Distribution of the Ratio of the Largest Eigenvalue to the Trace of Complex Wishart Matrices
Ayse Kortun, Mathini Sellathurai, Tharm Ratnarajah, Caijun Zhong · IEEE Transactions on Signal Processing · 2012
This correspondence investigates the statistical properties of the ratioT= λ1/Σi=1mλi, where are λ1≥ λ2≥ ··· ≥ λmthemeigenvalues of anm×mcomplex central Wishart matrixWwithndegrees of freedom. We derive new exact analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) ofTfor complex central Wishart matrices with arbitrary dimensions. We also formulate simplified statistics ofTfor the special case of dual uncorrelated and dual correlated complex central Wishart matrices (m= 2) . The investigated ratioTis 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.