Max-Min SNR Signal Energy based Spectrum Sensing Algorithms for Cognitive Radio Networks with Noise Variance Uncertainty

This paper proposes novel spectrum sensing algorithms for cognitive radio networks. By assuming known transmitter pulse shaping filter, synchronous and asynchronous receiver scenarios have been considered. For each of these scenarios, the proposed algorithm is explained as follows: First, by introducing a combiner vector, an over-sampled signal of total duration equal to the symbol period is combined linearly. Second, for this combined signal, the Signal-to-Noise ratio (SNR) maximization and minimization problems are formulated as Rayleigh quotient optimization problems. Third, by using the solutions of these problems, the ratio of the signal energy corresponding to the maximum and minimum SNRs are proposed as a test statistics. For this test statistics, analytical probability of false alarm ($P_f$) and detection ($P_d$) expressions are derived for additive white Gaussian noise (AWGN) channel. The proposed algorithms are robust against noise variance uncertainty. The generalization of the proposed algorithms for unknown transmitter pulse shaping filter has also been discussed. Simulation results demonstrate that the proposed algorithms achieve better $P_d$ than that of the Eigenvalue decomposition and energy detection algorithms in AWGN and Rayleigh fading channels with noise variance uncertainty. The proposed algorithms also guarantee the desired $P_f(P_d)$ in the presence of adjacent channel interference signals

A Convex Programming Approach to the Trace Quotient Problem

The trace quotient problem arises in many applications in pattern classification and computer vision, e.g., manifold learning, low-dimension embedding, etc. The task is to solve a optimization problem involving maximizing the ratio of two traces, i.e., maxiy Tr(f(W))/Tr(h(W)). This optimization problem itself is non-convex in general, hence it is hard to solve it directly. Conventionally, the trace quotient objective function is replaced by a much simpler quotient trace formula, i.e., maxw Tr (h(W)-1 f (W)), which accommodates a much simpler solution. However, the result is no longer optimal for the original problem setting, and some desirable properties of the original problem are lost. In this paper we proposed a new formulation for solving the trace quotient problem directly. We reformulate the original non-convex problem such that it can be solved by efficiently solving a sequence of semidefinite feasibility problems. The solution is therefore globally optimal. Besides global optimality, our algorithm naturally generates orthonormal projection matrix. Moreover it relaxes the restriction of linear discriminant analysis that the projection matrix's rank can only be at most c - 1, where c is the number of classes. Our approach is more flexible. Experiments show the advantages of the proposed algorithm

A convex programming approach to the trace quotient problem

© Springer The original publication can be found at www.springerlink.comThe trace quotient problem arises in many applications in pattern classification and computer vision, e.g., manifold learning, low-dimension embedding, etc. The task is to solve a optimization problem involving maximizing the ratio of two traces, i.e., maxiy Tr(f(W))/Tr(h(W)). This optimization problem itself is non-convex in general, hence it is hard to solve it directly. Conventionally, the trace quotient objective function is replaced by a much simpler quotient trace formula, i.e., maxw Tr (h(W)-1 f (W)), which accommodates a much simpler solution. However, the result is no longer optimal for the original problem setting, and some desirable properties of the original problem are lost. In this paper we proposed a new formulation for solving the trace quotient problem directly. We reformulate the original non-convex problem such that it can be solved by efficiently solving a sequence of semidefinite feasibility problems. The solution is therefore globally optimal. Besides global optimality, our algorithm naturally generates orthonormal projection matrix. Moreover it relaxes the restriction of linear discriminant analysis that the projection matrix's rank can only be at most c - 1, where c is the number of classes. Our approach is more flexible. Experiments show the advantages of the proposed algorithm. © Springer-Verlag Berlin Heidelberg 2007.Chunhua Shen, Hongdong Li and Michael J. Brook

A Convex Programming Approach to the Trace Quotient Problem

Abstract. The trace quotient problem arises in many applications in pattern classification and computer vision, e.g., manifold learning, low-dimension embedding, etc. The task is to solve a optimization problem involving maximizing the ratio of two traces, i.e., maxW Tr(f(W))/Tr(h(W)). This optimization problem itself is non-convex in general, hence it is hard to solve it directly. Conventionally, the trace quotient objective function is replaced by a much simpler quotient trace formula, i.e., maxW Tr h(W) −1 f(W) ¡, which accommodates a much simpler solution. However, the result is no longer optimal for the original problem setting, and some desirable properties of the original problem are lost. In this paper we proposed a new formulation for solving the trace quotient problem directly. We reformulate the original non-convex problem such that it can be solved by efficiently solving a sequence of semidefinite feasibility problems. The solution is therefore globally optimal. Besides global optimality, our algorithm naturally generates orthonormal projection matrix. Moreover it relaxes the restriction of linear discriminant analysis that the projection matrix’s rank can only be at most c − 1, wherecisthenumber of classes. Our approach is more flexible. Experiments show the advantages of the proposed algorithm.