Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unsourced Random Access in MIMO Quasi-Static Rayleigh Fading Channels with Finite Blocklength

Junyuan Gao, Yongpeng Wu, Giuseppe Caire, Wei Yang, Wenjun Zhang

TL;DR

The fundamental limits of unsourced random access with a random and unknown number of active users in MIMO quasi-static Rayleigh fading channels are explored and using codewords distributed on a sphere is shown to outperform the Gaussian random coding scheme in the non-asymptotic regime.

Abstract

This paper explores the fundamental limits of unsourced random access (URA) with a random and unknown number ${\rm{K}}_a$ of active users in MIMO quasi-static Rayleigh fading channels. First, we derive an upper bound on the probability of incorrectly estimating the number of active users. We prove that it exponentially decays with the number of receive antennas and eventually vanishes, whereas reaches a plateau as the power and blocklength increase. Then, we derive non-asymptotic achievability and converse bounds on the minimum energy-per-bit required by each active user to reliably transmit $J$ bits with blocklength $n$. Numerical results verify the tightness of our bounds, suggesting that they provide benchmarks to evaluate existing schemes. The extra required energy-per-bit due to the uncertainty of the number of active users decreases as $\mathbb{E}[{\rm{K}}_a]$ increases. Compared to random access with individual codebooks, the URA paradigm achieves higher spectral and energy efficiency. Moreover, using codewords distributed on a sphere is shown to outperform the Gaussian random coding scheme in the non-asymptotic regime.

Unsourced Random Access in MIMO Quasi-Static Rayleigh Fading Channels with Finite Blocklength

TL;DR

The fundamental limits of unsourced random access with a random and unknown number of active users in MIMO quasi-static Rayleigh fading channels are explored and using codewords distributed on a sphere is shown to outperform the Gaussian random coding scheme in the non-asymptotic regime.

Abstract

This paper explores the fundamental limits of unsourced random access (URA) with a random and unknown number of active users in MIMO quasi-static Rayleigh fading channels. First, we derive an upper bound on the probability of incorrectly estimating the number of active users. We prove that it exponentially decays with the number of receive antennas and eventually vanishes, whereas reaches a plateau as the power and blocklength increase. Then, we derive non-asymptotic achievability and converse bounds on the minimum energy-per-bit required by each active user to reliably transmit bits with blocklength . Numerical results verify the tightness of our bounds, suggesting that they provide benchmarks to evaluate existing schemes. The extra required energy-per-bit due to the uncertainty of the number of active users decreases as increases. Compared to random access with individual codebooks, the URA paradigm achieves higher spectral and energy efficiency. Moreover, using codewords distributed on a sphere is shown to outperform the Gaussian random coding scheme in the non-asymptotic regime.
Paper Structure (13 sections, 10 theorems, 79 equations, 3 figures)

This paper contains 13 sections, 10 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Estimation of the Number of Active Users
  4. Data Detection
  5. Achievability Bound
  6. Converse Bound
  7. Numerical Results
  8. Conclusion
  9. Proof of Theorem \ref{['Theorem_Ka']}
  10. Proof of Corollary \ref{['Theorem_Ka_asymn']}
  11. Proof of Theorem \ref{['Theorem_achievability']}
  12. Proof of Theorem \ref{['Theorem_converse_single_noKa']}
  13. Proof of Theorem \ref{['Theorem_converse_noCSI_Gaussian_cKa']}

Key Result

Theorem 1

Assume $K_a$ is fixed and unknown in advance. The error probability of estimating $K_a$ as $K'_a\neq K_a$ satisfies Here, $p_{0,K_a}= {K_a\Gamma \left(n, \frac{nP}{P'}\right)} /{\Gamma\left(n\right)}$ when the columns of $\mathbf{C} \in \mathbb{C}^{n\times M}$ are i.i.d. $\mathcal{CN} \left(0,P'\mathbf{I}_{n}\right)$ distributed, and $p_{0,K_a}=0$ when columns of $\mathbf{C}$ are drawn uniformly

Figures (3)

  • Figure 1: The achievability bound on $\mathbb{P}\left[ K_a \to K'_a \right]$ with $K_a=300$ and $K=600$: (a) $\mathbb{P}\left[ K_a \to K'_a \right]$ versus $P$ for different values of $K'_a$ with $n=1000$ and $L=64$; (b) $\mathbb{P}\left[ K_a \to K'_a \right]$ versus $n$ with $L=64$ and $P=-20$ dB.
  • Figure 2: The mean of ${\rm K}_a$ versus energy-per-bit with ${\rm K}_a \sim {\rm Binom}(K,0.5)$, $n=1000$, $J = 100$ bits, $L=128$, and $\epsilon_{\rm MD} = \epsilon_{\rm FA} = 0.001$.
  • Figure 3: Comparison of existing schemes and theoretical bounds in the case of $n=3200$, $J = 100$ bits, $L\in\{50,200\}$, and $\epsilon_{\rm MD} = \epsilon_{\rm FA} = 0.025$. (Let $K=K_a/0.4$ for the scenario with individual codebooks.)

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Lemma 1: quadratic_form1
  • Lemma 2: chisumprod
  • ...and 1 more