Energy-Efficient Power Optimization with Spectrum Sensing Errors in OFDMA Cognitive Opportunistic Relay Links

1. Introduction

Cognitive radio (CR) networks have emerged as an efficient solution to the problem of spectrum scarcity and its underutilization. This is achieved by granting SUs' opportunistic access to the white spaces within PUs' spectrum while controlling the interference to PUs. Orthogonal frequency-division multiplexing access (OFDMA) is recognized as an attractive technique for CR due to its spectrum shaping flexibility, adaptively in allocating vacant radio resources, and capability of analyzing the spectral activities of PUs [1]. Incorporating cooperation into cognitive radio networks results in substantial performance gains in terms of spectrum efficiency (SE) for both PUs and SUs [2]. Besides the SE, the EE becomes a key issue for future wireless networks since energy cost imposes both financial and ecological burden on its development. EE power allocation especially is of crucial importance for cognitive relaying network design [3]. This is because that high EE is a basic premise for SUs to achieve high utilization of the limited transmit power which is consumed to not only improve SE but also implement some additionally important functionalities, for example, spectrum sensing.

As a result, energy-efficient resource management has attracted attention in both industry and academia recently, especially for the OFDMA system which is the most popular modulation technique for current wireless networks. Different from the throughput-oriented optimization targets, energy-efficient resource management aims at maximizing the energy efficiency of the wireless system. One of the energy-efficiency metrics is called “bits-per-Joule,” which is defined as the system throughput to unit-energy consumption. For instance, the EE-maximization problem in an OFDMA system under a maximum total power constraint in frequency-selective channels is addressed [4]. In [5], the authors studied the tradeoff between EE and SE in the downlink of OFDMA networks. They showed that the EE is quasi-concave in the SE. Then based on this observation, a power and subcarrier allocation algorithm is proposed. In the uplink of an OFDMA system, the EE is addressed in [6]. Furthermore, in [7], the EE of two-way relaying was compared with those of the one-way relaying, showing that two-way relay transmission is not always more energy-efficient than one-way relay transmission. Nevertheless, [4–7] aim at maximizing the EE of system without taking the interference by SUs in CRNs into account. In [8], a method named as water-filling factor aided search (WFAS) was proposed to maximize EE under multiple constraints with perfect channel state information (CSI) at CR source, but relaying was not considered. In [9], the energy efficiency of an OFDM-based system is maximized, where multiple radio access technologies are employed for parallel transmission. In [10], an energy-efficient resource management scheme is developed for a downlink multiuser OFDM system with distributed antennas while considering proportional fairness among users. In our previous work [11], we propose an optimal power allocation scheme to maximize the EE of OFDMA opportunistic relay which is first proposed in [12] to better exploit the frequency-selective channels. However, [11] has not considered the peak primary PU's interference constraints and spectrum sensing errors. Note that [13] also studied the EE optimization problem in CR system with imperfect spectrum sensing, but they all focus on the frame design including optimal sensing duration and data-transmission duration, as well as the optimal transmission power instead of the power allocation among each subcarrier. Besides, the authors in [14–16] analyze the EE performance of CRNs with imperfect spectrum sensing while relay strategies are not applied. Although a solution for EE-maximization problem in relay-aided CRN is proposed in [17], the authors only consider the ideal situation, that is, ignoring sensing errors.

Based on research in CR relaying system, the motivation of this paper is expressed as follows. In order to further improve the SUs' performance in terms of EE metric, we introduce the opportunistic DF (Decode and Forward) relaying strategy [12] to CR relay-aided networks to better exploit the frequency-selective channels, unlike [5, 6] where always-relaying protocol was considered. On the other hand, we assume that the SUs can coexist with the PU in the presence of both idle and busy sensing decisions while adapting their transmission power according to the imperfect sensing results, which differs from [15, 18]. In fact, the perfect spectrum sensing results are unavailable in practice, which makes the past research too idealistic to achieve feasible schemes for real system. Therefore, it is of great importance to study the energy-efficient resource management scheme under imperfect channel sensing.

In this paper, adaptive power allocation is investigated to maximize EE for CORL with spectrum sensing errors considered in the system model. To achieve the optimal solution, the EE-maximization problem is simplified and transformed into an equivalent concave form, and then we use the Lagrangian technique to transform the equivalent problem into a corresponding dual problem. Finally, optimal allocation algorithm is proposed. To the best of our knowledge, adaptive EE power allocation is investigated to maximize EE for CORL considering spectrum sensing errors which has not been discussed in the literature. Our main contributions of this paper are summarized as follows: (i)

An EE optimization problem for CORL system with imperfect sensing results is established, subject to the individual power budget, peak PU's interference constraints, and circuit power consumption in the total power expenditure, as well as considering subcarrier transmission mode. Particularly, our model can be easily extended to many practical scenarios with necessary modifications.

The rest of this paper is organized as follows. Section 2 introduces the system model considering the imperfect sensing. The EE power allocation optimization problem is analyzed, and we outline the proposed algorithm for its solution in Sections 3 and 4. Finally, simulation results are presented in Section 5, and conclusions are drawn in Section 6.

2. System Model and Problem Formulation

2.1. System Model

We consider a scenario where a two-hop OFDM-based CR system coexists with a PU in the same geographical location. As shown in Figure 1(a), the system comprises one PU, one SU-transmitter (ST), one SU-relay (SR), and one SU-destination (SD) in the system. Let us denote the set of the PU's bands K = 1,2 , … , K including the occupied subcarriers set K O and spectrum holes (unoccupied subcarriers by PU) set K U . Thus, we can obtain K = K O ∪ K U , K U ≠ Φ . Each of PU's bands has a fixed bandwidth of Δ f  Hz. The opportunistic DF protocol in [12] which is used assists ST transmission to SD. It can be seen in Figure 1(b) that data frame structure for the considered CORL is different from the always relay-aided transmission protocol which is always idle for ST in the second slot. Specifically, every data-transmission session takes two consecutive equal-duration time slots ( T S 1 , T S 2 ) and OFDM with k ∈ K U subcarrier is used. In the first time slot, the ST radiates OFDM symbols using P s , k as the transmit power for subcarrier k while the SR and SD receive. The ST-to-SD and ST-to-SR channel coefficients for subcarrier k are h s d , k and h sr ⁡ , k , respectively. In the second time slot, we define the subcarriers transmission mode indicator θ k , which is a binary integer variable; that is, θ k ∈ { 0,1 } . θ k = 1 represents relay transmission mode (RTM) which means that the SR retransmit OFDM symbols using P r , k as the transmit power. The SR-to-SD channel coefficient is h rd ⁡ , k for subcarrier k. In band θ k = 0 represents direct transmission mode (DTM) which means transmission is solely undertaken by the ST in two successive time slots, and the SR is inactive for subcarrier k. Here, we define θ ≜ θ k ∣ k ∈ K U to facilitate further description. Based on the two signaling intervals, the SD exploits maximum ratio combining (MRC) to retrieve the message. We further assume noise variance within one OFDM subcarrier to be σ r 2 at SR and σ d 2 at SD. According to the Shannon capacity formula, the secondary achievable data rate for DTM and RTM over subcarrier k can be, respectively, expressed as

R sd ⁡ = log 2 ⁡ 1 + P s , k G s d , k ,

(1)

R co ⁡ s d = 1 2 min ⁡ lo g 2 ⁡ 1 + P s , k G sr ⁡ , k ︸ R sr ⁡ , log 2 ⁡ 1 + P s , k G s d , k + P r , k G rd ⁡ , k ︸ R co ⁡ srd ,

(2)

where G s d , k = h s d , k 2 / σ d 2 , G sr ⁡ , k = h sr ⁡ , k 2 / σ r 2 , and G rd ⁡ , k = h rd ⁡ , k 2 / σ d 2 denote the channel gain to noise ratio on the kth subchannel. Consequently, the achievable sum data rate for CORL can be derived as

R T P S , P R = ∑ k ∈ K U 1 - θ k R sd ⁡ P S + ∑ k ∈ K U θ k R co ⁡ sd ⁡ P S , P R ,

(3)

where P S , P R ≜ P s , k , P r , k ∣ k ∈ K U denotes the power allocation on ST and SR.

OPEN IN VIEWER

Figure 1

System model. (a) Transmission process of CORL. (b) A frame structure for one particular subcarrier in CORL.

2.2. Interference with Spectrum Sensing Errors

In CR system, PU can access the licensed spectrum at any time and the probability of PU using subcarrier j is denoted by P O j . Let Q f j and Q d j denote the false alarm and detection probability of subcarrier j, respectively. Since this paper is focused on the EE problem, we consider Q f j , Q d j independent to reduce the complexity. Therefore, Q f j and Q d j are given by [19]

Q f j = Q ε σ 2 - 1 τ f s , ∀ j = K O ∪ K U , Q d j = Q ε σ 2 - 1 - γ τ f s 2 γ + 1 , ∀ j = K O ∪ K U ,

(4)

where Q x = 2 / π ∫ x ∞ e - t 2 / 2 d t is the Gaussian tail probability, ε denotes a common threshold used across all subcarriers, σ 2 denotes the noise power received at each SU, and τ is the sensing time used by the SU when sensing the primary behavior in each frame. f s is the sampling frequency during the sensing time and γ is the PU's average SNR. Via (4), Q d j can be rewritten as

Q d j Q f j = Q 1 2 τ + 1 Q - 1 Q f j - γ τ f s , ∀ j = K O ∪ K U .

(5)

Let β O O j be the posterior probability that the SU detects subcarrier as being used by PU which is indeed occupied. Using Bayes' theorem and the law of total probability [20], β O O j can be derived as

β O O j = P r H j , 1 ∣ H ^ j , 1 = P r H ^ j , 1 ∣ H j , 1 P r H j , 1 ∑ Δ = 0,1 P r H ^ j , Δ ∣ H j , Δ P r H j , Δ = Q d j P O j Q f j 1 - P O j + Q d j P O j ,

(6)

where H j , 1 and H j , 0 represent the events that PU is active and idle on subcarrier j and H ^ j , 1 , H ^ j , 0 are the sensing results that subcarrier j is occupied or unoccupied by PU, respectively. β U U j is the posterior probability that the evidence subcarrier j is really unoccupied given that SU senses it to be unoccupied, which can be expressed as

β U U j = P r H j , 0 ∣ H ^ j , 0 = P r H ^ j , 0 ∣ H j , 0 P r H j , 0 ∑ Δ = 0,1 P r H ^ j , Δ ∣ H j , Δ P r H j , Δ = 1 - Q f j 1 - P O j 1 - Q d j 1 - P O j + 1 - Q f j P O j .

(7)

Note that for perfect sensing β O O j = 1 and β U U j = 1 . There exist two cases in which subcarrier k may introduce interference to PU. One is that subcarrier k is sensed correctly to be occupied by PU and the other is that subcarrier k is sensed incorrectly to be unoccupied by PU. Taking the above into account, the average interference introduced into the PU over subcarrier k with unit transmission power [21] can be written as

I k = ∑ j ∈ K O β O O j I k , j θ k + ∑ j ∈ K U 1 - β U U j I k , j θ k , k ∈ K U ,

(8)

where I k , j θ k indicates the interference introduced into PU on subcarrier j when ST or SR transmits on subcarrier k with unit transmission power [22], and it can be expressed as

I k , j θ k = ∫ j - 1 Δ f - k - 1 / 2 Δ f j Δ f - k - 1 / 2 Δ f 1 - θ k G sp , k + θ k G r p , k ϕ f d f ,

(9)

where G s p , k and G r p , k are, respectively, denoted as the channel gain from ST to PU and SR to PU over subcarrier k, respectively. ϕ f = T s s i n π f T s / π f T s 2 represents the power spectral density (PSD) of OFDM transmitted signal, and T s represents the duration of OFDM symbol.

2.3. Problem Formulation in CORL

The overall transmission power consumption in a unit frame contains the transmit power on ST and SR, which is calculated by

P tr ⁡ = 1 2 ∑ k ∈ K U P s , k + 1 2 ∑ k ∈ K U 1 - θ k P s , k + θ k P r , k .

(10)

To transmit data, the energy consumption consists of two parts: the energy consumption of power amplifier related to transmit power and the circuit energy consumption incurred by signal processing and active circuit blocks. We further assume that the circuit power consumption of equipment has nothing to do with the state of transmission system, and its average value is constant [13, 23]. In conclusion, the system total power consumption consists of circuit consumptions and overall transmit power. Therefore, considering power amplifier efficiency ρ ∈ 0,1 , the total circuit power consumption can be expressed as

P TC = P C S + P C R + 1 ρ ∑ k ∈ K U 1 2 2 - θ k P s , k + θ k P r , k ,

(11)

where P C S , P C R are denoted as the ST and SR circuit consumption which can be a constant. Like [14–18, 23], the EE measured by the “throughput-per-Joule” metric is defined as ratio of total throughput and total power; that is, E E ( θ ) ≜ R T P S , P R / P TC . Hence, maximizing the average EE metric for the CORL system can be written as

P S ∗ , P R ∗ = a r g max P S , P R ⁡ E E θ P S , P R ,

(12)

where P S ∗ , P R ∗ ≜ P s , k ∗ , P r , k ∗ ∣ k ∈ K U represents the optimal power allocation on ST and SR.

3. Problem Analysis on EE Power Allocation

To maximize the EE of the CORL network while guaranteeing that the interference to the PU receiver is maintained below a predefined threshold, we formulate a transmit power allocation optimization problem under some practical constraints. For simplicity, let us collect all indicator and power variables in θ, P S , and P R , respectively, and define D ≜ θ , P S , P R . Mathematically, we can formulate the EE-maximization problem for CORL as follows:

O P 1 : m a x D ∑ k ∈ K U 1 - θ k log 2 ⁡ 1 + P s , k G sd ⁡ , k + 1 / 2 ∑ k ∈ K U θ k min ⁡ R sr ⁡ , R co ⁡ srd P C S + P C R + 1 / 2 ρ ∑ k ∈ K U 2 - θ k P s , k + θ k P r , k ,

(13)

s.t. ∑ k ∈ K U P s , k ≤ P S m a x ,

(14)

∑ k ∈ K U P r , k ≤ P R m a x ,

(15)

∑ k ∈ K U P s , k I k θ k ≤ I P th ,

(16)

∑ k ∈ K U 1 - θ k P s , k + θ k P r , k I k θ k ≤ I P th ,

(17)

θ k ∈ 0,1 , k = K U ,

(18)

P s , k ≥ 0 , P r , k ≥ 0 , k ∈ K U ,

(19)

S D T = k ∣ G sd ⁡ , k > G sr ⁡ , k or G sd ⁡ , k < G sr ⁡ , k , G sd ⁡ , k > G rd ⁡ , k , S R T = k ∣ G sd ⁡ , k ≤ G sr ⁡ , k , G sd ⁡ , k ≤ G rd ⁡ , k . ⁡

(20)

According to Theorem 1, we have the following relationship:

P s , k G sr ⁡ , k = P s , k G sd ⁡ , k + P r , k G rd ⁡ , k if θ k = 1 .

(21)

O P 2 : m a x P S R T P S P C S + P C R + 1 / 2 ρ ∑ k ∈ S D T 2 P s , k + ∑ k ∈ S R T 1 + χ k P s , k ,

(22)

s.t. ∑ k ∈ S D T P s , k + ∑ k ∈ S R T P s , k ≤ P S m a x ,

(23)

∑ k ∈ S R T P s , k ≤ P R m a x ,

(24)

∑ k ∈ S D T P s , k I k + ∑ k ∈ S R T P s , k I k ≤ I P th ,

(25)

∑ k ∈ S R T 1 + χ k P s , k I k ≤ I P th ,

(26)

P s , k ≥ 0 , k ∈ S D T ∪ S R T ,

(27)

R T P S = ∑ k ∈ S D T R sd ⁡ P S + ∑ k ∈ S R T R co ⁡ sd ⁡ P S .

(28)

Proof.

From (3), R T P S can be expressed as

R T P S = 1 - θ R sd ⁡ P S + θ R co ⁡ sd ⁡ P S .

(29)

According to [24], it is easy to know that R sd ⁡ , R sr ⁡ , and R co ⁡ srd are monotonically increasing and strictly concave with respect to P S . On the other hand, we observe that the subcarriers transmission mode indicator θ = θ k ∈ { 0,1 } ∣ k ∈ K U ≥ 0 is defined as nonnegative integers, so we only need to prove that the second item in (29), that is, R co ⁡ sd ⁡ P S , is monotonically increasing and strictly concave with respect to P S . Considering the relationships in (21), we can rewrite (3) as R co ⁡ sd ⁡ P S = m i n R sr ⁡ P S , R co ⁡ srd ( χ k P S ) . Then we have

R co ⁡ sd ⁡ ξ p 1 + 1 - ξ p 2 = min ⁡ R sr ⁡ ξ p 1 + 1 - ξ p 2 , R co ⁡ srd ξ χ 1 p 1 + 1 - ξ χ 2 p 2 > min ⁡ ξ R sr ⁡ p 1 + 1 - ξ R sr ⁡ p 2 , ξ R co ⁡ srd χ 1 p 1 + 1 - ξ R co ⁡ srd χ 2 p 2 ≥ ξ min ⁡ R sr ⁡ p 1 , R co ⁡ srd χ 1 p 1 + 1 - ξ min ⁡ R sr ⁡ p 2 , R co ⁡ srd χ 2 p 2 = ξ R co ⁡ sd ⁡ p 1 + 1 - ξ R co ⁡ sd ⁡ p 2 ,

(30)

where 0 ≤ ξ ≤ 1 , ∀ p 1 , p 2 ∈ dom ⁡ R co ⁡ sd ⁡ ( P S ) . Hence, R T P S is monotonically increasing and strictly concave with respect to P S .

4. Adaptive Power Allocation to Maximize EE

It is noticeable that objective function equation (22) in O P 2 is not a concave function, and the solution for it will be of high complexity. In this section, we first use fraction programming [25] to transform the problem into an equivalent convex optimization problem and then use the Lagrangian technique to transform the equivalent problem into a corresponding dual problem. Subsequently, an optimal iterative algorithm is proposed.

4.1. Adaptive Power Allocation

From Theorem 2, we follow the fact that the numerator R T P S in (22) is a concave function, and the denominator of (22) is affine function of SUs' power. Besides, all the constraints are convex set. Inspired by Dinkelbach's algorithm in [25], we can transform this problem into a parameterized convex maximization problem. Primarily, a new objective function is defined as

T P S , q = ∑ k ∈ S D T R sd ⁡ P S + ∑ k ∈ S R T R co ⁡ sd ⁡ P S - q 1 2 ρ ∑ k ∈ S D T 2 P s , k + ∑ k ∈ S R T 1 + χ k P s , k + P C S + P C R ,

(31)

where q is a positive parameter and can be interpreted as a pricing factor for SUs' power consumption. Hence, another problem is formulated as

O P 3 : m a x P S , q T P S , q , s.t. 25 , 26 , 27 , 28 , 29 .

(32)

Let S denote the feasible region of O P 2 and O P 3 . Define F q = m a x P S T P S , q ∣ P S ∈ S as the maximum value of O P 3 with each fixed q. Then, the optimal value and solution of O P 3 can be defined as

f q = a r g m a x P S T P S , q ∣ P S ∈ S .

(33)

The following lemma introduced by Dinkelbach's algorithm [25] can relate O P 2 and O P 3 , and the detailed proof of Lemma 3 can also be found in [25].

Lemma 3.

The optimal solution P S ∗ achieves the optimal value q ∗ of O P 2 , that is, q ∗ = m a x P S T P S , q ∣ P S ∈ S = E E ( θ ) ( P S ∗ ) , if and only if

F q ∗ , P S ∗ = max P S ⁡ T P S , q ∗ ∣ P S ∈ S = 0 , f q ∗ = P S ∗ .

(34)

This lemma indicates that, at the optimal parameter q ∗ , the optimal solution to O P 3 is also the optimal solution to O P 2 . Hence, solving O P 2 can be realized by finding the optimal power allocation of O P 3 for a given q and then update q until (31) is established. For a given q, the optimal power allocation can be obtained using convex theory [24] because of the convex characteristic of O P 3 . Hence, the existing water-filling power allocation approach gives the solution to it [26]. However, besides adapting the power allocation on all subcarriers, we need to consider subcarrier transmission mode. The Lagrange function for O P 3 is constructed as

L P S , q , λ = - T P S , q + λ 1 ∑ k ∈ S D T P s , k + ∑ k ∈ S R T P s , k - P S m a x + λ 2 ∑ k ∈ S R T P s , k - P R m a x + λ 3 ∑ k ∈ S D T P s , k I k + ∑ k ∈ S R T P s , k I k - I P th + λ 4 ∑ k ∈ S R T 1 + χ k P s , k I k - I P th ,

(35)

where λ 1 , λ 2 , λ 3 , and λ 4 are the nonnegative Lagrangian dual variables for constraints (26)–(30), respectively. And the dual problem of O P 3 is given by

D = m i n λ i i = 1 4 max q , P S ⁡ L q , P s , k ∣ k ∈ K U , λ i i = 1 4 s . t . λ i i = 1 4 ≥ 0 .

(36)

Using the Karush-Kuhn-Tucker (KKT) conditions [24], we can deduce the optimal power allocation of problem O P 3 , which can be written as

P s , k ∗ = ρ log 2 ⁡ e q + ρ ξ k - 1 G sd ⁡ , k + , ∀ k ∈ S D T ,

(37)

P s , k ∗ = ρ log 2 ⁡ e q 1 + χ k + ρ ϑ k - 1 G sr ⁡ , k + , ∀ k ∈ S R T ,

(38)

where ξ k = λ 1 + λ 3 I k and ϑ k = 2 λ 1 + λ 2 + λ 3 I k + λ 4 I k 1 + χ k and x + denotes m a x 0 , x . In addition, subcarrier k is used for RTM communication; that is, k ∈ S R T . The corresponding relay transmission power at SR can be expressed as

P r , k ∗ = G sr ⁡ , k - G sd ⁡ , k ρ log 2 ⁡ e q G rd ⁡ , k + G sr ⁡ , k - G sd ⁡ , k + ρ ϑ k G rd ⁡ , k - G sr ⁡ , k - G sd ⁡ , k G rd ⁡ , k G sr ⁡ , k + , ∀ k ∈ S R T .

(39)

It can be found that the traditional water-filling method could not solve the primal problem directly, because the denominator of (37) and (38) contain a linear combination of the Lagrangian dual factors. Thus the incremental-update based subgradient method in [27] is introduced to derive the optimal dual factors λ ∗ = λ i ∗ i = 1 4 for power and interference constraints. For our problems, the corresponding iterative update λ is based on the following iteration procedure:

λ 1 l + 1 = λ 1 l - μ l P S m a x - ∑ k ∈ S D T P s , k - ∑ k ∈ S R T P s , k + λ 2 l + 1 = λ 2 l - μ l P R m a x - ∑ k ∈ S R T P s , k + λ 3 l + 1 = λ 3 l - μ l I P th - ∑ k ∈ S D T P s , k I k - ∑ k ∈ S R T P s , k I k + λ 4 l + 1 = λ 4 l - μ l I P th - ∑ k ∈ S R T 1 + χ k P s , k I k + ,

(40)

where l refers to the iteration index and μ l > 0 denotes a sufficiently small positive step size for the lth iteration, and it is a sequence of step size which is defined in many types in [27]. It should be mentioned that small step size leads to slow convergence. Besides, each element of the gradient depends on the corresponding subcarrier's channel gain, potentially differing from each other by orders of magnitude. Hence, a line search of the optimal step size needs to cover a large range to ensure global convergence on all subcarriers, which is computationally expensive. Therefore, in order to find the optimal step size, like [4], we define f l u = λ [ l ] - u ∇ L λ [ l ] + , which has also been proved to be concave in μ l , and can quickly obtain the optimal step value of u l ∗ by using bisection search algorithm summarized in [4]. After we obtain the optimal power assignment P S , P R for the given q, update q = E E ( θ ) P S , P R until Lemma 3 is satisfied; thus the equivalent optimal power assignment P S ∗ , P R ∗ of O P 3 can be achieved in the end. So, jointing the fractional programming and bisection method, an EE-oriented power allocation iterative optimization algorithm for CORL called ECORL is provided, which is described in Algorithm 1 to solve the power allocation of programming O P 1 .

Algorithm 1: ECORL.

Input: the error tolerance ς > 0 , δ > 0 and the maximum iteration number M a x I t e r ;

Output: optimal EE power allocation policy sets of O P 1;

Initialize maximum EE q 0 = q 0 , the iteration index n = 0 and l = 0 , dual variables λ 0 = λ initial ;

Compute   G sr ⁡ , k , G sd ⁡ , k , G rd ⁡ , k ∣ ∀ k ∈ K U , Then

Obtain subcarrier transmutation mode θ using (20);

While   ( F q n > ς and n ≤ M a x I t e r )   do

 Update λ via the subgradient method as follow:

 Repeat

  Step 1 Compute P s , k l , ∀ k ∈ S D T and P s , k l , ∀ k ∈ S R T through (37) and (38) respectively;

  Step 2 Find the optimal step size u l ∗ by using the bisection algorithm described in [27];

  Step 3 Update λ according to (40);

  Step 4 Calculate Δ λ = Δ λ l + 1 - Δ λ l and Set l = l + 1 ;

 Until 2-norm Δ λ < δ (λ converges)

 Set n = n + 1 and Obtain P S through (37) and (38);

 Calculate P R on SR by the corresponding (21);

 Update q n = E E ( θ ) P S , P R through (22);

End while

Return   q ∗ = q n , the ξ-optimal solution P S ∗ , P R ∗ of O P 1 and calculate the EE and capacity for CROL networks.

Remark 4.

Note that, in the case of q = 0 , EE-maximization problem is equivalent to SE-maximization. Consequently, for given maximum iteration number M a x I t e r and error tolerance ς and δ, the optimal EE and SE power allocation policy of O P 1 can be easily obtained by ECORL, which will be validated by the simulation in Section 5.

4.2. Coverage and Complexity Analysis

The proposed algorithm ECORL will always converge to optimum provided by Theorem 5, meaning that, for every ς, the power policy set P S that maximizes T P S , q is found. The algorithm execution stops if q is zero or less than the ς value.

Theorem 5.

The ECORL will always converge to optimum.

Proof.

To prove the convergence of ECORL, we let T P S , q = R T P S - q E P S , for ease of presentation, where E P S denotes the energy consumption in CORL system and can be written as

E P S = 1 2 ρ ∑ k ∈ S D T 2 P s , k + ∑ k ∈ S R T 1 + χ k P s , k + P C S + P C R .

(41)

Before going into the proof of convergence, we will first discuss an observation that F q is decreasing in q. For the sake of generality, we assume q 1 < q 2    q 1 , q 2 ∈ S ; then

F q 2 = max P S ⁡ R T P S - q 2 E P S ∣ P S ∈ S = R T P S ∗ - q 2 E P S ∗ < R T P S ∗ - q 1 E P S ∗ < max P S ⁡ R T P S - q 1 G P S ∣ P S ∈ S = F q 1 .

(42)

From (42), we know that F q is decreasing in q. To prove the convergence of ECORL, we only need to show that F q becomes smaller than ς with the number of iterations. We now show that q is nonincreasing in successive iterations of the algorithm. We have

T P S ∗ n - 1 , q n ≤ max ⁡ T P s ∗ n - 1 , q n ∣ P S ∈ S = F q n = R T P S ∗ n - q n E P S ∗ n = q n + 1 E P S ∗ n - q n E P S ∗ n = q n + 1 - q n E P S ∗ n .

(43)

Now, it follows that q n + 1 ≥ q n , because G P S ∗ n > 0 . From the above observation, we have that F q is decreasing in q and q is nonincreasing in successive iterations of ECORL. Therefore, F q is nonincreasing in successive iterations. Furthermore, F q n does become zero from Lemma 3, which follows that fact that F q n does become smaller than ς. Therefore, the ECORL algorithm will always converge to optimum.

In addition, the ECORL has two iterations: one is for the total individual power constraint and the other is for the Lagrange multiplier λ for the total power constraint. The complexity of ECORL can be expressed as O N f N s N c , where N f , N s , respectively, denote the iteration numbers of the while-loop and repeat-loop in ECORL and N c represent the number of subcarrier realizations. N f and N s especially rely on the choice of the corresponding error tolerance ς and δ, respectively. Hence, the EE optimization problem can be solved by the proposed algorithm with a considerable saving in terms of computational complexity, which is very applicable for the physical life.

5. Simulation Results

In this section, we present numerical results to illustrate the performance of the proposed EE-maximizing power adaptation methods. Consider the scenario as the following: the location of ST is ( 0,0 ) m and the coordinates of PU, SR, and SD are fixed on ( 500,288 ) m , ( 500 , - 288 ) m , and ( 1000,0 ) m , respectively. We consider the CR relay-aided system consists of K = 64 subcarriers; that is, K = 1,2 , … , 64 , the probability of spectrum holes ≥50%. The bandwidth of each subcarrier Δ f and the duration of the OFDM symbol T s in T S 1 , T S 2 are assumed as 0.3125  MHz and 4  us, respectively. Further define circuit power consumption P C S = 3  W, P C R = 2  W, f s = 100  MHz, τ = 2  ms, γ = - 5  dB, ρ = 0.5 , P S m a x = P R m a x = P tr ⁡ th , and the noise variance σ d 2 = σ r 2 = σ p 2 = 1 0 - 6 W . In the primary network, the central location of the PU is defined as c = 16 . Assume the activity probability P O j equals 0.8 · e x p - j - c 2 / 50 [14]. For ease of analysis, the detection Q d j is uniformly distributed over 0.90,0.99 . Channel complex gains h k are picked from a Rayleigh fading channel with the following distribution: h k = C N 0 , 1 / L 1 + d α [12], where the path loss exponent α is 3 , distance is i  m, and the number of taps L is 16 . Finally, given h k 2 , σ d 2 , σ r 2 , and σ p 2 the channel gains-to-noise-power ratios G sd ⁡ , k , G sr ⁡ , k , G rd ⁡ , k , G s p , k , and G r p , k are calculated as described in Section 2.