Analysis and Optimization of an Intelligent Reflecting - - PowerPoint PPT Presentation

Analysis and Optimization of an Intelligent Reflecting Surface-assisted System With Interference

Ying Cui

Department of Electrical Engineering Shanghai Jiao Tong University

• Sept. 2020

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Outline

Introduction System model Rate analysis Rate optimization Comparision with system without IRS Numerical results Conclusion

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Outline

Introduction System model Rate analysis Rate optimization Comparision with system without IRS Numerical results Conclusion

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Background

◮ Current 5G solutions require high hardware cost and energy

consumption

◮ Finding spectral and energy efficient, and yet cost-effective

solutions for 6G wireless networks is still imperative

◮ Intelligent Reflecting Surface (IRS) is envisioned to be a

promising solution

◮ An IRS consists of nearly passive, low-cost and reflecting

elements whose phase shifts can be adjusted independently by smart switches

◮ Signals reflected by an IRS can add constructively with those

from the other paths to enhance the desired signal power, or destructively to cancel the interference

◮ IRSs can be practically deployed and integrated in wireless

networks with low cost

◮ low profile, light weight, conformal geometry, and easy to

mount/remove them on/from the wall, ceiling, building, etc

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Typical IRS applications

(a) User at dead zone. (b) Physical layer security. (c) User at cel- l edge. (d) Massive D2D communications.

Figure: Typical IRS applications [Wu & Zhang (2020)]

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Previous work

◮ Consider optimal phase shift (and beamforming) design for IRS-assisted

systems where one BS serves one or multiple users with the help of one or multiple IRSs

◮ Instantaneous CSI-adaptive phase shift design: phase shifts are

adjusted based on instantaneous CSI (assumed known)

◮ Maximize the weighted sum rate [Nadeem et al. (2019); Yang

et al. (2019); Guo et al. (2019); Wu & Zhang (2019)], and energy efficiency [Yu et al. (2019b,a); Huang et al. (2019)]

◮ Minimize the transmission power [Wu & Zhang (2019); Jiang

& Shi (2019)]

◮ Quasi-static phase shift design: phase shifts are determined by CSI

statistics (Line-of-Sight (LoS) components and distributions of Non-Line-of-Sight (NLoS) components) and do not change with instantaneous CSI (assumed unknown)

◮ Consider slowly varying Non-line-of sight (NLoS) components,

and minimize the outage probability [Zhang et al. (2019),Guo et al. (2020)]

◮ Consider fast varying NLoS components, and maximize the

ergodic rate [Han et al. (2019); Nadeem et al. (2020)], [Hu et al. (2020)]

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Previous work

◮ Quasi-static phase shift design has less frequent phase

adjustment than instantaneous CSI-adaptive phase shift design

◮ All the aforementioned works ignore interference from other

transmitters

◮ However, interference usually has a severe impact, especially

in dense networks or for cell-edge users

◮ Consider optimal phase shift and beamforming design for

IRS-assisted systems where multiple BSs serve their own users with the help of one IRS

◮ Instantaneous CSI-adaptive phase shift design in the presence

• f interference

◮ Consider fast varying NLoS components and maximize the

weighted sum average rate [Pan et al. (2020)], [Xie et al. (2020); Ni et al. (2020)]

◮ It is highly desirable to obtain cost-efficient quasi-static design

for IRS-assisted systems with interference

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Outline

Introduction System model Rate analysis Rate optimization Comparision with system without IRS Numerical results Conclusion

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Network model

◮ A multi-antenna signal BS S, equipped with a URA of

MS × NS antennas, serves a single-antenna user U

◮ A multi-antenna interference BS I, equipped with a URA of

MI × NI antennas, serves a single-antenna user U′

◮ A multi-element IRS, equipped with a URA of MR × NR

antennas, is installed on the wall of a high-rise building

◮ Channels between the BSs and users follow Rayleigh fading

◮ scattering is often rich near the ground

◮ Channels between the IRS and BSs/user follow Rician fading

◮ scattering is much weaker far from the ground

• SJTU

Ying Cui 9 / 56

Channel model

◮ Rayleigh channels between the BSs and the users:

hH

i = √αi˜

hH

i , i = SU, IU, IU′

◮ αi > 0 is the distance-dependent path losses ◮ The elements of ˜

hH

i are i.i.d. according to CN(0, 1)

◮ Rician channels between the IRS and the BSs (users):

HcR =√αcR

• KcR

KcR + 1 ¯ HcR +

• 1

KcR + 1 ˜ HcR

• , c = S, I

hRU =√αRU

• KRU

KRU + 1 ¯ hRU +

• 1

KRU + 1 ˜ hRU

• ◮ αcR, αRU > 0 denote the distance-dependent path losses and

KcR, KRU ≥ 0 denote the Rician factors, where i = S, I

◮ ¯

HcR, ¯ hRU represent the deterministic normalized LoS components, with unit-modulus elements

◮ ˜

HcR, ˜ hRU represent the normalized NLoS components, with elements i.i.d. according to CN(0, 1)

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Channel model

◮ Define:

f (θ(h), θ(v), m, n) 2π d λ sin θ(v)((m − 1) cos θ(h) + (n − 1) sin θ(h)) Am,n(θ(h), θ(v), M, N)

• ejf (θ(h),θ(v),m,n)

m=1,...,M,n=1,...,N

a(θ(h), θ(v), M, N) rvec

• Am,n(θ(h), θ(v), M, N)
• ◮ λ denotes the wavelength of transmission signals

◮ d (≤ λ

2 ) denotes the distance between adjacent elements or

antennas in each row and each column of the URAs

◮ ¯

HcR and ¯ hH

RU are modeled as:

¯ HcR =aH(δ(h)

cR , δ(v) cR , MR, NR)a(ϕ(h) cR , ϕ(v) cR , Mc, Nc), c = S, I

¯ hH

RU =a(ϕ(h) RU, ϕ(v) RU, MR, NR)

◮ δ(h)

cR

• δ(v)

cR

• , ϕ(h)

cR

• ϕ(v)

cR

• and ϕ(h)

RU

• ϕ(v)

RU

• represent the

corresponding azimuth (elevation) angles

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Quasi-static phase shift design

◮ Phase shifts of the IRS φ (φm,n)m∈MR ,n∈NR with φm,n ∈ [0, 2π) is

fixed, where MR {1, 2, ..., MR}, NR {1, 2, ..., NR}

◮ Define Φ(φ) diag

• rvec
• ejφm,n

m∈MR ,n∈NR

• ∈ CMR NR ×MR NR

◮ Considering linear beamforming at BSs S, I, the signal received at user U:

Y

• PS(hH

RUΦ(φ)HSR + hH SU)wSXS +

• PI
• hH

RUΦ(φ)HIR + hH IU

• wIXI + Z

◮ wS ∈ CMS NS ×1 and wI ∈ CMI NI ×1 denote the normalized

beamforming vectors at BS S and BS I, where ||wS||2

2 = 1 and

||wI||2

2 = 1 ◮ XS and XI are the information symbols for user U and user U′,

respectively, with E

• |XS|2

= 1 and E

• |XI|2

= 1, and Z ∼ CN(0, σ2) is the additive white gaussian noise (AWGN)

◮ hH RUΦ(φ)HcR + hH cU represents the equivalent channel between BS c

and user U via the IRS

◮ Assume that user U knows (hH

RUΦ(φ)HSR + hH SU)wS, but does not know

• hH

RUΦ(φ)HIR + hH IU

• wI

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Instantaneous CSI case

◮ Assumptions:

◮ CSI of the equivalent channel between BS S and user U, i.e.,

hH

RUΦ(φ)HSR + hH SU, is known at BS S

◮ CSI of the channel between BS I and user U′, i.e., hIU′, is

known at BS I

◮ Consider instantaneous CSI-adaptive MRT beamformers:

w(instant)

S

=

• hH

RUΦ(φ)HSR + hH SU

H

• hH

RUΦ(φ)HSR + hH SU

• 2

, w(instant)

I

= hIU′ ||hIU′||2

◮ w(instant)

S

and w(instant)

I

are chosen to enhance the signals received at user U and user U′

◮ w(instant)

S

is optimal for the average rate maximization

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Instantaneous CSI case

◮ The SINR at user U:1

γ(instant)(φ) = PS

• hH

RUΦ(φ)HSR + hH SU

• 2

2

PIE

• (hH

RUΦ(φ)HIR + hH IU) hIU′ ||hIU′||2

• 2

+ σ2

◮ The average rate for the IRS-assisted system with

interference: C (instant)(φ) = E

• log2
• 1 + γ(instant)(φ)
• ◮ log2
• 1 + γ(instant)(φ)
• can be achieved by coding over one

coherence time interval

◮ C (instant)(φ) with PI = 0 reduces to the average rate in [Han

et al. (2019)]

1Treat

• hH

RUΦ(φ)HIR + hH IU

• wIXI ∼CN
• 0, E
• hH

RUΦ(φ)HIR + hH IU

• wI
• 2

, which corresponds to the worst-case noise.

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Statistic CSI case

◮ Assumptions:

◮ Only the CSI of the LoS components hH

RU, HSR are known at

BS S

◮ No channel knowledge is known at BS I

◮ Consider statistical CSI-adaptive MRT beamformers:

w(statistic)

S

= ¯ hH

RUΦ(φ)¯

HSR H

• ¯

hH

RUΦ(φ)¯

HSR

• 2

, w(statistic)

I

= 1 √MINI 1MI NI

◮ w(statistic)

S

is approximately optimal for the ergodic rate maximization (optimal for maximizing an upper bound)

◮ Any wI with ||wI||2

2 = 1 achieves the same ergodic rate for

user U′

◮ Have lower costs on channel estimation and beamforming

adjustment

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Statistic CSI case

◮ The SINR at user U:

γ(statistic)(φ) = PS

• hH

RUΦ(φ)HSR + hH SU

(¯

hH

RU Φ(φ)¯

HSR)

H

||¯

hH

RU Φ(φ)¯

HSR||2

• 2

PIE

• hH

RUΦ(φ)HIR + hH IU

• 1

√

MI NI 1

• 2

+ σ2

◮ The ergodic rate for the IRS-assisted system with interference:

C (statistic)(φ) = E

• log2
• 1 + γ(statistic)(φ)
• ◮ Code over a large number of channel coherence time intervals

◮ C (statistic)(φ) with PI = 0 is recently studied in [Hu et al.

(2020)]

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Outline

Introduction System model Rate analysis Rate optimization Comparision with system without IRS Numerical results Conclusion

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Notations

◮ Define:

τcRU KcRKRU (KcR + 1)(KRU + 1) θcRU,m,n f

• ϕ(h)

RU, ϕ(v) RU, m, n

• − f
• δ(h)

cR , δ(v) cR , m, n

• θIR,m,n f
• ϕ(h)

IR , ϕ(v) IR , m, n

• , m ∈ MR, n ∈ NR

◮ τcRU increases with KcR and KRU ◮ f

• ϕ(h)

RU, ϕ(v) RU, m, n

f

• ϕ(h)

IR , ϕ(v) IR , m, n

• represents the difference
• f the phase change over the LoS component between the (m, n)-th

element of the IRS (the (m, n)-th antenna of BS I) and user U (the IRS) and the phase change over the LoS component between the (1, 1)-th element of the IRS (the (1,1)-th antenna of BS I) and user U (the IRS)

◮ f

• δ(h)

cR , δ(v) cR , m, n

• represents the difference of the phase change
• ver the LoS component between BS c and the (m, n)-th element
• f the IRS and the phase change over the LoS component between

BS c and the (1, 1)-th element of the IRS

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Notations

◮ Define: ASRU,LoS PSMSNSαSRαRUτSRU, AIU PI αIU + σ2, A(Q)

SU

• PSMSNSαSU,

Q = instant, PSαSU, Q = statistic, A(Q)

IRU,LoS

• PI αIRαRUτIRU,

Q = instant, PI αIRαRUτIRU

yIR MI NI ,

Q = statistic, A(Q)

SRU,NLoS

• PSMSNSαSRαRUMRNR(1 − τSRU),

Q = instant, PSMSNSαSRαRUMRNR

• 1 − τSRU −

MS NS −1 MS NS (KSR +1)

• ,

Q = statistic, A(Q)

IRU,NLoS

• PI αIRαRUMRNR(1 − τIRU),

Q = instant, PI αIRαRUMRNR

• 1 − τIRU + τIRU (yIR −MI NI )

MI NI KRU

• ,

Q = statistic, yIR

• MI
• m=1

NI

• n=1

ejθIR,m,n

• 2

, ycRU(φ)

• MR
• m=1

NR

• n=1

ejθcRU,m,n+jφm,n

• 2

.

◮ McNcycRU(φ) represents the sum channel power of the LoS components

• f the indirect signal link between BS c and user U via the IRS

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Rate analysis

Theorem 1 (Upper Bound of average rate or ergodic rate)

For Q = instant or statistic, C (Q)(φ) ≤ log2

• 1+

ASRU,LoSySRU(φ) + A(Q)

SRU,NLoS + A(Q) SU

A(Q)

IRU,LoSyIRU(φ) + A(Q) IRU,NLoS + AIU

• γ(Q)

ub (φ)

• C (Q)

ub (φ).

◮ Proof: Jensen inequality ◮ C (Q) ub (φ) is a good approximation of C (Q)(φ), and can

facilitate the evaluation and optimization for it

◮ For all φ, C (Q) ub (φ) increases with PS, MS, NS, αSR and αSU ◮ For all φ, C (Q) ub (φ) decreases with PI, αIR, αIU and σ2

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Rate analysis

Corollary 1

(i) A(instant)

SRU,NLoS > A(statistic) SRU,NLoS and A(instant) SU

> A(statistic)

SU

(ii) If PI > 0 and yIR > MINI, A(instant)

IRU,LoS < A(statistic) IRU,LoS and A(instant) IRU,NLoS

< A(statistic)

IRU,NLoS

◮ Corollary 1 (i): the received signal power at user U in the

instantaneous CSI case always surpasses that in the statistical CSI case, at any phase shifts

◮ Corollary 1 (ii): if the placement of the URA at the

interference BS and the locations of the interference BS and IRS satisfy certain condition, the received interference power at user U in the instantaneous CSI case is weaker than that in the statistical CSI case, at any phase shifts

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Rate analysis

Corollary 2

(i) If PI < ε for some ε > 0, γ(instant)

ub

(φ) > γ(statistic)

ub

(φ), for all φ. (ii) If yIR > MINI, γ(instant)

ub

(φ) > γ(statistic)

ub

(φ), for all φ.

◮ Corollary 2 (i): if the interference is weak, the average rate in the

instantaneous CSI case is greater than the ergodic rate in the statistical CSI case, at any phase shifts

◮ Corollary 2 (ii): if the placement of the URA at the interference BS

and the locations of the interference BS and IRS satisfy certain condition, the average rate in the instantaneous CSI case is greater than the ergodic rate in the statistical CSI case, at any phase shifts

◮ Corollary 2 reveals the advantage2 of CSI of the NLoS components

in improving the receive SINR at user U

2γ(instant) ub

(φ) > γ(statistic)

ub

(φ) does not always hold, as the interference powers in the two cases are different.

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Outline

Introduction System model Rate analysis Rate optimization Comparision with system without IRS Numerical results Conclusion

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Problem formulation

Problem (Average or Ergodic Rate/SINR Maximization)

For Q = instant or statistic, max

φ

γ(Q)

ub (φ)

s.t. φm,n ∈ [0, 2π), m ∈ MR, n ∈ NR

◮ An optimal solution depends on the LoS components and the

distributions of the NLoS components

◮ It is a challenging non-convex problem

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Notations

◮ Define: η(Q) =A(Q)

SRU,LoS

• A(Q)

IRU,NLoS + A(Q) IU

• − A(Q)

IRU,LoS

• A(Q)

SRU,NLoS + A(Q) SU

• Λ(x) x − 2π

x 2π

• ,

x ∈ R B(t)

SRU,m,n 2ASRU,LoS

• k=m,l=n

ej

• φ(t)

k,l +θSRU,k,l

• B(Q,t)

S,m,n ASRU,LoS

 1 +

• k=m,l=n

ej

• φ(t)

k,l +θSRU,k,l

• 2

 + A(Q)

SRU,NLoS + A(Q) SU

B(Q,t)

IRU,m,n 2A(Q) IRU,LoS

• k=m,l=n

ej

• φ(t)

k,l +θIRU,k,l

• B(Q,t)

I,m,n A(Q) IRU,LoS

 1 +

• k=m,l=n

ej

• φ(t)

k,l +θIRU,k,l

• 2

 + A(Q)

IRU,NLoS + AIU

B(Q,t)

1,m,n B(Q,t) S,m,nB(Q,t) IRU,m,n cos B(t) ∠IRU,m,n − B(t) SRU,m,nB(Q,t) I,m,n cos B(t) ∠SRU,m,n

B(Q,t)

2,m,n B(Q,t) S,m,nB(Q,t) IRU,m,n sin B(t) ∠IRU,m,n − B(t) SRU,m,nB(Q,t) I,m,n sin B(t) ∠SRU,m,n SJTU Ying Cui 25 / 56

Optimal solutions in special cases

Theorem 2 (Optimal Solution in Special Case (i))

Suppose MR = NR = 1. Then, any φ(Q)∗ satisfying φm,n ∈ [0, 2π) is optimal, and ySRU

• φ(Q)∗

= yIRU

• φ(Q)∗

= 1.

◮ The phase shift of the single element has no impact on the

average or ergodic rate (as ySRU(φ) = yIRU(φ) = 1 for all φ)

◮ The channel between each BS and user U follows Rayleigh

fading

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Optimal solutions in special cases

Theorem 3 (Optimal Solution in Special Case (ii))

Suppose MRNR > 1, δ(h)

SR = δ(h) IR , δ(v) SR = δ(v) IR and η(Q) > 0. Then, any

φ(Q)∗ with φ(Q)∗

m,n = Λ (α − θIRU,m,n) , m ∈ MR, n ∈ NR, for all α ∈ R, is

• ptimal, and ySRU
• φ(Q)∗

= yIRU

• φ(Q)∗

= M2

RN2 R.

◮ The phase shifts that achieve the maximum sum channel

power of the LoS components of the indirect single and interference links, i.e., M2

RN2 R, also maximize the average or

ergodic rate

◮ ySRU(φ) = yIRU(φ) y(φ), γ(Q)

ub = ˜

γ(Q)

ub ◦ y, η(Q) reflects d˜ γ(Q)

ub

dy

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Optimal solutions in special cases

Theorem 4 (Optimal Solution in Special Case (iii))

Suppose MRNR > 1, δ(h)

SR = δ(h) IR , δ(v) SR = δ(v) IR and η(Q) ≤ 0. If NR 2 ∈ N, any

φ(Q)∗ satisfying φ(Q)∗

m,2i − φ(Q)∗ m,2i−1 = (2ki + 1)π − (θIRU,m,2i − θIRU,m,2i−1)

for some ki ∈ Z, m ∈ MR, i = 1, ..., NR

2 is optimal, ySRU(φ(Q)∗)

= yIRU(φ(Q)∗) = 0.

◮ The phase shifts that achieve the minimum sum channel

power of the LoS components of the indirect single and interference links, i.e., 0, also maximize the average or ergodic rate

◮ ySRU(φ) = yIRU(φ) y(φ), γ(Q)

ub = ˜

γ(Q)

ub ◦ y, η(Q) reflects d˜ γ(Q)

ub

dy

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Optimal solutions in special cases

Theorem 5 (Optimal Solution in Special Case (iv))

When PI = 0, any φ(Q)∗ with φ(Q)∗

m,n = Λ (α − θSRU,m,n) ,

m ∈ MR, n ∈ NR, for all α ∈ R, is optimal, and ySRU

• φ(Q)∗

= M2

RN2 R.

◮ The phase shifts that achieve the maximum sum channel

power of the LoS components of the indirect signal link, i.e., M2

RN2 R, also maximize the average rate or ergodic rate ◮ The optimization result for Q = instant recovers the one

under the ULA model for the multi-antenna BS and multi-element IRS in the instantaneous CSI case in [Han et al. (2019)]

◮ The optimization result for Q = statistic recovers the one

under the ULA model for the multi-antenna BS and multi-element IRS in the statistical CSI case in [Hu et al. (2020)]

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Stationary point in general case

◮ Propose an iterative algorithm based on parallel coordinate

descent (PCD), with each phase shift φm,n as one block

◮ At each iteration, maximize γ(Q,t) ub

(φ) w.r.t. each phase shift with the other phase shifts being fixed, in parallel

Problem (Block-wise Problem w.r.t. φm,n at Iteration t)

For Q = instant or statistic,

φ

(Q,t) m,n

arg max

φm,n∈[0,2π)

B(Q,t)

SRU,m,n cos(φm,n + B(t) ∠SRU,m,n) + B(Q,t) S,m,n

B(Q,t)

IRU,m,n cos(φm,n + B(t) ∠IRU,m,n) + B(Q,t) I,m,n

,

◮ A closed-form optimal solution is:

φ

(Q,t) m,n

= arctan B(Q,t)

1,m,n

B(Q,t)

2,m,n

−arccos B(t)

SRU,m,nB(Q,t) I,m,n sin(B(t) ∠SRU,m,n − B(t) ∠IRU,m,n)

• B(Q,t)

1,m,n

2 +

• B(Q,t)

2,m,n

2 +C, where C = 0 for B(Q,t)

1,m,n ≥ 0 and C = π for B(Q,t) 1,m,n < 0 SJTU Ying Cui 30 / 56

Stationary point in general case

Algorithm 1 (PCD Algorithm in General Case)

1: initialization: choose any φ(Q,0) ∈ [0, 2π), and set t = 0. 2: repeat 3:

for all m ∈ MR and n ∈ NR, compute φ

(Q,t) m,n .

4:

Update φ(Q,t+1)

m,n

= (1 − ρ(t))φ(t)

m,n + ρ(t)φ (Q,t) m,n , where ρ(t) satisfies

ρ(t) > 0, limt→∞ ρ(t) = 0, ∞

t=1 ρ(t) = ∞, ∞ t=1

• ρ(t)2 < ∞

5:

Set t = t + 1.

6: until some convergence criterion is met.

◮ Algorithm 1 has higher computation efficiency than the BCD

algorithm [Yu et al. (2019b); Pan et al. (2020)] where all blocks are sequentially updated in each iteration, at large MR and NR

◮ Algorithm 1 has a larger convergence rate than the MM algorithm

[Yu et al. (2019b); Pan et al. (2020)] where only an approximate problem is solved in each iteration, at large MR and NR

◮ Algorithm 1 is suitable for the cases not covered in Theorem 1

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Outline

Introduction System model Rate analysis Rate optimization Comparision with system without IRS Numerical results Conclusion

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System without IRS: instantaneous CSI case

◮ Assumptions:

◮ The CSI of the channel between BS S and user U is known at BS S ◮ The CSI of the channel between BS I and user U′ is known at BS I

◮ Consider instantaneous-CSI adaptive MRT beamformers:

w(instant)

no,S

= hSU ||hSU||2 , w(instant)

no,I

= hIU′ ||hIU′||2

◮ The average rate of the counterpart system without IRS:

without IRS:

C (instant)

no

= E

• log2
• 1 +

PSαSU||hSU||2

2

PI αIUE

• hH

IU hIU′

||hIU′||2

• 2

+ σ2

• γ(instant)

no

• ◮ An upper bound of C (instant)

no

: C (instant)

no

≤ log2

• 1 +

γ(instant)

no,ub A(instant)

SU

/A(instant)

IU

• C (instant)

no,ub SJTU Ying Cui 33 / 56

System without IRS: statistic CSI case

◮ Assumptions:

◮ No channel knowledge is known at BS S or BS I

◮ Consider statistical CSI-adaptive MRT beamformers:

w(statistic)

no,S

= 1 √MSNS 1MSNS, w(statistic)

no,I

= 1 √MINI 1MI NI

◮ The ergodic rate of the counterpart system without IRS: C (statistic)

no

=E

• log2
• 1 +

PS αSU MS NS

• hH

SU1MS NS

• 2

PI αIU MI NI E

• hH

IU1MI NI

• 2

+ σ2

• γ(statistic)

no

• ◮ An upper bound of C (statistic)

no

: C (statistic)

no

≤ log2

• 1 +

γ(statistic)

no,ub

• A(statistic)

SU

/A(statistic)

IU

• C (statistic)

no,ub

SJTU Ying Cui 34 / 56

Notations

◮ Define:

ξ(Q)

>

=

• A(Q)

SRU,LoSA(Q) IU − A(Q) IRU,LoSA(Q) SU

• M2

RN2 R

+ A(Q)

SRU,NLoSA(Q) IU − A(Q) SU A(Q) IRU,NLoS

ξ(Q)

<

=A(Q)

SRU,LoSA(Q) IU M2 RN2 R + A(Q) SRU,NLoSA(Q) IU − A(Q) SU A(Q) IRU,NLoS

ς(instant) αIU

• ASRU,LoSM2

RN2 R + A(instant) SRU,NLoS

• − A(instant)

SU

αIRαRU

• τIRUM2

RN2 R + MRNR(1 − τIRU)

• ς(statistic) αIU
• ASRU,LoSM2

RN2 R + A(statistic) SRU,NLoS

• − A(statisic)

SU

αIRαRU τIRUyIR MINI + MRNR (MINIKRU + τIyIR) MINIKRU(KIR + 1)

• SJTU

Ying Cui 35 / 56

Comparision

Theorem 6 (Comparision)

If ξ(Q)

>

> 0, then γ(Q)

ub (φ∗) > γ(Q) no,ub.

If ξ(Q)

<

< 0, then γ(Q)

ub (φ∗) < γ(Q) no,ub. ◮ The IRS-assisted system with the optimal quasi-static phase

shift design is effective

◮ The channel between BS S and the IRS is strong, the channel

between BS I and user U is strong, or the LoS components of the indirect signal link are dominant

◮ ξ(Q) >

(and ξ(Q)

< ) increases with αSR, αIU and τSRU ◮ The channel between BS I and the IRS is weak, the channel

between BS S and user U is weak, or the LoS components of the indirect interference link are not dominant

◮ ξ(Q) >

(and ξ(Q)

< ) decreases with αIR, αSU and τIRU ◮ PI is weak ◮ ξ(Q) >

(and ξ(Q)

< ) decreases with PI SJTU Ying Cui 36 / 56

Comparision

Corollary 3

If ς(Q) > 0, then γ(Q)

ub (φ∗) > γ(Q) no,ub.

If ς(Q) < 0 and PI ≤ ε for some ε > 0, then γ(Q)

ub (φ∗) > γ(Q) no,ub.

If ς(Q) < 0 and PI > ε for some ε > 0, then γ(Q)

ub (φ∗) < γ(Q) no,ub. ◮ The IRS-assisted system with the optimal quasi-static phase shift design

is effective at any PI

◮ The channel between BS S and the IRS is strong, the channel

between BS I and user U is strong, the LoS components of the indirect signal link are dominant

◮ ς(Q) increases with αSR, αIU and τSRU ◮ The channel between BS I and the IRS is weak, the channel

between BS I and user U is weak, the LoS components of the indirect interference link are not dominant

◮ ς(Q) decreses with αIR, αSU and τIRU SJTU Ying Cui 37 / 56

Comparision

Corollary 3

If ς(Q) > 0, then γ(Q)

ub (φ∗) > γ(Q) no,ub.

If ς(Q) < 0 and PI ≤ ε for some ε > 0, then γ(Q)

ub (φ∗) > γ(Q) no,ub.

If ς(Q) < 0 and PI > ε for some ε > 0, then γ(Q)

ub (φ∗) < γ(Q) no,ub. ◮ Otherwise, the IRS-assisted system with the optimal

quasi-static phase shift design is effective when PI is small enough, and is not effective when PI is large enough

◮ If PI = 0, the IRS-assisted system with the optimal quasi-static

phase shift design is always beneficial

SJTU Ying Cui 38 / 56

Outline

Introduction System model Rate analysis Rate optimization Comparision with system without IRS Numerical results Conclusion

SJTU Ying Cui 39 / 56

System model

• Figure: The IRS-assisted system [Pan et al. (2020)].

◮ Network topology:

◮ BS S, BS I, user U and IRS, locate at (0, 0), (600, 0),

(dSU, 0), (dR, dRU), respectively

◮ User U is in the line between BS S and BS I

◮ Path Loss model:

αi = −30 + 10¯ αi log10(di) (in dB), i = SU, IU, SR, IR, RU

◮ ¯

αSU = 3.7, ¯ αIU = 3.5 (extensive obstacles and scatters)

◮ Set ¯

αSR, ¯ αIR = 2 (the location of the IRS is usually carefully chosen)

◮ Set ¯

αRU = 3 (the IRS is usually close to the user with few

• bstacles)

SJTU Ying Cui 40 / 56

System model

◮ System parameters:

◮ Set d = λ

2 , MS = NS = 4, MI = NI = 4, MR = NR = 8,

PS = PI = 30dBm, σ2 = −104dBm, ϕ(h)

SR = ϕ(v) SR = π/3,

ϕ(h)

IR = ϕ(v) IR = π/8, ϕ(h) RU = ϕ(v) RU = π/6, dR = 250m,

dSU = 250m, dRU = 20m, unless otherwise stated

◮ Set δ(h)

SR = δ(v) SR = π/6, δ(h) IR = δ(v) IR = π/6 in Special Case (ii)

and Special Case (iii)

◮ Set KSR = KIR = KRU = 20dB in Special Case (ii) ◮ Set KSR = −20dB, KIR = KRU = 20dB in Special Case (iii) ◮ Set δ(h)

SR = δ(v) SR = π/6, δ(h) IR = δ(v) IR = π/8, KSR = KRU = 20dB,

KIR = 10dB in the general case

SJTU Ying Cui 41 / 56

Baselines

◮ Baseline 1 (Without Reflector) reflects the average rate and

ergodic rate of the counterpart system without IRS

◮ Baseline 2 (Random phase shifts) chooses the phase shifts

uniformly at random

◮ Baseline 3 (Solution in [Han et al. (2019)]) implements the

phase shifts for the IRS-assisted system without interference

◮ Baseline 3 is an extension of the optimal solution for the

instantaneous CSI case under the ULA model in [Han et al. (2019)] to the URA model

◮ Baseline 4 (Instantaneous CSI-adaptive phase shift design)

represents instantaneous CSI-adaptive phase shift design corresponding to a stationary point of the maximization of γ(instant)(φ)

◮ In the general case, we also evaluate the BCD algorithm and

the MM algorithm in [Yu et al. (2019b)]

SJTU Ying Cui 42 / 56

Numerical results for special cases

2 4 6 8 10 1 2 3 4 5 6

Average Rate

Instantaneous CSI-adaptive design Opt in Case (ii) Monte Carlo for Opt in Case (ii) PCD in case (ii) Solution in [21] in case (ii) Random Phase Shifts in case (ii) Opt in Case (iii) Monte Carlo for Opt in Case (iii) PCD in Case (iii) Solution in [21] in case (iii) Random Phase shifts in case (iii) Without Reflector

2 4 6 8 10 1 2 3 4 5 6

Ergodic Rate

Opt in Case (ii) Monte Carlo for Opt in Case (ii) PCD in case (ii) Random Phase Shifts in case (ii) Opt in Case (iii) Monte Carlo for Opt in Case (iii) PCD in Case (iii) Random Phase shifts in case (iii) Without Reflector

Figure: Average rate and ergodic rate versus MR (= NR) in special cases.

◮ Cub(φ) is a good approximation of C(φ) ◮ The PCD solution has near-optimal performance ◮ The rates of the proposed solutions and Random phase shifts

increase with MR (=NR), mainly due to the increment of reflecting signal power

SJTU Ying Cui 43 / 56

Numerical results for special cases

2 4 6 8 10 1 2 3 4 5 6

Average Rate

Instantaneous CSI-adaptive design Opt in Case (ii) Monte Carlo for Opt in Case (ii) PCD in case (ii) Solution in [21] in case (ii) Random Phase Shifts in case (ii) Opt in Case (iii) Monte Carlo for Opt in Case (iii) PCD in Case (iii) Solution in [21] in case (iii) Random Phase shifts in case (iii) Without Reflector

2 4 6 8 10 1 2 3 4 5 6

Ergodic Rate

Opt in Case (ii) Monte Carlo for Opt in Case (ii) PCD in case (ii) Random Phase Shifts in case (ii) Opt in Case (iii) Monte Carlo for Opt in Case (iii) PCD in Case (iii) Random Phase shifts in case (iii) Without Reflector

Figure: Average rate and ergodic rate versus MR (= NR) in special cases.

◮ The average rate of the solution in [Han et al. (2019)]

decreases with MR (= NR), revealing the penalty of ignoring interference in the instantaneous CSI case

◮ Instantaneous CSI-adaptive phase shift design achieves the

maximum average rate in the instantaneous CSI case, with the highest phase adjustment cost

SJTU Ying Cui 44 / 56

Numerical results for special cases

• 2

2 4 6 1 2 3 4 5 6 7 8

Average Rate

Instantaneous CSI-adaptive design Opt in Case (ii) Monte Carlo for Opt in Case (ii) PCD in case (ii) Solution in [21] in case (ii) Random Phase Shifts in case (ii) Opt in Case (iii) Monte Carlo for Opt in Case (iii) PCD in Case (iii) Solution in [21] in case (iii) Random Phase shifts in case (iii) Without Reflector

• 2

2 4 6 2 4 6

Ergodic Rate

Opt in Case (ii) Monte Carlo for Opt in Case (ii) PCD in case (ii) Random Phase Shifts in case (ii) Opt in Case (iii) Monte Carlo for Opt in Case (iii) PCD in Case (iii) Random Phase shifts in case (iii) Without Reflector

Figure: Average rate and ergodic rate versus PI in special cases.

◮ The rate of each solution decreases with PI

SJTU Ying Cui 45 / 56

Numerical results for general case

• 20
• 10

10 20 3 4 5 6 7 8

Average Rate

Instantaneous CSI-adaptive design BCD MM PCD Solution in [21] Random Phase Shifts Without Reflector

• 20
• 10

10 20 2 4 6

Ergodic Rate

BCD MM PCD Random Phase Shifts Without Reflector

Figure: Rate for the instantaneous/statistical CSI versus KSR (dB) in the general

case. ◮ The rate of the PCD solution increases with KSR

SJTU Ying Cui 46 / 56

Numerical results for general case

• 20
• 10

10 20 4 5 6 7 8

Average Rate

Instantaneous CSI-adaptive design BCD MM PCD Solution in [21] Random Phase Shifts Without Reflector

• 20
• 10

10 20 2 4 6

Ergodic Rate

BCD MM PCD Random Phase Shifts Without Reflector

Figure: Rate for the instantaneous/statistic CSI versus KRU (dB) in the general case.

◮ The rate of the PCD solution increases with KRU

SJTU Ying Cui 47 / 56

Numerical results for general case

200 225 250 275 300 4 5 6 7 8

Average Rate

Instantaneous CSI-adaptive design BCD MM PCD Solution in [21] Random Phase Shifts Without Reflector

200 225 250 275 300 1 2 3 4 5 6

Ergodic Rate

BCD MM PCD Random Phase Shifts Without Reflector

Figure: Rate for the instantaneous/statistical CSI versus dR (m) in the general case.

◮ The rate of the PCD solution increases with dR, due to the decrement of

the distance between the IRS and user U when dR < dSU

◮ The rate of the PCD solution decreases with dR, due to the increment of

the distance between the IRS and user U when dR > dSU

◮ The rate in the case of dR < dSU is greater than that in the case of

dR > dSU, at the same distance between the IRS and user U, due to smaller path loss between the IRS and BS S

SJTU Ying Cui 48 / 56

Numerical results for general case

200 225 250 275 300 2 4 6 8 10

Average Rate

Instantaneous CSI-adaptive design for dRU=20m BCD for dRU=20m MM for dRU=20m PCD for dRU=20m Solution in [21] for dRU=20m Random Phase Shifts for dRU=20m Without Reflector for dRU=20m BCD for dRU=30m MM for dRU=30m PCD for dRU=30m Solution in [21] for dRU=30m Random Phase Shifts for dRU=30m Without Reflector for dRU=30m

200 225 250 275 300 2 4 6 8

Ergodic Rate

BCD for dRU=30m MM for dRU=30m PCD for dRU=30m Random Phase Shifts for dRU=30m Without Reflector for dRU=30m BCD for dRU=40m MM for dRU=40m PCD for dRU=40m Random Phase Shifts for dRU=40m Without Reflector for dRU=40m

Figure: Rate for the instantaneous/statistical CSI versus dSU (m) in the general case.

◮ When dRU is small, the rate of the PCD solution increases with dSU when dSU < dR, mainly due to the decrement of dRU, and decreases with dSU when dSU > dR, due to the increment of both dSU and dRU ◮ When dRU is large, the rate of the PCD solution always decreases with dSU, mainly due to the increment of the distance between BS S and user U ◮ The proposed solution achieves a higher rate than the system without IRS (in accordance with the theoretical results) ◮ For each scheme, the average rate in the instantaneous CSI case is greater than the ergodic rate in the statistical CSI case, which is in accordance with Corollary 2

SJTU Ying Cui 49 / 56

Numerical results for general case

18 20 22 24 26 20 40 60 80

Computation Time (sec.)

BCD MM PCD (16 cores) PCD (24 cores)

18 20 22 24 26 20 40 60 80 100

Computation Time (sec.)

BCD MM PCD (16 cores) PCD (24 cores)

Figure: Rate for the instantaneous/statistical CSI versus dSU (m) in the general case.

◮ When MRNR is large, the gain of the proposed PCD algorithm in

computation time over the BCD and MM algorithms increases with the number of the cores on a server, due to its parallel computation mechanism

◮ In practical systems with multi-core processors, the value of the

PCD algorithm will be prominent, especially for large-scale IRS

SJTU Ying Cui 50 / 56

Outline

Introduction System model Rate analysis Rate optimization Comparision with system without IRS Numerical results Conclusion

SJTU Ying Cui 51 / 56

Conclusion

◮ Consider an IRS-assisted system with interference ◮ Obtain a tractable expression of the average rate (ergodic rate)

◮ Under certain conditions, the average rate in the instantaneous CSI

case is greater than the ergodic rate in the statistical CSI case

◮ Optimize the phase shifts to maximize the average rate (ergodic rate)

◮ Under certain system parameters, obtain a globally optimal solution

• f each non-convex problem

◮ Under arbitrary system parameters, propose parallel iterative

algorithm, to obtain a stationary point of each non-convex problem

◮ Characterize the average rate (ergodic rate) degradation caused by

the quantization error for the phase shifts

◮ Provide sufficient conditions under which the optimal quasi-static phase

shift design is beneficial with interference, compared to a counterpart system without IRS

◮ Numerical results verify analytical results and demonstrate notable gains

• f the proposed solutions over existing schemes

SJTU Ying Cui 52 / 56

Publications

◮ Y. Jia, C. Ye and Ying Cui, Analysis and optimization of an

intelligent reflecting surface-assisted system with interference, in Proc. of IEEE ICC, Jun. 2020, pp. 1-6.

◮ Y. Jia, C. Ye and Y. Cui, Analysis and Optimization of an

Intelligent Reflecting Surface-assisted System with Interference, to appear in IEEE Trans. Wirel. Commun., Aug. 2020.

SJTU Ying Cui 53 / 56

Reference I

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• ptimization for intelligent reflecting surface enhanced wireless networks. arXiv

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Hu, X., Wang, J., & Zhong, C. (2020). Statistical CSI based Design for Intelligent Reflecting Surface Assisted MISO Systems. arXiv e-prints, (p. arXiv:2008.08453). arXiv:2008.08453. Huang, C., Zappone, A., Alexandropoulos, G. C., Debbah, M., & Yuen, C. (2019). Reconfigurable intelligent surfaces for energy efficiency in wireless communication. IEEE Transactions on Wireless Communications, 18, 4157–4170. doi:10.1109/TWC.2019.2922609. Jiang, T., & Shi, Y. (2019). Over-the-air computation via intelligent reflecting

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Reference II

Nadeem, Q., Kammoun, A., Chaaban, A., Debbah, M., & Alouini, M. (2020). Asymptotic max-min sinr analysis of reconfigurable intelligent surface assisted miso

• systems. IEEE Transactions on Wireless Communications, (pp. 1–1).

Nadeem, Q.-U.-A., Kammoun, A., Chaaban, A., Debbah, M., & Alouini, M.-S. (2019). Large intelligent surface assisted mimo communications. arXiv preprint arXiv:1903.08127, . Ni, W., Liu, X., Liu, Y., Tian, H., & Chen, Y. (2020). Resource Allocation for Multi-Cell IRS-Aided NOMA Networks. arXiv e-prints, (p. arXiv:2006.11811). arXiv:2006.11811. Pan, C., Ren, H., Wang, K., Xu, W., Elkashlan, M., Nallanathan, A., & Hanzo, L. (2020). Multicell mimo communications relying on intelligent reflecting surfaces. IEEE Transactions on Wireless Communications, (pp. 1–1). Wu, Q., & Zhang, R. (2019). Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming. IEEE Transactions on Wireless Communications, (pp. 1–1). doi:10.1109/TWC.2019.2936025. Wu, Q., & Zhang, R. (2019). Weighted sum power maximization for intelligent reflecting surface aided swipt. arXiv preprint arXiv:1907.05558, . Wu, Q., & Zhang, R. (2020). Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network. IEEE Communications Magazine, 58, 106–112.

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Reference III

Xie, H., Xu, J., & Liu, Y.-F. (2020). Max-Min Fairness in IRS-Aided Multi-Cell MISO Systems with Joint Transmit and Reflective Beamforming. arXiv e-prints, (p. arXiv:2003.00906). arXiv:2003.00906. Yang, G., Xu, X., & Liang, Y.-C. (2019). Intelligent reflecting surface assisted non-orthogonal multiple access. arXiv preprint arXiv:1907.03133, . Yu, X., Xu, D., & Schober, R. (2019a). Enabling secure wireless communications via intelligent reflecting surfaces. In 2019 IEEE Global Communications Conference (GLOBECOM) (pp. 1–6). Yu, X., Xu, D., & Schober, R. (2019b). Miso wireless communication systems via intelligent reflecting surfaces : (invited paper). In 2019 IEEE/CIC International Conference on Communications in China (ICCC) (pp. 735–740). doi:10.1109/ICCChina.2019.8855810. Zhang, Z., Cui, Y., Yang, F., & Ding, L. (2019). Analysis and optimization of outage probability in multi-intelligent reflecting surface-assisted systems. arXiv preprint arXiv:1909.02193, .

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