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) 1 / 22
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 RIS Base Station Mobile Users � � � 2 / 22
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 3 / 22
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 RIS Control Link Wireless Link ˆ w w TX RX 4 / 22
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 RIS ˆ w w TX RX ( nR bits) 5 / 22
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 RIS Control Link (Rate = 1 / m ) ˆ w w θ θ θ ( 1 ) θ θ θ ( 2 ) · · · θ θ θ ( n m ) TX RX ( nR bits) block ( m symbols) codeword ( n symbols) 6 / 22
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 ∈ C K × 1 — channel from the transmitter to the RIS ◮ H ∈ C N × K — channel from the RIS to the receiver ◮ Full CSI: the transmitter and receiver know g and H RIS Control Link (Rate = 1 / m ) g H Wireless Link ˆ w w · · · θ θ θ ( n m ) θ θ θ ( 1 ) θ θ ( 2 ) θ TX RX ( nR bits) block ( m symbols) codeword ( n symbols) 7 / 22
System Model ◮ Signal received within each block Y ( t ) = HS ( t ) gx ( t ) + Z ( t ) ◮ x ( t ) = ( x 1 ( t ) , . . . , x m ( t )) ∈ B 1 × m — transmitted signal subject to the power constraint E [ | x i ( t ) | 2 ] ≤ P ◮ S ( t ) — RIS configuration matrix � e j θ 1 ( t ) , . . . , e j θ K ( t ) � S ( t ) = ❞✐❛❣ with θ θ θ ( t ) = ( θ 1 ( t ) , . . . , θ K ( t )) ∈ A K ◮ Z ( t ) ∈ C N × m — additive noise matrix whose elements are i.i.d. CN ( 0 , 1 ) RIS Control Link (Rate = 1 / m ) g H Wireless Link ˆ w w θ θ ( 1 ) θ θ θ ( 2 ) θ · · · θ θ θ ( n m ) TX RX ( nR bits) 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 RIS Control Link (Rate = 1 / m ) g H Wireless Link ˆ w w θ θ θ ( 1 ) θ θ θ ( 2 ) · · · θ θ θ ( n m ) TX RX ( nR bits) block ( m symbols) codeword ( n symbols) 9 / 22
Joint Encoding: Channel Capacity ◮ Channel Capacity 1 θ C ( g , H ) = max mI ( x ,θ θ ; Y ) p ( x ,θ θ ): θ E [ | x i | 2 ] ≤ P , x ∈B m , θ θ ∈A K θ ◮ 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 10 / 22
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 ) = HSg x ( t ) + z ( t ) , with z ( t ) ∼ CN ( 0 , I N ) ◮ Achievable rate R max-SNR ( g , H ) = max I ( x ; y ) p ( x ) , θ θ θ : E [ | x | 2 ] ≤ P , x ∈B , θ θ θ ∈A K ◮ Generally smaller than capacity 11 / 22
Joint Encoding: Channel Capacity Some Definitions ◮ Amplitude Shift Keying (ASK) constellation B ASK � { β, 3 β, . . . , ( 2 B − 1 ) β } 3 P / [ 3 + 4 ( B 2 − 1 )] � with β � ◮ Equivalent-input set � ⊺ � � θ ∈ A K � e j θ 1 , . . . , e j θ K x , x ∈ B 1 × m , θ C � C : C = θ ◮ 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 ) 12 / 22
Joint Encoding: Channel Capacity Proposition The high-SNR limit of the average capacity is given as P →∞ C = log 2 ( |C| ) lim . m Furthermore, for a given cardinality B = |B| of the constellation, the limit is maximized for the ASK constellation B ASK , yielding the limit P →∞ C = log 2 ( B ) + K log 2 ( A ) lim . m 13 / 22
Joint Encoding: Channel Capacity ◮ High-SNR regime ◮ The rate of the max-SNR scheme is limited to log 2 ( B ) ◮ Modulating the RIS state can be used to increase the achievable rate by K log 2 ( 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 ) 14 / 22
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 w 1 of rate R 1 is encoded by the phase shift vector θ θ θ ◮ Layer w 2 of rate R 2 is encoded by the transmitted signal x = ( x 1 , . . . , x m ) ◮ The first τ symbols x 1 , . . . , x τ , in vector x are fixed and used as pilots ◮ Successive cancellation decoding ◮ Step 1: The receiver decodes w 1 using the first τ vectors y 1 , . . . , y τ , in every received block Y = ( y 1 , . . . , y m ) ◮ Step 2: The receiver obtains vector θ θ θ and uses it as side information to decode w 2 15 / 22
Layered Encoding ◮ Achievable rate mR 1 ( g , H , τ ) + ˜ m − τ R layered ( g , H , τ ) = 1 R 2 ( g , H ) ˜ ˜ m m � max { τ + 1 , m } where ˜ ◮ Rate R 1 ( g , H , τ ) is derived from the capacity of a point-to-point Gaussian MIMO channel with PSK input [He and Georghiades ’05] ◮ Rate R 2 ( 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 R 1 ( g , H , τ ) and R 2 ( g , H ) can be found in the paper 16 / 22
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 , 2 π/ A , . . . , 2 π ( A − 1 ) / A } ◮ Input constellation ◮ Amplitude Shift Keying (ASK) B ASK = { β, 3 β, . . . , ( 2 B − 1 ) β } ◮ Phase Shift Keying (PSK) √ √ √ Pe j 2 π · 0 / B , Pe j 2 π · 1 / B , . . . , Pe j 2 π · ( B − 1 ) / B } B PSK = { ◮ All rates are averaged over the channel vector g ∼ CN ( 0 , I K ) and channel matrix H whose elements are i.i.d. as CN ( 0 , 1 ) and independent of g 17 / 22
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 3 . 5 Average rate [bits per channel use] 3 2 . 5 2 1 . 5 1 Joint Encoding (Capacity) 0 . 5 Layered Encoding Max-SNR 0 − 20 − 15 − 10 − 5 0 5 10 15 20 25 30 35 40 P [dB] 18 / 22
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 3 Joint Encoding (Capacity) 2 . 8 Average rate [bits per channel use] Max-SNR 2 . 6 Layered Encoding 2 . 4 2 . 2 2 1 . 8 1 . 6 1 . 4 1 . 2 1 0 . 8 1 2 3 4 5 6 7 m 19 / 22
Recommend
More recommend
