Abstract
It is well-known that the optimality of the Kalman filter relies on the
Gaussian distribution of the process and observation model errors, which
in many situations is well justified [1, 2, 3]. However, its
optimality is useless in applications that the distribution assumptions
of the model errors do not hold in practice. Even minor deviation from
the assumed (or nominal) distribution may cause the Kalman filter’s
performance to drastically degrade or to completely breakdown. In
particular, when dealing with perceptually important signals such as
speech, images, medical, campaign and ocean engineering, the
measurements have confirmed the presence of non-Gaussian impulsive
(heavy tailed) or Laplace noises [4]. Therefore, the classical
Kalman filter which is derived under nominal Gaussian probability model
is biased or even breaks down in such situations. This article presents
a simple modification approach to overcome this limitation of the Kalman
filter. We show that in the smoothing Wiener filter, the estimated state
is chosen to minimize the ℓ2-norm of the variation of
the system model error. The ℓ2-norm, however,
corresponds to Gaussian priors. It confirms the optimality of Wiener
filter and Kalman filter in linear Gaussian systems and explains why
they suffer from the sensitivity to Laplase or impulsive error
statistics. To address this limitation, we propose a variation on
smoothing Wiener filter which substitutes a sum of absolute values
(i.e., ℓ1-norm) for the sum of squares used in
ℓ2 smoothing Wiener filter to penalize variations in the
system model error. The proposed ℓ1 smoothing Wiener
filter is suitable for analyzing systems with impulsive or Laplace model
errors. The idea of ℓ1 smoothing Wiener filter is then
used to recast and correct the system (or process) model equation. The
modified model puts a Laplace or a sparse distribution on the system
model error to enforce the sparsity on the states of the system. Based
on the fact that the formulations of the optimum filter by Wiener and
Kalman are equivalent in discrete time steady state [5], the Kalman
filter can be used to estimate the desired states using the modified
model.