[Paper Review] Theoretical Accuracy Analysis of RSS-Based Range Estimation for Visible Light Communication

Changeez Amini, Paeiz Azmi+1 more|arXiv (Cornell University)|Nov 28, 2020

Optical Wireless Communication Technologies16 references4 citations

TL;DR

This paper proposes a refined theoretical model for RSS-based range estimation in visible light communication (VLC) by incorporating distance-dependent received signal noise (RSN) into the channel model. By deriving a closed-form Cramer-Rao Lower Bound (CRLB) that accounts for RSN's variance dependence on distance and transmitted power, the study demonstrates that RSN significantly degrades ranging accuracy—leading to higher CRLB values than previously assumed, especially at larger distances and lower transmitted powers.

ABSTRACT

In this paper, an improved channel model of visible light communication (VLC) for ranging in presented. For indoor channel model of VLC, distance is estimated based on received signal strength. In this model, received shot noise as a distance-dependent parameter is considered in range estimation accuracy. Moreover, based on this model, the Cramer-Rao lower bound is computed as the theoretical limits on the performance and accuracy of any unbiased estimator. In this way, the effects of horizontal and vertical distances are investigated. In addition, the transmitted power effect on RSN and accordingly on CRLB is demonstrated.

Motivation & Objective

• To address the lack of accurate theoretical modeling of noise in RSS-based VLC ranging, particularly the overlooked impact of received signal noise (RSN).
• To develop a more precise channel model that accounts for RSN's dependence on distance and transmitted power.
• To derive a closed-form Cramer-Rao Lower Bound (CRLB) that reflects the true theoretical limits of unbiased RSS-based range estimators in VLC.
• To investigate how geometric parameters—horizontal and vertical distances—and transmitted power affect ranging accuracy via CRLB.

Proposed method

• A two-dimensional VLC system model is defined with a line-of-sight (LOS) path, where the transmitter (LED) and receiver (photodetector) are positioned at different horizontal and vertical distances.
• The channel gain is modeled using Lambertian emission with order $ m $, and the total received optical power is expressed as $ x = R_p H(0) P_t + w $, where $ w $ includes all noise sources.
• The received signal noise (RSN) is explicitly modeled as distance-dependent, with its variance proportional to the received optical power, unlike constant noise sources.
• The Cramer-Rao Lower Bound (CRLB) is analytically derived by incorporating the RSN variance into the Fisher information matrix, yielding a closed-form expression for $ \text{CRLB}(d) $.
• Theoretical analysis is performed by varying parameters such as horizontal distance $ \ell $, vertical distance $ h $, transmitted power $ P_t $, and Lambertian order $ m $, while computing $ \sqrt{\text{CRLB}} $.
• Comparisons are made between the proposed RSN-aware CRLB and the conventional model that ignores RSN, using numerical simulations across different system configurations.

Experimental results

Research questions

• RQ1 How does the inclusion of distance-dependent received signal noise (RSN) affect the theoretical accuracy limit of RSS-based range estimation in VLC?
• RQ2 What is the closed-form expression for the Cramer-Rao Lower Bound (CRLB) when RSN is modeled as a function of distance and received power?
• RQ3 How do horizontal and vertical distances between the transmitter and receiver influence the CRLB in RSS-based VLC ranging?
• RQ4 What is the impact of transmitted optical power $ P_t $ on the CRLB, and how does this relationship change across different power regimes?
• RQ5 Does the Lambertian emission order $ m $ affect the CRLB, and if so, is there an optimal $ m $ that minimizes ranging error?

Key findings

• The inclusion of distance-dependent RSN increases the CRLB significantly compared to models that treat RSN as constant, with the difference becoming more pronounced at larger distances.
• For a given setup with $ P_t = 1\text{W} $, $ m = 1 $, and $ h = 1\text{m} $, $ \sqrt{\text{CRLB}} $ increases with horizontal distance $ \ell $, and the new model yields higher bounds than the conventional approach.
• The CRLB decreases with increasing transmitted power $ P_t $, but the rate of decrease slows at higher power levels, showing a transition from $ \propto 1/P_t $ to $ \propto 1/\sqrt{P_t} $ behavior.
• The proportion of the accurate CRLB to the inaccurate one increases with $ P_t $, indicating that the error from ignoring RSN grows substantially at higher power levels.
• The CRLB is a convex function of the Lambertian emission order $ m $, with a minimum at an optimal $ m_{\text{opt}} $, which depends on the angle $ \phi $, and this optimal value can be approximated analytically.
• The study confirms that RSN cannot be neglected in theoretical accuracy analysis, as its variance dependence on distance introduces a non-trivial and significant impact on ranging performance limits.