Robust spectrum sensing algorithm based on free probability theory

Abstract

In low signal-to-noise ratio (SNR) cases, the performance of spectrum sensing algorithms cannot meet the practical needs, which is a major problem faced by spectrum sensing technology in current cognitive radio field. Now, existing algorithms based on random matrix theory (RMT) have high sensing performance, but they require a large number of samples, which are very difficult to satisfy in practice. Free probability theory (FPT) is a main branch of RMT. It describes the asymptotic behavior of large random matrices and portrays a strong link between two matrices and their sum or product matrices. FPT can also be utilized to the digital communication system that can be modeled by random matrices and has been applied to spectrum sensing in simplified ideal channels, for example, additive white Gaussian noise channel. The most pivotal issue and difficulty of the FPT-based methods is to set up and solve the asymptotic freeness equation corresponding to a specific communication model. In this paper, FPT-based spectrum sensing schemes are proposed for some typical wireless communication systems, such as multiple-input multiple-output system, Rayleigh multipath fading system, and orthogonal frequency division multiplexing system. It is shown that the asymptotic freeness behavior of random matrices and the property of Wishart distribution can be used to assist spectrum sensing for these typical systems with low SNR and very limited samples. Simulation results demonstrate that compared with the existing RMT-based spectrum detection methods, for example, the maximum and minimum eigenvalue detectors, the proposed FPT-based schemes offer superior detection performance and are more robust to low SNR cases, especially for a small sample of observations. Robust spectrum sensing algorithm based on free probability theory L. Wang et al. Free probability theory (FPT) [27-31] is a valuable toolfor describing the asymptotic behavior of random matrices. It has grown into a main branch of RMT through the pioneering work of Voiculescu in the 1980s [32,33]. FPT portrays a strong link between two matrices and their sum matrix or product matrix, which is completely different from other branches of RMT. For the purpose of spectrum sensing, it can separate the eigenvalues of real signal matrix from that of the received sample covariance matrix, which includes both signal and noise matrices. This is almost impossible in traditional mathematics. The most pivotal issue of the FPT-based methods is to set up and solve the asymptotic freeness equation corresponding to a specific communication model. In some simplified communication environment, FPT has been managed to utilize in the field of spectrum sensing [34,35]. However, for some typical wireless communication scenarios, such as multiple-input multiple-output (MIMO) Rayleigh channels, MIMO multipath channels, and MIMO-orthogonal frequency division multiplexing (OFDM) channels, it needs to be further studied.

Precise Asymptotics for the Uniform Empirical Process and the Uniform Sample Quantile Process

“…Probability theory is a subject that studies the quantitative regularity of random phenomena everywhere. Probability is a method of thinking about the world [3].…”