Motivated by the above-mentioned recent works on NOMA and IRS, the proposed work considers NOMA-based wireless communication supported by IRS. The closed form expression of bit error probability in two-user downlink NOMA system is derived in [15]. Based on [15,21], the error probability of the two-user downlink IRS-NOMA system is derived in this paper. In the article [15], the near user signal is modulated using QPSK and BPSK modulation is used for far user. In this article, BPSK modulation is considered for both users. The main contributions of this article are

Closed-form expressions of average error probability of received signal at IRS-NOMA are derived by considering SIC mechanism under Rayleigh and Rician fading channel.

This article is organized as follows. IRS-NOMA system model is described in Section 2. In Section 3, the analytical expression of NOMA users is derived. The BER performance of the proposed IRS-NOMA system is discussed in Sections 4 and 5. The article is concluded in Section 6.

The detailed symbols and corresponding descriptions are listed in Tab. 1.

In the system proposed, IRS is kept in between the BS and users to aid NOMA transmission. The users are assumed to receive a signal from BS and IRS. IRS is a reflecting array made up of periodic metasurfaces. Metasurfaces are artificially composite materials, which have been utilized in wireless communication for introducing considerable phase shift on the incident signal [24]. The array elements are termed as cell fabricated with PIN or varactor diode to introduce phase shift on the RF signal. The unusual characteristics of metamaterials in IRS make it to interrelate with both electric and magnetic fields of the input signal. The collective effect of array elements is utilized to focus the energy in the directions of users. As illustrated in Fig. 1, the IRS receives the incoming superimposed signal (x) and introduces the phase shift of θ on the reflected signal. The users in the focusing area can receive the reflected signal from IRS. The phase shift θ can be tuned or adjusted to focus and de-focus the energy in the relevant and irrelevant directions.

This paper assumes that the users receive direct signal as well as reflected signal. We analyze the IRS aided NOMA transmission in both Rayleigh as well as Rician fading. The direct channel fading coefficients between BS-U₁ and BS-U₂ are denoted as h_D1 and h_D2. Let g denotes channel fading between BS and IRS, g₁ and g₂ be the channel coefficients between IRS and users (U₁,U₂). Therefore, the received signals at the end-users (U₁,U₂) are written as(2) y1=gH⁡θg1⁡x+hD1x+n1

(3) y2=gH⁡θg2⁡x+hD2x+n2

where h_D1x and h_D2x represent the direct signal received at the users from BS. gH⁡θg1⁡x and gH⁡θg2⁡x represent received signals at U₁ and U₂ via BS-IRS–user channels. θ is the phase shift matrix (NxN) introduced in the IRS to focus power towards users, which is given as diagonal matrix as θ=diag{β1ejφ1,β2ejφ2,…,βNejφN},βi(0,1),φi∈(0,π) . β_i and φ_i represent the amplitude of reflection and phase-shift at i^th reflecting unit of IRS. Fading coefficients of indirect channels (BS-IRS, IRS-users) are denoted by the vectors g ∈ C^Nx1,  g_k ∈ C^Nx1 where (k = 1, 2). In the case of the Rician channel fading model, the fading coefficients h_Dk, g and g_k are given by(4) hDk=K1K1+1h¯Dk+1K1+1h~Dk

(5) g=K2K2+1g¯+1K2+1g~

(6) gk=K3K3+1g¯k+1K3+1g~k

In (4), K₁ is the Rician factor of h_Dk, h¯Dk and h~Dk represent the dominant LoS and non-LoS (NLoS) terms. K₂ is the Rician factor of g. g¯∈CNx1 and g¯∈CNx1 indicates direct LoS component and NLoS terms. Similarly, K₃ is the Rician factor of g_k. g¯k∈CNx1 and g~k∈CNx1 are the LoS component and NLoS terms. We assume that Maximal Ratio Combiner (MRC) [25] is employed to combine the direct and indirect signals at users. The channel is considered as the Additive White Gaussian Noise (AWGN), where n_i = N(0, σ²_i) (i = 1, 2) and σ²_i is noise variance. In case of NOMA-based communication, far user U₂ decodes its information directly by considering other user’s (U₁) information as noise. The U₁ (near user) applies SIC mechanism to decode its symbol s₁. First, U₁ detects U₂, then subtracts the detected signal from the received superimposed signal y₁ and then decodes its own signal.

Introduction of phase shift θ at IRS can be coherent phase shifting and random phase shifting as mentioned in [21]. In the case of intelligent transmission, it is assumed that IRS has known the channel characteristics perfectly; the reflection angle θ between IRS and users could be adjusted to maximize the received signal strength at the user. The IRS introduces phase shift randomly in the random phase-shifting method. In this paper, both phase-shifting methods are considered at IRS to evaluate the BER performance.

For the BPSK modulation signal, the constellations of the received signal at U₁ and U₂ are as illustrated in Fig. 2. (b₁, b₂) represents the binary symbols of U₁ and U₂. The ML detector [26] is employed to decode the user symbols. The decision boundary depends on the amount of AWGN noise (n_i) present in the received signal. For BPSK signal, decision boundary could be yi>0or yi≤0(i=1,2) . The total error probability is calculated by summing up the error probability of every BPSK symbol multiplied by its prior probability. The prior probabilities of the superimposed signal in BPSK are 14 , since the probabilities of occurrence of all possible symbols (s1, s2) are equal. In BPSK modulation, the in-phase component of noise influences the detection boundary [27]. The error in the detection occurs whenever the noise is higher than the signal. From the constellation diagram, the error probability U₂ is expressed as P₂(e),(7) P2⁡(e)=12pr(y2>0|b2=−1)+12pr(y2≤0|b2=1)

Using the ML detection principle, the conditional error probability is written as(8) P2⁡(e)=12pr(n2≥(a1+a2)(gHθg2+hD2))+12pr(n2≥(a2−a1)(gHθg2+hD2))

The error probability expressed in (8) can be written using complex Q-function [[28], Eq. (7.108)] with noise variance of N02 .(9) P2⁡(e)=12[Q[(a1+a2)(gHθg2+hD2)No2]+Q[(a2−a1)(gHθg2+hD2)No2]] Eq. (9) can be represented using instantaneous SNR as(10) γ1=2(a1+a2)2|gHθg2+hD2|No2,γ1¯=2(a1+a2)2E[|gHθg2+hD2|2]No

(11) γ2=2(a2−a1)2|gHθg2+hD2|No2,γ2¯=2(a2−a1)2E[|gHθg2+hD2|2]No

In (10) and (11) γ₁ and γ₂ represent SNRs, γ¯1 and γ¯2 are the corresponding mean values. Where E [.] denotes the expectation operator. The expression (9) is rewritten in terms of SNR as(12) P2⁡(e)=12(Q(γ1)+Q(γ2))

Average error probability in AWGN over fading channel at far user is calculated by integrating the product of Gaussian Q-function and channel fading Probability Density Function (PDF) fγk(γk) , k = 1, 2 using [[29], Eq. (5.1)](13) P¯2(e)=12[∫0∞Q(γ1)fγ1Rayleigh(γ)dγ1+∫0∞Q(γ2)fγ2Rayleigh(γ)dγ2]

By substituting the Gaussian Q-function (10) can be rewritten using [[29], Eq. (5.2)] (14) P¯2(e)=12[1π∫0π2{∫0∞exp⁡(−γ12sin2θ)fγ1Rayleigh(γ)dγ1}dθ+1π∫0π2{∫0∞exp⁡(−γ22sin2θ)fγ2Rayleigh(γ)dγ2}dθ] where PDF of the SNRs for the Rayleigh distribution is given by [[29], Eq. (5.4)](15) fγ1Rayleigh(γ)=1γ¯1e−γγ¯1

(16) fγ2Rayleigh(γ)=1γ¯2e−γγ¯2,γ≥0

Using [[29], Eq. (5.3)], the term {∫0∞exp⁡(−γk2sin2θ)fγk(γk)dγk} , k = 1, 2 represents using Moment Generation Function (MGF) for the Rayleigh fading and (14) is expressed in terms of MGF as(17) P¯2Rayleigh(e)=12[1π∫0∞Mγ1Rayleigh(−12sin2θ)dθ+1π∫0∞Mγ2Rayleigh(−12sin2θ)dθ] Using [[29], Eq. (5.5)], MγkRayleigh k = 1, 2 for Rayleigh fading is expressed as(18) Mγ1Rayleigh(−s)=11+sγ¯1,Mγ2Rayleigh(−s)=11+sγ¯2

where s=−12sin2θ for BPSK modulation. With the use of [[29], Eq. (5.6)], by substituting (18) in (17), the error probability of far user becomes(19) P¯2Rayleigh(e)=14[(1−γ¯12+γ¯1)+(1−γ¯22+γ¯2)]

In NOMA system, the near user signal is decoded after canceling the far user information using SIC process. Far user information can be decoded correctly or erroneously during SIC process. Therefore, the error probability of near user is calculated as the sum of error probability under far user information decoded successfully and erroneously. The constellation map is shown in Fig. 3.(20) P1Rayleigh(e)=P1(e|U2successful)+P1(e|U2unsuccessful)

In (20), P₁(e) represents the probability of error at U₁, p₁(e|U₂successful) indicates error probability of U₁, while U₂ the signal is detected successfully. p₁(e|U₂unsuccessful) represents error probability of U₁, when U₂. signal is detected incorrectly. The constellation map of correct decoding and incorrect decoding of U₂ is illustrated in Fig. 3. U₁ signal is affected by channel fading and AWGN noise after SIC mechanism. During the detection of U₁ symbol, the error probability of successful detection of U₂ the symbol is considered as prior probability of different superimposed signals. Therefore, p₁(e|U₂successful) is expressed as(21) P1(e|U2successful)=12{[Pr(n1≤(a2−a1)(gHθg1+hD1))Pr(n1≤−a1(gHθg1+hD1)|n1≤(a2−a1)(gHθg1+hD1))]+[Pr(n1≤(a2+a1)(gHθg1+hD1))Pr(n1≥a1(gHθg1+hD1)|n1≤(a2+a1)(gHθg1+hD1))]}

Using conditional probability of two events A and B, P(A|B)=P(A∩B)P(B) , (21) is written as,(22) P1(e|U2successful)=12{Pr(n1≤−a1(gHθg1+hD1))+Pr(a1(gHθg1+hD1)≤n1≤(a2+a1)(gHθg1+hD1))}

The first term Pr(n1≤−a1(gHθg1+hD1)) and second terms Pr(a1(gHθg1+hD1)≤n1≤(a2+a1)(gHθg1+hD1)) of (22) can be represented in Q-function as(23) Pr(n1≤−a1(gHθg1+hD1))=Q(γ3)

(24) Pr(a1(gHθg1+hD1)≤n1≤(a2+a1)(gHθg1+hD1))=Q(γ3)−Q(γ4) where, γ3=2(a1)2|gHθg1+hD1|No2 , γ4=2(a1+a2)2|gHθg1+hD1|2No and the mean values are γ¯3=2(a1)2E[|gHθg1+hD1|2]No , γ¯4=2(a1+a2)2E[|gHθg1+hD1|2]No .

Therefore error probability of U₁ when s₂ detected correctly is calculated by substituting (23) and (24) in (22) as(25) P1(e|U2successful)=12[2Q(γ3)−Q(γ4)]

In case of incorrect detection of s_2, this is shown in Fig. 3b. When the symbol b₂=1 is transmitted for U₂ and is decoded as b₂= −1. Thus, the error probability of symbol s₁ at U₁ is calculated by taking the product of the error probability of every symbol in the constellation map shown in Fig. 3b. with respective prior error probability of U₂ symbol. From the constellation map, p₁(e|U₂unsuccessful) is expressed as(26) P1(e|U2unsuccessful)=12{[Pr(n1≥(a2−a1)(gHθg1+hD1))Pr(n1≤(2a2−a1)(gHθg1+hD1)|n1≥(a2−a1)(gHθg1+hD1))]+[Pr(n1≥(a2+a1)(gHθg1+hD1))Pr(n1≥(2a2+a1)(gHθg1+hD1)|n1≥(a2+a1)(gHθg1+hD1))]}

P₁(e|U₂unsuccessful) in (26) is rewritten using ML principle as(27) P1(e|U2unsuccessful)=12{[Pr((a2−a1)(gHθg1+hD1)≤n1≤(2a2−a1)(gHθg1+hD1))+Pr(n1≥(2a2+a1)(gHθg1+hD1))]}

The first Pr((a2−a1)(gHθg1+hD1)≤n1≤(2a2−a1)(gHθg1+hD1)) and second term Pr(n1≥(2a2+a1)(gHθg1+hD1)) of (27) are expressed in terms of Q-function as(28) Pr((a2−a1)(gHθg1+hD1)≤n1≤(2a2−a1)(gHθg1+hD1))=Q(γ5)−Q(γ6)

(29) Pr(n1≥(2a2+a1)(gHθg1+hD1))=Q(γ7) where γ5=2(a2−a1)2|gHθg1+hD1|No2 , γ6=2(2a2−a1)2|gHθg1+hD1|No2γ7=2(2a2+a1)2|gHθg1+hD1|No2 and their corresponding mean expressions are γ¯5=2(a2−a1)2E[|gHθg1+hD1|2]No , γ¯6=2(2a2−a1)2E[|gHθg1+hD1|2]Noγ¯7=2(2a2+a1)2E[|gHθg1+hD1|2]No

By substituting (28) and (29) in (27), P₁(e|U₂unsuccessful) is written as(30) p1(e|U2unsuccessful)=12[Q(γ5)+Q(γ7)−Q(γ6)]

By combining (25) and (30) and using (13), (14), (17) and (18) the probability of error for near user U₁ signal is written using as,(31) P¯1Rayleigh(e)=12(1−γ¯32+γ¯3)+14[γ¯42+γ¯4−γ¯52+γ¯5+γ¯62+γ¯6−γ¯72+γ¯7]

The average error probability of far users under the Rician fading channel is derived by substituting the PDF and MGF of Rician distribution in (14). Where PDF of Rician fading is expressed using [[29], Eq. (5.10)](32) fγkRician(γ)=(1+K)γ¯ke−Kexp⁡[−(1+K)γkγ¯k]Io[2K(1+K)γkγ¯k],k=1,2 where I_o [] represents the modified Bessel function. Similarly, the MGF of Rician fading using [[29], Eq. (5.11)] is represented by(33) MγkRician(−s)=(1+K)1+K+sγ¯kexp⁡[−Ksγ¯k1+K+sγ¯k],s>0,k=1,2

By substituting alternate representation of MGF for Rician fading in (17), the average error probability of far user is expressed as(34) P¯2(e)=12[1π∫0∞Mγ1Rician(−12sin2θ)dθ+1π∫0∞Mγ2Rician(−12sin2θ)dθ]

Therefore by substituting (34), the average error probability for the far user under Rician fading can be expressed as(35) P¯2Rician(e)=12{ 1π∫0π2(1+K)sin2θ(1+K)sin2θ+γ¯1exp[ −Kγ¯1(1+K)sin2θ+γ¯1 ]dθ+1π∫0π2(1+K)sin2θ(1+K)sin2θ+γ¯2exp[ −Kγ¯2(1+K)sin2θ+γ¯2 ]dθ}

Similarly, the error probability of successful detection of U₂ signal at near user under Rician fading can be expressed by introducing (32) and (33) in (25)(36) P¯1(e|U2successful)Rician(e)=12{ 2π∫0π2(1+K)sin2θ(1+K)sin2θ+γ¯3exp[ −Kγ¯3(1+K)sin2θ+γ¯3 ]dθ−1π∫0π2(1+K)sin2θ(1+K)sin2θ+γ¯4exp[ −Kγ¯4(1+K)sin2θ+γ¯4 ]dθ}

The error probability of unsuccessful detection of U₂ signal at near user for Rician fading is expressed by rewriting (30)(37) P¯1(e|U2unsuccessful)Rician(e)=12{ 1π∫0π2(1+K)sin2θ(1+K)sin2θ+γ¯5exp[ −Kγ¯5(1+K)sin2θ+γ¯5 ]dθ−1π∫0π2(1+K)sin2θ(1+K)sin2θ+γ¯6exp[ −Kγ¯6(1+K)sin2θ+γ¯6 ]dθ+1π∫0π2(1+K)sin2θ(1+K)sin2θ+γ¯7exp[ −Kγ¯7(1+K)sin2θ+γ¯7 ]dθ}

Thus the total average error probability at U₁ under Rician fading is calculated by summing (36) and (37)(38) P¯1Rician(e)=P¯1(e|U2successful)Rician(e)+P¯1(e|U2unsuccessful)Rician(e)

In the simulation results, near and far users are denoted as U₁ and U₂. The IRS-NOMA transmission is investigated at an indoor environment model [30] with a path loss exponent of 1.6. Far user is allocated with the factor of α =0.8, wherein near user with α =0.2. The derived analytical expressions for Rayleigh [(19) and (31)] and Rician fading [(35) and (38)] channel are verified via simulations as shown in Figs. 4a and 4b. The simulation results show the accuracy of the derived expression. We also compare the performance of the derived BER expressions of the proposed two-user IRS-NOMA system with simulation results.

Fig. 4b illustrates the analytical BER performance of the IRS-NOMA system with random phase shifting at IRS under Rayleigh and Rician fading. It has been shown that the BER of users at coherent phase-shifting outperforms the random phase-shifting method in low SNR regions.

The impact of IRS in the NOMA system is evaluated as shown in Fig. 5, it has been demonstrated that BER performance of IRS aided NOMA system outperforms the conventional NOMA system under both fading channels. The proposed IRS-NOMA system achieves an approximate 30 dB SNR gain to obtain 10⁻³ BER than NOMA system for both users from the simulation results. From the simulation analysis, the proposed IRS-NOMA outperforms NOMA, especially in the low SNR region. While comparing the BER-performance of the proposed IRS-NOMA system with the conventional NOMA system in [15], the former has a gain of 30 dB SNR at BER of 10⁻³ under Rayleigh fading channel. Wherein [15] considers QPSK modulation for near users and BPSK for far users. The performance comparison between the proposed IRS-NOMA system with reference to NOMA system [[15], Fig. 4a] is listed in Tab. 3. It has been shown that introducing IRS in NOMA system enhances the BER performance considerably than NOMA system. The proposed simulation work as shown in Fig. 5 is obtained by considering BPSK modulation for both users with the pathloss exponent of 1.6 and 32 number of reflective elements in IRS under Rayleigh fading channel.

The effect of power allocation on BER performance of IRS-NOMA system is investigated and illustrated as shown in Fig. 6. The variations of BER performance with power allocation for various transmit SNR are analyzed and illustrated as shown in Fig. 7. In case of U₁, though the power allocation is increased, the BER performance is limited because of SIC process. The better BER performance is achieved at α = 0.25with the transmit SNR of −10 dB under both fading channels.