On Non-Orthogonal Multiple Access with Finite-Alphabet Inputs in Z-Channels

# On Non-Orthogonal Multiple Access with Finite-Alphabet Inputs in Z-Channels

## Abstract

This paper focuses on the design of non-orthogonal multiple access (NOMA) in a classical two-transmitter two-receiver Z-channel, wherein one transmitter sends information to its intended receiver from the direct link while the other transmitter sends information to both receivers from the direct and cross links. Unlike most existing designs using (continuous) Gaussian input distribution, we consider the practical finite-alphabet (i.e., discrete) inputs by assuming that the widely-used quadrature amplitude modulation (QAM) constellations are adopted by both transmitters. To balance the error performance of two receivers, we apply the max-min fairness design criterion in this paper. More specifically, we propose to jointly optimize the scaling factors at both transmitters, which control the minimum Euclidean distance of transmitting constellations, to maximize the smaller minimum Euclidean distance of two resulting constellations at the receivers, subject to an individual average power constraint at each transmitter. The formulated problem is a mixed continuous-discrete optimization problem and is thus intractable in general. By resorting to the Farey sequence, we manage to attain the closed-form expression for the optimal solution to the formulated problem. This is achieved by dividing the overall feasible region of the original optimization problem into a finite number of sub-intervals and deriving the optimal solution in each sub-interval. Through carefully observing the structure of the optimal solutions in all sub-intervals, we obtain compact and closed-form expressions for the optimal solutions to the original problem in three possible scenarios defined by the relative strength of the cross link. Simulation studies are provided to validate our analysis and demonstrate the merits of the proposed design over existing orthogonal or non-orthogonal schemes.

## 1Introduction

Multiple access technologies have been playing an important role in determining the performance of each generation of mobile communication systems. Based on how the resources are allocated to users, multiple access technologies can generally be categorized into two types: orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) [1]. The current generation of cellular networks, known as 4G, and all previous generations have primarily adopted the OMA technologies, which include frequency-division multiple access (FDMA) for 1G, time-division multiple access (TDMA) for 2G, code-division multiple access (CDMA) for 3G, and orthogonal frequency-division multiple access (OFDMA) for 4G [2]. In these OMA schemes, the resource is partitioned into orthogonal blocks in time/frequency/code domain, and each resource block is then assigned to one single user exclusively. In this sense, there is no inter-user interference in OMA, leading to low-complexity receiver and scheduling algorithms. Moreover, after the resource allocation, the multiple-user problem is divided into several point-to-point problems such that the well-established single-user encoder/decoder techniques can be directly applied. However, early information-theoretic studies showed that compared with NOMA, OMA has lower spectral efficiency as it normally cannot achieve the multi-user channel capacity region [3]. Besides, OMA is not scalable as the total number of orthogonal resources and their granularity strictly limit the maximum number of served users.

Different from OMA, NOMA exploits the power domain to multiplex multiple users together such that they can be served in the same time/frequency/code resources [2]. As such, with proper multi-user detection techniques to deal with the inter-user interference at the receiver side (e.g., successive interference cancellation (SIC) [7]), NOMA is capable of achieving improved spectral efficiency and serving much more users simultaneously. In fact, the uplink and downlink versions of NOMA, well-known as multiple access channel (MAC) and broadcast channel (BC) respectively, have been intensively investigated for several decades in the information theory community, see, e.g., [8]. However, due to the high complexity of interference cancellation, these studies mainly lied in the theoretical aspects and their results were not implemented in practical communication systems. With the fast advances of hardware, the implementation of NOMA with interference cancellation becomes more affordable and feasible. Actually, NOMA has been regarded as a key enabling technology to meet the unprecedented requirements of 5G wireless networks due to its significant network throughput gain and great potential to support massive connectivity, low latency and user fairness [7]. Furthermore, a two-user downlink scenario of NOMA, termed multiuser superposition transmission (MUST), has been incorporated in the 3rd Generation Partnership Project (3GPP) Long Term Evolution-Advanced (LTE-A) [18].

Most conventional information-theoretic and recent studies on NOMA adopted the assumption of Gaussian input distribution [8]. Although the designs with Gaussian signaling can approach most of the known capacity inner bounds, such as in [8], their direct implementation in practical communication systems may lead to significant performance loss [26]. Moreover, Gaussian signaling will require unaffordable encoding and decoding efforts, which could lead to extremely high hardware cost, huge storage capability, high computational complexity, and long delay. Therefore, Gaussian inputs could arguably be infeasible for current hardware and it acts mostly as the theoretical benchmark. The inputs of practical wireless systems are actually drawn from finite constellations, such as phase shift keying (PSK) modulation or quadrature amplitude modulation (QAM), which are essentially different from the continuous Gaussian inputs. When it comes to a NOMA system with finite input constellations, the key design challenge is to guarantee that each user’s codeword can be uniquely decoded from their sum signal at the receiver side [27]. For the two-user MAC with finite-alphabet inputs, a constellation rotation (CR) scheme and a constellation power allocation (CPA) scheme were proposed in～[30] and～[31] to construct an unambiguous sum constellation at the receiver, respectively. This is achieved by strategically introducing certain angle of rotation between the input constellations in the CR scheme and appropriately controlling the transmit power of each user in the CPA scheme. The results in～[30] and～[31] have been extended to various multiple-antenna scenarios, see, e.g., [32] and references therein. The aforementioned NOMA designs were primarily for the PSK modulations by utilizing its circular structure. The studies on NOMA with QAM, another practical modulation scheme that has been widely adopted in cellular systems due to its higher spectral efficiency, are quite limited. Very recently, the mutual information were used as the performance metric to optimize NOMA systems with QAM in [33]. However, the optimal NOMA designs in [33] were achieved by numerical approaches with high computational complexities.

Motivated by this gap, in this paper we concentrate on the practical design of NOMA with QAM in a two-transmitter two-receiver ZC. It is worth emphasizing that the design of NOMA with QAM is much more challenging than that for PSK modulation. This is mainly because the unambiguity of sum QAM constellations is much harder to maintain since its signal points are distributed more evenly and there is a higher probability that more than one signal points coincide or close to each other on the sum constellation. The main contributions of this paper can be summarized as follows:

1. We, for the first time, develop a practical NOMA framework with QAM and max-min fairness in ZC. In our framework, we optimize the scaling factors of both transmitters, which adjust the minimum Euclidean distance of the transmitting constellations, to maximize the smaller minimum Euclidean distance among the resulting constellations at both receivers subject to an individual power constraint on each transmitter. Through our design, the average error performance of both transmitters in the considered ZC can be minimized with good user fairness, which is fundamentally different from the existing designs that mainly focused on the channel capacity maximization.

2. The formulated optimization problem is shown as a mixed continuous-discrete optimization problem, which is challenging to solve in general. By carefully observing the features of the formulated problem, we realize that the Farey sequence (also known as Farey series) [49] can be applied to resolve the problem. More specifically, by taking the advantage of Farey sequence associated with the finite-alphabet, we strategically partition the entire feasible region of the original optimization problem into a finite number of sub-intervals and attain the closed-form solution in each sub-interval. Then, by a careful observation on the structure of the solutions in all sub-intervals, the overall solution is obtained in a compact closed-form for three complementary scenarios divided by the relative strength of the cross link.

3. We verify the correctness of the analytical results by conducting simulations in both deterministic and random fading channels. Simulation results show that the sum constellation at the receiver side is still a regular QAM constellation with a larger size for most scenarios, but could be a hierarchical QAM with two-resolution [50], e.g., when the cross link is very strong relative to the direct link. We adopt the bit error rate (BER) as the performance metric to compare the proposed NOMA design with the existing OMA and NOMA schemes under random fading channels. The comparison illustrates that our scheme can achieve a significant lower BER performance than the benchmark schemes, which validates the effectiveness of our design.

## 2System Model of Complex Gaussian ZC with QAM Constellations

We consider a two-user complex Gaussian ZC consisting of two transmitters and , and two receivers and , as depicted in Fig.  ?. We consider that each node is equipped with a single antenna and works in a half-duplex mode. As per the ZC, only the cross link between and is assumed to be available. Moreover, sends one unicasting message to , while transmits one multicasting message to both and . and transmit their messages simultaneously using the same frequency band. The equivalent complex baseband signals observed at and can be given, respectively, by the following equations:

where , denotes the complex channel coefficients from transmitter to receiver . Hereafter, we call and the direct links, while is referred to as the cross link. In line with [35], all the channel links are assumed to be known perfectly at all the terminals. The additive noise processes are independent and identically distributed (i.i.d.) over time and are assumed to be circularly symmetric complex Gaussian (CSCG). Note that the case with different noise levels at receivers can be incorporated into our model by scaling operations. We suppose that QAM constellations are used by both transmitters since it is more spectrally efficient than other frequently-used modulation schemes such as phase-shift keying (PSK), and is also relatively easy to implement [51]. A predefined average transmitted power constraint is imposed to both transmitters1, i.e., and . In this paper, the system signal-to-noise ratio (SNR) is defined by .

For analytical simplicity, we decompose the considered complex Gaussian ZC given in into two parallel real scalar Gaussian ZCs [35], which are called the in-phase and quadrature components, respectively. We note that this method was commonly used in the study of IC and the real IC was studied directly in [52]. Actually, designing two-dimensional QAM constellations is an extremely challenging problem even for two-user MAC, see e.g., [30] and references therein. In this paper, instead of designing the two-dimensional QAM constellations directly, we propose a practical design that decomposes the complex Gaussian ZC into two parallel real scalar Gaussian ZCs such that we can split the two-dimensional QAM constellation into two one-dimensional PAM constellations. In fact, although we use complex baseband representation in , the actual modulated and demodulated signals are all real since the oscillator at the transmitter can only generate real sinusoids rather than complex exponentials, and the channel just introduces amplitude and phase distortion to the transmitted signals [51]. By this means, the original two-dimensional QAM constellation can be split into two one-dimensional pulse amplitude modulation (PAM) constellations for both the in-phase and quadrature components. Mathematically, for the complex Gaussian ZC described in , the in-phase and quadrature components can be attained by rotating and according to the instantaneous channel coefficients to compensate for the phase offset, and then taking the real and imaginary parts, respectively. First of all, we note that is equivalent to

Now, we set

where and are the real and imaginary parts of the complex number, respectively. We also assume that , , sent by , and , , transmitted by , are the information-bearing symbols, which are drawn from standard PAM constellation with equal probability, in which is a -ary PAM constellation set. Moreover, the scaling factors , , , and are real positive scalars that determine the minimum Euclidean distance of the corresponding PAM constellation set.

From , we obtain and , where

are - and -ary QAM constellations, respectively. If and , we call and symmetric QAM constellations. Otherwise, we are using unsymmetric signaling [45]. In addition, are i.i.d. real additive white Gaussian since the complex noise terms are assumed to be CSCG. Then, the in-phase and quadrature sub-channels of , as illustrated in Fig. ?, can be reformulated by

The transmitted signals over both subchannels should still be subject to average power constraints, i.e., , , , such that and . The following power allocation among the in-phase and quadrature components is normally performed to balance the minimum Euclidean distance of the two PAM constellations [51], i.e.,

It can be observed that, if square-QAM constellations are used at both transmitters with and , we have and .

An important problem for the considered ZC is that 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 both receivers, subject to the individual average power constraint at both transmitters. By leveraging the decomposable property of the complex Gaussian ZC and the symmetry of the two subchannels, we can simply focus on the design for one of the two real Gaussian ZCs with PAM constellation sets, which will be elaborated in the next section2.

## 3The Constellation Design for the Real Gaussian ZC

In this section, we consider the constellation design problem, i.e., finding the optimal values of and for the in-phase real Gaussian ZC characterized by . The optimal solution to the quadrature component can be obtained in a similar fashion and hence omitted for brevity. In particular, if and , then the two sub-channels are identical. It is worth noting that, similar design for BC or MAC can be included as a special case of our proposed design for the considered ZC.

### 3.1Problem Formulation

As the first effort towards the design of NOMA with finite-alphabet inputs in ZC, in this paper we concentrate on the case that and . As a result, we have . As and , we thus have and .

In our scheme, the transmitted signal from and are superimposed together at , which is inherently a non-orthogonal transmission. In line with [36], we use a joint decoding3

at the receiver since the error performance is dominated by the minimum Euclidean of the resulting sum-constellation at . We assume that each receiver uses a coherent maximum-likelihood (ML) detector to estimate the transmitted signals in a symbol-by-symbol fashion4. For receivers and , the estimated signals can be expressed as

By applying the nearest neighbor approximation method [51] at high SNRs for the 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. To balance the error performance of both receivers, in this paper, we aim to devise the optimal value of and by applying the max-min fairness criterion on the minimum Euclidean distance of the received signal constellation points among both receivers.

The Euclidean distance between two received signals and at and that between and at for the transmitted signal vectors and at and in the noise-free case are given, respectively, by

Note that , , and are all odd numbers, and we thus can let and , in which with denoting the set containing all the possible differences. Similarly, we also define , and . From the above definition, is equivalent to . Here, by , we mean that or . To proceed, we define

We are now ready to formally formulate the following max-min optimization problem:

Note that the inner optimization problem of finding the minimum Euclidean distances is a discrete one, while the outer optimization problem on is a continuous problem. In other words, Problem ? is a mixed continuous-discrete 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 literature, see e.g., [30] and references therein.

To optimally and systematically solve this problem, we now develop a novel framework based on the Farey sequence (also known as Farey series) [49], which can divide the entire feasible region of into a finite number of mutually exclusive sub-intervals. Then for each sub-interval, the formulated optimization problem can be solved optimally with a closed-form solution, and subsequently the overall maximum value of Problem ? can be attained by taking the maximum value of the objective function among all the sub-intervals.

For the inner optimization problem of given in , it can be observed that

However, for , we have

We should point out that the closed-form solution to the optimal is not trivial, since the solution depends on the values of and , which can span the whole positive real axis. Moreover, the value of and can not be determined beforehand. Actually, the problem in is essentially equivalent to finding a real rational number with finite order to approximate a real irrational number as closely as possible. This naturally leads us to resorting to the Farey sequence, which particularly plays a critical role in solving such kind of problems [49]. In the subsequent section, we will introduce the definition and some important properties of Farey sequence.

### 3.2Farey Sequence

The Farey sequence characterizes the relationship between two positive integers and the formal definition is given as follows:

By the definition, is a sequence of fractions such that and arranged in an increasing order, where denotes the largest common divider of non-negative integers . is the cardinality of with being the Euler’s totient function [49]. Some examples of Farey sequences are given as follows:

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 number sequence together with its reciprocal version as the extended Farey sequence which is formally defined as follows:

From the definition, we have with and . We have the following example:

It can be observed that the extended Farey sequence starts with number 0 (fraction ) and end with (fraction ).

The positive real axis can be divided by the extended Farey sequence into a finite number (i.e., ) of intervals. In this paper, we call the fractions consisting of adjacent terms in the extended Farey sequence as a Farey pair, and the interval between the Farey pair is referred to as a Farey interval. We then have the Farey interval set formally defined as follows:

By the definition, we have , where . Note that, with a slight abuse of notations, denotes the interval between end nodes and rather than a vector, and this will be clear from the context.

The Farey interval set can be further divided into two subsets and for such that and . In particular, and while and .

We now review some elementary properties of Farey sequences [49] which are also true for extended Farey sequences.

### 3.3The Minimum Euclidean Distance of the Received Signal Constellation Points

We are now ready to solve the problem in to find the constellation point pairs that have the minimum Euclidean distance. To that end, we first have the following preliminary propositions.

The proof is given in Appendix ?.

The proof can be found in Appendix ?.

The proof is provided in Appendix ?.

The proof is given in Appendix ?.

### 3.4Optimal Solution to Problem for |h21|w2|h11|w1 in Certain Farey Interval

In this section, we solve Problem ? by restricting into a certain Farey interval where a closed-form solution is attainable. We consider the Farey interval set given by where . Now we consider the case , and we aim to find the optimal such that

By applying the propositions in last subsections, we manage to attain the following lemma in terms of the optimal solution to problem .

The proof of Lemma 1 can be found in Appendix ?.

### 3.5The Optimal NOMA Design with PAM Constellation for the Gaussian ZC

Now we are ready to give the closed-form optimal solution of to Problem ? which maximizes the minimum Euclidean distance over all the Farey intervals for the aforementioned three scenarios.

For this case, we have . Consider the Farey interval set . By Property ?, we have , and therefore the positive real axis can be divided into the intervals in an increasing order:

In particular, we have and .

The proof of Theorem ? is provided in Appendix ?.

In this case, . We suppose that . Then, the optimal solution to Problem 1 can be obtained by considering the following two cases: the Farey interval set and . The whole discussions on them can be summarized as the following theorem.

The proof of Theorem 2 is very similar to that of Theorem 1 and the following Theorem 3 and thus, is omitted due to space limitation.

In this case, . Likewise, we consider the Farey interval set . Note that for . Then, by Property ? and Proposition ?, we have . As a result, the positive real axis can be divided into the following intervals in increasing order:

where .

The proof of Theorem ? is provided in Appendix ?.

## 4Simulation Results and Discussions

In this section, computer simulations are carried out to demonstrate the effectiveness of our proposed NOMA design under different channel configurations. More precisely, we compare our proposed NOMA design with CR based NOMA [30], time-division multiple access (TDMA) and frequency-division multiple access (FDMA) approaches. Without loss of generality, we set . For simplicity, in the simulations, we assume that the same square-QAM constellation is adopted by both users, i.e., and according to , we have and .

### 4.1The Resulting Optimal Sum Constellation at Receiver D1 For Several Deterministic Channels

Here, we consider several deterministic channels corresponding to the three scenarios of weak cross link, strong cross link and very strong cross link. We assume that 16-QAM constellations are used by both users with . We discuss the results of these cases one by one as follows:

In this case, we assume that , and . Based on the derived expressions for the optimal solution provided in the previous section, we can readily obtain that . The corresponding received constellation at is plotted in Fig.  ?. It can be observed that the sum-constellation at is a regular 256-QAM generated by the superposition of two 16-QAM. Hereafter, we call the signal constellation with smaller minimum Euclidean distance as the satellite constellation. By observing that and (i.e., ), we can deduce that the constellation used by forms the satellite constellation of the sum constellation at .

We investigate two channel realizations for this scenario. For the first realization, we let , and . We then have and the resulting sum-constellation at is also regular, as illustrated in Fig.  ?. Since and (i.e., ) in this case, the constellation used by forms the satellite constellation at . For the second realization, we set , and , leading to . The resulting constellation plotted in Fig.  ? is also uniform as in the previous two scenarios. We have and (i.e., ). Thus, the constellation used by the transmitter forms the satellite constellation at .

In this case, we suppose that , and , generating . The obtained constellation at is shown in Fig.  ?. In this case, we have and (i.e., ). However, it can be observed that , i.e., .

### 4.2Average Error Performance Comparison in Rayleigh Fading Channels

We now compare the average BER of the proposed NOMA scheme with that of three existing methods, including TDMA, FDMA, and NOMA with CR [30] methods, over Rayleigh fading channels with , and . Recall that we use error performance (i.e., BER) as the design criterion for the NOMA in ZCs with finite-alphabet inputs using fixed transmission rate (i.e., fixed constellation size). However, we are unable to compare the error performance of the considered system using finite-alphabet inputs with that of Gaussian inputs. This is because for Gaussian input, it is intractable to evaluate the BER for uncoded system since its input signal is continuous. Moreover, the BER for coded system with Gaussian input is hard to simulate due to the huge storage capacity requirement for the large codebook and the high computational complexity [59].

For TDMA, we assume that both users transmit alternatively by using half of the total time slots and thus no interference occurs at the destination side. More importantly, the individual instantaneous power constraints on both users and remain unchanged. On the other hand, for FDMA, each user uses only half of the available bandwidth. Due to the orthogonality between different frequency band, there is also no interference occurring at the destination side. Note that, in FDMA, the bandwidth occupied by each user is halved and the noise arises at the receiver is assumed to be white Gaussian. Therefore, the variance of the noise is also halved. In addition, for the CR based NOMA as proposed in [30], we let each user transmit at the maximum allowable power by using constellations and for user and , respectively.

In Fig.  ?, we consider that the variances of all channels are the same, i.e., , and the average BER over both receivers of all the methods are plotted against the SNR . In Fig.  ?, 16-QAM is used for the proposed NOMA scheme while 16-PSK is employed by the NOMA with CR. Since only half of the total time slots or total bandwidths are available for each transmitter, to maintain the same data rate for each user in each block compared with NOMA methods, we should increase the constellation size by using 256-QAM in both TDMA and FDMA. It can be observed that, our proposed method has significant BER gain over TDMA and FDMA methods, which confirms the effectiveness of the NOMA scheme. From the simulation results, we also find that FDMA has a smaller BER compared with TDMA. This is because the variance of the effective noise is smaller than that of TDMA. As the rotation based method uses the PSK constellation, which is not spectrally efficient, it has the worst BER performance. Then, in Fig.  ?, the average BER of all the cases are plotted against the SNR where 64-QAM is used by each user for the proposed NOMA and 64-PSK is used by CR based NOMA while 4096-QAM constellations are used by TDMA and FDMA methods. We also simulate another two cases with unequal channel variances. Specifically, the average BER of both receivers for all the considered methods is plotted in Fig.  ?, wherein the variances of three channels are set as . The case with the variances of three channels is provided in Fig.  ?. Similar observations can be seen as the previous case with equal channel variances, which further verifies the effectiveness of our proposed NOMA design.

## 5Conclusions

In this paper, we developed a novel and practical design framework for the non-orthogonal multiple access (NOMA) in a classical two-transmitter two-receiver Z-channel with widely-used quadrature amplitude modulation (QAM) and max-min user fairness. Specifically, we formulated a max-min optimization problem to jointly optimize the scaling factors at both transmitters to maximize the smaller minimum Euclidean distance among the two resulting signal constellations at both receivers, subject to the individual average power constraint at each transmitter. The formulated mixed continuous-discrete problem was successfully resolved in compact closed-form by strategically applying the Farey sequences and their unique properties. Simulation results verified the correctness of our analytical derivations and showed that the proposed NOMA design significantly outperforms the existing orthogonal multiple access and NOMA schemes, especially at high signal-to-noise ratio. Furthermore, the performance gap of the proposed scheme over its existing counterparts can be further enlarged when the size of constellations used at the transmitter side becomes larger.

## Appendix

### Footnotes

1. Our design can also be generalized to the case with peak power constraint straightforwardly.
2. It should be pointed out that this design is a practical but not necessarily optimal approach, which has been widely adopted in practice [36].
3. We note that, for ZIC, the joint decoder used by may not necessarily be the most efficient one. Instead, we should use a treat-interference-as-noise (TIN) receiver when the channel gain of the cross link is very low and use a successive-interference-cancellation (SIC) receiver when the channel gain of cross link is very strong compared with the direct link [44]. In general, a joint decoder can be used when the cross link is moderately strong [43], which will result in a similar design as our case. However, how to extend our design to ZIC with finite-alphabet input is still an open problem and has been left as a future work.
4. Since we are doing a symbol-by-symbol detection, the decoding complexity is with being the PAM constellation size of and , respectively. Although we can use the message passing algorithm (MPA) [58] to further decrease the decoding complexity, however, our method is feasible.

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