In this article, we use Intelligent Reflecting Surfaces (IRS) to improve the throughput of Non Orthogonal Multiple Access (NOMA) with Adaptive Transmit Power (ATP). The results are valid for Cognitive Radio Networks (CRN) where secondary source adapts its power to generate low interference at primary receiver. In all previous studies, IRS were implemented with fixed transmit power and previous results are not valid when the power of the secondary source is adaptive. In CRN, secondary nodes are allowed to transmit over the same band as primary users since they adapt their power to minimize the generated interference. Each NOMA user has a subset of dedicated reflectors. At any NOMA user, all IRS reflections have the same phase. CRN-NOMA using IRS offers 7, 13, 20 dB gain vs. CRN-NOMA without IRS for N = 8, 16, 32 reflectors. We also evaluate the effects of primary interference. The results are valid for any number of NOMA users, Quadrature Amplitude Modulation (QAM) and Rayleigh channels.
IRS6GCRNNOMAadaptive transmit power (ATP)Introduction
Intelligent Reflecting Surfaces (IRS) are a good candidate for sixth generation 6G networks [1–3]. IRS phases are adjusted so that reflections have a zero-phase at all users [4–7]. IRS has been suggested for optical communications [8–10] and Millimeter Wave (mmWave) systems [11,12]. Experimental results of IRS have been presented in [13–15]. A practical implementation of IRS can be found in [16,17] were phase shifts can be continuous or quantized. Machine learning algorithms can be used to optimize the performance of IRS [18,19]. A real time cutting model using finite element was proposed in [20]. A fast and accurate tissue simulation model was discussed in [21]. Device to Device (D2D) communications for the fifth generation and beyond was studied in [22].
IRS can be used to reflect the transmitted signal to NOMA users. The source combines of symbols of K NOMA users. This signal is reflected by RIS toward K users. The weakest user detects its signal and considers the rest of signals as noise. The strongest user detects weakest user signal. Then, it removes it and continue the detection process of remaining users that are ranked from the weakest to the strongest one. IRS was implemented when the transmitter has a fixed transmit power in [1–19]. In all previous studies [1–19], IRS were implemented with fixed transmit power and previous results are not valid when the power of the secondary source is adaptive. In CRN, secondary nodes are allowed to transmit over the same band as primary users since they adapt their power to minimize the generated interference. In this paper, we derive the throughput of NOMA using IRS and adaptive transmit power.
In this article, we propose to:
Compute the throughput of CRN using NOMA and IRS where the secondary source has an adaptive power. Each secondary NOMA user has a given set of reflectors.
We derive the statistics of Signal to Noise Ratio (SNR) as well as Signal to Interference plus Noise Ratio (SINR). We study the effects of primary interference. CRN-NOMA using IRS offers 7, 13, 20 dB gain versus CRN-NOMA without IRS for N = 8, 16, 32 reflectors.
Two algorithms are discussed to rank the NOMA users.
Next section gives the throughput when there are two users. Section 3 generalizes the results to CRN-NOMA with K users. Section 4 discusses the obtained results. The paper is concluded in last section.
CRN-NOMA with Two Users
Fig. 1 depicts the network model with two secondary users, a Source (SS), a Primary Receiver and Transmitter and PR and PT. SS adapts its power to have a small interference at PR. We consider Rayleigh channels. Let λak be the channel from SS to k-th reflector of IRS. λ = 1/dSS, IRSple dX, Y is the distance from X to Y and ple is the path loss exponent. We can write ak = cke−j Φk where ck = |ak|.
A network with two users
Let λ(i)bk(i) be the channel from k-th reflector to i-th user U(i). λ(i)=1/dIRS,U(i)ple. We can write bk(i)=ek(i)e−jfk(i) where ek(i)=|bk(i)|. Let I(i) be the set of reflector’s of U(i). The phase of k-th reflector dedicated to U(i) in set I(i) given byvk(i)=fk(i)+Φk,
The transmitted symbol by SS is written ass=po1s(1)+po2s(2),where s(i) is the symbol of U(i), poi is the power of U(i), po1 + po2 = 1 and 1 > po2 > po1 > 0.
The signal at U(i) given byr(i)=sλλ(i)ESS∑k∈I(i)akbk(i)ejvk(i)+n(i),
n(i) is an additive Gaussian r.v. with variance N0 and ESS is SS symbol energy defined asESS=min(Emax,I|gSSPR|2)
Emax is the maximum symbol energy, I is the interference threshold and gSSPR is channel coefficient between SS and PR. SS verifies interference constraints asESS|gSSPR|2≤I,
Using (1), we obtainr(i)=A(i)λλ(i)ESS[po1s(1)+po2s(2)]+n(i),whereA(i)=∑k∈I(i)ckek(i),
Weak user U(2) estimates s(2) with SINRΓ(2)=po2B(2)po1B(2)+N0,whereB(i)=[A(i)]2λλ(i)ESS,.
The probability of an outage event at U(2) is given byPoutage(2)(x)=PB(2)(N0xpo2−po1x)where the Cumulative Distribution Function (CDF) of B(i), PB(i)(x), is provided in Appendix A. U(1) detects s(2) as po2>po1 with SINRΓ(1)→(2)=po2B(1)po1B(1)+N0,
Then U(1) removes s(2) and demodulates s(1) with SNRΓ(1)→(1)=po1B(1)N0,
The probability of an outage event at U(1) is computed asPoutage(1)(x)=P(min[Γ(1)→(1),Γ(1)→(2)]≤x)=PB(1)(max[N0xpo1,N0xpo2−po1x])
The Packet Error Probability (PEP) of U(i) is given byPEP(i)(po1,po2)≤Poutage(i)(W0),whereW0=∫0+∞1−[1−SEP(w)]Ldw,
L is packet length andSEP(w)=2(1−1M)erfc(3wM−1),
The throughput of U(i) is given byThr(i)(po1,po2)=log2(M)[1−PEP(i)(po1,po2)],
The total throughput (TThr) is given byTThr(po1,po2)=Thr(1)(po1,po2)+Thr(2)(po1,po2)
We maximize the total throughput as followsTThrmaximized=max0<po1<po2<1TThr(po1,po2)
CRN-NOMA with K UsersRanking Using Average Gains
The network model is depicted in Fig. 2. It contains PT, PR, SS and K secondary NOMA users. U(i) has the i-th maximum average channel gain between SS and NOMA users. Let P be defined as
A network with K users
P=∑j=1KN(j),
N(j) is the number of IRS reflectors of U(j).
NOMA symbol is written ass=∑i=1Kpois(i),where po1 + po2 = 1 and 1>po2>po1 > 0.∑i=1Kpoi=1
The received signal at U(i) is written asr(i)=A(i)λλ(i)ESS[∑i=1Kpois(i)]+n(i),
U(i) detects sK as poK > poi with SINRΓ(i)→(K)=poKB(i)B(i)∑l=1K−1pol+N0,
Then U(i) performs Successive Interference Cancelation (SIC), removes sK to detect sK−1 with SINRΓ(i)→(K−1)=poK−1B(i)B(i)∑l=1K−2pol+N0,
U(i) detects sp with SINRΓ(i)→(p)=popB(i)B(i)∑l=1p−1pol+N0,
The probability of an outage event at U(i) is computed asPoutage(i)(x)=P(Γ(i)→(K)≤x,…,Γ(i)→(i)≤x)=PB(i)(max1≤p≤K[N0xpop−x∑l=1p−1pol]),
The PEP at U(i) is equal toPEP(i)(po1,…,poK)≤Poutage(i)(W0),where W0 is defined in (15)
The throughput of U(i) is given byThr(i)(po1,…,poK)=log2(M)[1−PEP(i)(po1,…,poK)],
The total throughput (TThr) is given byTThr(po1,…,poK)=∑i=1KThr(i)(po1,…,poK),
We maximize the total throughput as followsTThrmaximized=max0<po1<…<poK<1TThr(po1,…,poK).
Ranking Using Instantaneous Gains
Let Ui(1) be the strongest user with largest instantaneous channel gain B(i):Bi(1)=max1≤p≤KB(p),
Let Ui(K) be the weakest user :Bi(K)=min1≤p≤KB(p),
Let Ui(q) be q-th ranked user:Bi(q)=q−th−max1≤p≤KB(p),
The CDF of Bi(q) is given byPBi(q)(x)=∑j=1q∑m1,m2,…,mj−1∏l=1j−1[1−PB(ml)(x)]∑mj,…,mN∏p=jKPB(mp)(x)where 1 ≤ mi ≤ N for i = 1,…,N. m1¹m_2 ¹… mN, mq < mq + 1<…< mK and PB^(i)(x) is given in Appendix A.
The PEP and throughput can be computed as Section 3.1 where we have to replace PB(q)(x) by PBi(q)(x) given in (35).
Effects of Primary Interference
The SINR is computed asΓ(i)→(p)=popB(i)B(i)∑l=1p−1pol+N0+IPT,i,
The probability of an outage at U(i) is computed asPoutage(i)(x)=∫0+∞PB(i)(max1≤p≤K[(N0+y)xpop−x∑l=1p−1pol])pIPT,i(y)dy,wherepIPT,i(y)=e−yIPT,i¯IPT,i¯
IPT,i¯ is the average interference. The PEP and throughput are computed using (37).
Numerical Results
Fig. 3 shows the total throughput for CRN-NOMA for K = 2, for I = 1 16 Quadrature Amplitude Modulation (QAM), dIRSU(i) = 1,1.5 i = 1,2 Packet length is L = 200.. IRS allows 6, 12, 18 dB vs. CRN-NOMA without IRS for N = 8, 16, 32.
Total throughput for 2 users for 16QAM
Fig. 4 shows the throughput for different values of I, 16QAM modulation and N = 8 reflectors. As I increases as the throughput improves since SS can increase its power while verifying interference constraints.
Effect of interference threshold I: 2 users, 16QAM and N = 8 reflectors
Fig. 5 shows that NOMA with IRS for N = 64 offers better performance than Orthogonal Multiple Access (OMA) and NOMA without IRS for 16QAM and two users. At high average SNR, the throughput of OMA is half that of NOMA.
OMA and NOMA performance comparison
Fig. 6 depicts the effects of primary interference when there are two users, N = 8, 16 reflectors per user for 16QAM Modulation. The parameters are dPTU(i) = 1,0.9, 0.5,0.6. We notice that the performance degrades as PT is close to NOMA users since there is more interference.
Effects of primary interference on Total throughput of NOMA: 2 users, 16QAM modulation and N = 16
Fig. 7 depicts the total throughput for 16-QAM modulations for two users and N = 8, 16 reflectors. Ranking using instantaneous channel gains offers the best throughput.
Secondary throughput for 16QAM modulation and different ranking strategies: N = 8
Conclusions
In this article, we computed the PEP and throughput of NOMA with adaptive transmit power and IRS. IRS are deployed to enhance data reception at all users. CRN-NOMA using IRS offers 7, 13, 20 dB gain vs. the absence of IRS for N = 8, 16, 32. We have also derived the SNR and SINR statistics.
Funding Statement: The authors extend their appreciation to the Deanship of Scientific Research at Saudi Electronic University for funding this research work through the Project Number 8093.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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The variance and mean of A(i) are σA2=N(1−π216) and mA=Nπ/4. B(i) is equal toB(i)=[A(i)]2λλ(i)ESS
We deducePB(i)(x)=P(B(i)≤x|I|gSSPR|2<Emax)P(I|gSSPR|2<Emax)+P(B(i)≤x|T|gSSPR|2>Emax)P(I|gSSPR|2>Emax),whereP(I|gSSPR|2<Emax)=e−IλSSPREmax,where λSSPR = E(|gSSPR|2), E(.) is the expectation operation and gSSPR is the channel coefficient between SS and PR. When I/|gSSPR|2>Emax, we haveB(i)=Emax[A(i)]2λλ(i)andP(B(i)≤x|I|gSSPR|2>Emax)=1−Q0.5(mAσA,xEmaxλλ(i)σA2),
where Qm(.,.) is the Generalized Marcum Q-function.
When I/|gSSPR|2<Emax , ESS=I|gSSPR|2 and we haveP(B(i)≤x|I|gSSPR|2<Emax)=∫IEmax+∞1−Q0.5(mAσA,xyEmaxIλλ(i)σA2)e−ySSPR1λSSPRdy
PB(i)(x) is computed using (40),(41) and (43),(44).