Design and Optimal Configuration of Full-Duplex MAC Protocol for Cognitive Radio Networks Considering Self-Interference

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Received November 8, 2015, accepted December 10, 2015. Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.

Digital Object Identifier 10.1109/ACCESS.2015.2509449

Design and Optimal Configuration of Full-DuplexMAC Protocol for Cognitive Radio NetworksConsidering Self-InterferenceLE THANH TAN, (Member, IEEE), AND LONG BAO LE, (Senior Member, IEEE)Institut National de la Recherche Scientifique-Énergie, Matériaux et Télécommunications, Université du Québec, Montreal, QC J3X 1S2, Canada

Corresponding author: L. T. Le ([email protected])

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ABSTRACT In this paper, we propose an adaptive medium access control (MAC) protocol forfull-duplex (FD) cognitive radio networks in which FD secondary users (SUs) perform channel contentionfollowed by concurrent spectrum sensing and transmission, and transmission only with maximum powerin two different stages (called the FD sensing and transmission stages, respectively) in each contentionand access cycle. The proposed FD cognitive MAC (FDC-MAC) protocol does not require synchronizationamong SUs, and it efficiently utilizes the spectrum and mitigates the self-interference in the FD transceiver.We develop a mathematical model to analyze the throughput performance of the FDC-MAC protocol, whereboth half-duplex (HD) transmission and FD transmission modes are considered in the transmission stage.Then, we study the FDC-MAC configuration optimization through adaptively controlling the spectrumsensing duration and transmit power level in the FD sensing stage. We prove that there exists optimal sensingtime and transmit power to achieve the maximum throughput, and we develop an algorithm to configurethe proposed FDC-MAC protocol. Extensive numerical results are presented to illustrate the optimalFDC-MAC configuration and the impacts of protocol parameters and the self-interference cancellationquality on the throughput performance. Moreover, we demonstrate the significant throughput gains of theFDC-MAC protocol with respect to the existing HD MAC and single-stage FD MAC protocols.

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INDEX TERMS Asynchronous MAC, full-duplex MAC, full-duplex spectrum sensing, optimalsensing duration, throughput maximization, self-interference control, full-duplex cognitive radios,throughput analysis.

I. INTRODUCTION19

Engineering MAC protocols for efficient sharing of white20

spaces is an important research topic in cognitive radio21

networks (CRNs). One critical requirement for the cognitive22

MAC design is that transmissions on the licensed frequency23

band from primary users (PUs) should be satisfactorily pro-24

tected from the SUs’ spectrum access. Therefore, a cognitive25

MAC protocol for the secondary network must realize both26

the spectrum sensing and access functions so that timely27

detection of the PUs’ communications and effective spectrum28

sharing among SUs can be achieved. Most existing research29

works on cognitive MAC protocols have focused on the30

design and analysis of HD MAC (e.g., see [1]–[4] and the31

references therein).32

Due to the HD constraint, SUs typically employ a33

two-stage sensing/access procedure where they perform34

spectrum sensing in the first stage before accessing35

available spectrum for data transmission in the second 36

stage [5]–[11]. The HD constraint also requires SUs be syn- 37

chronized during the spectrum sensing stage, which could 38

be difficult to achieve in practice. In fact, spectrum sensing 39

enables SUs to detect white spaces that are not occupied by 40

PUs [2]–[8], [12], [13]; therefore, imperfect spectrum sens- 41

ing can reduce the spectrum utilization due to failure in 42

detecting white spaces and potentially result in collisions 43

with active PUs. Consequently, sophisticated design and 44

parameter configuration of cognitive MAC protocols must be 45

conducted to achieve good performance while appropriately 46

protecting PUs [1], [6]–[11], [14]. As a result, traditional 47

MAC protocols [15]–[19] adapted to the CRN may not 48

provide satisfactory performance. 49

In general, HD MAC protocols may not exploit white 50

spaces very efficiently since significant sensing time may 51

be required, which would otherwise be utilized for data 52

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L. T. Tan, L. B. Le: Design and Optimal Configuration of FD MAC Protocol for CRNs Considering Self-Interference

transmission. Moreover, SUs may not timely detect the PUs’53

activity during their transmissions, which can cause severe54

interference to active PUs. Thanks to recent advances on55

FD technologies, a FD radio can transmit and receive56

data simultaneously on the same frequency band [20]–[25].57

In fact, the FD technology can be integrated into58

next-generation wireless networks, e.g., small cell networks59

and heterogeneous wireless networks [26], [27], to signifi-60

cantly enhance the network performance. One of the most61

critical issues of wireless FD communication is the presence62

of self-interference, which is caused by power leakage from63

the transmitter to the receiver of a FD transceiver. The64

self-interference may indeed lead to serious communication65

performance degradation of FD wireless systems. Despite66

recent advances on self-interference cancellation (SIC) tech-67

niques [21]–[23] (e.g., propagation SIC, analog-circuit SIC,68

and digital baseband SIC), self-interference still exists due69

to various reasons such as the limitation of hardware and70

channel estimation errors.71

A. RELATED WORKS72

There are some recent works that propose to exploit the73

FD communications for MAC-level channel access in multi-74

user wireless networks [25]–[31]. In [25], the authors develop75

a centralized MAC protocol to support asymmetric data76

traffic where network nodes may transmit data packets of77

different lengths, and they propose to mitigate the hidden78

node problem by employing a busy tone. To overcome this79

hidden node problem, Duarte et al. propose to adapt the80

standard 802.11MAC protocol with the RTS/CTS handshake81

in [28]. Moreover, Goyal et al. in [29] extend this study82

to consider interference between two nodes due to their83

concurrent transmissions. Different from conventional84

wireless networks, designing MAC protocols in CRNs is85

more challenging because the spectrum sensing function86

must be efficiently integrated into the MAC protocol.87

In addition, the self-interference must be carefully addressed88

in the simultaneous spectrum sensing and access to mitigate89

its negative impacts on the sensing and throughput90

performance.91

The FD technology has been employed for more efficient92

spectrum access design in cognitive radio networks [32]–[35]93

where SUs can perform sensing and transmission simulta-94

neously. In [32], a FD MAC protocol is developed which95

allows simultaneous spectrum access of the SU and PU net-96

works where both PUs and SUs are assumed to employ the97

p-persistent MAC protocol for channel contention resolu-98

tion and access. This design is, however, not applicable99

to the hierarchical spectrum access in the CRNs where100

PUs should have higher spectrum access priority compared101

to SUs.102

In our previous work [33], we propose the FD MAC103

protocol by using the standard backoff mechanism as in104

the 802.11 MAC protocol where we employ concurrent FD105

sensing and access during the data transmission phase as106

well as frame fragmentation. Moreover, engineering of a107

cognitive FD relaying network is considered in [34] and [35], 108

where various resource allocation algorithms to improve the 109

outage probability are proposed. In addition, the authors 110

in [30] develop the joint routing and distributed resource 111

allocation for FD wireless networks. In [31], Choi et al. 112

study the distributed power allocation for a hybrid FD/HD 113

system where all network nodes operate in the HD mode 114

but the access point (AP) communicates by using the FD 115

mode. In practice, it would be desirable to design an adapt- 116

able MAC protocol, which can be configured to operate in 117

an optimal fashion depending on specific channel and net- 118

work conditions. This design will be pursued in our current 119

work. 120

B. OUR CONTRIBUTIONS 121

In this paper, we make a further bold step in designing, 122

analyzing, and optimizing an adaptive FDC–MAC protocol 123

for CRNs, where the self-interference and imperfect 124

spectrum sensing are explicitly considered. In particular, 125

the contributions of this paper can be summarized 126

as follows. 127

1) We propose a novel FDC–MAC protocol that can effi- 128

ciently exploit the FD transceiver for spectrum sens- 129

ing and access of the white space without requiring 130

synchronization among SUs. In this protocol, after the 131

p-persistent based channel contention phase, the 132

winning SU enters the data phase consisting of two 133

stages, i.e., concurrent sensing and transmission in the 134

first stage (called FD sensing stage) and transmission 135

only in the second stage (called transmission stage). 136

The developed FDC–MAC protocol, therefore, enables 137

the optimized configuration of transmit power level 138

and sensing time during the FD sensing stage to mit- 139

igate the self-interference and appropriately protect the 140

active PU. After the FD sensing stage, the SU can 141

transmit with the maximum power to achieve the 142

highest throughput. 143

2) We develop a mathematical model for throughput per- 144

formance analysis of the proposed FDC-MAC proto- 145

col considering the imperfect sensing, self-interference 146

effects, and the dynamic status changes of the PU. 147

In addition, both one-way and two-way transmission 148

scenarios, which are called HD transmission (HDTx) 149

and FD transmission (FDTx) modes, respectively, are 150

considered in the analysis. Since the PU can change its 151

idle/active status during the FD sensing and transmis- 152

sion stages, different potential status-change scenarios 153

are studied in the analytical model. 154

3) We study the optimal configuration of FDC-MAC 155

protocol parameters including the SU’s sensing dura- 156

tion and transmit power to maximize the achievable 157

throughput under both FDTx and HDTx modes. 158

We prove that there exists an optimal sensing time to 159

achieve the maximum throughput for a given transmit 160

power value during the FD sensing stage under both 161

FDTx and HDTx modes. Therefore, optimal protocol 162

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parameters can be determined through standard163

numerical search methods.164

4) Extensive numerical results are presented to illustrate165

the impacts of different protocol parameters on the166

throughput performance and the optimal configura-167

tions of the proposed FDC-MAC protocol. Moreover,168

we show the significant throughput enhancement169

of the proposed FDC-MAC protocol compared170

to existing cognitive MAC protocols, namely the171

HD MAC protocol and a single-stage FD MAC pro-172

tocol with concurrent sensing and access during the173

whole data phase. Specifically, our FDC-MAC pro-174

tocol achieves higher throughput with the increasing175

maximum power while the throughput of the176

single-stage FD MAC protocol decreases with the177

maximum power in the high power regime due178

to the self-interference. Moreover, the proposed179

FDC-MAC protocol significantly outperforms the HD180

MAC protocol in terms of system throughput.181

The remaining of this paper is organized as follows.182

Section II describes the system and PU models. FDC–MAC183

protocol design, and throughput analysis are performed in184

Section III. Then, Section IV studies the optimal con-185

figuration of the proposed FDC–MAC protocol. Section V186

demonstrates numerical results followed by concluding187

remarks in Section VI.188

II. SYSTEM AND PU ACTIVITY MODELS189

A. SYSTEM MODEL190

We consider a cognitive radio network where n0 pairs of191

SUs opportunistically exploit white spaces on one channel192

for communications. We assume that each SU is equipped193

with a FD transceiver; hence, the SUs can perform sens-194

ing and transmission simultaneously. However, the sensing195

performance of each SU is affected by the self-interference196

from its transmitter since the transmitted power is leaked197

into the received signal. We denote I (P) as the average self-198

interference power, which is modeled as I (P) = ζ (P)ξ [20]199

where P is the SU’s transmit power, ζ and ξ (0 ≤ ξ ≤ 1)200

are predetermined coefficients which represent the quality201

of self-interference cancellation (QSIC). In this work, we202

design a asynchronous cognitive MAC protocol where no203

synchronization is required among SUs and between SUs and204

the PU. We assume that different pairs of SUs can overhear205

transmissions from the others (i.e., a collocated network is206

assumed). In the following, we refer to pair i of SUs as SU i207

for brevity.208

B. PRIMARY USER ACTIVITY209

We assume that the PU’s idle/active status follows two210

independent random processes. We say that the channel is211

available and busy for SUs’ access if the PU is in the idle and212

active (or busy) states, respectively. LetH0 andH1 denote the213

events that the PU is idle and active, respectively. To protect214

the PU, we assume that SUs must stop their transmissions215

and evacuate from the busy channel within the maximum216

delay of Teva, which is referred to as channel evacuation 217

time. 218

Let τac and τid denote the random variables which repre- 219

sent the durations of active and idle channel states, respec- 220

tively. We denote probability density functions (pdf) of τac 221

and τid as fτac (t) and fτid (t), respectively. While most results 222

in this paper can be applied to general pdfs fτac (t) and fτid (t), 223

we mostly consider the exponential pdf in the analysis. 224

In addition, let P (H0) =τid

τid+τacand P (H1) = 1−P (H0) 225

present the probabilities that the channel is available and 226

busy, respectively where τid and τac denote the average values 227

of τac and τid, respectively. We assume that the probabilities 228

that τac and τid are smaller than Teva are sufficiently small 229

(i.e., the PU changes its status slowly) so that we can ignore 230

events with multiple idle/active status changes in one channel 231

evacuation interval Teva. 232

III. FULL-DUPLEX COGNITIVE MAC PROTOCOL 233

In this section, we describe the proposed FDC-MAC protocol 234

and conduct its throughput analysis considering imperfect 235

sensing, self-interference of the FD transceiver, and dynamic 236

status change of the PUs. 237

A. FDC-MAC PROTOCOL DESIGN 238

The proposed FDC-MAC protocol integrates three important 239

elements of a cognitive MAC protocol, namely contention 240

resolution, spectrum sensing, and access functions. 241

Specifically, SUs employ the p-persistent CSMA princi- 242

ple [17] for contention resolution where each SU with data 243

to transmit attempts to capture an available channel with 244

a probability p after the channel is sensed to be idle dur- 245

ing the standard DIFS interval (DCF Interframe Space). 246

If a particular SU decides not to transmit (with probability 247

of 1 − p), it will sense the channel and attempt to trans- 248

mit again in the next slot of length σ with probability p. 249

To complete the reservation, the four-way handshake with 250

Request-to-Send/Clear-to-Send (RTS/CST) exchanges [16] 251

is employed to reserve the available channel for transmission. 252

Specifically, the secondary transmitter sends RTS to the 253

secondary receiver and waits until it successfully receives 254

the CTS from the secondary receiver. All other SUs, which 255

hear the RTS and CTS exchange from the winning SU, defer 256

to access the channel for a duration equal to the data trans- 257

mission time, T . Then, an acknowledgment (ACK) from the 258

SU’s receiver is transmitted to its corresponding transmitter 259

to notify the successful reception of a packet. Furthermore, 260

the standard small interval, namely SIFS (Short Interframe 261

Space), is used before the transmissions of CTS, ACK, and 262

data frame as in the standard 802.11 MAC protocol [16]. 263

In our design, the data phase after the channel con- 264

tention phase comprises two stages where the SU performs 265

concurrent sensing and transmission in the first stage with 266

duration TS and transmission only in the second stage with 267

duration T − TS . Here, the SU exploits the FD capability of 268

its transceiver to realize concurrent sensing and transmission 269

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FIGURE 1. Timing diagram of the proposed full-duplex cognitive MAC protocol.

the first stage (called FD sensing stage) where the sensing270

outcome at the end of this stage (i.e., an idle or busy channel271

status) determines its further actions as follows. Specifically,272

if the sensing outcome indicates an available channel then273

the SU transmits data in the second stage; otherwise, it274

remains silent for the remaining time of the data phase with275

duration T − TS .276

We assume that the duration of the SU’s data phase T is277

smaller than the channel evacuation time Teva so timely278

evacuation from the busy channel can be realized with reli-279

able FD spectrum sensing. Therefore, our design allows280

to protect the PU with evacuation delay at most T if the281

MAC carrier sensing during the contention phase and282

the FD spectrum sensing in the data phase are perfect.283

Furthermore, we assume that the SU transmits at power levels284

Psen ≤ Pmax and Pdat = Pmax during the FD sensing285

and transmission stages, respectively where Pmax denotes286

the maximum power and the transmit power Psen in the287

FD sensing stage will be optimized to effectively mitigate288

the self-interference and achieve good sensing-throughput289

tradeoff. The timing diagram of the proposed FDC–MAC290

protocol is illustrated in Fig. 1.291

We allow two possible operation modes in the transmission 292

stage. The first is the HD transmission mode (HDTx mode) 293

where there is only one direction of data transmission from 294

the SU transmitter to the SU receiver. In this mode, there 295

is no self-interference in the transmission stage. The second 296

is the FD transmission mode (FDTx mode) where two-way 297

communications between the pair of SUs are assumed 298

(i.e., there are two data flows between the two SU nodes in 299

opposite directions). In this mode, the achieved throughput 300

can be potentially enhanced (at most doubling the throughput 301

of the HDTx mode) but self-interference must be taken into 302

account in throughput quantification. 303

Our proposed FDC–MAC protocol design indeed enables 304

flexible and adaptive configuration, which can efficiently 305

exploit the capability of the FD transceiver. Specifically, if the 306

duration of the FD sensing stage is set equal to the duration 307

of the whole data phase (i.e., TS = T ), then the SU per- 308

forms concurrent sensing and transmission for the whole data 309

phase as in our previous design [33]. This configuration may 310

degrade the achievable throughput since the transmit power 311

during the FD sensing stage is typically set smaller Pmax 312

to mitigate the self-interference and achieve the required 313

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sensing performance. We will refer the corresponding MAC314

protocol with TS = T as one-stage FD MAC in the sequel.315

Moreover, if we set the SU transmit power Psen in the316

sensing stage equal to zero, i.e., Psen = 0, then we317

achieve the traditional two-stage cognitive HDMAC protocol318

where sensing and transmission are performed sequentially319

in two different stages [6], [8]. Moreover, the proposed320

FDC–MAC protocol is more flexible than existing321

designs [6], [8], [33] since different existing designs can322

be achieved through suitable configuration of its protocol323

parameters. It will be demonstrated later that the proposed324

FDC–MAC protocol achieves significant better throughput325

than that of the existing cognitive MAC protocols. In the326

following, we present the throughput analysis based on which327

the protocol configuration optimization can be performed.328

B. THROUGHPUT ANALYSIS329

We now conduct the saturation throughput analysis for the330

secondary network where all SUs are assumed to always331

have data to transmit. The resulting throughput can be served332

as an upper bound for the throughput in the non-saturated333

scenario [16]. This analysis is performed by studying one334

specific contention and access cycle (CA cycle) with the335

contention phase and data phase as shown in Fig. 1. Without336

loss of generality, wewill consider the normalized throughput337

achieved per one unit of system bandwidth (in bits/s/Hz).338

Specifically, the normalized throughput of the FDC–MAC339

protocol can be expressed as340

NT =B

Tove + T, (1)341

where Tove represents the time overhead required for one342

successful channel reservation (i.e., successful RTS/CTS343

exchanges), T denotes the packet transmission time, and B344

denotes the amount of data (bits) transmitted in one CA cycle345

per one unit of system bandwidth, which is expressed in346

bits/Hz. To complete the throughput analysis, we derive the347

quantities Tove and B in the remaining of this subsection.348

1) DERIVATION OF Tove349

The average time overhead for one successful channel reser-350

vation can be calculated as351

Tove = T cont + 2SIFS + 2PD+ ACK , (2)352

where ACK is the length of an ACK message, SIFS is the353

length of a short interframe space, and PD is the propagation354

delay where PD is usually small compared to the slot size σ ,355

and T cont denotes the average time overhead due to idle356

periods, collisions, and successful transmissions of RTS/CTS357

messages in one CA cycle. For better presentation of the358

paper, the derivation of T cont is given in Appendix A.359

2) DERIVATION OF B360

To calculate B, we consider all possible cases that capture361

the activities of SUs and status changes of the PU in the362

FDC-MAC data phase of duration T . Because the PU’s activ-363

ity is not synchronizedwith the SU’s transmission, the PU can364

change its idle/active status any time. We assume that there 365

can be at most one transition between the idle and active states 366

of the PU during one data phase interval. This is consistent 367

with the assumption on the slow status changes of the PU as 368

described in Section II-B since T < Teva. Furthermore, we 369

assume that the carrier sensing of the FDC-MAC protocol 370

is perfect; therefore, the PU is idle at the beginning of the 371

FDC-MACdata phase. Note that the PUmay change its status 372

during the SU’s FD sensing or transmission stage, which 373

requires us to consider different possible events in the data 374

phase. 375

We use hij (i, j ∈ {0, 1}) to represent events capturing status 376

changes of the PU in the FD sensing stage and transmission 377

stage where i = 0 and i = 1 represent the idle and active 378

states of the PU, respectively. For example, if the PU is idle 379

during the FD sensing stage and becomes active during the 380

transmission stage, then we represent this event as (h00, h01) 381

where sub-events h00 and h01 represent the status changes in 382

the FD sensing and transmission stages, respectively. More- 383

over, if the PU changes from the idle to the active state during 384

the FD sensing stage and remains active in the remaining of 385

the data phase, then we represent this event as (h01, h11). 386

It can be verified that we must consider the following three 387

cases with the corresponding status changes of the PU during 388

the FDC-MAC data phase to analyze B. 389

• Case 1: The PU is idle for the whole FDC-MAC data 390

phase (i.e., there is no PU’s signal in both FD sensing 391

and transmission stages) and we denote this event as 392

(h00, h00). The average number of bits (in bits/Hz) trans- 393

mitted during the data phase in this case is denoted asB1. 394

• Case 2: The PU is idle during the FD sensing stage but 395

the PU changes from the idle to the active status in the 396

transmission stage. We denote the event corresponding 397

to this case as (h00, h01) where h00 and h01 capture the 398

sub-events in the FD sensing and transmission stages, 399

respectively. The average number of bits (in bits/Hz) 400

transmitted during the data phase in this case is repre- 401

sented by B2. 402

• Case 3: The PU is first idle then becomes active during 403

the FD sensing stage and it remains active during the 404

whole transmission stage. Similarly we denote this event 405

as (h01, h11) and the average number of bits (in bits/Hz) 406

transmitted during the data phase in this case is denoted 407

as B3. 408

Then, we can calculate B as follows: 409

B = B1 + B2 + B3. (3) 410

To complete the analysis, we will need to derive B1, B2, 411

and B3, which are given in Appendix B. 412

IV. FDC–MAC PROTOCOL CONFIGURATION FOR 413

THROUGHPUT MAXIMIZATION 414

In this section, we study the optimal configuration of the 415

proposed FDC–MAC protocol to achieve the maximum 416

throughput while satisfactorily protecting the PU. 417

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A. PROBLEM FORMULATION418

Let NT (TS , p,Psen) denote the normalized secondary419

throughput, which is the function of the sensing time TS ,420

transmission probability p, and the SU’s transmit power Psen421

in the FD sensing stage. In the following, we assume422

a fixed frame length T , which is set smaller the required423

evacuation time Teva to achieve timely evacuation from a424

busy channel for the SUs. We are interested in determining425

suitable configuration for p, TS and Psen to maximize the sec-426

ondary throughput,NT (TS , p,Psen). In general, the optimal427

transmission probability p should balance between reducing428

collisions among SUs and limiting the protocol overhead.429

However, the achieved throughput is less sensitive to the430

transmission probability p as will be demonstrated later via431

the numerical study. Therefore, we will seek to optimize the432

throughput over Psen and TS for a reasonable and fixed value433

of p.434

For brevity, we express the throughput as a function ofPsen435

and TS only, i.e.,NT (TS ,Psen). Suppose that the PU requires436

that the average detection probability is at leastPd . Then, the437

throughput maximization problem can be stated as follows:438

maxTS ,Psen

NT (TS ,Psen)439

s.t. Pd (ε,TS) ≥ Pd ,440

0 ≤ Psen ≤ Pmax, 0 ≤ TS ≤ T , (4)441

where Pmax is the maximum power for SUs, and TS is upper442

bounded by T . In fact, the first constraint on Pd (ε,TS)443

implies that the spectrum sensing should be sufficiently444

reliable to protect the PU which can be achieved with suf-445

ficiently large sensing time TS . Moreover, the SU’s transmit446

powerPsen must be appropriately set to achieve good tradeoff447

between the network throughput and self-interference448

mitigation.449

B. PARAMETER CONFIGURATION FOR FDC–MAC450

PROTOCOL451

To gain insights into the parameter configuration of the452

FDC–MAC protocol, we first study the optimization with453

respect to the sensing time TS for a given Psen. For any454

value of TS , we would need to set the sensing detection455

threshold ε so that the detection probability constraint is met456

with equality, i.e., Pd (ε,TS) = Pd as in [5] and [6]. Since457

the detection probability is smaller in Case 3 (i.e., the PU458

changes from the idle to active status during the FD sensing459

stage of duration TS ) compared to that in Case 1 and Case 2460

(i.e., the PU remains idle during the FD sensing stage) consid-461

ered in the previous section, we only need to consider Case 3462

to maintain the detection probability constraint. The average463

probability of detection for the FD sensing in Case 3 can be464

expressed as465

Pd =∫ TS

0P01d (t)fτid (t |0 ≤ t ≤ TS ) dt, (5)466

where t denotes the duration from the beginning of the FD467

sensing stage to the instant when the PU changes to the active468

Algorithm 1 FDC-MAC Configuration Algorithm1: for each considered value of Psen ∈ [0,Pmax] do2: Find optimal TS for problem (7) using the bisection

method as T S (Psen) = argmax0≤TS≤T

NT (T ,Psen).

3: end for4: The final solution

(T ∗S ,P

∗sen)

is determined as(T ∗S ,P

∗sen)= argmax

Psen,T S (Psen)

NT (TS (Psen) ,Psen).

state, and fτid (t |A ) is the pdf of τid conditioned on event A 469

capturing the condition 0 ≤ t ≤ TS , which is given as 470

fτid (t |A ) =fτid (t)Pr {A}

=

1τid

exp(− tτid)

1− exp(− TSτid). (6) 471

Note that P01d (t) is derived in Appendix C and fτid (t) is given 472

in (18). 473

We consider the following single-variable optimization 474

problem for a given Psen: 475

max0<TS≤T

NT (TS ,Psen) . (7) 476

We characterize the properties of function NT (TS ,Psen) 477

with respect to TS for a given Psen in the following theorem 478

whose proof is provided in Appendix D. For simplicity, the 479

throughput function is written as NT (TS ). 480

Theorem 1: The objective functionNT(TS) of (7) satisfies 481

the following properties 482

1) limTS→0

∂NT∂TS= +∞, 483

2) a) For HDTxmodewith ∀Psen and FDTxmodewith 484

Psen < Psen, we have limTS→T

∂NT∂TS

< 0, 485

b) For FDTx mode with Psen > Psen, we have 486

limTS→T

∂NT∂TS

> 0, 487

3) ∂2NT∂T 2

S< 0, ∀TS , 488

4) The objective functionNT(TS) is bounded from above, 489

where Psen = N0

[(1+ Pdat

N0+ζPξ

dat

)2

− 1

]is the critical 490

value of Psen such that limTS→T

∂NT∂TS= 0. 491

We would like to discuss the properties stated in 492

Theorem 1. For the HDTx mode with ∀Psen and FDTx mode 493

with low Psen, then properties 1, 2a, and 4 imply that there 494

must be at least one TS in [0,T ] that maximizes NT (TS). 495

The third property implies that this maximum is indeed 496

unique. Moreover, for the FDTx with high Psen, then 497

properties 1, 2b, 3 and 4 imply that NT(TS) increases 498

in [0,T ]. Hence, the throughput NT(TS) achieves its maxi- 499

mum with sensing time TS = T . We propose an algorithm 500

to determine optimal (TS ,Psen), which is summarized in 501

Algorithm 1. Here, we can employ the bisection scheme and 502

other numerical methods to determine the optimal value TS 503

for a given Psen. 504

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V. NUMERICAL RESULTS505

For numerical studies, we set the key parameters for the506

FDC–MAC protocol as follows: mini-slot duration is507

σ = 20µs; PD = 1µs; SIFS = 2σ µs; DIFS = 10σ µs;508

ACK = 20σ µs; CTS = 20σ µs; RTS = 20σ µs. Other509

parameters are chosen as follows unless stated otherwise:510

the sampling frequency fs = 6 MHz; bandwidth of PU’s511

signal 6 MHz; Pd = 0.8; T = 15 ms; p = 0.0022; the512

SNR of the PU signal at each SU γP =PpN0= −20 dB;513

varying self-interference parameters ζ and ξ . Without loss514

of generality, the noise power is normalized to one; hence,515

the SU transmit power Psen becomes Psen = SNRs; and we516

set Pmax = 15dB.517

We first study the impacts of self-interference parameters518

on the throughput performance with the following parameter519

setting: (τid, τac) = (1000, 100) ms, Pmax = 25 dB, Teva =520

40 ms, ζ = 0.4, ξ is varied in ξ = {0.12, 0.1, 0.08, 0.05},521

and Pdat = Pmax. Recall that the self-interference depends522

on the transmit power P as I (P) = ζ (P)ξ where P = Psen523

and P = Pdat in the FD sensing and transmission stages,524

respectively. Fig. 2 illustrates the variations of the throughput525

versus the transmission probability p. It can be observed that526

when ξ decreases (i.e., the self-interference is smaller), the527

achieved throughput increases. This is because SUs can trans-528

mit with higher power while still maintaining the sensing con-529

straint during the FD sensing stage, which leads to throughput530

improvement. The optimal Psen corresponding to these val-531

ues of ξ are Psen = SNRs = {25.00, 18.01, 14.23, 11.28} dB532

and the optimal probability of transmission is p∗ = 0.0022533

as indicated by a star symbol. Therefore, to obtain all other534

results in this section, we set p∗ = 0.0022.535

FIGURE 2. Normalized throughput versus transmission probability p forT = 18 ms, τid = 1000 ms, τac = 100 ms, and varying ξ .

Fig. 3 illustrates the throughput performance versus536

number of SUs n0 when we keep the same parameter set-537

tings as those for Fig. 2 and p∗ = 0.0022. Again, when538

ξ decreases (i.e., the self-interference becomes smaller), the539

achieved throughput increases. In this figure, the optimal540

FIGURE 3. Normalized throughput versus the number of SUs n0 forT = 18 ms, p = 0.0022, τid = 1000 ms, τac = 100 ms, and varying ξ .

FIGURE 4. Normalized throughput versus SU transmit power Psen andsensing time TS for p = 0.0022, τid = 500 ms, τac = 50 ms, n0 = 40,ξ = 1, ζ = 0.7 and FDTx with Pdat = 15 dB.

SNRs achieving the maximum throughput corresponding 541

to the considered values of ξ are Psen = SNRs = 542

{25.00, 18.01, 14.23, 11.28} dB, respectively. 543

We now verify the results stated in Theorem 1 for the FDTx 544

mode. Specifically, Fig. 4 shows the throughput performance 545

for the scenario where the QSIC is very lowwith large ξ and ζ 546

where we set the network parameters as follows: p = 0.0022, 547

τid = 500 ms, τac = 50 ms, n0 = 40, ξ = 1, ζ = 0.7, 548

and Pdat = 15 dB. Moreover, we can obtain Psen as in (42) 549

in Appendix D, which is equal to Psen = 6.6294 dB. In this 550

figure, the curve indicated by asterisks, which corresponds to 551

Psen = Psen, shows the monotonic increase of the through- 552

put with sensing time TS and other curves corresponding 553

to Psen > Psen have the same characteristic. In contrast, 554

all remaining curves (corresponding to Psen < Psen) first 555

increase to the maximum values and then decrease as we 556

increase TS . 557

Fig. 5 illustrates the throughput performance for the very 558

high QSIC with small ξ and ζ where we set the network 559

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FIGURE 5. Normalized throughput versus SU transmit power Psen andsensing time TS for p = 0.0022, τid = 500 ms, τac = 50 ms, n0 = 40,ξ = 1, ζ = 0.08 and FDTx with Pdat = 15 dB.

parameters as follows: p = 0.0022, τid = 500 ms,560

τac = 50 ms, n0 = 40, ξ = 1, ζ = 0.08, and561

Pdat = 15 dB. Moreover, we can obtain Psen as in (42) in562

Appendix D, which is equal to Psen = 19.9201 dB. We have563

Psen < Pmax = 15dB < Psen in this scenario; hence, all564

the curves first increases to the maximum throughput and565

then decreases with the increasing TS . Therefore, we have566

correctly validated the properties stated in Theorem 1.567

Nowwe investigate the throughput performance versus SU568

transmit power Psen and sensing time TS for the case of569

high QSIC with ξ = 0.95 and ζ = 0.08. Fig. 6 shows the570

throughput versus the SU transmit power Psen and sensing571

time TS for the FDTx mode with Pdat = 15 dB, p = 0.0022,572

τid = 150 ms, τac = 50 ms, and n0 = 40. It can be observed573

that there exists an optimal configuration of the SU transmit574

power P∗sen = 4.6552 dB and sensing time T ∗S = 2.44 ms to575

FIGURE 6. Normalized throughput versus SU transmit power Psen andsensing time TS for p = 0.0022, τid = 150 ms, τac = 50 ms, n0 = 40,ξ = 0.95, ζ = 0.08 and FDTx with Pdat = 15 dB.

achieve the maximum throughputNT(T ∗S ,P

∗sen)= 2.3924, 576

which is indicated by a star symbol. These results confirm that 577

SUs must set appropriate sensing time and transmit power for 578

the FDC–MAC protocol to achieve the maximize throughput, 579

which cannot be achieved by setting Ts = T as proposed in 580

existing designs such as in [33]. 581

FIGURE 7. Normalized throughput versus SU transmit power Psen andsensing time TS for p = 0.0022, τid = 150 ms, τac = 50 ms, n0 = 40,ξ = 0.95, ζ = 0.8 and FDTx with Pdat = 15 dB.

In Fig. 7, we present the throughput versus the SU transmit 582

power Psen and sensing time TS for the low QSIC sce- 583

nario where p = 0.0022, τid = 150 ms, τac = 50 ms, 584

Pmax = 15 dB, n0 = 40, ξ = 0.95, and ζ = 0.8. The 585

optimal configuration of SU transmit power P∗sen = 15 dB 586

and sensing time T ∗S = 15 ms to achieve the maximum 587

throughput NT(T ∗S ,P

∗sen)= 1.6757 is again indicated 588

by a star symbol. Under this optimal configuration, the FD 589

sensing is performed during the whole data phase (i.e., there 590

is no transmission stage). In fact, to achieve the maximum 591

throughput, the SU must provide the satisfactory sensing 592

performance and attempt to achieve high transmission rate. 593

Therefore, if the QSIC is low, the data rate achieved during 594

the transmission stage can be lower than that in the FD sens- 595

ing stage because of the very strong self-interference in the 596

transmission stage. Therefore, setting longer FD sensing time 597

enables to achieve more satisfactory sensing performance 598

and higher transmission rate, which explains that the optimal 599

configuration should set T ∗S = T for the low QSIC scenario. 600

This protocol configuration corresponds to existing design 601

in [33], which is a special case of the proposed FDC–MAC 602

protocol. 603

We now investigate the throughput performance with 604

respect to the SU transmit power Psen and sensing time TS 605

for the HDTx mode. Fig. 8 illustrates the throughput per- 606

formance for the high QSIC scenario with ξ = 0.95 and 607

ζ = 0.08. It can be observed that there exists an optimal 608

configuration of SU transmit power P∗sen = 5.6897 dB 609

and sensing time T ∗S = 3.5 ms to achieve the maximum 610

throughput NT(T ∗S ,P

∗sen)= 1.4802, which is indicated 611

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FIGURE 8. Normalized throughput versus SU transmit power Psen andsensing time TS for p = 0.0022, τid = 150 ms, τac = 50 ms, n0 = 40,ξ = 0.95, ζ = 0.08 and HDTx.

by a star symbol. The maximum achieved throughput of the612

HDTxmode is lower than that in the FDTxmode presented in613

Fig. 6. This is because with high QSIC, the FDTx mode can614

transmit more data than the HDTx mode in the transmission615

stage.616

In Fig. 9, we show the throughput versus the SU transmit617

power Psen for TS = 2.2 ms, p = 0.0022, τid = 1000 ms,618

τac = 50 ms, n0 = 40, ξ = 0.95, ζ = 0.08 and various619

values of T (i.e., the data phase duration) for the FDTx mode620

with Pdat = 15 dB. For each value of T , there exists the621

optimal SU transmit power P∗sen which is indicated by an622

asterisk. It can be observed that as T increases from 8 ms623

to 25 ms, the achieved maximum throughput first increases624

then decreases with T . Also in the case with T ∗ = 15 ms,625

the SU achieves the largest throughput which is indicated by626

a star symbol. Furthermore, the achieved throughput signifi-627

cantly decreases when the pair of (T ,Psen) deviates from the628

optimal values,(T ∗,P∗sen

).629

FIGURE 9. Normalized throughput versus SU transmit power Psen forTS = 2.2 ms, p = 0.0022, τid = 1000 ms, τac = 50 ms, n0 = 40, ξ = 0.95,ζ = 0.08, varying T , and FDTx with Pdat = 15 dB.

FIGURE 10. Normalized throughput versus Pmax for τid = 150 ms,τac = 75 ms, n0 = 40, ξ = 0.85, n0 = 40, ζ =

{0.2,0.7

}, and FDTx with

Pdat = Pmax dB.

Finally, we compare the throughput of our proposed 630

FDC-MAC protocol, the single-stage FD MAC protocol 631

where FD sensing (concurrent spectrum sensing and trans- 632

mission) is performed during the whole data phase [33] and 633

the HDMAC protocol which does not allow the transmission 634

during the spectrum sensing interval in Fig. 10. For brevity, 635

the single-stage FD MAC protocol is refereed to as FD 636

MAC in this figure. The parameter settings are as follows: 637

τid = 150 ms, τac = 75 ms, n0 = 40, ξ = 0.85, n0 = 40, 638

ζ = {0.2, 0.7}, and FDTx with Pdat = Pmax dB. For fair 639

comparison, we first obtain the optimal configuration of the 640

single-stage FD MAC protocol, i.e., then we use (T ∗, p∗) 641

for the HD MAC protocol and FDC-MAC protocol. For the 642

single-stage FD MAC protocol, the transmit power is set 643

to Pmax because there is only a single stage where the SU 644

performs sensing and transmission simultaneously during the 645

data phase. In addition, the HDMAC protocol will also trans- 646

mit with the maximum transmit power Pmax to achieve the 647

highest throughput. For both studied cases of ζ = {0.2, 0.7}, 648

our proposed FDC-MAC protocol significantly outperforms 649

the other two protocols. Moreover, the single-stage FD MAC 650

protocol [33] with power allocation outperforms the HD 651

MAC protocol at the corresponding optimal value of Pmax 652

required by the single-stage FD MAC protocol. However, 653

both single-stage FDC-MAC andHDMAC protocols achieve 654

increasing throughput with higher Pmax while the single- 655

stage FD MAC protocol has the throughput first increased 656

then decreased as Pmax increases. This demonstrates the neg- 657

ative self-interference effect on the throughput performance 658

of the single-stage FD MAC protocol, which is efficiently 659

mitigated by our proposed FDC-MAC protocol. 660

VI. CONCLUSION 661

In this paper, we have proposed the FDC–MAC protocol 662

for cognitive radio networks, analyzed its throughput per- 663

formance, and studied its optimal parameter configuration. 664

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The design and analysis have taken into account the665

FD communication capability and the self-interference of666

the FD transceiver. We have shown that there exists an667

optimal FD sensing time to achieve the maximum through-668

put. We have then presented extensive numerical results to669

demonstrate the impacts of self-interference and protocol670

parameters on the throughput performance. In particular, we671

have shown that the FDC–MAC protocol achieves signifi-672

cantly higher throughput than the HD MAC protocol, which673

confirms that the FDC–MAC protocol can efficiently exploit674

the FD communication capability. Moreover, the FDC–MAC675

protocol results in higher throughput with the increasingmax-676

imum power budget while the throughput of the single-stage677

FD MAC can decrease in the high power regime. This result678

validates the importance of adopting the two-stage procedure679

in the data phase and the optimization of sensing time and680

transmit power during the FD sensing stage to mitigate the681

negative self-interference effect.682

APPENDIX A683

DERIVATION OF T cont684

To calculate T cont, we define some further parameters as685

follows. Denote Tcoll as the duration of the collision and Tsucc686

as the required time for successful RTS/CTS transmission.687

These quantities can be calculated as follows [17]:688 {Tsucc = DIFS + RTS + SIFS + CTS + 2PDTcoll = DIFS + RTS + PD,

(8)689

where DIFS is the length of a DCF (distributed coordination690

function) interframe space, RTS and CTS denote the lengths691

of the RTS and CTS messages, respectively.692

As being shown in Fig. 1, there can be several idle periods693

and collisions before one successful channel reservation. Let694

T iidle denote the i-th idle duration between two consecutive695

RTS/CTS exchanges, which can be collisions or successful696

exchanges. Then, T iidle can be calculated based on its prob-697

ability mass function (pmf), which is derived as follows.698

In the following, all relevant quantities are defined in terms of699

the number of time slots. With n0 SUs joining the contention700

resolution, let Psucc, Pcoll and Pidle denote the probabilities701

that a particular generic slot corresponds to a successful702

transmission, a collision, and an idle slot, respectively. These703

probabilities can be calculated as follows:704

Psucc = n0p (1− p)n0−1 (9)705

Pidle = (1− p)n0 (10)706

Pcoll = 1− Psucc − Pidle, (11)707

where p is the transmission probability of an SU in a generic708

slot. In general, the interval Tcont, whose average value is709

T cont given in (2), consists of several intervals correspond-710

ing to idle periods, collisions, and one successful RTS/CTS711

transmission. Hence, this quantity can be expressed as712

Tcont =

Ncoll∑i=1

(Tcoll + T

iidle

)+ TNcoll+1

idle + Tsucc, (12)713

where Ncoll is the number of collisions before the success- 714

ful RTS/CTS exchange and Ncoll is a geometric random 715

variable (RV) with parameter 1 − Pcoll/P idle where 716

P idle = 1− Pidle. Therefore, its pmf can be expressed as 717

f NcollX (x) =

(Pcoll

P idle

)x (1−

Pcoll

P idle

), x = 0, 1, 2, . . . 718

(13) 719

Also, Tidle represents the number of consecutive idle slots, 720

which is also a geometric RV with parameter 1 − Pidle with 721

the following pmf 722

f TidleX (x) = (Pidle)

x (1− Pidle), x = 0, 1, 2, . . . (14) 723

Therefore, T cont (the average value of Tcont) can be written 724

as follows [17]: 725

T cont = N collTcoll + T idle(N coll + 1

)+ Tsucc, (15) 726

where T idle and N coll can be calculated as 727

T idle =(1− p)n0

1− (1− p)n0(16) 728

N coll =1− (1− p)n0

n0p (1− p)n0−1− 1. (17) 729

These expressions are obtained by using the pmfs of the 730

corresponding RVs given in (13) and (14), respectively [17]. 731

APPENDIX B 732

DERIVATIONS OF B1, B2, B3 733

We will employ a pair of parameters (θ, ϕ) to represent the 734

HDTX and FDTX modes where ((θ, ϕ) = (0, 1)) for HDTx 735

mode and ((θ, ϕ) = (1, 2)) for the FDTx mode. Moreover, 736

since the transmit powers in the FD sensing and transmis- 737

sion stages are different, which are equal to Psen and Pdat, 738

respectively, we define different SNRs and SINRs in these 739

two stages as follows: γS1 =PsenN0

and γS2 =PsenN0+Pp

are 740

the SNR and SINR achieved by the SU in the FD sensing 741

stage with and without the presence of the PU, respectively; 742

γD1 =PdatN0+θ I

and γD2 =Pdat

N0+Pp+θ Ifor I = ζPξdat are 743

the SNR and SINR achieved by the SU in the transmission 744

stage with and without the presence of the PU, respectively. It 745

can be seen that we have accounted for the self-interference 746

for the FDTx mode during the transmission stage in γD1 by 747

noting that θ = 1 in this case. The parameter ϕ for the 748

HDTx and FDTx modes will be employed to capture the 749

throughput for one-way and two-way transmissions in these 750

modes, respectively. 751

The derivations of B1, B2, and B3 require us to consider 752

different possible sensing outcomes in the FD sensing stage. 753

In particular, we need to determine the detection 754

probabilityP ijd , which is the probability of correctly detecting 755

the PU given the PU is active, and the false alarm probabil- 756

ity P ijf , which is the probability of the erroneous sensing 757

of an idle channel, for each event hij capturing the state 758

changes of the PU. In the following analysis, we assume 759

the exponential distribution for τac and τid where τac and τid 760

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denote the corresponding average values of these active and761

idle intervals. Specifically, let fτx (t) denote the pdf of τx762

(x represents ac or id in the pdf of τac or τid, respectively)763

then764

fτx (t) =1τx

exp(−tτx). (18)765

Similarly, we employ T ijS and T ijD to denote the number of766

bits transmitted on one unit of system bandwidth during the767

FD sensing and transmission stages under the PU’s state-768

changing event hij, respectively.769

We can now calculate B1 as follows:770

B1 = P (H0)

∫∞

t=Tove+TT 001 fτid (t)dt771

= P (H0)T 001 exp

(−Tove + Tτid

), (19)772

where P (H0) denotes the probability of the idle state of773

the PU, and P00f is the false alarm probability for event774

h00 given in Appendix C. Moreover, T 001 = P00

f T 00S +775

(1 − P00f )(T 00

S + T 00D ), T 00

S = TS log2 (1+ γS1), T00D =776

ϕ (T − TS) log2 (1+ γD1) where T 00S and T 00

D denote the777

number of bits transmitted (over oneHz of system bandwidth)778

in the FD sensing and transmission stages of the data phase,779

respectively. After some manipulations, we achieve780

B1 = Ke exp(T1τ

)[TS log2(1+ γS1)+ ϕ

(1− P00

f

)781

(T − TS) log2(1+ γD1)], (20)782

where Ke = P (H0) exp(−

(Toveτid+

Tτac

))and 1

1τ=783

1τac−

1τid.784

Moreover, we can calculate B2 as785

B2 = P (H0)

∫ Tove+T

t1=Tove+TS

∫∞

t2=Tove+T−t1786

T 012 (t1)fτid (t1)fτac (t2)dt1dt2, (21)787

where T 012 (t1) = P00

f T 00S + (1 − P00

f )(T 00S + T 01

D

(t1)),788

T 01D (t1) = ϕ

(T − TS − t1

)log2 (1+ γD2) + ϕ t1 log2789

(1+ γD1), and t1 = t1 − (Tove + TS). In this expression,790

t1 denotes the interval from the beginning of the CA cycle791

to the instant when the PU changes to the active state from792

an idle state. Again, T 00S and T 01

D denote the amount of data793

transmitted in the FD sensing and transmission stages for this794

case, respectively. After some manipulations, we achieve795

B2=Ke1τ

τid

{(exp

(T1τ

)− exp

(TS1τ

))796

×

[TS log2 (1+γS1)−ϕ1τ

(1−P00

f)log2

(1+γD11+γD2

)]797

+ϕ (T − TS)(1−P00

f

)798

×

[exp

(T1τ

)log2(1+γD1)−exp

(TS1τ

)log2(1+γD2)

]}.799

(22)800

Finally, we can express B3 as follows: 801

B3 = P (H0)

∫ Tove+TS

t1=Tove

∫∞

t2=Tove+T−t1802

×

[P01d(t1)T 01S(t1)+ (1−P01

d(t1))(T 01

S(t1)+ T 11

D )]

803

fτid (t1)fτac (t2)dt1dt2, (23) 804

where t1 = t1 − Tove, T 01S

(t1)= t1 log2 (1+ γS1) + 805(

TS − t1)log2 (1+ γS2), T

11D = ϕ (T − TS) log2 (1+ γD2), 806

and t1 is the same as in (21). Here, T 01S and T 11

D denote the 807

amount of data delivered in the FD sensing and transmis- 808

sion stages for the underlying case, respectively. After some 809

manipulations, we attain 810

B3 = Ke

∫ TS

t=0

[T 01S (t)+ T 11

D − P01d (t)T 11

D

]811

fτid (t) exp(

tτac

)dt = B31 + B32, (24) 812

where 813

B31 = Ke

∫ TS

t=0

[T 01S (t)+ T 11

D

]fτid (t) exp

(tτac

)dt 814

= Ke1τ

τid

{1τ

[(TS1τ−1)exp

(TS1τ

)+1]log2

(1+γS11+γS2

)815

+

[exp

(TS1τ

)− 1

] [T 11D + TS log2 (1+ γS2)

]}, 816

(25) 817

and 818

B32 = −KeT 11D T32, (26) 819

where T32 =∫ TSt=0 P

01d (t) fτid (t) exp

(tτac

)dt . 820

APPENDIX C 821

FALSE ALARM AND DETECTION PROBABILITIES 822

We derive the detection and false alarm probabilities for 823

FD sensing and two PU’s state-changing events h00 and h01 824

in this appendix. Assume that the transmitted signals from 825

the PU and SU are circularly symmetric complex 826

Gaussian (CSCG) signals while the noise at the secondary 827

receiver is independently and identically distributed CSCG 828

CN (0,N0) [5]. Under FD sensing, the false alarm probability 829

for event h00 can be derived using the similar method as in [5], 830

which is given as 831

P00f = Q

[(ε

N0 + I (Psen)− 1

)√fsTS

], (27) 832

whereQ (x) =∫+∞

x exp(−t2/2

)dt; fs,N0, ε, I (Psen) are the 833

sampling frequency, the noise power, the detection threshold 834

and the self-interference, respectively; TS is the FD sensing 835

duration. 836

The detection probability for event h01 is given as 837

P01d = Q

(

εN0+I (Psen)

−TS−tTSγPS − 1

)√fsTS√

TS−tTS

(γPS + 1)2 + tTS

, (28) 838

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where t is the interval from the beginning of the data phase to839

the instant when the PU changes its state, γPS =Pp

N0+I (Psen)840

is the signal-to-interference-plus-noise ratio (SINR) of the841

PU’s signal at the SU.842

APPENDIX D843

PROOF OF THEOREM 1844

The first derivative of NT can be written as follows:845

∂NT∂TS

=1

Tove + T

3∑i=1

∂Bi∂TS

. (29)846

We derive the first derivative of Bi (i = 1, 2, 3) in the847

following. Toward this end, we will employ the approxima-848

tion of exp (x) ≈ 1 + x, x = Txτx, Tx ∈ {T ,TS ,T − TS},849

τx ∈ {τid, τac,1τ } where recall that 11τ=

1τac−

1τid. This850

approximation holds under the assumption that Tx << τx851

since we can omit all higher-power terms xn for n > 1852

from the Maclaurin series expansion of function exp (x).853

Using this approximation, we can express the first derivative854

of B1 as855

∂B1

∂TS=Ke exp

(T1τ

){log2 (1+ γS1)856

−ϕ

[(T − TS )

∂P00f

∂TS+

(1−P00

f

)]log2(1+ γD1)

},857

(30)858

where∂P00

f∂TS

is the first derivative of P00f whose derivation is859

given in Appendix E.860

Moreover, the first derivative of B2 can be written as861

∂B2

∂TS= Ke

1τ

τid862

×

{[exp

(T1τ

)−

(1+

T1τ

)exp

(TS1τ

)]log2 (1+ γS1)863

−ϕ∂P00

f

∂TS

[1τ

(exp

(TS1τ

)−exp

(T1τ

))log2

(1+γD11+γD2

)864

+ (T−TS)(exp

(T1τ

)log2(1+γD1)−exp

(TS1τ

)log2(1+γD2)

)]865

+ϕ(1− P00

f

) [−T − TS1τ

exp(TS1τ

)log2 (1+ γD2)866

−

(exp

(T1τ

)log2(1+γD1)−exp

(TS1τ

)log2(1+γD2)

)867

+ exp(TS1τ

)log2

(1+ γD11+ γD2

)]}. (31)868

Finally, the first derivative of B3 can be written as869

∂B3

∂TS=∂B31

∂TS+∂B32

∂TS, (32)870

where 871

∂B31

∂TS= Ke

1τ

τid

{1τ

[1+

(TS1τ− 1

)exp

(TS1τ

)]872

log2

(1+ γS11+ γS2

)+

(exp

(TS1τ

)− 1

)873

[TS log2(1+ γS2)+ ϕ(T − TS ) log2(1+ γD2)

] }. 874

(33) 875

To obtain the derivative for B32, we note that 1 ≤ 876

exp(

tτac

)≤ exp

(TSτac

)for ∀t ∈ [0,TS ]. Moreover, 877

from the results in (5) and (6) and using the definition 878

of T32 in (26), we have Pd

(1− exp

(−TSτid

))≤ T32 ≤ 879

Pd

(1− exp

(−TSτid

))exp

(TSτac

). Using these results, the first 880

derivative of B32 can be expressed as 881

∂B32

∂TS= −KePdϕ

T − 2TSτid

log2 (1+ γD2) . (34) 882

Therefore, we have obtained the first derivative ofNT and 883

we are ready to prove the first statement of Theorem 1. Sub- 884

stitute TS = 0 to the derived ∂NT∂TS

and use the approximation 885

exp (x) ≈ 1 + x, we yield the following result after some 886

manipulations 887

limTS→0

∂NT∂TS

= −K0K1 limTS→0

∂P00f

∂TS, (35) 888

where K0 =1

Tove+TKe and 889

K1 = ϕ

[T(1+

T1τ

)+T 2

τid

]log2 (1+ γD1) 890

+ϕT1ττid

log2 (1+ γD2) . (36) 891

It can be verified that K0 > 0, K1 > 0 and limTS→0

∂P00f

∂TS= 892

−∞ by using the derivations in Appendix E; hence, we have 893

limTS→0

∂NT∂TS= +∞ > 0. This completes the proof of the first 894

statement of the theorem. 895

We now present the proof for the second statement of 896

the theorem. Substitute TS = T to ∂NT∂TS

and utilize the 897

approximation exp (x) ≈ 1+ x, we yield 898

limTS→T

∂NT∂TS

=1

Tove + T

3∑i=1

∂Bi∂TS

(T ), (37) 899

where we have 900

∂B1

∂TS(T ) = Ke

(1+

T1τ

)901

×

[log2 (1+ γS1)− ϕ

(1− P00

f (T ))log2 (1+ γD1)

]902

(38) 903

∂B2

∂TS(T ) = −Ke

Tτid

log2 (1+ γS1) (39) 904

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∂B31

∂TS(T ) = Ke

Tτid

[log2(1+γS1)(1+γS2)−ϕ log2(1+γD2)]905

(40)906

∂B32

∂TS(T ) = Keϕ

Tτid

Pd log2 (1+ γD2) .907

Omit all high-power terms in the expansion of exp(x)908

(i.e., xn with n > 1) where x = Txτx, Tx ∈ {T ,TS ,T − TS},909

τx ∈ {τid, τac,1τ }, we yield910

limTS→T

∂NT∂TS

≈1

Tove + T∂B1

∂TS(T ). (41)911

We consider the HDTx and FDTx modes in the fol-912

lowing. For the HDTx mode, we have ϕ = 1 and913

θ = 0. Then, it can be verified that limTS→T

∂NT∂TS

< 0914

by using the results in (38) and (41). This is because915

we have log2 (1+ γS1) −(1− P00

f (T ))log2 (1+ γD1) ≈916

log2 (1+ γS1) − log2 (1+ γD1) < 0 (since we have917

P00f (T ) ≈ 0 and γS1 ≤ γD1).918

For the FDTxmode, we have ϕ = 2, θ = 1, and also γS1 =919

PsenN0

and γD1 =Pdat

N0+I (Pdat)=

Pdat

N0+ζPξ

dat. We would like to920

define a critical value of Psen which satisfies limTS→T

∂NT∂TS= 0921

to proceed further. Using the result in (38) and (41) as well as922

the approximationP00f (T ) ≈ 0, and by solving lim

TS→T∂NT∂TS=923

0 we yield924

Psen = N0

(1+ Pdat

N0 + ζPξ

dat

)2

− 1

. (42)925

Using (38), it can be verified that if Psen > Psen then926

limTS→T

∂NT∂TS

> 0; otherwise, we have limTS→T

∂NT∂TS≤ 0. So927

we have completed the proof for the second statement of928

Theorem 1.929

To prove the third statement of the theorem, we derive the930

second derivative of NT as931

∂2NT∂T 2

S

=1

Tove + T

3∑i=1

∂2Bi∂T 2

S

, (43)932

where we have933

∂2B1

∂T 2S

= −Keϕ exp(T1τ

)log2 (1+ γD1)934

×

[(T − TS)

∂2P00f

∂T 2S

− 2∂P00

f

∂TS

], (44)935

where∂2P00

f

∂T 2S

is the second derivative of P00f and according to936

the derivations in Appendix E, we have∂2P00

f

∂T 2S> 0,

∂P00f

∂TS< 0,937

∀TS . Therefore, we yield∂2B1∂T 2

S< 0 ∀TS .938

Consequently, we have the following upper bound for ∂2B1∂T 2

S939

by omitting the term exp( T1τ

)> 1 in (44)940

∂2B1

∂T 2S

≤ Ke [h1(TS )+ h2(TS )] , (45)941

where 942

h1(TS ) = −ϕ (T − TS)∂2P00

f

∂T 2S

log2 (1+ γD1), 943

h2(TS ) = 2ϕ∂P00

f

∂TSlog2 (1+ γD1). 944

Moreover, we have 945

∂2B2

∂T 2S

=Ke1τ

τid

{−2+ TS

1τ

1τexp

(TS1τ

)log2 (1+ γS1) 946

−ϕ∂2P00

f

∂T 2S

[1τ

(exp

(TS1τ

)−exp

(T1τ

))log2

(1+γD11+γD2

)947

+ (T − TS)(exp

(T1τ

)log2(1+ γD1) 948

− exp(TS1τ

)log2(1+ γD2)

)]949

+ 2ϕ∂P00

f

∂TS

[(exp

(T1τ

)− exp

(TS1τ

))log2(1+ γD1) 950

+T − TS1τ

exp(TS1τ

)log2 (1+ γD2)

]951

−ϕ(1− P00

f

) T − TS1τ

exp(TS1τ

)log2 (1+ γD2) 952

+ϕ(1− P00

f

) 11τ

exp(TS1τ

)log2

(1+ γD11+ γD2

)}. 953

(46) 954

Therefore, we can approximate ∂2B2∂T 2

Sas follows: 955

∂2B2

∂T 2S

= Ke [h3(TS )+ h4(TS )+ h5(TS )], (47) 956

where 957

h3(TS ) = −

(2+ TS

1τ

) (1+ TS

1τ

)τid

log2 (1+ γS1) , 958

h4(TS ) = −ϕ(1− P00

f

) T−TSτid

(1+

TS1τ

)log2(1+γD2) 959

−ϕ∂2P00

f

∂T 2S

(T−TS)[Tτid

log2(1+γD1)−TSτid

log2(1+γD2)]

960

+ 2ϕ∂P00

f

∂TS

T − TSτid

[log2

(1+γD11+γD2

)+TS1τ

log2(1+γD2)], 961

h5(TS ) = ϕ(1− P00

f

) 1τid

(1+

TS1τ

)log2

(1+ γD11+ γD2

). 962

In addition, we have 963

∂2B31

∂T 2S

=Ke

τidexp

(TS1τ

){(1+

T1τ

)log2 (1+ γS1) 964

+ log2 (1+ γS2)+ ϕ(T − TS1τ

− 2)log2 (1+ γD2)

}. 965

(48) 966

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We can approximate ∂2B31∂T 2

Sas follows:967

∂2B31

∂T 2S

= Ke [h6(TS )+ h6(TS )] , (49)968

where969

h6(TS ) =1τid

log2 (1+γS1) (1+γS2) ,970

h7(TS ) = −2ϕτid

log2 (1+γD2) . (50)971

Finally, we have972

∂2B32

∂T 2S

= Keh8(TS ), (51)973

where974

h8(TS ) =2ϕPdτid

log2 (1+γD2) . (52)975

The above analysis yields ∂2NT∂T 2

S= Ke

∑8i=1 hi(TS ). There-976

fore, to prove that ∂2NT∂T 2

S< 0, we should prove that h(TS ) < 0977

since Ke > 0 where978

h(TS ) =8∑i=1

hi(TS ). (53)979

It can be verified that h1(TS ) < 0 and h4(TS ) < 0, ∀TS980

because∂2P00

f

∂T 2S

> 0,∂P00

f∂TS

< 0 according to Appendix E981

and γD2 < γD1. Moreover, we have982

h3(TS ) < −2τid

log2 (1+ γS1) , (54)983

and because γS1 > γS2, we have984

h3(TS ) < −1τid

log2 (1+ γS1) (1+ γS2) = −h6(TS ). (55)985

Therefore, we have h3(TS )+h6(TS ) < 0. Furthermore, we can986

also obtain the following result h7(TS )+ h8(TS ) ≤ 0 because987

Pd ≤ 1. To complete the proof, we must prove that h2(TS )+988

h5(TS ) ≤ 0, which is equivalent to989

−2τid∂P00

f

∂TS

log2 (1+ γD1)

log2(1+γD11+γD2

) ≥ (1− P00f

)(1+

TS1τ

), (56)990

where according to Appendix E991

∂P00f

∂TS= −

γ√fsTS

2√2πTS

exp

(−

(α + γ

√fsTS

)22

), (57)992

where α = (γ1 + 1)Q−1(Pd). It can be verified that (56)993

indeed holds because the LHS of (56) is always larger than994

to 2 while the RHS of (56) is always less than 2. Hence, we995

have completed the proof of the third statement of Theorem 1.996

Finally, the fourth statement in the theorem obviously997

holds because Bi (i = 1, 2, 3) are all bounded from above.998

Hence, we have completed the proof of Theorem 1.999

APPENDIX E 1000

APPROXIMATION OF P00f AND ITS 1001

FIRST AND SECOND DERIVATIVES 1002

We can approximate Pd in (5) as follows: 1003

Pd = Q[(

ε

N0 + I− γ − 1

) √fsTS

γ1 + 1

], (58) 1004

where γ and γ1 are evaluated by a numerical method. Hence, 1005

P00f can be calculated as we set Pd = Pd , which is given as 1006

follows: 1007

P00f = Q

(α + γ

√fsTS

), (59) 1008

where α = (γ1 + 1)Q−1(Pd). 1009

We now derive the first derivative of P00f as 1010

∂P00f

∂TS= −

γ√fsTS

2√2πTS

exp

(−

(α + γ

√fsTS

)22

). (60) 1011

It can be seen that∂P00

f∂TS

< 0 since γ > 0. Moreover, the 1012

second derivative of P00f is 1013

∂2P00f

∂T 2S

=γ√fsTS

4√2πT 2

S

(1+

12yγ√fsTS

)exp

(−y2

2

), (61) 1014

where y = α + γ√fsTS . 1015

We can prove that∂2P00

f

∂T 2S> 0 by considering two different 1016

cases as follows. For the first case with α2

γ 2fs≤ TS ≤ T (0 ≤ 1017

P00f ≤ 0.5), this statement holds since y > 0. For the second 1018

case with 0 ≤ TS ≤ α2

γ 2fs(0.5 ≤ P00

f ≤ 1), y ≤ 0, then 1019

we have 0 < y − α = γ√fsTS ≤ −α and 0 ≤ −y ≤ −α. 1020

By applying the Cauchy-Schwarz inequality, we obtain 0 ≤ 1021

−y(y − α) ≤ γ 2

4 < 1 < 2; hence 1 + 12yγ√fsTS > 0. This 1022

result implies that∂2P00

f

∂T 2S> 0. 1023

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LE THANH TAN (S’11–M’15) received the 1131

B.Eng. and M.Eng. degrees from the Ho Chi Minh 1132

City University of Technology, in 2002 and 2004, 1133

respectively, and the Ph.D. degree from the Institut 1134

National de la Recherche Scientifique–Énergie 1135

Matériaux Télécommunications, Canada, in 2015. 1136

He was a Lecturer with the Ho Chi Minh City 1137

University of Technical Education from 2002 to 1138

2010. He is currently a Post-Doctoral Research 1139

Associate with the École Polytechnique de 1140

Montréal, Canada. His current research activities focus on Internet of Things 1141

(IoT over LTE/LTE-A network, cyber-physical systems, big data, distributed 1142

sensing, and control), time series analysis and dynamic factor models 1143

(stationary and non-stationary), wireless communications and networking, 1144

cloud-RAN, cognitive radios (software defined radio architectures, proto- 1145

col design, spectrum sensing, detection, and estimation), statistical signal 1146

processing, random matrix theory, compressed sensing, and compressed 1147

sampling. He has served on TPCs of different international conferences, 1148

including the IEEE CROWNCOM, VTC, and PIMRC. 1149

AQ:4

LONG BAO LE (S’04–M’07–SM’12) received 1150

the B.Eng. degree in electrical engineering from 1151

the Ho Chi Minh City University of Technology, 1152

Vietnam, in 1999, the M.Eng. degree in telecom- 1153

munications from the Asian Institute of Technol- 1154

ogy, Thailand, in 2002, and the Ph.D. degree in 1155

electrical engineering from the University of Man- 1156

itoba, Canada, in 2007. He was a Post-Doctoral 1157

Researcher with the Massachusetts Institute of 1158

Technology (2008–2010) and the University of 1159

Waterloo (2007–2008). Since 2010, he has been with the Institut National 1160

de la Recherche Scientifique, Université du Québec, Montréal, QC, Canada, 1161

where he is currently an Associate Professor. He has co-authored the 1162

book entitled Radio Resource Management in Multi-Tier Cellular Wireless 1163

Networks (Wiley, 2013). His current research interests include smart grids, 1164

cognitive radio, radio resource management, network control and optimiza- 1165

tion, and emerging enabling technologies for 5G wireless systems. He is 1166

a member of the Editorial Board of the IEEE TRANSACTIONS ON WIRELESS 1167

COMMUNICATIONS, the IEEE COMMUNICATIONS SURVEYS AND TUTORIALS, and 1168

the IEEE WIRELESS COMMUNICATIONS LETTERS. He has served as a Technical 1169

ProgramCommittee Chair/Co-Chair for several IEEE conferences, including 1170

the IEEE WCNC, the IEEE VTC, and the IEEE PIMRC. 1171

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