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Reconfigurable Intelligent Surfaces Assisted Communications with Discrete Phase Shifts: How Many Quantization Levels are Required to Achieve Full Diversity?
Peng Xu, Gaojie Chen, Zheng Yang, Marco Di Renzo
TL;DR
Hardware limitations make continuous RIS phase shifts difficult to implement, and the diversity implications of discrete quantization levels were not established for arbitrary N ≥ 2. The paper analyzes diversity using outage-probability bounds and proves full diversity for L ≥ 3, while L = 2 cannot exceed (N+1)/2. It concludes that three levels are the minimum for full diversity.
Problem
Continuous RIS phase shifts are difficult to implement in practice, and the diversity order’s dependence on L for arbitrary N ≥ 2 had not been revealed.
Source: §I · p. 1Method
The paper studies quantized RIS phases and uses upper and lower outage-probability bounds to analyze achievable diversity.
Source: §I · p. 1Results
L ≥ 3 achieves diversity order N, whereas for L = 2 the diversity order is at most (N+1)/2.
Source: §III · p. 3, §IV · p. 4Takeaways & Limitations
Three quantization levels are the minimum required for full diversity, and simulations report negligible outage-performance loss for moderate quantization levels.
Source: §IV · p. 4, §VII · p. 5
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
RIS elements can passively co-phase reflected signals, but practical hardware limits make continuous phase shifts difficult to implement. Prior work studied quantization effects, while the diversity order’s relation to the number of levels remained unknown for arbitrary N ≥ 2.
I. INTRODUCTION
- Continuous phase shifts are difficult to implement in practice because of hardware limitations.RIS elements can control reflected-signal amplitude and phase to co-phase multiple reflections through passive beamforming.Source: §I · p. 1
- Prior analyses showed average SNR scales with N^2 for large N, but did not reveal how diversity order depends on L for arbitrary N ≥ 2.The cited prior work analyzed average SNR and achievable data rate under quantization errors.Source: §I · p. 1
- The letter targets the minimum number of phase-quantization levels needed to achieve full diversity in RIS-assisted transmission.It investigates systems with any N ≥ 2.Source: §I · p. 1
- The authors prove full diversity N for L ≥ 3 using an upper bound and mathematical induction, and bound diversity by (N+1)/2 for L = 2 using a lower bound.The lower-bound argument conditions on each phase error being close to a quantization boundary.Source: §I · p. 1
- For L ≥ 3, the reported outage-performance loss is negligible, offering a design insight for phase quantization in RIS systems.The numerical results are described as demonstrating this performance observation.Source: §I · p. 1
II. SYSTEM MODEL AND PRELIMINARIES
The model considers an RIS-aided single-antenna link with independent channel coefficients and discrete phase shifts chosen by quantizing each element’s optimal phase. The analysis uses outage-probability bounds because the effective-channel CDF is difficult to obtain, and the CLT approximation is unsuitable for diversity order.
II. SYSTEM MODEL AND PRELIMINARIES
- The direct source-destination link is assumed weak and omitted in the model; channel coefficients are mutually independent complex Gaussian variables.The source, destination, and RIS each use single antennas or N reflecting elements, respectively.Source: §II · p. 1
- Each element’s optimal phase is mapped to the nearest of L uniformly quantized phase values, yielding a uniformly distributed phase error in [-π/L, π/L].The number of quantization bits is log2 L.Source: §II · p. 2
- The central limit theorem approximation is unsuitable for diversity analysis because it has an outage-probability error floor as ρ tends to infinity.The authors instead derive lower and upper bounds for outage probability to study achievable diversity for different L.Source: §II · p. 2
- An exact outage-probability expression is generally unavailable because the real and imaginary parts of the effective channel are correlated, motivating bound-based analysis.The model assumes all reflecting elements share the same amplitude reflection coefficient.Source: §II · p. 2
III. DIVERSITY ORDER FOR L = 3
For L ≥ 3, the RIS-assisted system achieves the full diversity order N. The proof uses an upper bound on outage probability and mathematical induction.
III. DIVERSITY ORDER FOR L = 3
- The proof uses an upper bound on outage probability and mathematical induction to extend the diversity result to arbitrary N ≥ 2.For L = 3, each phase error lies in [−π/3, π/3].Source: §III · pp. 2–3
- For every L ≥ 3 and N ≥ 2, the diversity order is dN(L) = N.The proof establishes the N = 2 case and generalizes it by induction.Source: §III · p. 3
- The section frames L = 3 as a candidate minimum and analyzes L = 2 separately to determine whether fewer levels suffice.The passage introduces the L = 2 analysis as the next step in answering that question.Source: §III · p. 3
IV. DIVERSITY ORDER FOR L = 2
With L = 2, the diversity order is at most (N+1)/2, so two quantization levels do not achieve full diversity. The lower-bound proof conditions on phase errors near the quantization boundary.
IV. DIVERSITY ORDER FOR L = 2
- The lower-bound analysis conditions on the phase difference being close to π, where |G2| is minimized for fixed channel magnitudes.The proof generalizes the N = 2 argument to N ≥ 2.Source: §IV · pp. 3–4
- For L = 2 and N ≥ 2, the diversity order is bounded above by (N+1)/2.The lower bound on outage probability yields this upper bound on diversity order.Source: §IV · pp. 3–4
- Combining the L = 2 bound with the L ≥ 3 result identifies L = 3 as the minimum number of levels for full diversity.Source: §IV · p. 4
V. IMPACT OF DIRECT LINK
The direct link changes the phase setting and permits the diversity theorems to be adapted. With the direct link included, the paper states that the full-diversity result extends with order N+1, while the L=2 upper bound is modified as well.
V. IMPACT OF DIRECT LINK
- Theorem 2 is also modified for the direct-link case, with its diversity-order bound replaced by an upper bound involving N+1.The provided passage gives the replacement as (N+1)/2.Source: §V · p. 4
- The optimal reflecting phase is adjusted using the direct-link channel phase.The passage says the optimal phase value from Section II is modified when the direct link cannot be ignored.Source: §V · p. 4
- With the direct link included, Theorem 1 still holds after replacing diversity order N with N+1.The stated SNR lower bound includes both direct-link power and the RIS contribution.Source: §V · p. 4
VI. NUMERICAL RESULTS
Simulations examine how phase quantization affects outage probability and diversity order. They find full diversity for L≥3, while L=2 has a diversity-order ceiling; outage performance also improves as L increases.
VI. NUMERICAL RESULTS
- The simulations use no direct link, with ΩS=1, ΩI=0.5, and η=0.8.The channel coefficients are set to CN(0,1) and CN(0,0.5) for the source-to-RIS and RIS-to-destination links, respectively.Source: §VI · p. 4
- The figures compare outage probability across phase quantization levels and reflecting-element counts, including high-SNR slope guides.The slope guides shown are N+1/2 and N.Source: §VI · p. 4
- For L=2, the high-SNR diversity order does not exceed 1.5, 2.5, and 4.5 for N=2, 4, and 8; for L≥3, full diversity is achieved.The L=2 slope limits are attributed to phase errors near the quantization boundaries, where channel amplitudes can cancel.Source: §VI · p. 4
- The outage probability decreases with L and approaches the perfect-phase lower bound as L increases.For N=2, the curves approach that bound for L≥3; for N=5, the gap decreases as L increases.Source: §VI · p. 5
VII. CONCLUSION
The letter identifies three phase quantization levels as the minimum for full diversity in RIS-aided communication systems. Simulations verify the theory and indicate negligible outage-performance loss for moderate quantization levels.
VII. CONCLUSION
- Full diversity is achievable if and only if at least three phase quantization levels are used.The conclusion identifies this as the minimum required number of levels.Source: §VII · p. 5
- The simulations verify the theoretical finding about the full-diversity threshold.The conclusion also reports negligible outage-performance loss for moderate quantization levels.Source: §VII · p. 5
- The paper investigates diversity order in RIS-aided communication systems with discrete phase shifts.Source: §VII · p. 5
APPENDIX A PROOF OF LEMMA 2
The proof excerpts analyze distinct regimes for a and use asymptotic probability and series-expansion steps. They provide only fragments of the appendix argument.
APPENDIX A PROOF OF LEMMA 2
- For 0<a<cρ^-1/2, the excerpt bounds Pr{(|g_n|-a)^2<bρ^-1} by ρ^-1 asymptotically.The passage says the final step uses (3).Source: Appendix A · p. 5
- For a>0 with a≈ρ^0, the proof invokes a series representation of K̃_1(x) and expands around 2a.The excerpts reference [17, Eq. 8.446] for the series representation and use a Taylor expansion.Source: Appendix A · p. 5
- The resulting K̃_1 expansions are substituted into (24) to obtain the next expression.The excerpt identifies (26) and (28) as the expressions inserted.Source: Appendix A · p. 5