# Intelligent Reflecting Surfaces with Adaptive Transmit Power for Underlay Cognitive Radio Networks

Authors: Raed Alhamad; Hatem Boujemaa
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

In this article, we study the performance of intelligent reflecting surfaces (IRS) with adaptive transmit power (ATP). The power of secondary source is adapted to have a low interference at primary destination. IRS with ATP offers 14, 20, 26, 32, and 38 dB gains versus the absence of IRS forN=8,16,32,64,and128 reflectors. Rayleigh channels are studied with arbitrary positions of primary and secondary nodes.


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## Body

## 1. Introduction

In underlay cognitive radio networks, secondary users adapt their power to minimize the interference at primary nodes [1, 2]. Power adaptation allows joint transmission of primary and secondary nodes over the same band [2–5]. The system throughput is high in UCRN since it is the sum of primary and secondary throughputs. In non-UCRN, the throughput is equal to that of primary network and is therefore small than that of UCRN. In order to enhance data rates in UCRN, we suggest the use of reflecting surfaces [6–8]. The phase of each reflector is wisely chosen so that received signals have a zero phase at the destination. The phase shifts of IRS can be quantized over n bits [9–12]. The number of reflectors is N=8,16,⋯,1024. The diversity order is high since it is equal to N=8,16,⋯,1024 leading to significant performance enhancement. The throughput of IRS with constant transmit power was derived in [13–19]. Channel estimation for millimeter wave communications using IRS was studied in [20]. Joint power allocation and user association for millimeter wave communications using IRS were suggested in [21]. In all previous studies, IRS were studied with fixed transmit power [6–21]. In this paper, we derive the performance of IRS with adaptive transmit power. In fact, the secondary source adapts its power so that the interference at the primary destination is lower than T. The secondary source adapts its power, and its signal is reflected by IRS so that all reflections have a zero phase at secondary destination.When intelligent reflecting surfaces (IRS) are used in wireless systems with fixed transmit power, the signal-to-noise ratio (SNR) follows a chi-square distribution with one degree of freedom. IRS can be used in the secondary network of underlay CRN where the secondary source adapts its power to generate a low interference at primary destination. In this case, the SNR is the product of a chi-square random variable (r.v.) with one degree of freedom and the random adaptive transmit power. The statistics of SNR are not available when IRS are used with adaptive transmit power which is the first motivation of this research article. Besides, IRS were previously studied without interference. The second motivation of the paper is to derive the throughput of CRN using IRS and adaptive transmit power in the presence of interference from primary source.The main innovations of the article are as follows:(i)
We compute the throughput of IRS and adaptive transmit power. IRS is between secondary source and destination to improve the throughput. IRS with fixed constant power was studied in [6–22].(ii)
IRS with ATP offers 14, 20, 26, 32, and 38 dB gains versus the absence of IRS forN=8,16,32,64,and128 reflectors(iii)
We derive the CDF of SNR and SINR while taking account of primary interference. Our results are confirmed with computer simulationsSections2 and 3 derive the throughput when IRS is a reflector or a transmitter. Section 4 describes the results. Conclusions are given in Section 5.


## 2. SNR and SINR Analysis when IRS Is a Reflector

Figure1 depicts the network model containing a primary destination and source (PD, PS) and a secondary destination and source (SD, SS). SS has an ATP. Besides, an IRS is placed between SS and SD in order to improve the secondary throughput.Figure 1
Network model when IRS is a reflector.


### 2.1. SNR Analysis

Secondary sourceSS adapts its symbol energy E so that the interference at PD is small:
(1)EhSSPD2≤T.where hXY is the channel of the X-Y link.The adaptive transmit energy of a symbolE is defined as
(2)E=minEmax,ThSSPD2,where Emax is the maximum energy of a symbol.The adaptive transmit power (ATP) is equal to(3)P=ETs=minEmaxTs,TTshSSPD2,where Ts is the symbol period.Letβ1ak be the channel of SS-k-th reflector link. β1=1/dSSIRSPLE where dXY is the distance from X to Y, and PLE is the path loss exponent (PLE). We can write ak=bke−jck where bk=ak.Letβ2dk be the channel from k-th reflector to SD. β2=1/dIRSSDPLE. We can write dk=eke−jfk where ek=dk.The phase of shift ofk-th reflector is [1]:
(4)wk=ck+fk.The received signal atSD is
(5)rl=slEβ1β2∑k=1Nakdkejwk+nl,where sl is the l-th symbol and nl is a zero-mean Gaussian r.v. of variance N0.We replaceak=bke−jck and dk=eke−jfk by their expressions and use (4) to obtain
(6)rl=slEβ1β2∑k=1Nbkeke−jck−jfk+jwk+nl=slEβ1β2A+nl,where
(7)A=∑k=1Nbkek.Using the central limit theorem,A can be approximated by a Gaussian random variable with mean and variance denoted by mA=Nπ/4 and σA2=N1−π2/16. This approximation is valid for a large number of reflectors N≥8.The SNR atSD is expressed as
(8)ΓSSSD=Eβ1β2N0A2.Equation (8) shows that the SNR is the product of a chi-square r.v. A2 and Eβ1β2/N0 which is also random as E given in (2) depend on channel gain hSSPD. In case of fixed transmit power, Eβ1β2/N0 is deterministic, and the SNR follows a chi-square distribution as it is proportional to A2.The CDF of SNR is(9)FΓSSSDx=PΓSSSD<xThSSPD2<Emax×PThSSPD2<Emax+PΓSSSD<xThSSPD2<Emax×PThSSPD2>Emax.We have(10)PThSSPD2<Emax=e−T/EmaxλSSPD,where λSSPD=EhSSPD2 and hSSPD is the channel from SS to PD.When the adaptive transmit symbol energy is larger thanEmaxT/hSSPD2>Emax, E=Emax, and the SNR is expressed as
(11)ΓSSSDThSSPD2>Emax=Emaxβ1β2N0A2.WhenT/hSSPD2>Emax, we have
(12)PΓSSSD<xThSSPD2>Emax=1−Q0.5mAσA,xN0Emaxβ1β2σA2,where Qm∙,∙ is the generalized Marcum Q-function.WhenT/hSSPD2<Emax, E=T/hSSPD2, and the first term of (8) is written as
(13)PΓSSSD<xThSSPD2<Emax=PTA2β1β2hSSPD2N0<xhSSPD2>TEmax.We deduce(14)PΓSSSD<xThSSPD2<Emax=∫T/Emax+∞PA2<xyN0Tβ1β2e−y/λSSPDλSSPDdy.A has a Gaussian distribution, and we have
(15)PΓSSSD<xThSSPD2<Emax=∫T/Emax+∞1−Q0.5mAσA,xyN0σA2Tβ1β2e−y/λSSPDλSSPDdy.Letu=y, so that dy=2udu, and we deduce
(16)PΓSSSD<xThSSPD2<Emax=e−T/EmaxλSSPD−2∫TEmax+∞Q0.5mAσA,uxN0σA2Tβ1β2e−u2/λSSPDλSSPDudu.We use ([22], Equation (34)):
(17)∫c+∞e−ρ2u2/2QMb,auudu=e−ρ2c2/2ρ2QMb,ac−1ρ2a2ρ2+a2Me−b2ρ2/2ρ2+2a2QMaba2+ρ2,cρ2+a2,with M=0.5, c=T/Emax, ρ2=2/λSSPD, b=mA/σA, and xN0/Tσ12β1β2.Using (10), (11), (13), (17), and (18), we finally obtain the CDF of SNR of CRN using IRS and adaptive transmit power:
(18)FΓSSSDx=e−2T/EmaxλSSPD−λSSPDe−2T/EmaxλSSPD×Q0.5mAσA,xN0Emaxβ1β2σA2+e−2T/EmaxλSSPD0.5λSSPDxN02/λSSPDTσA2β1β2+xN0×e−2mA2/2σA2+xN0λSSPD/β1β2TσA2×Q0.5xN0mA22TσA4β1β2/λSSPD+xN0σA2,TEmax2λSSPD+xN0Tβ1β2σA2+1−e−T/EmaxλSSPD1−Q0.5mAσA,xN0Emaxβ1β2σA2.


### 2.2. SINR Analysis

The SINR atSD is equal to
(19)γSSSD=Eβ1β2A2N0+IPSSD,where IPSSD=EPShPSSD2 and EPS is PS symbol energy.The SINRγSSSD can be expressed in terms of SNR ΓSSSD (8) as follows:
(20)γSSSD=ΓSSSD1+IPSSD/N0.The CDF of SINR can be evaluated numerically as follows:(21)FγSSSDx=∫0+∞FΓSSSDx1+yN0fIPSSDydy,where FΓSSSDx is provided in (19) and
(22)fIPSSDy=1EPSλPSSDe−y/EPSλPSSD,where λPSSD=EhPSSD2.The integral (21) always converges as it computes the CDF of SINR defined in (20). The numerical evaluation of (21) was done by truncating the integral as follows:
(23)P=100,error=∫0PFΓSSSDx1+yN0fIPSSDydy−∫0P−1FΓSSSDx1+yN0fIPSSDydy,while error>0.001 is
(24)P=P+1,error=∫0PFΓSSSDx1+yN0fIPSSDydy−∫0P−1FΓSSSDx1+yN0fIPSSDydy.At the end, return the value of the integral as∫0PFΓSSSDx1+y/N0fIPSSDydy.


### 2.3. Throughput Analysis

The packet error probability (PEP) is given by [23].
(25)PEP<FΓSSSDλ,where
(26)λ=∫0+∞1−1−SEPuLdu.L is packet length and
(27)SEPu=41−1Merfc3uM−1,for M-QAM.ForM-PSK, we have
(28)SEPx=erfcsin2πMx.The throughput atSD is deduced from the PEP as follows:
(29)Thr=log2M×1−PEP.


## 2.1. SNR Analysis

Secondary sourceSS adapts its symbol energy E so that the interference at PD is small:
(1)EhSSPD2≤T.where hXY is the channel of the X-Y link.The adaptive transmit energy of a symbolE is defined as
(2)E=minEmax,ThSSPD2,where Emax is the maximum energy of a symbol.The adaptive transmit power (ATP) is equal to(3)P=ETs=minEmaxTs,TTshSSPD2,where Ts is the symbol period.Letβ1ak be the channel of SS-k-th reflector link. β1=1/dSSIRSPLE where dXY is the distance from X to Y, and PLE is the path loss exponent (PLE). We can write ak=bke−jck where bk=ak.Letβ2dk be the channel from k-th reflector to SD. β2=1/dIRSSDPLE. We can write dk=eke−jfk where ek=dk.The phase of shift ofk-th reflector is [1]:
(4)wk=ck+fk.The received signal atSD is
(5)rl=slEβ1β2∑k=1Nakdkejwk+nl,where sl is the l-th symbol and nl is a zero-mean Gaussian r.v. of variance N0.We replaceak=bke−jck and dk=eke−jfk by their expressions and use (4) to obtain
(6)rl=slEβ1β2∑k=1Nbkeke−jck−jfk+jwk+nl=slEβ1β2A+nl,where
(7)A=∑k=1Nbkek.Using the central limit theorem,A can be approximated by a Gaussian random variable with mean and variance denoted by mA=Nπ/4 and σA2=N1−π2/16. This approximation is valid for a large number of reflectors N≥8.The SNR atSD is expressed as
(8)ΓSSSD=Eβ1β2N0A2.Equation (8) shows that the SNR is the product of a chi-square r.v. A2 and Eβ1β2/N0 which is also random as E given in (2) depend on channel gain hSSPD. In case of fixed transmit power, Eβ1β2/N0 is deterministic, and the SNR follows a chi-square distribution as it is proportional to A2.The CDF of SNR is(9)FΓSSSDx=PΓSSSD<xThSSPD2<Emax×PThSSPD2<Emax+PΓSSSD<xThSSPD2<Emax×PThSSPD2>Emax.We have(10)PThSSPD2<Emax=e−T/EmaxλSSPD,where λSSPD=EhSSPD2 and hSSPD is the channel from SS to PD.When the adaptive transmit symbol energy is larger thanEmaxT/hSSPD2>Emax, E=Emax, and the SNR is expressed as
(11)ΓSSSDThSSPD2>Emax=Emaxβ1β2N0A2.WhenT/hSSPD2>Emax, we have
(12)PΓSSSD<xThSSPD2>Emax=1−Q0.5mAσA,xN0Emaxβ1β2σA2,where Qm∙,∙ is the generalized Marcum Q-function.WhenT/hSSPD2<Emax, E=T/hSSPD2, and the first term of (8) is written as
(13)PΓSSSD<xThSSPD2<Emax=PTA2β1β2hSSPD2N0<xhSSPD2>TEmax.We deduce(14)PΓSSSD<xThSSPD2<Emax=∫T/Emax+∞PA2<xyN0Tβ1β2e−y/λSSPDλSSPDdy.A has a Gaussian distribution, and we have
(15)PΓSSSD<xThSSPD2<Emax=∫T/Emax+∞1−Q0.5mAσA,xyN0σA2Tβ1β2e−y/λSSPDλSSPDdy.Letu=y, so that dy=2udu, and we deduce
(16)PΓSSSD<xThSSPD2<Emax=e−T/EmaxλSSPD−2∫TEmax+∞Q0.5mAσA,uxN0σA2Tβ1β2e−u2/λSSPDλSSPDudu.We use ([22], Equation (34)):
(17)∫c+∞e−ρ2u2/2QMb,auudu=e−ρ2c2/2ρ2QMb,ac−1ρ2a2ρ2+a2Me−b2ρ2/2ρ2+2a2QMaba2+ρ2,cρ2+a2,with M=0.5, c=T/Emax, ρ2=2/λSSPD, b=mA/σA, and xN0/Tσ12β1β2.Using (10), (11), (13), (17), and (18), we finally obtain the CDF of SNR of CRN using IRS and adaptive transmit power:
(18)FΓSSSDx=e−2T/EmaxλSSPD−λSSPDe−2T/EmaxλSSPD×Q0.5mAσA,xN0Emaxβ1β2σA2+e−2T/EmaxλSSPD0.5λSSPDxN02/λSSPDTσA2β1β2+xN0×e−2mA2/2σA2+xN0λSSPD/β1β2TσA2×Q0.5xN0mA22TσA4β1β2/λSSPD+xN0σA2,TEmax2λSSPD+xN0Tβ1β2σA2+1−e−T/EmaxλSSPD1−Q0.5mAσA,xN0Emaxβ1β2σA2.


## 2.2. SINR Analysis

The SINR atSD is equal to
(19)γSSSD=Eβ1β2A2N0+IPSSD,where IPSSD=EPShPSSD2 and EPS is PS symbol energy.The SINRγSSSD can be expressed in terms of SNR ΓSSSD (8) as follows:
(20)γSSSD=ΓSSSD1+IPSSD/N0.The CDF of SINR can be evaluated numerically as follows:(21)FγSSSDx=∫0+∞FΓSSSDx1+yN0fIPSSDydy,where FΓSSSDx is provided in (19) and
(22)fIPSSDy=1EPSλPSSDe−y/EPSλPSSD,where λPSSD=EhPSSD2.The integral (21) always converges as it computes the CDF of SINR defined in (20). The numerical evaluation of (21) was done by truncating the integral as follows:
(23)P=100,error=∫0PFΓSSSDx1+yN0fIPSSDydy−∫0P−1FΓSSSDx1+yN0fIPSSDydy,while error>0.001 is
(24)P=P+1,error=∫0PFΓSSSDx1+yN0fIPSSDydy−∫0P−1FΓSSSDx1+yN0fIPSSDydy.At the end, return the value of the integral as∫0PFΓSSSDx1+y/N0fIPSSDydy.


## 2.3. Throughput Analysis

The packet error probability (PEP) is given by [23].
(25)PEP<FΓSSSDλ,where
(26)λ=∫0+∞1−1−SEPuLdu.L is packet length and
(27)SEPu=41−1Merfc3uM−1,for M-QAM.ForM-PSK, we have
(28)SEPx=erfcsin2πMx.The throughput atSD is deduced from the PEP as follows:
(29)Thr=log2M×1−PEP.


## 3. SNR and SINR Analysis when IRS Is a Transmitter

Figure2 shows the network model when IRS are used as transmitter at SS. Let β3gk be the channel coefficient of k-th reflector to SD. We can write gk=eke−jhk where ek=gkβ3=1/dSSSDPLE.Figure 2
Network model when IRS is a transmitter.Letwk be k-th RIS phase [1].
(30)wk=hk+vm,where vm is the phase of symbol.The received signal atSD is equal to
(31)rl=Eβ3∑k=1Ngkejwk+nl=Eβ3ejvmB+nl,where
(32)B=∑k=1Nek.The SNR atSD is given by
(33)ΓSSSD=Eβ3B2N0.When IRS are used as a transmitter, the SNR is the product of a chi-square r.v.B2, and Eβ3/N0 where E given in (2) is also random as it depends on channel gain hSSPD. When IRS is used as a transmitter, the SNR is similar to the case where IRS is a reflector given in (8) where we have to replace β3 by β1β2 and B by A.The SINR atSD is expressed as:
(34)γSSSD=Eβ3B2N0+IPSSD=ΓSSSD1+IPSSD/N0.The CDF of SNR and SINR are computed as Equations (17) and (20). We have only to replace β1β2 by β3 and mA, σA by mB, σB. The throughput can be evaluated in Section 2.


## 4. Numerical Results

We used the MATLAB software to make the simulations and plot the numerical results. In this paper, Rayleigh channels were studied so that channel gains are complex random variables. We generatedak using the command randn+j×randn×sqrtc/2 where c is a constant to set the value of fix Eak2. Channel gains such as hSSPD, dk... were generated similarly. The other variables such as bk and ek were generated using bk=absak and ek=absdk. The parameters of the simulations are PLE=3, L=200, dSSIRS=1, dIRSSD=1.1, dSSSD=2.1, dSSPD=1, and dPSSD=1,1.5Figures3–5 show the throughput at SD for QPSK, 16QAM, and 64QAM when IRS is a reflector for T=1. Figure 5 shows that IRS allows 14, 20, 26, 32, and 38 dB gains versus the absence of IRS for N=8,16,32,64,and128. When the number of reflectors is doubled, we obtained 6 dB enhancement in throughput. The simulations results give a better throughput than theoretical curves at low SNR because we used the upper bound (26). At medium and large SNR, the theoretical curves are very close to simulation results.Figure 3
Throughput for QPSK: IRS is a reflector.Figure 4
Throughput for 16QAM: IRS deployed is a reflector.Figure 5
Throughput for 64QAM: IRS deployed is a reflector.Figure6 compares the throughput at SD for different values of interference threshold T=5 and T=1 for QPSK and N=16. The throughput at SD increases as T increases due to less constraints on the interference level. In fact, the secondary source can increase its power when T increases resulting in a larger throughput. The adaptive transmit power (3) increases when T increases so that the throughput improves.Figure 6
Throughput for different values of interference threshold: IRS deployed withN=16 reflectors.Figure7 compares the throughput at SD for 16PSK modulation when IRS is a reflector and a transmitter for N=8. We observe that IRS used as a transmitter offers the best performance. IRS used as a transmitter offers 1 dB gain with respect to IRS used as a reflector.Figure 7
IRS as a reflector or transmitter for 16PSK andN=8.Figure8 shows that the secondary throughput decreases as PS is near SD where dPSSD=1,1.5. This degradation in performance is expected due to interference term. In fact, when PS is close to SD, the interference term IPSSD in Equations (20) and (34) increases so that the SINR and throughput decrease.Figure 8
Effects of primary interference on secondary throughput: 64QAM withN=16 reflectors.


## 5. Conclusions

In this article, we derived the secondary throughput of underlay CRN using IRS with adaptive power. IRS allows 14, 20, 26, 32, and 38 dB gains versus the absence of IRS forN=8,16,32,64,and128 reflectors. We derived the throughput at the packet level as well as the CDF of SNR and SINR. IRS with fixed power was studied in previous papers while our results deal with adaptive power.



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