Fading Multiaccess Channels in the Wideband Regime:
the Impact of Delay Constraints
D.TuninettiandG.Caire
InstitutEurecom
Sophia-Antipolis,France
e-mail: name@eurecom.fr
S. Verdu
PrincetonUniversity
Princeton,NJ,08544USA
e-mail: verdu@ee.princeton.edu
Abstract | We consider a multiaccess Gaussian
block fading channel wheretransmitters have causal
state information. Variable-rate coding with input
powerconstraint enforcedonaper-codewordbasis is
examined. We nd the average capacity region and
average capacity region per unit energy. Moreover,
we study the wideband slope of spectral eÆciency
vs. (E
b
=N
0
)dB, and we quantify the bandwidth ex-
pansionfactorofTDMAoversuperpositioncodingin
the widebandregime.
I.Introductionand motivations
Theliterature onthe capacityof fading channels hasfol-
lowed two distinct approaches to characterize power con-
straints: A)Powerconstraintonaper-symbolbasis(averaged
overthecodebook); B) Powerconstraint ona per-codeword
basis(averagedoverthelengthofthecodewordandthecode-
book).
Basicinformationtheoryresults[1 ,2]haveshownthatthe
laxer constraint B oers no advantage in unfaded channels
or infading channels where the transmitterdoes not know
the channel. However, when the transmitter has instanta-
neousknowledgeofthechannelfadingcoeÆcients,constraint
Bleadsto strictlylarger capacity thanA because itenables
theuseof\powercontrol"whichavoidswastingpoweratsym-
bols where thechannel undergoes deepfades. Under B,the
optimumstrategyas shownin[3] is water-lling intime. In
this setting, the fading process is assumed to be stationary
andergodicandthecodewordsarelongenoughforthefading
distributiontobe revealedwithinthespanofonecodeword.
Ifthefadingdynamicsareslow,thisleadstointolerablylong
blocklength,andconsequentlydelay. Furthermore,waterll-
ingpowercontrolleadstoverylargepeak-to-averageratioof
the transmitted waveform, inthe low-SNR (or\wideband")
regime.
In the high Eb=N0 high spectral eÆciency regime, con-
straintsAandB,althoughleadingtodierentoptimumtrans-
mission strategies, achieve very similar single-user capacity.
Onlyinconjunctionwithmultiaccessandmultiuserdetection
do optimum power control strategies lead to noticeable ad-
vantagesinthe highSNR regime[4 ]. Ontheotherhand,in
the low spectral eÆciency regime, constraint B enables (for
fading distributionswithinnite support)reliablecommuni-
cationwithenergyperbitassmallasdesired,instarkcontrast
toconstraintAwhichrequiresaminimumtransmittedenergy
perbitthat isboundedawayfromzero. Therefore,itisnat-
uraltofocusthe analysisofcapacityunderdelayconstraints
inthewidebandregime.
Incorporating delay constraints in Shannontheoretic set-
tingsisaperennialchallenge. Infadingchannels,itisessen-
tial to specify: 1) the duration of a codeword with respect
to the fadingprocesscoherence time, and2) the timeinter-
valonwhichthe average input powerconstraint is enforced.
Although vanishing error probability is unattainable unless
thenumberofdegreesoffreedomgrowswithoutbound,that
numbergrows with theproductof timeduration and band-
width. Thus,inthewidebandregime,anasymptoticanalysis
isfeasibleeveninasettingofxedduration codebooks.
In[5 ]theconceptof \delay-limited"capacity regionfor a
multiaccessfadingchannelisintroduced. Inthissetting,each
codewordspansasinglefadingstate(i.e.,thefadingcoherence
timeismuchlongerthanthecodewordduration)buttheinput
powerconstraintisevenlaxerthanBgivenabove: itisdened
overanarbitrarilylongsequenceofcodewords(weshallrefer
to suchconstraint as long-term). Thedelay-limited capacity
regionisthesetofrateswhichcanbeachievedforallfading
states(uptoasetofmeasurezero),subjecttothelong-term
input constraint. Inotherwords: thecodingrates are xed
whilethetransmitpoweructuates.
Inthispaperwetakeasomewhatcomplementarypointof
view: weassumeablockfadingmodelwhereacodewordspans
anite numberof slots, withfading constant overeachslot
and varying independentlyfrom slot to slot, and the power
constraint isenforced onaper-codewordbasis (constraint B
above). However, weallow variableratecoding sothat users
can coordinate their rates in order to be always inside the
fading-dependentcapacity region. Here,the transmitpower
isxedwhilethecodingratesarerandom. Consequently,we
dene thelong-term averagecapacity regionas thesetofall
achievable rates averaged over an arbitrarily long sequence
of codewords. Finally, we assume that thefading statesare
revealedcausallytothetransmitters[6 ,7].
Intherestofthispaper,weborrowsometerminologyfrom
current wirelesscellular systems. Ablock-fading model(ad-
mittedly,highlyidealized)isassumed[8 ],withL1dimen-
sions per fading state. Blocks of L input symbols spanning
the samefading state are referred to as \slots". Codewords
spangroups ofN slots, referred to as \frames". Themodel
underinvestigationhasseveralinterestingfeatures: byrelax-
ingtheframe-by-framecodinganddecodingrequirement,(but
stillenforcingtheper-framepowerconstraint),thelong-term
average capacity region coincides with the standard ergodic
capacity regionunder the per-frame powerconstraint. Fur-
thermore,theparameterN governsthepeak-to-averageratio
of transmitpower. For N !1 the modelis equivalent to
theconventionalsettingB,wheretransmittersareallowedto
blast power on good fading states and spend no power on
badfadingstates. ForN =1constant powertransmission is
enforced and transmitterchannel stateinformation playsno
role.
Asshownrecentlyin[9 ],informationtheoreticperformance
inthewidebandregimeisnotonlycharacterizedbythemin-
the spectral-eÆciency curve as a function of Eb=N0 (dB)
(b/s/Hz/3 dB).Minimum energy perbit alone is unable to
giveanyindication aboutbandwidthrequirements. Accord-
ingly, our analysis focuses onboth fundamental limits. We
showthat a\one-shot" powerallocation policythat concen-
tratesthe whole transmitenergyoveroneoutof N slots, is
optimal intermsof minimum energyper bit. In thesingle-
usercase, thesamepolicyisalsooptimalwithrespecttothe
widebandslope. Sincesuchslot mustbechosenonthe basis
ofcausalfeedback, thetransmittercannotsimplychoose the
mostfavorableslot. Rather,thesolutionisobtainedthrough
dynamicprogrammingandhasthestructureofacomparison
withadecreasingthreshold[6 ].
Inthe multiaccesssetting, where users are subject to in-
dependentfading coeÆcients,wemakeuseofthe framework
developedin[10 ]forthecapacity-per-unitcostregionformul-
tiaccesschannelsaswellasresultsontheregionofachievable
widebandslopewehaveobtainedin[11 ]. Theone-shotpower
allocation policy is decentralized: i.e., each user needs the
knowledgeofitsownfadingstate sequenceonly. Thispolicy
inconjunctionwithTDMAsuÆcestoachievethecapacityre-
gionperunitenergy. However,weshowthatifthesamepower
allocation policy is used in conjunction with superposition
signaling and a successive interference cancellation receiver,
widebandslopesgenerallylargerthanTDMAareachievable.
II.System model andbasic definitions
We consider a block-fading Gaussian Multi-AccessChan-
nel(MAC)where Ktransmittersmustdelivertheirmessage
within N slots to the receiver by spending a xed maxi-
mum energy. Thenumberof complexdimensions perslot is
L=bWTc,whereT istheslotdurationandW isthechannel
bandwidth. Thebasebandcomplexreceivedvectorinslot n
is
y
n
= K
X
k =1 c
k ;n x
k ;n +z
n
(1)
where zn NC(0;IL) 1
is an i.i.d. Gaussian noise vector
withnormalized unitvariancepercomponent, x
k ;n 2 C
L
is
thesignalofuserktransmittedinslotn,ck ;n isthecomplex
fading coeÆcient for user k with powergain
k ;n
=jc
k ;n j
2
,
assumedi.i.d. withcontinuouscdfF
(x).
The receiver has perfect (non-causal) Channel State In-
formation (CSI) while the transmitters have perfect causal
CSI [6, 7 ], i.e., inslot n the transmitters know the channel
stateuptotimen,denedby
Sn
=fck ;i:k=1;;K ; i=1;;ng (2)
Eachtransmitterk issubjecttotheper-codewordinputcon-
straint(referredtoas\short-term"powerconstraint).
1
NL N
X
n=1 jx
k ;n j
2
k
(3)
where
k
isthe transmittedenergypersymbol, andbecause
of the noise variance normalization adopted here it has the
1
ThenotationaNC( ;R)indicatesthataisapropercomplex
GaussianrandomvectorwithmeanE[a]=andcovarianceE[(a
)(a ) H
]=R.I
L
denotestheidentitymatrixofdimensionLL.
lowingwewillusethenotation
k ;n
= 1
L jxk ;nj
2
(4)
fortheinstantaneousSNRoftransmitterk inslotn.
ForniteN andLnopositiverateisachievable. However,
we canconsider a sequence of channels indexedby the slot
lengthLandstudytheachievableratesinthelimitforL!1
andxedN. Thisisastandardmathematicalabstractionin
thestudyofthelimitperformanceofblock-fadingchannels[8]
anditismotivatedbythefactthat,inmanypracticalappli-
cations,theproductWT islargeandT ismuchsmallerthan
the fadingcoherence time. Eveninthelimit oflarge L, the
rateK-tupleatwhichreliablecommunicationispossibleover
aframe of N slots is arandom vector,because only axed
numberKN offadingcoeÆcientsaecteachframe.
Consider along sequence offrames, eachcomposed ofN
slots. In this setting, it is meaningful to study the largest
achievablelong-termaveragerateregion,subjecttotheshort-
termpowerconstraint (3). Moreover, inthe energy-limited
caseinvestigatedhere,ameaningfulsystemdesigncriterionis
to look for thelargest achievable long-term average capacity
per unit energy (bit/joule). Next, in analogy with [12 , 10 ],
wecharacterizethelong-termaveragecapacityregionandthe
long-termaveragecapacityperunitenergyforoursystem.
Variable-rate coding inour setting is essentially dierent
fromvariable-ratecodinginanergodicsetting,suchasin[12 ,
3]. Here, we assume that each transmitter has an innite
\bit-reservoir" and, depending on the fading instantaneous
realization,transmitsavariablenumberofbitsperframe. We
modelthissettingbyletting themessage setsize dependon
thefadingstate. Becauseofspacelimitation,weomitformal
denitionsof variable-rate codingscheme,long-term average
capacityregionandlong-termaveragecapacityregionperunit
energy(see[13 ]).
III. Long-term average capacity region
Wehavethefollowing:
Theorem 1. The long-term average capacity region is
givenby
CK;N()= [
2
K;N ()
CK;N(;) (5)
where
C
K;N (;)
= n
R2R K
+ :
P
k 2A R
k
E h
1
N P
N
n=1 log 1+
P
k 2A
k ;nk ;n(Sn)
i
;
8Af1;;Kgg (6)
whereexpectationiswithrespecttothechannelstateS
N and
where
K;N
() is the set of feasible causal powerallocation
policies=f
k ;n
:k=1;;K ;n=1;;Ngdenedas
K;N ()
= (
2R KN
+ :
1
N N
X
n=1
k ;nk; k ;n=k ;n(Sn) )
(7)
(
k ;n
=
k ;n
(Sn) indicatesthe causality constraint, i.e., that
k ;n
isafunctionofthechannelstateattimen).
Theexplicitcharacterizationof theboundaryofthelong-
termcapacity regionC
K;N
() canbe done by following the
K;N
closureofallK-tuplesR2R K
+
thatsolve
max
R2C
K;N ()
K
X
k =1
k R
k
(8)
for some = (1;;K) 2 R K
+
. A closed form solution
of (8)seems infeasible. In [6], the single-user long-term av-
erage capacity that, with aslight abuseof notation, will be
denotedby
C1;N()
= sup
2
1;N ()
E
"
1
N N
X
n=1
log(1+nn(Sn))
#
(9)
wascomputednumericallybyshowingthattheoptimalpower
allocationpolicy
b
=arg sup
2
1;N ()
E
"
1
N N
X
n=1
log(1+
n
n (S
n ))
#
(10)
isthesolutionofadynamicprogrammingproblem:
Theorem 2[6 ]. Denetherecursion
S
n (P)=
E
"
sup
p2[0;P]
f log(1+p)+S
n 1
(P p)g
#
(11)
forn=1;:::;N,withinitialconditionS0(P)=0. Thesingle
userlong-termaveragecapacityisgivenby
C
1;N ()=
1
N S
N
(N) (12)
Althoughwe cannot characterize explicitly the boundary
surface ofCK;N() for everyniteN, wecan prove thefol-
lowinglimittheorem:
Theorem 3. Inthelimit forlargeN,thelong-termaver-
age capacity region CK;N() tends to C (erg)
K
() ,the ergodic
capacityregiongivenin[12 ].
IV.Long-term average capacity region per
unit energy
A byproduct ofthe proof ofTheorem 1(see [13 ]) is that
thelong-termaveragecapacityregioncoincideswiththestan-
dard\ergodic" capacity regionoftheN-slot extensionchan-
nel, whichis frame-wisememoryless. Thefollowing theorem
is an immediateconsequence of this fact and of the general
theoryofcapacityperunitcost[10 ]:
Theorem 4. Thelong-termaverage capacity regionper
unitenergyisgivenby
UK;N= [
2R K
+ n
r2R K
+
:(1r1;;KrK)2CK;N() o
(13)
Inanalogywith[10 ],itiseasytoshowthefollowing:
Theorem 5. Thelong-termaverage capacity regionper
unitenergyisthehyper-cube
UK;N = n
r2R K
+ :r
k U1;N
o
(14)
U1;N =lim
!0 1
sup
2
1;N ()
E
"
1
N N
X
n=1
k ;nk ;n(Sn)
#
(15)
Theorem 6[6 ]. Denetherecursion
sn=
E
[maxfsn 1;g] (16)
forn=1;:::;N,withinitialconditions
0
=0. Thesingle-user
long-termaveragecapacityperunitenergyisgivenbyU1;N =
s
N
anditisachievedbythe\one-shot"powerallocationpolicy
denedby
?
n
=
N ifn=n
?
()
0 otherwise
(17)
wherewedenethe\level-crossing"time
n
?
()=minfn2f1;:::;Ng : nsN ng (18)
ThebehaviorofU1;N whenN growstoinnityisgivenby
thefollowing:
Theorem7. ForlargeN,U
1;N
tendstotheergodiccapac-
ityregionperunitenergydenedin[10 ]andgivenexplicitly
by
lim
N!1 U
1;N
=lim
!0 dC
(erg)
1 ()
d
=supfg (19)
(supfg denotes thesupremumof thesupportofthe proba-
bilitydistributionof).
V.Wideband performance
V.ABackground
Theoptimalityofacodingschemeinthewidebandregime
is dened andstudied for several input-constrained additive
noisechannels in[9 ]. LetC(SNR) bethecapacity expressed
in nat/dimensionas a functionof the (transmit) SNR, and
let C(E
b
=N
0
) denote thecorrespondingspectraleÆciencyin
bit/s/Hzasa functionofthe energyperbit vs. noisepower
spectral density, Eb=N0, given implicitly by the parametric
equation
(
E
b
N0
= SNRlog2
C(SNR)
C
E
b
N
0
=C(SNR)=log2
(20)
Thevalue (Eb=N0)min for whichC(Eb=N0) >0, Eb=N0 >
(E
b
=N
0 )
min
,isgivenby[9 ]
Eb
N0
min
= lim
SNR #0
SNRlog2
C(SNR)
= log2
_
C(0)
(21)
where _
C(0) is the derivative of the capacity function at
SNR = 0. From [10 ], we see immediately that the recipro-
calof (E
b
=N
0 )
min
isthe capacityperunitenergy(expressed
inbit/joule)ofthechannel.
Inthewideband(i.e.,vanishingSNR)regime,thebehavior
ofspectraleÆciencyina(right)neighborhoodof(E
b
=N0)min
isofgreatimportance,asitisabletoquantifythebandwidth
requirement for a given desired data rate (see the detailed
discussion in[9 ]). Thisbehavior is capturedbythe slope of
b 0 min
by(see[9,Theorem6])
S0= 2
_
C(0)
2
C(0)
(22)
where
C(0)denotesthesecondderivativeofthecapacityfunc-
tionatSNR=0. Asignalingstrategyissaidtoberst-order
optimalifitachieves(E
b
=N
0 )
min
andsecond-order optimalis
itachievesS0 [9 ].
In a multiple-access channel, the individual user energy
perbit over N0 are denedby Ek=N0
=klog2=Rk, where
k
isthe transmitSNR (energy/symbol) and R
k
is the rate
(in nat/symbol) of user k. In [11], the achievable slope re-
gionforthestandard2-userGaussianMACisstudiedandits
boundaryisexplicitlyparameterizedwithrespecttotheratio
=R1=R2.
InSectionIV wehaveshownthat theone-shotpoweral-
location
?
(inconjunction withGaussianvariable-ratecod-
ing)achievesthecapacityregionperunitenergy,i.e.,achieves
(Eb=N0)minforallusersfortheblock-fadingMACwithcausal
transmitterCSIconsideredinthispaper. Then,weconclude
thattheone-shotpolicyisrst-orderoptimalforanynumber
ofusersK. FromtheproofofTheorem5[13 ] itfollows that
rst-orderoptimality canbeobtained eitherby using super-
positioncodingorbyusingTDMAinsideeachslot. Next,we
shallstudythewidebandslopeperformanceof
?
.
V.BSecond-orderoptimality forK=1
Evenifwecannotgiveaclosedformfor eitherC1;N()or
thecorrespondingoptimalpowerallocationpolicy b
givenin
(10),thecharacterization ofC1;N()inthewidebandregime
andthesecond-orderoptimalityoftheone-shotpolicy
?
are
givenbythefollowing:
Theorem 8. (E
b
=N
0 )
min and S
0
for thesingle-userblock
fadingchannelwithcausaltransmitterCSIaregivenby
E
b
N0
min
= log2
sN
S0 =
2(sN) 2
N
SN(0)
(23)
where sN is given in(16), where the functionSN(P) is de-
nedin(11) and where
SN(0)denotesthe secondderivative
ofSN(P)atP=0,givenbytherecursion
Sn(0) = Pr (sn 1)E[
2
jsn 1]
Sn 1(0)Pr(<sn 1) (24)
for n=1;:::;N,with
S0(0)=0. Furthermore,theone-shot
powerallocationpolicy
?
alsoachieves(E
b
=N0)min andS0.
V.CAnachievable sloperegion andcomparisonwithTDMA
Weinvestigatetheslopesofthe userratesas functionsof
theindividualEk=N0 fortheparticularchoiceoftheone-shot
power allocation policy
?
dened in Theorem 6. Our ap-
proachfollowsthatof[11 ]. Wehavethefollowing:
Theorem9. Fixavectors2R K
+
suchthat P
K
k =1 k=1.
For vanishinguserrates while keeping xeduser rateratios
k j k j
codingtheachievablesloperegionisgivenbytheparametric
form
[
K
Y
k =1 8
<
: 0S
(k )
0
S0
1+K0 P
P
j<
1
(k )
j
k 9
=
; (25)
whereS0isthesingle-userslopegivenin(23),where
K0= 2
P
N
n=1 (E[
n 1fn
?
()=ng]) 2
P
N
n=1 E
[ 2
n 1fn
?
()=ng]
(26)
where P
denotes the sum over all permutations of
f1;:::;Kg and where = f
g are non-negative \time-
sharing" coeÆcients (indexed by the permutations ) such
that P
=1.
AsacorollaryofTheorem9wegetthatiftheuserratesare
veryimbalanced,moreprecisely,if
j
=k isvanishingforall
k=1;:::;Kand j<
1
(k),for somepermutation,then
alluserscanachievesingleuserslopes. Infact,itissuÆcient
tochoosethevertexcorrespondingtoandthedenominator
in(25)becomes1forallk. Theconditionforachievingsingle-
userslopesfor all users isthat thereexist a permutation
suchthatR
k 1
=o(R
k
)forall k=2;:::;K.
AsfarasTDMAisconcerned,becauseofsecond-orderop-
timalityof
?
inthesingle-usercase,thesameone-shotpolicy
achievesthemaximumpossible slopesunderTDMA. Thisis
givenbythefollowing:
Theorem10. ForanyarbitraryrateratiosR
k
=Rj,asthe
ratesvanish,thebestachievablesloperegionunderTDMAis
givenby
[
K
Y
k =1 n
0S (k )
0;tdma
k S
0 o
(27)
where =f
k
g arenon-negative \time-sharing" coeÆcients
suchthat P
K
k =1
k=1.
We can use Theorems 9 and 10 to determine the max-
minslopeofanequal-ratewidebandsystem. Forequalrates,
j=k=1for allk;j,andthedenominatorof(25) becomes
1+K0 X
X
j<
1
(k )
1 = 1+K0 X
1
(k) 1
= 1 K0+K0 X
1
(k)
As variesoverall K!permutations, 1
(k) takesoneach
value1;:::;K exactly(K 1)!times. Becauseofsymmetry,
maximizingtheminimumslope isachievedbylettingS (k )
0
=
const:,i.e.,=1=K!for all. Thisyieldsto themax-min
slope
maxmin
k S
(k )
0
=
S0
1+K0(K 1)=2
For TDMA, the max-minslope is obtained by letting k =
1=K, i.e., maxmin
k S
(k )
0;tdma
=S
0
=K. Foradesired userrate
Rb(inbit/s)commontoallusers,andassumingthatallusers
transmit with equal power, i.e., they have the same E
b
=N
0
suchthat(E
b
=N
0 )
dB ((E
b
=N
0 )
min )
dB
=,thesystemband-
widthisgivenapproximatelyby[9 ]
W
R
b
min
k S
(k )
0
Therefore, the bandwidth expansion factor of TDMA with
respecttosuperpositioncodingisgivenby
=
K
1+K
0
(K 1)=2
(28)
From(26) we haveimmediatelythat K0 <2(i.e.,TDMAis
strictlywideband-suboptimal) for anynon-degenerate fading
distribution. Noticealsothatthe caseofequalE
b
=N
0 forall
usersisthemostfavorableforTDMA[11].Asalreadynoticed,
foraveryimbalancedsystemthebandwidthexpansionfactor
canbemuchlargerthan(28).
Next,westudyinmoredetailthecase K=2. For super-
positioncoding,byletting=1=2,wehave
S (1)
0
=
S
0
1+K0(1 ) 1
S (2)
0
=
S0
1+K0
(29)
By eliminating the time sharing parameter we obtain the
sloperegionboundaryas
1
S (1)
0 1
S0
!
+ 1
S (2)
0 1
S0
!
1
= K0
S0
; 0S (k )
0 S0
(30)
WithTDMAweobtaintheboundaryS (1)
0;tdma +S
(2)
0;tdma
=S0.
Wemight wonderiffor some TDMAachievesthesame
slopetrade-oofsuperpositioncoding,i.e.,ifthetwobound-
ariesofthesloperegionsintersectsatsomepoint(S (1)
0
;S (2)
0 ).
By substitutingin (30) S (1)
0
=S
0 and S
(2)
0
= (1 )S
0 for
2[0;1], wend
1
1
+
1
1
1
1
=K
0
whichyields
=
2
2+K0 p
K 2
0 4
2
+K
0 +1
Again, for K
0
<2 (non-constant fading), TDMAis strictly
suboptimal,foranychoiceoftherateratio.
VI.Example withRayleigh fading
In order to illustrate the results of previous sections we
considerthecaseofi.i.d. Rayleighfading. Thechannelsgain
lawisF(x)=1 e x
forx0.Theone-shotpolicyisgiven
bythethreshold
sn=sn
1 +e
s
n 1
; n=1;:::;N (31)
withs0=0.
If we allow the input to depend onthe whole CSI SN in
anon-causalway,theoptimalpowerallocationwouldbewa-
terllingoverthegains=(1;:::;N). Itisimmediateto
showthat,underthenoncausalpolicy,
_
C
(noncausal)
1;N
(0) = N
X
n=1
N
n
( 1) n+1
1
n
C
(noncausal)
1;N
(0) = N N
X
n=1
N
n
( 1) n+1
2!
n 2
0 10 20 30
−7.6
−5.6
−3.6
−1.6
N (E b /N 0 ) min
causal non causal
Figure1:(E
b
=N
0 )
min
(dB) vs. N fortheRayleighfading
channel.
Hence,thecapacityperunitenergywithnoncausaltransmit-
terCSIis
U
(noncausal)
1;N
= N
X
n=1
N
n
( 1) n+1
1
n
(32)
andthespectraleÆciencyslopeis
S
(noncausal)
0
= 2
P
N
n=1
N
n
( 1) n+1
1
n
2
N P
N
n=1
N
n
( 1) n+1
2!
n 2
(33)
Figs.1and2show(E
b
=N
0 )
min andS
0
vs. N andforboth
thecausalandnoncausalknowledgeofthechannel.
Fig. 3 shows the bandwidthexpansionfactor vs. K of
TDMAwithrespecttosuperpositioncoding,forseveralvalues
of N, for anequal-rate equalEb=N0 system. Finally, Fig. 4
shows the 2-user achievable slope region with the one-shot
policyandsuperpositioncoding,fordierentrateratios. The
optimal region achievable by TDMA is shown for compari-
son. This gureclearly illustrates that even thoughTDMA
achievesthecapacityperunitenergy,itisactuallyverysubop-
timalinthewidebandregime,especiallyinafadingscenario.
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η
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