An innovations approach to least-squares estimation--Part VI: Discrete-time innovations representations and recursive estimation

The linear stochastic discrete-time realization problem is to find a white-noise driven finite-dimensional linear system whose output generates a specified separable covariance. The solution to this problem is presented in the form of a causal and causally invertible innovations representation (IR) whose existence depends only on the positive definite nature of the separable covariance. It is also shown that least-squares filtered and smoothed estimates of one process given observations of a related colored process can be expressed as linear combinations of the state vector of the IR of the observed process. The analogous continuous-time problems have been studied earlier, and it has been shown that an important role is played by what is known as the relative order of the covariance. Here this is defined as the number of differencing operations required to produce a delta function component in the differenced covariance. It is shown that, unlike the continuous-time case, the relative order of the covariance does not necessarily induce similar (relative order) constraints on the impulse response of all models whose responses to white noise have the given covariance. This fact is at the heart of certain differences between continuous-time and discrete-time results. It is shown, however, that the innovations representations obey a number of constraints equal to the relative order of the covariance.