  State estimation in heavy-tailed process and measurement noise is an
important challenge that must be addressed in, e.g., tracking scenarios with
agile targets and outlier-corrupted measurements. The performance of the Kalman
filter (KF) can deteriorate in such applications because of the close relation
to the Gaussian distribution. Therefore, this paper describes the use of
Student's t distribution to develop robust, scalable, and simple filtering and
smoothing algorithms.
  After a discussion of Student's t distribution, exact filtering in linear
state-space models with t noise is analyzed. Intermediate approximation steps
are used to arrive at filtering and smoothing algorithms that closely resemble
the KF and the Rauch-Tung-Striebel (RTS) smoother except for a nonlinear
measurement-dependent matrix update. The required approximations are discussed
and an undesirable behavior of moment matching for t densities is revealed. A
favorable approximation based on minimization of the Kullback-Leibler
divergence is presented. Because of its relation to the KF, some properties and
algorithmic extensions are inherited by the t filter. Instructive simulation
examples demonstrate the performance and robustness of the novel algorithms.
