## I Introduction

Non-orthogonal multiple access (NOMA) is being considered as one of the enabling technologies for the fifth generation (5G) wireless networks. With its two general power- and code-domain forms, NOMA can potentially pave the way toward higher throughput, lower latency, improved fairness, higher reliability, and massive connectivity [1]. Motivated by these fascinating advantages, extensive research activities have been carried out in the past few years to promote the NOMA advancement in diverse directions (see, e.g., [2] for a comprehensive survey).

In a variety of applications, there is a need to transmit the users’ data to a central unit (CU) or a wired base station (BS) while, given the limited power of the users, it is not feasible for the users to directly communicate with the relatively far destination. Motivated by this fact, several recent work have dealt with the relaying problem in downlink and uplink NOMA communications. In particular, capacity analysis of a simple cooperative relaying system, consisting of a source, a relay, and a destination is provided in [3]. The outage probabilities and ergodic sum rate of a downlink two-user NOMA system, with a full-duplex relay helping one of the users, is characterized in [4]. Performance of downlink NOMA transmission with an intermediate amplify-and-forward (AF) relay for multiple-antenna systems, and over Nakagami- fading channels is investigated in [5] and [6], respectively. The performance of coordinated direct and relay transmission for two-user downlink and uplink NOMA systems is investigated in [7] and [8], respectively. Hybrid decode-and-forward (DF) and AF relaying in NOMA systems is proposed in [9], and forwarding strategy selection is explored in [10]. Additionally, in order to enable NOMA technology for massive communications, primary work on power-domain NOMA can be mixed with the low-complexity recursive approach proposed in [11] based on the Kronecker product of NOMA pattern matrices.

All of the aforementioned work consider sub-6 GHz radio frequency (RF) band for the backhaul link through an AF or DF relay in the absence of any external multiuser interference to the NOMA users. However, the scarce available bandwidth in the sub-6 GHz band will not be able to support the users’ aggressive demand for the higher data rates, especially when NOMA is employed in the users-relay access links to provide higher throughput for the users. In this case, the relay-destination backhaul link can pose a severe bottleneck on the end-to-end performance and substantially negate the NOMA advantages through reducing the users’ achievable throughput and reliability which can in turn even increase their latency.

A potential approach to overcome the aforementioned drawback is moving to higher frequency bands, e.g., through the deployment of millimeter-wave [12] or free-space optics (FSO) backhaul links [13]. Millimeter-wave communication is usually preferred for relatively shorter communication lengths due to the severe propagation conditions at millimeter frequencies [14]. FSO links, on the other hand, can provide much more available bandwidth and support ranges on the order of several kilometers [15].

In this paper, we investigate the outage probability performance of uplink NOMA transmission over mixed RF-FSO systems when an AF relay is employed to forward the amplified received signal from the Rayleigh fading access links to the destination through an ultra-high-throughput directive interference-free FSO link subject to Gamma-Gamma (GG) fading with beam misalignment error. We consider a two-user uplink NOMA system where the communication is subject to the presence of multiuser interference from some independent users. Such interference can be induced, e.g., due the co-channel interference from some nearby users aiming to communicate with some other relays or destinations. We apply dynamic-order decoding, also employed very recently in [16, 17], to dynamically determine the detection order of the NOMA users at the destination, and then derive the closed-form expressions for the individual- and sum-rate outage probabilities. This paper can be considered as an initial attempt to incorporate power-domain NOMA in mixed RF-FSO systems.

The rest of the paper is organized as follows. In Section II, we describe the system model. In Section III, we derive the individual- and sum-rate outage probability closed-form formulas for uplink NOMA over mixed-RF-FSO system. Section IV provides the numerical results, and Section V concludes the paper.

## Ii System Model

Consider two RF users and grouped together for uplink NOMA transmission to an AF relay . Denote the composite channel gain of the link by , , where and are respectively the path loss gain and independent-and-identically-distributed (iid) Rayleigh fading coefficient of the RF link given by [18, Eq. (2)]. Furthermore, assume that the uplink transmission to the relay is affected by undesired multiuser interference from interfering users , , each with the transmit power , path loss gain , and iid Rayleigh fading coefficient . This interference can be from the users scheduled for the concurrent transmission to some other relays in the cellular network or any other non-vanishing interference during the desired transmission block. The received signal by the relay is then expressed as

(1) |

where and are the transmit symbols by and , respectively, and

is the additive white Gaussian noise (AWGN) of the relay receiver with mean zero and variance

. Moreover, and are the power allocation coefficients determining the portion of the total power assigned to each of the desired users. Note that for the users with independent Rayleigh fading, all fading gains ’s and’s have an exponential distribution with mean one (to ensure that fading neither amplifies nor attenuates the received power) as

, . Note that, in this paper, for the sake of simplicity, we do not consider the stochastic geographical positions of the nodes, and deal with the building part of the analysis, corresponding to a given time slot, where the nodes are at some fixed positions.The received signal at the relay is then converted to optical signal using intensity-modulation direct-detection (IM/DD), and is amplified with a constant gain to keep the disparity between the power level of different NOMA users for successive interference cancellation (SIC) detection at the destination. In this case, the transmitted optical signal by the relay toward the destination can be expressed as , where is the electrical-to-optical conversion coefficient [19]. The transmitted signal then undergoes the composite FSO channel gain of where is the path loss gain of the FSO backhaul link, with length , defined as where is the responsivity of the photodetector, and is the weather-dependent attenuation coefficient [18]. Moreover, is the total fading coefficient due to pointing error and optical turbulence . In the case of GG optical turbulence with beam misalignment, the distribution of can be expressed as [20]

(2) |

where and are the fading parameters of the GG distribution,

is the ratio of the equivalent beam radius and the pointing error displacement standard deviation (jitter) measured at the receiver,

is the Gamma function [21, Eq. (8.310)], and is the Meijer’s G-function [21, Eq. (9.301)]. Furthermore, is the geometric loss in the case of perfect beam alignment (zero radial displacement) defined as in which is the error function, is the receiver aperture radius, and is the transmitter beam divergence angle. The destination then filters out the direct current (DC) component of from to obtain the received signal as(3) |

where is the destination AWGN with mean zero and variance .

We assume that the NOMA users are indexed based on their path loss gains, i.e., , and the power allocation strategy proposed in [22] is adopted to determine and as where is the power back-off step; hence, , and . We further assume that the BS has perfect knowledge about the channel state information (CSI) and orders the NOMA users based on their instantaneous received power. In fact, based on the principles of uplink power-domain NOMA [22, 23], the BS orders the users based on their channel conditions from best to worst. Such a dynamically ordering the users can potentially prevent the possibility of firstly decoding the users with worse instantaneous channel conditions if they are fixedly ordered based on their channel statistics. Therefore, depending on the fading coefficients and , the detection order is either , meaning that the first user is decoded first, if or if otherwise.

## Iii Outage Probability Analysis

In this section, we characterize the individual- and sum-rate outage probability formulas for dual-hop uplink NOMA over mixed RF-FSO systems.

### Iii-a Individual-Rate Outage Analysis

(7) |

Note that if the detection order is , the SIC receiver first treats the signal from the second NOMA user as noise to decode with the signal-to-interference-plus-noise ratio (SINR) of

(4) |

and then, after removing the received power from the first user, decodes with the SINR of

(5) |

where . Similarly, when the detection order is the SINR values and can be obtained by appropriate change of indexes in (4) and (5).

The outage probability of the first user in achieving an individual rate of can be characterized as

(6) |

where step

follows from the law of total probability by defining

, , as the conditional outage probability of the first NOMA user given the decoding order . Moreover, in step , , and are the probabilities of having decoding orders , and , respectively. Furthermore, step follows, first, by defining , , as the probability of successfully achieving for conditioned on the decoding order , and then noting that correct detection of for the decoding order requires successful decoding of the preceding symbol , i.e., and where is the threshold SINR for an IM/DD FSO link to achieve the desired data rate , . In the following, we calculate the three joint probabilities in (III-A) to ascertain the outage probability of the first user .In order to calculate we first note that . Then using (4), can be calculated as (III-A) shown at the top of this page where, in step , is the sum of the power of multiuser interference and noise, at the relay, normalized to the average power of the first NOMA user. Moreover, step follows, first, by defining the constant for , and then noting that

for three random variables (RVs)

, , and can be calculated using the law of total probability as since . Finally, step follows by noting that for any constant(8) |

where step follows from the cumulative density function (CDF) of exponential distribution , and is obtained using the independency of and , and then applying the independency among ’s to get . Furthermore, in step , and .

We should further emphasize that (III-A) is obtained for . If , the upper limit of in (III-A) is always greater than its lower limit meaning that the condition holds for the all values of and there is no need to impose such an extra condition on the calculation of the corresponding probability. As a consequence, for can be calculated as (9) shown at the top of the next page.

(9) |

(10) |

(11) |

(12) |

Finally, using (III-A) for or (9) for , one can obtain the coverage probability of the first NOMA user given the decoding order as . However, the closed-form characterization of still requires the calculation of expressions of the form , where is a constant and is distributed according to (2). To do so, we first apply [24, Eq. (11)] and [21, Eq. (9.31.2)] to write in the form of Meijer’s G-function as . Then we can apply [24, Eq. (21)] to calculate the infinite integral of product of Meijer’s G-functions involved in as (10) shown at the top of the next page. For the ease of notation, hereafter, we denote by for any constant .

Similarly, the second probability term in (III-A) can be obtained, first, by writing . Then using the symmetry of the problem, it can be shown that for and can be obtained as (III-A) and (12), respectively, shown at the top of the next page, where is defined in (10), and is defined for . Also, and .

(13) |

Moreover, the last probability term in (III-A) can be calculated by first writing

(14) |

Then using a similar approach to (III-A), the closed-form expression for all values of can be expressed as (13) shown at the top of this page. This completes the closed-form characterization of the outage probability of the first NOMA user .

Finally, the outage probability of the second NOMA user can be characterized as

(15) |

where and have already been calculated, and can be obtained as

(16) |

This means that the outage probability of the second NOMA user can be characterized using the preceding analysis by substituting for and appropriate change of indexing . This is because the only difference between and is that the user with a lower average gain is labeled as the second user, i.e., .

### Iii-B Sum-Rate Outage Analysis

Assuming that the data rate of the -th NOMA user for the -th decoding order is related to the corresponding SINR as , , then it is easy to verify that the sum rate of the NOMA users, regardless of their decoding order, can be expressed as

(17) |

Denoting the fractional term of the logarithm argument in (17) by , the sum-rate outage probability defined as , where is the threshold equivalent SINR to achieve the desired sum-rate of , can be expressed as

(18) |

Let represent the event , and

(19) |

i.e., the sum-rate outage event defined in (III-B). Clearly, where is the complementary event of . Therefore, using the law of total probability, can be expressed as which is calculated in a closed-form as (III-B) at the top of the next page.

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