Exploring the Exponentially Decaying Merit of an Out-of-Sequence Observation

It is well known that in a Kalman filtering framework, all sensor observations or measurements contribute toward improving the accuracy of state estimation, but, as observations become older, their impact toward improving estimations becomes smaller to the point that they offer no practical benefit. In this paper, we provide an practical technique for determining the merit of an old observation using system parameters. We demonstrate that the benefit provided by an old observation decreases exponentially with the number of observations captured and processed after it. To quantify the merit of an old observation, we use the filter gain for the delayed observation, found by re-processing all past measurements between the delayed observation and the current time estimate, a high cost task. We demonstrate the value of the proposed technique to system designers using both nearly-constant position (random walk) and nearly-constant velocity (discrete white-noise acceleration, DWNA) cases. In these cases, the merit (that is, gain) of an old observation can be computed in closed-form without iteration. The analysis technique incorporates the state transition function, the observation function, the state transition noise, and the observation noise to quantify the merit of an old observation. Numerical simulations demonstrate the accuracy of these predictions even when measurements arrive randomly according to a Poisson distribution. Simulations confirm that our approach correctly predicts which observations increase estimation accuracy based on their delay by comparing a single-step out-of-sequence Kalman filter with a selective version that drops out-of-sequence observations. This approach may be used in system design to evaluate feasibility of a multi-agent target tracking system, and when selecting system parameters including sensor rates and network latencies.