A simple form for the two-dimensional Q-function suitable for performance evaluation of communication systems

Using a geometrical approach inspired by the method of Craig for the one-dimensional Gaussian integral (one-dimensional Q-function, 1DQ) and similar to that of Simon and Divsalar, we propose a new form for the two-dimensional Gaussian integral (two-dimensional Q-function, 2DQ). Unlike that of Simon and Divsalar, our alternate form is applicable to any set of arguments and is not limited to nonnegative arguments. This rewrites the infinite-range double-integral expression in the naive form of 2DQ in terms of simple finite-range single integrals. The expressions given are simpler than those obtained by writing the 2DQ for negative arguments in terms of a sum of a number of 2DQ terms with nonnegative arguments. In addition, for a fixed required accuracy, the numerical complexity of the method is smaller than the rational function approximations of Donnelly.