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This paper introduces the shifted Rayleigh mixture filter (SRMF), which is based on jump Markov linear systems. The formulation permits the presence of clutter. For bearings-only tracking problems involving maneuvering targets, the conditional density of the target state given the available measurements evolves as a growing mixture of probability density functions associated with a history of manoeuvre "modes." Similar to other "mixture" algorithms, the SRMF approximates this conditional density by a Gaussian mixture of fixed order. Unlike the extended or unscented Kalman filters, the shifted Rayleigh filter incorporates an exact calculation of the posterior density, when the prior is assumed to be Gaussian, given the latest bearings measurement. Computer simulations are provided to demonstrate the performance of the algorithm.