Non-linear Bayesian filtering by convolution method using fast Fourier transform

Using Kalman techniques, it is possible to perform optimal estimation in linear Gaussian state-space models. When the Gaussian assumptions are inadequate, the Kalman-type filters fail to be optimal. Classical filtering methods, such as the particle filter or Zakai filter can still be optimal as they provide not only the mean and covariance matrix estimations but also the conditional probability density of the state, given the observations. In this article, we propose a new method to calculate the filtering distribution. Our method is grid-based, and uses the convolution method to calculate the prediction step. The novelty of our approach is that we apply a fast Fourier transform technique to obtain a competitive numerical algorithm. Our approach is compared to classical methods such as UKF, EKF and particle filters.