Abstract
Conventional designs on OFDMbased underlay cognitive radio (CR) networks mainly focus on interference avoidance and spectral efficiency (SE) improvement. As green radio becomes increasingly important, this paper investigates energy efficient power allocation. Our aim is to maximize energy efficiency (EE), subject to the constraints on the total transmit power, the peak interference power, and the minimum data rate requirement. We first analyze the relationship between SE and EE and solve this optimization problem with the help of bisection search technique. However, the accuracy of the power allocation solution is dependent on the number of iterations. In order to achieve the exact optimal solution, a new energy efficient power allocation scheme is proposed to balance the tradeoff between SE and EE. Simulation results are provided to demonstrate the effectiveness of the proposed schemes.
1. Introduction
The rapid development of wireless communication applications has given rise to a tremendous demand for scarce spectrum resources and energy conservation [1]. Since a large number of licensed frequency bands are underutilized, cognitive radio (CR) has been viewed as a promising technology to improve spectral efficiency (SE) via spectrum sharing [2].
Orthogonal frequency division multiplexing (OFDM), which has been regarded as a leading technique for highdata rate transmission due to its inherent resistance to multipath fading and the flexibility in resource allocation, has received wide attention in the evolution and development of CR technology [3]. In order to protect primary user (PU) activity, the interference caused by secondary user (SU) transmitter (Tx) at PU receiver (Rx) should be maintained below a target threshold [4].
Power allocation is a critical design issue for CR networks. Conventional power allocation for OFDMbased CR networks mainly focused on SE [5–7], which measures how efficiently a spectrum band is utilized. However, maximizing the SE may not lead to a high energy efficiency (EE), which is defined as the number of transmitted bits per unit energy [8]. To this end, power allocation schemes to maximize EE have been studied. In [9], the authors discussed joint transmission duration and power allocation to achieve the maximum EE. The authors in [10] investigated energy efficient power allocation schemes in heterogeneous CR networks with femtocells. However, the quality of service (QoS) of SU is ignored in both of them. For an OFDMbased CR networks, several waterfilling factor aided search (WFAS) methods were derived in [11] to maximize EE under the minimum rate requirement constraint and average interference power constraint.
Although the above power allocation schemes are very efficient, they require to know perfect intersystem channel state information (CSI) and intrasystem CSI. The acquisition of the instantaneous intersystem CSI is difficult to obtain due to not only the lack of cooperation, but also wastes of network resources.
In this paper, we study the EE optimization problem for OFDMbased CR networks with partial intersystem CSI while ensuring a certain data rate of SU. We first analyze the relationship between SE and EE and solve this optimization problem with the help of bisection search technique. However, the accuracy of optimal solution is dependent on the number of iterations. In order to derive the exact optimal solution, we propose a new algorithm to maximize EE, which is different from all existing methods.
The remainder of the paper is organized as follows. Section 2 introduces an OFDMbased CR system model and provides the EE optimization problem. In Section 3, a bisection search aided energy efficient power allocation algorithm is described. To achieve the exact optimal solution, a new method with closedform expression is developed to obtain the maximum EE. In Sections 4 and 5, simulation results and conclusions are given, respectively.
2. System Model
Consider an OFDMbased CR network where an SU coexists with a PU over the same spectrum band with subchannels indexed by . Let be the instantaneous channel frequency response from SUTx to SURx on the th subchannel. In this paper, we assume that is perfectly known to the SUTx. However, compared with intrasystem CSI, the instantaneous channel frequency response between SUTx and PURx is difficult to obtain due to the lack of intersystem cooperation. We assume that only statistic distribution of intersystem CSI is available at the SUTx. Under an uncorrelated fading channel model [7], and are independent and identically distributed (i.i.d.) complex Gaussian random variables with and , respectively.
For a spectrum sharing scenario, the PU and SU will interfere with each other. In a practical network with a sufficiently large number of subchannels, the interference introduced by PUTx into SURx can be modeled as an additive white Gaussian noise (AWGN). This assumption is widely treated and justified in [7, 11]. Let be the transmit power on the th subchannel, and the data rate can be given by where is the bandwidth of each subchannel, is the channel gain to noise ratio (CNR) on the th subchannel and denotes the variance of complex white Gaussian noise.
Consequently, the overall data rate and transmit power can be expressed as respectively.
In the transmission mode of SUTx, 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. According to [12], the circuit power, , can be modeled as a linear function of data rate. We have where is the static circuit power and is the dynamic circuit power per unit data rate.
In order to prevent the PU from severe performance degradation, the interference power should be restricted by where is the interference threshold that the PU can tolerate.
Since the random variable follows the chisquare distribution with mean then the equivalent constraint of (5) can be expressed as
To maximize the EE of the CR 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. The optimization problem is formulated as follows:where (e.g., ) is the reciprocal of drain efficiency of power amplifier, is the minimum rate requirement, and is the maximum allowable transmit power. Equation (8b) is the minimum rate requirement constraint to ensure a certain QoS of SU. Equation (8c) is the interference constraint to prevent CR network from causing severe interference to PU. Equation (8d) is the total transmit power constraint at SUTX.
It is possible that problem P1 does not have any feasible solution if cannot be achieved subject to the constraints (8c) and (8d). In this case, the SUTx may have to decrease to make the solution feasible. For simplicity, we assume that the minimum rate requirement is achievable under the constraints (8c) and (8d).
3. Energy Efficient Power Allocation
After formulating the EE optimization problem, we investigate the relation between EE and SE and then propose two optimal power allocation schemes to maximize EE.
First we define a data rateadaptive (RA) optimization problem for the OFDMbased CR network as
It can be seen that the objectives in (8a)–(8d) and (9a)–(9c) conflict with each other sometimes. Without loss of generality, assume that . According to [4], the optimal power allocation is achieved through waterfilling method; that is, for all , where and is the water level.
Since is a strictly increasing function of in the feasible region, it is easy to see that has a unique value for all .
Lemma 1. For problem P2 without peak interference power constraint (9c), the optimal water level to maximize data rate is
Proof. Please see Appendix in [13].
Based on Lemma 1, the optimal water level corresponding to problem P2 can be obtained with iterative water filling.
If the overall data rate is fixed, the objective of problem P1 is equivalent to a transmit poweradaptive (PA) optimization problem. The optimization problem can be formulated as follows:
Lemma 2. For problem P3 without peak interference power constraint (12c), the optimal water level to minimize transmit power rate is
Proof. Please see Appendix in [13].
According to Lemma 2, we can derive the optimal water level to minimize using the similar way to problem P2.
3.1. Bisection Search Aided Algorithm (BSAA)
Lemma 3. For problem P1 without constraints (8b)–(8d), the EE, , is a quasiconcave function of water level .
Proof. Please see Appendix A.
Since EE is a quasiconcave function of without considering (8b)–(8d), there exists a globally optimal water level to maximize EE. More specifically speaking, is a monotonically increasing function of when and is a strictly decreasing function of when .
Based on the above discussion, both and can be acted as the lower bound and upper bound of feasible region for problem P1, respectively. Hence, the equivalent problem of P1 can be given as
The optimal water level can be solved by linesearch techniques, for example, bisection search, which requires a predictable number of iterations and is suitable for practical implementation. Here we list the bisection search aided algorithm (BSAA) in Algorithm 1.

The key work of Algorithm 1 is to determine the sign of . It can be calculated by where is an infinitely small positive constant. The number of iterations to determine the optimal water level is , where denotes the smallest integer not less than .
3.2. Exact Power Allocation Algorithm (EPAA)
Although the optimization problem P4 can be easily solved by the search algorithm, the accuracy of solution is primarily dependent on . In this subsection, a new energy efficient power allocation is proposed to achieve the exact optimal solution.
Lemma 4. For problem P4 without constraints (14b) and (14c), the optimal water level to maximize EE is where is the base of the natural logarithm, denotes the real branch of the Lambert function [14], and is the number of subcarriers with transmit power . and are defined as respectively, where .
Proof. Please see Appendix B.
However, the optimal unconstrained water level may lead to the transmit power of some subchannels exceeding peak power . To solve this problem, we should pour transmit power to these subcarriers in prior and keep the allocated power of all subchannels satisfying , . The pseudocode of this power allocation algorithm is presented in Algorithm 2.

4. Simulation Results
Simulation results are provided to corroborate our analysis and to demonstrate the performance of the proposed scheme. We evaluate the EE and SE performance of proposed schemes and compare them with conventional RA and PA power allocation schemes. The minimum SE requirement is 0.5 bits/s/Hz. The other simulation parameters are set as follows: kHz, , (we fix and vary ), W, W, , W, , and W/Mbps.
Figure 1 shows the EE performance of different schemes. In this simulation example, the interference tolerance is . Compared with conventional RA and PA power allocation schemes, the proposed schemes achieve the best EE performance. For different power allocation schemes, the EE increases with the relative channel gain ratio . Figure 2 plots the SEversus curves corresponding to Figure 1. It can be seen that the EE continuously increases as increases, which means as the CNR becomes larger, less transmit power has to be consumed to achieve the same data rate, and as a result, the EE increases. We can see that the proposed schemes can achieve higher EE with less power consumption at the cost of some SE loss when compared with the RA scheme.
Figures 3 and 4 show the impact of interference tolerance on the power allocation results where the relative channel gain ratio is dB. It can be seen that all schemes may have to operate exactly at the maximum allowable transmit power to meet the minimum rate requirement when is small. It indicates that for the different interference tolerance constraints, which are larger than or equal to a certain threshold, both BSAA and EPAA schemes lead to a same SE. However, for the interference tolerance that is smaller than the threshold, both the EE and SE of the proposed schemes increase with . Once the highest EE is achieved, the data rate as well as the transmit power consumption of the proposed schemes remains the same and no more transmit power will be used even when the interference tolerance continues to increase. In contrast to optimal power allocation schemes, the SE of RA scheme greedily grows with the interference tolerance all the time until the total transmit power is used up.
In Figures 5 and 6, we demonstrate the impact of the minimum SE requirement, , on the system performances in the case that dB and . We can divide the minimum SE region into two parts based on whether (8b) is active. It can be observed that for the different minimum throughput requirements, , which are smaller than or equal to a certain threshold (i.e., 1.8 bits/s/Hz), energy efficient designs lead to the same system performance. However, for the minimum SE requirement that is larger than the threshold, the proposed schemes are overlapped with the PA scheme. Furthermore, we can see that the optimal EE decreases with while SE increases in the case that . This is because the optimal energy efficient designs are to operate exactly at the required minimum data rate requirement. Hence, there exists a tradeoff between the EE and SE in the active region of . Note that once the highest achievable SE is obtained, the EE as well as the power consumption remains the same and no more power will be used even when the minimum SE requirement continues to increase for the energy efficient designs.
5. Conclusion
We consider the energy efficient power allocation problem of OFDMbased CR systems with statistical intersystem CSI available. We first introduce two conventional optimization problems: rateadaptive (RA) problem and poweradaptive (PA) problem. After analyzing the relationship between EE and SE, an efficient power allocation scheme with the help of bisection search method has been proposed to maximize EE. However, the accuracy of the solution is dependent on the number of iterations. To derive the exact optimal performance, we also propose a new power allocation algorithm which is different from all existing methods. Simulation results confirm considerable EE improvement by the proposed schemes.
In future, the energy efficient resource allocation problems for more complicated CR networks (e.g., multiuser scenario with imperfect channel state information) should be considered. We may jointly extend the proposed schemes and the methods in [15] to address these optimization problems.
Appendix
A. Proof of Lemma 1
From problem P1, the objective of EE optimization is modeled as Since , , , and , the EE is a strictly monotonic decreasing function of . Then the equivalent objective can be reformulated as For arbitrary , since is concave function of , we have
According to [13], we can conclude that is quasiconcave for . Since is a monotonic increasing function of water level without (7), hence, EE is a quasiconcave function of water level.
B. Proof of Lemma 4
According to Lemma 3, energy efficiency is a quasiconcave function of water level. Hence, a unique unconstrained optimal water level corresponding to P4 without (14b) and (14c) always exists. The true water level is assumed to be and the transmit power on the th subcarrier is where is the number of subcarriers with transmit power .
Denote that is the CNR set of all subcarriers. Substituting (B.1) into (2) and (3), we can obtain the overall data rate and total transmit power as respectively, where
Thus the objective function, (14a), can be given as
Differentiating the objective function, (B.5), with respect to , we have where is the base of the natural logarithm and
It is obvious that is a positive variable. Setting the derivative, (B.6), to zero, we can obtain
Let and ; then (B.8) can be expressed as Its solution is , where denotes the real branch of the Lambert function [14]. Thus, the optimal unconstrained water level is
Here Lemma 4 is proved.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work in this paper has been supported by funding from Beijing Natural Science Foundation under Grant no. 4122009.