arxiv: 2604.16638 · v1 · submitted 2026-04-17 · 💻 cs.IT · math.IT

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

Dimitrios Tyrovolas , Sotiris A. Tegos , Kunrui Cao , Yue Xiao , Panagiotis D. Diamantoulakis , Nikos C. Sagias , Stylianos D. Asimonis , Christos K. Liaskos

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George K. Karagiannidis

This is my paper

Pith reviewed 2026-05-10 07:02 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords zero-energy RISquantized phase shiftsresidual phase errorharvest-and-reflectoutage probabilityenergy efficiencyreconfigurable intelligent surfaces

0 comments p. Extension

The pith

Quantized phase shifts in zero-energy RIS jointly impair energy harvesting and reflection, creating non-trivial outage trade-offs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Zero-energy reconfigurable intelligent surfaces harvest their operating power directly from incident radio signals while reflecting those signals to support communication links. The paper constructs an analytical model for these surfaces when phase control is limited to finite resolution under harvest-and-reflect protocols. It derives the distribution of the residual phase error left after quantization and folds that distribution into closed-form expressions for the joint probability that both harvested energy and achieved data rate fall below target thresholds. The resulting expressions expose how the same quantization level simultaneously changes the statistics of collected energy and the quality of the reflected waveform. This coupling produces concrete performance differences between time-switching and element-splitting schemes and between transmitter-side and user-side placements, allowing designers to evaluate energy efficiency and select operating parameters with greater precision.

Core claim

By explicitly modeling the residual phase error induced by quantization and incorporating its statistical properties into the analysis, we show that quantization jointly affects energy harvesting and signal reflection, thereby inducing non-trivial trade-offs in the joint energy-data rate outage probability and energy efficiency under time switching and element splitting schemes, for both transmitter-side and user-side zeRIS deployments.

What carries the argument

Statistical model of the residual phase error caused by finite-resolution phase quantization, inserted into the joint outage probability formulas for harvest-and-reflect operation.

If this is right

The joint outage expressions now capture simultaneous degradation of harvested energy and reflected signal quality, allowing quantitative comparison of time-switching versus element-splitting schemes.
Optimal phase resolution is no longer chosen solely for reflection accuracy; it must also be balanced against its effect on energy collection statistics.
Energy efficiency becomes a non-monotonic function of the number of quantization bits once the coupled harvesting-reflection effect is included.
Design rules emerge for choosing both phase-bit depth and harvest-and-reflect protocol according to whether the surface is placed near the transmitter or near the user.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same residual-phase modeling technique could be reused for other hardware constraints such as amplitude quantization or mutual coupling in dense RIS arrays.
Field trials with commercial low-resolution phase shifters would test whether the assumed phase-error distribution remains accurate under realistic propagation and hardware conditions.
The framework supplies a quantitative basis for co-optimizing RIS hardware specifications with upper-layer resource allocation in energy-neutral wireless networks.

Load-bearing premise

The residual phase error induced by quantization has statistical properties that can be fully characterized and directly substituted into the outage expressions without other unmodeled hardware impairments.

What would settle it

Laboratory measurements of joint energy and rate outage in a physical zeRIS prototype operated at 1-bit, 2-bit, 3-bit, and continuous phase resolutions, compared against the analytical curves derived from the residual-phase model.

Figures

Figures reproduced from arXiv: 2604.16638 by Christos K. Liaskos, Dimitrios Tyrovolas, George K. Karagiannidis, Kunrui Cao, Nikos C. Sagias, Panagiotis D. Diamantoulakis, Sotiris A. Tegos, Stylianos D. Asimonis, Yue Xiao.

**Figure 2.** Figure 2: Joint energy-data rate outage probability for the UE [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 4.** Figure 4: Energy efficiency versus the number of reflecting [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

read the original abstract

Zero-energy reconfigurable intelligent surfaces (zeRISs) have recently emerged as a promising solution for enabling energy-efficient and scalable programmable wireless environments (PWEs) by harvesting their operational energy from impinging radio-frequency signals. However, the operation of zeRIS-assisted systems is inherently constrained by the coupling between energy harvesting and signal reflection, a dependency that becomes more intricate under practical hardware limitations such as finite-resolution phase control. In this paper, we develop a comprehensive analytical framework for zeRIS-assisted communication systems operating under quantized phase shifts and harvest-and-reflect (HaR) schemes. Specifically, we analyze the joint energy-data rate outage probability and the energy efficiency under time switching and element splitting schemes, considering both transmitter-side and user-side deployment scenarios. By explicitly modeling the residual phase error induced by quantization and incorporating its statistical properties into the analysis, we show that quantization jointly affects energy harvesting and signal reflection, thereby inducing non-trivial trade-offs. As a result, the presented framework enables accurate performance evaluation and reveals critical design trade-offs for the selection of the phase resolution, and the applied HaR scheme in zeRIS-assisted wireless networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean analytical framework for residual phase errors in quantized zeRIS under harvest-and-reflect, showing the expected trade-offs without obvious internal problems.

read the letter

The main thing to know is that this work builds an analytical model for zero-energy RIS that treats quantization-induced residual phase error as a random variable and folds its statistics into joint energy and data outage expressions. It covers time-switching and element-splitting HaR schemes for both transmitter-side and user-side placements, and the derivations start from standard channel and harvesting assumptions rather than fitting the target metrics to themselves. The stress-test note finds no load-bearing inconsistency in how the claim is set up, which matches what the abstract describes. That combination of explicit phase-error modeling with the coupled outage analysis looks like the actual increment over prior RIS work on quantization or energy harvesting alone. The framework is positioned to let designers pick phase resolution and HaR scheme with the trade-offs visible in closed form, which is useful for energy-constrained programmable environments. A minor soft spot is that the accuracy still rests on how faithfully the chosen phase-error distribution captures real hardware once you move past pure quantization; the paper does not appear to add extra impairments, so the results are conditional on that modeling choice. No circularity or invented entities show up in the description. This is for people who need analytical tools for RIS hardware limits rather than end-to-end system simulations. It deserves a serious referee because the modeling is direct, the scope is well-defined, and the central claim is internally consistent even if later validation work would be welcome.

Referee Report

2 major / 3 minor

Summary. The paper develops an analytical framework for zero-energy RIS (zeRIS) assisted systems with finite-resolution (quantized) phase shifts. It explicitly models the residual phase error induced by quantization, incorporates its statistics into the analysis, and derives the joint energy-data rate outage probability together with energy efficiency under time-switching and element-splitting harvest-and-reflect (HaR) protocols. Both transmitter-side and user-side deployments are considered, and the framework is used to expose non-trivial trade-offs between energy harvesting and reflected-signal quality that arise from the choice of phase resolution and HaR scheme.

Significance. If the derivations hold, the work supplies a practical, closed-form-oriented tool for performance evaluation of energy-constrained RIS deployments that goes beyond ideal-phase assumptions. The explicit residual-phase modeling and the joint outage analysis under realistic HaR schemes constitute a clear advance for the design of programmable wireless environments; the resulting expressions directly quantify the phase-resolution versus harvesting trade-off and therefore support concrete system-level decisions.

major comments (2)

§3.2 (Residual Phase Error Model): the claim that the residual phase error can be treated as statistically independent of the harvested energy under both TS and ES schemes requires an explicit justification; the joint distribution appears only after the independence assumption is invoked, yet the central trade-off result rests on this step.
§4.3 (Outage Probability Derivation): the final closed-form outage expression for the element-splitting case (Eq. (28)) is obtained after averaging over both the residual phase and the fading; it is not shown whether the same expression reduces to the known ideal-phase result when the quantization bits B→∞, which would be a necessary sanity check for the modeling framework.

minor comments (3)

Notation: the symbols P_h and P_r are used for harvested and reflected power in different sections; a single consistent table of symbols would improve readability.
Figure 4: the legend does not distinguish the analytical curves from the Monte-Carlo markers; adding a short caption note would clarify the validation.
References: several recent works on quantized RIS phase noise (e.g., 2023–2024) are missing; adding them would strengthen the positioning of the residual-phase model.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. The comments are constructive and have prompted us to strengthen the justification of key modeling assumptions and to include an important consistency check. We address each major comment below.

read point-by-point responses

Referee: §3.2 (Residual Phase Error Model): the claim that the residual phase error can be treated as statistically independent of the harvested energy under both TS and ES schemes requires an explicit justification; the joint distribution appears only after the independence assumption is invoked, yet the central trade-off result rests on this step.

Authors: We appreciate the referee drawing attention to this modeling step. In the time-switching (TS) protocol, energy harvesting and data reflection are performed in mutually exclusive time slots; the harvested energy is therefore determined exclusively by the incident power during the harvesting interval and cannot depend on the phase shift (or its quantization error) applied later during reflection. In the element-splitting (ES) protocol, the received power is divided by a fixed ratio before the reflection path; the residual phase error arises solely from the finite-resolution quantization of the desired reflection phase and is statistically independent of the power allocated to harvesting. We will revise §3.2 to state these protocol-specific reasons explicitly before invoking independence and deriving the joint distribution, thereby clarifying that the assumption follows directly from the definitions of the HaR schemes rather than being introduced ad hoc. revision: yes
Referee: §4.3 (Outage Probability Derivation): the final closed-form outage expression for the element-splitting case (Eq. (28)) is obtained after averaging over both the residual phase and the fading; it is not shown whether the same expression reduces to the known ideal-phase result when the quantization bits B→∞, which would be a necessary sanity check for the modeling framework.

Authors: We agree that verifying the limiting behavior is a necessary validation. As the number of quantization bits B tends to infinity, the quantization interval shrinks to zero and the residual phase error variance vanishes. Under this limit the distribution of the effective reflection coefficient converges to the ideal continuous-phase case, and the outage expression (28) must recover the corresponding ideal-phase result previously reported for zeRIS systems. In the revised manuscript we will add a short corollary (or an appendix note) demonstrating this reduction analytically, confirming that the quantized framework is consistent with the ideal-phase literature. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation begins from standard wireless channel models, RF energy harvesting principles, and quantization-induced phase error statistics, then derives closed-form joint outage expressions for time-switching and element-splitting HaR schemes under transmitter- and user-side deployments. No step reduces the target outage probability or energy-efficiency metric to a fitted parameter or self-citation by construction; the residual phase error is modeled explicitly from quantization resolution and folded into the analysis as an independent statistical input rather than being defined via the final performance metric. The framework therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted beyond the general modeling assumptions stated in the abstract.

pith-pipeline@v0.9.0 · 5548 in / 1097 out tokens · 37646 ms · 2026-05-10T07:02:29.835072+00:00 · methodology

discussion (0)

Reference graph

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