1460 IEEE SIGNAL PROCESSING LETTERS, VOL. 25, NO. 10, OCTOBER 2018 Bobrovsky–Zakai-Type Bound for Periodic Stochastic Filtering Eyal Nitzan , Student Member, IEEE, Tirza Routtenberg , Member, IEEE, and Joseph Tabrikian , Senior Member, IEEE Abstract—Mean-squared-error (MSE) lower bounds are com- monly used for performance analysis and system design. Recursive algorithms have been derived for computation of Bayesian bounds in stochastic filtering problems. In this letter, we consider stochas- tic filtering with a mixture of periodic and nonperiodic states. For periodic states, the modulo-T estimation error is of interest and the MSE lower bounds are inappropriate. Therefore, in this case, the mean-cyclic error and the MSE risks are used for estima- tion of the periodic and nonperiodic states, respectively. We derive a Bobrovsky-Zakai-type bound for mixed periodic and nonperi- odic stochastic filtering. Then, we derive a recursive computation method for this bound in order to allow its computation in dynamic settings. The proposed recursively-computed mixed Bobrovsky– Zakai bound is useful for the design and performance analysis of filters in stochastic filtering problems with both periodic and nonperiodic states. This bound is evaluated for a target tracking example and is shown to be a valid and informative bound for particle filtering performance. Index Terms—Bobrovsky–Zakai bound (BZB), mean-cyclic error, mean-squared error, periodic stochastic filtering, target tracking. I. INTRODUCTION N UMEROUS Bayesian mean-squared-error (MSE) lower bounds are available in the literature for offline estima- tion (see, e.g., [1]–[9]). In addition, several MSE bounds are proposed for stochastic filtering by deriving recursive imple- mentation methods of existing bounds with low computational complexity. In [10], the Bayesian Cram´ er–Rao bound (BCRB) for the stochastic filtering and its recursive computation are de- rived. Several extensions of this recursive BCRB are presented in [11]–[17] for various settings. Tighter recursively-computed MSE bounds for stochastic filtering from the Weiss–Weinstein family [2], are derived in [18]–[23]. In many stochastic filtering problems, some of the unknown states have a periodic nature. Examples for periodic states include direction-of-arrival (DOA) [24], [25], phase, and Manuscript received May 28, 2018; accepted July 13, 2018. Date of publi- cation July 31, 2018; date of current version August 20, 2018. This work was supported by the Israel Science Foundation under Grant 1160/15 and Grant 1173/16. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Ashish Pandharipande. (Corresponding author: Eyal Nitzan.) The authors are with the Department of Electrical and Computer Engineer- ing, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel (e-mail:, eyalni@ee.bgu.ac.il; tirzar@bgu.ac.il; joseph@bgu.ac.il). This letter has supplementary downloadable material available at http://ieeexplore.ieee.org, provided by the author. Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2018.2861464 frequency [26]–[29]. For estimation of T -periodic states, tra- ditional Bayes risks, such as the MSE, are inappropriate, since they ignore the periodicity and may give high penalty to small modulo-T errors. Consequently, existing Bayesian MSE bounds for filtering are inappropriate in this case. An appropriate risk for estimation of periodic states is the mean-cyclic error (MCE) [30]–[36]. Therefore, mixed MCE and MSE Bayesian lower bounds, which are computationally manageable, are useful for estimation of a mixed periodic and nonperiodic state vector. Recently, several lower bounds have been suggested for of- fline estimation involving periodic parameters. Non-Bayesian periodic lower bounds are proposed in [36] and [37]. For Bayesian estimation of the mixed periodic and nonperiodic pa- rameter vector, lower bounds are derived in [32] via modification of the Ziv–Zakai lower bound [5]–[7] and a new class of mixed lower bounds is derived in [31]. This class includes Bayesian Cram´ er–Rao-type and Bobrovsky–Zakai-type bounds denoted by the mixed BCRB and mixed Bobrovsky–Zakai bound (BZB), respectively. However, in practice, these bounds cannot be di- rectly computed in a dynamic setting due to high computational complexity. In [38] and [39], the mixed BCRB for stochastic filtering with both periodic and nonperiodic states is derived, as well as a recursive method for its computation. Similar to the conventional BCRB (see, e.g., [3], [40]), the mixed BCRB is informative only for high signal-to-noise ratios (SNRs). In this letter, we develop the mixed BZB for dynamic state space models with both periodic and nonperiodic states. The proposed bound is a large-error bound that is informative for any SNR. We prove that the mixed BZB is always tighter than or equal to the mixed BCRB and that the mixed BCRB for stochastic filtering can be derived from the proposed mixed BZB as a special case. Since direct computation of the mixed BZB is cumbersome in dynamic models, we derive a computa- tionally manageable recursive method for computing this bound by informed choice of the test-point vector, recursive computa- tion of the corresponding information matrix, and limited search across test points. The proposed recursively-implemented lower bound is valid for any stochastic filter. In the simulations, we consider the problem of target tracking and show that the pro- posed bound is valid and informative for performance analysis of a particle filter. II. MIXED STOCHASTIC FILTERING SETUP We consider the following nonlinear discrete-time state space model:  θ n = a n (θ n -1 , w n ) x n = h n (θ n , ν n ) ∀n ∈ N (1) 1070-9908 © 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information.