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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3, MARCH
2001 1083
Capacity and Optimal Resource Allocation for FadingBroadcast
Channels—Part I: Ergodic Capacity
Lifang Li, Member, IEEE,and Andrea J. Goldsmith, Senior Member,
IEEE
Abstract—In multiuser wireless systems, dynamic resource
al-location between users and over time significantly improves
effi-ciency and performance. In this two-part paper, we study
threetypes of capacity regions for fading broadcast channels and
obtaintheir corresponding optimal resource allocation strategies:
the er-godic (Shannon) capacity region, the zero-outage capacity
region,and the outage capacity region with nonzero outage. In Part
I, wederive the ergodic capacity region of an -user fading
broadcastchannel for code division (CD), time division (TD), and
frequencydivision (FD), assuming that both the transmitter and the
receivershave perfect channel side information (CSI). It is shown
that byallowing dynamic resource allocation, TD, FD, and CD
withoutsuccessive decoding have the same ergodic capacity region,
whileoptimal CD has a larger region. Optimal resource allocation
poli-cies are obtained for these different spectrum-sharing
techniques.A simple suboptimal policy is also proposed for TD and
CD withoutsuccessive decoding that results in a rate region quite
close to the er-godic capacity region. Numerical results are
provided for differentfading broadcast channels. In Part II, we
obtain analogous resultsfor the zero-outage capacity region and the
outage capacity region.
Index Terms—Broadcast channels, capacity region, fading
chan-nels, optimal resource allocation.
I. INTRODUCTION
T HE wireless communication channel for both point-to-point and
broadcast communications varies with timedue to user mobility,
which induces time-varying path loss,shadowing, and multipath
fading in the received signal power[1]. For these time-varying
channels, dynamic allocation ofresources such as power, rate, and
bandwidth can result in betterperformance than fixed resource
allocation strategies [2]–[5].Indeed, adaptive techniques are
currently used in both wirelessand wireline systems and are being
proposed as standards fornext-generation cellular systems.
By using an optimal dynamic power and rate allocationstrategy,
the ergodic (Shannon) capacity of a single-user fading
Manuscript received February 11, 1999; revised April 29, 2000.
This workwas supported by NSF Career Award NCR-9501452 and under a
grant fromPacific Bell, while L. Li pursued the Ph.D. degree under
the supervision of A. J.Goldsmith at the California Institute of
Technology, Pasadena, CA 91125 USA.The material in this paper was
presented in part at the 36th Allerton Conferenceon Communication,
Control, and Computing, Monticello, IL, September 1998.
L. Li is with the Exeter Group, Inc., Los Angeles, CA 90013 USA
(e-mail:[email protected]).
A. J. Goldsmith is with the Department of Electrical
Engineering, StanfordUniversity, Stanford, CA 94305-9515 USA
(e-mail: [email protected]).
Communicated by M. L. Honig, Associate Editor for
Communications.Publisher Item Identifier S
0018-9448(01)01353-0.
channel with channel side information (CSI) at both the
trans-mitter and the receiver is obtained in [6]. The
correspondingoptimal power allocation strategy is a water-filling
procedureover time or, equivalently, over the fading states. The
ergodiccapacity corresponds to the maximum long-term achievablerate
averaged over all states of the time-varying channel. Fora fading
multiple-access channel (MAC), assuming perfectCSI at the receiver
and at all transmitters, the optimal powercontrol policy that
maximizes the total ergodic rates of all usersis derived in [7].
The ergodic capacity region of this channeland the corresponding
optimal power and rate allocation areobtained in [8] using the
polymatroidal structure of the region.1
This ergodic capacity region is a multiuser generalization of
thetwo-user capacity region derived in [9] for the Gaussian MACwith
intersymbol interference (ISI), and the corresponding op-timal
power allocation is a multiuser version of the
single-userwater-filling procedure.
In [10], the zero-outage capacity region2 and the optimalpower
allocation for the fading MAC are derived under the as-sumption
that CSI is available at both the transmitters and thereceiver.
This capacity definition, in contrast with the ergodiccapacity
region, is important for delay-constrained applicationssuch as
voice and video, since it represents the maximum instan-taneous
data rate that can be maintained in all fading conditionsthrough
optimal power control. Under this adaptation policy, theend-to-end
delay is independent of the channel variation. By al-lowing some
nonzero transmission outage under severe fadingconditions, the
minimum outage probability for a given rateand the corresponding
optimal power allocation policy are de-rived for the single-user
fading channel in [11] and the capacitywith nonzero outage is
implicitly obtained. The correspondingoutage capacity region for
the fading MAC with nonzero outageis derived in [12].
In Part I of this paper, we first derive the ergodic capacity
ofan -user flat-fading broadcast channel with transmitter
andreceiver CSI3 and obtain the corresponding optimal resource
al-location strategy for code division (CD) with and without
suc-cessive decoding, time division (TD), and frequency
division(FD). The optimal power allocation that achieves the
boundaryof the ergodic capacity region is derived by solving an
optimiza-tion problem over a set of time-invariant additive white
Gaussiannoise (AWGN) broadcast channels with a total average
transmit
1The ergodic (Shannon) capacity of a fading channel is called
“throughputcapacity” in [8].
2The zero-outage capacity is called “delay-limited capacity” in
[10].3In practice, the CSI can be obtained either by estimating it
at the receiver and
sending it to the transmitter via a feedback path or through
channel estimationof the opposite link in a time-division duplex
system.
0018–9448/01$10.00 © 2001 IEEE
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1084 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3,
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power constraint. For CD with successive decoding, the
optimalpower allocation is similar to that of the parallel Gaussian
broad-cast channels discussed in [13], [14], and the solution uses
theresults therein. We solve the optimization problem for TD
andshow that one of the nonunique optimal power allocation
strate-gies is also optimal for CD without successive decoding,
andthe ergodic capacity regions for these two techniques are
thesame. TD and FD are equivalent in the sense that they have
thesame ergodic capacity region and the optimal power allocationfor
one of them can be directly obtained from that of the other[15].
Thus, we obtain the optimal resource allocation for FD aswell. For
TD and CD without successive decoding we also pro-pose a simple
suboptimal power allocation strategy that resultsin an ergodic rate
region close to their capacity region. These re-sults are then
extended to frequency-selective fading broadcastchannels that vary
randomly.
In Part II of this paper, we first obtain the zero-outage
ca-pacity regions and the associated optimal resource
allocationstrategies for an -user flat-fading broadcast channel
with TD,FD, and CD with and without successive decoding. We
thendetermine the outage probability region for a given rate
vectorof the users and derive the optimal power allocation
policythat achieves the boundaries of the outage probability
regionsfor these different spectrum-sharing techniques. The outage
ca-pacity regions are thus obtained implicitly from the outage
prob-ability regions for given rate vectors. These results are also
ex-tended to frequency-selective fading broadcast channels.
Part I of this paper is organized as follows. In Section II,
wepresent the flat-fading broadcast channel model. The ergodic
ca-pacity regions and the optimal resource allocation for CD
withand without successive decoding, TD, and FD, as well as
thesuboptimal power and time allocations for TD are obtained
inSection III. In Section IV, we extend our flat-fading model tothe
case of frequency-selective fading. Section V shows
variousnumerical results, followed by our conclusions in the last
sec-tion.
Notation: The prime is used to denote the derivative ofa
function throughout this paper except in the proofs aboutthe
convexity of a capacity region in the Appendix, SectionB, where the
prime or double prime of a symbol just denotesanother symbol.
II. THE FADING BROADCAST CHANNEL
We consider a discrete-time -user broadcast channel withflat
fading as shown in Fig. 1. In this model, the signal source
is composed of independent information sources, andthe broadcast
channel consists of independent flat-fadingsubchannels. The
time-varying subchannel gains are denoted as
and the Gaussian noises of these subchannels are denoted as. Let
be the total average transmit
power, the received signal bandwidth, and the noisedensity of ,
. Since the time-varyingreceived signal-to-noise ratio (SNR)
Fig. 1. AnM -user fading broadcast channel model.
Fig. 2. An equivalentM -user fading broadcast channel model.
if we define4 , we have .Therefore, for slowly time-varying
broadcast channels,
we obtain an equivalent channel model, which is shown inFig. 2.
In this model, the noise density of is ,
. We assume that are known to thetransmitter and all the
receivers at time. Thus, the trans-mitter can vary the transmit
power for each user relativeto the noise density vector ,subject
only to the average power constraint. For TD orFD, it can also vary
the fraction of transmission time or band-width assigned to each
user, subject to the constraint
for all . For CD, the superposition code canbe varied at each
transmission. Since every receiver knowsthe noise density vector ,
they can decode their individualsignals by successive decoding
based on the known resourceallocation strategy given the noise
densities. In practice, itis necessary to send the transmitter
strategy to each receiverthrough either a header on the transmitted
data or a pilot tone.We call the joint fading process and denote as
theset of all possible joint fading states. denotes a
givencumulative distribution function (cdf) on .
III. ERGODIC CAPACITY REGIONS
Under the assumption that both the transmitters and the
re-ceiver have perfect CSI, the ergodic capacity region of a
fading
4Note that wheng [i] = 0 for somej, n [i] = 1 and no information
canbe transmitted through thejth subchannel.
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LI AND GOLDSMITH: CAPACITY AND OPTIMAL RESOURCE ALLOCATION FOR
FADING BROADCAST CHANNELS—PART I 1085
MAC is derived in [8] by exploiting its special
polymatroidalstructure. In that work, the optimal resource
allocation schemeis obtained by solving a family of optimization
problems over aset of parallel Gaussian MACs, one for each fading
state. In thissection, we derive the ergodic capacity region for
the flat-fadingbroadcast channel under the assumption that the
transmitter andall receivers have perfect CSI. The corresponding
optimal re-source allocation strategy is obtained by optimizing
over a set ofparallel Gaussian broadcast channels for CD with and
withoutsuccessive decoding and for TD. For FD it is shown that
theergodic capacity region is the same as for TD and the
corre-sponding optimal power and bandwidth allocation policy canbe
derived directly from that of TD [15]. We will discuss exten-sions
of the results obtained in this section to the case of
fre-quency-selective fading channels in Section IV.
A. CD
We first consider superposition coding and successive de-coding
where, in each joint fading state, the-user broadcastchannel can be
viewed as a degraded Gaussian broadcastchannel with noise densities
and themultiresolution signal constellation is optimized relative
tothese instantaneous noise densities. Given a power
allocationpolicy , let be the transmit power allocated to Userfor
the joint fading state and denoteas the set of all possible power
policies satisfying the averagepower constraint where denotesthe
expectation function. For simplicity, we assume that thestationary
distributions of the fading processes have continuousdensities,5
i.e., , .
Theorem 1: The ergodic capacity region for the fadingbroadcast
channel when the transmitter and all the receiversknow the current
channel state is given by
(1)
where
(2)
, and denotes the indicator func-tion ( if is true and zero
otherwise). Moreover, thecapacity region is convex.
Proof: See the Appendix, Section A.
Since this capacity region is convex
5If Prfn = n g 6= 0 for somei; j then, in statennn, Useri and
Userj canbe viewed as a single user and superposition coding and
successive decodingare applied toM � 1 users. The information for
Useri and Userj are thentransmitted by time-sharing the
channel.
with , if a rate vector is a solution to the fol-lowing
maximization problem, it will be on the boundary sur-face of in
(1)
subject to (3)
The maximization problem in (3) is equivalent to
subject to:(4)
where and the objectivefunction
(5)
In (5), is the Lagrangian multiplier and can be viewed as
aweighting parameter (rate reward) proportional to the priority
ofUser . The problem in (4) is quite similarto that of the parallel
AWGN broadcast channels discussed in[13], [14] and its solution is
obtained by applying the resultstherein. For each fading state ,
let thepermutation be defined such that
. The optimal power allocation procedure for each stateas
derived in [13] is essentially water filling. We now describethis
optimal power allocation procedure in the following.
Initialization: Do not assign power to any user forwhich , with
. Remove these usersfrom further consideration.
Step 1: Denote the number of remaining users asand de-fine the
permutation such that
. Then, due to the removal criterion, wehave
i.e.,
Step 2: Define
where and assign powerto User . If
the total power for state has been allocated. If not,only the
power for User has been allocated. Inthis case, increase the
noisesby and do not assign power to any user
for which such that with
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1086 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3,
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. Remove these users from furtherconsideration. Also remove User
and return toStep 1.
In the above procedure, is the water-filling powerlevel that
satisfies the total average power constraint of theusers in (4)
instead of the total power constraint for a parallelAWGN broadcast
channel as in [13], [14]. In each iteration, thiswater-filling
procedure consists of selecting the best receiver ac-cording to a
modified noise criterion using the weighting param-eter for each
user , and adding power to the correspondingsubchannel until a
predetermined power is achieved. Note thatin each iteration, some
subchannels will be identified to hold nopower.
This optimal power allocation procedure can be obtained in
adifferent form through a greedy algorithm [14]. Specifically,
ineach fading state, define the utility function for Useras
and let
Obviously, and are all decreasingfunctions. Moreover, for any ,
the two curves and
will cross each other at most once. Now let andlet denote the
point where intersects
for some and in the set , assumingthat for . That is,denotes the
set of all intersection points for the utility functions
, and . Thencan be expressed as6
if
if .(6)
Then the power allocation for User in statethat maximizes (4)
is
if for some
otherwise.
This power allocation strategy is shown to be equivalent to
thewater-filling procedure [16] and it has a greedy interpretation
inthe following sense [14]: can be interpreted as the mar-ginal
weighted rate increase in the objective functionin (5) when a
marginal power is allocated to User at in-terference level . The
optimal solution to (4) can be obtainedgreedily by allocating
marginal power to the user with thelargest positive marginal
weighted rate increase
at each interference level . This power allocationprocess
continues until no user obtains a positive marginal
6From (6) we see thatz is defined as the smallestz for whichu
(z) = 0;if no such point exists, thenz = 1.
weighted rate increase with the addition of power [i.e.,], in
which case we allocate
no more power to state.In the special case where each user has
the same rate re-
ward , it is easily seen that the optimal power allocation
policyin a fading state is to assign power to User
and assign no power to any other user. Wewill see in the
following subsection that this is actually the sameas the optimal
TD policy when all users have the same rate re-ward .
B. TD
Now we consider the TD case where, in each fading state, the
information for the users will be divided and sent in
time slots which are functions of. For a given power and
timeallocation policy , let and bethe transmit power and fraction
of transmission time allocated toUser , respectively, for fading
state, andlet be the set of all such possible power and time
allocationpolicies satisfying
and
(7)
Theorem 2: The achievable rate region for the variable powerand
variable transmission time scheme is
(8)
where
(9)
Moreover, the rate region is convex.Proof: See the Appendix,
Section B.
Note that in this paper we will refer to this achievable
rateregion as the capacity region for TD, though we do not have
aconverse proof due to the fact that the converse only holds for
theoptimal transmission strategy for this channel, which,
accordingto Theorem 1, is CD with successive decoding.
Due to the convexity of this capacity region, with, if a rate
vector is a solution to
subject to (10)
it will be on the boundary surface of . Based on the ex-pression
for in (9) and the average total power constraint in(7), we can
decompose the maximization problem (10) into thefollowing two
problems.
1) Assuming that , is the total power assignedto the users,
i.e., , we mustdetermine how to distribute among the users so
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LI AND GOLDSMITH: CAPACITY AND OPTIMAL RESOURCE ALLOCATION FOR
FADING BROADCAST CHANNELS—PART I 1087
that the total weighted rate in stateis maximized. Thatis, we
must find
subject to
(11)
where with
and
(12)
2) After we obtain the expression for by solving (11),the
remaining problem is how to assign the total power
of the users for each state so that the totalweighted rate
averaged over all fading states as expressedin (10) is maximized.
That is,
subject to(13)
where is the Lagrangian multiplier.
We solve the maximization problems (11) and (13) for thetwo-user
case first and then generalize the results to the-usercase.
1) Two-User Case:Lemma 1: When , assuming that , the solu-
tion to (11) is
1) if , then , which isachieved when , , , and
;2) if , let
(14)
Then
if
if
if
(15)
where
and satisfies
(16)
in (15) is achieved by letting
if (17)
if (18)
if (19)
Proof: See the Appendix, Section C.
From (11) we see that when , is a linearcombination of . The
proof of Lemma 1 in theAppendix, Section C shows that if , then
,
and is simply ; if, then for , and will
cross each other once at some positive value , as shown inFig.
3.7 In this figure, the slope of the tangent line between thecurves
and is and it satisfies
i.e.,
Thus, in this case, is the continuous contour in Fig. 3which
consists of part of the curve , the tangent line,and part of the
curve , as indicated with the dash-dotted line which is offset
slightly for clarity. The expressionof is therefore as given in
(15). Note that the slope ofthe tangent of the curve is continuous
and it decreaseswith the increase of .
For a given fading state, from (13) we know that the
optimalpower satisfies
(20)
Therefore, for any given , is determined by thepoint(s) on the
curve whose tangent has a slope. Inthe case where and , since all
the pointson the tangent line between and in Fig. 3have the same
tangent slope, can be any value between
and : if , from Lemma 1we know that will be time-shared by the
two users; if
is simply chosen as or , then it is only as-signed to User 1 or
to User 2, respectively. In all other cases, thepoint that has a
tangent with slopeis unique and hence so is
. The unique choice of is then allocated to a singleuser based
on its relative value compared to and
7In this figure,P ,P ,P andP are all functions ofnnn. Their
explicit depen-dence onnnn is not shown to simplify the
notation.
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1088 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 3,
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Fig. 3. The functionsf (P ), f (P ), andJ(P ) when > .
as discussed in Lemma 1. Consequently, we have the
followingtheorem.
Theorem 3: When , assuming that , theoptimal power and time
allocation policy that achieves the TDcapacity region boundary for
each fading state is
1) if , then
and
2) if , then
a) if or if and
and
b) if and
and
c) if and
and
In the above expressions, is given in (14) andis the
water-filling power level. and satisfy
the total average power constraint
(21)
and they may not be unique.Proof: See the Appendix, Section
D.
When , the optimal power policy for User 1 and User2 is
similarly derived using appropriate substitutions for all
sub-scripts. When , the optimal power policy is simplifiedas
follows: if then
otherwise
That is, this is the same power allocation as for CD when.
Note that when the cdf is continuous, , theprobability measure
of the set
(22)
is zero. Since according to Theorem 3,is defined on a subsetof
and the probability measure of any subset ofmust alsobe zero, the
value of will not affect the power constraint (21)and is therefore
uniquely determined by (21). Moreover, inthis case, with
probability , at most a single user transmits ineach fading state.
If is not continuous, the set mayhave positive probability measure
and for any fading state inthis set, the broadcast channel can be
either time-shared by thetwo users, occupied by a single user, or
not used by any user ifthe fading is too severe.
2) -User Case: The optimal power and time allocationpolicy that
achieves the capacity region boundary for the two-user case can be
generalized to the-user case .In the -user case, the optimal power
allocation is again ob-tained based on the values of functions ,
.Specifically, for a given in (12), if such that
, , then we can show that will notappear in the expression of ,
. Thus, no re-sources should be assigned to Userin the state . That
is, theoptimal and are , . In thespecial case where each user has
the same rate reward, as-suming that , then ,
. This is because for any , ,. Therefore, it is clear that the
optimal power allocation
policy in a fading state is to assign power to Userand assign no
power to any other user, which is the same as
that for CD with successive decoding.For any assuming , we
know
that, as shown in the proof of Lemma 1, if then ,; if then and
will cross each
other once at some positive. In both cases, since , forlarge
enough ( ), . Thus, assuming withoutloss of generality (WLOG) that
, we firstremove any user [i.e., let , ] for which
, with or which satisfies and. For the remaining users, there
are still some users
whose corresponding may not appear in the expression of. For
example, assume WLOG that the remaining users
are User 1–User 4. Due to the removal criterion, we know
that
(23)
and
(24)
which means that their correspondingwill cross one another once
at some positive , and
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LI AND GOLDSMITH: CAPACITY AND OPTIMAL RESOURCE ALLOCATION FOR
FADING BROADCAST CHANNELS—PART I 1089
Fig. 4. The functionsJ(P ) andf (P ) = � ln[1 + ] (1 � j �
4)
which satisfy (23) and (24).
for large enough. Thus, if are as shownin Fig. 4,8 it is clear
that will not appear in the expressionof since, , the curve is
alwaysunder the dash-dotted contour of formed by part ofthe curves
, , , and the straight tan-gent lines between them. Note that the
dash-dotted curve in thisfigure is offset slightly for clarity. In
the following, we use aniterative procedure to find all the
usersamong the remainingusers whose corresponding will not appear
in the expres-sion of and identify all other users to whom the
re-sources will be allocated later. An interpretation of this
proce-dure based on Fig. 4 will then be given.
Initialization: Let .
Step 1: Denote the number of remaining users asanddefine the
permutation such that
Then, due to the removal criterion, we have
(25)
Step 2: Let . If , all the users whosecorresponding will not
appear in the expres-sion of have been removed and stop; if
, go toStep 3.Step 3: For , define
and let satisfy . Let, . If , re-
move those users for which[i.e., let , , and removethem from
further consideration]. Also remove User
. Increase by and return toStep 1.
8In this figure,P , P , andP (j = 1; 2) are all functions ofnnn.
Theirexplicit dependence onnnn is not shown to simplify the
notation.
In this procedure we observe that in the first iteration,of User
must be the first part of the curve
where is close to zero, since
and according to (25), for close to zero , it must betrue
that
For , satisfying correspondsto the slope of the common tangent
between the two curves
and . Since , if, all the curves will always be under the
contour formed by part of the curve , part of thecurve , and the
common tangent between them.Thus, no power should be assigned to
users , .In this case, we know that part of the curveof User as
well as the common tangent betweenthe curves and must be part
of
. For example, in Fig. 4, since the number of remainingusers is
and , if we drawthe common tangents between curves and ,
and , and also and , theslopes of which are , , and ,
respectively, then it is clearthat and , i.e., the slope of the
tan-gent between and is the largest among theslopes of the three
tangents. Thus, will not be partof but and the common tangent
between
and will.After removing those users , , and User ,
the number of remaining users is reduced fromtoand User in the
first iteration becomes Userin the second iteration. Similarly, in
the second iteration, bycomparing the slopes of the common tangents
between curves
and , , we may re-move more users and find a new User in this
iterationwhose corresponding as well as the commontangent between
curves and mustbe part of . This User becomes User inthe third
iteration and the iterative procedure goes on until allthe users
whose corresponding will be part ofhave been identified and all
other users have been removed.
Note that in each iteration, the value of is different. As-sume
that by the time the iteration stops, . If ,then is simply ; if ,
then iscomposed of as well as the common tangentsbetween curves and
, ,the slopes of which are . That is, by denoting
, , ,and letting and ( ) be the pointssatisfying
(26)
i.e.,
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we can express as
if
if
(27)
where . For example, in the case shown in Fig. 4,since only User
2 will be removed from further consideration,by the time the
iterative procedure stops, we have and
The and satisfying (26) are asshown in Fig. 4 and and denote the
slopes of thetangent lines for which ranges from to ,and from to ,
respectively. Thus, can beexpressed as
if
if
if
if
if .
Once we obtain the curve , similar to the two-usercase, fixed,
since the optimal power satisfies thecondition (20), i.e., , is
determined bythe point(s) on the curve whose tangent has a slope.If
equals any , since all the points onthe common tangent between
curves and sharethe same tangent, can be any value between and
: if , will be time-shared by the two users and ; if is
simplychosen as or , then it is only assigned to User
or User , respectively. If for all, then is uniquely determined
by the single
point on that has a tangent with slope. Therefore,based on the
expression of in (27) and the condition(20), for fixed, the optimal
power and time allocationpolicy for the remaining users is
a) if , then
b) if , by denoting , we know that forthe given , there exists a
such that
or , since
which results from the iterative procedure generatingand from
the fact that is an
increasing, concave function. If such that
from (27) we have , since onlywhen does the tangent slopeof
decrease from to . In this case,we set
which corresponds to transmitting the information ofUser only.
If such that , then
since only when does the tan-gent slope of equal . Therefore, as
in thetwo-user case, we can set
(28)
and
which indicates that the channel is time-shared by Userand User
if choosing ,
and is occupied by User or User aloneif choosing or ,
respectively.
In the above policy, and satisfy the average power
con-straint
(29)
and they may not be unique, sincecan be any value betweenand .
Notice that as in the two-user case, if the cdf is
continuous, , the probability measure of the set9
such that and (30)
is zero and is uniquely determined by (29). Moreover, in
thiscase, the above optimal power and time allocation policy forthe
-user broadcast channel implies that with probability,the
information of at most a single user is transmitted in eachfading
state. If is not continuous then the setmay havenonzero probability
measure and for , as discussed be-fore, the channel capacity region
is achieved by time-sharingbetween two users or dedicated
transmission to just one user;for any other channel state , the
information of atmost one user is transmitted. Thus, in all cases,
the capacity re-gion boundary of TD can be achieved by sending
informationto just one user in every fading state. This motivates
the subop-timal TD policy we propose in the next subsection.
C. Suboptimal TD Policy
The optimal power and time allocation policy in Section
III-Bindicates that the capacity region in (8) is achieved
bytransmitting the information of at most a single user in
eachfading state , although it can also be achieved by a
strategy
9Note that in (30),M is a function ofnnn, which results from the
iterativeprocedure described earlier in this section.
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LI AND GOLDSMITH: CAPACITY AND OPTIMAL RESOURCE ALLOCATION FOR
FADING BROADCAST CHANNELS—PART I 1091
that transmits the information of two users in some states
bytime sharing and assigns the channel to a single user or no
userin the other states. Based on this observation, we now proposea
suboptimal method for resource allocation. This method se-lects a
single user in each channel state and allocates appropriatepower to
him according to the fading state. We now describe ourmethod to
choose the single user and his corresponding power.
In each state , , define
and let
where is given in (12). Then, in the state, only the
infor-mation for User is sent with transmit power , wheresatisfies
the -user average power constraint (29).
For example, in the two-user case, by denoting
(31)
we can express the suboptimal power and time allocation policyas
follows:
a) if and , or ifand (i.e.,
), then
and
where satisfies the two-user average power constraint(21);
b) if and , or ifand (i.e.,
), then
and
In the special case where all users have the same rate
reward,i.e., , it is obvious that this suboptimal TDpower
allocation policy in a fading stateis to assign power
to User and assign no powerto any other user, which is the same
as the optimal TD policy.
Compared to the optimal power and time allocation policy,the
advantage of this suboptimal scheme is that it is much easierto
compute the water-filling power level using (29).As will be shown
in Section V, the resulting rate region of thetwo-user case comes
very close to that of the optimal TD policy.This is due to the fact
that the two policies are identical exceptover a small set of
fading states. Specifically, assuming WLOGthat , in the case where
the cdf is continuous,the detailed comparison of the optimal and
suboptimal decisionregions for the two-user fading broadcast
channel in the Ap-pendix, Section E shows that for a given , the
two policies
transmit the information of different users only in the rare
occa-sions when , where
and
with determined by:
and
(32)
D. CD Without Successive Decoding
For CD without successive decoding, each receiver treats
thesignals for other users as interference noise. For a given
powerallocation policy , by denoting as the transmit powerallocated
to User and as the set of all possible power policiessatisfying the
average power constraint ,we have the following theorem.
Theorem 4: The achievable rate region for CD without suc-cessive
decoding is given by
(33)
where
The proof of the achievability follows along the same linesas
that for the capacity region of CD with successive decodinggiven in
the Appendix, Section A and is therefore omitted. Notethat in this
paper, as in the case of TD, we refer to this achiev-able rate
region as the capacity region for CD without successivedecoding,
though we do not have a converse proof since the con-verse only
applies to the optimal transmission strategy, which isCD with
successive decoding.
In order to show that in (33) cannot be larger than thecapacity
region of TD in (8), we give the following lemma.
Lemma 2: , , ,
(34)
Proof: See the Appendix, Section F.
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Theorem 5:
(35)
where equality is achieved using the optimal TD policy withpower
allocated to at most one user in each fading state.
Proof: Recall that the optimal power and time allocationpolicy
discussed in Section III-B2) indicates that if is con-tinuous, then
with probability, no more than one user is usingthe broadcast
channel in each fading state; if is dis-continuous, in some fading
states with nonzero probability, wecan either choose two users and
transmit their information bytime-sharing the channel, or just
select one of them and transmithis information alone. Therefore, in
any case, the boundary ofthe capacity region in (8) can be achieved
by an optimalTD policy which transmits the information of at most
one userthrough the fading broadcast channel in each channel
state.Obviously, this optimal policy can be used as a power
allocationpolicy for CD without successive decoding to eliminate
inter-ference from all other users and, therefore, to achieve the
samecapacity region boundary as TD. Thus, we need only to showthat
the capacity region of CD without successive decoding in(33) cannot
be larger than the capacity region of TD in (8). Weuse Lemma 2 to
prove this as follows.
For CD without successive decoding, , denote
For , let
and let . Then according to Lemma 2, we have
(36)
Now consider the equal-power TD strategy which assigns thepower
to each user for a fraction of the total trans-mission time in the
state. By denoting
we have
(37)
since . Therefore, from (36) and (37), it is clearthat given a
fading state, , the capacity region ofthe equivalent AWGN broadcast
channel for CD without succes-sive decoding is within that for
equal-power TD and is, there-fore, within that for optimal TD.
Consequently, the capacity re-gion of CD without successive
decoding in (33) cannot be largerthan the capacity region of TD in
(8).
Note that since the suboptimal TD scheme proposed in Sec-tion
III-C indicates that the broadcast channel is used by no more
than one user, this suboptimal policy can also be applied to
CDwithout successive decoding.
IV. FREQUENCY-SELECTIVE FADING CHANNELS
In the previous sections we have considered a
flat-fadingbroadcast channel model which is appropriate for
narrow-bandapplications. For wide-band communication systems,
thetime-varying frequency-selective fading model is more
appro-priate. In this section, we will extend our previous results
tothis model.
First we consider an -user time-invariant spectral
Gaussianbroadcast channel with continuous noise spectra
, where ranges over the system band-width [13]. For CD with
successive decoding, given a powerallocation policy , can be viewed
as the transmit powerallocated to User at frequency , . Let
denotethe set of all power allocation policies satisfying the total
powerconstraint , i.e.,
Then it can be similarly shown as in Section III-A that the
ca-pacity region is
(38)
For TD, given a resource allocation policy, , andcan be viewed
as the transmit power and fraction of trans-
mission time allocated to Userat frequency . In this case,
theset is defined as
Then the achievable rate region using TD is
(39)
Note that the regions in (38) and (39) are actually identical
tothose in (1) and (8), respectively, with the role of the fading
state
now played by frequency. Therefore, the boundary surfacesof the
regions in (38) and (39) and the corresponding optimal re-source
allocation strategies can be obtained by using the resultsin
Section III. Moreover, it is clear that one of the nonuniqueoptimal
resource allocation strategies for TD can also be usedfor FD and CD
without successive decoding to achieve the samecapacity region as
TD.
In the general case, where the channel is time-varying, if
weassume as in [8] that the time variations are random and
ergodic,and the channel varies very slowly relative to the
multipath delayspread, then the channel can be decomposed into a
set of par-allel time-invariant spectral Gaussian broadcast
channels. In thiscase, for each fading state, let the continuous
noise spectra
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FADING BROADCAST CHANNELS—PART I 1093
Fig. 5. Two-user ergodic capacity region comparison: 3 dB SNR
difference.
of the users be . Thecapacity regions and the optimal resource
allocation strategiesfor CD, TD, and FD of this time-varying
channel can then beobtained from the results in Section III by
replacing the fadingstate with . That is, the average is now taken
on both fre-quency and fading state . Note that for most physical
chan-nels the time variations are correlated, not random. The
capacityregion for multiuser channels under this more realistic
channelmodel is unknown (see [17] for the capacity of a
single-usertime-varying frequency selective fading channel).
V. NUMERICAL RESULTS
In this section, we present numerical results for the
two-userergodic capacity regions of the Rician and Rayleigh
flat-fadingchannels under different spectrum-sharing techniques.
The ca-pacity regions obtained analytically in the previous section
leadto double-integral formulas that were solved numerically
usingMathematica to obtain the numerical results in this section.
Inthe figures below, as in [15], the equal power TD scheme refersto
the strategy that assigns the constant transmission powerand total
bandwidth to User 1 for a fraction of the total trans-mission time,
and then to User 2 for the remainder of the trans-mission. The
optimal TD scheme for both the AWGN channeland the fading channels
is obtained by allocating different power
to the two users. We refer to CD without successive decoding
asCDWO. Since TD and FD are equivalent in the sense that theyhave
the same capacity region, all results for TD in the figuresalso
apply for FD.
In Fig. 5, the ergodic capacity regions of the Rician
andRayleigh fading broadcast channels are compared to that of
theGaussian broadcast channel using the CD, TD, equal powerTD, and
CDWO techniques. The SNR difference between thetwo users is 3 dB (
and denote the average noise densitiesof User 1’s channel and User
2’s channel, respectively). Thetotal transmission power is 10 dB
and the signal band-width 100 kHz. The ratio of the direct-path
power to thescattered-path power in the Rician fading subchannels
is6 dB.
In this figure we see that while the single-user ergodic
rate(the -axis or -axis intercept) in fading is smaller than
therate in AWGN, the two-user capacity regions of both the Ri-cian
fading and the Rayleigh fading broadcast channels usingeither
optimal CD or suboptimal TD techniques in some placesdominate that
of the AWGN broadcast channel using optimalCD. That is, for the
fading broadcast channel, ergodic rate pairsbeyond the capacity
region of the nonfading broadcast channelcan be achieved by
applying optimal resource allocation overthe joint fading channel
states. However, as shown in Fig. 6,this is not true when the
difference between the average noise
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Fig. 6. Two-user ergodic capacity region comparison: 20 dB SNR
difference with strong signal power.
variances of the two users is quite large. An intuitive
analyticalexplanation of the two cases is given in the Appendix,
SectionG by comparing the sum rate on a fading broadcast channel
tothe sum rate on an AWGN channel, where the sum rate refers tothe
sum of the weighted rates in (3) with equal rate reward foreach
user.
In Fig. 5, for simplicity, we calculate the rate region of
thefading broadcast channel for TD by applying the simple
sub-optimal TD power allocation policy. The resulting rate
regionturns out to be very close to the capacity region for
optimalCD. Therefore, the capacity region using the optimal TD
powerpolicy will also come very close to that of optimal CD. This
ob-servation implies that, due to the small SNR difference
betweenthe two users, superposition encoding with successive
decodingis not necessary, since time sharing is near-optimal. For
theAWGN broadcast channel, the capacity region boundary of
theoptimal TD scheme is indistinguishable from the equal-powerTD
straight line, which means that when the two users have asimilar
channel noise power, constant power allocation is goodenough for
TD. The CDWO capacity region boundary (omittedfrom Fig. 5 but shown
in figures in [15]) includes the two end-points of the equal-power
TD line but is below this straight linedue to its convexity.
However, for the fading channels, optimalor suboptimal TD has a
much larger capacity region than equalpower TD and the capacity
region for CDWO is the same as thatfor optimal TD.
We show in Fig. 6 that when the SNR difference between thetwo
users is 20 dB and the total average power is 25 dB, theergodic
capacity region for CD in Rayleigh fading is now com-pletely within
the region for CD in AWGN. However, optimalTD in fading can achieve
some rate pairs far beyond the capacityregion of the AWGN broadcast
channel using optimal TD, andthe suboptimal TD power policy for
Rayleigh fading results ina rate region almost as large as the
capacity region with the op-timal TD power policy. For both AWGN
and fading channels,due to the large SNR difference between the two
users, the ca-pacity region for optimal CD is noticeably larger
than that foroptimal TD, and the capacity region for optimal TD is
notice-ably larger than that for equal power TD.
Fig. 7 shows the case where the SNR difference between thetwo
users is 20 dB and the total average power is only 10 dB. Un-like
the previous cases, we see here that the ergodic
single-usercapacity of User 2 for the Rayleigh fading channel is
largerthan that for the AWGN channel due to its very low
averageSNR. Thus, the optimal CD or suboptimal TD scheme for
fadingyields a large rate region that is not achievable for the
AWGNchannel. However, as in Fig. 6, due to the great SNR
differencebetween the two users, optimal CD results in a capacity
regionmuch larger than that for suboptimal TD and the capacity
regionfor optimal TD is significantly larger than that for equal
powerTD. This observation holds for both fading and AWGN
chan-nels.
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Fig. 7. Two-user ergodic capacity region comparison: 20-dB SNR
difference with weak signal power.
VI. CONCLUSION
We have obtained the ergodic capacity region and the op-timal
dynamic resource allocation strategy for fading broadcastchannels
with perfect CSI at both the transmitter and the re-ceivers. These
results are obtained for CD with and without suc-cessive decoding,
TD, and FD. Comparisons of the capacity re-gions show that CD with
successive decoding has the largest ca-pacity region, while TD and
FD are equivalent and they have thesame capacity region as CD
without successive decoding. ForCD without successive decoding, the
optimal power policy is totransmit the information of at most one
user in each joint fadingstate. This policy is also optimal for TD,
though other strategieswhich allow at most two users to time-share
the channel mayalso be optimal. When the average channel fading
condition foreach user is similar, the capacity regions for optimal
CD and TDare quite close to each other. However, when each user has
anaverage channel condition quite different from that of the
others,optimal CD can achieve a much larger ergodic capacity
regionthan the other techniques. In Part II of this paper, we will
de-rive the zero-outage capacity region and the capacity region
withnonzero outage for fading broadcast channels. For
narrow-bandapplications, since each rate vector in the outage
capacity regionmust be achievable in every fading stateunless an
outage isdeclared, we cannot average over different fading
conditions.
However, the outage and ergodic capacity regions exhibit
sim-ilar relative performance between the various
channel-sharingtechniques.
APPENDIX
A. Proof of Theorem 1
The convexity of the capacity region in (1) can be easilyproved
by using the idea of time-sharing [18, pp. 396–397]. Wenow prove
the achievability and the converse of this capacityregion.
1) Achievability: We prove the achievability of the
capacityregion by proving the achievability of in (2)for each given
power allocation policy . , theproof is similar to that of the
achievability of the single-userfading channel capacity [6]. The
main idea is a “time diversity”system with multiplexed input and
demultiplexed output.That is, we first discretize the range of the
time-varying noisedensity of each user into states. Therefore,
there are
joint channel states of the users and wedenote them as , the
probabilities of whichare , respectively. In each joint state,
thechannel can be viewed as a time-invariant AWGN broadcastchannel,
the capacity region of which is known [19]. Given a
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block length , we then design an encoder/decoder pair for
theusers in each state with codewords
of average power for each user which achieve rate, , where is
the maximum
achievable rate for Useron the equivalent AWGN broadcastchannel
corresponding to state, for largeenough, and
These encoder/decoder pairs correspond to a set of input
andoutput ports associated with each state. When the channel is
instate , the corresponding pair of ports are connected throughthe
channel. The codewords associated with each state are
thusmultiplexed together for transmission, and demultiplexed at
thechannel output. This effectively reduces the time-varying
broad-cast channel to a set of time-invariant broadcast channels
inparallel, where the th channel only operates when the
time-varying channel is in state . The average rate for each
user
is thus the sum of rates associated with each stateweighted by ,
. Details of the proof canbe found in [16].
2) Converse:Suppose that a rate vector is achievable,then we
need to prove that any sequence of
codes with average total powerand probability of erroras must
have where
is
We assume that the codes are designed witha priori knowledgeof
the joint channel state. Since the transmitter and receiversknow
state up to the current time, this assumption can onlyresult in a
higher achievable rate.
As in the Appendix, Section A1), we first discretize the rangeof
the time-varying noise density of each user intostates.
Specifically, define and wesay that theth subchannel is in state if
the time-varying noisedensity of User satisfies , whereThus, the
set discretizes the fading range of each sub-channel into states
and there are dis-crete joint channel states. We denote thethof
these states as
where is the base-expansion of , and is the th subchannel state.
That is,
for all and
Note that a channel state if and only if ,.
Over a given time interval , let be the number oftransmissions
during which the channel is in the state, let
be the random subset of at which times thechannel is in the
state , and let be uniformly distributed on
. By the stationarity and ergodicity of the channelvariation, as
. For , let
be the transmit power allocated to Userat time anddefine
Since for any message from the base station to theusers,there is
a power constraint on the corresponding codeword, itfollows that
for each
Therefore, for all such that , arebounded sequences in. Thus,
there exist a converging subse-quence and a limiting such thatas .
Moreover
(40)
For a given power allocation policy , we assume thatis the
transmit power assigned to Userwhen the
time-varying broadcast channel is in channel state. Letbe the
set of all the power allocation policies which are piece-wise
constant in each channel state and which satisfy the averagetotal
power constraint (40). Assuming that the noise densitiesof the
users in each channel state areconstants and are denoted as ,
thechannel states can be viewed as AWGNbroadcast channels where at
any given time only one of thesechannels is in operation and the
probability that theth broad-cast channel is in operation is given
by , .We call this the probabilistic broadcast channel. We show
inthe Appendix, Section A3) that if is achievable on
theprobabilistic broadcast channel consisting of the broad-cast
channels with probabilities
under the assumption of perfect trans-mitter and receiver CSI
(i.e., at each timeit is known at boththe transmitter and receivers
which broadcast channel is inoperation), then
(41)
where
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Define
Since for and
it is clear that
(42)
, let
According to the power constraint (40), it is obvious
thatsatisfies
Thus, and
(43)
where is given in (2). Taking the limit of the left-handside of
(43), we obtain
(44)
For the time-varying broadcast channel, ifis achievable,
then
(45)
From (42) and (45) we have
That is,
(46)
where denotes the capacity region of the time-varyingbroadcast
channel. Combining (44) and (46) with the achiev-ability result
which indicates that
Fig. 8. Probabilistic broadcast channel: Channel 1 in operation
withprobabilityp , and Channel 2 in operation with probabilityp
.
we obtain
Since the upper bound equals the lower bound by the
monotoneconvergence theorem [20], it is clear that
3) Capacity of a Probabilistic Broadcast Channel withCSI: In the
Appendix, Section A2), while proving the converseof the capacity
region in Theorem 1 we have used the capacityof a probabilistic
broadcast channel consisting ofdiscreteAWGN broadcast channels with
given probabilities under theassumption that perfect CSI is
available at both the transmitterand the receivers. We now show how
to prove the capacityformula of a probabilistic broadcast channel
composed of twoAWGN broadcast channels and two users. As will be
discussedlater, the results can be easily generalized to the case
ofchannels and users ( ).
Assume that two discrete degraded memoryless AWGNbroadcast
channels
and
are as shown in Fig. 8, where, , and are finitealphabets and
denotes the channel transition probabilityfunction. Note that each
broadcast channel has two outputsand . In the first channel
(Channel 1), the Gaussian noises aredenoted as and , the noise
variances of which areand , respectively. In the second channel
(Channel 2), theGaussian noises are and , the noise variances of
whichare and , respectively. Let and
. We define the probabilistic broadcast channelconsisting of two
AWGN broadcast channels as a channel whereChannel 1 operates with
probability and Channel 2 operateswith probability , and at any
time only one of thetwo channels is in operation.
Denote as the capacity of an AWGN channel withSNR , i.e., . Let
and bethe transmission rates of the particular information toand
,respectively. Here we do not consider the case where
commoninformation is transmitted. , let ,
. Assuming that the total average power is, forfixed , we first
divide the total power
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into the part used in Channel 1 and used in Channel2 with . Then
is divided into the power
used to transmit the information to and usedto transmit the
information to ; is divided into the power
used to transmit the information to and usedto transmit the
information to . Then, under the assumption ofperfect CSI at both
the transmitter and the receivers, the capacityregion of the
probabilistic broadcast channel in Fig. 8 is definedby
(47)
Note that this capacity region is similar to that of a
parallelbroadcast channel composed of two broadcast channels
[21],[22], except that in the parallel case, bothand equal
.Therefore, its proof can be obtained by following very
similarsteps as that for the parallel broadcast channel [21], [22],
the de-tails of which are given in [16].
The capacity region in (47) is equivalent to [21], [22]
(48)
since it is obvious that , and every vertex of the convexhull is
inside , which means . The convexityof is easily shown by using the
time-sharing technique. Theformulas (47) and (48) can be readily
generalized to the case of
channels and users by careful inspec-tion of their structures,
and the generalized formulas are alsoequivalent. Since we can
similarly prove that the generalizedformula of (48) is the capacity
region of the-channel, -userprobabilistic broadcast channel, the
generalized formula of (47)as used in the Appendix, Section A2)
must also be the capacityregion.
B. Proof of Theorem 2
1) Achievability of the Rate Region:The proof of the
achiev-ability follows along the same idea as that for the capacity
regionof CD with successive decoding discussed in the Appendix,
Sec-tion A1) and is therefore omitted.
2) Convexity of the Rate Region: , , , let
Since
(49)
(50)
(51)
from (49)–(51), we obtain the Hessian matrix
Therefore, is a concave function of . That is,, , ,
(52)
where , . Thisresult will be used in the following to prove the
convexity of thecapacity region.
, , we need to show that. Let and be the two power policies
corresponding to the rate vectorsand , respectively. In agiven
channel state, according to the two policies, the transmitpower and
fractions of transmission time allocated to User
are and , , and , re-spectively. Therefore, for the two power
policies, the achievablerates for User in the state are
(53)
and
(54)
respectively. Equations (53) and (54) can also be expressed
as[23]
where and ,.
If we define a third power policy such that in each channelstate
, the transmit power and fraction of transmissiontime allocated to
User are
and
respectively, then
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Since
we know that . Moreover, it is proved in (52) thatfor
is a concave function of , i.e.,
Therefore, .
C. Proof of Lemma 1
In the following, , , , , , , , andare all functions of . Their
explicit dependence onis not
shown to simplify the notation.Let and . Define
where is given in (12), then
(55)
1) If , then . Since byassumption, the numerator of (55) is
nonnegative for any
. Thus, , . Therefore, if ,, i.e., . For ,
and , we have
(56)
The last inequality in (56) is due to the concavity ofand the
equality is achieved when and
. Thus, in this case, the solution to (11) is.
2) If , from (55) we know that
if
if .(57)
Since , for large enough, .Thus, for large , , i.e., .
However,
. Using this and (57), it is clear that andwill cross each other
once at some positive value, as shownin Fig. 3. In that figure, and
are the points that satisfy
where is the slope of the common tangent line in the
figure.Since
and can also be expressed as
we have
(58)
(59)
and satisfies
i.e.,
(60)
where is given in (14). Therefore, if ,by time-sharing, in (11)
can achieve the values between
and on the straight line; if or, is simply or , respectively.
That
is, in this case, the solution to (11) is (15).
D. Proof of Theorem 3
For a given fading state, from (13) we know that the
optimalpower satisfies (20). Let and be the optimalpower and
fraction of transmission time allocated to user
at state , respectively. From Lemma 1, we know that
1) if , then . Thus
(61)
and , , , .Substituting (61) into (20), we have
Since must be nonnegative
2) if , as shown in Fig. 3, it is clear that
if
if
if
(62)
where , , and are all functions of and they aregiven in
(58)–(60), respectively. Since we know from (20)
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that , from (15), (17)–(19), and (62) wehave
a) if then and. Thus, .
Consequently
and
b) if then and. Thus, .
Consequently
and
c) if , then
and can be any value between and .Thus
and
where can be any value betweenand . More-over, and must satisfy
the total average powerconstraint (21).
Because , from (58) and (59), we have
(63)
Furthermore, since and
if
if
where
we have
A) if , or if and , then;
B) if and , then ;
C) if and , then .
Therefore, the second part of Theorem 3 is proved by
combininga), b), and c) with A), B), and C).
E. Decision Region Comparison for Optimal and SuboptimalTD
Schemes
For the suboptimal TD policy, define
where is given in (31). Let satisfy
then
(64)
(65)
Assuming that , from (64) and (65) we know that
1) if , then
and
2) if , then
and
Since for , , where de-notes the derivative of with respect to ,
is adecreasing function. Thus, when the cdf is continuous,the
suboptimal policy is equivalent to
a) if and or if and ,then
and
b) if and , then
and
From the proof of Theorem 3 in the Appendix, Section D, it
isclear that the optimal TD policy is equivalent to
a) if and or if and ,where is defined in (60), i.e., , then
and
b) if and , then
and
Hence, when , satisfies , from thedefinition of in (14), we
have
(66)
According to (63) and (66), it is clear that . Conse-quently, .
Therefore, for a given , the only dif-ference between the optimal
and suboptimal policy is that when
, where is given in (32), the suboptimal schemetransmits the
information of User 2, while the optimal schemetransmits the
information of User 1. This suboptimal allocationof resources
occurs only rarely. Note that the values ofin thetwo schemes
satisfying the two-user power constraint (21) arenot the same,
though they may be very close to each other.
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F. Proof of Lemma 2
For simplicity, we denote
When , let , then , .Let
then
If , then for , we haveand
(67)
If and , then
for
for
which implies
(68)
If and , then for, we have and
(69)
From (67)–(69), we conclude that
That is,
Now assume that (34) is true for
then for
Therefore, by induction, we know that (34) is true.
G. Comparison of the Sum Rates on a Fading BroadcastChannel and
on an AWGN Broadcast Channel
In the following we consider the sum rate for an-userbroadcast
system. Assume WLOG that User 1 has the smallestaverage noise
variance, i.e.,
(70)
Then for the time-invariant AWGN broadcast channel with thesame
noise variances, the maximum sum rate of theuserswill be
AWGN
(71)
where is the total transmit power of the users.For the fading
broadcast channel, it is shown that for both CD
and TD, the optimal power allocation strategy in each
fadingstate is to allocate power to User only,where , and
satisfies
. Therefore, the maximum sum rate of theusers is
fading
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which is equivalent to the capacity of a single-user
fadingchannel with average noise power [6]
For this equivalent single-user fading channel, given the
totalaverage power constraint, it is well known by Jensen’s
in-equality that [6]
fading (72)
In the case where the average noise power of each user in
abroadcast system with fading is dramatically different, most ofthe
time the user with the best average fading condition (User 1)is
chosen for transmission. Therefore, in this case, .Hence, from (71)
and (72) it is clear that
fading AWGN
In the case where the average noise power of each user is
quitesimilar, in each joint fading state, only the user with the
bestchannel condition is chosen for transmission. In this case,
sinceat any time, the channel chosen for transmission is the best
of the
channels with similar average fading conditions, the equiv-alent
single-user fading channel is much better on the averagethan any of
the individual fading channels, i.e., .Therefore, it is very likely
that
fading AWGN
ACKNOWLEDGMENT
The authors wish to thank the Editor and the anonymous
re-viewers for their very helpful suggestions and comments.
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