The identification of parameter in a finite set under one-bit quantized observations

Abstract

This paper studies the identification of finite impulse response models whose parameters belong to a finite set under only one-bit quantized observations. Distinct from traditional parameter identification, the target parameter in some practical tasks belongs to a known finite set, rather than a continuous interval. Although the prior set of the parameter carries more information, the identification problem becomes more complicated due to it transforming into a discrete optimization problem, and existing identification methods based on continuous optimization cannot be used directly. To overcome this difficulty, an empirical measures based discrete projection algorithm is proposed. The algorithm first constructs a parameter estimation in a continuous set that includes the prior parameter set, and then maps the estimation to the known discrete set. The proposed algorithm is proved to be convergent in the almost sure sense, and to achieve an exponential convergence rate in the mean square sense under some mild conditions. Compared to the Cramér–Rao lower bound where the unknown parameter resides in a continuous interval, the conclusions reveal a significant improvement in parameter identification performance. Finally, simulations are presented to demonstrate the feasibility of the proposed algorithm.