Identifiability of the stochastic semi-blind deconvolution problem for a class of time-invariant linear systems

Semi-blind deconvolution is the process of estimating the unknown input of a linear system, starting from output data, when the kernel of the system contains unknown parameters. In this paper, identifiability issues related to such a problem are investigated. In particular, we consider time-invariant linear models whose impulse response is given by a sum of exponentials and assume that smoothness is the sole available a priori information on the unknown signal. We state the semi-blind deconvolution problem in a Bayesian setting where prior knowledge on the smoothness of the unknown function is mathematically formalized by describing the system input as a Brownian motion. This leads to a Tychonov-type estimator containing unknown smoothness and system parameters which we estimate by maximizing their marginal likelihood/posterior. The mathematical structure of this estimator is studied in the ideal situation of output data noiseless with their number tending to infinity. Simulated case studies are used to illustrate the practical implications of the theoretical findings in system modeling. Finally, we show how semi-blind deconvolution can be improved by proposing a new prior for signals that are initially highly nonstationary but then become, as time progresses, more regular.