## I Introduction

With the advent of new use-cases requiring high reliability and low-latency, transmission with finite blocklength becomes inevitable to reduce latency. In contrast to classical information-theoretic principles, the use of finite blocklength results in a non-negligible decoder error probability. Hybrid automatic repeat request (HARQ) procedures are used to improve the accuracy in decoding by exploiting time-diversity at the expense of increased latency. Thus, achieving high reliability and low-latency are Pareto-optimal, which calls for a trade-off between the two. Concurrently, non-orthogonal multiple access (NOMA) has gained widespread attention in research due to the ability to outperform its counterpart, orthogonal multiple access (OMA) in terms of spectral efficiency and user fairness.

Studies on NOMA with HARQ can be found in the literature [2, 3, 1, 10]. Authors in [2] show that NOMA with successive interference cancellation (SIC) employing HARQ with incremental redundancy (HARQ-IR) can outperform OMA in outage probability. In [3], a power allocation strategy for HARQ-IR with NOMA is presented. The outage performance of NOMA with the HARQ-CC scheme has been studied in [1] and [10], by deriving closed-form approximations for outage probability. Analysis of HARQ based systems using finite blocklength have been presented in [4, 8, 7]. Authors in [4] investigate the blocklength which maximizes throughput and minimizes average delay while authors in [8] investigate power allocation for systems using type-I ARQ in finite blocklength. In [7], a closed-form derivation of the outage probabilities on HARQ-IR in finite blocklength is provided. A power allocation method for HARQ-CC with finite blocklength, which targets reliability constraints is proposed in [5]. Analysis of NOMA in the finite blocklength regime is reported in [11, 6]. Authors in [11] and [6] investigate the finite blocklength performance of a two-user downlink NOMA system for single and multiple antenna base station (BS) and demonstrate that having a common blocklength for both NOMA users is optimal and NOMA outperforms OMA in latency.

Improving reliability and minimizing the latency are two targets that are conflicting with each other to be achieved simultaneously. The reason is improving reliability would be supported by re-transmissions including the use of longer packets, which will then increase the latency. A trade-off between latency and reliability that fit different use-cases is required when considering ultra-reliable low-latency communications (URLLC). Some URLLC use-cases are remote surgery and factory automation which have stricter targets like with 1 ms latency and V2X communications, and tactile internet which have reliability around and latency requirements ranging from 1 ms to 100 ms. The motivation behind this work is to analyze the performance of a system comprising of the three enablers; NOMA combined with HARQ in finite blocklength. While NOMA allows higher spectral efficiency by utilizing the same frequency-time resource, the use of HARQ improves the reliability, and the use of short packets allows reducing latency. The main goal is to investigate the ability of NOMA to deliver ultra-reliability using HARQ combined with the use of short packets to reduce latency. The reliability is investigated by characterizing the average block error rate (BLER). Also, searching for the required number of channel uses or blocklength for NOMA which satisfies reliability targets for the two users and comparison of the performance with OMA is in interest.

The next sections are organized as follows. Section II describes the system model. In Section III analytical approximations for the average BLERs are derived for the two users. Section IV presents asymptotic BLER approximations considering high SNR conditions and an algorithm is devised to determine the required blocklength for the system to meet given reliability constraints for the two users. In Section V numerical results are provided to validate the derived approximations in Section III and comparison of the blocklength requirement between NOMA and OMA is provided. Section VI concludes the paper and proofs of the important expressions are provided in Appendices.

## Ii System Model

Consider a downlink power domain NOMA system that uses short-packets for communications. The system comprises of a single antenna base station and two users equipped with single antennas. Without loss of generality, assume that is located close to the BS, thus having a higher channel gain, while is located far from the BS with a lower channel gain. With the limited channel state information (CSI) available in the BS, the reliability of communication can be degraded. To overcome this, the system uses the HARQ-CC scheme. The BS serves the users following the NOMA principle. Let and be the unit energy messages to and , respectively. The BS encodes these messages using the superposition coding technique with power allocation coefficients and such that with a total power of . According to the NOMA principle, BS allocates more power to the far user by setting ensuring user fairness. Therefore, the transmitted signal can be expressed as

(1) |

The received signal at , in the transmission round can be expressed as

(2) |

where , models the quasi-static Rayleigh fading of with blocklength in the transmission round, is the distance between and the BS, is the path loss exponent and

is the additive white Gaussian noise (AWGN) with variance

.The far user, attempts to decode the received signal treating ’s signal as interference. Then the received signal-to-noise-plus-interference ratio (SINR) at for decoding its message at the t^{th} transmission round is

(3) |

where is the transmit SNR such that .

The near user, applies SIC in decoding the messages, which means decodes ’s message first and then its own message without interference. The SINRs for decoding at are given by

(4) |

In the HARQ-CC procedure, in case of a failure to decode its message, the user retains the received signal and sends a negative acknowledgement (NACK) to the BS. If a NACK is received to the BS from any of the two users, BS retransmits the same encoded signal. Users employ maximum ratio combining (MRC) for decoding by combining the received signals stored during previous rounds and the new signal received. In case of successful decoding, the user will send a positive acknowledgement (ACK). BS transmits a new signal when it receives ACKs from both users. This work assumes the feedback channel, which ACKs/NACKs are sent, to be a one-bit error-free channel. The number of transmission rounds is limited to a maximum of . The SINR for decoding ’s signal at where after rounds of transmissions is

(5) |

## Iii Average BLER of NOMA with HARQ-CC in Finite Blocklength

### Iii-a Preliminaries

Short-packets are used in the system for achieving low-latency in communications with a finite blocklength. Based on the recent work by Polykiansky et al.[9], the decoder error probability or the BLER of for decoding ’s information, in finite blocklength is given by

(6) |

where is the number of information bits transferred using a blocklength of channel uses, is the SINR, is the channel dispersion defined by , is the Q function. This approximation holds when is sufficiently large[6], such as .

The user uses SIC in decoding, so the instantaneous BLER depends on the two stages in the SIC procedure. The success of the first stage affects the BLER in decoding at the second stage. Therefore, the instantaneous BLER for is given by

(7) |

Here is the BLER resulting from the first stage of the SIC decoding and denotes the success in the first stage. The average BLER , results from the interference-free decoding in the second stage. These are respectively given by

(8) |

The user directly decodes its message, so the instantaneous BLER is

(9) |

Then the average BLERs at the two users are obtained by

(10) |

By taking the expectation of the instantaneous BLER over the SINR distribution average BLER is given as

(11) |

(12) |

where

is the probability density function (PDF) of the SINR

. Equation 12, does not have a closed form solution and based on work the by Makki et al.[7], can be approximated as(13) |

where

(14) | ||||

(15) |

Using this approximation in (12), the average BLER is given by

(16) |

where

is the cumulative distribution function(CDF) of the SINR

.### Iii-B Average BLER for Decoding Far User’s Information

Based on the work by Cai et al.[1], the CDF of the SINR for decoding of ’s message with HARQ-CC is derived as

(17) |

The description of the variables and functions is given under (18).The proof is provided in Appendix A as an extension of the work in [1].

(18a) | ||||

where, | ||||

(18b) | ||||

(18c) | ||||

(18d) | ||||

(18e) | ||||

(18f) | ||||

(18g) | ||||

(18h) |

is the exponential integral function and are complexity-accuracy trade-off parameters. The proof is provided in Appendix B.

### Iii-C Average BLER for Interference-free Decoding of the Near User’s Information

For decoding its information with HARQ-CC after transmissions, the SNR is given by (4) which is

where . Since , is an exponential variable,

is exponentially distributed such that

. The sum ofexponential random variables is a Gamma distributed random variable with

degrees of freedom. Therefore can be described as(19) |

Then the CDF of is,

(20) |

where is the Gamma function and is the lower incomplete Gamma function.

Therefore, can be computed using (16) resulting in

(21a) | ||||

where, | ||||

(21b) |

and the as defined before. The proof is provided in Appendix C.

If the users are served using OMA, the blocklength or the number of channels uses available for transmission, would be shared between the two users and their messages will be transmitted utilizing the full power for that particular number of channel uses without interference from the other user. Note that the average BLER for OMA will have the same form as in (21) with and will be replaced by for .

## Iv Blocklength and Power Allocation

### Iv-a Asymptotic BLER approximations

Due to the mathematical complexity of the derived expressions in Section III, asymptotic expressions are derived in high SNR conditions. In short packet communications, the rate is small[6], which leads to being smaller. Thus, the integration in (16) can be approximated using the Reimann integral approximation such that

(22) |

where the superscript denotes the asymptotic approximation.

The average BLER targets for ultra reliable communication are in the order of or lower and can be achieved with high transmit SNR. Therefore, in (10), which results in . Therefore, approximates to and can be obtained by .

### Iv-B Required blocklength and power allocation

The problem of finding the required blocklength , which guarantees the target BLERs can be stated as

find | (23a) | |||

s.t | (23b) | |||

(23c) | ||||

(23d) | ||||

(23e) | ||||

for given | (23f) |

where the required BLERs for the two users are and . The conditions in (23b) and (23c) ensure the reliability targets of the users while (23d) and (23e) arise from the NOMA principle. Since , can be omitted from the expressions.

According to Section IV-A, and can be expressed as,

(24) |

Since is the stronger user with high channel gain and according to NOMA principle more power is allocated to , average BLER for decoding ’s information in the first stage of SIC at is smaller than the average BLER for the second stage of SIC when interference-free decoding of is done. Therefore, for simplicity, the average BLER for the first stage of SIC in is considered as , where . Then, from (24) can be written as

(25) |

Using the approximation with the Reimann integral as in (22), and can be obtained as

(26) | ||||

(27) |

Therefore, from (26) the blocklength , which satisfies the reliability targets can be found as

(28) |

where is the inverse of the lower incomplete Gamma function. With the use of (28) in (27) the required can be found. Also, the required blocklength for OMA can be obtained by the addition of the blocklengths needed to achieve their reliability targets using a similar expression to (28), with and replaced by for .

## V Numerical Results

Monte Carlo simulations are carried out based on the results for the decoding error probability in short blocklengths to verify the accuracy of the approximations derived in Section III. In all the simulations, the complexity-accuracy parameters and to ensure the numerical accuracy. The path loss exponent while and .

Figure 1 shows the average BLERs plotted against the transmit SNR () for different maximum transmission rounds. The approximations derived match with the Monte Carlo simulation results, which prove the accuracy of the expressions in (18) and (21). According to Figure 1, the far user always has a smaller average BLER than the near user, . The reason is that higher power is allocated for the far user for user fairness in the NOMA principle. Also, with the increasing number of maximum transmission rounds allowed, the average BLER decreases for a particular transmit SNR.

In Figure 2, average BLER is plotted with the blocklength at dB, with power allocation set to 0.2 and and set to 300. The comparison with OMA is provided for two scenarios as 20%-80% and 50%-50% blocklength share for respectively. It is clear from Figure 2, when the blocklength increases the average BLERs in all scenarios decrease monotonically which is desirable. One interesting result is that the performance of in NOMA and OMA with 80% share is almost similar with increasing blocklength. However, has a lower average BLER when NOMA is used compared to OMA with share of blocklength. Nevertheless, as the blocklength increase, this difference in performance between NOMA and OMA decreases. For the second scenario, blocklength is shared equally between the two users. The performance of degrades significantly compared to the performance with share. However, achieves a lower BLER than NOMA since a higher number of channel uses is available to . Although has a lower average BLER with an equal share in OMA than NOMA, ’s average BLER degrades significantly. Therefore, the NOMA scheme delivers fairness to both users, unlike OMA, since the difference in average BLER between two users is smaller than in OMA while achieving considerable average BLER performance for both users.

Figure 3 shows the gap between OMA and NOMA for the required blocklength to achieve a given reliability target of for and varying for using Algorithm 1. The value of is set to . Here the gap is taken by subtracting the NOMA blocklength from the OMA blocklength. It can be seen from Figure 3, NOMA has a smaller blocklength than OMA for the given reliability targets since the gap is positive. The bold red curve represents both users having the same reliability target of and NOMA always has a lower blocklength requirement and this gap increases as the transmit SNR increases. For lower reliability target such as for , the gap is smaller as seen from the dashed curve. Thus, NOMA has a lower blocklength requirement than OMA which leads to having lower latency when the reliability targets are in the order of .

## Vi Conclusion

This paper analyzed the performance of NOMA with HARQ-CC for finite blocklength by deriving tight closed-form approximations for the average BLER for two users. The comparison with OMA was done proving that NOMA could meet lower average BLER requirements such as while ensuring user fairness better than OMA. Further, an algorithm to determine the blocklength required to meet the reliability requirements of the two users was developed based upon the asymptotic expressions considering high SNR conditions. Simulations proved that NOMA has a lower blocklength requirement in high SNR leading to lower latency compared to OMA when the reliability requirements are in the order of .

## Appendix A Proof of Equation (16)

The PDF of the SINR for decoding ’s signal at where after rounds of transmissions, is given by Equation (35) in [1]. The CDF is calculated by extending that work. By taking the integral over with respect to z as

By change of variables with integral in converts to,

where as defined in (18e) and is the exponential integral function defined by .

## Appendix B Proof of Equation (18)

## Appendix C Proof of Equation (21)

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