[Paper Review] On Kalman-Like Finite Impulse Response Filters

Lubin Chang|arXiv (Cornell University)|Jan 27, 2015

Target Tracking and Data Fusion in Sensor Networks16 references3 citations

TL;DR

This paper establishes an explicit analytical relationship between two prominent finite impulse response (FIR) filters: the Kalman-like unbiased FIR (UFIR) and the receding horizon Kalman FIR (RHKF). It reveals that their only difference lies in noise statistics ignorance and initial condition handling, enabling performance improvements by transferring design insights between the two, with implications for robustness and accuracy in state estimation without requiring statistical assumptions.

ABSTRACT

This note reveals an explicit relationship between two representative finite impulse response (FIR) filters, i.e. the newly derived and popularized Kalman-Like unbiased FIR filter (UFIR) and the receding horizon Kalman FIR filter (RHKF). It is pointed out that the only difference of the two algorithms lies in the noise statistics ignorance and appropriate initial condition construction strategy in UFIR. The revelation can benefit the performance improvement of one by drawing lessons from the other. Some interesting conclusions have also been drawn and discussed from this revelation.

Motivation & Objective

• To clarify the theoretical relationship between two widely used finite impulse response (FIR) filters: the unbiased FIR (UFIR) and the receding horizon Kalman FIR (RHKF).
• To identify the precise distinction between UFIR and RHKF in terms of noise modeling and initial condition construction.
• To enable performance enhancement of one filter by leveraging design principles from the other, particularly in scenarios with uncertain or unknown noise statistics.
• To provide insights into the robustness and optimality trade-offs inherent in FIR filtering approaches.

Proposed method

• Analytical derivation of the state estimation equations for both UFIR and RHKF filters.
• Comparison of the two filters under identical system dynamics and measurement models.
• Identification of the role of noise statistics in the RHKF formulation versus their absence in UFIR.
• Examination of initial condition construction strategies in UFIR as a key differentiator from RHKF.
• Use of system-theoretic analysis to demonstrate equivalence in structure when noise assumptions are removed from UFIR.
• Derivation of conditions under which UFIR can be interpreted as a noise-ignorant variant of RHKF.

Experimental results

Research questions

• RQ1 What is the fundamental difference between the unbiased FIR (UFIR) and the receding horizon Kalman FIR (RHKF) filters?
• RQ2 How does the absence of noise statistics in UFIR affect its performance compared to RHKF?
• RQ3 Can insights from RHKF's noise-aware design improve the robustness of UFIR?
• RQ4 What role does initial condition construction play in distinguishing UFIR from RHKF?
• RQ5 Under what conditions can UFIR be considered a special case of RHKF?

Key findings

• The only structural difference between UFIR and RHKF lies in the treatment of process and measurement noise statistics and the strategy for constructing initial conditions.
• UFIR can be interpreted as a noise-ignorant version of RHKF, where statistical assumptions are omitted.
• The initial condition construction in UFIR is critical and directly influences estimation accuracy, especially in transient phases.
• RHKF's performance is more sensitive to accurate noise statistics, while UFIR maintains robustness under uncertainty.
• Theoretical equivalence between the two filters is established when noise statistics are ignored in RHKF, making UFIR a special case.
• The findings suggest that RHKF can be adapted to improve UFIR performance by incorporating noise statistics, and vice versa, by refining initial condition strategies.