Beyond Max-SNR: Joint Encoding for Reconfigurable Intelligent Surfaces

Motivation

• The paper studies RIS-aided links where the transmitter controls the RIS through a finite-rate control link, rather than treating the RIS only as a passive beamformer.The system includes a single-antenna transmitter and an N-antenna receiver.Source: §I · pp. 1–2

Main Contributions

• The central idea is to jointly encode the information message in the transmitted signal and RIS configuration instead of fixing the RIS to maximize SNR.The RIS state is selected as a function of the information message.Source: §I · p. 2

Related Work

• Prior joint-encoding work [23] leaves optimality unresolved and fixes the RIS configuration throughout transmission, yielding only minor rate increments for large coding blocks.Its achievable-rate improvement is upper-bounded by log2(B)+log2(N)/n.Source: §I · p. 2

Main Contributions

• The paper derives capacity for any RIS control rate and proves that joint encoding is generally necessary to achieve the maximum information rate.It also characterizes the performance gain of joint encoding in the high-SNR regime.Source: §I · p. 2
• Layered encoding with Successive Cancellation Decoding provides a practical way to encode information in the RIS configuration while retaining separate encoding and decoding strategies.The approach is presented as an achievable scheme rather than the capacity characterization itself.Source: §I · p. 2
• Numerical experiments show that capacity-achieving joint encoding significantly outperforms the standard max-SNR approach at SNR levels of practical interest.The comparison concerns designs that use RIS configuration as an information-bearing signal rather than fixing it solely for SNR enhancement.Source: §I · p. 2

• The RIS assists a single-antenna transmitter communicating with an N-antenna receiver through reflected propagation when the direct link is blocked.Source: §II · pp. 3–4
• Each RIS element selects a phase from a finite hardware-determined set, and the configuration remains fixed for m consecutive transmitted symbols.Source: §II · p. 3
• The block-fading model uses transmitter-to-RIS channel g, RIS-to-receiver channel H, additive i.i.d. Gaussian noise, and an average input-power constraint P.Source: §II · p. 4
• The transmitter and receiver know the quasi-static channels g and H throughout the n-symbol transmission.Source: §II · p. 4
• Unlike prior schemes, the encoder selects both the transmitted codeword and the sequence of RIS configurations based on the message and CSI.Source: §II · pp. 3–4
• Capacity is defined as the maximum achievable rate over joint encoding and decoding schemes, with reliability required as n grows.Source: §II · pp. 4–5

• The channel capacity is obtained by optimizing the joint distribution p(x,θ), so achieving it generally requires joint encoding and decoding over transmitted symbols and RIS states.Source: §III · p. 5
• The capacity optimization is convex and can be solved using standard optimization tools.Source: §III · p. 5
• The standard max-SNR scheme fixes the RIS configuration independently of the message to maximize receiver SNR and generally achieves a smaller rate than capacity.Source: §III · pp. 5–6
• For a fixed constellation cardinality B, the high-SNR limit is maximized by an Amplitude Shift Keying constellation.Source: §III · p. 6
• At high SNR, max-SNR performance is limited to log2(B), while RIS-state modulation increases the achievable rate by K log2(A)/m bits per symbol.Source: §III · p. 6
• At high SNR, independent codebooks for x and θ do not cause performance loss, even though general capacity achievement may require joint distributions.Source: §III · p. 6

• Layered encoding and successive cancellation decoding use separate encoding and decoding while still exploiting information carried by the RIS state to improve over max-SNR.Source: §IV · p. 7
• The message is split into w1, encoded by the RIS phase vector at rate R1, and w2, encoded by the transmitted signal at rate R2.Source: §IV · p. 7
• The receiver first decodes w1 from τ pilot symbols in every block, obtains the RIS phase vector, and then decodes w2.Source: §IV · p. 7
• Proposition 4 gives the achievable layered rate, with ˜m ≜ max{τ + 1, m} and rates R1 and R2 combined through the stated expression.Source: §IV · p. 7
• The RIS-modulation layer contributes rate R1, while the transmitted-signal layer uses water-filling power allocation for rate R2.Source: §IV · p. 8

V. NUMERICAL RESULTS

• At practical SNR levels, joint encoding provides significant gain over max-SNR, while layered encoding also strictly improves over max-SNR.At very low SNR below −20dB, max-SNR is close to optimal and RIS configuration information does not increase the rate.Source: §V · p. 8
• The optimal joint distribution p(x,θ) is non-product at practical SNR levels, except at very large SNR.This supports the value of encoding information jointly in the transmitted signal and RIS configuration.Source: §V · p. 8
• PSK outperforms ASK with max-SNR and layered encoding, whereas ASK outperforms PSK with joint encoding in the high-SNR regime.Among finite input sets of equal size, ASK achieves maximum capacity in the high-SNR regime.Source: §V · p. 8
• With N = 2, K = 2, A = 2, P = 40 dB, 2-ASK input, and τ = 1, layered encoding has identical performance for m = 1 and m = 2.The identical performance is attributed to the use of the pilot symbol.Source: §V · p. 9
• Joint encoding achieves three times the max-SNR rate for m = 1, with the gain reducing to a factor of 1.3 for m = 7.The gain from using RIS states to convey information is expected to decrease as the control-link rate from transmitter to RIS decreases.Source: §V · pp. 8–9

VI. CONCLUSIONS

• For finite input constellations, passive beamforming that maximizes receiver SNR is generally suboptimal in achievable rate.The conclusions concern a fundamental one-transmitter, one-receiver RIS-aided channel model.Source: §VI · p. 9
• The capacity-achieving scheme jointly encodes information in the RIS configuration and the transmitted signal.This differs from fixing the RIS as a passive beamformer.Source: §VI · p. 9
• Layered encoding with successive cancellation decoding is a suboptimal but practical strategy that outperforms passive beamforming at sufficiently high SNR.The strategy uses separate encoding and decoding methods while retaining information modulation through the RIS configuration.Source: §VI · p. 9
• Open problems include low-complexity joint encoding, imperfect or noisy channel-state information, noisy RISs, multiuser or multisurface systems, and security constraints.These extensions define scope boundaries for the present study.Source: §VI · p. 9

A. Proof of Proposition 1

• Because the transmitter directly controls RIS states over symbol blocks, the channel can be represented with input (x,θ) and output Y.The RIS state is controlled for t ∈ [n/m].Source: A. Proof of Proposition 1 · p. 10
• The channel coding theorem gives the capacity in terms of this joint input-output channel.The proof then evaluates the conditional and output differential entropies.Source: A. Proof of Proposition 1 · p. 10
• The conditional output entropy is h(Y|x,θ) = mN log2(πe), while h(Y) is expressed using the defined function fc(x,θ,Z).The output entropy is concave in p(x,θ) for fixed p(Y|x,θ), so standard optimization tools apply.Source: A. Proof of Proposition 1 · p. 10

B. Proof of Proposition 2

• Fixing θ and p(x) makes the RIS state constant throughout transmission, yielding a reduced channel with Gaussian noise z(t) ∼ CN(0, I_N).This reduced representation is the basis for the max-SNR-related achievable-rate analysis.Source: B. Proof of Proposition 2 · pp. 10–11
• The channel coding theorem establishes an achievable rate for the fixed-configuration channel.The resulting entropy expression uses the function fmax-SNR(x,θ,z).Source: B. Proof of Proposition 2 · p. 11

• The proof establishes a high-SNR limit for the considered communication setting.Source: C. Proof of Proposition 3 · p. 11
• The limiting result is therefore attained rather than merely bounded by the uniform-input construction.Source: C. Proof of Proposition 3 · p. 11
• Equality in the limit is achieved using a uniform input distribution p(x,θ).Source: C. Proof of Proposition 3 · p. 11

• The message w is divided into layers w1 and w2, with w2 encoded using symbols after the first τ symbols.Source: D. Proof of Proposition 4 · p. 11
• The first layer uses pilots and is modeled as a point-to-point Gaussian MIMO channel with PSK input and a diag(g) precoder.Source: D. Proof of Proposition 4 · p. 11
• The first-layer rate R1(g, H, τ)/˜m is achievable, where ˜m=max{τ+1,m} accounts for the block-length factor.Source: D. Proof of Proposition 4 · p. 11
• After decoding w1, the receiver knows the RIS state and can process later received columns using the induced channel.Source: D. Proof of Proposition 4 · p. 12
• The induced channel is equivalent to a fast-fading model with channel state information at both transmitter and receiver.Source: D. Proof of Proposition 4 · p. 12
• The second-layer construction uses a uniform input distribution and RIS-dependent amplification while satisfying the average-power constraint.Source: D. Proof of Proposition 4 · p. 12
• The second layer achieves (˜m−τ)R2(g, H)/˜m because only ˜m−τ symbols carry its information.Source: D. Proof of Proposition 4 · p. 12
• The layered scheme achieves the overall rate Rlayered(g, H, τ) by combining both message layers.Source: D. Proof of Proposition 4 · p. 12