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The application of Kalman estimation techniques to adaptive filtering problems is briefly reviewed. In particular, three common scenarios are examined in detail. The themes that link these areas are that they involve rigorous or almost rigorous application of the Kalman filter, and they can be modelled by a finite-impulse-response (FIR) filter structure. Two new nonrigorous applications of the Kalman filter to the adaptive infinite-impulse-response (IIR) equaliser problem are presented. The first, a direct approach, involves a simultaneous parameter and state estimation algorithm. In the second, an indirect approach, a Kalman-based IIR system identification algorithm is used to estimate the parameters which define the closed-form optimum IIR equaliser. Finally, the convergence of the two algorithms is compared by computer simulation.