# Recursive Optimal Finite Impulse Response Filter and Its Application to Adaptive Estimation
## 1 Introduction  
The Kalman filter has been used as a standard tool to deal with state estimation of linear state-space models. However, since the Kalman filter has infinite impulse response (IIR) structure, which makes use of whole measurements from the initial time to the current, model uncertainties which come from the limited knowledge of the system model and the statistics of the noises and computation errors may accumulate in the estimated state. These could originate the divergence problem in the Kalman filter [1; 2; 3]. In order to prevent divergence problems, finite impulse response (FIR) filters have been used as an alternative to the Kalman filter [4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. Since FIR filters estimate the states by using finite measurements on the most recent time interval, these filters are known to be more robust against modeling uncertainties and numerical errors that cause of divergence problem in Kalman filter. Moreover, due to their FIR structure, FIR filters have good properties such as built-in bounded input/bounded output (BIBO) stability and fast tracking speed. However, despite the aforementioned advantages of FIR filters, their complicated derivation and hold form might lead to computational inefficiency and limitations in further developments. Since the Kalman filter is well-known and has recursive form, which can give effective computation methods to the FIR filter, the recursive form of FIR filters were introduced by modifying the Kalman filter [7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. In [7; 8], the receding horizon Kalman (RHK) filters, whose concept is introduced in Figure 1a, and its fast iteration method, are proposed for time-invariant systems. The filter equations of RHK filters are easy to understand and many useful Kalman filtering methods could be directly applied to FIR filtering problems for improving the performance of FIR filters because they are derived by combining the Kalman filter algorithm and receding horizon strategy. Since RHK filters have exactly the same structure as the Kalman filter on the finite estimation horizon, theRHK filtering problem can be thought as a recursive finite horizon Kalman filtering problem with special initial conditions. Thus, the initial state and its error covariance for RHK filters are very important factors for the performance of estimation. However, RHK filters were derived with heuristic assumptions on initial conditions such as the infinite initial error covariance. In the derivation of the RHK filter, inversion of the state transition matrix is used in the iterative calculation to estimate state and error covariance matrix. For an infinite covariance, the inverse matrix becomes singular and the estimation problem may not be feasible. Furthermore, the optimality of RHK filters is not clear in their derivations, and they cannot be applied to time-varying systems. On the other hand, Kalman-like unbiased FIR (KUFIR) filters, whose concept is introduced in Figure 1b, are also proposed for the recursive FIR filtering [9; 10; 11; 12; 13; 14; 15; 16]. KUFIR filtering is a recursive Kalman-like algorithm that ignores the noise statistics and initial conditions. These are ignored by determining the optimal horizon length which minimize the mean-square estimation error, then the recursive prediction and correction procedure are repeated without using noise statistics. Since the optimal horizon length is the only design parameter of the KUFIR filter, the determination of optimal horizon length is a major problem, hence, several algorithms have been developed to find the optimal horizon length. In [9], the optimal horizon length was derived for the \(l\)-degree polynomial model by minimizing the mean-square estimation error. In [11], it was measured using the correlation method, and in [12], it was determined by using a bank of KUFIR filters operating in parallel. For fast computation of the optimal horizon length, an adaptive KUFIR filter was also suggested for time-invariant systems in [15].