yw|H1)

fyw|H0(yw|H0)

D1

≷D0

Υ, (3.3)

where Υ = 1 due to the assumption of equal a priori probabilities of each hypoth-

esis. Here, D1 and D0 correspond to a decision in favor of hypothesis H1 and H0,

and fyw|H1(yw|H1) and fyw|H0(yw|H0) are the likelihood functions of Willie’s ob-

servation vectors for the considered slot, under hypothesis H1 and H0, respectively.

Under H0, the tag chooses a load impedance that is conjugate matched to the an-

tenna impedance, resulting in a reflection bearing no information. The baseband


Noise

signal received by Willie under H0 is given by

yw(i,H0) = hrwc(i) + Sw(i) + nw(i), (3.4)

where i = 1, . . . , n represents the symbol index. Here, c(i) is the ith symbol trans-

mitted by the reader, Sw(i) = hrthtwc(i) represents the structural mode scattering

component [103, 104] of the tag’s reflection received at Willie2, and nw(i) is Willie’s

receiver noise component.

Under H1, the tag modulates the reader’s signal by intentionally mismatching

its load impedance to the antenna impedance, causing a deliberate reflection of the

received signal back to the reader. In this case, Willie’s baseband signal is

yw(i,H1) = hrwc(i) + Sw(i) + Aw(i) + nw(i), (3.5)

where Aw(i) represents the antenna mode scattering component of the tag’s reflec-

tion received at Willie. The antenna mode component depends on the load chosen

by the tag via (3.1) and (3.2), and is given by Aw(i) = hrthtw|Γ|c(i)x(i). Owing

to its low complexity and ease of implementation, we assume in this work that

Willie uses a radiometer [38, 41] for the detection of any covert response from the

tag. Under this assumption, the average power received at Willie becomes a cru-

cial quantity. Based on Frii’s equation [105, 106], we have h2ab = GabK

2

d2ab

, where Gab

represents the combined transmitter-receiver antenna gain between users a and b,

and K = λ4π

is a constant dependent upon the carrier wavelength. Using (3.4), the

average received power at Willie in a slot under H0 can be calculated as

Pw(H0) = limn→∞

1

n

n∑i=1

[(yw(i,H0)

)2]

= limn→∞

1

n

n∑i=1

[(hrwc(i) + Sw(i) + nw(i)

)2]

= αPR + σ2w,

(3.6)

where

α =GrwK

2

d2rw

+GrtGtwK

4

d2rtd

2tw

, (3.7)

2Note that the tag gives a constant (structural mode) reflection even when no information issent. In the majority of backscatter literature, the term originating from the structural mode isgenerally ignored in the analysis, as it has no impact on the reader’s error probability [73].


and in deriving (3.6), we have used the fact that∑n

i=1 c2(i) corresponds to the

sum of n independent and squared Gaussians, each with variance PR, and this sum

of squared Gaussians results in a Chi-squared random variable. In (3.7), the first

term corresponds to the reader’s signal received directly by Willie and the second

term corresponds to the structural mode component of tag’s antenna scattering as

received by Willie.

Under H1, the power received at Willie includes an additional term, due to the

information-bearing reflection from the tag. Following steps similar to the analysis

of H0, the average power received at Willie in a slot under H1 is given by

Pw(H1) = βPR + σ2w, (3.8)

where

β =GrwK

2

d2rw

+GrtGtwK

4

d2rtd

2tw

+GrtGtwK

4|Γ|2d2rtd

2tw

. (3.9)

In the following, we derive the optimal threshold of Willie’s radiometer that mini-

mizes the detection error probability.

Proposition 3.1 Under the assumption of a radiometer, the optimal value of

threshold for Willie’s detector isγ∗ ∈ (αPmax + σ2

w, βPmin + σ2w), if αPmax < βPmin

γ∗ = αPmax + σ2w, otherwise,

(3.10)

where α and β are as defined in (3.7) and (3.9), respectively.

Proof

Willie compares the average received power to a threshold, γ, and decides on either

of the hypothesis, H0 or H1, being true. In order to minimize his detection error,

Willie considers the following optimization problem

minγ

PFA + PMD. (3.11)


Noise

Here, we have

PFA = P [D1|H0] = P [Pw > γ|H0] = P[αPR + σ2

w > γ]

= P[PR >

γ − σ2w

α

].

(3.12)

Since PR ∼ U(Pmin, Pmax),

PFA =

1, if γ−σ2

w

α≤ Pmin

Pmax−(γ−σ2

wα

)Pmax−Pmin

, if Pmin <γ−σ2

w

α≤ Pmax

0, if γ−σ2w

α> Pmax.

(3.13)

Similarly,

PMD = P [D0|H1] = P [Pw < γ|H1] = P[βPR + σ2

w < γ]

= P[PR <

γ − σ2w

β

]

=

0, if γ−σ2

w

β≤ Pmin(

γ−σ2w

β

)−Pmin

Pmax−Pmin, if Pmin <

γ−σ2w

β≤ Pmax

1, if γ−σ2w

β> Pmax.

(3.14)

Willie has to choose his threshold, γ, such that PFA + PMD is minimized. Using

(3.13) and (3.14), the crucial values on the γ axis are αPmin + σ2w, αPmax + σ2

w,

βPmin + σ2w and βPmax + σ2

w. From (3.13) and (3.14), it can also be seen that

choosing γ ≤ αPmin + σ2w or γ > βPmax + σ2

w results in PFA + PMD = 1. Thus

the best choice of γ for Willie lies in the interval αPmin + σ2w < γ ≤ βPmax + σ2

w.

From the system model, we know that β > α and Pmax > Pmin, resulting in

βPmax + σ2w > αPmin + σ2

w, but the relation between αPmax + σ2w and βPmin + σ2

w

can not be determined. To resolve this discrepancy in order to determine the best

choice of γ for Willie, we consider these two options in further detail.

Case - I : αPmax < βPmin

We have three different intervals for the choice of γ here, which are considered in

the following:


(1) αPmin + σ2w ≤ γ ≤ αPmax + σ2

w

In this case,

PFA + PMD =αPmax − γ + σ2

w

α(Pmax − Pmin), (3.15)

and ∂(PFA+PMD)∂γ

= −1α(Pmax−Pmin)

< 0, dictating that γ > αPmax + σ2w should be

chosen.

(2) βPmin + σ2w ≤ γ ≤ βPmax + σ2

w

In this case,

PFA + PMD =γ − σ2

w − βPmin

β(Pmax − Pmin), (3.16)


= 1β(Pmax−Pmin)

> 0, and resultantly, γ < βPmin + σ2w should be

chosen.

(3) αPmax + σ2w < γ < βPmin + σ2

w

In this case, PFA + PMD = 0, which means that a choice of γ in this interval will

have no detection errors at Willie.

Case - II : αPmax ≥ βPmin

Again, we have three different intervals for the choice of γ, as considered in the

following:

(1) αPmin + σ2w ≤ γ ≤ βPmin + σ2

w

In this case,


w

α(Pmax − Pmin), (3.17)



< 0, which dictates that γ > βPmin + σ2w should be

chosen.


Noise

(2) βPmin + σ2w < γ ≤ αPmax + σ2

w

In this case,


w

α(Pmax − Pmin)+γ − σ2

w − βPmin




+ 1β(Pmax−Pmin)

< 0, and resultantly, γ ≥ αPmax + σ2w

should be chosen.

(3) αPmax + σ2w < γ < βPmax + σ2

w

In this case,

PFA + PMD =γ − σ2

w − βPmin



= 1β(Pmax−Pmin)

> 0, which dictates that γ ≤ αPmax + σ2w should be

chosen.

Since α and β are fixed quantities determined by the system parameters and

fully known by Willie, the results of Case-I and Case-II complete the proof.

3.4 Reader’s Strategy for Covertness

Under the considered scheme, the reader looks to manipulate its transmit power for

achieving covertness. We first establish a condition on the parameters of reader’s

transmit power distribution such that there are detection errors at Willie. Next we

consider the condition on the reader’s transmit power to achieve a target covertness

level determined by ε.

Lemma 3.1 To cause any detection errors at Willie, the reader has to choose the

range of its transmit power i.e., Pmin and Pmax, such that

Pmax

Pmin

≥ β

α, (3.20)


Proof

3.5 Reader’s BER Analysis 53

The proof builds on the proof of Proposition 3.1, where the condition under which

Willie makes detection errors is derived in Case-II.

After having derived the condition under which Willie is forced to make detec-

tion errors, we now present the condition for achieving a target level of covertness.

Proposition 3.2 To achieve a covertness level of ε, the reader should choose the

range of its transmit power i.e., Pmin and Pmax, such that

Pmax

Pmin

≥ εβ

εβ − (β − α), (3.21)


Proof

Building on Proposition 3.1 and Lemma 3.1, Willie’s optimal choice of threshold,

γ, under the condition αPmax ≥ βPmin, is to choose γ = αPmax +σ2w. For this value

of threshold, PFA = 0, and we have

PFA + PMD = PMD =αPmax − βPmin

β(Pmax − Pmin). (3.22)

To achieve a target covertness of ε, we require

PFA + PMD =αPmax − βPmin

β(Pmax − Pmin)≥ 1− ε, (3.23)

and a simple rearrangement gives the desired result.

Remark 3.1 We note that condition (3.21) in Proposition 3.2 holds as long as

ε > 1 − αβ

, thus the achievable value of ε depends on the ratio αβ

. This condition

manifests in such a way that for given system parameters, covertness beyond a

certain ε in not achievable, regardless of the choice of Pmax

Pmin.

3.5 Reader’s BER Analysis

The reader can easily tell whether the tag has transmitted BPSK-modulated signal

by looking at its received power because it completely knows its transmit power

in any slot. The reader’s receiver looks to decide about the tag’s message symbol


Noise

x being +1 or −1 from the received signal. The baseband signal received at the

reader after being reflected from the tag is

yr(i,H1) = Ar(i) + Sr(i) + nr(i), (3.24)

where Sr(i) = hrthtrc(i) and Ar(i) = hrthtr|Γ|c(i)x(i) represent the structural and

antenna mode reflections from the tag at the reader, respectively. Having complete

knowledge of c(i), hrt and htr, the reader can perfectly cancel out the structural

mode component from the received signal. Resultantly

yr(i,H1) = Ar(i) + nr(i)

= hrthtr|Γ|c(i)x(i) + nr(i),(3.25)

as the received signal. Rewriting (3.25), we get

yr(i,H1) = x(i) +nr(i)

hrthtr|Γ|c(i), (3.26)

where we recall that nr ∼ N (0, σ2r) and c ∼ N (0, PR). The second term in (3.26)

results in a Cauchy distribution with a location parameter of l0 = 0 [107]. Thus

the maximum likelihood decision rule at the reader’s receiver isx(i) = +1, if yr(i,H1) > 0

x(i) = −1, else.(3.27)

Using the pdf of a Cauchy random variable, the BER for the reader, pbr, can be

obtained as

pbr =

∫ ∞−∞

1

2− 1

πarctan

1√σ2rd

4rt

|Γ|2GrtGtrK4z

fPR(z)dz, (3.28)

where the argument of arctan(·) is the square-root reciprocal of the received SNR

at the reader, and fPR(·) denotes the pdf of PR.


10-4 10-3 10-2 10-1

Covertness Parameter (ǫ)

0

5

10

15

20

25

30

P max

/ P

min

(dB

)

Reflection Coefficient |Γ| = 0.3Reflection Coefficient |Γ| = 0.5Reflection Coefficient |Γ| = 0.8

Figure 3.2: Ratio of Pmax and Pmin required for a target covertness.


In this section, we present numerical results to study the performance of our pro-

posed covert communication scheme. A UHF system with a carrier frequency of

915 MHz is considered. The reader-tag, tag-Willie and reader-Willie distances are

assumed to be 2 m, and all the users are assumed to have isotropic antennas. The

noise variance at Willie and reader’s receiver is −100dBm [96].

Fig. 3.2 shows the ratio of the support parameters of the reader’s transmit

power, Pmax

Pmin, plotted in dB against the covertness requirement, ε, for different values

of the reflection coefficient, |Γ|. For a given value of the reflection coefficient, the

required power ratio increases as the covertness requirement increases. Thus for

a given |Γ|, the reader needs to have higher variations in its transmit power to

achieve a better covert performance. However, as discussed in Remark 3.1, for

a given combination of the reflection coefficient and system parameters (antenna

gains, distances, carrier frequency), the achievable covertness does not increase

beyond a certain value. Reducing the reflection coefficient |Γ| helps to achieve a

lower ε, hence better covertness. However, lowering |Γ| reduces the received SNR at

the receiver, hence degrading the BER performance of backscatter communication.


Noise

0 10 20 30 40 50SNR (dB)

10-3

10-2

10-1

BER

Non-Covert Communication Covert Communication ε = 0.1Covert ε = 1.1x10-4

Figure 3.3: BER Comparison of non-covert and covert communication schemes.The tag’s reflection coefficient |Γ| = 0.8.

We note here that the achievable covert performance depends on Pmin and Pmax

only through the ratio Pmax

Pmin, not their individual values.

Fig. 3.3 plots the BER of a conventional non-covert communication, where the

reader transmits a constant-amplitude CW signal, and the BER of the proposed

covert communication with variable power at the reader. For the covert commu-

nication, we consider two covert requirements of ε = 0.1 and ε = 1.1 × 10−4. The

tag’s reflection coefficient is |Γ| = 0.8. Note that ε = 0.1 represents a poor covert

performance while ε = 1.1 × 10−4 represents almost the best possible covert per-

formance that can be achieved (see the curve for |Γ| = 0.8 in Fig. 3.2). The BER

is plotted against the received SNR at the reader. For the covert communication

with variable power, the distribution of transmit power (i.e., the values of Pmax and

Pmin) is set such that the average received SNR is the same as the received SNR in

the non-covert communication. Firstly, we observe a huge BER difference between

the non-covert and covert communication schemes. This is due to the difference

between constant-amplitude signaling and the proposed signaling scheme. As ex-

plained in Sec 3.2.1, the variation in reader’s transmit power is necessary to create

3.7 Conclusion 57

confusion at Willie, regardless of tag’s transmission state, as an essential design to

achieve covertness in the proposed scheme. Unfortunately, such a design pays a

significant price in terms of BER. Next, focusing on the covert communication, we

see that the BER gap between a poorly covert system (i.e., ε = 0.1) and a strongly

covert system (i.e., ε = 1.1×10−4) is small, roughly 1.5−2.5 dB. This tells us that

the price to pay for improving the covert performance from a poorly covert system

is reasonably small.

3.7 Conclusion

In this work, we showed how a backscatter communication system can achieve

covertness in the presence of a warden Willie. The proposed scheme requires the

reader to use a noise-like signal with variable transmit power drawn from a uniform

distribution. By controlling the maximum and minimum transmit powers of the

reader, the system is able to achieve a target level of covertness. Comparing with

a conventional backscatter system with no covertness, the BER degradation from

no covertness to some (poor) covertness is huge. Nevertheless, the additional BER

degradation for improving covert performance is much smaller.

Chapter 4

Covert Communications within a

Public Link under Channel

Uncertainty

In the previous two chapters, we considered the use of AN to cause uncertainties

at Willie for achieving covertness. We now turn out attention towards the second

source of confusion at Willie, i.e., uncertainty in his knowledge of channel from

the transmitter of covert information. In this chapter, we consider the scenario

where Willie gathers infinite observations to make a decision about Alice’s covert

transmission to Bob, while we look to hide this transmission under a legitimate

link to another user exploiting channel uncertainty.

4.1 Background

One of the main assumptions in most of covert communications literature is that

the channel state information (CSI) of both the covert link is perfectly known at

both the legitimate receiver and the transmitter, enabling secure encoding and ad-

vanced signaling. From the PLS perspective, increasing attention has been paid

to the impact of the uncertainty in the CSI of both legitimate receiver and eaves-

dropper’s channels at the transmitter, e.g., [108–112]. Usually, the CSI is obtained

at the receiver by channel estimation during pilot transmission. Then, a feedback

link (if available) is used to send the CSI to the transmitter. Hence, the accuracy

59

60 Covert Communications within a Public Link under Channel Uncertainty

Figure 4.1: Illustration of the Covert Communication Scenario

of the channel estimation at the receiver affects the quality of CSI at the transmit-

ter. However, in covert communication scenarios, transmitting pilots and acquiring

feedbacks is often infeasible, especially as the transmission of pilots will also en-

able the adversary to acquire channel information from the covert transmitter. In

this chapter, we consider the scenario where a public link is used to hide a cover

link while users including the legitimate and covert receivers and warden Willie

suffer from uncertainty in their channel knowledge from the transmitter. Under

this uncertainty scenario, we first derive Willie’s optimal detection performance

and building on that, we determine the optimal transmission rates for both the

legitimate and covert links under certain transmission outage probabilities.

The rest of this chapter is organized as follows: Section 4.2 presents our system

model, discussing in detail the adopted channel uncertainty model. In Section 4.3,

we present Willie’s detection analysis under channel uncertainty and also analyti-

cally derive the optimal choice of his detection threshold. Section 4.4 builds upon

Willie’s optimal detection performance, deriving the transmission outage proba-

bilities for the legitimate and covert receivers. Numerical results are provided in

Section 4.5, and we conclude the chapter in Section 4.6.

4.2 System Model

We consider a scenario, as shown in Fig. 4.1, where the transmitter (Alice) openly

transmits to the legitimate user (Carol), all the time. Alice also wants to transmit

to the covert user (Bob), but she wants to hide this communication from the warden

(Willie), using the transmission to Carol as her cover. Willie, being passive, silently

4.2 System Model 61

observes the communication environment, and tries to detect whether Alice is also

transmitting to Bob. It is assumed that Willie knows the transmit power used by

Alice, and adopts a radiometer (power detector) as his detector. The distances

from Alice-to-Carol, Alice-to-Bob, and Alice-to-Willie are denoted by dac, dab and

daw, respectively, and each user is equipped with a single antenna.

When Alice communicates with Carol or Bob, she transmits her message by

mapping it to the sequence xc = [x1c , x

2c , . . . , x

nc ] or xb = [x1

b , x2b , . . . , x

nb ], respec-

tively, where n is the number of channel uses. The average power per symbol in

xc and xb is normalized to 1. Alice employs zero mean Gaussian signalling with

variances (i.e., transmit powers) Pac and Pab for Carol and Bob’s transmission, re-

spectively. It should be noted here that Alice uses a constant transmit power to

Carol, as Carol is unaware of any covert transmission from Alice, and expects a

known power at her receiving terminal.

4.2.1 Channel Model

The effect of fading between Alice and user k is modelled by a fading coefficient

hak, where k is either b (Bob), c (Carol) or w (Willie). Here hak follows a circu-

larly symmetric complex Gaussian (CSCG) distribution with zero mean and unit

variance, i.e., hak ∼ CN (0, 1). We consider block fading channels, hence the fading

coefficients remain constant in one block and change independently from one block

to another. We adopt a commonly-used assumption that transmission of a message

is completed within one block, i.e., quasi-static fading channels are considered, and

the block boundaries are synchronized among all the users. Due to the indepen-

dent change of fading coefficients among blocks, we focus our analysis on one given

block, as the knowledge of previous blocks does not help Willie in improving his

detection performance.

While transmitting continuously to Carol, Alice potentially transmits to Bob

in a given block. Alice and Bob have a pre-shared secret which enables Bob to

know beforehand the block chosen by Alice. Analyzing his observations for a given

block, Willie has to decide whether Alice also covertly transmitted to Bob. The

null hypothesis (H0) states that Alice did not talk to Bob, while the alternative

hypothesis (H1) states that Alice did talk to Bob. The signal vector received at


user k is

yk =

hak√Pacxc

dα/2ak

+ hak√Pabxb

dα/2ak

+ vk, if H1 is true

hak√Pacxc

dα/2ak

+ vk, if H0 is true(4.1)

where α is the path-loss exponent, vk ∼ CN (0, σ2kIn) represents the user k’s re-

ceiver noise vector, the elements of which follow a CSCG distribution with zero

mean and variance σ2k. Here, In represents an n× n identity matrix.

Considering channel uncertainty, the channel coefficient hak is given by [113, 114]

hak = hak + hak, (4.2)

where hak and hak represent the known part and the uncertain part of hak at the

corresponding receiver, respectively, and they are zero-mean, independent, CSCG

random variables. The variance of the channel uncertainty for user k is denoted by

βk = E[|hak|2], 0 ≤ βk ≤ 1, and provides a measure of channel uncertainty at user

k. Accordingly, the variance of hak is 1− βk, since the variance of hak is 1.


From the i.i.d. nature of Willie’s received vector yw, given in (4.1), each element

(symbol) of yw i.e., yiw has a distribution given byCN (0, |haw|2ζ1 + |haw|2ζ1 + σ2w), if H1 is true

CN (0, |haw|2ζ0 + |haw|2ζ0 + σ2w), if H0 is true

(4.3)

where ζ0 , Pacdαaw

and ζ1 , Pac+Pabdαaw

. By application of the Neyman-Pearson crite-

rion, the optimal approach for Willie to minimize his detection error is to use the

following likelihood ratio test [102],

Λ(yw) =fyw|haw,H1

(yw|haw, H1)

fyw|haw,H0(yw|haw, H0)

D1

≷D0

Υ, (4.4)

where Υ = 1 due to the assumption of equal a priori probabilities of each hypoth-

esis. Here, D1 and D0 correspond to a decision in favor of hypothesis H1 and H0,


and fyw|haw,H1(yw|haw, H1) and fyw|haw,H0

(yw|haw, H0) are the likelihood functions

of Willie’s observation vectors for the considered block, under hypothesis H1 and

H0, respectively.

4.3.1 Detection using a Radiometer

We first substantiate that radiometer is indeed the optimal detector for Willie in

our system model. We then derive the optimal threshold of the radiometer that

minimizes the detection error at Willie.

Lemma 4.1 Under the considered system model, the optimal decision rule that

minimizes the detection error at Willie is

Pwn

D1

≷D0

λ, (4.5)

which corresponds to a threshold test on Pw, where Pw =∑n

i=1 |yiw|2 is the total

power received by Willie in a given block. Here, λ is the chosen threshold, and n is

the number of channel uses in a block.

Proof

The proof follows along the same lines as the proof of Lemma 2 in [38], where it

has been shown using the concepts of stochastic ordering [93] that a radiometer is

optimal for Willie under block fading channels.

4.3.2 Optimal Threshold for Willie’s Radiometer

After establishing the fact that the optimal strategy for Willie is to employ a

radiometer, we next evaluate the optimal setting of his radiometer’s threshold.

Theorem 4.1 Using a radiometer for detecting Alice-Bob covert transmission, the

optimal value of threshold for Willie’s detector is

λ∗ =

λ†, if |haw|2 < λ†−σ2

w

ζ1

|haw|2ζ1 + σ2w, otherwise

(4.6)

where λ† = ζ1ζ0βwζ1−ζ0 log

[ζ1ζ0

exp(

(ζ1−ζ0)σ2w

ζ1ζ0βw

)], and haw is Willie’s known part of his

channel from Alice.


Proof

To find the optimal threshold, we consider the optimization problem

minλ

PFA + PMD. (4.7)

From Lemma 4.1, the decision at Willie’s detector regarding Alice’s transmission

to Bob is given by (4.5), where Pw is a sufficient statistic for Willie’s detector test.

The probabilities of detection error at Willie are given by

PFA = P [Pw/n > λ |H0] = P[(σ2

w + |haw|2ζ0 + |haw|2ζ0)χ2

2n

n> λ

], (4.8)

and

PMD = P [Pw/n < λ |H1] = P[(σ2

w + |haw|2ζ1 + |haw|2ζ1)χ2

2n

n< λ

], (4.9)

where χ22n represents a chi-squared random variable with 2n degrees of freedom.

From the Strong Law of Large Numbers, we know that χ22n/n converges to 1, almost

surely. The Lebesgue’s Dominated Convergence Theorem [115] allows us to directly

replace χ22n/n by 1, as n→∞. Thus for a given realization of haw, we have

PFA = P[(σ2

w + |haw|2ζ0 + |haw|2ζ0) > λ]

= P

[|haw|2 >

λ− σ2w − |haw|2ζ0

ζ0

]

=

exp(|haw|2ζ0+σ2

w−λζ0βw

), if λ−σ2

w−|haw|2ζ0ζ0

≥ 0

1, otherwise

(4.10)

and

PMD = P[(σ2

w + |haw|2ζ1 + |haw|2ζ1) < λ]

= P

[|haw|2 <

λ− σ2w − |haw|2ζ1

ζ1

]

=

1− exp(|haw|2ζ1+σ2

w−λζ1βw

), if λ−σ2

w−|haw|2ζ1ζ1

≥ 0

0, otherwise.

(4.11)


Following (4.10) and (4.11), we have

PFA + PMD =

1, if λ < |haw|2ζ0 + σ2

w

κ0, if |haw|2ζ0 + σ2w ≤ λ ≤ |haw|2ζ1 + σ2

w

κ, if λ > |haw|2ζ1 + σ2w

(4.12)

where

κ = 1− κ1 + κ0, κ0 , exp

(|haw|2ζ0 + σ2

w − λζ0βw

), κ1 , exp

(|haw|2ζ1 + σ2

w − λζ1βw

).

(4.13)

We next analyze the three possible cases in (4.12) separately, and find the

optimal value of λ that minimizes PFA + PMD.

Case I : λ < |haw|2ζ0 + σ2w

As long as λ < |haw|2ζ0 + σ2w, PFA + PMD = 1, and cannot be minimized.

Case II : |haw|2ζ0 + σ2w ≤ λ ≤ |haw|2ζ1 + σ2

w

Here, PFA+PMD is a decreasing function of λ, hence Willie chooses the highest pos-

sible value of λ, which is |haw|2ζ1 +σ2w, leading to PFA +PMD = exp

(|haw|2(ζ0−ζ1)

ζ0βw

).

Case III : λ > |haw|2ζ1 + σ2w

In order to determine the optimal value of λ in this case, we set the first derivative

of PFA + PMD w.r.t. λ equal to zero, which results in

∂(PFA + PMD)

∂λ=

1

ζ1βwexp

(|haw|2ζ1 + σ2

w − λζ1βw

)

− 1

ζ0βwexp

(|haw|2ζ0 + σ2

w − λζ0βw

)= 0.

(4.14)

After a few simple manipulations, the optimal value of λ in this case is given by

λ† ,ζ1ζ0βwζ1 − ζ0

log

[ζ1

ζ0

exp

((ζ1 − ζ0)σ2

w

ζ1ζ0βw

)]. (4.15)


We note that λ† is independent of the channel realization haw, and represents the

inflection point of PFA + PMD. It can be verified through simple calculations that∂(PFA+PMD)

∂λ> 0 for λ > λ†, and ∂(PFA+PMD)

∂λ< 0 for λ < λ†. The second derivative

of PFA + PMD w.r.t λ is

∂2(PFA + PMD)

∂λ2=− 1

ζ21β

2w

exp

(|haw|2ζ1 + σ2

w − λζ1βw

)

+1

ζ20β

2w

exp

(|haw|2ζ0 + σ2

w − λζ0βw

),

(4.16)

which is strictly positive as long as the chosen λ† satisfies

λ† <ζ1ζ0βwζ1 − ζ0

log

[ζ2

1

ζ20

exp

((ζ1 − ζ0)σ2

w

ζ1ζ0βw

)], (4.17)

where the requirement in (4.17) follows by simply considering the fact that ζ1 > ζ0.

Thus λ† represents the optimal threshold value for Willie, as long as it satisfies the

condition λ† > |haw|2ζ1 + σ2w. If λ† does not satisfy this, then using the monotonic

increase in PFA + PMD for λ > λ†, the minimum value of λ is chosen that satisfies

λ ≥ |haw|2ζ1 + σ2w.

4.4 Performance of Covert Communication

Knowing the best detection at Willie, we now consider the overall performance of

the covert communication system. We first derive Willie’s average detection error

probability from Alice’s perspective, which will be used to quantify the covertness.

Next, we derive the communication outage probabilities at Carol and Bob, which

are used to determine the feasible regime of the transmission rates.

4.4.1 Average Detection Error Probability

Using the optimal value of λ from (4.6), we have

PFA + PMD =

1− κ†1 + κ†0, if |haw|2 < λ†−σ2w

ζ1

κ†, otherwise(4.18)

4.4 Performance of Covert Communication 67

where

κ† , exp

(|haw|2(ζ0 − ζ1)

ζ0βw

),

κ†1 , exp

(|haw|2ζ1 + σ2

w − λ†ζ1βw

),

κ†0 , exp

(|haw|2ζ0 + σ2

w − λ†ζ0βw

).

(4.19)

Since haw is unknown to Alice, she has to rely on the average measure of Willie’s

performance to assess the possible covertness. We use PwE to denote the average

PFA + PMD over all realizations of haw.

Proposition 4.1 The average detection error probability at Willie is

PwE =

[1− exp

(σ2w − λ†

(1− βw)ζ1

)]×[1− βw

2βw − 1exp

(σ2w − λ†βwζ1

)+

βw2βw − 1

exp

(σ2w − λ†βwζ0

)]+ exp

(σ2w − λ†

(1− βw)ζ1

)[ζ0βw

(1− βw)ζ1 + (2βw − 1)ζ0

].

(4.20)

Proof

Relying on Willie’s knowledge of channel’s known part, and based on the law of

total expectation, we have

PwE = E|haw|2 [PFA + PMD]

= E|haw|2[PFA + PMD

∣∣∣|haw|2 < λ† − σ2w

ζ1

]P[|haw|2 <

λ† − σ2w

ζ1

]+ E|haw|2

[PFA + PMD

∣∣∣|haw|2 ≥ λ† − σ2w

ζ1

]P[|haw|2 ≥

λ† − σ2w

ζ1

],

(4.21)

and evaluating this expression completes the proof.

To achieve covertness, Alice chooses her transmit power levels to Carol and Bob

such that.

PwE ≥ 1− ε. (4.22)


4.4.2 Outage Probabilities at Carol and Bob

Proposition 4.2 Under hypothesis H1, the outage probability at Carol for a rate

Rc is

δc(H1) = 1−P βc∆c

βc∆c(Pac + Pab) + P βc∆c

exp

(−∆cd

αacσ

2c

P βc∆c

), (4.23)

where P βc∆c

, (1− βc) [Pac − Pab∆c], and ∆c , 2Rc − 1.

Proof

Under H1, the signal vector received at Carol is

yc =hac

√Pacxc

dα/2ac

+ hac

√Pacxc

dα/2ac

+ hac

√Pabxb

dα/2ac

+ hac

√Pabxb

dα/2ac

+ vc, (4.24)

and the signal-to-noise ratio (SNR) is

SNRcH1

=|hac|2Pac

|hac|2Pab + |hac|2(Pac + Pab) + dαacσ2c

. (4.25)

The outage probability at Carol is

δc(H1) = P[log2(1 + SNRc

H1) < Rc

]= P

[|hac|2Pac

|hac|2Pab + |hac|2(Pac + Pab) + dαacσ2c

< ∆c

]

= P

|hac|2 < ∆c

[|hac|2(Pac + Pab) + dαacσ

2c

]Pac − Pab∆c

,(4.26)

where ∆c , 2Rc − 1. Since hac and hac are independent, thus

δc(H1) =

∫ ∞0

1− exp

−∆c

[|hac|2(Pac + Pab) + dαacσ

2c

](1− βc)(Pac − Pab∆c)

f|hac|2(|hac|2)d|hac|2,

(4.27)

and the solution of this integration gives the desired result.

It is important to note here that under H0, the outage probability at Carol is

δc(H0) = 1− (1− βc)βc∆c + (1− βc)

exp

(− ∆cd

αacσ

2c

(1− βc)Pac

), (4.28)


which has a value lower than δc(H1) in (4.23), due to no interference from Alice-Bob

transmission. Thus Carol’s performance deteriorates under hypothesis H1.

Proposition 4.3 Under hypothesis H1. the outage probability at Bob for a rate

Rb is

δb(H1) = 1−P βb∆b

βb∆b(Pab + Pac) + P βb∆b

exp

(−∆bd

αabσ

2b

P βb∆b

)(4.29)

where P βb∆b

, (1− βb) [Pab − Pac∆b], and ∆b , 2Rb − 1.

Proof

The proof follows along the same lines as the proof of Proposition 2.

For given outage constraints, e.g. δc ≤ 0.1 and δb ≤ 0.1, the achievable rates for

Carol and Bob, under H1, can be numerically calculated using (4.23) and (4.29).

For Carol, any achievable rate that satisfies the outage constraint under H1 will

naturally satisfy the outage constraint under H0. Hence the focus is on the perfor-

mance of Carol and Bob under H1.


In this section, we present the numerical results to show the effect of covertness

requirement (ε) and channel uncertainty (β) on the achievable rate region for Carol

and Bob. The noise variance of all the users is assumed to be normalized to 1, and a

total transmit power constraint of 30dB is considered at Alice. For these numerical

results, we have considered βw = βc = βb , β, while the outage probability

constraints at Carol and Bob are δc ≤ 0.1 and δb ≤ 0.1, respectively.

Fig. 4.2 shows the achievable rate region for Carol and Bob, under the effect of

changing β, for a fixed ε = 0.2. The solid lines, indicated by arrows with varying

values of β, determine the rate region without any covert requirement. It can be

observed from the figure that increasing the value of β allows Alice to use more

power, Pab, for transmission to Bob, hence there is an increase in Bob’s achievable

rate. This increase in feasible Pab is due to the increased channel uncertainty at

Willie, causing his detection performance to deteriorate. On the other hand , there

is an adverse effect on the overall rate region for Carol and Bob, since the increase


0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.1

0.2

0.3

0.4

0.5

0.6β = 0.1

β = 0.2

β = 0.3

β = 0.1

β = 0.2

β = 0.3

Rb

Rc

Figure 4.2: The achievable rate region for Carol and Bob under the effect of varyingchannel uncertainty, β. Other parameters are ε = 0.2, α = 3 and daw = dac = dab =5.

in channel uncertainty variance affects their decoding performance. Thus for a

fixed covert requirement, increasing the value of β in a reasonable range incurs a

rate loss for Carol, but increases the achievable rate for Bob.1

Fig. 4.3 shows the achievable rates for Carol and Bob, under the effect of

changing ε, for a fixed β = 0.2. For a fixed channel uncertainty, relaxing ε from

0.1 to 0.3 shows an increase in the feasible rate region. Since relaxing ε allows a

direct increase in feasible Pab for a given Pac, we can clearly see an expansion in

the achievable rate region, in favor of Bob.

4.6 Conclusion

In this work, we examined how to achieve covert communication in a public and

legitimate communication link when users have uncertainty about their channels.

We first derived a closed-form expression for the optimal threshold of of Willies op-

1It should be noted here that a value of β = 0.3 or ε = 0.3 is quite large from the practicalperspective. We consider such values in our numerical results to illustrate the effect of theseparameters on the achievable rate region.

4.6 Conclusion 71

0.00 0.02 0.04 0.06 0.08 0.10

0.1

0.2

0.3

0.4 ε = 0.1

ε = 0.2

ε = 0.3

Rb

Rc

Figure 4.3: The achievable rate region for Carol and Bob under the effect of varyingcovertness requirement, ε. Other parameters are β = 0.2, α = 3 and daw = dac =dab = 5.

timal detector. Next, we quantified the achievable outage rate region for Carol and

Bob. Our results showed that the presence of channel uncertainty at Willie allows

Alice to achieve a certain amount of covertness, while this channel uncertainty also

affects the achievable rates for Carol and Bob.

Chapter 5

Covert Communications with

Channel Training and Finite

Blocklength

In Chapter 4, we considered exploiting channel uncertainty at Willie under the

case of asymptotically infinite blocklength where a legitimate link is used to hide

the covert transmission. In this chapter, we focus on the scenario where Willie’s

number of observations is finite. This introduces an inherent uncertainty at Willie

in comparison to the case of infinite blocklength where Willie will have no ambiguity

in his noise variance as his number of observations increases. We thus look to

exploit this fundamental uncertainty alongwith the channel uncertainty to hide

Alice’s transmission to Bob.

5.1 Background

In this chapter, we consider achieving covert communications under finite block-

length where both Bob and Willie have imperfect knowledge of their respective

channels from Alice. To help Bob estimate his channel, Alice transmits publicly

known pilot symbols. Intuitively, it is clear that the higher the training budget,

the lower will be the channel estimation error, resulting in a higher throughput.

This pilot transmission, on the other hand, also enables Willie to estimate his chan-

nel from Alice, improving his capability to detect any covert transmission. While

73

74 Covert Communications with Channel Training and Finite Blocklength

the impact of imperfect channel knowledge on the throughput performance is well

known from numerous prior works, e.g., [116–118], the impact on Willie’s detection

performance is much less understood.

Covert communications under imperfect channel knowledge has been previously

considered in [77], where under asymptotically infinite blocklength, the variance of

channel uncertainty at the users has been incorporated in the analysis. The authors

in [119] present a scheme where covertness is achieved with the help of a full-duplex

relay, and users suffer from channel uncertainty. More recently, [120] presented an

analysis of channel estimation design in covert communications, where the number

of training channel uses to maximize the effective signal-to-noise ratio at the covert

link is optimized. While [77] and [119] present their analysis under infinite block-

length assumptions, additional sources of uncertainty in the form of an additional

information receiver and an artificial noise transmitting relay, respectively, have

been considered in these works to achieve covertness. Although similar to our con-

sidered scenario, [120] presents the analysis under a finite blocklength, the authors

consider an AWGN channel for Willie, whereas we consider quasi-static fading

channels for both Bob and Willie. Furthermore, [120] advocates the use of equal

powers during the training and data transmission phases, while we first establish

the best detection performance at Willie and then optimize the data transmission

power to maximize the covert throughput under certain covertness requirements.

While the above mentioned works specifically rely on and exploit the channel un-

certainty at Willie to achieve covertness, we show that in scenarios pertinent to

covert communications, where the transmit power levels are generally low, Willie’s

channel knowledge does not play as an important role as considered in the prior

work, and hence, we are able to provide a unified approach to covert transmission

design regardless of the exact amount of channel uncertainty at Willie.

The rest of this chapter is organized as follows: Section 5.2 provides details of

our communication scenario, considered channel estimation and training and the

assumptions used in this paper. Section 5.3 explains the detection at Willie under

perfect CSI and CDI only scenarios, and establishes the equivalence of these two

cases for low transmit powers at Alice. In Section 5.4, we analyze the covertness

achieved by Alice, addressing the optimal design of data transmit powers and chan-

nel uses to maximize the covert throughput under a given covertness constraint.

5.2 System Model 75

Alice

Bob Willie

Com

mun

ication

Detection

Figure 5.1: Covert communications model under consideration.

Section 5.5 provides numerical results validating our analysis and giving further

design insights. Finally, the chapter is concluded in Section 5.6.

5.2 System Model

We consider a covert communication scenario, as shown in Fig. 5.1, where the

transmitter, Alice, desires to send information to the receiver, Bob, in presence of

an adversary, Willie, whose job is to detect whether any transmission by Alice took

place or not. Alice, Bob and Willie are assumed to have a single antenna each. The

complex Gaussian noise at Bob and Willie’s receivers is denoted by nb ∼ CN (0, σ2b )

and nw ∼ CN (0, σ2w), respectively. We follow the common assumption that a

secret is shared between Alice and Bob [25, 41], which is unknown to Willie but

lets Bob know when Alice transmits a covert message. Employing random coding

arguments, Alice generates codewords by independently drawing symbols from a

zero-mean complex Gaussian distribution, where the codebook is known to Alice

and Bob only. We define a communication slot as a block of time in which the

transmission of a message from Alice to Bob is complete. When Alice transmits in

a slot, she transmits the codeword corresponding to her covert message.

5.2.1 Channel Model

We consider the channels from Alice to Bob, and Alice to Willie to be quasi-static

Rayleigh fading channels, where the effect of fading is modelled by a fading co-

efficient, hk, and k is either b (Bob), or w (Willie). Here, hk follows a circularly

symmetric complex Gaussian (CSCG) distribution, with zero mean and unit vari-

ance, i.e., hk ∼ CN (0, 1). Due to the quasi-static fading assumption, the fading


coefficients remain constant in one slot (i.e., one coherence interval), and change

independently from one slot to the next. It is assumed here that the slot bound-

aries are synchronized among all parties. Due to the independent change of fading

coefficients among slots, the focus is on one given slot, as the knowledge of previous

slots does not help Willie in improving his detection performance [38, 77]. We also

note here that for quasi-static fading channels, the decoding error events at the

information receiver (i.e., Bob) are dominated by channel outage events [121, 122].

5.2.2 Training-Based Transmission and Channel Estima-

tion

It is assumed that Alice transmits publicly known pilot symbols periodically at

the beginning of every slot, whereas, covert data transmission only occurs in a

secretly chosen slot, which is only known to Alice and Bob. Each slot consists of

N symbols, where the first NT symbols serve as the pilots, and are transmitted

using power PT . Depending on whether or not covert data transmission happens

in the current slot, data symbols or nothing is transmitted over the remaining ND

symbol periods, i.e., N = NT +ND. During the training phase, the signal received

by Bob for the ith channel use is

yT (i) =√PThbxT (i) + nb(i) (5.1)

where hb is the channel coefficient from Alice to Bob, and xT (i) is the normalized

training signal transmitted by Alice. It is assumed that Bob uses the minimum

mean square error (MMSE) technique [123] to estimate his channel from Alice.

The estimation of channel coefficient and corresponding estimation error at Bob

are denoted by hb and hb, respectively. Thus

hb = hb + hb, (5.2)

where hb and hb follow zero mean CSCG distributions [124]. Furthermore, since

yT is a linear function of the channel coefficient, the linear MMSE estimation

becomes the optimal MMSE estimation, and the orthogonality principle implies

that E [|hb|2] = E[|hb|2] + E[|hb|2]. Based on LMMSE, the estimate of hb is given

5.2 System Model 77

by [124]

hb =

√PT

σ2b +NTPT

yTx†T . (5.3)

We define βb as the variance of the channel estimation error at Bob, i.e, βb =

E[|hb|2], and resultantly, E[|hb|2] = 1− βb, where [125]

βb =σ2b

σ2b +NTPT

. (5.4)

Since Bob is aware of the slot in which Alice transmits the covert data, he performs

channel estimation only in such a slot and then uses the obtained channel estimate

to perform data detection.

5.2.3 Performance Metrics

We assume that Willie uses a radiometer to detect any covert transmission by Alice.

Since Willie is unaware of the slot in which Alice transmits data, he observes all

the slots, where in each slot, he makes use of the first NT pilot symbols to learn the

channel coefficient from Alice, and collects the remaining ND symbols for detection

of possible data transmission. Under these assumptions, the average power received

over ND potential data symbols at Willie serves as the detection test statistic, given

by

T (yw) =1

ND

ND∑i=1

|yw(i)|2. (5.5)

Here, yw(i) represents Willie’s observation for the ith symbol duration of the po-

tential data transmission phase, given by

yw(i) =

nw(i), H0

√PDhwxD(i) + nw(i), H1,

(5.6)

where xD represents Alice’s transmit symbols and PD is Alice’s data transmit

power. From Alice and Bob’s perspective, it is imperative to force ε to be small,

to have large detection errors at Willie, i.e., to achieve strong covertness.


5.3 Detection Analysis at Willie

We note that Willie’s detection performance relies on his knowledge of the chan-

nel from Alice, hw. Here, we analyze Willie’s detection performance under two

extreme cases, i.e., when perfect CSI knowledge is available at Willie and when

only CDI is available. These two cases provide the bounds on Willie’s detection

performance under the case where he looks to utilize the publicly known pilot sym-

bols transmitted by Alice to learn the channel coefficients. We further show that

the detection performances of these extreme cases are asymptotically the same in

the large detection error regime, which is of interest to us for achieving a strong

covertness.

5.3.1 Detection under Perfect CSI Knowledge

Here, we consider the scenario when the instantaneous channel realization is pe-

fectly known at Willie. The probability of False Alarm and Missed Detection events

at Willie is given by

PFA = P

[1

ND

ND∑i=1

|yw(i)|2 > λ|H0

]= P

[χ2

2ND>NDλ

σ2w

]= 1−

γ(ND,

NDλσ2w

)Γ(ND)

,

(5.7)

and

PMD = P

[1

ND

ND∑i=1

|yw(i)|2 ≤ λ|H1

]= P

[χ2

2ND≤ NDλ

|hw|2PD + σ2w

](5.8)

=γ(ND,

NDλ|hw|2PD+σ2

w

)Γ(ND)

, (5.9)

respectively, where χ22ND

represents a chi-square random variable with 2ND degrees

of freedom, Γ(x) = (x− 1)! is the complete Gamma function, γ(·, ·) represents the

lower incomplete Gamma function, given by

γ(a, b) =

∫ b

0

e−xxa−1dx, (5.10)

5.3 Detection Analysis at Willie 79

and importantly, λ is the radiometer’s detection threshold chosen by Willie. The

detection error probability at Willie is thus given as

ζw = PFA + PMD = 1−γ(ND,

NDλσ2w

)Γ(ND)

+γ(ND,

NDλ|hw|2PD+σ2

w

)Γ(ND)

. (5.11)

We next present the optimal choice of Willie’s detection threshold and the

resulting minimum detection error probability.

Lemma 5.1 Under the assumption of perfect CSI knowledge, the optimal detection

threshold of Willie’s radiometer for a given channel realization, hw, is

λ∗CSI =σ2w(|hw|2PD + σ2

w)

|hw|2PDln

( |hw|2PD + σ2w

σ2w

), (5.12)

while the resulting minimum detection error probability is given by

ζ∗w,CSI = 1−γ(ND, ND

(1 + σ2

w

|hw|2PD

)ln( |hw|

2PDσ2w

+ 1))

Γ(ND)

+γ(ND,

NDσ2w

|hw|2PD ln( |hw|2PDσ2w

+ 1))

Γ(ND). (5.13)

Proof

To minimize the detection error probability, Willie considers the problem:

minλ

ζw = PFA + PMD. (5.14)

From the definition of upper and lower incomplete Gamma functions, Γ(s) =

Γ(s, x) + γ(s, x), where Γ(·, ·) is the corresponding upper incomplete Gamma func-

tion. Thus, we can write

ζw = 1− 1

Γ(ND)

[Γ

(ND,

NDλ

|hw|2PD + σ2w

)− Γ

(ND,

NDλ

σ2w

)]. (5.15)

Setting ∂ζw∂λ

= 0 and some algebraic manipulations give the optimal value of λ,

where we use derivative property of the upper incomplete Gamma function, given


by:

∂Γ(s, f(x))

∂x= −(f(x))s−1e−f(x)∂f(x)

∂x. (5.16)

Next, putting in the value of λ∗CSI into the expression for ζw in (5.11) gives the

desired result for ζ∗w,CSI .

5.3.2 Detection under Knowledge of CDI only

In this subsection, we consider the scenario where Willie does not know the channel

coefficient, and only the channel distribution information is available to Willie.

The detection error probability at Willie still has the same expression as given in

(5.11). However, since Willie is unaware of his instantaneous channel realizations

from Alice, the optimal detection threshold at Willie in this case is given by

λ∗CDI = arg minλ

E|hw|2 [ζw,CDI ] , (5.17)

where the average is considered over all possible realizations of hw.

5.3.3 Performance Comparison between CSI and CDI Cases

From Alice and Bob’s perspective, achieving strong covertness implies having large

detection errors at Willie which, in turn, requires Alice to transmit at very low

powers. Here, we show that for these low transmit power levels, as is customary in

covert communication scenarios, the detection error probabilities at Willie under

the perfect CSI case and CDI only case are indistinguishable. To show this, we

first present linear approximations of Willie’s detection error probability in the

asymptotically low power regime ( i.e., around PD → 0 ) under perfect CSI and

CDI only cases, and then establish the equivalence of these linear approximations.

Lemma 5.2 The linear approximation of ζ∗w,CSI for a given channel realization in

the asymptotically low power regime, is given as

limPD→0

ζ∗w,CSI ≈ 1− |hw|2NND

D e−ND

σ2wΓ(ND)

PD. (5.18)

Proof

5.4 Covertness under Channel Uncertainty 81

See Appendix B.1.

We next present a linear approximation for ζ∗w,CDI , which is Willie’s optimal

detection error probability under the case where only CDI is available to Willie.

Lemma 5.3 The linear approximation of ζ∗w,CDI for a given channel realization,

in the asymptotically low power regime, is given as

limPD→0

ζ∗w,CDI ≈ 1− |hw|2NND

D e−ND

σ2wΓ(ND)

PD. (5.19)

Proof

See Appendix B.2.

Proposition 5.1 For a given channel realization, the linear approximation of Willie’s

detection error probability under perfect CSI, ζ∗w,CSI , and under CDI only, ζ∗w,CDI ,

are equivalent in the asymptotically low power regime.

Proof

Results of Lemma 2 and Lemma 3 provide the desired equivalence.

Remark 5.1 From Proposition 1, Willie’s optimal (minimum) detection error

probabilities under the cases of perfect CSI and CDI only are asymptotically in-

distinguishable in the large detection error regime. The numerical validation of

this equivalence is provided in Fig. 5.2. This equivalence implies that the accuracy

of CSI at Willie does not change his detection performance that much as long as

Willie’s detection error probability is forced to be close to 1. From Alice and Bob’s

perspective, they are unaware of the CSI’s accuracy at Willie and want to ensure

large detection errors. Therefore, we use ζ∗w,CSI as the detection error probability at

Willie under training. Although this constitutes a worst case scenario from the per-

spective of covert communication design, it does yield a more robust, yet accurate,

approach.

5.4 Covertness under Channel Uncertainty

In this section, we first consider a system metric that affects the covert communi-

cation performance, and then find the optimal solution to the covertness problem


at Alice. We allow Alice to choose different power levels for pilot and data trans-

mission. For simplicity, the training duration is fixed to one symbol which is in

agreement with previous works on training-based communications [126, 127]. In

addition, the power of the pilot symbol is set to the maximum allowable transmit

power, i.e., PT = Pmax, Under this setup, the problem at Alice is of finding the

optimal power for data transmission and the number of symbols used for data in

order to maximize the covert throughput under a given covertness constraint. We

note here that as per Remark 5.1, it is desirable from Alice and Bob’s perspective

to keep Willie in the large detection error regime for achieving strong covertness.

5.4.1 Covert Connection Probability

During the covert data transmission, Alice transmits at a fixed, pre-determined,

rate to Bob which is denoted by R. Due to the random nature of fading channels

from Alice to Bob, a transmission outage occurs from Alice to Bob whenever C ≤ R,

where C is the capacity of the Alice to Bob channel, and in that case, Bob is

unable to reliably decode the information transmitted by Alice. Here, we derive the

complement of the outage probability, defined as the covert connection probability,

which is the probability that Bob can reliably decode a covert message from Alice,

transmitted at a fixed rate R. The covert connection probability, Pcc, is given by1

Pcc = 1− P [log2(1 + γb) ≤ R] (5.20)

where γb denotes the signal-to-noise ratio at Bob which, under the considered

channel uncertainty model, is given by [127]

γb =|hb|2PD

|hb|2PD + σ2b

. (5.21)

In the following, we present the desired covert connection probability.

Lemma 5.4 The covert connection probability for Alice to Bob transmission at a

1In our outage probability formulation, we use the Shannon capacity instead of the finite-blocklength capacity, because the channel dispersion, generally associated with finite blocklengthcommunication, is zero for quasi-static fading channels, and the decoding error events at theinformation receiver are dominated by channel outage [121, 122, 128].


fixed rate R, and under channel uncertainty at Bob, is given by

Pcc =1− βb

(1− βb) + βb(2R − 1)e− σ2

b (2R−1)

(1−βb)PD , (5.22)

where PD is Alice’s transmit power during data transmission and βb is the variance

of channel estimation error at Bob, as defined in (5.4).

Proof

Putting in the expression for γb into the expression of Pcc, we have

Pcc = 1− P

[log2(1 +

|hb|2PD|hb|2PD + σ2

b

) ≤ R

]

= 1− P

[|hb|2 ≤

(2R − 1)(|hb|2PD + σ2b )

PD

]. (5.23)

Then using the exponential distributions of |hb|2 and |hb|2 gives

Pcc =1

βb

∫ ∞0

e− (2R−1)(|hb|

2PD+σ2b )

(1−βb)PD− |hb|

2

βb d|hb|2

=1

βbe− (2R−1)σ2

b(1−βb)PD

∫ ∞0

e−|hb|2

((1−βb)PD+βbPD(2R−1))βb(1−βb)PD d|hb|2

=(1− βb)PD

(1− βb)PD + βbPD(2R − 1)e− σ2

b (2R−1)

(1−βb)PD , (5.24)

which concludes the proof.

5.4.2 Optimization of Transmit Power and Number Of Trans-

mit Symbols

As discussed in Remark 1, we consider ζ∗w,CSI provided in (5.13) as the minimum

detection error probability at Willie under channel uncertainty, simply denoting it

by ζ∗w. Since Alice is unaware of her channel realization to Willie, she considers

the expected value of ζ∗w over all possible realizations of her channel to Willie as

the detection metric. Here, Alice looks to maximize her covert throughput to Bob

while ensuring that Willie’s average detection error probability satisfies a given

covertness constraint. Owing to delay requirements, we assume in this work that

the transmitted signals are constrained by a maximum blocklength, ND,max, thus


the number of Alice’s covert data symbols is limited by ND ≤ ND,max. On the

other hand, there exists a limit on the minimum number of symbols Alice can use

due to the channel coding requirements for short-packet communications [128, 129],

and this limit is denoted by ND,min. In regards to the transmit power, a maximum

transmit power constraint at Alice is considered, given by Pmax. As mentioned

previously, Alice uses the maximum allowed transmit power, Pmax, for the pilot

symbol.

The design problem at Alice is to optimally choose the data transmission power

and the number of data symbols for covert communication, stated as

P5.1 maximizePD,ND

NDRPcc

subject to E|hw|2 [ζ∗w] ≥ 1− ε (5.25a)

PD ≤ Pmax (5.25b)

ND,min ≤ ND ≤ ND,max, (5.25c)

where NDRPcc is the throughput from Alice to Bob, and the design parameters PD

and ND refer to Alice’s data transmission power and the number of symbols used

for data transmission, respectively. Here, ε signifies the desired level of covertness,

whereas ζ∗w is as given in (5.13), and in the statement of P5.1, PT = Pmax is

assumed. The solution to this problem is stated in the following.

Lemma 5.5 Alice’s optimal transmit power for data transmission, as a function

of ND, is given by

P ∗D =

P†D(ND), If P †D(ND) ≤ Pmax

Pmax, Otherwise,(5.26)

where P †D(ND) is the solution to E|hw|2 [ζ∗w] = 1 − ε for a given ND. The optimal

number of data symbols transmitted by Alice is given by

N∗D =

ND,min, If N †D ≤ ND,min

N †D, If ND,min < N †D ≤ ND,max

ND,max, Otherwise,

(5.27)


where N †D is the solution for ND to

maximizeND

NDRPcc, (5.28)

and Pcc is a function of ND in terms of PD.

Proof

We first note that for a fixed PT = Pmax, the covert connection probability, Pcc, is an

increasing function of PD. On the other hand, E|hw|2 [ζ∗w] is a decreasing function

of PD, hence a given solution will satisfy the constraint at equality. From the

constraint at equality and a given ND, the solution for PD, as indicated by P †D(ND),

can be obtained. This results in the one-dimensional optimization problem in

(5.28), which can be solved by performing a numerical search over all possible

values of ND. Incorporating the maximum and minimum value of PD and ND

gives the desired result.

We note here that the optimal solution presented in Lemma 5.5 does not yield

a closed form expression for P ∗D and N∗D. Rather, the solution relies on numerical

search methods [130] to solve the optimization problem in (5.28). We next present

a suboptimal closed-form solution to this problem.

5.4.3 Suboptimal Solution

Based on the linear approximation in the asymptotically low power regime (small

ε regime) developed earlier, we present here a suboptimal solution to find closed

form expressions for P ∗D and N∗D. Using the linear approximation for ζ∗w, we rewrite

the problem at Alice as

P5.1a maximizePD,ND

NDRPcc

subject to E|hw|2 [ζ∗w] ≥ 1− ε (5.29a)

PD ≤ Pmax (5.29b)

ND,min ≤ ND ≤ ND,max, (5.29c)


where now,

ζ∗w ≈ 1− |hw|2NND

D e−ND

σ2wΓ(ND)

PD. (5.30)

The solution to this problem is presented in the following.

Lemma 5.6 In the asymptotically small ε regime, Alice’s optimal transmit power

for data transmission is given by

P ∗D =

P‡D, If P ‡D ≤ Pmax

Pmax, Otherwise,(5.31)

where

P ‡D =εσ2

wΓ(N∗D)

(N∗D)N∗De−N

∗D, (5.32)

and the optimal number of data symbols transmitted by Alice is N∗D = ND,min.

Proof

Under the exponential distribution of |hw|2, the expectation is calculated as

E|hw|2 [ζ∗w] = 1− NNDD e−ND

σ2wΓ(N)

PD, (5.33)

and the covertness constraint then gives

PD ≤εσ2

wΓ(ND)

(ND)NDe−ND. (5.34)

We note that Pcc is an increasing function of PD while the covertness constraint

puts an upper bound on PD, hence a given solution will satisfy the constraint at

equality. This results in the optimization problem given as

maximizeND

NDRPcc, (5.35)

where Pcc is now a function of ND. Considering the partial derivative w.r.t. ND,


we have

∂(NDRPcc)

∂ND

=− (1− βb)R(1− βb) + βb(2R − 1)

e−e−ND

(NDe

NDΓ(ND)+ANNDD

)Γ(ND)

×[ANND+1

D (ln(ND)− ψ(ND))− eNDΓ(ND)

Γ(ND)

], (5.36)

which is strictly negative for ND ≥ 1. Here A =σ2b (2R−1)

σ2w(1−βb)ε and ψ(x) is the Digamma

function defined as ψ(x) = Γ′(x)Γ(x)

. Thus the value of ND maximizing the throughput

is the minimum allowed ND, i.e., ND,min. This concludes the proof.


In this section, we present the numerical results and study the performance of

the considered covert communication scenario under given covertness constraints.

Unless stated otherwise, we consider a pre-determined rate for Alice to Bob trans-

mission of R = 1, the variance of Willie’s receiver noise is set to σ2w = 0.05, while

the variance of Bob’s receiver noise is set to σ2b = 0.01. We consider a maximum

power constraint of Pmax = 1 at Alice, while Nmin and Nmax are set to be 50 and

100, respectively.

We first provide a numerical validation for the equivalence of Willie’s detection

error probability under the cases of perfect CSI and CDI only in the large detection

error regime, as derived in Proposition 5.1, and also explained in Remark 5.1. In

Fig. 5.2, we plot these detection error probabilities at Willie against a range of

Alice’s data transmit power, PD, for different numbers of data transmit symbols,

ND. We first note that as ND or PD increases, Willie’s detection performance

improves. More importantly, Willie’s detection performances are indistinguishable

between the perfect CSI case and the CDI only case in the large detection error

regime, e.g., ζ∗w ≥ 0.9. The detection performances of the two cases are still very

close to each other even at ζ∗w = 0.8. These results validate our analysis and the

conclusion that Willie’s detection performance is extremely insensitive to the CSI’s

accuracy as long as the detection error probability is forced by Alice and Bob to

be fairly close to 1.

We next present the optimal choice of Alice’s data transmit power and the


0 1 2 3 4 5 6 7 8 9 10

·10−3

0.5

0.6

0.7

0.8

0.9

1.0

Alice’s Data Transmit Power

Willie’sMinim

um

Detection

Error

Probab

ility

ND = 50, Perfect CSIND = 50, CDI OnlyND = 100, Perfect CSIND = 100, CDI OnlyND = 200, Perfect CSIND = 200, CDI Only

Figure 5.2: Willie’s minimum detection error probability, ζ∗w, vs. Alice’s datatransmit power, PD, under perfect CSI and CDI only cases for varying ND.

optimal number of data transmit symbols under given covertness constraints in

Fig. 5.3 and Fig. 5.4, respectively, where we also plot the best choice for Al-

ice’s parameters under the proposed suboptimal scheme. We show these results

for two different sets of noise variances at Willie for ease of exposition. Firstly,

for the optimal data power values, we see that since a higher noise power causes

an increased uncertainty in Willie’s observations, Alice can transmit to Bob us-

ing a higher transmit power. Secondly, the proposed suboptimal scheme performs

very close to the optimal one, especially in the low transmit power regime. We

also note here that since the proposed suboptimal scheme is based on the linear

approximation of Willie’s detection performance around PD → 0, the curves for

optimal transmit power deviate further from each other as the covertness require-

ment is relaxed. Regarding the optimal number of data transmit symbols at Alice,

both the optimal and suboptimal scheme provide the same solution, i.e., to use the

minimum possible number of transmit symbols, ND,min.

It is important to highlight that the optimal (and suboptimal) solution of only

using the minimum number of transmit symbols is in sharp contrast to the previ-

ously established result for non-fading AWGN channels in [54]. Specifically, it was


0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

1

2

3

4

5

6

7

8·10−3

Covertness Requirement

Optimal

DataTransm

itPow

er

Optimal Solution, σ2w = 0.1

Suboptimal Solution, σ2w = 0.1



Figure 5.3: Comparison of the optimal data transmit power at Alice, P ∗D, underthe optimal and suboptimal solution vs. the covertness requirement, ε.

shown in [54] that it is optimal to use the maximum allowable number of trans-

mit symbols to maximize the covert throughput. This comparison demonstrates

a fundamental difference in the covert transmission design between the non-fading

AWGN channel and the quasi-static fading channel. To further illustrate the im-

portance of appropriate design, we investigate the advantage of using the optimal

and suboptimal solutions over a scheme where Alice uses the maximum allowable

number of symbols in a communication slot (with optimized data transmit power).

Fig. 5.5 shows the covert throughput achieved under the optimal and subopti-

mal solutions with N∗D = ND,min, and the covert throughput achieved by using

ND = ND,max. We note a significant difference in the achieved throughput between

the use of ND,min and ND,max. Specifically, we see that the optimal (and subopti-

mal) solution achieves 110-fold more throughput than that achieved by using the

maximum number of data symbols. We also observe that the throughput of the

suboptimal solution is roughly 20% lower than that of the optimal solution, due to

the small but non-negligible difference in the transmit power designs. Hence, such

a moderate performance reduction is the price to pay for using the closed-form


0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.1848

50

52


Optimal

No.

ofDataSymbols





Figure 5.4: The optimal number of data symbols used by Alice, N∗D, under theoptimal and suboptimal solution vs. the covertness requirement, ε. Note that allfour curves in this figure overlap completely.

suboptimal design with minimum complexity.

5.6 Conclusion

In this chapter, we have considered covert communications under the scenario

where users suffer from channel uncertainty while Alice uses pilot symbols to help

the intended receiver estimate their channel. We have derived the optimal detec-

tion threshold for Willie and the resulting minimum detection error probability

under the extreme cases of the availability of complete CSI and CDI only at Willie.

It has been shown that in the low transmit power regime, the two extreme cases

are indistinguishable and hence, the quality of channel knowledge at Willie does

not improve his detection performance as long as it is forced to stay in the large

detection error regime. From the covert communications pair perspective, we pro-

vide the optimal choice for data transmit power and the optimal number of data

transmit symbols that maximize the covert throughput. Our results show that

in contrast to AWGN channels, under quasi-static fading scenarios, Alice should

5.6 Conclusion 91

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18

0.5

1.0

1.5

2.0

2.5

3.0


Optimal

Through

put(bits)

Optimal SolutionSuboptimal SolutionUsing ND,max

Figure 5.5: The optimal throughput from Alice to Bob, N∗DRPcc, under the opti-mal approach, suboptimal approach and case of using ND,max vs. the covertnessrequirement, ε.

utilize the minimum allowed number of transmit symbols to maximize the covert

throughput to Bob.

Chapter 6

Conclusions and Future Research

Directions

In this chapter, we summarize the general conclusions drawn from this thesis. We

also outline some possible future research directions related to this work.

6.1 Conclusions

This thesis has investigated positive rate covert communications in wireless sce-

narios by introducing and exploiting uncertainties at the adversary in the form of

AN and channel knowledge uncertainty, and provided detailed analysis and design

guidelines in these scenarios. We have introduced the use of AN for achieving

covertness, causing confusion at Willie in determining the transmission state of Al-

ice. The scenarios of AN use are considered under the worst case assumption that

Willie is fully aware of the channel information from the covert transmitter, which

then motivates us to consider the case where Willie is uncertain of his channel

information and only possesses part of the channel knowledge. Both of these sce-

narios also consider the impact of these imperfections on the covert communication

pair as well, helping quantify the covert performance from a realistic perspective.

In the first half of the thesis, we focus on the use of AN with a varying transmit

power to induce uncertainty in the received power at Willie, causing uncertainties

in his received signal statistics. Chapter 2 has considered the potential of achieving

covert communication using a full-duplex receiver that generates artificial noise to

cause detection errors at a watchful adversary Willie. Considering a radiometer

93

94 Conclusions and Future Research Directions

as the detector of choice at Willie, we have analyzed the conditions under which

Willie makes detection errors, and characterized Willie’s optimal detection per-

formance conditioned over the fading channel realizations. From the perspective

of covert communication pair, we have provided design guidelines for the optimal

choice of transmit power of full-duplex receiver’s artificial noise. Owing to the

self-interference of the full-duplex receiver, these power levels need to be controlled

carefully, otherwise they affect the transfer of any covert information. We have also

shown that contrary to a commonly adopted assumption, the a priori transmission

probabilities of 0.5 are not always the optimal choice to achieve the best possible

covertness. In Chapter 3, we have shown how a backscatter communication system

can achieve covertness in the presence of a warden Willie. The proposed scheme

requires the reader to use a noise-like signal with variable transmit power drawn

from a uniform distribution. By controlling the maximum and minimum transmit

powers of the reader, the system is able to achieve a target level of covertness.

Comparing with a conventional backscatter system with no covertness, the BER

degradation from no covertness to some (poor) covertness is huge. Nevertheless,

the additional BER degradation for improving covert performance is much smaller.

The second half of the thesis consider scenarios where users suffer from un-

certainty in their channel knowledge and we consider achieving covertness under

this lack of channel knowledge under both finite and infinite blocklength cases.

In Chapter 4, we examined how to achieve covert communication in a public and

legitimate communication link when users have uncertainty about their channels.

We first derived a closed-form expression for the optimal threshold of of Willies

optimal detector. Next, we quantified the achievable outage rate region for Carol

and Bob. Our results showed that the presence of channel uncertainty at Willie al-

lows Alice to achieve a certain amount of covertness, while this channel uncertainty

also affects the achievable rates for Carol and Bob. In Chapter 5, we have con-

sidered covert communications under the scenario where users suffer from channel

uncertainty while Alice uses pilot symbols to help the intended receiver estimate

their channel. We have derived the optimal detection threshold for Willie and the

resulting minimum detection error probability under the extreme cases of the avail-

ability of complete CSI and CDI only at Willie. It has been shown that in the low

transmit power regime, the two extreme cases are indistinguishable and hence, the

6.2 Future Research Directions 95

quality of channel knowledge at Willie does not improve his detection performance

as long as it is forced to stay in the large detection error regime. From the covert

communications pair perspective, we provide the optimal choice for data trans-

mit power and the optimal number of data transmit symbols that maximize the

covert throughput. Our results show that in contrast to AWGN channels, under

quasi-static fading scenarios, Alice should utilize the minimum allowed number of

transmit symbols to maximize the covert throughput to Bob. We note that there

exists a strong contrast in the two scenarios of finite and infinite observations at the

adversary under channel uncertainty considerations. Under infinite observations,

channel uncertainty alone cannot achieve covertness as there is no uncertainty in

the noise variance at Willie, and any transmission by Alice, is successfully detected

by Willie despite being uncertain of the channel coefficient from Alice. Resultantly,

a public transmission is used to hide the covert transmission. On the other hand,

under finite observations at Willie, no additional source of uncertainty is required

as the uncertainty in channel and receiver decision statistics are sufficient to achieve

covertness.

Overall, this thesis has addressed the generation and exploitation of uncertain-

ties at the adversary for achieving covert communication subject to certain covert-

ness and throughput constraints. We have focused on two important aspects in this

regard, namely the use of AN and channel uncertainty to achieve the subject pur-

pose. Detailed performance analysis has been given in all the considered scenarios,

providing design guidelines for achieving covertness in practical situations.

6.2 Future Research Directions

The field of covert wireless communications is a vastly rich research area with

tremendous potential, and applications not only of interest to military and law

enforcement agencies but general public as well. The following major research

directions could form the focus of future work:

• Covert schemes generally require Alice to transmit at a low power, result-

ing in Bob experiencing a low SNR for signal decoding. In majority of the

recent literature on covert communications including this work, the effect of

choice of modulation on the achieved covertness and Alice to Bob throughput

96 Conclusions and Future Research Directions

has not been considered. Focusing on the low SNR regime, the throughput

performance of different modulation schemes is very important in covert back-

ground. This choice of a lower or higher order modulation scheme presents

an interesting tradeoff since a lower order modulation can provide a higher

reliability while higher order modulation can offer an increased throughput.

On the other hand, bringing in the covert requirements, higher order mod-

ulation generally leads to a higher detection error probability at Willie due

to Gaussian mixture distribution resulting from increased number of signal

components. A detailed analysis depending upon the covert requirements is

needed to find the appropriate scheme.

• The scenarios considered in this work have focused on a single-hop com-

munication between Alice and Bob. However, due to low transit powers of

Alice, the communication range in covert scenarios is essentially reduced,

and in many applications where the end-to-end distance is large, multi-hop

communications become essential, effectively increasing the communication

distance. The tradeoff between the communication distance of hops and num-

ber of hops in achieving a long-distance covert communication has not been

analyzed before, although an initial study on multi-hop covert communica-

tions in presented in [50]. It is not yet clear as to whether more small distance

hops or a few large distance hops are better to achieve a higher covertness.

• Although radiometers (power detectors) have been shown to be the optimal

detectors of choice in majority of the literature on covert communications, it

is known that their detection performance starts to deteriorate as the receive

SNR level decreases [131]. As evidenced by the literature on detection of

secondary users in cognitive radio, other types of detectors such as based

on cyclostationarity [132–134] and matched filtering can be used to improve

the detection performance. While an initial analysis of detection of covert

transmissions using cyclostationary detectors is presented in [135], further

investigation in this regard is warranted which will result in more robust

covert schemes.

• Covert communication scenarios are generally analysed under a discrete-time

model, assuming that the transmissions from Alice to Willie and Alice to Bob

6.2 Future Research Directions 97

are synchronized and that Willie samples at the symbol rate at the perfect

time, with the implicit implication that the results will be similar on the true

continuous-time model of the physical channel. In reality, under continuous

channel model considerations, timing offsets are highly probable, resulting in

deterioration of the covert scheme under consideration. An initial work in this

regard has been considered in [136], where it has been shown that in presence

of an uninformed jammer, timing offsets between Alice’s signal and that of

the jammer allow for the application of co-channel interference mitigation

techniques at Willie’s detector, which greatly question the results on the

improvement in covert throughput with the help of the jammer. Further

work in this case is required to analyse and compensate for these errors.

• A general assumption in the literature on covert communications is the inde-

pendence of channels from Alice to Bob and Alice to Willie. If this assumption

does not hold, since in general Willie is a silent observer and hence can be

present anywhere in the environment, these channels can be considered corre-

lated especially if Willie is in close vicinity to Bob. Under such circumstances,

Alice faces an interesting tradeoff as whether to transmit when the channel is

good or bad, as an increased throughput also results in a high chance of get-

ting detected by Willie. While parallels to this important problem of channel

correlation exist in the literature on PLS [137–139], this avenue is still to be

explored in covert communications background.

Appendix A

This appendix contains the proofs needed in Chapter 2.

A.1 Proof of Proposition 2.1

Using the definition of incorrect decisions at Willie, we have

PFA = P [D1|H0] = P [Pw ≥ γ |H0] = P[|hbw|2Pb + σ2

w ≥ γ]

= P[Pb ≥

γ − σ2w

|hbw|2]

=

1, if γ−σ2

w

|hbw|2 ≤ Pmin

|hbw|2Pmax+σ2w−γ

|hbw|2(Pmax−Pmin), if Pmin <

γ−σ2w

|hbw|2 ≤ Pmax

0, else,

(A.1)

and

PMD = P [D0|H1] = P [Pw < γ |H1] = P[|hbw|2Pb + |haw|2Pa + σ2

w < γ]

= P[Pb <

γ − |haw|2Pa − σ2w

|hbw|2]

=

0, if ν ≤ Pmin

γ−|haw|2Pa−|hbw|2Pmin−σ2w

|hbw|2(Pmax−Pmin), if Pmin < ν ≤ Pmax

1, else.

(A.2)

where ν , γ−σ2w−|haw|2Pa|hbw|2 , haw and hbw denote the channels from Alice and Bob

to Willie, respectively, P[·] denotes the probability measure and we have used the

conditioning over the uniform distribution of Bob’s transmit power, i.e., Pb ∼

99

100

|hbw|2Pmin + σ2w

|hbw|2Pmax + σ2w

|hbw|2Pmin + |haw|2Pa + σ2w

|hbw|2Pmax + |haw|2Pa + σ2w

γInterval for PFA No Error Interval for PMD

Figure A.1: Case-I : |hbw|2Pmax + σ2w < |hbw|2Pmin + |haw|2Pa + σ2

w

U(Pmin, Pmax). Since Willie has to choose the threshold of his detector, γ, such

that the probability of error at Willie, PE = π0PFA + π1PMD, is minimized, thus

Willie considers the following:

minimizeγ

π0PFA + π1PMD, (A.3)

where the expressions for incorrect decisions for individual slots at Willie are as

defined earlier in (A.1) and (A.2). Willie chooses his detector’s threshold, in the

intervals marked by the quantities given by |hbw|2Pmin +σ2w, |hbw|2Pmin + |haw|2Pa+

σ2w, |hbw|2Pmax + σ2

w and |hbw|2Pmax + |haw|2Pa + σ2w. We also note that

• |hbw|2Pmin + σ2w ≤ |hbw|2Pmax + |haw|2Pa + σ2

w, but the relationship between

|hbw|2Pmin + |haw|2Pa + σ2w and |hbw|2Pmax + σ2

w is unclear.

• For a choice of γ < |hbw|2Pmin + σ2w, PFA = 1, PMD = 0 and hence PE = π0.

• For a choice of γ > |hbw|2Pmax + |haw|2Pa + σ2w, PFA = 0, PMD = 1 and hence

PE = π1.

In the following, we analyse the error probability at Willie under the two differ-

ent cases of |hbw|2Pmin +|haw|2Pa+σ2w ≤ |hbw|2Pmax +σ2

w and |hbw|2Pmin +|haw|2Pa+

σ2w > |hbw|2Pmax + σ2

w.

Case - I : |hbw|2Pmax + σ2w < |hbw|2Pmin + |haw|2Pa + σ2

w

This case is graphically shown in Fig. A.1 and we have three intervals for the choice

of γ.

A.1 Proof of Proposition 2.1 101

(1) |hbw|2Pmin + σ2w < γ ≤ |hbw|2Pmax + σ2

w

In this case, PMD = 0, and

PE = π0PFA = π0

[ |hbw|2Pmax + σ2w − γ

|hbw|2 (Pmax − Pmin)

], (A.4)

which has a decreasing partial derivative with respect to (w.r.t.) γ, given by

−π0

|hbw|2(Pmax−Pmin), thus γ = |hbw|2Pmax + σ2

w should be chosen.

(2) |hbw|2Pmin + |haw|2Pa + σ2w < γ ≤ |hbw|2Pmax + |haw|2Pa + σ2

w

In this case, PFA = 0, and

PE = π1PMD = π1

[γ − |haw|2Pa − |hbw|2Pmin − σ2

w


], (A.5)

which has an increasing partial derivative w.r.t. γ, given by π1

|hbw|2(Pmax−Pmin), thus

γ = |hbw|2Pmin + |haw|2Pa + σ2w should be chosen.

(3) |hbw|2Pmax + σ2w < γ ≤ |hbw|2Pmin + |haw|2Pa + σ2

w

In this case, PFA = 0 and PMD = 0, which means that a choice of γ in this interval

will have no detection errors at Willie.

Case - II : |hbw|2Pmax + σ2w ≥ |hbw|2Pmin + |haw|2Pa + σ2

w

This case is graphically shown in Fig. A.2 and we have three intervals for the choice

of γ.

(1) |hbw|2Pmin + σ2w < γ ≤ |hbw|2Pmin + |haw|2Pa + σ2

w

In this case, PMD = 0 and

PE = π0PFA = π0



], (A.6)

which has a decreasing partial derivative w.r.t. γ, given by −π0


γ = |hbw|2Pmin + |haw|2Pa + σ2w should be chosen.

102

|hbw|2Pmin + σ2w

|hbw|2Pmin + |haw|2Pa + σ2w

|hbw|2Pmax + σ2w

|hbw|2Pmax + |haw|2Pa + σ2w

γInterval for PFA

Interval for PMD

Figure A.2: Case-II : |hbw|2Pmax + σ2w ≥ |hbw|2Pmin + |haw|2Pa + σ2

w

(2) |hbw|2Pmax + σ2w < γ ≤ |hbw|2Pmax + |haw|2Pa + σ2

w

In this case, PFA = 0, and

PE = π1PMD = π1


w


], (A.7)

which has an increasing partial derivative w.r.t. γ, given by π1


γ = |hbw|2Pmax + σ2w should be chosen.

(3) |hbw|2Pmin + |haw|2Pa + σ2w < γ ≤ |hbw|2Pmax + σ2

w

In this case, we have

PE = π0PFA + π1PMD

= π0



]+ π1


w


],

(A.8)

and

∂PE∂γ

=π1 − π0

|hbw|2 (Pmax − Pmin)=

≥ 0, if π1 ≥ π0

< 0, otherwise.(A.9)

Based on the knowledge of π0 and π1, Willie can choose the optimal value of γ. The

corresponding PE for the choice of optimal threshold, γ∗, can be found by using

the appropriate expressions of PE from Case-II, hence concluding the proof.

A.2 Proof of Lemma 2.2 103

A.2 Proof of Lemma 2.2

For the case of π1 ≥ π0, and under the condition of Willie making detection errors,

given by |hbw|2Pmax + σ2w ≥ |hbw|2Pmin + |haw|2Pa + σ2

w, we have

P∗E = π0

{∫ ∞0

∫ |hbw|2(Pmax−Pmin)

Pa

0

P∗E f|haw|2(x)f|hbw|2(y)dxdy

}, (A.10)

which, using the law of total expectation, can also be written as

P∗E = π0

{P[|haw|2 ≤


Pa

]× E

[P∗E∣∣∣|haw|2 ≤ |hbw|2 (Pmax − Pmin)

Pa

]},

(A.11)

where

P[|haw|2 ≤


Pa

]

=

∫ ∞0


Pa

0

f|haw|2(x)f|hbw|2(y)dxdy

=

∫ ∞0

[1− exp

(−λaw (Pmax − Pmin) y

Pa

)]λbw exp (−λbwy) dy

=λaw (Pmax − Pmin)

λbwPa + λaw (Pmax − Pmin),

(A.12)

and

E[P∗E∣∣∣[|haw|2 ≤ |hbw|2 (Pmax − Pmin)

Pa

]= 1− Pa

(Pmax − Pmin)E[ |haw|2|hbw|2

∣∣∣ |haw|2 ≤ |hbw|2 (Pmax − Pmin)

Pa

]

= 1− Pa(Pmax − Pmin)

∫ ∞0


Pa

0

x

yf|haw|2(x)f|hbw|2(y)dxdy

= 1− λbwPaλaw (Pmax − Pmin)

[ln

(1 +

λaw (Pmax − Pmin)

λbwPa

)− λaw (Pmax − Pmin)

λbwPa + λaw (Pmax − Pmin)

],

(A.13)

and putting in these expressions into (A.11) gives the desired result.

The case for π0 > π1 follows along the same lines, hence concluding the proof.

104


We first consider the maximization of P∗E, where the optimal choice of Pmin should

maximize κ(t) = 1 + t ln t − t2 under both cases of π1 ≥ π0 and π1 < π0, as per

(2.15). To first determine the monotonicity of P∗E w.r.t. Pmin, we consider the

derivatives of κ(t) w.r.t. t, given by ∂κ(t)∂t

= 1 + ln t− 2t and ∂2κ(t)∂t2

= 1t− 2. Since

Pmax ≥ Pmin, we have

∂t

∂Pmin

=λawλbwPa

[λbwPa + λaw(Pmax − Pmin)]2≥ 0. (A.14)

For t ∈ [0, 1), the first derivative of κ(t) w.r.t. t increases for 0 ≤ t < 1/2

and decreases for 1/2 ≤ t < 1, with the maximum value of − ln 2, occurring at

t = 1/2. Using this and the fact that ∂t∂Pmin

≥ 0, it can be concluded that κ(t), and

resultantly, P∗E is a decreasing function of Pmin. Hence, the optimal choice in this

regard is the minimum possible value of Pmin, which is zero.

We next consider the covert rate constraint, where the outage probability δab is

represented as

δab = 1− λbb exp(−λabµσ2b )v(x), (A.15)

and here

v(x) =1

y − x ln

(λbb + y

λbb + x

), (A.16)

x , λabφµPmin ≥ 0, y , λabφµPmax ≥ 0, and y ≥ x. Considering the first derivative

of v(x) w.r.t. x, we have

∂v(x)

∂x=

1

(y − x)2

[ln

(λbb + y

λbb + x

)− y − xλbb + x

]. (A.17)

Here, ∂v(x)∂x

depends on

l(x) , ln

(λbb + y

λbb + x

)−(y − xλbb + x

)= ln

(1 +

y − xλbb + x

)−(y − xλbb + x

)≤ 0, (A.18)

where the second line in (A.18) is due to the logarithmic inequality, ln(1 + a) ≤

A.4 Proof of Proposition 2.4 105

a, ∀a ≥ −1. Thus v(x) is always a decreasing function of x, and resultantly, δab

is always an increasing function of Pmin. From the covert rate constraint, we can

write

δab ≤ 1− τ

π1Rab

, (A.19)

and hence, to satisfy this constraint, Pmin is upper bounded by a value which can

be found by solving (A.19) at equality. This concludes the proof.


We first show the monotonicity of δab w.r.t Pmax. Here, δab can be written as

δab = 1− λbb exp(−λabµσ2b )u(x), (A.20)

where u(x) , 1x

ln(λbb+xλbb

)and x , λabφµPmax ≥ 0. We note that

∂u(x)

∂x=

1

x2

(x

λbb + x− ln

(λbb + x

λbb

)), (A.21)

which depends on m(x) , xλbb+x

− ln(λbb+xλbb

). Here, m(0) = 0 and ∂m(x)

∂x=

− x(λbb+x)2 ≤ 0, thus m(x) decreases monotonically with x, giving m(x) ≤ m(0)

for x ≥ 0, and resultantly, ∂u(x)∂x≤ 0. As a result, δab is a monotonically increasing

function of Pmax.

Next, we consider the optimal choice of Pmax under the two cases of P∗E, keeping

in view the change in δab w.r.t. Pmax.

Case-I: τRab(1−δab) ≤

12

We have

δab ≤ 1− 2τ

Rab

. (A.22)

Due to a monotonic increase in δab w.r.t. Pmax, the optimal value of Pmax has to

satisfy Pmax ≤ P †max, where P †max is the solution of (A.22) at equality. Combining

with the average power constraint, we have Pmax ≤ min(2Pavg, P

†max

). Now we

106

consider the monotonicity of P∗E(π∗1) = 12(1+s ln s−s2) w.r.t Pmax. Here,

∂P∗E(π∗1)

∂s=

12

(1 + ln s− 2s) and∂P∗E(π∗1)2

∂2s= 1

2s− 1, where s = λbwPa

λbwPa+λawPmax. Also,

∂s

∂Pmax

= − λawλbwPa

(λbwPa + λawPmax)2 ≤ 0. (A.23)

We note here that s ∈ [0, 1),∂P∗E(π∗1)

∂sincreases for 0 ≤ s < 1/2 and decreases

for 1/2 ≤ s < 1, with a maximum value of −12

ln 2. Since ∂s∂Pmax

≤ 0, P∗E(π∗1) is

an increasing function of Pmax, and hence the best possible choice in this case is

Pmax = min(2Pavg, P

†max

).

Case-II: τRab(1−δab) >

12

We have

δab > 1− 2τ

Rab

, (A.24)

and in this case,

P∗E(π∗1) =

(1− τ

Rab(1− δab)

)(1 + s ln s− s2

). (A.25)

Since δab increases monotonically in Pmax, hence to satisfy (A.24), Pmax > P †max,

and resultantly, the optimal choice lies between P †max and 2Pavg, where P †max is as

defined earlier. Let P∗E(π∗1) = p(x)q(x), where p(Pmax) =(

1− τRab(1−δab(Pmax))

)and

q(Pmax) = (1 + s ln s− s2). We note here that P∗E(π∗1) is not a monotonic function

of Pmax, since as Pmax increases, p(Pmax) decreases while q(Pmax) increases. Thus

there may exist an optimal value of Pmax that maximizes P∗E(π∗1), which motivates

the optimization

P ‡max = maximizePmax

P∗E(π∗1). (A.26)

We note that the optimization problem in (A.26) is of one dimension and can be

solved by methods of efficient numerical search.

Combining the two cases, the optimal value for Pmax is found, thus completing

the proof.

Appendix B

This appendix contains the proofs needed in Chapter 5.

B.1 Proof of Lemma 5.2

We note that for PD = 0, the expression of ζ∗w,CSI gives a value of 1. This is expected

since in case of no transmission by Alice, Willie is unable to distinguish between

the two hypotheses. This value also serves as the intercept of the linear (first order)

approximation of ζ∗w,CSI as a function of PD. To complete the approximation, we

need to find the slope of ζ∗w,CSI as PD → 0, i.e, limPD→0

∂ζ∗w,CSI∂P

. Using the relationship

of upper and lower incomplete Gamma functions given by Γ(a) = Γ(a, b) + γ(a, b),

we have

ζ∗w,CSI = 1−Γ(ND,

NDσ2w


+ 1))

Γ(ND)

+Γ(ND, ND

(1 + σ2

w

|hw|2PD

)ln( |hw|

2PDσ2w

+ 1))

Γ(ND). (B.1)

To calculate the desired derivative, we consider the terms in ζ∗w,CSI separately,

where we rely on the derivative property of upper incomplete Gamma function,

given by

∂Γ (s, f(x))

∂x= − (f(x))s−1 e−f(x)∂f(x)

∂x. (B.2)

107

108

The derivative for the second term of ζ∗w,CSI in (B.1) is calculated as

1

Γ(ND)

∂Γ(ND,

NDσ2w


+ 1))

∂PD

=− 1

Γ(ND)

[NDσ

2w

|hw|2PDln

( |hw|2PDσ2w

+ 1

)]ND−1

× e−NDσ

2w

|hw|2PDln

(|hw|2PD

σ2w

+1

)∂

∂PD

[NDσ

2w

|hw|2PDln

( |hw|2PDσ2w

+ 1

)]=− NND

D

Γ(ND)

[σ2w

|hw|2PDln

( |hw|2PDσ2w

+ 1

)]ND−1

× e−NDσ

2w

|hw|2PDln

(|hw|2PD

σ2w

+1

) σ2w

PD(|hw|2PD + σ2w)−σ2w ln

(|hw|2PDσ2w

+ 1)

|hw|2P 2D

.(B.3)

Similarly, the derivative for the third term of ζ∗w,CSI in (B.1) is calculated as

1

Γ(ND)

∂Γ(ND, ND

(1 + σ2

w

|hw|2PD

)ln( |hw|

2PDσ2w

+ 1))

∂PD

=− 1

Γ(ND)

[ND

(1 +

σ2w

|hw|2PD

)ln

( |hw|2PDσ2w

+ 1

)]ND−1

× e−ND(

1+σ2w

|hw|2PD

)ln

(|hw|2PD

σ2w

+1

)

× ∂

∂PD

[ND

(1 +

σ2w

|hw|2PD

)ln

( |hw|2PDσ2w

+ 1

)]=− NND

D

Γ(ND)

[(1 +

σ2w

|hw|2PD

)ln

( |hw|2PDσ2w

+ 1

)]ND−1

× e−ND(

1+σ2w

|hw|2PD

)ln

(|hw|2PD

σ2w

+1

) |hw|2PD − σ2w ln

(|hw|2PDσ2w

+ 1)

|hw|2P 2D

. (B.4)

The next step is to apply the limit as PD → 0. Thus

limPD→0

∂ζ∗w,CSI∂PD

= limP→0

1

Γ(ND)

[∂Γ(ND,

NDσ2w


+ 1))

∂PD

−∂Γ(ND, ND

(1 + σ2

w

|hw|2PD

)ln( |hw|

2PDσ2w

+ 1))

∂PD

],

(B.5)

B.1 Proof of Lemma 5.2 109

where, using the law of products for limits, we calculate the limit at each factor of

the above derivatives separately as follows.

For the first factor in (B.3),

limPD→0

(σ2w

|hw|2PDln

( |hw|2PDσ2w

+ 1

))ND−1

=

(limPD→0

σ2w

|hw|2PDln

( |hw|2PDσ2w

+ 1

))ND−1

= 1ND−1 = 1 (B.6)

where we have used L’Hopital rule to find the internal limit. For the second factor

in (B.3),

limPD→0

e− NDσ

2w

|hw|2PDln

(|hw|2PD

σ2w

+1

)= lim

PD→0

( |hw|2PDσ2w

+ 1

)− NDσ2w

|hw|2PD

=

limPD→0

( |hw|2PDσ2w

+ 1

)− σ2w

|hw|2PD

ND

=[e−1]ND = e−ND (B.7)

where we have used the Euler’s identity, given by [94]

ex = limn→∞

(1 +

x

n

)n, (B.8)

to calculate the internal limit. For the third factor in (B.3), repeated application

of L’Hopital rule yields

limPD→0

σ2w

PD(|hw|2PD + σ2w)−σ2w ln

(|hw|2PDσ2w

+ 1)

|hw|2P 2D

= −|hw|2

2σ2w

. (B.9)

Hence, overall for the first term on RHS of (B.5), we have

limPD→0

1

Γ(ND)

∂Γ(ND,

NDσ2w


+ 1))

∂PD= −N

NDD e−ND |hw|22σ2

wΓ(ND). (B.10)

110

Similarly, for the first factor in (B.4),

limPD→0

((1 +

σ2w

|hw|2PD

)ln

( |hw|2PDσ2w

+ 1

))ND−1

=

(limPD→0

(1 +

σ2w

|hw|2PD

)ln

( |hw|2PDσ2w

+ 1

))ND−1

= 1ND−1 = 1 (B.11)

where we have again used L’Hopital rule to find the internal limit. For the second

factor in (B.4),

limPD→0

e−(

1+NDσ

2w

|hw|2PD

)ln

(|hw|2PD

σ2w

+1

)= lim

PD→0

( |hw|2PDσ2w

+ 1

)−ND(1+

σ2w

|hw|2PD

)

=

limPD→0

( |hw|2PDσ2w

+ 1

)−(1+σ2w

|hw|2PD

)ND

=[e−1]ND = e−ND (B.12)

where we have again used the Euler’s identity to calculate the internal limit. For

the third factor in (B.4),

limPD→0

|hw|2PD − σ2w ln

(|hw|2PDσ2w

+ 1)

|hw|2P 2D

=|hw|22σ2

w

. (B.13)

Hence, overall for the second term on RHS of (B.5), we have

limPD→0

1

Γ(ND)

∂Γ(ND, ND

(1 + σ2

w

|hw|2PD

)ln( |hw|

2PDσ2w

+ 1))

∂PD=NNDD e−ND |hw|22σ2

wΓ(ND).

(B.14)

Combining the results in (B.10) and (B.14), we have

limPD→0

∂ζ∗w,CSI∂PD

= −NNDD e−ND |hw|2σ2wΓ(ND)

, (B.15)

which is the slope of the first order approximation, hence completing the proof.

B.2 Proof of Lemma 5.3 111

B.2 Proof of Lemma 5.3

The problem at Willie is of finding ζ∗w,CDI , given by

ζ∗w,CDI = minλ

E|hw|2 [ζw,CDI ] . (B.16)

Using the relationship of incomplete and complete Gamma functions given by

Γ(a) = Γ(a, b) + γ(a, b), (B.17)

we can rewrite ζw,CDI of (5.11) as

ζw,CDI = 1 +Γ(ND,

NDλσ2w

)Γ(ND)

−Γ(ND,

NDλ|hw|2PD+σ2

w

)Γ(ND)

. (B.18)

Here, we consider a linear approximation of ζw,CDI using Taylor series expansion.

where the first two terms of the expansion around PD = 0 are considered, and

these two terms are given by [f(0) + Pf ′(0)], where f(PD) is given by (B.18) above.

We first note that here, f(0) = 1. To calculate the derivative of f(PD), we use

the derivative property of upper incomplete Gamma function and the required

derivative is calculated as

∂f(PD)

∂PD= −

[− 1

Γ(ND)

(NDλ

|hw|2PD + σ2w

)ND−1

e− NDλ

|hw|2PD+σ2w

(− NDλ|hw|2

(|hw|2PD + σ2w)2

)]

= − 1

Γ(ND)

(NDλ|hw|2

(|hw|2PD + σ2w)2

)(NDλ

|hw|2PD + σ2w

)ND−1

e− NDλ

|hw|2PD+σ2w

(B.19)

which for PD = 0 becomes

∂f(PD)

∂PD

∣∣∣∣∣PD=0

= − 1

Γ(ND)

(NDλ|hw|2

(σ2w)2

)(NDλ

σ2w

)ND−1

e−NDλ

σ2w . (B.20)

Hence, we have the linear approximation for ζw,CDI as

ζw,CDI ≈ 1− PDΓ(ND)

(NDλ|hw|2

(σ2w)2

)(NDλ

σ2w

)ND−1

e−NDλ

σ2w . (B.21)

112

To find the best threshold for Willie under this approximation, we consider

λ∗CDI = arg minλ

E|hw|2 [ζw,CDI ] , (B.22)

where due to E [|hw|2] = 1, we have

E|hw|2 [ζw,CDI ] ≈ 1−(

NDλPD(σ2

w)2Γ(ND)

)(NDλ

σ2w

)ND−1

e−NDλ

σ2w (B.23)

Differentiating this quantity w.r.t λ gives

∂E|hw|2 [ζw,CDI ]

∂λ= − NND

D PD

Γ(ND) (σ2w)ND+1

[NDλ

ND−1e−NDλ

σ2w − NDλ

ND

σ2w

e−NDλ

σ2w

]. (B.24)

Setting the above derivative equal to zero and some further simplifications give

λ∗CDI = σ2w, (B.25)

Using this value of λ∗CDI in the linear approximation of ζw,CDI completes the proof.

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