## 1 Introduction

Non-orthogonal multiple access (NOMA) is considered as one of the most prominent solutions for G systems due to its increased spectral efficiency over conventional orthogonal multiple access. To avoid jointly using time, frequency and/or code resources, power-domain NOMA has gained considerable attention via superposing multiple users’ signal in the power domain. The received signals create mutual interference. Yet, the detector can employ successive interference cancellation (SIC), i.e., it decodes the strongest signal, and then subtracts it from the remaining signal, such that the next detection stage is interference-free. Doing so, NOMA can increase the system spectral efficiency and multiuser diversity as well as reduce latency. Nowadays, various Internet-of-things (IoT) applications utilize short packet transmission since connectivity and low-latency rather than high throughput is of prime importance. More so, ultra-reliable and low-latency communication (URLLC), which is at the forefront of wireless communications, defines another quite interesting and exciting paradigm that utilizes short packet transmission.

Due to the complementary benefits of NOMA and short packet transmission, their joint investigation has attracted a lot of research interest lately Amjad18 -HuangYang19 . In particular, coded NOMA was studied in Amjad18 -Ghanami21 when all the considered nodes are equipped with single antennas. In Tran21 -HuangYang19

, the case when a multi-antenna receiver is employed was analytically studied. Nevertheless, all the aforementioned works have not considered user mobility, which is present in various modern practical networking setups (e.g., vehicles and moving devices). To our knowledge, the scenario of NOMA short packet transmission systems with user mobility has not been investigated to date. Capitalizing on the mentioned observations, we analytically study a two-user uplink NOMA setup operating in the short packet transmission regime. The users support mobility and the time variation as well as correlation of the channel is modeled by the second-order statistic of level crossing rate (LCR) with respect to either the signal-to-interference-plus-noise ratio (SINR) or signal-to-noise ratio (SNR). Single-antenna mobile users and a multi-antenna base station are adopted, while the received signals undergo independent Rayleigh channel fading conditions. Unlike most previous research works, we capitalize on the fact that SINR (and SNR) is a stationary stochastic process and, doing so, a two-state Markov model is used to analyze the packet error rate (PER) for finite (short) packet transmissions. In addition, we show that the optimal NOMA power allocation can be numerically computed in a straightforward and cost-effective manner based on the derived PER expressions.

## 2 System Model

Consider an uplink wireless communication setup, where two single-antenna users transmit to a base station equipped with antenna elements. Power-domain NOMA is used for the signal transmission so as to achieve spectral efficiency, while both signals undergo independent Rayleigh channel fading conditions. Let and denote the channel fading and transmitted symbols of the user, respectively, with . The time autocorrelation of each channel path of the user is given by , where denotes the zeroth-order Bessel function, is the time difference between two correlated samples and stands for the maximum Doppler frequency associated with this channel path. Also, and represent the relative mobile speed of the user and carrier wavelength, correspondingly.

The received signal reads as

(1) |

where

is the received signal vector,

is the transmit power andstands for the additive white Gaussian noise vector with its elements having zero-mean and variance

. Also, defines the NOMA power allocator of the user, such that and . Further, represents the received channel fading vector with complex Gaussian entries having zero-mean and unit-variance. Upon the signal reception, SIC is used for detection.Without loss of generality, we assume that is firstly detected by applying the weight vector to the received signal; i.e., , where the superscript denotes the Hermitian transpose operator and stands for the Euclidean (vector) norm. Afterwards, in the case when is correctly decoded and removed, the same procedure follows for to the remaining received signal via the weight vector , i.e., , where is the remaining received signal after the contributing part of is stripped off. Thereby, assuming a unit-power signal and perfect channel state information at the receiver, the SINR and SNR at the and SIC detection stage is given, respectively, as

(2) |

and

(3) |

where , present in the denominator of (2), denotes the absolute (scalar) value operator.

## 3 Performance Metrics

We commence by analyzing PER of the considered uplink NOMA short packet transmission setup. First, we assume that a packet error occurs only when for any time instance during an entire packet duration with denoting a certain data rate threshold (in bps/Hz). This process is modeled by using the two-state Markov model analyzed in (Fukawa12, , §III.B). Thereupon, PER at the detection stage becomes (Fukawa12, , Eq. (58))

(4) |

where , and

denote the packet transmission time interval, LCR and cumulative distribution function (CDF) of

, respectively. Notably, the defined PER considers the time variations and correlations of the channel, captured by the second-order LCR statistic. Moreover, according to the adopted SIC-enabled reception approach, conditioned on the successful decoding of the stage, the stage is being performed. Thus, the instantaneous packet error at the stage can be modeled by , where andare the instantaneous error probability at the

detection stage and conditional error probability at the stage given that the stage was error-free. Note that the conditional (average) PER of the stage, i.e., is computed as per (4), by simply substituting and with and , correspondingly. Thereby, the following union bound on the unconditional PER at the stage yields as(5) |

Typically, error rates remain considerably low in ultra-reliable applications, i.e., ; hence, the latter union bound is quite sharp, yielding .

According to (2) and without delving into details, it is straightforward to show that the corresponding CDF of SINR is given by

(6) |

where is the upper incomplete Gamma function. Also, and

denote the Erlang CDF and exponential probability density function (PDF), respectively. In a similar basis, according to (

3), it holds that(7) |

On another front, LCR is an important second-order statistic that showcases the rate of fading occurrence within a certain time interval. According to the structure of in (2), its corresponding LCR is expressed as (AliTorlak17, , Eq. (21))

(8) |

Finally, the LCR of is presented as (Beaulieu03, , Eq. (17))

(9) |

## 4 Engineering Insights

In the short packet transmission regime, recall that remains quite low. In addition, when ultra-high reliability is required in the considered NOMA setup, it is obvious that a high SNR is a requisite. Capitalizing on the said observations and applying the first-order McLaurin series to the exponential function within (4) (i.e., as ), we arrive at

(10) |

Likewise, PER at the stage approaches

(11) |

It is noteworthy that the PER is lower bounded by the corresponding outage probabilities at each detection stage (say, ), whereon an ‘*extra penalty*’ is added so as to reach the total PER, which is reflected by the corresponding LCR performance of .

The optimal power allocation, defined by , defines a key performance indicator for the considered NOMA setup. Specifically, the optimization problem can be designed such that PER at the stage should be minimized under the constraint of PER at the stage not exceeding a predetermined threshold value . It can be formulated as

(12) | ||||

Unfortunately, is a non-convex optimization problem since both the objective and constraint function (regarding PER) are non-convex functions. However, can be directly obtained by a numerical search over the real line in the range for arbitrary antenna arrays and user mobility profiles.

## 5 Numerical Results

The derived analytical results are verified via numerical validation where they are cross-compared with corresponding Monte-Carlo simulations. The Rayleigh faded channels are generated by using the sum-of-sinusoids method Patzold09 , which is a modified Jakes model. In what follows, for ease of presentation and without loss of generality, we assume that ; namely, the two users have an identical mobility profile (i.e., the same relative speed). The packet duration is set to be millisecond, appropriate for low-latency short packet transmission.

In Fig. 1, the maximum Doppler frequency is set to be Hz, which corresponds to a vehicular speed of km/hr regarding a cellular system with carrier frequency GHz. We model a base station equipped with antenna elements operating in a high SNR region. The PER of the detection stage is depicted as well as the conditional PER at the stage (given that the stage is correctly decoded), which can be directly computed by setting in (5). The performance gap between the two stages is evident and gets even more emphatic for an increasing transmit SNR . Further, it can be seen that PER of the stage dominates the overall system performance for relatively low values in comparison to PER at the stage; and vice versa for relatively high values. This is a reasonable outcome since lower values typically reduce in (2) as well as increase in (3), while quite the opposite result holds as increases.

In Fig. 2, the role of multiple antennas at the receiver is highlighted, by considering a conventional base station with antennas compared to a massive antenna array of antennas. The union bound of PER at the stage is illustrated. It is noteworthy that the massive antenna scenario outperforms the conventional case only when the power allocation factor is carefully selected. Obviously, the PER performance is not affected by an increase of for a relatively low . On the other hand, there is a certain range of values, where the presence of a vast antenna array is greatly beneficial. The optimal (namely, ) is lying in this range, which can be quite easily computed as per . Also, as expected, PER is dramatically reduced for a lower rate threshold, . Notice that a relatively low data rate requirement is the typical case of various URLLC applications since high data rates are being sacrificed so as to guarantee connectivity and ultra-high reliability.

## 6 Conclusion

An uplink two-user NOMA communication system was considered, which operates under independent Rayleigh faded channels. The case when the users and receiver are equipped with a single-antenna and multi-antenna elements, respectively, was studied. Moreover, user mobility was supported reflecting several practical applications, e.g., vehicle-to-vehicle and device-to-device networking setups. Particularly, we focused on the short packet transmission regime, which relates to the rather timely URLLC applications. Closed-form expressions regarding the system PER were derived under arbitrary user mobility profiles and various antenna array ranges. Finally, some new useful engineering insights were obtained, while the optimal NOMA power allocation per user was formulated based on the derived PER results.

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