Bayesian Estimation for Linear Systems with Nonlinear Output and General Noise Distribution Using Fourier Basis Functions

Bayesian estimation can be used to estimate the state of dynamical systems, but its applicability is hampered due to the curse of dimensionality. This paper aims to mitigate this bottleneck for a relevant class of systems consisting of a linear plant with bounded input, driven by stochastic disturbances with non-linear noisy output; the distributions of the disturbances and noise have a bounded support but are otherwise general. Using a frequency-domain interpretation of the operations of the Bayes' filter, we show that, under mild assumptions, exact Bayesian estimation can be pursued in a countable space of Fourier series coefficients, rather than in the usual functional space of probability densities. This fact leads to a natural approximate method, where the Fourier series coefficients corresponding to high frequencies are discarded. For this approximate method, the complexity of the conditioned state distribution, measured by the number of Fourier coefficients, remains constant at prediction steps and grows only linearly at each update step. The applicability of the results is illustrated in the context of electron microscopy, where a residual error analysis indicates that the approximation is accurate.