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

IEEEProof
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-Duplex
MAC Protocol for Cognitive Radio Networks
Considering Self-Interference
LE 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 (long.le@emt.inrs.ca)
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ABSTRACT In this paper, we propose an adaptive medium access control (MAC) protocol for
full-duplex (FD) cognitive radio networks in which FD secondary users (SUs) perform channel contention
followed by concurrent spectrum sensing and transmission, and transmission only with maximum power
in two different stages (called the FD sensing and transmission stages, respectively) in each contention
and access cycle. The proposed FD cognitive MAC (FDC-MAC) protocol does not require synchronization
among 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, where
both 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 spectrum
sensing duration and transmit power level in the FD sensing stage. We prove that there exists optimal sensing
time and transmit power to achieve the maximum throughput, and we develop an algorithm to configure
the proposed FDC-MAC protocol. Extensive numerical results are presented to illustrate the optimal
FDC-MAC configuration and the impacts of protocol parameters and the self-interference cancellation
quality on the throughput performance. Moreover, we demonstrate the significant throughput gains of the
FDC-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, optimal
sensing 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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1

IEEEProof
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 802.11 MAC 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
2 VOLUME 3, 2015

IEEEProof
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. Let H0 and H1 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+¯τac
and 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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IEEEProof
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
4 VOLUME 3, 2015

IEEEProof
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 HD MAC 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, we will 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 = Tcont + 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 Tcont 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 Tcont 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 synchronized with 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-MAC data phase. Note that the PU may 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 as B1. 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
VOLUME 3, 2015 5

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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 of Psen435
and TS only, i.e., NT (TS, Psen). Suppose that the PU requires436
that the average detection probability is at least Pd . Then, the437
throughput maximization problem can be stated as follows:438
max
TS ,Psen
NT (TS, Psen)439
s.t. ˆPd (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 ˆPd (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
power Psen 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., ˆPd (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
ˆPd =
TS
0
P01
d (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 Algorithm
1: for each considered value of Psen ∈ [0, Pmax] do
2: Find optimal TS for problem (7) using the bisection
method as TS (Psen) = argmax
0≤TS ≤T
NT (T, Psen).
3: end for
4: The final solution T∗
S , P∗
sen is determined as
T∗
S , P∗
sen = argmax
Psen,TS (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}
=
1
¯τid
exp(− t
¯τid
)
1 − exp(−TS
¯τid
)
. (6) 471
Note that P01
d (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
max
0<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 function NT(TS) of (7) satisfies 481
the following properties 482
1) lim
TS →0
∂NT
∂TS
= +∞, 483
2) a) For HDTx mode with ∀Psen and FDTx mode with 484
Psen < Psen, we have lim
TS →T
∂NT
∂TS
< 0, 485
b) For FDTx mode with Psen > Psen, we have 486
lim
TS →T
∂NT
∂TS
> 0, 487
3) ∂2NT
∂T2
S
< 0, ∀TS, 488
4) The objective function NT(TS) is bounded from above, 489
where Psen = N0 1 + Pdat
N0+ζP
ξ
dat
2
− 1 is the critical 490
value of Psen such that lim
TS →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
6 VOLUME 3, 2015

IEEEProof
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 =
Pp
N0
= −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 for
T = 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 for
T = 18 ms, p = 0.0022, ¯τid = 1000 ms, ¯τac = 100 ms, and varying ξ.
FIGURE 4. Normalized throughput versus SU transmit power Psen and
sensing 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 low with 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
VOLUME 3, 2015 7

IEEEProof
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
Now we 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
ξ = 0.95, ζ = 0.08 and FDTx with Pdat = 15 dB.
achieve the maximum throughput NT 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
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
8 VOLUME 3, 2015

IEEEProof
ξ = 0.95, ζ = 0.08 and HDTx.
by a star symbol. The maximum achieved throughput of the612
HDTx mode is lower than that in the FDTx mode 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 for
TS = 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 HD MAC 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 HD MAC 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 and HD MAC 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
VOLUME 3, 2015 9

IEEEProof
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 increasing max-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 Tcont, 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 + 2PD
Tcoll = 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
Ti
idle denote the i-th idle duration between two consecutive695
RTS/CTS exchanges, which can be collisions or successful696
exchanges. Then, Ti
idle 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
Tcont 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 + Ti
idle + T
Ncoll+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/Pidle where 716
Pidle = 1 − Pidle. Therefore, its pmf can be expressed as 717
f
Ncoll
X (x) =
Pcoll
Pidle
x
1 −
Pcoll
Pidle
, 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
Tidle
X (x) = (Pidle)x
(1 − Pidle), x = 0, 1, 2, . . . (14) 723
Therefore, Tcont (the average value of Tcont) can be written 724
as follows [17]: 725
Tcont = NcollTcoll + Tidle Ncoll + 1 + Tsucc, (15) 726
where Tidle and Ncoll can be calculated as 727
Tidle =
(1 − p)n0
1 − (1 − p)n0
(16) 728
Ncoll =
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 = Psen
N0
and γS2 = Psen
N0+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 = Pdat
N0+θI and γD2 = Pdat
N0+Pp+θI for I = ζP
ξ
dat 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
probability P
ij
d , which is the probability of correctly detecting 755
the PU given the PU is active, and the false alarm probabil- 756
ity P
ij
f , 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
10 VOLUME 3, 2015

IEEEProof
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) =
1
¯τx
exp(−
t
¯τx
). (18)765
Similarly, we employ T
ij
S and T
ij
D 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+T
T00
1 fτid
(t)dt771
= P (H0) T00
1 exp −
Tove + T
¯τid
, (19)772
where P (H0) denotes the probability of the idle state of773
the PU, and P00
f is the false alarm probability for event774
h00 given in Appendix C. Moreover, T00
1 = P00
f T00
S +775
(1 − P00
f )(T00
S + T00
D ), T00
S = TS log2 (1 + γS1), T00
D =776
ϕ (T − TS) log2 (1 + γD1) where T00
S and T00
D denote the777
number of bits transmitted (over one Hz 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
T
τ
TS log2(1 + γS1)+ ϕ 1 − P00
f781
(T − TS) log2(1 + γD1) , (20)782
where Ke = P (H0) exp − Tove
¯τid
+ T
¯τac
and 1
τ =783
1
¯τac
− 1
¯τid
.784
Moreover, we can calculate B2 as785
B2 = P (H0)
Tove+T
t1=Tove+TS
∞
t2=Tove+T−t1
786
T01
2 (t1)fτid
(t1)fτac (t2)dt1dt2, (21)787
where T01
2 (t1) = P00
f T00
S + (1 − P00
f )(T00
S + T01
D
¯t1 ),788
T01
D (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, T00
S and T01
D denote the amount of data793
transmitted in the FD sensing and transmission stages for this794
case, respectively. After some manipulations, we achieve795
B2 = Ke
τ
¯τid
exp
T
τ
− exp
TS
τ
796
× TS log2 (1+γS1)−ϕ τ 1−P00
f log2
1+γD1
1+γD2
797
+ ϕ (T − TS) 1−P00
f798
× exp
T
τ
log2(1+γD1)−exp
TS
τ
log2(1+γD2) .799
(22)800
Finally, we can express B3 as follows: 801
B3 = P (H0)
Tove+TS
t1=Tove
∞
t2=Tove+T−t1
802
× P01
d ¯t1 T01
S ¯t1 + (1−P01
d ¯t1 )(T01
S ¯t1 + T11
D ) 803
fτid
(t1)fτac (t2)dt1dt2, (23) 804
where ¯t1 = t1 − Tove, T01
S
¯t1 = ¯t1 log2 (1 + γS1) + 805
TS − ¯t1 log2 (1 + γS2), T11
D = ϕ (T − TS) log2 (1 + γD2), 806
and t1 is the same as in (21). Here, T01
S and T11
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
T01
S (t) + T11
D − P01
d (t) T11
D 811
fτid
(t) exp
t
¯τac
dt = B31 + B32, (24) 812
where 813
B31 = Ke
TS
t=0
T01
S (t) + T11
D fτid
(t) exp
t
¯τac
dt 814
= Ke
τ
¯τid
τ
TS
τ
−1 exp
TS
τ
+1 log2
1+γS1
1+γS2
815
+ exp
TS
τ
− 1 T11
D + TS log2 (1 + γS2) , 816
(25) 817
and 818
B32 = −KeT11
D
¯T32, (26) 819
where ¯T32 =
TS
t=0 P01
d (t) fτid
(t) exp t
¯τac
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
P00
f = Q
N0 + I(Psen)
− 1 fsTS , (27) 832
where Q (x) =
+∞
x exp −t2/2 dt; fs, N0, , 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
P01
d = Q


N0+I(Psen) − TS −t
TS
γPS − 1
√
fsTS
TS −t
TS
(γPS + 1)2
+ t
TS

, (28) 838
VOLUME 3, 2015 11

IEEEProof
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 = Tx
τx
, Tx ∈ {T, TS, T − TS},849
τx ∈ {¯τid, ¯τac, τ} where recall that 1
τ = 1
¯τac
− 1
¯τid
. 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
T
τ
log2 (1 + γS1)856
−ϕ (T − TS)
∂P00
f
∂TS
+ 1−P00
f log2(1 + γD1) ,857
(30)858
where
∂P00
f
∂TS
is the first derivative of P00
f whose derivation is859
given in Appendix E.860
Moreover, the first derivative of B2 can be written as861
∂B2
∂TS
= Ke
τ
¯τid
862
× exp
T
τ
− 1 +
T
τ
exp
TS
τ
log2 (1 + γS1)863
− ϕ
∂P00
f
∂TS
τ exp
TS
τ
−exp
T
τ
log2
1+γD1
1+γD2
864
+ (T−TS) exp
T
τ
log2(1+γD1)−exp
TS
τ
log2(1+γD2)865
+ ϕ 1 − P00
f −
T − TS
τ
exp
TS
τ
log2 (1 + γD2)866
− exp
T
τ
log2(1+γD1)−exp
TS
τ
log2(1+γD2)867
+ exp
TS
τ
log2
1 + γD1
1 + 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
τ
¯τid
τ 1 +
TS
τ
− 1 exp
TS
τ
872
log2
1 + γS1
1 + γS2
+ exp
TS
τ
− 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 t
¯τac
≤ exp TS
¯τac
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 −TS
¯τid
≤ ¯T32 ≤ 879
Pd 1 − exp −TS
¯τid
exp TS
¯τac
. Using these results, the first 880
derivative of B32 can be expressed as 881
∂B32
∂TS
= −KePd ϕ
T − 2TS
¯τid
log2 (1 + γD2) . (34) 882
Therefore, we have obtained the first derivative of NT 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
lim
TS →0
∂NT
∂TS
= −K0K1 lim
TS →0
∂P00
f
∂TS
, (35) 888
where K0 = 1
Tove+T Ke and 889
K1 = ϕ T 1 +
T
τ
+
T2
¯τid
log2 (1 + γD1) 890
+ ϕ
T τ
¯τid
log2 (1 + γD2) . (36) 891
It can be verified that K0 > 0, K1 > 0 and lim
TS →0
∂P00
f
∂TS
= 892
−∞ by using the derivations in Appendix E; hence, we have 893
lim
TS →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
lim
TS →T
∂NT
∂TS
=
1
Tove + T
3
i=1
∂Bi
∂TS
(T), (37) 899
where we have 900
∂B1
∂TS
(T) = Ke 1 +
T
τ
901
× log2 (1 + γS1) − ϕ 1 − P00
f (T) log2 (1 + γD1) 902
(38) 903
∂B2
∂TS
(T) = −Ke
T
¯τid
log2 (1 + γS1) (39) 904
12 VOLUME 3, 2015

IEEEProof
∂B31
∂TS
(T) = Ke
T
¯τid
[log2(1+γS1)(1+γS2)−ϕ log2(1+γD2)]905
(40)906
∂B32
∂TS
(T) = Keϕ
T
¯τid
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 = Tx
τx
, Tx ∈ {T, TS, T − TS},909
τx ∈ {¯τid, ¯τac, τ}, we yield910
lim
TS →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 lim
TS →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
P00
f (T) ≈ 0 and γS1 ≤ γD1).918
For the FDTx mode, we have ϕ = 2, θ = 1, and also γS1 =919
Psen
N0
and γD1 = Pdat
N0+I(Pdat) = Pdat
N0+ζP
ξ
dat
. We would like to920
define a critical value of Psen which satisfies lim
TS →T
∂NT
∂TS
= 0921
to proceed further. Using the result in (38) and (41) as well as922
the approximation P00
f (T) ≈ 0, and by solving lim
TS →T
∂NT
∂TS
=923
0 we yield924
Psen = N0

 1 +
Pdat
N0 + ζP
ξ
dat
2
− 1

. (42)925
Using (38), it can be verified that if Psen > Psen then926
lim
TS →T
∂NT
∂TS
> 0; otherwise, we have lim
TS →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
∂T2
S
=
1
Tove + T
3
i=1
∂2Bi
∂T2
S
, (43)932
where we have933
∂2B1
∂T2
S
= −Keϕ exp
T
τ
log2 (1 + γD1)934
× (T − TS)
∂2P00
f
∂T2
S
− 2
∂P00
f
∂TS
, (44)935
where
∂2P00
f
∂T2
S
is the second derivative of P00
f and according to936
the derivations in Appendix E, we have
∂2P00
f
∂T2
S
> 0,
∂P00
f
∂TS
< 0,937
∀TS. Therefore, we yield ∂2B1
∂T2
S
< 0 ∀TS.938
Consequently, we have the following upper bound for ∂2B1
∂T2
S
939
by omitting the term exp T
τ > 1 in (44)940
∂2B1
∂T2
S
≤ Ke [h1(TS) + h2(TS)] , (45)941
where 942
h1(TS) = −ϕ (T − TS)
∂2P00
f
∂T2
S
log2 (1 + γD1), 943
h2(TS) = 2ϕ
∂P00
f
∂TS
log2 (1 + γD1). 944
Moreover, we have 945
∂2B2
∂T2
S
=
Ke τ
¯τid
−
2 + TS
τ
τ
exp
TS
τ
log2 (1 + γS1) 946
− ϕ
∂2P00
f
∂T2
S
τ exp
TS
τ
−exp
T
τ
log2
1+γD1
1+γD2
947
+ (T − TS) exp
T
τ
log2(1 + γD1) 948
− exp
TS
τ
log2(1 + γD2) 949
+ 2ϕ
∂P00
f
∂TS
exp
T
τ
− exp
TS
τ
log2(1 + γD1) 950
+
T − TS
τ
exp
TS
τ
log2 (1 + γD2) 951
− ϕ 1 − P00
f
T − TS
τ
exp
TS
τ
log2 (1 + γD2) 952
+ ϕ 1 − P00
f
1
τ
exp
TS
τ
log2
1 + γD1
1 + γD2
. 953
(46) 954
Therefore, we can approximate ∂2B2
∂T2
S
as follows: 955
∂2B2
∂T2
S
= Ke [h3(TS) + h4(TS) + h5(TS)], (47) 956
where 957
h3(TS) = −
2 + TS
τ 1 + TS
τ
¯τid
log2 (1 + γS1) , 958
h4(TS) = −ϕ 1 − P00
f
T−TS
¯τid
1 +
TS
τ
log2(1+γD2) 959
− ϕ
∂2P00
f
∂T2
S
(T −TS)
T
¯τid
log2(1 +γD1)−
TS
¯τid
log2(1 +γD2) 960
+ 2ϕ
∂P00
f
∂TS
T − TS
¯τid
log2
1+γD1
1+γD2
+
TS
τ
log2(1+γD2) , 961
h5(TS) = ϕ 1 − P00
f
1
¯τid
1 +
TS
τ
log2
1 + γD1
1 + γD2
. 962
In addition, we have 963
∂2B31
∂T2
S
=
Ke
¯τid
exp
TS
τ
1 +
T
τ
log2 (1 + γS1) 964
+ log2 (1 + γS2) + ϕ
T − TS
τ
− 2 log2 (1 + γD2) . 965
(48) 966
VOLUME 3, 2015 13

IEEEProof
We can approximate ∂2B31
∂T2
S
as follows:967
∂2B31
∂T2
S
= Ke [h6(TS) + h6(TS)] , (49)968
where969
h6(TS) =
1
¯τid
log2 (1+γS1) (1+γS2) ,970
h7(TS) = −
2ϕ
¯τid
log2 (1+γD2) . (50)971
Finally, we have972
∂2B32
∂T2
S
= Keh8(TS), (51)973
where974
h8(TS) =
2ϕ ¯Pd
¯τid
log2 (1+γD2) . (52)975
The above analysis yields ∂2NT
∂T2
S
= Ke
8
i=1 hi(TS). There-976
fore, to prove that ∂2NT
∂T2
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
∂T2
S
> 0,
∂P00
f
∂TS
< 0 according to Appendix E981
and γD2 < γD1. Moreover, we have982
h3(TS) < −
2
¯τid
log2 (1 + γS1) , (54)983
and because γS1 > γS2, we have984
h3(TS) < −
1
¯τid
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
−2¯τid
∂P00
f
∂TS
log2 (1 + γD1)
log2
1+γD1
1+γD2
≥ 1 − P00
f 1 +
TS
τ
, (56)990
where according to Appendix E991
∂P00
f
∂TS
= −
¯γ
√
fsTS
2
√
2πTS
exp −
¯α + ¯γ
√
fsTS
2
2
, (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 P00
f
AND ITS 1001
FIRST AND SECOND DERIVATIVES 1002
We can approximate ˆPd in (5) as follows: 1003
ˆPd = Q
N0 + I
− ¯γ − 1
√
fsTS
¯γ1 + 1
, (58) 1004
where ¯γ and ¯γ1 are evaluated by a numerical method. Hence, 1005
P00
f can be calculated as we set ˆPd = Pd , which is given as 1006
follows: 1007
P00
f = Q ¯α + ¯γ fsTS , (59) 1008
where ¯α = ( ¯γ1 + 1) Q−1 Pd . 1009
We now derive the first derivative of P00
f as 1010
∂P00
f
∂TS
= −
¯γ
√
fsTS
2
√
2πTS
exp −
¯α + ¯γ
√
fsTS
2
2
. (60) 1011
It can be seen that
∂P00
f
∂TS
< 0 since ¯γ > 0. Moreover, the 1012
second derivative of P00
f is 1013
∂2P00
f
∂T2
S
=
¯γ
√
fsTS
4
√
2πT2
S
1 +
1
2
y ¯γ fsTS exp −
y2
2
, (61) 1014
where y = ¯α + ¯γ
√
fsTS. 1015
We can prove that
∂2P00
f
∂T2
S
> 0 by considering two different 1016
cases as follows. For the first case with ¯α2
¯γ 2fs
≤ TS ≤ T (0 ≤ 1017
P00
f ≤ 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 + 1
2 y ¯γ
√
fsTS > 0. This 1022
result implies that
∂2P00
f
∂T2
S
> 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
Program Committee Chair/Co-Chair for several IEEE conferences, including 1170
the IEEE WCNC, the IEEE VTC, and the IEEE PIMRC. 1171
1172
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Design and Optimal Configuration of Full-Duplex MAC Protocol for Cognitive Radio Networks Considering Self-Interference

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