A KALMAN-LIKE FIR ESTIMATOR IGNORING NOISE AND INITIAL CONDITIONS Yuriy S. Shmaliy Electronics Department, Guanajuato University Ctra. Salamanca-Valle, 3.5+1.8km, Palo-Blanco, 36855, Salamanca, Mexico phone: + 52 (464) 647-01-95, email: shmaliy@ugto.mx, web: www.ingenierias.ugto.mx ABSTRACT A p-shift finite impulse response (FIR) unbiased estimator (UE) is addressed for linear discrete time-varying filtering ( p = 0), p-step prediction ( p > 0), and p-lag smoothing ( p < 0) of signal models in state space with no require- ments for initial conditions and zero mean noise. A solution is found in a batch form and represented in a computation- ally efficient iterative Kalman-like one. It is shown that the Kalman-like FIR UE is able to overperform the Kalman filter if the noise covariances and initial conditions are not known exactly, noise is not white, and both the system and measure- ment noise components need to be filtered out. Otherwise, the errors are similar. 1. INTRODUCTION For such unsuited applications of the Kalman filter (KF) [1] as estimation of nonlinear models, under unknown initial conditions, and in the presence of nonwhite or multiplica- tive noise sources, the Kalman-like one is often designed to save the recursive structure, while connecting the algorithm components with the model in different ways. Because there can be found an infinity of Kalman-like solutions depend- ing on applications, we meet a number of propositions sug- gesting some new qualities while saving (or not deteriorating substantially) the advantages of KF: fast computation and ac- curacy. Cox in [2] and others have derived the extended KF (EKF) for nonlinear models by a linearization of the state- space equations. Referring to the fact that EKF can give par- ticularly poor performance when the model is highly nonlin- ear [3], Julier and Uhlmann employed in [4] the unscented transform and proposed the unscented KF (UKF). Both EKF and UKF have then been used extensively and the former was developed in [5] to the invariant EKF for nonlinear systems possessing symmetries (or invariances). For high- dimensional systems, the ensemble KF was proposed by Evensen in [6] and, for systems with sparse matrices, the fast KF applied by Lange in [7]. Applications has also found the robust Kalman-type filter designed by Masreliez [8] [9] for linear state-space relations with non-Gaussian noise referred to as heavy tailed noise or Gaussian one mixed with outliers. Useful Kalman-like algorithms can also be found in works by Nahi [10], Basseville et al. [11], Baccarelli and Cusani [12], Ait-El-Fquih and Desbouvries [13], Carmi et al. [14], Ste- fanatos and Katsaggelos [15], and the list can be extended. In spite of great efforts in extending the applications and improving the performance of KF, its structure still remains recursive thus with the infinite impulse response (IIR). In- vestigating in [3] both the IIR and finite impulse response (FIR) filters, Jazwinski resumed that the limited memory fil- ter (having FIR) appears to be more robust against the un- bounded perturbation in the system. Referring to [3], opti- mal FIR filtering has been developed by W. H. Kwon et al. in [16]. There were also proposed several Kalman-like FIR estimators by Kwon et al. in [17], Han et al. in [18], and Shmaliy in [19]. A distinctive feature of such algorithms is that white Gaussian noise in the convolution-based estimate obtained over N past measured points is reduced as a recipro- cal of N [20] disregarding the model [19]. Moreover, the un- biased and optimal FIR estimates typically become strongly consistent if N occurs to be large [21] or the mean square ini- tial state function dominates the noise covariance functions in the order of magnitudes [19]. It is also known that the op- timal horizon N opt makes the FIR estimate (optimal or unbi- ased) similar or even better than the Kalman one [16,19–23]. Owing to the exciting engineering features of the Kalman-like FIR algorithms uniting advantages of KF and inherent properties of FIR structures such as the bounded input/bounded output (BIBO) stability as well as better ro- bustness against temporary model uncertainties, non Gaus- sian noise, and round-off errors, it may be expected that the FIR unbiased estimator (UE) ignoring noise and initial con- ditions will serve efficiently instead of optimal filters in many applications. 2. SIGNAL MODEL Consider a class of discrete time-varying (TV) linear state- space models represented with the state and observation equations, respectively, x n = A n x n-1 + B n w n , (1) y n = C n x n + D n v n , (2) where x n ∈ ℜ K and y n ∈ ℜ M are the state and observation vectors, respectively. Here, A n ∈ ℜ K×K , B n ∈ ℜ K×P , C n ∈ ℜ M×K , and D n ∈ ℜ M×M . The vectors w n ∈ ℜ P and v n ∈ ℜ M are zero mean, E {w n } = 0 and E {v n } = 0. It is implied that w n and v n are mutually uncorrelated and independent processes, E {w i v T j } = 0, having arbitrary distributions and known covariances Q w (i, j) = E {w i w T j } , (3) Q v (i, j) = E {v i v T j } , (4) for all i and j, to mean that w n and v n should not obligatorily be Gaussian and delta-correlated. Following the strategies of the recursive KF [1] and it- erative Kalman-like FIR unbiased filter (UF) [19], the TV 19th European Signal Processing Conference (EUSIPCO 2011) Barcelona, Spain, August 29 - September 2, 2011 © EURASIP, 2011 - ISSN 2076-1465 985