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Asymptotically Efficient Recursive Identification Under One-Bit Communications Achieving Original CRLB

Xingrui Liu, Jieming Ke, Mingjie Shao, Yanlong Zhao

Abstract

This paper develops an asymptotically efficient recursive identification algorithm for autoregressive systems with exogenous inputs under one-bit communications. In particular, the proposed method asymptotically achieves the Cramer-Rao lower bound (CRLB) based on the original data before quantization (original CRLB), whereas existing approaches typically attain only the CRLB corresponding to the quantized observations. The primary reason is that the existing methods quantize only the current system output, resulting in non-negligible information loss under one-bit quantization. To overcome this challenge, we present a novel quantization method that integrates both current and historical system outputs and inputs to provide richer parameter information in one-bit data, allowing the information loss caused by quantization to become a minor term relative to the original CRLB. Based on this technique, a corresponding remote estimation algorithm is further proposed. To address the convergence analysis challenge posed by the non-independence of the one-bit data, we establish a new framework that analyzes the tail probability of integrated data formed by combining current and historical system outputs and inputs before quantization, thereby eliminating the need for the traditional independence assumption on the quantized data. It is proven that the remote estimate achieves asymptotic normality, and the error covariance matrix converges to the original CRLB, confirming its asymptotic efficiency. Compared to existing identification algorithms under one-bit data, this method reduces the asymptotic mean squared error by at least 36%. Several numerical examples are simulated to show the effectiveness of the proposed algorithm.

Asymptotically Efficient Recursive Identification Under One-Bit Communications Achieving Original CRLB

Abstract

This paper develops an asymptotically efficient recursive identification algorithm for autoregressive systems with exogenous inputs under one-bit communications. In particular, the proposed method asymptotically achieves the Cramer-Rao lower bound (CRLB) based on the original data before quantization (original CRLB), whereas existing approaches typically attain only the CRLB corresponding to the quantized observations. The primary reason is that the existing methods quantize only the current system output, resulting in non-negligible information loss under one-bit quantization. To overcome this challenge, we present a novel quantization method that integrates both current and historical system outputs and inputs to provide richer parameter information in one-bit data, allowing the information loss caused by quantization to become a minor term relative to the original CRLB. Based on this technique, a corresponding remote estimation algorithm is further proposed. To address the convergence analysis challenge posed by the non-independence of the one-bit data, we establish a new framework that analyzes the tail probability of integrated data formed by combining current and historical system outputs and inputs before quantization, thereby eliminating the need for the traditional independence assumption on the quantized data. It is proven that the remote estimate achieves asymptotic normality, and the error covariance matrix converges to the original CRLB, confirming its asymptotic efficiency. Compared to existing identification algorithms under one-bit data, this method reduces the asymptotic mean squared error by at least 36%. Several numerical examples are simulated to show the effectiveness of the proposed algorithm.
Paper Structure (21 sections, 7 theorems, 75 equations, 8 figures, 1 algorithm)

This paper contains 21 sections, 7 theorems, 75 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introducation
  2. Background and motivations
  3. Related literature
  4. Main contributions
  5. Overview and notations
  6. Preliminaries
  7. Problem formulation
  8. Assumptions
  9. Algorithm design
  10. Main results
  11. The difference between the remote estimate and the RLS estimate
  12. Almost sure and $L^{p}$ convergence of the RLS-SA algorithm
  13. The asymptotic efficiency of the RLS-SA algorithm
  14. Numerical examples
  15. Convergence properties of the RLS-SA algorithm
  16. ...and 6 more sections

Key Result

Lemma A.1

(Theorem 4.5.2 in Chung2001) If the random process $\{X_{k}\}$ converges to $X$ in distribution, and for some $p > 0$, $\sup_{k}\mathbb{E}[\Vert X_{k} \Vert^{p}] < \infty$, then, for each $r < p$,

Figures (8)

  • Figure 1: Block diagram of the identification problem for ARX systems under one-bit communications
  • Figure 2: Trajectories of estimates given by the RLS-SA algorithm
  • Figure 3: Almost sure convergence of the RLS-SA algorithm
  • Figure 4: Asymptotic efficiency of the RLS-SA algorithm
  • Figure 5: Trajectories of $k$ times MSE by the RLS-SA algorithm and the sign-error type algorithm csaji2012recursive
  • ...and 3 more figures

Theorems & Definitions (7)

  • Lemma A.1
  • Lemma A.2
  • Lemma A.3
  • Lemma A.4
  • Lemma B.1
  • Lemma B.2
  • Lemma B.3