Averaging Q(‖X‖) for a Complex Gaussian Random Vector X: A Novel Approach

In this paper, we compute E[Q(‖X‖], where X is an n × 1 complex circularly Gaussian vector, ‖X‖ is the L norm of X and E[ ] is the expectation operator. This is done by finding the characteristic function of the decision variable and subsequently applying the inversion formula to obtain a one dimensional real integral expression. This integral is then converted to a contour integral which is evaluated using a variant of the Cauchy’s integral formula to obtain an expression for E[Q(‖X‖]. We then provide some applications of the above result by obtaining expressions for error probabilities in fading channels.