Standard Condition Number Distributions of Finite Wishart Matrices for Cognitive Radio Networks

The standard condition number (SCN) in random matrix theory (RMT) has widely been employed in cognitive radio networks (CRNs). The paper studies the SCN distributions of central and complex Wishart matrices with finite dimensions. The exact and closed-form expressions of SCN distributions are first proposed in sum of multiple polynomials. The proposed formulations provide an efficient way to determine probability density function (PDF) and cumulative distribution function (CDF) of the SCN in finite RMT (FRMT). In particular, the specific SCN distributions of triple Wishart matrix are initially presented. The theoretical and precise sensing thresholds for cooperative spectrum sensing (CSS) systems in CRNs are calculated using the proposed SCN distributions. Numerical results indicate that the proposed theoretical SCN distributions match the empirical SCN distributions very well. Furthermore, the exact SCN-based CSS systems are able to obtain higher sensing performance than the asymptotic SCN-based schemes because precise sensing thresholds can be determined.