This manuscript addresses the problem formulation in an optimization context for IRS aided wireless network.

1. Applied Optimization in IRS Aided Wireless Network
Domain: 5G Beyond Wireless Communication Research
Dr. Varun Kumar
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2. Outlines
1 Introduction to Intelligent Reflecting Surface (IRS)
2 Mathematical Model for IRS Assisted Wireless Network
3 Problem Formulation in Optimization Context
4 References
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3. Intelligent Reflecting Surface (IRS):
Intelligent Reflecting Surface (IRS)
⇒ Intelligent reflecting surface (IRS) is a cost-effective solution for
achieving high spectrum and energy efficiency.
⇒ It consists of massive low-cost passive elements that are able to
reflect the signals with adjustable phase shifts.
⇒ It minimize the transmit power at the access point (AP).
⇒ SNR is maximized by passive beamforming through the IRS.
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4. System Model for IRS Assisted Wireless Network
Let a IRS assisted wireless network has N reflecting surface element, whereas
base station (BS) has M antenna. This BS assist K users and analysis is done for
the downlink scenario.
System Model:
1 BS to users links are supposed to be in deep fade.
2 BS to users link experience the active beamforming.
3 IRS unit to users link experience the passive beamforming.
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5. Mathematical model for IRS assisted wireless network
⇒ hd,k → Direct channel gain vector of size 1 × M from BS to kth user
equipment, ∀ k = 1, 2, ...K.
⇒ hr,k → Channel gain vector of size 1 × N from IRS to kth user
equipment, ∀ k = 1, 2, ...K
⇒ G → Channel matrix of size N × M from BS to IRS unit.
⇒ θn → Angular shift of the incident EM wave from the nth IRS
reflecting surface element.
Received signal by kth user in IRS aided wireless network
yk = hH
d,kx
| {z }
Direct−link
+ hH
r,kΘH
Gx
| {z }
IRS−aided−link
+ nk (1)
⇒ nk is the additive noise experienced by kth user.
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6. Continued–
⇒ Θ is the reflection coefficient matrix of size N × N.
⇒ The composite transmit data symbol can be expressed as
x =
K
X
k=1
wksk (2)
⇒ The size of x is M × 1.
⇒ xk → Transmit symbol for kth user (size 1 × 1)
⇒ wk → Beamforming vector of size M × 1.
⇒ Using (2), the eqn (1) can also be expressed as
yk =
n
hH
d,k + hH
r,kΘH
G
o K
X
k=1
wksk + nk (3)
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7. Details about Θ
⇒ Θ is the reflection coefficient matrix of size N × N.
⇒ IRS will calculate this Θ matrix, so that the received SNR across UE
could be maximized.
⇒ Let the Θk is diagonal phase matrix of size N × N.
Θk =





ejφ1,k , 0, 0, ....., 0
0, ejφ2,k , 0, ....., 0
0, 0, ....,
... , 0
0, ................ ejφN,k





where ejφ1,k → Angle between 1st IRS element and kth user.
⇒ Similarly for jth user, the phase matrix should be
Θj = diag(eφ1,j , eφ2,j , ...., eφN,j )
⇒ Theoretically, NK number of phase needs for doing the passive beam
formation.
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8. SINR calculation for kth
user in DL scenario
⇒ Let yk be the received signal of kth user
yk =
n
hH
d,k + hH
r,kΘH
G
o
wksk
| {z }
Signal
+
n
hH
d,k + hH
r,kΘH
G
o K
X
i=1,i6=k
wi si
| {z }
Interference
+ nk
|{z}
Noise
(4)
⇒ Let γk be the signal to interference plus noise ratio (SINR) of the kth
user. Mathematically, it can be expressed as
γk =
|(hH
d,k + hH
r,kΘHG)wksk|2
|(hH
d,k + hH
r,kΘHG)
PK
i=1,i6=k wi si |2 + σ2
(5)
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9. Details about beamforming vector
⇒ Here, wk is beamforming vector that is estimated by the BS.
⇒ hd,k channel coefficient can be easily estimated.
⇒ BS should know hd,k, hr,k, Θ, and G.
wk = f (hd,k, hr,k, Θ, G)
⇒ BS knows the position of IRS.
⇒ BS provides the detail of mobile users.
⇒ IRS will calculate the Θ.
⇒ IRS has phase shifter that only reflect the incident EM wave at an
certain angle.
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10. Weighted sum-rate
Weighted sum-rate for downlink system
R =
K
X
k=1
Ωk log2
1 +
|(hH
d,k + hH
r,kΘHG)wksk|2
|(hH
d,k + hH
r,kΘHG)
PK
i=1,i6=k wi si |2 + σ2
(6)
⇒ Ωk is the weight of kth user.
Problem formulation in optimization context
⇒ R is the objective function and it is a maximization problem.
⇒ W = [w1, w2, ...., wk], Θ are the two unknown.
⇒
PK
k=1 k wk k2≤ PT → Optimal power control→ Constraints
⇒ Θ → Let L be the phase resolution of a discrete phase shifter.
⇒ Minimum phase resolution of phase shifter is 2π
L and phase will be the
integral multiple of 2π
L by IRS elements.
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11. Problem formulation in optimization context
Iterative process
⇒ If IRS is a flat and rectangular surface then 0 θn π ∀ n = 1, 2, ..N
⇒ (6) is a non-convex problem.
⇒ Unknown W can be calculated for fixed Θ and vice versa.
⇒ Θ = diag[θ1, θ2, ......., θn] is a diagonal matrix of size N × N.
N number of angle is required for N− IRS elements over NL number of
total search space by discrete phase shifter.
L be total number of phase angle across each IRS element.
⇒ W is matrix of size K × M.
⇒ At constant Θ there is a need for calculating the KM number
beamforming elements.
⇒ Minimum SNR is essential criteria for each mobile user, i.e γk γTh
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12. References
M. Di Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.-S. Alouini, C. Yuen,
V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis, H. Gacanin et al., “Smart
radio environments empowered by reconfigurable ai meta-surfaces: An idea whose
time has come,” EURASIP Journal on Wireless Communications and Networking,
vol. 2019, no. 1, pp. 1–20, 2019.
H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate optimization
for intelligent reflecting surface enhanced wireless networks,” arXiv preprint
arXiv:1905.07920, 2019.
Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via
joint active and passive beamforming,” IEEE Transactions on Wireless
Communications, vol. 18, no. 11, pp. 5394–5409, 2019.
J.-P. Niu and G. Y. Li, “An overview on backscatter communications,” 2019.
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