The Exponential Correlation Matrix: Eigen-Analysis and Applications

Abstract

We consider a complex-valued $L \times L$ exponential correlation matrix. Such a matrix has unit diagonal elements; each lower off-diagonal element is the correlation coefficient raised to the power of the modulus of the difference of the row and column indices, while each upper off-diagonal element is the complex conjugate of the correlation coefficient raised to the power of the modulus of the difference of the row and column indices. This makes it a Hermitian Toeplitz matrix. Analytical expressions for the eigenvectors of the exponential correlation matrix are presented, and closed form approximations of the eigenvalues for the low and high correlation cases and for the cases of linear interpolation and large matrix size are derived. Closed form expressions for the eigenvalues of exponential correlation matrices of sizes ranging from 3 to 8 in terms of the correlation coefficient, by a novel method of transformation of the characteristic polynomial and subsequent factorization of the transformed characteristic polynomial, are also derived. Furthermore, applications of the results obtained to the performance evaluation of wireless communication systems employing diversity are presented.