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Broadcast Channels with Heterogeneous Arrival and Decoding Deadlines: Second-Order Achievability

Homa Nikbakht, Malcolm Egan, Jean-Marie Gorce, H. Vincent Poor

TL;DR

The work tackles two-user URLLC over Gaussian broadcast channels with overlapping transmission windows due to heterogeneous arrival times and decoding deadlines. It introduces a Marton-inspired coding scheme with power-sharing and auxiliary codebooks, and derives finite-blocklength second-order bounds for the achievable rates of both messages. The results connect to and recover known normal-approximation bounds in point-to-point and parallel-channel settings, and show that the proposed scheme yields significant gains over time-sharing for short packets in the broadcast setting. This provides a practically relevant framework for reliable low-latency communications to multiple receivers with staggered deadlines and arrivals, including coherent handling of overlapping transmissions and causality in encoding.

Abstract

A standard assumption in the design of ultra-reliable low-latency communication systems is that the duration between message arrivals is larger than the number of channel uses before the decoding deadline. Nevertheless, this assumption fails when messages arrive rapidly and reliability constraints require that the number of channel uses exceed the time between arrivals. In this paper, we consider a broadcast setting in which a transmitter wishes to send two different messages to two receivers over Gaussian channels. Messages have different arrival times and decoding deadlines such that their transmission windows overlap. For this setting, we propose a coding scheme that exploits Marton's coding strategy. We derive rigorous bounds on the achievable rate regions. Those bounds can be easily employed in point-to-point settings with one or multiple parallel channels. In the point-to-point setting with one or multiple parallel channels, the proposed achievability scheme is consistent with the normal approximation. In the broadcast setting, our scheme agrees with Marton's strategy for sufficiently large numbers of channel uses and shows significant performance improvements over standard approaches based on time sharing for transmission of short packets.

Broadcast Channels with Heterogeneous Arrival and Decoding Deadlines: Second-Order Achievability

TL;DR

The work tackles two-user URLLC over Gaussian broadcast channels with overlapping transmission windows due to heterogeneous arrival times and decoding deadlines. It introduces a Marton-inspired coding scheme with power-sharing and auxiliary codebooks, and derives finite-blocklength second-order bounds for the achievable rates of both messages. The results connect to and recover known normal-approximation bounds in point-to-point and parallel-channel settings, and show that the proposed scheme yields significant gains over time-sharing for short packets in the broadcast setting. This provides a practically relevant framework for reliable low-latency communications to multiple receivers with staggered deadlines and arrivals, including coherent handling of overlapping transmissions and causality in encoding.

Abstract

A standard assumption in the design of ultra-reliable low-latency communication systems is that the duration between message arrivals is larger than the number of channel uses before the decoding deadline. Nevertheless, this assumption fails when messages arrive rapidly and reliability constraints require that the number of channel uses exceed the time between arrivals. In this paper, we consider a broadcast setting in which a transmitter wishes to send two different messages to two receivers over Gaussian channels. Messages have different arrival times and decoding deadlines such that their transmission windows overlap. For this setting, we propose a coding scheme that exploits Marton's coding strategy. We derive rigorous bounds on the achievable rate regions. Those bounds can be easily employed in point-to-point settings with one or multiple parallel channels. In the point-to-point setting with one or multiple parallel channels, the proposed achievability scheme is consistent with the normal approximation. In the broadcast setting, our scheme agrees with Marton's strategy for sufficiently large numbers of channel uses and shows significant performance improvements over standard approaches based on time sharing for transmission of short packets.
Paper Structure (36 sections, 12 theorems, 34 equations, 7 figures)

This paper contains 36 sections, 12 theorems, 34 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Finite Blocklength Regime
  4. Heterogeneous Decoding Deadlines
  5. Heterogeneous Arrival Times
  6. Broadcast Channels in the Finite Blocklength Regime
  7. Main Contributions
  8. Organization
  9. Notation
  10. Problem setup
  11. Random Coding Scheme
  12. Encoding
  13. Transmitting only $m_1$
  14. Transmitting both $m_1$ and $m_2$
  15. Transmitting only $m_2$
  16. ...and 21 more sections

Key Result

Theorem 1

Given $n_{1,1}, n_{1,2}, n_{2,2}$ and $\mathsf{P}$, for each $i \in \{1,2\}$, let $j \in \{1,2\} \backslash i$, $\log L_i := n_{i,j} R_{L_i}$ and $\log M_i := R_i \sum_{j =1}^ 2n_{i,j}$. For each $i \in \{1,2\}$, we then have the following upper bound: subject to where $C(x): = \frac{1}{2} \log (1 + x)$, $V_{i,i}(x) := \frac{x(2+x)}{2(1+x)^2}$, $V_{i,j}(x) = \frac{x(2+x)}{2(1+x)^2} + \tilde{V}_i

Figures (7)

  • Figure 1: System model. (a) Arrival times and decoding deadlines of $m_1$ and $m_2$, (b) Problem setup with one transmitter and two receivers.
  • Figure 2: Converse and achievability bounds on $R_1$ versus $n_{1,1}$ in the point-to-point case with $\Omega_{1,1} = 0$ dB, and $\epsilon_1 = 10^{-3}$. The value of $n_{1,1}$ varies from $70$ to $2000$ with a step size of $10$.
  • Figure 3: Converse and achievability bounds on $R_1$ versus $n_{1,1} + n_{1,2}$ in the two parallel Gaussian channel case with $\Omega_{1,1} = 20$ dB, $\Omega_{1,1} = \nu \Omega_{1,2}$ and $\epsilon_1 = 10^{-6}$. The value of $n = n_{1,1} + n_{1,2}$ varies from $20$ to $2000$ with a step size of $10$.
  • Figure 4: Effect of the correlation parameter $\rho$ on the bounds in Theorem \ref{['Th1:bounds']} and Proposition \ref{['prop2']}.
  • Figure 5: Example of comparing our scheme with the time-sharing scheme for $n = 300$ and $n_{1,2}$ taking values from $200$ to $0$ with a step size of $50$.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Remark 2
  • Proposition 1
  • Theorem 2: P2P, Only $m_1$
  • Remark 3
  • Theorem 3: P2P, Parallel Channels, Only $m_1$
  • Remark 4
  • Theorem 4: Marton's bound
  • ...and 9 more