Title :
Distribution of the Ratio of the Largest Eigenvalue to the Trace of Complex Wishart Matrices
Author :
Kortun, Ayse ; Sellathurai, Mathini ; Ratnarajah, Tharm ; Caijun Zhong
Author_Institution :
ECIT, Queen´s Univ. Belfast, Belfast, UK
Abstract :
This correspondence investigates the statistical properties of the ratio T = λ1/Σi=1mλi , where are λ1 ≥ λ2 ≥ ··· ≥ λm the m eigenvalues of an m × m complex central Wishart matrix W with n degrees of freedom. We derive new exact analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) of T for complex central Wishart matrices with arbitrary dimensions. We also formulate simplified statistics of T for the special case of dual uncorrelated and dual correlated complex central Wishart matrices (m = 2) . The investigated ratio T is the most important ratio in blind spectrum sensing, since it represents a sufficient statistics for the generalized likelihood ratio test (GLRT). Thus, the derived analytical results are used to find the exact decision threshold for the desired probability of false alarm for Blind-GLRT (B-GLRT) detector. It is shown that the exact decision threshold based B-GLRT detector gives superior performance over the asymptotic decision threshold schemes proposed in the literature, which leads to efficient spectrum usage in cognitive radio.
Keywords :
cognitive radio; eigenvalues and eigenfunctions; matrix algebra; maximum likelihood estimation; probability; statistical analysis; CDF; PDF; blind spectrum sensing; blind-GLRT detector; cognitive radio; cumulative distribution function; decision threshold based B-GLRT detector; degree of freedom; dual uncorrelated complex central Wishart matrices; eigenvalue; false alarm probability; generalized likelihood ratio test; probability density function; statistical properties; Closed-form solutions; Covariance matrix; Educational institutions; Eigenvalues and eigenfunctions; Noise; Sensors; Uncertainty; Complex central Wishart matrices; eigenvalue distribution; eigenvalue-based detection; spectrum sensing;
Journal_Title :
Signal Processing, IEEE Transactions on
DOI :
10.1109/TSP.2012.2205922