Signal Shaping for Two-User Gaussian Multiple Access Channels with Computation: Beyond the Cut-Set Bound

0.17 MB·

by  Zhiyong Chen

Signal Shaping for Two-User Gaussian Multiple
Access Channels with Computation: Beyond the
Cut-Set Bound
Zhiyong Chen, and Hui Liu
Cooperative Medianet Innovation Center, Shanghai Jiao Tong University, Shanghai, P. R. China
Email: zhiyongchen, huiliu @sjtu.edu.cn
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1
0 Abstract—In this paper, we investigate the signal shaping z
Jan 2 iaopncpecrteiamtsrsiatznwcinhsomga-uniitsnnteeeprrluit(dsMipbsArceorCybeo)tanebdwiltitiithtmhyeecdcouimsmtt-respimebututobatrtioyoiuolnenns,.dst.hIItGenaitscruoassnnshitsaormnwasinstmswitouhintlathtirptablhteyee- wA A xA y D wA¯wB
single-userdiscretememorylesschannel,theMaxwell-Boltzmann
5 distribution is no longer a good approximation to the optimal wB B xB
2
input probability distribution for this discrete-time Gaussian

MAC with computation. Specifically, we derive and analyze the
] Fig. 1. A two-user discrete-time memoryless Gaussian multiple access
T mutualinformationforthischannel.Becauseofthecomputation
channel withcomputation.
I in the destination, the mutual information is not concave in
. general on the input probability distribution, and then primal-
s
c dual interior-point method is used to solve this non-convex
[ problem. Finally, some good input probability distributions for be regarded as the multiple access phase in two-way relaying
16-ary pulse amplitude modulation (PAM) constellation are channels [1], [2]. To the best of my knowledge, the capacity

1 obtained and achieve 4.0349 dB gain over the cut-set bound of such channels is still unknown, but it is upper bounded by
v for the target transmission rate 3.0067 bits/(channel use).
4 1 P
9 I. INTRODUCTION, SIGNAL MODELAND MAIN RESULTS Ccs = 2log2(1+ σ2), (1)
3
Inthispaper,westudyatwo-userdiscrete-timememoryless per transmitter based on the cut-set bound [1], [2]. Here, we
7
0 Gaussian multiple access channel (MAC) with computation. use σ2 =N0/2 as the noise variance per dimension. As well
. The two-user discrete-time memorylessGaussian MAC is de- known, this upper bound is attained by continuous Gaussian
1
70 fiYn,eadnbdyatwcoonidniptiuotnaallphparobbetasbXiliit,yid=istAri,bBut,ioanndPoru(tYpu|XtaAlp,XhaBbe).t iisnpthuet.oFpotrimthailssidginscarlelitnegdsitsrtaritbeugtyioPnXininpuotsrdXerAtoanmdaxXimBi,zewhthaet
1 Here, let Xi and Y be random variables taking values in Xi mutual information I(WC;Y) in this two-user discrete time
v: aZnd∼Y, re(s0p,eNctiv)eilsy.aWGeaucsosnisaindenroYise=asXshAow+nXFBig+. 1Z. Zanids memoryless Gaussian MAC with computation Pr(y|xA,xB)?
i 0
X indepeCnNdent of X , i = A,B. Furthermore, we consider A. Main Results
i A
X
r and are one-dimensional signal constellations, but is In this paper, we answer this fundamental problem in
a orthoXgBonalwith . For example, we use = a MXAand some cases. In particular, we consider a real-value 2m-ary
XB XA { i}i=1
XityBP= {=√−p1aiM}Mi=,1w=he√re−a1XisArewailthnucmorbreesr,pion=di1n,g..p.,rNob.abil- apulcsoemapmlepxli-tvuadleuem2omdu-laartyionPA(PMAMc)oncsotneslltaetliloantionfofror XA, ia.ne.d,
NoXw, w{eid}ei=sc1ribe the ciomputation operation. Source i M = 2m and a = M + 2i 1 for i = X1B,...,M.
i
− −
has source bit messages W , i = A,B. W and W are With this orthogonalconstellations,the destination can dothe
i A B
independent.Byusinglinearmodulation,W ismappedtothe ambiguity-free detection, which means w can be uniquely
i C
signal symbol X , i = A,B. Thus, X is also independent decoded.
i A
of X . Different from the conventional MAC, the goal of 1) GoodInputProbabilityDistributions: Wefirstderivethe
B
the destination in this paper is to compute a target mod- mutualinformationI(W ;Y)forthetransmissionofarbitrary
C
2 sum of the messages from the received signals Y, i.e., and withP .Theoptimalsignallingstrategyproblem
A B X
X X
W =W W .Inthiscontext,differentx +x valuescan for an optimal choice P∗ to maximize I(W ;Y) is then for-
C A⊕ B A B X C
representthesamew value,duetothemany-to-oneoperation mulated. In contrast with the single-user discrete memoryless
C
of computation in the destination. channel (DMC) where the problem of capacity computation
Consider the probability restriction M p = 1 and 0 is convex, this convexity is missing. As a result, we use
Pi=1 i ≤
p 1 for i, and the average transmission power constraint primaldualinterior-pointmethodtocarryoutthisoptimization
i
E[≤x 2] =∀E[x 2] = M p a 2 P. This model can problem and obtain good input probability distributions.
| A| | B| Pi=1 i| i| ≤
TABLEI
GOODINPUTPROBABILITYDISTRIBUTIONSPX∗ OF16-PAM • The uniform distribution suffers a large shaping loss.
CONSTELLATIONFORGAUSSIANMACOVERCOMPUTATIONWITH For example, when the achievable rate Rt is 3.0067
CONSTRAINTSONTHEDIFFERENTTARGETTRANSMISSIONRATERt bits/(channel use), the uniform distribution is far away
(BITS/CHANNELUSE).FOREACHINPUTDISTRIBUTIONTHESNR
THRESHOLD(P )∗ISGIVEN.BESIDES,APPLYINGTHECUT-SETBOUND from the cut-set bound by 3.9901 dB, which points out
σ2
(1)FORTHETARGETRATERt,WECANGET(σP2)cs.WEALSOAPPLYTHE the importance of signal shaping.
MBDISTRIBUTIONANDUNIFORMDISTRIBUTIONTOTHE • In contrast with the single-user DMC, the Maxwell-
CONSTELLATION,ANDOBTAIN(σP2)MB AND(σP2)uf FORRt, Boltzmann distribution is no longer a good approx-
RESPECTIVELY.
imation to the optimal distribution for discrete-time
Rt 3.0067 1.9724 0.9846 0.5239 Gaussian MAC with computation. Similarly, consider-
p1 0.0002276 0.0000896 0.0000140 0.0184170 ing Rt = 1.9724 bits/(channel use), the proposed good
p2 0.0933530 0.0000637 0.0000495 0.0540220 input probability distribution P∗ achieves 2.5893 dB
p3 0.0001605 0.0001011 0.0250460 0.0003871 X
gain compared with the optimized MB distribution with
p4 0.0001664 0.0845740 0.0464800 0.0560510
p5 0.1692600 0.1336100 0.0062702 0.0000330 λ∗ =0.115.
p6 0.1833400 0.0001079 0.0229170 0.0000371
p7 0.0002105 0.2013800 0.0000193 0.0000140 2) Beyond the cut-set bound: Surprisingly, we find that in
p8 0.2352551 0.3054700 0.7495156 0.0000319 discrete-time Gaussian MAC with computation, the trans-
p9 0.0001460 0.0000463 0.0000159 0.7945245 mission rate with the proposed good input probability
p10 0.0001526 0.0000470 0.0000211 0.0000120
distributioncanbeatthecut-setbound!Basedonthecut-set
p11 0.2007700 0.1881500 0.0001262 0.0000012
p12 0.0002296 0.0001013 0.0903270 0.0000113 bound(1), thethreshold(σP2)cs onthischannelis18.0349dB
p13 0.0002384 0.0860131 0.0589980 0.0000281 for C = 3.0067 bits/(channel use), but we can get a much
cs
p14 0.1157200 0.0000909 0.0001697 0.0004899 better threshold (P )∗ = 14 dB based on the proposed good
p15 0.0001761 0.0000615 0.0000167 0.0256960 σ2
p16 0.0005942 0.0000936 0.0000138 0.0502440 inputprobabilitydistributionPX∗.Similarly,theproposedgood
(P )∗ dB 14 9 3 -1 inputdistributioncanoutperformthecut-setboundby2.5834,
σ2
(P )cs dB 18.0349 11.5834 4.6471 0.2831 1.6471 and 1.2831 dB for Rt = 1.9724,0.9846 and 0.5239
σ2
(P )MB dB 17.7858 11.5893 5.5165 2.3252 bits/(channel use), respectively.
σ2
(P )uf dB 22.0250 16.9335 10.1647 5.8724 Because I(W ;Y) for this channelcannotbe calculated in
σ2 C
closed form, we cannot prove this phenomenon theoretically.
However, we can express this observation from the mismatch
The Maxwell-Boltzmann(MB) distributionand the uniform of the cut-set bound for discrete-time Gaussian MAC with
distribution are considered as two benchmarks in this paper. computation. Accordingly, the cut-set bound is defined as
The Maxwell-Boltzmann distribution provides a very good C = max I(X ;Y X ), which means the maximum
cs PX A | B
approximation to the optimal distribution obtained from the rate achievable from source A to the destination when source
Blahut-Arimoto algorithm for the single-user DMC [3]–[5], B is not sending any information. In this context, when the
and can be written as transmission rate is below C , X can be reliably transmit-
cs A
pi = Peixepx(cid:0)p−(λ−|λa|ia|2i(cid:1)|2), (2) tHtheodawnweivintehdri,vaitrdhbueitardlaerwsiltAyinasotmiroanwllBwe.arrnDotrsueptortooobbatbhtaieilnitcyowmPApr(u⊕wtaAwtioB6=n,,rmwaˆtAhoer)er.
where the parameterλ characterizesthe trade-offbetween the than one superposition signals x +x are mapped to one
A B
mavienriamguempoawveerraPgeaennderegnytrwophyichHm(Weacn)s. Wthee mcainnicmounmsumsigentahle- wwA⊕=wwB′,.yTiehludsi,ngthewrAe m⊕awybBe=exwisA′t s⊕omweB′ cfaosreswsAuc6=hwthA′atatnhde
B 6 B
to-noise (SNR) to achieve a given transmission rate Rt by transmissionratepertransmittermaybecanexceedCcs,yield-
selecting λ properly. Taking Rt = 3.0067 (bits per channel ingPr(wC =wˆC) 0,althoughwehavePr(wA =wˆA)=0
use) for example, the optimized value λ∗ = 0.0295 for 16- or Pr(w =6 wˆ )=→0. 6 6
B B
6 6
PAM. We have pi =1/M for the uniform distribution. Besides,thiscut-setboundC canbeattainedbyGaussian
cs
The resulting input probability distributions of 16- input x N(0,P). In general, there is no solution to
A
PAM1 (M = 16) are shown in Table I for Rt = calculate I(∼W ;Y) with X N(0,P) and X N(0,P).
c A B
3.0067,1.9724,0.9846and0.5239(bitsperchanneluse).Each Butaccordingto(2),theMBd∼istributionisadiscre∼teGaussian
column corresponds to on particular input probability distri-
distribution. We can use the MB distribution to verify the
bution ϕ∗p as well as the corresponding SNR ((σP2)∗ (dB)). above-mentionedphenomenon.From Table I, we can see that
Here,thethreshold(P )∗ isthesmallestrequired P suchthat the transmission rate of the MB distribution can also beat
σ2 σ2
the transmission achieves a given rate Rt. Likewise, we have the cut-set bound at some SNR values. For example, the MB
the thresholds (σP2)MB and (σP2)uf for the MB distribution distribution with λ∗ = 0.0295 achieves 0.2491 dB gain over
andtheuniformdistribution,respectively.Interestingly,wecan the cut-set bound for rate R =3.0067 bits/(channel uses),
t
observe from Table I that:
Accordingly, we conjecture that the cut-set bound is not
tight for the discrete-time Gaussian MAC with computa-
1It is easily to obtain good input probability distributions with any n or
otherconstellations, e.g.,quadrature amplitude modulation (QAM). tion, and can be exceeded.
B. Related Work Ω =M and the probability of wi can be calculated as
| i| C
For the single-user DMC, the mutual information is a con-
Pr(wi )= p p , (3)
cave function of the input probability distribution and Kuhn- C X k l
Tucker condition is necessary and sufficient for a distribution (k,l)∈Ωi
to maximize the mutual information. The Blahut-Arimoto for i=1, ,M.
···
algorithm is then developed to compute the optimal input Remark 1: Therefore, the entropy of W is
C
probabilitydistribution[3], [4], and can be approachedby the
M
MB distribution [5]. Indeed, by selecting constellation points
H(W )= p p log p p . (4)
properly based on the MB distribution at any dimension, the C −X X k l 2 X k l
ultimate shaping gain (1.53 dB) can be achieved [5]. i=1(k,l)∈Ωi (k,l)∈Ωi
For discrete-time memoryless Gaussian MAC, the mutual The first- and second-order derivatives of H(W ) are
C
information (e.g., I(X ,X ;Y)) is not concave on the input
A B
probability distribution. However, with the binary input, the ∂H(WC) = 2 2 M pi log p p , (5)
total capacity can be calculated for a two-user discrete-time ∂p −ln2 − X j 2 X k l
memoryless Gaussian MAC [6]. Then, a generalized Blahut- i j=1  (k,l)∈Ωj 
Arimotoalgorithmhasbeendevelopedforcomputationofthe
totalcapacityofdiscrete-timememorylessGaussianMAC[7]. ∂2H(W ) 4 p p
C i n
= 2log p p ,
More recently, a two-user Gaussian MAC under peak power ∂p ∂p − 2 X k l− ln2 p p
constraintsatthetransmittersisaddressedin[8],whichproves i n (k,l)∈Ωj′ P(k,l)∈Ωj′ k l
that discrete distributions with a finite number of mass points M 4pipn
j j
, (6)
can achieve anypointon the boundaryof the capacity region. − X ln2 p p
Instead of reconstructingall the signals of each transmitter, j=1,j6=j′ P(k,l)∈Ωj k l
the destination only reconstructs a function of sources in a where pi is the probability of a or √ 1a , yielding a +
MAC over computation [9]. The work of [9] presents that j k − k k
√ 1a Ω or a +√ 1a Ω for given i, respectively.
i j i k j
structured codes can achieve higher computation rates for − ∈ − ∈
Ωj′ denotes the index pairs set of (xA,xB) including of
computing the modulo-sum of two messages in Gaussian
(k,l) = (i,n) or (k,l) = (n,i). We can see from (6) that
MAC over computation. Accordingly, [1] achieves a rate ∂2H(WC) isdependentofP ,sothattheconvexityofH(W )
of 1log(1 + P ) by using lattice coding in the multiple ∂pi∂pn X C
2 2 σ2 on P is missing.
access phase of the two-way relaying channels. The results X
Fortunately, because x is orthogonal with x , we have
are effective in understanding specific features of different A B
Pr(y x ,x )=Pr(y x )forall(x ,x ) .The
computation functions inherent the model. However, there | A B | AB A B ∈XA×XB
mutual information I(W ;Y) is then given by the following
remain much fundamental problems to be done, e.g., the C
theorem.
optimal input probability distribution for discrete memoryless
Theorem 1: The mutual information I(W ;Y) of a
Gaussian MAC with computation. C
discrete-time memoryless Gaussian MAC with computation
II. PROBLEM FORMULATIONAND ANALYSIS Pr(y|xA,xB) is shown in (7).
Proof: Given wi , i = 1, ,M, the conditional proba-
Definition 1: LetxAB =xA+xB denotethesuperimposed bility Pr(y wi ) canCbe written··a·s
signal without noise in the destination. Let = xi wi | C
denote the signal set with respect to a giVvein co{mApBut|atiCo}n Pr(wi y)Pr(y) Pr(xi (k,l)y)Pr(y)
Pr(y wi )= C| = AB |
messages wi , for i = 1, ,M. We use natural mapping | C Pr(wi ) X Pr(wi )
for 2m-ary PCAM in this pap··e·r, e.g., w3 =0010 for M =16. C (k,l)∈Ωi C
There are M different pairs of (wA,wCB) associated with the = Pr(y|xiAB(k,l))Pr(xiAB(k,l))
same wi . Thus, the cardinality of the signal set is =M. X Pr(wi )
DefinCition 2: Let Ω = (k,l) : xi (k,l) |V=i| a + (k,l)∈Ωi C
√−1al ∈Vi,1≤k ≤Mi ,1≤{l ≤M} denAoBte the index pkairs (=i) X pkplPr(y|xiABp(kp,l)) (8)
setof(xA,xB)ofassociatedwiththesamewCi .Wethenhave (k,l)∈Ωi P(k,l)∈Ωi k l
I(W ;Y)= M  p p Pr(y wi )log Pr(y|wCi ) (7)
C Xy Xi=1(k,Xl)∈Ωi k l | C PMj=1(cid:16)P(k′,l′)∈Ωjpk′pl′(cid:17)Pr(y|wCj )
M M p p Pr(y xi (k,l))
=  p p log p p +  p p Pr(y xi (k,l))log P(k,l)∈Ωi k l | AB .
−Xi=1(k,Xl)∈Ωi k l (k,Xl)∈Ωi k l Xy Xi=1(k,Xl)∈Ωi k l | AB  PMj=1(cid:16)P(k′,l′)∈Ωjpk′pl′ Pr(y|xjAB(k′,l′))(cid:17)
where Steps (i) is based on Pr(xi (k,l)) = p p . Accord-
AB k l
ingly, we have Theorem 1. 4
Cut−set bound: 1/2×log(1+P/σ2)
According to (8), Pr(y wi ) is dependent on P given the 2
| C X 3.5 Good input distribution, 16−PAM
channel transform matrix Pr(y x ,x ). If P is uniform
d1istribution, iP.er.(,yPxrii =(k,lM1)).fo|rAall Bi, Pr(y|wXCi ) becomes nel use) 3 MUnBi fdoirsmtr idbiusttiroibnu, t1io6n−,P 1A6M−PAM
M P(k,l)∈Ωi | AB an
canRemMaarlksPo2r:(wTihbe)ePr(ymwwuriittut)ealnlog iansforPmr(ayIt|(iwWoCin)C;Y)I(.WC;Y=) bits per ch2.25
PAcycoPrdii=n1g to Cthe con|caCvity oPf Mj=m1uPtru(walCi)iPnrf(oyr|mwCia)tion, for ate (
fifisuxnPecdrti(oyPnxra(lyo,|fwxPCi)r),(,wniCot).=PHr(oy1w,we.v..e,)rM.,th,eIc(hWanCn;eYlt)ranissfoarmcomnactarvixe nsmission r1.15 3.35
A B C a
There|fore, the optimal i|nput probability distribution prob- Tr
0.5
lem for a discrete-time memoryless Gaussian MAC with
2.5
computation Pr(y xA,xB) can be formulated as 0 16 18 20
| −5 0 5 10 15 20 25 30
max I(W ;Y) P/σ2 (dB)
C
PX
M Fig.2. Transmissionratepertransmitterof16-PAMwithdifferentdistribu-
s.t. p =1 tionsindiscrete timeGaussianmultiple access channelwithcomputation.
X i
i=1
p 0, i=1,...M
i
≥ the gap at 2.5 (bits/channel use) between the cut-set bound
M
p a 2 P (9) and the achievable rate with uniform distribution is 4.67 dB.
X i| i| ≤ Thus, it is very necessary to do the signal shaping. Moreover,
i=1
the results show that the proposed good input probability
Based on M p = 1, P is located on an (M 1)-
Pi=1 i X − distribution has significantly shaping gain compared with the
dimensional simplex D . Similar to [6], I(W ;Y) is not
p C uniform distribution, e.g., 8.38 dB in 2.5 bits/(channel use).
in general concave on the input probability distribution PX. Interestingly,itcanalsobeseenthatforfixed P ( 5dB
Because ∂Pr∂(pyi|wCj) does not equal 0 and I(WC;Y) is a σP2 ≤ 25 dB), the achievable rate per transmiσtt2er−with P≤X∗
function of PX PX, it is difficult to evaluate obtain the based 16-PAM is larger than the cut-set bound. For example,
N
necessary condition for the optimal problem based on Kuhn- for rate-2.5 bit per channeluse, P∗ providesgain of 3.71 dB
X
Tucker condition. Here, denotes Kronecker product. comparedthe cut-set bound.Notice also that the transmission
N
For this non-convex problem, we can use some stochas- rate based the MB distribution is very close to the cut-set
tic optimization algorithm to solve this non-convex problem bound, even more than the cut-set bound for high SNR (11.5
[10], e.g, genetic algorithm or annealing algorithm. However, dB P 21.5 dB). The reason for this observation is
≤ σ2 ≤
stochastic algorithms have high complexity and cannot guar- presented in Section I-A.
antee a global optimal solution. In this paper, we use primal- Fig. 3 plots the good distribution and the corresponding
dualinterior-pointalgorithmwithrandominitialvaluestofind probability distribution of W for R =3.0067 bits/(channel
C t
theoptimalsolution[11]. Primal-dualinterior-pointmethodis use). It is clear from this figure that this P has more
X
a deterministic optimization algorithm, where every feasible volatilitythanPr(W ).Thisbehaviorimpliesthecomputation
C
initialvaluesisrelatedtoa localoptimalsolution.Asaresult, operationcansmooththepeak-to-averageprobability.Thiscan
we can use different initial values to approach the global be very useful to improve the entropy. With this probability
optimal solution. distribution, we have H(X ) = H(X ) = 2.5455 bits, but
A B
the entropy of W is increased to H =3.7959 bits.
III. NUMERICAL RESULTS C WC
We consider16-PAMconstellationinthesimulation.Based IV. EXTENSIONAND DISCUSSION
onTheorem1andprimal-dualinterior-pointalgorithm,wecan
A. Differentinput probabilitydistributions for differenttrans-
search good inputprobabilitydistributionP∗ and then obtain
X mitters
the transmission rate at each P per transmitter. Fig. 2 plots
σ2
the transmission rate per transmitter for increasing P with Inprevioussections,twoorthogonal1-Dconstellationswith
σ2
differentPX, wheresomecorrespondingproposedgoodinput thesameinputprobabilitydistributionPX areused.Obviously,
probability distributions P∗ are listed in Table I. it can be extended to two orthogonal 1-D constellations with
X
From the simulation results, it is clearly show that with the differentinputprobabilitydistributionPX andQX.LetQX =
uniform distribution, the system suffers a large shaping loss, {qi}Mi=1 betheinputprobabilitydistributionofXB.Therefore,
where the shaping loss is larger than 1.53 dB. For example, (3) becomes P(k,l)∈Ωipkql and then we can get I(WC;Y)
TABLEII
0.25 GOODINPUTPROBABILITYDISTRIBUTIONPX∗ OF16-QAM
CONSTELLATIONWITHGRAYMAPPINGFORGAUSSIANMACOVER
p
i COMPUTATIONFORDIFFERENTTARGETTRANSMISSIONRATESRt
Pr(wi) (BITS/CHANNELUSE).
C
0.2
Rt 3.9176 3.2494 2.7475 1.5412
( P )∗ dB 14 9 7 3
0.15 (2Pσ2)cs dB 11.4958 9.2991 7.5706 2.8112
2σ2
pi (2Pσ2)MB dB 14.025 9.012 7.011 3.0002
( P )uf dB 15.0681 10.9920 9.1750 5.0278
0.1 2σ2
0.05 V. CONCLUSIONS
We address the optimization problem of the input proba-
bility distribution to maximize the mutual information for a
0
2 4 6 8 10 12 14 16 two-userGaussian MACwith computation.We formulateand
Index i
analyzetheoptimizationproblem,andthenusetheprimal-dual
interior-pointalgorithmtosearchtheoptimalinputprobability
Fig.3. Angoodinputprobability distribution PX ={pi}1i=61 for16-PAM distribution. The main results are summarized as follows:
andtheprobability ofWC atRt=3.0067bits/(channel use).
• The uniform distribution suffers a large shaping loss
comparedwith the cut-setbound,where the shapingloss
by substituting Pr(wCi ) into (7). QX is also located on an is larger than 1.53 dB.
(M 1)-dimensionalsimplex Qp, and Dp Qp is a domain • The Maxwell-Boltzmann distribution also suffers a large
of P−X QX. As a result, we need to searNch PX∗ and Q∗X in performanceloss compared with the optimal input prob-
N
Dp Qp to maximize I(WC;Y). By increasing the search ability distribution.
dimNension, I(WC;Y) based on PX∗ and Q∗X is no less than • Theproposedinputprobabilitydistributioncanachievea
that based on P∗ and P∗. significant gain compared with the cut-set bound.
X X
Taking16-PAMconstellationforexample,forgiven P =9
σ2
dB,I(W ;Y)canachieve2.9762bits/(channeluse)basedon REFERENCES
C
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X X
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by using 2-D constellation, one signal x can be mapped
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C
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A B ′ ′ ′ AB ′ 6092,Oct2014.
[12], i.e. xA +xB = xA +xB with (xA 6= xA,xA 6= xB). [9] B.NazerandM.Gastpar,“Computationovermultiple-accesschannels,”
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Consider 16 quadratureamplitude modulation(QAM) with
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P∗ also outperform the cut-set bound by 0.3 and 0.57 dB at IEEETrans.Commun.,vol.62,no.6,pp.1803–1817, June2014.
X
R =3.2494andR =2.7475bits/(channeluse),respectively.
t t