On the Connection between Gaussian Q-Functions and a Class of Hypergeometric Functions: Application to LCR of Lognormal Processes

In this letter, we derive a closed-form expression for the two-dimensional joint Gaussian Q-function in terms of a class of hypergeometric functions. Particularly, we show that the 2-D Gaussian Q-function can be expressed in terms of the bivariate confluent hypergeometric function Φ₁, which is well studied in classical books of integrals, special functions and Laplace transforms. As a direct application, we derive a new closed-form expression for the level crossing rate (LCR) and the average fade duration (AFD) of sampled lognormal processes with arbitrary correlation profile.