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Location Information Aided Multiple Intelligent Reflecting Surface Systems

Xiaoling Hu, Caijun Zhong, Yu Zhang, Xiaoming Chen, Zhaoyang Zhang

arXiv:2008.09248v1cs.ITeess.SP

TL;DR

The paper addresses the practical requirement for instantaneous CSI in IRS beamforming by using imperfect location information to estimate LOS-path angles and design low-complexity beams. It derives achievable-rate expressions and reports power scaling with antennas and reflecting elements, including stronger scaling when NLOS paths vanish.

  • Problem

    IRS beamforming in practice requires instantaneous CSI.

  • Method

    The scheme estimates LOS-path angle information from imperfect location information and uses it for low-complexity beamforming and achievable-rate analysis.

  • Results

    The proposed beamforming scheme is superior to the joint optimization scheme in, with individual SINR proportional to M and to M^2 when NLOS paths vanish.

  • Takeaways & Limitations

    The analysis links IRS-system performance to the number of reflecting elements and the presence of NLOS components.

Abstract

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This paper proposes a novel location information aided multiple intelligent reflecting surface (IRS) systems. Assuming imperfect user location information, the effective angles from the IRS to the users are estimated, which is then used to design the transmit beam and IRS beam. Furthermore, closed-form expressions for the achievable rate are derived. The analytical findings indicate that the achievable rate can be improved by increasing the number of base station (BS) antennas or reflecting elements. Specifically, a power gain of order $N M^2$ is achieved, where $N$ is the antenna number and $M$ is the number of reflecting elements. Moreover, with a large number of reflecting elements, the individual signal to interference plus noise ratio (SINR) is proportional to $M$, while becomes proportional to $M^2$ as non-line-of-sight (NLOS) paths vanish. Also, it has been shown that high location uncertainty would significantly degrade the achievable rate. Besides, IRSs should be deployed at distinct directions (relative to the BS) and be far away from each other to reduce the interference from multiple IRSs. Finally, an optimal power allocation scheme has been proposed to improve the system performance.

I. INTRODUCTION

The paper addresses the high training, update, and BS–IRS communication overhead required by instantaneous CSI for multi-IRS beamforming. It proposes location-aided beamforming using estimated LOS angles and derives closed-form performance expressions and power control.

  • Challenges: Instantaneous CSI is difficult to acquire because cascaded-channel training overhead becomes prohibitively high as reflecting elements increase.Grouping elements reduces training overhead but forces identical phase shifts within groups, degrading passive beamforming performance.
  • Motivation and framework: Location information enables multi-IRS transmit and phase-shift beamforming using statistical CSI instead of full instantaneous CSI.The framework is designed to reduce training overhead, frequent updates, and the capacity required for BS–IRS information exchange.
  • Proposed design: Imperfect location information is used to estimate LOS-path angles and characterize the resulting estimation error.These estimates support a low-complexity beamforming design with closed-form transmit and phase-shift beams.
  • Results: The proposed beamforming scheme is superior to the joint optimization scheme in when individual user rate requirements are low.This comparison is reported for the proposed low-complexity beamforming scheme.
  • Analysis and optimization: Closed-form achievable-rate expressions enable analysis of location accuracy, reflecting-element count, antenna count, and the Rician K-factor.An optimal power-control scheme minimizes total transmit power subject to individual user rate constraints.

II. SYSTEM MODEL

The system comprises a multi-IRS downlink with a multi-antenna BS, single-antenna users, and IRS-assisted links modeled using angle-domain Rician fading. Each user is associated with an IRS, while the BS and IRSs use ULAs and exchange information through low-capacity hardware links.

  • The model uses one BS with N antennas to serve K single-antenna users, each assisted by an IRS with M reflecting elements.
  • The analysis assumes ULAs aligned along the y axis, no direct BS–user links, and low-capacity BS–IRS connections that exchange information such as CSI and phase shifts.
  • The BS–IRS and IRS–user channels include LOS and NLOS components represented through angle-domain Rician fading.
  • When users are numerous, assigning every user a unique IRS is infeasible, motivating assignments to poorly covered users or effective user scheduling.
  • The BS broadcasts user symbols using transmit beamforming vectors, while each IRS applies a diagonal phase-shift matrix to its reflecting elements.

III. LOCATION BASED ANGLE INFORMATION ACQUISITION

The paper uses estimated user locations to obtain effective IRS–user angles for beam design, while explicitly modeling location uncertainty. The resulting angle-estimation error depends on both uncertainty radius and IRS–user distance.

  • CSI acquisition is avoided by exploiting user location information and known IRS positions to estimate the angles needed for beam design.
  • User location errors are modeled within a sphere of radius Υ centered at the GPS-provided estimated location.
  • The m-th IRS calculates its effective departure angle toward user k from the estimated IRS–user geometry.
  • The angle-estimation error variance increases with the ratio of location uncertainty squared to IRS–user distance squared, so greater distance can compensate for uncertainty.

IV. DESIGN OF TRANSMIT AND PHASE SHIFT BEAMS

The paper designs transmit and IRS phase-shift beams from estimated angles using separate low-complexity local optimizations rather than solving the coupled global problem. It then derives a closed-form achievable-rate expression and identifies how system parameters affect performance.

  • Because transmit and phase-shift beams are coupled and the global optimization is non-convex, the paper uses separate low-complexity local optimizations with closed-form solutions.
  • The transmit beam aligns each user’s signal with its associated IRS using BS–IRS angle information and a power-control coefficient.
  • The IRS phase-shift beam is designed from estimated IRS–user angles to maximize the received signal through the associated IRS.
  • Theorem 2 provides a closed-form achievable-rate expression that quantifies effects from antenna number, user number, reflecting elements, Rician K-factor, and location uncertainty.
  • The individual achievable rate decreases with more users and greater location uncertainty, while increasing with the Rician K-factor because weaker K-factors produce more NLOS interference.
  • IRSs should use distinct BS-relative directions to reduce interference from multiple IRSs.

A. Ideal directions (relative to the BS) of IRSs

The ideal-direction analysis characterizes how IRS placement affects interference and signal strength. It shows that suitable directions can eliminate inter-IRS interference, while separation between IRSs and increased BS antenna count improve desired-signal conditions.

  • A. Ideal directions (relative to the BS) of IRSs: Proper BS-relative IRS directions can eliminate interference from other IRSs.
  • A. Ideal directions (relative to the BS) of IRSs: The desired signal is strengthened when users associate with nearby IRSs, whereas other IRSs placed farther from the user contribute less interference.
  • A. Ideal directions (relative to the BS) of IRSs: The desired signal power is proportional to the number of BS antennas, demonstrating the benefit of multiple antennas.
  • A. Ideal directions (relative to the BS) of IRSs: With vanishing location uncertainty, the achievable rate converges to a limit and the desired signal power scales as N M^2.

1) A large number of reflecting elements:

The achievable rate increases with reflecting elements and BS antennas, but antenna gains converge while large-element gains become less significant; without NLOS paths, SINR scaling improves.

  • 1) A large number of reflecting elements:: The achievable rate increases with both the number of reflecting elements and BS antennas.
  • 1) A large number of reflecting elements:: With many reflecting elements, SINR is proportional to the number of reflecting elements, while rate growth becomes logarithmic at large scale.
  • 1) A large number of reflecting elements:: As BS antennas grow, achievable rate converges to a limit mainly determined by reflecting elements and the IRS-user-channel Rician K-factor.
  • 1) A large number of reflecting elements:: Increasing antenna number removes BS-to-IRS NLOS interference, while increasing reflecting elements requires balancing deployment cost against achievable-rate gains.
  • 1) A large number of reflecting elements:: Without NLOS paths, SINR is mainly determined by reflecting elements and antenna number, with scaling that becomes stronger than in the general case.

4) The impact of user directions relative to a IRS:

User directions relative to different IRSs strongly affect interference, while the proposed power-control method minimizes transmit power under individual rate requirements using statistical CSI.

  • 4) The impact of user directions relative to a IRS:: An IRS causes interference nearly proportional to M^2 when users assisted by different IRSs have similar relative directions.
  • 4) The impact of user directions relative to a IRS:: Users in similar directions relative to an IRS suffer more interference from that IRS.
  • 4) The impact of user directions relative to a IRS:: The proposed low-complexity power-control algorithm minimizes transmit power subject to individual user-rate requirements.
  • 4) The impact of user directions relative to a IRS:: The power-control problem becomes infeasible above a rate threshold, and greater location uncertainty lowers that threshold because of increased interference.
  • 4) The impact of user directions relative to a IRS:: The power-control algorithm requires only statistical CSI, reducing CSI-acquisition overhead relative to methods requiring instantaneous CSI.

VII. SIMULATION RESULTS

Simulations validate the analytical results and show that location uncertainty, IRS geometry, reflecting elements, antennas, and Rician fading strongly affect sum rate. The proposed beamforming and power-control scheme outperforms the benchmark at modest rate constraints but degrades under stringent interference-limited requirements.

  • The numerical results match the analytical curves, validating the derived sum-rate expressions.
  • Impact of location uncertainty: 16 bits/s/Hz at Υ = 0.5 falls to 5.5 bits/s/Hz at Υ = 2 under ρd = 40dBm and vB2I = vI2U = 5.
  • Impact of IRS geometry: Orthogonal IRS directions achieve much higher sum rate than non-orthogonal directions because interference decreases.
  • Impact of IRS geometry: The worst sum rate occurs when IRSs lie midway between BS and users, while moving them toward either endpoint improves performance.
  • System scaling: The sum rate grows logarithmically with reflecting elements and increases with BS antennas before converging to a limit determined mainly by reflecting elements and the Rician K-factor.
  • Beamforming and power control: The proposed scheme is superior at modest desired rates but becomes inferior, and potentially infeasible, at stringent rates because its beamforming does not directly suppress interference.

VIII. CONCLUSION

The paper develops a location-information-aided multi-IRS system using estimated effective angles to design low-complexity beams and derive achievable-rate expressions. Its findings quantify scaling with antennas and reflecting elements, sensitivity to location uncertainty, IRS deployment guidance, and power control.

  • VIII. CONCLUSION: The estimated effective angles support low-complexity transmit-beam and IRS phase-shift designs with closed-form expressions for achievable rate.The approach derives angle-estimation statistics before designing the beams.
  • VIII. CONCLUSION: Achievable rate degrades significantly as user location uncertainty increases.The conclusion identifies location uncertainty as a major performance sensitivity.
  • VIII. CONCLUSION: IRSs should use distinct BS-relative directions and be deployed far apart to reduce interference among multiple IRSs.The analytical findings connect IRS direction correlation and separation with interference behavior.
  • VIII. CONCLUSION: An optimal power-control scheme minimizes total transmit power under individual rate constraints.The scheme is based on the proposed beamforming design.
  • VIII. CONCLUSION: Future work includes robust beamforming under location uncertainty and user scheduling or IRS-user pairing when users outnumber IRSs.These extensions target worst-case quality-of-service guarantees and multi-user resource assignment.

APPENDIX I

Appendix I derives the effective AOD estimation error from three-dimensional user-location errors and characterizes its distribution. It then obtains the mean and variance of the error under a spherical uncertainty model.

  • APPENDIX I: The effective AOD from an IRS to a user is expressed using the IRS and user locations, with coordinate errors defined along the x, y, and z axes.The distance and coordinate-error terms provide the geometric basis for the angle-error derivation.
  • APPENDIX I: The effective AOD estimation error is represented as a linear combination of the three coordinate errors.The coefficients determine how each location-error component contributes to the angular error.
  • APPENDIX I: Assuming location errors are uniformly distributed within a sphere of radius Υ, the appendix derives the error CDF and PDF and then obtains its mean and variance.The derivation proceeds through geometric integration over the spherical uncertainty region.

APPENDIX III

Appendix III derives the distribution and expectation of a composite phase-error term used in the analytical performance calculation. Combining intermediate results yields the desired result.

  • APPENDIX III: The composite error term ǫk,mn,sl is expressed as a linear combination of the user’s three coordinate-location errors.The coefficients capture the contribution of each coordinate to the composite error.
  • APPENDIX III: The appendix obtains the PDF of ǫk,mn,sl by following the derivation used for the earlier effective-AOD error distribution.This reuses the preceding distributional analysis rather than introducing a separate procedure.
  • APPENDIX III: Using the error distribution, the expectation ζy-I2U,k,mn,sl = E {ejπǫk,mn,sl} is calculated and combined with earlier results to complete the derivation.The intermediate expectations are assembled into the desired analytical result.
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