Computing the bivariate Gaussian probability integral

In signal processing applications, it is often required to compute the integral of the bivariate Gaussian probability density function (PDF) over the four quadrants. When the mean of the random variables are nonzero, computing the closed form solution to these integrals with the usual techniques of integration is infeasible. Many numerical solutions have been proposed; however, the accuracy of these solutions depends on various constraints. In this work, we derive the closed form solution to this problem using the characteristic function method. The solution is derived in terms of the well-known confluent hypergeometric function. When the mean of the random variables is zero, the solution is shown to reduce to a known result for the value of the integral over the first quadrant. The solution is implementable in software packages such as MAPLE.