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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

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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

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Motivation

 Existing technologies for capacity enhancement: Massive MIMO (↑ SNR), mmWave (↑ bandwidth), full-duplex radio (↑ time), etc.

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 Increasingly high capacity demand for 5G and beyond (e.g., peak speed ~20 Gbps, edge area ~100 Mbps)

𝑫 = 𝑪𝑼𝐦𝐩𝐡(𝟐 + 𝑰 𝟑𝑸/𝝉𝟑)

 Can we alter the wireless channel 𝐼 as a new degree-of-freedom?

Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

Virtual/Augmented Reality (VR/AR) Mobile Ultra-High Definition (UHD) Video Streaming (e.g., 4K, 8K) Cloud Conferencing

(Image source: google search)

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Intelligent Reflecting Surface (IRS)

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  • 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)  Intelligent Reflecting Surface (Large Intelligent Surface / Reconfigurable Intelligent Surface)

Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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How to Alter Channel: IRS Reflection Optimization

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 Baseband equivalent IRS reflection model: 𝑧𝑛 = 𝛽𝑛𝑦𝑛 = (𝛾𝑛𝑓𝑘𝜄𝑛)𝑦𝑛, 𝑛 = 1, … , 𝑁

  • 𝛽𝑛 = 𝛾𝑛𝑓𝑘𝜄𝑛 ∈ ℂ: Reflection coefficient at element 𝑛
  • 𝛾𝑛: Reflection amplitude, 𝛾𝑛 ∈ [0,1].
  • Usually set as 1 due to practical difficulty to jointly tune the phase

shift and amplitude at the same time [Yang’16].

  • 𝜄𝑛: Reflection phase shift, 𝜄𝑛 ∈ [0,2𝜌).
  • 𝑁: # of IRS reflecting elements

Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

. . . . . . . . . Impinging signal: 𝑦𝑛 ∈ ℂ Reflected signal: 𝑧𝑛 ∈ ℂ

[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

IRS element 𝑛

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How to Alter Channel: IRS Reflection Optimization

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 Single-input single-output (SISO) system:

  • Effective channel: ෨

ℎ = ത ℎ + σ𝑛=1

𝑁

𝛽𝑛ℎ𝑛𝑕𝑛 = ത ℎ + σ𝑛=1

𝑁

𝑓𝑘𝜄𝑛ℎ𝑛𝑕𝑛

  • Optimal IRS reflection phase shifts: 𝜄𝑛

⋆ = 𝑓𝑘 (arg{ഥ ℎ}−arg{ℎ𝑛𝑕𝑛})

(Align each of the 𝑁 reflected channels with the direct channel)

  • Optimized effective channel: ෨

ℎ⋆ = ( ത ℎ + σ𝑛=1

𝑁

|ℎ𝑛||𝑕𝑛|)𝑓𝑘 arg{ഥ

ℎ}

  • IRS is effective in enhancing SISO channel capacity.

Transmitter Receiver IRS

ത ℎ {ℎ𝑛} {𝑕𝑛}

. . . . . . . . .

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How to Alter Channel: IRS Reflection Optimization

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 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?

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System Model

 Direct channel from transmitter to receiver:  Channel from transmitter to IRS:  Channel from IRS to receiver:

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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System Model

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 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:

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System Model

 Transmitted signal vector:

  • Transmit covariance matrix:
  • Transmit power constraint:

 Received signal vector:

  • CSCG noise vector:

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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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
  • f IRS reflection matrix 𝝔 and transmit covariance matrix 𝑹.

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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Problem Formulation

 Capacity maximization of IRS-aided MIMO system via joint IRS reflection and transmit covariance optimization:

  • Challenge 1: Non-convex problem
  • Objective function (channel capacity) is not concave over 𝝔 and 𝑹
  • Non-convex unit-modulus constraints on IRS reflection coefficients
  • Challenge 2: 𝝔 and 𝑹 are coupled in the objective function

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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Proposed Solution: Alternating Optimization

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 Motivation: Decouple the optimization of 𝛽𝑛’s and 𝑹  Alternating optimization framework: Iteratively optimize one IRS reflection coefficient 𝛽𝑛 or the transmit covariance matrix 𝑹 with other variables being fixed.

  • Sub-problem 1: Given {𝛽𝑛}𝑛=1

𝑁

, optimize 𝑹

  • Sub-problem 2: Given 𝑹 and {𝛽𝑗, 𝑗 ≠ 𝑛}𝑗=1

𝑁 , optimize the remaining IRS

reflection coefficient 𝛽𝑛

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Sub-Problem 1: Optimization of 𝑹

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 (P1) with given IRS reflection coefficients {𝛽𝑛}𝑛=1

𝑁

(and consequently given ෩ 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):
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Sub-Problem 2: Optimization of 𝛽𝑛

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 (P1) with given 𝑹 and 𝑁 − 1 IRS reflection coefficients {𝛽𝑗, 𝑗 ≠ 𝑛}𝑗=1

𝑁 :

 Explicit expression of IRS-aided MIMO channel over each IRS reflection coefficient 𝛽𝑛: Summation of direct channel and 𝑁 IRS-reflected channels

  • Channel from transmitter to IRS element 𝑛:
  • Channel from IRS element 𝑛 to receiver:
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Equivalent Reformulation of (P1-m)

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 Re-expression of channel capacity

  • Define eigenvalue decomposition (EVD) of 𝑹: as
  • Define and
  • Channel capacity can be re-expressed as
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Equivalent Reformulation of (P1-m)

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 For convenience, define

  • 𝑩𝑛 and 𝑪𝑛 are independent of 𝛽𝑛

 Channel capacity can be expressed as a function of 𝛽𝑛:  (P1-m) is equivalent to:

  • Still non-convex
  • Optimal solution in closed-form by exploiting problem structure
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Optimal Solution to (P1-m)

 Based on Lemma 1, channel capacity can be simplified as

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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 In the following, we optimize 𝛽𝑛 in two cases

  • Case I: Diagonalizable 𝑩𝑛

−1𝑪𝑛 (EVD exists)

  • Case II: Non-diagonalizable 𝑩𝑛

−1𝑪𝑛 (EVD does not exist)

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Optimal Solution to (P1-m)

ISIT 2020 Shuowen Zhang and Rui Zhang, National University of Singapore

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Optimal Solution to (P1-m): Case I

Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 Define the EVD of 𝑩𝑛

−1𝑪𝑛 as

 Matrix manipulations:

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Optimal Solution to (P1-m): Case II

Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

Independent of 𝛽𝑛  tr 𝑩𝑛

−1𝑪𝑛 = 0, expressed as 𝑩𝑛 −1𝑪𝑛 = 𝒗𝑛𝒘𝑛 𝐼 , 𝒗𝑛 ∈ ℂ𝑂𝑠×1, 𝒘𝑛 ∈ ℂ𝑂𝑠×1

 Matrix manipulations:

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 By combining Case I and Case II, the optimal solution to Problem (P1-m) is  The optimal value of Problem (P1-m) is

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Optimal Solution to (P1-m)

Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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Overall Algorithm for (P1)

 Initialization:

  • Randomly generate 𝑀 independent realizations of {𝛽𝑛}𝑛=1

𝑁

, obtain

  • ptimal 𝑹 for every realization.
  • Select the set of {𝛽𝑛}𝑛=1

𝑁

and 𝑹 with largest achievable rate as initial point.  Repeat:

  • For 𝑛 = 1 → 𝑁, optimize 𝛽𝑛 by solving (P1-m)
  • Optimize 𝑹 by solving sub-problem 1

 Until no rate improvement can be made by optimizing any variable

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

 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, 𝑂𝑢, and 𝑂𝑠

[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.
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 Convergence of proposed alternating optimization algorithm (𝑁 = 40, 𝑂𝑢 = 𝑂𝑠 = 4):

  • Monotonic and fast convergence (~5 outer iterations)
  • Significant rate gain as compared to initialization

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Numerical Examples

Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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  • MIMO channel capacity can be enhanced by deploying IRS, enhancement

increases with number of IRS reflecting elements.

  • Proposed solution achieves best performance among various benchmarks.
  • Various key parameters of MIMO channel can be improved, e.g., channel

power, condition number, rank.

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Numerical Examples

 Achievable rate versus number of reflecting elements:

Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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Conclusions

 First work on capacity maximization for IRS-aided MIMO systems  Open Problems:

  • 1. Can IRS enhance the MIMO channel capacity? Yes
  • 2. How to design the IRS reflection to maximally enhance the MIMO

channel capacity?

  • Alternating optimization based algorithm that finds a locally
  • ptimal solution in polynomial time
  • Closed-form optimal solution of each IRS reflection coefficient (may

also be used under other system setups!)

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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Extensions

 Journal version:

  • S. Zhang and R. Zhang, “Capacity characterization for intelligent reflecting

surface aided MIMO communications,” to appear in IEEE J. Sel. Areas Commun., [Online]. arXiv preprint: 1910.01573  New results:

  • Lower-complexity algorithms tailored for IRS-aided MISO/SIMO systems
  • Lower-complexity algorithms tailored for IRS-aided MIMO systems in low-

SNR and high-SNR regimes

  • Capacity maximization for IRS-aided MIMO-OFDM systems under

frequency-selective fading channels

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020

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Thank you!

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Shuowen Zhang and Rui Zhang, National University of Singapore ISIT 2020