Analysis, design, and performance limitations of H₂ optimal filtering in the presence of an additional input with known frequency

The standard $H_2$ optimal filtering problem considers the estimation of a certain output based on the measured output when the input is a zero mean white noise stochastic process of known intensity. In this paper, the inputs are considered to be of two types. The first type of input, as in standard $H_2$ optimal filtering, is a zero mean wide sense stationary white noise, while the second type is a linear combination of sinusoidal signals each of which has an unknown amplitude and phase but known frequency. Under such inputs, a generalized $H_2$ optimal filtering problem is formulated here. As in the standard $H_2$ optimal filtering problem, the generalized $H_2$ optimal filtering problem seeks to find a linear stable unbiased filter (called the generalized $H_2$ optimal filter) that estimates a desired output while utilizing the measured output such that the $H_2$ norm of the transfer matrix from the white noise input to the estimation error is minimized. The analysis, design, and performance limitations of generalized $H_2$ optimal filters are presented here.