Recursive projection-free identification with binary-valued observations

Abstract

This paper is concerned with parameter identification problem for finite impulse response (FIR) systems with binary-valued observations under low computational complexity. Most of the existing algorithms under binary-valued observations rely on projection operators, which leads to a high computational complexity of much higher than $O\left(n^2\right)$. In response, this paper introduces a recursive projection-free identification algorithm that incorporates a specialized cut-off coefficient to fully utilize prior information, thereby eliminating the need for projection operators. The algorithm is proved to be mean square and almost surely convergent. Furthermore, to better leverage prior information, an adaptive accelerated coefficient is introduced, resulting in a mean square convergence rate of $O\left(\frac{1}{k}\right)$, which matches the convergence rate with accurate observations. Inspired by the structure of the Cram$\acute{e}$r–Rao lower bound, the algorithm can be extended to an information-matrix projection-free algorithm by designing adaptive weight coefficients. This extension is proved to be asymptotically efficient for first-order FIR systems, with simulations indicating similar results for high-order FIR systems. Finally, numerical examples are provided to demonstrate the main results.