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A KALMAN-LIKE FIR ESTIMATOR IGNORING NOISE AND INITIAL CONDITIONS

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A KALMAN-LIKE FIR ESTIMATOR IGNORING NOISE AND INITIAL CONDITIONS
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  AKALMAN-LIKEFIRESTIMATORIGNORINGNOISEANDINITIALCONDITIONS Yuriy S. Shmaliy Electronics Department, Guanajuato UniversityCtra. Salamanca-Valle, 3.5+1.8km, Palo-Blanco, 36855, Salamanca, Mexicophone: + 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 solutionis found in a batch form and represented in a computation-ally efficient iterative Kalman-like one. It is shown that theKalman-like FIR UE is able to overperform the Kalman filterif the noise covariances and initial conditions are not knownexactly, 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 initialconditions, and in the presence of nonwhite or multiplica-tive noise sources, the Kalman-like one is often designed tosave the recursive structure, while connecting the algorithmcomponents with the model in different ways. Because therecan 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 deterioratingsubstantially) 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 unscentedtransform and proposed the unscented KF (UKF). Both EKFand UKF have then been used extensively and the formerwas developed in [5] to the invariant EKF for nonlinearsystems possessing symmetries (or invariances). For high-dimensional systems, the ensemble KF was proposed byEvensen in [6] and, for systems with sparse matrices, the fastKF applied by Lange in [7]. Applications has also found therobust Kalman-type filter designed by Masreliez [8] [9] forlinear state-space relations with non-Gaussian noise referredto as heavy tailed noise or Gaussian one mixed with outliers.UsefulKalman-likealgorithmscanalsobefoundinworksbyNahi [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 andimproving the performance of KF, its structure still remainsrecursive 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 FIRestimators by Kwon et al . in [17], Han et al . in [18], andShmaliy in [19]. A distinctive feature of such algorithms isthat white Gaussian noise in the convolution-based estimateobtained 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 stronglyconsistent if  N  occurs to be large [21] or the mean square ini-tial state function dominates the noise covariance functionsin 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 theKalman-like FIR algorithms uniting advantages of KF andinherent properties of FIR structures such as the boundedinput/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 theFIR unbiased estimator (UE) ignoring noise and initial con-ditionswillserveefficiently insteadofoptimal filtersinmanyapplications. 2. SIGNAL MODEL Consider a class of discrete time-varying (TV) linear state-space models represented with the state and observationequations, 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 observationvectors, 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 impliedthat w n and v n are mutually uncorrelated and independentprocesses, E  { w i v T  j } = 0 , having arbitrary distributions andknown 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 obligatorilybe 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 2076-1465985  A n Gain Actual signal Time A n –   N  +2 A n –   N  +2 Gain Averaging horizon of  N  points A n      E    s      t      i    m    a      t     e  Recursive s trategy of the KF Iterative s trategy of the Kalman - like FIR UF n N+ - 1 n opt Figure 1: Strategies of the recursive KF and iterativeKalman-like FIR UF algorithms.estimates of  x n can be obtained as shown in Fig. 1. Here,KF starts at some initial point n −  N  + 1, where N   2, withknown initial conditions and recursively produces estimatesat each subsequent point up to n . The estimate is formed intwo steps. First, the system matrix A n −  N  + 2 makes a projec-tion from n −  N  + 1 to n −  N  + 2 and then the Kalman gainadjusts the result to be the estimate at n −  N  + 2. The pro-cedure repeats recursively, provided the covariances of whitenoise sequences.The Kalman-like FIR UF does not require the noise per-formance and initial conditions [24], but needs an optimalaveraging horizon N  opt [25] in order for the estimate to havethe minimum mean square error (MSE) and be consistent tothe Kalman one [19,22]. This filter starts with any unknownvalue at n −  N  + 1 and iteratively produces estimates at sub-sequent points in two steps similarly to KF, although the truevalue is taken only at n when N  = N  opt . The algorithm oper-ates in any noise environment that makes it highly attractivefor engineering applications. 3. TIME-VARYING BATCH UNBIASED FIRESTIMATOR In order to find the FIR UE, (1) and (2) can be extended on ahorizon of  N  points from m = n −  N  + 1 to n following [26]and similarly to [21] as, respectively, X n , m = A n , m x m + B n , m W n , m , (5) Y n , m = C n , m x m + G n , m W n , m + D n , m V n , m , (6)where X n , m ∈ ℜ KN  , Y n , m ∈ ℜ  MN  , W n , m ∈ ℜ PN  , and V n , m ∈ ℜ  MN  are specified by, respectively, X n , m =  x T n x T n − 1 ... x T m  T  , (7) Y n , m =  y T n y T n − 1 ... y T m  T  , (8) W n , m =  w T n w T n − 1 ... w T m  T  , (9) V n , m =  v T n v T n − 1 ... v T m  T  , (10)and A n , m ∈ ℜ KN  × K  , C n , m ∈ ℜ  MN  × K  , G n , m ∈ ℜ  MN  × PN  , and D n , m ∈ ℜ  MN  ×  MN  are given with, respectively, A n , m =  A   m + 1 T  n A   m + 1 T  n − 1 ... A T m + 1 I  T  , (11) C n , m =  C n A   m + 1 n C n − 1 A   m + 1 n − 1 ... C m + 1 A m + 1 C m  , (12) G n , m = ¯ C n , m B n , m , (13) D n , m = diag  D n D n − 1 ... D m       N   , (14)where we assigned A   n − gn − h = g ∏ i = h A n − i ,¯ C n , m = diag  C n C n − 1 ... C m       N   , and performed B n , m ∈ ℜ KN  × PN  as B n , m =  B n A n B n − 1 ... A   m + 2 n B m + 1 A   m + 1 n B m 0 B n − 1 ... A   m + 2 n − 1 B m + 2 A   m + 1 n − 1 B m + 1 ............... 0 0 ... B m + 1 A m + 1 B m 0 0 ... 0 B m  , (15)The model, (5) and (6), suggests that the state equation atthe initial point m is x m = x m + B n w m that for B n special-ized with (15) can uniquely be satisfied with w m zero-valued.The initial state x m must thus be known a priori or estimatedoptimally a posteriori as shown in [19].By the convolution, the estimate 1 ˜ x n +  p | n of  x n can nowbe obtained if we assign a K  ×  MN  gain matrix H n , m (  p ) andclaim that˜ x n +  p | n = H n , m (  p ) Y n , m (16a) = H n , m (  p )( C n , m x m + G n , m W n , m + D n , m V n , m ) . (16b)The estimate (16a) will be unbiased if and only if thefollowing unbiasedness condition is satisfied  E  { ˜ x n +  p | n } = E  { x n +  p } , (17)where E  means averaging of the succeeding relation.Averaging in (16b), by (17), means removing thezero mean noise matrices that gives us E  { ˜ x n +  p | n } = ¯ H n , m (  p ) C n , m x m , where¯ H n , m (  p ) is the FIR UE gain. In turn,  E  { x n } can be substituted with the first vector row of (5) byremoving noise as E  { x n } = A   m + 1 n x m . Since n can be arbi-trary, one can substitute it with n + p and write  E  { x n +  p } = A   m + 1 n +  p x m . (18)Equating E  { ˜ x n +  p | n } to (18) leads to the unbiasednessconstraint for TV models A   m + 1 n +  p = ¯ H n , m (  p ) C n , m . (19) 1 ˜ x n +  p | n is an estimate at n +  p via measurement from the past to n ; ˆ x n +  p | n mean optimal and ¯ x n +  p | n unbiased. 986  If we further multiply (19) from the right hand sides withthe identity matrix ( C T n , m C n , m ) − 1 C T n , m C n , m and then remove C n , m from both sides, we go to¯ H n , m (  p ) = A   m + 1 n +  p ( C T n , m C n , m ) − 1 C T n , m (20)representing the gain of the TV FIR UE. It can easily be ver-ified that (20) becomes that derived in [19] for time-invariant(TI) models.Provided (20), the TV batch FIR UE is specified by thefollowing theorem, which proof belongs to (5)-(20). Theorem 1 Given (1) and (2) with zero mean mutually un-correlated and independent  w n and  v n having arbitrary dis-tributions and known covariance functions. Then, filter-ing (  p = 0 )  , p-lag smoothing (p < 0 ), and p-step predic-tion (p > 0 ) are provided at n + p using data taken fromm = n −  N  + 1 to n by the batch FIR UE as ¯ x n +  p | n = ¯ H n , m (  p ) Y n , m (21a) = A   m + 1 n +  p ( C T n , m C n , m ) − 1 C T n , m Y n , m , (21b) where C n , m is given by (12) and  Y n , m is the data vector (8). 4. TIME-VARYING KALMAN-LIKE ESTIMATOR Although theorem 1 establishes an exact convolution-basedrule to estimate unbiasedly the TV state at n as shown in Fig.1, the computational problem arises when N   1 owing tolarge dimensions of all of the matrices and vectors. For fastcomputation, the batch FIR UE (21b) can be represented inan iterative Kalman-like form stated by the following theo-rem, which proof is similar to that given in [19]. Theorem 2 Given the batch FIR UE (theorem 1). Then itsiterative Kalman-like form is the following: ¯ x l +  p | l = A l +  p ¯ x l +  p − 1 | l − 1 + A l +  p Υ − 1 l (  p ) F l C T l × [ y l − C l Υ l (  p ) ¯ x l +  p − 1 | l − 1 ] , (22) in which Υ l (  p ) =  A   l −|  p | l , p  − 1 ( smoothing ) A l , p = 0 (  filtering ) I , p = 1 (  prediction )  p − 1 ∏ i = 1 A − 1 l − i , p > 1 (  prediction ) , F l = [ C T l C l +( A l F l − 1 A T l ) − 1 ] − 1 , (23)¯ x s +  p | s = A   m + 1 s +  p PC T s , m Y s , m , (24) F s = A   m + 1 s P A   m + 1 T  s , (25) P = ( C T s , m C s , m ) − 1 , (26) where s = m + K  − 1  , m = n −  N  + 1  , and an iterative variablel ranges from m + K to n, because C T l , m C l , m is singular withl < m + K. The true estimate corresponds to l = n. As can be seen, (22) is the Kalman estimate, in which A l +  p Υ − 1 l (  p ) F l C T l plays the role of the Kalman gain that,however, does not depend on noise and initial conditions.The algorithm has two batch forms, (24) and (25), which canbe computed fast for small K  .Table 1: Full-Horizon TV Kalman-Like FIR UE AlgorithmStageGiven: K  , p , n  K  Set: Υ n (  p ) by (23) P = ( C T K  − 1 , 0 C K  − 1 , 0 ) − 1 F K  − 1 = A   1 K  − 1 P A   1 T  K  − 1 ¯ x K  +  p − 1 | K  − 1 = A   1 K  +  p − 1 PC T K  − 1 , 0 Y K  − 1 , 0 Update: F n = [ C T n C n +( A n F n − 1 A T n ) − 1 ] − 1 ¯ x n +  p | n = A n +  p ¯ x n +  p − 1 | n − 1 + A n +  p Υ − 1 n (  p ) F n C T n × [ y n − C n Υ n (  p ) ¯ x n +  p − 1 | n − 1 ] 4.1 Full-HorizonTime-VaryingKalman-LikeEstimator In special cases when noise is nonstationary or both the sys-tem and measurement noise components need to be filteredout, all the data available should be processed. By letting  N  = n + 1 and l = n  K  in (22)–(26), the relevant full-horizon algorithm becomes as shown in Table 1 and, for TImodels, simplifies to that proposed in [19]. As can be seen,the algorithm (Table 1) requires only K  and p , thus has ex-tremely strong engineering features. 4.2 Error Bound Provided¯ H n , m (  p ) , the estimate error bound can be ascer-tained via the noise power gain (NPG) in the three-sigmasense as follows:  EB k  ( vg ) ( n ,  N  ,  p ) = 3 σ  k  K  1 / 2 k  ( vg ) ( n ,  N  ,  p ) (27)where σ  k  is the noise variance of the measurement of the k  thstateand K  k  ( vg ) ( n ,  N  ,  p ) isthe ( vg ) thcomponentofthesquareNPG matrix K k   K k  ( n ,  N  ,  p ) specialized as K k  = H   k  H   T k  .Here the thinned K  ×  N  gain H   k    ¯ H n , m (  p )  k  is composedby K  th columns of ¯ H n , m (  p ) starting with the k  th one. 5. EXAMPLES OF APPLICATIONS Below, we provide filtering with p = 0 and prediction with  p > 0 of the two-state polynomial model, (1) and (2), speci-fied with B n = I , D n = I , C n = [ 1 0 ] , and A n =  1 ( 1 + d  n ) τ  0 1  , (28)where d  n temporary takes different values. Such a situa-tion occurs in oscillators undergoing temporary frequency“jumps” or in moving vehiculars with velocity “jumps”. ForTI filtering, d  n represents uncertainty and, in the TV case, d  n is supposed to be known exactly. We mostly allow noise tobe white Gaussian, noticing that the relevant investigationsfor the uniformly distributed and highly intensive sawtoothnoise were provided in [20–22,24,29,30]. 987  REFERENCES [1] R. E. Kalman, “A new approach to linear filtering andprediction problems,” J. Basic Engineer. , vol. 82, pp.35–45, Mar. 1960.[2] H. Cox, “On the estimation of state variables and pa-rameters fornoisy dynamic systems,”  IEEE Trans. Au-tom. Contr. , vol. 9, pp. 5–12, Jan. 1964.[3] A. H. Jazwinski, Stochastic Processes and FilteringTheory , New York: Academic Press, 1970.[4] S. J. Julier and J. K. Uhlmann, “A new extension of theKalmanfiltertononlinearsystems”, Proc.ofSPIE  ,vol. 3068, pp. 182-193, 1997.[5] S. Bonnabel, Ph. Martin and E. Sala¨un, “InvariantExtended Kalman Filter: theory and application to avelocity-aided attitude estimation problem”, in Proc.48th IEEE Conf. Decision Contr. , pp. 1297–1304,2009.[6] G. Evensen, “Sequential data assimilation with non-linear quasi-geostrophic model using Monte Carlomethods to forecast error statistics,” Journal of Geo- physical Research , vol. 99, pp. 143–162, May 1994.[7] A. A. Lange, “Simultaneous statistical calibration of the GPS signal delay measurements with related mete-orological data”, Physics and Chemistry of the Earth,Part A: Solid Earth and Geodesy , vol. 26, pp. 471–473, 2001.[8] C. J. Masreliez, “Approximate non-Gaussian filter-ing with linear state and observation relations,” IEEE Trans. Autom. Contr. , vol. AC-20, pp. 107–110, Feb1975.[9] R. D. Martin and C.J. Masreliez, “Robust estimationvia stochastic approximation,” IEEE Trans. Inform.Theory , vol. IT-21, pp. 263–271, May 1975.[10] N. E. Nahi, “Optimal recursive estimation with uncer-tain observation,” IEEE Trans. Inf. Theory , vol. IT-15,pp. 457–462. Jul. 1969.[11] M. Basseville, A. Benveniste, K. C. Chou, S. A.Golden, R. Nikoukhah, and A. S. Willsky, “Model-ing and estimation of multiresolution stochastic pro-cesses,” IEEE Trans. Inf. Theory , vol. 38, pp. 766-784,Mar. 1992.[12] E. Baccarelli and R. Cusani, “Recursive Kalman-typeoptimal estimation and detection of hidden Markovchains,” Signal Process. , vol. 51, pp. 55–64. May1996.[13] B. Ait-El-Fquih and F. Desbouvries, “Kalman Filter-ing in Triplet Markov Chains,” IEEE Trans. SignalProcess. , vol. 54, pp. 2957–2963, Aug. 2006.[14] A. Carmi, P. Gurfil, and D. Kanevsky, “Methods forsparse signal recovery using Kalman filtering withembedded pseudo-measurement norms and quasi-norms”, IEEE Trans. on Signal Process. , vol. 58, pp.2405–2409, Apr. 2010.[15] S. Stefanatos and A. K. Katsaggelos, “Joint data de-tection and channel tracking for OFDM systems withphase noise,” IEEE Trans. Signal Process. , vol. 56, pp.4230–4243, Sep. 2008.[16] W. H. Kwon and S. Han, Receding horizon control:model predictive control for state models , London:Springer, 2005.[17] W. H. Kwon, P. S. Kim, and P. Park, “A receding hori-zon Kalman FIR filter for discrete time-invariant sys-tems,” IEEE Trans. Autom. Contr. vol. 44, pp. 1787–1791, Sep. 1999.[18] S. H. Han, W. H. Kwon, and P. S. Kim, “Quasi-deadbeat minimax filters for deterministic state-spacemodels,” IEEE Trans. Autom. Contr. , vol. 47, pp.1904–1908, Nov. 2002.[19] Y. S. Shmaliy, “Linear optimal FIR estimation of dis-crete time-invariant state-space models,” IEEE Trans.Signal Process. , vol. 58, pp. 3086–3096, Jun. 2010.[20] Y. S. Shmaliy, “An unbiased FIR filter for TIE modelof a local clock in applications to GPS-based time-keeping,” IEEE Trans. on Ultrason., Ferroel. and Freq. Contr  ., vol. 53, pp. 862–870, May 2006.[21] Y. S. Shmaliy, “Optimal gains of FIR estimators for aclass of discrete-time state-space models,” IEEE Sig-nal Process. Letters , vol. 15, pp. 517–520, 2008.[22] P. S. Kim, “An alternative FIR filter for state estima-tionindiscrete-timesystems,”  DigitalSignalProcess., vol. 20, pp. 935–943, May 2010.[23] W. H. Kwon, P. S. Kim, and S. H. Han, “A reced-ing horizon unbiased FIR filter for discrete-time statespace models,” Automatica , vol. 38, pp. 545–551,Mar. 2002.[24] Y.S.Shmaliy, “UnbiasedFIRfilteringofdiscrete-timepolynomial state-space models,” IEEE Trans. SignalProcess. , vol. 57, pp. 1241–1249, Apr. 2009.[25] Y. S. Shmaliy, J. Mu˜noz-Diaz, and L. Arceo-Miquel,“Optimal horizons for a one-parameter family of unbi-ased FIR filters,” Digital Signal Process., vol. 18, pp.739–750, Sep. 2008.[26] H. Stark and J. W. Woods, Probability, Random Pro-cesses, and Estimation Theory for Engineers , 2 nd ed.,Upper Saddle River, NJ: Prentice Hall, 1994.[27] Y. S. Shmaliy, O. Ibarra-Manzano, L. Arceo-Miquel,and J. Mu˜noz-Diaz, “A thinning algorithm for GPS-based unbiased FIR estimation of a clock TIE model,”  Measurement, vol. 41, pp. 538–550, Jun. 2008.[28] M. Blum, “On the mean square noise power of an op-timum linear discrete filter operating on polynomialplus white noise input,” IRE Trans. Information The-ory , vol. 3, pp. 225–231, Dec. 1957.[29] Y. S. Shmaliy and O. Ibarra-Manzano, “Optimal FIRfiltering of the clock time errors,” Metrologia , vol. 45,pp. 571–576, Sep. 2008.[30] Y. S. Shmaliy, “A simple optimally unbiased MA fil-ter for timekeeping,” IEEE Trans. Ultrason. Ferroel.Freq. Control , vol. 49, pp. 789–797, Jun. 2002.[31] Y. S. Shmaliy, “Linear unbiased prediction of clock errors,” IEEE Trans. Ultrason. Ferroel. Freq. Control ,vol. 56, pp. 2027–2029, Sep. 2009.[32] G. H. Golub and G. F. van Loan, Matrix Computa-tions, 3 rd Ed., Baltimore: The John Hopkins Univ.Press, 1996. 989
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