On the Capacity of Intelligent Reflecting Surface Aided MIMO - PowerPoint PPT Presentation

On the Capacity of Intelligent Reflecting Surface Aided MIMO Communication Shuowen Zhang and Rui Zhang Department of Electrical and Computer Engineering National University of Singapore ISIT 2020 0

Shuowen Zhang and Rui Zhang, National University of Singapore Motivation  Increasingly high capacity demand for 5G and beyond (e.g., peak speed ~20 Gbps, edge area ~100 Mbps) Virtual/Augmented Mobile Ultra-High Definition (UHD) Cloud Conferencing Reality (VR/AR) Video Streaming (e.g., 4K, 8K) (Image source: google search) 𝑫 = 𝑪𝑼𝐦𝐩𝐡(𝟐 + 𝑰 𝟑 𝑸/𝝉 𝟑 )  Existing technologies for capacity enhancement: Massive MIMO ( ↑ SNR), mmWave ( ↑ bandwidth), full-duplex radio ( ↑ time), etc.  Can we alter the wireless channel 𝐼 as a new degree-of-freedom? 1 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Intelligent Reflecting Surface (IRS)  Intelligent Reflecting Surface (Large Intelligent Surface / Reconfigurable Intelligent Surface)  Massive low-cost passive reflecting elements mounted on a planar surface  Collaboratively alter the propagation channel via joint signal reflection (amplitude and phase shift), also called passive beamforming  Low energy consumption (without use of any transmit RF chains), high spectrum efficiency (full-duplex, noiseless reflection) 2 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore How to Alter Channel: IRS Reflection Optimization IRS element 𝑛 . . . . . . . . . Impinging signal: 𝑦 𝑛 ∈ ℂ Reflected signal: 𝑧 𝑛 ∈ ℂ  Baseband equivalent IRS reflection model: 𝑧 𝑛 = 𝛽 𝑛 𝑦 𝑛 = (𝛾 𝑛 𝑓 𝑘𝜄 𝑛 )𝑦 𝑛 , 𝑛 = 1, … , 𝑁  𝛽 𝑛 = 𝛾 𝑛 𝑓 𝑘𝜄 𝑛 ∈ ℂ : Reflection coefficient at element 𝑛 o 𝛾 𝑛 : Reflection amplitude, 𝛾 𝑛 ∈ [0,1] . o Usually set as 1 due to practical difficulty to jointly tune the phase shift and amplitude at the same time [Yang’16]. o 𝜄 𝑛 : Reflection phase shift, 𝜄 𝑛 ∈ [0,2𝜌) .  𝑁 : # of IRS reflecting elements [Yang’ 16] H. Yang et al ., “Design of resistor-loaded reflectarray elements for both amplitude and phase control,” IEEE Antennas Wireless Propag. Lett. , vol. 16, pp. 1159 – 1162, Nov. 2016 3 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore How to Alter Channel: IRS Reflection Optimization  Single-input single-output (SISO) system: IRS . . . . . . {𝑕 𝑛 } {ℎ 𝑛 } . . . Receiver Transmitter ത ℎ  Effective channel: ෨ ℎ = ത 𝛽 𝑛 ℎ 𝑛 𝑕 𝑛 = ത 𝑁 𝑁 𝑓 𝑘𝜄 𝑛 ℎ 𝑛 𝑕 𝑛 ℎ + σ 𝑛=1 ℎ + σ 𝑛=1 ⋆ = 𝑓 𝑘 (arg{ഥ  Optimal IRS reflection phase shifts: 𝜄 𝑛 ℎ}−arg{ℎ 𝑛 𝑕 𝑛 }) (Align each of the 𝑁 reflected channels with the direct channel) ℎ ⋆ = ( ത |ℎ 𝑛 ||𝑕 𝑛 |)𝑓 𝑘 arg{ഥ  Optimized effective channel: ෨ 𝑁 ℎ} ℎ + σ 𝑛=1  IRS is effective in enhancing SISO channel capacity. 4 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore How to Alter Channel: IRS Reflection Optimization  Multiple-input multiple-output (MIMO) system:  Every IRS reflection coefficient affects multiple transmit-receive channel pairs  IRS reflection needs to strike a balance between multiple spatial data streams  Open Problems:  1. Can IRS enhance the MIMO channel capacity?  2. How to design the IRS reflection to maximally enhance the MIMO channel capacity? 5 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore System Model  Direct channel from transmitter to receiver:  Channel from transmitter to IRS:  Channel from IRS to receiver: 6 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore System Model  Reflection coefficient of IRS element 𝑛 :  Maximum reflection amplitude:  Reflection phase flexibly tunable within [0,2𝜌)  IRS reflection matrix:  Effective channel from transmitter to receiver: 7 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore System Model  Transmitted signal vector:  Transmit covariance matrix:  Transmit power constraint:  Received signal vector:  CSCG noise vector: 8 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore MIMO Channel Capacity  MIMO channel capacity:  Optimal transmit covariance matrix depends on IRS reflection matrix  Fundamental capacity limit of IRS-aided MIMO channel: Joint optimization of IRS reflection matrix 𝝔 and transmit covariance matrix 𝑹 . 9 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Problem Formulation  Capacity maximization of IRS-aided MIMO system via joint IRS reflection and transmit covariance optimization:  Challenge 1: Non-convex problem o Objective function (channel capacity) is not concave over 𝝔 and 𝑹 o Non-convex unit-modulus constraints on IRS reflection coefficients  Challenge 2: 𝝔 and 𝑹 are coupled in the objective function 10 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Proposed Solution: Alternating Optimization  Motivation: Decouple the optimization of 𝛽 𝑛 ’s and 𝑹  Alternating optimization framework: Iteratively optimize one IRS reflection coefficient 𝛽 𝑛 or the transmit covariance matrix 𝑹 with other variables being fixed. 𝑁  Sub-problem 1: Given {𝛽 𝑛 } 𝑛=1 , optimize 𝑹 𝑁 , optimize the remaining IRS  Sub-problem 2: Given 𝑹 and {𝛽 𝑗 , 𝑗 ≠ 𝑛} 𝑗=1 reflection coefficient 𝛽 𝑛 11 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Sub-Problem 1: Optimization of 𝑹 (and consequently given ෩ 𝑁  (P1) with given IRS reflection coefficients {𝛽 𝑛 } 𝑛=1 𝑰 ):  Optimal transmission: Eigen-mode transmission + water-filling power allocation  Number of data streams:  Truncated singular value decomposition (SVD) of ෩ 𝑰 : ,  Optimal transmit covariance matrix 𝑹 to (P1): 12 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Sub-Problem 2: Optimization of 𝛽 𝑛 𝑁 :  (P1) with given 𝑹 and 𝑁 − 1 IRS reflection coefficients {𝛽 𝑗 , 𝑗 ≠ 𝑛} 𝑗=1  Explicit expression of IRS-aided MIMO channel over each IRS reflection coefficient 𝛽 𝑛 : Summation of direct channel and 𝑁 IRS-reflected channels  Channel from transmitter to IRS element 𝑛 :  Channel from IRS element 𝑛 to receiver: 13 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Equivalent Reformulation of (P1-m)  Re-expression of channel capacity  Define eigenvalue decomposition (EVD) of 𝑹 : as  Define and  Channel capacity can be re-expressed as 14 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Equivalent Reformulation of (P1-m)  For convenience, define  𝑩 𝑛 and 𝑪 𝑛 are independent of 𝛽 𝑛  Channel capacity can be expressed as a function of 𝛽 𝑛 :  (P1-m) is equivalent to:  Still non-convex  Optimal solution in closed-form by exploiting problem structure 15 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Optimal Solution to (P1-m)  Based on Lemma 1, channel capacity can be simplified as 16 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Optimal Solution to (P1-m)  In the following, we optimize 𝛽 𝑛 in two cases −1 𝑪 𝑛 (EVD exists)  Case I: Diagonalizable 𝑩 𝑛 −1 𝑪 𝑛 (EVD does not exist)  Case II: Non-diagonalizable 𝑩 𝑛 17 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Optimal Solution to (P1-m): Case I −1 𝑪 𝑛 as  Define the EVD of 𝑩 𝑛  Matrix manipulations: 18 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Optimal Solution to (P1-m): Case II 𝐼 , 𝒗 𝑛 ∈ ℂ 𝑂 𝑠 ×1 , 𝒘 𝑛 ∈ ℂ 𝑂 𝑠 ×1 −1 𝑪 𝑛 = 0 , expressed as 𝑩 𝑛 −1 𝑪 𝑛 = 𝒗 𝑛 𝒘 𝑛  tr 𝑩 𝑛  Matrix manipulations: Independent of 𝛽 𝑛 19 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Optimal Solution to (P1-m)  By combining Case I and Case II, the optimal solution to Problem (P1-m) is  The optimal value of Problem (P1-m) is 20 ISIT 2020

Shuowen Zhang and Rui Zhang, National University of Singapore Overall Algorithm for (P1)  Initialization: 𝑁  Randomly generate 𝑀 independent realizations of {𝛽 𝑛 } 𝑛=1 , obtain optimal 𝑹 for every realization. 𝑁  Select the set of {𝛽 𝑛 } 𝑛=1 and 𝑹 with largest achievable rate as initial point.  Repeat:  For 𝑛 = 1 → 𝑁 , optimize 𝛽 𝑛 by solving (P1-m)  Optimize 𝑹 by solving sub-problem 1  Until no rate improvement can be made by optimizing any variable  Monotonic convergence since optimal solution derived for every sub-problem  Locally optimal solution since no coupling of variables in constraints [Solodov’98]  Polynomial complexity over 𝑁 , 𝑂 𝑢 , and 𝑂 𝑠 [Solodov’98] M . V. Solodov , “On the convergence of constrained parallel variable distribution algorithm,” SIAM J. Optim. , vol. 8, no. 1, pp. 187 – 196, Feb. 1998. 21 ISIT 2020