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Two-Timescale Channel Estimation for Reconfigurable Intelligent Surface Aided Wireless Communications

Chen Hu, Linglong Dai

arXiv:1912.07990v2eess.SPcs.IT

TL;DR

RIS channel estimation faces prohibitively high pilot overhead, especially for high-dimensional cascaded channels. The paper proposes a two-timescale framework using dual-link pilots and coordinate descent for the quasi-static BS-RIS channel while frequently estimating lower-dimensional mobile channels. Simulations show accurate channel estimation with low pilot overhead.

  • Problem

    RIS-aided channel estimation is challenging because the cascaded channel has many coefficients and passive RISs cannot transmit or receive pilots, making pilot overhead prohibitively high.

  • Method

    The framework estimates the high-dimensional quasi-static BS-RIS channel on a large timescale using dual-link pilots and coordinate descent, while estimating mobile low-dimensional channels on a small timescale.

  • Results

    Simulation results show that the proposed framework achieves accurate channel estimation with low pilot overhead.

  • Takeaways & Limitations

    Exploiting the channel’s two-timescale property significantly reduces pilot overhead by estimating the quasi-static BS-RIS channel less frequently.

Abstract

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Channel estimation is challenging for the reconfigurable intelligent surface (RIS)-aided wireless communications. Since the number of coefficients of the cascaded channel among the base station (BS), the RIS and the user equipments (UEs) is the product of the number of BS antennas, the number of RIS elements, and the number of UEs, the pilot overhead can be prohibitively high. In this paper, we propose a two-timescale channel estimation framework to exploit the property that the BS-RIS channel is high-dimensional but quasi-static, while the RIS-UE channel is mobile but low-dimensional. Specifically, to estimate the quasi-static BS-RIS channel, we propose a dual-link pilot transmission scheme, where the BS transmits downlink pilots and receives uplink pilots reflected by the RIS. Then, we propose a coordinate descent-based algorithm to recover the BS-RIS channel. Since the quasi-static BS-RIS channel is estimated less frequently than the mobile channel be, the average pilot overhead can be reduced from a long-term perspective. Although the mobile RIS-UE channel has to be frequently estimated in a small timescale, the associated pilot overhead is low thanks to its low dimension. Simulation results show that the proposed two-timescale channel estimation framework can achieve accurate channel estimation with low pilot overhead.

I. INTRODUCTION

RIS-aided channel estimation is difficult because passive RISs cannot transmit or receive pilots, while conventional cascaded-channel estimation requires prohibitively high pilot overhead. Existing overhead-reduction methods rely on conditions such as low SNR tolerance, low-rank structure, or sparsity that may not hold generally.

  • Accurate CSI is important for designing the BS precoding matrix and RIS reflection coefficients.
  • Passive RISs cannot transmit or receive pilots, so existing methods typically estimate only the BS-RIS-UE cascaded channel using active BS and UE transceivers.
  • The cascaded-channel pilot overhead scales with the number of RIS elements and UEs and can reach hundreds to thousands in practice.
  • Methods estimating individual cascaded channels can require pilot overhead equal to the number of RIS elements multiplied by the number of UEs, limiting large-system use.
  • Some existing methods reduce overhead through reference-channel estimation, but accuracy degrades at low SNR.
  • Sparse, low-rank, compressive-sensing, and deep-learning approaches are not applicable in the general scenario where the channel is neither low-rank nor sparse.

B. Contributions

The paper exploits different channel timescales by estimating the high-dimensional, quasi-static BS-RIS channel less frequently and the mobile, low-dimensional channels more frequently. It combines dual-link pilots and coordinate descent for the BS-RIS channel with low-overhead estimation of mobile channels.

  • The framework exploits a quasi-static BS-RIS channel alongside mobile RIS-UE and BS-UE channels caused by UE mobility.
  • The high-dimensional BS-RIS channel is estimated on a large timescale, whereas low-dimensional mobile channels are estimated on a small timescale.
  • The dual-link pilot scheme has the BS transmit downlink pilots and receive uplink pilots reflected by the RIS.
  • A coordinate descent-based algorithm recovers the BS-RIS channel after dual-link pilot transmission.
  • The quasi-static stage is estimated less frequently, reducing its average pilot overhead from a long-term perspective.
  • Mobile RIS-UE and BS-UE channels can use existing least-square solutions, and their lower dimension keeps associated pilot overhead low despite frequent estimation.
  • The system model comprises a BS with M antennas, an RIS with N elements, and K UEs, with cascaded channel C_k defined as Gdiag(f_k).

III. THE PROPOSED TWO-TIMESCALE CHANNEL ESTIMATION FRAMEWORK

The framework separates estimation across two timescales: a high-dimensional, quasi-static BS-RIS channel is estimated infrequently, while low-dimensional mobile BS-UE and RIS-UE channels are estimated frequently. This design reduces average pilot overhead while retaining low-dimensional channel estimation for mobile links.

  • The dual-link scheme has the BS transmit downlink pilots to the RIS while receiving RIS-reflected uplink pilots at other BS antennas.
  • The mobile RIS-UE and BS-UE channels can be estimated with conventional uplink pilots and an LS-based algorithm because they are low-dimensional.
  • Less frequent estimation of the quasi-static BS-RIS channel reduces its long-term average pilot overhead, and the required pilot overhead can be significantly reduced overall.
  • The BS-RIS channel is high-dimensional but quasi-static, whereas the BS-UE and RIS-UE channels are mobile and low-dimensional.
  • (M + N)K coefficients are estimated for the BS-UE and RIS-UE channels instead of (M + MN)K coefficients required by cascaded-channel estimation.
  • The frame first estimates the quasi-static BS-RIS channel using dual-link pilots, then estimates mobile BS-UE and RIS-UE channels from uplink pilots before data transmission.

IV. QUASI-STATIC BS-RIS CHANNEL ESTIMATION

The paper addresses non-identifiability and the RIS’s lack of active transceivers with a dual-link pilot transmission scheme. A full-duplex BS sends pilots toward the RIS and receives their reflections, enabling BS-RIS channel estimation.

  • Conventional uplink pilots cannot uniquely estimate the BS-RIS channel and RIS-UE channels because different cascaded-channel decompositions can produce the same received pilots.
  • A. Dual-link pilot transmission: The proposed scheme does not require UEs to transmit or receive pilots for BS-RIS channel estimation.
  • A. Dual-link pilot transmission: The BS transmits pilots to the RIS through the downlink, and the RIS reflects them back through the uplink to BS receiving antennas.
  • A. Dual-link pilot transmission: The dual-link frame contains (N + 1) sub-frames of L time slots, with RIS reflection vectors varied across sub-frames.
  • The full-duplex implementation introduces self-interference, which can be severe before mitigation and increases BS hardware complexity after suppression.

B. Proposed coordinate descent-based channel estimation algorithm

Given the received dual-link pilots, the BS-RIS estimation algorithm proceeds through problem division, initialization, and iterative coordinate-descent refinement.

  • The algorithm estimates the quasi-static BS-RIS channel in three stages: independent subproblem formation, initial estimation, and iterative optimization.
  • Each subproblem estimates the channel between the BS and one RIS element, separating the high-dimensional recovery task into smaller problems.
  • Initial estimates are coarse and easy to calculate, helping the later iterative refinement converge faster.
  • The BS-RIS channel estimates are iteratively optimized using a coordinate descent approach.

1) Problem division:

The quasi-static BS-RIS estimation problem decomposes into N independent subproblems, one for each RIS element, and each subproblem is solved by minimizing its objective over the associated channel coefficients.

  • For a fixed RIS element n, the variables am1,m2,n depend only on the BS-to-element channel coefficients {gm,n}, not on other RIS elements.
  • Consequently, estimating the quasi-static BS-RIS channel divides into N independent subproblems.
  • The n-th subproblem estimates the channel coefficients related to the n-th RIS element from the corresponding observations.
  • Solvability requires |S| ≥ M, which implies L ≥ 2 for the pilot-index construction.
  • The algorithm formulates each subproblem as minimizing Jn over g1,n through gM,n, after which initial coarse estimates support iterative refinement.

3) Coordinate descent-based iterative refinement:

The proposed coordinate descent-based algorithm refines coarse BS-RIS channel estimates by repeatedly updating individual coefficients while holding the others fixed. It decomposes the estimation into independent RIS-element subproblems and iterates until convergence or a preset limit.

  • The algorithm refines coarse channel estimates to obtain accurate BS-RIS channel estimates.
  • Each outer iteration performs M inner iterations, refining the M coefficients sequentially from the first to the last.
  • During each inner iteration, one coefficient is optimized while the remaining M −1 coefficients remain fixed.
  • The coefficient update uses a closed-form solution to a univariate optimization problem.
  • The quasi-static estimation problem is divided into N independent subproblems, each corresponding to one RIS element.
  • Outer iterations stop when a well-fit solution is found or the maximum number of iterations Imax is reached.
  • After estimating the quasi-static BS-RIS channel, the framework estimates mobile RIS-UE and BS-UE channels using uplink pilots and an LS-based algorithm.

A. Uplink pilot transmission

The uplink pilot scheme uses multiple sub-frames with RIS reflection patterns and orthogonal UE pilot sequences to separate and collect channel observations. The number of received pilots is chosen to cover the mobile-channel dimension.

  • The uplink pilot frame contains τ0 sub-frames, each lasting K time slots.
  • In each sub-frame, the RIS uses a randomly generated reflection coefficient vector with independently uniform random phases.
  • The UEs transmit pilot sequences over the K time slots, with orthogonal sequences assigned to different UEs.
  • The received pilot matrix Yt has dimension M×K, with each column representing pilots received in one time slot and Nt denoting noise.
  • Right-multiplying by the conjugate pilot sequences separates the channel observations for different UEs.
  • The number of received pilots must be no smaller than the mobile-channel dimension, requiring τ0M ≥ M + N.

B. LS-based channel estimation algorithm

The LS-based stage forms a mobile-channel observation model using the estimated quasi-static BS-RIS channel, then applies least squares to estimate the RIS-UE and BS-UE channels. Averaging the infrequent BS-RIS estimation over its longer coherence time reduces pilot overhead.

  • The observation matrix A is determined by the exact BS-RIS channel G and the RIS reflection coefficients.
  • Because G is unknown, the method constructs an estimated matrix ˆA using the estimated quasi-static channel ˆG.
  • The mobile RIS-UE and BS-UE channels are estimated using the least square estimate based on ˆA and the equivalent received pilots.
  • The quasi-static BS-RIS channel is estimated on a large timescale, while mobile channels are estimated on a small timescale.
  • The large-timescale coherence time is modeled as TL = αTS with α >> 1, where TS is the small-timescale coherence time.
  • The proposed pilot overhead is lower than (M + N)K in [5] and lower than K + N + max in [6] when α > 2.
  • The coordinate descent algorithm has computational complexity O(MLN log N + MLNImax).

VII. SIMULATIONS

Simulations evaluate convergence, pilot overhead, channel-estimation accuracy, and sum-rate performance for the proposed two-timescale framework. The framework reduces pilot overhead while achieving lower NMSE than [6] and improved sum-rate performance over [6].

  • Simulation setup: With M = 32, N = 64, and K = 8, simulations model the proposed estimator under specified distances, transmit powers, and fading exponents.The BS-RIS, BS-UE, and RIS-UE distances are 20 m, 30 m, and 20 m; transmit powers are PBS = 10 W and PUE = 1 W.
  • Convergence: The coordinate descent-based estimator converges to a stable mean with low standard deviation within no more than 5 outer iterations.The simulations therefore set Imax = 5 for Algorithm 1.
  • Pilot overhead: The proposed two-timescale framework has lower pilot overhead than the multi-user method in [6] and significantly lower overhead than the MVU estimator in [5].The comparison leverages the distinction between large-timescale and small-timescale channel estimation.
  • Channel-estimation accuracy: The proposed method achieves lower NMSE than [6], whereas MVU estimation is more accurate at approximately 32 times the pilot overhead.NMSE is evaluated for both the cascaded channel and the BS-UE direct channel against uplink SNR.
  • Sum-rate performance: The sum-rate comparison uses cross-entropy-based joint BS precoding and RIS reflection optimization from CSI estimated by each scheme.Perfect CSI is included as an upper-bound reference, and the proposed method outperforms [6].

VIII. CONCLUSIONS

The proposed framework exploits different channel timescales: the high-dimensional BS-RIS channel is quasi-static, while RIS-UE and BS-UE channels are mobile but low-dimensional. This reduces pilot overhead while maintaining accurate channel estimation, with future work extending the approach to different channel models and scenarios.

  • The framework leverages the two-timescale property of RIS-aided channels for channel estimation.The BS-RIS channel is high-dimensional and quasi-static, whereas the RIS-UE and BS-UE channels are mobile but low-dimensional.
  • Pilot overhead can be significantly reduced by exploiting the different channel timescales and dimensions.The quasi-static BS-RIS channel is estimated less frequently, while frequently estimated mobile channels have smaller dimensions than the cascaded channel.
  • The quasi-static BS-RIS channel benefits from reduced average pilot overhead over the long term because it is not estimated frequently.
  • The mobile RIS-UE and BS-UE channels require frequent small-timescale estimation but incur lower pilot overhead because of their smaller dimensions.
  • Simulation results show accurate channel estimation with low pilot overhead.
  • Future research should extend the two-timescale channel estimation framework to different channel models and scenarios.
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