Analysis and Optimization of an Intelligent Reflecting...

$: Analysis and Optimization of an Intelligent Reflecting ...iwct.sjtu.edu.cn/personal/yingcui/Slides/Paper/TWC20.pdf · 5D\OHLJKIDGLQJ 5LFLDQIDGLQJ SJTUYing Cui9 / 56. Channel model$
Analysis and Optimization of an IntelligentReflecting Surface-assisted System With

Interference

Ying Cui

Department of Electrical EngineeringShanghai Jiao Tong University

Sept. 2020

SJTU Ying Cui 1 / 56

Outline

Introduction

System model

Rate analysis

Rate optimization

Comparision with system without IRS

Numerical results

Conclusion


Outline

Introduction

System model

Rate analysis

Rate optimization


Numerical results

Conclusion


Background

I Current 5G solutions require high hardware cost and energyconsumption

I Finding spectral and energy efficient, and yet cost-effectivesolutions for 6G wireless networks is still imperative

I Intelligent Reflecting Surface (IRS) is envisioned to be apromising solution

I An IRS consists of nearly passive, low-cost and reflectingelements whose phase shifts can be adjusted independently bysmart switches

I Signals reflected by an IRS can add constructively with thosefrom the other paths to enhance the desired signal power, ordestructively to cancel the interference

I IRSs can be practically deployed and integrated in wirelessnetworks with low cost

I low profile, light weight, conformal geometry, and easy tomount/remove them on/from the wall, ceiling, building, etc


Typical IRS applications

(a) User at deadzone.

(b) Physical layersecurity.

(c) User at cel-l edge.

(d) Massive D2Dcommunications.

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


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

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

I Instantaneous CSI-adaptive phase shift design: phase shifts areadjusted based on instantaneous CSI (assumed known)

I Maximize the weighted sum rate [Nadeem et al. (2019); Yanget al. (2019); Guo et al. (2019); Wu & Zhang (2019)], andenergy efficiency [Yu et al. (2019b,a); Huang et al. (2019)]

I Minimize the transmission power [Wu & Zhang (2019); Jiang& Shi (2019)]

I Quasi-static phase shift design: phase shifts are determined by CSIstatistics (Line-of-Sight (LoS) components and distributions ofNon-Line-of-Sight (NLoS) components) and do not change withinstantaneous CSI (assumed unknown)

I Consider slowly varying Non-line-of sight (NLoS) components,and minimize the outage probability [Zhang et al. (2019),Guoet al. (2020)]

I Consider fast varying NLoS components, and maximize theergodic rate [Han et al. (2019); Nadeem et al. (2020)], [Huet al. (2020)]


Previous work

I Quasi-static phase shift design has less frequent phaseadjustment than instantaneous CSI-adaptive phase shift design

I All the aforementioned works ignore interference from othertransmitters

I However, interference usually has a severe impact, especiallyin dense networks or for cell-edge users

I Consider optimal phase shift and beamforming design forIRS-assisted systems where multiple BSs serve their own userswith the help of one IRS

I Instantaneous CSI-adaptive phase shift design in the presenceof interference

I Consider fast varying NLoS components and maximize theweighted sum average rate [Pan et al. (2020)], [Xie et al.(2020); Ni et al. (2020)]

I It is highly desirable to obtain cost-efficient quasi-static designfor IRS-assisted systems with interference


Outline

Introduction

System model

Rate analysis

Rate optimization


Numerical results

Conclusion


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

MS × NS antennas, serves a single-antenna user UI A multi-antenna interference BS I , equipped with a URA of

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

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

antennas, is installed on the wall of a high-rise buildingI Channels between the BSs and users follow Rayleigh fading

I scattering is often rich near the groundI Channels between the IRS and BSs/user follow Rician fading

I scattering is much weaker far from the ground


Channel modelI Rayleigh channels between the BSs and the users:

hHi =√αi h

Hi , i = SU, IU, IU ′

I αi > 0 is the distance-dependent path lossesI The elements of hHi are i.i.d. according to CN (0, 1)

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

HcR =√αcR

(√KcR

KcR + 1HcR +

√1

KcR + 1HcR

), c = S , I

hRU =√αRU

(√KRU

KRU + 1hRU +

√1

KRU + 1hRU

)I αcR , αRU > 0 denote the distance-dependent path losses and

KcR ,KRU ≥ 0 denote the Rician factors, where i = S , II HcR , hRU represent the deterministic normalized LoS

components, with unit-modulus elementsI HcR , hRU represent the normalized NLoS components, with

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


Channel modelI Define:

f (θ(h), θ(v),m, n) ,2πd

λsin θ(v)((m − 1) cos θ(h) + (n − 1) sin θ(h))

Am,n(θ(h), θ(v),M,N) ,(e jf (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)

)I λ denotes the wavelength of transmission signalsI d (≤ λ

2 ) denotes the distance between adjacent elements orantennas in each row and each column of the URAs

I HcR and hHRU 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

hHRU =a(ϕ(h)RU , ϕ

(v)RU ,MR ,NR)

I 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


Quasi-static phase shift design

I Phase shifts of the IRS φ , (φm,n)m∈MR ,n∈NRwith φm,n ∈ [0, 2π) is

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

I Define Φ(φ) , diag(

rvec((

e jφm,n)m∈MR ,n∈NR

))∈ CMRNR×MRNR

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

Y ,√

PS(hHRUΦ(φ)HSR + hH

SU)wSXS +√

PI

(hHRUΦ(φ)HIR + hH

IU

)wIXI + Z

I wS ∈ CMSNS×1 and wI ∈ CMINI×1 denote the normalizedbeamforming vectors at BS S and BS I , where ||wS ||22 = 1 and||wI ||22 = 1

I 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)I hH

RUΦ(φ)HcR + hHcU represents the equivalent channel between BS c

and user U via the IRS

I Assume that user U knows (hHRUΦ(φ)HSR + hH

SU)wS , but does not know(hHRUΦ(φ)HIR + hH

IU

)wI


Instantaneous CSI case

I Assumptions:I CSI of the equivalent channel between BS S and user U, i.e.,

hHRUΦ(φ)HSR + hHSU , is known at BS SI CSI of the channel between BS I and user U ′, i.e., hIU′ , is

known at BS I

I Consider instantaneous CSI-adaptive MRT beamformers:

w(instant)S =

(hHRUΦ(φ)HSR + hHSU

)H∣∣∣∣hHRUΦ(φ)HSR + hHSU∣∣∣∣

2

, w(instant)I =

hIU′

||hIU′ ||2

I w(instant)S and w

(instant)I are chosen to enhance the signals

received at user U and user U ′

I w(instant)S is optimal for the average rate maximization


Instantaneous CSI case

I The SINR at user U:1

γ(instant)(φ) =PS

∣∣∣∣hHRUΦ(φ)HSR + hHSU∣∣∣∣2

2

PIE[∣∣∣(hHRUΦ(φ)HIR + hHIU) hIU′

||hIU′ ||2

∣∣∣2]+ σ2

I The average rate for the IRS-assisted system withinterference:

C (instant)(φ) = E[log2

(1 + γ(instant)(φ)

)]I log2

(1 + γ(instant)(φ)

)can be achieved by coding over one

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

et al. (2019)]

1Treat(hHRUΦ(φ)HIR + hH

IU

)wIXI ∼CN

(0,E

[∣∣(hHRUΦ(φ)HIR + hH

IU

)wI

∣∣2]),

which corresponds to the worst-case noise.SJTU Ying Cui 14 / 56

Statistic CSI case

I Assumptions:I Only the CSI of the LoS components hHRU , HSR are known at

BS SI No channel knowledge is known at BS I

I Consider statistical CSI-adaptive MRT beamformers:

w(statistic)S =

(hHRUΦ(φ)HSR

)H∣∣∣∣hHRUΦ(φ)HSR

∣∣∣∣2

, w(statistic)I =

1√MINI

1MINI

I w(statistic)S is approximately optimal for the ergodic rate

maximization (optimal for maximizing an upper bound)I Any wI with ||wI ||22 = 1 achieves the same ergodic rate for

user U ′

I Have lower costs on channel estimation and beamformingadjustment


Statistic CSI case

I The SINR at user U:

γ(statistic)(φ) =

PS

∣∣∣∣(hHRUΦ(φ)HSR + hH

SU

) (hHRUΦ(φ)HSR)H

||hHRUΦ(φ)HSR ||2

∣∣∣∣2PIE

[∣∣∣∣(hHRUΦ(φ)HIR + hH

IU

)1√MINI

1

∣∣∣∣2]

+ σ2

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

C (statistic)(φ) = E[log2

(1 + γ(statistic)(φ)

)]I Code over a large number of channel coherence time intervalsI C (statistic)(φ) with PI = 0 is recently studied in [Hu et al.

(2020)]


Outline

Introduction

System model

Rate analysis

Rate optimization


Numerical results

Conclusion


NotationsI 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

I τcRU increases with KcR and KRU

I f(ϕ

(h)RU , ϕ

(v)RU ,m, n

) (f(ϕ

(h)IR , ϕ

(v)IR ,m, n

))represents the difference

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

I f(δ

(h)cR , δ

(v)cR ,m, n

)represents the difference of the phase change

over the LoS component between BS c and the (m, n)-th elementof the IRS and the phase change over the LoS component betweenBS c and the (1, 1)-th element of the IRS


NotationsI Define:

ASRU,LoS ,PSMSNSαSRαRUτSRU , AIU , PIαIU + σ2,

A(Q)SU ,

{PSMSNSαSU , Q = instant,

PSαSU , Q = statistic,

A(Q)IRU,LoS ,

{PIαIRαRUτIRU , Q = instant,

PIαIRαRUτIRUyIR

MINI, Q = statistic,

A(Q)SRU,NLoS ,

{PSMSNSαSRαRUMRNR(1− τSRU), Q = instant,

PSMSNSαSRαRUMRNR

(1− τSRU − MSNS−1

MSNS (KSR+1)

), Q = statistic,

A(Q)IRU,NLoS ,

{PIαIRαRUMRNR(1− τIRU), Q = instant,

PIαIRαRUMRNR

(1− τIRU + τIRU (yIR−MINI )

MINIKRU

), Q = statistic,

yIR ,

∣∣∣∣∣∣MI∑m=1

NI∑n=1

e jθIR,m,n

∣∣∣∣∣∣2

, ycRU(φ) ,

∣∣∣∣∣∣MR∑m=1

NR∑n=1

e jθcRU,m,n+jφm,n

∣∣∣∣∣∣2

.

I McNcycRU(φ) represents the sum channel power of the LoS componentsof the indirect signal link between BS c and user U via the IRS


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).

I Proof: Jensen inequality

I C(Q)ub (φ) is a good approximation of C (Q)(φ), and can

facilitate the evaluation and optimization for it

I For all φ, C(Q)ub (φ) increases with PS , MS , NS , αSR and αSU

I For all φ, C(Q)ub (φ) decreases with PI , αIR , αIU and σ2


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

I Corollary 1 (i): the received signal power at user U in theinstantaneous CSI case always surpasses that in the statisticalCSI case, at any phase shifts

I Corollary 1 (ii): if the placement of the URA at theinterference BS and the locations of the interference BS andIRS satisfy certain condition, the received interference powerat user U in the instantaneous CSI case is weaker than that inthe statistical CSI case, at any phase shifts


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.

I Corollary 2 (i): if the interference is weak, the average rate in theinstantaneous CSI case is greater than the ergodic rate in thestatistical CSI case, at any phase shifts

I Corollary 2 (ii): if the placement of the URA at the interference BSand the locations of the interference BS and IRS satisfy certaincondition, the average rate in the instantaneous CSI case is greaterthan the ergodic rate in the statistical CSI case, at any phase shifts

I Corollary 2 reveals the advantage2 of CSI of the NLoS componentsin 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.SJTU Ying Cui 22 / 56

Outline

Introduction

System model

Rate analysis

Rate optimization


Numerical results

Conclusion


Problem formulation

Problem (Average or Ergodic Rate/SINR Maximization)


maxφ

γ(Q)ub (φ)

s.t. φm,n ∈ [0, 2π), m ∈MR , n ∈ NR

I An optimal solution depends on the LoS components and thedistributions of the NLoS components

I It is a challenging non-convex problem


NotationsI 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 6=m,l 6=n

ej(φ

(t)k,l

+θSRU,k,l

)∣∣∣∣∣∣B

(Q,t)S,m,n ,ASRU,LoS

1 +

∣∣∣∣∣∣∑

k 6=m,l 6=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 6=m,l 6=n

ej(φ

(t)k,l

+θIRU,k,l

)∣∣∣∣∣∣B

(Q,t)I ,m,n ,A

(Q)IRU,LoS

1 +

∣∣∣∣∣∣∑

k 6=m,l 6=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 cosB

(t)∠IRU,m,n − B

(t)SRU,m,nB

(Q,t)I ,m,n cosB

(t)∠SRU,m,n

B(Q,t)2,m,n ,B

(Q,t)S,m,nB

(Q,t)IRU,m,n sinB

(t)∠IRU,m,n − B

(t)SRU,m,nB

(Q,t)I ,m,n sinB

(t)∠SRU,m,n


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.

I The phase shift of the single element has no impact on theaverage or ergodic rate (as ySRU(φ) = yIRU(φ) = 1 for all φ)

I The channel between each BS and user U follows Rayleighfading



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

optimal, and ySRU(φ(Q)∗) = yIRU

(φ(Q)∗) = M2

RN2R .

I The phase shifts that achieve the maximum sum channelpower of the LoS components of the indirect single andinterference links, i.e., M2

RN2R , also maximize the average or

ergodic rate

I ySRU(φ) = yIRU(φ) , y(φ), γ(Q)ub = γ

(Q)ub ◦ y , η(Q) reflects

dγ(Q)ub

dy



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.

I The phase shifts that achieve the minimum sum channelpower of the LoS components of the indirect single andinterference links, i.e., 0, also maximize the average or ergodicrate

I ySRU(φ) = yIRU(φ) , y(φ), γ(Q)ub = γ

(Q)ub ◦ y , η(Q) reflects

dγ(Q)ub

dy



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

RN2R .

I The phase shifts that achieve the maximum sum channelpower of the LoS components of the indirect signal link, i.e.,M2

RN2R , also maximize the average rate or ergodic rate

I The optimization result for Q = instant recovers the oneunder the ULA model for the multi-antenna BS andmulti-element IRS in the instantaneous CSI case in [Han et al.(2019)]

I The optimization result for Q = statistic recovers the oneunder the ULA model for the multi-antenna BS andmulti-element IRS in the statistical CSI case in [Hu et al.(2020)]


Stationary point in general caseI Propose an iterative algorithm based on parallel coordinate

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

I 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)


φ(Q,t)

m,n , arg maxφm,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

,

I A closed-form optimal solution is:

φ(Q,t)

m,n = arctanB

(Q,t)1,m,n

B(Q,t)2,m,n

−arccosB

(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


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.

I Algorithm 1 has higher computation efficiency than the BCDalgorithm [Yu et al. (2019b); Pan et al. (2020)] where all blocks aresequentially updated in each iteration, at large MR and NR

I Algorithm 1 has a larger convergence rate than the MM algorithm[Yu et al. (2019b); Pan et al. (2020)] where only an approximateproblem is solved in each iteration, at large MR and NR

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


Outline

Introduction

System model

Rate analysis

Rate optimization


Numerical results

Conclusion


System without IRS: instantaneous CSI caseI Assumptions:

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

I Consider instantaneous-CSI adaptive MRT beamformers:

w(instant)no,S =

hSU||hSU ||2

, w(instant)no,I =

hIU′

||hIU′ ||2I The average rate of the counterpart system without IRS:

without IRS:

C(instant)no = E

[log2

(1 +

PSαSU ||hSU ||22

PIαIUE

[∣∣∣∣hHIU hIU′

||hIU′ ||2

∣∣∣∣2]

+ σ2

︸︷︷︸,γ(instant)

no

)]

I 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


System without IRS: statistic CSI caseI Assumptions:

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

I Consider statistical CSI-adaptive MRT beamformers:

w(statistic)no,S =

1√MSNS

1MSNS, w

(statistic)no,I =

1√MINI

1MINI

I The ergodic rate of the counterpart system without IRS:

C(statistic)no =E

[log2

(1 +

PSαSUMSNS

∣∣hHSU1MSNS

∣∣2PIαIUMINI

E[∣∣hHIU1MINI

∣∣2]+ σ2︸︷︷︸,γ(statistic)

no

)]

I 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


Notations

I Define:

ξ(Q)> =

(A

(Q)SRU,LoSA

(Q)IU − A

(Q)IRU,LoSA

(Q)SU

)M2

RN2R

+ A(Q)SRU,NLoSA

(Q)IU − A

(Q)SU A

(Q)IRU,NLoS

ξ(Q)< =A

(Q)SRU,LoSA

(Q)IU M2

RN2R + A

(Q)SRU,NLoSA

(Q)IU − A

(Q)SU A

(Q)IRU,NLoS

ς(instant) ,αIU

(ASRU,LoSM

2RN

2R + A

(instant)SRU,NLoS

)− A

(instant)SU αIRαRU

(τIRUM

2RN

2R + MRNR(1− τIRU)

)ς(statistic) ,αIU

(ASRU,LoSM

2RN

2R + A

(statistic)SRU,NLoS

)− A

(statisic)SU αIRαRU

(τIRUyIRMINI

+MRNR (MINIKRU + τI yIR)

MINIKRU(KIR + 1)

)


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.

I The IRS-assisted system with the optimal quasi-static phaseshift design is effective

I The channel between BS S and the IRS is strong, the channelbetween BS I and user U is strong, or the LoS components ofthe indirect signal link are dominant

I ξ(Q)> (and ξ

(Q)< ) increases with αSR , αIU and τSRU

I The channel between BS I and the IRS is weak, the channelbetween BS S and user U is weak, or the LoS components ofthe indirect interference link are not dominant

I ξ(Q)> (and ξ

(Q)< ) decreases with αIR , αSU and τIRU

I PI is weakI ξ

(Q)> (and ξ

(Q)< ) decreases with PI


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.

I The IRS-assisted system with the optimal quasi-static phase shift design

is effective at any PI

I The channel between BS S and the IRS is strong, the channelbetween BS I and user U is strong, the LoS components of theindirect signal link are dominant

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

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

I ς(Q) decreses with αIR , αSU and τIRU


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.

I Otherwise, the IRS-assisted system with the optimalquasi-static phase shift design is effective when PI is smallenough, and is not effective when PI is large enough

I If PI = 0, the IRS-assisted system with the optimal quasi-staticphase shift design is always beneficial


Outline

Introduction

System model

Rate analysis

Rate optimization


Numerical results

Conclusion


System model

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

I Network topology:I BS S , BS I , user U and IRS, locate at (0, 0), (600, 0),

(dSU , 0), (dR , dRU), respectivelyI User U is in the line between BS S and BS I

I Path Loss model:αi = −30 + 10αi log10(di ) (in dB), i = SU, IU,SR, IR,RU

I αSU = 3.7, αIU = 3.5 (extensive obstacles and scatters)I Set αSR , αIR = 2 (the location of the IRS is usually carefully

chosen)I Set αRU = 3 (the IRS is usually close to the user with few

obstacles)


System model

I System parameters:I 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 statedI 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)I Set KSR = KIR = KRU = 20dB in Special Case (ii)I Set KSR = −20dB, KIR = KRU = 20dB in Special Case (iii)I 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


Baselines

I Baseline 1 (Without Reflector) reflects the average rate andergodic rate of the counterpart system without IRS

I Baseline 2 (Random phase shifts) chooses the phase shiftsuniformly at random

I Baseline 3 (Solution in [Han et al. (2019)]) implements thephase shifts for the IRS-assisted system without interference

I Baseline 3 is an extension of the optimal solution for theinstantaneous CSI case under the ULA model in [Han et al.(2019)] to the URA model

I Baseline 4 (Instantaneous CSI-adaptive phase shift design)represents instantaneous CSI-adaptive phase shift designcorresponding to a stationary point of the maximization ofγ(instant)(φ)

I In the general case, we also evaluate the BCD algorithm andthe MM algorithm in [Yu et al. (2019b)]


Numerical results for special cases

2 4 6 8 101

2

3

4

5

6A

vera

ge R

ate

Instantaneous CSI-adaptive designOpt 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 8 100

1

2

3

4

5

6

Erg

odic

Rat

e

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.

I Cub(φ) is a good approximation of C (φ)

I The PCD solution has near-optimal performance

I The rates of the proposed solutions and Random phase shiftsincrease with MR (=NR), mainly due to the increment ofreflecting signal power



2 4 6 8 101

2

3

4

5

6A

vera

ge R

ate


2 4 6 8 100

1

2

3

4

5

6

Erg

odic

Rat

e


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

I The average rate of the solution in [Han et al. (2019)]decreases with MR (= NR), revealing the penalty of ignoringinterference in the instantaneous CSI case

I Instantaneous CSI-adaptive phase shift design achieves themaximum average rate in the instantaneous CSI case, withthe highest phase adjustment cost



-2 0 2 4 61

2

3

4

5

6

7

8

Ave

rage

Rat

e


-2 0 2 4 60

2

4

6

Erg

odic

Rat

e


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

I The rate of each solution decreases with PI


Numerical results for general case

-20 -10 0 10 203

4

5

6

7

8

Ave

rage

Rat

e

Instantaneous CSI-adaptive designBCDMMPCDSolution in [21]Random Phase ShiftsWithout Reflector

-20 -10 0 10 200

2

4

6

Erg

odic

Rat

e

BCDMMPCDRandom Phase ShiftsWithout Reflector

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

I The rate of the PCD solution increases with KSR



-20 -10 0 10 204

5

6

7

8

Ave

rage

Rat

e


-20 -10 0 10 200

2

4

6

Erg

odic

Rat

e BCDMMPCDRandom Phase ShiftsWithout Reflector

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

I The rate of the PCD solution increases with KRU



200 225 250 275 3004

5

6

7

8A

vera

ge R

ate


200 225 250 275 3001

2

3

4

5

6

Erg

odic

Rat

e

BCDMMPCDRandom Phase ShiftsWithout Reflector

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

I The rate of the PCD solution increases with dR , due to the decrement ofthe distance between the IRS and user U when dR < dSU

I The rate of the PCD solution decreases with dR , due to the increment ofthe distance between the IRS and user U when dR > dSU

I The rate in the case of dR < dSU is greater than that in the case ofdR > dSU , at the same distance between the IRS and user U, due tosmaller path loss between the IRS and BS S



200 225 250 275 3002

4

6

8

10A

vera

ge R

ate

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



200 225 250 275 3000

2

4

6

8

Erg

odic

Rat

e

BCD for dRU=30m

MM for dRU=30m

PCD for dRU=30m



BCD for dRU=40m

MM for dRU=40m

PCD for dRU=40m



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

I When dRU is small, the rate of the PCD solution increases with dSU whendSU < dR , mainly due to the decrement of dRU , and decreases with dSU whendSU > dR , due to the increment of both dSU and dRU

I 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

I The proposed solution achieves a higher rate than the system without IRS (inaccordance with the theoretical results)

I For each scheme, the average rate in the instantaneous CSI case is greater thanthe ergodic rate in the statistical CSI case, which is in accordance withCorollary 2



18 20 22 24 260

20

40

60

80C

ompu

tatio

n T

ime

(sec

.) BCDMMPCD (16 cores)PCD (24 cores)

18 20 22 24 2620

40

60

80

100

Com

puta

tion

Tim

e (s

ec.) BCD

MMPCD (16 cores)PCD (24 cores)

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

I When MRNR is large, the gain of the proposed PCD algorithm incomputation time over the BCD and MM algorithms increases withthe number of the cores on a server, due to its parallel computationmechanism

I In practical systems with multi-core processors, the value of thePCD algorithm will be prominent, especially for large-scale IRS


Outline

Introduction

System model

Rate analysis

Rate optimization


Numerical results

Conclusion


Conclusion

I Consider an IRS-assisted system with interference

I Obtain a tractable expression of the average rate (ergodic rate)

I Under certain conditions, the average rate in the instantaneous CSIcase is greater than the ergodic rate in the statistical CSI case

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

I Under certain system parameters, obtain a globally optimal solutionof each non-convex problem

I Under arbitrary system parameters, propose parallel iterativealgorithm, to obtain a stationary point of each non-convex problem

I Characterize the average rate (ergodic rate) degradation caused bythe quantization error for the phase shifts

I Provide sufficient conditions under which the optimal quasi-static phaseshift design is beneficial with interference, compared to a counterpartsystem without IRS

I Numerical results verify analytical results and demonstrate notable gainsof the proposed solutions over existing schemes


Publications

I Y. Jia, C. Ye and Ying Cui, Analysis and optimization of anintelligent reflecting surface-assisted system with interference,in Proc. of IEEE ICC, Jun. 2020, pp. 1-6.

I Y. Jia, C. Ye and Y. Cui, Analysis and Optimization of anIntelligent Reflecting Surface-assisted System withInterference, to appear in IEEE Trans. Wirel. Commun., Aug.2020.


Reference I

Guo, C., Cui, Y., Yang, F., & Ding, L. (2020). Outage probability analysis andminimization in intelligent reflecting surface-assisted miso systems. IEEECommunications Letters, (pp. 1–1).

Guo, H., Liang, Y.-C., Chen, J., & Larsson, E. G. (2019). Weighted sum-rateoptimization for intelligent reflecting surface enhanced wireless networks. arXivpreprint arXiv:1905.07920 , .

Han, Y., Tang, W., Jin, S., Wen, C., & Ma, X. (2019). Large intelligentsurface-assisted wireless communication exploiting statistical csi. IEEE Transactionson Vehicular Technology , 68 , 8238–8242. doi:10.1109/TVT.2019.2923997.

Hu, X., Wang, J., & Zhong, C. (2020). Statistical CSI based Design for IntelligentReflecting 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 reflectingsurfaces. In 2019 IEEE Global Communications Conference (GLOBECOM) (pp.1–6).


http://dx.doi.org/10.1109/TVT.2019.2923997

http://arxiv.org/abs/2008.08453

http://dx.doi.org/10.1109/TWC.2019.2922609

Reference IINadeem, Q., Kammoun, A., Chaaban, A., Debbah, M., & Alouini, M. (2020).

Asymptotic max-min sinr analysis of reconfigurable intelligent surface assisted misosystems. 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 preprintarXiv:1903.08127 , .

Ni, W., Liu, X., Liu, Y., Tian, H., & Chen, Y. (2020). Resource Allocation forMulti-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 networkvia joint active and passive beamforming. IEEE Transactions on WirelessCommunications, (pp. 1–1). doi:10.1109/TWC.2019.2936025.

Wu, Q., & Zhang, R. (2019). Weighted sum power maximization for intelligentreflecting 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 CommunicationsMagazine, 58 , 106–112.



http://dx.doi.org/10.1109/TWC.2019.2936025

Reference IIIXie, 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 assistednon-orthogonal multiple access. arXiv preprint arXiv:1907.03133 , .

Yu, X., Xu, D., & Schober, R. (2019a). Enabling secure wireless communications viaintelligent reflecting surfaces. In 2019 IEEE Global Communications Conference(GLOBECOM) (pp. 1–6).

Yu, X., Xu, D., & Schober, R. (2019b). Miso wireless communication systems viaintelligent reflecting surfaces : (invited paper). In 2019 IEEE/CIC InternationalConference 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 outageprobability in multi-intelligent reflecting surface-assisted systems. arXiv preprintarXiv:1909.02193 , .



http://dx.doi.org/10.1109/ICCChina.2019.8855810

Analysis and Optimization of an Intelligent Reflecting...

Documents

Transcript of Analysis and Optimization of an Intelligent Reflecting...