[Paper Review] Reconfigurable Intelligent Surfaces Assisted Communications with Discrete Phase Shifts: How Many Quantization Levels are Required to Achieve Full Diversity?
This paper investigates the minimum number of phase quantization levels required for reconfigurable intelligent surfaces (RIS) to achieve full diversity order in wireless communications. It proves that at least three quantization levels (L ≥ 3) are necessary and sufficient to achieve the full diversity order N, while L = 2 limits the diversity to at most (N+1)/2. Simulation results confirm that performance loss is negligible when L ≥ 3.
Due to hardware limitations, the phase shifts of the reflecting elements of reconfigurable intelligent surfaces (RISs) need to be quantized into discrete values. This letter aims to unveil the minimum required number of phase quantization levels $L$ in order to achieve the full diversity order in RIS-assisted wireless communication systems. With the aid of an upper bound of the outage probability, we first prove that the full diversity order is achievable provided that $L$ is not less than three. If $L=2$, on the other hand, we prove that the achievable diversity order cannot exceed $(N+1)/2$, where $N$ is the number of reflecting elements. This is obtained with the aid of a lower bound of the outage probability. Therefore, we prove that the minimum required value of $L$ to achieve the full diversity order is $L=3$. Simulation results verify the theoretical analysis and the impact of phase quantization levels on RIS-assisted communication systems.
Motivation & Objective
- To determine the minimum number of phase quantization levels required for RIS-aided systems to achieve full diversity order.
- To analyze the impact of discrete phase shifts on outage probability and diversity gain in RIS-assisted communications.
- To establish theoretical bounds on diversity order based on the number of quantization levels L.
- To provide design guidelines for practical RIS hardware with limited phase resolution.
Proposed method
- Derives an upper bound on outage probability to prove that full diversity order N is achievable when L ≥ 3.
- Uses mathematical induction based on the upper outage bound to establish the achievability of full diversity for L ≥ 3.
- Derives a lower bound on outage probability conditioned on phase errors near quantization boundaries to analyze performance limits when L = 2.
- Analyzes the asymptotic behavior of outage probability at high SNR to characterize diversity order.
- Employs complex Gaussian channel models for RIS links and assumes independent, identically distributed fading coefficients.
- Validates theoretical findings via Monte Carlo simulations with varying L and N.
Experimental results
Research questions
- RQ1What is the minimum number of phase quantization levels L required to achieve the full diversity order N in RIS-aided systems?
- RQ2How does the diversity order scale when only two phase levels (L = 2) are used?
- RQ3Can the full diversity order be preserved when phase shifts are quantized, and if so, under what conditions?
- RQ4What is the impact of phase quantization error distribution on outage performance at high SNR?
- RQ5How does the number of reflecting elements N interact with the number of quantization levels L in determining achievable diversity?
Key findings
- The full diversity order N is achievable if and only if the number of phase quantization levels L is at least 3.
- When L = 2, the achievable diversity order is at most (N+1)/2, which is strictly less than N for N > 1.
- For L ≥ 3, the outage probability curves are parallel to those with perfect phase shifts, indicating no diversity gain loss.
- The outage performance loss due to phase quantization is negligible when L ≥ 3, even for moderate N.
- When L = 2, the diversity order is limited because phase errors near ±π/2 cause deep fading due to constructive and destructive signal cancellation.
- Simulation results confirm that L ≥ 3 ensures near-perfect diversity gain, while L = 2 results in a significant performance floor at high SNR.
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This review was created by AI and reviewed by human editors.