A proper interpretation of "white Gaussian noise" in the context of optimum detection theory

Through a purely formal manipulation, one can show that the optimum detection of a zero-mean Gaussian signal in "white Gaussian noise" is achieved by the following decision rule: the signal is present if (x,Hx) ?? c, the signal is absent otherwise, where x is the observable waveform, H is a solution of an integral equation H + SH = S, with S(t,s) being the signal covariance, and c is a preset threshold. Since the white Gaussian noise is a mathematical fiction, neither the quadratic form (x,Hx) nor the whole detection problem has any meaning. With the use of the Wiener process, however, we rigorously show that the above decision rule can be regarded as an approximate solution to a well-defined realistic optimum detection problem. By comparing with the exact solution, we give a qualitative argument that it is a good approximation.