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Intelligent Reflecting Surface vs. Decode-and-Forward: How Large Surfaces Are Needed to Beat Relaying?
Emil Björnson, Özgecan Özdogan, Erik G. Larsson
TL;DR
The paper asks when an IRS can outperform classic DF relaying in wireless transmission. It develops a fair comparison by optimizing transmit power, IRS size, and energy efficiency under models favoring the IRS, and finds that very high rates or large surfaces are generally required. The practical boundary is constrained by the IRS’s total physical size and associated pathloss assumptions.
Problem
The paper addresses how large an IRS must be to outperform conventional DF relaying, a stronger benchmark than amplify-and-forward relaying.
Method
The study analytically and numerically compares optimized IRS-supported transmission and repetition-coded DF relaying, including transmit power and energy efficiency.
Results
Very high rates and/or large IRSs are needed to outperform DF relaying in transmit-power and energy-efficiency comparisons.
Takeaways & Limitations
DF relaying or a system switching between SISO and DF modes is preferable except when very high rates are required.
Takeaways & Limitations
The conclusion that hundreds of elements are needed relies on ideal phase shifting and frequency-flat channels that favor the IRS, while total IRS size determines pathloss.
Abstract
from arXiv · showhide
The rate and energy efficiency of wireless channels can be improved by deploying software-controlled metasurfaces to reflect signals from the source to the destination, especially when the direct path is weak. While previous works mainly optimized the reflections, this letter compares the new technology with classic decode-and-forward (DF) relaying. The main observation is that very high rates and/or large metasurfaces are needed to outperform DF relaying, both in terms of minimizing the total transmit power and maximizing the energy efficiency, which also includes the dissipation in the transceiver hardware.
I. INTRODUCTION
The paper introduces IRS-supported transmission as a passive, beamforming alternative to relaying and compares it fairly with repetition-coded DF relaying. It optimizes transmit powers and IRS size to determine when an IRS can outperform conventional relaying.
- I. INTRODUCTION: An IRS adapts the propagation environment by configuring sub-wavelength elements to beamform the received signal toward the destination.Each element scatters and phase-shifts the incident wave, while the overall phase pattern determines the reflected beam direction.
- I. INTRODUCTION: Unlike an IRS, a relay actively processes and retransmits the received signal, achieving higher SNR at the cost of a two-hop pre-log penalty.DF relaying is used as the stronger benchmark because it is known to outperform amplify-and-forward relaying in achievable rate.
- I. INTRODUCTION: The study fairly compares IRS-supported transmission with repetition-coded DF relaying by optimizing transmit powers and the IRS element count.The objective is to determine how large an IRS must be to outperform conventional relaying.
II. SYSTEM MODEL
The system model considers a single-antenna source and destination communicating over a deterministic flat-fading SISO channel. Additional equipment is modeled through IRS reflection or DF relaying, with achievable rates derived and optimized for comparison.
- II. SYSTEM MODEL: The baseline channel is a deterministic flat-fading SISO link between a single-antenna source and destination.The received signal uses transmit power, a unit-power information signal, and receiver noise; antenna gains are included in the channel.
- II. SYSTEM MODEL: The paper compares two enhancements: an IRS configured to reflect the signal toward the destination and a relay operating in DF mode.Their achievable rates, or spectral efficiencies, are derived before analytical optimization.
- II. SYSTEM MODEL: The deterministic flat-fading model favors the IRS because IRS-supported transmission is less capable of handling channel estimation and frequency-selective fading than relays.
A. IRS-supported Transmission
The IRS setup uses N discrete sub-wavelength elements whose reflection amplitudes and phase shifts determine the reflected channel. Perfect channel knowledge enables phase optimization, yielding an analytically characterized capacity.
- A. IRS-supported Transmission: The IRS contains N discrete sub-wavelength elements represented by source-to-IRS and IRS-to-destination channel vectors.Each element scatters the incoming signal with approximately constant gain in the relevant directions.
- A. IRS-supported Transmission: The IRS is modeled by a diagonal reflection matrix with fixed amplitude coefficient α and optimizable phase shifts θ1, …, θN.
- A. IRS-supported Transmission: With deterministic channels known perfectly, the phase shifts align all reflected terms with the direct-channel phase to maximize the rate.The resulting capacity is stated in Lemma 1.
B. Relay-supported Transmission
The relay setup uses half-duplex repetition-coded DF relaying over two equal phases. The relay decodes and re-encodes the source information before retransmission, and the achievable rate follows from combining the two received signals.
- B. Relay-supported Transmission: The relay is deployed at the IRS location and uses a half-duplex repetition-coded DF protocol divided into two equal-sized transmission phases.
- B. Relay-supported Transmission: The DF relay decodes the source information from its received signal and then encodes it again for transmission in the second phase.A compact relay can integrate an antenna, transceiver chains, and a baseband unit in a small mobile-phone-sized device.
- B. Relay-supported Transmission: The destination receives transmissions from the source and relay, then uses maximum ratio combining to obtain the achievable DF rate.The analysis assumes deterministic channels, with extension to fading channels with perfect channel knowledge by taking expectations of the rate expressions.
III. ANALYTICAL PERFORMANCE COMPARISON
The analytical comparison shows that IRS and DF performance depends on direct, source–relay, and relay–destination channel gains, transmit SNR, and IRS size. IRS rate advantages can require very large element counts, especially at low SNR.
- Rate comparison: IRS-supported transmission always increases or preserves the SISO rate, while comparison with DF relaying requires optimizing the relay’s two-phase power allocation.The IRS rate is increasing in N, whereas DF uses powers p1 and p2 under a fixed average-power constraint.
- Rate comparison: When βsd > βsr, SISO outperforms DF relaying for every power allocation, making DF relaying suboptimal.The relay-supported system therefore needs to switch between SISO and DF modes depending on the relative channel gains.
- Rate comparison: When βsd ≤ βsr, DF achieves its optimized rate through a specific allocation of p1 and p2, while a different allocation leaves the relay unused and favors SISO.The optimized DF powers are p1 = 2pβrd/(βsr+βrd−βsd) and p2 = 2p(βsr−βsd)/(βsr+βrd−βsd).
- IRS advantage: For βsd > βsr, IRS has the highest rate for any N ≥ 1, but its gain over SISO is small because each reflected path has gain βsrβrd.The appreciable IRS gain occurs mainly when the direct path is no stronger than the source–relay path.
- IRS advantage: At low SNR, N > 963 is required for α = 1, βsd = −110 dB, βsr = −80 dB, and βrd = −60 dB.At high SNR, the IRS achieves the largest rate for any N, whereas the required element count can be very large as transmit power approaches zero.
- Operating regime: The choice among SISO, DF, and IRS depends jointly on SNR and the number of IRS elements.The asymptotic rate comparisons motivate assessing whether practical deployments operate in low- or high-SNR regimes.
A. Transmit Power Minimization Under Rate Constraints
For a required data rate, the paper derives the transmit power for SISO, IRS, and relay-supported transmission, with the relay system selecting its better operating mode.
- Power expressions: The rate expressions identify the transmit power required by SISO, IRS-supported transmission, and relay-supported transmission to achieve a target rate.These power expressions are used for direct comparison under the same rate constraint.
- Mode selection: The relay-supported system minimizes transmit power by switching between SISO and DF relaying, using pDFmode = min(pSISO, pDF).This mode selection accounts for whichever relay-supported operating point requires less power at the target rate.
B. Total Power Minimization Under Rate Constraints
Total power minimization includes both transmit power and hardware dissipation, so the comparison accounts for amplifier efficiency and fixed source and destination consumption.
- Power model: Total SISO power combines amplifier-adjusted transmit power with hardware dissipation at the source and destination.The model is Ptotal^SISO = pSISO/ν + Ps + Pd, where ν is amplifier efficiency.
P IRS
The IRS total-power model includes relay hardware costs, element-level phase-shifting dissipation, and an optimization over the number of elements. The resulting objective is convex when βIRS is independent of N.
- Power model: In relaying, source activity is halved and total hardware dissipation includes 2Ps + Pd + Pr.Pr denotes hardware-dissipated power at the relay.
- IRS optimization: When βIRS is independent of N, IRS total power for a given rate is convex in N and has an analytically characterized minimizer.The minimizer is obtained by setting the derivative of total power with respect to N to zero.
- IRS optimization: The continuous optimum for N generally must be rounded to a neighboring integer, and a negative optimum makes SISO with N = 0 optimal.This behavior occurs, for example, under the line-of-sight model with βIRS = βsrβrd.
IV. NUMERICAL PERFORMANCE COMPARISON
The numerical comparison evaluates SISO, DF relaying, and IRS transmission under deterministic UMi channels as destination distance and target rate vary. DF relaying generally requires less transmit power, while IRS competitiveness improves with more elements and higher rates.
- Simulation setup: The simulations use deterministic 3GPP UMi LOS/NLOS channel gains at 3 GHz, with shadow fading neglected.The source and IRS/relay have fixed locations, while destination distance d1 varies; antenna gains are 5 dBi at the source and IRS/relay, and 0 dBi at the handset destination.
- Transmit power: At 4 bit/s/Hz, DF relaying requires the least transmit power at all considered destination locations, while SISO requires the most.For the IRS, required power decreases as N increases, and its gap to DF relaying is smallest when the destination is close to the source or IRS.
- Transmit power: N > 164 is required for the IRS to outperform DF relaying when d1 = 80 m at 4 bit/s/Hz.
- Transmit power: At 6 bit/s/Hz, the IRS becomes more competitive and requires the least power when the destination is close to the source.At d1 = 80 m, N > 76 is needed for the IRS to outperform relaying because DF’s 1/2-pre-log penalty makes its required power grow faster with rate.
A. Energy Efficiency
The energy-efficiency comparison optimizes IRS element count for each target rate while accounting for total power. SISO or DF relaying is preferable up to high rates, and the IRS surpasses DF only above 8.41 bit/s/Hz.
- Energy-efficiency definition: Energy efficiency is defined as B · ¯R/Ptotal and is evaluated with optimized IRS element count N.The analysis uses ν = 0.5, Ps = Pd = Pr = 100 mW, Pe = 5 mW, and d1 = 70 m.
- Energy-efficiency comparison: SISO provides the highest energy efficiency for ¯R ∈(0, 3.47] bit/s/Hz, while DF relaying leads for ¯R ∈(3.47, 8.41] bit/s/Hz.
- Energy-efficiency comparison: The IRS has N opt > 0 only for ¯R > 4 bit/s/Hz and exceeds DF relaying in energy efficiency only for ¯R > 8.41 bit/s/Hz.Thus, switching between SISO and DF relaying is preferable for minimizing transmit power and maximizing energy efficiency except at very high rates.
V. CONCLUSION AND DISCUSSION
The conclusion finds that IRSs require hundreds of reconfigurable elements to compete with repetition-coded DF relaying, despite idealized assumptions favoring IRS performance. IRSs avoid ideal-form power amplifiers but still incur pathloss and practical element-power costs.
- Conclusion and discussion: An IRS needs hundreds of reconfigurable elements to be competitive with repetition-coded DF relaying.The conclusion notes that this remains true under ideal phase shifting and frequency-flat channels, assumptions that benefit the IRS.
- Conclusion and discussion: The IRS’s per-element channel gain is small because the signal traverses source-to-IRS and IRS-to-destination channels, yielding βsrβrd per element.DF instead transmits sequentially over channels with gains βsr and βrd.
- Conclusion and discussion: An IRS avoids power amplifiers in its ideal form, but practical adaptive phase shifting requires active components whose aggregate dissipation is nonnegligible.
- Conclusion and discussion: Separating the source and destination from the IRS enables propagation-environment control but also creates large pathlosses.For the simulated sizes, hundreds of sub-wavelength elements can remain physically small; total IRS size, rather than element count, determines pathloss.