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Beyond Massive-MIMO: The Potential of Positioning with Large Intelligent Surfaces

Sha Hu, Fredrik Rusek, Ove Edfors

arXiv:1705.06860v1cs.IT

TL;DR

The paper studies LIS-based terminal positioning and derives closed-form CRLBs on the CPL, with accurate approximations elsewhere. It finds that CRLB scaling depends on terminal position and unknown phase uncertainty.

  • Problem

    The paper examines the potential of using large intelligent surfaces for terminal positioning beyond massive MIMO.

  • Method

    It derives closed-form Fisher-information and CRLB expressions for terminals on the central perpendicular line and accurate CRLB approximations for general terminal positions.

  • Results

    CRLBs generally decrease quadratically with LIS surface area, while the z-dimension CRLB on the CPL decreases linearly; unknown phase increases CRLBs and changes the scaling to third order.

  • Takeaways & Limitations

    Positioning accuracy depends on both terminal location relative to the CPL and phase knowledge in the LIS analog circuits.

Abstract

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We consider the potential for positioning with a system where antenna arrays are deployed as a large intelligent surface (LIS), which is a newly proposed concept beyond massive-MIMO where future man-made structures are electronically active with integrated electronics and wireless communication making the entire environment \lq\lq{}intelligent\rq\rq{}. In a first step, we derive Fisher-information and Cramér-Rao lower bounds (CRLBs) in closed-form for positioning a terminal located perpendicular to the center of the LIS, whose location we refer to as being on the central perpendicular line (CPL) of the LIS. For a terminal that is not on the CPL, closed-form expressions of the Fisher-information and CRLB seem out of reach, and we alternatively find approximations of them which are shown to be accurate. Under mild conditions, we show that the CRLB for all three Cartesian dimensions ($x$, $y$ and $z$) decreases quadratically in the surface-area of the LIS, except for a terminal exactly on the CPL where the CRLB for the $z$-dimension (distance from the LIS) decreases linearly in the same. In a second step, we analyze the CRLB for positioning when there is an unknown phase $\varphi$ presented in the analog circuits of the LIS. We then show that the CRLBs are dramatically increased for all three dimensions but decrease in the third-order of the surface-area. Moreover, with an infinitely large LIS the CRLB for the $z$-dimension with an unknown $\varphi$ is 6 dB higher than the case without phase uncertainty, and the CRLB for estimating $\varphi$ converges to a constant that is independent of the wavelength $λ$. At last, we extensively discuss the impact of centralized and distributed deployments of LIS, and show that a distributed deployment of LIS can enlarge the coverage for terminal-positioning and improve the overall positioning performance.

I. INTRODUCTION

The paper examines LIS-based terminal positioning beyond massive MIMO, deriving CRLB results for central and general terminal locations, phase uncertainty, and deployment layouts. It shows that positioning accuracy scales favorably with LIS area, while unknown phase and deployment choices materially affect performance.

  • LIS concept: A Large Intelligent Surface extends massive MIMO by making the surrounding environment electronically active for communication and sensing.LIS structures can transmit and receive signals while focusing energy in three-dimensional space.
  • Positioning analysis: The paper derives closed-form Fisher-information and CRLB expressions for terminals on the central perpendicular line, and accurate approximations for other locations.The CPL is perpendicular to the LIS and crosses its center; general off-CPL closed forms appear out of reach.
  • Scaling laws: Under mild conditions, CRLBs generally decrease quadratically with LIS surface-area, while a CPL terminal’s z-dimension CRLB decreases linearly.The wavelength impact scales as approximately λ2, and a larger aperture can compensate for LIS’s comparatively large wavelength.
  • Phase uncertainty: Unknown phase ϕ in LIS analog circuits increases CRLBs for all dimensions, whose general scaling changes to the third order of surface-area.For an infinitely large LIS, the z-dimension CRLB with unknown ϕ is 6 dB higher than with known ϕ, while the phase CRLB converges to a constant.
  • Deployment layouts: Distributed LIS deployments can extend positioning range and improve average CRLB relative to a centralized deployment under the stated distance condition.Splitting one LIS into four smaller LISs can provide these benefits; further splitting into sixteen improves CRLB but may increase cooperation overhead.

II. SIGNAL MODEL WITH LIS

The signal model represents a terminal’s received signal across the LIS under ideal free-space propagation and integrates information over the surface. Fisher information is then obtained from spatial derivatives and area integrals, with closed forms generally limited to CPL placement.

  • The model assumes perfect line-of-sight, isotropic terminal radiation, narrow-band signaling, and ideal free-space propagation to every LIS point.
  • The received signal at each LIS location is modeled as a noiseless signal plus zero-mean white Gaussian noise.
  • The LIS integrates received signals over its full area, allowing the model to apply in both near-field and far-field scenarios.
  • Fisher-information entries are computed as double integrals over a disk-shaped LIS, with radius R and noise scaling normalized by setting N0 = 2.
  • Closed-form expressions are generally unavailable for arbitrary terminal positions, but exist when the terminal lies on the CPL; non-CPL cases use approximations.

III. CRLB OF A TERMINAL ON THE CPL

For terminals on the CPL, the paper derives closed-form Fisher-information and CRLB expressions and characterizes their dependence on wavelength, distance, and LIS surface-area. The scaling differs between lateral and distance coordinates.

  • The CPL case places the terminal at coordinates (0, 0, z0), where the Fisher-information matrix and CRLBs can be obtained in closed form.
  • When the terminal approaches the LIS, Fisher information becomes infinitely large and the CRLB becomes 0 for all dimensions.
  • The CRLBs for all dimensions depend on wavelength λ and the normalized surface-area parameter τ.
  • When the LIS radius is much larger than the terminal distance, the asymptotic CRLBs for all three dimensions become identical and depend solely on λ.
  • As τ is proportional to R2, the CRLBs for x and y decrease quadratically in LIS surface-area, while the z CRLB decreases linearly.

IV. CRLB OF A TERMINAL NOT ON THE CPL

For terminals away from the CPL, the paper replaces intractable closed-form expressions with approximations based on the CPL analysis. These approximations reveal degraded distance estimation and quadratic surface-area scaling across all dimensions.

  • For non-CPL terminals, complicated integrals make exact closed-form CRLB expressions unavailable, so the paper develops approximations based on CPL Fisher-information results.
  • The approximated Fisher-information and CRLB matrices are closed form, and the approximation is exact when x0 = y0 = 0.
  • Under the stated conditions, terminals at equal radial distance from the CPL have approximately equal x and y CRLBs and matching all-dimensional CRLBs for a given z0.
  • For small R, moving away from the CPL dramatically degrades the z-dimension CRLB, which can exceed the x- and y-dimension CRLBs.
  • The CRLBs for all three dimensions decrease quadratically in LIS surface-area for non-CPL terminals.

B. CRLB for AoA and Radius Estimations

The paper transforms Cartesian-position CRLBs into bounds for estimating radius and angular parameters, while noting a singularity at φ=ψ=0 for Cψ.

  • Coordinate transformation: The spherical parameters are radius z1 and angles φ and ψ, represented as a function g of Cartesian coordinates.The Jacobian ∇g maps derivatives with respect to (x0, y0, z0) into the spherical-parameter representation.
  • CRLB transformation: The CRLBs for z1, φ, and ψ are obtained from the Cartesian CRLBs through the Jacobian transformation.The resulting expressions use Cx and Cz from Property 2.
  • Scaling behavior: The CRLBs for z1, φ, and ψ generally decrease quadratically with the LIS surface-area.This conclusion is stated for the transformed spherical parameters in the general case.
  • Singularity: At φ=ψ=0, Cψ has a singularity.This point is identified as a special case in the angular-parameter analysis.

V. CRLB WITH PHASE UNCERTAINTY IN ANALOG CIRCUITS OF THE LIS

The paper analyzes positioning when the LIS analog circuits contain an unknown phase, extending the Fisher-information matrix and deriving its effects on Cartesian CRLBs. Phase uncertainty degrades positioning precision, changes surface-area scaling, and produces a 6 dB asymptotic penalty for z on the CPL.

  • Model and Fisher information: An unknown phase ϕ expands the Fisher-information matrix to four dimensions and degrades the CRLBs for all three Cartesian dimensions.The additional parameter introduces cross-terms between position and phase.
  • Model and Fisher information: Theorem 2 expresses the unknown-phase Fisher-information matrix using the known-phase position information and position–phase cross-terms.The vector i contains the cross-terms, while I0 is the known-phase Fisher information for x, y, and z.
  • General CRLB effect: With unknown ϕ, the Cartesian CRLBs acquire the additional term C0i^T C0Cϕ and are therefore dramatically degraded.The CRLBs for position remain independent of the true value of ϕ.
  • CPL case: On the CPL, unknown ϕ leaves the x- and y-dimension CRLBs unchanged but increases the z-dimension CRLB.Property 4 gives closed-form expressions for the CPL case.
  • CPL case: The asymptotic z-dimension CRLB with unknown ϕ is four times the known-phase value, corresponding to a 6 dB degradation.This result applies to a terminal on the CPL with an infinitely large LIS.
  • Surface-area scaling: For sufficiently small τ, the z-dimension CRLB decreases in the third order of LIS surface-area under unknown phase.The slope relative to surface-area lies between 1 and 3 across the stated cases.
  • Surface-area scaling: The z-dimension CRLB with phase uncertainty is independent of wavelength, so decreasing λ does not improve distance-estimation precision.This differs from the known-phase CRLB.
  • Phase estimation: The phase CRLB can be around 4000 times the z-dimension CRLB when λ=0.1 m and τ is small.The comparison is given as approximately 4π^2/λ^2=4000 under the stated conditions.

VI. DEPLOYMENT OF THE LIS

The paper compares centralized and distributed LIS deployments with equal total surface-area. Distributed layouts can improve horizontal positioning when separated sufficiently and improve overall robustness and average performance.

  • Deployment setup: The analysis compares centralized and distributed LIS deployments over a surface of width W and length H.The deployments are evaluated with the same total surface-area and under far-field assumptions.
  • Deployment setup: The distributed deployment splits the LIS into four smaller surfaces centered at (±W/4, ±H/4), each with radius R/2.The Fisher-information matrices of the four components are summed using LIS symmetry.
  • Horizontal positioning: In the far field, distributed deployment improves x- and y-dimension CRLBs when the four LISs are sufficiently far apart relative to radius R.Otherwise, centralized deployment gives lower x- and y-dimension CRLBs.
  • Horizontal positioning: With distributed deployment, x- and y-dimension CRLBs decrease linearly with surface-area rather than quadratically when R≪D.The result describes the large-separation regime for the distributed layout.
  • Overall performance: More small pieces can form an ultra-densely distributed deployment, whose overall positioning performance is more robust than centralized deployment.The paper reports improved average positioning performance in later numerical simulations.

VII. NUMERICAL RESULTS

The numerical-results section introduces tests used to illustrate the paper’s theories and conclusions, with fixed noise and unit conventions.

  • Experimental setup: The numerical results illustrate the theories and conclusions developed in the preceding sections.The section presents numerical tests rather than introducing a new analytical result.
  • Experimental setup: The tests use noise spectral density N0=2 unless otherwise specified.This fixes the noise level for the reported numerical evaluations.
  • Experimental setup: Coordinates, wavelength λ, and LIS radius R are measured in meters, while CRLB is measured in m2.These unit conventions apply unless explicitly stated otherwise.

A. Exact-CRLB Evaluations

The evaluations confirm the predicted CRLB scaling for terminals on and away from the CPL, while showing accurate closed-form approximations for off-CPL positioning. Unknown phase substantially worsens positioning bounds and changes their surface-area scaling.

  • Exact-CRLB Evaluations: When τ is small, CRLBs for x and y decrease quadratically with LIS surface area, whereas z decreases linearly.These results align with the corresponding analytical properties.
  • Exact-CRLB Evaluations: The z-dimension CRLB increases dramatically when the terminal is away from the CPL.
  • Exact-CRLB Evaluations: Normalized approximation errors remain below 0.5% for x and y and close to 1% for z.The approximations are compared with numerical integrations.
  • Exact-CRLB Evaluations: For terminals on a fixed parallel circle, z-dimension CRLB is angle-invariant, while x- and y-dimension CRLBs are almost identical.This corroborates Corollary 1.
  • Exact-CRLB Evaluations: With unknown phase ϕ, all CRLBs increase and begin decreasing in the third order of surface area beyond a threshold.For off-CPL terminals, the CRLB curves have similar shapes across all three dimensions.

D. CRLB with Centralized and Distributed Deployments of the LIS

Distributed LIS deployments improve positioning bounds relative to centralized deployment in the evaluated settings, particularly for small surface radii and uniformly distributed terminals. Their benefits include lower average CRLBs, more concentrated CRLB distributions, and potential coverage extension, but further splitting can increase calibration and cooperation requirements.

  • Centralized versus distributed deployments: For a terminal on the CPL with R≤2.31, distributed deployments lower x- and y-dimension CRLBs while z-dimension CRLB remains unchanged.Beyond this threshold, distributed deployment is worse for x and y but slightly better for z.
  • Average positioning performance: Distributed deployments significantly improve average CRLBs for all three Cartesian dimensions.The evaluation uses 1000 terminals uniformly distributed in x and y over [-2, 2], with z0=12.
  • Average positioning performance: Four small LISs with radius 0.005 match the average CRLB of one LIS with R=0.2 using only 0.25% of the centralized surface area.This comparison applies when R is small.
  • Deployment trade-offs: Further splitting four small LISs into sixteen smaller LISs provides marginal gains but likely requires stricter phase calibration and cooperation among the small LISs.
  • CRLB distributions: With four small LISs, CRLB values become more concentrated, indicating improved overall positioning performance.The x- and y-dimension CRLBs are relatively larger than the z-dimension CRLBs but more concentrated.
  • Paper-level conclusions: The paper reports closed-form CRLBs on the CPL and accurate closed-form approximations elsewhere; off-CPL CRLBs scale quadratically with surface area in all Cartesian dimensions.For CPL terminals, the z-dimension CRLB scales linearly instead.
  • Unknown phase: Unknown phase dramatically increases CRLBs, with all dimensions decreasing in the third order of surface area beyond a threshold.For infinitely large LISs, the unknown-phase z-dimension CRLB is 6 dB higher than with known phase, and phase-CRLB convergence is wavelength-independent.
  • Paper-level conclusions: Distributed LIS deployments have potential to extend terminal-positioning coverage and provide better average CRLBs for all dimensions.

APPENDIX A: PROOF OF THEOREM 1

For a terminal on the CPL, symmetry simplifies the Fisher-information analysis: cross-terms vanish and the matrix becomes diagonal. The theorem proof then derives the dimension-specific Fisher-information expressions by integrating the resulting terms.

  • APPENDIX A: PROOF OF THEOREM 1: On the CPL, x0=y0=0, and the even symmetry of η in x and y makes cross-terms vanish.
  • APPENDIX A: PROOF OF THEOREM 1: The Fisher-information matrix is therefore diagonal, with its diagonal entries corresponding to the positioning dimensions.
  • APPENDIX A: PROOF OF THEOREM 1: The proof calculates the dimension-specific Fisher-information terms using the CPL derivatives and the results in equations (12) and (13).
  • APPENDIX A: PROOF OF THEOREM 1: For off-CPL coordinates, the proof approximates the Fisher-information using the corresponding CPL result and symmetry-based integral comparisons.
  • APPENDIX A: PROOF OF THEOREM 1: Odd-function terms integrate to zero, and combining the resulting relations yields the Fisher-information matrix in equation (38).

APPENDIX C: PROOF OF THEOREM 2

The unknown phase does not alter the direct Fisher information for x, y, and z, but introduces cross-terms with the phase parameter. Substituting these terms produces the CRLB expressions for z and phase.

  • APPENDIX C: PROOF OF THEOREM 2: Because unknown phase ϕ appears only in exponential terms of the first-order derivatives, it does not appear in the Fisher-information matrix.
  • APPENDIX C: PROOF OF THEOREM 2: The Fisher-information for x, y, and z remains unchanged when phase ϕ is unknown.
  • APPENDIX C: PROOF OF THEOREM 2: The proof then computes cross-terms between ϕ and the Cartesian dimensions using arguments analogous to the off-CPL approximation analysis.
  • APPENDIX C: PROOF OF THEOREM 2: Substituting the relevant Fisher-information terms into the CRLB expressions yields the z-dimension and phase CRLBs.
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