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Intelligent Reflecting Surface Meets OFDM: Protocol Design and Rate Maximization

Yifei Yang, Beixiong Zheng, Shuowen Zhang, Rui Zhang

arXiv:1906.09956v2cs.IT

TL;DR

The paper addresses IRS-enhanced OFDM over frequency-selective channels, where practical channel estimation and joint rate design are challenging. It groups adjacent IRS elements, estimates group channels, and alternately optimizes transmit power and IRS coefficients. The proposed design improves achievable rates and reveals an optimal grouping size balancing training overhead with beamforming flexibility.

  • Problem

    IRS research largely assumes frequency-flat channels and perfect transmitter CSI, motivating practical estimation and rate design for frequency-selective OFDM.

  • Method

    Adjacent IRS elements share reflection coefficients and combined channels are estimated, followed by alternating optimization of transmit power and IRS coefficients with customized initialization.

  • Results

    The proposed protocol and algorithm improve achievable rate over a straightforward no-IRS counterpart under various practical setups.

  • Takeaways & Limitations

    IRS grouping provides a practical rate-enhancement design that trades training overhead against passive beamforming flexibility.

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) is a promising new technology for achieving both spectrum and energy efficient wireless communication systems in the future. However, existing works on IRS mainly consider frequency-flat channels and assume perfect knowledge of channel state information (CSI) at the transmitter. Motivated by this, in this paper we study an IRS-enhanced orthogonal frequency division multiplexing (OFDM) system under frequency-selective channels and propose a practical transmission protocol with channel estimation. First, to reduce the overhead in channel training and estimation and to exploit the channel spatial correlation, we propose a novel IRS elements grouping method, where each group consists of a set of adjacent IRS elements that share a common reflection coefficient. Based on this grouping method, we propose a practical transmission protocol where only the combined channel of each group needs to be estimated, thus substantially reducing the training overhead. Next, with any given grouping and estimated CSI, we formulate the problem to maximize the achievable rate by jointly optimizing the transmit power allocation and the IRS passive array reflection coefficients. Although the formulated problem is non-convex and thus difficult to solve, we propose an efficient algorithm to obtain a high-quality suboptimal solution for it, by alternately optimizing the power allocation and the passive array coefficients in an iterative manner, along with a customized method for the initialization. Simulation results show that the proposed design significantly improves the OFDM link rate performance as compared to the case without using IRS. Moreover, it is shown that there exists an optimal size for IRS elements grouping which achieves the maximum achievable rate due to the trade-off between the training overhead and IRS passive beamforming flexibility.

I. INTRODUCTION

The paper develops an IRS-enhanced OFDM design for frequency-selective channels with practical channel estimation, grouping, and joint rate optimization. Simulations report improved rates and an optimal grouping size balancing training overhead against beamforming flexibility.

  • Motivation: IRS offers reconfigurable passive reflection for enhancing wireless links without consuming energy or additional time/frequency resources for signal retransmission.Its elements can adjust phase shifts and attenuations to constructively enhance received power or destructively suppress interference.
  • Research gap: Existing IRS studies mainly assume perfect CSI, while passive IRS channel acquisition remains challenging and training can become costly with many elements.Without IRS sensors, individual BS–IRS and IRS–user channels cannot be explicitly obtained, and codebook-based alternatives may require time-consuming searches.
  • Research gap: Frequency-selective broadband operation requires common IRS reflection coefficients to accommodate multiple delayed reflected and direct paths, while rate also depends jointly on OFDM power allocation.These coupled design variables create an optimization problem whose size increases with the number of IRS elements and OFDM subcarriers.
  • Proposed design: Adjacent IRS elements are grouped to share a reflection coefficient, so only each group’s combined channel is estimated, reducing training overhead and coefficient design cost.The grouping exploits spatial correlation among tightly packed adjacent elements and supports a practical pilot-training protocol.
  • Optimization: The achievable-rate problem jointly optimizes transmit power and IRS coefficients using an iterative alternating algorithm with customized initialization for the non-convex formulation.The alternating procedure is reported to converge and to produce high-quality suboptimal solutions.
  • Results: The proposed designs improve achievable rates over systems without IRS or with random reflection coefficients under both perfect and estimated CSI.An optimal grouping size is reported because training overhead and passive beamforming flexibility must be balanced for a given channel coherence time.

II. SYSTEM MODEL AND IRS ELEMENTS GROUPING DESIGN

The paper models a single-user IRS-enhanced OFDM link under quasi-static frequency-selective multipath channels and introduces adjacent-element grouping to reduce CSI acquisition overhead while retaining configurable reflections.

  • System model: The system uses single-antenna BS and user nodes, an IRS with M reflecting elements, and N orthogonal OFDM subcarriers.The direct and reflected links are represented with multipath channel taps and a cyclic prefix covering the maximum channel length.
  • System model: Each IRS element applies a complex coefficient φ_m = β_me^jθ_m with amplitude β_m ∈ [0, 1] and phase shift θ_m ∈ [−π, π).The reflected composite channel combines the BS–IRS channel, element reflection, and IRS–user channel.
  • Motivation: Accurate CSI for the direct channel h_d and composite reflected channel V is needed for coherent detection and joint rate design, but estimating V becomes costly as M grows.The composite-channel dimension scales linearly with the number of IRS elements, which can reach thousands.
  • IRS elements grouping: Adjacent, tightly packed IRS elements are grouped into K blocks that share a common reflection coefficient, with equal group size B = M/K and grouping ratio ρ = K/M.A smaller grouping ratio means more elements per group.
  • IRS elements grouping: Grouping changes the estimated reflected-channel matrix from size N × M to N × K, which is smaller whenever K ≤ M.The resulting grouped formulation equals the ungrouped model when K = M.

III. TRANSMISSION PROTOCOL WITH IRS ELEMENTS GROUPING

The proposed transmission protocol divides each channel-coherence block into channel estimation, parameter optimization, and data transmission phases using the grouped IRS representation.

  • Protocol overview: Each channel-coherence block contains three transmission phases, with duration normalized by the OFDM symbol duration.The block is characterized by channel coherence time T_c.
  • Channel estimation: During the first phase, K + 1 pilot symbols let the BS estimate the direct channel and each group’s composite reflected channel.The grouped channel estimates replace per-element reflected-channel estimation.
  • Optimization and feedback: In the second phase, the BS uses the estimated channels to optimize OFDM transmit powers and IRS group reflection coefficients for achievable rate.The optimized parameters are then fed back to the user and IRS controller.
  • Data transmission: The third phase performs data transmission using the optimized power allocation, group reflections, and estimated channels.

A. Pilot Transmission and Channel Estimation

Pilot training activates IRS groups in separate configurations so the BS can estimate the direct and grouped reflected channels with reduced training overhead.

  • Pilot transmission: The first pilot symbol switches all IRS elements off to isolate the BS–user direct channel.The pilot is received in the frequency domain after cyclic-prefix removal.
  • Pilot transmission: Each subsequent pilot activates only one IRS group with full reflection while the other groups remain off, producing K group-specific observations.The active-group configuration is represented by the unit vector e_k.
  • Channel estimation: Least-squares estimation recovers the direct channel and grouped composite reflected channels from the pilot observations.
  • Channel-estimation accuracy: The channel-estimation MSE is proportional to K and decreases with higher training transmit power or larger group size.Thus, fewer groups can reduce estimation error while also reducing training duration.
  • Training overhead: The proposed grouping reduces pilot training to T_p = K + 1 OFDM symbols instead of M + 1 when K < M.

B. Processing and Feedback

After estimating CSI, the BS optimizes the transmit powers and IRS group coefficients, then feeds these parameters back before data transmission.

  • Feedback: The optimized power vector and group reflection coefficients are fed back to the user and IRS controller, respectively.
  • Processing: Using estimated ĥ_d and V̂′, the BS optimizes transmit power allocation and IRS group reflection coefficients to maximize achievable rate.
  • Assumptions: The protocol accounts for processing and feedback delay τ_D while ignoring estimation error during parameter optimization.Estimation error is considered in the subsequent effect on transmission performance.

C. Data Transmission

Given grouping and channel estimates, the design jointly optimizes transmit power and IRS reflection coefficients to maximize achievable rate. Grouping trades estimation overhead and data-transmission time against reflection-design flexibility.

  • Channel-estimation error lowers the actual data-transmission achievable rate relative to the rate obtained without accounting for that error.
  • Reducing the number of groups decreases estimation error and increases effective data-transmission time, but reduces IRS reflection-coefficient design flexibility.
  • The optimal grouping ratio depends on channel coherence time because grouping balances training overhead against reflection-design flexibility.
  • The proposed alternating optimization iteratively updates power allocation and IRS coefficients, using a customized initial reflection-coefficient solution.
  • The achievable-rate problem jointly optimizes transmit power allocation and IRS reflection coefficients under a given grouping and estimated CSI.
  • The joint problem is non-convex because the objective is non-concave in the IRS coefficients and couples them with power allocation.

B. Power Allocation Optimization Given IRS Coefficients

With fixed IRS coefficients, the power-allocation subproblem uses water-filling; the IRS-coefficient subproblem is handled through successive convex approximation within an alternating algorithm.

  • Given IRS coefficients and CSI estimates, the optimal BS transmit power allocation follows the water-filling solution.The effective channel-to-noise power ratio is denoted by c_n, with c_u defining the cut-off CNR.
  • Each convex approximation is solved efficiently via CVX, and successive updates continue until the objective converges.
  • The IRS-coefficient subproblem is non-convex, so successive convex approximation is used to obtain a locally optimal solution.
  • The approximation introduces auxiliary variables and replaces a convex differentiable function with a first-order lower bound.
  • The convex subproblem has 2K + 3N optimization variables and overall complexity O(K^4.5N^3.5).
  • The overall alternating algorithm updates water-filling power allocation and IRS coefficients until convergence.
  • The objective is non-decreasing and upper-bounded, so the algorithm converges to at least a locally optimal solution.
  • Performance depends critically on the initial IRS reflection coefficients, motivating a customized initialization method.

D. Initialization Method

The initialization method maximizes effective channel power using semidefinite relaxation or lower-complexity successive alignment. Successive alignment achieves similar performance to SDR with generally lower complexity.

  • The IRS can increase link rate by strengthening the effective BS-IRS-user channel through an additional reflected channel path.
  • The initialization problem is a non-convex QCQP, approximated using semidefinite relaxation.
  • The relaxed problem is a convex SDP with complexity O(K^4.5), but solving it can be practically costly for large K.
  • When the relaxed solution has rank greater than one, Gaussian randomization generates candidate reflection coefficients and selects the one with the largest objective value.
  • The resulting initialization is supplied as the starting point for the alternating optimization algorithm.
  • Successive alignment optimizes one reflecting element's phase at a time while fixing the others, updating all elements sequentially.
  • The successive-alignment initialization has complexity O(I_SAKN) and generally lower complexity than SDR initialization.

V. NUMERICAL RESULTS

The numerical evaluation uses an OFDM system and frequency-selective channel models with specified delay spreads, fading assumptions, IRS geometry, and averaged channel realizations.

  • The evaluation studies the proposed protocol and algorithm designs through numerical results for downlink and uplink OFDM communications.
  • The OFDM system uses N = 64 subcarriers, while the direct BS-user link is modeled as Rayleigh fading.
  • The direct link has maximum delay spread L = 16 taps with zero-mean CSCG tap coefficients and a uniform power-delay profile.
  • The BS-IRS and IRS-user channels use maximum delay spreads L1 = 4 and L2 = 13 taps, respectively, with their first taps representing LoS paths.
  • The model sets ζBI = 3 dB and ζIu = −20 dB for the LoS-to-NLoS power ratios of the BS-IRS and IRS-user links.
  • The IRS model uses correlated element phases, element separation d = 0.01 m, and carrier wavelength λ = 0.0857 m at 3.5 GHz.
  • The direct-link power is normalized to P_d = 1, and α denotes the reflected-element average power relative to the direct link.
  • Results use Γ = 8.8 dB, Q = 50 randomizations, and averages over 100 independent channel realizations.

A. Performance of Proposed Algorithms with Perfect CSI

The proposed algorithms improve achievable rate over no-IRS and random-phase benchmarks under perfect CSI. Their convergence and gains increase with favorable IRS-assisted channel conditions.

  • Monotonic convergence is observed for all initialization methods, with the proposed CPM-based methods converging faster than random phases.The CPM-based methods and random-phase method ultimately reach the same converged rate of 1.3685 bps/Hz.
  • The SA-based CPM initialization with ISA = 1 converges in 41 iterations versus 42 for SDR-based initialization, at much lower complexity.Using ISA = 10 improves the initial rate and convergence rate to 9 iterations.
  • All IRS schemes outperform the scheme without IRS across the evaluated SNR values.The improvement is attributed to IRS-enhanced average channel power between the BS and user.
  • The proposed iterative and CPM-initialized schemes achieve higher achievable rates than the random-phase scheme.Careful reflection-coefficient design superposes the direct and reflected channels more constructively.
  • The proposed schemes’ rate gains over no IRS or random phase become more pronounced as the number of reflecting elements increases.Properly designed reflection coefficients harvest passive beamforming gain, whereas random phase grows more slowly with M.
  • As the reflected-to-direct link power ratio α increases, the proposed schemes’ gains over benchmarks increase drastically and exceed no-IRS performance when α is large.A nearby IRS can enhance the link rate for a cell-edge user even when the user is far from the BS.

B. Performance of Proposed Algorithms with Imperfect Channel Estimation

Under imperfect CSI, grouping affects the balance among training overhead, channel-estimation accuracy, and IRS beamforming flexibility. The best grouping ratio depends on SNR and channel coherence time, while proper grouping outperforms random phase.

  • At grouping ratio ρ = 1, channel-estimation errors cause greater sensitivity, especially at low-to-medium SNR.As SNR increases, the estimated-CSI rate approaches the perfect-CSI rate at ρ = 1.
  • At ρ = 1/25, the estimated-CSI performance gap from perfect CSI is much smaller across both high- and low-SNR regimes.Grouping activates more IRS elements per channel-estimation step, increasing receive SNR and estimation accuracy.
  • At high SNR, achievable rate generally first increases and then decreases with grouping ratio because beamforming flexibility is eventually outweighed by training overhead.For Tc = 2100, overhead is less significant and achievable rate keeps increasing with grouping ratio.
  • At low SNR, increasing grouping initially improves rate through higher IRS beamforming diversity, but larger ratios suffer from estimation error and overhead.The passive-beamforming gain is overwhelmed at large grouping ratios, producing a sharp rate decrease.
  • At high SNR, the optimal grouping ratio increases with channel coherence time.Longer coherence time compensates for longer training through more effective coherent path combination.
  • With proper grouping, the proposed algorithm outperforms random phase across SNR regimes, including extremely small coherence times.At low SNR, grouping is desirable because additional beamforming gain is limited relative to overhead and estimation error.

VI. CONCLUSION

The paper develops a grouped-IRS transmission protocol for single-user OFDM and efficient rate optimization under estimated channels. Numerical results show rate gains over no IRS, while several grouping, pilot, and multiuser extensions remain open.

  • The protocol groups adjacent IRS elements, estimates each group’s combined channel, and uses a common reflection coefficient per group.This design targets reduced training overhead while retaining IRS-assisted OFDM transmission.
  • For a given grouping, the paper jointly optimizes transmit power allocation and IRS reflection coefficients using estimated channels.The proposed optimization methods seek high-quality suboptimal solutions efficiently.
  • Numerical results show that IRS boosts the achievable rate of a cell-edge user.The proposed protocol outperforms the counterpart without IRS elements grouping under various practical setups.
  • The CPM-based initialization achieves rate performance close to the iterative method with much lower implementation complexity.This supports its suitability as a practical implementation choice.
  • The study fixes the grouping and training overhead during optimization rather than jointly optimizing grouping strategy and training sequence.More specialized pilot designs for IRS-aided systems are also identified as future work.
  • The paper considers a single-user system, leaving extensions to multiuser OFDMA with joint multiuser resource allocation for future work.
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