Optimal Minimum Euclidean Distance-Based Precoder for...

SPECIAL SECTION ON ADVANCES IN SIGNAL PROCESSINGFOR NON-ORTHOGONAL MULTIPLE ACCESS

Received March 18, 2019, accepted March 29, 2019, date of publication April 3, 2019, date of current version April 16, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2908893

Optimal Minimum Euclidean Distance-BasedPrecoder for NOMA With Finite-Alphabet InputsWENLONG XIA 1, (Member, IEEE), YUANPING ZHOU1, GUIDE YANG 1,AND RAY T. CHEN2, (Fellow, IEEE)1School of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China2Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78758, USA

Corresponding author: Wenlong Xia ([email protected])

This work was supported by the China Scholarship Council under Grant 201706245027.

ABSTRACT Non-orthogonal multiple access (NOMA) scheme with Gaussian input signals, which have noconstellation constraint on the input alphabets, has been investigated intensively, whereas the research onfinite input constellations is relatively few. This paper focuses on the power allocation in practical NOMAapplication with finite-alphabet inputs.We propose a practical power allocation scheme for downlinkNOMAscenario based on the merit of maximizing the minimum Euclidean distance (MMED) by adopting theconstellation-constrained (CC) capacity. Applying the nearest neighbor approximation and Jensen’s inequal-ity, we prove that optimal CC capacity can be achieved under the MMED criterion at high signal-to-noiseratio (SNR). Instead of updating power allocation coefficient instantaneously, a preset fixed power factorαMMED derived from the MMED criterion is preferred and thus reduces the complexity of overall systemimplementation. We take two cases into consideration, namely, Gaussian broadcast channel (GBC) andfading broadcast channel (FBC).Wemodel close channel conditions, which are challenging for conventionalNOMA as the GBC scenario, and model near-far effect as well Rayleigh fading as the FBC scenario. In thesetwo cases, we show that the optimal performance can be guaranteed at high SNR without the knowledgeof channel state information (CSI). We study the power coefficient pairs for the most commonly used M-ary quadrature amplitude modulation (M-QAM) constellations based on the MMED criterion. We explorethe relationship between the power allocation coefficients and the sizes of constellations. The numericalsimulations on CC sum capacity and capacity regions are provided to validate our analysis.

INDEX TERMS Constellation constrained capacity, finite-alphabet inputs, minimum Euclidean distance,non-orthogonal multiple access.

I. INTRODUCTIONOrthogonal multiple access (OMA) technologies, such astime-division multiple access (TDMA), frequency-divisionmultiple access (FDMA), code-division multiple access(CDMA) and orthogonal frequency-division multiple access(OFDMA), have played a significant role in securing thesuccess of modern communication for the past few decades[1]–[3]. For OMA schemes, each user data is assigned toa dedicated channel of time-slots, frequency bands, codesequences or their combinations which are orthogonal.Consequently, receivers can separate their own messagesfrom others by taking the advantage of this orthogonality.

The associate editor coordinating the review of this manuscript andapproving it for publication was Yuanwei Liu.

However, with the development of next generation commu-nications, enormous user equipments for internet of things,cloud-based applications and wireless access terminals withmassive data traffic pose huge challenges for the require-ments of the fifth generation (5G) wireless communication.Some potential technologies, such as device-centric archi-tectures, millimeter wave, massive multiple-input multiple-output (MIMO), smarter devices, and machine-to-machinecommunications [4], will lead to fundamental changes in 5Gto meet these explosive demands. Moreover, non-orthogonalmultiple access (NOMA) scheme has been proposed as apromising radio access technology for its advantages ofimproved spectral efficiency, massive connections and lowlatency [5]–[10]. UnlikeOMA, inwhich the number of servedusers is restricted by the corresponding number of orthogonal

VOLUME 7, 20192169-3536 2019 IEEE. Translations and content mining are permitted for academic research only.

Personal use is also permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

45123

https://orcid.org/0000-0002-9737-1566

https://orcid.org/0000-0002-9603-3194

W. Xia et al.: Optimal Minimum Euclidean Distance-Based Precoder for NOMA With Finite-Alphabet Inputs

resources, NOMA enables multiple users to transmit simul-taneously by sharing the same time/frequency/code/spatialchannel. Therefore, NOMA scheme can support more userequipments than OMA, which is significant for massive con-nectivity required in 5G.

Although there are many variant NOMA-related tech-nologies emerging, such as interleave division multipleaccess (IDMA) [11], power-domain NOMA [12], sparsecode multiple access (SCMA) [13], bit division multiplexing(BDM) [14], multiuser shared access (MUSA) [15], patterndivision multiple access (PDMA) [16] and network-codedmultiple access (NCMA) [17], [18], NOMA technologies canbe categorized as two main solutions in general [6], [19]:power-domain NOMA and code-domain NOMA. In thispaper, wemainly focus on power-domain NOMA. For power-domain NOMA, a key feature is that superposition codingis used in power domain at the transmitter and successiveinterference cancellation (SIC) detector [6], [10] or MLdetector [20] is used at the receiver. The decoding perfor-mance depends largely on the power allocation schemes forthe users who share the same channel. Therefore, power andresource scheduling [21]–[23] is the key issue to be tackledbefore putting NOMA technology into reality. In [24], twoscenarios, users’ targeted data rates determined by qualityof service (QoS) and users’ rates allocated opportunistically,are investigated in downlink with randomly deployed users.NOMA with fixed power allocation (F-NOMA) and NOMAwith cognitive-radio-inspired power allocation (CR-NOMA)are studied in [25], which a hybrid multiple access combin-ing NOMA with OMA is recommended. A dynamic userclustering and power allocation for uplink and downlinkNOMA system based on sum-throughput maximization isproposed in [26]. Yang et al. [27] proposed a cognitive-radioinspired NOMA with dynamic QoS at multicast users andthe unicast user as well. A linearly precoded Rate-SplittingMultiple Access (RSMA) is investigated in [28]. RSMA hasthe capability of partially decoding the interference and par-tially treating the interference as noise. By combing NOMAand zero-forcing beamforming (ZFBF), a Hybrid NOMA(H-NOMA) of multi-user multiple-input-single-output(MISO) based on quasi-degradation is proposed in [29].

However, most aforementioned work studied a NOMAsystem with an unrestricted input signals, i.e., infinite Gaus-sian input signals which has no constellation constraint onthe input alphabets and can reach most of the known capacityregions [30]–[33]. Whereas, the implementation of Gaussianinput is unaffordable in practice and might result in inevitablehigh computational complexity and extremely large decodingdelay [34]. In addition, the infinite inputs are known to bea continuous Gaussian distribution [35], when it comes topractical communication system, the inputs are generallyuniformly distributed and are constrained to finite alphabets.As a consequence, the known solutions for continuous inputscannot be applied directly to finite constellations [36].

Therefore, some researchers have been motivated to studyNOMA with finite input constellations. The constrained

capacity with phase-shift keying (PSK) for MIMO fad-ing channel is investigated in [37]. The constellation con-strained (CC) capacity region of two-user GMAC is analyzedin [38] by introducing constellation rotation between twousers. A constellation power allocation (CPA) varying the twousers’ transmit power is reported in [36]. Optimization basedon mutual information for NOMA with quadrature ampli-tude modulation (QAM) constellation is studied in [39]–[41].A max-min fairness optimization applying Farey sequencefor NOMAwith QAM is reported both in [42] for Z-channel,and in [20] for uplink scenario.

The abovementioned work tried to enlarge and optimizethe CC capacity. These schemes have three challenges: 1) Per-fect CSI or partial channel state information (CSI) is assumedto be known at the transmitter; 2) As shown in the follow-ing description in Section II, the expression of CC capacityentails the expectation which is time-consuming, especiallywhen a multitude of users are served; 3) The users’ powercoefficients need to be instantaneously updated according tothe computing of CC capacity and the varying of channelconditions. Nevertheless, with the aim to serve high-speeddata transmission and massive connectivity, dynamic userscheduling and power allocation are very challenging due tocomputational complexity in 5G system. Hence, it is mostimportant to investigate how to balance the performance andcomplexity in a practical system.

It is known that maximizing the mutual information canbe approximated to maximize the minimum distance betweenthe constellation vectors for MIMOwith arbitrary inputs [43]and with statistical CSI [44]. However, a low-complexitypower scheduling is not well studied for finite-alphabet inputsin absence of CSI for NOMA system to the best of ourknowledge. In this paper, we focus our work on simplifyingthe power allocation in real NOMA application with M -aryquadrature amplitude modulation (M -QAM) constellation,which is more spectrally efficient than other digital modu-lations such as pulse-amplitude modulation (PAM) and PSK.The main contributions in this paper can be summarized as,

1) We propose a low-complexity power allocation schemefor a downlink NOMA design based on the criterion of max-imizing the minimum Euclidean distance (MMED). Apply-ing nearest neighbor approximation and Jensen’s inequality,we prove that optimal CC capacity can be accomplishedat high SNR for our NOMA scheme. Instead of updatingpower allocation coefficient continuously, we prefer a presetfixed factor αMMED derived from MMED criterion with twobenefits: no CSI required (a brief description can be found inSection IV) and no complicated computation of CC capacity.Comparably, MMED criterion can dramatically reduce thecomputational complexity while providing optimal perfor-mance.

2) Two channel conditions are introduced, namely,Gaussian Broadcast Channel (GBC) and Fading BroadcastChannel (FBC). As it is well known that it is a big challengingwhen channel conditions are very similar in NOMA [9], [18],we refer the GBC as the model of power-balanced channel,

45124 VOLUME 7, 2019


which can work well for some complicate scenarios suchas unmanned aerial vehicle (UAV) communication [45] andhigh-speed railway (HSR) communications [46]. In thesecases, instantaneous CSI is hard to obtain due to the rapidlychanging channel. For GBC, we show that the optimal perfor-mance can still be guaranteed for the close channel conditionsin which there is no near-far effect. This reveals that theissue of user grouping is not a big concern when properpower allocation scheme is adopted in practical NOMA sys-tem given the proper SNR is met. We exploit the FBC tomodel the near-far and Rayleigh fading scenario. In bothcases, our proposed scheme can achieve the optimal CC sumcapacity in comparison with the maximizing entropy strategyat high SNR.

3) Based on the MMED criterion, we study the variouspower allocation coefficient pairs for multipleM -QAMpairs.We uncover the relationship between the sizes of constel-lations and the power allocation coefficients. This result isbeneficial when adaptive modulation and coding (AMC) isintroduced in NOMA scheme. Furthermore, we calculate theminimum Euclidean distance dmin for different constella-tion pairs. We show that some pairs, such as (4, 64)-QAM,(16, 16)-QAM and (64, 4)-QAM, have the same dmin becausethe sum constellations are identical after the superpositioncoding of two users, even though the power coefficients aredifferent.

FIGURE 1. Broadcast channel/downlink channel model.

II. SYSTEM MODEL AND GAUSSIANBROADCAST CHANNELIn this section, we consider a hybrid downlink NOMA [25]which users are divided into different groups and NOMAscheme is implemented among users within a group, whereasOMA is adopted among different groups. This downlinkNOMA with one single-antenna BS and two single-antennausers in one group is shown in Fig. 1. The transmitted signalsets are finite alphabets with constellations S1 = {x1(n)|n = 1, 2, . . . ,N1} and S2 = {x2(n)|n = 1, 2, . . . ,N2}, andthe alphabets in the constellations are uniformly distributed,where N1 and N2 are the size of the constellations. Symbollevel synchronization is assumed. Then the superposed signaltransmitted by the BS is described as

s (t) =√αPx1 (t)+

√(1− α)Px2 (t) (1)

where α(0 ≤ α ≤ 1) is the power allocation coefficientand P denotes the transmit power of the BS. x1 (t) ∈ S1

and x2 (t) ∈ S2 are the symbols of user1 and user2 withunit average energy, i.e.,E

[|xk (t)|2

]= 1, k ∈ {1, 2},

respectively. The received signals are corrupted by AWGNvariable z with z∼ CN(0,σ 2), where CN( 0,σ 2) denotes thecircular symmetric complex Gaussian noise with zero meanand variance of σ 2. Then the kth (k ∈ {1, 2}) user’s receivedsignal is formulated as

yk = hk (√αPx1 (t)+

√(1− α)Px2 (t))+ zk (2)

where hk and zk is the channel gain and noise variable forthe kth user. Note that z1and z2 in Fig. 1 have the samedistribution, we use z instead of z1and z2 for simplicity.

The CC capacity expressions in terms of mutual informa-tion between the input message and the output message foruser1 and user2 are denoted as I (x1 : y) and I (x2 : y|x1),respectively. For Gaussian Broadcast channel without fading,i.e., h1 = h2 = 1, then we have [35], [38], [41]

I (√αPx1 : y) = H (y)− H (y|x1) (3)

H (y) = −∫p (y) log2 (p (y)) dy (4)

H (y | x1) =1N1

∑N1

n=1H (y | x1 = x1 (n)) (5)

where p (y) is the probability density function (PDF) of y,given by

p (y) =1

N1N2

N1∑n=1

N2∑m=1

p(y|x1 = x1 (n) , x2 = x2 (m)) (6)

p (y|x1 = x1 (n) , x2 = x2 (m))

=1(√πσ)2 exp−

∣∣∣y−√αPx1 (n)−√(1− α)Px2 (m)∣∣∣2σ 2

(7)

H (y | x1 = x1 (n))

= −

∫p (y | x1 = x1 (n)) log2 (p (y | x1 = x1 (n))) dy

(8)

Using above equations, we have I (√αPx1 : y) and

I(√(1− α)Px2 : y |

√αPx1

)given in (9) and (10), as shown

at the top of the next page, where d (α, x1, x2, n1, n2,m1,m2)

=√αPx1 (n1) +

√(1− α)Px2 (n2) − (

√αPx1 (m1) +√

(1− α)Px2 (m2)), e (α, x2, n2,m2) =√(1− α)Px2 (n2)−√

(1− α)Px2 (m2), respectively. Then the CC sum capacitycan be given by

I(√αPx1 +

√(1− α)Px2 : y

)= I

(√αPx1 : y

)+ I

(√(1− α)Px2 : y|

√αPx1

)(11)

Apparently, the CC sum capacity is a function of powercoefficient α. Specifically, we use Cs (α) to denote the CC

VOLUME 7, 2019 45125


I(√αPx1 : y

)= log2 (N1)−

1N1N2

N1∑n1=1

N2∑n2=1

E

[log2

[∑N1m1=1

∑N2m2=1

exp(− |d (α, x1, x2, n1, n2,m1,m2)+ z|2 /σ 2

)∑N2m2=1

exp(− |e (α, x2, n2,m2)+ z|2 /σ 2

) ]](9)

I(√

(1− α)Px2 : y|√αPx1

)= log2 (N2)−

1N2

N2∑n2=1

E

log2∑N2

m2=1exp

(− |e (α, x2, n2,m2)+ z|2 /σ 2

exp(−|z|2

σ 2

) (10)

Cs (α) = log2 (N1N2)−1

N1N2

N1∑n1=1

N2∑n2=1

E

[log2

[∑N1m1=1

∑N2m2=1

exp(− |d (α, x1, x2, n1, n2,m1,m2)+ z|2 /σ 2

)exp

(− |z|2 /σ 2

) ]](12)

Q (α) =N1∑n1=1

N2∑n2=1

log2 N1∑m1=1

N2∑m2=1

exp(− |d (α, x1, x2, n1, n2,m1,m2)|

2/σ 2) (15)

sum capacity, which is given in (12), as shown at the top ofthis page. Then we can maximize Cs (α) by optimizing α as

αopt = argmaxCs (α) s.t. 0 < α < 1 (13)

However, Cs (α) is an expectation of random noise z andthe optimal power coefficient that maximizes Cs (α) has no aclosed-form expression. But at high SNR, it can be approxi-mated to the following expression [36]

αopt = arg min0<α<1

Q(α) (14)

whereQ (α) is given by (15), as shown at the top of this page,which is independent of the variable z. Note that Q (α) is afunction of the distance distribution of superposed symbols.Though (14) provides an easier way to compute the optimalα, the solution of α is still not straightforward. In addition,computing α varies with SNR and needs to search all possiblevalues with a small step at different SNR. So the receivermust estimate the SNR values and send them back to the BSinstantaneously. With the fact thatQ (α) is determinate by thedistance distribution of all superposed symbols, Theorem 1provides a low complexity algorithm to solve the powercoefficient α to maximize the CC capacity. More specifically,we propose a criterion to compute the power coefficientsby maximizing the minimum Euclidean distance [20], [47]among superposed symbols.Theorem 1: The optimal power allocation coeffi-

cient α to maximize the CC capacity at high SNR isapproximated by

αopt = arg max0<α<1

dmin (16)

where dmin is the minimum Euclidean distance of the twousers’ superposed symbols.Proof: Consider any two points pi and pj (i 6= j) in a

superposed constellation with M = N1 × N2 points in total.

Let us define a functionL (D (i, j))with respect to the squaredEuclidean distance D (i, j) as

L (D (i, j)) =M∑

i=1,i 6=j

log2 M∑j=1

exp(−D (i, j)

/σ 2)

(17)

where D (i, j) =∣∣pi − pj∣∣2 and pi =

√αPx1 (i1) +

√(1− α)Px2 (i2), pj = (

√αPx1 (j1) +

√(1− α)Px2 (j2),

i1, j1 ∈ {1, . . . ,N1} , i2, j2 ∈ {1, . . . ,N2 and (i1, j1) 6=(i2, j2). At high SNR, we have the approximation which iscalled nearest neighbor approximation [1, ch.5],

L (D (i, j))≈M∑

i=1,i 6=j

log2∑j∈R

exp(−Dmin (i, j)

/σ 2)(18)

where R denotes the set of nearest neighbor pointsof pi, and Dmin (i, j) is the minimum distance between piand pj.

We assume f (y) = exp(−y/σ 2), since the second

derivative of f (y) is f′′

(y) = 1σ 4exp(−y

/σ 2), for all y,

f′′

(y) > 0, so f (y) is convex. Then applying Jensen’sinequality,

f(∑

l wlyl∑l wl

)≤

∑l wl f (yl)∑

l wl(19)

Let w1 = w2 = . . . = σ2 and yl = Dmin (·, l), we have the

following inequality for pi

1W

∑j∈R

exp(−Dmin (i, j)

/σ 2)≥ exp

−∑j∈R

Dmin (i, j)

Wσ 2

(20)

45126 VOLUME 7, 2019


whereW is the number of elements in setR. Since logarithmis monotonically increasing, we have (21), as shown at thebottom of this page. For all pi, i ∈ {1, · · · ,M} and i 6= j,we readily have (22), as shown at the bottom of this page.

Only when y1 = y2 = . . . = yl = . . ., the equality in(19) holds, i.e., when all Dmin (i, j) are equal,L (D (i, j)) isminimized. Then from (22), dmin = Dmin (i, j) is maximized.It is worth noting that (15) and (17) are equivalent, hence (14)holds and the CC sum capacity Cs (α) is maximized. Thiscompletes the proof.

It is worthy to note that this GBC without fading withh1 = h2 does not reflect the practical operating conditions ofwireless communication systems due to the random channelsvariation over time. However, in some cases, such as in [18] togroup two cell-edge users (weak users) into a NOMA group,and in [45] to employNOMA for high-speed unmanned aerialvehicle (UAV) communication in the absence of instanta-neous CSI due to quickly moving of the UAV, it has beendemonstrated in [45] that allocating equal power is a robustway against unknown channel conditions. Thus, these equal-power superposition coding might be formulated as the GBCmodel when the channel conditions is challenging to obtain.

FIGURE 2. Modified fading channel.

III. FADING CHANNEL AND CONSTELLATIONCONSTRAINED CAPACITYIn this section, we take the path loss and Rayleigh fadinginto account. Then Fig. 1 can be modified as Fig. 2 asdescribed in [34].We assume that z1 and z2 in Fig. 2 is AWGNand have the same distribution: z1, z2∼ CN(0,σ 2), then wecan use z∼ CN(0,σ 2) instead of z1and z2 for the sake ofsimplicity.

This equivalent channel can be viewed as a degraded Gaus-sian broadcast channel with noise variance nk at user k , wherenk = σ 2

/|hk |2 , k = 1, 2. Then we have the Theorem 2.

Theorem 2: At high SNR, the optimal power allocationcoefficient α to maximize the CC capacity for fading chan-nel, assuming that hk is known to the receivers, is approxi-mated by

αopt = argmax0<α<1dmin (23)

where dmin is the minimum Euclidean distance and hk isknown to the receivers via channel estimation. The proof ofthis theorem is trivial and thus is omitted for brevity since it isvery similar to that of Theorem 1 due to the equivalent chan-nel model used. To make it more clear, we denote the channelgains for near user and far user as hnand hf , respectively. Themodified noise variances of z/|hf | and z/|hn|can be processedin the same way as in [36, Section III-A]. We define the pathloss ratio (PLR) β = 10 lg

(h2n/h

2f

). The transmitted signal

sets are denoted as Sn = {xn(k) | k = 1, 2, . . . ,Nn}, Sf ={xf (m) |m = 1, 2, . . . ,Nf }. The subscript n and f representnear user and far user, respectively. The SIC is used at nearuser and the far user’s signal is decoded treating the nearuser’s signal as noise. Then the CC capacities for far user andnear user are respectively given by (24) and (25), as shownat the bottom of the next page, where e(α, xn, n1,m1) =√αPxn(n1)−

√αPxn(m1) and d(α, xf , xn, n1, n2,m1,m2) =√

(1− α)P(xf (n2) − xf (m2)) +√αP(xn(n1) − xn(m1)).

Combing (24) and (25), the sum capacity for fading channelis given by

Csf (α) = Cf(xf |α, hf ,Nn,Nf

)+Cn

(xn | xf , α, hn,Nn,Nf

)(26)

For the Rayleigh channel with channel gain |hk |2 , k =1, 2, the probability density function (PDF) of hk is

f (hk ) =1

2σ 2 exp(− |hk |2 /2σ 2) (27)

Note that (12) varies with SNR, then the ergodic capacityat a certain SNR γ is given,

E[Cs (α, γ )] =1

2σ 2

∫+∞

0Cs (α, γ )exp(− |hk |2 /2σ 2)dhk

(28)

The calculation of ergodic capacity is not easy to conduct.We use the Gauss-Hermite quadrature approximation [48] toefficiently evaluate the performance of our proposed schemefor Rayleigh fading.

log2

∑j∈R

exp(−Dmin (i, j)

/σ 2)≥log2

W · exp−

∑j∈R

Dmin (i, j)

Wσ 2

= log2W −

∑j∈R

Dmin (i, j)

/(Wσ 2

· ln 2) (21)

L (D (i, j))≥M∑

i=1,i 6=j

log2W −∑j∈R

Dmin (i, j)

/(Wσ 2

· ln 2)

= (M2−M

)log2W −

∑Mi=1,i 6=j

∑j∈R Dmin (i, j)

Wσ 2 · ln 2(22)

VOLUME 7, 2019 45127


FIGURE 3. Illustration of the minimum Euclidean distance dintra, dinterfor superposed constellation with N1= 4 and N2 = 16.

IV. OPTIMAL MINIMUM EUCLIDEAN DISTANCEWithout loss of generality, let d0,1 and r0,1, d0,2 and r0,2denote the minimum Euclidean distance and the maximumamplitude for user1 (near user) with constellations S1 anduser2 (far user) with constellation S2, respectively. Theassigned powers for user1 and user2 are P1 and P2, whereP1 = αP,P2 = (1 − α)P. The maximum amplitudes(the distances from the furthest points of the constellationsto the origin) and the minimum distances for user1 anduser2 in the superposed constellation are denoted as r1 andr2, d1 and d2 as shown in Fig. 3, respectively. From (2),if the point of user1 (denoted as x1 (t)) is fixed, it can beseen that the constellation for user2 is moved to this pointto form a new constellation centered at x1 (t) and scaledby√(1− α)P. We define this new scaled constellation as a

cluster. So each point of user1’s constellation has a cluster as

shown in Fig.3. Then, for the superposed constellation, wehave

dinter = d1 −√2 r2 = d0,1

√P1 −

√2r2 (29)

dintra = d2 = d0,2√P2 (30)

where dintra and dinter denote theminimumEuclidean distanceamong the points within the cluster and that of between twoneighbor clusters as shown in Fig. 3, respectively.

Note that each point’s amplitude is scaled respectively with√P1 and

√P2 for user1 and user2, we have r2 = r0,2

√P2,

then the optimal power allocation precoder by maximizingthe minimum Euclidean distance can be described as

αopt = arg max0<α<1

dmin

= arg max0<α<1

min {d inter , d intra } (31)

where dmin denotes the minimum Euclidean distance of over-all superposed points. Appling Theorem 1, when dinter =dintra, dmin is maximized. So we have

d0,2√P2 = d0,1

√P1 −

√2(r0,2

√P2) (32)

From (32), the power allocation ratio for user1 and user2 canbe given by

P1P2=

αMMED

1− αMMED=

(√2r0,2 + d0,2d0,1

)2

(33)

where αMMED is the optimal power allocation coefficient forour MMED approach. Then we have,

αMMED =

(√2r0,2 + d0,2

)2(d0,1

)2+

(√2r0,2 + d0,2

)2 (34)

Cf(xf |α, hf ,Nn,Nf

)= log2

(Nf)−

1NnNf

Nn∑n1=1

Nf∑n2=1

E

log2∑Nn

m1=1∑Nf

m2=1exp

(−∣∣hf {d (α, xf , xn, n1, n2,m1,m2

)} + z

∣∣2/σ 2)

∑Nnm1=1

exp(−∣∣hf√αP(xn (n1)− xn (m1))+ z

∣∣2 /σ 2)

= log2(Nf)−

1NnNf

Nn∑n1=1

Nf∑n2=1

E

log2∑Nn

m1=1∑Nf

m2=1exp

(−∣∣d (α, xf , xn, n1, n2,m1,m2

)+ z/

∣∣hf ∣∣∣∣2/σ 2)

∑Nnm1=1

exp(−∣∣√αP(xn (n1)− xn (m1))+ z/

∣∣hf ∣∣∣∣2 /σ 2)

(24)

Cn(xn | xf , α, hn,Nn,Nf

)= log2 (Nn)−

1Nn

Nn∑n1=1

E

log2∑Nn

m1=1exp(− |hn {e (α, xn, n1,m1)} + z|2 /σ 2

exp(−|z|2

σ 2

)

= log2 (Nn)−1Nn

Nn∑n1=1

E

log2∑Nn

m1=1exp(− |e (α, xn, n1,m1)+ z/ |hn||2 /σ 2

exp(−|z/|hn||2

σ 2

) (25)

45128 VOLUME 7, 2019


For square M -QAM, d0,k =√2 r0,k/

(√M − 1

), k ∈

{1, 2}. When the alphabets of in-phase and quadrature com-ponents are chose from the set of ±{(2m − 1)}M/2m=1, themodulated signals are power normalized with the factor2(M − 1)/3 [49], for example, a typical 16-QAM withthe alphabets {±1+±1i,±1+±3i,±3+±1i,±3+±3i},a normalizing factor

√10 is applied to ensure the average

energy is one. αMMED for some commonly used M -QAM isshown in Table 1.

TABLE 1. Optimal αMMED and dmin for 4, 16, and 64-QAM modulationpairs.

From Table 1, we can see that the power allocation coeffi-cients vary with the size of constellations. This observation isespecially crucial for adaptive modulation and coding (AMC)employed in NOMA system where power coefficients needto be carefully allocated according each user’s constellation.It also can be seen that for some pairs, such as (4,16)-QAM and (16,4)-QAM, have the same dmin. As well, pairs(4,64)-QAM, (16,16)-QAM and (64,4)-QAM have the sameminimum Euclidean distance even though the power coeffi-cients are different. This is because the sum constellations areidentical after the superposition of users’ symbols. So theymight be expected to have the same performance with properdecoding design.

For downlink scenario, the channel gain just serves asa scale to the Euclidean distance of two users’ superposedsymbols. At the receiver, the CSI is known via channel esti-mation such as maximum-likelihood (ML) estimator or linearminimum-mean-square-error (LMMSE) estimator proposedin [50]. Once obtained the channel coefficient hk , the nor-malized received symbol from (2) is given by

yk =(√αPx1 (t)+

√(1− α)Px2 (t)

)+zkhk

(35)

Then the impact of channel fading is eliminated. Therefore,the dmin can remain the same at the two users (receivers) givenα = αMMED and P is normalized. When SIC is applied,two users both decode the strong signal first treating theweak signal as noise. Then we can identify to which userthe decoded data belongs if either user signatures (eachuser is assigned a unique signature sequence similar to

CDMA system) are adopted in the synchronizationframe or the power allocation coefficients are sent to usersthrough signaling channels. In these ways, for our proposedscheme, we might not need any feedback of CSI at thetransmitter side because the power allocation coefficientsretain unchanged regardless of the varying of channels oncethe user pair is chosen.

It is worth mentioning that, when our MMED criterionand the power coefficients in Table 1 are employed for(N1,N2)-QAM pairs, the superposed constellation is also asquared (N1 × N2)-QAM. This features our scheme a verysimple conventional QAM decoder rather than SIC or MLdecoder, hence further reduces the complexity of a NOMAimplementation for receivers.

V. NUMERICAL RESULTSIn this section, we show some of the numerical results forboth GBC and FBCwith constellation constrained from threeaspects: capacity regions, CC sum capacity versus powerallocation coefficients, and maximizing entropy versus max-imizing minimum Euclidean distance.

A. CAPACITY REGIONS FOR DIFFERENTCONSTELLATIONS PAIRSFirstly, the capacity region is given in Fig. 4, Fig. 5 andFig. 6 for different constellations pairs at differentSNRs. We do not list all the pairs to save space.Here, R1 = I (

√αPx1:y), R2 = I (

√(1− α)Px2:y |

√αPx1)

stated in (9) and (10), Rn= Cn(xn | xf , α, hn,Nn,Nf ),Rf = C f (xf |α, hf ,Nn,Nf ) stated in (24) and (25), respec-tively. Unconstrained Gaussian refers to Gaussian capacityfor NOMA with infinite Gaussian inputs. For OMA, we useTDMA as a comparison as mentioned in [20], [25], [45]because the outcomes are straightforward. All the rate pairs(R1,R2) or (Rn,Rf ) are obtained by varying the power coef-ficient α from 0 to 1 in steps of 0.01. We use the definitionSNR = P/σ 2 assuming σ 2

= 1.From Fig. 4-6, we can see that CC capacity for NOMA

(NOMA-CC), OMA (OMA-CC) and Gaussian capacity arealmost the same at low SNRs. However, CC capacity forNOMA is superior to that of OMA at moderate SNR andat high SNR. It also can be seen from the figures thathigher order modulation can reach closer to the Gaussiancapacity.

It is worth noting that from Fig.5 (b) and Fig.5 (c), thecurve of CC capacity region for (16,4)-QAM is closer to thatof Gaussian region comparing to (4,16)-QAM. Obviously,for fading channel, better performance can be achieved withthe scheme that near user adopts higher order modulationwhen near user and far user have the different sizes ofconstellations.

Note that in Figure 4(a) and Figure 6(a), the rate pairs athigh SNR are not smooth. For Gaussian Broadcast channelwithout fading, the rates decrease when P1 = P2 = 0.5

VOLUME 7, 2019 45129


FIGURE 4. Capacity regions for user pair (4, 4)-QAM at SNR = 0, 3 and7 dB for both GBC and FBC. (a) N1 = 4,N2 = 4, capacity regions for GBC.(b) Nn = 4,Nf = 4 with β = 10dB.

because symbols from both users cancel out each other. Thisdecrease is more obvious at high SNR.

As can be seen from the above capacity regions, Gaussiancapacity with infinite Gaussian inputs is the upper bound ofthe constellation constrained capacity with finite input. Since‘infinite Gaussian inputs’ means a multitude of independentrandom inputs, when the number of the independent randomvariables are large enough, the distribution is Gaussian dis-tribution, which has the maximum capacity for memorylesschannel. While ‘constellation constraint’ means finite inputsand certainly results in capacity loss.

B. CC SUM CAPACITY VERSUS POWERALLOCATION COEFFICIENTFig. 7 demonstrates the CC sum capacity varying with powercoefficient allocation from 0.5 to 1 in steps of 0.1 for twousers NOMA at various constellation pairs, to name a few,(16,4)-QAM, (16,16)-QAM and (16, 64)-QAM. There is

FIGURE 5. Capacity regions for user pair (4, 16)-QAM at SNR = 0, 5 and10 dB for GBC and FBC. (a) N1 = 4,N2 = 16, capacity regions for GBC.(b) Nn = 4,Nf = 16 FBC with β = 10 dB. (c) Nn = 16,Nf = 4 FBC withβ = 10 dB.

no significant difference on capacity at low SNRs, when itcomes to high SNR, maximizing the minimum Euclideandistance merit is optimal for different constellation pairs.

45130 VOLUME 7, 2019


FIGURE 6. Capacity regions for user pair (16, 16)-QAM at SNR = 5, 10 and15 dB for GBC and FBC. (a) N1 = 16,N2 = 16, capacity regions for GBC.(b) Nn = 16,Nf = 16 FBC with β = 10 dB.

Here, αMMED is chosen according to the constellation pairsin Table 1. When two users both exploit high order mod-ulation, for example, N1 = 16, N2 = 64, the capacitydifference is enlarged. It is worth noting that even whenthe two channel gains are close, as presented by Gaussianbroadcast channel (GBC), optimal CC capacity can still beachieved with finite-alphabet inputs. This means that highchannel difference required in Gaussian inputs might beunnecessary for uniformly distributed finite input alphabetsapplication.

Fig. 8 demonstrates the CC sum capacity varying withpower coefficient allocation for FBC with power loss ratioβ = 10dB. Here, we only present 3 cases to save space.These 3 cases cover the typical situation Nn > Nf , Nn =Nf and Nn < Nf . Obviously, MMED criterion has thebest performance at high SNR versus the other power coef-ficients. Note that the CC sum capacity worsens at lowSNR since far user’s signal deteriorates due to near-fareffect.

FIGURE 7. CC sum capacity versus power coefficients for GBC.(a) N1 = 16, N2 = 4. (b) N1 = 16,N2 = 16. (c) N1 = 16,N2 = 64.

C. MAXIMIZING ENTROPY VERSUS MAXIMIZINGMINIMUM EUCLIDEAN DISTANCE (MMED)In Fig. 9, Fig. 10, and Fig. 11, we verify the results ofTheorem 1 by plotting the CC sum capacity of different

VOLUME 7, 2019 45131


FIGURE 8. CC sum capacity versus power coefficients for FBC, β= 10dB.(a) Nn = 16,Nf = 4. (b) Nn = 16,Nf = 16. (c) Nn = 16,Nf = 64.

constellation pairs with maximum entropy αopt and maximiz-ing minimum Euclidean distance αMMED. Maximum entropyis obtained by varying the power coefficient αfrom 0 to 1 in

FIGURE 9. CC sum capacity comparison with αopt and αMMED for (4,4),(4,16) and (4,64) constellation pairs.

FIGURE 10. CC sum capacity comparison with αopt and αMMED for (16,4),(16,16) and (16,64) constellation pairs.

steps of 0.01. With this setup, the computation cost for maxi-mum entropy criterion is 100 times of MMED criterion. As amatter of fact, there is no need to compute the CC capacitywhen a preset power coefficient is used for our proposedmethod. From these figures, it can be seen that MMED canwell approximate the CC sum capacity while reducing thecomplexity with fixed coefficients. Moreover, we present theαopt for different constellation pairs in Table 2, which showsthat, αopt varies with the SNR values. This means αopt needto be updated instantly as long as SNR changes.

Fig. 12-16 are presented to verify the results of Theorem 2.The CC sum capacities of maximum entropy and MMEDare plotted for several constellation pairs with path loss ratio5dB in Fig. 12 and Fig. 13, 10 dB in Fig. 14 and Fig. 15,respectively. Maximizing minimum Euclidean distance canstill well approximate the CC sum capacity at high SNR.However, the performance degrades at low SNR because faruser’s signal deteriorates due to channel fading, just as been

45132 VOLUME 7, 2019


FIGURE 11. CC sum capacity comparison with αopt and αMMED for (64,4),(64,16), and (64,64) constellation pairs.

TABLE 2. Numerically computed αopt with respect to SNR for GBC.

FIGURE 12. CC sum capacity comparison with αopt and αMMED for (4,4),(4,16), and (4,64) constellation pairs for FBC with β = 5dB.

seen in Fig. 8. This performance deterioration expands asthe path loss ratio increases. Furthermore, the αopt for someconstellation pairs is listed in Table 3 with β= 10dB, whichshows that, αopt varies with the SNR values and needs to



TABLE 3. Numerically computed αopt with respect to SNR for FBC withβ = 10 dB.

be updated instantly. Fig. 16 is presented to demonstrate theergodic capacity of our proposed scheme comparing to themaximum entropy in the Rayleigh fading channel. It showsthat αMMED can well approximate αopt for the maximum

VOLUME 7, 2019 45133



FIGURE 16. Ergodic capacity comparison with αopt and αMMED for (4,4),(4,16), and (16,4) constellation pairs for Rayleigh fading channel.

entropy in Rayleigh fading channel while dramaticallyreducing the complexity.

VI. CONCLUSIONIn this paper, we present a low complexity power alloca-tion approach for downlink NOMA with finite constellationinputs in both cases of GBC and FBC. The trade-off betweenthe performance and the complexity has been obtained.Specifically, by formulating to maximize the sum capac-ity, we approximate the solution to maximize the minimumEuclidean distance of superposed symbols at the transmitterat high SNR without the knowledge of channel conditions.The optimal power allocation coefficients are calculated forthe most commonly-used constellation pairs. Instead of com-puting the CC sum capacity and updating power coefficientsimultaneously, a fixed optimal coefficient pair is used todramatically decrease the complexity while provides the opti-mal performance at high SNR. We compute the CC capacity

and the corresponding regions for two users with 4, 16,64-QAM constellations to compare with the regions for infi-nite Gaussian inputs. It is worth noting that even when thetwo channel gains are close, optimal CC capacity can still beachievedwith finite-alphabet inputs for our proposed scheme.This means that we can group two users with similar channelconditions, such as pair two strong users or pair two weakusers as long as the outage probability can be satisfied.This observation is different from that of the conventionalGaussian input signals. On the other hand, the scheduling ofpower coefficients for different users must take the size ofconstellations into consideration in practical system. Highercapacity can be achieved for fading channel when near useremploys higher order modulation given two users have differ-ent constellation sets. Hence, it might be better to pair a higherorder modulation for near user with a lower order modulationfor far user with the FBC scenario to reach the capacityregions. While for GBC, user pairing is immaterial due to thesame channel gains for two users. We can pair any of twousers as needed. With our proposed MMED approach, thesuperposed constellation is also a squaredQAM. This enablesour scheme to the conventional QAM decoder rather thanSIC or ML at receivers and further reduces the complexityof NOMA implementation. Note that our proposed MMEDapproach might be applied to MISO downlink NOMA aswell. Since the MMED scheme tries to enlarge the distancebetween two constellations regardless of CSI at the transmit-ter. We hope that these results might shed light on practicalNOMA applications in the coming future.

ACKNOWLEDGMENTThe authors would like to thank Robert Heath (Electricaland Computer Engineering, University of Texas at Austin)for the valuable inputs on the power allocation schemes andconstellation constrained capacity during discussions.

REFERENCES[1] A. Goldsmith, Wireless Communications. Cambridge, U.K.: Cambridge

Univ. Press, 2005.[2] Further Advancements for E-UTRA Physical Layer Aspects,

document V. 3GPP TR, Mar. 2010.[3] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.

Cambridge, U.K.: Cambridge Univ. Press, 2005.[4] F. Boccardi, R. W. Heath, A. Lozano, T. L. Marzetta, and P. Popovski,

‘‘Five disruptive technology directions for 5G,’’ IEEE Commun. Mag.,vol. 52, no. 2, pp. 74–80, Feb. 2014.

[5] L. Dai, B. Wang, Y. Yuan, S. Han, C. L. I, and Z. Wang, ‘‘Non-orthogonal multiple access for 5G: Solutions, challenges, opportuni-ties, and future research trends,’’ IEEE Commun. Mag., vol. 53, no. 9,pp. 74–81, Sep. 2015.

[6] S. M. R. Islam, N. Avazov, O. A. Dobre, and K.-S. Kwak, ‘‘Power-domainnon-orthogonal multiple access (NOMA) in 5G systems: Potentials andchallenges,’’ IEEE Commun. Surveys Tuts., vol. 19, no. 2, pp. 721–742,2nd Quart., 2017.

[7] W. Shin, M. Vaezi, B. Lee, D. J. Love, J. Lee, and H. V. Poor,‘‘Non-orthogonal multiple access in multi-cell networks: Theory, perfor-mance, and practical challenges,’’ IEEE Commun. Mag., vol. 55, no. 10,pp. 176–183, Oct. 2017.

[8] Z. Ding et al., ‘‘Application of non-orthogonal multiple access in LTEand 5G networks,’’ IEEE Commun. Mag., vol. 55, no. 2, pp. 185–191,Feb. 2017.

45134 VOLUME 7, 2019


[9] Z. Ding, X. Lei, G. K. Karagiannidis, R. Schober, J. Yuan, and V. Bhargava,‘‘A survey on non-orthogonal multiple access for 5G networks: Researchchallenges and future trends,’’ IEEE J. Sel. Areas Commun., vol. 35, no. 10,pp. 2181–2195, Oct. 2017.

[10] S. Vanka, S. Srinivasa, Z. Gong, P. Vizi, K. Stamatiou, and M. Haenggi,‘‘Superposition coding strategies: Design and experimental evaluation,’’IEEE Trans. Wireless Commun., vol. 11, no. 7, pp. 2628–2639, Jul. 2012.

[11] L. Ping, L. Liu, K. Wu, and W. K. Leung, ‘‘Interleave division multiple-access,’’ IEEE Trans. Wireless Commun., vol. 5, no. 4, pp. 938–947,Apr. 2006.

[12] Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, andK. Higuchi, ‘‘Non-orthogonal multiple access (NOMA) for cellular futureradio access,’’ in Proc. 77th IEEE VTC-Spring, Dresden, Germany,Jun. 2013, pp. 1–5.

[13] H. Nikopour and H. Baligh, ‘‘Sparse code multiple access,’’ in Proc. IEEE24th Annu. Int. Symp. Pers., Indoor, Mobile Radio Commun. (PIMRC),Sep. 2013, pp. 332–336.

[14] J. Huang, K. Peng, F. Yang, C. Pan, and H. Jin, ‘‘Scalable video broad-casting using bit division multiplexing,’’ IEEE Trans. Broadcast., vol. 60,no. 4, pp. 701–706, Dec. 2014.

[15] Z. Yuan, G. Yu, and W. Li, ‘‘Multi-user shared access for 5G,’’ Telecom-mun. Netw. Technol., vol. 5, no. 5, pp. 28–30, May 2015.

[16] S. Kang, X. Dai, and B. Ren, ‘‘Pattern division multiple access for 5G,’’Telecommun. Netw. Technol., vol. 5, no. 5, pp. 43–47, May 2015.

[17] L. Lu, L. You, and S. C. Liew, ‘‘Network-coded multiple access,’’ IEEETrans. Mobile Comput., vol. 13, no. 12, pp. 2853–2869, Dec. 2014.

[18] H. Pan, L. Lu, and S. C. Liew, ‘‘Practical power-balanced non-orthogonalmultiple access,’’ IEEE J. Sel. Areas Commun., vol. 35, no. 10,pp. 2312–2327, Oct. 2017.

[19] L. Dai, B. Wang, Z. Ding, Z. Wang, S. Chen, and L. Hanzo, ‘‘A surveyof non-orthogonal multiple access for 5G,’’ IEEE Commun. Surveys Tuts.,vol. 20, no. 3, pp. 2294–2323, 3rd Quart.,2018.

[20] Z. Dong, H. Chen, J. K. Zhang, L. Huang, and B. Vucetic, ‘‘Uplinknon-orthogonal multiple access with finite-alphabet inputs,’’ IEEE Trans.Wireless Commun., vol. 17, no. 9, pp. 5743–5758, Sep. 2018.

[21] S. Timotheou and I. Krikidis, ‘‘Fairness for non-orthogonal multiple accessin 5G systems,’’ IEEE Signal Process. Lett., vol. 22, no. 10, pp. 1647–1651,Oct. 2015.

[22] M. S. Ali, H. Tabassum, and E. Hossain, ‘‘Dynamic user clustering andpower allocation for uplink and downlink non-orthogonal multiple access(NOMA) systems,’’ IEEE Access, vol. 4, pp. 6325–6343, 2016.

[23] W. Bao, H. Chen, Y. Li, and B. Vucetic, ‘‘Joint rate control and powerallocation for non-orthogonal multiple access systems,’’ IEEE J. Sel. AreasCommun., vol. 35, no. 12, pp. 2798–2811, Dec. 2017.

[24] Z. Ding, Z. Yang, P. Fan, and H. V. Poor, ‘‘On the performance ofnonorthogonal multiple access in 5G systems with randomly deployedusers,’’ IEEE Signal Process. Lett., vol. 21, no. 12, pp. 1501–1505,Dec. 2014.

[25] Z. Ding, P. Fan, and H. V. Poor, ‘‘Impact of user pairing on 5G nonorthog-onal multiple-access downlink transmissions,’’ IEEE Trans. Veh. Technol.,vol. 65, no. 8, pp. 6010–6023, Aug. 2016.

[26] Z. Yang, Z. Ding, P. Fan, and N. Al-Dhahir, ‘‘A general power allo-cation scheme to guarantee quality of service in downlink and uplinkNOMA systems,’’ IEEE Trans. Wireless Commun., vol. 15, no. 11,pp. 7244–7257, Nov. 2016.

[27] Z. Yang, J. A. Hussein, P. Xu, Z. Ding, and Y. Wu, ‘‘Power alloca-tion study for non-orthogonal multiple access networks with multicast-unicast transmission,’’ IEEE Trans. Wireless Commun., vol. 17, no. 6,pp. 3588–3599, Jun. 2018.

[28] Y. Mao, B. Clerckx, and V. O. K. Li, ‘‘Rate-splitting multiple accessfor downlink communication systems: Bridging, generalizing, and out-performing SDMA and NOMA,’’ EURASIP J. Wireless Comput. Netw.,vol. 2018, p. 133, 2018. doi: 10.1186/s13638-018-1104-7.

[29] Z. Chen, Z. Ding, X. Dai, and G. K. Karagiannidis, ‘‘On the applicationof quasi-degradation to MISO-NOMA downlink,’’ IEEE Trans. SignalProcess., vol. 64, no. 23, pp. 6174–6189, Dec. 2016.

[30] T. Cover, ‘‘Broadcast channels,’’ IEEE Trans. Inf. Theory, vol. 18, no. 1,pp. 2–14, Jan. 1972.

[31] G. Caire and S. Shamai, ‘‘On the achievable throughput of a multiantennaGaussian broadcast channel,’’ IEEE Trans. Inf. Theory, vol. 49, no. 7,pp. 1691–1706, Jul. 2003.

[32] A. D. Wyner, ‘‘Recent results in the Shannon theory,’’ IEEE Trans. Inf.Theory, vol. IT-20, no. 1, pp. 2–10, Jan. 1974.

[33] R. S. Cheng and S. Verdu, ‘‘Gaussian multiaccess channels with ISI:Capacity region and multiuser water-filling,’’ IEEE Trans. Inf. Theory,vol. 39, no. 3, pp. 773–785, May 1993.

[34] L. Li and A. J. Goldsmith, ‘‘Capacity and optimal resource allocationfor fading broadcast channels-part I: Ergodic capacity,’’ IEEE Trans. Inf.Theory, vol. 47, no. 3, pp. 1083–1102, Mar. 2001.

[35] T. M. Cover, and J. A. Thomas, Elements of Information Theory, 2nd ed.Hoboken, NJ, USA: Wiley, 2006.

[36] J. Harshan and B. S. Rajan, ‘‘A Novel Power Allocation Scheme forTwo-User GMAC with Finite Input Constellations,’’ IEEE Trans. WirelessCommun., vol. 12, no. 2, pp. 818–827, Feb. 2013.

[37] W. He and C. N. Georghiades, ‘‘Computing the capacity of aMIMO fadingchannel under PSK signaling,’’ IEEE Trans. Inf. Theory, vol. 51, no. 5,pp. 1794–1803, May 2005.

[38] J. Harshan and B. S. Rajan, ‘‘On two-user Gaussian multiple accesschannels with finite input constellations,’’ IEEE Trans. Inf. Theory, vol. 57,no. 3, pp. 1299–1327, Mar. 2011.

[39] J. Choi, ‘‘On the power allocation for a practical multiuser superposi-tion scheme in NOMA systems,’’ IEEE Commun. Lett., vol. 20, no. 3,pp. 438–441, Mar. 2016.

[40] S.-L. Shieh and Y.-C. Huang, ‘‘A simple scheme for realizing the promisedgains of downlink nonorthogonal multiple access,’’ IEEE Trans. Commun.,vol. 64, no. 4, pp. 1624–1635, Apr. 2016.

[41] T. Yazaki and Y. Sanada, ‘‘Throughput performance of joint detectionin non-orthogonal multiple access schemes,’’ IEICE Trans. Commun.,vol. E100-B, no. 2, pp. 344–353, Feb. 2017.

[42] Z. Dong, H. Chen, J.-K. Zhang, and L. Huang, ‘‘On non-orthogonalmultiple access with finite-alphabet inputs in Z-channels,’’ IEEE J. Sel.Areas Commun., vol. 35, no. 12, pp. 2829–2845, Dec. 2017.

[43] F. Perez-Cruz, M. R. D. Rodrigues, and S. Verdu, ‘‘MIMO GaussianChannels with arbitrary inputs: Optimal precoding and power allocation,’’IEEE Trans. Inf. Theory, vol. 56, no. 3, pp. 1070–1084, Mar. 2010.

[44] W. Zeng, C. Xiao, M. Wang, and J. Lu, ‘‘Linear precoding for finite-alphabet inputs over MIMO fading channels with statistical CSI,’’ IEEETrans. Signal Process., vol. 60, no. 6, pp. 3134–3148, Jun. 2012.

[45] H. Pan, S. C. Liew, J. Liang, Y. Shao, and L. Lu, ‘‘Network-coded multipleaccess on unmanned aerial vehicle,’’ IEEE J. Sel. Areas Commun., vol. 36,no. 9, pp. 2071–2086, Sep. 2018.

[46] X. Ren, M. Tao, and W. Chen, ‘‘Compressed channel estimation withposition-based ICI elimination for high-mobility SIMO-OFDM systems,’’IEEE Trans. Veh. Technol., vol. 65, no. 8, pp. 6204–6216, Aug. 2016.

[47] L. Collin, O. Berder, P. Rostaing, and G. Burel, ‘‘Optimal minimumdistance-based precoder for MIMO spatial multiplexing systems,’’ IEEETrans. Signal Process., vol. 52, no. 3, pp. 617–627, Mar. 2004.

[48] T. Ketseoglou and E. Ayanoglu, ‘‘Linear precoding gain for large MIMOconfigurations with QAM and reduced complexity,’’ IEEE Trans. Com-mun., vol. 64, no. 10, pp. 4196–4208, Oct. 2016.

[49] J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication,3rd ed. New York, NY, USA: Academic, 2004.

[50] Y. Tan, J. Zhou, and J. Qin, ‘‘Novel channel estimation for non-orthogonalmultiple access systems,’’ IEEE Signal Process. Lett., vol. 23, no. 12,pp. 1781–1785, Dec. 2016.

WENLONG XIA (M’18) received the B.Sc. andM.Sc. degrees in communication engineering andthe Ph.D. degree in signal and information pro-cessing from Sichuan University (SCU), Chengdu,China, in 2000, 2003, and 2014, respectively.He is currently a Visiting Scholar with the Depart-ment of Electrical and Computer Engineering,The University of Texas at Austin, USA. He isalso a Lecturer with the School of Electron-ics and Information Engineering, SCU. He has

authored or coauthored many papers and holds eight patents. Hisresearch interests include wireless communication, blind source separation,space–time coding, software-defined radio, and advanced multiple accesstechniques.

VOLUME 7, 2019 45135

http://dx.doi.org/10.1186/s13638-018-1104-7


YUANPING ZHOU received the B.S. degree in radio engineering fromChongqing University, Chongqing, China, in 1982, the M.S. degree in elec-trical engineering from the University of Illinois at Chicago, USA, in 1986,and the Ph.D. degree in electrical engineering from the Georgia Instituteof Technology, Atlanta, USA, in 1999. From 2000 to 2002, he was withMotorola, Inc., as a Lead Systems Engineer. He is currently a Professor withthe School of Electronics and Information Engineering, Sichuan University,Chengdu, China. His research interests include space–time signal processing,MIMO systems, adaptive antennas, interference suppression, andmodulationmethods.

GUIDE YANG received the B.S. and Ph.D. degreesfrom the School of Electronics and InformationEngineering, Sichuan University, Chengdu, China,in 2013 and 2018, respectively. From 2013 to2018, he was a Graduate Student majoring in sig-nal and information processing. He holds threearticles and five patents. His research interestsinclude massive MIMO, wireless communication,and intelligent optimization algorithm.

RAY T. CHEN (M’91–SM’98–F’04) received theB.S. degree in physics from National Tsing HuaUniversity, Taiwan, in 1980, and the M.S. degreein physics and the Ph.D. degree in electrical engi-neering from the University of California at SanDiego, USA, in 1983 and 1988, respectively.

He joined The University of Texas at Austin,in 1992, where he is currently the Keys and JoanCurry/Cullen Trust Endowed Professor of electri-cal and computer engineering. He is the Direc-

tor of the Nanophotonics and Optical Interconnects Research Laboratory,Microelectronics Research Center. He is also the Director of the AFOSRMURI Center for Silicon Nanomembrane involving faculty from Stanford,UIUC, Rutgers, and UT Austin. He then served as the CTO, the Founder,and the Chairman of the Board of Radiant Research, Inc. He also serves asthe Founder and the Chairman of the Board of Omega Optics Inc., since itsinitiation, in 2001. He has authored or coauthored more than 800 publishedpapers, is an editor or a co-editor for over 20 books, and holds morethan 35 patents. His research topics are focused on nano-photonic passiveand active devices for bio- and EM-wave sensing, thin-film guided-waveoptical interconnection and packaging for 2D and 3D laser beam routing andsteering, true time delay (TTD) wideband phased array antenna (PAA), andhigh-speed communication and interconnect application.

Dr. Chen is a Fellow of OSA and SPIE. He was a recipient of the 1987 UCRegent’s Dissertation Fellowship and the 1999 UT Engineering FoundationFaculty Award, for his contributions in research, teaching, and services.He received the Honorary Citizenship Award, in 2003, from the Austin CityCouncil for his contribution in community service. He was also a recipientof the 2008 IEEE Teaching Award, the 2010 IEEE HKN Loudest ProfessorAward, and the 2013 NASA Certified Technical Achievement Award for hiscontribution on moon surveillance conformable phased array antenna.

45136 VOLUME 7, 2019

Optimal Minimum Euclidean Distance-Based Precoder for...

Documents

Transcript of Optimal Minimum Euclidean Distance-Based Precoder for...