## I Introduction

NOMA has recently been recognized as a promising multiple access technology for 5G wireless networks and beyond, as it can meet the ubiquitous and heterogeneous demands on low latency and high reliability, and can support massive connectivity by providing high throughput and better spectral efficiency [1]. It enables multiple users to simultaneously share a time slot, a frequency channel and/or a spreading code, via multiplexing them in the power domain at the transmitter and using successive interference cancellation (SIC) at the receiver to remove messages intended for other users.

An interesting application of NOMA for a power-domain multiplexed system using cooperative relaying in Rayleigh distributed block fading channels was proposed in [2], where the source was able to deliver two data symbols to the destination in two time slots with the help of a relay. The advantage of such a system can easily be seen in terms of throughput, compared to the conventional OMA relaying system where a single symbol is delivered to the destination in two time slots. In particular, closed-form expressions for the average achievable sum rate and for near-optimal power allocation were derived in [2]. A performance analysis of the CRS-NOMA system over Rician fading channels was presented in [3], where the authors developed an analytical framework for the average achievable sum-rate and also proposed a method to calculate the approximate achievable rate by using Gauss-Chebyshev integration.

In this paper, we investigate the performance of the CRS-NOMA system for the case when the relay and the destination are equipped with multiple receive antennas. We consider two different diversity combining techniques at the relay and destination receivers – SC and MRC. We derive closed-form expressions for the average achievable sum-rate and outage probability for the CRS-NOMA system for the cases of SC and MRC receivers. For the purpose of comparison, we present numerical results for the achievable rate of the CRS-OMA system with SC and MRC. In order to have a better insight into the system performance, we present the diversity analysis for the CRS-NOMA system and prove analytically that the system achieves full diversity order for both SC and MRC schemes.

## Ii System Model

Consider the CRS-NOMA model shown in Fig. 1, which consists of a source with a single transmit antenna, a relay with receive antennas and a single transmit antenna, and a destination with receive antennas. All nodes are assumed to be operating in half-duplex mode and all wireless links are assumed to be independent and Rayleigh distributed. The channel coefficient between the source and the relay antenna is denoted by and has mean-square value for any value of , while that between the source and the destination antenna is denoted by and has the mean-square value for any value of . Similarly, the channel coefficient between the relay and the destination antenna is denoted by and has mean-square value for any value of . Furthermore, it is assumed that the channels between the source and the destination are on average weaker than those between the source and the relay, i.e., .

In the CRS-NOMA scheme, the source broadcasts to both relay and destination, where and are the data-bearing constellation symbols which are multiplexed in the power domain , is the the total power transmitted from the source, and and are power weighting coefficients satisfying the constraints and . Upon reception, the destination decodes symbol treating interference from as additional noise, while the relay first decodes symbol and then applies SIC to decode symbol

. In the second time slot, the source remains silent and only the relay transmits its estimate of symbol

, denoted by , to the destination with full transmit power . In this manner, two different symbols are delivered to the destination in two time slots.In contrast to this, in the conventional OMA scheme, the source broadcasts symbol with power to both relay and destination in the first time slot and the relay retransmits the estimate of symbol , denoted by , to the destination in the second time slot. The destination then combines both copies of symbol and in this manner only a single symbol is delivered to the destination in two time slots.

## Iii Performance Analysis

In this section, we present the achievable sum-rate, outage probability and diversity analysis of the CRS-NOMA system with two different receive diversity combining techniques, namely SC and MRC.

### Iii-a Reception using SC for CRS-NOMA

The signal received at the relay (resp. destination) in the first time slot is given by

where (resp. ) and . Moreover,

denotes complex additive white Gaussian noise (AWGN) with zero mean and variance

. The received instantaneous signal-to-interference-plus-noise ratio (SINR) at the relay for decoding symbol and the instantaneous signal-to-noise ratio (SNR) for decoding symbol (assuming the symbol is decoded correctly) are and , respectively, where . Similarly, the received instantaneous SINR at the destination for the decoding of symbol is given by , where . In the next time slot, the relay transmits the decoded symbol to the destination with power . The received signal at the destination is given bywhere and is zero-mean complex AWGN with variance . The received instantaneous SNR at the destination while decoding the symbol is given by , where . Since the symbol should be correctly decoded at the destination as well as at the relay for SIC, the average achievable rate for the symbol is given by (c.f. [3])

(1) |

where is the transmit SNR, and

denotes the cumulative distribution function (CDF) of the random variable

.###### Theorem 1.

A closed-form expression for the average achievable rate for symbol in Rayleigh fading using SC in CRS-NOMA is given by

(2) |

where and denotes the upper-incomplete Gamma function.

*Proof*: See Appendix A.

###### Proposition 1.

A closed-form expression for the average achievable rate for symbol in Rayleigh fading using SC in CRS-NOMA is given by

(4) |

where .

###### Proof.

### Iii-B Reception using SC for CRS-OMA

The signal received at the relay (resp. destination) in the first time slot is given by

where (resp. ) and . In the next time slot, the relay forwards its estimate of , denoted by , to the destination. The signal received at the destination is given by

The average achievable rate for the symbol is given by (c.f. [5])

(7) |

where^{1}^{1}1The rate calculation is based on the assumption that the destination performs SC in the first and the second time slots and then applies MRC on the resulting signals from the two time slots. In the case where the destination applies SC on the resulting signals instead of MRC, will instead be defined as , and this will result in a performance degradation with respect to the system described here. and denotes the expectation with respect to the random variable . Since the focus of this paper is on the NOMA-based systems, we do not present a closed-form analysis for CRS-OMA.

### Iii-C Outage probability for CRS-NOMA using SC

In this subsection, we will characterize the outage probability of symbols and for the CRS-NOMA using selection combining in Rayleigh fading. We define as the outage event for symbol using SC, i.e., the event where either the relay or the destination fails to decode successfully. Hence the outage probability for the symbol is given by

(8) |

where is the instantaneous achievable rate of symbol in CRS-NOMA using SC in Rayleigh fading, is the target data rate for the symbol , and . The system design must ensure that , otherwise the outage probability for symbol will always be 1 as noted in [6]. The closed-form expressions for and are given in Appendix A. Next, we define as the outage event for symbol using SC. This outage event can be decomposed as the union of the following disjoint events: (i) symbol cannot be successfully decoded at the relay; (ii) symbol is successfully decoded at the relay, but symbol cannot be successfully decoded at the relay; and (iii) both symbols are successfully decoded at the relay, but symbol cannot be successfully decoded at the destination. Therefore, the outage probability for the symbol may be expressed as

(9) |

where is the target data rate for the symbol , , and . The closed-form expressions for and are given in Appendix A.

### Iii-D Diversity analysis for CRS-NOMA using SC

From (23), we have

(Using [4, eqn. (0.154-3), p. 4]) | |||

(10) |

where is the Landau symbol. Hence it is clear from (10) that decays as as . Similarly, it can be easily shown that decays as and decays as as . Therefore it is straightforward to conclude using (8) that the diversity order of the symbol is . Following similar arguments, it can be shown that the diversity order of the symbol is .

### Iii-E Reception using MRC for CRS-NOMA

The signal received at the relay (resp. destination) in the first time slot is given by

where (resp. ), , , is the Hermitian operator and

is the transpose operator. The elements in the vector

are independent and distributed as and the elements in are independent and distributed according to .The received instantaneous SINR at the relay for decoding symbol and instantaneous SNR for decoding symbol (assuming the symbol is decoded correctly) are obtained as and , respectively, where . Similarly, the received instantaneous SINR at the destination while decoding is given by , where . In the next time slot, the relay transmits the decoded symbol to the destination with power . The received signal at the destination (after applying MRC) is given by

where with independent elements each distributed as and with independent elements each distributed according to . The received instantaneous SNR at the destination while decoding the symbol is , where . The average achievable rate for the symbol is given by (c.f. [3])

(11) |

where .

###### Theorem 2.

A closed-form expression for the average achievable rate for symbol for CRS-NOMA using MRC in Rayleigh fading is given by

(12) |

where and denotes the Gamma function.

*Proof*: See Appendix B.

###### Proposition 2.

The closed-form expression for the average achievable rate for symbol for CRS-NOMA using MRC in Rayleigh fading is obtained as

(14) |

where .

###### Proof.

### Iii-F Reception using MRC for CRS-OMA

The signals received in the first time slot at the relay (resp. destination) is given by

where (resp. ). In the next time slot, the relay forwards its estimate of , denoted by , to the destination. The signal received at the destination is given by

Similar to the case of CRS-OMA using SC, the average achievable rate for symbol in CRS-OMA using MRC is given by (c.f. [5])

(17) |

where .

### Iii-G Outage probability for CRS-NOMA using MRC

Similar to the CRS-NOMA using SC, we define as the event that the symbol is in outage in the CRS-NOMA using MRC in Rayleigh fading. Hence,

(18) |

where is the instantaneous achievable rate for symbol in CRS-NOMA using MRC in Rayleigh fading. Similarly, we define as the event that the symbol is in outage in the CRS-NOMA using MRC in Rayleigh fading. Therefore,

(19) |

The closed-form expressions for , , and can be found using the fact that , and

are Gamma distributed random variables.

### Iii-H Diversity analysis of CRS-NOMA using MRC

Since is Gamma distributed with shape and scale we have

(20) |

where is lower-incomplete Gamma function. Using the series expansion of the lower-incomplete Gamma function as given in [7, eqn. 8.11.4, p. 180],

Using the series expansion of the exponential function and replacing by yields

(21) |

It is clear from (21) that decays as as . Similarly, it can be proved that decays as as . Also, using the series expansion of the lower-incomplete Gamma function and the exponential function, we have

(22) |

Hence it is straightforward to conclude using (18), (21) and (22) that the diversity order for the symbol is . Analogously, by representing and in (19) in terms of the lower-incomplete gamma function, it can be shown that the diversity order for the symbol is .

## Iv Results and Discussion

In this section we present analytical and numerical^{2}^{2}2We do not realize the actual scenario for numerical computation, but rather generate the random variables and then evaluate (6), (7), (16) and (17). results for the average achievable rate and the outage probability for the cooperative relaying system. We consider the CRS system where , and . For all NOMA-based systems, we consider and bps/Hz. Fig. 2 shows a comparison of the average achievable rate for the CRS-NOMA (both numerical and analytical results) and CRS-OMA (numerical results) systems. It is clear from the figure that for low transmit SNR , the CRS-NOMA system performs worse compared to the conventional CRS-OMA system in terms of achievable rate, but as the transmit SNR becomes large, the CRS-NOMA system outperforms its OMA counterpart for both SC and MRC schemes. It is evident from Fig. 1(a) that the CRS-NOMA using SC with achieves the same spectral efficiency as that of the CRS-OMA using SC with at high transmit SNR. Also, the CRS-NOMA using SC with achieves higher spectral efficiency as compared to CRS-OMA using SC with at high SNR. From Fig. 1(b), it is clear that the CRS-NOMA using MRC with achieves the same spectral efficiency as that of the CRS-OMA using MRC with at high transmit SNR. It can also be noted that the CRS-NOMA system using MRC results in a higher average achievable sum-rate as compared to the CRS-NOMA system using SC.

Fig. 3 shows the outage probability of the symbols and with varying transmit SNR for the CRS-NOMA system using SC. It is clear that the diversity order for both symbols is as derived in Section III-D.

Fig. 4 shows the outage probability of the symbols and with varying transmit SNR for the CRS-NOMA system using MRC. It is evident from the figure that the diversity order for both symbols is as proved analytically. It can also be noted that the outage probabilities for symbols and are lower for the CRS-NOMA system using MRC as compared to the corresponding probabilities for the CRS-NOMA system using SC.

## V Conclusion

In this paper, we provided a comprehensive achievable sum-rate analysis of a CRS-NOMA system with receive diversity. We considered two different diversity combining schemes – SC and MRC. It was shown that the CRS-NOMA system outperforms its OMA-based counterpart by achieving higher spectral efficiency. Our analysis also confirms that the CRS-NOMA can achieve the same rate as CRS-OMA, but with a smaller number of receive antennas. We also presented the outage probability analysis of the CRS-NOMA system. Diversity analysis of the CRS-NOMA system confirms that the system achieves full diversity order of for both SC and MRC schemes.

## Appendix A Proof of Theorem 1

Since is Rayleigh distributed for every , the CDF of is given by

Therefore, the CDF of can be obtained as

(23) |

The CDF of (resp. ) can be found by replacing by (resp. ), while also replacing by , in (23). The CDF of can be found as^{3}^{3}3Given two independent random variables and

with probability density functions (PDFs)

and respectively, and CDFs and respectively, the PDF of is given by and the CDF of is given by
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