### I. Introduction

### II. System and Channel Models

*S*communicates with two users, a near user

*U*

_{1}and a far user

*U*

_{2}, with the help of an IRS attached to a wall. All nodes are single-antenna devices. The NOMA technique is used for data transmission for the users. The IRS includes

*N*RTs,

*R*

*,*

_{n}*n*= 1,…,

*N*, and each RT is a reconfigurable metasurface and can independently direct the signal to a user under the control of a programmable controller. The channel model proposed for indoor IRS-aided wireless communications in [10, 13] is adopted to characterize the wireless channels of our system. For instance, the received signal at

*U*

*,*

_{m}*m*∈ 1, 2, comprises of (i) the signals propagating over the line-of-sight (LoS) (or the direct link) and reflecting links, both are the dominant links, and (ii) scattering signals caused by multi-path fading or multiple reflections on the IRS. Let ℛ

*be the set of RTs allocated to*

_{m}*U*

*,*

_{m}*m*= {1, 2}; hence, the channel between

*S*and

*U*

*is expressed as*

_{m}*S*and

*U*

*and for the reflecting link between*

_{m}*S*and

*U*

*via*

_{m}*R*

*, i.e., the*

_{n}*S*–

*R*

*–*

_{n}*U*

*link, respectively, and*

_{m}*G*

_{S}(

*θ*,

*φ*) and

*G*

_{Um}(

*θ*,

*φ*), respectively denote directivity patterns of the transmit antenna of

*S*and the receive antenna of

*U*

*where*

_{m}*θ*and

*φ*are respectively the zenith and azimuth angles. For given locations of

*S*and

*U*

*and given antenna’s orientations, we can calculate the values for*

_{m}*θ*and

*φ*, hence obtaining the values for transmit/receive directivities for wireless links. Let

*S*to a given point

*X*, and let

*Y*to

*U*

*. Using these transmit/receive directivities and the free-space path loss (FSPL) model, we can determine*

_{m}*λ*

_{0};

*d*

_{SUm}is the distance between

*S*and

*U*

*;*

_{m}*d*

_{SRnUm}

*= d*

_{SRn}+

*d*

_{RnUm}is the distance of the

*S*–

*R*

*–*

_{n}*U*

*link;*

_{m}*d*

_{SRn}and

*d*

_{Rn}

_{Um}are the distances between

*S*and

*R*

*and between*

_{n}*R*

*and*

_{n}*U*

*, respectively;*

_{m}*ψ*

_{Rn}=

*β*

_{Rn}

*e*

^{jϕRn}is the reflecting coefficient of

*R*

*with*

_{n}*β*

_{Rn}≤ 1 and

*ϕ*

_{Rn}∈ [0, 2

*π*]; and

*d*

_{SUm}and

*d*

_{SRnUm}.

*h*

*is a Rice distributed RV. Let*

_{m}*h ~*Rice(Ω,

*K*) be a general Rice distributed RV, then the cumulative distribution function (CDF) of |

*h*|

^{2}is expressed as [10]:

*S*transmits a combined signal

*x*

_{1}, E{|

*x*

_{1}|

^{2}} =

*P*

_{0}, and

*x*

_{2}, E{|

*x*

_{2}|

^{2}} =

*P*

_{0}, are respectively the desired signals for

*U*

_{1}and

*U*

_{2};

*α*

_{1}and

*α*

_{2}are the PA factors of NOMA satisfying

*α*

_{1}+

*α*

_{2}= 1. When

*x*

_{s}is reflected from the RTs, the PD caused by the non-ideal RTs make it distorted. The effect of non-ideal hardware on system performance has been widely studied in [14–16] and is characterized as a complex Gaussian noise

*τ*

_{Rn}≪ 1. Using (1), the received signal for

*U*

*in the presence of PD is given by*

_{m}*n*

*~*

_{m}*CN*(0,

*N*

_{0}) is the additive white Gaussian noise (AWGN) at the receive antenna, and

*R*

*. The factor*

_{n}*ĥ*

*~ Rice(Ω̂*

_{m}*,*

_{m}*K̂*

*) is still a Rice distributed RV, and*

_{m}*,*

_{m}*K̂*

*, and*

_{m}*ϕ*

_{Rn}, plays an important role in system performance. It defines whether the received signals at

*U*

*via the dominant links are constructive or destructive. Let us consider the distance-propagation phase shift of the LoS link, i.e,*

_{m}*ϕ*

_{SUm}, as the standard reference in the phase shift at

*U*

*, the signal-combining phase shift denoted by*

_{m}*S*-

*R*

*-*

_{n}*U*

*link and*

_{m}*ϕ*

_{SUm}.

*ψ*

_{Rn},

*and*

_{m}*K̂*

*using a similar approach in [10]:*

_{m}##### (8)

### III. SINR Distribution Characterization and Average Achievable Rate Analysis

*x*

_{2}at both users, i.e.,

*U*

*,*

_{m}*m*= {1, 2}, are given by

*x*

_{1}at

*U*

_{1}is given by

*x*

_{2}at both users as

*x*

_{1}at

*U*

_{1}as

*Y*= min(

*Y*

_{1},

*Y*

_{2}) is the condition for guaranteeing successfully decoding

*x*

_{2}at both users (note that

*U*

_{1}must decode

*x*

_{2}before decoding its information

*x*

_{1}), and

*g*~ CN (0,

*λ*

_{ImSIC}) in (13) represents the residual signal of

*x*

_{2}caused by the imperfect SIC receiver at

*U*

_{1}with variance

*λ*

_{SICIm}=

*ζ*E{|

*h*

_{SU1}|

^{2}} and the SIC imperfection factor ζ ≤ 1.

### Proposition 1

*X*denoted by

*F*

*(*

_{X}*t*) = Pr(

*X*<

*t*) is approximated as

*ρ*

*=*

_{m}*μ*/(

*α*

_{1}

*γ*

*), and*

_{m}*η*=

*λ*

_{ImSIC}

*α*

_{1}/(

*μα*

_{2}).

#### Proof

*X*is calculated as

*F*

*(*

_{X}*t*) ≈ Pr(|

*ĥ*

_{1}|

^{2}< (1 +

*γ*

_{1}|

*g*|

^{2}

*α*

_{2})

*t*/(

*γ*

_{1}

*α*

_{1})). Using the probability density function (PDF) of the exponential distribution given by

*f*

_{|g|}_{2}(

*t*) =

*λ*

_{ImSIC}exp (−

*λ*

_{ImSIC}

*t*)and (4), we obtain

*F*

*(*

_{X}*t*) via solving the following equation.

### Proposition 2

*Y*denoted by

*F*

*(*

_{Y}*t*) = Pr(

*Y*<

*t*) is equal to one if

*t*>

*α*

_{2}/

*α*

_{1}, and otherwise, it is calculated as

### Proposition 3

*x*

_{1}and

*x*

_{2}are respectively given in the equations (19) and (20) shown in the top of the next page, where

### IV. Numerical Results and Discussion

*ϕ*

_{Rn}= 2

*π*× mod(

*d*

_{SRn}+

*d*

_{RnUm}–

*d*

_{SUm},

*λ*

_{0})/

*λ*

_{0}to gain the strongest received signal strengths (RSSs) at the users,

*β*

_{Rn}= 0.9 and

*τ*

_{Rn}= 0.1 where

*n*= 1,..., 2

*N*

_{col}. The coordinate setups are shown in Fig. 2, and the setup of other parameters is listed in Table 1 [20]. The transmit antenna is a half-wavelength dipole with its orientation illustrated in Fig. 2, and its directivity pattern is given in Table 1. The receive antennas have the same directivity for all directions. The numerical results include the theoretical AARs obtained using (19) and (20), and the simulation results are obtained by evaluating a simulation system using MATLAB. The good match between the theoretical and the simulation results confirms the accuracy of our study.

*m*

_{1},

*m*

_{2}],

*m*

_{1},

*m*

_{2}∈{1, 2}, which indicates that the first and the second RTs are allocated to

*U*

_{m}_{1}and

*U*

_{m}_{2}, respectively. The AARs for

*x*

_{1},

*x*

_{2}and total AAR,

*C̄*

_{∑}=

*C̄*

_{1}+

*C̄*

_{2}, for the ideal-IRS scenario (without the presence of the PD) are shown in Fig. 3(a)–3(c), respectively, and those of the non-ideal-IRS scenario (with the presence of the PD) are shown in Fig. 3(d)–3(f), respectively. The AARs are displayed as increasing functions of

*P*

*. Without the IRS, the received signal at each user propagates on the direct link and multi-path fading channel. Therefore, the AARs are very low, as shown as the lowest curves in Fig. 3. When the IRS is utilized, the AARs are significantly enhanced even with the presence of the PD. Although different levels of AAR enhancement are observed, these results still indicate the important role of the IRS in improving the RSSs at the users. As shown in Fig. 3(c) and 3(f), the total AARs for the cases of IRS[1,2] and IRS[2,1] (in these cases, the IRS assists the communication for both users) achieve better values than other cases.*

_{S}*P*

*= 0 dBm with*

_{S}*τ*

_{PD}= 0.1 (found in Fig. 3, i.e., IRS[2,1]) and then investigate the effect of the PA factors on the total AAR. The advantages of the NOMA scheme are confirmed by investigating a reference orthogonal multiple access (OMA) scheme. Due to the advantageous location of the near user, the system achieves a highest or lowest total-AAR when allocating all resources to the near user or the far user, respectively. When the SIC imperfection factor

*ζ*increases, the total AAR becomes worse. For a performance comparison, we use a conventional OMA scheme, which defines the AAR for

*U*

*,*

_{m}*m*= {1, 2}, as

*υ*

*∈ (0, 1),*

_{m}*υ*

_{1}+

*υ*

_{2}= 1, is the resource allocation (RA) factors for OMA (e.g., time allocation factor for TDMA or bandwidth allocation factor for FDMA). Compared to OMA, NOMA can achieve a higher total AAR in the presence of PD and imperfect-SIC. When

*ζ*is sufficiently large, NOMA yields a lower performance compared to OMA. In Fig. 5, we show the joint effect of the RT-allocation strategy and PA factors, i.e.,

*α*

_{1}and

*α*

_{2}, on the AAR. Because the size of each RT is relatively small compared to the wave-propagation distances, the difference in distance between reflecting links is trivial. It is reasonable to assume that the signal attenuations of reflecting links toward a user are similar. For this reason, the RT allocation strategy is as follows. Let

*N*

_{U}_{1}be the number of RTs allocated to

*U*

_{1}. These

*N*

_{U}_{1}RTs are assigned from left to right and from top to bottom. The other

*N*

_{U}_{2}=

*N*–

*N*

_{U}_{1}RTs are assigned to

*U*

_{2}. For a given value of

*N*

_{U}_{1},

*C̄*

_{1}increases, while

*C̄*

_{2}decreases when we increase

*α*

_{1}and vice versa. This is because the power resource allocated for the users is defined by

*α*

_{1}. For given PA factors,

*C̄*

_{1}is shown as an increasing function of

*N*

_{U}_{1}, whereas

*C̄*

_{2}improves with the first increase in

*N*

_{U}_{1}and then reaches the optimal value. Finally, it degrades with further increases in

*N*

_{U}_{1}. The trend of

*C̄*

_{2}is explained as follows. Since both users need to decode

*x*

_{2}successfully, the IRS needs assist both users to gain the highest value of

*C̄*

_{2}. Therefore, using very high or very low values of

*N*

_{U}_{1}does not yield good values for

*C̄*

_{2}. Although the highest total AAR is observed at (

*N*

_{U}_{1}= 10,

*α*

_{1}= 1), the NOMA system does not use this configuration. The reason is that at this point, all system resources, including

*P*

_{S}and RTs, are allocated to

*U*

_{1}; hence,

*U*

_{2}is disconnected from the network. For our proposed NOMA system, we focus on the points lying on the intersection curve of the two user-AAR surfaces denoted by

^{*}. The trends of ℓ

^{*}and

*N*

_{col}RTs and then plot the optimal NOMA-AAR as shown in Fig. 6(a) and its respective optimal solution, i.e., optimal PA factors as shown in Fig. 6(b) and optimal RT-allocation as shown in Fig. 6(c) and 6(d), under different parameters of

*P*

*and*

_{S}*ζ*. Fig. 6(a) shows that the more RTs are used, the higher value for optimal NOMA-AAR is achieved. When the IRS’s size is sufficiently large, the increase in size does not gain much improvement in optimal NOMA-AAR. Therefore, the size of the IRS should be chosen carefully to achieve a reasonable trade-off between performance and cost. For instance, at

*ζ*= 0.04 and

*P*

*= 0 dBm, the 2-by-1 and 2-by-4 IRSs can support 1 bit/s/Hz and 2 bit/s/Hz optimal NOMA-AAR, respectively. In Fig. 6(b)–6(d), we show the trends of the obtained optimal solutions. Regarding the NOMA principle, first, both users need to decode the signal allocated with stronger power, which is*

_{S}*x*

_{2}for our proposed system, and consider the rest signals as interference, as shown in (12). After successfully decoding

*x*

_{2},

*U*

_{1}eliminates the signal

*x*

_{2}from its received signal and then decodes its desired signal

*x*

_{1}, which is allocated with lower power. The allocated power for each signal is defined by the PA factors. Compared to

*x*

_{2}, the decoding process of

*x*

_{1}deals with lower interference, which is caused by the imperfect-SIC decoder as shown in (13). For this reason, the AAR for

*x*

_{2},

*C̄*

_{2}, significantly depends on the PA factors, and

*C̄*

_{2}increases when

*C̄*

_{1}significantly depends on the RSS at

*U*

_{1}. Hence,

*C̄*

_{1}is enhanced when more RTs are allocated to

*U*

_{1}. Using the above assessments, it is easy to explain the trends of

*U*

_{1}and using higher values for

*N*

_{col}, as shown in Fig. 6(b), whereas

*N*

_{col}, as shown in Fig. 6(c) and 6(d). Moreover,

*P*

*or the decrease in the SIC-imperfection factor*

_{S}*ζ*also give the same effect on the signal quality at the users.

*P*

*increases or*

_{S}*ζ*decreases.

*τ*

_{PD}and the reflection amplitude

*β*

_{Rn}on the system performance for two cases: with and without the presence of the LoS links. The results for the non-LoS case are obtained by set the source-to-user distances in (8) by very large values, i.e.,

*d*

_{SUm}→ ∞. The optimal NOMA-AAR drops around 0.8 bits/s/Hz when the LoS links are not available. This indicates the importance of the LoS links; however, without the LoS links, the system can achieve a moderately optimal NOMA-AAR with the help of the reflecting link and the multi-path fading channel. When

*τ*

_{PD}increases, the total noise at each user increases. Hence, the optimal NOMA-AAR is displayed as a decreasing function of

*τ*

_{PD}as shown in Fig. 7(a). With the presence of a multi-path fading channel and/or LoS links, the influence of the PD on the optimal NOMA-AAR can be reduced; however, when the user is blocked and its communication only relies on the reflecting links, the value of

*τ*

_{PD}becomes a crucial factor to performance. Fig. 7(b) presents the optimal NOMA-AAR for different values of

*β*

_{Rn}. When

*β*

_{Rn}increases, the stronger received signal strengths are achieved at the users. Therefore, the optimal NOMA- AAR is shown as an increasing function of

*β*

_{Rn}. Moreover, the optimal NOMA-AAR for the case using the IRS is higher than that for the non-IRS case. This indicates the effectiveness of the IRS in improving system performance. Since

*β*

_{Rn}only affects the quality of the reflecting links, the influence of low

*β*

_{Rn}values can be reduced in the case of the presence of the LoS links or the stronger multi-path fading channel.