3092 IEEE TRANSACHONS ON SIGNAL PROCESSING, VOL. 42, NO. I I NOVEMBER 1994 Deinterleaving Pulse Trains Using Discrete-Time Stochastic Dynamic-Linear Models John B. Moore, Fellow, IEEE, and Vikram Krishnamurthy, Member, IEEE Abstract—Pulse trains from a number of different sources are often received on the one communication channel. It is then of interest to identify which pulses are from which source, based on different source characteristics. This sorting task is termed dein- terleaving. In this paper we next propose time-domain techniques for deinterleaving pulse trains from a finite number of periodic sources based on the time of arrival (TOA) and pulse energy, if available, of the pulses received on the one communication channel. We formulate the pulse train deinterleaving problem as a stochastic discrete-time dynamic linear model (DLM), the “discrete-time” variable k being associated with the kth received pulse. The time-varying parameters of the DLM depend on the se- quence of active sources. The deinterleaving detection/estimation task can then be done optimally via linear signal processing using the Kalman filter (or recursive least squares when the source periods are constant) and tree searching. The optimal solution, however, is computationally infeasible for other than small data lengths since the number of possible sequences grow exponentially with data length. Here we propose and study two of a number of possible suboptimal solutions: 1) Forward dynamic programming with fixed look-ahead rather than total look-ahead as required for the optimal scheme; 2) a probabilistic teacher Kalman filtering for the detection/estimation task. In simulation studies we show that when the number of sources is small, the proposed suboptimal schemes yield near-optimal estimates even in the presence of relatively large jitter noise. Also, issues of robustness and generalizations of the approach to the case of missing pulses, unknown source number, and non-Gaussian jitter noise are addressed. NOMENCLATURE N Ti, ~i tk !/k yk=~l...,yk Sk=i xk=x~....,xk r; Number of signal sources. Period and phase of ith source. Noise-free TOA of kth pulse. Observed TOA of kth pulse (2.2). If ith source generated the kth pulse. State of process; Xk = e~ if sk =’i (2.1). Last time source i was active up to and including arrival of kth pulse (2.3). Manuscript received April 6, 1993; revised February 12, 1994. This work was supported by the Cooperative Research Center for Robust and Adaptive Systems. The associate editor coordinating the review of this paper and aPProvint? it for publication was Prof. Douglas Williams. The authors are wi[h the CRASyS, Department of Systems Engineering, Research School of Physical Sciences and Engineering, Australian National University, Canberra ACT 2601. Australia. IEEE Log Number 9404769. ~(xk), G(Xk), ~(xk) DLM FDP KF MAP PT RLS TOA WGN State space representation of DLM (2.7). Dynamic linear model. Forward dynamic programming. Kalman filter. Maximum a po.steriori. Probabilistic teacher. Recursive least squares. Time of arrival. White Gaussian noise. L INTRODUCTION P ULSE trains from a number of different sources are often received on the one communication channel. It is then of interest to identify which pulses are from which source, based on the assumption that the different sources have different characteristics. This sorting task is termed deinterleaving. It has applications in radar detection and potential applications in computer communications and neural systems. In this paper we address, in the first instance, the problem of deinterleaving time-interleaved pulse trains from a finite krzowvt number of periodic sources, We assume that observations of the time of arrival of the pulses are obtained in additive white Gaussian noise without any information of the pulse amplitudes and phases. The aim is to deinterleave the received signal, i.e., to detect which source is responsible for each received pulse. From this it is trivial to estimate the periods and phases of the periodic pulse-train sources, although the detection and estimation tasks are intimately linked. A number of suboptimal heuristic solutions have been proposed for deinterleaving, e.g., histogramming [ 1], folding [2]. These techniques work well when the jitter noise is small. In addition they require prior information about the periods of the sources to select appropriate initial conditions. In this paper, we first formulate the pulse-train deinterleav- ing problem as a stochastic discrete-time dynamic linear model (DLM) (see pp. 212-215 in [3], [4]). A DLM is a time-varying linear system formulated in state space form with the state matrix and observation matrix at each time instant belonging to a finite set of possible values. In the deinterleavin,g case. the discrete-time instants are not the pulse times of arrival but rather integers indicating the pulses number. Thus the “time” instant k indicates the arrival of the kth pulse. Then the state and observation matrices at each “time” instant k, termed here pulse instant k, depend on which source is active to generate the Lth pulse. The state at each pulse instant consists of the 1053-587X/94$04.00@ 1994lEEE