# Lossy State Communication over Fading Multiple Access Channels

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- Analysis of the State of the art and Research gaps: We identify the following crucial aspects.

- We note that none of the works above consider joint state estimation along with message communication over state-dependent multi-terminal settings with noncausal transmitter state information, which is highly relevant in applications like the joint sensing and communication setting in Figure 1. This is addressed in this paper.
- While there exist system-level guidelines and waveform design specifications for such systems, a network information-theoretic analysis of the absolute performance capabilities of joint sensing and communication systems that take into account practical limitations has not been addressed in the literature, which we undertake here.
- Moreover, none of the works on joint communication and estimation mentioned above take fading links into account. Fading is an impairment that must be accounted for in practical wireless communication channel models, such as the joint sensing and communication application shown in Figure 1. This is another gap in the literature that this paper seeks to fill by investigating joint communication and estimation over state-dependent multi-user fading channels, the point-to-point counterpart of which was addressed by the author in [35].

- The key novelty of our work is that it is the first instance where joint communication and estimation have been considered in a multiple-user setting that also accounts for fading links, as opposed to previous works, which focused only on non-fading links.
- Moreover, it is the first work that considers non-causal state information (as opposed to causal or strictly causal) at the transmitter in a fading multi-user scenario which is practically relevant as described in the sensor network example from Figure 1.
- Furthermore, we undertake a comprehensive network information-theoretic study of the fundamental performance limits of such joint communication and estimation settings, which is lacking in the literature. Please refer to Table 1, which highlights our contributions in this paper with respect to the existing works.

- Summary of contributions: We list them below. See also the contribution summary Table 1, which emphasizes the novelty of our work with respect to the existing works.

- One of our main contributions in the paper is a complete characterization (Theorem 1) of the rate-distortion trade-off region for joint state estimation and communication over a two-user fading GMAC with the state. The key non-trivial part is the proof of converse, which is given in Section 5.
- We prove that the optimal strategy for the setting under consideration involves uncoded transmissions to amplify the state, along with the superposition of the digital message streams using appropriate Gaussian codebooks and DPC.
- We prove the optimality of uncoded state amplification in the special case where there are no messages to communicate—please refer to the section on implications of our result given after the statement of Theorem 1 for the details.
- Our study gives a network information-theoretic analysis of the fundamental performance limits of joint sensing and communication systems that take into account practical limitations such as fading. This acts as a design directive for realistic systems using joint sensing and transmission anticipated in upcoming wireless standards and points to the relative merits of uncoded communications and joint source-channel coding in such systems.

## 3. System Model and Results

**Definition 1.**

**Theorem 1.**

**Proof.**

**Remark 1.**

**Remark 2.**

**Remark 3.**

**Remark 4.**

## 4. Achievability

## 5. Converse

#### 5.1. Outer Bound

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Lemma 3.**

**Proof.**

**Case 1**- ($\lambda \le 1\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\eta \ge 1$): In this regime, Lemma 2 directly gives a bound on the weighted sum rate.
**Case 2**- ($\lambda \ge \eta \phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\eta \ge 1$): Since $\eta \ge 1$, we have$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \eta {R}_{1}+{R}_{2}+\frac{\lambda}{2}log\frac{{\sigma}_{S}^{2}}{{D}_{n}}\le \eta {R}_{1}+\eta {R}_{2}+\frac{\lambda}{2}log\frac{{\sigma}_{S}^{2}}{{D}_{n}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\eta ({R}_{1}+{R}_{2})+\frac{\lambda}{2}log\frac{{\sigma}_{S}^{2}}{{D}_{n}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\eta \left({R}_{1}+{R}_{2}+\frac{1}{2}log\frac{{\sigma}_{S}^{2}}{{D}_{n}}\right)+\frac{\lambda -\eta}{2}log\frac{{\sigma}_{S}^{2}}{{D}_{n}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \stackrel{\left(a\right)}{\le}\eta \left({R}_{sum}(0,0)+\frac{1}{2}log\frac{{\sigma}_{S}^{2}}{D(0,0)}\right)+\frac{\lambda -\eta}{2}log\frac{{\sigma}_{S}^{2}}{{D}_{n}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \stackrel{\left(b\right)}{\le}0+\frac{\eta}{2}log\frac{{\sigma}_{S}^{2}}{D(0,0)}+\frac{\lambda -\eta}{2}log\frac{{\sigma}_{S}^{2}}{D(0,0)},\hfill \end{array}$$
**Case 3**- ($1\le \lambda \le \eta \phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\eta \ge 1$): Since $\lambda \ge 1$, we have

#### 5.2. Equivalence of Inner and Outer Bounds

**Lemma 4.**

**Proof.**

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Lemma 1

#### Appendix A.2. Proof of Lemma 2

#### Appendix A.3. Proof of Lemma 4

**Lemma A1.**

**Proof.**

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**Figure 2.**State estimation over a fading Gaussian MAC with state, without fading knowledge at the transmitters.

**Table 1.**Summary of paper contributions. Note that single-user (noncausal) refers to a point-to-point state-dependent channel with noncausal transmitter state information, BC (causal) refers to a state-dependent broadcast channel with causal transmitter state information, while MAC (noncausal) refers to a state-dependent multiple access channel with noncausal state information at all the transmitters.

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Ramachandran, V. Lossy State Communication over Fading Multiple Access Channels. *Entropy* **2023**, *25*, 588.
https://doi.org/10.3390/e25040588

**AMA Style**

Ramachandran V. Lossy State Communication over Fading Multiple Access Channels. *Entropy*. 2023; 25(4):588.
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**Chicago/Turabian Style**

Ramachandran, Viswanathan. 2023. "Lossy State Communication over Fading Multiple Access Channels" *Entropy* 25, no. 4: 588.
https://doi.org/10.3390/e25040588