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Channel Estimation for Intelligent Reflecting Surface Assisted Multiuser Communications: Framework, Algorithms, and Analysis

Zhaorui Wang, Liang Liu, Shuguang Cui

arXiv:1912.11783v4cs.ITeess.SP

TL;DR

IRS-assisted communications require estimating many concatenated user-IRS-BS channel coefficients, creating substantial training overhead. This paper proposes a correlation-exploiting three-phase framework and derives pilot lengths and LMMSE estimators for noise-free and noisy settings.

  • Problem

    IRSs generally lack RF chains, so traditional training cannot separately estimate user-IRS and IRS-BS channels, while the concatenated channels involve MK+MKN coefficients.

  • Method

    The proposed three-phase framework estimates direct channels with the IRS off, reflected channels for one typical user, then leverages correlations among users’ reflected channels.

  • Results

    K+N+max(K−1, ⌈(K−1)N/M⌉) pilot symbols are theoretically minimal for perfect noise-free estimation, and noisy-case LMMSE estimators are derived.

  • Takeaways & Limitations

    The required pilot length generally decreases with M, showing that massive MIMO can reduce IRS channel-estimation time while estimating KMN+KM coefficients with scalable pilot overhead.

  • Takeaways & Limitations

    Phase II requires N time slots, which may be long for IRSs with many reflecting elements, and errors from Phases I and II propagate into Phase III.

Abstract

from arXiv · show

In intelligent reflecting surface (IRS) assisted communication systems, the acquisition of channel state information (CSI) is a crucial impediment for achieving the beamforming gain of IRS because of the considerable overhead required for channel estimation. Specifically, under the current beamforming design for IRS-assisted communications, $KMN+KM$ channel coefficients should be estimated, where $K$, $N$ and $M$ denote the numbers of users, IRS reflecting elements, and antennas at the base station (BS), respectively. To accurately estimate such a large number of channel coefficients within a short time interval, we propose a novel three-phase pilot-based channel estimation framework in this paper for IRS-assisted uplink multiuser communications. Under this framework, we analytically prove that a time duration consisting of $K+N+\max(K-1,\lceil (K-1)N/M \rceil)$ pilot symbols is sufficient for the BS to perfectly recover all the $KMN+KM$ channel coefficients for the case without receiver noise at the BS. In contrast to the channel estimation for conventional uplink communications without IRS where the minimum channel estimation time is independent of the number of receive antennas at the BS, our result reveals the crucial role of massive MIMO (multiple-input multiple-output) in reducing the channel estimation time for IRS-assisted communications. Further, for the case with receiver noise, the user pilot sequences, IRS reflecting coefficients, and BS linear minimum mean-squared error (LMMSE) channel estimators are characterized in closed-form, and the corresponding estimation mean-squared error (MSE) is quantified.

I. INTRODUCTION

IRS-assisted multiuser communications require CSI for beamforming, but estimating the concatenated user-IRS-BS channels involves a large number of coefficients and substantial training overhead. The paper introduces a three-phase passive-pilot framework that exploits shared IRS-to-BS channels across users.

  • IRS beamforming depends critically on CSI, while the IRS’s lack of RF chains generally prevents separate estimation of user-IRS and IRS-BS channels.
  • The proposed framework exploits the fact that each IRS element reflects signals from different users to the BS through the same channel.
  • K + N + max(K −1, ⌈(K −1)N/M⌉) pilot symbols suffice for perfect estimation without receiver noise.
  • With receiver noise, the paper derives closed-form pilot sequences, IRS reflection coefficients, BS LMMSE estimators, and corresponding channel-estimation MSEs.
  • When M > N, the minimum pilot length is 2K + N −1, so one additional user requires only 2 extra pilot symbols for its MN + M coefficients.
  • The system requires estimating MK + MKN channel coefficients, which can be large when the BS has many antennas and serves many users.

III. THREE-PHASE CHANNEL ESTIMATION PROTOCOL

The protocol separates direct-channel estimation, typical-user reflected-channel estimation, and estimation of other users’ reflected channels. It reduces Phase III complexity by representing each other user’s reflected channel through scalar scaling factors.

  • The framework exploits shared IRS-to-BS channel correlations without requiring the IRS to know those channels.
  • Phase I: Phase I switches off the IRS so the BS can estimate the users’ direct channels from their pilot sequences.
  • Phase II: Phase II activates all IRS elements and lets one typical user transmit non-zero pilots to estimate that user’s IRS-reflected channels.
  • Phase III: Phase III has users 2 through K transmit pilots, while their reflected channels are modeled as scaled versions of the typical user’s channels.
  • Phase III: With the typical user’s reflected channels known, each other user’s M-dimensional channel vector requires estimating only the scalar λk,n.

IV. PERFORMANCE LIMITS FOR CASE WITHOUT NOISE

The paper first analyzes the proposed three-phase protocol under the idealized assumption of no receiver noise at the BS. This setting is used to characterize perfect-estimation limits.

  • The no-noise analysis assumes z(i) = 0 for every time slot.

A. Phase I: Direct Channel Estimation

In Phase I, the BS estimates users’ direct channels using orthogonal pilot sequences while the IRS is switched off. At least K pilot symbols are sufficient for this perfect estimation in the no-noise case.

  • Without receiver noise, each user transmits pilot symbols to support direct-channel estimation in Phase I.
  • The direct channels can be perfectly estimated when the users’ pilot sequences are mutually orthogonal.
  • K pilot symbols are sufficient to design orthogonal pilot sequences for the users.
  • The BS then obtains the direct channels by solving the received-signal system using the orthogonal pilots.

B. Phase II: Reflecting Channel Estimation for Typical User

The framework estimates the typical user’s reflected channels with structured IRS pilots, then reduces Phase III overhead for the remaining users by exploiting shared channel structure and BS antennas.

  • Phase II: Phase II isolates user 1 after direct-channel interference cancellation, enabling estimation of its reflected channels from the effective BS signal.Only user 1 transmits during this phase, while the direct channels estimated in Phase I are canceled.
  • Phase II: The IRS reflection matrix can be constructed from a DFT matrix with rank N, while satisfying the IRS coefficient constraints.The resulting matrix obeys ΦII(ΦII)H = τ2I.
  • Phase III: Phase III uses an equivalent model with (K−1)N effective users, whose pilot dimensions are expanded from τ3 to Mτ3 by the BS’s M antennas.This antenna-induced expansion enables shorter training for the remaining users.
  • Phase III: For M ≥ N, Theorem 1 gives the minimum τ3 needed for perfect estimation of the remaining users’ reflected-channel parameters.The theorem characterizes the minimum directly, while the construction uses pilot and IRS designs achieving it.
  • Phase III: For M < N, Theorem 2 partitions IRS elements across users and time slots to achieve the minimum Phase III duration.The procedure first estimates subsets with M active IRS elements, then cancels known interference to recover the remaining coefficients.
  • Phase III: The proposed design reduces Phase III training from (K−1)N symbols and requires no channel feedback from the BS to users or the IRS.The pilot and IRS reflection designs are independent of the typical user’s reflected channels.

D. Overall Channel Estimation Overhead

The three-phase protocol achieves a closed-form minimum overhead for perfect noiseless channel recovery, with massive MIMO making the overhead scale linearly in users and IRS elements.

  • Overall overhead: The minimum noiseless pilot length equals the sum of the direct-channel, typical-user reflected-channel, and remaining-user Phase III durations.The overall result covers perfect estimation of all direct and reflected channels.
  • Massive MIMO regime: 2K + N − 1 pilot symbols suffice in the massive-MIMO regime as M approaches infinity.This is linear in K and N.
  • Massive MIMO regime: Massive MIMO enables scalable estimation of KMN + KM coefficients, unlike conventional uplink estimation whose minimum time is independent of receive-antenna count.The contrast concerns the dependence of training time on M.

V. CHANNEL ESTIMATION FOR CASE WITH NOISE

With receiver noise, the paper extends the three-phase protocol by using at least the noiseless minimum duration in each phase and develops estimators for the resulting noisy signals.

  • Noisy case: With BS noise, the protocol uses τ1 ≥ τ̃1, τ2 ≥ τ̃2, and τ3 ≥ τ̃3 in Phases I, II, and III, respectively.The noisy-case treatment follows the ideal-case minimum durations as lower bounds.

A. Phase I: Direct Channel Estimation

Phase I estimates users’ direct channels from noisy uplink pilots, using orthogonal sequences and an MMSE estimator whose error contributes to later phases.

  • Phase I: The noisy Phase I received signal combines the users’ direct channels with pilot transmissions and additive noise.The model stacks the received vectors across τ1 time slots.
  • Phase I: Orthogonal user pilots are optimal for Phase I, and they can be assigned when τ1 ≥ K.The orthogonality condition separates the users’ direct-channel observations.
  • Phase I: The Phase I MMSE estimator and the corresponding MSE quantify direct-channel recovery under receiver noise.These estimates are subsequently used for interference cancellation.
  • Phase I: Phase I estimation errors remain in general and become part of the effective noise when estimating reflected channels in Phase II.The effective noise combines direct-channel estimation error with AWGN.
  • Phase II: Because the typical user’s reflected channels are not modeled as Rayleigh fading, Phase II uses an LMMSE estimator with designed user pilots and IRS coefficients.The paper derives the estimator and its MSE under this design.

C. Phase III: Reflecting Channel Estimation for Other Users

Phase III estimates the reflected channels of other users through orthogonal user transmission and IRS reflection, while addressing noise-related self-interference through an accurate Phase II estimate. The resulting LMMSE estimators and MSE expressions characterize noisy channel recovery.

  • LMMSE estimation: The LMMSE design is difficult when imperfect g1,n estimates make λk,n’s contribute to the noise used to estimate themselves.The framework therefore assumes g1,n − ˆg1,n = 0 and suggests increasing τ2 so Phase II estimation is sufficiently accurate.
  • Orthogonal transmission and reflection: At each Phase III slot, one user transmits while at most M IRS elements reflect its pilot, enabling estimation of the corresponding reflected channels.The active IRS elements are selected as a set ∆i with Mi = |∆i| ≤ M.
  • Orthogonal transmission and reflection: τ3 = (K −1)⌈N/M⌉ time slots estimate the reflected channels gk,n’s for all k ≥2 and n.Each user receives ⌈N/M⌉ slots: M IRS elements are active in each early slot, with the remainder active in the last.
  • LMMSE estimation: The LMMSE channel estimator in Phase III is constructed for each realization of G1,i assumed perfectly estimated in Phase II.The estimator is given after the effective received signal is reduced under the Phase II estimation assumption.
  • MSE characterization: The overall MSE for estimating λki,i’s is obtained by averaging the per-realization MSE across G1,i realizations.The section gives both the per-λki,i MSE and the aggregate εIII expression.

D. Overall Channel Estimation Strategy

The overall strategy summarizes the noisy-BS channel estimation protocol and evaluates it numerically under specified propagation and noise settings. A benchmark keeps Phases I and II unchanged but does not exploit channel correlations in Phase III.

  • Overall strategy: The overall channel estimation strategy for the case with BS noise is summarized in Table I.The table is presented as Algorithm I for the proposed protocol.
  • Numerical examples: The numerical evaluation uses an IRS with N = 32 reflecting elements and specified path-loss, correlation, transmit-power, bandwidth, and AWGN settings.The setup includes 33 dBm user transmit power, 1 MHz bandwidth, and −169 dBm/Hz AWGN power spectral density.
  • Benchmark scheme: The benchmark matches the proposed framework in Phases I and II but estimates each other user’s reflected channels as for user 1 in Phase III.It therefore does not exploit the channel correlations in (21) to improve channel estimation performance.
  • Numerical examples: The numerical examples evaluate the proposed framework for cases both without and with noise at the BS.The evaluation is introduced as a verification of the three-phase protocol’s effectiveness.

A. The Case without Noise at the BS

Without receiver noise, the proposed framework reduces the pilot length needed for perfect estimation and benefits strongly from more BS antennas. With receiver noise, its DFT-based and overall estimation performance also outperforms benchmark strategies.

  • Minimum estimation time: The proposed framework’s minimum pilot length grows much more slowly with users than the benchmark’s K + KN requirement.This improvement exploits correlation among IRS-reflected channels.
  • Minimum estimation time: Increasing BS antennas from M = 8 to M = 32 decreases the proposed framework’s minimum pilot length rapidly, whereas the benchmark is antenna-independent.
  • Noisy channel estimation: The DFT-based Phase II solution achieves much lower normalized MSE than single-element-on and random-phase alternatives.The comparison is made for estimating g_1,n’s with receiver noise at the BS.
  • Noisy channel estimation: Theoretical Phase III MSE matches Monte Carlo simulations under perfect Phase II estimation; imperfect Phase II estimation creates a small mismatch that decreases with more Phase II time.
  • Noisy channel estimation: Under the proposed framework, normalized Phase III MSE is below 10^-2 when τ_3 ≥ 12, while the benchmark exceeds 0.3 for τ_3 from 7 to 32.The benchmark’s poor performance is attributed to its 224-symbol minimum noiseless pilot length.
  • Noisy channel estimation: The proposed scheme significantly improves overall channel-estimation MSE compared with the benchmark scheme.The study also compares allocating extra time to Phase I, Phase II, or evenly across all three phases.

APPENDIX

The appendix establishes the minimum Phase III pilot duration needed for unique recovery of the remaining channel parameters, with different bounds depending on the relationship between BS antennas and IRS elements.

  • Case M ≥ N: τ3 ≥ K − 1 is necessary when M ≥ N, and τ3 = K − 1 is sufficient for perfect estimation of λ.The construction uses the pilot and IRS coefficient designs in (39) and (40).
  • Case M < N: When M < N, unique recovery requires τ3 ≥ ⌈(K − 1)N/M⌉ because the system has (K − 1)N variables and τ3M equations.The proposed construction partitions estimation across time and IRS-element subsets.
  • Case M < N: For M < N, the first time instants estimate M unique λk,n values without interference using one active user and M active IRS elements.Any M of the N vectors g1,n are linearly independent with probability one, enabling unique recovery.
  • Case M < N: For later M < N time instants, multiple users transmit simultaneously, while previously estimated terms are canceled to recover the remaining λk,n values.The construction also uses disjoint parameter subsets across simultaneously scheduled users.
  • Case M < N: The M < N construction perfectly estimates all λk,n values, with the final time instant handling the remaining parameters and yielding the minimum τ3 in (49).This completes the proof of Theorem 2.
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