Source-linked AI summary
Reconfigurable Intelligent Surfaces assisted Communications with Limited Phase Shifts: How Many Phase Shifts Are Enough?
Hongliang Zhang, Boya Di, Lingyang Song, Zhu Han
TL;DR
Practical RISs use limited phase shifts, motivating analysis of their effect on uplink achievable data rate. The paper derives an achievable-rate approximation and phase-shift design under a degradation constraint, finding that required coding bits decrease as RIS size grows.
Problem
Most works assume continuous phase shifts, while practical RISs have limited phase shifts; the paper therefore studies their impact on achievable data rate.
Method
The paper derives an achievable data-rate expression, proposes an optimal limited-phase-shift design, and defines a degradation ratio relative to continuous phase shifts.
Results
3 bits are required for N = 3, 2 bits for N = 300, and 1 bit for N = 3 ∗ 10^70 when κ = 4 and ϵ_0 = 0.9.
Takeaways & Limitations
The required number of phase shifts decreases as RIS size grows; the conclusions state that 2 phase shifts are enough when RIS size is infinite.
Abstract
from arXiv · showhide
Reconfigurable intelligent surface~(RIS) has drawn a great attention worldwide as it can create favorable propagation conditions by controlling the phase shifts of the reflected signals at the surface to enhance the communication quality. However, the practical RIS only has limited phase shifts, which will lead to the performance degradation. In this letter, we evaluate the performance of an uplink RIS assisted communication system by giving an approximation of the achievable data rate, and investigate the effect of limited phase shifts on the data rate. In particular, we derive the required number of phase shifts under a data rate degradation constraint. Numerical results verify our analysis.
I. INTRODUCTION
RIS is presented as a low-cost way to improve wireless-link quality, while practical systems motivate studying how limited phase shifts affect achievable data rate.
- RIS elements control electromagnetic responses to reflect incident signals and generate directional beams for improved link quality and coverage.
- Prior work optimized continuous RIS phase shifts, but most existing studies assume phase control that is difficult to implement practically.
- The paper studies an uplink cellular network with a deeply faded direct BS-user link and uses a limited-phase RIS to reflect the user signal toward the BS.
- Its analysis derives achievable data-rate behavior with continuous phase shifts and examines the effect of limited phase shifts on that rate.
- Numerical results are used to validate the analysis after the paper discusses the system model, achievable rate, and limited-phase-shift impact.
II. SYSTEM MODEL
The system is a narrow-band uplink with one base station and one user, where an RIS redirects the user’s signal when the direct link is unreliable.
- The network contains one base station and one cellular user communicating over a narrow-band uplink.
- Unexpected fading and obstacles can destabilize or completely outage the direct line-of-sight user-to-base-station link.
- The RIS reflects the user signal toward the base station to improve the received quality of service.
A. RIS assisted Communication Model
The RIS uses electrically controlled elements whose phase shifts are implemented through PIN diodes and restricted to uniformly spaced K-bit configurations.
- The RIS contains M × N electrically controlled elements, with each element adjusting its reflected-signal phase through a PIN diode.
- PIN diodes switch between ON and OFF states through bias voltage, allowing the metal plate to impose different reflected-signal phase shifts.
- A K-bit-coded RIS provides 2^K uniformly spaced phase-shift patterns with interval Δθ = 2π/2^K.
- Each element’s reflection factor is denoted by Γ_m,n, with constant reflection amplitude Γ ∈ [0, 1].
B. Reflection Dominant Channel Model
The channel model treats the RIS-mediated BS-user path as reflection-dominant and Rician, then relates channel gain to RIS geometry and transmission distance.
- Directional RIS reflections make the BS-RIS-user path stronger than multipath effects and the degraded direct BS-user link.
- The Rician factor κ denotes the ratio of the LoS component to the NLoS component.
- For RIS element (m, n), D_m,n and d_m,n are the BS-element and element-user distances, and L_m,n = D_m,n + d_m,n is the reflected transmission distance.
- The channel-gain formulation assumes RIS-element distances to the BS and user greatly exceed adjacent-element horizontal and vertical spacings.
- With fixed total distance D_m,n + d_m,n = L, channel gain first decreases and then increases as the RIS moves farther from the BS.
III. ACHIEVABLE DATA RATE ANALYSIS
The achievable data rate is obtained by maximizing received SNR through RIS phase-shift optimization. The analysis bounds performance across pure LoS and Rayleigh channels, showing stronger asymptotic scaling for LoS propagation.
- The received SNR is maximized by optimizing the response of each RIS element, yielding the achievable data rate.
- Continuous phase shifts must align the reflected signal phases across all RIS elements to maximize the data rate.
- O(M^2N^2) asymptotic received power gain forms the pure LoS upper-bound regime as κ →∞.
- O(MN) asymptotic squared received power gain forms the Rayleigh lower-bound regime as κ →0.
- The data rate grows with κ as received SNR increases from O(MN) to O(M^2N^2).
IV. ANALYSIS ON THE NUMBER OF PHASE SHIFTS
The limited-phase-shift analysis selects the available phase shift nearest to the continuous optimum and derives coding-bit requirements under a data-rate degradation constraint. Required coding bits decrease with RIS size, reaching one bit asymptotically.
- Limited-phase-shift operation selects the finite phase shift closest to the optimal continuous phase shift.
- The phase-shift error with K coding bits is bounded by −2π/2^(K+1) ≤ δ_m,n < 2π/2^(K+1).
- The degradation ratio ε compares data rate with limited phase shifts against data rate with continuous phase shifts and must satisfy ε ≥ ε0, with ε0 < 1.
- The required number of coding bits is derived from the degradation constraint and expressed in equation (18).
- The required coding bits decrease as RIS size MN increases; when MN →∞, one bit is sufficient for the performance threshold.
- With continuous phase shifts, achievable data rate increases with RIS size N because more energy is reflected.
V. SIMULATION RESULTS
The simulations validate the achievable-rate analysis and show how RIS size, channel conditions, phase resolution, and RIS placement affect performance. Larger RISs require fewer coding bits, while location-induced degradation varies non-monotonically with distance.
- Channel conditions: Data rate increases with the Rician factor κ, and the pure-LoS and Rayleigh asymptotic slopes are 4 and 2, respectively.These slopes correspond to received SNR scaling with the squared number and number of RIS elements, respectively.
- Limited phase shifts: With κ = 4 and degradation threshold ϵ0 = 0.9, the required coding resolution falls from 3 bits at N = 3 to 2 bits at N = 300 and 1 bit at N = 3 * 10^70.Thus, the required coding bits decrease as the RIS size grows.
- RIS placement: With fixed transmission distance D0 + d0 = 160, data-rate degradation first decreases and then increases as the BS–RIS distance D0 grows.The location-related variance becomes smaller for larger RISs, so location may affect coding-bit requirements mainly when the RIS is small.
VI. CONCLUSIONS AND FUTURE WORKS
The paper derives the achievable uplink rate, analyzes limited phase-shift design, and establishes coding-bit requirements under a degradation threshold. Its conclusions relate RIS size and channel model to asymptotic SNR and phase-shift needs, while future work extends the setting to several RIS-enabled applications.
- Conclusions: The paper derives the achievable data rate and analyzes limited phase-shift design for an RIS-assisted uplink cellular network.It also proposes an optimal phase-shift design that maximizes data rate under a predefined degradation threshold.
- Conclusions: Pure-LoS channels achieve asymptotic SNR proportional to the squared RIS-element count, whereas Rayleigh-faded channels achieve proportionality to the element count.These conclusions are stated for a fixed RIS location.
- Conclusions: The required number of phase shifts decreases as RIS size grows; the conclusion states that 2 phase shifts are enough when RIS size is infinite.The requirement is evaluated under a data-rate degradation threshold.
- Future works: Future directions include RIS-assisted D2D heterogeneous networks, energy cooperation, and RF sensing.The paper associates these extensions with higher data rates, more effective energy harvesting, and accurate object tracking, respectively.
APPENDIX A PROOF OF PROPOSITION 1
Appendix A sets up the RIS geometry and uses distance approximations to derive channel-gain behavior. It also expresses phase-shift degradation as a function of channel gain for fixed coding bits and RIS size.
- Geometry: The RIS principal directions are defined from the x-, y-, and z-axis directions using its angle θR to the x-y plane.The horizontal principal direction is nh = nx cos θR + ny sin θR, while nv = nz.
- Geometry: The position of RIS element (m, n) is denoted by cm,n, while D0,0 is the BS–RIS projected distance on the y-axis.These quantities provide geometric coordinates for the channel derivation.
- Degradation relation: For fixed coding bit K and RIS size, degradation ϵ is the ratio log2(1 + g′x) / log2(1 + gx), where g′ < g reflects limited phase shifts and x is channel gain.The ratio increases with x according to the cited derivative argument.
- Channel approximation: When RIS-to-BS and RIS-to-user distances greatly exceed adjacent-element separations, pathloss through different RIS elements can be treated as constant.The approximation assumes Dm,n, dm,n ≫ dh, dv and uses the small-element-spacing condition relative to the BS–user distance.
APPENDIX B DERIVATIONS OF EQUATION (10)
Appendix B derives the expected received SNR by reducing the rate expression to the expected channel-related term. Independence and zero mean of NLoS components eliminate the final cross term.
- Derivation: The logarithmic-function property is used as the starting point for the derivation.The passage introduces the applicable transformation before subsequent expectations are evaluated.
- Derivation: Because Pσ^2 is constant, the derivation focuses on obtaining E[γ].The constant factor is separated from the expected SNR term.
- Cross-term simplification: The final term in (25) vanishes because the NLoS component has zero mean and is independent across distinct RIS elements.This independence yields the stated subsequent equation.
- Derivation: The appendix concludes after establishing the required derivation.