A projection method for signal detection in colored Gaussian noise

The problem of the detection of known signals in colored Gaussian noise is usually studied through infinite-series representations for the signals and noise. In particular, the Karhunen-Loève (K-L) expansion is often used for this purpose. Such infinite-series methods, while elegant, often introduce mathematical complications because they raise questions of convergence, interchange of orders of integration, etc. The resolution of these problems is difficult and has led, when the K-L expansion is used, to the introduction of subsidiary conditions whose physical meaning is often unclear. We present a method of reducing the detection problem to a finite-dimensional form where many of the difficulties with the infinite-series K-L expansion do not arise. The resulting simplicity provides more direct derivations of and more physical insights into several earlier results. It has also suggested some new results. The method is essentially based on the use of a projection in a special kind of Hilbert space called a reproducing kernel Hilbert space.