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Joint Activity-Delay Detection and Channel Estimation for Asynchronous Massive Random Access: A Free Probability Theory Approach

Xinyu Bian, Yuyi Mao, Jun Zhang

TL;DR

The paper addresses asynchronous grant-free massive RA where $N$ users, $M$ BS antennas, and $K$ active users per block require joint activity, delay, and channel estimation despite non-i.i.d. pilot matrices caused by random delays. It proposes an OAMP-based receiver that leverages common sparsity across antennas to improve detection/estimation, and a low-complexity FPAMP receiver that uses rectangular free cumulants from free probability theory to handle general pilot matrices without matrix inversions. The results show that FPAMP achieves about a 40% complexity reduction with comparable accuracy to OAMP, and both methods substantially outperform baseline approaches, enabling support for more active users under strict false-alarm and miss-detection constraints. These findings highlight the value of incorporating high-order pilot-matrix statistics and cross-antenna sparsity for efficient asynchronous massive RA in practical mMTC deployments.

Abstract

Grant-free random access (RA) has been recognized as a promising solution to support massive connectivity due to the removal of the uplink grant request procedures. While most endeavours assume perfect synchronization among users and the base station, this paper investigates asynchronous grant-free massive RA, and develop efficient algorithms for joint user activity detection, synchronization delay detection, and channel estimation. Considering the sparsity on user activity, we formulate a sparse signal recovery problem and propose to utilize the framework of orthogonal approximate message passing (OAMP) to deal with the non-independent and identically distributed (i.i.d.) Gaussian pilot matrices caused by the synchronization delays. In particular, an OAMP-based algorithm is developed to fully harness the common sparsity among received pilot signals from multiple base station antennas. To reduce the computational complexity, we further propose a free probability AMP (FPAMP)-based algorithm, which exploits the rectangular free cumulants to make the cost-effective AMP framework compatible to general pilot matrices. Simulation results demonstrate that the two proposed algorithms outperform various baselines, and the FPAMP-based algorithm reduces 40% of the computations while maintaining comparable detection/estimation accuracy with the OAMP-based algorithm.

Joint Activity-Delay Detection and Channel Estimation for Asynchronous Massive Random Access: A Free Probability Theory Approach

TL;DR

The paper addresses asynchronous grant-free massive RA where users, BS antennas, and active users per block require joint activity, delay, and channel estimation despite non-i.i.d. pilot matrices caused by random delays. It proposes an OAMP-based receiver that leverages common sparsity across antennas to improve detection/estimation, and a low-complexity FPAMP receiver that uses rectangular free cumulants from free probability theory to handle general pilot matrices without matrix inversions. The results show that FPAMP achieves about a 40% complexity reduction with comparable accuracy to OAMP, and both methods substantially outperform baseline approaches, enabling support for more active users under strict false-alarm and miss-detection constraints. These findings highlight the value of incorporating high-order pilot-matrix statistics and cross-antenna sparsity for efficient asynchronous massive RA in practical mMTC deployments.

Abstract

Grant-free random access (RA) has been recognized as a promising solution to support massive connectivity due to the removal of the uplink grant request procedures. While most endeavours assume perfect synchronization among users and the base station, this paper investigates asynchronous grant-free massive RA, and develop efficient algorithms for joint user activity detection, synchronization delay detection, and channel estimation. Considering the sparsity on user activity, we formulate a sparse signal recovery problem and propose to utilize the framework of orthogonal approximate message passing (OAMP) to deal with the non-independent and identically distributed (i.i.d.) Gaussian pilot matrices caused by the synchronization delays. In particular, an OAMP-based algorithm is developed to fully harness the common sparsity among received pilot signals from multiple base station antennas. To reduce the computational complexity, we further propose a free probability AMP (FPAMP)-based algorithm, which exploits the rectangular free cumulants to make the cost-effective AMP framework compatible to general pilot matrices. Simulation results demonstrate that the two proposed algorithms outperform various baselines, and the FPAMP-based algorithm reduces 40% of the computations while maintaining comparable detection/estimation accuracy with the OAMP-based algorithm.
Paper Structure (21 sections, 4 theorems, 101 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 21 sections, 4 theorems, 101 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works and Motivations
  3. Contributions
  4. Organization
  5. Notations
  6. System Model
  7. OAMP-based Algorithm for Asynchronous Massive RA
  8. OAMP Iteration for Each BS Antenna
  9. LE
  10. NLE
  11. OAMP-based Algorithm for Multi-antenna BSs
  12. A Low-Complexity FPAMP Approach for Asynchronous Massive RA
  13. Free Probability Theory
  14. The Proposed FPAMP-based Algorithm
  15. Onsager coefficients
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\psi$: $\mathbb{C}^{2i+1}\rightarrow \mathbb{C}$ and $\phi$: $\mathbb{C}^{2i+2}\rightarrow \mathbb{C}$ be any pseudo-Lipschitz functions of order 2. For $i = 1, 2, \cdots$, it is almost sure that where $H_{m}$, $H^{(i)}_{m}$, $\tilde{H}^{(i)}_{m}$, $S^{(i)}_{m}$, $R^{(i)}_{m}$, and $Y_{m}$ are defined in the state evolution of the FPAMP-based algorithm as follows: In particular, $\mathbf{h}

Figures (5)

  • Figure 1: Frame structure of asynchronous grant-free massive RA, where a guard interval spanning $T=3$ symbol periods are inserted between pilot and data symbols.
  • Figure 2: Probability of missed detection versus probability of false alarm.
  • Figure 3: Delay detection error probability versus the number of active users $K$.
  • Figure 4: NMSE of channel estimation versus transmit power.
  • Figure 5: Probability of missed detection versus probability of false alarm with different maximum synchronization delays.

Theorems & Definitions (13)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • proof : Proof Sketch
  • Theorem 2
  • proof
  • Lemma 1
  • ...and 3 more