Paper— A Robust Power Allocation Scheme in Ad-Hoc Cognitive Radio Networks A Robust Power Allocation Scheme in Ad-Hoc Cognitive Radio Networks

— We investigate the secondary users (SUs) power allocation in ad-hoc cognitive radio networks, considering an underlay paradigm where SUs can access the licensed spectrum owned by the primary users (PUs), if the total power to SUs and the interference power threshold from PUs and the signal-to-interference-noise-ratio (SINR) requirements of SUs are guaranteed. While prevalent works about distributed resources allocation algorithms are mainly focused on power allocation under the consideration of many practical limitations, which may not be the case in practice. We assume that SUs also are needed to satisfy heterogenous constraints with different positions by taking the fairness of SUs into account. Moreover, we develop a weighted robust distributed power allocation scheme to guarantees the fairness of SUs. Techniques from robust optimization theory are used to study an objective formulation with infinite numbers of constraints. The optimal solution is obtained by using Lagrange dual method, and the Lagrange multiplier is solved by the sub-gradient update algorithm. The feasibility and availability of our proposed power allocation scheme is confirmed by numerical results.


Introduction
With the development of cognitive technology, the last decade has witnessed a growing demand for wireless radio spectrum. The concept of cognitive radio is proposed to address the issue of spectrum efficiency and has been receiving an increasing attention in recent years, since it equips SUs the capability to optimally adapt their operating parameters according to the interactions with the surrounding radio environment [1].The Federal Communications Commission (FCC) found the utilization of the spectrum resource is low most of time. Thus, the technology of cognitive radio networks is proposed to solve the problem of spectrum scarcity and improve spectrum efficiency.
In the cognitive radio networks, there are two approaches for dynamic spectrum sharing, spectrum underlay and spectrum overlay. The overlay approach increases the spectrum efficiency by granting unused frequency bands of PUs when the frequency communication between SUs is ensured, but also cognitive system can achieve more green communication.
The rest of the paper is organized as follows. In section 2, we introduce the system model and the minimum total transmit power allocation (MTTPA) problem, and transfer the MTTPA problem into a robust formulation problem. In section 3, a weighted robust distributed power allocation algorithm is proposed, and Lagrange dual method [15] is introduced to solve above problem. In section 4, the performance of our proposed scheme is evaluated. Finally, we draw the conclusions in section 5.

System Model and Problem Formulation
In this section, we firstly describe the system model. Then, we provide the MTTPA problem scheme for the robust formulation in the cognitive radio networks.

System Model
This paper considers an ad-hoc cognitive radio network (AHCRN) as shown in Fig.1, which consists of multiple distributed cognitive links. PBS and SBS refer to the PUs' base station and SUs' base station, respectively. Both !! !!!"#!! represent channel gain between users. Each link consists of a transmitter node and a receiver node. Consider the AHCRN with ! SUs, coexisting with ! PUs in a licensed system. In order to further expand the scope of application of the distributed resource allocation method, we consider the fairness of resource allocation among SUs. It is assumed that each SU can acquire its location information perfectly by sensing method. And these information are sent to the corresponding transmitter via a feedback channel.
As mentioned in the introduction, we consider the underlay paradigm where the instantaneous interference power introduced to the PUs receiver should below a certain threshold. Considering the fairness of the SUs, we have where ! ! ! ! !, ! ! is a weighted coefficient, ! !! is the tolerable interference power threshold, ! ! is the channel gain from SU transmitter!! to PU receiver and ! ! represents the transmit power of SU transmitter !.
In order to administrate the interference among SUs and avoid intolerable interference to PUs, SUs cannot increase its total transmission power beyond a certain level. Considering the fairness of the SUs, we have that is, the amount of SUs' power should not exceed ! ! , where ! ! is the total power budget of SUs.
To guarantee SUs' QoS, the SINR of each SU must be no less than a certain target value to guarantee a reliable transmission. Thus, the SINR should be restricted by where ! ! !"#$ is the SINR of SU!, ! ! !"# is theminimum SINR requirement. And the ! ! !"#$ can be represented by (4) where! !! represents the channel gain from SU transmitter!to SU receiver! and! !" represents the channel gain from SU transmitter !to SU receiver!, ! ! represents the transmit power of SU transmitter !. And, ! !" is interference power caused by PU to SU!!, ! !" ! ! ! ! ! , ! ! represents the transmit power of PU transmitter, ! ! represents the channel gain from PU transmitter to SU receiver !, and ! ! is the background noise.
The SINR requirement problem can be written as follows where

MTTPA Problem
The objective is to design a MTTPA problem under the constraints of the interference power threshold from PU, the transmitting power budget of each SU and the SINR requirement of each SU. The optimization problem is formulated as

Robust Formulation
In this section, we consider the uncertainty in channel gain matrix due to fluctuation of the channels. The specific shape of the uncertainty set depends on the underlying channel model and the sources of uncertainty. Then, we use ellipsoid to characterize the uncertainty set of channel gain, which succinctly describe a set of discrete points in Euclidean geometry.
The interference channel gain denotes ! ! ! ! , which describes the perturbation of interference channel gain at link !. In order to incorporate the effect of imperfect channel gain, we write ! ! ! ! ! ! !! ! , where ! ! is the nominal value. And !! ! means the channel error, which is the perturbation part. Using the additive uncertainty description method, we can obtain an ellipsoidal set to describe the uncertainty of interference channel gains as where ! refers to the Euclidean norm, and! ! is the!"!row of!and!! ! is the perturbation part. ! ! ! ! denotes the upper bound on the ellipsoidal set. ! !" is determined by Similarly, the channel gain ! !" is written as ! !" ! ! !" ! !! !" , with ! !" represents the nominal value of ! !" , and !! !" is the corresponding perturbation part. Besides, ! !" can capture the perturbation of interfering channel gains relative to the main channel gain of link !.Then the uncertainty set under ellipsoid approximation can be represented by where ! ! describes the perturbation of channel gains, and ! ! ! !!is the upper bound on the ellipsoidal set. ! ! and ! ! denote the size of uncertainty region respectively. The bigger ! ! and ! ! is, the more uncertainty there is in channel. Based on the ellipsoidal sets (7) and (9), the weighted robust distributed power allocation problem can be formulated as following The formulation (10) is a minimization of concave function with linear constraints, which cannot be solved by standard convex optimization scheme due to the infinity of objective constraint C3. In other words, the formulation is subject to a NP-hard problem [16] respect to sets ! ! , for all ! and ! ! .

Weighted Robust Distributed Power Allocation Algorithm
In this section, we present the weighted robust distributed power allocation algorithm, after introducing the worst estimation error and the Lagrange dual approach for problem formulation.

Solution of the Optimization Problem
The problem (10) belongs to a semi-infinite programming (SIP) owing to the constraint C2 and C3, which make it difficult to handle. In order to get the optimal transmit power, the problem (10) is transformed into a deterministic optimization problem under the condition of limited constraints. From Cauchy-Schwarz inequality, we have By applying (9), we introduce an equation in (11) based on the worst estimation error as follows Similarly, taking all parameters' uncertainties into consideration and from the perspective of worst case optimization, the weighted robust distributed power allocation problem is rewritten as follows ! , ! ! is the maximum uncertainty of the channel gain from PU transmitter to SU receiver.

Weighted Robust Distributed Algorithm
We can see that the problem (13) is a convex optimization problem [16]. The optimal transmission power for this problem can be calculated by Lagrange dual approach. To apply the dual decomposition to the problem (13), additional auxiliary variables and corresponding equality constraints need to be introduced owing to the specific nature of problem, in which the coupling exists in both objective functions and the system wide constraints [17]. According to the Lagrange principle, the Lagrange function as follows where ! ! ! !! ! ! ! ! and ! ! ! ! are the Lagrange multipliers.
Due to the powers of SUs are coupled above problem, the problem can be decomposed into the minimization power problem and the dual optimization problem of Lagrange multipliers by the Lagrange dual decomposition method. The dual function of (14) can be expressed as: where each sub-optimization problem ! ! represents the resource allocation problem in each SU link since the decomposability of the Lagrange function. We have By applying dual decomposition method, we obtain the corresponding dual problem of (15) is According to the sub-gradient updating algorithm, the Lagrange multipliers updating function as follows where !! !! !"#!!arestep-size, ! ! ! !"# !! ! and ! is time gap. The corresponding Lagrange sub-gradient !!! ! ! !!! ! ! !"#!!!! ! can then be expressed as: Based on the above description and deduction, the optimal transmit power of each SU transmitter can be obtained individually according to the Karush Kuhn Tucker (KKT) conditions [18] through the following equation: and the optimal solution of robust power control scheme for a given ! ! ! ! ! ! ! ! is We adopt sub-gradient method to solve the dual problem (17). Since the subgradient method is able to converge to the dual optimal solution, we can obtain the optimal the power control solution through (21). To apply the optimal solution, it needs a central controller that obtains the location distribution information and the number of SUs by sensing method, and broadcasts Lagrange multipliers!! ! ! ! ! ,!! ! ! ! ! ,!! ! ! ! ! and!! ! !"# ! ! ! . Such a system may not be practical because of infrastructure coat, or the difficultly to build control channel to exchange parameters information among SUs. Therefore, we derive a distributed mechanism in AHCRN that can utilize the inherent PU feedback information. We summarize the weighted robust distributed power allocation algorithm as below.

4
Simulation Results and Discussion

Simulation Parameters and Assumptions
The numerical results of the proposed weighted robust distributed power allocation algorithm are presented in this section. Our numerical experiments are performed to support the theoretical discussions of previous sections. In order to illustrate the effectiveness and convergence of the algorithm, and the convenience of calculation mathematically, we consider the scenario in the AHCRN, wherein three SUs and one PU share the same spectrum with each other. The value of ! ! ! ! ! !!"#!! !! are assumed to be 1.7mW, 2mW and 0.3mW, respectively. The background noise ! ! is generated by random function, and the interval of random function is !!! !!!!. The minimum SINR requirement values ! ! !"# are assumed to be 6.85dB, 7.60dB and 8.29dB, respectively. Furthermore, the weighted constraint values ! ! are assumed to be 0.25, 0.35 and 0.40.

Performance Analysis
In Fig.2, simulation results are provided to evaluate the interference performance of the proposed weighted robust scheme (PWRS) compared with the proposed weighted non-robust scheme (PWNRS) under the AHCRN. The PWNRS refers to the scheme which does not take channel uncertainties into account and directly utilizes the estimates of all channels. Similarly, the PWRS investigates the scheme in practical system that the perturbation value ! is assumed to be 5%, 7.5%, 10%, respectively. And ! include three parts, ! ! ! ! ! !"#!! ! .
In this scenario, the interference incurred by SUs has the lowest level which can guarantee constraint C2 in (13) under PWNRS, but it may be violate this constraint when they adopt the optimal transmit power to achieve MTTPA without considering the uncertainty in practical system. In order to show the effect of parameter's uncertainty on interference more clearly, we change the uncertainties of parameters. In Fig.2, the PWRS (! ! !"! ! ! !!!") can guarantee constraint C2 in (13) while the PWRS (! ! !"#) violates this constraint.
In Fig.3, we can also obtain that the interference to PU increases linearly with parameter perturbation for PWRS because the algorithm requires more transmit power to suppress the impact of parameter perturbation on the system, which guarantee that the actual SINR of SU receiver is not below the target value. This phenomenon shows the trade-off between performance (i.e., the interference to PU) and robustness (!).
In Fig.4 (a), simulation results are provided to evaluate the SU3' SINR of PWRS compared with PWNRS under the AHCRN,!! ! !"# is the SINR of SU3. As we see, the PWNRS and PWRS can guarantee the SINR of SU3 no less than the minimum target SINR, but the PWRS below the PWNRS owing to the parameter perturbation. In Fig.4 (b), SUs' SINR under PWRS with the variation of ! is evaluated. SUs' SINR decreases with the growing of !, since the SINR is no-increasing function with respect to the parameters' perturbations. The larger ! means that there is a lager ! perturbation in the system. This phenomenon shows the trade-off between SUs' SINR and robustness (!). Thus, we can confirm that the PWRS is efficient compared with the PWNRS. In Fig.5, we compare the total power, the capacity, the interference performance and the SINR experienced by SUs of different schemes: PWRS and traditional robust scheme (TRS) [19]- [21] under the same parameter perturbation (! ! !"). Fig.5 (a) and Fig.5 (b) show histograms with the total power and total capacity of SUs with different schemes. We note that, with the increasing of total power for above schemes, the total power for PWRS!! ! !") is greatly reduced when compared with TRS as in (1) and (2) which do introduce weights into its constraints. And the total power of TRS has reached the total power limit (! ! ! !!!!"). It is found that the total power of PWNRS is the smallest, which means that it consumes minimal energy under the same data transmission, and the total of PWRS increase with the augment of !. For the dynamic cognitive radio communication system, the PWNRS is obviously powerless. These two examples point that the PWRS consumes smallest power and the consumed power is increased with the disturbance amplitude. Similarly, we found that the capacity of PWNRS and TRS are larger than the capacity of PWRS as in Fig.5 (b). Fig.5 (c) and Fig.5 (d) show the interference performance and the SINR of SUs for both PWRS and TRS. We note that the SINR of PWRS no more than the SINR of TRS as in Fig.5(d),which can not only ensure the seamless communication of the system, but also improve the energy efficiency of the system. The interference caused by SUs for TRS is greatly increased when compare with PWRS as in Fig.5(c). Furthermore, the interference of TRS achieves nearly 80% more than the interference power threshold.
A good candidate energy efficiency function is one that measure the system throughout for unit-energy consumption [22]. In [23]- [25], the energy efficiency function is the cost of users to gain their benefits or the QoS of the network. Fig.6 also shows the value of energy efficiency function (the sum of reliable data transmission divided by the total of transmit power consumed) toward increasing the parameter The performance comparison of the algorithms, including the total power, the capacity, the interference and the SINR perturbation. It can be observed that energy efficiency function for cognitive system is decreased by increasing the parameter perturbation. Therefore their SINR also decrease, which means the throughput will be decreased considerably as in Fig.5 (b).
We can obtain that the energy efficiency function of PWRS is always located above the TRS because the TRS ignores the fairness of communication among SUs, indicting our proposed algorithm performs quit well for the consider problem. While ensuring the fairness of the communication, the strategy also can prevent some edge users increasing the transmission power. Hence reducing the energy consumption, we can achieve a more green communication. http://www.i-joe.org

Conclusions
In this paper, we propose a weighted robust distributed power allocation algorithm based on underlay paradigm in ad-hoc cognitive radio networks. Our model is general and covers many practical constrains, including total transmission power constraint, the interference power constraint and required SINR constraint. Simulation results show that PWRS can achieve more rational allocation of power for resource sharing compared with other algorithms since we take the information state and the fairness of SUs into account. This scheme allows SUs consume less power to attain the basic SINR requirements. Either total transmission power or interference caused by SUs is decreased when compare with other algorithm without affecting the performance of the primary network. Additionally, the significant promotion in the energy efficiency of PWRS gives the highest preference to apply it in cognitive radio senor networks and green cognitive radio networks.