Short-Packet Downlink Transmission with Non-Orthogonal Multiple Access

# Short-Packet Downlink Transmission with Non-Orthogonal Multiple Access

Xiaofang Sun,  Shihao Yan,  Nan Yang,
Zhiguo Ding,  Chao Shen,  and Zhangdui Zhong,
X. Sun, C. Shen, and Z. Zhong are with the State Key Lab of Rail Traffic Control and Safety, and the Beijing Engineering Research Center of High-speed Railway Broadband Mobile Communications, Beijing Jiaotong University, Beijing 100044, China (emails: {xiaofangsun, chaoshen, zhdzhong}@bjtu.edu.cn). S. Yan is with the School of Engineering, Macquarie University, Sydney, NSW, Australia (email: shihao.yan@mq.edu.au). N. Yang is with the Research School of Engineering, Australian National University, Canberra, ACT 2601, Australia (email: nan.yang@anu.edu.au). Z. Ding is with the School of Electrical and Electronic Engineering, University of Manchester, Manchester, UK (email: Zhiguo.ding@gmail.com).This paper has been presented in part at IEEE ICC workshop 2018 [1].
###### Abstract

This work introduces downlink non-orthogonal multiple access (NOMA) into short-packet communications. NOMA has great potential to improve fairness and spectral efficiency with respect to orthogonal multiple access (OMA) for low-latency downlink transmission, thus making it attractive for the emerging Internet of Things. We consider a two-user downlink NOMA system with finite blocklength constraints, in which the transmission rates and power allocation are optimized. To this end, we investigate the trade-off among the transmission rate, decoding error probability, and the transmission latency measured in blocklength. Then, a one-dimensional search algorithm is proposed to resolve the challenges mainly due to the achievable rate affected by the finite blocklength and the unguaranteed successive interference cancellation. We also analyze the performance of OMA as a benchmark to fully demonstrate the benefit of NOMA. Our simulation results show that NOMA significantly outperforms OMA in terms of achieving a higher effective throughput subject to the same finite blocklength constraint, or incurring a lower latency to achieve the same effective throughput target. Interestingly, we further find that with the finite blocklength, the advantage of NOMA relative to OMA is more prominent when the effective throughput targets at the two users become more comparable.

Non-orthogonal multiple access (NOMA), short-packet communications, finite blocklength, optimization, Internet of Things.

## I Introduction

### I-a Background and Motivation

In the fifth generation (5G) wireless ecosystem, the majority of wireless connections will most likely be originated by autonomous machines and devices [2]. Against this background, machine-type communications (MTC) are emerging to constitute the basic communication paradigm in the Internet of Things (IoT) [3]. The requirement on latency is stringent in some MTC scenarios [2], e.g., intelligent transportation, factory automation (FA), and industry control systems, since low latency is pivotal to ensure the real-time functionality in interactive communications of machines. Specifically, in some FA applications, the end-to-end latency at the application layer is required to be less than 1 ms [4]. This leads to a more strict transmission latency requirement at the physical layer.

To support low-latency communications, short-packet with finite blocklength codes is considered to reduce the transmission latency [5]. Specifically, in short-packet communications, as pointed out by [6], the decoding error probability at a receiver is not negligible since the blocklength is particularly small. This is different from Shannon’s capacity theorem, in which the decoding error probability is negligible as the blocklength approaches infinity. Taking into account the effect of a finite blocklength, the achievable rate was examined in [6] to approximate the information-theoretic limit, which brings novelty in short-packet system design. This pioneering work serves as the foundation in examining the performance of short-packet communications. Triggered by [6], the impact of finite blocklength on different communication systems has been widely studied. For example, the achievable transmission rate in quasi-static multiple-input multiple-output (MIMO) fading channels was examined in [7]. The latency-critical packet scheduling was investigated in [8]. In terms of the channel coding schemes with finite blocklength, [9] and [10] proposed optimal coding schemes in terms of the rate distortion function constrained to channel capacity under additive white Gaussian noise (AWGN) broadcast channel and medium access control (MAC) channel, respectively. In particular, these works showed that uncoded transmission schemes are optimal when the SNR is below some specific values. [11] proposed a joint source-channel coding scheme for a given finite blocklength and evaluated the performance of the scheme by mean square error distortion. [12] proposed a joint source-channel coding scheme with arbitrary blocklength on a Gaussian MAC channel and discussed the impact of the blocklength on the performance of the proposed coding scheme.

Furthermore, one challenge in MTC is the scalable and efficient connectivity for a massive number of devices sending short packets [13]. To address this challenge, different types of radio access technologies are investigated in the context of MTC [14, 15]. Particularly, non-orthogonal multiple access (NOMA) has attracted sharply increasing research interests as a promising technique for providing superior spectral efficiency [16]. The concept of downlink NOMA stems from the superposition coding on the degraded broadcast channel [17], which has been widely studied in the literature from information theory perspective, e.g., [18, 19]. By leveraging NOMA, we can allocate more power to the users with poor channel qualities to ensure the achievable target rates at these users, thus striking a balance between network throughput and user fairness [20]. Specifically, NOMA can significantly increase the number of connected devices. This is due to the fact that NOMA allows for overloading spectrum by multiplexing users in the power domain [21], which yields a higher flexibility and a more efficient use of spectrum and energy. In order to unlock the benefit of NOMA, successive interference cancellation (SIC) is normally adopted at some users such that they can remove the co-channel interference caused by other users in NOMA and decode the desired signals successively [22].

Meanwhile, the benefit of NOMA has been widely examined in various wireless communications, such as broadcast channels [23, 20], full-duplex communications [24], and physical layer security [25]. Advocated by the unique benefit of multi-antenna systems, the application of MIMO techniques to NOMA was addressed in [20, 26]. Driven by the ever-increasing demand of high spectral and energy efficiencies, NOMA has also been applied to multi-cell networks [27, 28]. However, the potential benefit of NOMA in terms of latency reduction in the context of short-packet communications and the impact of finite blocklength on the performance of NOMA have rarely been examined. These leave an important gap in understanding on the benefit of NOMA in the context of short-packet communications and the impact of finite blocklength on NOMA, which motivate this work. Moreover, the number of users is generally large in IoT scenarios. To facilitate NOMA transmission, the users are scheduled to different clusters first. Then, each cluster can perform random access during an allowable time slot [29]. However, the clustering problem is an NP-hard problem. To reduce its complexity, some suboptimal solutions have been investigated, e.g., [30, 31], but at the cost of system performance. Hence, it is pivotal to improve the spectral efficiency within a user cluster by resource allocation. This is also one of the motivations of this work.

### I-B Our Main Contributions

In this work, we introduce NOMA into short-packet communications and thoroughly examine its benefits in achieving a higher effective throughput relative to OMA subject to the same blocklength constraint, which in turn demonstrates the benefits of NOMA in latency reduction111The latency considered in this work is directly related to the blocklength. The transmitter encodes the data bits to be delivered into the codeword with symbols. Each symbol duration is defined as . As such, the latency considered in this work is , which is proportional to the blocklength.. We consider a specific MTC scenario, i.e., an FA scenario where an access point (AP) (e.g., a radio coordinator) has to transmit a certain amount of information to two stationary users (e.g., a process logic controller (PLC) and an actuator) within a short time period (requiring a low latency), enabling them to cooperatively perform some real-time functionalities. The two users are assumed to have been scheduled to one cluster and allocated to a resource block. We optimally design the transmission strategy for this cluster. Hence, this work serves as an important step for the further investigation of a massive-user scenario. To facilitate the optimal design, we assume that the channel gains are available at the AP. To fully exploit the performance gain of NOMA over orthogonal multiple access (OMA), we assume that the channel gain disparity between the two users is large. Based on these assumptions, we examine how NOMA helps the AP communicating to the two users with a low latency, while keeping the OMA scheme as a benchmark. Different from the case of NOMA with infinite blocklength where perfect SIC can be always guaranteed [21, 32], the consideration of finite blocklength leads to the fact that the perfect SIC may not be guaranteed. This brings about new challenges in the optimal design of the NOMA scheme, which have been addressed in this work. To take into account the impact of the non-zero decoding error probability caused by finite blocklength, we adopt the effective throughput as the metric to evaluate the system performance.

Different from our previous work [1], we provide insights and analysis on the constraints and determine the optimal solution to the optimization problem in this work. Moreover, we propose a fixed-point iteration algorithm which allows us to seek the optimal transmission rate for the weak channel user with a higher computational efficiency than one-dimensional search in this work, but not [1]. The main contributions of this work are summarized as below.

• We explicitly determine the optimal design of the NOMA scheme, in which the transmission rates and power allocation are optimized. To strike a balance between the system throughput and user fairness, this optimization maximizes the effective throughput of the user with a higher channel gain while guaranteeing a certain effective throughput target at the other user. To address the challenges caused by the complex capacity formula and unguaranteed SIC, we first analytically prove that the equality in the power constraint is active and the effective throughput target imposed at the user with a lower channel gain is always ensured. Then, we detail the steps to achieve the optimal transmission rates and power allocation. Besides, a computationally efficient algorithm, i.e., fixed-point iteration algorithm, is proposed for seeking the optimal transmission rate for the weak channel user.

• In order to explicitly demonstrate the benefit of NOMA in the context of short-packet communications, we take the optimal design of the OMA scheme as a benchmark, in which the optimal time slot allocation has to be determined on top of the optimal transmission rates and power allocation.

• Considering practical application scenarios with a finite blocklength, a thorough comparison between the NOMA and OMA schemes is provided. Our examination indicates that the NOMA scheme can significantly outperform the OMA scheme in terms of achieving a higher effective throughput at one user (subject to the same constraint on the effective throughput at the other user) with the same latency or incurring a lower latency to achieve the same effective throughput targets. Interestingly, the advantage of NOMA relative to OMA is more dominant when the effective throughput targets at the two users become more comparable, which is different from their comparison result with an infinite blocklength.

The rest of the paper is organized as follows. Section II presents the system model and formulates the optimization problem. Section III details the transmission strategy in the NOMA scheme. In Section IV, the optimal design of the NOMA scheme is provided. Section V presents the optimal design of the OMA scheme. Numerical results are presented in Section VI to draw useful insights. Finally, Section VII concludes this work.

## Ii System Model and Problem Formulation

### Ii-a System Model

In this work, we consider a downlink broadcast FA scenario as depicted in Fig. 1, in which a single-antenna AP (e.g., a radio coordinator) serves two stationary single-antenna users (e.g., a PLC and an actuator) within a finite blocklength of symbol periods. As the AP is equipped with a single antenna, it serves the two users either over orthogonal resource blocks with the OMA scheme, or over the same resource block with the NOMA scheme. In this work, we mainly focus on the NOMA scheme, while the OMA scheme serves as a benchmark. In the NOMA scheme, the two users are assumed to have been scheduled to one cluster and allocated to a resource block. Since the users’ locations are fixed, the channel gains vary slowly in time. We assume that the channel gains from the AP to the users are available at the AP and users222To estimate the channel gains, we follow [15] and assume that the users can perform the channel estimation by using regular pilot signals transmitted by the AP. The channel estimation method proposed in [15] is particularly advantageous for fixed-location applications.. To fully exploit the benefit of NOMA over OMA, we assume that the channel gains of the two users are significantly different. Without loss of generality, we define the user with the higher channel gain (the norm of the channel is higher) as user 1 (denoted by u) and the other one as user 2 (denoted by u). The channel coefficients from the AP to u and from the AP to u are denoted by and , respectively. We assume that and are subject to independent quasi-static Rayleigh fading with equal blocklength . Given the channel gain relationship between u and u, we have .

### Ii-B Achievable Transmission Rate with Finite Blocklength

As per Shannon’s coding theorem, the decoding error probability at the receiver becomes negligible as the blocklength approaches infinity [33]. Differently, short-packet communication aims to achieve low latency (e.g., short delay), in which the blocklength needs to be finite and typically small. As pointed out by [6], the decoding error probability at the receiver is non-negligible when the blocklength is finite. Specifically, perfect SIC cannot be guaranteed at the receiver side in this work, due to the finite blocklength constraint. Taking into account the impact of the non-zero error probabilities on successively decoding the signals at the receivers, we introduce the effective error probability, which is defined by , to denote the actual error probability at u. Furthermore, considering the trade-off between error probability and transmission rate, we adopt the effective throughput as the metric to evaluate the system performance with finite blocklength. Mathematically, the effective throughput at u is defined by

 ¯¯¯¯Ti=NiNRi(1−¯ϵi) (bps/Hz), (1)

where is blocklength allocated to u and is the transmission rate in the finite blocklength regime at u.

Based on [6, 34], with finite blocklength for a given decoding error probability at user , , can be accurately approximated by

 Ri≈log2(1+γi)−√ViNiQ−1(ϵi)ln2, (2)

where denotes the received signal-to-noise ratio (SNR) at user , is the inverse function of , and is the channel dispersion. Specifically, based on the results in [7], the expression for for the single-antenna quasi-static Rayleigh fading channel is .

Notably, the authors in [6] have compared the performance of a certain family of multiedge LDPC codes decoded via a low-complexity belief-propagation decoder against this finite blocklength fundamental limit. The simulation results showed that the relative gap to the finite blocklength fundamental limit is approximately constant. Furthermore, LDPC, Reed-Muller, Polar, and BCH codes have been investigated in [35, 36]. The aforementioned studies have verified (2). Therefore, we adopt (2) as a preliminary result for our work.

For a given transmission rate , the decoding error probability at user is approximated by

 ϵi≈Q(f(γi,Ni,Ri)), (3)

where , and is for the sake of practical reliable communication.

### Ii-C Optimization Problem

In the considered system, the AP needs to serve u and u within symbol periods (i.e., the finite blocklength is ). In addition, the AP fully consumes the symbol periods to transmit signals for ensuring the reliability. The ultimate goal is to achieve the maximum effective throughput at u, while guaranteeing a specific constraint on the effective throughput at u subject to a total power constraint. Mathematically, the optimization problem for the AP is formulated as

 maxΔ ¯¯¯¯T1 (4a) s.t. ¯¯¯¯T2≥T0, (4b) P1N1+P2N2≤PN, (4c) (N1,N2)∈{{(N1,N2)|N1=N2=N},for NOMA,{(N1,N2)|N1+N2=N},for OMA, (4d)

where represents the variable set that needs to be determined at the AP. It is noted that based on the expressions for the transmission rate and error probability in (2) and (3), respectively, we find that the transmission rate, error probability, and the allocated power are coupled together. Given any two of them, the third one is uniquely determined. This indicates that any two of them can be viewed as optimization variables. We further note that the transmission rates and power allocation are determined by the AP. Thus, we adopt the two variables to facilitate the system design and control the decoding error probabilities at the receiver side. is the minimum required effective throughput at u, and are the allocated transmit powers to u and u, respectively, is the average transmit power within one fading block. (4d) imposes the finite blocklength on the two users. As per (4), we need to design the symbol period allocation at the AP on top of determining the power allocation and transmission rates for u and u, such that is maximized subject to the given constraints.

## Iii Transmission Strategies with NOMA

In this section, we focus on the design of NOMA transmission. We first specify the optimization problem given in (4) for NOMA. Then, we detail the transmissions to user 1 and user 2 and mathematically characterize the optimization problem associated with the NOMA transmission.

### Iii-a Optimization Problem in NOMA

In NOMA, superposition coding (SC) is employed in the transmission such that the AP is able to transmit signals to u and u simultaneously at different power levels. As such, we have in the NOMA transmission strategy. Then, when NOMA is adopted at the AP, the optimization problem given in (4) yields

 maxΔn ¯¯¯¯T1 (5a) s.t. P1+P2≤P, (5b) ¯¯¯¯T2≥T0, (5c)

where is the variable set that needs to be determined at AP for NOMA transmission. To facilitate the optimal design, we then detail the NOMA transmission strategy and derive the expressions for and in the following two subsections, respectively.

### Iii-B Transmission to User 1

In the NOMA transmission, the transmitted signal at the AP is given by

 x=√P1x1+√P2x2, (6)

where and represent the information bearing signals to u and u, respectively, following circularly symmetric complex Gaussian (CSCG) distribution with zero mean and variance one. It is assumed that and are independent and identically distributed.

Then, the received signal at u at each symbol period is given by

 y1=~h1x+n1=~h1(√P1x1+√P2x2)+n1, (7)

where denotes the AWGN at u following CSCG distribution with zero mean and variance . Due to , we consider that SIC is employed at u to remove the interference caused by . To this end, u first decodes while the interference caused by is treated as noise (e.g., [21]). Following (7), the signal-to-interference-plus-noise ratio (SINR) of at u, denoted by , is given by

 γ12=P2h1P1h1+1, (8)

where denotes the normalized channel gain from AP to u.

For the sake of clarity, we denote as the event that is correctly decoded at u, , while denote as the event that is incorrectly decoded at u. The probability of an event occurring is denoted by . Following (3), the decoding error probability of at u (the outage probability of SIC) for a given , denoted by , is approximated by

 ϵ12=P(D12=1)=Q(f(γ12,N2,R2)). (9)

Accordingly, the probability that is correctly decoded and completely canceled at u is . This indicates that perfect SIC may not be guaranteed in NOMA with finite blocklength, differing from that in NOMA with infinite blocklength where the perfect SIC can always be guaranteed [21]. Taking into account the impact of the non-zero error probabilities for successively decoding the signals at u, the effective decoding error probability of at u is achieved by the marginal probability, which is given by

 ¯ϵ1=P(D11=1)=1∑i=0P(D11=1|D12=i)P(D12=i). (10)

To address this issue, we next derive and for .

If SIC succeeds, i.e., , (which occurs with the probability ), following (7) the SNR of at u, denoted by , is given by

 γ1=P1h1. (11)

Accordingly, the decoding error probability of at u for a given conditioned on the guaranteed SIC (i.e., ), denoted by , is approximated by

 ϵ1=P(D11=1|D12=0)=Q(f(γ1,N1,R1)). (12)

In this case, the throughput at u, denoted by , is given by .

Alternately, if SIC fails, i.e., , (which occurs with the probability ), u has to decode directly subject to the interference caused by . Correspondingly, following (7) the SINR of at u, denoted by , is given by

 γ′1=P1h1P2h1+1. (13)

In general, is significantly less than given in (11), since more power is allocated to u. As such, we may have in the design of NOMA. Considering the case with , the decoding error probability of for a given conditioned on the failed SIC (i.e., ), denoted by , is approximated by

 ϵ′1 =P(D11=1|D12=1) ={Q(f(γ′1,N1,R1)),if  R1≤log2(1+γ′1),1,if  R1>log2(1+γ′1). (14)

Then, the throughput at u, denoted by , is given by .

Following (10), we obtain the effective decoding error probability of at u as

 ¯ϵ1=ϵ1−ϵ1ϵ12+ϵ12ϵ′1. (15)

As such, the effective throughput at u is given by

 ¯¯¯¯T1 =R1(1−ϵ1+ϵ1ϵ12−ϵ12ϵ′1). (16)

### Iii-C Transmission to User 2

Following (6), the received signal at u, when AP adopts NOMA, is given by

 y2=~h2x+n2=~h2(√P1x1+√P2x2)+n2, (17)

where denotes the AWGN at u with zero mean and variance .

Due to , SIC is not conducted at u. As such, u decodes its own signal directly subject to the interference caused by . Following (17), the SINR of at u, denoted by , is given by

 γ2=P2h2P1h2+1, (18)

where denotes the normalized channel gain from AP to u. Accordingly, the decoding error probability of at u for given , denoted by , is approximated by

 ϵ2=P(D22=1)=Q(f(γ2,N2,R2)). (19)

Since there only exists one decoding strategy at u, the decoding error probability is actually the effective decoding error probability at u, i.e., . Then, the effective throughput at u, , is given by

 ¯¯¯¯T2=R2(1−¯ϵ2)=R2(1−ϵ2). (20)

With the expressions for and given in (16) and (20), respectively, the optimization problem given in (5) is well defined. We next tackle this optimization problem in the following section.

## Iv Design of Transmission Rates and Power Allocation in NOMA

In this section, we focus on the design of the transmission rates and power allocation in the NOMA transmission, i.e., focus on solving the optimization problem given in (5). To this end, we first provide analysis and insights on the constraints and then find the optimal solution.

### Iv-a Equalities in Constraints

In this subsection, we tackle the two constraints given in (5b) and (5c) in order to facilitate solving the optimization problem given in (5).

For the sake of clarity, in the NOMA scheme, we replace both and with , and we substitute for in (3), since we have in NOMA. In order to simplify constraint given in (5b), we first examine the monotonicity of the error probability given in (3) with respect to the corresponding SNR/SINR in the following theorem.

###### Proposition 1

The decoding error probability given in (3) is a monotonically decreasing function of the corresponding SNR/SINR.

###### Proof:

The detailed proof is provided in Appendix A. \qed

We note that in Proposition 1, the decoding error probability can be any one of , , , and . Based on Proposition 1, we next prove in the following lemma that the equality in (5b) is always guaranteed.

###### Lemma 1

The equality in the power constraint (5b), i.e., , is always guaranteed in order to maximize subject to .

###### Proof:

The detailed proof is provided in Appendix B. \qed

Lemma 1 indicates that the AP fully consumes the maximum transmit power to maximize subject to . This lemma significantly facilitates the power allocation at the AP, since it shows that as long as we can determine the power allocation to one user, all the remaining power needs to be allocated to the other user.

In the following lemma, we prove that the equality in the constraint (5c) is also always guaranteed.

###### Lemma 2

The equality in the effective throughput constraint (5c) is always guaranteed, i.e., , in order to maximize subject to .

###### Proof:

To facilitate the proof, we first exploit the monotonicity of the decoding error probability given in (3) with respect to . To this end, the partial derivative of with respect to is given by

 ∂ϵi∂Ri=√Nln2√2π√1−(1+γi)−2e−f2(γi,Ri)2, (21)

which is always larger than zero. Accordingly, we have that is a monotonically increasing function of .

We next prove by contradiction that the equality in (5c) is active at the optimal solution. We first suppose that in the optimal solution we have , where the maximum value of is . Then, we can reduce to by using a smaller while keeping the power allocation fixed, since is a continuous function of and when . By doing so, the outage probability of SIC, i.e., , decreases, since monotonically increases with as per (21). As we proved in Appendix B that monotonically increases with , increases as decreases. This contradicts to the claim of optimality that is the maximum value of . Therefore, the equality in the effective throughput constraint must hold for the optimal design. \qed

Based on this lemma, the constraint uniquely determines the one-to-one relationship between the transmit power and the corresponding optimal transmission rate , which will be discussed in the next subsection.

### Iv-B Optimal Transmission Design

Following the aforementioned analysis, in this subsection we determine the optimal solution to the optimization problem given in (5).

Based on (9), the perfect SIC cannot be always guaranteed in the NOMA scheme with a finite blocklength. As such, taking into account the outage probability of SIC, i.e., , leads to a fact that the objective function does not always increase with subject to . As increases with based on Proposition 1. But on the other hand, the decoding error probabilities and at u decrease with . Accordingly, there is a non-trivial trade-off between the effective throughput and . This fact is different from the NOMA scheme with an infinite blocklength, where monotonically increases with subject to . This fact also brings in challenges in the design of the optimal power allocation and transmission rates in the NOMA scheme with a finite blocklength.

To address the previous issue, we next detail the main steps to determine the optimal design of the power allocation and transmission rates in the NOMA scheme.

Step 1: Determine for a feasible .

As is directly decoded at u by treating as noise, for a given power allocation the effective throughput achieved at u is independent of . Inspired by this, we first determine the value of that maximizes for given feasible and . The feasibility of will be discussed in Step 3.

We define the effective throughput in terms of by . To facilitate the design of , we examine the monotonicity and concavity of with respect to in the following lemma.

###### Lemma 3

does not monotonically increase with but is concave with respect to .

###### Proof:

The detailed proof is provided in Appendix C. \qed

Following Lemma 2, we note that for a feasible , the optimal value of satisfies . Lemma 3 indicates that for a given feasible , there are two values of that satisfy . The smaller one that satisfies , denoted by , is the optimal value that maximizes . This is due to the fact that monotonically decreases with , which is proved in Lemma 1. However, is an input argument of -function for given by (20). This prevents us from deriving a closed-form expression for . To address this issue, a fixed-point iteration algorithm [37] is proposed in the following proposition for seeking .

###### Proposition 2

The solution of to can be obtained by the fixed-point iteration

 R2:=F(R2)=T01−Q(f(γ2,R2)). (22)
###### Proof:

Based on the Theorem 2.1 in [37], the iteration converges to a fixed point if is guaranteed for that satisfies , i.e., , where denotes the value that satisfies . Notably, guarantees that the iteration converges to the smaller value that satisfies . To verify the convergence of the iteration, the first derivative of with respect to is derived as

 F′(R2)=T0√2πbe−f2(γ2,R2)2(1−Q(f(γ2,R2)))−2≥0, (23)

where . monotonically increases with , as . The proof of is omitted here due to page limit.

Then, we have for . Based on (39), we have . Substituting for in (23), we have

 F′(R‡2)=T0R‡2(1−Q(f(γ2,R‡2))). (24)

Notably, for any feasible and , to ensure the effective throughput target can be achieved at u based on Step 3, is guaranteed. This indicates that , otherwise is the solution to .

Therefore, we prove that holds for that satisfies .

\qed

Step 2: Determine for given , , and .

Following the previous optimal design for , we then determine the value of that maximizes for given , , and in the following theorem.

###### Proposition 3

The value of that maximizes the effective throughput for given , , and is given by

 R†1=⎧⎨⎩¯¯¯¯R†1,if  0≤R1≤log2(1+γ′1),˜R†1,if  log2(1+γ′1)

where is the unique solution to . is the unique solution to . And function is defined by

 U(x,y)≜1−Q(f(x,y))+y∂f(x,y)∂y√2πe−f2(x,y)2, (26)

where .

###### Proof:

The detailed proof is provided in Appendix D. \qed

Based on Appendix D, we find that the partial derivative of the effective throughput with respect to is monotonically decreasing in terms of , as its second partial derivative is less than zero. Thus, the one-dimensional search based on bisection search can be introduced to find the optimal .

We note that for a given power allocation we can first determine the value of as per Step 1 and then obtain the value of with the aid of Proposition 3. The final hurdle for the optimal design arises from the power allocation. To address this issue, we adopt the one-dimensional numerical search to find the optimal power allocation. In addition, for ensuring the effective throughput target achieved at u, we next determine the feasible set for power allocation by calculating the lower bound on .

Step 3: Determine a strict lower bound on .

###### Lemma 4

The strict lower bound on , denoted by , is the unique solution to

 R‡2(1−Q(f(γl2,R‡2)))=T0, (27)

where is satisfies , which is defined in (26), and .

###### Proof:

The detailed proof is provided in Appendix E. \qed

For a similar reason as the previous steps, we cannot derive a closed-form expression for the lower bound on . Hence, the one-dimensional search based on bisection search is introduced to obtain the lower bound on .

Following Lemma 4, we note that the feasible value range of is . This is due to the fact that cannot be guaranteed when for any possible value of subject to .

Step 4: Determine the optimal power allocation.

Following the aforementioned three steps, the problem given in (5) can be simplified to the following optimization problem

 maxP1 R†1(1−¯ϵ1) (28a) s.t. P1∈P, (28b)

where , which can be solved by the one-dimensional line search algorithm over the feasible set of . However, we can further draw the following lemma to improve the computational efficiency.

###### Lemma 5

The optimization problem (28) is strictly convex in if

 log2(1+γ′1)
###### Proof:

The detailed proof is provided in Appendix F. \qed

It is noted that due to the assumption of large channel disparity between and for fully exploiting the benefit of NOMA, we have that . Furthermore, is negligible since is assumed in the fundamental work [6] for accuracy and a high channel gain is assumed at u. Thus, the sufficient condition, i.e., , can be guaranteed with a high probability based on the fact that .

This lemma shows that the optimal can be found by golden section search instead of line search when and satisfy (29).

To facilitate the optimal power allocation design, we first choose a feasible value of that guarantees and then determine for the chosen as per Step 1. After that, is determined as per Step 2 with , since is always guaranteed as proved in Lemma 1. Finally, we can calculate the achieved for the chosen and then we repeat the above steps until we find the optimal value of that achieves the maximum value of , which is denoted by . Once the optimal value of , denoted by , is determined, the optimal values of , , and can be determined accordingly, which are denoted by , , and , respectively.

## V Design of OMA with a Finite Blocklength

In this section, we present the OMA scheme as the benchmark, where the two users are served in different (orthogonal) time slots and hence we have .

### V-a Transmission to Two Users with OMA

When the AP adopts OMA to serve the two users in orthogonal time slots, the received signal at u, is given by

 yi=√Pi~hixi+ni. (30)

Due to the orthogonal transmissions to u and u, there is no interference at u (or u) caused by (or ). As such, the SNR at u of is given by

 γi=Pihi, (31)

where denotes the noise normalized channel gain from AP to u, .

Accordingly, the decoding error probability of at u for given is approximated by given in (3). In addition, the effective decoding error probability is in OMA, since u decodes its own message independently in different time slots. Then, the effective throughput achieved by u is given by

 ¯¯¯¯Ti=NiNRi(1−Q(f(γi,Ni,Ri))). (32)

We note that given a time slot allocation, is a function of only and , while is a function of only and . This is different from the case in the NOMA scheme, where is a function of , , , and . The design of OMA is detailed in the following subsection.

### V-B Optimal Design of OMA

In the OMA scheme, we have . As such, for the OMA scheme the optimization problem given in (4) can be rewritten as

 maxΔo ¯¯¯¯T1 (33a) s.t. ¯¯¯¯T2≥T0, (33b) N1P1+N2P2≤NP, (33c) N1+N2=N, (33d)

where is the variable set that needs to be determined at the AP with OMA transmission. We note that the AP has to determine the time slots allocation on top of optimizing the transmission rates and the power allocation in the OMA scheme.

In order to solve (33), we first clarify that the equalities in (33b) and (33c) are always guaranteed. This is due to the fact that following (32) for any given , , , and , the effective throughput and are monotonically increasing functions of and respectively. We now briefly outline the steps to solve the optimization problem given in (33).

Step 1: Determine and for a given .

For a given time slot allocation (i.e., ), the optimal value of is the minimum one that guarantees , where the optimal value of is the one that maximizes for a given . This is due to the fact that monotonically decreases with due to and is independent of .

Step 2: Determine and for given and .

Once is determined in Step 1, we have , since the equality in (33c) is always guaranteed. Then, for given and , the optimal value of is the one that maximizes , since is independent of .

Step 3: Search for the optimal time slot allocation.

Following the aforementioned two steps, we can see that the optimization problem given in (33) can be simplified to a one-dimensional numerical search problem in order to determine the optimal values of and subject to , which can be easily solved. Specifically, we first choose a value of that guarantees and then determine and as per Step 1. After that, we can determine and as per Step 2. Finally, we can calculate the achieved for the chosen value of and then we repeat the above steps until we find the optimal value of that achieves the maximum , denoted by . Once the optimal value of , denoted by , is determined, the optimal values of , , , , and can be determined accordingly, which are denoted by , , , , and , respectively.

## Vi Numerical Results

In this section, we present numerical results to examine the performance of the proposed NOMA scheme with the OMA scheme as the benchmark by considering the finite blocklength. We assume that the AP and the users are located in a 30m 80m factory. The distances between AP and the two users are set to be m and m, respectively. Unless otherwise stated, we set the noise power at each user to unit, i.e., . We also define the average transmit SNR as . Besides, we set to zero if the optimization problem in (4) is infeasible, i.e., if the constraint cannot be guaranteed, to incorporate the penalty of failure.

### Vi-a Numerical Results Based on Fixed Channel Gains

In order to provide the insight into the relationships between the objective function and the parameters of interest, throughout this subsection, the channel gains of the two users are set to be fixed.

In Fig. 2, we plot the effective throughput of u achieved by the NOMA scheme, i.e., , versus , while other parameters (e.g., , , ) are optimized accordingly. In this figure, we first observe that the lower bound on , i.e., , increases with . When is smaller than , u cannot achieve the target effective throughput , which leads to that is set to be zero when . We also observe that the optimal that maximizes , i.e., , is not . This is caused by the high-order terms of the effective error probability in (15). This observation also verifies our analysis presented in Section IV-B that may not be a monotonically increasing function of , which is significantly different from the case with an infinite blocklength. The observation indicates that in the NOMA scheme with a finite blocklength, more power is allocated to u relative to in NOMA with an infinite blocklength, which enables us to adopt a smaller in order to reduce the outage probability of SIC in the NOMA scheme with a finite blocklength (SIC can be guaranteed in NOMA with an infinite blocklength). To confirm this, we examine the optimal value of in the following figure.

In Fig. 3, we plot the effective throughput achieved at u, i.e.,