Is TDMA optimal in the low power regime?

(1)

Is TDMA Optimal in the Low Power

Regime?

Sergio Verdu GiuseppeCaire DanielaTuninetti

Princeton University Institut Eurecom

Princeton, New Jersey 08544 USA F-06904Sophia Antipolis,France

Abstract|We consider two-user additive Gaussian noise

multiple-access and broadcast channels. Although theset

of rate pairs achievable by time-division multiple-access

(TDMA) is not equal to the capacity region (achieved by

superposition),asthepowerdecreases,theTDMA achiev-

able regionconvergestothecapacityregion. Furthermore,

TDMA achieves the same minimumenergy per bit as su-

perposition.

DespitethosefeaturesofTDMA,thispaperanswersthe

questionin thetitle negativelyexcept intwospecialcases:

multiaccess channels where the users' energy per bit are

identical and broadcast channels where the receivers have

identicalsignal-to-noiseratios.

Oneofthesimplestwaystoengineermultipoint-to-point

(multiaccess)orpoint-to-multipoint(broadcast)linksisto

useTime-DivisionMultiple Access(TDMA),bymeansof

whicheachuserisassignednonoverlappingtimeslotsdur-

ingwhichtheyaretheonlyactivetransmitters. Thismu-

tiaccess technology leads to very simple receiver design.

However,multiuserinformationtheoryhasshownthatsu-

perpositionstrategieswhereuserstransmitsimultaneously

intimeandfrequencycausingmutualinterferenceoersin

generalhighercapacityprovidedtheinter-userinterference

istakenintoaccountatthereceiver. Forexample:

Thecapacityregion(setof achievablerates)ofmultiac-

cess channels and of broadcastchannels are notachieved

byTDMA[1].

Whenusersareaectedbyindependentfading,theycan

achievehigheraggregateratewithsuperpositionthanwith

TDMA,asasimpleconsequenceoftheconcavityofchannel

capacityas afunction ofsignal-to-noiseratio[2].

In cellularmodels where each base station neglects the

structureoftheout-of-cellinterference,superpositioncod-

ing (possibly coupledwith so-calledintercell time-sharing

protocols,iftheout-of-cellinterferenceissuÆcientlyhigh)

oershighercapacitythanTDMAinthepresenceoffading

[3].

Moreover, in practical implementations TDMA suers

from performance-limiting multiuser interference because

of nonideal eects such as channel distortion and out-of-

cellinterference.

The most common practical embodiment of superposi-

tion multiaccess strategies is CDMA. Thus, in practice,

superpositionisparticularlyrelevantinthewidebandlow-

powerregimewherethereceivedenergyperinformationbit

maynotbefarfromitsminimumvalue. Therefore,itisof

considerable practicalinterestto comparethecapabilities

of TDMA to the capabilities of superposition in the low-

powerregime. Tomakethecomparisonascrispaspossible

andinordernotto incorporatefeaturessuchascommuni-

cationinacellularenvironmentorinthepresenceoffading,

whichaswesawbeforemaytiltthecomparisoninfavorof

superposition strategies, welimitour analysis to additive

white Gaussian noise channels not subject to fading. In

thissummarysubmittedtoISITweincludebothmultiple-

access channels and broadcastchannels in their simplest

possiblesetting: theclassicaltwo-user scalarwhiteGaus-

siannoisemodel. Wehavealsoobtainedgeneralizationsto

K-userchannelswithfading.

I. The MultipleAccessChannel

Weconsiderthecomplex-valuedmultiple-accesschannel

Y =X

1 +X

2

+N (1)

where N is Gaussian with independent real and imagi-

nary components and E[jNj 2

] = 2

, E[jX

1 j

2

] P

1 and

E[jX

2 j

2

] P

2

. The capacity region is the Cover-Wyner

pentagon[1]:

fR

1

log

2

1+ P

1

2

R

2

log

2

1+ P

2

R

1 +R

2

log

2

1+ P

1 +P

2

g (2)

Inparticular, wecanconcludefrom(2) thecelebratedre-

sultthatthetotalcapacity(maximumsumofrates)ofthe

multiaccesschannelisequaltothecapacityofasingle-user

channelwhosepowerisequaltothesumoftheindividual

powers,namely

log

2

1+ P

1 +P

2

:

Asis wellknown,theboundary(or,moreprecisely, the

Pareto-optimal points) of the capacity region is achieved

bysuperposition. Incontrast,TDMA achievestheregion

describedastheunionofrectangles:

[

01 fR

1

log

2

1+ P

1

2

R

2

(1 )log

2

1+ P

2

(1 )

2

g (3)

wheretheparameterisequaltothefractionoftimethat

therstuserisactive. Bylettingthetimesharingparam-

eterbeequalto

= P

1

P +P

;

(2)

we obtainthat the total capacity achieved by (3) is also

equalto(cf. Figure1)

log

2

1+ P

1 +P

2

:

0.5 1 1.5 2

0.2 0.4 0.6 0.8 1

Fig.1. MultiaccesschannelcapacityregionandTDMAachievable

regionwithwithP1=

2

=4andP2=

2

=1.

0.1 0.2 0.3 0.4

0.02 0.04 0.06 0.08 0.1 0.12 0.14

Fig.2. MultiaccesschannelcapacityregionandTDMAachievable

regionwithwithP1=

2

=0:4andP2=

2

=0:1.

In particular, ifP

1

=P

2 and R

1

=R

2

, then TDMAis

optimal.

Moreover,as grows,weoperatepredominantlyin the

linear region of the logarithm. Roughly speaking, as the

background noise grows, the multiaccess interference be-

comesasecondaryfactorandtheachievableratesbecome

decoupled. ThisisillustratedbycomparingFigures1and

2,whereweseethattheTDMAachievablerateregionoc-

cupiesanincreasinglylargefractionofthecapacityregion

as 2

increases. This can be formalized by showing that

theTDMAachievableregionconvergesto therectangle

fR

1

log

2

1+ P

1

2

R

2

log

2

1+ P

2

g (4)

inthefollowingsense:

lim

!1 log

2 1+

P

1

2

log

2 1+

P1

2

+

(1 )log

2

1+ P2

(1 ) 2

log

2 1+

P2

2

=2: (5)

Denethe(received)energyperinformationbitrelative

tothenoisespectrallevelofuseri=1;2as

E

i

N

0

= P

i

R

i

2

: (6)

Note thatsometimesa\system"energy perbit is consid-

eredinsteadoftheindividualper-user energiesperbitde-

nedin(6). Forexample,whenalltheper-symbolenergies

areidentical,[4]usesasystemenergyperbitwhichisequal

totheharmonicmeanoftheindividualenergiesperbit.

Oneofthefundamentallimitsofinterestinthispaperis

theminimumenergyperinformationbit,whichisobtained

withasymptoticallylowpower[5]. Tothatend,wecanap-

plythegeneralframeworkofcapacityregionperunit cost

developedin[5]. However,intheparticularcaseathandit

isinstructivetogiveaself-containedderivation. Severalof

theperformancemeasures wewill encounter laterdepend

ontheratiowithwhich bothratesgoto0. Asthefollow-

ing result shows, this is not the case for the multiaccess

minimumenergyperbit.

Theorem 1: For all R

1

=R

2

, the minimum energies per

information bit for the multiple-access channel are equal

to

E

1

N

0

= E

2

N

0

=log

e

2= 1:59dB: (7)

Furthermore,(7)isachievedbyTDMA.

Proof: Forasingle-userchanneltheminimumenergyper

bitiscomputedfrom thecapacity-SNRfunctionC(SNR)(in

bits):

E

b

N

0

min

= lim

SNR!0 SNR

C(SNR )

(8)

= log

e 2

_

C(0)

(9)

where _

C(0)=derivativeat 0ofC(SNR)computedinnats.

Let us apply this framework to TDMA. First, consider

axed time-sharingparameter0 < <1. Using (3) we

obtain

E

1

N

0

= lim

P1!0 P

1

= 2

log

2 1+

P1

2

=log

e

2: (10)

and

E

2

N

0

= lim

P

2

!0

P

2

= 2

(1 )log

2

1+ P2

(1 ) 2

=log

e

2: (11)

Sincetheconvergenceofthelimits(10)and(11)isuniform

over,wecanconcludethat theresultholds even ifis

(3)

VERDU-CAIRE-TUNINETTI:ISTDMAOPTIMALINTHELOWPOWERREGIME? 3

notheldxedandvarieswiththesignal-to-noiseratio. (For

example,inorderto enforceaconstraintonR

1

=R

2 .)

Since TDMA achieves a subset of the capacity region,

and since in noninterfering single-userchannels the mini-

mum energies per bit are also equal to (7), we conclude

thatforthemultiaccesschannel,TDMAachievesthesame

minimumenergiesperbitassuperposition.

Fromalltheevidence wehaveseensofar, wewouldbe

justied to suspect that the purported advantage of su-

perposition over TDMA may actually vanish in the low

power regime. If this is thecase, then theincrease in re-

ceivercomplexityrequiredtorealizethecapacityachieved

by superposition would behardly justiedunless someof

the other factorsmentioned above come into play. How-

ever, this isnot thecase. Theminimum valuesof energy

perbit areobtainedinthelimitofinnitebandwidthand

therefore imply zero spectral eÆciency. As explained in

the recent work [6], the key performance measure in the

wideband regime is the slopeof the spectraleÆciency vs

Eb

N0

curve(b/s/Hz/3dB)at Eb

N0

min :

S

0 def

= lim

E

b

N

0

# E

b

N

0

min

C(

Eb

N0 )

10log

10 Eb

N

0

10log

10 Eb

N

0min 10log

10 2

= 2

h

_

C(0) i

2

C(0)

(12)

Anumberofwell-knownconclusionsmadein thelitera-

turebasedonthe(innitebandwidth)analysisof E

b

N0

min are

shownin[6]tonolongerapplyinthewidebandlow-power

regimewherebandwidthisniteandthespectraleÆciency

isnonzero. Inparticular,unlike E

b

N

0

min

, theslopeS

0 gives

an indication of the bandwidth requirements for a given

datarate. Notethatintheparticularcaseofthesingle-user

additivewhiteGaussian noisechannel, C(x)=log(1+x)

andS

0

=2.

Whereas the conventional capacity region supplies the

tradeo of rates for xed powers, we can dene a corre-

sponding\sloperegion"that givesthetradeoofindivid-

ual userslopesforaxed ratiowith which theindividual

ratesvanish. Althoughformula(12)appliestosingle-user

channels it turns out to be suÆcient for our analysis of

bothmultiaccessandbroadcastchannels.

Theorem2: ForallR

1

=R

2

, themultiaccesssloperegion

achievedbyTDMAis:

f(S

1

;S

2

): 0S

1

; 0S

2

; S

1 +S

2 2g:

Proof: Fix01. Applying (12)to theindividual

rateconstraintequationsin (3),

C

1 (P

1

) = log

2

1+ P

1

2

C

2 (P

2

) = (1 )log

2

1+ P

2

(1 )

2

g (13)

weobtain

_

C

1 (0) =

1

2

_

C

2 (0) =

1

2

C

1 (0) =

1

4

C

2 (0) =

1

(1 )

4

(14)

Thus,iftheratepairbelongstotheboundaryoftheachiev-

ableregion,thenS

1

=2andS

2

=2 2.

Theorem 3: LettheratesvanishwhilekeepingR

1

=R

2

=

. Theoptimummultiaccesssloperegion(achievedbysu-

perposition)is:

S()=f(S

1

;S

2

) : 0S

1

2; 0S

2 2;

1

2

1+

2

1

S

1 +

1

1+

2

1

S

2 g:

(15)

Furthermore,

closure (

[

>0 S()

)

=f(S

1

;S

2

): 0S

1

2; 0S

2

2g:(16)

Proof: AsweshowedinTheorem1,asweletthepowers

andratesvanish,bothenergiesperbitE

1 andE

2

approach

thesamevalue,andtherefore(6)impliesthatinthelimit

P

1

P

2

= R

1

R

2

=:

In the rest of the proof we will assume that P

2

= P

1

=.

While this is only required in the limit, we can handle

themoregeneralcaseinvokinguniformconvergenceinthe

samewayasintheproofofTheorem1.

Wecanre-write(2)as

[

01 fR

1

log

2

1+ P

1

2

+(1 )log

2

1+ P

1

P

2 +

2

R

2

log

2

1+ P

2

P

1 +

2

+(1 )log

2

1+ P

2

:g

(17)

Let us apply (12) to the individual rate constraints of

(17)(assumingweareoperatinginthePareto-optimalseg-

ment of the Cover-Wynerpentagon). Forxed and ,

theindividualmaximalachievableratesbecome

C

1 (P

1

) = log

2

1+ P

1

2

+(1 )log

2

1+ P

1

P

1

=+ 2

C

2 (P

2

) = log

2

1+ P

2

P

2 +

2

+(1 )log

2

1+ P

2

:g

(18)

Therstand second derivativesareequalto (inthe limit

asP

1

!0):

_

C

1 (0)=

_

C

2 (0)=

1

2

;

(4)

C

1 (0)=

1

4

1+

2(1 )

:

C

2 (0)=

1

4

(2+1):

Pluggingtheseresultsinto(12)weobtain

S

1

=

2

2 2+

(19)

S

2

= 2

1+2

: (20)

We cansolvefor in (19) and (20)and subtract the re-

sultingequationsin ordertoobtain:

1=

S

1

S

2 +

1

S

2 1

2

; (21)

whichisequivalenttotheboundaryconditionin(15). The

conditions S

1

2, S

2

2 follow immediately from the

fact thattheexistenceofaninterferercannotimprovethe

rate. Moreover,thepoints at which the linesS

1

=2and

S

2

=2intersect with (21)correspondto =1 and =

0, respectively, i.e. to the vertices to the Cover-Wyner

pentagon.

Toshow(16) note that as either ! 0or ! 1 the

third constraintin(15)becomesredundant.

0.25 0.5 0.75 1 1.25 1.5 1.75 2 slope user 1

0.25 0.5 0.75 1 1.25 1.5 1.75 2

slope user 2

Fig.3. SloperegionsinmultiaccesschannelwithTDMAandsuper-

position=1and=2.

Theorem3showsthatbothuserscanachieveslopesthat

arearbitrarilyclosetothesingle-userslopesprovidedthey

usesuperposition,optimumdecoding,andtheirpowersare

suÆcientlyunbalanced. Therefore,eveninthesimpleset-

tingofthetwo-useradditiveGaussianmultiaccesschannel

thelow-powercapabilities ofTDMA aremarkedlysubop-

timal. Asaconcreteexample,supposeweconstrainuser1

tohaveasmallrateR

1

=and

E

1

N

0

=(3:01 1:59)dB

whereas

E

2

N

0

=(3:01

4

1:59)dB;

thenthehighestrateachievablebyTDMAis

R T

2

=

4

whereassuperpositionachieves

R

2

=

2

;

operatingatthepoint=2,S

1

=1,S

2

=2(Figure3).

Theorem 4: If both users are constrained to have the

sameenergyperbit,thenTDMAachievesoptimumslopes.

Proof: Identicalenergiesperbitimplythat

S

1

S

2

= R

1

R

2

=

which,whensubstitutingthevaluesfoundin(20),requires

= 1=2. On the other hand, for every value of the

superposition sloperegion\touches"theTDMAregionat

onepoint (Figure3), which corresponds tothe midpoint

=1=2inthePareto-optimalsegmentoftheCover-Wyner

pentagon. To see this, note that = 1=2 achieves the

minimumsum(equal to2)oftheslopesin(20):

2

2 2+ +

2

1+2 :

II. Broadcast Channels

Weconsiderthesimplecomplex-valuedtwo-userbroad-

cast Gaussian channel where users 1 and 2 receive the

samesignalfromthetransmitterembeddedinindependent

Gaussiannoisewithdierentpowers:

Y

1

= X+N

1

Y

2

= X+N

2

(22)

whereE[jXj 2

]P,E[jN

i j

2

] 2

i

. Wewillassume 2

1

<

2

as in the case 2

1

= 2

2

TDMA is trivially optimal. The

capacityregionof thischannel (achievedbysuperposition

andstripping)isequalto[1]

[

01 fR

1

log

2

1+ P

2

1

R

2

log

2

1+

(1 )P

P+ 2

g (23)

(5)

VERDU-CAIRE-TUNINETTI:ISTDMAOPTIMALINTHELOWPOWERREGIME? 5

whereastheregionachievablebyTDMAis

[

01 fR

1

log

2

1+ P

2

1

R

2

(1 )log

2

1+ P

2

g (24)

0.5 1 1.5 2

0.2 0.4 0.6 0.8 1

Fig.4. CapacityregionandTDMA-achievablerateregionofbroad-

castchannelwithP= 2

1

=4andP=

2

=1.

0.1 0.2 0.3 0.4

0.02 0.04 0.06 0.08 0.1 0.12 0.14

Fig.5. CapacityregionandTDMA-achievablerateregionofbroad-

castchannelwithP= 2

1

=0:4andP=

2

=0:1.

As for multiaccess channels, it appears from Figures 4

and 5that asthe powerdecreases, theTDMA-achievable

region occupies a larger fraction of the capacity region.

Thisfacthasbeenformalizedin[7]showingthatforevery

pair(R

1

;R

2

)in thebroadcastcapacityregion,

limsup

P!0

R

1

log

2

1+ P

2

1

+

R

2

log

2

1+ P

2

1: (25)

Analogouslyto(6)wedene

E

i

N

0

= P

R

i

2

i

: (26)

Theorem 5: Suppose that R

1

=R

2

=. Then, themini-

mumenergiesperbitachievedbybothTDMAandsuper-

positionare:

E

1

N

0

=

1+

2

1

log

e

2 (27)

E

2

N

0

=

1+

2

1

2

log

e

2 (28)

Proof: LetusstartwithTDMA.Enforcingtheconstraint

on the ratio of the rates in (24), pins down the value of

the time-sharing parameter and we obtain that the rate

achievedbyuser1is

R

1

= log

2

1+ P

2

1

log

2

1+ P

2

log

2

1+ P

2

1

+log

2

1+ P

2

: (29)

Thereciprocalof the derivativeof(29)with respect to P

atP =0isequalto=(

2

1 +

2

)and(27)follows. Formula

(28)isobtainedinanentirelyanalogouswayorsimplyby

noticingfrom (26)that

E

1

N0

E

2

N0

=

2

1

: (30)

Letusanalyzenowthecapacityregion(23). Dene

(P)

tobethesolutionto

log

2

1+

(P)P

2

1

=log

2

1+

(1

(P))P

(P)P+ 2

2

: (31)

Although we are unable to nd an explicit solution for

(P),wewillbeableto computethederivativewith re-

specttoP atP =0.Equatingthederivativesofbothsides

of(31),weobtain

(P)

2

1 +

P_

(P)

2

1

1+ P

(P)

2

1

=

1 (P)

2

2 +P

(P)

P(P_ )

2

2 +P

(P)

P(1 (P))((P)+P(P_ )

( 2

2 +P

(P))

2

1+ P(1

(P))

2

2 +P

(P)

(32)

Letting P = 0 in (32) we get that the derivative at 0 is

equalto

(0)

2

1 and

(0)=

2

1

2

+

2

1

: (33)

Takingthereciprocalofthederivativeandmultiplyingby

log

e 2=

2

1

,(27)follows,andso does(28)byapplying (30).

Letusdirectourattentionto theanalysisoftheslopes.

Theorem 6: LettheratesvanishwhilekeepingR

1

=R

2

=

. ThebroadcastsloperegionachievedbyTDMAis:

f(S

1

;S

2

): 0S

1

2

; 0S

2

g: (34)

(6)

Proof: As intheproofofTheorem5,enforcingthecon-

straintR

1

=R

2

,weobtainthevalueofthetimesharing

parameterandthevalueoftheindividualrates. Applying

formula(12)to (29)and afterconsiderablealgebra(omit-

tedinthissummary)weobtainthedesiredresult. Thefact

that ifweoperateontheboundaryofthecapacityregion

wegetS

1

=S

2

canbereadilyseenfromthefactthatthe

numerator in thedenition of slope hasa factor of be-

cause R

1

=R

2

, whereas thedenominators are identical:

the Eb

N

0

's dier by amultiplicative constant (30) which is

immaterialinthedenitionofslope(leftsideof(12)).

Theorem7: LettheratesvanishwhilekeepingR

1

=R

2

=

. The optimum broadcast sloperegion (achieved by su-

perposition)is:

f(S

1

;S

2

): 0 S

1

2(+ 2

2

= 2

1 )

2

+2+ 2

2

= 2

1

;

0 S

2

2(+ 2

2

= 2

1 )

2

+2+ 2

2

= 2

1

g: (35)

Proof: ItissuÆcienttojustifytheslopeofuser1because

as we saw above S

1

= S

2

. As in the proof of Theorem

5, we need to work with anexpression forthe achievable

rate (31)which dependson animplicitly-dened function

(P). Equating the second derivatives of both sides in

(31)weget

1

0

@

(P)

2

1 +

P_

(P)

2

1 2

1+ P

(P)

2

1 2

+ 2_

(P)

2

1 +

P

(P)

2

1

1+ P(P)

2

1 1

A

=

1 (P)

2

+P(P)

P(P_ )

2

+P(P)

P(1 (P))((P)+P(P_ ))

( 2

2

+P(P)) 2

2

1+ P(1

(P))

2

2 +P

(P)

2

+

1

1+ P(1

(P))

2

2 +P

(P)

2_

(P)

2

2 +P

(P)

2(1

(P))(

(P)+P_

(P))

( 2

2 +P

(P))

2

+

2P_

(P)(

(P)+P_

(P))

( 2

2 +P

(P))

2

+

2P(1

(P))(

(P)+P_

(P))

2

( 2

2 +P

(P))

3

P

(P)

2

2 +P

(P)

P(1

(P))(2_

(P)+P

(P))

( 2

2 +P

(P))

2

Substituting thevalueof

(0) found in (33), andletting

P =0,weobtain

_

(0)=

2

1 (2

2

1 +

2

2 )

2(

2

+ 2

) 3

: (36)

Therefore, the rst and second derivatives are now com-

putableexplicitly. Applying(12),weobtaintheformulain

(35).

ComparingtheresultsofTheorems6and7weseethat

unless 2

1

= 2

2

(in which caseTDMAisoptimal),TDMA

iswastefulofchannel resources.

Figure6plotstheratiobetweentheslopesachievedwith

superposition andTDMAasafunction of. Thisquanti-

es thebandwidth expansionfactor that TDMA requires

inthelow-powerwidebandregime.

2 4 6 8 10

1.2 1.4 1.6 1.8 2

Fig.6. BandwidthfactorpenaltyincurredbyTDMAasafunction

ofR

1

=R

2

=for 2

2

=10 2

1

Reference [7] devoted to an analysis of the broadcast

channel in the wideband regime states \Our results thus

suggest that for high bandwidth channels, time-sharing

maybe verycloseto optimal,and that forsuch channels,

verylittleistobegainedfromusingthecomplicatedcoding

schemes that are usuallyneeded to beat thesimpletime-

sharingstrategy." Thisisanotherexampletoaddtothose

inreference[6]whereitwasshownthatinnite-bandwidth

analyses maylead to misleading conclusions in thewide-

bandregime.

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