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Intelligent Reflecting Surface-Assisted Millimeter Wave Communications: Joint Active and Passive Precoding Design
Peilan Wang, Jun Fang, Xiaojun Yuan, Zhi Chen, Huiping Duan, Hongbin Li
TL;DR
MmWave links provide high rates but are vulnerable to path loss and blockage, motivating IRS-assisted reflected paths. The paper jointly optimizes BS and IRS precoding, deriving closed-form or near-optimal analytical solutions from mmWave channel structure. The solutions exhibit quadratic scaling with reflecting elements, while simulations show near-optimality and improved blockage robustness.
Problem
MmWave communications are vulnerable to severe path loss and blockage, motivating effective reflected paths for reliable coverage.
Method
The paper jointly optimizes the BS transmit precoder and IRS phase shifts for single- and multi-IRS systems using mmWave channel characteristics.
Results
The analysis shows received signal power increases quadratically with reflecting elements, while simulations verify single-IRS optimality and multi-IRS near-optimality.
Takeaways & Limitations
IRSs can create effective virtual LOS paths and substantially improve robustness against mmWave blockages.
Abstract
from arXiv · showhide
Millimeter wave (MmWave) communications is capable of supporting multi-gigabit wireless access thanks to its abundant spectrum resource. However, the severe path loss and high directivity make it vulnerable to blockage events, which can be frequent in indoor and dense urban environments. To address this issue, in this paper, we introduce intelligent reflecting surface (IRS) as a new technology to provide effective reflected paths to enhance coverage of mmWave signals. In this framework, we study joint active and passive precoding design for IRS-assisted mmWave systems, where multiple IRSs are deployed to assist the data transmission from a base station (BS) to a single-antenna receiver. Our objective is to maximize the received signal power by jointly optimizing the transmit precoding vector at the BS and the phase shift parameters used by IRSs for passive beamforming. Although such an optimization problem is generally non-convex, we show that, by exploiting some important characteristics of mmWave channels, an optimal closed-form solution can be derived for the single IRS case and a near-optimal analytical solution can be obtained for the multi-IRS case. Our analysis reveals that the received signal power increases quadratically with the number of reflecting elements for both the single IRS and multi-IRS cases. Simulation results are included to verify the optimality and near-optimality of our proposed solutions. Results also show that IRSs can help create effective virtual LOS paths and thus substantially improve robustness against blockages in mmWave communications.
I. INTRODUCTION
MmWave communications offer high data rates but are vulnerable to severe path loss and blockage. This paper studies IRS-assisted joint precoding and exploits mmWave channel structure to derive analytical solutions and improve blockage robustness.
- Motivation: MmWave communication offers gigabits-per-second rates through abundant bandwidth but suffers severe path loss and narrow-beam blockage.Small obstacles can block links, especially indoors and in dense urban environments.
- Motivation: IRSs use software-controlled passive elements with independently adjustable phase shifts to coherently enhance desired reflected signals.The surfaces are planar arrays of reconfigurable elements made from metamaterials.
- Approach: The paper maximizes received signal power by jointly designing the BS transmit precoder and IRS phase shifts for single- and multi-IRS systems.It focuses on single-stream transmission, including scenarios involving a single user antenna or rank-deficient cascade channels.
- Research gap: Prior joint BS-IRS optimization methods for conventional microwave systems are generally sub-optimal, lack analytical solutions, and can have high computational complexity.The paper revisits this problem for mmWave systems with multiple IRSs assisting a single-antenna user.
- Approach: An approximately rank-one BS-IRS channel enables a closed-form optimal solution for one IRS and a near-optimal analytical solution for multiple IRSs.The rank-one approximation can achieve nearly the same received signal power as a model including NLOS paths.
III. JOINT PRECODING DESIGN FOR SINGLE IRS
For a single IRS, the paper exploits the rank-one BS-IRS channel to decouple phase optimization from the BS precoder. This yields a closed-form optimal design whose computation scales with the larger of the IRS-element and BS-antenna counts.
- Rank-one reformulation: Substituting the rank-one channel G = λabT transforms the single-IRS objective into a form suitable for closed-form optimization.The derivation separates the IRS phase vector from the remaining scalar phase and BS precoder variables.
- IRS phase design: The optimal normalized IRS phase vector is independent of the scalar phase α and BS precoding vector w.Its entries align the relevant complex terms so the objective reaches its maximum ∥g∥1.
- BS precoding design: For fixed α, maximum ratio transmission gives the optimal BS precoding vector w.The resulting problem is then reduced to optimizing α.
- Closed-form solution: The optimal scalar phase α is obtained from the reduced one-dimensional optimization, after which the optimal precoder and diagonal IRS phase matrix follow by substitution.The final IRS matrix is constructed from the optimized phase vector and scalar phase.
- Complexity: O(max(M, N)) computational complexity is sufficient to calculate the closed-form solution under the rank-one BS-IRS assumption.Only bT h_d and g need to be computed.
B. Power Scaling Law
Under a rank-one BS-IRS channel model, the optimal single-IRS beamforming solution yields an average received power that scales quadratically with reflecting elements M.
- Assumptions: The rank-one geometric BS-IRS channel model assumes correlated fading for the IRS-user and direct BS-user channels.The model specifies hr ∼ CN(0,̺ 2 rI), hd ∼ CN(0,̺ 2 dI), and a rank-one BS-IRS channel.
- Optimal solution: The optimal solution provides a closed-form expression for the average received power at the user.The expression is given under the stated rank-one channel assumptions.
- Scaling law: The average received signal power scales quadratically with the number of reflecting elements M.The squared improvement combines an IRS-user transmit beamforming gain of M with an additional BS-IRS coherent collection gain of M.
- Implication: Increasing M is identified as a promising way to compensate for significant mmWave path loss.This conclusion follows from the quadratic received-power scaling under the optimal beamforming solution.
IV. JOINT PRECODING DESIGN FOR MULTIPLE IRSS
For multiple IRSs, the joint active and passive precoding problem is non-convex, but mmWave channel structure enables a near-optimal analytical design.
- Problem formulation: Multiple IRSs require jointly designing the BS precoding vector and the phase-shift matrices of all K IRSs.The multi-IRS problem is more challenging than the single-IRS case because these variables must be optimized jointly.
- Passive precoding: The passive phase design can be decomposed into independent IRS subproblems after aligning the relevant complex phases.The optimization over each IRS phase vector reaches its maximum when the involved complex-number arguments are identical.
- Analytical solution: Near-orthogonality among BS array response vectors enables a near-optimal analytical solution for the multi-IRS problem.The resulting phase vector is used to obtain the BS precoder and each IRS phase-shift matrix.
- Active precoding: For fixed passive phases, the optimal BS precoder is the maximum-ratio transmission solution.The resulting precoder is substituted into the objective to reduce the optimization problem.
- Non-convex formulation: The multi-IRS optimization is non-convex because the phase vector entries are constrained to the unit circle.The problem is formulated as a non-convex QCQP after introducing an auxiliary variable.
1) A SDR-Based Approach for Solving (30):
The SDR approach converts the non-convex unit-modulus phase problem into a convex semidefinite program, while the paper also motivates an analytical alternative using array-response orthogonality.
- SDR formulation: The unit-modulus phase constraint is lifted by defining V = ¯v¯vH and relaxing its rank-one constraint.This transforms the original formulation into a semidefinite relaxation.
- SDR properties: The relaxed problem is a standard convex semidefinite program solvable with tools such as CVX.Its computational complexity is of order O((K + 1)^6), and the relaxed optimum is not generally rank one.
- Solution recovery: A rank-one solution can be recovered from the higher-rank SDR solution using procedures described in prior work.The relaxation therefore requires an additional recovery step when the SDP solution is not rank one.
- Analytical alternative: Near-orthogonality of steering vectors is used to develop a near-optimal analytical solution without the SDR’s computational expense.For a uniform linear array, the steering-vector inner product depends on antenna spacing, wavelength, and departure angle.
- Implementation: The analytical design computes u = Φh_d and g_k, with dominant complexity O(max(KN, M)).The resulting phase vector determines the BS precoder and the IRS phase-shift matrices.
B. Power Scaling Law
For multiple IRSs, the proposed near-optimal design preserves quadratic received-power scaling with reflecting elements, while finite phase resolution introduces a quantization-dependent constant loss.
- Assumptions: The multi-IRS analysis assumes correlated IRS-user fading and rank-one geometric BS-IRS channels.The BS-IRS model uses LOS gains and normalized IRS and BS array-response vectors.
- Power expression: The near-optimal analytical solution has an average received-power expression under these channel assumptions.The proposition characterizes the power achieved in the general multi-IRS setup.
- Scaling law: The average received signal power scales quadratically with the number of reflecting elements M.The multi-IRS result matches the quadratic scaling established for the single-IRS case.
- Multiple-IRS effect: Multiple IRSs contribute additively to average received signal power, indicating improved performance from deploying multiple IRSs.The received power is described as a sum of the powers contributed by multiple IRSs.
- Discrete phases: Finite phase shifters are modeled by selecting each IRS phase from a discrete set rather than allowing arbitrary values.Each phase is chosen as the discrete value closest to its optimal or near-optimal continuous-phase value.
- Resolution dependence: η(b) increases monotonically with b and approaches 1 as b →∞.Thus, increasing phase-shifter resolution reduces the constant quantization loss.
- Quantization impact: η(1) = 0.4053, η(2) = 0.8106 and η(3) = 0.9496 quantify the receive-power factor for b-bit phase shifters.Compared with infinite-resolution phase shifters, the proposed discrete-phase solution loses a constant factor depending on b.
VI. SIMULATION RESULTS
The simulations model a BS with a ULA and IRSs with URAs, using geometric millimeter-wave channel models for the links. Results are averaged over 1000 random channel realizations, with fixed noise power relating received SNR to received signal power.
- Simulation setup: The BS uses a ULA with N antennas, while each IRS uses a URA with M = MyMz reflecting elements.My and Mz denote the horizontal and vertical element counts, respectively.
- Channel models: The BS-user channel is generated using a geometric channel model with multipath gains, departure angles, and normalized transmit array responses.The complex path gain parameters follow a complex Gaussian distribution.
- Channel models: The IRS-user and BS-IRS channels use geometric SV models for LOS scenarios, incorporating LOS and NLOS paths with normalized array responses.The BS-IRS model includes azimuth and elevation arrival angles, departure angles, and LOS-associated gains.
- Channel models: The BS-IRS channel uses LOS measurement parameters a = 61.4, b = 2, and σξ = 5.8dB.These values are stated as coming from real-world LOS channel measurements.
- Evaluation procedure: All simulation results are averaged over 1000 random channel realizations, and fixed noise power makes average received SNR differ from average received signal power by a constant.
A. Results for Single IRS
For a single IRS, the proposed continuous-phase solution nearly reaches the receive-SNR upper bound, while IRS deployment improves coverage and the SNR scales quadratically with reflecting elements. Low-resolution phase shifts retain much of the continuous-phase performance.
- Simulation setup: The simulations evaluate average receive SNR against BS-user distance and reflecting-element count under a single-IRS geometric-channel setup.The IRS is positioned with BS-IRS horizontal distance d1 = 119 m and vertical separation dv = 0.6 m.
- Optimality and coverage: The continuous-phase solution nearly achieves the upper bound of average receive SNR, validating the closed-form solution's optimality.The comparison also suggests that ignoring NLOS paths in the BS-IRS channel has little performance impact.
- Optimality and coverage: IRSs substantially enhance signal coverage as BS-user distance increases, compared with the rapidly declining SNR of a system without IRSs.The conventional system uses optimal maximum-ratio transmission, whereas the IRS-assisted schemes use the proposed precoding solutions.
- Phase-shift resolution: 2-bit phase shifts achieve average receive SNR close to infinite-precision phase shifters in the distance experiment.The analyzed average SNR losses are η(1) = −3.9224 dB and η(2) = −0.9121 dB, with simulations matching the theoretical result.
- Scaling with reflecting elements: 6 dB: increasing reflecting elements from M = 300 to M = 600 produces an approximately 6 dB receive-SNR gain, consistent with quadratic scaling.The experiment fixes My = 20 and increases Mz at d = 119 m.
APPENDIX A PROOF OF PROPOSITION 1
The proof derives the received signal power expression by evaluating the expectations of intermediate random quantities and combining the resulting identities. It concludes by reaching the stated proposition.
- The proof begins from the received signal power expression under the optimal active and passive beamforming solution.
- It evaluates the mean and variance of the modulus of entries of h_r under a circularly symmetric complex Gaussian model.
- The derivation separately computes E[z], E[z^2], and E[b^T h_d] before combining the intermediate results.
- Combining the referenced identities yields the proposition and completes the proof.
APPENDIX B PROOF OF PROPOSITION 2
The proof analyzes the received signal power for the analytical active and passive beamforming solution. It derives expectations for the relevant random variables and combines them to obtain the stated result.
- The proof starts from the received signal power expression associated with the analytical beamforming solution.
- It computes the expectation of |z_k| using the Gaussian model for h_rk.
- The derivation then evaluates the remaining intermediate quantities, including |u_k|, through successive identities.
- Combining the intermediate equations yields the stated proposition and completes the proof.
APPENDIX C PROOF OF PROPOSITION 3
The proof rewrites the analytical solution, models phase-shift discretization errors, and evaluates the resulting average received power. It then derives the asymptotic ratio in equation (45).
- The proof rewrites equation (43) to facilitate analysis of the analytical beamforming solution.
- The average received signal power is expanded using definitions for the intermediate diagonal quantities and the entries of s.
- Discretization errors are modeled as independent uniformly distributed random variables because the discrete phase values are uniformly spaced.
- The proof computes the first and second moments of the resulting intermediate variables before obtaining the average received power.
- The phase-shift relation is specified as k = −arg(u_k) = −arg(s_k).
- It concludes by deriving the ratio of γ(b) to γ(∞) as M approaches infinity.