Asymptotically Efficient Recursive Identification of FIR Systems With Binary-Valued Observations

This paper considers the identification problem of finite impulse response (FIR) systems with binary-valued observations under the assumption of fixed threshold and bounded persistently excitations. A recursive projection algorithm is constructed to estimate the unknown parameter. For first-order FIR systems, the convergence properties of the algorithm are analyzed theoretically. With mild conditions on the weight coefficients in the parameter update, the algorithm is proved to be convergent in mean square and the convergence rate can be the reciprocal of the number of observations, which has the same order as the optimal estimation when the system output is exactly known. Furthermore, it is also shown that the Cramér–Rao (CR) lower bound is achieved asymptotically with proper weight coefficients, which indicates that the algorithm is optimal in the sense of asymptotic efficiency. Some numerical examples are simulated to demonstrate the effectiveness of the proposed algorithm in both first-order and high-order FIR systems.