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  IEEE ACCESS 1 Designand Optimal Configuration of Full–Duplex MAC Protocol for Cognitive Radio Networks Considering Self-Interference Le Thanh Tan, Member, IEEE, Long Bao Le, Senior Member, IEEE 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 syn- chronization among SUs and it efficiently utilizes the spectrum and mitigates the self-interference in the FD transceiver. We then develop a mathematical model to analyze the throughput performance of the FDC-MAC protocol where both half-duplex (HD) transmission (HDTx) and FD transmission (FDTx) 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 where 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 il- lustrate the characteristic of 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 existing half-duplex MAC (HD MAC) and single-stage FD MAC protocols. Index Terms—General asynchronous MAC, full-duplex MAC, full-duplex spectrum sensing, optimal sensing duration, through- put maximization, self-interference control, full-duplex cognitive radios, throughput analysis. I. INTRODUCTION Engineering MAC protocols for efficient sharing of white spaces is an important research topic in cognitive radio net- works (CRNs). One critical requirement for the cognitive MAC design is that transmissions on the licensed frequency band from primary users (PUs) should be satisfactorily pro- tected from the SUs’ spectrum access. Therefore, a cognitive MAC protocol for the secondary network must realize both the spectrum sensing and access functions so that timely detection of the PUs’ communications and effective spectrum sharing among SUs can be achieved. Most existing research works Manuscript received November 08, 2015; accepted December 10, 2015. The editor coordinating the review of this paper and approving it for publication is Dr. Wei Wang. The authors are with the Institut National de la Recherche Scientifique– ´Energie, Mat´eriaux et T´el´ecommunications, Universit´e du Qubec, Montr´eal, QC J3X 1S2, Canada (e-mail: [email protected]; [email protected]) on cognitive MAC protocols have focused on the design and analysis of HD MAC (e.g., see [1]–[4] and the references therein). Due to the HD constraint, SUs typically employ a two- stage sensing/access procedure where they perform spectrum sensing in the first stage before accessing available spectrum for data transmission in the second stage [5]–[11]. This con- straint also requires SUs be synchronized during the spectrum sensing stage, which could be difficult to achieve in practice. In fact, spectrum sensing enables SUs to detect white spaces that are not occupied by PUs [2]–[8], [12], [13]; therefore, imperfect spectrum sensing can reduce the spectrum utilization due to failure in detecting white spaces and potentially result in collisions with active PUs. Consequently, sophisticated design and parameter configuration of cognitive MAC protocols must be conducted to achieve good performance while appropriately protecting PUs [1], [6]–[11], [14]. As a result, traditional MAC protocols [15]–[19] adapted to the CRN may not provide satisfactory performance. In general, HD MAC protocols may not exploit white spaces very efficiently since significant sensing time may be required, which would otherwise be utilized for data transmission. Moreover, SUs may not timely detect the PUs’ activity during their transmissions, which can cause severe interference to active PUs. Thanks to recent advances on FD technologies, a FD radio can transmit and receive data simultaneously on the same frequency band [20]–[25]. One of the most critical issues of wireless FD communication is the presence of self- interference, which is caused by power leakage from the trans- mitter to the receiver of a FD transceiver. The self-interference may indeed lead to serious communication performance degra- dation of FD wireless systems. Despite recent advances on self-interference cancellation (SIC) techniques [21]–[23] (e.g., propagation SIC, analog-circuit SIC, and digital baseband SIC), self-interference still exists due to various reasons such as the limitation of hardware and channel estimation errors. A. Related Works There are some recent works that propose to exploit the FD communications for MAC-level channel access in multi-user wireless networks [25]–[29]. In [25], the authors develop a centralized MAC protocol to support asymmetric data traffic where network nodes may transmit data packets of different
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  IEEE ACCESS 2 lengths,and they propose to mitigate the hidden node problem by employing a busy tone. To overcome this hidden node prob- lem, Duarte et al. propose to adapt the standard 802.11 MAC protocol with the RTS/CTS handshake in [26]. Moreover, Goyal et al. in [27] extend this study to consider interference between two nodes due to their concurrent transmissions. Dif- ferent from conventional wireless networks, designing MAC protocols in CRNs is more challenging because the spectrum sensing function must be efficiently integrated into the MAC protocol. In addition, the self-interference must be carefully addressed in the simultaneous spectrum sensing and access to mitigate its negative impacts on the sensing and throughput performance. The FD technology has been employed for more efficient spectrum access design in cognitive radio networks [30]–[33] where SUs can perform sensing and transmission simultane- ously. In [30], a FD MAC protocol is developed which allows simultaneous spectrum access of the SU and PU networks where both PUs and SUs are assumed to employ the p- persistent MAC protocol for channel contention resolution and access. This design is, therefore, not applicable to the hierarchical spectrum access in the CRNs where PUs should have higher spectrum access priority compared to SUs. In our previous work [31], we propose the FD MAC protocol by using the standard backoff mechanism as in the 802.11 MAC protocol where we employ concurrent FD sensing and access during data transmission as well as frame fragmentation. Moreover, engineering of a cognitive FD relay- ing network is considered in [32], [33], where various resource allocation algorithms to improve the outage probability are proposed. In addition, the authors in [28] develop the joint routing and distributed resource allocation for FD wireless networks. In [29], Choi et al. study the distributed power allocation for a hybrid FD/HD system where all network nodes operate in the HD mode but the access point (AP) communicates by using the FD mode. In practice, it would be desirable to design an adaptable MAC protocol, which can be configured to operate in an optimal fashion depending on specific channel and network conditions. This design will be pursued in our current work. B. Our Contributions In this paper, we make a further bold step in designing, analyzing, and optimizing an adaptive FDC–MAC protocol for CRNs, where the self-interference and imperfect spectrum sensing are explicitly considered. In particular, the contribu- tions of this paper can be summarized as follows. 1) We propose a novel FDC–MAC protocol that can effi- ciently exploit the FD transceiver for spectrum spectrum sensing and access of the white space without requiring synchronization among SUs. In this protocol, after the p-persistent based channel contention phase, the winning SU enters the data phase consisting of two stages, i.e., concurrent sensing and transmission in the first stage (called FD sensing stage) and transmission only in the second stage (called transmission stage). The developed FDC–MAC protocol, therefore, enables the optimized configuration of transmit power level and sensing time during the FD sensing stage to mitigate the self-interference and appropriately protect the active PU. After the FD sensing stage, the SU can transmit with the maximum power to achieve the highest throughput. 2) We develop a mathematical model for throughput per- formance analysis of the proposed FDC-MAC proto- col considering the imperfect sensing, self-interference effects, and the dynamic status changes of the PU. In addition, both one-way and two-way transmission scenarios, which are called HD transmission (HDTx) and FD transmission (FDTx) modes, respectively, are considered in the analysis. Since the PU can change its idle/active status during the FD sensing and transmission stages, different potential status-change scenarios are studied in the analytical model. 3) We study the optimal configuration of FDC-MAC proto- col parameters including the SU’s sensing duration and transmit power to maximize the achievable throughput under both FDTx and HDTx modes. We prove that there exists an optimal sensing time to achieve the maximum throughput for a given transmit power value during the FD sensing stage under both FDTx and HDTx modes. Therefore, optimal protocol parameters can be determined through standard numerical search methods. 4) Extensive numerical results are presented to illustrate the impacts of different protocol parameters on the throughput performance and the optimal configurations of the proposed FDC-MAC protocol. Moreover, we show the significant throughput enhancement of the pro- posed FDC-MAC protocol compared to existing cogni- tive MAC protocols, namely the HD MAC protocol and a single-stage FD MAC protocol with concurrent sensing and access during the whole data phase. Specifically, our FDC-MAC protocol achieves higher throughput with the increasing maximum power while the throughput of the single-stage FD MAC protocol decreases with the maximum power in the high power regime due to the self-interference. Moreover, the proposed FDC- MAC protocol significantly outperforms the HD MAC protocol in terms of system throughput. The remaining of this paper is organized as follows. Section II describes the system and PU models. FDC–MAC protocol design, and throughput analysis for the proposed FDC–MAC protocol are performed in Section III. Then, Section IV studies the optimal configuration of the proposed FDC–MAC protocol to achieve the maximum secondary throughput. Sec- tion V demonstrates numerical results followed by concluding remarks in Section VI.
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  IEEE ACCESS 3 II.SYSTEM AND PU ACTIVITY MODELS A. System Model We consider a cognitive radio network where n0 pairs of SUs opportunistically exploit white spaces on one channel for communications. We assume that each SU is equipped with a FD transceiver; hence, the SUs can perform sensing and trans- mission simultaneously. However, the sensing performance of each SU is affected by the self-interference from its transmitter since the transmitted power is leaked into the received signal. We denote I(P) as the average self-interference power, which is modeled as I(P) = ζ (P) ξ [20] where P is the SU’s transmit power, ζ and ξ (0 ≤ ξ ≤ 1) are predetermined coefficients which represent the quality of self-interference cancellation (QSIC). In this work, we design a asynchronous cognitive MAC protocol where no synchronization is required among SUs and between SUs and the PU. We assume that different pairs of SUs can overhear transmissions from the others (i.e., a collocated network is assumed). In the following, we refer to pair i of SUs as SU i for brevity. B. Primary User Activity We assume that the PU’s idle/active status follows two independent random processes. We say that the channel is available and busy for SUs’ access if the PU is in the idle and active (or busy) states, respectively. Let H0 and H1 denote the events that the PU is idle and active, respectively. To protect the PU, we assume that SUs must stop their transmissions and evacuate from the busy channel within the maximum delay of Teva, which is referred to as channel evacuation time. Let τac and τid denote the random variables which represent the durations of active and idle channel states, respectively. We denote probability density functions (pdf) of τac and τid as fτac (t) and fτid (t), respectively. While most results in this paper can be applied to general pdfs fτac (t) and fτid (t), we mostly consider the exponential pdf in the analysis. In addition, let P (H0) = ¯τid ¯τid+¯τac and P (H1) = 1 − P (H0) present the probabilities that the channel is available and busy, respectively where ¯τid and ¯τac denote the average values of τac and τid, respectively. We assume that the probabilities that τac and τid are smaller than Teva are sufficiently small (i.e., the PU changes its status slowly) so that we can ignore events with multiple idle/active status changes in one channel evacuation interval Teva. III. FULL-DUPLEX COGNITIVE MAC PROTOCOL In this section, we describe the proposed FDC-MAC proto- col and conduct its throughput analysis considering imperfect sensing, self-interference of the FD transceiver, and dynamic status change of the PUs. A. FDC-MAC Protocol Design The proposed FDC-MAC protocol integrates three impor- tant elements of a cognitive MAC protocol, namely contention resolution, spectrum sensing, and access functions. Specifi- cally, SUs employ the p-persistent CSMA principle [17] for contention resolution where each SU with data to transmit attempts to capture an available channel with a probability p after the channel is sensed to be idle during the standard DIFS interval (DCF Interframe Space). If a particular SU decides not to transmit (with probability of 1 − p), it will sense the channel and attempt to transmit again in the next slot of length σ with probability p. To complete the reserva- tion, the four-way handshake with Request-to-Send/Clear-to- Send (RTS/CST) exchanges [16] is employed to reserve the available channel for transmission. Specifically, the secondary transmitter sends RTS to the secondary receiver and waits until it successfully receives the CTS from the secondary receiver. All other SUs, which hear the RTS and CTS exchange from the winning SU, defer to access the channel for a duration equal to the data transmission time, T. Then, an acknowledgment (ACK) from the SU’s receiver is transmitted to its correspond- ing transmitter to notify the successful reception of a packet. Furthermore, the standard small interval, namely SIFS (Short Interframe Space), is used before the transmissions of CTS, ACK, and data frame as in the standard 802.11 MAC protocol [16]. In our design, the data phase after the channel contention phase comprises two stages where the SU performs concurrent sensing and transmission in the first stage with duration TS and transmission only in the second stage with duration T − TS. Here, the SU exploits the FD capability of its transceiver to realize concurrent sensing and transmission the first stage (called FD sensing stage) where the sensing outcome at the end of this stage (i.e., an idle or busy channel status) determines its further actions as follows. Specifically, if the sensing outcome indicates an available channel then the SU transmits data in the second stage; otherwise, it remains silent for the remaining time of the data phase with duration T − TS. We assume that the duration of the SU’s data phase T is smaller than the channel evacuation time Teva so timely evacuation from the busy channel can be realized with reliable FD spectrum sensing. Therefore, our design allows to protect the PU with evacuation delay at most T if the MAC carrier sensing during the contention phase and the FD spectrum sensing in the data phase are perfect. Furthermore, we as- sume that the SU transmits at power levels Psen ≤ Pmax and Pdat = Pmax during the FD sensing and transmission stages, respectively where Pmax denotes the maximum power and the transmit power Psen in the FD sensing stage will be optimized to effectively mitigate the self-interference and achieve good sensing-throughput tradeoff. The timing diagram of the proposed FDC–MAC protocol is illustrated in Fig. 1. We allow two possible operation modes in the transmission stage. The first is the HD transmission mode (HDTx mode) where there is only one direction of data transmission from the SU transmitter to the SU receiver. In this mode, there is no self-interference in the transmission stage. The second is the FD transmission mode (FDTx mode) where two-way
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  IEEE ACCESS 4 DATAtime CTS RTS/CTS exchange DIFS RTS SIFS SIFS DATA SIFS ACK Data Transmission DATA 1 DATA 2 FD Channel is available DATA 1 FD Channel is not available I C . . . C I U Contention and Access cycle . . . DATA RTSDIFS Collision . . . Collision (C) Idle (I) Successful channel reservation (U) Contention and Access cycle evaT TevaT T oveT PU activity 1t Data phaseContention phase PU activity 1t PU activity 00h 00h 01h 11h 00h 01h ST TST T Sensing stage Tx stage Sensing stage Tx stage Case 1
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  00 00,h h Case2
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  00 01,h h Case3
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  01 11,h h Fig.1. Timing diagram of the proposed full-duplex cognitive MAC protocol. communications between the pair of SUs are assumed (i.e., there are two data flows between the two SU nodes in opposite directions). In this mode, the achieved throughput can be potentially enhanced (at most doubling the throughput of the HDTx mode) but self-interference must be taken into account in throughput quantification. Our proposed FDC–MAC protocol design indeed enables flexible and adaptive configuration, which can efficiently ex- ploit the capability of the FD transceiver. Specifically, if the duration of the FD sensing stage is set equal to the duration of the whole data phase (i.e., TS = T), then the SU performs concurrent sensing and transmission for the whole data phase as in our previous design [31]. This configuration may degrade the achievable throughput since the transmit power during the FD sensing stage is typically set smaller Pmax to mitigate the self-interference and achieve the required sensing performance. We will refer the corresponding MAC protocol with TS = T as one-stage FD MAC in the sequel. Moreover, if we set the SU transmit power Psen in the sensing stage equal to zero, i.e., Psen = 0, then we achieve the traditional two-stage cognitive HD MAC protocol where sensing and transmission are performed sequentially in two different stages [6], [8]. Moreover, the proposed FDC–MAC protocol is more flexible than existing designs [31], [6], [8] since different existing designs can be achieved through suitable configuration of its protocol parameters. It will be demonstrated later that the proposed FDC–MAC protocol achieves significant better throughput than that of the existing cognitive MAC protocols. In the following, we present the throughput analysis based on which the protocol configuration optimization can be performed. B. Throughput Analysis We now conduct the saturation throughput analysis for the secondary network where all SUs are assumed to always have data to transmit. The resulting throughput can be served as an upper bound for the throughput in the non-saturated scenario [16]. This analysis is performed by studying one specific contention and access cycle (CA cycle) with the contention phase and data phase as shown in Fig. 1. Without loss of generality, we will consider the normalized throughput achieved per one unit of system bandwidth (in bits/s/Hz). Specifically, the normalized throughput of the FDC–MAC
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  IEEE ACCESS 5 protocolcan be expressed as NT = B Tove + T , (1) where Tove represents the time overhead required for one successful channel reservation (i.e., successful RTS/CTS ex- changes), T denotes the packet transmission time, and B denotes the amount of data (bits) transmitted in one CA cycle per one unit of system bandwidth, which is expressed in bits/Hz. To complete the throughput analysis, we derive the quantities Tove and B in the remaining of this subsection. 1) Derivation of Tove: The average time overhead for one successful channel reservation can be calculated as Tove = Tcont + 2SIFS + 2PD + ACK, (2) where ACK is the length of an ACK message, SIFS is the length of a short interframe space, and PD is the propagation delay where PD is usually small compared to the slot size σ, and Tcont denotes the average time overhead due to idle periods, collisions, and successful transmissions of RTS/CTS messages in one CA cycle. For better presentation of the paper, the derivation of Tcont is given in Appendix A. 2) Derivation of B: To calculate B, we consider all possible cases that capture the activities of SUs and status changes of the PU in the FDC-MAC data phase of duration T. Because the PU’s activity is not synchronized with the SU’s transmission, the PU can change its idle/active status any time. We assume that there can be at most one transition between the idle and active states of the PU during one data phase interval. This is consistent with the assumption on the slow status changes of the PU as described in Section II-B since T Teva. Furthermore, we assume that the carrier sensing of the FDC- MAC protocol is perfect; therefore, the PU is idle at the beginning of the FDC-MAC data phase. Note that the PU may change its status during the SU’s FD sensing or transmission stage, which requires us to consider different possible events in the data phase. We use hij (i, j ∈ {0, 1}) to represent events captur- ing status changes of the PU in the FD sensing stage and transmission stage where i = 0 and i = 1 represent the idle and active states of the PU, respectively. For example, if the PU is idle during the FD sensing stage and becomes active during the transmission stage, then we represent this event as (h00, h01) where sub-events h00 and h01 represent the status changes in the FD sensing and transmission stages, respectively. Moreover, if the PU changes from the idle to the active state during the FD sensing stage and remains active in the remaining of the data phase, then we represent this event as (h01, h11) It can be verified that we must consider the following three cases with the corresponding status changes of the PU during the FDC-MAC data phase to analyze B. • Case 1: The PU is idle for the whole FDC-MAC data phase (i.e., there is no PU’s signal in both FD sens- ing and transmission stages) and we denote this event as (h00, h00). The average number of bits (in bits/Hz) transmitted during the data phase in this case is denoted as B1. • Case 2: The PU is idle during the FD sensing stage but the PU changes from the idle to the active status in the transmission stage. We denote the event corresponding to this case as (h00, h01) where h00 and h01 capture the sub- events in the FD sensing and transmission stages, respec- tively. The average number of bits (in bits/Hz) transmitted during the data phase in this case is represented by B2. • Case 3: The PU is first idle then becomes active during the FD sensing stage and it remains active during the whole transmission stage. Similarly we denote this event as (h01, h11) and the average number of bits (in bits/Hz) transmitted during the data phase in this case is denoted as B3. Then, we can calculate B as follows: B = B1 + B2 + B3. (3) To complete the analysis, we will need to derive B1, B2, and B3, which are given in Appendix B. IV. FDC–MAC PROTOCOL CONFIGURATION FOR THROUGHPUT MAXIMIZATION In this section, we study the optimal configuration of the proposed FDC–MAC protocol to achieve the maximum throughput while satisfactorily protecting the PU. A. Problem Formulation Let NT (TS, p, Psen) denote the normalized secondary throughput, which is the function of the sensing time TS, trans- mission probability p, and the SU’s transmit power Psen in the FD sensing stage. In the following, we assume a fixed frame length T, which is set smaller the required evacuation time Teva to achieve timely evacuation from a busy channel for the SUs. We are interested in determining suitable configuration for p, TS and Psen to maximize the secondary throughput, NT (TS, p, Psen). In general, the optimal transmission prob- ability p should balance between reducing collisions among SUs and limiting the protocol overhead. However, the achieved throughput is less sensitive to the transmission probability p as will be demonstrated later via the numerical study. Therefore, we will seek to optimize the throughput over Psen and TS for a reasonable and fixed value of p. For brevity, we express the throughput as a function of Psen and TS only, i.e., NT (TS, Psen). Suppose that the PU requires that the average detection probability is at least Pd. Then, the throughput maximization problem can be stated as follows: max TS ,Psen NT (TS, Psen) s.t. ˆPd (ε, TS) ≥ Pd, 0 ≤ Psen ≤ Pmax, 0 ≤ TS ≤ T, (4) where Pmax is the maximum power for SUs, and TS is upper bounded by T. In fact, the first constraint on ˆPd (ε, TS) implies
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  IEEE ACCESS 6 thatthe spectrum sensing should be sufficiently reliable to protect the PU which can be achieved with sufficiently large sensing time TS. Moreover, the SU’s transmit power Psen must be appropriately set to achieve good tradeoff between the network throughput and self-interference mitigation. B. Parameter Configuration for FDC–MAC Protocol To gain insights into the parameter configuration of the FDC–MAC protocol, we first study the optimization with respect to the sensing time TS for a given Psen. For any value of TS, we would need to set the sensing detection threshold ε so that the detection probability constraint is met with equality, i.e., ˆPd (ε, TS) = Pd as in [5], [6]. Since the detection probability is smaller in Case 3 (i.e., the PU changes from the idle to active status during the FD sensing stage of duration TS) compared to that in Case 1 and Case 2 (i.e., the PU remains idle during the FD sensing stage) considered in the previous section, we only need to consider Case 3 to maintain the detection probability constraint. The average probability of detection for the FD sensing in Case 3 can be expressed as ˆPd = TS 0 P01 d (t)fτid (t |0 ≤ t ≤ TS ) dt, (5) where t denotes the duration from the beginning of the FD sensing stage to the instant when the PU changes to the active state, and fτid (t |A) is the pdf of τid conditioned on event A capturing the condition 0 ≤ t ≤ TS, which is given as fτid (t |A) = fτid (t) Pr {A} = 1 ¯τid exp(− t ¯τid ) 1 − exp(−TS ¯τid ) . (6) Note that P01 d (t) is derived in Appendix C and fτid (t) is given in (18). We consider the following single-variable optimization problem for a given Psen: max 0TS ≤T NT (TS, Psen) . (7) We characterize the properties of function NT (TS, Psen) with respect to TS for a given Psen in the following theorem whose proof is provided in Appendix D. For simplicity, the throughput function is written as NT (TS). Theorem 1: The objective function NT(TS) of (7) satisfies the following properties 1) lim TS →0 ∂N T ∂TS = +∞, 2) a) For HDTx mode with ∀Psen and FDTx mode with Psen Psen, we have lim TS →T ∂N T ∂TS 0, b) For FDTx mode with Psen Psen, we have lim TS →T ∂N T ∂TS 0, 3) ∂2 N T ∂T 2 S 0, ∀TS, 4) The objective function NT(TS) is bounded from above, where Psen = N0 1 + Pdat N0+ζP ξ dat 2 − 1 is the critical value of Psen such that lim TS →T ∂N T ∂TS = 0. We would like to discuss the properties stated in Theorem 1. For the HDTx mode with ∀Psen and FDTx mode with low Psen, then properties 1, 2a, and 4 imply that there must be at least one TS in [0, T] that maximizes NT (TS). The third property implies that this maximum is indeed unique. Moreover, for the FDTx with high Psen, then properties 1, 2b, 3 and 4 imply that NT(TS) increases in [0, T]. Hence, the throughput NT(TS) achieves its maximum with sensing time TS = T. We propose an algorithm to determine optimal (TS, Psen), which is summarized in Algorithm 1. Here, we can employ the bisection scheme and other numerical methods to determine the optimal value TS for a given Psen. 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,T S (Psen) NT (TS (Psen) , Psen). V. NUMERICAL RESULTS For numerical studies, we set the key parameters for the FDC–MAC protocol as follows: mini-slot duration is σ = 20µs; PD = 1µs; SIFS = 2σ µs; DIFS = 10σ µs; ACK = 20σ µs; CTS = 20σ µs; RTS = 20σ µs. Other parameters are chosen as follows unless stated otherwise: the sampling frequency fs = 6 MHz; bandwidth of PU’s signal 6 MHz; Pd = 0.8; T = 15 ms; p = 0.0022; the SNR of the PU signal at each SU γP = Pp N0 = −20 dB; varying self- interference parameters ζ and ξ. Without loss of generality, the noise power is normalized to one; hence, the SU transmit power Psen becomes Psen = SNRs; and we set Pmax = 15dB. We first study the impacts of self-interference parameters on the throughput performance with the following parameter setting: (¯τid, ¯τac) = (1000, 100) ms, Pmax = 25 dB, Teva = 40 ms, ζ = 0.4, ξ is varied in ξ = {0.12, 0.1, 0.08, 0.05}, and Pdat = Pmax. Recall that the self-interference depends on the transmit power P as I(P) = ζ (P) ξ where P = Psen and P = Pdat in the FD sensing and transmission stages, respectively. Fig. 2 illustrates the variations of the throughput versus the transmission probability p. It can be observed that when ξ decreases (i.e., the self-interference is smaller), the achieved throughput increases. This is because SUs can transmit with higher power while still maintaining the sensing constraint during the FD sensing stage, which leads to through- put improvement. The optimal Psen corresponding to these values of ξ are Psen = SNRs = {25.00, 18.01, 14.23, 11.28} dB and the optimal probability of transmission is p∗ = 0.0022 as indicated by a star symbol. Therefore, to obtain all other results in this section, we set p∗ = 0.0022.
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  IEEE ACCESS 7 10 −3 10 −2 10 −1 0.6 0.8 1 1.2 1.4 1.6 Throughputvs probability of transmission Probability of transmission (p) Throughput(NT,bits/s/Hz) ξ = 0.12, 0.1, 0.08, 0.05 Fig. 2. Normalized throughput versus transmission probability p for T = 18 ms, ¯τid = 1000 ms, ¯τac = 100 ms, and varying ξ. 1 100 0.8 1 1.2 1.4 1.6 Throughput vs number of users Number of users (n0) Throughput(NT,bits/s/Hz) ξ = 0.12, 0.1, 0.08, 0.05 Fig. 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. Fig. 3 illustrates the throughput performance versus number of SUs n0 when we keep the same parameter settings as those for Fig. 2 and p∗ = 0.0022. Again, when ξ decreases (i.e., the self-interference becomes smaller), the achieved throughput increases. In this figure, the optimal SNRs achieving the maximum throughput corresponding to the considered values of ξ are Psen = SNRs = {25.00, 18.01, 14.23, 11.28} dB, respectively. We now verify the results stated in Theorem 1 for the FDTx mode. Specifically, Fig. 4 shows the throughput performance for the scenario where the QSIC is very low with large ξ and ζ where we set the network parameters as follows: p = 0.0022, ¯τid = 500 ms, ¯τac = 50 ms, n0 = 40, ξ = 1, ζ = 0.7, and Pdat = 15 dB. Moreover, we can obtain Psen as in (42) in Appendix D, which is equal to Psen = 6.6294 dB. In this figure, the curve indicated by asterisks, which corresponds to Psen = Psen, shows the monotonic increase of the throughput 0 0.005 0.01 0.015 0 0.5 1 1.5 2 2.5 3 3.5 Throughput vs TS TS (s) Throughput(NT,bits/s/Hz) ¯Psen = 6.6294 dB Psen = −15 : 1.0344 : 15 dB Fig. 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. 0 0.005 0.01 0.015 0 1 2 3 4 Throughput vs TS TS (s) Throughput(NT,bits/s/Hz) P sen = −15 : 1.0344 : 15 dB Fig. 5. 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 = 0.08 and FDTx with Pdat = 15 dB. with sensing time TS and other curves corresponding to Psen Psen have the same characteristic. In contrast, all remaining curves (corresponding to Psen Psen) first increase to the maximum values and then decrease as we increase TS. Fig. 5 illustrates the throughput performance for the very high QSIC with small ξ and ζ where we set the network parameters as follows: p = 0.0022, ¯τid = 500 ms, ¯τac = 50 ms, n0 = 40, ξ = 1, ζ = 0.08, and Pdat = 15 dB. Moreover, we can obtain Psen as in (42) in Appendix D, which is equal to Psen = 19.9201 dB. We have Psen Pmax = 15dB Psen in this scenario; hence, all the curves first increases to the maximum throughput and then decreases with the increasing TS. Therefore, we have correctly validated the properties stated in Theorem 1. Now we investigate the throughput performance versus SU transmit power Psen and sensing time TS for the case of high QSIC with ξ = 0.95 and ζ = 0.08. Fig. 6 shows the
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  IEEE ACCESS 8 0 0.005 0.01 −10 0 10 0 1 2 3 TS(s) Throughput vs Psen and TS Psen (dB) Throughput(NT,bits/s/Hz) 0.5 1 1.5 2 NT * (2.44 ms, 4.6552 dB) = 2.3924 Fig. 6. Normalized throughput versus SU transmit power Psen and sensing 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. 0 0.005 0.01 −10 0 10 0 0.5 1 1.5 2 TS (s) Throughput vs Psen and TS Psen (dB) Throughput(NT,bits/s/Hz) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 NT * (15 ms, 15 dB) = 1.6757 Fig. 7. Normalized throughput versus SU transmit power Psen and sensing 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. throughput versus the SU transmit power Psen and sensing time TS for the FDTx mode with Pdat = 15 dB, p = 0.0022, ¯τid = 150 ms, ¯τac = 50 ms, and n0 = 40. It can be observed that there exists an optimal configuration of the SU transmit power P∗ sen = 4.6552 dB and sensing time T∗ S = 2.44 ms to achieve the maximum throughput NT (T∗ S , P∗ sen) = 2.3924, which is indicated by a star symbol. These results confirm that SUs must set appropriate sensing time and transmit power for the FDC–MAC protocol to achieve the maximize throughput, which cannot be achieved by setting Ts = T as proposed in existing designs such as in [31]. In Fig. 7, we present the throughput versus the SU transmit power Psen and sensing time TS for the low QSIC scenario where p = 0.0022, ¯τid = 150 ms, ¯τac = 50 ms, Pmax = 15 dB, n0 = 40, ξ = 0.95, and ζ = 0.8. The optimal configuration of 0 0.005 0.01 −10 0 10 0 0.5 1 1.5 TS (s) Throughput vs Psen and TS Psen (dB) Throughput(NT,bits/s/Hz) 0.2 0.4 0.6 0.8 1 1.2 1.4NT * (3.5 ms, 5.6897 dB) = 1.4802 Fig. 8. Normalized throughput versus SU transmit power Psen and sensing time TS for p = 0.0022, ¯τid = 150 ms, ¯τac = 50 ms, n0 = 40, ξ = 0.95, ζ = 0.08 and HDTx. SU transmit power P∗ sen = 15 dB and sensing time T∗ S = 15 ms to achieve the maximum throughput NT (T∗ S , P∗ sen) = 1.6757 is again indicated by a star symbol. Under this optimal configuration, the FD sensing is performed during the whole data phase (i.e., there is no transmission stage). In fact, to achieve the maximum throughput, the SU must provide the satisfactory sensing performance and attempt to achieve high transmission rate. Therefore, if the QSIC is low, the data rate achieved during the transmission stage can be lower than that in the FD sensing stage because of the very strong self- interference in the transmission stage. Therefore, setting longer FD sensing time enables to achieve more satisfactory sensing performance and higher transmission rate, which explains that the optimal configuration should set T∗ S = T for the low QSIC scenario. This protocol configuration corresponds to existing design in [31], which is a special case of the proposed FDC– MAC protocol. We now investigate the throughput performance with respect to the SU transmit power Psen and sensing time TS for the HDTx mode. Fig. 8 illustrates the throughput performance for the high QSIC scenario with ξ = 0.95 and ζ = 0.08. It can be observed that there exists an optimal configuration of SU transmit power P∗ sen = 5.6897 dB and sensing time T∗ S = 3.5 ms to achieve the maximum throughput NT (T∗ S , P∗ sen) = 1.4802, which is indicated by a star symbol. The maximum achieved throughput of the HDTx mode is lower than that in the FDTx mode presented in Fig. 6. This is because with high QSIC, the FDTx mode can transmit more data than the HDTx mode in the transmission stage. In Fig. 9, we show the throughput versus the 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 and various values of T (i.e., the data phase duration) for the FDTx mode with Pdat = 15 dB. For each value of T, there exists the optimal SU transmit power P∗ sen which is indicated by an asterisk. It can be observed that as T increases from 8 ms to 25 ms, the
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  IEEE ACCESS 9 −15−10 −5 0 5 10 15 3 3.5 4 4.5 5 Throughput vs Psen Psen (dB) Throughput(NT,bits/s/Hz) T * = 0.015 T* = 0.020 T * = 0.025 T * = 0.010 T* = 0.008 Fig. 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. 5 10 15 20 25 0 0.5 1 1.5 2 Throughput vs Pmax Pmax (dB) Throughput(NT,bits/s/Hz) ζ = 0.2 ζ = 0.7 FDC-MAC HD MAC FD MAC Fig. 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. achieved maximum throughput first increases then decreases with T. Also in the case with T∗ = 15 ms, the SU achieves the largest throughput which is indicated by a star symbol. Furthermore, the achieved throughput significantly decreases when the pair of (T, Psen) deviates from the optimal values, (T∗ , P∗ sen). Finally, we compare the throughput of our proposed FDC- MAC protocol, the single-stage FD MAC protocol where FD sensing (concurrent spectrum sensing and transmission) is performed during the whole data phase [31] and the HD MAC protocol which does not allow the transmission during the spectrum sensing interval in Fig. 10. For brevity, the single- stage FD MAC protocol is refereed to as FD MAC in this figure. The parameter settings are as follows: ¯τid = 150 ms, ¯τac = 75 ms, n0 = 40, ξ = 0.85, n0 = 40, ζ = {0.2, 0.7}, and FDTx with Pdat = Pmax dB. For fair comparison, we first obtain the optimal configuration of the single-stage FD MAC protocol, i.e., then we use (T∗ , p∗ ) for the HD MAC protocol and FDC-MAC protocol. For the single-stage FD MAC protocol, the transmit power is set to Pmax because there is only a single stage where the SU performs sensing and transmission simultaneously during the data phase. In addition, the HD MAC protocol will also transmit with the maximum transmit power Pmax to achieve the highest throughput. For both studied cases of ζ = {0.2, 0.7}, our proposed FDC- MAC protocol significantly outperforms the other two proto- cols. Moreover, the single-stage FD MAC protocol [31] with power allocation outperforms the HD MAC protocol at the corresponding optimal value of Pmax required by the single- stage FD MAC protocol. However, both single-stage FDC- MAC and HD MAC protocols achieve increasing throughput with higher Pmax while the single-stage FD MAC protocol has the throughput first increased then decreased as Pmax increases. This demonstrates the negative self-interference effect on the throughput performance of the single-stage FD MAC protocol, which is efficiently mitigated by our proposed FDC-MAC protocol. VI. CONCLUSION In this paper, we have proposed the FDC–MAC protocol for cognitive radio networks, analyzed its throughput per- formance, and studied its optimal parameter configuration. The design and analysis have taken into account the FD communication capability and the self-interference of the FD transceiver. We have shown that there exists an optimal FD sensing time to achieve the maximum throughput. We have then presented extensive numerical results to demonstrate the impacts of self-interference and protocol parameters on the throughput performance. In particular, we have shown that the FDC–MAC protocol achieves significantly higher throughput than the HD MAC protocol, which confirms that the FDC– MAC protocol can efficiently exploit the FD communication capability. Moreover, the FDC–MAC protocol results in higher throughput with the increasing maximum power budget while the throughput of the single-stage FD MAC can decrease in the high power regime. This result validates the importance of adopting the two-stage procedure in the data phase and the optimization of sensing time and transmit power during the FD sensing stage to mitigate the negative self-interference effect. APPENDIX A DERIVATION OF Tcont To calculate Tcont, we define some further parameters as follows. Denote Tcoll as the duration of the collision and Tsucc as the required time for successful RTS/CTS transmission. These quantities can be calculated as follows [17]: Tsucc = DIFS + RTS + SIFS + CTS + 2PD Tcoll = DIFS + RTS + PD, (8)
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  IEEE ACCESS 10 whereDIFS is the length of a DCF (distributed coordination function) interframe space, RTS and CTS denote the lengths of the RTS and CTS messages, respectively. As being shown in Fig. 1, there can be several idle periods and collisions before one successful channel reservation. Let Ti idle denote the i-th idle duration between two consecutive RTS/CTS exchanges, which can be collisions or successful exchanges. Then, Ti idle can be calculated based on its prob- ability mass function (pmf), which is derived as follows. In the following, all relevant quantities are defined in terms of the number of time slots. With n0 SUs joining the contention resolution, let Psucc, Pcoll and Pidle denote the probabilities that a particular generic slot corresponds to a successful transmission, a collision, and an idle slot, respectively. These probabilities can be calculated as follows: Psucc = n0p (1 − p) n0−1 (9) Pidle = (1 − p) n0 (10) Pcoll = 1 − Psucc − Pidle, (11) where p is the transmission probability of an SU in a generic slot. In general, the interval Tcont, whose average value is Tcont given in (2), consists of several intervals corresponding to idle periods, collisions, and one successful RTS/CTS transmission. Hence, this quantity can be expressed as Tcont = Ncoll i=1 Tcoll + Ti idle + TNcoll+1 idle + Tsucc, (12) where Ncoll is the number of collisions before the successful RTS/CTS exchange and Ncoll is a geometric random variable (RV) with parameter 1 − Pcoll/Pidle where Pidle = 1 − Pidle. Therefore, its pmf can be expressed as fNcoll X (x) = Pcoll Pidle x 1 − Pcoll Pidle , x = 0, 1, 2, . . . (13) Also, Tidle represents the number of consecutive idle slots, which is also a geometric RV with parameter 1 − Pidle with the following pmf fTidle X (x) = (Pidle) x (1 − Pidle) , x = 0, 1, 2, . . . (14) Therefore, Tcont (the average value of Tcont) can be written as follows [17]: Tcont = NcollTcoll + Tidle Ncoll + 1 + Tsucc, (15) where Tidle and Ncoll can be calculated as Tidle = (1 − p) n0 1 − (1 − p) n0 (16) Ncoll = 1 − (1 − p) n0 n0p (1 − p) n0−1 − 1. (17) These expressions are obtained by using the pmfs of the corresponding RVs given in (13) and (14), respectively [17]. APPENDIX B DERIVATIONS OF B1, B2, B3 We will employ a pair of parameters (θ, ϕ) to represent the HDTX and FDTX modes where ((θ, ϕ) = (0, 1)) for HDTx mode and ((θ, ϕ) = (1, 2)) for the FDTx mode. Moreover, since the transmit powers in the FD sensing and transmission stages are different, which are equal to Psen and Pdat, respectively, we define different SNRs and SINRs in these two stages as follows: γS1 = Psen N0 and γS2 = Psen N0+Pp are the SNR and SINR achieved by the SU in the FD sensing stage with and without the presence of the PU, respectively; γD1 = Pdat N0+θI and γD2 = Pdat N0+Pp+θI for I = ζPξ dat are the SNR and SINR achieved by the SU in the transmission stage with and without the presence of the PU, respectively. It can be seen that we have accounted for the self-interference for the FDTx mode during the transmission stage in γD1 by noting that θ = 1 in this case. The parameter ϕ for the HDTx and FDTx modes will be employed to capture the throughput for one-way and two-way transmissions in these modes, respectively. The derivations of B1, B2, and B3 require us to consider different possible sensing outcomes in the FD sensing stage. In particular, we need to determine the detection probability Pij d , which is the probability of correctly detecting the PU given the PU is active, and the false alarm probability Pij f , which is the probability of the erroneous sensing of an idle channel, for each event hij capturing the state changes of the PU. In the following analysis, we assume the exponential distribution for τac and τid where ¯τac and ¯τid denote the corresponding average values of these active and idle intervals. Specifically, let fτx (t) denote the pdf of τx (x represents ac or id in the pdf of τac or τid, respectively) then fτx (t) = 1 ¯τx exp(− t ¯τx ). (18) Similarly, we employ Tij S and Tij D to denote the number of bits transmitted on one unit of system bandwidth during the FD sensing and transmission stages under the PU’s state-changing event hij, respectively. We can now calculate B1 as follows: B1 = P (H0) ∞ t=Tove+T T00 1 fτid (t)dt = P (H0) T00 1 exp − Tove + T ¯τid , (19) where P (H0) denotes the probability of the idle state of the PU, and P00 f is the false alarm probability for event h00 given in Appendix C. Moreover, T00 1 = P00 f T00 S + (1 − P00 f )(T00 S + T00 D ), T00 S = TS log2 (1 + γS1), T00 D = ϕ (T − TS) log2 (1 + γD1) where T00 S and T00 D denote the number of bits transmitted (over one Hz of system bandwidth) in the FD sensing and transmission stages of the data phase,
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  IEEE ACCESS 11 respectively.After some manipulations, we achieve B1 = Ke exp T ∆τ [TS log2 (1 + γS1) + ϕ 1 − P00 f (T − TS) log2 (1 + γD1) , (20) where Ke = P (H0) exp − Tove ¯τid + T ¯τac and 1 ∆τ = 1 ¯τac − 1 ¯τid . Moreover, we can calculate B2 as B2 = P (H0) Tove+T t1=Tove+TS ∞ t2=Tove+T −t1 T01 2 (t1)fτid (t1)fτac (t2)dt1dt2, (21) where T01 2 (t1) = P00 f T00 S + (1 − P00 f )(T00 S + T01 D (¯t1)), T01 D (t1) = ϕ (T − TS − ¯t1) log2 (1 + γD2) + ϕ¯t1 log2 (1 + γD1), and ¯t1 = t1 − (Tove + TS). In this expression, t1 denotes the interval from the beginning of the CA cycle to the instant when the PU changes to the active state from an idle state. Again, T00 S and T01 D denote the amount of data transmitted in the FD sensing and transmission stages for this case, respectively. After some manipulations, we achieve B2 = Ke ∆τ ¯τid exp T ∆τ − exp TS ∆τ × TS log2 (1+γS1)−ϕ∆τ 1−P00 f log2 1+γD1 1+γD2 +ϕ (T − TS) 1−P00 f × exp T ∆τ log2(1+γD1)−exp TS ∆τ log2 (1+γD2) . (22) Finally, we can express B3 as follows: B3 = P (H0) Tove+TS t1=Tove ∞ t2=Tove+T −t1 P01 d (¯t1) T01 S (¯t1) + (1 − P01 d (¯t1))(T01 S (¯t1) + T11 D ) fτid (t1)fτac (t2)dt1dt2, (23) where ¯t1 = t1 − Tove, T01 S (¯t1) = ¯t1 log2 (1 + γS1) + (TS − ¯t1) log2 (1 + γS2), T11 D = ϕ (T − TS) log2 (1 + γD2), and t1 is the same as in (21). Here, T01 S and T11 D denote the amount of data delivered in the FD sensing and transmission stages for the underlying case, respectively. After some ma- nipulations, we attain B3 = Ke TS t=0 T01 S (t) + T11 D − P01 d (t) T11 D fτid (t) exp t ¯τac dt = B31 + B32, (24) where B31 = Ke TS t=0 T01 S (t) + T11 D fτid (t) exp t ¯τac dt = Ke ∆τ ¯τid ∆τ TS ∆τ −1 exp TS ∆τ +1 log2 1+γS1 1+γS2 + exp TS ∆τ − 1 T11 D + TS log2 (1 + γS2) , (25) and B32 = −KeT11 D ¯T32, (26) where ¯T32 = TS t=0 P01 d (t) fτid (t) exp t ¯τac dt. APPENDIX C FALSE ALARM AND DETECTION PROBABILITIES We derive the detection and false alarm probabilities for FD sensing and two PU’s state-changing events h00 and h01 in this appendix. Assume that the transmitted signals from the PU and SU are circularly symmetric complex Gaussian (CSCG) signals while the noise at the secondary receiver is independently and identically distributed CSCG CN (0, N0) [5]. Under FD sensing, the false alarm probability for event h00 can be derived using the similar method as in [5], which is given as P00 f = Q ǫ N0 + I(Psen) − 1 fsTS , (27) where Q (x) = +∞ x exp −t2 /2 dt; fs, N0, ǫ, I(Psen) are the sampling frequency, the noise power, the detection threshold and the self-interference, respectively; TS is the FD sensing duration. The detection probability for event h01 is given as P01 d = Q   ǫ N0+I(Psen) − TS −t TS γP S − 1 √ fsTS TS −t TS (γP S + 1) 2 + t TS   , (28) where t is the interval from the beginning of the data phase to the instant when the PU changes its state, γP S = Pp N0+I(Psen) is the signal-to-interference-plus-noise ratio (SINR) of the PU’s signal at the SU. APPENDIX D PROOF OF THEOREM 1 The first derivative of NT can be written as follows: ∂NT ∂TS = 1 Tove + T 3 i=1 ∂Bi ∂TS . (29) We derive the first derivative of Bi (i = 1, 2, 3) in the following. Toward this end, we will employ the approximation of exp (x) ≈ 1 + x, x = Tx τx , Tx ∈ {T, TS, T − TS}, τx ∈ {¯τid, ¯τac, ∆τ} where recall that 1 ∆τ = 1 ¯τac − 1 ¯τid . This approximation holds under the assumption that Tx τx since we can omit all higher-power terms xn for n 1 from the Maclaurin series expansion of function exp (x). Using this approximation, we can express the first derivative of B1 as ∂B1 ∂TS = Ke exp T ∆τ {log2 (1 + γS1) −ϕ (T − TS) ∂P00 f ∂TS + 1 − P00 f log2 (1 + γD1) , (30)
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  IEEE ACCESS 12 where ∂P00 f ∂TS isthe first derivative of P00 f whose derivation is given in Appendix E. Moreover, the first derivative of B2 can be written as ∂B2 ∂TS = Ke ∆τ ¯τid × exp T ∆τ − 1 + T ∆τ exp TS ∆τ log2 (1 + γS1) −ϕ ∂P00 f ∂TS ∆τ exp TS ∆τ −exp T ∆τ log2 1+γD1 1+γD2 + (T−TS) exp T ∆τ log2(1+γD1)−exp TS ∆τ log2(1+γD2) +ϕ 1 − P00 f − T − TS ∆τ exp TS ∆τ log2 (1 + γD2) − exp T ∆τ log2(1+γD1)−exp TS ∆τ log2(1+γD2) + exp TS ∆τ log2 1 + γD1 1 + γD2 . (31) Finally, the first derivative of B3 can be written as ∂B3 ∂TS = ∂B31 ∂TS + ∂B32 ∂TS , (32) where ∂B31 ∂TS =Ke ∆τ ¯τid ∆τ 1+ TS ∆τ −1 exp TS ∆τ log2 1+γS1 1+γS2 + exp TS ∆τ −1 [TSlog2(1+γS2)+ϕ(T −TS)log2(1+γD2)] . (33) To obtain the derivative for B32, we note that 1 ≤ exp t ¯τac ≤ exp TS ¯τac for ∀t ∈ [0, TS]. Moreover, from the results in (5) and (6) and using the definition of ¯T32 in (26), we have Pd 1 − exp −TS ¯τid ≤ ¯T32 ≤ Pd 1 − exp −TS ¯τid exp TS ¯τac . Using these results, the first derivative of B32 can be expressed as ∂B32 ∂TS = −KePdϕ T − 2TS ¯τid log2 (1 + γD2) . (34) Therefore, we have obtained the first derivative of NT and we are ready to prove the first statement of Theorem 1. Substitute TS = 0 to the derived ∂N T ∂TS and use the approximation exp (x) ≈ 1 + x, we yield the following result after some manipulations lim TS →0 ∂NT ∂TS = −K0K1 lim TS →0 ∂P00 f ∂TS , (35) where K0 = 1 Tove+T Ke and K1 = ϕ T 1 + T ∆τ + T2 ¯τid log2 (1 + γD1) +ϕ T∆τ ¯τid log2 (1 + γD2) . (36) It can be verified that K0 0, K1 0 and lim TS →0 ∂P00 f ∂TS = −∞ by using the derivations in Appendix E; hence, we have lim TS →0 ∂N T ∂TS = +∞ 0. This completes the proof of the first statement of the theorem. We now present the proof for the second statement of the theorem. Substitute TS = T to ∂N T ∂TS and utilize the approximation exp (x) ≈ 1 + x, we yield lim TS →T ∂NT ∂TS = 1 Tove + T 3 i=1 ∂Bi ∂TS (T), (37) where we have ∂B1 ∂TS (T) = Ke 1 + T ∆τ × log2 (1 + γS1) − ϕ 1 − P00 f (T) log2 (1 +γD1) (38) ∂B2 ∂TS (T) = −Ke T ¯τid log2 (1 + γS1) (39) ∂B31 ∂TS (T) = Ke T ¯τid [log2(1+γS1) (1+γS2)−ϕlog2(1+γD2)] (40) ∂B32 ∂TS (T) = Keϕ T ¯τid Pd log2 (1 + γD2) . Omit all high-power terms in the expansion of exp(x) (i.e., xn with n 1) where x = Tx τx , Tx ∈ {T, TS, T − TS}, τx ∈ {¯τid, ¯τac, ∆τ}, we yield lim TS →T ∂NT ∂TS ≈ 1 Tove + T ∂B1 ∂TS (T). (41) We consider the HDTx and FDTx modes in the fol- lowing. For the HDTx mode, we have ϕ = 1 and θ = 0. Then, it can be verified that lim TS →T ∂N T ∂TS 0 by using the results in (38) and (41). This is because we have log2 (1 + γS1) − 1 − P00 f (T) log2 (1 + γD1) ≈ log2 (1 + γS1)−log2 (1 + γD1) 0 (since we have P00 f (T) ≈ 0 and γS1 ≤ γD1). For the FDTx mode, we have ϕ = 2, θ = 1, and also γS1 = Psen N0 and γD1 = Pdat N0+I(Pdat) = Pdat N0+ζP ξ dat . We would like to define a critical value of Psen which satisfies lim TS →T ∂N T ∂TS = 0 to proceed further. Using the result in (38) and (41) as well as the approximation P00 f (T) ≈ 0, and by solving lim TS →T ∂N T ∂TS = 0 we yield Psen = N0   1 + Pdat N0 + ζPξ dat 2 − 1   . (42) Using (38), it can be verified that if Psen Psen then lim TS →T ∂N T ∂TS 0; otherwise, we have lim TS →T ∂N T ∂TS ≤ 0. So we have completed the proof for the second statement of Theorem 1. To prove the third statement of the theorem, we derive the second derivative of NT as ∂2 NT ∂T2 S = 1 Tove + T 3 i=1 ∂2 Bi ∂T2 S , (43)
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  IEEE ACCESS 13 wherewe have ∂2 B1 ∂T2 S = −Keϕ exp T ∆τ log2 (1 + γD1) × (T − TS) ∂2 P00 f ∂T2 S − 2 ∂P00 f ∂TS , (44) where ∂2 P00 f ∂T 2 S is the second derivative of P00 f and according to the derivations in Appendix E, we have ∂2 P00 f ∂T 2 S 0, ∂P00 f ∂TS 0, ∀TS. Therefore, we yield ∂2 B1 ∂T 2 S 0 ∀TS. Consequently, we have the following upper bound for ∂2 B1 ∂T 2 S by omitting the term exp T ∆τ 1 in (44) ∂2 B1 ∂T2 S ≤ Ke [h1(TS) + h2(TS)] , (45) where h1(TS) = −ϕ (T − TS) ∂2 P00 f ∂T2 S log2 (1 + γD1) , h2(TS) = 2ϕ ∂P00 f ∂TS log2 (1 + γD1) . Moreover, we have ∂2 B2 ∂T2 S = Ke∆τ ¯τid − 2 + TS ∆τ ∆τ exp TS ∆τ log2 (1 + γS1) −ϕ ∂2 P00 f ∂T2 S ∆τ exp TS ∆τ −exp T ∆τ log2 1+γD1 1+γD2 + (T−TS) exp T ∆τ log2(1+γD1)−exp TS ∆τ log2(1+γD2) +2ϕ ∂P00 f ∂TS exp T ∆τ −exp TS ∆τ log2(1+γD1) + T−TS ∆τ exp TS ∆τ log2(1+γD2) −ϕ 1 − P00 f T−TS ∆τ exp TS ∆τ log2(1+γD2) +ϕ 1 − P00 f 1 ∆τ exp TS ∆τ log2 1 + γD1 1 + γD2 . (46) Therefore, we can approximate ∂2 B2 ∂T 2 S as follows: ∂2 B2 ∂T2 S = Ke [h3(TS) + h4(TS) + h5(TS)] , (47) where h3(TS) = − 2 + TS ∆τ 1 + TS ∆τ ¯τid log2 (1 + γS1) , h4(TS) = −ϕ 1 − P00 f T−TS ¯τid 1 + TS ∆τ log2(1+γD2) −ϕ ∂2 P00 f ∂T2 S (T −TS) T ¯τid log2(1 +γD1)− TS ¯τid log2(1 +γD2) +2ϕ ∂P00 f ∂TS T − TS ¯τid log2 1+γD1 1+γD2 + TS ∆τ log2(1+γD2) , h5(TS) = ϕ 1 − P00 f 1 ¯τid 1 + TS ∆τ log2 1 + γD1 1 + γD2 . In addition, we have ∂2 B31 ∂T2 S = Ke ¯τid exp TS ∆τ 1 + T ∆τ log2 (1 + γS1) + log2 (1 + γS2) + ϕ T − TS ∆τ − 2 log2 (1 + γD2) . (48) We can approximate ∂2 B31 ∂T 2 S as follows: ∂2 B31 ∂T2 S = Ke [h6(TS) + h6(TS)] , (49) where h6(TS) = 1 ¯τid log2 (1+γS1) (1+γS2) , h7(TS) = − 2ϕ ¯τid log2 (1+γD2) . (50) Finally, we have ∂2 B32 ∂T2 S = Keh8(TS), (51) where h8(TS) = 2ϕ ¯Pd ¯τid log2 (1+γD2) . (52) The above analysis yields ∂2 N T ∂T 2 S = Ke 8 i=1 hi(TS). There- fore, to prove that ∂2 N T ∂T 2 S 0, we should prove that h(TS) 0 since Ke 0 where h(TS) = 8 i=1 hi(TS). (53) It can be verified that h1(TS) 0 and h4(TS) 0, ∀TS because ∂2 P00 f ∂T 2 S 0, ∂P00 f ∂TS 0 according to Appendix E and γD2 γD1. Moreover, we have h3(TS) − 2 ¯τid log2 (1 + γS1) , (54) and because γS1 γS2, we have h3(TS) − 1 ¯τid log2 (1 + γS1) (1 + γS2) = −h6(TS). (55) Therefore, we have h3(TS) + h6(TS) 0. Furthermore, we can also obtain the following result h7(TS) + h8(TS) ≤ 0
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  IEEE ACCESS 14 because¯Pd ≤ 1. To complete the proof, we must prove that h2(TS) + h5(TS) ≤ 0, which is equivalent to − 2¯τid ∂P00 f ∂TS log2 (1 + γD1) log2 1+γD1 1+γD2 ≥ 1 − P00 f 1 + TS ∆τ , (56) where according to Appendix E ∂P00 f ∂TS = − ¯γ √ fsTS 2 √ 2πTS exp − ¯α + ¯γ √ fsTS 2 2 , (57) where ¯α = (¯γ1 + 1) Q−1 Pd . It can be verified that (56) indeed holds because the LHS of (56) is always larger than to 2 while the RHS of (56) is always less than 2. Hence, we have completed the proof of the third statement of Theorem 1. Finally, the fourth statement in the theorem obviously holds because Bi (i = 1, 2, 3) are all bounded from above. Hence, we have completed the proof of Theorem 1. APPENDIX E APPROXIMATION OF P00 f AND ITS FIRST AND SECOND DERIVATIVES We can approximate ˆPd in (5) as follows: ˆPd = Q ǫ N0 + I − ¯γ − 1 √ fsTS ¯γ1 + 1 , (58) where ¯γ and ¯γ1 are evaluated by a numerical method. Hence, P00 f can be calculated as we set ˆPd = Pd, which is given as follows: P00 f = Q ¯α + ¯γ fsTS , (59) where ¯α = (¯γ1 + 1) Q−1 Pd . We now derive the first derivative of P00 f as ∂P00 f ∂TS = − ¯γ √ fsTS 2 √ 2πTS exp − ¯α + ¯γ √ fsTS 2 2 . (60) It can be seen that ∂P00 f ∂TS 0 since ¯γ 0. Moreover, the second derivative of P00 f is ∂2 P00 f ∂T2 S = ¯γ √ fsTS 4 √ 2πT2 S 1 + 1 2 y¯γ fsTS exp − y2 2 , (61) where y = ¯α + ¯γ √ fsTS. We can prove that ∂2 P00 f ∂T 2 S 0 by considering two different cases as follows. For the first case with ¯α2 ¯γ2fs ≤ TS ≤ T (0 ≤ P00 f ≤ 0.5), this statement holds since y 0. For the second case with 0 ≤ TS ≤ ¯α2 ¯γ2fs (0.5 ≤ P00 f ≤ 1), y ≤ 0, then we have 0 y − ¯α = ¯γ √ fsTS ≤ −¯α and 0 ≤ −y ≤ −¯α. By applying the Cauchy-Schwarz inequality, we obtain 0 ≤ −y(y − ¯α) ≤ ¯γ2 4 1 2; hence 1 + 1 2 y¯γ √ fsTS 0. This result implies that ∂2 P00 f ∂T 2 S 0. REFERENCES [1] C. Cormiob and K. R. 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  IEEE ACCESS 15 transceiver,”IEEE J. Sel. Areas Commun., vol. 32, no. 9, pp. 1674– 1687, June 2014. [24] J. I. Choi, M. Jain, K. Srinivasan, P. Levis, and S. Katti, “Achieving single channel, full duplex wireless communications,” in Proc. ACM MobiCom’2010. [25] M. Jain et al., “Practical, real-time, full duplex wireless,” in Proc. ACM MobiCom’2011. [26] M. Duarte, A. Sabharwal, V. Aggarwal, R. Jana, K. Ramakrishnan, C. Rice, and N. Shankaranarayanan, “Design and characterization of a full-duplex multiantenna system for WiFi networks,” IEEE Trans. Veh. Tech., vol. 63, no. 3, pp. 1160–1177, Mar. 2014. [27] S. Goyal, P. Liu, O. Gurbuz, E. Erkip, and S. Panwar, “A distributed MAC protocol for full duplex radio,” Proc. Asilomar Conf. Signals, Syst. Comput. 2013. [28] D. Ramirez, and B. Aazhang, “Optimal routing and power allocation for wireless networks with imperfect full-duplex nodes,” IEEE Trans. Wireless Commun., vol. 12, no. 9, pp. 4692–4704, Sept. 2013. [29] W. Choi, H. Lim, and A. Sabharwal, “Power-controlled medium access control protocol for full-duplex WiFi networks,” IEEE Trans. Wireless Commun., vol. 14, no. 7, pp. 3601–3613, July 2015. [30] W. Cheng, X. Zhang, H. Zhang, “Full-duplex spectrum-sensing and mac-protocol for multichannel non-time-slotted cognitive radio networks,” IEEE J. Sel. Areas Commun., vol. 33 , no. 5, pp. 820–831, April 2015. [31] L. T. Tan and L. B. Le, “Distributed MAC protocol design for full– duplex cognitive radio networks,” in Proc. IEEE’GLOBECOM’2015. [32] H. Kim, S. Lim, H. Wang, and D. Hong, “Optimal power allocation and outage analysis for cognitive full duplex relay systems,” IEEE Trans. Wireless Commun., vol. 11, no. 10, pp. 3754–3765, Sep. 2012. [33] J. Bang, J. Lee, S. Kim, and D. Hong, “An efficient relay selection strategy for random cognitive relay networks,” IEEE Trans. Wireless Commun., vol. 14, no. 3, pp. 1555–1566, Nov. 2015. [34] G. Zheng, I. Krikidis, and B. Ottersten, “Full-duplex cooperative cognitive radio with transmit imperfections,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 2498–2511, May 2013. Le Thanh Tan (S’11–M’15) received the B.Eng. and M.Eng. degrees from Ho Chi Minh City Uni- versity of Technology in 2002 and 2004, respec- tively and the Ph.D. degree from Institut National de la Recherche Scientifique–´Energie, Mat´eriaux et T´el´ecommunications (INRS–EMT), Canada in 2015. He is currently a Postdoctoral Research Associate at ´Ecole Polytechnique de Montr´eal, Canada. Be- fore that he worked as a lecturer at Ho Chi Minh City University of Technical Education from 2002 to 2010. His current research activities focus on internet of things (IOT over LTE/LTE–A network, cyber–physical systems, big data, distributed sensing and control), time series analysis and dynamic factor models (stationary and non–stationary), wireless communications and networking, Cloud–RAN, cognitive radios (software defined radio architec- tures, protocol design, spectrum sensing, detection, and estimation), statistical signal processing, random matrix theory, compressed sensing, and compressed sampling. He has served on TPCs of different international conferences including IEEE CROWNCOM, VTC, PIMRC, etc. He is a Member of the IEEE. Long Le (S’04–M’07–SM’12) received the B.Eng. degree in Electrical Engineering from Ho Chi Minh City University of Technology, Vietnam, in 1999, the M.Eng. degree in Telecommunications from Asian Institute of Technology, Thailand, in 2002, and the Ph.D. degree in Electrical Engineering from the University of Manitoba, Canada, in 2007. He was a Postdoctoral Researcher at Massachusetts Institute of Technology (2008–2010) and University of Wa- terloo (2007–2008). Since 2010, he has been with the Institut National de la Recherche Scientifique (INRS), Universit´e du Qu´ebec, Montr´eal, QC, Canada where he is currently an associate professor. His current research interests include smart grids, cognitive radio, radio resource management, network control and optimization, and emerging enabling technologies for 5G wireless systems. He is a co- author of the book Radio Resource Management in Multi-Tier Cellular Wireless Networks (Wiley, 2013). Dr. Le is a member of the editorial board of IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, IEEE COMMUNICATIONS SURVEYS AND TUTORIALS, and IEEE WIRELESS COMMUNICATIONS LETTERS. He has served as a technical program com- mittee chair/co-chair for several IEEE conferences including IEEE WCNC, IEEE VTC, and IEEE PIMRC.