Beyond Max-SNR: Joint Encoding for Reconfigurable Intelligent - - PowerPoint PPT Presentation

Beyond Max-SNR: Joint Encoding for Reconfigurable Intelligent Surfaces

Roy Karasik∗, Osvaldo Simeone†, Marco Di Renzo‡, and Shlomo Shamai (Shitz)∗

∗Technion - Israel Institute of Technology †King’s College London ‡CentraleSupélec

ISIT 2020

Supported by the European Research Council (ERC) and by the Information and Communication Technologies (ICT)

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Reconfigurable Intelligent Surface (RIS)

◮ “Anomalous Mirror”

◮ Reflect impinging radio waves towards arbitrary angles ◮ Apply phase shifts ◮ Modify polarization

◮ Considered for future wireless networks as means to shape the wireless propagation channel

Base Station RIS

• Mobile Users

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Related Work

◮ Most prior work proposed to use the RIS as a passive beamformer

◮ Fixed RIS configuration

◮ Examples

◮ Jointly optimizing precoding and beamforming [Wu and Zhang ’18], [Zhang and Zhang ’19], [Perović et al ’19] ◮ Maximizing weighted sum-rate [Guo et al ’19] ◮ Minimizing energy consumption [Huang et al ’18], [Han et al ’19]

◮ Index modulation [Basar ’19]

◮ The receiver antenna for which Signal-to-Noise-Ratio (SNR) is maximized encodes the information bits ◮ The RIS configuration is fixed ◮ Provides minor rate increments for large coding blocks

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Main Contributions

◮ Generalize the RIS-aided communication model

◮ Finite-rate RIS control link

◮ Prove that joint encoding of transmitted signals and RIS configuration is generally necessary to achieve capacity ◮ Characterize the performance gain of joint encoding in the high-SNR regime ◮ Propose an achievable scheme based on layered encoding and successive cancellation decoding

◮ Enables information encoding in the RIS configuration ◮ Supports standard separate encoding and decoding strategies

TX RX RIS Wireless Link w ˆ w Control Link

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System Model

◮ RIS of K elements

◮ Each element applies phase shift to the impinging wireless signal

◮ Single-antenna transmitter ◮ Receiver equipped with N antennas ◮ A message w of nR bits with rate R [bits/symbol] ◮ Transmitted symbols are taken from a constellation B of B = |B| points

TX RX RIS w (nR bits) ˆ w

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System Model

◮ The phase shift applied by each element is chosen from a finite set A of A = |A| distinct values ◮ The transmitter controls the state of the RIS via a finite-rate control link

◮ The phase shifts θ θ θ(t) are fixed for blocks of m consecutive transmitted symbols TX RX RIS w (nR bits) ˆ w Control Link (Rate = 1/m) θ θ θ(1) θ θ θ(2) · · · θ θ θ( n

m)

block (m symbols) codeword (n symbols)

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System Model

◮ No direct link between transmitter and receiver ◮ Quasi-static fading channels

◮ Channel coefficients remain fixed throughout the coding block of n symbols ◮ g ∈ CK×1 — channel from the transmitter to the RIS ◮ H ∈ CN×K — channel from the RIS to the receiver

◮ Full CSI: the transmitter and receiver know g and H

TX RX RIS g H Wireless Link w (nR bits) ˆ w Control Link (Rate = 1/m) θ θ θ(1) θ θ θ(2) · · · θ θ θ( n

m)

block (m symbols) codeword (n symbols)

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System Model

◮ Signal received within each block Y(t) = HS(t)gx(t) + Z(t)

◮ x(t) = (x1(t), . . . , xm(t)) ∈ B1×m — transmitted signal subject to the power constraint E[|xi(t)|2] ≤ P ◮ S(t) — RIS configuration matrix S(t) = ❞✐❛❣

• ejθ1(t), . . . , ejθK(t)

with θ θ θ(t) = (θ1(t), . . . , θK(t)) ∈ AK ◮ Z(t) ∈ CN×m — additive noise matrix whose elements are i.i.d. CN(0, 1)

TX RX RIS g H Wireless Link w (nR bits) ˆ w Control Link (Rate = 1/m) θ θ θ(1) θ θ θ(2) · · · θ θ θ( n

m)

block (m symbols) codeword (n symbols) 8 / 22

System Model

◮ The transmitter encodes message w jointly into the transmitted signal and the configuration of the RIS ◮ Rate R(g, H) is said to be achievable if Pr(ˆ w = w) → 0 as n → ∞ ◮ Capacity definition C(g, H) sup{R(g, H) : R(g, H) is achievable}

◮ The supremum is taken over all joint encoding and decoding schemes

TX RX RIS g H Wireless Link w (nR bits) ˆ w Control Link (Rate = 1/m) θ θ θ(1) θ θ θ(2) · · · θ θ θ( n

m)

block (m symbols) codeword (n symbols) 9 / 22

Joint Encoding: Channel Capacity

◮ Channel Capacity C(g, H) = max

p(x,θ θ θ): E[|xi|2]≤P, x∈Bm, θ θ θ∈AK

1 mI(x,θ θ θ; Y)

◮ The optimization over the joint distribution p(x,θ θ θ) is a convex problem and can be solved using standard tools ◮ Explicit expression for the mutual information can be found in the paper

◮ Capacity achieved by

◮ jointly encoding the message over the phase shift vector θ θ θ and the transmitted signal x ◮ Maximum-likelihood joint decoding at the receiver

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Joint Encoding: Channel Capacity

◮ Standard max-SNR approach

◮ Fixed phase shift vector θ θ θ selected to maximize SNR at the receiver ◮ The channel can be restated as y(t) = HSgx(t) + z(t), with z(t) ∼ CN(0, IN) ◮ Achievable rate Rmax-SNR(g, H) = max

p(x), θ θ θ: E[|x|2]≤P, x∈B, θ θ θ∈AK

I(x; y) ◮ Generally smaller than capacity

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Joint Encoding: Channel Capacity

Some Definitions

◮ Amplitude Shift Keying (ASK) constellation BASK {β, 3β, . . . , (2B − 1)β} with β

• 3P/[3 + 4(B2 − 1)]

◮ Equivalent-input set C

• C : C =
• ejθ1, . . . , ejθK

⊺ x, x ∈ B1×m, θ θ θ ∈ AK ◮ Average rate and capacity R E[R(g, H)] and C E[C(g, H)] where the average is taken with respect to the CSI (g, H)

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Joint Encoding: Channel Capacity

Proposition

The high-SNR limit of the average capacity is given as lim

P→∞ C = log2(|C|)

m . Furthermore, for a given cardinality B = |B| of the constellation, the limit is maximized for the ASK constellation BASK, yielding the limit lim

P→∞ C = log2(B) + K log2(A)

m .

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Joint Encoding: Channel Capacity

◮ High-SNR regime

◮ The rate of the max-SNR scheme is limited to log2(B) ◮ Modulating the RIS state can be used to increase the achievable rate by K log2(A)/m bits per symbol ◮ ASK modulation is optimal ◮ Choosing independent codebooks for input x and RIS configuration θ θ θ does not cause any performance loss p(x,θ θ θ) = p(x)p(θ θ θ)

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Layered Encoding

◮ Achievable scheme based on layered encoding and successive cancellation decoding

◮ Enables information encoding in the RIS configuration ◮ Supports standard separate encoding and decoding strategies

◮ The message w is split into two layers

◮ Layer w1 of rate R1 is encoded by the phase shift vector θ θ θ ◮ Layer w2 of rate R2 is encoded by the transmitted signal x = (x1, . . . , xm) ◮ The first τ symbols x1, . . . , xτ, in vector x are fixed and used as pilots

◮ Successive cancellation decoding

◮ Step 1: The receiver decodes w1 using the first τ vectors y1, . . . , yτ, in every received block Y = (y1, . . . , ym) ◮ Step 2: The receiver obtains vector θ θ θ and uses it as side information to decode w2

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Layered Encoding

◮ Achievable rate Rlayered(g, H, τ) = 1 ˜ mR1(g, H, τ) + ˜ m − τ ˜ m R2(g, H) where ˜ m max{τ + 1, m}

◮ Rate R1(g, H, τ) is derived from the capacity of a point-to-point Gaussian MIMO channel with PSK input [He and Georghiades ’05] ◮ Rate R2(g, H) follows from the “water-filling” power allocation scheme used to achieve the capacity of a fast-fading Gaussian channel with CSI at the transmitter and receiver [Goldsmith and Varaiya ’97] ◮ Explicit expressions for R1(g, H, τ) and R2(g, H) can be found in the paper

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Numerical Results

◮ Compare the capacity with the rates achieved by the max-SNR approach and by the layered-encoding strategy ◮ Uniformly spaced phases A {0, 2π/A, . . . , 2π(A − 1)/A} ◮ Input constellation

◮ Amplitude Shift Keying (ASK) BASK = {β, 3β, . . . , (2B − 1)β} ◮ Phase Shift Keying (PSK) BPSK = { √ Pej2π·0/B, √ Pej2π·1/B, . . . , √ Pej2π·(B−1)/B}

◮ All rates are averaged over the channel vector g ∼ CN(0, IK) and channel matrix H whose elements are i.i.d. as CN(0, 1) and independent of g

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Numerical Results

◮ Average rate as a function of the average power P

◮ Parameters: N = 2, K = 3, A = 2, m = 2, and τ = 1 ◮ Solid and dashed lines are for 4-ASK and QPSK input constellations, respectively

−20 −15 −10 −5 5 10 15 20 25 30 35 40 0.5 1 1.5 2 2.5 3 3.5 P [dB] Average rate [bits per channel use] Joint Encoding (Capacity) Layered Encoding Max-SNR

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Numerical Results

◮ Average rate as a function of the RIS control rate factor m

◮ Parameters: N = 2, K = 2, A = 2, P = 40 dB, τ = 1, and 2-ASK input constellation

1 2 3 4 5 6 7 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 m Average rate [bits per channel use] Joint Encoding (Capacity) Max-SNR Layered Encoding

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Summary

◮ The common approach of using the RIS as a passive beamformer to maximize the SNR at the receiver was shown to be generally suboptimal ◮ The capacity-achieving scheme was proved to jointly encode information in the RIS configuration as well as in the transmitted signal ◮ A practical strategy based on layered encoding and successive cancellation decoding was demonstrated to

• utperform passive beamforming for sufficiently high SNR

levels ◮ Outlook

◮ Capacity for channels with imperfect a priori CSI ◮ Extensions to RIS systems with multiple users/surfaces ◮ Low-complexity joint encoding and decoding strategies that approach capacity

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Contact Info

Thank you

◮ Roy Karasik: royk@campus.technion.ac.il ◮ Osvaldo Simeone: osvaldo.simeone@kcl.ac.uk ◮ Marco Di Renzo: marco.direnzo@l2s.centralesupelec.fr ◮ Shlomo Shamai (Shitz): sshlomo@ee.technion.ac.il

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References

◮ Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network: Joint active and passive beamforming design,” in Proc. IEEE Global Conf. Communications (GLOBECOM), Dec 2018, pp. 1–6. ◮ S. Zhang and R. Zhang, “Capacity characterization for intelligent reflecting surface aided MIMO communication,” arXiv preprint arXiv:1910.01573, 2019. ◮ N. S. Perović, M. Di Renzo, and M. F. Flanagan, “Channel capacity optimization using reconfigurable intelligent surfaces in indoor mmWave environments,” arXiv preprint arXiv:1910.14310, 2019. ◮ H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate optimization for intelligent reflecting surface enhanced wireless networks,” arXiv preprint arXiv:1905.07920, 2019. ◮ C. Huang, G. C. Alexandropoulos, A. Zappone, M. Debbah, and C. Yuen, “Energy efficient multi-user MISO communication using low resolution large intelligent surfaces,” in Proc. IEEE Global Conf. Communications (GLOBECOM), Dec 2018, pp. 1–6. ◮ H. Han, J. Zhao, D. Niyato, M. Di Renzo, and Q.-V. Pham, “Intelligent reflecting surface aided network: Power control for physical-layer broadcasting,” arXiv preprint arXiv:1910.14383, 2019. ◮ E. Basar, “Large intelligent surface-based index modulation: A new beyond MIMO paradigm for 6G,” arXiv preprint arXiv:1904.06704, 2019 ◮ W. He and C. N. Georghiades, “Computing the capacity of a MIMO fading channel under PSK signaling,” IEEE Trans. Inf. Theory, vol. 51, no. 5, pp. 1794–1803, May 2005. ◮ A. J. Goldsmith and P. P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inf. Theory, vol. 43, no. 6, pp. 1986–1992, Nov 1997.

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