Fourier Filters, Grid Filters, and the Fourier-Interpreted Grid Filter

While estimation problems on periodic manifolds are inherently nonlinear, filters can benefit from the boundedness of the domain. In this paper, we describe how densities on the unit circle are represented in four recently proposed filters that take advantage of the finite size of the domain. Specifically, we address representations based on trigonometric polynomials, piece-wise constant functions, and probability mass functions. Further, we propose a novel filter based on function values on a grid. To provide a continuous density function, we propose two interpolations. The first involves a trigonometric polynomial with the function values at the grid points. The second ensures a nonnegative approximation of the density by interpolating the square roots and then squaring the interpolation. Prediction and update steps are provided for the filter, both of which are independent of the interpolation employed. In an evaluation of the accuracy of the resulting probability density, the newly proposed filter outperforms the filter based on piece-wise constant functions and achieves results comparable to those of the previously proposed filters based on trigonometric polynomials.