On Performance of Intelligent Reflecting Surface aided Wireless Powered IoT Network with HIs

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Item Type Article

Authors Kumar, Deepak;Singya, Praveen Kumar;Krejcar, Ondrej;Choi, Kwonhue;Bhatia, Vimal

Citation Kumar, D., Singya, P. K., Krejcar, O., Choi, K., & Bhatia, V. (2022).

On Performance of Intelligent Reflecting Surface aided Wireless Powered IoT Network with HIs. IEEE Communications Letters, 1–

1. https://doi.org/10.1109/lcomm.2022.3228907 Eprint version Post-print

DOI 10.1109/LCOMM.2022.3228907

Publisher IEEE

Journal IEEE Communications Letters

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On Performance of Intelligent Reflecting Surface aided Wireless Powered IoT Network with HIs

Deepak Kumar¹, Praveen Kumar Singya², Ondrej Krejcar³, Kwonhue Choi⁴, Senior Member, IEEE, and Vimal Bhatia⁵, Senior Member, IEEE

Abstract—Intelligent reflecting surface (IRS) is envisioned as a key technology for the next-generation wireless communication systems that enhances coverage and performance by reconfig- uring the wireless propagation environment. In this letter, the performance of an IRS-aided wireless-powered internet-of-things (IoT) network over Rayleigh fading channels is investigated that consists a power station, an IRS, an access point, and IoT devices.

In particular, the impact of transceiver hardware impairments (HIs) is considered. An IoT node selection strategy is adopted that maximizes the harvested energy and improves the system performance. The closed-form expressions of outage probability (OP), ergodic capacity, and symbol error rate are derived by using the Gaussian Chebyshev Quadrature method. Further, the closed-form expressions of the asymptotic OP and asymptotic ergodic capacity are derived and the diversity order of the considered network is obtained. The impact of HIs, overall system ceiling effect, IoT devices, reflecting elements, and various system parameters on the considered network are highlighted. Finally, the Monte-Carlo simulations are performed to verify the derived closed-form expressions.

Index Terms—Intelligent reflecting surface (IRS), IoT network, reflecting element (RE), hardware impairments (HIs), symbol error rate (SER).

I. INTRODUCTION

The internet-of-things (IoT) network is a prominent com- ponent for the deployment of the next-generation wireless communication system that demands massive connectivity with high-data rate. The IoT network consists of an access point (AP) and multiple IoT devices. The IoT devices may suffer from the energy-constrained problem due to limited

This work was supported in part by the COMET TiH IIITB Smart Radio Environment project, and in part by Grant Agency of Excellence, No.

2204/2022, University of Hradec Kralove, Faculty of Informatics and Man- agement, Czech Republic. This work was also supported by the NRF grant funded by the Korean government (MSIT) under Grant 2021R1A2C1010370.

(Corresponding Author: Kwonhue Choi.)

1D. Kumar is with the Department of Electrical Engineering, In- dian Institute of Technology Indore, Indore 453552, India (e-mail:

[email protected]).

2P. K. Singya is with the CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955, Saudi Arabia (e-mail:

[email protected]).

3O. Krejcar is with Faculty of Informatics and Management, University of Hradec Kralove, 50003 Hradec Kralove, Czech Republic and with Malaysia- Japan International Institute of Technology, Universiti Teknologi Malaysia Kuala Lumpur, Jalan Sultan Yahya Petra, Kuala Lumpur 54100, Malaysia (e-mail: [email protected]).

4K. Choi is with the Department of Information and Communication Engineering, Yeungnam University, Gyeongsan 38541, South Korea (e-mail:

[email protected])

5V. Bhatia is with the Department of Electrical Engineering, Center for Advanced Electronics, Indian Institute of Technology Indore, Indore 453552, India, and also with the Faculty of Informatics and Management, University of Hradec Kralove, 50003 Hradec Kralove, Czech Republic (e-mail: vbha- [email protected]).

battery lifetime and their locations may not allow a direct power connection. To overcome this problem, the wireless- powered communication is a promising solution that harvests the energy from the dedicated energy source or power station (PS) via the received radio-frequency (RF) signal by utilizing the wireless energy transfer (WET) protocol [1].

Recently, intelligent reflecting surface (IRS) has emerged as a promising technology for the deployment of next-generation wireless communication systems due to its capability to mod- ify the propagation environment [2]. Typically, the IRS consists of a large number of reconfigurable reflecting elements (REs) that adjusts the phase of the incident signal in the desired direction and is controlled by an IRS controller. In [3]–[7], the performance of an IRS-aided wireless-powered IoT networks are studied. In [8], the performance of an IRS-aided wireless- powered IoT network with a nonlinear energy harvesting model is studied. In [9], the impact of transceiver hardware impairments (HIs) and phase shift error on the performance of a reconfigurable intelligent surface aided wireless-powered IoT network are studied.

In recent years, the deployment of the IoT network in smart environments such as smart cities, smart health, smart homes, etc. is possible because of the rapid advances in wireless communication technology [10]. However, the IoT network experiences attenuation because of the fading channels that degrades the overall system performance. To overcome this problem, the IRS is a promising technology to improve the performance of wireless IoT network over faded channels.

Motivated by the above discussions, in this letter, the performance of an IRS-aided wireless-powered IoT network is investigated that consists a PS, an IRS, an AP, and IoT devices.

The considered network can be applied in various practical applications like wireless surveillance, event detection for emergency services, environment monitoring, etc. In practice, the transceiver may suffer from several kinds of impairments, such as in-phase/quadrature-phase imbalance, phase noise, etc. that affects the overall performance [11]. We consider the impact of transceiver HIs which arise due to low-cost devices. The IRS-aided wireless communications are usually characterized by line-of-sight (LoS) connections, however there are cases where LoS communication is not possible due

N otations : f_Z(·) and F_Z(·) denote the probability density function (PDF) and cumulative distribution function (CDF) of a random variable (RV) Z, respectively. Also,E{·}and Var{·}denote the statistical expectation and variance operator, respectively. Further, Pr[·], Γ(·), and Υ(·,·) denote the probability, complete gamma function, and lower incomplete gamma function, respectively.

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to deep fade, shadowing effect, or both [12]. Hence, such non LoS (NLoS) links are statistically characterized with Rayleigh fading. Considering this, the main contributions of this letter are as follows:

• We derive the closed-form expression of outage probability (OP) by using the Gaussian Chebyshev quadrature (GCQ) method. We also derive the asymptotic OP at high signal-to-noise ratio (SNR) and obtain the diversity order of the considered network.

• We obtain the ergodic capacity and asymptotic ergodic capacity of the considered network in terms of the Meijer- G function by using the GCQ method.

• We derive the closed-form expression of symbol error rate (SER) by using the GCQ method.

• Finally, we highlight the impact of the RE, IoT devices, HIs, overall system ceiling (OSC), and various system parameters on the OP, asymptotic OP, ergodic capacity, asymptotic ergodic capacity, and SER performance.

II. SYSTEMMODEL

Fig. 1 illustrates an IRS-aided IoT network that consists of a power station (PS), N IoT devices Mn (n = 1, ..., N), an IRS, an access point (AP), and an IRS controller. The IoT devices are wirelessly charged by using WET protocol through the received RF signal from the dedicated PS and the IoT device M_n transmits its information signal to the AP via the IRS. The IRS is installed on a tall building and equipped with L REs while the other devices (i.e., PS, AP, and IoT devices) are equipped with a single antenna [4], [6].

The reflection configuration at the IRS is controlled by an IRS controller. The direct link between the IRS and AP is not available. We consider that the IoT devices and AP are impacted by transceiver HIs and thus, a distortion noise is introduced in the system. The fading coefficients of PS→Mn, Mn → IRS, and IRS → AP links are denoted as hsn, hnq, and hqb, respectively, where s, q, b ∈ {PS, IRS, AP} and q= 1, ..., L. The fading coefficients follows independent and non-identically distributed (i.n.i.d) Rayleigh fading and are quasi-static.

We adopt a harvest-then-transmit strategy at the energy con- straint IoT nodes, where the IoT node first harvests the energy through the received RF signal from the PS by using the WET protocol. The IoT node utilizes the harvested energy for the wireless information transmission. The AP is connected with a dedicated power source. The amount of harvested energy at the n^th IoT node is given asE_n =ηT P_s|h_sn|², where η, T, and P_s denote the energy conversion efficiency at M_n, the time period, and transmit power at the PS, respectively. Further, we adopt an IoT node selection strategy that maximizes the harvested energyE_n to improve the system performance [13].

Specifically, the best IoT node denoted as n^⋆ is selected such that n^⋆ = arg max

n∈{1,...,N}{|hsn|²}. With such IoT node selection strategy, the harvested energy atMn^⋆node is given as En^⋆=ηT Ps|hsn^⋆|². We adopt a generic harvest-then-transmit protocol, and consider the whole time period T = 1 [6]. The transmit power at theMn^⋆ node is given asPn^⋆ =ηPs|hsn^⋆|². The IoT node M_n^⋆ transmits its information signal to the AP

M1

Mn

MN

Access Point Power Station (AP)

(PS)

IRS IRS Controller

IoT Devices (Mn) Wireless energy transfer Wireless information transfer

hn*q hqb

hs¹

hsn

hsN

Fig. 1: Intelligent reflecting surface aided IoT network.

by utilizing the harvested energy. Hence, the signal received at the AP is given as

y_b=p P_n^⋆

L

X

q=1

h_n^⋆_qe^jθ^qh_qb(x_n^⋆+ Φ_n^⋆_b) +w_b, (1) where θq ∈ [0,2π] denotes the phase-shift at the q^th RE, xn^⋆ is the unit energy symbol transmitted by nodeMn^⋆, and wb∼ CN(0, σ²)is the additive white Gaussian noise (AWGN) at the AP. The term Φn^⋆b ∼ CN(0, ϕ²) in (1) denotes the distortion noise due to the HIs at both the IoT and AP, such thatϕ=p

ϕ²_n⋆+ϕ²_b [11]. Here,ϕcharacterizes the aggregate level of HIs, andϕ²_n⋆ andϕ²_b represent the level of HIs in the transmitter and the receiver at the IoT and AP, respectively.

The received instantaneous signal-to-noise-plus distortion ratio (SNDR) at the AP is given as

Λb= ηζ|hsn^⋆|²|PL

q=1h_n^⋆_qe^jθ^qh_qb|² ηζ|h_sn^⋆|²|PL

q=1h_n^⋆_qe^jθ^qh_qb|²ϕ²+ 1, (2) where ζ = _σ^P2^s denotes the transmit signal-to-noise ratio (SNR).

III. PERFORMANCEANALYSIS

A. Outage Probability

For the considered network, the outage event is declared when the received instantaneous SNDR at the AP is lower than a pre-defined thresholdΛ_th which is given as

Pout(Λth) =Pr[Λb<Λth], (3) where Λ_th = 2^r^th −1 and rth denotes the threshold data- rate. Further, the cummulative distribution function (CDF) FΛ_b(Λ_th)^∆=Pout(Λ_th)is expressed as

FΛ_b(Λth) =

(ψout(Λth) , for Λth< _ϕ¹2

1 , for Λ_th≥ _ϕ¹2. (4) The termψ_out(Λ_th) is given as

Ψ_out(Λ_th)≈

V

X

m=1 N

X

p=0

N p

(−1)^pπp 1−x²_m 4λ^αϕ²VΓ (α) (√

y_m)^α−2

×e⁻

√ym

λ − ^pΛ^th

ηζΩ_sn⋆(1−ϕ2 Λ_th)ym, (5) whereα = _16−π^Lπ²₂, λ= ⁴_π−^π₄ p

Ω_n^⋆_qΩ_qb, y_m = ^(1+x_2ϕ₂^m⁾, andxm=cos_(2m−1)π

2V

. P roof: See Appendix A.

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B. Asymptotic Outage Probability

The asymptotic OP is carried out to obtain the diversity order of the considered network which can be evaluated by approximating the OP at high SNR. By employing the high SNR approximation of the lower incomplete Gamma function Υ (c, x) ≈

x→0(^x_c^c)[14], asymptotic OP is derived as P^∞out(Λ_th)≈

N

X

q=0

N q

(−1)^q+1 α _π⁴ −^π₄^α

Γ (α) 1

Ωn^⋆qΩqb

^α₂

× q

Ωsn^⋆

^α₂ Γ

−α

2 + 1 Λ_th ηζ(1−ϕ²Λth)

^α₂

. (6) From (6), we observe that the diversity order of the considered network is1.5α, i.e.,1.5

Lπ² 16−π²

, which depends on the RE.

C. Ergodic Capacity

The ergodic capacity of the considered network is expressed as [14]

C_e=E{log₂(1 + Λ_b)}= Z ∞

0

log₂(1 +z)fΛ_b(z)dz. (7) The term log₂(1 +z)can be represented in terms of Meijer- G function as log₂(1 +z) = G^1,2_2,2h

z|^1,1_1,0i

. Differentiating FΛ_b(z), the probability density function (PDF) of Λb is obtained as

fΛ_b(z) =

V

X

m=1 N

X

p=0

N p

(−1)^p+1πp

1−x²_m √ ym^α−4

p 4λ^αϕ²VΓ (α)ηζΩ_sn^⋆

×e⁻

√ym

λ 1

(1−ϕ²z)²e⁻

pz

ηζΩ_sn⋆ ym(1−ϕ2z). (8) Substituting (8) in (7), the ergodic capacity is given as Ce=

V

X

m=1 N

X

p=0

N p

(−1)^p+1πp

1−x²_m √ ym

α−4

p 4λ^αϕ²VΓ (α)ηζΩ_sn^⋆ e⁻

√ym λ

× Z _ϕ¹₂

0

1 (1−ϕ²z)²e⁻

pz

ηζΩ_sn⋆ ym(1−ϕ2z)G^1,2_2,2h z|^1,1_1,0i

dz. (9) The assessment of (9) is mathematically intractable. Hence, we apply GCQ method to obtain the closed-form expression of ergodic capacity which is given as

C_e≈

U

X

u=1 V

X

m=1 N

X

p=0

N p

(−1)^p+1πpp

1−x²_mp 1−w²_m 8λ^αϕ⁴U VΓ (α)ηζΩsn^⋆

×e⁻

√ym λ

√y_mα−4

(1−ϕ²z_u)²e⁻

pzu

ηζΩ_sn⋆ ym(1−ϕ2zu)G^1,2_2,2h zu|^1,1_1,0i

, (10) wherewu=cos_(2u−1)π

2U

andzu= ^(1+w_2ϕ2^u⁾. D. Asymptotic Ergodic Capacity

The ergodic capacity at high SNRs can be approximated as [15]

C_e^∞≈E{log₂(Λ_b)}=log₂ Z ∞

0

zf_Λ_b(z)dz

. (11)

Substituting (8) in (11), the asymptotic ergodic capacity is given as

C_e^∞≈log₂

V

X

m=1 N

X

p=0

N p

(−1)^p+1πp

1−x²_m √ ymα−4

4λ^αϕ²VΓ (α)ηζΩ_sn^⋆

×pe⁻

√ym λ

Z ¹

ϕ2 0

z (1−ϕ²z)²e⁻

pz

ηζΩ_sn⋆ ym(1−ϕ2z)dz

! .

(12) The assessment of (12) is mathematically intractable. Hence, we apply GCQ method to obtain the closed-form expression of asymptotic ergodic capacity which is given as

C_e^∞≈log₂

U

X

u=1 V

X

m=1 N

X

p=0

N p

(−1)^p+1π²pp

1−x²_me⁻

√ym λ

8λ^αϕ⁴U VΓ (α)ηζΩsn^⋆

×(√

ym)^α−4p

1−w²_m zu

(1−ϕ²zu)²e⁻

pzu ηζΩ_sn⋆ ym(1−ϕ2zu)

! . (13) From (13), we observe that the diversity order of the considered network is1.5α, i.e.,1.5

Lπ² 16−π²

, which depends on the RE.

E. Symbol Error Rate

The SER for various modulation schemes is given as [15]

Pe=a√

√ b π

Z ∞

0

e^−bz

√z Pout(z)dz, (14) wherea= 1andb=sin²(π/M)forM-ary phase shift keying (PSK), a = 2(1−(1/√

M)) and b = 3/(2 (M −1)) for rectangular M-ary quadrature amplitude modulation (QAM), and M denotes the constellation size. Substituting (5) into (14), we obtain

Pe= a√

√ b π

V

X

m=1 N

X

p=0

N p

(−1)^pπp 1−x²_m 4λ^αϕ²VΓ (α) (√

ym)^α−2

×e⁻

√ym λ

Z _ϕ¹₂

0

z⁻¹²e^−bz−

pz

ηζΩ_sn⋆(1−ϕ2z)ymdz. (15) The assessment of (15) is mathematically intractable. Hence, we apply GCQ method to obtain the closed-form expression of the SER which is given as

Pe≈ a√

√b π

U

X

u=1 V

X

m=1 N

X

p=0

N p

(−1)^pπ²p 1−x²_m 8λ^αϕ⁴V UΓ (α) (√

y_m)^α−2

×e⁻

√ym

λ p

1−w_u²z⁻

1 u2e^−bz^u⁻

pzu

ηζΩ_sn⋆(1−ϕ2zu)ym. (16) IV. SIMULATION ANDNUMERICALRESULTS

In this Section, we present the numerical and simulation results of OP, asymptotic OP, ergodic capacity, asymptotic ergodic capacity, and SER for the considered network. We consider a 3D coordinate system to describe the system model, where the PS, IoT deviceMn^⋆, the IRS, and the AP are located at(−10,0,0),(0,0,0),(−2,6,0), and(10,0,0), respectively [6]. We consider a path-loss model, where Ωij = x^−v_ij , with v= 3as path-loss exponent. Here,xijdenotes the distance between i and j nodes, where{i,j} ∈ {PS, AP, IRS, IoT device}.

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-20 -10 0 10 20 30 SNR (dB)

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Outage Probability

= 0.2, = 0.3, r_th = 3 = 0.2, = 0.3, r_th = 2 = 0.2, = 0.3, r_th = 1 = 0.2, = 0.7, r_th = 1 = 0.1, = 0.7, r_th = 1 = 0, = 0.7, r_th = 1 Asymptotic

Simulation 20.6 20.8

2.6 2.8

3 10^-4 N = 4, L = 2

(a)

-20 -10 0 10 20 30

SNR (dB) 10^-6

10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

Analytical Simulation L = 2, 4, 8, 16, 32, 64

N = 4 N = 5

L = 64

N = 3

(b)

0 2 4 6 8 10

rth ( bps/Hz ) 10^-5

10^-4 10^-3 10^-2 10^-1 10⁰

= 0 = 0.1 = 0.2 = 0.3 Simulation L = 4, N = 4, = 0.7 SNR = 10 dB

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Fig. 2: (a) OP versus SNR. (b) Impact of the REs on the outage performance. (c) Impact of the OSC effect on the OP.

-10 -5 0 5 10 15 20

SNR (dB) 0

1 2 3 4 5 6 7

Ergodic Capacity = 0.5, = 0.2, N = 3, L = 3

= 0.8, = 0.2, N = 3, L = 3 = 0.8, = 0.1, N = 3, L = 3 = 0.8, = 0.1, N = 4, L = 3 = 0.8, = 0.1, N = 4, L = 4 Asymptotic Simulation

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-20 -10 0 10 20 30

SNR (dB) 10^-5

10^-4 10^-3 10^-2 10^-1 10⁰

Symbol Error Rate 256-QAM

64-QAM 16-QAM QPSK QPSK QPSK Simulation

= 0.7, L = 8, = 0.1

N = 10 N = 6

N = 4

(b)

-20 -15 -10 -5 0 5 10 15 20

SNR (dB) 10^-5

10^-4 10^-3 10^-2 10^-1 10⁰

Symbol Error Rate

= 0.3, L = 8 = 0.7, L = 8 = 0.7, L = 10 = 0.7, L = 12 = 0.7, L = 14 = 0.7, L = 16 Simulation

= 0.1, N = 4, 16-QAM

(c)

Fig. 3: (a) Ergodic capacity versus SNR. (b) Symbol error rate versus SNR. (c) Impact of the RE on the SER.

Further, we considerU=V = 100to obtain the numerical results of OP, ergodic capacity, asymptotic ergodic capacity, and SER with acceptable accuracy and considerable computational complexity.

Fig. 2(a) depicts the OP versus SNR curves for N = 4 and L = 2. We observe that the numerical and simulation results match perfectly, which shows the accuracy of the GCQ method. Further, the asymptotic OP curves match with the OP curves at high SNRs. The outage performance improves with the decrease in threshold data rate rth. This is because lower threshold data rates result in lower threshold SNRs. Hence, the probability of outage occurrence reduces at lower threshold SNRs. To attain an OP of 10⁻², we achieve approximately 4.63 and 9.74 dB gains when rth decreases from 3 to 2 and 3 to 1 bps/Hz, respectively. Further, we observe that the outage performance improves with the increase in the energy conversion efficiency η because the amount of harvested energy increases withη. To attain an OP of10⁻², we achieve approximately 3.82 dB gain whenη increases from 0.3 to 0.7.

Furthermore, we observe that the outage performance degrades with the increase in the level of HIsϕsince the distortion noise increases withϕ.

Fig.2(b) depicts the impact of REs and the IoT device on the outage performance of the considered network. We observe that the outage performance improves with the increase in REs because system collaboratively achieves fine-grained reflect beamforming [1]. To attain an OP of 10⁻², we achieve

approximately 10.87, 19.33, 26.55, 33.12, and 39.36 dB gains whenLincreases from 2 to 4, 8, 16, 32, and 64, respectively.

Further, we observe that the outage performance improves with N as the amount of harvested energy increases with the IoT nodes.

Fig.2(c)showcases the impact of OSC (caused due to HIs ϕ) on the OP of the considered network. Fig. 2(c), gives an indication about the maximum data, the system can achieve without entering into outage. We observe that for a fixed ϕ, OSC occurs at a specific threshold. This specifies the level of HIs that the system can handle and the maximum data rate the system can provide. Further, we observe that the OSC occurs at approximately 3.85, 4.84, and 6.55 bps/Hz threshold data rates forϕ= 0.1,0.2,and 0.3, respectively.

Fig. 3(a) depicts the ergodic capacity versus SNR curves of the considered network. The asymptotic ergodic capacity curves match with the ergodic capacity curves at high SNRs.

We observe that the ergodic capacity improves withηbecause the harvested energy increases with the efficiency of the energy harvesting circuitry. Further, the ergodic capacity improves with the decrease in the level of HIs ϕ. To attain an ergodic capacity of 4 bps/Hz, we achieve approximately 6.26 dB gain when ϕ decreases from 0.2 to 0.1. Further, we observe that the ergodic capacity improves with the increase inN andL.

Fig. 3(b) depicts the SER versus SNR curves of the considered network. We observe that the SER improves with the

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constellation size M. To attain an SER of10⁻², we achieve approximately 4.92 and 16.96 dB gains when M decreases from 256 to 64 and 16-QAM, respectively. Further, we observe that the SER improves with N. To attain an SER of 10⁻² for quadrature PSK (QPSK) modulation scheme, we achieve approximately 1.37 and 2.69 dB gains whenN increases from 4 to 6 and 10, respectively.

Fig. 3(c) depicts the impact of the RE and the energy conversion efficiency on the SER of the considered network.

We observe that the SER improves with the increase in L because the system collaboratively achieves fine-grained reflect beamforming at the destination node. To attain an SER of10⁻², we achieve approximately 2.43, 5.43, 9.37, and 17.52 dB gains when L increases from 8 to 10, 12, 14, and 16, respectively. Further, the SER improves with η.

V. CONCLUSION

In this letter, the performance of an IRS-aided wireless- powered IoT network with HIs is investigated. The closed- form expressions of OP, ergodic capacity, and SER have been derived by using the GCQ method. Further, the closed-form expressions of asymptotic OP and asymptotic ergodic capacity have been derived and the diversity order of the considered network is obtained. It has been found that the OSC effect can limit the performance of the considered network, whereas increasing the number of REs and IoT devices improves the performance significantly.

APPENDIXA

Let X =^∆|hsn^⋆|², then the CDF FX(x) =h 1−e⁻

x Ω_sn⋆i^N

. Further, let √

Y =^∆ |PL

q=1h_n^⋆_qe^jθ^qh_qb|. For an opti- mal phase shift (i.e., θ_q = 0), √

Y can be approximated as PL

q=1|h_n^⋆_qh_qb| [2]. Therefore, for large L, the corresponding CDF of the RV √

Y is approximated as F^√_Y (y) = 1 − e⁻

y Ω_n⋆_qΩ_qb

[3]. The mean and the variance of Y are computed as E{√

Y} = Lπp

Ωn^⋆qΩqb/4 and Var{√

Y}=L 1−π²/16

Ωn^⋆qΩqb, respectively. Since

√Y is the product of two i.n.i.d Rayleigh RVs, it can be approximated as Gamma distribution [2]. Therefore by utilizing moment-matching method, shape parameter α and scale parameter λ of the Gamma distribution are given as α =

E{√ Y}2

/Var{√

Y} = Lπ²/ 16−π²

and λ = Var{√

Y}/E{√

Y} = (4/π−π/4)p

Ωn^⋆qΩqb, respectively.

Let Z = W², then the CDF of Z can be deduced from W as FZ(w) = FW(√

w). Therefore, the CDF and PDF of RV Y are given as FY (y) = (1/Γ (α)) Υ α,√

y/λ and fY (y) = √

yα−2

e⁻^√^y/α

/(2Γ (α)λ^α), respectively. On invoking (2) into (3), we obtain

Pout(Λth) =Pr[Λb<Λth] =Pr

X < Λth

ηζY (1−ϕ²Λ_th)

. (17) By utilizing (4), the termψout(Λth)is evaluated as

ψ_out(Λ_th) = Z ∞

0

F_X

Λ_th ηζ(1−ϕ²Λ_th)y

f_Y(y)dy. (18) On substituting the CDF FX(·) and PDF fY (·), (18) is express as

ψout(Λth) =

N

X

p=0

N p

(−1)^p 2λ^αΓ (α)

Z ¹

ϕ2 0

(√ y)^α−2

×e⁻

√y

λ − ^pΛ^th

ηζΩ_sn⋆(1−ϕ2 Λ_th)ydy. (19) The assessment of (19) is mathematically intractable. Hence, we apply the GCQ method which is given as [16]

Z b

a

f(y)dy≈(b−a)π 2V

V

X

m=1

p1−x²_mf(y_m), (20) wherey_m= ^(b−a)x₂ ^m +^(b+a)₂ ,x_m =cos_(2m−1)π

2V

, and V is a finite value. Finally, by employing the GCQ method in (19), we obtain the closed-form expression of OP which is given in (5).

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