FIR filters and recursive forms for discrete-time state-space models

Abstract An FIR (finite impulse response) filter and an FIR smoother are introduced for discrete-time state-space models with system noise. The FIR structure not only guarantees both the BIBO stability and the robustness to some parameter changes, but also improves the filter divergence problem. It is shown in this paper that impulse responses of both the FIR filter and the FIR smoother are easily obtained from Riccati-type difference equations. Especially for time-invariant state-space models, they are also to be time-invariant and finally determined by simple equations on a finite interval. For the purpose of implementation, recursive forms of the FIR filter and smoother are derived using each other as adjoint variables. Improved characteristics of the FIR filter in the divergence problem are demonstrated via a simulation.

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