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Robust resource allocation for cognitive relay networks with multiple primary users
EURASIP Journal on Wireless Communications and Networking volume 2017, Article number: 107 (2017)
Abstract
A robust resource allocation (RA) algorithm for cognitive relay networks with multiple primary users considering joint channel uncertainty and interference uncertainty is proposed to maximize the capacity of the networks subject to the interference threshold limitations of primary users’ receivers (PURXs) and the total power constraint of secondary user’s transmitter and relays. Ellipsoid set and interval set are adopted to describe the uncertainty parameters. The robust relay selection and power allocation problems are separately formulated as semiinfinite programming (SIP) problems. With the worstcase approach, the SIP problems are transformed into equivalent convex optimization problems and solved by Lagrange dual decomposition method. Numerical results show the impact of channel uncertainties and validation of the proposed robust algorithm for strict guarantee the interference threshold requirements at different PURXs.
Introduction
Cognitive radio (CR) is a promising technology to alleviate the looming spectrum scarcity crisis by opportunistic accessing the licensed spectrum [1]. In underlay mode of CR networks (CRNs), secondary users (SUs) can share the same radio spectrums allocated to primary users (PUs), provided that the interference to PUs’ receivers (PURXs) generated by SUs’ transmitters (SUTXs) are below the specified interference thresholds [2]. At the same time, orthogonal frequency division multiplexing (OFDM) is widely accepted as a potential air interface physical layer technology for CRNs owing to its high flexibility for allocating radio resources [3]. In OFDMbased CRNs, SUs adopt OFDM modulation and cannot cause mutual interference. However, there exists the scenario of severe channel attenuation between the pair of SUs, and large power requirement for reliable data transmission can bring harmful interference to PURXs and degrade system performance in both CRNs and the licensed networks. Therefore, end to end transmission for conventional CRNs cannot satisfy the quality of service (QoS) requirements of PUs.
In order to increase spectrum efficiency, extend coverage area, and achieve spatial diversity gain, cooperative relay technique is introduced into CRNs [4–6]. Decodeandforward (DF) [7] and amplifyandforward (AF) [8] are most widely known relay transmission protocols. In DF case, relay decodes the signal received from cognitive source, encodes it, and forwards the regenerated signal to cognitive destination. As for AF protocol, relay amplifies the signal received from cognitive source with amplify factor and then forwards it to cognitive destination. With the help of cooperative relays, the transmit power of cognitive source is reduced and the interference to PURXs is mitigated, which can prohibit the performance degradation of primary network.
Resource allocation (RA) is a very important issue for CRNs with cooperative relays, which can further improve the system performance. Most existing literatures on RA for OFDMbased relay CRNs assume that perfect channel state information (CSI) is available [9, 10]. In [9], a joint subcarrier selection and power allocation scheme using variational inequality in OFDMbased cognitive relay networks is proposed to maximize the system throughput. In [10], an asymptotically optimal subcarrier pairing, relay selection, and power allocation scheme is obtained in DF relayed OFDMbased cognitive system under the individual power constraints in source and relays. These optimization problems with perfect CSI are referred as the nominal problems [11]. Due to time varying and random nature of wireless system, this assumption is invalid. Therefore, the solutions to nominal problems are not robust.
Robust optimization theory has been applied in CRNs to overcome uncertainty parameters in nominal problems in recent years [12, 13]. Worstcase approach [14] and Bayesian approach [15] are widely used in robust optimization problems. For worstcase approach, uncertainty parameters belong to a predefined uncertainty region, so that this approach can strictly guarantee QoS of PURXs for all realizations of the uncertainty in that region. Currently, most works on worstcase approach focus on how to find a proper set to describe the characteristics of the uncertainty parameters. However, there is no unified way to define an uncertainty set, and the main principle is to preserve the convexity of the original nominal problem and obtain a solvable optimization problem. In [16], the authors study robust beamforming and power allocation problem in cognitive relay networks with imperfect CSI modeled by Gaussian random variables. In [17], the robust ergodic uplink resource allocation problem for secure communication with imperfect CSI in relayassisted CRNs is investigated. The channel uncertainty set is represented by an ellipsoid uncertainty region. In [18], the authors study robust worstcase interference control and use general norm and polyhedron model to describe channel uncertainty set in underlay CRNs. Bayesian approach handles uncertainty parameters with certain statistical properties and satisfies QoS of PURXs in a probabilistic manner. Generally, worstcase approach is more appealing since the interference to PURXs should be below the interference threshold under any circumstances, even in the presence of the worstcase scenario.
Recently, some research works have been conducted on robust RA in cognitive relay networks considering the impacts of uncertainty parameters [19–21]. In [19], the authors propose a joint relay precoder and power allocation design to minimize the sum meansquare error (MSE) of the transceiver node with channel uncertainties between SU and SU. In [20], the outage performance of cognitive relay networks is analyzed with the channel uncertainties between SU and PU. In [21], robust resource allocation problem for cooperative CRNs is optimized with the channel uncertainties between SU and SU, SU and PU. However, the interference uncertainty is not considered in these works.
Different from most existing literatures, this paper investigates the robust RA problem for OFDMbased cognitive relay network with multiple PUs, where channel uncertainty and interference uncertainty are all considered. Ellipsoid set and interval set are jointly used to characterize the uncertainty region sets. Under the total power budget constraint of SUTX and relays, the optimization objective is to maximize capacity of the cognitive relay network while maintaining the interference to PURXs under their prescribed thresholds. The major contributions of this work are as follows.

1.
Robust RA problem is solved through robust relay selection and power allocation separately under the consideration of both channel uncertainty and interference uncertainty characterized by ellipsoid set and interval set.

2.
Robust relay selection and power allocation problems are formulated as semiinfinite programming (SIP) problems. And they are converted into equivalent convex optimization by worstcase approach and solved by Lagrange dual decomposition method.

3.
Closed form analytical solutions to robust relay selection and power allocation have been derived. Numerical results show that the proposed robust algorithm outperforms the nonrobust ones for ensuring QoS of multiple PURXs under the influence of the uncertainties.
The rest of this paper is organized as follows. The system model and nominal RA problem formulation are described in Section 2. The robust RA problem is proposed and solved in Section 3. The performance of the proposed robust algorithm is illustrated with simulation results in Section 4. Finally, conclusions are drawn and future works are given in Section 5.
System model and nominal RA problem formulation
System model
We consider an OFDMbased cognitive relay network with a primary network as shown in Fig. 1. The related symbol explanations are given in Table 1. It is assumed that there is no direct link between the cognitive source (S) and the cognitive destination (D), so that S and D communicate with each other through multiple relays. We also assume that there are K relays and N subcarriers. The relays operate in time division halfduplex mode with DF protocol. Signal transmissions are conducted in two time slots. In the first time slot, S transmits signals to relay node R _{ k },k∈[1:K]. While in the second time slot, R _{ k } forwards the regenerated signals to D through the subcarrier which has the same sequence number as the first time slot. The capacity of the i ^{th}, i∈[1:N] subcarrier in the two time slots can be expressed as
where \(C_{SR_{k}}^{i}\) and \(C_{R_{k}D}^{i}\) represent the capacity of the i ^{th} subcarrier transmitted from S to R _{ k } (S R _{ k } link) and from R _{ k } to D (R _{ k } D link), respectively. L is the number of primary users. \(J_{ik}^{l}\) and \(J_{iD}^{l}\) can be modeled as the additive white Gaussian noise (AWGN) described in [22]. In order to simplify mathematical analysis, we assume that the interference power from PUTXs plus the noise power to the i ^{th} subcarrier in the first time slot and the second time slot are equal, which can be denoted by \(\sigma _{i}^{2}=\sigma _{ik}^{2}+\sum _{l=1}^{L}{J_{ik}^{l}}=\sigma _{iD}^{2}+\sum _{l=1}^{L}{J_{iD}^{l}}\) as described in [23]. In fact, the achievable rate of each subcarrier is the minimum rate of the two time slots. Therefore, the capacity of the i ^{th} subcarrier for the cognitive relay network can be given as
As for DF protocol, \(C_{k}^{i}\) can achieve maximum capacity when the signal to interference plus noise ratio (SINR) at R _{ k } equals to that at D, then we can have the following relation for the i ^{th} subcarrier as [23]
where \(a_{k}^{i}=\frac {\left  h_{SR_{k}}^{i}\right ^{2}}{\sigma _{i}^{2}}, b_{k}^{i}=\frac {\left  h_{R_{k}D}^{i}\right ^{2}}{\sigma _{i}^{2}}\). We define \( H_{k}^{i}=\frac {\left  h_{SR_{k}}^{i}\right ^{2}\left  h_{R_{k}D}^{i}\right ^{2}}{\left  h_{SR_{k}}^{i}\right ^{2}+\left  h_{R_{k}D}^{i}\right ^{2}}\) and \(\alpha _{k}^{i}=\frac {H_{k}^{i}}{\sigma _{i}^{2}}\) to represent the normalized channel gain and the equivalent channel gain of the i ^{th} subcarrier in the two time slots, respectively. The total transmit power on the i ^{th} subcarrier in the two time slots is
According to (4) and (5), the maximum capacity of the i ^{th} subcarrier can be rewritten as
Note that in cognitive relay network, both SUTX and relays have to take the interference thresholds of different PURXs into account. Therefore, the interferences introduced to the l ^{th} PURX in the two time slots are
where \(I_{SP_{l}}\) and \(I_{RP_{l}}\) are the interference to the l ^{th} PURX from the total subcarriers transmitted between S and all relays (SR link) and between all relays and D (RD link), respectively. \(\rho _{k}^{i}\) is a binary factor indicating whether the i ^{th} subcarrier is allocated to R _{ k }. And \(I_{th}^{l}\) is the interference threshold of the l ^{th} PURX.
According to the definition of \(H_{k}^{i}\) and \(P_{k}^{i}\), (7) and (8) can be rewritten as
where \(G_{S,k,l}^{i}=\frac {\left  h_{SP_{l}}^{i}\right ^{2}}{\left  h_{SR_{k}}^{i}\right ^{2}}\) is defined to represent the normalized channel gain of the i ^{th} subcarrier for S P _{ l } link and S R _{ k } link, and \(G_{k,l,D}^{i}=\frac {\left  h_{R_{k}P_{l}}^{i}\right ^{2}}{\left  h_{R_{k}D}^{i}\right ^{2}}\) is defined to describe the normalized channel gain of the i ^{th} subcarrier for R _{ k } P _{ l } link and R _{ k } D link.
Nominal RA problem formulation
The optimization objective is to maximize the capacity of the cognitive relay network while maintaining the interference to PURXs below their interference thresholds under the total power constraint. Mathematically, this optimization can be formulated as
Nominal RA problem (P0):
subject to
where C1 and C2 are the interference constraints of the l ^{th} PURX in the first time slot and the second time slot, respectively. C3 is the transmit power constraint, and P _{total} is the maximum total transmit power. C4 and C5 are the relay selection constraints to indicate that only one relay is selected by each subcarrier to guarantee the exclusiveness of subcarrier. If \(\rho _{k}^{i}=1\), R _{ k } is allocated to the i ^{th} subcarrier, otherwise not.
Apparently, P0 is a mixed binary integer programming problem, and it is difficult to achieve a joint relay selection and power allocation solution. Therefore, we will solve this problem by dividing P0 into two subproblems, i.e., relay selection and power allocation separately. Firstly, we allocate subcarriers to relays through supposing uniform power distribution over subcarriers (\(\textrm {i.e.,\,\,}P_{k}^{i}=\frac {P_{\text {total}}}{N}\)). Secondly, power allocation is carried out under the given relay selection scheme. Note that relay selection scheme needs to solve a discrete optimization problem, which is NPhard. Thus, we relax the integrality constraint on \(\rho _{k}^{i}\) from \(\rho _{k}^{i}\in \left \{0,1\right \}\) to \(\rho _{k}^{i}\in \left (0,1\right ]\) and relax \(\sum _{k=1}^{K}{\rho _{k}^{i}}=1\) to \(\sum _{k=1}^{K}{\rho _{k}^{i}}\le 1\), which can transform the discrete optimization problem into a continuous linear optimization problem [24].
Nominal relay selection problem (P1):
subject to
This time P1 is a convex problem, and we can obtain an optimal solution to \(\rho _{k}^{i}\) after applying KarushKuhnTucker (KKT) conditions [24]. Therefore, R _{ k } with maximum \(\rho _{k}^{i}\) is allocated to the i ^{th} subcarrier, which can be expressed as
where K(i)∈[1:K] presents the relay selected by the i ^{th} subcarrier. Under this relay selection scheme, P0 can be converted into
Nominal power allocation problem (P2):
subject to
P2 is also a convex optimization problem with linear constraints so that an optimal power allocation analytical solution can also be obtained by using KKT conditions. We define this nominal relay selection and power allocation algorithm under the assumption of the perfect CSI as nonrobust algorithm.
Robust RA algorithm
In cognitive relay networks, channel gain is very important information for RA. However, perfect channel gain is extremely difficult to obtain due to estimation errors, quantization errors, and feedback delays. The interference power at relay and SURX are also inaccuracy due to the measurement errors. The assumption of perfect CSI is so idealistic that it can lead to system performance degradation. In this paper, we study robust RA problem with joint channel uncertainty and interference uncertainty. Based on the robust optimization theory, the uncertainty parameter is modeled by the sum of the estimated value (i.e., nominal value) and additive error (i.e., perturbation part). Mathematically, the uncertainty parameter can be expressed as \(g=\bar {g}+\varDelta g\), where g is the actual value, \(\bar {g}\) is the estimated value, and Δg is the estimated error. We adopt ellipsoid set to describe the channel uncertainty and interval set to describe interference uncertainty respectively. According to the definition of the ellipsoid uncertainty, \(H_{k}^{i}\), \(G_{S,k,l}^{i}\), and \(G_{k,l,D}^{i}\) can be formulated as
where \(\mathcal {H}\), \(\mathcal {G}\), and \(\mathcal {F}\) denote the channel uncertainty sets. ε _{ l }∈[0,1), η _{ l }∈[0,1), and δ _{ l }∈[0,1) are the maximum deviations of \(H_{k}^{i}\), \(G_{S,k,l}^{i}\), and \(G_{k,l,D}^{i}\), respectively. We define ψ as the uncertainty set of \( \alpha _{k}^{i} \). Based on the definition of the interval uncertainty, \( \alpha _{k}^{i} \) can be formulated as
where \(\xi _{k}^{i}\in [0,1)\) is the upper bound of the uncertainty that determines the size of the uncertainty region. Afterwards, we study robust RA problem with these uncertainty parameters. According to (18)–(21), P1 can be transformed as
Robust relay selection problem (P3):
subject to
P3 is a SIP problem [25] subject to an infinite number of constraints with respect to the sets of \(\mathcal {H},\,\,\mathcal {G},\,\,\mathcal {F}\,\,\), and ψ. Through worstcase approach, we can make all the constraints be in the worstcase scenario, and P3 is transformed into an equivalent problem with finite constraints. We first transform (22) by \(\varDelta \alpha _{k}^{i}=\xi _{k}^{i}\bar {\alpha }_{k}^{i}\), which denotes the worstcase equivalent channel gain. Then, according to the CauchySchwartz inequality [26], we have
Similarity, C1 and C2 in P3 can be further formulated as
We can see that the transformed P3 is convex and it can be solved by the Lagrange dual decomposition method [27]. The optimal robust relay selection factor \(\rho _{k}^{i}\) can be derived in the form of
where
\([x]^{+}\triangleq \max (0,x)\). \(\varphi _{l}\geqslant 0\), \(\beta _{l}\geqslant 0\) and \(\lambda _{i}\geqslant 0\) are the Lagrange multipliers updated by the subgradient method with recursive forms
where t is the iteration number, \(s_{1}^{l}\),\(s_{2}^{l}\), and \(s_{3}^{i}\) are the small positive step sizes. In the sequel, we optimize the robust power allocation problem after obtaining the robust relay selection factor.
Robust power allocation problem (P4):
subject to
Obviously, P4 is still a SIP problem, so we need to transform it into an equivalent optimization problem limited by the finite constraints with the worstcase approach. First, (31) can be reformulated as \(\underset {\,\,P_{K(i)}^{i}\geqslant 0}{\max }\sum _{i=1}^{N}{\frac {1}{2}\log _{2}\left (1+P_{K(i)}^{i}\bar {\alpha }_{K(i)}^{i}\left (1\xi _{K(i)}^{i}\right)\right)}\). Then C1 and C2 in P4 can be formulated as
We can see that the transformed P4 is also convex and its Lagrange function is
where \(\mu _{l}\geqslant 0\), \(\nu _{l}\geqslant 0\), and \(\omega \geqslant 0\) are the Lagrange multipliers. (35) can be solved by the dual decomposition method, i.e.,
The dual function of (35) is defined as
Substituting (35) to (37), we can get
where
We can get the optimal solution of (38) though optimizing (40) by solving \(\frac {\partial D(P_{K(i)}^{i})}{\partial P_{K(i)}^{i}}=0\), the optimal robust power allocation is derived in the form of
where
The Lagrange multipliers can be updated by the subgradient method as
where \(s_{4}^{l}\), \(s_{5}^{l}\), and s _{6} are the small positive step sizes. The iterations are repeated until the robust power allocation process converges. We define this robust relay selection and power allocation algorithm under the influence of different uncertainty parameters as robust algorithm.
The computational complexity of the proposed robust algorithm can be counted roughly as follows. We address the robust resource allocation problem with two steps, i.e., relay selection and power allocation separately. First, we allocate subcarriers to relays assuming uniform power distribution over subcarriers. Each subcarrier selects the best relay (i.e., the relay with highest \({{\rho {{\!~\!}_{k}^{i}}}^{*}}\)) for itself with computational complexity K. We assume T _{1} is the number of iterations with subgradient method to get \({{\rho {{\!~\!}_{k}^{i}}}^{*}}\). After N subcarriers all finish the relay selection, the computational complexity of this relay selection procedure is \(\mathcal {O}(NK{T_{1}})\). Then, power allocation is carried out under the given relay selection scheme. The power allocation problem is decomposed into N parallel power allocation subproblems. In every subproblem, we assume T _{2} is the number of iterations to obtain \({P{{\!~\!}_{K(i)}^{i}}^{*}}\) with subgradient method. After N evaluations, the computational complexity of the power allocation is \(\mathcal {O}(N{T_{2}})\). To sum up, the overall computational complexity is \(\mathcal {O}({N({K{T_{1}} + {T_{2}}})})\), which is linear to N.
Simulation results
In this section, we use Monte Carlo simulation with MATLAB software to show the effectiveness and the outperformance of our proposed robust algorithm. Setting of simulation parameters are described in Table 2. Channel gains \(h_{SR_{k}}^{i}\), \(h_{R_{k}D}^{i}\), \(h_{SP_{l}}^{i}\), and \(h_{R_{k}P_{l}}^{i}\) are assumed to be frequency flat Rayleigh fading channels. They are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian (CSCG) random variables (RVs) and distributed as \(h\sim \mathcal {C}\mathcal {N}\left (0,\,\frac {1}{\left (1+d\right)^{\tau }}\right)\), where τ is the path loss coefficient and d is the distance among different nodes in the system. Without lose of generality and to make the realization of the proposed algorithm simple and better understood, we assume \(\sigma ^{2}=\sigma _{i}^{2}\), \(\xi =\xi _{k}^{i}\), \(\varepsilon _{l}=\varepsilon \bar {H}_{k}^{i}\), \(\eta _{l}=\eta \bar {G}_{S,k,l}^{i}\,\), and \(\delta _{l}=\delta \bar {G}_{k,l,D}^{i}\), respectively. For notational brevity, let ζ=ξ=ε=δ=η and ζ∈[0,1) represent the normalized upper bound for the all uncertainty regions. The simulations have been conducted for 20,000 independent channel realizations.
Figure 2 shows the relay selection factors of all subcarriers in both robust algorithm and nonrobust algorithm. The relays in these two algorithms are distinguished by different colors. Based on the relay selection scheme in these two algorithms, R _{ k } with maximum \(\rho _{k}^{i}\) is allocated to the i ^{th} subcarrier. In other words, each subcarrier selects the best relay for itself. As for nonrobust algorithm, we can observe that the relays selected by subcarriers are R _{1},R _{2},R _{2},R _{1},R _{1}, and R _{1} in sequence. When ζ=0.1, the robust relay selection results are the same as the nonrobust algorithm which validates the robustness of the proposed robust relay selection scheme.
Under the aforementioned relay selection scheme, Fig. 3 illustrates the power allocation schemes to subcarriers for the two algorithms. We can see that the power allocated to subcarriers in the nonrobust algorithm is higher than that in robust algorithm. According to C1 and C2 in P4, we can find that the transmit power in the robust algorithm has to be reduced to keep the interference thresholds of PURXs below their preset limit values under the influence of the uncertainties.
Figure 4 shows the saved power versus the increase number of iterations. We define the saved power as the difference value of the total transmit power P _{total} and the actual transmit power \(\sum _{i=1}^{N}{P_{K(i)}^{i}}\). We find that the saved power in nonrobust algorithm converges to zero, which indicates that the nonrobust algorithm makes full use of the total transmit power. However, due to the existing uncertainties, the robust algorithm cannot take advantage of the total transmit power. Moreover, the saved power increases with the increase of uncertainties, since the robust algorithm has to reduce more transmit power to cope with the increasing uncertainties.
Figure 5 demonstrates the interference power to each PURX from SR link and RD link in the proposed robust algorithm, nonrobust algorithm, method in [9], and method in [10]. In this paper, we assume different PURX has different interference threshold. It is also more suitable to practical systems than the assumption that the interference constraints of different PURX under the same interference level. Figure 5 a, b shows that the proposed robust algorithm can strictly guarantee the interference threshold of each PURX, and the interference to each PURX in nonrobust algorithm, method in [9]and method in [10] are all exceed their interference thresholds, which means that the nonrobust algorithm cannot satisfy the QoS of PURXs with uncertainty parameters.
Figure 6 shows the capacity convergence results of the two algorithms with the increase number of iterations. We can see that the two algorithms can quickly converge to the equilibrium points. It can also be observed that, although we have maximized the capacity of the cognitive relay network through the robust algorithm with uncertainty parameters, the capacity of the robust algorithm is still lower than that of the nonrobust algorithm with perfect CSI. The reason is that the nonrobust algorithm can make full use of the total transmit power regardless of the influence of the uncertainty parameters on the interference power to PURXs, but the robust algorithm reduces the transmit power to overcome the uncertainties for the QoS of PURXs. In addition, the capacity of the robust algorithm decreases with the increase of the uncertainty parameter. Figures 5 and 6 indicate that the robust algorithm outperforms the nonrobust algorithm for the perspective of ensuring the QoS of PURXs at the expense of system capacity loss.
Figure 7 illustrates the capacity of the nonrobust algorithm and the robust algorithm versus the interference threshold under the influence of uncertainties. In order to make simulations more simplified, we assume that the interference thresholds of different PURXs are the same. From Fig. 7, we can see that the capacity of the robust algorithm is lower than that of the nonrobust one and decreases with the increase of the uncertainties in accordance with Fig. 6. Moreover, we can find that the capacity of these two algorithms increases with the increase of the interference threshold at first. When power allocation reaches its maximum in the two algorithms, the capacity does not change and becomes constant regardless of the increase of interference threshold.
Figure 8 shows the capacity of the nonrobust algorithm and the robust algorithm versus the power budget with the assumption of \(I_{th}^{}=I_{th}^{1}=I_{th}^{2}=0.8\times 10^{12}W\). As expected, the capacity of the robust algorithm is lower than that of the nonrobust algorithm and decreases with the increase of the uncertainties. We can also observe that the capacity of the two algorithms increases with the increasing power budget at the beginning. However, when the interference to PURXs reaches the preset thresholds, the capacity stops increasing and remains constant despite of the increase of power budget.
Figure 9 gives the effect of the interferencethreshold and the power budget on the capacity under the assumption of ζ=0.2.With the increase of the interference threshold, the capacity increases in both robust and nonrobust algorithm initially. However, when interference threshold becomes larger to some extent, interference constraint becomes inactive. Power budget becomes the limiting constraint and plays a dominant role. We can also observe that the higher power budget is, the larger the capacity is.
Conclusions
In this paper, we have studied robust RA problem in the underlay OFDMbased cognitive relay network with joint channel uncertainty and interference uncertainty. Based on worstcase approach and Lagrange dual decomposition method, closed form analytical solutions to robust relay selection and power allocation have been derived. Numerical results demonstrate the robustness of the proposed robust relay selection scheme. The robust algorithm is superior to the nonrobust algorithms in terms of guaranteeing the interference thresholds of different PURXs, but the capacity of the robust algorithm is little lower than that of nonrobust algorithm for overcoming the uncertainties and also decreases with the increase of the uncertainties. In our future works, we will extend this frame work to twoway cognitive radio networks with multiple SUs or multiple antennas.
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Acknowledgements
The work of this paper is supported by the National Natural Science Foundation of China under grant No. 61571209.
Authors’ contributions
WY contributed in the conception of the study and design of the study and wrote the manuscript. Furthermore, WY carried out the simulation and revised the manuscript. XH helped to perform the analysis with constructive discussions and helped to draft the manuscript. All authors read and approved the final manuscript.
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The authors declare that they have no competing interests.
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Yang, W., Zhao, X. Robust resource allocation for cognitive relay networks with multiple primary users. J Wireless Com Network 2017, 107 (2017). https://doi.org/10.1186/s1363801708934
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DOI: https://doi.org/10.1186/s1363801708934
Keywords
 Cognitive relay networks
 Robust resource allocation
 Channel and interference uncertainty
 Worstcase approach
 Lagrange dual decomposition method