[Paper Review] Cramér-Rao Lower Bounds for Positioning with Large Intelligent Surfaces

Sha Hu, Fredrik Rusek+1 more|arXiv (Cornell University)|Feb 10, 2017

Advanced Wireless Communication Technologies7 references6 citations

TL;DR

This paper derives closed-form Cramér-Rao Lower Bounds (CRLB) for positioning accuracy using Large Intelligent Surfaces (LIS) in a line-of-sight, isotropic radiating environment. It shows that CRLB decreases quadratically with LIS surface area for x and y dimensions, and linearly for z when the terminal is on the central perpendicular line (CPL), with accurate approximations provided for off-CPL positions.

ABSTRACT

We consider the potential for positioning with a system where antenna arrays are deployed as a large intelligent surface (LIS). We derive Fisher-informations and Cramér-Rao lower bounds (CRLB) in closed-form for terminals along the central perpendicular line (CPL) of the LIS for all three Cartesian dimensions. For terminals at positions other than the CPL, closed-form expressions for the Fisher-informations and CRLBs seem out of reach, and we alternatively provide approximations (in closed-form) which are shown to be very accurate. We also show that under mild conditions, the CRLBs in general decrease quadratically in the surface-area for both the $x$ and $y$ dimensions. For the $z$-dimension (distance from the LIS), the CRLB decreases linearly in the surface-area when terminals are along the CPL. However, when terminals move away from the CPL, the CRLB is dramatically increased and then also decreases quadratically in the surface-area. We also extensively discuss the impact of different deployments (centralized and distributed) of the LIS.

Motivation & Objective

• To analyze the fundamental limits of positioning accuracy using Large Intelligent Surfaces (LIS) in a line-of-sight, isotropic propagation environment.
• To derive closed-form Cramér-Rao Lower Bounds (CRLB) for terminal positioning in three-dimensional space relative to the LIS.
• To provide accurate closed-form approximations for Fisher information and CRLB when the terminal is not on the central perpendicular line (CPL).
• To compare centralized versus distributed LIS deployments and evaluate their impact on positioning accuracy.
• To establish scaling laws for CRLB in terms of surface area, wavelength, and terminal position relative to the LIS.

Proposed method

• Derives the noiseless signal model for a terminal radiating isotropically to the LIS using free-space path loss and phase shift based on distance.
• Computes the Fisher information matrix by taking first-order derivatives of the signal with respect to the terminal's x, y, and z coordinates.
• Uses polar coordinate transformation and symmetry properties to simplify integrals over the LIS aperture for terminals on the CPL.
• Applies closed-form expressions for integrals of the form ∫∫ x² η^(-n/2) dx dy and ∫∫ η^(-n/2) dx dy over circular regions to derive exact CRLB for CPL positions.
• Develops effective approximations for off-CPL positions using the CPL results, validated numerically to show high accuracy.
• Compares centralized and distributed LIS deployments by modeling the LIS as a single large aperture or multiple smaller apertures, analyzing their impact on CRLB via numerical integration and approximation.

Experimental results

Research questions

• RQ1 What is the theoretical lower bound on positioning accuracy for a single-antenna terminal using a Large Intelligent Surface (LIS) in a line-of-sight environment?
• RQ2 How do the Cramér-Rao Lower Bounds (CRLB) scale with the surface area of the LIS for different terminal positions?
• RQ3 What are the fundamental differences in positioning accuracy between centralized and distributed LIS deployments?
• RQ4 How accurate are closed-form approximations for CRLB when the terminal is not located on the central perpendicular line (CPL) of the LIS?
• RQ5 What scaling laws govern the dependence of CRLB on wavelength, surface area, and terminal distance from the LIS?

Key findings

• For terminals on the central perpendicular line (CPL), the CRLB for the x and y dimensions decreases quadratically with increasing LIS surface area.
• For terminals on the CPL, the CRLB for the z-dimension (distance from LIS) decreases linearly with surface area.
• When terminals move off the CPL, the CRLB increases significantly but still decreases quadratically with surface area, indicating a strong benefit from larger apertures even off-axis.
• The CRLB scales approximately as λ² with wavelength, favoring LIS over optical systems despite larger λ.
• Distributed LIS deployments can outperform centralized ones for x and y dimensions when the total surface area is below a certain threshold, due to improved spatial diversity.
• Approximations for off-CPL positions based on CPL results show negligible normalized error, confirming high accuracy for practical use.