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.

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)