Wireless Communications and Mobile Computing

Wireless Communications and Mobile Computing / 2020 / Article
Special Issue

Recent Advances in Green Communications and Networks

View this Special Issue

Research Article | Open Access

Volume 2020 |Article ID 5104605 | https://doi.org/10.1155/2020/5104605

Li Feng, Jirong Sun, Hua Sun, Shengju Tang, "Censoring-Based Relay Transmission for Achieving Energy Efficiency in Cognitive Radio Networks", Wireless Communications and Mobile Computing, vol. 2020, Article ID 5104605, 14 pages, 2020. https://doi.org/10.1155/2020/5104605

Censoring-Based Relay Transmission for Achieving Energy Efficiency in Cognitive Radio Networks

Academic Editor: Yujin Lim
Received22 Feb 2020
Revised31 Jul 2020
Accepted06 Aug 2020
Published08 Sep 2020

Abstract

Energy efficiency (EE) is critical to achieve cooperative sensing and transmission in relay-assisted cognitive radio networks (CRNs) with limited battery capacity. This paper proposes an energy-efficient cooperative transmission strategy with combined censoring report and spatial diversity in cooperative process, namely, censoring-based relay transmission (CRT). Specifically, secondary relays (SRs) take part in cooperative sensing with differential censoring to reduce energy consumption, and the best SR assists secondary transmission to enhance communication quality in transmission stage and thus to improve secondary transmission EE. First, we derived generalized-form expressions for detection probability, reporting probability, sensing energy, and expected throughput for CRT. Second, we investigate a mean EE-oriented maximization nonconvex problem by joint optimizing sensing duration and power allocation for secondary users under secondary outage probability and sensing performance constraints. With the aid of Jensen’s inequality, an efficient cross-iteration algorithm with low complexity is proposed to obtain the suboptimal solutions, which is developed by golden segmentation search method. Finally, extensive simulations are conducted to evaluate the performance of CRT. The results show significant improvements of SUs’ EE compared with traditional noncooperation single cognitive transmission schemes, which demonstrate the benefits of our proposed cooperative strategy in conserving energy for secondary transmission.

1. Introduction

The energy consumption of communication systems has increased dramatically, making it an urgent issue in the information world to improve energy efficiency (EE) of mobile communications. The high energy consumption and exponential growth in wireless communication networks face serious challenges to the design of more EE and spectrum efficiency (SE) green communications that should deal with the scarcity of radio resources. A promising approach technique called cognitive radio networks (CRNs) is proposed as a key design to improve spectral efficiency of wireless communication. Especially, advancements in miniaturization of devices with higher computational capabilities and ultralow power communication technologies are driving forces for the ever growing deployment of embedded devices in our surroundings; how to achieve energy-efficient transmission becomes even more critical [13]. Cognitive relay networks have received a lot of interest to improve network performance by expanding the coverage and increasing SE [4]. It is well known that cooperative relaying promotes a diversity gain, improving spectrum efficiency and reducing the time used to deliver a message through distributed transmission and signal processing [5, 6]. At the same time, the flexibility of cognitive radios is important to address the challenges and trade-offs between EE performance and practicality in [2, 7]. As important means for improving SE, cognitive radio and cooperative transmission are also considered potential techniques to achieve green communication. By introducing cooperative transmission into CRNs, cooperative CRNs can not only improve SE but also overcome fading effect and thus attract much attention from researchers [410]. In this context, cooperation and cognition techniques can bring invaluable contributions to development of energy efficient networks.

In short, in this paper, our goal is to improve network EE while ensuring performance of secondary transmitting system. We propose an innovative best-relay-assisted transmission with censoring scheme to provide energy-efficient cooperative transmission in overlay CRNs. Our main contributions of this paper are summarized as follows: (i)We propose a novel energy-efficient and best-relay-assisted cooperative transmission scheme called CRT to reduce sensing energy overhead on sensing stage and enhance communication quality in transmission stage. CRT naturally integrates the censoring report, best-relay-assisted transmission, and time and power optimization in cooperative process, which can effectively improve secondary transmission EE(ii)On the basis of CRT scheme, we derive the generalized-form expressions for reporting probability, average energy, and average throughput for CRT. We formulate a mean EE-oriented maximization nonconvex problem by joint optimizing sensing duration and power allocation for secondary users under constraints of minimal sensing performance and secondary outage probability. Theoretical analysis demonstrates that the EE function of CRT scheme has a globally unique optimal solution. And then, an efficient cross-iteration algorithm was proposed to obtain the suboptimal solutions(iii)Finally, we conduct extensive simulations to evaluate EE performance of CRT which is also compared with that of the noncooperative scheme. It is shown that CRT scheme can remarkably improve EE metrics of secondary transmission compared to the noncooperative case

The rest of this paper is structured as follows. Section 2 briefly describes any related work. The system model and transmission description is described in Section 3. In Section 4, the proposed optimization problem is formulated, and sensing time and power allocation strategy and the algorithm are presented in Section 5. Simulation results are discussed in Section 6. Finally, the conclusions are drawn in Section 7.

As discussed above, fast increasing popularity of powerful intelligent devices and vast penetration of mobile internet business cause explosive volume of wireless traffic, which in turn makes spectrum even more crowded and communication systems more energy-intensive. Therefore, EE has been considered one of the key features in the 5th-generation mobile communication and has attracted many research interests [1, 4, 6]. As important means for improving SE, cognitive radio and cooperative transmission are also considered potential techniques to achieve green communication. By introducing cooperative transmission into cognitive radio networks, cooperative CRNs can not only improve SE but also overcome fading effect and thus attract much attention from researchers.

Most recent research works on cooperative CRNs are mainly focused on improvement of access opportunities and transmission throughput; few of them are concerning on EE. However, it is necessary to study impacts on cooperative spectrum sensing (CSS) and cooperative transmission in cooperative CRNs when EE is concerned to meet low-carbon communication requirement. Given the importance of this issue, recent studies have focused on energy-efficient secondary transmission in CRNs from different aspects and indicated that cognitive and cooperation technology can reduce network energy consumption in CRNs [818]. Collaborative sensing also known as CSS is an efficient spectrum sensing technique to improve the sensing accuracy in cognitive radio. However, it brings extra collaborative sensing overhead due to mutual exchange of large information among cognitive users. In CSS, the number of cooperative users, fusion rule, transmission power, and sensing time affects the energy efficiency (EE) of the CSS. The authors in [8, 9], investigate subcarrier transmission path selection and power allocation optimization problem with the goal of maximizing system EE in cognitive relaying links considering secondary QoS and total power budget, and a step-by-step strategy based on binary search-assisted ascent method is proposed. In [11], the authors proposed a cluster-based CSS strategy to maximize EE by optimizing the fusion rule, whereby the transmission power and sensing time were presented by the joint optimization problem. In [13], a spectrum sharing strategy is proposed via multiwinner auction with multiple bands to increase throughput of secondary system under constraint of minimum EE requirement for primary system. In [15], an EE-oriented “win-win” cognitive spectrum sharing scheme is proposed in heterogeneous cognitive radio networks. The authors in [5] investigated a trade-off between the secrecy throughput and EE in CRNs and proposed a cooperative spectrum sharing paradigm to improve both the secrecy throughput and the energy efficiency of primary users. Comprehensively considering sensing energy and detection performance, a differential reporting adaptive energy-efficient spectrum detection scheme was proposed [10], which can remarkably reduce sensing energy consumption with only slightly degrading detection performance. In [16], the authors modeled power allocation problem with the goal of maximizing the average EE of cognitive users in the fast-fading scene subject to a transmission interruption probability constraint and proposed an efficient power allocation strategy by using the fractional programming and Lagrange dual method. Similar to [16], the authors in [17] investigated EE maximization problem with sensing time and transmitting power as optimization variables. Different from noncooperative single cognitive transmission (NST) in [1618] studied a weighted convex objective function of relay numbers with the consideration of global detection probability and bit error rate of cooperative transmission which is solved by applying numerical analysis to obtain the optimal relay numbers.

Based on recent research in energy-efficient cooperative sensing and transmission in CRNs, the motivation of this paper is expressed as follows. Compared with single cognitive transmission [16, 17, 19], CSS improves spectrum detection accuracy of a cognitive radio network. Clearly, more secondary relays (SRs) involving in spectrum detection require more energy consumption. On the other hand, when secondary system needs to meet higher rate requirements, longer transmission distance will inevitably consume a large amount of transmitting power and appear to be inefficient. In [14], a combined censoring scheme is proposed with the goal of minimizing the network energy consumption subject to a specific detection performance constraint. It is shown that such a system can attain high energy savings. Notice that the essence of censoring mechanism is by decreasing the number of decision reporting information to reduce energy consumption on CSS. In other words, the result is only transmitted, if it is deemed to be informative. Moreover, [15, 18] also studied the maximization of energy-efficient transmission problem considering the energy consumption of sensing and transmission two stages, but they did not utilize censoring report. Besides, we investigate secondary transmission performance characterized by constraint of cognitive transmission outage probability, which differs from previous works [1518] without QoS requirements.

Note that, in this paper, denotes the expectation operator; represents the probability of a random variable; denotes the inverse function of ; denotes the number of set ; represents the feasible region of programming; and stands for the definition operator.

3. System Model and Transmission Description

3.1. System Model

Consider a cognitive radio system with the coexistence of a pair of primary users (PU) , a pair of secondary users ST-SD, and secondary relay users that are available to assist secondary sensing and transmission [4, 17, 18]. Primary spectrum owned by the primary network is divided into several narrowbands with fixed bandwidth. Like [2, 17], without loss of generality, we assume that the channels are modeled as independent Rayleigh fading, where denotes the fading coefficient. Besides, the additive white Gaussian noise (AWGN) received at is denoted as which is with zero mean and power spectral density . From [16], when PU’s signal is a complex PSK, we know that the false alarm probability and detection probability of energy detection (ED) with predefined detection threshold for cognitive users are, respectively, written as and , where is the sampling frequency, ,, , and . It is noted that, in this paper, the AND rule is used for decision fusion criterion to improve secondary system transmission opportunity as well as satisfy detection threshold . For implementation simplicity, we assume that the received signal-to-noise ratio at are the same value, i.e., . We let (i.e., ) and (i.e., ) denote two standard hypotheses used in spectrum sensing corresponding to PU’s absence and presence, respectively. According to [13, 15], the global detection probability and false alarm probability of CSS based on AND fusion rule can be, respectively, written as

Here, it should be pointed out that the secondary transmission opportunity is generally satisfied in a cognitive radio network. So we can easily obtain from . On the other hand, to avoid interference with PU’s transmission, the global detection probability needs to be not lower than a predetermined detection probability value , and then, we can obtain the relation . Following, we have by analyzing . To facilitate the upcoming analysis, the main parameters used are listed in Table 1 that are used in our analytical model. Based on the above discussion, we propose the CRT scheme to achieve energy-efficient transmission for overlay CRNs.


SymbolDescription

Channel coefficient between the link
Time slot duration
Sensing phase duration of MAC frame
Received signal-to-noise ratio at
Sensing time allocation factor
Number of available secondary relays
Secondary source, relay circuit power consumptions, and total transmission power
False alarm probability and detection probability of each SU
Transmitting power allocation on and
Transition matrix of PU’s state
Global detection probability and false alarm probability of CSS
Sensing, decision reporting consumption power for each SU
Average energy of CSS
Secondary transmitting average throughput
Secondary transmitting average energy consumption
Mean energy efficiency
Stationary probabilities for primary band to be busy and idle
Secondary outage probability
Minimum global detection probability of secondary transmission
Maximum outage probability of secondary transmission
Average busy and idle periods
Noise power
Transmission data rate of the link

3.2. Scheme Description

As illustrated in Figure 1, CRT scheme adopts both censoring report and best-relay-assisted strategies, which is completely different from the traditional noncooperative single cognitive transmission (NST). Each media access control (MAC) frame of SU is composed of sensing phase duration and transmission phase of duration [510]. We assume that SRs operate in a fixed time division multiple access (TDMA) manner which is commonly considered in existing studies [16, 17], where each MAC frame is composed of two consecutive durations called spectrum sensing phase and data transmission phase. CRT is periodically executed both in sensing and transmission phase at the beginning of each MAC frame. Besides, the following transmission phase is the same as conventional packet transmission processes [517]. The communication process of CRT scheme is described in detail below. (i)Cooperative sensing with censoring: As can be seen from Figure 1, each sensing process with DCR consists of two essential parts: local decision phase and decision report phase occupying and fractions of , respectively. The decision report phase is further split into equal segments. Then, each decision report phase is organized as subslots occupying fraction of . In DCR, ST holds a bits buffer denoted as to store all ’s latest reporting decision. Accordingly, needs a 1-bit buffer denoted as to save previous local decision result. makes its local decision first and sends only when . Hence, differential censoring report strategy in CRT can be formulated as where means the current sensing result differs from its buffer content in . Meanwhile, ST attempts to receive the indicator from in . In this case, if ’s indicator is received successfully, ST inverses the bit of buffer and makes a decision by . Otherwise, ST makes a decision by directly. Specific ideas of DCR strategy can be found in [5], which is omitted here due to space limitation(ii)Cooperative transmission by best-relay assisted: From Figure 1, transmission phase in CRT is further divided into two equal subslots, which are, respectively, denoted as and satisfying . In , ST’s data will broadcast to all secondary relay users with transmit power in the idle band. Then, with the largest end-to-end SNR is selected as best-relay-assisted user in cooperative secondary relay set and . The selection criterion of best-relay SUs can be described as . Motivated by the best-relay selection mechanism in [17], forward received data with transmitting power to SD by decode and forward protocol in . It is worth mentioning that cooperative sensing and cooperative transmission stages are all assisted by secondary relays in CRT. In actuality, to save energy, ST can select an appropriate set of cooperative users based on instantaneous channel state [18]

4. Secondary Sensing and Transmission Optimization

4.1. Cooperative Sensing and Transmission Energy Overhead

Our main goal is to reduce the energy consumptions of sensing stage and satisfy sufficient accuracy. Next, we will examine the average energy of CSS, denoted as . Without loss of generality, we assume that the circuit energy consumption of the secondary source users and the cooperative relay users are and and further assume that sensing and decision reporting consumption power for each cognitive user are denoted as and [10]. Note that we ignore the constant of circuit consumptions in analysis of ; i.e., consists of and only, due to the circuit power consumptions for each cognitive user which is always equal both in traditional and CRT schemes. By definition, false alarm means that PU is detected under and detection indicates that PU is detected under . Besides, miss detection is defined as that PU is undetected under . Like [5, 12, 15], as shown in Figure 2, whether PU is communicating or not can be modeled as a renewal process which alternates between busy and idle states. The busy and idle periods can be assumed to be exponentially distributed which can be expressed as and , where is the transition rate from busy to idle state and is the transition rate from idle to busy state.

Accordingly, the average busy and idle periods are and . Hence, the stationary probabilities for primary band to be busy and idle are given by and . From [16], the transition matrix is given by

For the purpose of comparison, the cooperative sensing energy consumption of traditional scheme without censoring can be derived as where is the average decision reporting probability by a traditional dedicated reporting channel [17]. It should be mentioned that, in order to fully guarantee primary transmission QoS, secondary transmission opportunity is generally satisfied in a cognitive radio network. Analyzing (5), we can get where indicates the cooperative sensing energy consumption when condition (I) in is satisfied.

Theorem 1. The average reporting probability by DCR is , where

Proof. According to the initial status of PU, sensing result, and PU’s transition cases in time slot , the SR decision censoring report scheme analysis for DCR can be, respectively, given as the following four possible scenarios as shown in Table 2.


PU’s initial statePU’s transition cases in SRi’s censoring report occurrence event

Busy⟶busyPU is undetected under (misdetection)
Busy⟶idlePU is undetected under (correct detection)
Idle⟶idlePU is detected under (false alarm)
Idle⟶busyPU is detected under (detection)

From Table 2, decision censoring report probability for in DCR can be calculated as

Therefore, substituting (4) into (8), after some algebra,is proved.

In what follows, represents the differential censoring sensing energy consumption in CRT, which can be calculated as

Next, we analyze the transmission rate and energy consumption for CRT scheme based on link status of and global decision of . For ease of analysis, let and represent the secondary transmission rate and energy consumption in CRT. As shown in Figure 1, the throughput and power consumption in transmission stage can be discussed four scenarios as follows: (1)Presence detection with probability : primary link is busy while sensing result is correct. In this case, in order to avoid interference to the primary system, the secondary system is prohibited transmission. Consequently, transmission rate is zero, and energy consumption in this case can be calculated as , where (2)Miss detection with probability : primary link is busy while sensing result is incorrect. In this case, like [12, 13], PU’s signal usually has a large interference to secondary transmission, which causes SU to be unable to correctly decode data, so secondary transmission rate is zero and secondary energy consumption can be expressed as (3)False alarm with probability : primary link is idle while sensing result is incorrect. There is no transmission with energy consumption (4)Absence detection with probability : primary link is idle while sensing result is correct. Secondary transmission is successful with a rate of , where and represents transmission data rate under bandwidth , , indicates the expectation operation, and secondary energy consumption in this case is calculated as

Based on above four cases, the expected energy consumption for CRT can be expressed as where . As discussed above, . usually needs to reach a predetermined detection value , and then, . So, we can approximate in [13] as follows: where , , and .

Similarly, the expected secondary throughput in the CRT scheme can be derived as

As mentioned in Section 3.1, secondary link channel coefficient obeys exponential distribution with mean , so it is impossible to obtain the closed expression in . However, we observed that is a concave function. Therefore, is still a concave function due to the concavity of min(log) function. In order to obtain the closed expression , like [13], is approximated by Jensen inequality relationship, and it can be described as where . Furthermore, the expected secondary throughput in CRT can be expressed as

In summary, according to the primary link state and global decision probability, the secondary throughput and energy consumption expressions under a corresponding situation are discussed.

4.2. Energy-Efficient Resource Optimization Problem in CRT

Similar to [12, 13, 17], mean EE is defined as a ratio between average throughput and average energy consumption, i.e., . Resource allocation in the CRT maximizes the mean EE as an optimization goal while considering secondary transmission QoS requirements. It is noted that the outage probability is introduced here to characterize secondary transmission QoS. According to the Shannon channel coding theorem, when channel capacity is lower than transmission rate required, it is considered a communication interruption. Thus, the secondary outage probability can be expressed as where and denotes the minimum secondary throughput. The EE maximization problem for CRT is denoted as P1, which is a resource allocation problem by joint optimizing sensing duration and power allocation for SU under constraints of sensing performance and minimal secondary outage probability. Mathematically, the optimization problem is modeled as

In P1, , , and , respectively, denote the minimum global detection probability, the maximum outage probability, and power consumption budgets of SUs. The constraint (16).C1 indicates that the global detection probability in CRT is not lower than a predefined detection threshold , thereby avoiding missed detection and causing interference to primary transmission. (16).C2 indicates that secondary outage probability is not greater than the present value , and thus, secondary transmission quality is guaranteed. In addition, constraint relationship (16).C3 indicates that secondary total energy consumption needs to be no greater than total power consumption budgets of SUs . It is noted that, like [5, 17], the saved sensing energy can be used for possible interweave transmission. Similar to the detection probability, can be minimized by optimizing in CRT due to the fundamental tradeoff in sensing time allocation. So, it can be formulated as P2: . Actually, minimum sensing energy can be achieved by optimizing in CRT, which will be validated by the simulations in Section 6. In addition, both the objective function and constraints are very complicated, which makes solving P1 with a low complexity resource allocation strategy difficult. However, we observe that and in (13) have the same optimal sensing duration, which accounts for and that do not consist . Based on this analysis, the original optimization problem P1 can be decoupled into P3 for optimizing sensing time and P4 for optimizing power allocation separately. Specifically, secondary EE is only determined by sensing time at a fixed power . Consequently, P3 is described as follows: where and represent system throughput and energy consumption for a given transmission power . Similarly, given a sensing time , EE is only determined by SU’s transmitting power in turn. Therefore, P4 can be expressed as where and denote the secondary throughput and energy consumption in CRT scheme for a given sensing time , respectively.

5. Joint Sensing Time and Power Resource Allocation

In this section, a joint sensing time and power resource allocation for CRT is discussed. The following lemmas are provided to solve P3 and P4. Hereafter, notations and are defined the first derivative and the second derivative of with respect to for ease of presentation.

Theorem 2. The objective function in P3 is a quasiconcave function with respect to .

Proof. From (14), we can derive the first derivative of as follows: where can be formulated as and the second-order derivative of can be further derived as where and are, respectively, written as

As discussion in Section 3.1, we have , , and . Analyzing (20) and (22), we have and , which leads to and , then , which means is concave function with respect to . Obviously, in (11) is an affine function. It is not difficult to analyze that in (11) is a convex function and monotonically increasing on , and is a convex function and monotonically decreasing on . From [19], is a convex function with respect to . Consequently, we can obtain that is convex function on . From which is a concave function with respect to , we can ready analyze that is a quasiconcave function with respect to by the definition of quasiconcave function in [20] (see Appendix A for detail).

Afterward, analyzing constraint (17).C3, we can obtain the mathematical characteristics described in Theorem 2.

Theorem 3. The constraint (17).C1 is equivalent to between , and is concave with respect to ; , , and in are, respectively, expressed as

Proof. See Appendix B.

According to Theorem 2 and Theorem 3, the optimal sensing duration time strategy is described in Theorem 4 specifically.

Theorem 4. The optimal sensing time allocation strategy for P3 can be characterized as where , , , and are, respectively, written as where , , and denotes the inverse function of .

Proof. The first derivative of and on are, respectively, derived as

From (27), we can get and. Examining (28) leads to , which indicates that is a monotonically increasing function on , so constraint (16).C1 in can be formulated as . Additionally, in Theorem 3 shows that is a monotonically increasing function on . We have as and as . In other words, monotonically decreases with respect to in and monotonically increases with respect to in . Therefore, the necessary and sufficient condition of (17).C1 existing a feasible in is . Furthermore, one of the following cases may occur for : (1) when,monotonically decreases onin, and, which indicates that there exists a uniquesatisfying; (2) when , similarly, we can get monotonically decreases on in , and , which indicates that there exists a unique . Conversely, if , there is no sensing time to satisfy . Above all, the optimal sensing duration is obtained at the boundary point or the stagnation point. This depends on the zero point of the first derivative of (17).C1.

Similar to Theorem 2, of P4 can be proved a quasiconcave function with respect to . We will not go into details here to save space. Before solving P4, Theorem 5 is given.

Theorem 5. In the CRT scheme, when secondary EE achieves maximum, the power allocation in P4 should be satisfied .

Proof. Prove by contradiction. According to the objective function in (16), secondary EE is not only related to but also limited by. Suppose that secondary system EE achieves optimum, there exists a , satisfying . From (12), it can be seen that the secondary throughput indicates that its rate depends on the smaller rate, which means one can reduce the transmit power of the larger rate hop in and , leading to . By the definition of EE function, i.e., , reducing transmission power of the larger rate hop will increase system EE under the condition that the system rate is a constant. Obviously, the original hypothesis leads to contradiction. Therefore, the original proposition is true.

Theorem 5 shows that power allocation when secondary transmission EE achieves the maximum must satisfy the following relationship . So, the optimal power for P4 is described in Theorem 6 specifically.

Theorem 6. The optimal power allocation strategy for P3 is where and , .

Proof. According to (18).C2 in P4, the upper bound of is obtained as And from (18).C2 in P4, the lower bound of P4 can be derived as

As discussed above, the optimal transmit power for P4 can be obtained either at both ends of the boundary or taken at the stagnation point . This is depending on the constraints (18).C1 and (18).C2 in P4. In addition, when secondary system EE achieves maximum, each rate for two hops should be equal. It indicates that the optimal power allocation satisfies relationship . So Theorem 6 is proved.

1: Initialize: Let , , tolerance
 and be the interval of GSS;
2: Compute ;
3: Compute ;
4: Repeat
5:    Find optimal sensing time allocation factor for a given ;
6:    Calculate ;
7:    If then
8:      Obtain , , ;
9:      Calculate ;
10:    Else
11:      Obtain , , ;
12:      Calculate ;
13:    Endif
14: Until converges.
15: Output the optimal time allocation factor of P2.
Algorithm 1: Obtain optimal sensing time allocation factor for CRT.

Theorem 6 shows that the optimal power allocation problem can be obtained according to (18) for a given sensing time . In what follows, aiming to maximize secondary EE, a joint resource allocation algorithm for sensing time and transmission power on and is proposed. We know that closed-form expressions of false alarm and detection probabilities for CRT are available in (1) and (2), which only require average channel gains and thus can be estimated in prior. Consequently, to solve P2, we develop linear search methods by golden section search (GSS) method [20]. The steps of optimal sensing time allocation factor of P2 is expressed in Algorithm 1.

The pseudocode for CRT is presented in Algorithm 2. It is not difficult to observe that the key basis in Algorithm 2 is and alternating iterations according to Theorem 4 and Theorem 6 until or does not satisfy the constraints of P3 and P4. For ease of analysis, we assume that sensing duration and transmission power after iteration are denoted as . Theorem 6 guarantees that is available during the th iteration, and substituting into (29) has . Similarly, can be calculated by Theorem 4, that is, substituting