Outage Performance in Secure Cooperative NOMA

02/03/2019 ∙ by Milad Abolpour, et al. ∙ Sharif Accelerator 0

Enabling cooperation in a NOMA system is a promising approach to improve its performance. In this paper, we study the cooperation in a secure NOMA system, where the legitimate users are distributed uniformly in the network and the eavesdroppers are distributed according to a homogeneous Poisson point process. We consider a cooperative NOMA scheme (two users are paired as strong and weak users) in two phases: 1) Direct transmission phase, in which the base station broadcasts a superposition of the messages, 2) Cooperation phase, in which the strong user acts as a relay to help in forwarding the messages of the weak user. We study the secrecy outage performance in two cases: (i) security of the strong user, (ii) security of both users, are guaranteed. In the first case, we derive the exact secrecy outage probability of the system for some regions of power allocation coefficients and a lower bound on the secrecy outage probability is derived for the other regions. In the second case, the strong user is a relay or a friendly jammer (as well as a relay), where an upper bound on the secrecy outage probability is derived at high signal-to-noise-ratio regimes. For both cases, the cooperation in a two-user paired NOMA system necessitate to utilize the joint distribution of the distance between two random users. Numerical results shows the superiority of the secure cooperative NOMA for a range of the cooperation power compared to secure non-cooperative NOMA systems.

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I Introduction

RECENTLY, non-orthogonal multiple access (NOMA) systems have been popular in fifth-generation (5G) networks due to their high power spectral efficiency. In NOMA systems, users are classified into orthogonal multiple access (OMA) groups. In each group, by accommodating several users within the same resource blocks, such as frequency and time, the significant bandwidth and also the latency of users are decreased. Base station (BS) superimposes the messages and the stronger user exploits successive interference cancelation (SIC)

[1]. Ding et al. in [2], investigated the performance of a NOMA system with random deployed users. In this scheme, outage probability (OP) was used to demonstrate that under a condition on power allocation coefficients and users’ targeted data rates, NOMA can achieve a diversity order as an orthogonal multiple access system and in the case of ergodic sum rates, NOMA has a better performance than OMA systems.

In NOMA systems, existence of weak users degrades the outage performance of the system. Exploiting device-to-device (D2D) transmission capability of 5G users, enabling cooperative NOMA, enhances the efficiency of NOMA systems. Effects of the cooperative transmission in NOMA, by applying outage probability as a metric, was investigated by Ding et al. [3], where demonstrated that due to the complexity limitations of the system, utilizing all the users in a cooperative NOMA system is not an efficient way but pairing users with more distinctive channel coefficients provides higher gain.

In wireless networks, signals are transmitted in open to all network users, so the security of the users must be provided. Using the physical layer capabilities is a promising way to maintain the security of the users. Signals are overheard by external or internal eavesdroppers. Secrecy performance of a cellular NOMA network in cases of single and multiple-antenna BS was investigated in [4], where a random number of external passive eavesdroppers are distributed according to a homogeneous Poisson point process (PPP) across a circular area. Lei et al. investigated the secrecy outage probability (SOP) of a NOMA system containing two users, multiple-antenna BS and an external eavesdropper, who overhears only one of the users [5]. The case of internal eavesdroppers in a cooperative NOMA system is studied in [6].

Although cooperation in NOMA systems enhances the outage performance, it gives more opportunity to the eavesdroppers to overhear the messages of the weaker users in the network. Chen et al. studied the secrecy of a cooperative NOMA system with a Decode-and-Forward (DF) and an Amplify-and-Forward (AF) relay in existence of one eavesdropper. It was shown that at high signal-to-noise -ratio (SNR) regimes, DF and AF relays have the same performances [7]. Zheng et al. investigated the secrecy in a network consists of two users, a relay and some eavesdroppers [8], in which the relay transmits the messages and generate an artificial noise in order to decrease the SNR of the eavesdroppers to decode the messages of the legitimate users (LUs).

Though using external relay nodes in order to realize cooperation in secure NOMA systems has been studied in some works [7-8], this cooperation is also possible by using internal nodes, where the strong user acts as a relay to help in forwarding the signal of the weak user as studied in [9] for a simple network with a Base station, two LUs and one eavesdropper at high SNR regimes [9].

In this paper, we investigate the secrecy performance of a cooperative NOMA network with many LUs in existence of a random number of external passive eavesdroppers. We assume that every LUs are paired randomly (called strong and weak users) and we analyze the secrecy performance of one pair. We consider two cases: the security is provided (1) for the message of the strong user, when the targeted data rate of the strong user is greater than the weak user; (2) for the messages of both users. In case 1, we derive a lower bound on the SOP, which is tight in some regions of the power allocation coefficients and users’ targeted data rates. In case 2, we derive an upper bound on the SOP of the system at high SNR regimes. In this case, we propose two strategies: in the first strategy the strong user acts a relay and allocates all of its transmitting power for sending the message of the weak user, while in the second strategy, the strong user acts as a friendly jammer (as well as a relay), where it allocates a proportion of its transmitting power to send noise and the rest of its transmitting power is allocated to send the message of the weak user.

Notation: and

are the cumulative distribution function (CDF) and the probability density function (PDF) of random variable

, respectively. is the complementary event of the event , where , is the gamma function, where and is the upper incomplete gamma function, where .

Ii System Model

We consider as the eavesdropper-free zone with radius , and as the user zone with radius to , and as the eavesdropper zone with radius to , as depicted in Fig. 1. Our system consists of a single antenna base station which is located at the center of the , LUs distributed uniformly in the user zone and a random number of the eavesdroppers distributed according to a homogeneous PPP, which is denoted by with the density , in the eavesdropper zone. The system model resembles the one in [4].

Channel state information (CSI) of each LU is known at the BS, the eavesdroppers and the other LUs but CSI of the eavesdroppers are unknown at the BS and LUs.

Fig. 1: Network model for secure cooperative NOMA transmission

All channels assumed to experience quasi-static Rayleigh fading, where the channel coefficients are constant for each transmission block but vary independently between different blocks. LUs are ordered according to their channel coefficients as , where , in which denotes the Rayleigh fading channel gain between user and the , denotes the distance between user and the and also is the path-loss exponent. In this model, every two users are paired randomly with each other and they make cooperative NOMA systems and we investigate the secrecy performance of one pair. We remark that each resource block is devoted to two paired users based on the NOMA scheme. Transmitting the messages of the users contains two phases: 1) Direct transmission phase, 2) Cooperation phase.

Consider a strong user, , is paired with a weak user, . We study two different cases. First, the message of is sent securely and second, the messages of both users are sent securely.

Ii-a Direct Transmission Phase

In this phase, the broadcasts a linear combination of the messages of and with total power of as:

where denote the power allocation coefficients for and , respectively and are the messages of and , respectively. By following the NOMA protocols, we assume that and .

The received signals at LUs and the eavesdroppers are:

where

is a zero-mean additive white Gaussian noise (AWGN) with variance

, denotes the channel coefficient, in which is the fading channel gain between the and the eavesdroppers and is the distance between the and eavesdroppers. At the end of this phase, performs SIC and decodes its own message. The SNR of the and the maximum SNR of an eavesdropper to decode are shown in the following:

Ii-B Cooperation Phase

Case 1: Maintaining Secrecy at the Stronger User

If is not able to perform SIC, system stops working. When is able to decode , this phase starts and transmits the message with the power . Therefore, the transmitted signal of is shown below:

(1)

The Received signal at equals to:

(3)

where is the channel coefficient between and

with exponential distribution with parameter

, denotes the distance between and and is a zero-mean AWGN with variance . uses the maximum ratio combining (MRC) receivers [10] to decode , therefore the signal-to-noise-plus-interference (SINR) of to decode equals to:

where , , and .

Case 2: Maintaining Secrecy at both Users

In this subsection, we study the secure transmission of the messages of both users. We consider two strategies in which acts as a relay or acts as a friendly jammer and a relay, simultaneously (FJR). and the eavesdroppers are using the MRC receivers to decode .

is a relay: In this phase, acts as a relay and allocates all of its transmitting power () to send the message of . So the transmitted signal of is as (1) and the received signals at the eavesdroppers and are as:

where and is a zero-mean AWGN with variance , , such that and denote the channel gain and distance between and the eavesdroppers, respectively. Now we write the maximum SINR of an eavesdropper and the SINR of to decode as shown in the following:

where , , .

is a FJR: In this phase, allocates a proportion of its transmitting power for sending the message of and the rest of the transmitting power is allocated for sending the noise-like signal. So the transmitted power by is as:

where denotes the noise-like signal with power 1. Therefore, the received signals at the eavesdroppers and is obtained as:

where and is a zero-mean AWGN with variance . Now we write the maximum SINR of an eavesdropper and the SINR of as obtained:

where .

Iii Secrecy Performance Analysis

We use SOP as the metric to evaluate the secrecy performance of the system for both cases. For simplicity, we assume that .

Case 1: Maintaining Secrecy at the Stronger User

In this subsection, only the security of the message of is provided , where the targeted data rate of is greater than the targeted data rate of , i.e. , . As described, when is able to decode , we have a cooperative NOMA system, otherwise system goes to outage, i.e. , SOP=1. The outage event occurs if can not decode or can not decode securely or can not decode . So we write the outage event as:

where , and .

Theorem 1

If , the SOP of the system is as:
Condition 1) When , then:

Condition 2) When , then:

where

Proof:

The proof is provided in Appendix A.

Remark 1

In NOMA systems, must be less than so that the strong user can perform SIC.

In the following, first we derive .

(2)

where holds due to the independence of and . For calculating and we follow a similar approach as [4].

is written at the bottom of this page in (3), where and

The is obtained as [4, eq(4)] :

(4)
(5)
(8)

where is a parameter defined to guarantee the complexity-accuracy trade off, and , and Now by substituting (3) and (4) into (2), is derived as illustrated at the bottom of this page in (III).

The last step for finding the SOP is deriving The is obtained as [4, eq(5)].

where , and

According to [11], is a random variable with the probability density function as:

(6)

where . By using Gaussian-Chebyshev quadrature method, and are derived as the method in [2]. The details of the proof are provided in Appendix B.

(7)

where and . Due to the independence of the and , is derived as the bottom of this page in (III).

Case 2: Maintaining Secrecy at both Users

In this subsection, the messages of two users are sent securely, where the targeted data rates of and are and , respectively. The only effect of cooperation on is to divide the transmission duration by 2, thus in this subsection, we investigate the SOP of only and for avoiding untractable calculations, we derive an upper bound on the SOP of at high SNR regimes as [7] and [9], where . We remark that at high SNR regimes, on the condition , is able to carry out the SIC with probability one. So the SOP of the system is obtained as:

(9)

where equals to which is derived in (III). So we write at high SNR regimes as:

where and is used to differentiate between two cases, the relay (shown by ) and the FJR (shown by ).

Iii-1 is a relay

In this subsection, acts as a relay and allocates all of the to send . SOP of is as the following lemma. (Proof is provided in Appendix C.)

Lemma 1

Let , if , then , otherwise we have:

Now we calculate the and in order to derive the , by following a similar way as [4].

Lemma 2

Let . The and are as following: (Proof is provided in Appendix D.)

Iii-2 is a FJR

In this subsection, we investigate the at high SNR regimes, while is acting as a friendly jammer and a relay, simultaneously. So we derive the by using the following lemma. (Proof is provided in Appendix C.)

Lemma 3

If , then otherwise we have:

Now we find the terms of the as obtained in the following.

Lemma 4

Let , the and equal to: (Proof is provided in Appendix E.)

and is written at the bottom of this page in (9).

By following a similar approach as (7) and using Gaussian-Chebyshev quadrature method, the is derived as:

Iv Numerical Results

Number of iterations in Monte Carlo simulations
The radius of the eavesdropper zone
The radius of the user zone
The radius of the eavesdropper-free zone
Path-loss exponent
Order of the users
Users’ targeted data rates
The power allocation coefficients
The density of the eavesdroppers
The complexity-vs-accuracy coefficient
TABLE I: Parameters Of The Simulations
Fig. 2: SOP of the system versus for different values with: and

In this section, we present numerical and simulation results, where the parameters of the simulations are shown in the Table I.

Case 1: Maintaining Secrecy at the Strong User

Fig. 2 illustrates that SOP of the system decreases by increasing the radius of the eavesdropper-free zone , thanks to increasing the distance of the eavesdroppers. Also as expected, by increasing SOP of the system increases and the simulation results confirm our analytical results.

Fig. 3: Comparison of the SOP of the cooperative NOMA systems with the eavesdroppers (Sec+Coop+NOMA), cooperative NOMA systems without the eavesdroppers (Coop+NOMA) and NOMA systems with the eavesdroppers (Sec+NOMA).

Fig. 3 shows the comparison of NOMA systems and cooperative NOMA systems with and without the eavesdroppers. We see that at high SNR regimes, the SOP of the system doesn’t vary by increasing . Since at high SNR regimes, the outage probability of the weak user is almost zero and the SOP of the system equals to the SOP of the strong user which is almost independent of and only depends on and . This is due to the increase occurred in by increasing . So it depends on the users’ targeted data rates and power allocation coefficients in order to determine the usefulness of the cooperation. AS observed in Fig. 3, for lower , Coop+NOMA+Sec outperforms NOMA+Sec and thus cooperation is beneficial, while for high using the cooperation degrades the performance of the system. This is due to the half duplex property of the relay, in which half of the time resource is allocated to relaying in cooperative NOMA and thus the rate of the strong user is divided by 2.

Case 2: Maintaining Secrecy at both users

When we have secrecy at both users, the SOP of the strong user is the same as the case of the maintaining secrecy at the strong user and therefore we only investigate the SOP of the weak user, while the strong user is a relay or a FJR, at high SNR regimes . As Fig. 4 indicates, for , the SOP of goes to one. Since by increasing , the received power at the and eavesdroppers increases and therefore the eavesdroppers would be able to decode for with probability one. For the lower , the increment of the received power at is greater than the increment of the received power at the eavesdroppers, thus the SOP of the decreases by increasing . When is very close to zero, would not be able to decode the . Moreover, we see that for , the FJR strategy has a better secrecy performance than the relaying strategy for the high value of . Besides, at the low value of , it is better to choose the relaying instead of FJR strategy, for the given users’ targeted data rates and power allocation coefficients.

Fig. 4: SOP of the (weak user) for different values.

V Conclusion

We studied the secrecy performance of a cooperative NOMA system with many legitimate users in existence of a random number of external passive eavesdroppers in two cases: either security of the strong user or both users were provided, while the strong user was a relay or a friendly jammer. In case 1, we derived a lower bound on the SOP which is tight in some regions of the power allocation coefficients and users’ targeted data rates. In case 2, we derived an upper bound on the SOP of the system at high SNR regimes. Our results showed that the amount of power must be allocated to send jamming noise has an optimal value that might be derived in a future work.

References

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Appendix A Proof Of Theorem 1

First, we provide a lemma that we use it for deriving the SOP the system for the case of maintaining the secrecy at the strong user.

Lemma 5

If and , then we have

Proof:

We use proof by contradiction. Our contradiction assumption is :

which implies that:

(10)

Also, on the condition

which implies that:

(11)

By substituting (11) into (10) we have:

(12)

By using the assumptions and we know