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Channel Estimation for Intelligent Reflecting Surface Assisted Multiuser Communications

Zhaorui Wang, Liang Liu, Shuguang Cui

arXiv:1911.03084v2cs.ITeess.SPmath.IT

TL;DR

IRS channel estimation is difficult because passive IRSs cannot estimate their BS and user channels, while beamforming requires KMN + KM coefficients. The paper proposes a three-phase passive-pilot protocol that exploits shared IRS-to-BS channels and shows that, without noise, all coefficients can be perfectly recovered in K + N + max(K −1, ⌈(K −1)N/M⌉) pilot symbols. The result highlights the role of BS antennas in reducing training time.

  • Problem

    Passive IRSs lack RF chains for channel estimation, yet beamforming requires estimating KMN + KM coefficients in large-MIMO, massive-connectivity systems.

    Source: §II · p. 3

  • Method

    The paper develops a three-phase passive-pilot protocol that estimates direct channels, one user's reflected channels, and the remaining users' channels through shared-channel correlations.

    Source: §I · p. 1, §III · p. 3, C. Phase III: Reflecting Channel… · p. 4

  • Results

    Without BS noise, perfect recovery of all KMN + KM coefficients requires K + N + max(K −1, ⌈(K −1)N/M⌉) pilot symbols.

    Source: §I · pp. 1–2

  • Takeaways & Limitations

    Massive MIMO can reduce IRS-assisted channel-training time, contrasting with conventional uplink estimation whose minimum pilot length is independent of receive-antenna count.

    Source: §I · p. 2, C. Phase III: Reflecting Channel… · pp. 4, 6

Abstract

from arXiv · show

In the intelligent reflecting surface (IRS) assisted communication systems, the acquisition of channel state information (CSI) is a crucial impediment for achieving the passive 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 if the passive IRS cannot estimate its channels with the base station (BS) and users due to its lack of radio frequency (RF) chains, where $K$, $N$ and $M$ denote the number of users, reflecting elements of the IRS, and antennas at the BS, respectively. This number can be extremely large in practice considering the current trend of massive MIMO (multiple-input multiple-output), i.e., a large $M$, and massive connectivity, i.e., a large $K$. To accurately estimate such a large number of channel coefficients within a short time interval, we devote our endeavour in this paper to investigating the efficient pilot-based channel estimation method in IRS-assisted uplink communications. Building upon the observation that the IRS reflects the signals from all the users to the BS via the same channels, we analytically verify 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 in the case without noise. In contrast to the conventional uplink communications without IRS in which the minimum pilot sequence length for channel estimation is independent with the number of receive antennas, our study reveals the significant role of massive MIMO in reducing the channel training time for IRS-assisted communication systems.

I. INTRODUCTION

IRS-assisted communications can improve throughput by shaping wireless channels, but realizing this gain requires costly CSI acquisition because passive IRS hardware cannot estimate its own channels. This paper targets efficient multiuser uplink estimation by exploiting shared IRS-to-BS channels.

  • The IRS enhances throughput by independently phase-shifting incident signals to make BS-user channels more favorable –.Source: §I · p. 1
  • Passive IRSs generally lack RF chains, so they cannot estimate channels with the BS and users.Source: §I · p. 1
  • The beamforming design requires estimating KMN + KM channel coefficients, a burden amplified by massive MIMO and massive connectivity.Source: §I · p. 1
  • The proposed passive-pilot approach perfectly recovers all coefficients without noise using K + N + max(K −1, ⌈(K −1)N/M⌉) pilot symbols.Source: §I · p. 1
  • Using shared IRS reflection channels, Phase I estimates direct channels, Phase II estimates one user's reflected channels, and Phase III estimates the remaining users' channels.Source: §I · pp. 1–2
  • When M > N, the minimum duration becomes 2K + N −1, so adding one user requires only two additional pilot symbols for MN + M new coefficients.Source: §I · p. 2

II. SYSTEM MODEL

The system is a narrow-band IRS-assisted uplink with K single-antenna users, an M-antenna BS, and N passive IRS elements. The BS estimates direct and reflected channels from pilot observations while IRS coefficients shape the received superposed signals.

  • The model contains K users, M BS antennas, and N passive IRS elements in a quasi-static block-fading uplink.Source: §II · p. 2
  • Each IRS element dynamically adjusts its reflection coefficient to alter the phase of its incident signal and combine useful and interfering components constructively or destructively.Source: §II · p. 2
  • The BS receives a superposition of users' direct links and IRS-reflected links, with additive white Gaussian noise and effective reflected channels.Source: §II · p. 2
  • A coherence block divides into τ pilot symbols for channel estimation and T −τ symbols for data transmission.Source: §II · p. 2
  • The BS estimates all direct and reflected channels from received pilot signals and known user pilots and IRS reflection coefficients.Source: §II · p. 2
  • Joint BS receive beamforming and IRS reflection optimization requires MK + MKN coefficients, creating substantial overhead in large-array, heavily connected networks.Source: §II · pp. 2–3

III. THREE-PHASE CHANNEL ESTIMATION PROTOCOL

The proposed protocol reduces multiuser IRS channel-training overhead by exploiting the common IRS-to-BS channel across users. It uses three phases: direct-channel estimation, one-user reflected-channel estimation, and scalar recovery for the remaining users.

  • Shared IRS-to-BS channels create correlations that reduce the effective degrees of freedom below the KMN reflected-channel unknowns, despite the IRS lacking RF chains.Source: §III · p. 3
  • Phase I switches off the IRS to estimate direct channels, while Phase II uses only one typical user to estimate its N reflected channels.Source: §III · p. 3
  • For each remaining user, gk,n = λk,ng1,n, so known typical-user channels reduce recovery to estimating scalar coefficients λk,n.Source: §III · p. 3
  • The protocol is analyzed primarily for the ideal noiseless BS case, while practical noisy channel estimation is deferred to future work.Source: §III · p. 3

IV. PERFORMANCE LIMITS FOR THE CASE WITHOUT RECEIVER NOISE

The performance-limit analysis considers the ideal case without receiver noise and evaluates perfect estimation under the proposed protocol.

  • The analysis assumes zero BS receiver noise when deriving perfect channel-estimation limits.Source: §IV · p. 3
  • Under this assumption, the three-phase protocol is the basis for the performance-limit characterization.Source: §IV · p. 3
  • The stated scope is an ideal noiseless scenario rather than practical noisy operation.Source: §IV · p. 3

A. Phase I: Direct Channel Estimation

Phase I switches off the IRS and uses user pilot sequences to estimate the direct channels. Orthogonal pilots make the direct channels perfectly recoverable, with pilot length at least K.

A. Phase I: Direct Channel Estimation

B. Phase II: Reflecting Channel Estimation for Typical User

Phase II switches on the IRS and estimates the reflected channels of a typical user using interference cancellation and an IRS reflection matrix with full rank. A DFT-based construction makes the reflected channels perfectly recoverable.

B. Phase II: Reflecting Channel Estimation for Typical User

C. Phase III: Reflecting Channel Estimation for Other Users

Phase III estimates the remaining users’ IRS-reflected channels by exploiting their shared-channel structure, converting the problem into an equivalent linear model whose effective observation dimension grows with the BS antennas. Proper pilot and IRS-reflection designs then minimize the required Phase III duration.

Motivation and equivalent model

  • Exploiting correlations among reflected channels reduces Phase III training from the straightforward (K −1)N-symbol scheme by using the shared BS–IRS channels across users.The direct-channel interference is first cancelled, and the remaining estimation problem leverages the fact that each IRS element uses the same BS-side channel for all users.Source: C. Phase III: Reflecting Channel… · pp. 4–5
  • The Phase III observations form an equivalent linear model with (K −1)N effective users, while M BS antennas expand the effective channel-estimation dimension to Mτ3.The pilot and IRS-reflection design seeks the smallest τ3 for which the corresponding matrix V has rank (K −1)N.Source: C. Phase III: Reflecting Channel… · p. 4

Minimum Phase III duration

  • For M < N, Theorem 2 handles the antenna-limited regime by partitioning IRS elements and designing pilots and reflections across the resulting groups.The construction uses ρ = ⌊N/M⌋ and υ = N − Mρ to organize the element sets used during Phase III.Source: C. Phase III: Reflecting Channel… · p. 5

Overall training consequence

  • As M →∞, the total pilot length becomes 2K + N −1, making the training overhead linear in K and N despite KMN + KM unknown channel coefficients.This contrasts with conventional uplink channel estimation without IRS, whose minimum pilot length is independent of the number of receive antennas.Source: C. Phase III: Reflecting Channel… · pp. 5–6

V. NUMERICAL EXAMPLES

The numerical examples compare the proposed three-phase protocol with a two-phase benchmark under a fixed IRS-user geometry and channel model. As the BS antenna count increases, the proposed strategy substantially reduces the pilot sequence length, especially at M = 64.

Simulation setup

  • The simulations use K = 8 users and an IRS with N = 32 reflecting elements.Source: §V · p. 6
  • The numerical setup models path loss for direct, user-to-IRS, and IRS-to-BS channels using factors α1 = 4.2, α2 = 2.1, and α3 = 2.2.Source: §V · p. 6

Benchmark strategy

  • The benchmark estimates direct channels first, then estimates each user's reflected channel separately without exploiting the relation in (9).Source: §V · p. 6

Performance comparison

  • The proposed strategy substantially shortens the minimum pilot sequence length relative to the benchmark when M = 64.The comparison varies the BS antenna count from M = 1 to M = 64.Source: §V · p. 6

VI. CONCLUSION

The paper proposes a three-phase uplink protocol that estimates many IRS-assisted channel coefficients with few pilots by exploiting the common BS-side IRS reflection channel. Channel reciprocity further supports obtaining downlink CSI through TDD operation.

VI. CONCLUSION

  • The three-phase protocol accurately estimates a large number of IRS-assisted uplink channel coefficients using merely a small number of pilot symbols.Its key enabling observation is that each IRS element reflects all users' signals to the BS through the same channel.Source: §VI · p. 6
  • TDD is favorable because channel reciprocity allows downlink CSI to be obtained from the proposed uplink channel estimation strategy.Source: §VI · p. 6

A. Proof of Theorem 1

For M ≥ N, the proof establishes that Phase III needs exactly K − 1 pilot symbols to uniquely recover the remaining reflected-channel parameters.

A. Proof of Theorem 1

  • With τ3 = K − 1, the received observations can be written as [g1,1, · · · , g1,N]λk for users k = 2, · · · , K.Source: A. Proof of Theorem 1 · p. 6
  • The estimation equations become a linear system whose unknowns are recovered using the linear independence of g1,n with probability one.The pseudo-inverse of [g1,1, · · · , g1,N] exists under M ≥ N.Source: A. Proof of Theorem 1 · p. 6
  • When M ≥ N, a unique solution requires τ3 ≥ K − 1, and τ3 = K − 1 is sufficient for perfect estimation.The sufficient construction sets ak,i and φn,i according to (30) and (31).Source: A. Proof of Theorem 1 · p. 6

B. Proof of Theorem 2

For M < N, the proof uses the BS antennas to recover groups of reflected-channel parameters across Phase III, requiring τ3 = ceil((K − 1)N/M) pilot symbols.

B. Proof of Theorem 2

  • The lower bound follows because the linear system has (K − 1)N variables but only τ3M equations.Source: B. Proof of Theorem 2 · p. 7
  • The construction first schedules individual users with M active IRS elements, then jointly transmits selected users' pilots while canceling already estimated interference.Source: B. Proof of Theorem 2 · p. 7
  • The grouped construction relies on any M of the N vectors g1,n being linearly independent with probability one under i.i.d. Rayleigh fading.Source: B. Proof of Theorem 2 · p. 7
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