## I Introduction

Since non-orthogonal multiple access (NOMA) serves multiple users at the same time and frequency resources, using NOMA in fifth-generation (5G) networks enhances the spectral efficiency. NOMA uses superposition at the base station (BS) and successive interference cancelation (SIC) at the strong users [1]. The outage probability (OP) of a randomly deployed users NOMA is studied in [2], which is shown that NOMA enhances the outage performance of the system in comparison with orthogonal multiple access systems. There exist many works that investigate the OP of the different types of NOMA systems such as cooperative NOMA in [3], relay assisted NOMA in [4, 5] and NOMA with energy harvesting in [6, 7]. Another system performance metric is the ergodic sum-rate, which is shown to be enhanced in a NOMA system in [2]. The ergodic rates of the users in different NOMA scenarios have been also studied in [8, 9].

As the messages are sent in the open medium to all users in wireless networks, the secrecy of the messages of the users must be provided against the internal and external eavesdroppers. Utilizing the physical layer capabilities of wireless networks is a promising way to enhance the secrecy performance of the system. Secrecy outage probability (SOP) of a NOMA system with the external eavesdroppers is studied in [10, 11]. SOP of the other various NOMA systems in existence of the external eavesdroppers have been investigated in different scenarios such as secure cooperative NOMA systems in [12, 13, 14] and relay assisted NOMA networks in [15, 16]. The ergodic secrecy rate of the users in NOMA systems with the external eavesdropper is also investigated in [17, 18, 19].

As users that share the same resource blocks in NOMA systems may not trust each other, at least some level of secrecy must be provided against the internal users. In power domain NOMA systems, the near users first carry out the SIC and decode the messages of the far users, then they decode their own messages. Therefore, they are aware of the messages of the far users, which means following the NOMA protocol forces the far users to trust the near users. But in some scenarios, it is necessary to maintain the secrecy at the near users against the far users. First in [20], the secrecy performance of a NOMA system with an internal eavesdropper is investigated, in which the SOP of the near user and the OP of the system are derived.

In this paper, we study the effects of the existence of both external and internal passive eavesdroppers on the secrecy performance of a NOMA system. We consider a system consisting of a single antenna base station, two legitimate users and an external passive eavesdropper. The users are called the near and far users according to their distances to the base station. The external eavesdropper is interested in overhearing the messages of both users. The far user trusts the near user in order to follow the power domain NOMA protocol, while it is interested in overhearing the message of the near user. We derive the ergodic secrecy rate of the users and the SOP of the system in a closed form. Finally, by ignoring the existence of the external eavesdropper, we reduce our system to the one in [20] and derive the ergodic secrecy rate of the near user and also the ergodic rate of the far user, not studied in [20]. Moreover, we provide the closed-form SOP of both users. Compared to the results of [20], which is the SOP of the near user and OP of both users, deriving the SOP of both users is more complex due to the correlation between the channel coefficients. In presence of the external eavesdropper, we use Gaussian-Chebyshev quadrature method to present an approximation for the SOP. Our simulations confirm the accuracy of this approximation.

Notation:

In this paper, we denote the cumulative distribution function and probability density function of a random variable

as and , respectively. shows the complementary event of event and is the exponential integral, where . Moreover, is the unit step function.## Ii System Model

Our system consists of a single antenna BS, an external passive eavesdropper, , and two legitimate users, and , while it is assumed that is nearer than to the BS. The channel between , and and BS are rayleigh fading channels with coefficients as , and , respectively. The , and are exponentially random variables distributed with parameter . Also, , and are the distances between , and to the BS, respectively. Secrecy at both users are guaranteed against the and also secrecy at the near user is provided against the .

As depicted in Fig. 1, BS with power , transmits a superposition of the messages of both users, called and . The allocated power coefficients to and are denoted as and , respectively. So the transmitted signal of the BS is as:

(1) |

By following the NOMA protocol, we allocate more power to the far user, and thus . At the receivers, and observe the signals as:

(2) |

where , the path-loss is denoted as and

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

decodes by and then by subtracting it from observes a signal as:(3) |

Simultaneously, decodes from , while it considers as noise.

The received signal at the external eavesdropper is as:

(4) |

where is the zero mean AWGN with variance one. first decodes from and then decodes from .

### Ii-a No external eavesdropper

As illustrated in Fig. 2, by ignoring the external eavesdropper and assuming , the far user overhears the message of the near user in a system as [20]. The transmitted signal of the BS is as (1) and the received signals of the near and far users are as (2). We note that in Fig. 1 and Fig. 2, , and are denoting the decoded messages at , and , respectively, where .

## Iii Secrecy Outage Probability

The secrecy of the message of the far user must be provided against the and also the secrecy of the message of the near user must be maintained against the and . We assume , in which and . By defining and as the targeted data rates of and , the SOP event occurs if either is not able to decode , i.e. ,

(5) |

or is not able to decode , i.e. ,

(6) | ||||

or is not able to decode its own message, i.e. ,

(7) |

Therefore, the SOP event is defined as:

(8) |

According to (6) and (7), in order to avoid occurring secrecy outage, if we have and , then we are sure that , which means:

(9) |

Here, we rewrite the SOP as the summation of the probabilities of two distinct events as:

(10) |

where and are defined at the bottom of this page in (11) and (12), respectively. According to (12), and , therefore and thus . The last step for deriving the SOP is calculating . By defining , and , we have:

(11) |

(12) |

(13) | ||||

By following (13), we know that and are greater than zero, therefore must be lower than , otherwise and SOP of the system becomes one. Now we rewrite (13) as:

(14) | ||||

First we derive in order to calculate . So we have:

(15) | ||||

By taking derivative of (15), is obtained as:

(16) | ||||

By substituting (16) into (14) and after some mathematical calculations, is obtained as:

(17) | ||||

where . We use Gaussian-Chebyshev quadrature method to approximate as:

(18) |

where , , and is the complexity-vs-accuracy coefficient. Finally by substituting (18) into (17) the SOP of the system is obtained as:

(19) |

### Iii-a No external eavesdropper

By assuming , we reduce our system to a system in [20]. We consider a more general definition on the SOP, which includes the secrecy outage probability of and total outage probability of the system, considered in [20].

## Iv Ergodic Secrecy Rate

In this section, we study the ergodic secrecy rates of the users. First we derive the achievable rate of each user without secrecy constraints and then we subtract the leakage rate from its achievable rate to obtain secure achievable rate, as:

(25) |

where and are the achievable and leakage rates of the user , . Finally, the expected value of presents the ergodic secrecy rate of the user , as:

(26) |

Since the message of is overheard by and , so the rate of the leakage information of is as:

(27) | ||||

Now, we find in order to derive the expected value of (27) as:

(28) | ||||

By taking derivative of (28), equals to:

(29) | ||||

Now is obtained as:

(30) | ||||

The achievable rate of is as:

(31) |

By following the approach of equations (27) and (28) in [19], and after some mathematical calculations, the expected value of achievable rate of equals to:

(32) |

Finally, by substituting (30) and (32) into (26), the ergodic secrecy rate of is obtained as:

(33) | ||||

The leakage rate of is:

(34) |

Therefore by following the approach of [19], the expected value of the leakage rate of the far user is obtained as:

(35) | ||||

The achievable rate of is the minimum of the achievable rate of the near user for performing SIC and the achievable rate of the far user in order to decode its own message, which means:

(36) | ||||

The ergodic rate of is the expected value of , which is shown in the following:

(37) | ||||

In order to calculate (37), we derive . So we have:

(38) | ||||

where (a) holds due to the independence of the channel coefficients and also we define as . So by taking derivative we have:

(39) |

After applying (37) and (39) and some mathematical calculations and following the approach of [19], the ergodic rate of is derived as:

(40) | ||||

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