A Systematic Approach to Deriving the Covariance Matrix of Correlated Nakagami-$m$ Fading Channels

The covariance matrix plays an important role in second-order statistics, enabling numerous signal processing techniques. By exploiting the multivariate moment-generating function (MGF), this paper analytically derives an exact, closed-form joint probability density function (JPDF) of the envelopes of two correlated Nakagami-m channels and their phase difference when m ≥ 1.0. By exploiting a common-variance approximation, an approximate, closed-form JPDF is derived for 0.5 ≤ m <; 1.0 because of the in-phase/quadrature-phase (I/Q) imbalance property of Hoyt fading channels. According to the derived JPDFs, the covariance matrix can be calculated in a form consisting of only one definite integral. Simulations are conducted to confirm the analytical results in terms of the covariance matrix and the PDF of the phase difference.