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On the Rate Region of I.I.D. Discrete Signaling and Treating Interference as Noise for the Gaussian Broadcast Channel

Yujie Shao, Min Qiu

Abstract

We revisit the Gaussian broadcast channel (GBC) and explore the rate region achieved by purely discrete inputs with treating interference as noise (TIN) decoding. Specifically, we introduce a simple scheme based on superposition coding with identically and independently distributed (i.i.d.) inputs drawn from discrete constellations, e.g., pulse amplitude modulations (PAM). Most importantly, we prove that the resulting achievable rate region under TIN decoding is within a constant gap to the capacity region of the GBC, where the gap is independent of all channel parameters. In addition, we show via simulation that the weak user can achieve a higher rate with PAM than with Gaussian signaling in some cases.

On the Rate Region of I.I.D. Discrete Signaling and Treating Interference as Noise for the Gaussian Broadcast Channel

Abstract

We revisit the Gaussian broadcast channel (GBC) and explore the rate region achieved by purely discrete inputs with treating interference as noise (TIN) decoding. Specifically, we introduce a simple scheme based on superposition coding with identically and independently distributed (i.i.d.) inputs drawn from discrete constellations, e.g., pulse amplitude modulations (PAM). Most importantly, we prove that the resulting achievable rate region under TIN decoding is within a constant gap to the capacity region of the GBC, where the gap is independent of all channel parameters. In addition, we show via simulation that the weak user can achieve a higher rate with PAM than with Gaussian signaling in some cases.

Paper Structure

This paper contains 10 sections, 3 theorems, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Proposed Scheme And The Main Result
  4. Proposed Signaling and Minimum Distance Analysis
  5. Main Result
  6. Proof of Theorem 1
  7. Achievable Rate Analysis
  8. Gap Analysis for Power Allocation Regime $\alpha \in (0,\alpha^\ast]$
  9. Gap Analysis for the Time Sharing Rate Region
  10. Conclusion

Key Result

Lemma 1

Consider the superimposed constellation in superposition with $\mathsf{X}_k$ uniformly distributed over a normalized zero mean $M_k\mathrm{-PAM}$ for $k\in\{1,2\}$. If $0 < \alpha \le \alpha^\ast \triangleq \tfrac{M_1^2-1}{M_1^2M_2^2-1}$, then $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: An example of the superimposed constellation in \ref{['superposition']} with $\mathsf{X}_1 \sim \mathrm{PAM}\!\left(6,d_{\min}(\mathsf{X}_1)\right)$ and $\mathsf{X}_2 \sim \mathrm{PAM}\!\left(2,d_{\min}(\mathsf{X}_2)\right)$. The intra-cluster and inter-cluster distances are indicated.
  • Figure 2: Achievable rate region of the proposed scheme for $(\mathrm{SNR}_1,\mathrm{SNR}_2)=(22,12)$ in dB. The rate pairs achieved with $\alpha^\ast$ are represented by blue boxes.
  • Figure 3: Comparison between user 2's achievable rate and $C_2(\alpha)$.
  • Figure 4: Illustration of achievable rate points.

Theorems & Definitions (8)

  • Lemma 1
  • Definition 1
  • Theorem 1
  • Remark 1
  • Example 1
  • Remark 2
  • Example 2
  • Lemma 2: Proposition 1 of DytsoTIN2016