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Joint Channel Estimation and Signal Recovery for RIS-Empowered Multi-User Communications

Li Wei, Chongwen Huang, Qinghua Guo, Zhaohui Yang, Zhaoyang Zhang, George C. Alexandropoulos, Merouane Debbah, Chau Yuen

arXiv:2106.14391v2cs.ITmath.IT

TL;DR

Channel acquisition in RIS-assisted systems is challenging because many passive elements create unknown channel variables, and prior work often separates channel estimation from signal recovery. This paper develops two joint message-passing schemes; simulations report benchmark advantages, with BUTAMP more robust to damping-factor changes.

  • Problem

    RIS channel acquisition is challenging with many passive elements, while joint channel estimation and signal recovery has received less study than separate or two-stage approaches.

    Source: §I · pp. 1–2

  • Method

    The paper formulates joint channel and signal recovery with factor graphs, using Taylor expansion, Gaussian approximation, and bidirectional BAMP and BUTAMP schemes.

    Source: §I · p. 2

  • Results

    At pilot length Tp = 100, the proposed schemes show about an 18 dB gap over the baseline method, and their NMSE performance is similar.

    Source: §IV · p. 7, §V · p. 9

  • Takeaways & Limitations

    BUTAMP is more robust to damping factors than BAMP, while both proposed schemes show superiority over the benchmark scheme.

    Source: §IV · p. 8, §V · p. 9

Abstract

from arXiv · show

Reconfigurable intelligent surfaces (RISs) have been recently considered as a promising candidate for energy-efficient solutions in future wireless networks. Their dynamic and lowpower configuration enables coverage extension, massive connectivity, and low-latency communications. Due to a large number of unknown variables referring to the RIS unit elements and the transmitted signals, channel estimation and signal recovery in RIS-based systems are the ones of the most critical technical challenges. To address this problem, we focus on the RIS-assisted wireless communication system and present two joint channel estimation and signal recovery schemes based on message passing algorithms in this paper. Specifically, the proposed bidirectional scheme applies the Taylor series expansion and Gaussian approximation to simplify the sum-product procedure in the formulated problem. In addition, the inner iteration that adopts two variants of approximate message passing algorithms is incorporated to ensure robustness and convergence. Two ambiguities removal methods are also discussed in this paper. Our simulation results show that the proposed schemes show the superiority over the state-of-art benchmark method. We also provide insights on the impact of different RIS parameter settings on the proposed schemes.

I. INTRODUCTION

RISs are motivated by energy-efficient, low-power communications, but their many passive elements make channel acquisition challenging. The paper develops joint channel estimation and signal recovery for a multi-user RIS-assisted system, using a beam-domain model and message-passing methods.

I. INTRODUCTION

  • Most RIS configuration studies assume perfect channel state information, while estimating cascaded channels is challenging and joint channel estimation with signal recovery remains underexplored.Earlier separate-stage methods also incur high training overhead, and AMP-related algorithms can diverge with ill-conditioned measurement matrices.Source: §I · pp. 1–2
  • The proposed factor-graph approach jointly estimates two cascaded channels and the transmitted signal, simplifying message updates with Taylor expansion and Gaussian approximation.BAMP and BUTAMP are presented to improve convergence; the paper also investigates ambiguity removal and computational complexity.Source: §I · p. 2

II. SYSTEM MODEL

  • The system model considers uplink communication from multiple single-antenna users to a multi-antenna base station through a discrete-element RIS, with no direct path assumed.The paper notes that a direct link can be estimated using conventional methods with RIS elements off.Source: §I · p. 2, §II · p. 3
  • In the beam domain, the equivalent channels are represented as sparse, and the objective is to jointly estimate both channels and the transmitted signal during transmission.The model uses a beam-domain input-output relationship for the received signal.Source: §II · p. 3

III. PROBLEM FORMULATION AND MESSAGE PASSING ALGORITHM

The system is formulated as a two-layer estimation problem and represented with a factor graph for message passing. The first layer maps the transmitted signal to an intermediate variable, while the received signal reflects the second-layer channel output and noise.

A. Problem Formulation and Factor Graph Representation

1) The First Layer:

  • In the first layer, the message from funt to xmt is specified alongside the message from xmt to funt in the next iteration.Source: 1) The First Layer · p. 4

A. Problem Formulation and Factor Graph Representation

  • The factor graph represents joint channel estimation and signal recovery, and Table I is identified as containing its message definitions.Source: Figure 3 · p. 4, Table I · p. 4

1) The First Layer:

  • The first-layer message definitions also include messages from funt to hb nm and from unt to funt.Source: 1) The First Layer · p. 4

2) The Second Layer:

3) The Output Layer:

The output layer specifies message exchanges among akt, fykt, and ykt. The estimated-variable beliefs are approximate posterior distributions for xmt, hb nm, and qkn.

3) The Output Layer:

4) Beliefs of Estimated Variables:

B. Derivation and Approximation of Message Passing

The paper simplifies loopy sum-product message passing with Gaussian approximation and Taylor expansion, then presents bidirectional BAMP and BUTAMP schemes. BAMP refines Q but is sensitive to damping; BUTAMP uses a transformed relationship for AMP-based Q estimation to improve robustness.

B. Derivation and Approximation of Message Passing

  • Gaussian approximation and Taylor series expansion simplify loopy belief-propagation messages for efficient joint channel estimation and signal recovery.The exact sum-product implementation is described as impractical because of numerous loops and discrete and continuous variables.Source: B. Derivation and Approximation of… · p. 5

C. Two Proposed Schemes

  • The bidirectional algorithm alternates backward estimation of Q and U before X and Hb with forward estimation of X and Hb before Q.Its inner iteration for estimating Q yields the BAMP and BUTAMP schemes.Source: C. Two Proposed Schemes · p. 5

1) BAMP Two-layer Algorithm:

  • BAMP refines Q using pilot-initialized Xp, then feeds the estimated U = HbX into subsequent Q estimation; damping addresses divergence associated with ill-conditioned U.The passage identifies BAMP as sensitive to the damping factor.Source: 1) BAMP Two-layer Algorithm · p. 5

2) BUTAMP Two-layer Algorithm:

  • BUTAMP responds to BAMP’s damping sensitivity by using an SVD-based transformed measurement relationship and AMP with unitary transformation to estimate Q.The transformed measurement matrix is UΛ and the transformed received signal is R = YV^H; the passage describes the resulting estimation as low computational cost.Source: 2) BUTAMP Two-layer Algorithm · pp. 5–6

3) Ambiguities Removal:

The section describes inherent ambiguity in recovering the channel and signal factors and presents measures to remove or mitigate it. Its lower-cost approach estimates a portion of Hr from pilots for use in the joint recovery algorithms.

3) Ambiguities Removal:

  • Factorization ambiguities undermine estimation performance, and the proposed scheme’s three unknown matrices make the ambiguity issue more complex than in single-layer bilinear recovery.Source: 3) Ambiguities Removal · p. 6
  • Scaling ambiguity is removed by assuming the first Kp rows of Hr are known, while phase ambiguity is addressed with a full-rank transmitted signal.The paper notes that obtaining Kp known rows is unreasonably costly in a large system, motivating a lower-cost removal method.Source: 3) Ambiguities Removal · p. 6
  • The lower-cost method estimates a portion of Hr from pilots using PARAFAC or BiG-AMP, then uses that estimate in joint channel and signal recovery.PARAFAC uses multiple phase matrices; BiG-AMP estimates the involved channels simultaneously.Source: 3) Ambiguities Removal · p. 7

D. Complexity Analysis

The proposed algorithms have lower computational cost and favorable benchmark performance, with pilot length, RIS size, and damping affecting their behavior. The simulations also show stronger damping robustness and faster convergence for BUTAMP than BAMP.

D. Complexity Analysis

  • The proposed BAMP algorithm has total computational cost O((KNT + NMT)L), with L denoting the iteration number.The first layer costs O(NMT), and the second costs O(KNT).Source: D. Complexity Analysis · p. 7

IV. SIMULATION RESULTS

  • About 18 dB separates the proposed schemes from BiGAMP+LS at Tp = 100; with Tp = 150 for the baseline, the gap is about 4 dB for Hb and larger for X.The two proposed schemes have similar NMSE performance.Source: Figure 4 · p. 7, §IV · p. 7
  • A pilot ratio Tp/T = 0.3 achieves performance similar to higher ratios; increasing Kp improves estimates, with X gaps of about 9 dB from 120 to 150 and 1.5 dB from 150 to 180.The improvement for Hr is reported as even larger than for X.Source: §IV · p. 7, Figure 5 · p. 8, Figure 6 · p. 8
  • Increasing RIS elements from N = 150 to 300 increases performance loss, with the gap growing as SNR increases; the X NMSE gap is 5 dB between N = 150 and 200.The paper attributes this loss to the increase in unknown variables to estimate.Source: §IV · pp. 7–8, Figure 7 · p. 8
  • BUTAMP is more robust to damping: it converges across tested factors, whereas BAMP diverges at β = 0.7 and β = 1.For BUTAMP, β = 0.7 and β = 1 give similar NMSE performance; the X-estimate gap between β = 0.3 and β = 1 is about 7 dB.Source: §IV · p. 8, Figure 8 · p. 8, Figure 9 · p. 8
  • At SNR = 20 dB, BUTAMP converges in 18 iterations, compared with 28 for BAMP.Both algorithms require fewer iterations as SNR increases.Source: §IV · pp. 8–9, Figure 10 · p. 9

V. CONCLUSION

  • The conclusion reports similar performance for the proposed schemes, BUTAMP's greater damping robustness, benchmark superiority, and sensitivity to pilot length and RIS-element count.The algorithms target uplink multiple single-antenna users; the paper says they can also be applied to uplink multi-antenna users and downlink single multi-antenna-user cases.Source: §V · p. 9

APPENDIX

The appendix outlines message passing over two factor-graph layers and derives approximations for first-layer factor-to-variable messages. Its approximations use Taylor expansion and discard infinitesimal terms in the large-system limit.

APPENDIX

  • The appendix derives message passing between variables and factor nodes in two layers, simplifying intermediate variables.Source: APPENDIX · p. 9

1) The First Layer (Approximated Factor-to-Variable Messages):

  • The derivation drops infinitesimal terms under a large-system approximation and substitutes the message mean for its directional-message mean.Source: 1) The First Layer (Approximated… · p. 9
  • For first-layer approximated factor-to-variable messages, the derivation applies a Taylor expansion to the logarithm of a message.The expansion uses derivatives of G with respect to its arguments and retains terms through the displayed approximation order.Source: 1) The First Layer (Approximated… · p. 9

2) The First Layer (Approximated Variable-to-Factor Messages):

The first-layer variable-to-factor update uses the prior on each signal entry and an expectation-propagation message for discrete-valued entries. A Gaussian approximation to the Gaussian-mixture prior is used for low-complexity implementation.

2) The First Layer (Approximated Variable-to-Factor Messages):

3) The Second Layer (Approximated Factor-to-Variable Messages):

The second-layer derivation simplifies factor-to-variable messages and describes an approximate posterior distribution. It uses incoming-message means and variances and derivatives of G in the update expressions.

3) The Second Layer (Approximated Factor-to-Variable Messages):

4) The Second Layer (Approximated Variable-to-Factor Messages):

The subsection presents an expression for the message from u_nt to f_akt and gives belief expressions for u_nt and q_kn.

5) Simplications of Intermediate Variables:

The subsection simplifies messages between layers using Gaussian products and Gaussian approximations. It also specifies a message for the ℓ-th iteration.

6) Message Between Different Layers:

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