Recursive nonlinear filtering for angular data based on circular distributions

Abstract

Estimation of circular quantities is a widespread problem that occurs in many tracking and control applications. Commonly used approaches such as the Kalman filter, the extended Kalman filter (EKF), and the unscented Kalman filter (UKF) do not take periodicity explicitly into account, which can result in low estimation accuracy. We present a filtering algorithm for angular quantities in nonlinear systems that is based on circular statistics. The new filter switches between three different representations of probability distributions on the circle, the wrapped normal, the von Mises, and a Dirac mixture density. It can be seen as a systematic generalization of the UKF to circular statistics. We evaluate the proposed filter in simulations and show its superiority to conventional approaches.

Spatial correlation in 3D MIMO channels using fourier coefficients of power spectrums

“…g t (φ, θ, θ tilt ) ≈ g t,H (φ)g t,V (θ, θ tilt ), and the properties just described, we expand (15) and take the deterministic terms out of the expectation operator. After some trigonometric manipulations, we get (19). The same approach would yield a similar expression for ρ r (u − u ) with g t (φ, θ, θ tilt ) replaced by g r (ϕ, ϑ) and the AoDs replaced by AoAs.…”

“…The expansion in (19) exhibits several difficulties in deriving a closed-form expression for the SCF. The random variables, AoD and AoA, with respect to which the expectations need to be computed appear as the arguments of Legendre and associated Legendre polynomials.…”

“…Using the relations given in (20)-(22), the expectations in (19) can be expressed analytically as a linear combination of the FS coefficients of PAS and PES. This is made more explicit by defining the FS coefficients of PAS and PES given in (4) and (6) respectively as,…”

The partially wrapped normal distribution for SE(2) estimation

“…This is easy to verify because not even the special case of a WN distribution is closed under multiplication [23]. In order to perform a Bayes update in the context of a stochastic filter, approximations similar to [23] or sampling and reweighting schemes similar to [26] or [27] can be applied.…”

“…The formula for the first component is identical to formula used for WN parameter estimation in [23]. In order to obtain the entries of C that encode the linear-circular correlation, we use Lemma 6.…”