Application of the Volterra Functional Expansion in the Detection of Nonlinear Functions of Gaussian Processes

Detection of a memoryless nonlinear functional of a Gaussian process in additive Gaussian white noise is considered. The Volterra functional expansion for the likelihood ratio, and two examples of calculating the kernels are presented. It is shown that kernels up to third order can be obtained for a hard-limited Gaussian process and for the absolute value of a Gaussian process. For the case of hard limiting, the kernels are nonlinear functions of the autocorrelation of the Gaussian process. For the absolute value case, the kernels are nonlinear functions of the kernel derived for the linear problem. A Monte Carlo simulation of receiver performance is presented for the case of detection of the absolute value of a first-order Butterworth process in additive Gaussian white noise. The suboptimal detector is obtained by truncating the log likelihood ratio to second order.