An optimal adaptive filtering algorithm with a polynomial prediction model

A new approach to the optimal adaptive filtering is proposed in this paper. In this approach, a polynomial prediction model is used to describe the time-variant/invariant impulse response coefficients of an identified system. When the polynomial prediction model is viewed as the state equations of the identified impulse response coefficients and the relationships between the inputs and outputs of the system are regarded as the measurements of the states, our adaptive filtering can be achieved in the framework of the Kalman filter. It is understood that Kalman filter is optimal in the sense of the MAP (maximum a posteriori), ML (most likelihood) and MMSE (minimum mean square error) under the linear and Gaussian white noise conditions. As a result, our algorithm is also optimal in the statistical senses as Kalman filter does, provided that the impulse response coefficients can be modeled by a polynomial. Not only do the analytical results of the algorithm but also the simulation results show that our algorithm outperforms the traditional known algorithms.