Non Orthogonal Multiple Access for Millimeter Wave Communications Using Intelligent Reflecting Surfaces

Abstract

In this paper, we suggest the use of Intelligent Reflecting Surfaces (IRS) for Non Orthogonal Multiple Access (NOMA) using millimeter wave communications. The source sends a combination of K symbols dedicated to K users. The received signal at relay node R is affected with P interferers. The relay node detects the symbol of the weak user as it is transmitted with the largest power. Then, it uses Successive Interference Cancelation (SIC) to remove the contribution of the signal of the weak user to detect the symbol of the second weakest user. The rest of the detections are performed similarly until relay node detects all K symbols. The relay node sends a combination of the detected symbols. The transmitted signal by the relay node is reflected by different sets of IRS reflectors. The reflected signals reach the i-th user with the same phase. The phase shift of i-th IRS reflector depends on the phase of channel coefficient between relay and IRS as well as the phase of channel coefficient between IRS and user. The i-th strong user uses SIC to detect the symbols of $K-i-1$ weak users to be able to detect its symbol. We optimize the fraction of powers allocated to NOMA users at the source and relay node to maximize the total throughput. The results are valid for Nakagami channels and any number of interferers at the relay and NOMA users. When there are two users, a total throughput of 3.5 bit/s/Hz is reached for 16QAM modulation and average SNR per bit equal to $-22.7$ dB, $-19.7$ dB, $-16.6$ dB, $-13.6$ dB, $-10.4$ dB, $-7.2$ dB, $-3.8$ dB and 6.5 dB respectively for a number of reflectors per user $N=512,256,128,64,32,16,8$ and when there is no IRS. For $N=32$ reflectors, optimal power allocation allows 2.1 dB gain with respect to fixed power allocation. When there are three users, a total throughput of 2.5 bit/s/Hz is reached for QPSK modulation and average SNR per bit equal to $-10.7$ dB, $-7.6$ dB, $-4.3\,dB$ and 6.9 dB for a number of reflectors per user $N=32,16,8$ and when there is no IRS.

1 Introduction

Millimeter wave communications (mmwave) offer data rates of the order of 10 Gb/s since the used bandwidth is going from 30 to 300 GHZ [1,2,3,4]. Millimeter wave signals cannot penetrate through walls and building and relaying techniques are mandatory [1,2,3,4,5,6,7]. Multihop relaying can be used when the source is far from the destination [6,7,8,9,10,11,12]. When multiple branches are available, we can activate the branch with the highest end-to-end Signal to Interference plus Noise Ratio (SINR). Both Amplify and Forward (AF) and Decode and Forward (DF) relaying can be used [8]. AF relays amplify the received signal and forward it to the next relay. AF relaying uses a constant amplification gain and these are known as blind relays. Otherwise, non-blind relays use an adaptive amplification factor that depends on the channel coefficient. DF relays decode and regenerate the transmitted signal [9, 10]. In DF relaying, the relay node transmits only if it has correctly detected the received packet. The use of relay nodes is complex to implement since relay selection techniques require many signalization. Besides, the throughput is low if all relays transmit without relay selection since multiple orthogonal channels are required.

Intelligent Reflecting Surfaces (IRS) have been suggested to enhance the throughput of wireless communications [11,12,13,14,15]. The transmitted signal by the source is reflected by IRS reflectors and reaches the destination with a null phase. The phase shift of i-th reflector depends on the phase of channel coefficient between the source and i-th IRS reflector as well as the phase of channel coefficient between i-th reflector and destination [16,17,18]. Different sets of reflectors are dedicated to different NOMA users [19]. The reflected signals over each set of reflectors reach the corresponding user with a null phase [19]. IRS have been recently suggested for mmwave communications [20, 21]. However, the derived results in [20, 21] correspond to Orthogonal Multiple Access (OMA) that offers lower data rates than NOMA. Exact and asymptotic performance analysis of wireless communications using IRS were derived in [22,23,24]. Some experimental results of wireless networks using IRS have been discussed in [25].

Millimeter wave communications using multiple antennas for NOMA systems were proposed in [26, 27]. Relaying techniques for millimeter wave communications using NOMA were studied in [28]. The coverage of millimeter wave communications using NOMA was studied in [29]. Precoding and combining techniques for millimeter wave communications were suggested in [30]. Deep learning for millimeter wave communications was suggested in [31]. A high gain antenna was used in [32] to improve the throughput of millimeter wave communications.

To the best of our knowledge, mmwave communications using NOMA and IRS have not been yet suggested and analyzed in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]. The contributions of the paper are;

• We suggest the use of Intelligent Reflecting Surfaces (IRS) for NOMA systems using millimeter wave communications. The source sends a combination of K symbols dedicated to K users. The received signal by the relay node is affected by P interferers. The relay node uses SIC to detect the symbols of K users. Then, it sends a combination of detected symbols. The transmitted signal by the relay node is reflected by different sets of IRS reflectors dedicated to different users. We show that the reflected signals have the same phase at each NOMA user.

• The proposed NOMA using IRS offers 10, 13, 16, 20, 24, 27 and 30 dB gain with respect to conventional NOMA using millimeter wave communications without IRS for a number of reflectors $N=8,16,32,64,128,256,512$. Besides, mmwave using NOMA offers larger data rates than OMA as suggested in [20, 21].

• We also suggest optimizing the powers allocated to NOMA users at the source and relay node to enhance the total throughput. The paper contains six sections. Next section studies the performance of millimeter wave link in the presence of P interferers at the relay node. Section 3 evaluates the outage probability at any user using intelligent reflecting surfaces. Section 4 optimizes the total throughput by adjusting the fraction of powers dedicated to K NOMA users. Section 5 gives some theoretical and simulation results. Section 6 concludes the paper.

2 Millimeter Wave Link

The system model provided in Fig. 1 contains a source S, a relay node R and K users $U_i$, $i=1,\ldots K$. $U_i$ is the i-th strong user. We assume Nakagami channel between all nodes and we denote by M the m-fading figure. The source sends a combination of K symbols $s_i$, $i=1,2,\ldots K$ dedicated to K users:

$$\begin{aligned} s=\sqrt{E_S}\sum _{i=1}^K\sqrt{C_i}s_i \end{aligned}$$

(1)

where $E_S$ is the Transmitted Energy per Symbol (TES) of the source, $0<C_i<1$ is the fraction of power allocated to user $U_i$. Less power is allocated to the strong user: $0<C_1<C_2<\cdots <C_K$. We have

$$\begin{aligned} \sum _{i=1}^KC_i=1. \end{aligned}$$

(2)

Fig. 1

Millimeter wave communications using IRS

We assume that the received signal at R is affected by P interferers: [1,2,3,4,5]

$$\begin{aligned} r=fs+I=f\left[ \sqrt{E_s}\sum _{i=1}^K\sqrt{C_i}s_i\right] +I. \end{aligned}$$

(3)

where f is the channel coefficient between S and R, I is the interference term at R composed of P interferers:

$$\begin{aligned} I=\sum _{q=1}^{P} E_q |i_q|^2 \end{aligned}$$

(4)

$E_q$ is the TES of q-th interferer, $i_q$ is the channel coefficient between q-th interferer and R.

The relay node detects the symbol of user $U_K$ since it is received with the largest power. The SINR is equal to

$$\begin{aligned} \Gamma _{R,K}=\frac{C_KE_S|f|^2}{I+E_S|f|^2\sum _{i=1}^{K-1}C_i} \end{aligned}$$

(5)

Then, relay R removes the contribution of $s_{K}$ using SIC to detect $s_{K-1}$ with SINR

$$\begin{aligned} \Gamma _{R,K-1}=\frac{C_{K-1}E_S|f|^2}{I+E_S|f|^2\sum _{i=1}^{K-2}C_i} \end{aligned}$$

(6)

The remaining detections are made similarly, R will detect $s_p$ $p=K,K-1,\ldots ,1$ with SINR

$$\begin{aligned} \Gamma _{R,p}=\frac{C_{p}E_S|f|^2}{I+E_S|f|^2\sum _{i=1}^{p-1}C_i} \end{aligned}$$

(7)

The Cumulative Distribution Function (CDF) of $\Gamma _{R,p}$ is equal to

$$\begin{aligned} F_{\Gamma _{R,p}}(x)&= P\left( \frac{C_{p}E_S|f|^2}{I+E_S|f|^2\sum _{i=1}^{p-1}C_i} \le x\right) =P\left( \frac{C_pU}{1+U\sum _{i=1}^{p-1}C_i} \le x\right) \\ &= F_U\left( \frac{x}{C_p-x\sum _{i=1}^{p-1}C_i}\right) \end{aligned}$$

(8)

where $F_U(x)$ is the CDF of U where

$$\begin{aligned} U=\frac{E_{S}|f|^2}{I} \end{aligned}$$

(9)

There is no outage at R if all SINR $\Gamma _{R,1}$, $\Gamma _{R,2},\ldots ,\Gamma _{R,K}$ are larger than x:

$$\begin{aligned} P_{{\textit{outage}},R}(x)&= 1-P(\Gamma _{R,1}>x,\ldots ,\Gamma _{R,K}>x) \\ &= F_{U}\left( \underset{1\le p\le K}{\hbox {max}}\left( \frac{x}{C_p-x\sum _{i=1}^{p-1}C_i}\right) \right) \end{aligned}$$

(10)

For Nakagami fading channels, $X=E_S |f|^2$ follows a Gamma distribution $Gamma(M,\beta )$ defined as

$$\begin{aligned} f_{X}(x)=\frac{x^{M}e^{-\frac{x}{\beta }}}{\Gamma (M)\beta ^{M}} \end{aligned}$$

(11)

$M>0.5$ is the m-fading figure of Nakagami channel,

$$\begin{aligned} \beta =\frac{E_SE(|f|^2)}{M} \end{aligned}$$

(12)

E(.) is the expectation operator.

The interference I at the relay node is expressed as

$$\begin{aligned} I=\sum _{q=1}^{P} I_q \end{aligned}$$

(13)

where $I_q=E_q |i_q|^2$

$I_q$ has a Gamma distribution $Gamma(M,\alpha )$

$$\begin{aligned} \alpha =\frac{E(I_q)}{M} \end{aligned}$$

(14)

The sum of P independent Gamma random variables (r.v.) $I_{q}$ is a Gamma r.v. ${\textit{Gamma}}(PM,\alpha )$. We deduce that $U=\frac{X}{I}$ is the quotient of two Gamma r.v. that has a general prime distribution and Probability Density Function (PDF) [33]:

$$\begin{aligned} f_{U}(x)=\frac{\Gamma (M+PM)\alpha ^Mx^{M-1}(1+\frac{x\alpha }{\beta })^{-PM-M}}{\beta ^M\Gamma (M)\Gamma (PM)} \end{aligned}$$

(15)

The proof is provided in “Appendix A”.

We have [34]

$$\begin{aligned} _{2}F_1(a,\,b;\,c;\,z)= \frac{\Gamma (c)}{\Gamma (b)\Gamma (c-b)} \times \int _0^1 x^{b-1}(1-x)^{c-b-1}(1-zx)^{-a}dx \end{aligned}$$

(16)

where $_{2}F_1(a,b;c;z)$ is the hypergeometric function. We use (16) to write the CDF of SINR as

$$\begin{aligned} F_{U}(x)= _{2}F_1\left( PM+M,M;M+1,\frac{-x\alpha }{\beta }\right) \times \frac{\Gamma (M+PM)(x\alpha )^{M}}{\Gamma (M)\Gamma (PM)\beta ^{M}} \end{aligned}$$

(17)

When there is Additive White Gaussian Noise (AWGN), the SINR can be upper bound by

$$\begin{aligned} \Gamma _{R,p,AWGN}=\frac{C_{p}E_S|f|^2}{N_0+I+E_S|f|^2\sum _{i=1}^{p-1}C_i}<\Gamma _{R,p}=\frac{C_{p}E_S|f|^2}{I+E_S|f|^2\sum _{i=1}^{p-1}C_i} \end{aligned}$$

(18)

The derived outage probability in (10) is a lower bound.

When there is AWGN, we can derive numerically the outage probability. In fact, we can write

$$\begin{aligned} \Gamma _{R,p,AWGN}=\frac{C_{p}E_S|f|^2}{N_0+I+E_S|f|^2\sum _{i=1}^{p-1}C_i}=\frac{C_pX}{N_0+I+X\sum _{i=1}^{p-1}C_i} \end{aligned}$$

(19)

The CDF of $\Gamma _{R,p,AWGN}$ can be computed numerically as

$$\begin{aligned} F_{\Gamma _{R,p,AWGN}}(x)=\int _0^{+\infty }F_{X}\left( \frac{x(N_0+y)}{C_p-x\sum _{i=1}^{p-1}C_i}\right) f_{I}(y)dy \end{aligned}$$

(20)

$F_{X}(z)$ is the CDF of X given by

$$\begin{aligned} F_{X}(x)=\frac{\gamma \left( M,\frac{x}{\beta }\right) }{\Gamma (M)} \end{aligned}$$

(21)

$\Gamma (M)$ is the Gamma function and

$$\begin{aligned} \gamma (M,x)=\int _0^xt^{M-1}e^{-t}dt, \end{aligned}$$

(22)

and $f_{I}(y)$ is the PDF of I given by

$$\begin{aligned} f_{I}(y)=\frac{y^{PM-1}e^{-\frac{y}{\alpha }}}{\Gamma (MP)\alpha ^{MP}} \end{aligned}$$

(23)

Therefore, in the presence of AWGN, the outage probability at R is similarly as

$$\begin{aligned} P_{{\textit{outage}},R,AWGN}(x)=\int _0^{+\infty }F_{X}\left( \underset{1\le p\le K}{\hbox {max}}\left( \frac{x(N_0+y)}{C_p-x\sum _{i=1}^{p-1}C_i}\right) \right) f_{I}(y)dy \end{aligned}$$

(24)

3 IRS Link

To extend the coverage of millimeter wave communications and to serve K NOMA users, IRS is placed between relay node R and users $U_1$, $U_2,\ldots ,U_K$. We define $h_k$ as the channel coefficient between R and k-th IRS reflector with average power $E(|h_k|^2)=\frac{1}{d^{ple}}$ where E(.) is the expectation operator, ple is the path loss exponent and d is the distance between R and IRS. We define $g_k \in I_i$ as the channel coefficient between k-th reflector of IRS and user $U_i$ with average power $E(|g_k|^2)=\frac{1}{d_i^{ple}}$ where $d_i$ is the distance between IRS and $U_i$. $I_i$ is the set of IRS reflectors dedicated to user $U_i$.

Let $a_k$ and $b_k$ be the absolute value and phase of $h_k=a_ke^{-jb_k}$. For Nakagami channels, $a_k$ has a Gamma distribution with $E(a_k)=\frac{\Gamma (M+0.5)}{\Gamma (M)}\sqrt{\frac{1}{Md_1^{ple}}}$ and $E\left( a_k^2\right) =E(|h_k|^2)=\frac{1}{d^{ple}}$ [36]. Let $c_k$ and $d_k$ be the absolute and phase of $g_k=c_ke^{-jd_k}$. We have $E(c_k)=\frac{\Gamma (M+0.5)}{\Gamma (M)}\sqrt{\frac{1}{Md_i^{ple}}}$ and $E\left( c_k^2\right) =E(|g_k|^2)=\frac{1}{d_i^{ple}}$ [36].

IRS optimizes the phase $\phi _k$ of k-th reflector as follows

$$\begin{aligned} \phi _k=b_k+d_k. \end{aligned}$$

(25)

The received signal at $U_i$ is equal to

$$\begin{aligned} r=S\sqrt{E_R}\sum _{q\in I_i}h_qg_qe^{j\phi _q}+n \end{aligned}$$

(26)

where $I_i$ is the set of reflectors dedicated to $U_i$, S is the transmitted NOMA symbol by relay node R, n is zero-mean Gaussian r.v. with variance $N_0$ and $E_R$ is the TES of R.

Let $\widehat{s_i}$ the i-th detected symbol at relay node R. When the detection is successful at the relay $\widehat{s_i}=s_i$. The transmitted NOMA symbol S by relay R is written as

$$\begin{aligned} S=\sum _{l=1}^K\sqrt{D_l}\widehat{s_l} \end{aligned}$$

(27)

$D_l$ is the fraction of power dedicated at $U_l$ at relay R. Less power is dedicated to strong user: $0<D_1<D_2<\cdots <D_K$. The sum of fraction of powers dedicated to of all users is one: $\sum _{l=1}^KD_l=1$.

Using (26) and (27), we have

$$\begin{aligned} r=\sqrt{E_R}\left[ \sum _{l=1}^K\sqrt{D_l}\widehat{s_l}\right] \sum _{q\in I_i}h_qg_qe^{j\phi _q}+n \end{aligned}$$

(28)

Using (25) and (28), we obtain

$$\begin{aligned} r=\sqrt{E_R}A_i \left[ \sum _{l=1}^K\sqrt{D_l}\widehat{s_l}\right] +n, \end{aligned}$$

(29)

where

$$\begin{aligned} A_i=\sum _{q\in I_i}a_qc_q. \end{aligned}$$

(30)

Using the Central Limit Theorem (CLT), $A_i$ can be approximated by a Gaussian r.v. with mean $m_{A_i}=\frac{N_i \Gamma (M+0.5)^2}{M\Gamma (M)^2d^{ple/2}d_i^{ple/2}}$ and variance $\sigma _{A_i}^2=\frac{N_i}{d_i^{ple}d^{ple}}\left[ 1-\frac{ \Gamma (M+0.5)^4}{M^2\Gamma (M)^4}\right]$, $N_i=|I_i|$ is the number of IRS reflectors dedicated to $U_i$,.

The CDF of $A_i^2$ is written as

$$\begin{aligned} F_{A_i^2}(x)&= P\left( A_i^2\le x\right) =P(-\sqrt{x}\le A_i\le \sqrt{x}) \\ &\simeq 0.5{\textit{erfc}}\left( \frac{-\sqrt{x}-m_{A_i}}{\sqrt{2}\sigma _{A_i}}\right) -0.5{\textit{erfc}}\left( \frac{\sqrt{x}-m_{A_i}}{\sqrt{2}\sigma _{A_i}}\right) \end{aligned}$$

(31)

User $U_i$ detects first the symbol of user $U_K$ as $D_K>D_i$ $\forall i\ne K$. The corresponding SINR is expressed as

$$\begin{aligned} \Gamma _{U_i,K}=\frac{E_RA_i^2D_K}{N_0+E_RA_i^2\sum _{l=1}^{K-1}D_l} \end{aligned}$$

(32)

User $U_i$ removes the signal of $U_K$ using SIC and detect the symbol of $U_{K-1}$ with SINR

$$\begin{aligned} \Gamma _{U_i,K-1}=\frac{E_RA_i^2D_{K-1}}{N_0+E_RA_i^2\sum _{l=1}^{K-2}D_l} \end{aligned}$$

(33)

User $U_i$ detects the symbols of users $U_p$ $p=K,K-1,\ldots ,i$ with SINR

$$\begin{aligned} \Gamma _{U_i,p}=\frac{E_RA_i^2D_p}{N_0+E_RA_i^2\sum _{l=1}^{p-1}D_l} \end{aligned}$$

(34)

There is no outage at $U_i$ when all SINRs are larger than x:

$$\begin{aligned} P_{{\textit{outage}},U_i}(x)&= 1-P(\Gamma _{U_i,K}>x,\Gamma _{U_i,K-1}>x,\ldots ,\Gamma _{U_i,i}>x) \\ &= F_{A_i^2}\left( \underset{i\le p\le K}{\hbox{max}}\left( \frac{N_0x}{E_RD_p-xE_R\sum _{l=1}^{p-1}D_l}\right) \right) \end{aligned}$$

(35)

3.1 IRS Link Analysis in the Presence of $P_i$ Interferers at User $U_i$

Figure 2 depict the system model in the presence of $P_i$ interferers user $U_i$. User $U_i$ detects the symbols of users $U_p$ $p=K,K-1,\ldots ,i$ with SINR

$$\begin{aligned} \Gamma _{U_i,p}=\frac{E_RA_i^2D_p}{N_0+J_i+E_RA_i^2\sum _{l=1}^{p-1}D_l} \end{aligned}$$

(36)

where $J_i$ is the interference at $U_i$ expressed as

$$\begin{aligned} J_i=\sum _{q=1}^{P_i} J_{i,q} \end{aligned}$$

(37)

$J_q=E_q'|j_q|^2$, $E'_q$ is the TES of q-th interferer and $j_q$ is the channel coefficient between q-th interferer and D. $J_q$ are assumed to be i.i.d so that the $J_i$ follows a Gamma distribution $Gamma(P_iM,\zeta _i)$ written as

$$\begin{aligned} f_{J_i}(y)=\frac{y^{P_iM-1}e^{-\frac{y}{\zeta _i}}}{\Gamma (MP_i)\zeta _i^{MP_i}} \end{aligned}$$

(38)

where $\zeta _i=\frac{E(J_{i,q})}{M}$.

Fig. 2

Millimeter wave communications using IRS in the presence of $P_i$ interferers at user $U_i$

In the presence of $P_i$ interferers at $U_i$, the outage probability at $U_i$ is computed numerically as follows

$$\begin{aligned} P_{{\textit{outage}},U_i}(x)=\int _{0}^{+\infty }F_{A_i^2}\left( \underset{i\le p\le K}{\hbox {max}}\left( \frac{(N_0+y)x}{E_RD_p-xE_R\sum _{l=1}^{p-1}D_l}\right) \right) f_{J_i}(y)dy, \end{aligned}$$

(39)

where $f_{J_i}(y)$ is the PDF of interference at $U_i$ (38) and $F_{A_i^2}(x)$ is provided in (31).

4 Throughput Optimization

The end-to-end (e2e) outage probability at user $U_i$ is computed as

$$\begin{aligned} P_{{\textit{outage}},e2e,U_i}(x)=1-\left[ 1-P_{{\textit{outage}},U_i}(x)\right] \left[ 1-P_{{\textit{outage}},R}(x)\right] . \end{aligned}$$

(40)

There is no outage at user $U_i$ if there is no outage event in the first hop at R and in the second hop.

An upper bound of Packet Error Probability (PEP) is expressed as [35]

$$\begin{aligned} {\textit{PEP}}_i\left( D_1,D_2,\ldots ,D_K,C_1,C_2,\ldots ,C_K\right) <P_{{\textit{outage}},e2e,U_i}(W_O) \end{aligned}$$

(41)

where $W_0$ is a waterfall threshold evaluated as [35]

$$\begin{aligned} W_0=\int _0^{+\infty }1-\left[ 1-2\left( 1-\frac{1}{\sqrt{Q}}\right) {\textit{erfc}}\left( \sqrt{\frac{3y}{Q-1}}\right) \right] ^{L}dy \end{aligned}$$

(42)

and L is packet length in Quadrature Amplitude Modulation (QAM) symbols, Q is the constellation size.

We deduce the throughput at user $U_i$:

$$\begin{aligned} {\textit{Thr}}_i(D_1,D_2,\ldots ,D_K,C_1,C_2,\ldots ,C_K)=0.5 \log _2(Q)[1-{\textit{PEP}}_i(D_1,D_2,\ldots ,D_K,C_1,C_2,\ldots ,C_K)] \end{aligned}$$

(43)

The total throughput is given by

$$\begin{aligned} {\textit{Thr}}(D_1,D_2,\ldots ,D_K,C_1,C_2,\ldots ,C_K)=\sum _{i=1}^K {\textit{Thr}}_i(D_1,D_2,\ldots ,D_K,C_1,C_2,\ldots ,C_K) \end{aligned}$$

(44)

We optimize the fraction of powers allocated to users at the source ($C_i$) and relay ($D_i$) to enhance the total throughput

$$\begin{aligned} {\textit{Thr}}^{\mathrm{max}}=\underset{0<D_1<D_2<\cdots<D_K<1,0<C_1<C_2<\cdots<C_K<1}{\hbox {max}} {\textit{Thr}}(D_1,D_2,\ldots ,D_K,C_1,C_2,\ldots ,C_K) \end{aligned}$$

(45)

under constraints $\sum _{i=1}^KC_i=1$ and $\sum _{i=1}^KD_i=1$.

5 Theoretical and Simulation Results

This section provides some theoretical and simulation results when there are $K=2$ and $K=3$ NOMA users and $ple=3$. The distance between S and R is 1.5 and the distance between R and IRS is $d=2$. The distance between IRS and users are $d_1=1$, $d_2=1.5$ and $d_3=2$. The number of interferers at R is $P=3$. The number of interferers at users is $P_1=2$, $P_2=3$ and $P_3=4$. The m-fading figure is $M=2$. The same powers were allocated to S and R, i.e. $E_S=E_R=E_s/2$ where $E_s$ is the TES.

Figures 3 and 4 depict the throughput at weak and strong users for $K=2$ and 16QAM modulation. The fraction of powers allocated to users are $C_1=0.4=1-C_2$ and $D_1=0.4=1-D_2$. The number of reflectors per user is $N=N_1=N_2=8,16,32$. The proposed NOMA system using IRS offers 10, 13, 16 dB gain with respect to conventional NOMA using millimeter wave communications without IRS.

Fig. 3

Throughput at weak user for 16QAM modulation: 2 users

Fig. 4

Throughput at strong user for 16QAM modulation: 2 users

Figure 5 depicts the total throughput for 16QAM modulation. For fixed power allocation given by $C_1=0.4=1-C_2$ and $D_1=0.4=1-D_2$, the proposed NOMA system using IRS offers 10, 13, 16 dB gain with respect to conventional NOMA using millimeter wave communications. For an Optimal Power Allocation (OPA) maximizing the total throughput (45), the proposed NOMA using IRS offers 19 dB gain for a number of reflectors per user $N=N_1=N_2=32$.

Fig. 5

Total throughput for 16QAM modulation: 2 users

Figure 6 depicts the total throughput when there are two NOMA users for 16QAM modulation. Figure 6 shows the total throughput for a large number of reflectors $N=N_1=N_2=64,128,256,512$. The proposed NOMA system using IRS offers 20, 24, 27 and 30 dB gain with respect to conventional NOMA using millimeter wave communications for $N=N_1=N_2=64,128,256,512$.

Fig. 6

Total throughput for 16QAM modulation: $N=64,128,256,512$

The required $E_b/N_0$ in dB to reach a total throughput of 3.5 bit/s/Hz is provided in Table 1 for different values of N. We used the results of Figs. 5 and 6 with two NOMA users and 16QAM modulation.

Table 1 Required $E_b/N_0$ in dB to reach a total throughput of 3.5 bit/s/Hz: two users and 16QAM modulation

Figure 7 depicts the total throughput for QPSK modulation when there are three users, i.e. $K=3$. The fractions of allocated powers are ($C_1=0.2$, $C_2=0.3$, $C_3=0.5$) and ($D_1=0.2$, $D_2=0.3$, $D_3=0.5$). The proposed NOMA system using IRS offers 12, 14, 16 dB with respect to conventional NOMA system without IRS for a number of reflectors $N=N_1=N_2=N_3=8,16,32$. IRS with OPA (45) and $N=N_1=N_2=N_3=32$ reflectors offers 19 dB gain with respect to conventional NOMA using millimeter wave communication without IRS.

Fig. 7

Total throughput for QPSK modulation with optimal power allocation: 3 users

The required $E_b/N_0$ in dB to reach a total throughput of 2.5 bit/s/Hz is provided in Table 2 for different values of N. We used the results of Fig. 7 with three NOMA users and QPSK modulation.

Table 2 Required $E_b/N_0$ in dB to reach a total throughput of 2.5 bit/s/Hz: three users and QPSK modulation

6 Conclusions and Perspectives

In this paper, we used intelligent Reflecting Surfaces (IRS) to enhance the throughput of NOMA systems using millimeter wave communications. The source sends a combination of symbols to K users. The received signal at the relay node is affected with P interferers. The results are valid for any number of interferers at the relay and users. The relay node uses SIC to detect the symbols of all K NOMA users. Then, the relay node sends a combination of K detected symbols. The transmitted signal by relay node is reflected by different sets of reflectors dedicated to K users. The reflected signals by a set of IRS reflectors have the same phase at each user. The proposed NOMA using IRS offers 10, 13, 16, 20, 24, 27 and 30 dB gain with respect to conventional NOMA using millimeter wave communications without IRS for a number of reflectors $N=8,16,32,64,128,256,512$. We also suggested optimizing the powers dedicated to NOMA users at the source and relay node to enhance the total throughput. As a perspective, we can consider millimeter wave communications with multihop relaying and IRS.

Appendix A

Let $X=E_S|f|^2$ and $I=\sum _{q=1}^PI_q$. U is defined as $U=\frac{X}{I}$. X and I are independent Gamma r.v. with PDF [36]

$$\begin{aligned} f_{X,I}(x,y)=\frac{x^{M-1}y^{PM-1}e^{-\frac{x}{\beta }}e^{-\frac{y}{\alpha }}}{\Gamma (M)\Gamma (MP)\beta ^{M}\alpha ^{MP}} \end{aligned}$$

(46)

Let $U=\frac{X}{I}$ and $W=X+I$, the determinant of Jacobian matrix is

$$\begin{aligned} |J|=\begin{vmatrix} \frac{\partial U}{\partial X}&\frac{\partial U}{\partial I} \\ \frac{\partial W}{\partial X}&\frac{\partial W}{\partial I} \\ \end{vmatrix}=\begin{vmatrix} \frac{1}{I}&\frac{-X}{I^2} \\ 1&1 \\ \end{vmatrix}=\frac{X+I}{I^2}=\frac{(1+U)^2}{W} \end{aligned}$$

(47)

We can write $I=\frac{W}{1+U}$ and $X=\frac{UW}{1+U}$. We deduce the PDF of (U, W)

$$\begin{aligned} f_{U,W}(u,w)&= \frac{f_{X,I}(x,y)}{|J|}=\frac{w}{(1+u)^2}\left( \frac{w}{1+u}\right) ^{PM-1}\left( \frac{wu}{1+u}\right) ^{M-1} \\ &\quad\times \frac{e^{-\frac{wu}{(1+u)\beta }}e^{-\frac{w}{(1+u)\alpha }}}{\Gamma (M)\Gamma (MP)\beta ^{M}\alpha ^{MP}} \end{aligned}$$

(48)

We can write

$$\begin{aligned} f_U(u)=\int _0^{+\infty }f_{U,W}(u,w)dw. \end{aligned}$$

(49)

We have [34]

$$\begin{aligned} \int _0^{+\infty }e^{-Aw}w^Bdw=\frac{\Gamma (B+1)}{A^{B+1}} \end{aligned}$$

(50)

Equations (29–31) give

$$\begin{aligned} f_{U}(u)=\frac{\Gamma (M+PM)\alpha ^Mu^{M-1}}{\beta ^M\Gamma (M)\Gamma (PM)}\left( 1+\frac{u\alpha }{\beta }\right) ^{-PM-M} \end{aligned}$$

(51)