Non-commutative large entries for cognitive radio applications

Abstract

Cognitive radio has been proposed as a solution for the problem of underutilization of the radio spectrum. Indeed, measurements have shown that large portions of frequency bands are not efficiently assigned since large pieces of bandwidth are unused or only partially used. In the last decade, studies in different areas, such as signal processing, random matrix theory, information theory, game theory, etc., have brought us to the current state of cognitive radio research. These theoretical advancements represents a solid base for practical applications and even further developments. However, still open questions need to be answered. In this study, free probability theory, through the free deconvolution technique, is used to attack the huge problem of retrieving useful information from the network with a finite number of observations. Free deconvolution, based on the moments method, has shown to be a helpful approach to this problem. After giving the general idea of free deconvolution for known models, we show how the moments method works in the case where scalar random variables are considered. Since, in general, we have a situation where more complex systems are involved, the parameters of interest are no longer scalar random variables but random vectors and random matrices. Random matrices are non-commutative operators with respect to the matrix product and they can be considered elements of what is called non-commutative probability space. Therefore, we focus on the case where random matrices are considered. Concepts from combinatorics, such as crossing and non-crossing partitions are useful tools to express the moments of Gaussian and Vandermonde matrices, respectively. Our analysis and simulation results show that free deconvolution framework can be used for studying relevant information in cognitive radio such as power detection, users detection, etc.