Energy-Efficient Optimal Power Allocation for Fading ... - IEEE Xplore

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 4, APRIL 2016

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Energy-Efficient Optimal Power Allocation for Fading Cognitive Radio Channels: Ergodic Capacity, Outage Capacity, and Minimum-Rate Capacity Fuhui Zhou, Student Member, IEEE, Norman C. Beaulieu, Fellow, IEEE, Zan Li, Senior Member, IEEE, Jiangbo Si, Member, IEEE, and Peihan Qi Abstract—Green communications is an inevitable trend for future communication network design, especially for a cognitive radio network. Power allocation strategies are of crucial importance for green cognitive radio networks. However, energyefficient power allocation strategies in green cognitive radio networks have not been fully studied. Energy efficiency maximization problems are analyzed in delay-insensitive cognitive radio, delaysensitive cognitive radio, and simultaneously delay-insensitive and delay-sensitive cognitive radio, where a secondary user coexists with a primary user and the channels are fading. Using fractional programming and convex optimization techniques, energyefficient optimal power allocation strategies are proposed subject to constraints on the average interference power, along with the peak/average transmit power. It is shown that the secondary user can achieve energy efficiency gains under the average transmit power constraint, in contrast to the peak transmit power constraint. Simulation results show that the fading of the channel between the primary user transmitter and the secondary user receiver and the fading of the channel between the secondary user transmitter and the primary user receiver are favorable to the secondary user with respect to the energy efficiency maximization of the secondary user, whereas the fading of the channel between the secondary user transmitter and the secondary user receiver is unfavorable to the secondary user. Index Terms—Cognitive radio, energy efficiency, ergodic capacity, fading channels, interference power constraint, minimum rate capacity, outage capacity, power allocation strategies, spectrum sharing, transmit power constraint. Manuscript received June 7, 2015; revised October 23, 2015; accepted December 11, 2015. Date of publication December 17, 2015; date of current version April 7, 2016. This work was supported in part by the Fundamental Research Funds for the Central Universities under Grant 7215433803, Grant 7214433802, and Grant 7214514902, in part by the 863 Project under Grant 2014AA8098080E, in part by the 111 Project under Grant B08038, and in part by the Natural Science Foundation of China under Grant 61301179, Grant 61401323, Grant 61401338, Grant 61471395, and in part by a scholarship from China Scholarship Council. The associate editor coordinating the review of this paper and approving it for publication was H. Wymeersch. (Corresponding author: Zan Li). F. Zhou, Z. Li, J. Si, and P. Qi are with the State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an 710071, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). N.C. Beaulieu is with the School of Information and Communication Engineering, Beijing University of Posts and Telecommunications (BUPT), Beijing 100876, China, and also with the Beijing Key Laboratory of Network System Architecture and Convergence, Beijing University of Posts and Telecommunications (BUPT), Beijing 100876, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2015.2509069

I. I NTRODUCTION

C

OGNITIVE radio (CR) is one of the most promising technologies that aim for efficient spectrum utilization and alleviating the spectrum scarcity problem [1]–[3]. In CR, a secondary user (SU) is allowed access to primary user (PU) spectrum bands on the condition that the interference caused to PUs is tolerable. There are several paradigms in CR, such as opportunistic spectrum access and spectrum sharing [4]. According to opportunistic spectrum access, the SU is able to access a PU’s spectrum band only when the PU is detected idle. However, it is difficult to rapidly and precisely detect an idle spectrum band. Alternatively, spectrum sharing is considered to achieve higher spectrum efficiency (SE) since the SU can coexist with the PU, as long as the interference from the SU is tolerable [5]. In this paper, we focus on a CR under spectrum sharing due to its easy implementation and higher SE. Energy efficient communication networks are becoming an inevitable trend for future wireless network design since they can ameliorate energy consumption and provide users a more satisfactory experience [6]–[8]. The diverse and ubiquitous wireless services and the tremendous increase of mobile devices have contributed to the sharp growth of energy consumption and greenhouse gas emission. According to the results in [6]–[10], 2% to 10% of global energy consumption and 2% of the greenhouse gas are generated by information and communication technologies. Thus, energy efficient transmission is of crucial significance for CR design since it not only strictly requires utilizing limited transmission power, but also considers operational expenditure and the greenhouse problem. However, investigation into the energy efficiency (EE) issue in CR has been limited up to now [6]–[10]. In CR, an optimal design of the SU’s transmission power strategy not only provides the SU a reasonably high transmission rate with limited power, but well protects the PU from harmful interference. Moreover, an energy-efficient optimal power allocation strategy cuts deployment costs to enable economical green CR and reduces environmental impact. Thus, it is important to design energy-efficient optimal power allocation strategies for green CR. Although many works have addressed the power allocation strategy in non-CR and CR, the optimal power allocation strategies designed for non-CR and CR may not be optimal in green CR in terms of EE maximization.

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A. Related Work and Motivation Optimal power allocation strategies have been well studied for conventional non-CR and CR. Recently, some investigations have been dedicated to design energy-efficient optimal power allocation strategies for green non-CR and green CR in order to maximize the achievable EE. Those works can be summarized as follows. • Non-CR networks: A well-known “water-filling” power allocation strategy was proposed to maximize the achievable ergodic capacity (EC) in a fading channel [11]. In [12], under the perfect channel state information (CSI) assumption, the minimization of the outage probability (OP) problem with delay-constrained transmission has been studied in block-fading channels. The results show that the optimal power allocation has a “truncated channel inversion” structure. A class of low-complexity power allocation strategies based on a subchannel grouping technique were presented in [13]. Recently, the EE instead of the capacity was maximized by the proposed energy-efficient optimal power allocation strategy in [14]–[16]. Energy-efficient optimal power allocation strategies that maximize the instantaneous EE and the ergodic EE under the average/peak power constraint were designed in [14]. In [15], besides a power constraint, the instantaneous EE was maximized on the condition that a minimum rate was guaranteed. Energy-efficient optimal power allocation strategies for maximizing the ergodic EE were analyzed in the low rate regime in [16]. • CR: Different from non-CR networks, in CR, the design of power allocation strategies of the SU should consider the interference caused to the PU in order to protect the quality of service (QoS) of the PU. In [17], the authors have studied whether a peak interference power (PIP) constraint can better protect PUs, or whether an average interference power (AIP) constraint can better protect PUs. It was shown that the AIP constraint can better protect PUs and simultaneously provide SUs higher transmit data rates. The maximum ergodic throughputs of CR operating on wideband sensing-based spectrum sharing together with wideband opportunistic spectrum access schemes were realized based on the design of the optimal sensing time and optimal transmit power, subject to average transmit power (ATP) and AIP constraints in [18]. In [19], the optimal power design was extended into a sensing-enhanced spectrum sharing CR while continuous power allocation strategies based on sensing the PU channels in multiband CR have been proposed in [20]. In spectrum sharing CR, the optimal power allocation strategies were presented under different power constraints in [21] and [22]. A PU OP constraint instead of power constraints was imposed to protect the QoS of the PU and further improve the flexibly of optimal power allocation strategies of the SU in [23]. The joint optimization of bandwidth and power allocation for CR were first investigated in [24]. Optimal bandwidth allocation was derived for any given power constraint and the structures of optimal power allocations were analyzed

under several possible combinations of power constraints. Under assumption of knowledge of statistical CSI or partial CSI, the authors proposed the optimal power control strategies and analyzed the performance of spectrum sharing CR in [25] and [26]. In [27], the OP of a SU in spectrum sharing CR was minimized based on an optimal transmit power allocation scheme where the SU only has quantized channel state information. Optimal power control strategies that maximize the achievable rates of CR with arbitrary input distributions were studied in [28]. • Green CR: In green CR, the maximization of the EE instead of the capacity of the SU is the object. Energy-efficient optimal power allocation strategies were designed to maximize the EE of the SU in CR with the opportunistic spectrum access paradigm in [29]–[31] and in CR with the spectrum sharing paradigm in [32]– [39]. In [31], the soft-sensing information, adaptive power and adaptive sensing thresholds were designed to achieve green CR. Ref. [32] proposed a low complexity method to obtain the transmit power level that maximizes the EE of CR. In [33], the energy-efficient power allocation problem in heterogeneous CR with femtocells was formulated as a Stackelberg game. Energy-efficient power allocation strategies for OFDM-based CR were studied in [34] and [35]. In [36], the energy-efficient resource allocation problem was formulated as a sum-EE maximization problem. A preference-based scheme was proposed obtain the energy-efficient power allocation strategy. An energyefficient joint relay selection and power allocation scheme was proposed in [37]. The optimal relay selection and power allocation policy was decided in a distributed way. The design of joint power allocation and transmission beamforming strategy for MIMO CR was proposed to maximize the EE under constraints on transmit power and interference power in [38]. All the above-mentioned works for green CR [29]–[38] consider static optimization on the basis of instantaneous CSI. However, it is inappropriate for the fading channel since the metrics related to the protection of the PU’s quality of services (QoS) and the EE should be based on all the fading states. Thus, the average SUs’ EE and average metrics related to the protection of the PU’s QoS, such as AIP, should be considered in fading channels. Although average metrics have been considered in non-CR [14]–[16], unique challenges need to be addressed for green CR. In particular, the interference caused to the PU from the SU and the mutual interferences between the PU and the SU should be considered in green CR. To the best of the authors’ knowledge, the only existing works considering the average EE of green CR are [30] and [39]. In [30], the average EE of CR operating on the opportunistic spectrum access paradigm was analyzed and the energy-efficient optimal power allocation strategy was proposed under imperfect spectrum sensing. However, the impact of the fading CSI between the PU’s transmitter and the SU’s receiver on the SU’s average EE was not fully considered. Although this impact on the SU’s average EE of CR with the spectrum sharing paradigm has been considered in [39], it only takes the PU’s OP constraint into

ZHOU et al.: ENERGY-EFFICIENT OPTIMAL POWER ALLOCATION FOR FADING COGNITIVE RADIO CHANNELS

consideration. Moreover, [39] only considered the EE maximization problem in delay-insensitive CR. However, in wireless systems that are delay sensitive, such as voice and video applications, the OC of CR is a more appropriate optimization object. Furthermore, in wireless systems that simultaneously transmit delay-sensitive and delay-insensitive messages [40], [41], the minimum-rate capacity is a more appropriate metric. Thus, it is an interesting, although challenging problem, to study EE maximization problems in those three green CRs. On the other hand, an energy-efficient optimal power allocation strategy not only provides the SU a reasonably high transmission rate, but well protects the PU from intolerable interference. Moreover, it can maximize the EE of the SU in green CR. In this paper, the AIP is considered as the metric of PU protection, which is different from [39]. To our best knowledge, it is the first time that energy-efficient optimal power allocation strategies are designed to maximize the EE of the SU subject to the AIP constraint in fading channels. EE maximization problems are formulated in delay-insensitive CR, delay-sensitive CR, and simultaneously delay-insensitive and delay-sensitive CR under spectrum sharing. B. Contributions and Organization Motivated by the works in [29]–[39], this paper is devoted to designing energy-efficient optimal power allocation strategies for green CR in fading channels. Different from the works in [29]–[38], the fading channel and the average EE of the CR are considered. The AIP is taken as the PU protection metric, which is different from the work in [39]. EE maximization problems are studied in CRs where the SU is delayinsensitive, delay-sensitive, or simultaneously delay-insensitive and delay sensitive. To this end, the EE maximization problem is first formulated in delay-insensitive CR. Then, EE definitions are extended to delay-sensitive CR and simultaneously delayinsensitive and delay-sensitive CR. The framework for EE maximization problems is formulated. The main contributions of this work are summarized as follows. 1) Energy-efficient optimal power allocation strategies are designed that maximize the EE of the SU in delayinsensitive CR with the spectrum sharing paradigm. Different from the work in [39], which takes a constraint on the maximum tolerable OP of the PU into consideration, a constraint on AIP is identified as the protection metric of the PU in this paper. The PTP/ATP limit is the constraint of the SU transmit power. It is proved that the EE maximization problem is a nonlinear concave fractional programming problem. Based on fractional programming and convex optimization techniques, energy-efficient optimal power strategies are proposed to maximize the EE of the SU. It is shown that the energyefficient optimal power allocation strategies are required to take the EE of the SU and the power amplifier coefficient into consideration. Simulation results show that the SU can achieve an EE gain under the ATP constraint in contrast to the PTP constraint. 2) The EE maximization problem is first studied in delaysensitive CR with the spectrum sharing paradigm.

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Although the EE maximization problem in delaysensitive CR is not a nonlinear concave fractional programming problem under the power constrains, Dinkelbach’s method and convex optimization techniques still can be applied to solve the EE maximization problem and energy-efficient optimal power allocation strategies are derived. It is proved that energy-efficient optimal power allocation strategies have similar forms as the well-know truncated channel inversion power allocation strategies. Simulation results show that the fading of the channel between the PU-Tx and the SU-Rx is more favorable to the SU than the fading of the channel between the SU-Tx and the PU-Rx with respect to EE maximization of the SU, whereas the fading of the channel between the SU-TX and the SU-Rx is unfavorable to the SU. 3) The EE maximization problem is extended to simultaneously delay-insensitive and delay-sensitive CR. It is shown that the energy-efficient optimal transmit power levels of the SU in this CR that maximize the EE of the SU depend on the tradeoff between the achieved transmit rate of the SU and the outage cost of the SU. Simulation results show that the maximization of the EE of the SU in the simultaneously delay-insensitive and delay-sensitive CR converges to that of the SU achieved in delay-insensitive CR when the AIP constraint becomes looser. The rest of this paper is organized as follows. Section II presents the system model and power constraints. The EE maximization problem is formulated in delay-insensitive CR in Section III. Section IV presents the EE maximization problem in delay-sensitive CR. The EE maximization problem is examined in simultaneously delay-insensitive and delay-sensitive CR in Section V. Section VI presents simulation results. The paper concludes with Section VII.

II. S YSTEM M ODEL AND P OWER C ONSTRAINTS A. System Model The purpose of this paper is to investigate the EE problem. Considering a simplified CR network will help to clarify the issues pertaining to EE and permit reaching meaningful insights into the EE problem. Note that the simplified CR network has been widely used [23], [25], [26], and [39]. As shown in Fig. 1, the simplified CR network model consists of one secondary link and one primary link. The secondary link coexists with the primary link under the spectrum sharing paradigm. The primary link has one PU transmitter (PU-Tx) and PU receiver (PU-Rx) pair. The secondary link consists of a SU transmitter (SU-Tx) and a SU receiver (SU-Rx). It is assumed that the PU and SU transmitter transmit signals in the same narrowband frequency channel. All the terminals have one antenna. The instantaneous channel power gains for the secondary link, the link from SU-Tx to SU-Rx, the primary link, and the link from PU-Tx to PU-Rx at fading state ν are denoted by gss (ν), gsp (ν), h pp (ν), and h ps (ν), respectively, where ν denotes the fading index for all related channels. It is assumed that all the channels involved are block fading and the channel power gains

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the interference link from SU-Tx to PU-Rx, and the link from PU-Tx to SU-Rx. From the perspective of the PU, an interference power constraint should be imposed on the SU-Tx communication in order to protect the PU from intolerable interference. It was shown that the AIP constraint not only provides better protection of the PU, but provides the SU higher capacity in [17]. Thus, the AIP constraint is applied here, given as (2) E gsp (ν) Ps (ν) ≤ PI n , ∀ν where PI n represents the maximum AIP that is tolerable for the PU-Rx link. III. E NERGY E FFICIENT E RGODIC C APACITY

Fig. 1. The system model.

are independent identically distributed (i.i.d.) ergodic and stationary random variables with continuous probability density functions. The noise at the SU-Rx is a circularly symmetric complex additive white Gaussian noise (AWGN) with mean zero and variance σw2 . It is assumed that the PU employs non-adaptive power transmission with constant power, denoted by Pp . Note that this assumption has been widely used in [23], [25], [26] and [39], and leads to a theoretical limit to performance. For the purpose of the analysis on the average EE limits in fading channels, the SU is assumed to have perfect knowledge of all involved instantaneous CSI at all fading states. Note that the CSI of the secondary link can be obtained by estimating it at the SU-Rx and then sending it back to the SU-Tx through a feedback link (assumed error-free in this simplified model). The interference CSI of the link from SU-Tx to PU-Rx and the link from PU-Tx to SU-Rx can be obtained at the SU-Tx through cooperation of the PU [42], or can be obtained from a third party such as a manager center [43]. B. Power Constraints According to the works [17]–[28], from the perspective of the SU-Tx communication, there are two kinds of power constrains. One is the PTP constraint which is related to the nonlinearity of power amplifiers. The other is the ATP constraint for the aim of satisfying the long-term power budget of the SU. The PTP constraint and the ATP constraint can be given as Ps (ν) ≤ Pth , ∀ν, E {Ps (ν)} ≤ Pth , ∀ν Ps (ν) ≥ 0, ∀ν

(1a) (1b) (1c)

where Ps (ν) denotes the transmit power of the SU at fading state ν. Pth and Pth denote the PTP and the maximum ATP of the SU, respectively; E (·) denotes the expectation over the different fading states, including the CSI of the second link,

According to [29]–[31], EE in fading channels is defined as the ratio of the average rate to the average power consumption. In conventional delay-insensitive CR, the EC is an appropriate metric to evaluate the performance of the SU, which is achieved by averaging over all states of an ergodic fading channel. Thus, EE in delay-insensitive green CR is the ratio of the EC to the average power consumption. In order to maximize the EE of the SU, energy-efficient optimal power allocation strategies are designed under the AIP constraint and the PTP/ATP constraint. A. Average Transmit Power Constraint In this subsection, the green CR scenario where the SU is delay-insensitive in fading channels is considered under the average power constraints. It is assumed that the PU uses a Gaussian codebook and the SU experiences interference from the PU as additional Gaussian noise. Thus, the EE maximization problem under the average power constraints can be formulated as the following problem, P1 , given as (ν)Ps (ν) E log2 1 + h gss(ν)P 2 ps p +σw P1 : max η E E (Ps (ν)) = Ps (ν) E {ζ Ps (ν) + PC } (3a) s.t. (1b) , (1c) , and (2) are satisfied,

(3b)

where η E E (Ps (ν)) denotes the EE of the SU. ζ and PC represent the amplifier coefficient1 and the constant circuit power consumption of the SU-Tx link, respectively. In Appendix A, the EE maximization problem, P1 , is proved to be a nonlinear concave fractional programming problem. Let S1 denote the set S1 = {Ps (ν) |Ps (ν) ∈ (3b)}. Since problem P1 is a nonlinear concave fractional programming problem, Lemma 1-3 can be proved. Lemma 1: The objective function η E E (Ps (ν)) of the EE maximization problem (P1 ) is strictly quasiconcave on S1 since the numerator of η E E (Ps (ν)) is strictly concave. Proof: See Appendix B. Lemma 2: η E E (Ps (ν)) of the EE maximization problem (P1 ) is strictly pseudoconcave on S1 since the numerator and denominator of η E E (Ps (ν)) are differentiable and the numerator is strictly concave. 1 1/ζ is also known as the power efficiency of the power amplifier.


Proof: See Appendix C. Lemma 3: In a concave fractional programming problem, P1 , any local maximum is a global maximum, and P1 has at most one maximum since P1 is strictly quasiconcave. Proof: It is straightforward to prove Lemma 3. Based on Lemma 1 and Lemma 2, η E E (Ps (ν)) ≤ η E E (Pυ (ν)) holds at υ (ν)) = 0. Thus, η E E (Pυ (ν)) is the global any Ps (ν) if dη EdEP(P υ (ν) maximum. In order to solve problem P1 and obtain the global maximum, observe that problem P1 can be equivalent to a parameter problem based on Dinkelbach’s method [47], denoted by P2 , given as

gss (ν) Ps (ν) P2 : max f (η) = E log2 1 + Ps (ν)∈S1 h ps (ν) Pp + σw2 −ηE {ζ Ps (ν) + PC }

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TABLE I E NERGY-E FFICIENT O PTIMAL P OWER A LLOCATION A LGORITHM

(4)

where η is a non-negative parameter. It is easy to prove that problem P2 is a convex problem. Thus, problem P2 can be solved by using the Lagrange duality method and the duality gap is zero [44]. The Lagrangian with respective to the transmit power Ps (ν) of P2 is given as

gss (ν) Ps (ν) L (Ps (ν) , τ, μ) = E log2 1 + h ps (ν) Pp + σw2 − ηE {ζ Ps (ν) + PC } − τ E {Ps (ν)} − Pth − μ E gsp (ν) Ps (ν) − PI n (5) where τ and μ are the non-negative dual variables related to eq. (1b) and eq. (2), respectively. Thus, the Lagrange dual function of P2 can be presented as g (τ, μ) =

max

0≤Ps (ν),∀ν

L (Ps (ν) , τ, μ).

(6)

Similar to [17] and [23], the problem given by eq. (6) can be decoupled into parallel subproblems on the basis of the Lagrange dual-decomposition method [44]. Those subproblems have the same structure for each fading state. Thus, the corresponding subproblem for a particular fading state can be given as

gss Ps max y (Ps ) = log2 1 + 0≤Ps h ps Pp + σw2 − ηζ Ps −τ Ps − μgsp Ps .

(7)

Note that the fading state index ν is dropped. P2 can be solved by iteratively solving eq. (7) for all fading states with respect to fixed τ and μ, and updating τ and μ by using the subgradient method [44]. Since y (Ps ) is a concave function related to Ps , the energy-efficient optimal power allocation strategy of P2 , opt denoted by Ps , can be given as in Theorem 1. Theorem 1: The energy-efficient optimal power allocation strategy for P2 is given by +

h ps Pp + σw2 1 opt Ps = (8) − gss ηζ + τ + μgsp ln 2 where [a]+ = max (a, 0) and max (a, 0) denotes the maximum between a and 0.

Remark 1: The design of an energy-efficient optimal power allocation strategy for maximizing the achievable EE is required to take the EE of the SU and the power amplifier coefficient into consideration, which is different from the traditional CR. When η = 0, P2 degenerates to the EC maximization problem and the energy-efficient optimal power allocation strategy is similar to the form given in [22, eq. (14)]. It is seen that P2 can be efficiently solved by using the proposed energy-efficient optimal power allocation strategy given by eq. (8) for a given η. In order to solve problem P1 and obtain the maximum EE, the well-known Dinkelbach’s method is applied. It has been proved that it converges to the optimal solution with a superlinear convergence rate [48]. An optimal power allocation algorithm based Dinkelbach’s method is

on proposed to solve P1 . When f ηnE E = 0, the energy-efficient optimal transmit power and the maximum EE are obtained. Otherwise, an ξ -optimal solution with an error tolerance ξ is adopted. Inother words, the optimal solution is obtained when f ηnE E ≤ ξ . The optimal power allocation algorithm based on Dinkelbach’s method, denoted by Algorithm 1, is summarized in Table I.

B. Peak Transmit Power Constraint In this subsection, the PTP constraint is considered for the SU. In this case, the EE maximization problem can be formulated as the following problem, P3 , given as

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P3 :


max η E E (Ps (ν)) =

E log2 1 +

Ps (ν)

gss (ν)Ps (ν) h ps (ν)Pp +σw2

E {ζ Ps (ν) + PC }

(9a) s.t. (1a) , (1c) and (2) are satisfied.

(9b)

Let S2 denote the set S2 = {Ps (ν) |Ps (ν) ∈ (9b)}. It is easy to prove that the set S2 is a convex set. Similar to P1 , P3 is also a concave fractional programming problem. Thus, using Dinkelbach’s method, problem P3 is equivalent to the parameter optimization problem, P4 , given as

gss (ν) Ps (ν) P4 : max f (η) = E log2 1 + Ps (ν)∈S2 h ps (ν) Pp + σw2 −ηE {ζ Ps (ν) + PC }

(10)

where η is a nonnegative parameter. Similar to P2 , the Lagrange duality method is applied to solve P4 . Let μ denote the nonnegative dual variable with respect to eq. (2). Thus, P4 can be decomposed into parallel subproblems having the same structure for all fading states, given as

gss Ps − ηζ Ps max y (Ps ) = log2 1 + 0≤Ps ≤Pth h ps Pp + σw2 −μgsp Ps .

(11)

For brevity, the fading state index ν is dropped. Similar to P2 , P4 can be solved iteratively by solving eq. (11) for all fading states with respect to fixed μ, and updating μ by using the subgradient method. Since y (Ps ) is a concave function with respect to Ps , the energy-efficient optimal power alloopt cation strategy for problem P4 , denoted by Ps , is given in Theorem 2. Theorem 2: The energy-efficient optimal power allocation strategy for P4 is given by + h ps Pp + σw2 1 2 = P (12a) − gss ηζ + μgsp ln 2

opt 2 , Pth . Ps = min P (12b) Remark 2: In delay-insensitive green CR, the energyefficient optimal power allocation strategy for maximizing the achievable EE under constraints on the PTP and the AIP has a similar form to that given in [22, eq. (12)] when η = 0. For a given η, P4 can be efficiently solved by using the proposed strategy given by eq. (12). Algorithm 1 can be modified to solve P3 . In this case, only μ is required to be updated and opt the energy-efficient optimal power Ps is calculated by using eq. (12). For brevity, the details are not given here. IV. E NERGY E FFICIENT O UTAGE C APACITY In delay-sensitive CR where the SU is sensitive to delay, such as voice and video, the OC instead of the EC is a more appropriate metric to evaluate the performance of the SU. The OC is defined as the constant rate that is achieved with an OP less than a threshold [40]–[42]. Thus, in delay-sensitive green CR, the EE of the SU is the ratio of the product of the constant OC and the

non-outage probability to the average power consumption. In this section, the EE maximization problems are formulated in delay-sensitive CRs. The energy-efficient optimal power allocation strategies are proposed to maximize the EE of the SU under the AIP constraint, subject to either the ATP constraint or the PTP constraint. A. Average Transmit Power Constraint In this subsection, the ATP constraint is considered. Motivated by the OP minimization problem proposed in [21]– [25], the EE maximization problem under constraints on the AIP and the ATP in delay-sensitive CR can be formulated as the problem, P5 , expressed as P5 :

rs E {1 − χs (v)} E {ζ Ps (ν) + PC } s.t. (1b) , (1c) , and (2) are satisfied max η E E (Ps (ν)) = Ps (ν)

(13a) (13b)

where χs (v) denotes an indicator function for the outage event of the SU at fading state ν, given as (ν)Ps (ν) < rs 1, log2 1 + h gss(ν)P 2 ps p +σw χs (v) = (14) 0, other wise where rs is the prescribed OC of the SU. Since χs (v) is not a concave function with respect to Ps , P5 is not a concave fractional programming problem. However, P5 can still be solved by using Dinkelbach’s method [47]. The reason is that the numerator and the denominator of η E E (Ps (ν)) are continuous and satisfy E {ζ Ps (ν) + PC } > 0 and E {1 − χs (v)} ≥ 0 for all Ps (ν) ∈ S1 . Thus, problem P5 is equivalent to the parameter optimization problem, P6 , given as P6 : max f (η) = rs E {1 − χs (v)} − ηE {ζ Ps (ν) + PC } Ps (ν)∈S1

(15) where η is a non-negative parameter. Using a similar method as used for P2 and P4 , P6 can be decomposed into parallel subproblems having the same structure for all fading states. For brevity, the fading state index ν is dropped. Let τ and μ denote the non-negative dual variables associated with eq. (1b) and eq. (2), respectively. The subproblem for a particular state can be given as max y (Ps ) = −rs χs (Ps ) − ηζ Ps − τ Ps − μgsp Ps

0≤Ps

(16)

where the indicator function χs (Ps ) is expressed as a function of Ps . Then, P6 can be solved by iteratively solving eq. (16) for all fading states with fixed τ and μ, and updating τ and μ by using the subgradient method. χs (Ps ) is a step function with respect to Ps . Let y denote the turning point of χs (Ps ), given as

(2rs − 1) h ps Pp + σw2 (17) y= gss where y ≥ 0. It is seen that the minimum power required for opt the SU to guarantee the OC rs is y. Let Ps denote the


energy-efficient optimal transmit power where y (Ps ) achieves the maximum. Since χs (Ps ) = 1 for Ps < y and χs (Ps ) = 0 for Ps ≥ y, the

maximum of y (Ps ) may be −rs when Ps = 0 or may be − ηζ + μgsp + τ y when Ps = y, and depends on their relationship. This is formalized in Theorem 3. Theorem 3: The energy-efficient optimal power allocation strategy for P6 is given by rs 0, y > ηζ +μg opt sp +τ Ps = (18) rs y, y ≤ ηζ +μgsp +τ . Remark 3: The EE maximization problem in delay-sensitive green CR degenerates into the OP minimization problem when η = 0. In this case, the energy-efficient optimal power allocation strategy given by eq. (18) is similar to the form given in [22, eq. (30)]. It is seen that P6 can be efficiently solved by using the energy-efficient optimal power allocation strategy given by eq. (18) for a given η. P5 can be solved by using the modified Algorithm 1. In this case, the energy-efficient

optimal power, opt Ps , is calculated by using eq. (18). f ηnE E , and EE, ηnE E , are calculated by using eq. (15) and eq. (13a), respectively. For brevity, the details are not given here.

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Case 2: y ≤ Pth . In this case, the maximumof y (Ps ) may be −rs when Ps = 0 or may be − ηζ + μgsp y when Ps = y; which is the maximum depends on their relationship. When y > rs / ηζ + μgsp , the required power to maintain the OC rs of the SU is very large, and the SU stops transmitting in order opt to save power and Ps = 0. Otherwise, the SU transmits with opt Ps = y. Theorem 4: The energy-efficient optimal power allocation strategy for P8 is given by ⎧ ⎪ ⎨ 0, y >r Pth s opt Ps = 0, ηζ +μgsp < y ≤ Pth (22) ⎪ rs ⎩ y, y ≤ Pth , y ≤ ηζ +μgsp . Remark 4: Theorem 4 is similar to the form given in [22, eq. (26)] when η = 0. Similar to P5 , Algorithm 1 can be modified to solve P7 . In this case, only μ is required to be updated and the energy-efficient optimal power is calculated based on eq. (22).

f ηnE E , and EE, ηnE E , are calculated by using eq. (20) and eq. (13a), respectively. For brevity, the details are not given here. V. E NERGY E FFICIENT M INIMUM R ATE C APACITY

B. Peak Transmit Power Constraint In this subsection, the AIP constraint and the PTP constraint are considered. In this case, the EE maximization problem in delay-sensitive CR can be formulated as problem, P7 , given as P7 :

rs E {1 − χs (v)} E {ζ Ps (ν) + PC } s.t. (1a) , (1c) , and (2) are satisfied. max η E E (Ps (ν)) = Ps (ν)

(19a) (19b)

It is seen that P7 is a general fractional programming problem. Similar to P5 , Dinkelbach’s method can be applied to solve P7 . Thus, the equivalent parameter optimization problem (P8 ) of P7 can be given as P8 : max f (η) = rs E {1 − χs (v)} − ηE {ζ Ps (ν) + PC }

As previously stated, the EC is an appropriate metric to evaluate the performance of the SU in delay-insensitive CR. The OC is more efficient in delay-sensitive CR since the SU transmits more power in channels with poorer quality in order to maintain a constant rate. However, neither the EC or the OC are the optimal metric to evaluate the performance of the SU in simultaneously delay-insensitive and delay-sensitive CR. In this CR, the minimum rate capacity is more appropriate to evaluate the performance of the SU since the long-term average rate is maximized as long as a minimum rate is guaranteed for a certain time [40], [41]. In this section, the EE maximization problem is formulated in simultaneously delay-insensitive and delay-sensitive green CR. Energy-efficient optimal power allocation strategies are found to maximize the EE under the AIP constraint and the ATP/PTP constraint.

Ps (ν)∈S2

(20) where η is a non-negative parameter. Let μ denote the nonnegative dual variable related to eq. (2). Using a similar method as used for P6 , P8 can be decomposed into parallel subproblems having the same structure for all different fading states. For convenient analysis, the fading state indicator is dropped and the subproblems can be given as max y (Ps ) = −rs χs (Ps ) − ηζ Ps − μgsp Ps

0≤Ps ≤Pth

opt

(21)

where χs (Ps ) is an explicit function of Ps . Let Ps denote the energy-efficient optimal power of P8 . The following results can be obtained by solving eq. (21). Case 1: y > Pth , where y is given by eq. (17). In this case, the required power to maintain the OC rs of the SU is larger than the PTP constraint. The SU is always in outage. Thus, opt Ps = 0.

A. Average Transmit Power Constraint In this subsection, the ATP constraint is considered. Motivated by the traditional minimum rate capacity maximization problem, the EE maximization problem with respect to the minimum rate capacity under constraints on the AIP and ATP in simultaneously delay-insensitive and delay-sensitive green CR can be formulated as problem, P9 , given as (ν)Ps (ν) E log2 1 + h gss(ν)P 2 ps p +σw P9 : max η E E (Ps (ν)) = Ps (ν) E {ζ Ps (ν) + PC } (23a)

gss (ν) Ps (ν) s.t. Pr log2 1 + < rs ≤ Pout h ps (ν) Pp + σw2 (23b) (1b) , (1c) , and (2) are satisfied

(23c)

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where rs is the prescribed OC of the SU, and Pout is the OP threshold. Note that there is no feasible power allocation when all the power allocation levels satisfy constraints given by eq. (1b), eq. (1c) and eq. (2) but can not satisfy the constraint given by eq. (23b). Similar to Section III and IV, Dinkelbach’s method is applied to solve P9 . Let S3 denote the set S3 = {Ps (ν) |Ps (ν) ∈ (23b) , Ps (ν) ∈ (23c)}. Then, P9 can be equivalent to the parameter optimization problem (P10 ), given as

gss (ν) Ps (ν) P10 : max f (η) = E log2 1 + Ps (ν)∈S3 h ps (ν) Pp + σw2 − ηE {ζ Ps (ν) + PC }

(24)

where η is a non-negative parameter. For convenient notation, the fading state index ν is dropped. Using similar approaches to those used in Section IV and Section V, P10 can be decomposed into parallel subproblems having the same structure. For a particular state, the subproblem (P11 ) can be given as

gss Ps − ηζ Ps − τ Ps P11 : max log2 1 + 0≤Ps h ps Pp + σw2 − μgsp Ps − λ {χs (Ps )}

(25)

where τ , μ and λ denote the non-negative dual variables associated with eq. (1b), eq. (2) and eq. (23b), respectively. χs (Ps ) is an explicit function of Ps . χs (Ps ) = 1 for Ps < y and χs (Ps ) = 0 for Ps ≥ y, where y is given by eq. (17). In order to solve P11 , let P12 be defined as

gss Ps P12 : max H (Ps ) = log2 1 + 0≤Ps h ps Pp + σw2 − ηζ Ps − τ Ps − μgsp Ps .

(26)

It is seen that the object function of P11 is H (Ps ) − λ {χs (Ps )}. s denote the optimal solution of P12 . Since P12 is equivaLet P s can be given as lent to the problem given in eq. (7), P +

h ps Pp + σw2 1 s = P . (27) − gss ηζ + τ + μgsp ln 2 opt

Let Ps denote the optimal solution of P10 . The following results can be obtained by solving eq. (25). s = 0, and P11 s . In this case, χs P Case 1: 0 ≤ y ≤ P opt s . In this achieves the maximum at Ps = Ps . Thus, Ps = P scenario, the minimum rate always can be maintained. P9 is thus equivalent to P1 .

s − λ < H (y), s . In this case, when H P Case 2: y > P opt opt s . Ps = y; otherwise, Ps = P Theorem 5: The energy-efficient optimal power allocation strategy for P10 is given by ⎧ s , 0 ≤ y ≤ P s ⎨P

opt s − λ ≥ H (y) Ps = Ps , y > Ps , H P (28) ⎩ s − λ < H (y). y, y > Ps , H P s ), the SU Remark 5: When the constraints are tense (y > P may be in an outage event. In this case, whether the SU transs or y depends on the tradeoff between achieved mits with P

s − λ. transmit rate (equal to H (y)) and the outage cost H P It is found that the energy-efficient optimal power allocation strategy given in Theorem 5 is equivalent to that strategy given in Theorem 1 when the AIP constraint and the ATP constraint s always are sufficiently loose, which can guarantee that y ≤ P holds. In order to obtain the maximum EE and solve P9 , Algorithm 1 can be applied. In this case, three dual variables (τ , μ and λ) are required to be updated and the energy-efficient optimal power can be calculated using eq. (28). B. Peak Transmit Power Constraint Under constrains on the AIP and PTP, the EE maximization problem in simultaneously delay-insensitive and delaysensitive CR can be formulated as P13 , given as (ν)Ps (ν) E log2 1 + h gss(ν)P 2 ps p +σw P13 :max η E E (Ps (ν)) = Ps (ν) E {ζ Ps (ν) + PC } (29a) s.t. (1a) , (1c) , (2) , and (23b) are satisfied.

(29b)

Note that there is no feasible power allocation when all power allocation levels satisfy constraints given by eq. (1a), eq. (1c) and eq. (2) but can not satisfy the constraint given by eq. (23b). Let S4 denote the set S4 = {Ps (ν) |Ps (ν) ∈ (29b)}. Similar to P9 , P13 can be solved by solving problem (P14 ), given as

gss (ν) Ps (ν) P14 : max f (η) = E log2 1 + Ps (ν)∈S4 h ps (ν) Pp + σw2 − ηE {ζ Ps (ν) + PC }

(30)

where η is a non-negative parameter. Applying similar techniques as used in solving P10 , P14 can be solved by solving the parallel subproblems for different fading states. For brevity, the fading state index is dropped. For a particular state, the subproblem (P15 ) is given as

gss Ps P15 : max − ηζ Ps log2 1 + 0≤Ps ≤Pth h ps Pp + σw2 − μgsp Ps − λ {χs (Ps )} .

(31)

In order to solve P15 , the optimization problem (P16 ) is given as

gss Ps P16 : max H (Ps ) = log2 1 + 0≤Ps ≤Pth h ps Pp + σw2 − ηζ Ps − μgsp Ps .

(32)

Note that P16 is equivalent to the problem given by eq. (11). Let opt optimal solution of P14 . Let Ps denote

the energy-efficient

s is given s , Pth and z 2 = max P s , Pth , where P z 1 = min P as + h ps Pp + σw2 1 s = P . (33) − gss ηζ + μgsp ln 2 Case 1: 0 ≤ y ≤ z 1 , where y is given by eq. (17). In this case, χs (z 1 ) = 0. The SU will never be in an outage event. opt Thus, the optimal power is Ps = z 1 .


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Case 2: y > z 2 . In this case, χs (z 1 ) = 1 and there is no feasible power level in the set consisting of constraints given by opt eq. (29b). Thus, Ps = ∅, where ∅ denotes no feasible power level. Case 3: z 1 < y ≤ z 2 . In this case, χs (z 1 )=1 and χs (z 2 )= 0. s , there is no feasible power level in the If Pth < y ≤ P opt s < y ≤ Pth , set. Thus, the optimal power is Ps = ∅. If P opt s when there power is Ps = P two cases. The optimal opt

are H Ps − λ ≥ H (y); otherwise, Ps = y. Theorem 6: The energy-efficient optimal power allocation strategy for P14 is given by ⎧ z1, 0 ≤ y ≤ z1 ⎪ ⎪ ⎪ ⎪ ∅, Pth < y ≤ P s ⎨

opt s < y ≤ Pth , H P s , P Ps = P (34)

s − λ ≥ H (y) ⎪ ⎪ y, P s − λ < H (y) s < y ≤ Pth , H P ⎪ ⎪ ⎩ ∅, y > z 2 . Remark 6: When the minimum OC can always be guaranteed, the EE maximization problem P13 degenerates to P3 . When the required power to guarantee the minimum OC is very large, the SU stops transmitting in order to save energy. s or y depends on Otherwise, whether the SU transmits with P the tradeoff between transmit rate (equal to H (y)) and

achieved s − λ. the outage cost H P P13 can be solved by using the modified Algorithm 1. In this case, the dual variables (μ and λ) are required to be updated by opt using the subgradient method, and Ps is calculated by using eq. (34). VI. N UMERICAL R ESULTS In this section, simulation results are presented to evaluate the EE of the SU with the proposed optimal power allocation strategies. The amplifier coefficient and the constant circuit power, ζ and PC , are set to be 0.2 and 0.05 W . The variance of the noise is 0.01. Pp is set as 60 mW . All iterative step sizes of the subgradient method for updating τ , μ and λ, are set as 0.1. ξ , ξ1 , ξ2 and ξ3 are set as 10−4 , where ξ3 is used for computing λ. It is seen from Theorems 1-6 that the EE optimal power level is dependent on the CSI of the second link, the interference link from SU-Tx to PU-Rx, and the link from PU-Tx to SU-Rx. Thus, the EE and capacities are evaluated by using 105 channel realizations. The average power constraint given by eq. (1b) is the result obtained by averaging over all fading states. In the simulation, Rayleigh fading channels and Nakagami-m fading channels are chosen as examples. The CSIs of the second link, the interference link from SU-Tx to PU-Rx, and the link from PU-Tx to SU-Rx of each channel realization are specialized according to the chosen fading channels. A. EE and EE Ergodic Capacity of the SU In this subsection, simulation results are given to investigate the performance of the EE of the SU in delay-insensitive CR under spectrum sharing. It is assumed that all the channels involved are Rayleigh block fading. gss , gsp , and h ps , are exponential distributions with means 1, 0.5, and 0.5, respectively.

Fig. 2. (a) The EE of the SU versus the AIP constraint for EE maximization or for conventional EC maximization under the PTP/ATP constraint, Pth = Pth = 100 mW ; (b) The EC of the SU versus the AIP constraint for EE maximization or for conventional EC maximization under the PTP/ATP constraint, Pth = Pth = 100 mW .

Fig. 2 presents the comparison of the EE and the EC of the SU achieved for the EE maximization with that achieved for the conventional EC maximization. The PTP/ATP constraint is set as Pth = Pth = 100 mW . From Fig. 2, several interesting results can be observed. It is seen that the achievable EE and EC of the SU under the ATP constraint are larger than those achieved under the PTP constraint regardless of the EE maximization or the EC maximization. This result for the EC maximization is consistent with the result shown in [17]–[22]. Note the interesting results in Fig. 2 that the EE of the SU achieved for the EE maximization is not less than that achieved for the EC maximization, and that the EC of the SU achieved for EE maximization is not larger than the EC of the SU obtained for the EC maximization. This is due to the fact that EE maximization is the objective of the green CR, whereas EC maximization is the goal of conventional CR. Thus, the conventional power allocation strategy for maximizing the EC of the SU is not optimal in delay-insensitive green CR in terms of EE maximization. As seen in Fig. 2, the EE and EC of the SU first increase with PI n . However, when the AIP constraint is sufficiently loose compared with the PTP/ATP constraint, i.e., PI n = 50 mW , the maximum EE and EC are unchanged since the AIP constraint given in eq. (2) is satisfied and the maximum EE and EC only depend on the PTP/ATP constraint. Fig. 3(a) shows the EE of the SU versus the AIP constraint under different PTP/ATP constraints. The PTP/ATP constraints

2750


B. EE and EE Outage Capacity of the SU

Fig. 3. (a) The EE of the SU versus the AIP constraint under different PTP/ATP constraints; (b) The EE of the SU versus numbers of iterations for the for-loop under different PTP/ATP constraints, along with different AIP constraints.

are set as Pth = Pth = 50 mW , Pth = Pth = 100 mW and Pth = Pth = 150 mW , respectively. It is seen that when AIP constraints are very tense, the EEs of the SU are almost the same. The reason is that the AIP constraints require that the SU can only transmit signal with low power levels. It is also seen that the EEs of the SU under the ATP constraint are larger than those of the SU under the PTP constraint. This can be explained by the fact that the power allocation strategy is more flexible for exploiting all the available CSI under the ATP constraint. When PI n becomes sufficiently large, the EE of the SU depends on the transmit power constraints since the transmit power constraints are the dominant constraints. The EE of the SU increases with Pth and Pth . This is because the transmit power of the SU can increase due to the increasingly loose constraints. In Fig. 3(b), the EE of the SU versus numbers of iterations is presented for the for-loop, subject to different PTP/ATP constraints, along with different AIP constraints. The PTP/ATP constraints are set as Pth = Pth = 100 mW and Pth = Pth = 150 mW , respectively. The PI n are set to be 10 mW and 50 mW , respectively. It is seen that Algorithm 1 is convergent regardless of how loose the constraints are. The convergence rate of Algorithm 1 is dependent on the constraints, the CSI and the optimal power levels. This can be seen in Algorithm 1.

In this subsection, simulation results for the EE and OP of the SU in delay-sensitive CR are presented. The mean values of the power gains, gss , gsp , and h ps , are assumed to be 2, 1.5, and 1.5, respectively. For Nakagami-m fading channels, m is chosen as 0.5. When the channel is an AWGN channel, the channel has constant power gains. The prescribed OC of the SU, rs , is set to be 1 bit/s/Hz. Fig. 4 shows the maximum EE of the SU versus the AIP constraint for different fading channel models. The PTP/ATP constraint is set to be Pth = Pth = 100 mW . It is seen that the maximum EE of the SU under the ATP constraint is larger than that of the SU achieved under the PTP constraint. This phenomenon can be seen in all fading channel models. It can also be explained by the fact that the transmit power of the SU is more flexible under the ATP constraint in contrast to the PTP constraint. It is interesting to see that the maximum EEs of the SU when gss is the AWGN channel power gain are larger than those of the SU when gss is the Rayleigh channel power gain. This indicates that the fading of the channel between the SU-Tx and the SU-Rx is an unfavorable factor with respect to maximizing the EE of the SU. It is also interesting to note that the maximum EE of the SU when gsp models the Nakagami-m fading channel is larger than that when gsp models the Rayleigh fading channel, and that the maximum EE of the SU when h ps is the Nakagami-m fading channel power gain is larger than that when h ps is the Rayleigh fading channel power gain. It is well known that the Nakagami-m fading channel with m = 0.5 is more severe fading than when the channel is the Rayleigh fading channel [49], [50]. This illustrates that the fading of the channel between the SU-Tx and the PU-Rx, and the fading of the channel between the PU-Tx and SU-Rx, are favorable to the SU with respect to the achievable EE maximization of the SU. The reason is that the constraints can still be satisfied even when the transmit power of the SU increases since the fading of the corresponding channel is more severe. Moreover, it is seen that the fading of the channel between the PU-Tx and SURx is more favorable to the SU than the fading of the channel between the SU-Tx and the PU-Rx. This can be explained by the fact that the fading of the channel between the PU-Tx and SU-Rx directly influences the EE of the SU, whereas the fading of the channel between the SU-Tx and the PU-Rx influences the transmit power of the SU and indirectly influences the EE of the SU, which is shown by eq. (13). Finally, as seen in Fig. 4, when the AIP constraint is sufficiently loose, the maximum EE of the SU when gsp models the Nakagami-m fading channel is equal to that when gsp models the Rayleigh fading channel. The reason is that the AIP constraint is inactive, and thus the fading of the channel between the SU-Tx and the PU-Rx has no influence on the SU. Fig. 5 shows the EE and the OP of the SU versus the AIP constraint for the EE maximization or the conventional OP minimization under the PTP/ATP constraint, respectively. The ATP/PTP constraint is set as Pth = Pth = 100 mW . All the involved channels are Rayleigh fading. The channels’ power gains, gss , gsp , and h ps , are exponential distributions with means 2, 1.5, and 1.5, respectively. It is seen in Fig. 5(b) that


2751

Fig. 4. The EE of the SU versus the AIP constraint for different fading channel models.

the OP of the SU achieved under the ATP constraint is lower than that achieved under the PTP constraint. A similar results has been obtained for CR with the spectrum sharing paradigm for the conventional OP minimization in [21] and [22]. It can be similarly explained by the fact that the ATP constraint can provide more flexibility for the transmit power of the SU compared with the PTP constraint. It is rather interesting to see in Fig. 5(a) that the EE of the SU obtained for EE maximization is not less than that achieved for OP minimization. In Fig. 5(b), it is also interesting to note that the OP of the SU achieved for EE maximization is not less than that achieved for OP minimization in conventional CR. This gives two insights. One is that the conventional optimal power allocation strategies for OP minimization are not always optimal for EE delay-sensitive CR in terms of EE maximization. The other is that the proposed optimal power allocation strategies for EE maximization can guarantee that the SU achieves the maximum EE but may increase the OP of the SU. As seen in Fig. 5, when the AIP constraint is sufficiently loose, the EE and OP of the SU attained for EE maximization are equal to those attained for the OP minimization under the PTP constraint. This is because the energy efficient optimal transmit power is equal to the transmit power for the OP minimization when the AIP constraint is inactive and Pth ≤ 1/ ηζ + μgsp . This is in complete agreement with our analysis in Subsection B of Section IV. Fig. 6 illustrates the EE of the SU versus the PTP/ATP constraint with different AIP constraints. The AIP constraints are set as PI n = 10 mW and PI n = 50 mW . All the channels involved are Rayleigh fading. The maximum EEs of the SU achieved with PI n = 10 mW and PI n = 50 mW are equal when the PTP/ATP constraints are very tight. The reason is that the AIP constraints are inactive when the PTP/ATP constraints are very tight, and thus the maximum EE of the SU only depends on the PTP/ATP constraints. As seen in Fig. 6, the maximum EE of the SU first increases with the PTP/ATP and becomes unchanged when the PTP/ATP constraints are very loose. This can be explained by the fact that the PTP/ATP constraints are first active and become inactive when the PTP/ATP

Fig. 5. (a) The EE of the SU versus the AIP constraint for EE maximization or OP minimization under the PTP/ATP constraint, Pth = Pth = 100 mW ; (b) The OP of the SU versus the AIP constraint for EE maximization or OP minimization under the PTP/ATP constraint, Pth = Pth = 100 mW .

Fig. 6. The EE of the SU versus the PTP/ATP constraint with different AIP constraints, PI n = 10 mW and PI n = 50 mW .

constraints are very loose, and thus the maximum EE of the SU only depends on the AIP constraint. C. EE and EE Minimum Rate Capacity of the SU In this subsection, simulation results are presented to evaluate the EE of the SU in simultaneously delay-insensitive and

2752


delay-sensitive CR. The parameters are set the same as in Subsection A of this Section in order to perform comparisons of the maximum EEs of the SU achieved in those two settings. The OC of the SU is set as rs = 1 bit/s/Hz. The maximum OP corresponding to the OC of the SU, Pout , is set to 0.3. The mean values of the power gains, gss , gsp , and h ps , are assumed to be 1, 0.5, and 0.5, respectively. For Nakagami-m fading channels, m is chosen as 0.5. Fig. 7(a) shows the EE of the SU versus the AIP constraint for EE maximization or the conventional minimum rate maximization. The PTP/ATP constraint is set as Pth = Pth = 150 mW . All the channels involved are Rayleigh fading. It is seen that the EE of the SU under the ATP constraint can be obtained when the AIP is larger than 20 mW , whereas the EE of the SU under the PTP constraint can be obtained when the AIP is larger than 60 mW . The reason is that there is no feasible power allocation when the AIP is below 20 mW under the ATP constraint, or when the AIP is below 60 mW under the PTP constraint since the OP of the SU is always larger than 0.3 with respect to the OC being 1 bit/s/Hz. It is also seen that the EE of the SU achieved under the ATP constraint is larger than that achieved under the PTP constraint. It is interesting to note that the EE of the SU achieved for EE maximization is not less than that achieved for the conventional minimum rate maximization irrespective of the PTP/ATP constraint. This indicates that the proposed energy-efficient optimal power allocation strategies can guarantee that the SU achieves the maximum EE, whereas the conventional optimal power allocation strategies that maximize the minimum rate of the SU can not guarantee that the SU obtains the maximum EE. The reason is that the EE of the SU is the maximization objective of the green CR, whereas the minimum rate maximization of the SU is the goal of the conventional CR. A comparison of the maximum EE of the SU achieved in simultaneously delay-insensitive and delay-sensitive CR with that obtained in delay-insensitive CR is presented in Fig. 7(b). The PTP/ATP constraint is set to be Pth = Pth = 150 mW . All the channels involved are assumed to be block Rayleigh fading channels. It is seen that the maximum EE of the SU achieved in simultaneously delay-insensitive and delay-sensitive CR is lower than that achieved in delay-insensitive CR when the AIP constraint is tense. This can be seen for both the PTP constraint and the ATP constraint. It is rather interesting to see that the maximum EE of the SU achieved in simultaneously delayinsensitive and delay-sensitive CR converges to that obtained in delay-insensitive CR when the AIP constraint becomes increasingly loose. This can be explained by the fact that the SU is required to guarantee the OC of the SU in simultaneously delayinsensitive and delay-sensitive CR when the AIP constraint is tense, and the OP constraint is inactive and, thus, EE maximization problems in those two CRs are equivalent when the AIP constraint and the PTP/ATP constraint are loose. Fig. 8 shows the EE of the SU versus the AIP constraint for different fading channel models. The ATP constraint is applied and is set as Pth = 150 mW . It is seen that the fading of the channel between the SU-Tx and the SU-Rx is unfavorable to the SU in terms of EE maximization of the SU. Similarly, an interesting observation that the fading of the channel between the

Fig. 7. (a) The EE of the SU versus the AIP constraint for EE maximization or minimum rate maximization under the PTP/ATP constraint; (b) The EE of the SU versus the AIP constraint for minimum rate or EC under the PTP/ATP constraint.

Fig. 8. The EE of the SU versus the AIP constraint for different fading channel models.

PU-Tx and the SU-Rx is more favorable to the SU in contrast to the fading of the channel between the SU-Tx and the PURx with respect to the EE maximization of the SU can be also made in simultaneously delay-insensitive and delay-sensitive CR, which is shown in Fig. 8.


VII. C ONCLUSION The EE maximization problem was studied in delayinsensitive CR, delay-sensitive CR and simultaneously delayinsensitive and delay-sensitive CR employing the spectrum sharing paradigm. Based on fractional programming and convex optimization techniques, energy-efficient optimal power allocation strategies that maximize the EE of the SU in fading channels were proposed subject to constraints on the AIP, along with the ATP/PTP. It was proved that the design of the energy efficient optimal power allocation strategies for green CR should take the EE of the SU and power amplifier coefficient into consideration. It was shown that the SU can achieve EE gain under the ATP constraint in contrast to the PTP constraint. An interesting observation that the fading of the channel between the PU-Tx and the SU-Rx is more favorable to the SU compared with the fading of the channel between the SU-Tx and the PU-Rx, whereas the fading of the channel between the SU-TX and the SU-Rx is unfavorable to the SU in terms of EE maximization was made.

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η E E Ps2 . Since R EECE (Ps ) is a strictly concave function of Ps , one has R EECE λPs1 + (1 − λ) Ps2 > λR EECE Ps1 + (1 − λ) R EECE Ps2

R EECE Ps1 EC 1

PE E Ps2 > λR E E Ps + (1 − λ) PE E Ps1

R EC P 1 = E E s1 λPE E Ps1 + (1 − λ) PE E Ps2 . (38) PE E Ps

Since λPE E Ps1 + (1 − λ) PE E Ps2 = PE E λPs1 + +PE E (1 − λ) Ps2 , one has

R EECE λPs1 + (1 − λ) Ps2 R EECE Ps1 >

. (39) PE E λPs1 + (1 − λ) Ps2 PE E Ps1 Thus, η E E (Ps (ν)) is strictly quasiconcave on S1 . The proof is complete.

A PPENDIX A For brevity, the index ν for the fading state is dropped. Let Ps2 ∈ S1 and 0 ≤ λ ≤ 1. One has E λPs1 + (1 − λ) Ps2 = λE Ps1 + (1 − λ) Ps2

Ps1 ,

≤ λPth + (1 − λ) Pth = Pth 1 E λgsp Ps + (1 − λ) gsp Ps2 = λE gsp Ps1 + (1 − λ) Egsp Ps2 ≤ λPI n + (1 − λ) PI n = PI n .

(35)

(36)

According to eq. (35) and eq. (36), S1 is proved to be a convex set. Let R EECE (Ps ) and PE E (Ps ) denote the respecnumerator and the denominator of η E E (Ps (ν)), gss Ps EC and tively, namely, R E E (Ps ) = E log2 1 + h P +σ 2 ps

PE E (Ps ) = E{ζ Ps + PC }.

gss Ps h ps Pp +σw2

REECE

p

w

(Ps ) is a concave function of

is a concave function of Ps [44]. Ps since log2 1 + PE E (Ps ) is a convex function of Ps and also is a concave function of Ps since ζ Ps + PC is an affine function of Ps . Note that PE E (Ps ) > 0. According to the definition of the concave fractional programming problem in [46] and [48], the EE maximization problem (P1 ) in delay-insensitive CR under the AIP constraints is a nonlinear concave fractional programming problem. A PPENDIX B P ROOF OF L EMMA 1 The definition of a strictly quasiconcave function is presented as follows. Let S be a nonempty convex set. f is a strictly quasiconcave function if for each x1 , x2 ∈ S with f (x1 ) = f (x2 ), one has f [λx1 + (1 − λ) x2 ] > min { f (x1 ) , f (x2 )} , for each λ ∈ (0, 1) .

(37)

For brevity, the index ν for the fading state is dropped.

Let Ps1 , Ps2 ∈ S1 and 0 ≤ λ ≤ 1. It is assumed that η E E Ps1