Distance and Position Estimation in Visible Light Systems with RGB LEDs: A CRLB and ML Analysis

Table of Contents

1. 1. Introduction & Overview
2. 2. System Models and Scenarios
   1. 2.1 Scenario 1: Synchronous System with Known Channel Model
   2. 2.2 Scenario 2: Asynchronous System with Known Channel Model
   3. 2.3 Scenario 3: Synchronous System with Unknown Channel Model
3. 3. Theoretical Accuracy Limits: Cramér-Rao Lower Bound
4. 4. Practical Estimators: Maximum Likelihood Approach
5. 5. Results and Performance Analysis
6. 6. Core Insight & Analyst's Perspective
7. 7. Technical Details & Mathematical Framework
8. 8. Analysis Framework: A Conceptual Case Study
9. 9. Future Applications & Research Directions
10. 10. References

1. Introduction & Overview

This work investigates fundamental accuracy limits for distance and position estimation in Visible Light Positioning (VLP) systems that utilize Red-Green-Blue Light Emitting Diodes (RGB LEDs). The core contribution is a rigorous theoretical and practical analysis across three distinct operational scenarios, evaluating performance via the Cramér-Rao Lower Bound (CRLB) and deriving corresponding Maximum Likelihood (ML) estimators. The study provides critical insights into when and how RGB LEDs offer advantages over single-color LEDs for localization.

2. System Models and Scenarios

The analysis is structured around three key scenarios that represent common practical constraints in VLP deployment.

2.1 Scenario 1: Synchronous System with Known Channel Model

Assumes perfect synchronization between transmitter and receiver, and perfect knowledge of the channel attenuation formula (e.g., Lambertian model). This represents a theoretical best-case scenario where both Time-of-Arrival (TOA) and Received Signal Strength (RSS) information can be fully exploited.

2.2 Scenario 2: Asynchronous System with Known Channel Model

No synchronization is available between transmitter and receiver. The receiver must rely solely on RSS information for estimation, but the channel model is known. This is a more practical but challenging scenario common in cost-sensitive deployments.

2.3 Scenario 3: Synchronous System with Unknown Channel Model

While synchronization is available (enabling TOA use), the exact channel attenuation characteristics are unknown to the receiver. This models situations with unpredictable environmental factors or uncalibrated hardware.

3. Theoretical Accuracy Limits: Cramér-Rao Lower Bound

The CRLB provides a fundamental lower bound on the variance of any unbiased estimator. For a parameter vector $\boldsymbol{\theta}$ (e.g., distance or 2D/3D position), based on observation vector $\mathbf{x}$, the CRLB is given by the inverse of the Fisher Information Matrix (FIM) $\mathbf{I}(\boldsymbol{\theta})$:

$\text{Var}(\hat{\theta}_i) \geq [\mathbf{I}^{-1}(\boldsymbol{\theta})]_{ii}, \quad \text{where} \quad [\mathbf{I}(\boldsymbol{\theta})]_{ij} = -E\left[ \frac{\partial^2 \ln p(\mathbf{x}; \boldsymbol{\theta})}{\partial \theta_i \partial \theta_j} \right]$

The paper derives explicit CRLB expressions for distance and position estimation in each scenario. A key finding is that the CRLB for distance estimation in Scenario 1 is inversely proportional to the square of the effective bandwidth $\beta^2$ of the transmitted optical signal: $\text{CRLB}(d) \propto 1/\beta^2$. This highlights the critical role of signal design in synchronous systems.

4. Practical Estimators: Maximum Likelihood Approach

For each scenario, the corresponding ML estimator is derived. The ML estimator for the distance $d$ in Scenario 1, under an additive white Gaussian noise (AWGN) assumption, involves solving:

$\hat{d}_{\text{ML}} = \arg\min_d \sum_{k=1}^{K} \left( r_k - \alpha \frac{P_t}{d^2} s(t_k - \tau(d)) \right)^2$

where $r_k$ are received samples, $P_t$ is transmit power, $\alpha$ is channel gain, $s(\cdot)$ is the transmitted waveform, and $\tau(d)$ is the TOA. The paper shows that these ML estimators can asymptotically achieve the CRLB under high signal-to-noise ratio (SNR) conditions.

5. Results and Performance Analysis

The theoretical and simulation results demonstrate several key trends:

• Scenario Comparison: Scenario 1 (synchronous, known channel) provides the best accuracy, followed by Scenario 3 (synchronous, unknown channel), with Scenario 2 (asynchronous) showing the highest error bounds, especially at lower bandwidths.
• RGB LED Advantage: The use of RGB LEDs is shown to improve estimation accuracy. This is intuitively explained by the diversity gain—the independent signals from the R, G, and B channels provide multiple, slightly uncorrelated observations of the same geometric parameters (distance/position), effectively averaging out noise.
• Bandwidth vs. Power Trade-off: In synchronous systems, increasing the signal's effective bandwidth $\beta$ significantly reduces the CRLB, often more effectively than simply increasing optical power. This has important implications for system design, favoring sophisticated modulation over brute-force power increase.
• ML Performance: The derived ML estimators are shown via simulation to approach their respective CRLBs at sufficiently high transmit optical powers, validating their practical optimality in high-SNR regimes.

6. Core Insight & Analyst's Perspective

Core Insight: Demirel and Gezici's work is not just another VLP paper; it's a rigorous deconstruction of the value proposition of RGB LEDs in localization. The core insight is that the benefit of RGB goes beyond color or data transmission—it's a form of implicit spatial diversity. By providing three parallel, physically colocated but spectrally distinct channels, an RGB LED inherently offers a 3x observational redundancy for geometric parameters, directly attacking the noise-limited nature of RSS and TOA measurements. This is analogous to using multiple antennas in RF systems but achieved through a cheap, illumination-centric hardware modification.

Logical Flow: The paper's logic is impeccably clean. It starts by defining the battlefield (three realistic scenarios), establishes the ultimate performance limits (CRLB) as the gold standard, and then builds practical soldiers (ML estimators) to see how close they can get to that limit. The comparison across scenarios is particularly powerful. It quantitatively shows that synchronization is worthless below a certain bandwidth threshold—a crucial design rule often missed in practice. If your signal's effective bandwidth is low, you might as well save the cost and complexity of synchronization and stick to asynchronous RSS-based methods.

Strengths & Flaws: The strength is in its foundational, math-first approach. It doesn't propose a heuristic hack; it derives the fundamental bounds, making its conclusions universally applicable. The use of CRLB provides an unchallengeable benchmark. However, the analysis has the classic flaw of many theoretical works: it leans heavily on the AWGN assumption and known channel models like the Lambertian model. Real-world VLP is plagued by multipath, shadowing, non-Lambertian reflections (from glossy surfaces), and ambient light noise—factors that can severely degrade performance from these theoretical bounds, as noted in experimental studies like those from the University of California's Visible Light Communication Consortium. The paper acknowledges unknown channel models in Scenario 3 but treats it as a parametric uncertainty. The more disruptive challenge is a non-parametric, dynamic channel, which is where data-driven and machine learning approaches, inspired by works like CycleGAN for domain adaptation, are now heading.

Actionable Insights: For system architects, this paper offers clear directives: 1) Prioritize Bandwidth: If you're building a synchronous system, invest in high-bandwidth drivers and modulation schemes (e.g., OFDM) before cranking up the optical power. 2) Justify RGB: Use the diversity argument to justify the marginally higher cost of RGB LEDs over single-color LEDs for high-accuracy positioning applications. 3) Choose Your Battleground: For large-scale, low-cost indoor tracking (e.g., warehouse inventory), an asynchronous RSS-based system with RGB LEDs might offer the best cost-accuracy trade-off. For surgical robot guidance, go synchronous and spare no expense on bandwidth. 4) The Next Frontier is Robustness: The theoretical bounds are now well understood. The next wave of innovation, as seen in recent arXiv preprints and IEEE journals, will focus on making these estimators robust to the messy realities of indoor propagation, likely fusing model-based approaches (like this paper's) with learning-based techniques for channel resilience.

7. Technical Details & Mathematical Framework

The received optical power $P_r$ from an LED is typically modeled by the Lambertian formula:

$P_r = \begin{cases} \frac{m+1}{2\pi d^2} A \cos^m(\phi) \cos(\psi) P_t, & 0 \le \psi \le \Psi_c \\ 0, & \psi > \Psi_c \end{cases}$

where $d$ is distance, $A$ is detector area, $\phi$ is irradiance angle, $\psi$ is incidence angle, $\Psi_c$ is the receiver's field-of-view, $m$ is the Lambertian order, and $P_t$ is transmit power. For an RGB LED, this model applies independently to each color channel (R, G, B), with potentially different $P_t$ per channel.

The Fisher Information for distance $d$ in Scenario 1, considering both TOA and RSS, and aggregating information from $N_c$ color channels (e.g., 3 for RGB), can be expressed as:

$I(d) = \sum_{c=1}^{N_c} \left( \frac{2 \beta_c^2 \text{SNR}_c}{c^2} + \frac{4 \text{SNR}_c}{d^2} \right)$

where $\beta_c$ is the effective bandwidth of channel $c$, $c$ is the speed of light, and $\text{SNR}_c$ is the signal-to-noise ratio for that channel. The first term inside the summation comes from the TOA information and depends on $\beta_c^2$. The second term comes from the RSS information. The summation clearly shows the diversity gain from using multiple channels.

8. Analysis Framework: A Conceptual Case Study

Scenario: Designing a VLP system for automated guided vehicle (AGV) navigation in a smart factory.

Framework Application:

1. Requirement Analysis: Target positioning accuracy < 10 cm in 3D. Environment has high ceilings (5m), machinery causing occasional occlusion, and fluorescent ambient lighting.
2. Scenario Selection: High accuracy requirement pushes towards a synchronous system (Scenario 1 or 3). However, the unknown and variable occlusion profile suggests the channel model won't be perfectly known at all times, arguing for Scenario 3 analysis.
3. Technology Choice: Use RGB LEDs for ceiling fixtures. The analysis from this paper justifies the choice: the diversity gain helps mitigate accuracy loss when one color channel is blocked or heavily attenuated by an occluding object.
4. Parameter Design: To achieve the CRLB-derived accuracy, calculate the required effective bandwidth $\beta$. The paper's formulas indicate that with RGB diversity, the required $\beta$ (and thus the system cost/complexity) for a given accuracy is lower than for a single-color system.
5. Estimator Implementation: Implement the ML estimator for Scenario 3. Use a calibration phase to build an initial channel model, but allow the estimator to adapt by treating some channel parameters as unknown (per the paper's framework).
6. Validation: Compare the real-world AGV positioning error against the CRLB predicted for the system's SNR and bandwidth. A significant gap would indicate unmodeled effects (e.g., multipath), prompting a move towards more robust, hybrid model-based/data-driven methods.

9. Future Applications & Research Directions

The foundational work presented opens doors to several advanced applications and research avenues:

• 6G Integrated Sensing and Communication (ISAC): VLP is a natural candidate for ISAC in next-generation networks. RGB LEDs can simultaneously provide illumination, high-speed data communication (Li-Fi), and precise positioning, as explored in research from institutions like PureLiFi and the University of Edinburgh.
• Augmented Reality (AR) & Metaverse: Sub-centimeter indoor positioning is critical for seamless AR experiences. RGB VLP systems embedded in room lighting could provide the necessary precision for object anchoring and user tracking without external sensors.
• Robotics and Drone Navigation: In GPS-denied environments like warehouses, mines, or indoor farms, VLP with RGB LEDs offers a reliable and infrastructure-based navigation solution. The diversity gain is crucial for dealing with robot/drone orientation changes.
• Biomedical and Healthcare Monitoring: Patient and asset tracking in hospitals with high reliability and no RF interference.
• Research Directions:
  1. Machine Learning for Channel Agnostic Positioning: Developing deep learning estimators (e.g., using convolutional neural networks on received signal patterns) that are robust to completely unknown and dynamic channels, moving beyond the parametric unknown model of Scenario 3.
  2. Hybrid RF-VLC Systems: Fusing VLP with UWB or WiFi positioning to cover each technology's blind spots, leveraging the high accuracy of VLP in open spaces and the penetration capability of RF.
  3. Energy-Harvesting VLP Receivers: Designing receivers that can perform positioning using the harvested optical energy itself, enabling perpetual IoT sensor nodes.
  4. Standardization: Pushing for industry-wide standards on modulation, coding, and protocols for VLP, similar to IEEE 802.15.7 for VLC, to ensure interoperability.

10. References

1. Demirel, I., & Gezici, S. (2021). Distance and Position Estimation in Visible Light Systems with RGB LEDs. arXiv preprint arXiv:2106.00396.
2. Kahn, J. M., & Barry, J. R. (1997). Wireless infrared communications. Proceedings of the IEEE, 85(2), 265-298.
3. Zhuang, Y., Hua, L., Qi, L., Yang, J., Cao, P., Cao, Y., ... & Thompson, J. (2018). A survey of positioning systems using visible LED lights. IEEE Communications Surveys & Tutorials, 20(3), 1963-1988.
4. Visible Light Communication Consortium (VLCC). (2023). Research on Practical VLP Impairments. [Online]. Available: http://www.vlcc.net
5. Isola, P., Zhu, J. Y., Zhou, T., & Efros, A. A. (2017). Image-to-image translation with conditional adversarial networks. Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 1125-1134). (Relevant for data-driven channel adaptation methods).
6. PureLiFi. (2023). Li-Fi for Integrated Sensing and Communication. [White Paper].
7. IEEE Standard for Local and Metropolitan Area Networks–Part 15.7: Short-Range Wireless Optical Communication Using Visible Light. (2018). IEEE Std 802.15.7-2018.