The innovations approach to detection and estimation theory

Given a stochastic process, its innovations process will be defined as a white Gaussian noise process obtained from the original process by a causal and causally invertible transformation. The significance of such a representation, when it exists, is that statistical inference problems based on observation of the original process can be replaced by simpler problems based on white noise observations. Seven applications to linear and nonlinear least-squares estimation. Gaussian and non-Gaussian detection problems, solution of Fredholm integral equations, and the calculation of mutual information, will be described. The major new results are summarized in seven theorems. Some powerful mathematical tools will be introduced, but emphasis will be placed on the considerable physical significance of the results.