Inverse filtering and deconvolution

This paper studies a well known problem called inverse filtering or deconvolution. The basic problem is to estimate the intended inputs of a linear time invariant system from its measured outputs. In doing so, one would like to make sure that unintended or disturbance inputs do not adversely affect the estimates of the inputs. The unintended inputs arise due to various reasons, such as measurement noise, external disturbances, model uncertainties that act on the system, etc. We make an attempt to define notions of exact, almost, optimal, and suboptimal deconvolution problems in the presence of disturbance inputs. Our study has been guided by three important perspectives: 1) obtaining the solvability conditions both necessary and sufficient; 2) obtaining the optimal deconvolution or filtering performance whenever it applies; and 3) developing sound methodologies to design and construct appropriate filters. Both continuous- and discrete-time systems are considered. Also, all the filter construction problems for deconvolution are posed in a deterministic setting.