# On the Secrecy Unicast Throughput Performance of NOMA Assisted Multicast-Unicast Streaming With Partial Channel Information

This paper considers a downlink single-cell non-orthogonal multiple access (NOMA) network with uniformly deployed users, while a mixed multicast and unicast traffic scenario is taken into account. By guaranteeing the quality of service (QoS) of the multicast traffic, the multicast outage and secrecy unicast throughput performance is evaluated. In particular, two types of partial channel state information (CSI), namely the imperfect CSI and CSI based on second order statistics (SOS), are investigated in the analytical framework. For both two cases, the closed-form approximations for the multicast outage probability and secrecy unicast throughput are derived except that the approximation for the secrecy unicast throughput in SOS-based CSI only concerns two users. As to the multicast outage probability, the simulation results demonstrate that the NOMA scheme considered in both two cases achieves superior performance compared to the traditional orthogonal multiple access (OMA) scheme, while for the secrecy unicast throughput, the NOMA scheme shows great advantages over the OMA scheme in good condition (high signal-to-noise ratio (SNR)), but is inferior to the OMA scheme in poor condition (low SNR). Moreover, both two cases achieve similar performance except that the NOMA scheme with imperfect CSI obtains larger secrecy unicast throughput than that based on SOS under high SNR condition. Finally, the provided numerical results also confirm that the derived approximations of both two cases for the multicast outage probability and secrecy unicast throughput match well with the Monte Carlo simulations.

Comments

There are no comments yet.

## Authors

• 43 publications
• 15 publications
• 8 publications
• 1 publication
• ### Wireless Powered Cooperative Relaying using NOMA with Imperfect CSI

The impact of imperfect channel state (CSI) information in an energy har...
09/29/2018 ∙ by Dinh-Thuan Do, et al. ∙ 0

read it

• ### Outage Performance of A Unified Non-Orthogonal Multiple Access Framework

In this paper, a unified framework of non-orthogonal multiple access (NO...
01/24/2018 ∙ by Xinwei Yue, et al. ∙ 0

read it

• ### Secrecy Outage Probability Analysis for RIS-Assisted NOMA Systems

In this paper, the physical layer security (PLS) for a novel reconfigura...
07/31/2020 ∙ by Liang Yang, et al. ∙ 0

read it

• ### Hardware and Interference Limited Cooperative CR-NOMA Networks under Imperfect SIC and CSI

The conflation of cognitive radio (CR) and nonorthogonal multiple access...
04/22/2020 ∙ by Sultangali Arzykulov, et al. ∙ 0

read it

• ### Secure Communications in a Unified Non-Orthogonal Multiple Access Framework

This paper investigates the impact of physical layer secrecy on the perf...
04/02/2019 ∙ by Xinwei Yue, et al. ∙ 0

read it

• ### On the Secrecy Performance of Random VLC Networks with Imperfect CSI and Protected Zone

This paper investigates the physical-layer security for a random indoor ...
10/13/2019 ∙ by Jin-Yuan Wang, et al. ∙ 0

read it

• ### Secrecy Analysis of Ambient Backscatter NOMA Systems under I/Q Imbalance

We investigate the reliability and security of the ambient backscatter (...
04/30/2020 ∙ by Xingwang Li, et al. ∙ 0

read it

##### This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

## I Introduction

Non-orthogonal multiple access (NOMA) has been recognized as a promising technology to satisfy the challenging requirements of the fifth generation (5G) wireless networks, such as high data speed, massive connectivity, and low latency[1, 2]. On the other hand, downlink NOMA has been introduced in 3rd generation partnership project (3GPP)-long term evolution-advance (LTE-A) systems[3], aiming to improve the spectral efficiency. The principle behind NOMA is that multiple users’ signals are superimposed at the base station (BS) with different power level, and the user decodes its own signal by removing signals of poor channel conditions’ users, which is called successive interference cancellation (SIC) technique[4]. Note that the key feature of the downlink NOMA is to take user fairness into consideration as it allocates more power to users with worse channel conditions than those with better channel conditions, which realizes an improved trade-off between user fairness and system throughput.

Based on the perfect channel state information (CSI), performance analysis of the downlink NOMA is extensively studied in[5, 6, 7, 8, 9]. The outage and sum-rate performance of NOMA in a cellular downlink scenario with randomly deployed users was analyzed in [5]. Meanwhile, [6] investigated the outage balancing problem while the issues of power allocation, decoding order selection, and user grouping were taken into account. As to the downlink and uplink NOMA scenarios with two users for flexible quality of service (QoS) requirements, [7] developed a novel power allocation scheme and established the exact expressions of the outage probability and average rate. As research goes deep, by taking the cooperative communication into consideration, the closed-form expressions for the outage probability and ergodic sum rate were derived in a downlink NOMA system with cooperative half-duplex relaying[8]. Finally, in a multiple-input multiple output (MIMO) system, [9] studied sum channel capacity and ergodic sum capacity performance of NOMA in detail.

However, in practice, perfect CSI at the transmitter is usually not available, because obtaining the perfect CSI consumes a significant system overhead, especially for the wireless network with a large number of users. Furthermore, one of key features towards 5G is highly mobile, leading to rapidly changing channel, which makes it greatly challenging to achieve perfect CSI as the transmitter. Therefore, assuming partial CSI, several related works have been investigated in[10, 11, 12, 13] recently. In [10], the outage performance of the downlink NOMA was studied for the case where each user feeds back only one bit of its CSI to the BS. Based on average CSI, optimal power allocation was analyzed while fairness for the downlink users were ensured in[11]. Meanwhile, [12] solved the ergodic capacity maximization problem of a MIMO-NOMA system with second order statistical (SOS) CSI at the transmitter. Emphasize that both [11] and [12] focused on the case where user location are fixed, namely the distances and path loss are deterministic. Therefore, considering a downlink single-cell NOMA network with uniformly deployed users, an analytical framework to evaluate the outage and sum rate performance was developed in[13].

Note that all the works mentioned above considered only single unicast traffic. Meanwhile, Several works have studied the downlink NOMA with a mixed multicast and unicast traffic scenario[14, 15, 16]. In [14], Multicast beamforming with superposition coding (SC) was investigated for multi-resolution broadcast where both data streams of high priority (HP) and low priority (LP) were to be transmitted for a near user, while only data stream of LP was to be transmitted to a far user. [15] developed a novel beamforming and power allocation scheme while the unicast performance was improved and the reception reliability of the multicast was maintained. What’s more, [15] also analyzed how the use of NOMA can prevent those multicast receivers intercepting the unicast messages and proposed the secrecy unicast rate metric. Finally, [16] introduced a novel NOMA unicast-multicast system, where a number of unicast users and a group of multicast users shared the same wireless resource.

However, to date few work has made performance analysis where both partial CSI condition and mixed unicast-multicast traffic are taken into account. Hence, in this paper, we consider a downlink single-cell NOMA network, where the users are uniformly distributed in a disk and the BS is located at the center. Meanwhile a mixed multicast and unicast traffic scenario is taken into account, where the BS transmits two types of data streams, one for multicast and one for unicast. Assuming partial CSI at the BS, by guaranteeing the QoS of the multicast traffic, the multicast outage and secrecy unicast throughput performance are investigated. In particular, similar to

[13], we consider two types of partial CSI, namely imperfect CSI and SOS-based CSI, which are defined as follows.

• Imperfect CSI: According to [13, 18, 19]

, a channel estimation error model is assumed, where the BS and the users estimate the channel with a priori knowledge of the variance of estimation error. More specifically, we concentrate on the minimum mean square error (MMSE) channel estimation error model

[20, 21, 22].

• SOS-based CSI: Only the distances between the BS and the users are known at the BS. The motivation for studying such SOS-based CSI is analyzed below. First, distance information is easy to obtain in practical communications. Second, small-scaling fading only weekly changes the large-scale fading, which means that large-scale fading is dominant in CSI[17].

In particular, the contribution of this paper is three-fold:

• We develop a novel performance analysis framework in a downlink single-cell NOMA network with uniformly deployed users, where only partial CSI is available at the BS and a mix multicast-unicast traffic is considered. We guarantee the QoS of the multicast, which means that the minimum rate of the multicast message is ensured, and the corresponding outage performance is studied. Meanwhile, since the unicast message is broadcast to all the users, the secrecy unicast throughput is also investigated, where the power allocation and unicast user selection problem is properly formulated and mathematically solved.

• Two types of partial CSI are considered, namely imperfect CSI and SOS-based CSI. For both two cases, employing the probability theory, order statistics theory and multi-binomial theorem, the closed-form approximate expressions for the outage probability and secrecy unicast throughput are derived. Note that as to the SOS-based CSI case, due to significant complication of calculating the secrecy unicast throughput , we only focus on two users. Numerical results are provided to demonstrate that the derived approximate expressions of both two cases for the outage probability and secrecy unicast throughput match well with the Monte Carlo simulations.

• The corresponding orthogonal multiple access (OMA) scheme with partial CSI is developed and a performance comparison between the NOMA and OMA schemes is made. Simulation results show that as to the multicast outage probability, the NOMA scheme in both two cases achieves superior performance compared to the OMA scheme, while for the secrecy unicast throughput, the NOMA scheme shows great advantages over the OMA scheme in good condition (high signal-to-noise ratio (SNR)), but is slightly inferior to the OMA scheme in bad condition (low SNR). Moreover, a comparison between the NOMA scheme with imperfect CSI and the NOMA scheme with SOS-based CSI is presented. Numerical results suggest that both two NOMA schemes obtain similar performance except that the NOMA scheme with imperfect CSI achieves larger secrecy unicast throughput than that based on SOS in high SNR.

The rest of this paper is organized as follows. In Section II, we introduce the system model and formulate the corresponding power allocation problem. In Section III, we first mathematically solve the resource optimization problem. Second, by assuming two types of partial CSI, the multicast outage and secrecy unicast throughput performance are analyzed in detail. Third, the corresponding OMA scheme with partial CSI is provided as a benchmark. Section IV presents the simulation results and the conclusions are drawn in Section V.

## Ii System Model and Problem Formulation

In this section, we consider a downlink single-cell NOMA network with uniformly deployed users while a mixed multicast and unicast traffic scenario is takeing into account. Aiming to maximize the secrecy unicast throughput, the power allocation and unicast user selection problem is correspondingly formulated.

### Ii-a System Model

Consider a single-cell downlink wireless network with one BS communicating with users. The location of users are uniformly distributed in a disc with radius , which is denoted as , while the BS is located in the center of the disc. Here all the users and the BS are equipped with single antenna. Furthermore, we assume that all users share the same wireless channel resource. The channel between the BS and user , , is modeled as , where is the Rayleigh fading coefficient, is the distance between BS and user , and is the path loss exponent. Here we assume that (

) follows complex normal distribution with zero mean and unit variance, that is

. Denote as the channel gain, namely . Furthermore, the background noise is modeled as the additive white Gaussian noise (AWGN) with zero mean and variance .

Note that the optimal performance can be achieved with perfect CSI. However, perfect CSI is usually not available in practical communications as the significant consumed overhead and great challenge in achieving CSI exactly. Motivated by this fact, this paper consider two types of practical channel models: imperfect CSI and SOS-based CSI, which will be fully discussed in Section III.

This work focuses on a mixed multicast and unicast traffic scenario, i.e., the BS has two messages to send. The multicast traffic is to be received by all the users, while the unicast traffic is intended to a particular user. Here we assume that the BS has unlimited multicast data to transmit, and infinite unicast data for each user to send. For a particular time slot, based on the partial CSI, the BS can select a specific user to transmit the unicast traffic, while the secrecy unicast throughput is maximized and the QoS of the multicast traffic is guaranteed.

### Ii-B Problem Formulation

The transmit data combines the multicast and unicast streaming. The multicast message is intended to all the users, whereas the unicast message is to be received by a particular user. Then, employing the NOMA transmission technique, the transmit data , sent by the BS, can be denoted as

 x=√PT(√θMsM+√θUsU), (1)

where is the total transmit power, and are the power allocation coefficients for the multicast and unicast messages, respectively. Note that and are designed to satisfy

 θM+θU≤1,θM≥0,θU≥0. (2)

The received data at user is

 yk=hkx+nk. (3)

According to the principle of the NOMA, here detects by treating as noise. Therefore, the signal-interference-noise ratio (SINR) for decoding at can be calculated as

 SINRMk=θMαkθUαk+1ρ, (4)

where denotes the transmit SNR, namely . Once is successfully decoded, SIC will be carried out at and is totally removed for detecting . Hence, the SNR for decoding is given by

 SNRUk=ρθUαk. (5)

According to the shannon capacity theory, the achievable rate of the multicast and unicast streams at can be obtained as

 RMk=log2(1+SINRMk),RUk=log2(1+SNRUk), (6)

where the transmit bandwidth is normalized here.

The optimization problem can be mathematically formulated as follows. First, the QoS of the multicast stream is guaranteed, namely the minimum rate of the multicast traffic is ensured, that is

 RMk≥RM,∀k∈K, (7)

where is the minimum rate of the multicast traffic.

Similar to [15], the secrecy unicast throughput is defined as

 RUS≜[RUj−max{RUk,k∈K∖{j}}]+, (8)

where . Note that for a particular time slot, unicast traffic receiver should be carefully selected as discussed above.

Therefore, the power allocation and unicast user selection problem considered in this paper can be written as

 maxj,θM,θURUS, (9)

subject to

 (???),(???). (10)

## Iii Performance analysis

In this section, the analytical framework to evaluate the multicast outage and secrecy unicast throughput performance is developed. First, the optimization problem (9) is fully solved. Second, based on two types of partial CSI, namely imperfect CSI and SOS-based CSI, the closed-form approximate expressions for the multicast outage probability and secrecy unicast throughput are mathematically derived. Third, the corresponding OMA scheme with partial CSI is provided as a benchmark.

### Iii-a Solution of problem (9)

Here we denote as a permutation of the user indices, and if , then the user has the -th best channel gain. In other words, the channel gains are sorted as .

According to the definition of in (8), it is easy to derive that . As , in order to maximize the , the BS should choose as user . Then . Therefore, increases with , which accords with common sense. Furthermore, for the given and , it is easy to find that if . Thus constraint (7) can be simplified as

 RMπK≥RM. (11)

Hence, Problem (9) can be equally expressed as

 maxθM,θURUS=log2(1+ρθUαπ11+ρθUαπ2), (12)

subject to

 (???),(???). (13)
###### Theorem 1.

The optimal solution of problem (12) can be obtained while both (2) and (11) achieve equality.

###### Proof.

See Appendix A. ∎

Based on Theorem 1, the optimal solution should satisfy

 θ∗M+θ∗U=1 (14) RMπK(θ∗M,θ∗U)=RM.

Therefore, we can obtain , where .

###### Remark 1.

If , a multicast outage happens, that is the QoS of the multicast traffic can not be satisfied even if all the power is allocated to .

In conclusion, the optimal solution of problem (9) is .

### Iii-B Performance analysis for imperfect CSI

In this subsection, based on the imperfect CSI, we analyze the multicast outage and secrecy unicast throughput performance in detail.

Here we assume that the channel feedback to the transmitter is instantaneous and error free, which means that CSI is also achievable at the transmitter whatever CSI the receiver has. As discussed above, such assumption is widely applied in the literature[13, 18, 19, 20, 21, 22]. Define as the estimation for channel , and estimated channel gain can be correspondingly calculated as . Similar to [20, 21, 22], assuming the MMSE estimation error, it holds that

 αk=^αk+ζ, (15)

where

is the channel estimation error, which follows a complex Gaussian distribution with mean

and variance , namely . Thus the estimated channel gain follows a complex Gaussian distribution with mean and variance [20, 21, 22]. Note that indicates the quality of channel estimation.

Correspondingly, the optimal solution of problem (9) is .

Next, the multicast outage probability and statistical expectation of secrecy unicast throughput will be analyzed as follows.

###### Theorem 2.

As to , it can be approximated as

 P1out≈1−[πcDc∑i=1|sin2i−12cπ|xiexp(−ϵMρ(x−ηi−σ2ζ))]K, (16)

where , and is the number of terms included in the summation, which controls the approximation accuracy, due to the use of Gauss-Chebyshev integration [25].

###### Proof.

See Appendix B. ∎

###### Remark 2.

Here we evaluate the computational complexity as the number of loops in computation. Then it is easy to obtain that the complexity of is .

###### Remark 3.

The multicast outage expression (16) is accurate for whole SNR. Furthermore, is a important parameter, which affects the accuracy of analytical results. However, as shown by the simulation results, (16) achieves an accurate approximation even with a small ().

###### Remark 4.

Under high SNR, namely is large enough, it is easy to see that , then

 (P1out)∞≈1−[πcDc∑i=1|sin2i−12cπ|xi]K. (17)

According to (17), we can find that there is no diversity gain, which is in line with our system design, that is all the users independently decode the and .

In addition, the case with perfect CSI is also worth studying as it provides a multicast outage performance upper bound, where the loss due to imperfect CSI can be clearly demonstrated. Fortunately, if perfect CSI is available, namely , an exact closed-form expression for the multicast outage performance can be derived as follows.

###### Proposition 1.

For the case with perfect CSI, namely , the multicast outage probability can be exactly obtained as

 (P1out)σ2ζ=0=1−[2η(ϵMρ)2ηD2γ(2η,ϵMDηρ)]K, (18)

where is a lower incomplete gamma function.

###### Proof.

See Appendix C. ∎

###### Remark 5.

For the perfect CSI, the complexity of obtaining the exact closed-form expression for in (18) is , which is acceptable for large .

###### Remark 6.

According to in[26], namely , when , can be expressed as

 \footnotesize(P1out)∞σ2ζ=0≈1−W(η), (19)

where is a positive constant related to . From (19), we can conclude that even for the perfect CSI, there is also no diversity gain, which accords with our system design, that is all the users independently decode the and .

###### Theorem 3.

As to the average secrecy unicast throughput , it can be approximated as

 RUS,1 ≈(1−P1out)Kπρ2mln2m∑u=1|sin2u−12mπ| (20) ⎧⎨⎩1ρτu−∑r0+r1+⋯+rn=K−1A(r1,⋯,rn)H(r1,⋯,rn,u)⎫⎬⎭,

where

 A(r1,⋯,rn)=(K−1)!(K−1−∑nt=1rt)!r1!⋯rn!(−πnD)r1+r2+⋯+rn n∏t=1(|sin2t−12nπ|xt)rt,
 τu=12(cos2u−12mπ+1),xt=D2(1+cos2t−12nπ),
 H(r1,⋯,rn,u)=−πnDρτun∑t=1|sin2t−12nπ|xtG(r1,⋯,rn,u,t) +I2(r1,⋯,rn),
 I2(r1,⋯,rn)={1ρτu,r1=r2=⋯=rn=00,othwise.
 G(r1,⋯,rn,u,t)=1−B(r1,⋯,rn,u)ρτu¯μ1+ νeν¯μ1Ei(−ν¯μ1)[¯μ1−B(r1,⋯,rn,u)ρτu],
 B(r1,⋯,rn,u)=τu(n∑t=1rtx−ηt−σ2ζ),
 ¯μ1△=μ1(r1,⋯,rn,u,t)=B(r1,⋯,rn,u)+1x−ηt−σ2ζρτu, (21) ν=1+ϵM.

Note that , and has been derived in (16). Similarly, and controls the approximation accuracy due to the use of Gauss-Chebyshev integration[25].

###### Proof.

See Appendix D. ∎

###### Remark 7.

The computational complexity of can be analyzed as follows. Note that the complexity of calculating is and the possibilities of is . Next, the complexity of calculating is , while that of obtaining is . Hence, the complexity of can be easily achieved as . Here an important tradeoff between the accuracy and computational complexity should be carefully considered. Note that the complexity dramatically increases with due to . However, a small will lead to a large estimation gap. According to our simulation results, is proper as both the complexity and the accuracy are acceptable.

###### Remark 8.

Although the average secrecy unicast throughput is derived under high SNR, numerical results show that (20) is also accurate for low and moderate SNR.

###### Remark 9.

For the case with perfect CSI, namely , unfortunately the exact closed-form expression for can not be achieved because the integration over lower incomplete gamma function is difficult to obtain.

### Iii-C Performance analysis for SOS-based CSI

In this subsection, the multicast outage and secrecy unicast throughput performance for the SOS-based CSI are analyzed in detail.

Here the BS only knows the distance information. Without loss of generality, we assume that . The meaning of this case is that only the distance can be available in certain practical communications. As the channel gain is modeled as , we can find that are not necessarily ordered, that is might be larger than for . However, the channel gain is mainly dominated by the large-scale fading , which is also the motivation why we order the users according to their distances.

As to the optimization problem (9), since the perfect channel gains are not available at the BS, BS should set .

Next, the multicast outage probability and statistical expectation of secrecy unicast throughput will be analyzed as follows.

###### Theorem 4.

As to , the exact closed-form expression can be achieved as

 P2out=1−K∏k=1 (22) ⎡⎢ ⎢ ⎢⎣2k(Kk)K−k∑j=0(K−kj)(−1)jD2(k+j)(ϵMρ)−2(k+j)ηηγ(2(k+j)η,ϵMDηρ)⎤⎥ ⎥ ⎥⎦.

Note that is a lower incomplete gamma function.

###### Proof.

See Appendix E. ∎

###### Remark 10.

It is easy to obtain that the complexity of is . Compared to (18), the complexity of achieving the multicast outage probability in SOS-based CSI is larger than that of perfect CSI, and it is also larger than that of imperfect CSI (shown in (16)) in our simulation settings.

###### Remark 11.

The multicast outage expression (22) is accurate for whole SNR, which is similar to that in imperfect CSI.

###### Remark 12.

Under high SNR, namely is large enough, according to in[26], can be expressed as

 (P2out)∞≈1−Z(η), (23)

where is a positive constant related to . Therefore, we can conclude that there is no diversity gain for SOS-based CSI, which is similar to that in imperfect CSI.

As to the average secrecy unicast throughput , due to that BS only knows the distance information, we can not guarantee that channel gains are sorted according to their distances. For example, there are two users and in the cell, with . It is possible that . Note that the achieved security throughput depends on the order of joint channel gain. Hence, in general, when , it is difficult to obtain a closed-form expression for . In this paper, we only focus on the special case of .

###### Theorem 5.

When , can be approximated as

 RUS,2 =4πlηD4ln2l∑i=1|sin(2i−12lπ)|κ2η−1i (24) {J(i)lnν−ln(ν+ϵM)γ(4η,(κi+1)ϵMDηρ)η(κi+1)[(κi+1)ϵMρ]4/η +Dπ2qq∑j=1|sin(2j−12qπ)|xj¯J(i,xj)},

where

 J(i)=D42e−ϵMDηρκi−γ(4η,ϵMκiDηρ)η[ϵMκiρ]4/η+D44(κi+1)
 ¯J(i,x)=−eνxηρEi(−νxηρ)(D2e−ϵMDηρκi−x2e−ϵMxηρκi) −x2κi+1eν(κi+1)xηρ[Ei(−ν(κi+1)xηρ)−Ei(−(ν+ϵM)(κi+1)xηρ)] −D2[xηlnνDηκi+xη−xηDηκi+xηeν(Dηκi+xη)ρEi(−ν(Dηκi+xη)ρ)],
 κi=12(1+cos(2i−12lπ)),xj=D2(1+cos(2j−12qπ)). (25)

Note that , , and is a lower incomplete gamma function. Here and are the number of terms included in the summation, which controls the approximation accuracy, due to the use of Gauss-Chebyshev integration[25].

###### Proof.

See Appendix F. ∎

###### Remark 13.

The computational complexity of can be easily obtained as , which is smaller than that of in our simulation settings.

###### Remark 14.

Although the average secrecy unicast throughput is obtained under high SNR, simulation results show that (24) is also accurate for low and moderate SNR.

### Iii-D Performance analysis in OMA systems

In this subsection, the multicast outage and average secrecy unicast throughput achieved by the OMA scheme with partial CSI are correspondingly derived. Note that the OMA scheme is provided as a benchmark in our paper. As to the OMA scheme, a frame is equally divided into time slots, while the first one is used for the multi-cast traffic only, and the second one is singly for the unicast traffic. Then the achievable multicast rate in user can be expressed as

 RMk=12log2(1+αkρ), (26)

where denotes the time slot ratio for the multicast traffic. In the following, two partial CSI cases, namely imperfect CSI and SOS-based CSI, are carefully considered.

#### Iii-D1 Imperfect CSI

The multicast outage of the OMA scheme can be denotes as

 (P1out)OMA=1−Pr{^αk≥λMρ,∀k∈K}, (27)

where . Therefore, can be easily approximated as

 (P1out)OMA=1−[1−F^α(λMρ)]K (28) (a)≈1−[πcDc∑i=1|sin2i−12cπ|xiexp(−λM/ρx−ηi−σ2ζ)]K,

where derivation results from (41).

###### Remark 15.

The computational complexity of is , which is same with that of . As and , then , namely the NOMA scheme achieves a lower multicast outage probability (better outage preformance) than the OMA scheme.

###### Remark 16.

As to the perfect CSI, namely , can be exactly obtained as

 (P1out)OMAσ2ζ=0=1−[2η(λMρ)2ηD2γ(2η,λMDηρ)]K. (29)
###### Remark 17.

Similarly, there is no diversity gain for both the imperfect CSI OMA and perfect CSI OMA schemes.

As to the average security rate , it can be easily expressed as

 (RUS,1)OMA=12E{log2(1+^απ1ρ)−log2(1+^απ2ρ)}. (30)

In the similar way of calculating in (47), can be approximated as

 (RUS,1)OMA≈Kπρ4mln2m∑u=1|sin2u−12mπ| (31) ⎧⎨⎩1ρτu−∑r0+r1+⋯+rn=K−1A(r1,⋯,rn)¯H(r1,⋯,rn,u)⎫⎬⎭,

where

 ¯H(r1,⋯,rn,u)=−πnDρτun∑t=1|sin2t−12nπ|xt¯G(r1,⋯,rn,u,t) +I2(r1,⋯,rn),
 ¯G(r1,⋯,rn,u,t) =1−B(r1,⋯,rn,u)ρτu¯μ1 (32) +e¯μ1Ei(−¯μ1)[¯μ1−B(r1,⋯,rn,u)ρτu].
###### Remark 18.

The complexity of is , which is close to that of . Meanwhile, is accurate for whole SNR range.

#### Iii-D2 SOS-based CSI

Similarly, The multicast outage of the OMA scheme can be exactly obtained as

 (P2out)OMA=1−Pr{αk≥λMρ,∀k∈K}(a)=1−K∏k=1 (33) ⎡⎢ ⎢ ⎢ ⎢⎣2k(Kk)K−k∑j=0(K−kj)(−1)jD2(k+j)(λMρ)−2(k+j)ηηγ(2(k+j)η,λMDηρ)⎤⎥ ⎥ ⎥ ⎥⎦,

where derivation results from (58).

###### Remark 19.

The computational complexity of is , which is same with . Note that is accurate for whole SNR range. Meanwhile, under high SNR condition, we can find that there is also no diversity gain.

As to the average security rate , we also consider the special case of , that is

 (RUS,2)OMA=12E[log2(1+ρ|g1|2dη1)−log2(1+ρ|g2|2dη2)||g1|2dη1≥|g2|2dη2]. (34)

In the similar way of deriving , can be calculated as

 (RUS,2)OMA=πqD3ln2q∑j=1|sin(2j−12qπ)|xj¯Jo,1(xj)+π2qlηD3ln2 (35) l∑i=1|sin(2i−12lπ)|κ2η−1i{q∑j=1|sin(2j−12qπ)|xj¯Jo,2(i,xj)},

where

 ¯Jo,1(x) =−exηρEi(−xηρ)(D2−x2) (36) ¯Jo,2(i,x) =D2xηDηκi+xηe(Dηκi+xη)ρEi(−(Dηκi+xη)ρ)].
###### Remark 20.

The computational complexity of is , which is close to that of . Meanwhile, is accurate for the whole SNR range.

## Iv Numerical Results

In this section, numerical results are presented to validate the analytical expressions derived in this paper. In particular, the simulation parameters are set as follows. The disk radius is , the path loss factor is , and the number of users is . Meanwhile, as to the number of included terms in the Gaussian-Chebyshev integration approximation, we set . The channel gain model has been discussed in subsection II-A, and the channel estimation error in imperfect CSI is valued as in general if it is not specially pointed out. Note that all the numerical results are acquired by averaging over random channel gains realizations. Finally, in the following figures, “sim” and “cal” denote simulation results and derived analytical expression values, respectively.

Fig. 1 investigates the analytical performance with imperfect CSI as a function of the SNR , where and are considered. On the one hand, the multicast outage probability is shown in , and we can find that the derived expressions, namely (16) for the NOMA scheme and (28) for the OMA scheme, match exactly with the Monte Carlo simulations. Furthermore, as increases, the multicast outage probability correspondingly decreases, that is the outage performance improves, which accords with common sense. In addition, a large will lead to a large