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IRSA-based Random Access over the Gaussian Channel

Velio Tralli, Enrico Paolini

TL;DR

This work develops an IRSA-based framework for synchronous grant-free random access over the Gaussian MAC, framing IRSA as unsourced slotted codes formed by a frame-level MAC layer and slot-level PHY coding. It derives density-evolution equations to characterize asymptotic packet-loss behavior and average load thresholds, and introduces a convergence boundary that bounds achievable performance under slot-decoding errors. The paper evaluates two PHY-layer options—an ideal MPR-based scheme and a BPR/BCH-based scheme—and quantifies achievable energy efficiency via $E_b/N_0$ alongside spectrum efficiency, showing IRSA can approach random-coding bounds (within a few dB) for large activity, while suggesting that the BPR-based practical scheme offers limited gains at higher spectrum efficiencies. Numerical results illustrate the energy-spectral tradeoffs and confirm the framework’s predictive capability, highlighting substantial energy gains over slotted ALOHA under realistic conditions. The work thereby provides a rigorous, scalable method for evaluating and designing IRSA-based unsourced random access in noisy multi-access environments, with clear avenues for extending to CSA, fading channels, and finite SIC iterations.

Abstract

A framework for the analysis of synchronous grant-free massive multiple access schemes based on the irregular repetition slotted ALOHA (IRSA) protocol and operating over the Gaussian multiple access channel is presented. IRSA-based schemes are considered here as an instance of the class of unsourced slotted random access codes, operating over a frame partitioned in time slots, and are obtained by concatenation of a medium access control layer code over the entire frame and a physical layer code over each slot. In this framework, an asymptotic analysis is carried out in presence of both collisions and slot decoding errors due to channel noise, which allows the derivation of density-evolution equations, asymptotic limits for minimum packet loss probability and average load threshold, and a converse bound for threshold values. This analysis is exploited as a tool for the evaluation of performance limits in terms of minimum signal-to-noise ratio required to achieve a given packet loss probability, and also provides convergence boundary limits that hold for any IRSA scheme with given physical layer coding scheme. The tradeoff between energy efficiency and spectrum efficiency is numerically evaluated comparing some known coding options, including those achieving random coding bounds at slot level. It is shown that IRSA-based schemes have a convergence boundary limit within few dB from the random coding bound when the number of active transmitters is sufficiently large.

IRSA-based Random Access over the Gaussian Channel

TL;DR

This work develops an IRSA-based framework for synchronous grant-free random access over the Gaussian MAC, framing IRSA as unsourced slotted codes formed by a frame-level MAC layer and slot-level PHY coding. It derives density-evolution equations to characterize asymptotic packet-loss behavior and average load thresholds, and introduces a convergence boundary that bounds achievable performance under slot-decoding errors. The paper evaluates two PHY-layer options—an ideal MPR-based scheme and a BPR/BCH-based scheme—and quantifies achievable energy efficiency via alongside spectrum efficiency, showing IRSA can approach random-coding bounds (within a few dB) for large activity, while suggesting that the BPR-based practical scheme offers limited gains at higher spectrum efficiencies. Numerical results illustrate the energy-spectral tradeoffs and confirm the framework’s predictive capability, highlighting substantial energy gains over slotted ALOHA under realistic conditions. The work thereby provides a rigorous, scalable method for evaluating and designing IRSA-based unsourced random access in noisy multi-access environments, with clear avenues for extending to CSA, fading channels, and finite SIC iterations.

Abstract

A framework for the analysis of synchronous grant-free massive multiple access schemes based on the irregular repetition slotted ALOHA (IRSA) protocol and operating over the Gaussian multiple access channel is presented. IRSA-based schemes are considered here as an instance of the class of unsourced slotted random access codes, operating over a frame partitioned in time slots, and are obtained by concatenation of a medium access control layer code over the entire frame and a physical layer code over each slot. In this framework, an asymptotic analysis is carried out in presence of both collisions and slot decoding errors due to channel noise, which allows the derivation of density-evolution equations, asymptotic limits for minimum packet loss probability and average load threshold, and a converse bound for threshold values. This analysis is exploited as a tool for the evaluation of performance limits in terms of minimum signal-to-noise ratio required to achieve a given packet loss probability, and also provides convergence boundary limits that hold for any IRSA scheme with given physical layer coding scheme. The tradeoff between energy efficiency and spectrum efficiency is numerically evaluated comparing some known coding options, including those achieving random coding bounds at slot level. It is shown that IRSA-based schemes have a convergence boundary limit within few dB from the random coding bound when the number of active transmitters is sufficiently large.
Paper Structure (23 sections, 11 theorems, 76 equations, 17 figures)

This paper contains 23 sections, 11 theorems, 76 equations, 17 figures.

Table of Contents

  1. Introduction
  2. System model
  3. Slotted Random Access Codes
  4. Decoding of Slotted Random Access Codes
  5. MAC Layer Code Performance
  6. Frame-Based SA with and without MPR Capability
  7. IRSA-based Random Access Codes over a Noiseless Channel
  8. IRSA-based Random Access over GMAC: Asymptotic Threshold
  9. IRSA-based Random Access over GMAC: Convergence Boundary
  10. PHY Layer Encoding and Decoding Schemes
  11. Option 1: Optimum Coding for MPR with Binary Error Detection Coding
  12. Option 2: BPR and Binary Linear Concatenated Coding with Error Detection Coding
  13. Achievable $E_b/N_0$
  14. Achievable $E_b/N_0$ for IRSA on the Convergence Boundary
  15. Tradeoff between Energy Efficiency and Spectrum Efficiency
  16. ...and 8 more sections

Key Result

Lemma 1

Let $N_{\mathrm{s}} \rightarrow \infty$ and $\mathbb{E}[K_{\mathrm{a}}] \rightarrow \infty$ for constant $\mathsf{G}=\mathbb{E}[K_{\mathrm{a}}]/N_{\mathrm{s}}=\pi \alpha$. Let $\eta=1/\bar{d}$ be the efficiency of the IRSA protocol as defined in eq:csa_efficiency. Then, at the $\ell$-th iteration of where the starting point of the recursion is $p_0=1-\exp ( - \mathsf{G} \bar{d} ) \sum_{t = 0}^{T -

Figures (17)

  • Figure 1: An example of encoding function for IRSA. A nonzero message $m\in [M]$ is mapped onto a sequence $f_M(m)$ of length $N_{\mathrm{s}}$, where the number $D$ of copies of message $m$ and the positions of them in the sequence are all functions of the message $m$ only. Each message in $f_M(m)$ is encoded into a slot codeword which can be $\mathbf{x}=f_P(m)$ or $\mathbf{0}=f_P(0)$.
  • Figure 2: Graphical representation of IRSA access with $K=11$ users ($K_{\mathrm{a}}=7$ active) and $N_{\mathrm{s}}=13$ slots. Light-blue circles are active users and blank circles idle users. Assuming $T=2$, light-blue squares are resolvable slots, light-red squares unresolvable slots, and blank squares empty slots.
  • Figure 3: EXIT chart for IRSA protocol obtained with $d_{\max}=4$, $\mathbf{\Lambda}=\{0, 0.5102, 0, 0.4898\}$, $T=2$, $P_{\text{E}|1}=P_{\text{E}|2}=0.2$ and different values of $\mathsf{G}$.
  • Figure 4: Packet loss probability versus the average load $\mathsf{G}$ for an IRSA protocol with $d_{\max}=4$, $\mathbf{\Lambda}=\{0, 0.5102, 0, 0.4898\}$. Solid: asymptotic packet loss probability for $P_{\text{E}|t}=0.2, \;\forall t \leqslant T$. Dashed: asymptotic packet loss probability for $P_{\text{E}|t}=0, \;\forall t \leqslant T$. Markers: Finite frame size Monte Carlo simulation for $N_{\mathrm{s}}=400$ and $P_{\text{E}|t}=0.2, \;\forall t \leqslant T$.
  • Figure 5: Asymptotic packet loss probability $P_L$ versus the average load $\mathsf{G}$ for an IRSA protocol with $d_{\min}=1$, $d_{\max}=2$, $\mathbf{\Lambda}=\{0.1382, 0.8618\}$. Solid: noisy channel with $P_{\text{E}|t}=0.2, \;\forall t \leqslant T$. Dashed: noiseless channel with $P_{\text{E}|t}=0, \;\forall t \leqslant T$.
  • ...and 12 more figures

Theorems & Definitions (20)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Corollary 1
  • Example 1
  • Theorem 2
  • Remark 1
  • Definition 1
  • Example 2
  • Remark 2
  • ...and 10 more