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Compressive Sensing Based Adaptive Active User Detection and Channel Estimation: Massive Access Meets Massive MIMO

Malong Ke, Zhen Gao, Yongpeng Wu, Xiqi Gao, Robert Schober

arXiv:1906.09867v4eess.SPcs.IT

TL;DR

Massive access in massive MIMO is constrained by limited orthogonal resources, motivating compressive-sensing methods that exploit sporadic traffic and channel sparsity. The paper develops structured-sparsity-based detection and estimation algorithms with adaptive access latency, and reports lower latency and improved performance than baseline schemes.

  • Problem

    Massive numbers of potential UEs exceed available orthogonal channels, while uplink massive MIMO OFDM requires active user detection and channel estimation under sporadic traffic.

  • Method

    The paper designs DCS-based non-orthogonal pilots and uses GMMV-AMP, Turbo-GMMV-AMP, and adaptive access to exploit structured channel sparsity for simultaneous detection and estimation.

  • Results

    The proposed schemes outperform baseline algorithms; for M = 16 and eP = 1, BI-AD reduces the required access latency from G = 72 to G = 58, approximately 19%.

  • Takeaways & Limitations

    Adaptive access adjusts overhead to the uplink channel sparsity level, with 88.6% of runs requiring G ∈[12, 18] and corresponding MSE performance better than Schemes 1–3.

Abstract

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This paper considers massive access in massive multiple-input multiple-output (MIMO) systems and proposes an adaptive active user detection and channel estimation scheme based on compressive sensing. By exploiting the sporadic traffic of massive connected user equipments and the virtual angular domain sparsity of massive MIMO channels, the proposed scheme can support massive access with dramatically reduced access latency. Specifically, we design non-orthogonal pseudo-random pilots for uplink broadband massive access, and formulate the active user detection and channel estimation problems as a generalized multiple measurement vector compressive sensing problem. Furthermore, by leveraging the structured sparsity of the uplink channel matrix, we propose an efficient generalized multiple measurement vector approximate message passing (GMMV-AMP) algorithm to realize simultaneous active user detection and channel estimation based on a spatial domain or an angular domain channel model. To jointly exploit the channel sparsity presented in both the spatial and the angular domains for enhanced performance, a Turbo-GMMV-AMP algorithm is developed for detecting the active users and estimating their channels in an alternating manner. Finally, an adaptive access scheme is proposed, which adapts the access latency to guarantee reliable massive access for practical systems with unknown channel sparsity level. Additionally, the state evolution of the proposed GMMV-AMP algorithm is derived to predict its performance. Simulation results demonstrate the superiority of the proposed active user detection and channel estimation schemes compared to several baseline schemes.

I. INTRODUCTION

The paper addresses unreliable massive access in uplink massive MIMO-OFDM by exploiting sporadic UE activity and structured channel sparsity. It develops compressive-sensing methods for active user detection and channel estimation, including adaptive latency control.

  • Motivation: Massive connectivity requires reliable low-latency access, but existing wireless networks do not support reliable massive access for billions of connected UEs.Grant-free access avoids permission signaling, while orthogonal pilots become infeasible when potential UEs are numerous and only a small subset is active.
  • Contributions: AUD and CE are formulated as a GMMV compressive-sensing problem, enabling simultaneous detection and estimation with the structured uplink channel matrix.The GMMV-AMP algorithm supports spatial-domain or angular-domain channel models and learns prior hyper-parameters and noise variance using EM.
  • Contributions: The paper designs DCS-based pseudo-random pilots for broadband OFDM massive access, extending beyond prior frequency-flat narrow-band, single-carrier settings.The design leverages structured channel sparsity across multiple subcarriers for active user detection and channel estimation.
  • Contributions: Turbo-GMMV-AMP alternates active user detection and channel estimation to jointly exploit spatial- and angular-domain channel sparsity.The approach is proposed to improve performance and reduce access latency relative to simultaneous GMMV-AMP processing and state-of-the-art solutions.
  • Contributions: The adaptive access scheme adjusts pilot-transmission latency to the actual uplink channel sparsity level for reliable AUD and CE.The paper defines access latency here as the time-slot overhead for transmitting access pilots.

B. Space-Frequency Structured Sparsity in Massive Access

Massive access channels exhibit shared sparsity across users, antennas, subcarriers, and angular components, enabling compressed formulations for active-user detection and channel estimation.

  • Space-Frequency Structure: Because only a small number of UEs are active, channel matrices are sparse across users and share support across receive antennas and subcarriers.This is termed space-frequency structured sparsity.
  • Angular-Domain Structure: Small angular spread and large antenna arrays make virtual angular-domain channel vectors sparse and clustered.The angular spread is approximately arctan(r/R) when scatterers surround a UE at distance R.
  • Angular-Domain Structure: Angular-domain channel vectors associated with different subcarriers share a common sparsity pattern because subchannels experience similar spatial propagation characteristics.This structure is called angular-frequency structured sparsity.
  • Compressed Formulation: The joint space-frequency and angular-frequency sparsity properties are exploited to achieve low-latency and highly reliable detection and channel estimation.The proposed schemes transform the received-signal models into compressed-sensing problems with G much less than K.
  • Alternating Processing: The alternating scheme uses the original-domain model for detection and the angular-domain model for channel estimation to exploit both structured and enhanced sparsity.The angular-domain matrix is sparser, while the original-domain model preserves common sparsity across columns.

A. DCS-Based Pilot Design for Broadband Massive Access

The paper designs diverse non-orthogonal pilots for broadband massive access and solves the resulting generalized multiple-measurement-vector compressed-sensing problems with structured-sparsity-aware AMP.

  • Pilot Design: Pilot matrices are measurement matrices in the compressed-sensing models, so their design is crucial for reliable recovery of sparse channel matrices.The scheme uses pilot entries drawn from a standard complex Gaussian distribution.
  • Pilot Design: Different pilot matrices are assigned across pilot subcarriers to create diversity and improve detection and channel-estimation performance under DCS theory.This differs from conventional MMV designs that reuse identical pilots.
  • AMP Inference: In the large-system limit, AMP decouples matrix estimation into scalar estimation problems, while the paper notes good practical performance for medium-sized systems.The paper assumes a Gaussian channel-gain prior and uses a spike-and-slab prior to represent channel sparsity.
  • GMMV-AMP: The GMMV-AMP algorithm jointly estimates subchannel matrices across pilot subcarriers while learning unknown channel hyper-parameters and noise variance through EM.Damping is used to prevent divergence, and incremental EM updates parameters one at a time.
  • GMMV-AMP: Structured sparsity is incorporated by assigning channel elements associated with the same UE a common sparsity ratio.This refines the sparsity-ratio updates beyond independent updates for each subcarrier, user, and antenna.
  • Activity Detection: For the spatial and angular formulations, the more reliable activity decision rule differs: BI-AD is preferred for Scheme 1, whereas CG-AD is preferred for Scheme 2.This distinction is reported from simulations.

C. Alternating AUD and CE Schemes

The alternating scheme addresses the inability of simultaneous processing to fully exploit both angular sparsity and common sparsity patterns.

  • Alternating AUD and CE: Turbo-GMMV-AMP alternates between the spatial and angular channel models to jointly exploit enhanced angular sparsity and shared sparsity structure.This alternating processing supports adaptive access when the channel sparsity level is unknown.

1) Turbo-GMMV-AMP Algorithm (Scheme 3):

Turbo-GMMV-AMP alternates activity detection and angular-domain channel estimation, progressively refining reliable users and channel estimates while controlling signal removal to avoid divergence.

  • Turbo-GMMV-AMP Algorithm: The algorithm uses separate activity-detection and virtual-angular-domain channel-estimation modules executed iteratively.The first iteration obtains a rough active-user estimate before angular-domain channel estimation.
  • Turbo-GMMV-AMP Algorithm: Two activity sets use different belief-indicator thresholds: a lower threshold obtains a rough set with low missed detection, while a higher threshold yields a reliable subset with fewer false alarms.The reliable set is contained within the rough set.
  • Turbo-GMMV-AMP Algorithm: Channel estimation is reduced to the rough active-user set, whose angular-domain channel matrix remains sparse because of angular-frequency structured sparsity.Signals from a subset of reliable users are removed to further enhance sparsity.
  • Turbo-GMMV-AMP Algorithm: Only a fraction of selected-user signals is removed, with λ_aus less than 1, to prevent GMMV-AMP from diverging.The paper gives λ_aus = 0.8 as an example.
  • Turbo-GMMV-AMP Algorithm: Iterative re-estimation continually refines the reliable-user set and channels, enabling more reliable detection and estimation with significantly smaller G and reduced access latency.The alternating approach is reported to outperform simultaneous processing in latency requirements.
  • Adaptive Access: The adaptive scheme varies access latency with channel sparsity: sparse matrices require less overhead, whereas denser matrices require larger G for reliable recovery.This addresses time-varying user activity and channel environments.

2) CS-Based Adaptive AUD and CE (Scheme 4):

Scheme 4 adaptively adjusts access latency by collecting non-orthogonal pilot observations until AUD and CE reliability meets a prescribed criterion. Turbo-GMMV-AMP alternately estimates active users and their channels, enabling data transmission without scheduling permission once the criterion is satisfied.

  • Adaptive access: Scheme 4 begins with an initial time-slot overhead G0 and iteratively increases G when the reliability criterion is not met.The update is Gi+1 = Gi + 1, with the iteration index incremented accordingly.
  • Adaptive access: In each time slot, all active UEs transmit predesigned non-orthogonal RA pilots known to the system.The pilots are designed using DCS theory.
  • Joint estimation: Turbo-GMMV-AMP uses accumulated observations over successive time slots to alternately estimate the active-user set and corresponding CSI.The scheme also evaluates estimation reliability using a pre-specified criterion.
  • Adaptive access: When reliability is sufficient, the BS informs all UEs, allowing active UEs to stop pilot transmission and send data without scheduling permission.Otherwise, the scheme repeats pilot transmission and signal collection until the criterion is met.
  • Adaptive access: The initial overhead is selected according to G0 ≥ E[|supp{[Wp]:,m}|c], with ϵ = 0.8 suggested for the reliability criterion.This choice is motivated by compressive-sensing theory.

D. Computational Complexity Analysis

The complexity analysis compares the proposed GMMV-AMP methods with greedy CS recovery algorithms for AUD and CE. Greedy methods are dominated by matrix inversion, whereas the proposed methods scale linearly with key system dimensions and can be more efficient for massive access.

  • Comparison: Table I compares GMMV-AMP and Turbo-GMMV-AMP with GSP, SOMP, and DSAMP by complex multiplications per AUD-and-CE iteration.The comparison concerns computational complexity in large-scale massive-access systems.
  • Greedy methods: Matrix inversion for least-squares operations contributes most of the computational complexity in the three greedy CS recovery algorithms.The affected algorithms are GSP, SOMP, and DSAMP.
  • Proposed methods: The proposed GMMV-AMP and Turbo-GMMV-AMP complexities increase linearly with K, G, M, and P.This scaling contrasts with the dependence of the greedy methods on the number of active UEs Ka.
  • Implication: For massive-access scenarios with large Ka, the proposed algorithms can be more computationally efficient.This conclusion follows the reported complexity scaling comparison.

IV. STATE EVOLUTION

The paper derives state evolution to characterize GMMV-AMP mean-square-error performance in the large-system limit. Because scalar analysis misses structured channel sparsity, the proposed evaluation uses Monte Carlo state evolution that tracks the algorithm’s hyper-parameter updates.

  • State-evolution framework: State evolution analyzes the MSE performance of GMMV-AMP as K tends to infinity.It is framed as a large-system-limit analysis of AMP algorithms.
  • Scalar interpretation: For each pilot subcarrier, GMMV-AMP decouples matrix estimation into KM independent scalar estimation problems.The scalar formulation uses an equivalent measurement and effective noise for each channel coefficient.
  • Scalar interpretation: The scalar equivalent measurement is modeled as Cq = x0 + ñq, with ñq distributed as complex Gaussian noise of variance Dq.The posterior distribution of x0 is then used in the state-evolution updates.
  • Monte Carlo procedure: The Monte Carlo procedure generates samples from the prior and iteratively updates hyper-parameters until the iteration limit or convergence condition is reached, returning MSESE.The inputs include γ = Ka/K, κ = G/K, M, P, Tamp, ρ, and η.
  • Scope and limitation: Scalar state evolution cannot accurately analyze GMMV-AMP MSE because the prior p0(X) omits structured sparsity, so Monte Carlo simulation incorporates that structure and tracks hyper-parameter updates.This distinguishes the proposed state evolution from conventional AMP analyses that assume full prior and noise-variance knowledge.

V. SIMULATION RESULTS

The simulations evaluate pilot design, AUD, CE, and adaptive access across spatial and angular channel models. The proposed schemes generally reduce required measurements or improve detection and estimation, with adaptive access handling varying activity.

  • Pilot design: The DCS-based pilot design improves success rate over identical pilots, with further gains from massive MIMO and larger eP.These gains are attributed to exploiting structured sparsity across the channel matrix.
  • AUD performance: For M = 16 and eP = 1, Scheme 1 with BI-AD reaches Pe ≤10^-5 at G = 58, versus G = 72 for GSP, reducing access latency by approximately 19%.Larger M and/or eP can further improve detection by enhancing space-frequency structured sparsity, although gains become negligible when both are sufficiently large.
  • CE performance: When G < 70, Scheme 1 outperforms three baseline GMMV-CS algorithms in CE MSE, while its performance is accurately predicted by state evolution.For G < Ka, reliable CE is unavailable for both Scheme 1 and oracle LS, motivating Scheme 2.
  • Spatial and angular models: Scheme 2 improves AUD and CE over Scheme 1 for M = 32 and G > 28 because angular-domain channel sparsity is lower than spatial-domain sparsity.For M = 16, Scheme 1 performs better because it benefits from the stronger common sparsity pattern across antenna columns.
  • Adaptive access: Scheme 3 provides better AUD and CE than Schemes 1 and 2 at very low overheads, while Scheme 4 adaptively adjusts overhead for satisfactory performance.Scheme 4 achieves much better MSE than Schemes 1–3 in 88.6% of runs requiring G ∈[12, 18].
  • Adaptive access: Scheme 4 maintains reliable AUD and CE across different Ka, whereas fixed-overhead Scheme 3 performs poorly as Ka increases.The fixed overhead can prevent some active UEs from accessing the network.

VI. CONCLUSION

The paper combines sporadic activity and massive-MIMO channel sparsity to reduce access latency. Its alternating Turbo-GMMV-AMP and adaptive access schemes jointly target reliable AUD and CE across spatial, angular, and varying-activity settings.

  • VI. CONCLUSION: Spatial-domain space-frequency sparsity improves AUD, while angular-domain angular-frequency sparsity improves CE.Using only one channel model cannot fully exploit both sparsity structures.
  • VI. CONCLUSION: Turbo-GMMV-AMP alternates spatial-domain AUD and angular-domain CE to exploit both structured sparsity forms.The alternating procedure achieves a significant performance improvement according to the conclusion.
  • VI. CONCLUSION: The adaptive AUD and CE scheme adjusts time-slot overhead when active-UE numbers are unknown, targeting ultra-reliable low-latency massive access.This addresses practical systems with time-varying UE activity.

APPENDIX A PROOF OF THE PROPOSITION 1

The proof motivates approximating sum-product messages as Gaussian densities and then simplifying updates in the large-system limit. It derives posterior mean and variance expressions used by GMMV-AMP, including convergence behavior when the channel estimate is reliable.

  • Motivation: High-dimensional message integrals make the sum-product algorithm unacceptably complex for large-scale systems.The message count also scales with the number of potential UEs K.
  • Gaussian approximation: Gaussian message approximations enable simplified variable-node and factor-node update rules as K approaches infinity.The derivation uses the large-system limit and neglects terms approximated as zero.
  • Posterior calculation: The posterior distribution of each variable is obtained from the simplified message-passing updates.The proof introduces a family of densities and derives corresponding mean and variance expressions.
  • Convergence: For a reliable estimate of Xp after GMMV-AMP convergence, the posterior variance v∞_k,m tends to zero.This expresses concentration of the posterior around the estimated channel coefficient.
  • Assumptions: The derivation assumes i.i.d. standard complex Gaussian pilots, which supports the limiting variance approximation used in the proof.The pilot coefficients satisfy s_g,k ∼ CN(s_g,k; 0, 1).

APPENDIX C PROOF OF THE PROPOSITION 3

The proof extends the single-subgraph derivation to the broader channel model by using large-system approximations. It establishes Gaussian behavior for the residual-related variable and completes Proposition 2.

  • Derivation setup: The derivation focuses on one antenna-indexed subgraph and states that it extends readily to model (12).The antenna index is omitted for notational simplicity.
  • Large-system approximation: In the large-system limit, the interference-related term is treated as approximately independent of the factor index g.This approximation is used in deriving the message-update expression.
  • Distributional result: By the central limit theorem, r_k is modeled as complex Gaussian when K approaches infinity.This follows from independent complex Gaussian pilot and noise variables.
  • Conclusion: The proof substitutes the derived expressions into the relevant posterior quantities to obtain the mean and variance of r^q.It then obtains D^q_k and concludes the proposition.
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