Quantized Zero-Energy RIS: Residual Phase Modeling and Outage Analysis

1. Problem Statement

The paper addresses the performance analysis of Zero-Energy Reconfigurable Intelligent Surfaces (zeRISs). Unlike traditional RISs powered by external sources, zeRISs harvest energy from impinging radio-frequency (RF) signals to power their control circuitry and reflection elements.

The core challenge identified is the inherent coupling between energy harvesting and signal reflection. A zeRIS must split the incident signal: part is absorbed to generate power, and the remainder is reflected to manipulate the wireless channel. This trade-off is further complicated by practical hardware limitations, specifically finite-resolution phase control.

• Quantization Effects: Real-world zeRIS elements use discrete phase shifts (quantized) rather than ideal continuous ones, introducing residual phase errors.
• The Gap: Existing literature often assumes ideal continuous phase control or treats quantization errors in isolation, failing to account for how quantization simultaneously degrades signal reflection and alters energy harvesting efficiency (due to increased power consumption for higher resolution). There is a lack of analytical frameworks that jointly model these effects under realistic propagation conditions (e.g., Line-of-Sight vs. Fading).

2. Methodology

The authors develop a comprehensive analytical framework to evaluate zeRIS-assisted downlink communication systems.

• System Model:

  • Deployment Scenarios: Two configurations are analyzed: Transmitter-side (Tx-side) where the zeRIS is near the source (LoS link to Tx, fading to User), and User-side (UE-side) where the zeRIS is near the receiver (fading from Tx, LoS to User).
  • Harvest-and-Reflect (HaR) Schemes:
    1. Time Switching (TS): The zeRIS alternates between harvesting energy and reflecting signals over time.
    2. Element Splitting (ES): A subset of elements is dedicated to harvesting, while the rest reflect signals simultaneously.
  • Channel Model: Uses Nakagami-$m$ fading for fading links and deterministic Line-of-Sight (LoS) for the other link. The phase distribution follows a Tikhonov-like distribution to model non-isotropic scattering.

• Statistical Characterization of Residual Phase Error:

  • The paper derives the Probability Density Function (PDF) of the residual phase error ($\epsilon$) induced by $q$-bit quantization.
  • Crucially, the model accounts for the deterministic LoS component, showing that the error distribution is not uniform but contains a Dirac delta term (representing perfect alignment for LoS components) and a continuous part (for fading components).
  • Circular Moments: The authors derive the $n$-th circular moments of the residual phase error. These moments are essential for calculating the statistical behavior of the coherent sum of reflected signals. They prove that the infinite series required for these moments converges rapidly, allowing for accurate truncation.

• Performance Metrics:

  • The primary metric is the Joint Energy–Data Rate Outage Probability. An outage occurs if the system fails to harvest enough energy to sustain operation or if the achievable data rate falls below a threshold.
  • Energy Efficiency (EE): Calculated as the ratio of successful data bits to total energy consumed.

3. Key Contributions

1. Novel Analytical Framework: The paper presents the first framework that explicitly models the residual phase error caused by quantization and integrates its statistical properties into the joint analysis of energy harvesting and communication performance.
2. Closed-Form Expressions: The authors derive closed-form approximations for the joint outage probability under both TS and ES schemes for both Tx-side and UE-side deployments. They utilize moment-matching techniques to approximate the sum of random variables with a Gamma distribution.
3. Quantization Trade-off Analysis: The study reveals a non-trivial trade-off: increasing phase resolution ($q$) improves signal alignment but increases power consumption (more diodes per element). The paper demonstrates that higher resolution does not always yield better performance due to this energy penalty.
4. Critique of Existing Models: The paper shows that previous models (e.g., assuming uniform phase error distribution) significantly overestimate the negative impact of quantization in LoS-dominant scenarios and can lead to overly pessimistic design conclusions.

4. Key Results

Numerical results, validated by Monte Carlo simulations, highlight several critical findings:

• Impact of Deployment:
  • UE-side deployment is significantly more challenging than Tx-side. Because the energy harvesting link suffers from fading and higher path loss, UE-side systems require a much larger number of reflecting elements ($N$) to achieve the same reliability.
• Quantization Resolution ($q$):
  • Contrary to intuition, lower resolution ($q=1$) often outperforms higher resolution ($q=2$) in terms of outage probability and energy efficiency. The power consumed by the extra diodes required for $q=2$ outweighs the marginal gains in beamforming gain, especially in scenarios with strong LoS components where quantization error is naturally small.
• HaR Scheme Comparison:
  • The Element Splitting (ES) scheme consistently outperforms the Time Switching (TS) scheme. ES achieves higher energy efficiency and requires fewer elements to meet outage constraints because it utilizes the full time block for reflection while dedicating specific hardware to harvesting.
• Model Accuracy:
  • The proposed model (accounting for LoS-induced phase concentration) matches simulations closely.
  • Benchmark models (ignoring LoS effects) incorrectly predict that zeRIS systems cannot operate reliably with low resolution ($q=1$) in certain scenarios, whereas the proposed model shows reliable operation is achievable.

5. Significance and Implications

This work provides essential design guidelines for the next generation of sustainable wireless networks:

• Hardware Design: It suggests that for zeRISs, especially in LoS-rich environments, low-resolution phase shifters may be optimal. Designers should not blindly pursue high-resolution control if the energy cost of the control circuitry negates the communication gains.
• System Architecture: The Element Splitting (ES) scheme is identified as a superior strategy for balancing energy and data rates compared to time-switching.
• Placement Strategy: Network planners must account for the severe energy constraints of UE-side deployments, which may necessitate significantly larger surface areas or higher transmit power compared to Tx-side deployments.
• Theoretical Contribution: By rigorously modeling the interaction between quantization, residual phase error, and energy harvesting, the paper prevents the adoption of suboptimal designs based on inaccurate theoretical assumptions.

In summary, the paper establishes that the design of zero-energy RISs requires a joint optimization of phase resolution, harvesting strategy, and deployment location, where "more bits" does not necessarily mean "better performance."