Outage probability minimising joint channel and power allocation for cognitive radio networks D. Xu, Z.Y. Feng, Y. Liu and P. Zhang The problem of joint channel and power allocation in a downlink spec- trum sharing cognitive radio network is investigated. The objective is to minimise the average outage probability of the secondary users (SUs). In this regard, a joint channel and power allocation scheme is proposed. The obtained numerical results show that the proposed joint channel and power scheme greatly improves the performance of the SUs. Introduction: In spectrum sharing (SS) cognitive radio (CR) networks [1], resource allocation for secondary users (SUs) is used for restricting the interference caused by the SUs to the primary users (PUs), as well as improving the performance of the SUs. For real-time services, outage capacity, which is defined as the maximum rate that can be supported with a given outage probability, is a good performance limit indicator. However, there are few studies in the literature that focus on resource allocation for the SUs targeting at maximising outage capacity. Notably, in [2–4], the authors studied the problem of optimal power allocation to minimise the outage probability for a given rate, which is equivalent to maximising the outage capacity for a given outage prob- ability. However, only a single SU and single channel is considered in [2–4]. In this Letter, we study the problem of joint channel and power allocation to minimise the average outage probability of multiple SUs in multichannel CR networks, which, to our best knowledge, has not been touched upon in the literature. System model and problem formulation: Consider a downlink SS CR network with one cognitive base station (CBS) and a set of SUs that shares spectrum with a primary network. The spectrum of interest is divided into N independent, orthogonal channels. We assume that each SU can use one channel and each channel can be allocated to one SU at a time. Thus, an admission control should be performed to choose N SUs to share N channels with the PU. Since our focus is on joint channel and power allocation, we assume the admission control has been made. The channel power gains on channel n from CBS to SU k and CBS to the PU at fading state n are denoted by g ss n,k (n) and g sp n (n), respectively, where n denotes the joint fading state for all the channels involved. All the channels are assumed to be independent flat fading channels. The white Gaussian noise power is denoted by s 2 . Denote p n,k (n) as the transmit power allocated to SU k on channel n at fading state n, and p(n) (where p(n)=(p 1 (n)), ... , p N (n))) as a permutation of {1, ..., N} such that channel p k (n) is allocated to SU k at fading state n, i.e. p pk (n),k ′ (n)= 0, ∀k ′ = k . The average transmit power of the CBS is restricted as E ∑ N k=1 p pk (n),k (n)   ≤ P (1) where E denotes the statistical expectation, and P is the prescribed limit for the average transmit power of the CBS. In addition, to protect the PU, the average interference power introduced to the PU band is restricted as follows E ∑ N k=1 p pk (n),k (n)g sp pk (n)   ≤ Q (2) where Q is the prescribed limit for the average interference power. The transmission outage probability of SU k is given by P k = Pr log 2 1 + P pk (n),k (n)g ss pk (n),k (n) s 2   , R k   (3) where R k is the target rate of SU k. Then the problem of joint channel and power allocation to minimise the average outage probability of the SUs is formulated as follows P1 : min p (n),p(n) 1 N ∑ N k=1 P k s.t. (1), (2) (4) where p(n) is a vector of transmit power allocation for the SUs, which is given by ( p p1(n),1 (n), ... , p pN (n),N (n)). Joint channel and power allocation algorithm: To solve P1, we intro- duce the following indicator function for the outage event of SU k at fading state n as I k (n)= 1, log 2 1 + p p k (n),k (n)g ss p k (n) ,k (n) s 2   , R k 0, otherwise ⎧ ⎨ ⎩ (5) First we assume p(n) is fixed. Then P1 is transformed to a power allo- cation problem as follows P2 : min p(n) 1 N ∑ N k=1 E{I k (n)} s.t. (1), (2) (6) The Lagrangian function of P2 is written as L(p(n), l, m)= 1 N ∑ N k=1 E{I k (n)} + l E ∑ N k=1 p pk (n),k (n)   − P   + m E ∑ N k=1 p pk (n),k (n)g sp pk (n) (n)   − Q   (7) where l and m are the dual variables associated with the constraints in (1) and (2), respectively. The dual function is then expressed as G(l, m)= min p(n) L(p(n), l, m) = E ∑ N k=1 G k (l, m)   − lP − mQ (8) where G k (l, m)= min p p k (n),k (n) 1 N I k (n)+ (l + mg sp pk (n) (n))p pk (n),k (n) (9) It is observed that, by using the Lagrange dual decomposition, G(l, m) is obtained by solving N independent problems for each fading state n, as given by (9). It is also noted that I k (n) is a step function where the crit- ical value is x k (n)= s 2 (2 R k−1 ) g ss p k (n),k (n) . Then, it is easy to obtain the optimal solution, p pk (n),k (n), for the problem (9) as follows p pk (n),k (n)= x k (n), x k (n)≤ 1 N(l+mg sp p k (n) (n)) 0, otherwise  (10) The above power allocation belongs to the well-known truncated channel inversion with fixed rate (TIFR) power allocation. Then, G k (l, m) is obtained by inserting (10) into the objective function in (9) for each fading state n and each of the N SUs. Recall that at each fading state n, for all the N SUs, there are N! possible channel allocation choices p(n)’s, which results in N! possible values of S N k=1 G k (l, m). Thus, we transform the channel allocation into a weighted bipartite matching and then find a minimal weighted match. The weighted bipartite graph is formed as follows. First, represent the N SUs by a set of vertices, which is connected to another set of vertices representing N channels. An edge exists between the vertex representing SU k and the vertex representing channel n if and only if channel n is allocated to SU k. For each edge, assign a weight that is equal to G k (l, m). Therefore, the problem of channel allocation to find the minimal S N k=1 G k (l, m) among N! possible channel allocation choices is equival- ent to the problem of finding a minimal weighted matching for the cor- responding weighted bipartite graph, which can be efficiently solved by existing fast algorithms [5]. Thus, for each fading state n, p(n) is obtained correspondingly. For the fixed l and m, after S N k=1 G k (l, m) has been obtained for each fading state n by solving the minimal weighted matching, G(l, m) can be obtained correspondingly by (8). Finally, it remains to find l ≥ 0 and m ≥ 0 that maximises G(l, m). This can be done efficiently by a subgra- dient-based method that iteratively updates l and m by the subgradient of G(l, m) until convergence. ELECTRONICS LETTERS 8th December 2011 Vol. 47 No. 25