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State-Dependent Fading Gaussian Channel with Common Reconstruction Constraints

Viswanathan Ramachandran

TL;DR

This paper analyzes joint message transmission and common reconstruction of a Gaussian state over a state-dependent fading channel, with noncausal state information available at the transmitter and perfect fading knowledge at both ends. Using a Gaussian Gel’fand–Pinsker coding scheme and a CR-appropriate state decomposition, it derives a parametric characterization of the achievable rate–distortion region, expressed via $(\rho_1,\rho_2,d) \in \kappa$ with $\rho_1^2+\rho_2^2 \le 1$ and $0\le d \le Q$, and the bound $R \le \mathbb{E}_G[R(\rho_1,\rho_2,d)]$, $D \ge d$. A matching converse is established through a covariance-based outer bound, confirming that the region is tight and given by the convex hull of the union over admissible parameters. Numerical examples with Rayleigh fading illustrate the rate–distortion and power–distortion tradeoffs, showing a fading penalty under the common reconstruction constraint. The results extend CR and GP coding frameworks to fading channels and pave the way for future multi-user and adaptive power-control extensions.

Abstract

The task of jointly communicating a message and reconstructing a common estimate of the channel state is examined for a fading Gaussian model with additive state interference. The state is an independent and identically distributed Gaussian sequence known noncausally at the transmitter, and the instantaneous fading coefficient is perfectly known at both the transmitter and the receiver. The receiver is required to decode the transmitted message and, in addition, reconstruct the state under a common reconstruction constraint ensuring that its estimate coincides with that at the transmitter. A complete characterization of the optimal rate distortion tradeoff region for this setting is the main result of our work. The analytical results are also validated through numerical examples illustrating the rate distortion and power distortion tradeoffs.

State-Dependent Fading Gaussian Channel with Common Reconstruction Constraints

TL;DR

This paper analyzes joint message transmission and common reconstruction of a Gaussian state over a state-dependent fading channel, with noncausal state information available at the transmitter and perfect fading knowledge at both ends. Using a Gaussian Gel’fand–Pinsker coding scheme and a CR-appropriate state decomposition, it derives a parametric characterization of the achievable rate–distortion region, expressed via with and , and the bound , . A matching converse is established through a covariance-based outer bound, confirming that the region is tight and given by the convex hull of the union over admissible parameters. Numerical examples with Rayleigh fading illustrate the rate–distortion and power–distortion tradeoffs, showing a fading penalty under the common reconstruction constraint. The results extend CR and GP coding frameworks to fading channels and pave the way for future multi-user and adaptive power-control extensions.

Abstract

The task of jointly communicating a message and reconstructing a common estimate of the channel state is examined for a fading Gaussian model with additive state interference. The state is an independent and identically distributed Gaussian sequence known noncausally at the transmitter, and the instantaneous fading coefficient is perfectly known at both the transmitter and the receiver. The receiver is required to decode the transmitted message and, in addition, reconstruct the state under a common reconstruction constraint ensuring that its estimate coincides with that at the transmitter. A complete characterization of the optimal rate distortion tradeoff region for this setting is the main result of our work. The analytical results are also validated through numerical examples illustrating the rate distortion and power distortion tradeoffs.
Paper Structure (5 sections, 2 theorems, 26 equations, 3 figures)

This paper contains 5 sections, 2 theorems, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Channel Model and Key Results
  3. Achievability proof of Theorem \ref{['thm:mainN']}
  4. Converse proof of Theorem \ref{['thm:mainN']}
  5. Conclusion

Key Result

Theorem 1

The capacity region $\mathcal{C}_{\text{CR}}^{\text{fad}}(\mathrm{P})$ is completely characterized by the convex hull in $\mathbb R_+^2$ of where the expectation is over the fading distribution, along with the power constraint $\mathbb{E}_{G}[P(G)] \leq \mathrm{P}$.

Figures (3)

  • Figure 1: State-dependent fading channel with Common Reconstructions
  • Figure 2: Illustration of the rate-distortion region in Theorem \ref{['thm:mainN']}.
  • Figure 3: Power-distortion trade-off for various rates.

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof