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Rate-Reliability Tradeoff for Deterministic Identification over Gaussian Channels

Pau Colomer, Christian Deppe, Holger Boche, Andreas Winter

TL;DR

This work extends the rate–reliability analysis of deterministic identification to general linear Gaussian channels, the first treatment for continuous-output models. It develops a metric-based framework linking output statistics to input distances, and derives both converse and achievability bounds that reveal a fundamental trade-off: if identification errors decay exponentially, the DI rate becomes at most linear in the blocklength, whereas sub-exponential decay with $E_i(n)=\Omega(1/n)$ recovers the known linearithmic capacity bound $\dot{C}_{\mathrm{DI}}(\mathcal{G}) \le 1/2$. The authors provide explicit distance-decoding code constructions achieving these regimes, including a linear rate with exponential errors and a linearithmic rate with sub-exponential errors, and they show that both error exponents must vanish slowly to preserve the linearithmic scaling. The results illuminate the DI performance in Gaussian models, offer practical coding insights for future networks, and suggest that similar rate–reliability phenomena may extend to broader continuous-channel classes once an appropriate metric is identified.

Abstract

We extend the recent analysis of the rate-reliability tradeoff in deterministic identification (DI) to general linear Gaussian channels, marking the first such analysis for channels with continuous output. Because DI provides a framework that can substantially enhance communication efficiency, and since the linear Gaussian model underlies a broad range of physical communication systems, our results offer both theoretical insights and practical relevance for the performance evaluation of DI in future networks. Moreover, the structural parallels observed between the Gaussian and discrete-output cases suggest that similar rate-reliability behaviour may extend to wider classes of continuous channels.

Rate-Reliability Tradeoff for Deterministic Identification over Gaussian Channels

TL;DR

This work extends the rate–reliability analysis of deterministic identification to general linear Gaussian channels, the first treatment for continuous-output models. It develops a metric-based framework linking output statistics to input distances, and derives both converse and achievability bounds that reveal a fundamental trade-off: if identification errors decay exponentially, the DI rate becomes at most linear in the blocklength, whereas sub-exponential decay with recovers the known linearithmic capacity bound . The authors provide explicit distance-decoding code constructions achieving these regimes, including a linear rate with exponential errors and a linearithmic rate with sub-exponential errors, and they show that both error exponents must vanish slowly to preserve the linearithmic scaling. The results illuminate the DI performance in Gaussian models, offer practical coding insights for future networks, and suggest that similar rate–reliability phenomena may extend to broader continuous-channel classes once an appropriate metric is identified.

Abstract

We extend the recent analysis of the rate-reliability tradeoff in deterministic identification (DI) to general linear Gaussian channels, marking the first such analysis for channels with continuous output. Because DI provides a framework that can substantially enhance communication efficiency, and since the linear Gaussian model underlies a broad range of physical communication systems, our results offer both theoretical insights and practical relevance for the performance evaluation of DI in future networks. Moreover, the structural parallels observed between the Gaussian and discrete-output cases suggest that similar rate-reliability behaviour may extend to wider classes of continuous channels.
Paper Structure (8 sections, 4 theorems, 59 equations, 1 figure)

This paper contains 8 sections, 4 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Channel model
  3. Preliminaries
  4. Rate–Reliability Tradeoff Analysis
  5. Tradeoff in the symmetric error regime
  6. Tradeoff in asymmetric error regimes
  7. Achievability of linear rate and exponentially small errors
  8. Conclusions

Key Result

Theorem 2

For any DI code of block length $n$ over the general linear Gaussian channel described in section sec:channel, and with positive error exponents, $E_1(n), E_2(n) \geq E(n) \geq \frac{\ln 16}{n}$, the linear rate $R(n)=\frac{1}{n} \log N$ is upper-bounded as where $P$ is the transmission power constraint in Eq. eq:power_constraint, $\nu_{\max}$ the maximum eigenvalue of the matrix $M=A^\top\Sigma^

Figures (1)

  • Figure 1: A packing of $N$ small spheres $\mathcal{S}_i(n,r)$ centred inside the bigger $S_0(n,\sqrt{nP})$, and all contained by the biggest $S_+(n,\sqrt{nP}+r)$.

Theorems & Definitions (10)

  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Remark 5
  • Theorem 6
  • proof