双变量高斯概率积分的计算

在信号处理的应用中 ,经常需要计算双变量高斯概率密度函数在四个象限上的积分值 .当随机变量的均值不为零时 ,用通常的积分方法计算这些积分的闭合形式解是行不通的 .人们曾经提出过许多种数值解法 ,然而 ,这些解法的准确度都受到各种条件的制约 .在本文中 ,用特征函数解法推导出了这个问题的闭合形式解 .这个解法是以著名的合流超几何函数的形式推出的 .当随机变量的均值为零时 ,这个解在第一象限上的积分值可简化为一个已知的结论 .这个解可用某些软件包 (如MAPLE)来解 .

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 random variables is zero, the solution is shown to reduce a known result for the value of the integral over the first quadrant. The solution is implementable in software packages such as MAPLE.