On Non-Orthogonal Multiple Access with Finite-Alphabet Inputs in Z-Channels
Zheng Dong He Chen Jian-Kang Zhang Lei Huang
cs.IT math.IT









On Non-Orthogonal Multiple Access with Finite-Alphabet Inputs in Z-Channels
    Zheng Dong, He Chen, Jian-Kang Zhang, and Lei Huang
	
	
		
		
		Z. Dong is with Shenzhen University, Shenzhen 518060, China, and also with McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail: dongz3@mcmaster.ca).

        H. Chen is with The University of Sydney, Sydney, NSW 2006, Australia (e-mail: he.chen@sydney.edu.au).

       J.-K. Zhang is with McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail: jkzhang@mail.ece.mcmaster.ca).

       L. Huang is with Shenzhen University, Shenzhen 518060, China (e-mail: lhuang@szu.edu.cn).

       
	
	

	
	
	
		
    May 29, 2018
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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.










Non-orthogonal multiple access (NOMA), Z-channel, finite-alphabet inputs, quadrature amplitude modulation, max-min fairness, Farey sequence.





















§ INTRODUCTION

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)  <cit.>. 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 <cit.>. 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 <cit.>. 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 <cit.>. 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) <cit.>), 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., <cit.>. 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 <cit.>. 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) <cit.>.

Most conventional information-theoretic and recent studies on NOMA adopted the assumption of Gaussian input distribution <cit.>. Although the designs with Gaussian signaling can approach most of the known capacity inner bounds, such as in <cit.>, their direct implementation in practical communication systems may lead to significant performance loss <cit.>.  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 <cit.>. For the two-user MAC with finite-alphabet inputs, a constellation rotation (CR) scheme and a constellation power allocation (CPA) scheme were proposed in～<cit.> and～<cit.> 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～<cit.> and～<cit.> have been extended to various multiple-antenna scenarios, see, e.g., <cit.> 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 <cit.>. However, the optimal NOMA designs in <cit.> were achieved by numerical approaches with high computational complexities.

We also notice that the existing studies on NOMA mainly focused on MAC and BC, due to their wide applications in centralized systems like cellular networks. Also there are some initial efforts considering the NOMA design for the interference channel (IC) <cit.>.  With recent advances in non-centralized networks (e.g., wireless sensor networks and ad hoc networks), Z-channel (ZC) was proposed in <cit.> and attracted considerable attention in the past decade <cit.>. As a special case of the classical two-user IC <cit.>, a two-user ZC consists of two transmitters and two receivers, wherein one transmitter sends information to its intended receiver from the direct link without causing interference to the unintended receiver, while the other transmitter sends information to both receivers from the direct and cross links. In this sense, the ZC is a general channel model that includes MAC and BC as special cases. The ZC is also closely related to the Z-interference channel (ZIC) <cit.>, wherein each transmitter transmits information to its corresponding receiver only from the direct link and the received signal from the  cross link carries no desired information, and thus is treated as interference at the receiver side. It is also worth emphasizing that there are two messages transmitted from the two direct links in ZIC, while three messages are sent via the two direct links and the cross link in ZC.

Both ZC and ZIC are proper models for the multi-cell downlink transmission, where one user is located near to the cell edge and thus can receive signals from both transmitters, while the other user is near the cell center and suffers from no interference. Another example corresponds to the two-user IC, where one cross link is blocked by obstacles with large pathloss such as tall buildings or thick walls, while the other user is still exposed to interference <cit.>. Despite of the existing great efforts on ZC, to our best knowledge, the design of NOMA with practical finite-alphabet inputs in ZC is still an open problem in the literature.

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:

	
  * 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.
	
  * 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) <cit.> 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.
	
  * 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 <cit.>, 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.
	
	





§ SYSTEM MODEL OF COMPLEX GAUSSIAN ZC WITH QAM CONSTELLATIONS













We consider a two-user complex Gaussian ZC consisting of two transmitters S_1 and S_2, and two receivers D_1 and D_2, as depicted in Fig. <ref>. 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 S_2 and D_1 is assumed to be available.
Moreover, S_1 sends one unicasting message x_1 to D_1, while S_2 transmits one multicasting message x_2 to both D_1 and D_2. S_1 and S_2 transmit their messages simultaneously using the same frequency band. The equivalent complex baseband signals observed at D_1 and D_2 can be given, respectively, by the following equations:

	
    z_1    =h_11 x_1+ h_21 x_2 +ξ_1,
    
    	z_2    =h_22 x_2 +ξ_2,


where h_kℓ∈ℂ, k,ℓ∈{1,2} denotes the complex channel coefficients from transmitter S_k to receiver D_ℓ. Hereafter, we call h_11 and h_22  the direct links, while h_21 is referred to as the cross link. In line with <cit.>, all the channel links  are assumed to be known perfectly at all the terminals. The additive noise processes ξ_1, ξ_2 ∼𝒞𝒩(0, 2σ^2) 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 <cit.>. A predefined average transmitted power constraint is imposed to both transmitters[Our design can also be generalized to the case with peak power constraint straightforwardly.], i.e., 𝔼[|x_1|^2]≤ P_1 and 𝔼[|x_2|^2] ≤ P_2. In this paper, the system signal-to-noise ratio (SNR) is defined by ρ=1/2σ^2.

For analytical simplicity, we decompose the considered complex Gaussian ZC given in (<ref>) into two parallel real scalar Gaussian ZCs <cit.>, 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 <cit.>.

Actually, designing two-dimensional QAM constellations is an extremely challenging problem even for two-user MAC, see e.g., <cit.> 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 (<ref>), 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 <cit.>. 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 (<ref>), the in-phase and quadrature components can be attained by rotating x_1 and x_2 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 (<ref>) is equivalent to

	
    z_1 =|h_11|exp(j(h_11)) x_1+|h_21|exp(j(h_21)) x_2+ξ_1 
       exp(j ( h_21/h_22))z_2=|h_22| exp(j(h_21)) x_2 
                               +exp(j ( h_21/h_22))ξ_2.



Now, we set


    y_1= Re(z_1), y_2= Re(exp (j ( h_21/h_22))z_2),
       w_1 s_1= Re(exp(j(h_11))x_1),
       w_2 s_2= Re(exp (j (h_21))x_2 ),
       n_1= Re(ξ_1),   n_2=  Re(exp(j ( h_21/h_22)) ξ_2); 
       y_1'= Im(z_1),   y_2'= Im(exp (j ( h_21/h_22))z_2),
       w_1' s_1'= Im(exp(j(h_11))x_1), 
       w_2' s_2'= Im(exp (j (h_21))x_2 ),
       n_1'= Im(ξ_1),   n_2'= Im(exp(j ( h_21/h_22)) ξ_2),


where Re(·) and Im(·) are the real and  imaginary parts of the complex number, respectively.
We also assume that s_1 ∈𝒜_M_1, s_1' ∈𝒜_M_1', sent by S_1, and s_2 ∈𝒜_M_2,  s_2' ∈𝒜_M_2', transmitted by S_2, are the information-bearing symbols, which are drawn from standard PAM constellation with equal probability, in which 𝒜_M ≜{± (2k-1)}_k=1^M/2 is a M-ary PAM constellation set. Moreover, the scaling factors w_1, w_2, w_1', and w_2' are real positive scalars that determine the minimum Euclidean distance of the corresponding PAM constellation set.

From (<ref>), we obtain exp(j(h_11)) x_1=  (w_1 s_1 + w_1' s_1'j) ∈𝒬_1 and exp(j(h_21)) x_2= (w_2 s_2 + w_2' s_2'j) ∈𝒬_2, where

	
    𝒬_1 ≜{± w_1 (2k-1) ± w_1' (2ℓ-1)j: 
                    k=1,…, M_1/2, ℓ=1,…, M_1'/2},
       𝒬_2 ≜{± w_2 (2k-1) ± w_2' (2ℓ-1)j:
                    k=1,…, M_2/2, ℓ=1,…, M_2'/2},

	
are M_1 M_1'- and M_2 M_2'-ary QAM constellations, respectively.
If w_1=w_1' and w_2=w_2', we call 𝒬_1 and 𝒬_2 symmetric QAM constellations. Otherwise, we are using unsymmetric signaling <cit.>. In addition, n_1, n_2, n_1', n_2'∼𝒩(0, σ^2) 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 (<ref>), as illustrated in Fig. <ref>, can be reformulated by


    y_1    = |h_11|w_1 s_1 + |h_21| w_2 s_2 +n_1, 
    
    y_2    = |h_22|w_2 s_2 +n_2,
    	
    y_1'   = |h_11|w_1' s_1' + |h_21| w_2' s_2' +n_1',
    
    y_2'   = |h_22|w_2' s_2' +n_2'.



The transmitted signals over both subchannels should still be subject to average power constraints, i.e., 𝔼[w_1^2 |s_1|^2]≤ p_1, 𝔼[w_2^2|s_2|^2]≤ p_2,  𝔼[w_1'^2 |s_1'|^2]≤ p_1', 𝔼[w_2'^2|s_2'|^2]≤ p_2' such that p_1+p_1'=P_1 and p_2+p_2'=P_2.
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 <cit.>, i.e.,

	
    p_1   =(M_1^2-1)P_1/M_1^2+M_1'^2-2, 
    	p_2=(M_2^2-1)P_2/M_2^2+M_2'^2-2,
    
    	p_1'   =(M_1'^2-1)P_1/M_1^2+M_1'^2-2, 
    	p_2'=(M_2'^2-1)P_2/M_2^2+M_2'^2-2.


It can be observed that, if square-QAM constellations are used at both transmitters with M_1=M_1' and M_2=M_2', we have p_1=p_1'=P_1/2 and p_2=p_2'=P_2/2.

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 w_1, w_2, w_1' and w_2' 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 section[It should be pointed out that this design is a practical but not necessarily optimal approach, which has been widely adopted in practice <cit.>.].













§ THE CONSTELLATION DESIGN FOR THE REAL GAUSSIAN ZC

In this section, we consider the constellation design problem, i.e., finding the optimal values of w_1 and w_2 for the in-phase real Gaussian ZC characterized by (<ref>). The optimal solution to the  quadrature component can be obtained in a similar fashion and hence omitted for brevity. In particular, if M_1=M_1' and M_2=M_2', 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.



 §.§ Problem 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 M_1=M_2=M and M_1'=M_2'=M'.

As a result, we have s_1, s_2 ∈𝒜_M={± (2k-1)}_k=1^M/2. As 𝔼[w_1^2 |s_1|^2]≤ p_1 and 𝔼[w_2^2|s_2|^2]≤ p_2, we thus have  0< w_1 ≤√(3 p_1/M^2-1) and 0< w_2 ≤√(3 p_2/M^2-1).


In our scheme, the transmitted signal from S_1 and S_2 are superimposed together at D_1, which is inherently a non-orthogonal transmission. In line with <cit.>, we use a joint decoding[We note that, for ZIC, the joint decoder used by D_1 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 <cit.>. In general, a joint decoder can be used when the cross link is moderately strong <cit.>, 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.]
 at the receiver D_1 since the error performance is dominated by the minimum Euclidean of the resulting sum-constellation at D_1.
We assume that each receiver uses a coherent maximum-likelihood (ML) detector to estimate the transmitted signals in a symbol-by-symbol fashion[Since we are doing a symbol-by-symbol detection, the decoding complexity is 𝒪(M^2) with M being the PAM constellation size of S_1 and S_2, respectively. Although we can use the message passing algorithm (MPA) <cit.> to further decrease the decoding complexity, however, our method is feasible.].
For receivers D_1 and D_2, the estimated signals can be expressed as

    (ŝ_1,ŝ_2)=min_(s_1, s_2) |y_1 -(|h_11|w_1 s_1 + |h_21| w_2 s_2)|,
       ŝ_2' =min_s_2 |y_2 -|h_22|w_2 s_2|.


By applying the nearest neighbor approximation method <cit.> 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 w_1 and w_2 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 y_1(s_1,s_2) and y_1(s̃_1, s̃_2) at D_1 and that between y_2(s_2) and y_2(s̃_2) at D_2
 for the transmitted signal vectors (s_1, s_2) and (s̃_1, s̃_2) at S_1 and S_2 in the noise-free case are given, respectively, by

	
    |y_1(s_1, s_2) -y_1(s̃_1, s̃_2)|=||h_11| w_1(s_1 -s̃_1)
                                 -|h_21| w_2 (s̃_2-s_2 ) |,
       |y_2(s_2) -y_2(s̃_2)|=|h_22|w_2 |s_2 -s̃_2|.



Note that s_1, s̃_1, s_2 and s̃_2 are all odd numbers, and we thus can let s_1 -s̃_1=2 n and s̃_2-s_2 =2 m, in which m, n∈ℤ_M-1 with ℤ_M-1≜{0,± 1,…, ± M-1}  denoting the set containing all the possible differences.  Similarly, we also define ℤ_M-1^2≜{(a,b):a,b ∈ℤ_M-1}, ℕ_M-1≜{0,1,⋯,M-1} and ℕ_M-1^2 ≜{(a,b): a,b ∈ℕ_M-1}. From the above definition, (s_1, s_2) ≠ (s̃_1, s̃_2) is equivalent to (m,n) ≠ (0,0). Here, by (m,n) ≠ (0,0), we mean that m ≠ 0 or n ≠ 0. To proceed, we define


    d_1(m,n)=1/2|y_1(s_1, s_2) -y_1(s̃_1, s̃_2)|
         =||h_11|w_1 n -  |h_21| w_2 m|,  for (m,n) ∈ℤ^2_M-1∖{(0,0)},
       d_2(m)=1/2|y_2(s_2) -y_2(s̃_2)|
         =|h_22| w_2 |m|,    for m ∈ℤ_M-1∖{0}.



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

[Optimal Design of NOMA in real scalar ZC with PAM constellation]
Find the optimal value of (w_1^*,w_2^*) subject to the individual average power constraint such that the minimum value of the minimum Euclidean distances of the received signal constellation points over both received signals is maximized, i.e.,
	
	
    (w_1^*, w_2^*)=max_(w_1, w_2) min{min_ (m,n) ∈ℤ_M-1^2 ∖{(0,0)} d_1(m,n)_T_1,
                                 min_m ∈ℤ_M-1∖{0} d_2(m)_T_2}
        s.t.  0< w_1 ≤√(3 p_1/M^2-1)  and  0< w_2 ≤√(3 p_2/M^2-1).

	
	▪


Note that the inner optimization problem of finding the minimum Euclidean distances is a discrete one, while the outer optimization problem on (w_1, w_2) is a continuous problem. In other words, Problem <ref> 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., <cit.> 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) <cit.>, which can divide the entire feasible region of (w_1, w_2) 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 <ref> can be attained by taking the maximum value of the objective function among all the sub-intervals.

For the inner optimization problem of T_2 given in (<ref>), it can be observed that

    min_m ∈ℤ_M-1∖{0} d_2(m)=min_m ∈ℤ_M-1∖{0} |h_22| w_2|m|
        =|h_22| w_2,  with m=1.

However, for T_1, we have

    min_ (m,n)∈ℤ_M-1^2 ∖{(0,0)} d_1(m,n) 
        = min_   (m,n)∈ℤ_M-1^2 ∖{(0,0)}||h_11|w_1 n -  |h_21| w_2 m|.

We should point out that the closed-form solution to the optimal (m,n) is not trivial, since the solution depends on the values of |h_11| and |h_22|, which can span the whole positive real axis. Moreover, the value of w_1 and w_2 can not be determined beforehand. Actually, the problem in (<ref>)  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 <cit.>. In the subsequent section, we will introduce the definition and some important properties of Farey sequence.



 §.§ Farey Sequence

The Farey sequence characterizes the relationship between two positive integers and the formal definition is  given as follows:
<cit.>
The Farey sequence 𝔉_K of order K is the ascending sequence of irreducible fractions between 0 and 1 whose denominators are less than or equal to K.
	

By the definition, 𝔉_K=(b_k/a_k)_k=1^|𝔉_K| is a sequence of fractions b_k/a_k such that 0≤ b_k≤ a_k≤ K and ⟨ a_k, b_k ⟩ =1  arranged in an increasing order, where ⟨ a,b⟩ denotes the largest common divider of non-negative integers a,b. |𝔉_K| =1+∑_m=1^K φ(m) is the cardinality of 𝔉_K with φ(·) being the Euler's totient function <cit.>.  Some examples of Farey sequences are given as follows:

    𝔉_5 is the ordered sequence
	
    (0/1, 1/5, 1/4, 1/3, 2/5,1/2, 3/5, 2/3,  3/4, 4/5, 1/1).









It can be observed that each Farey sequence begins with number 0 (fraction 0/1) and ends with 1 (fraction 1/1). The series of breakpoints after 1/1 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:

The extended Farey sequence 𝔖_K of order K is the sequence of ascending irreducible fractions, where the maximum value of the numerator and denominator do not exceed K.

	
From the definition, we have 𝔖_K=(b_k/a_k)_k=1^|𝔖_K| with ⟨ a_k,b_k⟩=1 and  |𝔖_K|=1+2∑_m=1^K φ(m). We have the following example:

	𝔖_5 is the sequence
	
    (0/1, 1/5, 1/4, 1/3, 2/5,1/2, 3/5, 2/3,  3/4, 4/5, 1/1,5/4,4/3,3/2,5/3,2/1,5/2,3/1,4/1,5/1,1/0).


It can be observed that the extended Farey sequence starts with number 0 (fraction 0/1) and end with ∞ (fraction 1/0).

The positive real axis can be divided by the extended Farey sequence 𝔖_K into a finite number (i.e., |𝔖_K|-1=2∑_m=1^K φ(m))  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:

A Farey interval set 𝒮_K of order K is the set containing all the Farey intervals generated by the Farey pair of the extended Farey sequence 𝔖_K.	

By the definition, we have 𝒮_K={(b_k/a_k, b_k+1/a_k+1)}_k=1^|𝒮_K|, where |𝒮_K|=|𝔖_K|-1=2∑_m=1^K φ(m). Note that,  with a slight abuse of notations, (b_k/a_k, b_k+1/a_k+1) denotes the interval between end nodes b_k/a_k and b_k+1/a_k+1 rather than a vector, and this will be clear from the context.

The	Farey interval set 𝒮_5 is the set given by
	
    {(0/1, 1/5), (1/5, 1/4), (1/4, 1/3), …, (3/1, 4/1), (4/1, 5/1),(5/1, 1/0)}.




The Farey interval set can be further divided into two subsets 𝒰_K^L={(b_k'/a_k', b_k+1'/a_k+1')∈𝒮_K: b_k' +b_k+1' ≥ L}  and 𝒱_K^L={(b_k”/a_k”, b_k+1”/a_k+1”)∈𝒮_K: b_k” +b_k+1” < L} for L=1,2,…, 2K such that 𝒮_K=𝒰_K^L ∪𝒱_K^L and 𝒰_K^L ∩𝒱_K^L=∅. In particular, 𝒰_K^1 = 𝒮_K and 𝒱_K^1=∅ while 𝒰_K^2K=∅ and 𝒱_K^2K=𝒮_K.


For the	Farey interval set 𝒮_5, we have

    𝒰_5^4=   {(1/2, 3/5),
    (3/5, 2/3),
    (2/3, 3/4),
    (3/4, 4/5),
    (4/5, 1/1),
         (1/1, 5/4),
    (5/4, 4/3),
    (4/3, 3/2),
    (3/2, 5/3),
    (5/3, 2/1),
           (2/1, 5/2),
    (5/2, 3/1),
    (3/1, 4/1),
    (4/1, 5/1),
    (5/1, 1/0)
    },

and

    𝒱_5^4=   {(0/1, 1/5),
    (1/5, 1/4),
    (1/4, 1/3),
    (1/3, 2/5),
    (2/5, 1/2)
    }.













We now review some elementary properties of Farey sequences <cit.> which are also true for extended Farey sequences.

If n_1/m_1 and n_2/m_2 are  two adjacent terms (Farey pairs) of 𝔖_K with K≥ 1, such that n_1/m_1< n_2/m_2, then,

	
  * m_1 n_2 - m_2 n_1=1.
	
  * n_1 + n_2/m_1 + m_2∈(n_1/m_1, n_2/m_2), m_1+m_2/n_1 + n_2∈(m_2/n_2, m_1/n_1). ▪





	If n_1/m_1, n_2/m_2  and n_3/m_3 are three consecutive terms of 𝔖_K with K≥ 1 such that n_1/m_1< n_2/m_2<n_3/m_3, then, n_2/m_2=n_1 + n_3/m_1 + m_3.
	 ▪



Given K≥ 2, we assume n_1/m_1, n_2/m_2, n_3/m_3, n_4/m_4∈𝔖_K and n_1/m_1< n_2/m_2<n_3/m_3<n_4/m_4. If n_2/m_2 and n_3/m_3 form one Farey pair, then n_1+n_3/m_1+m_3≤n_2/m_2 and n_3/m_3≤n_2+n_4/m_2+m_4.
 ▪




 §.§ The Minimum Euclidean Distance of the Received Signal Constellation Points

We are now ready to solve the problem in (<ref>) to find the constellation point pairs (m,n) that have the minimum Euclidean distance.
To that end, we first have the following preliminary propositions.


Let 𝔽_K^2 ={(m,n): n/m∈𝔖_K}, where 𝔖_K is the extended Farey number sequence of order K, and then we have

    min_   (m,n)∈ℤ_K^2 ∖{(0,0)} d_1(m,n)=min_   (m,n)∈𝔽_K^2  d_1(m,n).

▪

The proof is given in Appendix<ref>.


Consider the Farey interval (n_1/m_1, n_2/m_2) ∈𝒮_K, with K≥ 1, and n_1/m_1< n_2/m_2. Then, for |h_21| w_2/|h_11|w_1∈ (n_1/m_1, n_2/m_2) and d_1(m,n) =||h_11|w_1 n-|h_21| w_2 m|, we have	
	
		
  * If |h_21| w_2/|h_11|w_1= n_1 +n_2/m_1 + m_2, then d_1(m_1, n_1)=d_1(m_2, n_2);
		
  * If |h_21| w_2/|h_11|w_1∈(n_1/m_1, n_1 +n_2/m_1 + m_2), then d_1(m_1, n_1) <d_1(m_2, n_2);
		
  * If  |h_21| w_2/|h_11|w_1∈(n_1 +n_2/m_1 + m_2, n_2/m_2), then d_1(m_1, n_1)>d_1(m_2, n_2).  ▪
	

The proof can be found in Appendix<ref>.

Consider n_1/m_1,n_2/m_2,n_3/m_3,n_4/m_4∈𝔖_K, with K≥ 2, such that n_1/m_1<n_2/m_2<n_3/m_3<n_4/m_4 where n_2/m_2, n_3/m_3 form one Farey pair,

	
  * If |h_21| w_2/|h_11|w_1∈ (n_2/m_2,n_2+n_3/m_2+m_3), then
	
    min_(m,n) ∈𝔽_M-1^2 d_1(m,n)=d_1(m_2, n_2)
    
    		=|h_21|w_2 m_2- |h_11| w_1 n_2.








	
  * If  |h_21| w_2/|h_11|w_1∈ (n_2+n_3/m_2+m_3,n_3/m_3), then
	
    min_(m,n) ∈𝔽_M-1^2 d_1(m,n)=d_1(m_3, n_3)
    
    		 =|h_11| w_1 n_3 - |h_21|w_2 m_3.








 ▪	


The proof is provided in Appendix<ref>.
	
For the Farey interval set 𝒮_M-1={(b_k/a_k, b_k+1/a_k+1)}_k=1^|𝒮_M-1|, if |h_21|/|h_22|≥ b_k+b_k+1, then a_k+1/b_k+1+|h_22| /b_k+1|h_21|≤a_k+a_k+1/b_k+b_k+1≤a_k/b_k-|h_22| /b_k|h_21|. ▪	

The proof is given in Appendix<ref>.


 §.§ Optimal Solution to Problem <ref> for  |h_21| w_2/|h_11|w_1 in Certain Farey Interval














In this section, we solve Problem <ref> by restricting |h_21|w_2 /|h_11| w_1 into a certain Farey interval where a closed-form solution is attainable. We consider the Farey interval set 𝒮_M-1 given by 𝒮_M-1={(b_k/a_k,b_k+1/a_k+1)}_k=1^|𝒮_M-1| where |𝒮_M-1|=2∑_m=1^M-1φ(m). Now we consider the case |h_21|w_2 /|h_11| w_1∈(b_k/a_k, b_k+1/a_k+1),     k=1,2,…,|𝒮_M-1| and we aim to find the optimal
(w_1^*(k), w_2^*(k)) such that


    g(b_k/a_k, b_k+1/a_k+1) = max_(w_1, w_2) min{min_(m,n) ∈𝔽_M-1^2 d_1(m,n),
                                   min_m ∈ℤ∖{0} d_2(m)}
        s.t. b_k/a_k< |h_21|w_2 /|h_11| w_1≤b_k+1/a_k+1,0< w_1 ≤√(3 p_1/M^2-1)
                             and  0 < w_2 ≤√(3 p_2/M^2-1).



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

The optimal solution to (<ref>) is given as follows.
	
		
  * If |h_21|/|h_22|≤ b_k +b_k+1, the following statements are true:
		
			
  * If |h_11|/|h_21|≥√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1), then g(b_k/a_k, b_k+1/a_k+1)=|h_21|/b_k+b_k+1√(3 p_2/M^2-1) and
			(w_1^*(k), w_2^*(k))=((a_k + a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)).
			
  * If |h_11|/|h_21|<√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1),  then g(b_k/a_k, b_k+1/a_k+1)=|h_11|/a_k+ a_k+1√(3 p_1/M^2-1) and (w_1^*(k), w_2^*(k))=(√(3 p_1/M^2-1), (b_k +b_k+1)|h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)) .
		
		
  * If |h_21|/|h_22| > b_k +b_k+1, we have the following results:
		
			
  * If |h_11|/|h_21|≥√(p_2)/√(p_1)(a_k+1/b_k+1
			+|h_22| /b_k+1|h_21|),
			then  g(b_k/a_k, b_k+1/a_k+1)=|h_22|√(3 p_2/M^2-1) and  (w_1^*(k), w_2^*(k))=(a_k+1 |h_21|+|h_22| / b_k+1 |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)).
			
  * If |h_11|/|h_21|<√(p_2 )/√(p_1)(a_k+1/b_k+1
			+|h_22| /b_k+1|h_21|),
			then g(b_k/a_k, b_k+1/a_k+1)=b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1) and (w_1^*(k), w_2^*(k))=(√(3 p_1/M^2-1), b_k+1 |h_11|/a_k+1 |h_21|+|h_22|√(3 p_1/M^2-1)) .
		
	▪

The proof of Lemma 1 can be found in Appendix<ref>.
 We have the following insights from the above lemma,


  * It can be observed from Lemma <ref> that at least one transmitter should employ the maximum allowable power, since otherwise we could scale up both transmitted power without violating the power constraint such that the minimum Euclidean distance is enlarged.

  * We can see from Lemma <ref> that the optimal value of the objective function together with (w_1^*(k), w_2^*(k))  substantially depend on the relative strength of the channel coefficients. Inspired by <cit.>, we divide the considered real Gaussian ZC into three scenarios:
	
	
  * Gaussian ZC with weak cross link, i.e., |h_21|/|h_22|∈ (0,1];
	    
  * Gaussian ZC with strong cross link, i.e., |h_21|/|h_22|∈ (1,2M);
	
  * Gaussian ZC with very strong cross link, i.e., |h_21|/|h_22|∈ [2M, ∞).
	
Then, a compact closed-form expression for Problem <ref> can be established for three complimentary scenarios, which constitutes main contents of the subsequent subsection. 




 §.§ The Optimal NOMA Design with PAM Constellation for the Gaussian ZC

Now we are ready to give the closed-form optimal solution of (w_1^*, w_2^*)  to Problem <ref> which maximizes the minimum Euclidean distance over all the Farey intervals for the aforementioned three scenarios.



  §.§.§ Scenario 1: ZC with Weak Cross Link

For this case, we have |h_21|/|h_22|∈ (0,1]. Consider the Farey interval set 𝒮_M-1={(b_k/a_k, b_k+1/a_k+1)}_k=1^|𝒮_M-1|. By Property <ref>, we have a_k+1+a_k+2/b_k+1+b_k+2<a_k+1/b_k+1 <a_k+a_k+1/b_k+b_k+1, and therefore the positive real axis can be divided into the |𝒮_M-1|+1 intervals in an increasing order:

    {(0, a_|𝒮_M-1|+a_|𝒮_M-1|+1/b_|𝒮_M-1|+b_|𝒮_M-1|+1),(a_|𝒮_M-1|+a_|𝒮_M-1|+1/b_|𝒮_M-1|+b_|𝒮_M-1|+1,
                   a_|𝒮_M-1|-1 +a_|𝒮_M-1|/b_|𝒮_M-1|-1 +b_|𝒮_M-1|), ⋯,(a_1 +a_2/b_1 +b_2,∞)}.

In particular, we have a_1 +a_2/b_1 +b_2=M and a_|𝒮_M-1|+a_|𝒮_M-1|+1/b_|𝒮_M-1|+b_|𝒮_M-1|+1=1/M.





Suppose that |h_21|/|h_22|∈ (0,1].

Then, the optimal power scaling factors to Problem <ref> are explicitly determined as follows:
	
		
  * If |h_11|/|h_21|≤√(p_2)/M√(p_1), then, we have (w_1^*, w_2^*)=(√(3 p_1/M^2-1), M |h_11|/|h_21|√(3 p_1/M^2-1));
		
		
  * If |h_11|/|h_21|≥M √( p_2)/√(p_1), then, we have  (w_1^*, w_2^*)=(M |h_21|/|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1));
		
		
  * Let |h_11|/|h_21| ∈(√(p_2 )(a_ℓ_1+1 +a_ℓ_1+2)  /√(p_1)(b_ℓ_1+1 +b_ℓ_1+2), √(p_2)(a_ℓ_1 +a_ℓ_1+1) /√(p_1)(b_ℓ_1 +b_ℓ_1+1)) for some ℓ_1=1, ⋯,   |𝒮_M-1|-1. If we let ℓ̃_a=min_k {(a_1+a_2), ⋯, (a_ℓ_1+ a_ℓ_1+1) } and
		ℓ̃_b=min_k {(b_ℓ_1+1+b_ℓ_1+2), ⋯, (b_|𝒮_M-1|+b_|𝒮_M-1|+1)}, then we have
		
    (w_1^*, w_2^*)=(√(3 p_1/M^2-1),(b_ℓ̃_a +b_ℓ̃_a+1)|h_11|/(a_ℓ̃_a+a_ℓ̃_a+1)|h_21|√(3 p_1/M^2-1)),
              if |h_11|/|h_21|≥√( p_2)(a_ℓ̃_a+ a_ℓ̃_a+1) /√( p_1)(b_ℓ̃_b+b_ℓ̃_b+1);
    ((a_ℓ̃_b + a_ℓ̃_b+1)|h_21|/(b_ℓ̃_b+b_ℓ̃_b+1) |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)),
              if |h_11|/|h_21| < √( p_2)(a_ℓ̃_a+ a_ℓ̃_a+1) /√( p_1)(b_ℓ̃_b+b_ℓ̃_b+1).

	
		
The proof of Theorem <ref> is provided in Appendix<ref>.


  §.§.§ Scenario 2: ZC with Strong Cross Link

In this case,  |h_21|/|h_22|∈ (1, 2M). We suppose that L-1 <|h_21|/|h_22|≤ L.
Then, the optimal solution to Problem 1 can be obtained by considering the following two cases: the Farey interval set 𝒰_M-1^L={(b_k'/a_k', b_k+1'/a_k+1')∈𝒮_M-1: b_k' +b_k+1' ≥ L} and 𝒱_M-1^L={(b_k”/a_k”, b_k+1”/a_k+1”)∈𝒮_M-1: b_k” +b_k+1” < L}. The whole discussions on them can be summarized as the following theorem.
 The optimal solution to Problem <ref> in this case can be obtained by finding the maximum value of the objective functions in the following two cases which are explicitly attained as follows:


  * 




Let (b_k'/a_k', b_k+1'/a_k+1')∈𝒰_M-1^L. Then, the following three statements are true.	
		
  * If |h_11|/|h_21|≤√(p_2)(a_|𝒰_M-1^L|' +a_|𝒰_M-1^L|+1' )/√(p_1)(b_|𝒰_M-1^L|' +b_|𝒰_M-1^L|+1'), then we have (w_u_1^*, w_u_2^*)=(√(3 p_1/M^2-1), (b_ℓ̃_c'+b_ℓ̃_c+1') |h_11|/(a_ℓ̃_c'+a_ℓ̃_c+1')|h_21|√(3 p_1/M^2-1)), where ℓ̃_c=min_k{(a_1'+a_2'), …, (a_|𝒰_M-1^L|'+a_|𝒰_M-1^L|+1') }.
		
		
  * If |h_11|/|h_21|≥ M√(p_2)/√(p_1), then we have  (w_u_1^*, w_u_2^*)=(M |h_21|/|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)).
		
  * Suppose that |h_11|/|h_21| ∈(√(p_2)(a_ℓ_2+1' +a_ℓ_2+2')  /√(p_1 )(b_ℓ_2+1' +b_ℓ_2+2'), √(p_2)(a_ℓ_2' +a_ℓ_2+1') /√(p_1)(b_ℓ_2' +b_ℓ_2+1')) for some ℓ_2'=1, ⋯,  |𝒰_M-1^L|. If we let ℓ̃_d=min_k {(a_1+a_2), …, (a_ℓ_2+ a_ℓ_2+1) } and
		ℓ̃_e=min_k {(b_ℓ_2+1+b_ℓ_2+2),…, (b_|𝒰_M-1^L|+b_|𝒰_M-1^L|+1)}, then we have
		
    (w_u_1^*, w_u_2^*)=(√(3 p_1/M^2-1),(b_ℓ̃_d' +b_ℓ̃_d+1')|h_11|/(a_ℓ̃_d'+a_ℓ̃_d+1')|h_21|√(3 p_1/M^2-1)),
     if |h_11|/|h_21|≥√( p_2)(a_ℓ̃_d'+ a_ℓ̃_d+1') /√( p_1)(b_ℓ̃_e'+b_ℓ̃_e+1');
    ((a_ℓ̃_e' + a_ℓ̃_e+1')|h_21|/(b_ℓ̃_e'+b_ℓ̃_e+1') |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1))
     if |h_11|/|h_21| < √( p_2)(a_ℓ̃_d'+ a_ℓ̃_d+1') /√( p_1)(b_ℓ̃_e'+b_ℓ̃_e+1').

	
	
	
  * Let (b_k-1”/a_k-1”, b_k”/a_k”) ∈𝒱_M-1^L. Then, the following two statements are true.	








	
		
  * If |h_11|/|h_22|≤√(p_2)/√(p_1), then we have (w_v_1^*, w_v_2^*)=( √(3 p_1/M^2-1), b_|𝒱_M-1|+1” |h_11||h_22|/a_|𝒱_M-1|+1” |h_21|+|h_22|√(3 p_1/M^2-1)).
		
  * If 
		|h_11|√(p_1)/|h_21| √(p_2)∈(a_ℓ_3+1”/b_ℓ_3+1”+|h_22| /b_ℓ_3+1” |h_21|, a_ℓ_3”/b_ℓ_3”+|h_22| /b_ℓ_3” |h_21|) for some ℓ_3=1,⋯, |𝒱_M-1|, then we have
		
    (w_v_1^*,w_v_2^*)=(√(3 p_1/M^2-1), b_ℓ_3”|h_11|/a_ℓ_3”|h_21|+|h_22|√(3 p_1/M^2-1)),
     if |h_11|/|h_21|≥√(p_2 )/√(p_1 )(a_ℓ_3”/b_ℓ_3”+|h_22| /b_ℓ_3”|h_21|);
    (a_ℓ_3+1”|h_21|+|h_22|/b_ℓ_3+1” |h_11| √(3 p_2/M_2^2-1), √(3 p_2/M_2^2-1))
     if |h_11|/|h_21|< √(p_2)/√(p_1)(a_ℓ_3”/b_ℓ_3”+|h_22| /b_ℓ_3”|h_21|).
	
	

 
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.


  §.§.§ Scenario 3: ZC with Very Strong Cross Link

In this case, |h_21|/|h_22|∈ [2M, ∞). Likewise, we consider the Farey interval set 𝒮_M-1={(b_k/a_k, b_k+1/a_k+1)}_k=1^|𝒮_M-1|. Note that |h_21|/|h_22|≥ 2M> b_k+b_k+1 for k=1,…,|𝒮_M-1|. Then, by Property <ref> and Proposition <ref>, we have
a_k+1/b_k+1+|h_22| /b_k+1|h_21|≤a_k+a_k+1/b_k+b_k+1< a_k/b_k <a_k/b_k+|h_22| /b_k|h_21|. As a result, the positive real axis can be divided into the following intervals in increasing order:


    {(0, a_|𝒮_M-1|+1/b_|𝒮_M-1|+1+|h_22| /b_|𝒮_M-1|+1|h_21|),
    …, (a_2/b_2+|h_22| /b_2 |h_21|, a_1/b_1+|h_22| /b_1 |h_21|)},


where a_1/b_1+|h_22| /b_1 |h_21|=∞.

	Let |h_21|/|h_22|∈ [2M, ∞). Then, the optimal solution to Problem <ref> is given below:
	
		
  * If |h_11|/|h_22|≤√(p_2)/√(p_1), then we have (w_1^*, w_2^*)=(√(3 p_1/M^2-1),  |h_11|/|h_22|√(3 p_1/M^2-1)).
		
  * If |h_11|√(p_1)/|h_21| √(p_2)∈(a_ℓ_4+1/b_ℓ_4+1+|h_22| /b_ℓ_4+1 |h_21|, a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4 |h_21|) for some ℓ_4=1,…, |𝒮_M-1|, then we have
		
    (w_1^*,w_2^*)=(√(3 p_1/M^2-1), b_ℓ_4|h_11|/a_ℓ_4|h_21|+|h_22|√(3 p_1/M^2-1)),
     if |h_11|/|h_21|≥√(p_2)/√(p_1 )(a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4|h_21|);
    (a_ℓ_4+1|h_21|+|h_22|/b_ℓ_4+1 |h_11| √(3 p_2/M^2-1), √(3 p_2/M^2-1)),
     if |h_11|/|h_21|< √(p_2)/√(p_1)(a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4|h_21|).

	
The proof of Theorem <ref> is provided in Appendix<ref>.






























































































































































§ SIMULATION 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 <cit.>, time-division multiple access (TDMA) and frequency-division multiple access (FDMA) approaches. Without loss of generality, we set P_1=P_2=1. For simplicity, in the simulations, we assume that the same square-QAM constellation is adopted by both users, i.e., M=M' and  according to (<ref>), we have p_1=p_1'=P_1/2=1/2 and p_2=p_2'=P_2/2=1/2.




 §.§ The Resulting Optimal Sum Constellation at Receiver D_1 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 M=M'=4. We discuss the results of these cases one by one as follows:


  §.§.§ Weak Cross Link

In this case, we assume that h_11=1, h_21=1/2 and h_22=1. Based on the derived expressions for the optimal solution provided in the previous section, we can readily obtain that (w_1^*, w_2^*)=(0.4472, 0.2236).
The corresponding received constellation at D_1 is plotted in Fig. <ref>.
It can be observed that the sum-constellation at D_1 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 w_1^* h_11=0.4472 and w_2^* h_21=0.1118 (i.e., w_2^* h_21=1/4w_1^* h_11), we can deduce that the constellation used by S_2 forms the satellite constellation of the sum constellation at D_1.


  §.§.§ Strong Cross Link

We investigate two channel realizations for this scenario. For the first realization, we let h_11=1, h_21=3/2 and h_22=1. We then have (w_1^*, w_2^*)=(0.1677, 0.4472) and the resulting sum-constellation at D_1 is also regular, as illustrated in Fig. <ref>. Since w_1^* h_11=0.1677 and w_2^* h_21=0.6708 (i.e., w_2^* h_21=4 w_1^* h_11) in this case, the constellation used by S_1 forms the satellite constellation at D_1. For the second realization, we set h_11=1, h_21=3 and h_22=1, leading to (w_1^*, w_2^*)=(0.3354, 0.4472). The resulting constellation plotted in Fig. <ref> is also uniform as in the previous two scenarios. We have w_1^* h_11=0.3354 and w_2^* h_21=1.3416 (i.e., w_2^* h_21=4 w_1^* h_11). Thus, the constellation used by the transmitter S_1 forms the satellite constellation at D_1.




  §.§.§ Very Strong Cross Link

In this case, we suppose that h_11=1, h_21=5 and h_22=1/2, generating (w_1^*, w_2^*)=(0.2236, 0.4472). The obtained constellation at D_1 is shown in Fig. <ref>. In this case, we have w_1^* h_11=0.2236 and w_2^* h_21=2.236 (i.e., w_2^* h_21 =10 w_1^* h_11).

However, it can be observed that w_2^* h_22=0.2236, i.e., w_1^* h_11=w_2^* h_22.




 §.§ Average 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 <cit.> methods, over Rayleigh fading channels with h_11∼𝒞𝒩(0, δ_11^2), h_21∼𝒞𝒩(0, δ_21^2) and h_22∼𝒞𝒩(0, δ_22^2).
 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 <cit.>.  


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 S_1 and S_2 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 <cit.>, we let each user transmit at the maximum allowable power by using constellations {exp(j 2π k/N)}_k=0^N-1 and {exp(j 2πℓ +jπ/N)}_ℓ=0^N-1 for user S_1 and S_2, respectively.






In Fig. <ref>, we consider that the variances of all channels are the same, i.e., (δ_11^2, δ_21^2, δ_22^2)=(1,1,1), and the average BER over both receivers of all the methods are plotted against the SNR ρ=1/2σ^2. In Fig. <ref>, 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. <ref>, 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. <ref>, wherein the variances of three channels are set as (δ_11^2, δ_21^2, δ_22^2)=(1,1/4,1). The case with the variances of three channels (δ_11^2, δ_21^2, δ_22^2)=(1,4,1) is provided in Fig. <ref>. Similar observations can be seen as the previous case with equal channel variances, which further verifies the effectiveness of our proposed NOMA design.



§ CONCLUSIONS

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




 §.§ Proof of Proposition <ref>
	
We first divide the feasible region ℤ_K^2 ∖{(0,0)} into four subsets given by 𝕊_K,1^2={(m,n): m,n ∈ℕ_K, (m,n) ≠ (0,0)}, 𝕊_K,2^2={(-m,n): m,n ∈ℕ_K, (m,n) ≠ (0,0)}, 𝕊_K,3^2={(m,-n): m,n ∈ℕ_K,  (m,n) ≠ (0,0)}, 𝕊_K,4^2={(-m,-n): m,n ∈ℕ_K, (m,n) ≠ (0,0)}. Then clearly we have ℤ_K^2 ∖{(0,0)}  = ∪_k=1^4 𝕊_K,k^2. For (-m,n) ∈𝕊_K,2^2 with m,n≥ 0,  we can always find (m,n) ∈𝕊_K,1^2 such that
d_1(m,n)=||h_11|w_1 n -  |h_21| w_2 m| ≤||h_11|w_1 n -  |h_21| w_2 (-m)|= d_1(-m,n). Hence,

min_(m,n)∈𝕊_K,1^2 d_1(m,n)≤min_(m,n)∈𝕊_K,2^2 d_1(m, n).

By a similar argument on 𝕊_K,3^2 and 𝕊_K,4^2, it follows that

min_(m,n) ∈𝕊_K,1^2 d_1(m,n)=min_(m,n) ∈ℤ_K,1^2 ∖{(0,0)} d_1(m,n).


Then, we further divide 𝕊_K,1^2 into 𝔽^2_K and 𝕊_K,1^2 ∖𝔽^2_K.

In what follows, we will show that

min_   (m,n)∈𝔽_K^2  d_1(m,n)=min_   (m,n)∈𝕊_K,1^2  d_1(m,n).

This can be proved by contradiction. Suppose that

min_   (m,n) ∈𝕊_1^2∖𝔽_K^2  d_1(m,n)=min_   (m,n) ∈𝕊_K,1^2  d_1(m,n),

where the minimum is achieved by (m^*, n^*) ∈𝕊_K,1^2 ∖𝔽_K^2 such that ⟨ m^*,n^*⟩=ℓ>1 by the definition of Farey sequences. Then, we can find (m^*/ℓ,n^*/ℓ) ∈𝔽_K^2 such that d_1(m^*/ℓ,n^*/ℓ)=1/ℓ d_1(m^*,n^*)<d_1(m^*,n^*), which contradicts the assumption.  This completes the proof. □



 §.§ Proof of Proposition <ref>








Recall that d_1(m,n) =||h_11|w_1 n-|h_21| w_2 m|. Therefore, for |h_21| w_2/|h_11|w_1∈(n_1/m_1,n_2/m_2), we have:

d_1(m_1, n_1) -d_1(m_2, n_2)


=  (m_1+m_2) |h_11|w_1 (|h_21| w_2/ |h_11|w_1 - n_1+n_2/ m_1+m_2).

The results presented in the proposition can be readily obtained, and we complete the proof.






□



 §.§ Proof of of Proposition <ref>

We first consider the case |h_21| w_2/|h_11|w_1∈ (n_2/m_2,n_2+n_3/m_2+m_3). By using Proposition <ref>, we have d_1(m_2, n_2)<d_1(m_3, n_3).
Then, with the help of Property <ref> and <ref>, we have |h_21| w_2/|h_11|w_1∈ (n_2/m_2, n_2+n_3/m_2+m_3)⊂  (n_2/m_2, n_3/m_3) ⊆ (n_2/m_2, n_2+n_4/m_2+m_4) and using Property <ref> again, we have d_1(m_2, n_2)≤ d_1(m_4, n_4).
On the other hand, by Property <ref> and <ref>, we have |h_21| w_2/|h_11|w_1∈ (n_2/m_2, n_2+n_3/m_2+m_3)⊂  (n_2/m_2, n_3/m_3) ⊆ (n_1+n_3/m_1+m_3, n_3/m_3) and using Property <ref> again, we have d_1(m_3, n_3)≤ d_1(m_1, n_1).









As n_1/m_1 and n_4/m_4 are randomly picked entry in 𝔖_K, or equivalently (m_1, n_1) and (m_4, n_4) are randomly picked in 𝔽_M-1^2 ∖{(m_2, n_2),(m_3,n_3)}, this proves that for |h_21| w_2/|h_11|w_1∈ (n_2/m_2,n_2+n_3/m_2+m_3),
min_(m,n) ∈𝔽_M-1^2 d_1(m,n)=d_1(m_2, n_2)= |h_21| w_2 m_2-|h_11|w_1 n_2.
The other case can also be proved by a similar argument and hence omitted.
We completes the proof.




 §.§ Proof of Proposition <ref>

First of all, we calculate the following difference
a_k+a_k+1/b_k+b_k+1 -a_k+1/b_k+1-|h_22| /b_k+1|h_21|


=|h_21| - |h_22|(b_k+b_k+1)/(b_k+b_k+1)b_k+1 |h_21|.
Hence, if |h_21|/|h_22|≥ b_k+b_k+1, then a_k+1/b_k+1+|h_22| /b_k+1|h_21|≤a_k+a_k+1/b_k+b_k+1.	
Similarly, we have a_k/b_k-|h_22| /b_k|h_21| -a_k+a_k+1/b_k+b_k+1


=|h_21| - |h_22|(b_k+b_k+1) /b_k (b_k+b_k+1) |h_21|.	
As a result, if |h_21|/|h_22|≥ b_k+b_k+1, then a_k+a_k+1/b_k+b_k+1≤a_k/b_k-|h_22| /b_k|h_21|.	
This completes the proof.	


 §.§ Proof of Lemma <ref>
 
According to proposition <ref> and notice that (b_k/a_k,b_k+1/a_k+1)=(b_k/a_k, b_k+b_k+1/a_k+a_k+1)∪(b_k+b_k+1/a_k+a_k+1,b_k+1/a_k+1), problem in (<ref>) can be further divided into the following two sub-problems and the overall solution is the maximum value of the two problems:
[Sub-problem 1]
	We aim to solve the following optimization problem:
	
		
    g_1(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2 min {|h_11| w_1 b_k+1
                            - |h_21|w_2 a_k+1,|h_22|w_2 }
        s.t. b_k+b_k+1/a_k+a_k+1≤|h_21|w_2 /|h_11| w_1≤b_k+1/a_k+1,
          0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1).

		▪
		

[Sub-problem 2]
	The optimization problem is stated as follows:
	
		
    g_2(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2min{|h_21|w_2 a_k 
                            - |h_11| w_1 b_k, |h_22|w_2 }
        s.t. b_k/a_k≤|h_21|w_2 /|h_11| w_1 < b_k+b_k+1/a_k+a_k+1,
          0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1).

		▪
		








We first consider Sub-problem 1 and Problem <ref>, which can be divided into the following two case:	

	
  * Case 1: Receiver 1 has smaller Euclidean distance.
	
		
    g_11(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2   b_k+1 |h_11| w_1 -a_k+1 |h_21|w_2
        s.t.  b_k+1|h_11| w_1  -a_k+1 |h_21|w_2  ≤ |h_22|w_2,
         b_k+b_k+1/a_k+a_k+1≤|h_21|w_2 /|h_11| w_1≤b_k+1/a_k+1,
          0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1).

		
	
  * Case 2: Receiver 2 has smaller Euclidean distance.
	
		
    g_12(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2 |h_22|w_2 
        s.t.  |h_22|w_2<b_k+1 |h_11| w_1-a_k+1|h_21|w_2,
         b_k+b_k+1/a_k+a_k+1≤|h_21|w_2 /|h_11| w_1<b_k+1/a_k+1,
          0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1).

		


For Case-1 of Sub-problem 1, constraint (<ref>) is equivalent to b_k+1 |h_11| /a_k+1|h_21|+|h_22|w_1 ≤ w_2 and  constraint (<ref>) means (b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21| w_1≤ w_2≤b_k+1 |h_11|/a_k+1 |h_21| w_1.
Also we notice that if |h_21|/|h_22|≥ b_k+b_k+1 then (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|w_1  ≤ b_k+1 |h_11| /a_k+1|h_21|+|h_22| w_1.


















Hence, the optimization problem can be further divided into two cases:

	
  * If |h_21|/|h_22|≤ b_k +b_k+1, then
	
		
    g_11(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2  b_k+1 |h_11| w_1-a_k+1|h_21|w_2
        s.t. (b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21| w_1≤ w_2≤b_k+1 |h_11|/a_k+1 |h_21|,
          0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1).

	
	We let w_2=(b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21| w_1, then the objective function is |h_11| w_1/a_k+a_k+1.
	
	
	
	
	
	
	
	
	In this case, (<ref>) is equivalent to w_1 ≤(a_k+a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1) and w_1 ≤√(3 p_1/M^2-1) and hence, we have g_11(b_k/a_k, b_k+1/a_k+1)=min{|h_11|/a_k+a_k+1(a_k+a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1), |h_11|/a_k+a_k+1√(3 p_1/M^2-1)}. As a consequence, we have
	
		
    g_11(b_k/a_k, b_k+1/a_k+1)
       =|h_21|/b_k+b_k+1√(3 p_2/M^2-1), where 
    
    			(w_1, w_2)= ((a_k+a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)),
         if |h_11|/|h_21|≥√(p_2)(a_k+ a_k+1)/√(p_1) (b_k+b_k+1);
    |h_11|/a_k+ a_k+1√(3 p_1/M^2-1),
    				 where 
    
    				(w_1, w_2)=(√(3 p_1/M^2-1),(b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)), 
         if |h_11|/|h_21|< √(p_2)(a_k+ a_k+1)/√(p_1) (b_k+b_k+1),

	
	
	
	
	
	
	
	
	
  * If |h_21|/|h_22| > b_k +b_k+1, then
	
		
    g_11(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2  b_k+1 |h_11| w_1-a_k+1|h_21|w_2 
        s.t.b_k+1 |h_11| /a_k+1|h_21|+|h_22|w_1≤ w_2≤b_k+1 |h_11|/a_k+1 |h_21| w_1,
          0< w_1 ≤√(3 p_1/M^2-1), 0< w_2 ≤√(3 p_2/M^2-1).

	

We first notice that b_k+1 |h_11| /a_k+1|h_21|+|h_22|w_1<b_k+1 |h_11|/a_k+1 |h_21| w_1 and hence the problem is always feasible.
By letting w_2=b_k+1 |h_11| /a_k+1|h_21|+|h_22|w_1, the objective function can be written by b_k+1 |h_11||h_22|/a_k+1|h_21|+|h_22|w_1.







In this case, the constraints in (<ref>) are equivalent to w_1 ≤√(3 p_1/M^2-1) and w_1 ≤a_k+1|h_21|+|h_22|/b_k+1 |h_11| √(3 p_2/M^2-1).
Therefore, the solution is

	
    g_11(b_k/a_k, b_k+1/a_k+1) 
    
    		   =b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1),
    			 where 
    
    			(w_1, w_2)=(√(3 p_1/M_1^2-1), b_k+1|h_11|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1)),
    			
         if |h_11|/|h_21|≤√(p_2)/√(p_1)(a_k+1/b_k+1
    			+|h_22| /b_k+1|h_21|);
    
    			|h_22|√(3 p_2/M_2^2-1),  where 
    
    			(w_1, w_2)=(a_k+1|h_21|+|h_22|/b_k+1 |h_11| √(3 p_2/M^2-1), √(3 p_2/M^2-1)),
         if |h_11|/|h_21|> √(p_2)/√(p_1)(a_k+1/b_k+1
    			+|h_22| /b_k+1|h_21|).









For Case-2 of Sub-problem 1,









constraint (<ref>) is equivalent to w_2< b_k+1 |h_11| /a_k+1|h_21|+|h_22| w_1 and constraint (<ref>) implies (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|w_1≤ w_2 ≤b_k+1 |h_11|/a_k+1|h_21| w_1. By noticing that b_k+1 |h_11| /a_k+1|h_21|+|h_22|< b_k+1 |h_11|/a_k+1|h_21|,







the problem is equivalent to

	
    g_12(b_k/a_k, b_k+1/a_k+1)=max_w_1, w_2  |h_22|w_2 
         s.t. (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|w_1≤ w_2 ≤ b_k+1 |h_11| /a_k+1|h_21|+|h_22| w_1,
          0< w_1 ≤√(3 p_1/M^2-1),0< w_2 ≤√(3 p_2/M^2-1).


Constraint (<ref>) is feasible if |h_21|/|h_22|≥ b_k+b_k+1.











In this case, the solution is

	
    g_12(b_k/a_k, b_k+1/a_k+1) 
    
    		   =b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1),
    			 where 
    
    			 (w_1, w_2)=(√(3 p_1/M^2-1),b_k+1|h_11|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1)),
    			
         if |h_11|/|h_21|≤√(p_2)/√(p_1)(a_k+1/b_k+1
    			+|h_22| /b_k+1|h_21|);
    
    			|h_22|√(3 p_2/M_2^2-1),
    			 where 
    
    			 (w_1, w_2)=(a_k+1|h_21|+|h_22|/ b_k+1 |h_11| √(3 p_2/M_2^2-1),√(3 p_2/M_2^2-1)),
    			
         if |h_11|/|h_21|> √(p_2)/√(p_1)(a_k+1/b_k+1
    			+|h_22| /b_k+1|h_21|).






















































































































































































































































The solution of Sub-problem 2 can be attained in a similar fashion as Sub-problem 1, and hence is omitted.
Then, for the subinterval division |h_21|w_2 /|h_11| w_1∈(b_k/a_k, b_k+1/a_k+1), the solution to both Sub-problems can be summarized as follows:

	
  * Scenario 1, b_k+b_k+1/a_k+a_k+1≤|h_21|w_2 /|h_11| w_1≤b_k+1/a_k+1:
	
	
	
		
  * If |h_21|/|h_22|≤ b_k +b_k+1
		
			
    g_11(b_k/a_k, b_k+1/a_k+1)
    
    				   =|h_21|/b_k+b_k+1√(3 p_2/M^2-1), where 
    
    				  (w_1, w_2) = ((a_k+a_k+1)|h_21|/(b_k+b_k+1) |h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)),
         if |h_11|/|h_21|≥√(p_2)(a_k+ a_k+1)/√(p_1) (b_k+b_k+1);
    |h_11|/a_k+ a_k+1√(3 p_1/M^2-1), where 
    
    					 (w_1, w_2)=(√(3 p_1/M^2-1), (b_k+b_k+1) |h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)), 
         if |h_11|/|h_21|< √(p_2)(a_k+ a_k+1)/√(p_1) (b_k+b_k+1),

		
		
  * If |h_21|/|h_22| > b_k +b_k+1
		
			
    g_11(b_k/a_k, b_k+1/a_k+1)
    
    				    =b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1),
    					 where  (w_1, w_2)=
    (√(3 p_1/M^2-1),b_k+1|h_11|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1)),
    					
         if |h_11|/|h_21|≤√(p_2)/√(p_1)(a_k+1/b_k+1
    					+|h_22| /b_k+1|h_21|);
    
    					|h_22|√(3 p_2/M^2-1),  where (w_1, w_2)=
    (a_k+1|h_21|+|h_22|/b_k+1 |h_11| √(3 p_2/M^2-1), √(3 p_2/M^2-1)),
         if |h_11|/|h_21|> √(p_2)/√(p_1)(a_k+1/b_k+1
    					+|h_22| /b_k+1|h_21|).

		
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
  * Scenario 2, b_k/a_k≤|h_21|w_2 /|h_11| w_1≤b_k+b_k+1/a_k+a_k+1:
	
		
  * Case 1: Receiver 1 has smaller Euclidean distance.
		
			
  * If |h_21|/|h_22|≤ b_k+b_k+1, the solution is
			
				
    g_21(b_k/a_k, b_k+1/a_k+1)
    
    					   =|h_21|/b_k+b_k+1√(3 p_2/M^2-1),  where (w_1, w_2)=
    ((a_k+a_k+1)|h_21|/(b_k+b_k+1)|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)),	
         if |h_11|/|h_21|≥√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1);
    						
    |h_11|/a_k+a_k+1√(3 p_1/M^2-1),  where (w_1, w_2)=
    (√(3 p_1/M^2-1), (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)),
         if |h_11|/|h_21|< √(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1).

			
			
  * If |h_21|/|h_22|≥ b_k+b_k+1,
			
				
    g_21(b_k/a_k, b_k+1/a_k+1)
       =|h_22|√(3 p_2/M^2-1),  where (w_1, w_2)=
    (|h_21| a_k-|h_22|/b_k |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)),
         if |h_11|/|h_21|≥√(p_2)/√(p_1)( a_k/b_k -|h_22|/|h_21|b_k);
    b_k |h_11||h_22| /|h_21| a_k-|h_22|√(3 p_1/M^2-1),  where (w_1, w_2)=
    (√(3 p_1/M^2-1), b_k |h_11|/|h_21| a_k-|h_22|√(3 p_1/M^2-1)),
         if |h_11|/|h_21|<√(p_2)/√(p_1)( a_k/b_k -|h_22|/|h_21|b_k).

			
			
			
			
			
			
			
			
			
			
		
		
  * Case 2: Receiver 2 has smaller Euclidean distance.
		
			
  * If |h_21|/|h_22|≥ b_k +b_k+1, the solution is
			
				
    g_22(b_k/a_k, b_k+1/a_k+1)
    
    					    =(b_k+b_k+1)|h_11||h_22|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1),  where (w_1, w_2)
    = (√(3 p_1/M^2-1), (b_k+b_k+1)|h_11|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1)),
       if |h_11| /|h_21|≤√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1);
    
    						|h_22|√(3 p_2/M^2-1),  where (w_1, w_2)=
    ((a_k+a_k+1)|h_21|/(b_k+b_k+1)|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)),
         if |h_11| /|h_21|>√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1).

			
			
			
			
			
			
			
			
			
			
		
	

Now, we aim to combine g_21(b_k/a_k, b_k+1/a_k+1) and g_22(b_k/a_k, b_k+1/a_k+1).

By Proposition <ref>, for |h_21|/|h_22|≥ b_k +b_k+1, we have a_k/b_k -|h_22|/|h_21|b_k≥a_k+a_k+1/b_k+b_k+1.
Also, for |h_11|/|h_21|≤√(p_2)/√(p_1)(a_k/b_k-|h_22|/b_k |h_21|), we have √(3 p_2/M^2-1)≥b_k |h_11| /|h_21| a_k-|h_22|√(3 p_1/M^2-1). In addition, for |h_21|/|h_22|≥ b_k+b_k+1, we have (b_k+b_k+1)/(a_k+a_k+1)|h_21|≥b_k /|h_21| a_k-|h_22|.
Hence, we have g_22(b_k/a_k, b_k+1/a_k+1) ≥ g_21(b_k/a_k, b_k+1/a_k+1).
We notice that for |h_21|/|h_22|≤ b_k +b_k+1, g_11(b_k/a_k, b_k+1/a_k+1)=g_21(b_k/a_k, b_k+1/a_k+1). Then for |h_21|/|h_22|≥ b_k +b_k+1, we combine g_11(b_k/a_k, b_k+1/a_k+1) and g_22(b_k/a_k, b_k+1/a_k+1).











































By Proposition <ref>, for |h_21|/|h_22|≥ b_k+b_k+1, we have √(p_2)/√(p_1)(a_k+1/b_k+1
+|h_22| /b_k+1|h_21|) ≤√(p_2)(a_k+a_k+1)/√(p_1)(b_k+b_k+1).
















Also, for |h_21|/ |h_22|≥  b_k +b_k+1,  we attain b_k+1|h_11||h_22|/a_k+1|h_21|+|h_22|√(3 p_1/M^2-1)≥(b_k+b_k+1)|h_11||h_22|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1). In addition, for |h_11|/|h_21|≤√(p_2) (a_k+a_k+1)/√(p_1)(b_k+b_k+1), we have |h_22|√(3 p_2/M^2-1)≥(b_k+b_k+1)|h_11||h_22|/(a_k+a_k+1)|h_21|√(3 p_1/M^2-1). In conclusion, for |h_21|/|h_22|≥ b_k +b_k+1, g_11(b_k/a_k, b_k+1/a_k+1) ≥ g_22(b_k/a_k, b_k+1/a_k+1).

With the above discussion, we have the result in Lemma <ref> and we complete the proof.




















































 §.§ Proof of Theorem <ref>

For the weak cross link, we have |h_21|/|h_22|≤ 1 ≤ b_k +b_k+1 for k=1, …, |𝒮_M-1|.

	
  * If |h_11|/|h_21|≤√(p_2)/M √(p_1), we have |h_11|/|h_21| ≤√(p_2) (a_k +a_k+1) /√(p_1)(b_k +b_k+1), and  then by Lemma <ref>, we attain g(b_k/a_k, b_k+1/a_k+1)=
	|h_11|/a_k + a_k+1√(3 p_1/M^2-1)  for  k=1, …, |𝒮_M-1|.
	Also, note that |𝒮_M-1| =min_k {(a_1+ a_2), …, (a_ |𝒮_M-1|+ a_ |𝒮_M-1|+1)}, then max {|h_11|/a_1+ a_2√(3 p_1/M^2-1),⋯, |h_11|/a_ |𝒮_M-1|+ a_ |𝒮_M-1|+1√(3 p_1/M^2-1)}
		
		=|h_11|√(3 p_1/M^2-1),
	with (w_1^*, w_2^*)=(√(3 p_1/M^2-1), M |h_11|/|h_21|√(3 p_1/M^2-1)).
	
	
  * If |h_11|/|h_21|≥M√(p_2)/√(p_1), we have |h_11|/|h_21| ≥(a_k +a_k+1) √(p_2)/(b_k +b_k+1)√(p_1) and then, by  using Lemma <ref>, we have g(b_k/a_k, b_k+1/a_k+1)= |h_21|/b_k + b_k+1√(3 p_2/M_2^2-1) for k=1, …, |𝒮_M-1|. Also, note that 1=min_k {(b_1+ b_2), …, (b_ |𝒮_M-1|+ b_ |𝒮_M-1|+1)}.  Then,
	
    max {|h_21|/b_1+ b_2√(3 p_2/M^2-1),⋯, 
       |h_21|/b_ |𝒮_M-1|+ b_ |𝒮_M-1|+1√(3 p_2/M^2-1)}
    		
    		=|h_21|√(3 p_2/M^2-1),

	with (w_1^*, w_2^*)=(M |h_21|/|h_11|√(3 p_2/M^2-1),√(3 p_2/M^2-1)).
	
  * If |h_11|/|h_21| ∈(√(p_2)(a_ℓ_1+1 +a_ℓ_1+2)  /√(p_1)(b_ℓ_1+1 +b_ℓ_1+2), √(p_2) (a_ℓ_1 +a_ℓ_1+1)/√(p_1)(b_ℓ_1 +b_ℓ_1+1)), for ℓ_1=1, …,   |𝒮_M-1|-1, then, with the help of Lemma <ref>, we have
	
    g(b_k/a_k, b_k+1/a_k+1)
       =|h_11|/a_k + a_k+1√(3 p_1/M^2-1)  for k=1, …, ℓ_1, 
    |h_21|/b_k+b_k+1√(3 p_2/M^2-1)  for  k=ℓ_1+1, …,|𝒮_M-1|.

	
	Note that, ℓ̃_a=min_k {(a_1+a_2), …, (a_ℓ_1+ a_ℓ_1+1) } and
	ℓ̃_b=min_k {(b_ℓ_1+1+b_ℓ_1+2), …, (b_|𝒮_M-1|+b_|𝒮_M-1|+1)}, hence we have

	
    max {|h_11|/a_1+ a_2√(3 p_1/M^2-1),⋯, |h_11|/a_ℓ_1+ a_ℓ_1+1√(3 p_1/M^2-1),
       |h_21|/b_ℓ_1+1+b_ℓ_1+2√(3 p_2/M^2-1), …,|h_21|/b_|𝒮_M-1|+b_|𝒮_M-1|+1√(3 p_2/M^2-1)}
       =max {|h_11|/a_ℓ̃_a+ a_ℓ̃_a+1√(3 p_1/M^2-1),|h_21|/b_ℓ̃_b+b_ℓ̃_b+1√(3 p_2/M^2-1)}.


	Therefore, if |h_11|/|h_21|≥√( p_2)(a_ℓ̃_a+ a_ℓ̃_a+1) /√( p_1)(b_ℓ̃_b+b_ℓ̃_b+1), we have
	|h_11|/a_ℓ̃_a+ a_ℓ̃_a+1√(3 p_1/M^2-1)≥|h_21|/b_ℓ̃_b+b_ℓ̃_b+1√(3 p_2/M^2-1) and hence,
	(w_1^*, w_2^*)=(√(3 p_1/M^2-1),(b_ℓ̃_a +b_ℓ̃_a+1)|h_11|/(a_ℓ̃_a+a_ℓ̃_a+1)|h_21|√(3 p_1/M-1)) and else we have (w_1^*, w_2^*) = ((a_ℓ̃_b + a_ℓ̃_b+1)|h_21|/(b_ℓ̃_b+b_ℓ̃_b+1) |h_11|√(3 p_2/M^2-1), √(3 p_2/M^2-1)).




This completes the proof.











 §.§ Proof of Theorem <ref>


	
  * If |h_11| ≤√(p_2)/√(p_1) |h_22|, we have |h_11|/|h_21|≤√(p_2 )/√(p_1)|h_22| /|h_21|=√(p_2)/√(p_1)(a_|𝒮_M-1|+1/b_|𝒮_M-1|+1+|h_22| /b_|𝒮_M-1|+1|h_21|) ≤√(p_2)/√(p_1)( a_k+1/b_k+1+|h_22| /b_k+1|h_21|) for k=1,…, |𝒮_M-1|.
	According to Lemma <ref>, we have g(b_k/a_k, b_k+1/a_k+1)= b_k+1 |h_11||h_22|/a_k+1 |h_21|+|h_22|√(3 p_1/M^2-1), for k=1,…, |𝒮_M-1|. We also have
	
    max {b_2 |h_11||h_22|/a_2 |h_21|+|h_22|√(3 p_1/M^2-1) , …,
    b_|𝒮_M-1|+1 |h_11||h_22|/a_|𝒮_M-1|+1 |h_21|+|h_22|√(3 p_1/M^2-1)}
    		=|h_11| √(3 p_1/M^2-1),

	with (w_1^*, w_2^*)=(√(3 p_1/M^2-1),  |h_11|/|h_22|√(3 p_1/M^2-1)).
	
  * If |h_11|/|h_21| ∈√(p_2)/√(p_1)(a_ℓ_4+1/b_ℓ_4+1+|h_22| /b_ℓ_4+1 |h_21|, a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4 |h_21|) for ℓ_4=1,…, |𝒮_M-1|.

Then, according to Lemma <ref>, we have g(b_k/a_k, b_k+1/a_k+1)=b_k+1 |h_11||h_22|/a_k+1 |h_21|+|h_22|√(3 p_1/M^2-1), for k=1,…, ℓ_4-1 and  g(b_k/a_k, b_k+1/a_k+1)=|h_22|√(3 p_2/M^2-1) for k=ℓ_4, …, |𝒮_M-1|.

    g(b_k/a_k, b_k+1/a_k+1)=b_k+1 |h_11||h_22|/a_k+1 |h_21|+|h_22|√(3 p_1/M^2-1) for  k=1,…, ℓ_4-1,
    
    		|h_22|√(3 p_2/M^2-1) for  k=ℓ_4, …, |𝒮_M-1|.

Then, we have,

    max {b_2 |h_11||h_22|/a_2 |h_21|+|h_22|√(3 p_1/M^2-1) , …, 
           b_ℓ_4 |h_11||h_22|/a_ℓ_4 |h_21|+|h_22|√(3 p_1/M^2-1),  |h_22|√(3 p_2/M^2-1)}
    
    	=   max {b_ℓ_4 |h_11||h_22|/a_ℓ_4 |h_21|+|h_22|√(3 p_1/M^2-1), |h_22|√(3 p_2/M^2-1)}.

As a result, if |h_11|/|h_21|≥√(p_2)/√(p_1)(a_ℓ_4/b_ℓ_4+|h_22| /b_ℓ_4|h_21|), then b_ℓ_4 |h_11||h_22|/a_ℓ_4 |h_21|+|h_22|√(3 p_1/M^2-1)≥ |h_22|√(3 p_2/M^2-1)
and (w_1^*,w_2^*)=(√(3 p_1/M^2-1), b_ℓ_4|h_11|/a_ℓ_4|h_21|+|h_22|√(3 p_1/M^2-1)). Else, we can attain (w_1^*, w_2^*)=(a_ℓ_4+1|h_21|+|h_22|/b_ℓ_4+1 |h_11| √(3 p_2/M^2-1), √(3 p_2/M^2-1)). This completes the proof.


ieeetr






























