Unsourced Random Access in MIMO Quasi-Static Rayleigh Fading Channels: Finite Blocklength and Scaling Law Analyses

TL;DR

This work analyzes unsourced random access over MIMO quasi-static Rayleigh fading channels with a random, unknown number of active users ${\rm K}_a$ and finite blocklength. It develops non-asymptotic bounds for estimating ${\rm K}_a$ and for data detection, including both per-user misdetection/false-alarm metrics and minimum energy-per-bit $E_b^*$, using sphere-codebooks and a two-stage MAP decoding approach. It also derives asymptotic scaling laws showing how the minimum energy and the required number of receive antennas scale with blocklength and activity, and demonstrates substantial energy efficiency gains from MIMO and sphere coding, while revealing gaps to practical schemes, especially at large ${\mathbb E}[{\rm K}_a]$. Numerical results corroborate the theory, illustrating energy-per-bit reductions with increasing $L$ and the advantage of sphere codes over Gaussian coding in the finite-blocklength regime. Overall, the findings underscore MIMO’s potential to enable low-cost, reliable URA under finite-blocklength constraints and uncertain user activity.

Abstract

This paper considers the unsourced random access (URA) problem with a random and unknown number of active users in multiple-input multiple-output (MIMO) quasi-static Rayleigh fading channels. We derive non-asymptotic achievability bounds on the probability of incorrectly estimating the number of active users, and provide scaling laws on the gap between the estimated and true numbers of active users. We prove that the error probability reaches a plateau as the power $P$ and blocklength $n$ increase, whereas it decays exponentially with the number $L$ of receive antennas and eventually vanishes. Then, we explore the fundamental limits of URA by deriving non-asymptotic achievability bounds and converse bounds (including two single-user converse bounds and one multi-user ensemble converse bound) on the minimum energy-per-bit required by each active user to transmit $J$ bits with blocklength $n$ under misdetection and false-alarm constraints. Numerical results show that the extra required energy-per-bit due to the uncertainty in the number ${\rm{K}}_a$ of active users decreases as $L$ and $\mathbb{E}[{\rm{K}}_a]$ increase and the error requirement becomes milder. In the non-asymptotic regime, using codewords distributed on a sphere outperforms Gaussian random coding. Existing schemes are shown to exhibit a large gap to our bounds when the number of active users is large, calling for more advanced schemes that perform energy-efficiently in this case. In the asymptotic regime with $n\to\infty$, we establish scaling laws on the minimum required $P$ and $L$ to reliably support ${\rm{K}}_a$ active users as functions of $n$, which highlight the potential of MIMO in enabling low-cost communication and indicate that it is possible for the minimum required $P$ and $L$ to remain on the same order when the number of active users increases but stays below a threshold.

Figures (8)

• 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 $K'_a$ with $L \in \{16,32\}$, $P \in \{-20,-30\}$ dB, and $n\in\{1000,5000\}$; (b) $\mathbb{P}\left[ K_a \to K'_a \right]$ versus $L$ for different values of $K'_a$ with $n=1000$ and $P=-20$ dB; (c) $\mathbb{P}\left[ K_a \to K'_a \right]$ versus $P$ for different values of $K'_a$ with $n=1000$ and $L=64$; (d) $\mathbb{P}\left[ K_a \to K'_a \right]$ versus $n$ for different values of $K'_a$ with $L=64$ and $P=-20$ dB.
• Figure 2: The mean of the number of active users versus $E_b/N_0$ 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: The mean of the number of active users versus $E_b/N_0$ in the setting where ${\rm K}_a$ follows the binomial distribution with active probability $p_a\in\{0.5,0.05\}$, $n=1000$, $J = 100$ bits, $L=128$, and $\epsilon_{\rm MD} = \epsilon_{\rm FA} = 0.001$.
• Figure 4: Per-user probabilities of misdetection and false-alarm versus $E_b/N_0$ with ${\rm K}_a \sim {\rm Binom}(200,0.5)$, $n=1000$, $J =100$ bits, and $L = 128$.
• Figure 5: The required $E_b/N_0$ versus the number of BS antennas with $n=1000$, $J = 100$ bits, and $\epsilon_{\rm MD} = \epsilon_{\rm FA} = 0.001$: (a) ${\rm K}_a\sim{\rm Binom}(200,0.5)$; (b) ${\rm K}_a\sim{\rm Binom}(1000,0.5)$.

Theorems & Definitions (21)

• Definition 1
• Theorem 1
• Corollary 1
• Theorem 2
• Corollary 2
• Corollary 3
• Theorem 3
• Theorem 4
• Corollary 4
• Theorem 5