On the Expected Value and Higher-Order Moments of the Euclidean Norm for Elliptical Normal Variates

We consider the Euclidean norm of a bivariate normal vector, equivalently, the amplitude or envelope of a complex Gaussian variable with correlated real and imaginary parts - known also as Hoyt or Nakagami-q distribution. Such a model, popular in fading and other communication contexts, is also relevant for the analysis of spatial errors in positioning-related applications, namely localization. In this paper an efficient expression in terms of complete elliptic integrals is given for the expected value, and good approximations in simple algebraic form are derived. Results are generalized to moments of any order.