An RKHS approach to detection and estimation problems-- III: Generalized innovations representations and a likelihood-ratio formula

The concept of a white Gaussian noise (WGN) innovations process has been used in a number of detection and estimation problems. However, there is fundamentally no special reason why WGN should be preferred over any other process, say, for example, an nth-order stationary autoregressive process. In this paper, we show that by working with the proper metric, any Gaussian process can be used as the innovations process. The proper metric is that of the associated reproducing kernel Hilbert space. This is not unexpected, but what is unexpected is that in this metric some basic concepts, like that of a causal operator and the distinction between a causal and a Volterra operator, have to be carefully reexamined and defined more precisely and more generally. It is shown that if the problem of deciding between two Gaussian processes is nonsingular, then there exists a causal (properly defined) and causally invertible transformation between them. Thus either process can be regarded as a generalized innovations process. As an application, it is shown that the likelihood ratio (LR) for two arbitrary Gaussian processes can, when it exists, be written in the same form as the LR for a known signal in colored Gaussian noise. This generalizes a similar result obtained earlier for white noise. The methods of Gohberg and Krein, as specialized to reproducing kernel spaces, are heavily used.