Optimal filter design for signal estimation based on linear time-variant system theory

Optimal Wiener filtering is a popular method for the estimation of stationary processes which can be completely derived system-theoretically. Although there exist several optimal filtering concepts for non-stationary processes, there is still a lack of fundamental time-variant system theory that describes the problem statement for non-stationary process estimation. This paper provides an intuitively understandable theory for the fundamental optimal filtering concept. By interpreting cross- and autocorrelations as a time-variant impulse response of a linear system, the problem statement can be illustrated with a network of linear systems. This paper introduces a double periodic system model that approximates a time-variant transfer function in both time and frequency domain which leads to an analytic solution for the optimal time-variant filter. The presented results are generally applicable and degenerate for simplified process properties (e.g. stationarity) to the well known results. We also present how the problem statement can easily be extended due to the fundamental and uniform theoretical approach.