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Cooperative multiple access in cognitive radios to enhance QoS for cell edge primary users: asynchronous algorithm and convergence

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Abstract

Cognitive radio (CR) systems have been proposed for efficient usage of spare spectrum licensed to primary systems. This leads to the issue of providing as much spectrum to CR users as possible while not degrading the quality of service (QoS) of primary users of the spectrum. This paper proposes a novel cooperation scheme between primary and CR users to guarantee QoS of primary users up to the cell edge while making the licensed spectrum available for opportunistic access by the CR users. We suggest that the primary users at the cell edge, who have poor QoS, should allow secondary users to access their spectrum, while at the same time, the secondary users would help to enhance the primary users QoS using superposition coding on the primary users transmissions. Thus the proposed method can provide a so called “win-win strategy” by benefiting both primary and CR users. The proposed cooperative access scheme in cognitive radios solves efficiently the sum-rate maximization problem on cognitive Gaussian Multiple Access Channels (GMACs) for power allocation of primary systems that cooperates with CR systems in a distributed fashion. We solved the problem using iterative Jacobian method in a distributed manner. A totally asynchronous distributed power allocation for sum-rate maximization on cognitive GMACs is suggested. Numerical results show that the QoS of primary users at the cell edge is improved by the proposed cooperative access scheme.

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Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2013427). Parts of this work had been done while the first author was with The School of Information and Communication Technology, KTH Royal Institute of Technology, Kista, Sweden.

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Correspondence to Hoon Kim.

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This paper was presented in part at the IFIP WMNC 2013, Dubai, April 2013 [12].

Appendix : Lagrangian relaxation of sum-rate maximization on cognitive GMAC

Appendix : Lagrangian relaxation of sum-rate maximization on cognitive GMAC

Lagrangian duality provides lower bounds on the optimal value of the problem and simplifies the computation of the lower bound (or optimal in some cases) on the optimal value of the nonconvex QCQP [10, 17].

Lagrangian of (8) is,

$$\begin{aligned} L(\mathbf{x},\lambda , \mu ) = {\mathbf{x}}^{{T}} ({\mathbf{H}} + \lambda {\mathbf{G}}){\mathbf{x}} + 2(\lambda {\mathbf{b}}+\frac{{\mu }}{2})^T {\mathbf{x}} + \lambda c -\mu ^T \mathbf{1 }+ z, \end{aligned}$$

and the dual function is,

$$\begin{aligned} g(\lambda , \mu )= & {} \mathop {\inf }\limits _\mathbf{{x \ge 0}} \mathrm{{ }}L(x,\lambda , \mu ) \\= & {} \left\{ {\begin{array}{c@{\quad }c} {y - \mathbf {w}^T (\mathbf{{H}} + \lambda \mathbf{{G}})^{-1} \mathbf {w} } &{} {{\mathbf{H}} + \lambda \mathbf{{G}} \succeq 0}, \\ &{} \mathbf{w} \in \mathcal {R(\mathbf {H}} + \lambda \mathbf{{G})} \\ { - \infty } &{} {\mathrm{{otherwise,}}}\\ \end{array}} \right. \end{aligned}$$

where \(y=\lambda c-\mu ^T \mathbf 1 +z\) and \(\mathbf w =\lambda \mathbf b +\frac{\mu }{2}\).

Then, the use of Schur complement allows the dual problem to be expressed as an SDP:

$$\begin{aligned}&\text{ maximize } \; \phi + \lambda c -\mu ^T \mathbf 1 + z \nonumber \\&\quad {\hbox {subject to}} \nonumber \\&\qquad \lambda \ge \mathrm{{0 }},~~ \mu \succeq 0 \nonumber \\&\qquad \left[ {\begin{array}{c@{\quad }c} {{\mathbf{H + }}\lambda \mathbf{{G}}} &{} {(\lambda \mathbf b +\frac{\mu }{2})\mathrm{{/2}}} \\ {(\lambda \mathbf b +\frac{\mu }{2})^T\mathrm{{/2}}} &{} -\phi \nonumber \\ \end{array}} \right] \succeq 0, \nonumber \\&\qquad \phi ,\lambda \in {\mathcal {R}}~~\hbox {and}~ \mu \in {\mathcal {R}}^N. \end{aligned}$$
(27)

This result is also known as S-procedure in control theory. The main insight is that while the original problem is possibly nonconvex and numerically hard to solve, its dual can be expressed as an SDP and is easy to solve. Regarding the complexity, it is known that SDR has a complexity order of O(\(N^3\)) where \(N\) is the number of variables in problems [19]. The number of variables for sum-rate maximization problem on Gaussian cognitive MAC problem addressed in this paper is \(N\) which denotes the number of users. While as in the previous approach [8], the problem is solved using a heuristic search algorithm which requires considerations of values (0 or not) for all \(x_k\) which leads the complexity order to O(\(2^N\)), an NP hard problem.

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Han, Sw., Kim, H., Han, Y. et al. Cooperative multiple access in cognitive radios to enhance QoS for cell edge primary users: asynchronous algorithm and convergence. Telecommun Syst 61, 631–643 (2016). https://doi.org/10.1007/s11235-015-0058-x

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