Uplink NonOrthogonal Multiple Access with FiniteAlphabet Inputs
Abstract
This paper focuses on the nonorthogonal multiple access (NOMA) design for a classical twouser multiple access channel (MAC) with finitealphabet inputs. In contrast to most of existing NOMA designs using continuous Gaussian input distributions, we consider practical quadrature amplitude modulation (QAM) constellations at both transmitters, the sizes of which are assumed to be not necessarily identical. We propose to maximize the minimum Euclidean distance of the received sumconstellation with a maximum likelihood (ML) detector by adjusting the scaling factors (i.e., instantaneous transmitted powers and phases) of both users. The formulated problem is a mixed continuousdiscrete optimization problem, which is nontrivial to resolve in general. By carefully observing the structure of the objective function, we discover that Farey sequence can be applied to tackle the formulated problem. However, the existing Farey sequence is not applicable when the constellation sizes of the two users are not the same. Motivated by this, we define a new type of Farey sequence, termed punched Farey sequence. Based on this, we manage to achieve a closedform optimal solution to the original problem by first dividing the entire feasible region into a finite number of Farey intervals and then taking the maximum over all the possible intervals. The resulting sumconstellation is proved to be a regular QAM constellation of a larger size, and hence a simple quantization receiver can be implemented as the ML detector for the demodulation. Moreover, the superiority of NOMA over timedivision multiple access (TDMA) in terms of minimum Euclidean distance is rigorously proved. Furthermore, the optimal rate allocation among the two users is obtained in closedform to further maximize the obtained minimum Euclidean distance of the received signal subject to a total rate constraint. An asymptotic solution is also derived to reveal more insights on how to allocate the rate to each user. Finally, simulation results are provided to verify our theoretical analysis and demonstrate the merits of the proposed NOMA over existing orthogonal and nonorthogonal designs.
I Introduction
The forthcoming fifth generation (5G) cellular systems are envisioned to support three generic services, including extreme mobile broadband (eMBB), massive machinetype communications (mMTC), and ultrareliable and lowlatency communications (uRLLC) [1, 2]. These diverse services, driven by the explosive growth of mobile data traffic and expected wide rollout of Internet of Things (IoT), pose challenging requirements for the air interface of wireless networks where enhanced multiple access technologies are essential. Apart from several other potential technologies such as massive multipleinput multipleoutput (MIMO) and millimeter wave (mmWave) communications, nonorthogonal multiple access (NOMA) has recently emerged as a key enabling radio access technology to meet these unprecedented requirements of 5G networks, due to its inherent advantages of high spectral efficiency, massive connectivity, and low transmission latency [3, 4, 5, 6, 7]. The concept of NOMA has multiple variants, such as powerdomain NOMA, sparse code multiple access, pattern division multiple access, low density spreading, and lattice partition multiple access [5]. In this paper, we mainly consider the powerdomain NOMA.
The basic principle of NOMA is to serve more than one user with distinct channel conditions simultaneously in the same orthogonal resource block along the time, frequency, or code axes. This can be achieved by applying the superposition coding (SC) at the transmitter as well as multiuser detector (e.g., successive interference cancellation (SIC)) at the receiver side to distinguish the cochannel users. As such, NOMA is fundamentally different from conventional orthogonal multiple access (OMA) methods primarily used in the previous generations of mobile systems, where each user is allocated to one dedicated orthogonal radio resource block exclusively. This in turn means that, in OMA, multiuser communication can be decomposed into several parallel singleuser ones free of interuser interference, and then the wellestablished singleuser encoding/decoding methods can be directly applied with a reasonable tradeoff between network throughput and implementation complexity [8, Ch. 14].
Although the OMA schemes have been widely used in the past several decades, they generally cannot achieve the whole multiuser capacity region and thus tend to have a lower spectral efficiency than NOMA approaches [3, 5, 9, 10]. For example, in OMA, a resource block allocated to a user with a poor channel condition cannot be reused by another user with a much stronger channel state. Apart from that, OMA is in general not scalable. This is because the amount of resource blocks as well as the granularity of user scheduling strictly limit the number of users that can be supported at the same time. On the contrary, by breaking the orthogonality of the radio resource allocation, NOMA has been shown to be able to provide better user fairness and improve physical layer security in addition to the advantages mentioned above [3, 5].
Ia Related Work
Despite the fact that the deployment of NOMA as a new radio access technology in nextgeneration mobile systems is relatively new, the performance of NOMA has been studied extensively in the information theory society for various channel topologies such as broadcast channel (BC) [11, 12, 13, 14], multiple access channel (MAC) [15, 16, 17, 18], and interference channel (IC) [19, 20, 21, 22, 23]. However, these results concentrated mainly on the study of the channel capacity region with the assumption of unlimited encoding/decoding complexity, and therefore lie mostly in the theoretical aspects due to their extremely high implementation cost. Thanks to the rapid progress of the radio frequency (RF) chain and the processing capability of mobile devices in the past decades, the implementation of NOMA is becoming more and more feasible and thus has drawn tremendous attention from both academia and industry very recently [5]. More specifically, by taking practical constraints on user fairness and/or radio resource management into consideration, NOMA has been investigated in various wireless systems, such as cognitive radio [24, 25], cooperative communications [26, 27], cellular uplink [28, 29], cellular downlink [30, 31, 32, 33, 34], and multicell networks [35, 36]. In fact, a twouser downlink scenario of NOMA, known as multiuser superposition transmission (MUST), has already been incorporated in the 3rd Generation Partnership Project (3GPP) Long Term EvolutionAdvanced (LTEA) [37, 38].
We note that, up to now, the vast majority of existing NOMA designs assumed the use of Gaussian input signals [11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 6, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35]. Although the Gaussian input is of great significance both theoretically and practically, its implementation in reality will require huge storage capacity, unaffordable computational complexity and extremely long decoding delay [9, Ch. 9]. More importantly, the actual transmitted signals in real communication systems are drawn from finitealphabet constellations, such as pulse amplitude modulation (PAM), quadrature amplitude modulation (QAM), and phaseshift keying (PSK) [8, Ch. 5]. Applying the results derived from the Gaussian inputs to the signals with finitealphabet inputs can lead to significant performance loss [39]. In this sense, Gaussian input serves mostly as the theoretical benchmark.
Motivated by the above facts, the NOMA design with finitealphabet inputs is of utmost importance and has attracted considerable efforts, see e.g., [40, 41, 42, 43, 44, 36] and references therein. The main principle^{1}^{1}1Note that the principle was originally proposed in the seminal work [45, 46, 47], wherein the finitelength codeword design problem in the binary domain were considered from an informationtheoretical perspective. of these efforts is to ensure that the signal originated from each user can be uniquely decoded from the received sumsignal at the receiver side. By using mutual information as a performance measure, references [40, 41] considered the NOMA design in an ideal twouser Gaussian MAC with finiteinput constellations under individual power constraint on each user. Specifically, NOMA was realized by strategically introducing certain constellation rotations (CR) to the adopted PSK signals in [40] or using proper power control in [41]. However, only numerical solutions to the optimal NOMA designs were provided in [40, 41]. Moreover, linear precoders were considered for the MIMO MAC in [42], where the expression of the weighted sumrate was asymptotic and the optimal solution was also numerical. Besides, the downlink NOMA system with discrete input distributions was studied in [43], where the solution is intuitive based on the deterministic approximation of the actual fading channel. The discrete input alphabets were also considered for a twouser interference channel to evaluate the capacity inner bound in [44]. In other words, all NOMA designs provided in [40, 41, 42, 43, 44] used mutual information as the performance measure, where the solutions were numerical and limited insights on the relationship between the sumconstellation and each user’s constellation can thus be drawn from the obtained solutions.
IB Motivation and Contributions
Inspired by the aforementioned work, in this paper we target a closedform NOMA design for a classical twouser Gaussian MAC with finitealphabet inputs and an optimal maximum likelihood (ML) detector at the receiver, where the two users are allowed to transmit simultaneously in the same frequency band. Finding the capacity bound of a Gaussian MAC with Gaussian inputs and adaptive power control has always been a classic problem, see e.g., [15, 22, 23, 16, 17, 18, 48, 29]; the optimal power control scheme for the Gaussian MAC with finitealphabet inputs, however, is still an open problem and only numerical solutions are available [40, 41, 38, 49]. To fill this gap, in this paper we, for the first time, investigate the optimal power control problem for the twouser Gaussian MAC with finite square QAM constellations that maximizes the minimum Euclidean distance of the received signals with the maximum likelihood (ML) detector. Note that QAM signaling is more spectrally efficient than other commonlyused constellations such as PSK signaling. Nevertheless, the NOMA design with QAM is more challenging than that with PSK since in QAM both the amplitude and the phase of the modulated signal vary, while in PSK only the phase is different, and thus the unambiguity of the sumconstellation at the receiver side is much more difficult to maintain. Here, it is worth pointing out that in our previous work [36], the NOMA design for the Gaussian Zchannel with QAM constellations was investigated, which incorporates the considered twouser MAC as a special case. In particular, to resolve the formulated problem, Farey sequence [50] was introduced to characterize the minimum Euclidean distance of the sumconstellation. However, due to the inherent symmetric structure between numerators and denominators of the conventional Farey sequence, our results presented in [36] refer to the case where both transmitters need to use an identical constellation size implying the same transmission rate. However, the transmission rates of the users are not necessarily the same in practice due to their distinct quality of service (QoS) requirements. To our best knowledge, the NOMA design in terms of power control at users for the Gaussian MAC with not necessarily identical QAM constellations still remains an open problem.
The main contributions of this paper can be summarized as follows:

We develop a practical NOMA design for the classical twouser complex Gaussian MAC, where the two users are allowed to adopt not necessarily the same QAM constellations. In our design framework, we aim to maximize the minimum Euclidian distance of the received sumconstellation at the receiver side, which dominates the error performance of the considered system, by adjusting the transmit power and phase of each user. To this end, we first decompose the complex MAC design problem into two real MAC design problems by strategically rotating the phase of the input signals at the two users. Nevertheless, the decomposed problems are still nontrivial due to their mixed continuousanddiscrete feature. Furthermore, our Farey sequencebased design framework developed in [36] can no longer be applied here due to the fact that the two users may use different QAM constellations.

To address this challenging problem, we define a new type of Farey sequence, termed punched Farey sequence, which is essential for our NOMA design with not necessarily the same QAM constellations. This concept is even mathematically new to the best of our knowledge [50]. We identify and rigourously prove several important properties of the punched Farey sequence in parallel to the conventional Farey sequence. Based on the punched Farey sequence and its important properties, we manage to resolve the above decomposed problem for each channel branch by providing a neat closedform optimal solution, which reveals that the optimal sumconstellation is a regular QAM constellation of a larger size. Due to this nice structure of the sumconstellation, a simple quantization decoder can be employed to implement the ML detector.

Based on the obtained closedform solution, we prove the superiority of this NOMA design over the timedivision multiple access (TDMA) approach in terms of the minimum Euclidean distance at the receiver for arbitrary given channel realization and rate allocation. Actually, this is a surprising result since the new NOMA method can achieve a better error performance than TDMA in a high SNR regime even if there is no nearfar effect. Furthermore, we also address the optimal rateallocation problem among the two users to maximize the minimum Euclidean distance of the received sumconstellation subject to a total rate constraint. More importantly, we derive a highrate approximate solution to the optimal rateallocation problem, which uncovers a lot of insights on the practical system designs.
Ii TwoUser Gaussian MultipleAccess Channel
We consider a twouser Gaussian MAC given by
(1) 
where is the received signal at the base station (BS), denotes the complex channel coefficient between the transmitter and BS for , and is the additive zeromean, circularly symmetric complex Gaussian (CSCG) noise with variance , i.e., . We assume that perfect channel state information (CSI) is available to all the nodes^{2}^{2}2The optimal design can also be performed at the BS which sends the results back to the transmitters via the forward links. In this case, only BS needs to know the full CSI. and symbol synchronization is maintained at BS. The transmitted symbols are superimposed at the receiver in a NOMA manner which are chosen randomly, independently and equally likely from the (finite) square QAM constellation , and are subject to individual average power constraint , i.e., for .
Although we use a complex baseband representation in (1), the modulated and demodulated signals are real since the oscillator at the transmitter can only generate real sinusoids rather than complex exponentials, and the channel then introduces amplitude and phase distortion to the transmitted signals [8]. As such, we follow [41] to decompose the considered complex Gaussian MAC given in (1) into two parallel realscalar Gaussian MACs, which are called the inphase and quadrature components, respectively [8]. This means that the original twodimensional QAM constellation can be split into two onedimensional PAM constellations to be transmitted via the inphase and quadrature branches. Besides, since the inphase and quadrature components of the sumconstellation are separable, they can be decoded independently at the receiver, thereby reducing the decoding complexity. Mathematically, we notice that (1) is equivalent to
(2) 
To simplify the subsequent expressions, we let , , , , , , and , where and are the real and imaginary parts of the complex number, respectively. Besides, , , , and are the real nonnegative scalars determining the minimum Euclidean distance of the actual transmitted PAM constellation sets, which are referred to as the weighting coefficients throughout this paper. Now, the inphase and quadrature branches of (1) can be reformulated by
(3a)  
(3b) 
where are independent and identically distributed (i.i.d.) real additive white Gaussian components since the complex noise term is assumed to be CSCG noise.
Without loss of generality, we assume that and , where and are  and ary square QAM constellations ( and are both no less than 2 but not necessarily equal to each other), respectively, given by and . As a result, the informationbearing symbols , sent by , and , transmitted by , are drawn from the standard PAM constellations with equal probability. We consider that an equal power allocation between two branches is performed to balance the minimum Euclidean distance of the two PAM constellations [8, Ch. 6.1.4] and the transmitted signals over both subchannels should still be subject to average power constraints, i.e., , , , and .
An important problem for the considered MAC is, for any given QAM constellation sizes of both messages, how to optimize the values of scaling coefficients , , and to minimize the average error probability at the receiver, subject to the individual average power constraints at both transmitters. As the inphase and quadrature subchannels are symmetric, if the same algorithm is applied to both branches, we will expect to have and , and we call and the symmetric square QAM constellations. It is worth mentioning that our framework can be readily extended to unsymmetric signaling [51, 52], i.e., unequal power allocation between the two branches. By leveraging the decomposable property of the complex Gaussian MAC and the symmetry of the two subchannels, we can simply focus on the design for one of the two realscalar Gaussian MACs with PAM constellation sets, which will be elaborated in next section^{3}^{3}3It should be pointed out that designing two PAM constellations for both subchannels separately is a practical but not necessarily optimal approach. In fact, this approach has been widely adopted in literature, such as in [53, 54, 52, 55, 56, 51]. How to design a twodimensional complex constellation directly for the Gaussian MAC has been left as a future work..
Iii The Weighting Coefficients Design for the RealScalar Gaussian MAC
In this section, we consider the constellation design problem, i.e., finding the optimal weighting coefficients and , for the inphase realscalar Gaussian MAC. As the two subchannels are symmetric, the optimal solution to the quadrature component can be obtained in exactly the same way and hence is omitted for brevity.
Iiia Problem Formulation
Recall that , , and hence , . For notation simplicity, we set , and
(4) 
where and . The received signal in (3a) can thus be rewritten as
(5) 
We assume that a coherent maximumlikelihood (ML) detector is used by BS to estimate the transmitted signals in a symbolbysymbol fashion^{4}^{4}4Since we perform a symbolbysymbol detection, the decoding complexity is at most with and being the PAM constellation size of and , respectively. . Mathematically, the estimated signals can be expressed as
By applying the nearest neighbour approximation method [8, Ch.6.1.4] at high SNRs for ML receiver, the average error rate is dominated by the minimum Euclidean distance of the received constellation points owing to the exponential decaying of the Gaussian distribution. As such, in this paper, we aim to devise the optimal value of (or equivalently constellations and ) to maximize the minimum Euclidean distance of constellation points of the received signal. The Euclidean distance between the two received signals and at the receiver for and in the noisefree case is given by
(6) 
Note that , , and are all odd numbers, and thus we can let and , in which and with denoting the set containing all the possible differences. Similarly, we also define , and where . From the definitions above, is equivalent to (i.e., or ). To proceed, we define
(7) 
where . We are at a point to formally formulate the following maxmin optimization problem,
Problem 1
Power Control of NOMA in realscalar MAC with PAM constellation: Find the optimal value of subject to the individual average power constraint such that the minimum Euclidean distance of the received signal constellation points is maximized, i.e.,
(8a)  
(8b) 
Note that the inner optimization variable of finding the minimum Euclidean distances is discrete, while the outer one is continuous. In other words, Problem 1 is a mixed continuousdiscrete optimization problem and it is in general hard to solve. To the best of our knowledge, only numerical solutions to such kind of problems are available in the open literature [40, 41, 38, 49]. To optimally and systematically solve this problem, we now develop a design framework based on the Farey sequence [50], in which the entire feasible region of is divided into a finite number of mutually exclusive subregions. Then, for each subregion, the formulated optimization problem can be solved optimally with a closedform solution, and subsequently the overall maximum value of Problem 1 can be attained by taking the maximum value of the objective function among all the possible subregions. We first consider the inner optimization problem in (8) given by:
Problem 2
Finding differential pairs with the minimum Euclidean distance:
(9) 
We should point out that finding the closedform solution to the optimal for (9) is not trivial since the solution depends on the values of and , which can span the whole positive real axis. Moreover, the values of and will be optimized later and cannot be determined beforehand. It is worth mentioning here that a similar optimization problem was formulated and resolved for a Gaussian Z channel in our previous work [36]. In [36], we resorted to the existing Farey sequence to solve the formulated problem. However, due to the inherent symmetric structure between numerators and denominators of the conventional Farey sequence, our results presented in [36] refers only to the case where both transmitters need to use exactly identical constellation size (i.e., the same transmission rate) and thus cannot be applied to the problem in this paper with and not necessarily the same. Motivated by this, in this paper we define a new type of Farey sequence, termed punched Farey sequence. In the subsequent section, we will introduce the definition and some important properties of the original Farey sequence and the developed punched Farey sequence.
IiiB Farey Sequence
The Farey sequence characterizes the relationship between two positive integers and the formal definition is given as follows:
Definition 1
Farey sequence [50]: The Farey sequence is the ascending sequence of irreducible fractions between and whose denominators are less than or equal to .
By the definition, is a sequence of fractions such that and arranged in an increasing order, where denotes the largest common divider of nonnegative integers . In addition, is the cardinality of with being the Euler’s totient function [50]. An example of Farey sequence is given as follows:
Example 1
is the ordered sequence .
It can be observed that each Farey sequence begins with number 0 (fraction ) and ends with 1 (fraction ). The series of breakpoints after is the reciprocal version of the Farey sequence. We call the Farey sequence together with its reciprocal version as the extended Farey sequence which is formally defined as follows:
Definition 2
Extended Farey sequence: The extended Farey sequence of order is the sequence of ascending irreducible fractions, where the maximum value of the numerator and denominator do not exceed .
From the definition, we have with and . We have the following example:
Example 2
is the sequence .
It can be observed that the extended Farey sequence starts with number 0 (fraction ) and end with (fraction ). We now propose a new definition called Punched Farey sequence in number theory as follows.
Definition 3
Punched Farey sequence: The punched (extended) Farey sequence is the ascending sequence of irreducible fractions whose denominators are no greater than and numerators are no greater than .
Example 3
is the ordered sequence .
From Definition 3, when , degenerates into Farey sequence , i.e., . We can also observe that each punched Farey sequence begins with number 0 (fraction ) and ends with (fraction ).
We now develop some elementary properties of the punched Farey sequence in line with Farey sequences [50]. It is worth pointing out that, although for some properties, we can find the counterparts in conventional Farey sequences, the extension to the punched Farey sequences is nontrivial and the following results are new.
Property 1
If and are two adjacent terms (called Farey pairs) in () such that , then, 1) , ; 2) ; 3) If , then and if , then ; 4) where the equality is attained if and only if and . Likewise, where the equality is attained if and only if and .
The proof is given in AppendixA.
Property 2
If , and are three consecutive terms in with such that , then .
The proof is provided in AppendixB.
Property 3
Consider with , such that where are successive in , then and .
The proof is provided in AppendixC.
IiiC The Minimum Euclidean Distance of the Constellation Points of the Received Signal
We are now ready to solve Problem 2 to find the differential pairs having the minimum Euclidean distance. To this end, we first introduce the following preliminary propositions.
Proposition 1
Let , and then
The proof is similar to [36, App.A] and hence is omitted for brevity.
Proposition 2
Let and be two terms of such that . Then, for and , we have 1) If , then ; 2) If , then ; 3) If , then .
The proof can be found in AppendixD.
Proposition 3
For any with , such that , and are successive in , we have 1) If , then ; 2) If , then .
The proof is given in AppendixE.
IiiD ClosedForm Optimal Solution to Problem 1
With the propositions presented in the previous subsection, we now can solve Problem 1 by restricting into a certain punched Farey interval determined by the corresponding Farey pair where a closedform solution is attainable. More specifically, we consider the punched Farey sequence given by , where . Now, assume that where is the th punched Farey interval for , and we aim to find the optimal such that
(10a)  
(10b) 
By applying the propositions in last subsections, we obtained the following lemma related to the optimal solution to problem (10).
The proof of Lemma 1 can be found in AppendixF.
Now, we are ready to present the closedform optimal solution to Problem 1 in terms of instead of defined in (4) for clarity, which maximizes the minimum Euclidean distance of the sumconstellation, denoted by , over the entire feasible region.
Theorem 1
Closedform optimal weighting coefficients: The optimal solution to Problem 1 in terms of is given by:
(11) 
The resulting minimum Euclidean distance in each case is: