Power Allocation in Underlay Cognitive Radio Systems with Feasibility Detection Sudhir Singh * , Paul D. Teal † , Pawel A. Dmochowski † and Alan J. Coulson * * Industrial Research Limited, Lower Hutt, New Zealand † School of Engineering and Computer Science, Victoria University of Wellington, Wellington, New Zealand Email:{s.singh,a.coulson}@irl.cri.nz, {paul.teal,pawel.dmochowski}@vuw.ac.nz Abstract—We consider a cognitive radio system with N sec- ondary user (SU) pairs and a pair of primary users (PU). The SU power allocation problem is formulated as a capacity maximisation problem under PU and SU quality of service and SU peak power constraints. The problem is formulated as a geometric program, solved for both low- and high- signal-to- interference-and-noise ratio (SINR) regimes. We present a novel method of detecting and removing infeasible SU quality of service constraints from the SU power allocation problem that results in considerably improved SU performance. Capacity cumulative distribution functions for Rayleigh fading channels are produced. I. I NTRODUCTION A large number of papers have appeared on various aspects of cognitive radio (CR) systems, including fundamental infor- mation theoretic capacity limits (see, for example, [1–7]). In an underlay CR system the secondary users (SUs) protect the pri- mary user (PU) by regulating their transmit power to maintain the PU receiver interference below a well defined threshold level. The limits on this received interference level at the PU receiver can be imposed by an average/peak constraint [2], or a minimum value for its signal-to-interference-and-noise ratio (SINR) [4]. Although it imposes an additional requirement that channel state information (CSI) be available, the advantage of using an SINR-based PU protection mechanism is that it removes the constant interference threshold, thus benefiting the SUs when the PU link is strong. Power control in conventional wireless networks has been extensively studied in the literature [8–10]. Power control in CR systems presents its own unique challenges. In spectrum sharing applications, SU power must be allocated in a manner that achieves the goals of the CR system while not adversely affecting the operation of the PU. Generally the goals of the CR are not compatible with the goals of the PU; for instance, increasing SU power to increase SU capacity will tend to increase interference to the PU. There is a growing body of lit- erature on power control and capacity of CR systems. In [11], soft sensing information was used for optimal power control to maximise capacity of one SU pair coexisting with one PU pair. The impacts of SU transmission power on the occurrence of spectrum opportunities and the reliability of opportunity detection was analysed in [12]. In [13], dynamic programming was used to develop a power control strategy for one SU pair under a Markov model of the PU’s spectrum usage. Optimal power allocation strategies to achieve the ergodic capacity and the outage capacity of one SU pair coexisting with one PU pair under different types of power constraints and fading channel models were obtained in [6]. Power control using game-theoretic approaches have been proposed in [14, 15]. Power control for CR systems using geometric programming have been proposed in [16–18]. A minimax approach was used in [18] to minimise the maximum transmit power for a CR system coexisting with a PU-Rx. The interference caused by a PU-Tx to the SU-Rxs was not considered in the problem formulation of [18]. In [16], a distributed approach was used for power allocation to maximise SU sum capacity under a peak interference constraint, but the approach did not include the interference caused by the PU-Tx in the analysis and the problem was only analysed for a high SINR scenario. Convex optimisation methods are widely used in the design and analysis of communications systems. Many problems that arise in communications signal processing can be cast or converted into convex optimisation problems which allow an- alytical or numerical solutions to be calculated easily [19]. In [20], several problems for designing optimal dynamic resource allocation in CR systems are formulated and the key role that convex optimisation plays in finding the optimal solutions is demonstrated. In [21], we formulated the SU power allocation problem as a capacity maximisation problem under PU and SU quality of service (QoS) and SU peak power constraints and showed that it can be solved using geometric programming and convex optimisation techniques. Unlike in [16–18], where the PU interference at each SU-Rx is neglected, the effect of the PU interference is evaluated in [21] by explicitly including it in the formulations. In this paper we extend the work of [21] and present a novel method of detecting and removing infeasible SU quality of service constraints from the SU power allocation problem that results in considerably improved SU performance. Solutions for both low and high SINR scenar- ios are presented. Capacity cumulative distribution functions (CDFs) for various channel conditions are obtained through solution of our optimisation problems. II. SYSTEM MODEL As shown in Fig. 1, we consider a cognitive radio system with a single PU and N SU transmitters communicating simultaneously over a common channel to their respective receivers. Independent, point-to-point, flat Rayleigh fading channels are assumed for all links in the network. Let g p =