Robust power allocation for massive connections underlaying cognitive radio networks with channel uncertainty
 Le Wang^{1}Email author,
 Guochun Ren^{1},
 Jin Chen^{1}Email author,
 Guoru Ding^{1},
 Zhen Xue^{1} and
 Haichao Wang^{1}
https://doi.org/10.1186/s1363801606420
© Wang et al. 2016
Received: 26 November 2015
Accepted: 1 June 2016
Published: 8 June 2016
Abstract
This paper investigates a robust power allocation scheme for a cognitive radio network (CRN) with channel uncertainty, where a large number of secondary connections (SCs) share the same frequency spectrum with a primary user (PU). Specifically, considering the fact that the channel gain estimates from the secondary transmitters (STx) to the primary receiver (PRx) are typically uncertain and cannot be perfectly known in practice, this paper advocates a robust interference constraint. Meanwhile, we take the STx’s transmit power limitation and interference between SCs into account. The optimization problem is a nonconvex and nonlinear program (NCNLP) with an outage probability constraint. Both the original objective function and the outage probability constraint are converted into manageable forms via the mathematical transformation. Then, we introduce the convex optimization theory to solve the intractable problem. Additionally, we develop an efficient algorithm to obtain the nearoptimal solution based on the interior point method, where we take advantage of the Newton algorithm to make the search of the feasible solution simple and effective. Finally, we conduct the indepth simulations under various parameter configurations, which demonstrate that our proposed robust power allocation scheme can achieve higher performance in comparison to the existing scheme in the literature.
Keywords
1 Introduction
Since wireless personal communications have been in widespread applied to exchange information among individuals so far, not only are massive wireless devices growing up but also requirement of ubiquitous communications among individuals drives the development of abundant advanced wireless technologies [1]. In addition to this, the increasing demand for scarce frequency spectrum brings an exciting prospect to the development of the cognitive radio (CR) technology [2]. CR is a highly promising wireless technology which can mitigate the scarcity of spectral resources. In a cognitive radio network (CRN), there is usually a primary user (PU) and massive secondary connections (SCs) sharing the same spectrum, i.e., they coexist with each other. There is a great deal of coexistence modes for a CRN. In general, we classify the transmission modes into three types: overlay, interweave, and underlay modes. In the overlay mode, it is assumed that SCs have to acquire the perfect knowledge of the PU’s messages. The secondary transmitters (STxs) send out their own signals along with the signal received from PU. The interference inflicted upon PU from SCs obtains the compensation via SC relaying the PU’s signal. The interweave spectrum sharing mode is also called “opportunistic spectrum access” [3]. There is no cooperation between SCs and PU in the interweave mode. SC is allowed to occupy the channel if and only if the transmitter channels of SC and PU are mutually orthogonal. In time domain, SC should abandon the spectrum resource once the PU returned to utilize the channel occupied by SC. In the underlay mode, SCs are permitted to transmit concurrently with PU sharing the same spectrum bands, provided that the interference experienced by PU is controlled less than a certain threshold which PU can tolerate to [3].
This paper focuses on the underlay mode. It should be noted that the massive SCs will interfere with the communication of PU in general. A significant design challenge related to this network is how to maximize the throughput capacity of SCs with the quality of service (QoS) of the PU being guaranteed in the underlay mode. Moreover, when we consider the uncertainty of the channel between the STx and the primary receiver (PRx), this problem becomes more troublesome.
The optimal or suboptimal power allocation policies in the underlay mode have been already investigated widely [4–20]. A heuristic algorithm in CR systems under OFDMA has been reported in [4]. Ghasemi and Sousa [4] do not take the transmit power constraint of SCs into account. However, the transmit power constraint of SCs cannot be ignored due to the hardware of the STx. In [5], ergodic and outage capacity of SCs over Rayleigh faded channel based on MMSE estimation is evaluated. The transmit power of the PRx is under the constraint of peak received power. In [6], transmission capacity of SCs is evaluated under the interference constraint of the PU. The SCs serve as cooperative relays in spectrum sharing environment. It is notable that perfect channel state information (CSI) is assumed in these works [4–11]. However, the CSI between SC and PU is usually difficult to be estimated accurately in actual situations mainly due to the lack of signaling cooperation between them. Therefore, power allocation with channel uncertainty should be considered in CRN [12–20]. A worstcase formulation is utilized in [12] for beamforming design, in which an ellipsoidal channel uncertainty region is assumed. Dall’Anese et al. [13] considers the probabilistic interference constraints for a CR power control problem, where uncertainty in composite fading channels comprising shadowing and Nakagami fading is accounted for. Power allocation problems for generic OFDMA systems with channel uncertainty have also been investigated extensively [14–16]. Ma et al. [17] shows the availability of imperfect CSI that includes probabilistic uncertainty and considers its worst effect when performing resource allocation. Different small cell networks is considered in [18], and there are tight interference constraints on PU. Power allocation along with relay selection schemes is optimized especially considering different channel uncertainty models in [18]. It is worth noting that [12–19] consider the channel uncertainty without the interference between massive SCs. Son et al. [20] study the power allocation problem via the waterfilling algorithm. Furthermore, Son et al. [20] consider neither a great deal of SCs nor the interference constraints between SCs in the system. However, to the best of our knowledge, power allocation involving both channel uncertainty and mutual interference between massive SCs has not been addressed in the existing literature, which is the motivation of this work.

Formulate an optimization problem which takes the SC’s mutual interference constraints and channel uncertainty between the STx and PRx into account, analyze the control of STx’s peak transmitpower constraints and adopt average interference to the probabilistic model for the QoS of the PU.

Convert the original objective function and the outage probability constraint into a form which is easy to handle by some mathematical transformation and derive the analytical expressions via the convex optimization.

Develop an efficient algorithm to obtain the nearoptimal solution based on the interior point method under the strict interference constraint of the PU. Furthermore, the Newton algorithm is set for searching viable power value because it uses the second derivative information in the calculation of lowering direction and has a faster rate of decline.

Conduct the indepth simulations under various parameter configurations, such as outage probability and the number of SCs, which demonstrate that our proposed robust power allocation scheme can achieve higher performance in comparison to the stateoftheart work in the literature.
The remainder of this paper is organized as follows. Section 2 presents our system model, and the optimization problem is formulated in Section 3, along with formulation analysis as well as problem transformation. The proposed power allocation algorithm is derived in Section 4. The results of extensive numerical simulations and the performance evaluation are presented in Section 5, followed by conclusions in Section 6.
2 System model
where h _{1} and h _{2} are two independent Gaussian variables which follow the distribution of N∼(0,δ ^{2}).
The channel gain g _{PU}≥0 between the primary transmitter (PTx) and PRx is assumed to have been easily acquired by some conventional channel estimation techniques [22]. Similarly, the channel gain g _{ i,j } between the ith STx and the jth SRx is also known accurately, i.e., the STx has perfect channel state information [20]. Because SC and PU rarely have any cooperation during the process of their transmissions, the channel gain \(g_{i}^{\text {PU}}\) between them can be hardly estimated precisely unlike the channel gain between SCs.
where the channel gain \(g_{i}^{\text {PU}}\) is an independent and identically distributed exponential random variable with mean θ and ε is the outage probability.
In (10), the goal of the optimal problem \((\mathcal {P})\) is to find the optimal power allocation method to maximize the total throughput capacity of the system. For the sake of the description convenience, we regard \((\mathcal {P})\) as the original problem. Among the three constraints, the first one is to restrict the transmit power of STx, the second one ensures the coexistence of the communication between the SCs, and the final one focuses on the protection for the PU. However, since the objective function is nonlinear and it is not even convex, it will become very complicated if we directly solve the optimization problem of the above form. Additionally, the mutual influence actually exists between multiple variables. Therefore, not only does the problem become tricky but also the complexity will rise steeply as the number of the variables increases. Paper [23, 24] suggest that a large number of iterations is needed in order to tackle with this problem. Because of the structure of the function, this problem cannot directly use the methods proposed by [25, 26]. As a result, intractability and unavailability of the uncertain channel gain along with SCs’ interference make the nonlinear and nonconvex expression mathematically difficult to handle. In addition, there is not yet any satisfactory solution to the open issue so far. In the following part, we try to introduce a feasible way to tackle with the difficulty, which takes advantage of the convex optimization theory.
3 Problem transformation
where all of f _{0},⋯,f _{ q } are convex functions, f _{ i }(x) is the constraint of inequality, and \({h_{j}}(x) = {a_{j}^{T}}x  {b_{j}}\) is the constraint of equality [27]. But as it is shown in problem (10), neither optimization objective nor the constraints is convex function. The convex optimization cannot be used straightly. So, we first investigate the primal objective function and approximate it with a series of convex ones.
It is can be seen that −f _{0}(p ^{(k)}) provides a lower bound. Maximizing function (12) is equivalent to maximizing the initial function. Through the above analysis, the original problem is nonconvex, but we can approximate it with convex functions.
Nevertheless, for constraint C3, the uncertainty of the channel between STx and PRx makes C3 intractable, i.e., C3 has no idea to satisfy the requirement of a convex function. As a result, we take different attitudes towards the three constraints. For the first two, we can acquire the goal through above steps (17). As to C3, we have to study its transformation as the following description.
How to test and verify the constraint (19)? The key lies in the distribution of X. To determine the distribution explicitly, we use Gaussian approximation based on the following Lemma 1 [28].
Lemma 1.
where \({B_{n}^{2}} = \sum \limits _{k = 1}^{n} {{\sigma _{k}^{2}}}\).
Due to the above Lemma 1, the Lyapunovcondition should be satisfied as follows to apply the Lyapunov’s (CLT):
where r _{ i } is the thirdorder center distance of X _{ i }, i.e., E[(X _{ i }−m _{ i })^{3}] and m _{ i } and \({\sigma _{i}^{2}}\) are the mean and variance of the exponential distributed random variable X _{ i }, respectively. We can easily check this condition [20].
where F _{ N }(·) is the cumulative distribution function (CDF) of a normal distribution with mean m and variance σ ^{2} and \(\text {erfc}(z) = \frac {2}{{\sqrt \pi }}\int _{z}^{\infty } {{e^{ {t^{2}}}}dt}\). For (24), we assume that \({f_{3}}\left (\mathbf {p} \right) = \frac {1}{2}\text {erfc}(\frac {{I_{\max }^{p}  m}}{{\sqrt 2 \sigma }})  \varepsilon \). It is obvious that it is not a standard convex function at all which needs separate treatment. Inspired by the scheme proposed in [20], once a power allocation is given from the first two constraints, we can check whether it satisfies the outage constraint (24) or not.
If we acquire an optimal solution p from \((\mathcal {SP}1)\), we will examine whether it meets or not by substituting into \((\mathcal {SP}2)\). In another word, the input of \((\mathcal {SP}2)\) is the output of \((\mathcal {SP}1)\).
If there is no lower bound of the Lagrangian, the dual function value is −∞.
Theorem 1.
For a given I _{max}>0,p _{max}>0, problem \((\mathcal {SP}1')\) is infeasible if and only if there exists λ ^{′}≥0,ν ^{′}≥0 such that G ^{′}(λ ^{′},ν ^{′})>0; otherwise, \((\mathcal {SP}1')\) is feasible.
Proof.
See Appendix for details.
For the second part \((\mathcal {SP}2)\), we can check whether the optimal value p t ^{∗} obtained by \((\mathcal {SP}1)\) satisfies the outage constraint (27) or not. If it does, we think it is the optimal solution to \(\mathcal {P}1\); otherwise, all the transmit power decreases by a step γ and we search for the optimal one again.
Since the original complex problem is transformed into a more tractable form, we obtain the nearoptimal solution along with acceptable computational complexity. Detailed algorithm design will be provided in the following section. □
4 Power allocation algorithm
In the above section, we transfer the constraint C3 into a tractable form and approximate the original problem with convex functions. As is mentioned, once a power allocation is given, we can check whether it satisfies the outage constraint C3 or not. So our power allocation algorithm consists of two steps. Firstly, we apply the convex optimization to getting over the problem \((\mathcal {SP}1)\) based on the interior method. Note that the optimal solution must satisfy the constraints C1 and C2. Then, we should check the optimal solution whether satisfies the outage constraint C3 or not. If it does, the optimal solution of problem \((\mathcal {SP}1)\) is also the optimal solution of the original problem; otherwise, all the transmit power decreases by a step γ until the constraint C3 is satisfied.
where the inequality constraints must meet the condition strictly. Since \((\mathcal {SP}1)\) is convex as is mentioned above and we can learn from Slater’s theorem that the Lagrangian dual function (29) satisfies the strong duality, so it also satisfies the following important theorem [27] which is its sufficient and necessary condition.
Lemma 2.
The above formulation is only one of the KKT conditions. According to the Lemma 2 mentioned above, we can ensure that the optimal duality gap is zero. So when p ^{∗} is optimal, we can attain the dual optimum as well as λ ^{∗},ν ^{∗}. Nevertheless, we can hardly tackle with the KKT conditions analytically of the intractable problem in this paper. As a result, we introduce the interior point method to deal with the problem together with the KKT conditions.
It can be seen that the nonconvex optimization problem \((\mathcal {P})\) has already been approximated by \((\mathcal {P}1)\) in which only the third constraint is not convex. Then, we research for the existence of an optimal solution. Moreover, the optimality conditions (KKT conditions) are captured. In this section, we attempt to develop a quick and effective iterative algorithm. The barrier method is introduced to search for the global optimal solution throughout the previous analysis. As is well known, the barrier method is a representative technique of the class of interior point methods. It usually contains a logarithmic barrier function and the Newton method for minimization without constraints [21]. The advantage of the barrier method is that it is simple to implement and it can surely outperform other methods in performance.
Newton algorithm is a universal descent method, where the iteration consists of damping Newton stage and quadratic convergence stage, i.e., pure Newton stage. The trail of steps is the direction of searching in Newton algorithm. The backtracking line search is applied to the iteration, which is a kind of nonaccurate line search approach. Newton algorithm plays a role as the internal iteration of the interior point method. In terms of searching the feasible solution, it works simply and effectively [27].
where \(\phi \left (\mathbf {p} \right) =  \sum \limits _{i = 1}^{N} {\log \left ({{f_{1,i}}\left (\mathbf {p} \right)  {f_{2,i}}\left (\mathbf {p}\right)}\right)} \) is called the logarithmic barrier function of problem \((\mathcal {SP}1)\). Because −(1/t) log(−u) is convex, differentiable, and increasing when u<0, Newton algorithm can be applied to searching for the minimum of (35) within p _{ i }∈[0,p _{max}]. Why do we select Newton algorithm? As is mentioned above, it uses the second derivation information when computing the descent direction. It exhibits the quite quick convergence rates. As [29] suggests, the approximation quality will improve apparently as the parameter t grows.
Finally, the overall procedure for solving the optimization problem is presented in the following algorithm.
In order to verify its effectiveness, the proposed algorithm is compared with the algorithm in paper [20] and this will be discussed further with the simulation results.
5 Performance evaluation
System parameters used in simulations
Parameter  Value  Comments 

f  700 MHz  Center frequency of transmitted signal 
\({\sigma _{0}^{2}}\)  −130 dBm  Noise power 
α  4  Path loss exponent 
l _{ T },l _{ R }  2 m  Antenna height of the STx, SRx 
p _{max}  25 dBm  The maximum power of the STx 
\({I_{\max }},I_{\max }^{p}\)  −120 dBm  The interference threshold value of SC,PU 
The presented results are obtained by simulating 1000 independent trials and then taking the expected value. To benchmark performance of the proposed nearoptimal scheme, a simple suboptimal algorithm in paper [20] is considered. In paper [20], Son et al. investigate power allocation algorithms for OFDMbased cognitive radio systems, where the intrasystem channel state information (CSI) of the SC is perfectly known and the intersystem CSI is only partially available to the STx. They develop a suboptimal power allocation algorithm, which repeatedly solves a subproblem having only a transmitpower constraint and then adjusts the available transmit power until the desired outage probability is achieved. In this paper, we regard this algorithm as a comparison with our proposed algorithm. More information can be referred to paper [20].
6 Conclusions
In this paper, the robust power allocation scheme solved the problem on how to maximize the throughput capacity of SCs on the condition that the quality of service of the PU would be ensured. This problem was formulated for a CRN with massive connections. Due to the uncertainty present in the channel between the STx and the PRx, a robust interference constraint with probability was imposed to protect the PU system, which utilized the convex optimization theory. In this paper, we also introduced the iterated algorithm including Newton algorithm based on the interior point method and compared it with the algorithm proposed in [20]. We evaluated the spectral efficiency through extensive simulations and showed that the SCs can achieve higher performance in the proposed algorithm than that in the algorithm showed in [20]. As future work, a subject of extension to more general channel models including correlation or feedback delay might be researched further.
7 Appendix
7.1 The proof of Theorem 1
and thus \(\mathop {\max }\limits _{\mathbf {\lambda },\mathbf {\nu }} G'\left ({\mathbf {\lambda '},\mathbf {\nu '}} \right) \le 0\), which contradicts with the given assumption that there exists an λ ^{′}≥0,ν ^{′}≥0 such that G ^{′}(λ ^{′},ν ^{′})>0. The part is thus proved.
We thus have G ^{′}(λ ^{′},ν ^{′})≤L ^{′}(p,λ ^{′},ν ^{′})≤0.
This contradicts G ^{′}(λ ^{′},ν ^{′})>0, and thus, problem \((\mathcal {SP}1')\) is feasible if G ^{′}(λ ^{′},ν ^{′})<0. The “only if” part is thus proved.
Combining the above proofs of both “if” and “only if” parts, Theorem 1 thus follows.
Declarations
Acknowledgements
This work is supported in part by the National Natural Science Foundation of China under Grant No. 61501510 and No. 61301160, and Natural Science Foundation of Jiangsu Province under Grant No. BK20150717, and Jiangsu Planned Projects for Postdoctoral Research Funds.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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