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

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.

💡 Research Summary

The paper tackles the fundamental problem of identifying the parameters of an autoregressive system with exogenous inputs (ARX) when only one‑bit quantized data can be transmitted from a remote sensor to a central estimator. Existing one‑bit identification schemes typically quantize only the current output and transmit its sign, which incurs a substantial information loss; consequently, the asymptotic error covariance of such estimators is at best π/2 times larger than the Cramér‑Rao lower bound (CRLB) that would be achievable with the original, unquantized measurements.

To overcome this limitation, the authors propose a joint quantizer‑estimator design that embeds both current and historical system outputs and inputs into the one‑bit message. On the sensor side, a conventional recursive least‑squares (RLS) algorithm runs in real time, producing a local parameter estimate that exploits the full regression vector (including past outputs). The quantizer then computes the difference between this local estimate and the remote estimator’s current estimate, and transmits only the sign of that difference (a single bit). Because the local RLS already incorporates the entire history of regressors, the transmitted bit implicitly carries far richer information than a simple sign of the current output.

The remote estimator updates its parameter vector using a stochastic‑approximation (SA) rule driven by the received bits. The key technical contribution is a new convergence‑analysis framework that does not rely on the traditional independence assumption for quantized data. Instead, the authors define an “integrated data” sequence that combines the pre‑quantized outputs and inputs, and they derive tail‑probability bounds for this sequence despite its strong temporal dependence. These bounds enable rigorous proof of (i) almost‑sure convergence of the remote estimate to the true parameter, (ii) Lp convergence for any positive integer p, and (iii) asymptotic normality with covariance equal to the original CRLB. In other words, the estimator is asymptotically efficient both in the distributional sense (√k(θ̂k−θ) ⇒ N(0, Σ̄CR)) and in the covariance sense (k E