Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

5.26 EXPECTATIONS OF NONLINEAR TRANSFORMATIONS

Generally, it is difficult to obtain a closed-form expression for the expectation of nonlinear functions of several random variables. However, if the random variables are jointly Gaussian with nonzero cross-correlation, then it is possible to compute useful expectations when the nonlinear functions are memoryless. This is convenient because memoryless nonlinearities often arise in engineering problems, and Gaussian noise is inevitably present. The following theorem provides an interesting result relating the derivative of a cross-correlation function to derivatives of the nonlinear functions. Thus, if differentiating a nonlinear function leads to a simpler expression, the expectation is derived more easily. Although the theorem has been proved in general for N jointly Gaussian random variables, we provide the result only for the case of n = 2, and without proof (Price, 1958; McMahon, 1964; Pawula, 1967).

Theorem 5.13 (Price and an extension). Let X1 and X2 be jointly Gaussian random variables with correlation coefficient ρ, and suppose they are transformed via memoryless nonlinear functions: g1(X1) and g2(X2). Define the following cross-correlation of the transformed random variables:

(5.356)

Then

(5.357)

For the more general function ...