Gaussian Broadcast Channel
Giuseppe Caire and Shlomo Shamai (Shitz)
July 23, 2001
Abstract
AGaussianbroadcastchannelwithr single-antennareceiversandtantennasat
the transmitter is considered. Both transmitter and receivers have perfect knowl-
edge of the channel. Despite apparent simplicity, this model is in general a non-
degraded broadcast channel,for which the capacity regionis not fully known. We
proposeanoveltransmissionscheme basedon \ranked knowninterference"(RKI).
In brief,thetransmitterdecomposes thechannelintoan ordered(orranked)setof
interferencechannelsforwhichtheinterferencesignalinthei-thchannelis alinear
combination of the signals transmitted in channels j < i. Since all transmitted
signalsaregeneratedbythetransmitter,interference ineachchannelisknownnon-
causally. Hence, known techniques of coding for non-causally known interference
can be applied to make the interference in each channel harmless without further
powerpenalty. Weshowthattheproposedschemeisthroughputwiseasymptotically
optimalforbothlowand highSNR,and wecomparethethroughput achievable by
RKIwiththethroughputachievablebymoreconventionalzero-forcingbeamforming
(space-division multiple access) and with the throughput of a single-user multiple
antenna system obtained by allowing the receivers to cooperate. Also, we provide
a modicationof the basicRKI scheme which achieves optimalthroughput for all
SNRs in the special case of two users. For independent Rayleigh fading, closed-
form throughputexpressionsareobtainedforthebasicRKIstrategyand arbitrary
t and r. Numerical resultsare shown forniter;t and inthe large-systemlimitof
r;t!1 withxedratio =r=tuserspertransmitantenna.
Keywords: Gaussianbroadcast channel,multiple-antennasystems.
1 Introduction
We consider a wirelesscommunication system with one transmitter and r receivers. The
transmitter has t antennas, while the receivers have one antenna each. The transmitter
has todelivertoeachreceiverindependentinformation,asinthe downlinkof asingle-cell
system where the base-station is equipped with anantenna array of t elements.
Weconsider thecasewherethe channeltransferfunctionbetween thetransmitterand
all receivers is perfectlyknown toall terminals. Our main results are independent of the
statistics of the channel and they can be directly applied to the more general multi-cell
downlink scenario with full cooperation between all base-station antennas. However, in
thisworkwedonotconsiderexplicitlytheeectofthespatialdistributionofthetransmit
and receive antennas, asfor example inthe cellularmodelof [1, 2].
This channel is referred to inthe following as the t1: r GBC (to be read \ttimes
1 to r Gaussian Broadcast Channel "), in order to stress the fact that the r receivers
must process their signals separately, as opposed to a t r multi-antenna single-user
system where the signals at the r receive antennas can be processed jointly [3]. Despite
apparentsimplicity,thismodelisingeneralanon-degraded broadcast channel,forwhich
the capacity regionis not fully known.
Conventional approaches to the t1 : r GBC have been proposed by many authors
under dierent assumptions on the channel knowledge at the transmitter. They are all
based on some form of linear precoding of the user signals. In [4], the multipath channel
knowledge atthe transmitterisused topre-lterthe signalof eachuser by itsown space-
time matched lter, in order to induce at the receiver the Maximal-Ratio Combining
(MRC) [5] of the multipath without having to implementthe matched lter in the user
terminal. We refer to this approach as MRC beamforming.
1
In [6, 7, 8, 9, 10, 11] joint
linear precoders are foundinorder tomaximizethe signal-to-interferenceplus noise ratio
attheuserterminalssubjecttovariousconstraints. Theseschemes,usuallyproposedina
CDMAmultiple-antennasingle-celldownlinkscenario,arebasedonlineartransformations
of the individual user signals which essentially null-out interference while combining the
multipath of the useful signal for each user. In our frequency-at unspread channel
model, this reduces to inverting the channel at the transmitter by using the Moore-
Penrose pseudoinverse [12] of the channel matrix. This approach shall be referred to in
the following asZero-Forcing(ZF)beamforming.
An upper bound on the achievable throughput of the t 1 : r GBC is obtained by
letting the receivers cooperate. In this case, the channelreduces to asingle-user multiple
antenna situation, for which the capacity with perfect channel knowledge at both ends
is well-known [3, 13]: the channel can be decomposed into a set of parallel channels
corresponding to its eigenmodes without loss of information. It is worthwhile to notice
that for both ZF beamforming and the cooperative system the optimal coding strategy
reduces tostandard single-user Gaussiancoding, and the multiple-inputmultiple-output
natureofthesystemiscapturedbysimplelinearsignalprocessingatthetransmitteronly
(for ZF) orat both transmitter and receiver(for cooperative).
A quitedierent coding strategy is proposed here for the t1:r GBC.Our scheme,
referred to as Ranked Known Interference (RKI), makes use of linear precoding in order
to decompose the channel into a ranked set of interference channels for which the inter-
ference signal in the i-th channel is a linear combination of the signals transmitted in
channels j < i. Since all signals are generated by the same transmitter, interference in
1
IntheCDMAliteraturethisis alsoknownas\pre-rake".
each channel is known non-causally. Then, known techniques of coding for non-causally
known interference [14, 15, 16, 17, 18] can be applied to make the interference in each
channel harmless without further power penalty. In particular, the recently proposed
scheme of [17] based on modulo-lattice precoding can be applied here. Interestingly, a
version ofour RKIscheme with suboptimal one-dimensionallatticeprecoding,analogous
to Tomlinson-Harashima precoding (see [19] and references therein), has been recently
and independently proposed in [20, 21] for the transmitter-based cancellation of far-end
cross-talk inDSL.
We show that the RKI strategy is throughputwise asymptotically optimal for both
low and high SNR, and it is equivalent to MRC beamforming to the best user only for
low SNR and to the cooperative scheme for high SNR. Also, we provide a modication
of the basic RKI scheme which achieves optimal throughput for all SNRs in the special
case of twousers. In the case of Rayleighfading, closed-formthroughput expressions are
obtained for the basic RKI strategy for both nite r;t and in the large-system limit of
r;t!1 with xed ratio =r=t users pertransmitantenna.
The paper is organized as follows. In Section 2 the channel model is dened. In
Section 3 the basic RKI strategy and its properties are illustrated. Section 4 presents
upperand lowerbounds onthe achievable throughputof the t1:r GBCand Section5
shows that a modication of the basic RKI strategy is actually optimal for the t1: 2
GBC. In Section 6 results for the basic RKI scheme in independent Rayleighfading are
given. Finally,Section 7summarizes our conclusions.
Notation.
LetAbeamatrix. Thei-throw,j-thcolumnand(i;j)-thelementofAaredenoted
by a i
, a
j
and a
i;j
or equivalently by [A]
i;j
, respectively.
The submatrixobtained by the rows of A numbered by i 2S, where S isan index
set, is denoted by A[S].
Superscripts T
and H
denote transpose and Hermitiantranspose, respectively.
zN
C
(;) indicatesthat z isacomplex circularly-symmetricGaussianrandom
vector with mean and covariancematrix .
[x]
+
=maxf0;xg.
2 System model
Weconsider a discrete-timecomplexbaseband channel modeland assumethat the prop-
agation channel between each transmit-receive antenna pair is frequency non-selective.
Extension to the time-invariantnite-memory frequency-selective case is easilyobtained
in the frequency domain(see [22] andreferences therein) and willbe briey addressed in
Appendix A. The t1:r GBC isdescribed by
y =Hx +z (1)
where x
i 2 C
t
is the transmitted symbol vector at time i, y
i
and z
i
N
C
(0;I) are the
corresponding vectors of received signal and noise and H 2 C rt
is the channel matrix,
where h
k;`
is the complex channel gain from transmit antenna ` to the receiver antenna
of user k.
The input isconstrained tosatisfy
1
n n
X
i=1 jx
i j
2
A (2)
whereAisthemaximumallowedtotaltransmitenergyperchanneluse. Sinceweconsider
normalized unit noise variance, A takes onthe meaning of totaltransmitSNR.
Foragivenblocklengthn, acode C
n
forthe input-constrainedt1:rGBCisdened
by a codebook of exp (n P
r
k=1 R
k
) code arrays of the form X= [x
1
;:::;x
n ]2 C
tn
, such
that (2)is satisedforeach array,byanencodingfunction mappingr-tuplesofindexes
(w
1
;:::;w
r
) (where w
k
2 f1;2;:::;exp (nR
k
)g) onto the code words, and by r decoding
functions
1
;:::;
r
, such that
k : C
n
! f1;2;:::;exp (nR
k
)g. The received signal at
the k-th receiver antenna is given by the k-th row y k
of the array Y = HX+Z. The
error probability for the k-th user is given by
k
= Pr(
k (y
k
) 6= w
k
). A r-tuple of rates
R=(R
1
;:::;R
r
)isachievableifthereexistasequenceofcodesC
n
withratesapproaching
R and vanishing
k
(for all k = 1;:::;r), as n increases. The system throughput R ,
measured in bit/channeluse (or bit/s/Hz),is dened as the rate sum
R= r
X
k=1 R
k
(3)
We assume perfect Channel State Information at the transmitter (CSIT) and at all re-
ceivers (CSIR), i.e., the channel matrix H is known to everybody, and consider the fol-
lowingscenarios:
1. H isdeterministic and xed.
2. H is xed during the transmission of each code array, but it is randomly and in-
dependently selectedaccording toagiven probabilitydistribution(compositechan-
nel). In thiscase [23, 24], wecan consider both the short-termconstraint(2)orthe
long-term constraint
E
H
"
1
n n
X
i=1 jx
i j
2
#
A (4)
where E
H
denotes expectationwith respect to H.
3. H is generated by an ergodic matrix random process and varies during the trans-
mission of eachcode word so that the channel is informationstable [24, 25].
Since R depends on H, the instantaneous throughput of the composite channel (case 2)
is a random variable. In this case we consider the average throughput
R =E [R ]. This
is achievable by a variable-rate coding scheme which adjusts its instantaneous through-
put according to the channel realization H. By the law of large numbers, the average
throughputachievableoveralongsequence of channelrealizationsisgiven by
R . More in
general,the sameresult appliestothe case ofablock-fadingchannelwhere Hisconstant
over blocks of n consecutive channel uses and changes from block to block according to
an ergodic matrix randomprocess [24].
Withperfect CSIT and CSIR, it turns out that the information stable channel (case
3) has the same averagethroughput of the composite channel with long-termpowercon-
straint(4)(althoughcodinganddecodingstrategiesanderrorexponentsforthesechannel
are generallydierent[24]). Therefore, weshall focus on cases 1 and 2only.
The 11:r GBC coincides with the classical degraded Gaussianbroadcast channel,
whose capacity region is well-known (see [26] in the deterministic case and [27, 28] in
the composite or ergodic cases). However, the t1 : r GBC for t > 1 is in general a
non-degraded broadcast channel, for which the capacity region is not fully known, and
cannot be reduced to anequivalent set of parallel degraded broadcast channels (studied
in [29, 30, 31, 27, 22, 28]). Obviously,an inner bound tothe capacity region is provided
by exhibiting an explicit coding scheme for which the error probabilities vanish as the
block lengthincreases.
It is interesting to notice here that perfect CSIT is a key fact in our model. For
example, under the assumption of no CSIT and of symmetric ergodicchannel, where H
is random with independently and identically distributed rows, the marginal transition
pdfs p(y
k;i
;Hjx
i
) are identical for all k = 1;:::;r, and the resulting t1 : r GBC is
stochastically degraded [26] also for t > 1. In this case, the capacity region is trivially
obtained by time-sharingand the optimalthroughput coincides with the t1\transmit
diversity" capacity obtained in[3], explicitlygiven by
R tx div
=E[log(1+jh 1
j 2
A=t)] (5)
Wehastento saythat the cases ofperfectCSIT and of symmetricchannelwithoutCSIT
are not the only possibilities. Other settings might also yield new and interesting prob-
lems.
3 The RKI coding strategy
Let H=GQ be a QR-type decomposition[12] obtained by applyingthe Gram-Schmidt
orthogonalization procedure [12] to the rows of H. Letm =rank(H), then G2 C rm
is
lowertriangular(i.e.,ithaszerosaboveitsmaindiagonal)andQ2C mt
hasorthonormal
rows. By letting the transmitted i-th signal vector be given by x
i
= Q H
u
i
, the original
channelis turnedintothe set of m interference channels
y
k;i
=g
k;k u
k;i +
X
j<k g
k;j u
j;i +z
k;i
; k =1;:::;m (6)
whilenoinformationissenttousersm+1;:::;r. Theinputconstrainttranslates directly
to the new input signal u
i
, infact,
1
n n
X
i=1 jx
i j
2
= 1
n n
X
i=1 u
H
i QQ
H
u
i
= 1
n n
X
i=1 ju
i j
2
(7)
The interference signal s
k;i
= P
j<k g
k;j u
j;i
in the k-th interference channel (6) is given
by a linear combination of the transmitted signals u
j;i
in channels j < k. Since all
these signals are generated at the transmitter, and the coeÆcients g
k;j
are known, the
interference signal ineach channel isknown non-causally by the encoder (if the channels
are considered inthe order1;:::;m dictated by the QRdecomposition). Forthis reason,
we refer to this scheme as Ranked Known Interference strategy. We have the following
result:
Theorem 1. Forgiven H=GQ, with d
k
=jg
k;k j
2
,the maximumthroughput achieved
by the RKI strategy isgiven by
R rki
= m
X
k=1
[log(d
k )]
+
(8)
where is the solution of the waterllingequation [32]
m
X
k=1
[ 1=d
k ]
+
=A (9)
Proof. See Appendix B.
Corollary 1. For the composite channel we have
R rki
= E
H [R
rki
] where solves the
short-terminputconstraint(9),orthelong-terminputconstraintE
H [
P
m
k=1
[ 1=d
k ]
+ ]=
A.
Remark: on the capacity ofchannelswith interference knownto the transmit-
ter. InAppendixB,weoutlineacodingschemebasedonmodulo-latticeprecoding,rst
introducedby Erez Shamai and Zamir[17], whichis able toachieve the RKI throughput
of Theorem 1 and Corollary 1.
Also,inAppendix B we investigate the relationbetween the RKIscheme and Costa's
\writingondirtypaper"capacityresult[15],andweshowthatthethroughputofTheorem
1andCorollary1canbealsoachieved byasimplegeneralizationofCosta'sscheme,which
is quite dierent from the lattice precoding scheme of [17]. The main dierence between
the two schemes is that Costa's relies on the fact that the known interference signal is
Gaussian, i.i.d., with given (known) variance, while lattice precoding is universal in the
sense that it works for any arbitrary interference signal sequence, provided that it is
interference realization, but need not know the interference statistic). The link between
Costa's \writing on dirty paper" capacity and the lattice precoding scheme is provided
in Appendix B by combining a well-known result of Ahlswede on the AVC with random
parameters [16] non-causally known atthe transmitter witha recent resultof Cohenand
Lapidoth [18].
Forthe sake of comparison,we review here the ZF beamformingand the cooperative
schemes. ZF beamforming consists of inverting the channelmatrix atthe transmitter in
order to create orthogonal channels between the transmitter and the receivers without
receivers cooperation. Let S f1;:::;rg be a subset of cardinality m for which the
corresponding submatrixH[S] isfull row-rank, i.e., rank(H[S])=jSj. Let
H +
S
=H[S]
H
H[S]H[S]
H
1
(10)
betheMoore-Penrosepseudoinverse [12]ofH[S]. InZFbeamforming,thetransmitsignal
is obtained as x
i
=H +
S u
i
and yieldsthe set of parallelchannels
y
k;i
=u
k;i +z
k;i
; k 2S (11)
while no information is sent to users k 2= S. Clearly, the maximum ZF throughput is
obtained by u
i
Gaussianwith independent components.
The (short-term)input constraint isgiven by
1
n n
X
i=1
trace H +
S u
i u
H
i (H
+
S )
H
= trace (H +
S )
H
H +
S 1
n n
X
i=1 u
i u
H
i
!
! jSj
X
k=1
2
u;k
b
k
(12)
where we letE[u
i u
H
i
]=diag(
2
u;1
;:::; 2
u;jSj
) and where we dene
b
k
=
1
(H[S]H[S]
H
) 1
k;k
(13)
The maximum ZF throughput is obtained asthe solutionof the optimization problem
(
max P
jSj
k=1
log(1+ 2
u;k )
subjectto P
jSj
k=1
2
u;k
b
k
=A
which can be conveniently reparameterizedby lettinga
k
=
2
u;k
b
k
and yields
R zf
= jSj
X
k=1
[log(b
k )]
+
(14)
where is the solution of P
jSj
[ 1=b
k ]
+
=A.
The above ZF maximum throughput can be further optimized with respect to the
choice of the active user set S. In particular, by optimizing over the sets of cardinality
one, we obtaina system basedon MRC beamformingtothe user with highestindividual
channel capacity, whose throughputis given by
R mrc
=log 1+jh max
j 2
A
(15)
where h max
is the row of H with largest 2-norm. Interestingly, for t>1 this choice does
not yield necessarilythe largest throughput, in sharp contrast to the standard degraded
broadcast channel(t=1) forwhich the throughput ismaximized by transmitting tothe
best user only [27].
Now, we consider a single-user multiple antenna system which is obtained from our
systembyallowingtherreceivers tocooperate[3]. LetH=USV H
betheSingularValue
Decomposition[12]ofH,whereU2C rr
andV 2C tt
areunitaryandSisdiagonalwith
non-negative diagonalelements,the rst m of which are strictly positive and denoted by
p
c
1
p
c
m
. Bylettingx
i
=Vu
i
and v
i
=U H
y
i
the channel(1)is diagonalizedas
v
i
=Su
i +z
0
i
(16)
where z 0
i
= U H
z
i
has the same statistics of z
i
. The cooperative throughput is also
maximized by u
i
Gaussian with independent components and itis given by [3]
R coop
= m
X
k=1
[log(c
k )]
+
(17)
where is the solution of P
jSj
k=1
[ 1=c
k ]
+
=A.
The ZF and cooperative throughputs for the composite channel (with short or long-
term power constraints)are immediatelyobtained from(14) and(17) by taking expecta-
tion with respect to H.
Remark: on theuserordering problem. SinceforanyunitarymatrixQthematrix
QHhasthesamesingularvaluesofH,R coop
isobviouslyindependentoftheuserordering
(permutation matrices are unitary). On the contrary, R zf
depends on the choice of the
unorderedactive userset S, andR rki
depend onthe orderedactive userset S(whoserows
are considered in order to perform Gram-Schmidt orthogonalization). In order to stress
this dependence, weintroducethe following notation
R rki max
= max
S R
rki
R zf max
= max
S R
zf
(18)
where in the rst lineS ranges over the ordered user sets with cardinality m, and inthe
second lineS ranges over the unordered user sets with cardinalityjSjm.
Ifrank(H)=m thenR rki max
isachieved by anorderedset of musers, forevery SNR
set S 0
of cardinalityk <m,such that H[S 0
]=G 0
Q 0
. Then, there exists an ordered set of
users S =S 0
[fi
1
;:::;i
m k
g such that H[S] =GQ where jg
i;i j
2
= jg 0
i;i j
2
for i = 1;:::;k
and jg
i;i j
2
>0 for i=k+1;:::;m. Therefore, the RKI strategy applied to H[S 0
]and to
H[S] yields the same throughput.
Also,theuserorderingisirrelevantfortheRKIstrategyforasymptoticallylargeSNR
ifHisfullrow-rank,i.e.,ifm=r. Infact,letfd
1
;:::;d
r
gandfd 0
1
;:::;d 0
r
gbethetwosets
ofordered squareddiagonalelementsofthe matricesGandG 0
intheQRdecompositions
H = GQ and H = G 0
Q 0
, respectively, where is a rr permutation matrix. We
have that
r
Y
i=1 d
i
=jdet(G)j 2
=det (HH H
)=det (HH H
H
)=jdet(G 0
)j 2
= r
Y
i=1 d
0
i
Dene the following arithmetic means
M
a
= 1
r r
X
i=1 1
d
i
; M
0
a
= 1
r r
X
i=1 1
d 0
i
and the geometric mean M
g
=( Q
r
i=1 1=d
i )
1=r
=( Q
r
i=1 1=d
0
i )
1=r
. There existA
0
<1 such
that the equation
r
X
i=1
[ 1=d
i ]
+
=A
0
has solution
0
=A
0
=r+M
a
and the equation
r
X
i=1
[ 1=d 0
i ]
+
=A
0
has solution 0
0
=A
0
=r+M 0
a
. Then, for allA A
0
, the RKI throughputs corresponding
to the original and permuted row orders are given by rlog
A=r+Ma
Mg
and by rlog A=r+M
0
a
Mg ,
respectively, implying that their dierence vanish asA !1.
WithZFbeamforming,foragivenSNRAthemaximumthroughputR zf max
mightbe
achieved by auser subset Sof cardinality strictly less than m=rank(H). However, it is
easytoseefromthepropertiesofthewaterllingpowerallocationin(14)thatthereexists
a nite value A
0
(whichdepends onH) forwhich, forall AA
0 , R
zf max
isachieved by
a subset of cardinality m.
Sincebyconstrainingthereceivers toprocesstheirsignalsindependentlythethrough-
put cannotbeincreased,R coop
upperboundsboth R zf max
andR rki max
. The RKIscheme
yields generally a larger maximal throughput than ZF beamforming, as stated in the
following:
Theorem 2. Forany channel matrix H,R rki max
R zf max
.
Proof. Assume that,afterasuitablerowpermutation,therst krows ofHarelinearly
independent, choose the user subset S=f1;:::;kg. The columns v
i of H
+
S
satisfy
h j
v
i
=Æ
i;j
; j =1;:::;k
Therefore, v H
i
must lie in the orthogonal complement of the subspace V
i
= spanfh j
:
j = 1;:::;k;j 6= ig. Let P
?
i
be the orthogonal projector [12] on V
?
i
. From the above
orthonormalitycondition we get
v H
i
= h
i
P
?
i
h i
P
?
i (h
i
) H
The inverse of the i-th diagonalelement of (H[S]H[S]
H
) 1
=(H +
S )
H
H +
S
is given by
b
i
= 1
jv
i j
2
= jh
i
P
?
i (h
i
) H
j 2
h i
P
?
i (h
i
) H
=jh i
P
?
i j
2
(19)
where we used the fact that orthogonal projectors are idempotents [12]. The rows q i
of
Q inthe QRdecomposition H =GQ are obtained by applying Gram-Schmidt orthogo-
nalizationto the ordered rows h 1
;h 2
;:::;h k
. We obtain
h i
= q
h ie
P
?
i (h
i
) H
q i
+ i 1
X
j=1 h
i
(q j
) H
q j
where e
P
?
i
istheorthogonalprojectorontheorthogonalcomplementof e
V
i
=span fh 1
;:::;h i 1
g.
From the denition of d
i
in(8)and the formulaabove we obtain
d
i
=h i
e
P
?
i (h
i
) H
=jh i
e
P
?
i j
2
(20)
SinceV
i
e
V
i
,then b
i d
i
foralli=1;:::;k. Finally,since bothk andthe userordering
were arbitrary, this implies that R zf max
R rki max
.
Weconclude that R coop
R rki max
R zf max
holdsfor anychannelmatrix. The next
result makes this statement stronger in the limitsfor high and lowSNR.
Theorem 3. Forany channel matrix Hwith full row-rank,
lim
A!1 R
coop
R rki max
=0 (21)
For any channel matrix H,
lim
A!0 R
rki max
R zf max
=1 (22)
Proof. Consider rst (21). Letc
1
;:::;c
r
denote the (non-zero) squared singularvalues
of H and d
1
;:::;d
r
the (non-zero) squared diagonalelements of G in the QR decompo-
sition H=GQ. Dene the followingarithmetic means
M
a
= 1
r r
X
i=1 1
c
i
; f
M
a
= 1
r r
X
i=1 1
d
i
and the geometric mean M
g
= ( Q
r
i=1 1=c
i )
1=r
= ( Q
r
i=1 1=d
i )
1=r
, where the last equality
follows from
r
Y
i=1 c
i
=det(HH H
)=jdet(G)j 2
= r
Y
i=1 d
i
It is immediate tosee that there existA
0
<1 suchthat the equation
r
X
i=1
[ 1=c
i ]
+
=A
0
has solution
0
=A
0
=r+M
a
and the equation
r
X
i=1
[ 1=d
i ]
+
=A
0
has solution e
0
=A
0
=r+ f
M
a
. Then, for all A A
0
, the maximum throughputs can be
written as
R coop
= rlog
A=r+M
a
M
g
R rki
= rlog
A=r+ f
M
a
M
g
(23)
By substituting these expressions in the limit(21) we obtain
lim
A!1 rlog
1+rM
a
=A
1+r f
M
a
=A
=0
In order to show (22), consider rst the case where there is a single row h max
in H
of maximum squared Euclidean norm. We notice both RKI and ZF achieve the MRC
throughput R mrc
=log(1+jh max
j 2
A), by choosing anactiveuser set containing onlythe
user correspondingtothe rowh max
. LetSdenotes anarbitraryuser subsetof cardinality
k for which H[S] has rank k,let
b
i (S)=
1
(H[S]H[S]
H
) 1
i;i
and let R zf
(S) denote the maximum of P
k
i=1
log(1 +b
i (S)a
i
) subject to P
k
i=1 a
i A,
a
i
0. There exists A
1
(S) >0 such that, forall AA
1 (S),
R zf
(S)=log(1+max
i fb
i
(S)gA)
Hence, by denition of h max
, for all A A
1
(S) we have R zf
(S) R mrc
. By considering
all possible subsets S of cardinality k = 1;:::;m we conclude that R zf max
= R mrc
for
0<Amin
S A
1 (S).
Similarly,consider an ordered set S of cardinality m, let d
1
(S);:::;d
m
(S) denote the
squared diagonal elements of G in the QR decomposition H[S] = GQ and let R rki
(S)
denote the maximum of P
m
i=1
log(1 +d
i (S)a
i
) subject to P
m
i=1 a
i
= A. There exists
A
2
(S) >0such that, forall AA
2 (S),
R rki
(S)=log(1+max
i fd
i
(S)gA)
Hence,bydenitionofh max
,forallA A
2
(S)wehaveR rki
(S)R mrc
. Byconsideringall
possiblesuchsubsets Sweconclude that R rki max
=R mrc
for 0<Amin
S A
2
(S). Then,
there exists anA
3
>0 such that for A2[0;A
3
] wehave R rki max
=R zf max
=R mrc
.
In the case where H has more than one row with maximum squared Euclidean norm
we have to distinguish the case where there exists a subset of mutually orthogonal rows
withmaximalnormfromthecasewhereanysubsetofthemaximalnormrowsismutually
non-orthogonal. In the latter case, the above proof stillholds, and the MRC throughput
can be obviously achieved by transmitting to anyone of the users corresponding to the
maximalnormrows. Intheformercase, itisnotdiÆculttoshowthat thereexists A
4
>0
for which for every A 2 [0;A
4
] both the ZF and the RKI throughputs are maximized
by transmitting with equal power to the users corresponding to the subset of mutually
orthogonal rows with maximalnorm. This concludes the proof.
4 Throughput bounds for the t 1 : r GBC
Inthis sectionweconsider Hdeterministicand xedandwend upperandlowerbounds
to the maximum achievable throughput R of the corresponding t 1 : r GBC. These
allowus toestablishthe asymptoticoptimalityof the RKIscheme for highand lowSNR
provided that the channel matrix has full row-rank. For nite and non-vanishing SNR
these bounds are generally diÆcultto evaluate explicitly,however, they will beuseful to
prove the maximum throughputfor the t1:2 GBC (two-users case) inSection 5.
An upperbound on R is obtained by noticing that the capacity region of a general
broadcast channel depends only on the marginal transition probabilities fp(y
k
jx) : k =
1;:::;rg and not on the joint transition probability p(yjx) [33, 26]. In our case, the
marginaltransitionpdfs are given by
p(y
k jx)=
1
e jy
k h
k
xj 2
; k =1;:::;r
Any set of marginaltransitionpdfs
p 0
(y
k jx)=
1
k e
jy
k h
k
xj 2
=
k
; k =1;:::;r
with
k
1 yields a GBC capacity region containing that of the original GBC, since
any user k in the new channel can emulate articially the corresponding output of the
originalchannelby addingindependent Gaussiannoisewith variance1
k
. This implies
that the channelsinthe family(1)for given Hand with zN
C (0;
z
), where
z
isany
non-negativedeniteHermitianmatrixwhosediagonalelementsarenotlargerthan1(we
refer to this constraint as the sub-unit diagonal constraint), have all broadcast capacity
region containing the region of the original GBC (and therefore throughput not smaller
thanR ). Inorder totighten thecooperativethroughputbound,wecan choose theworst-
case (cooperative throughput-wise) channelin this family. By usingthe generalcapacity
formulafor the cooperativethroughput [3], we have:
Lemma 1. For any channel matrix H,
R max
x min
z log
det H
x H
H
+
z
det
z
(24)
where minimizationis over allnoise covariances
z
satisfying the sub-unitdiagonalcon-
straintand maximizationis overallinput covariancematrices
x
satisfyingtrace(
x )=
A.
From Theorem 3and Lemma1 we can prove the following
Theorem 4. If Hhas full row-rank, then
lim
A!1
(R R rki max
)=0 (25)
and
lim
A!0 R =R
rki max
=1 (26)
Proof. ItisclearthatR rki max
andR coop
arealowerandanupperboundonR . Therst
statement follows directly from the rst part of Theorem 3, since R rki max
R R
coop
and lim
A!1 (R
coop
R rki max
)=0imply the statement.
Inorder toprovethe second statement,let bethe rr permutationmatrix which
sorts the rows of H such that jh 1
j jh r
j, and consider the QR decomposition
H=GQ. We apply Lemma 1by choosing as noise covariance
z
=GD 2
G H
, where
D =diag(jh 1
j;:::;jh r
j). By construction,
z
is positivedenite (recall that Hhas rank
r)and satisesthe sub-unit diagonalconstraint,in fact,
[
z ]
k;k
= k
X
jg
k;j j
2
jh j
j 2
k
X
jg
k;j j
2
jh k
j 2
=1
With this choice, the RHS in(24) becomes
log
det GQ
x Q
H
G H
+GD 2
G H
det GD 2
G H
=log
det Q
x Q
H
+D 2
detD 2
From Hadamard inequality [26] we obtain the maximizing signal covariance in the form
x
=Q H
diag(a
1
;:::;a
r
)Q, whichyields the bound
R r
X
k=1
[log(a
k )]
+
(27)
where is the solution of P
r
k=1
[ 1=jh
k j
2
]
+
=A.
Ifjh 1
j>jh 2
j,thenthereexists avalue0<A
1
<1suchthat,forallAA
1
,theRHS
in (27) is equal tolog(1+jh 1
j 2
A)=R mrc
. This is clearly achievable by RKI and by ZF,
thereforeforA2[0;A
1
]theRKI(andZF)strategyisoptimal(notonlyasymptoticallyfor
A !0). If there exist >1rows with maximal 2-normjh max
j, then there exists a value
0<A
2
<1suchthat, forallA A
2
,the RHSin(27) is equaltolog(1+jh max
j 2
A=).
In this case,
lim
A!0
log(1+jh max
j 2
A=)
log(1+jh max
j 2
A)
=1
and the statementof Theorem 4 stillholds (but only inthe limitfor vanishing A).
Remark: downlinkstrategies. Theorem4showsaninterestingfeatureofthet1:r
GBC and of the RKI strategy, which might have a relevant impact on the design of the
downlinkofwirelesscommunicationsystems. Ifthebase-stationisstronglypowerlimited,
then thethroughput-maximizingstrategyconsistsofMRCbeamformingtothebestuser,
whichis the same optimalstrategy for the standard degraded GBC(t =1). Inthis case,
the transmit antenna array is used to enhance the received SNR of the best user but
does not expand the useful dimensions for transmission. Practical downlink protocols
for high-rate packet communications are currently proposed and implemented according
to this principle: only one user in each time-slotis served according to a channel-driven
schedulingallocatingthechanneltotheuserenjoyingtheinstantaneoushighestindividual
capacity [34, 35].
2
Onthe contrary,ifthe base-stationcan transmitatlargepower, thesame throughput
of a single-user multiple-antenna system can be approached even if the receivers cannot
cooperate. In particular, by lettingthe numberof served users per time-slotequaltothe
numberof transmitantennas, under mildconditionson the channel matrix statisticsthe
slope of the throughput as a function of SNR in dB is equal to t. Hence, in the GBC
setting, the \capacity boost" typical of multiple-antenna systems depends strongly on
the available transmit power. Notice also that the same throughput slopes for low and
2
Inpractice,theschedulermustmaximizethecellthroughputsubjecttosomefairnessconstraint[34,
35], thereforethechannelallocationrulediersfrom thesimple\bestuser" rule.
high SNR are obtained by the conventional ZF beamforming,although this is generally
asymptoticallysuboptimal for high SNR.
Assume that H has rank m r and, after a suitable row permutation , can be
partitioned as
H=
H
1
H
2
(28)
where H
1 2C
mt
has rank m and H
2 2C
(r m)t
. Any row of H
2
can be expressed as a
linear combinationof rows of H
1
, i.e., we can write
H
2
=BH
1
where B =H
2 H
+
1
. The noise vector is also partitioned as z =[z T
1
;z T
2 ]
T
, where
z1 and
z2
are the mm upper left and (r m)(r m) lower right diagonal blocks of
z
(both
z1
and
z2
must satisfy the sub-unit diagonal constraint). Then, we have the
following:
Lemma 2. If
z2
B
z1 B
H
is non-negative denite, then R R coop
1
, where R coop
1 is
the cooperativethroughput of the tm channely
1
=H
1 x+z
1 .
Proof. Dene the auxiliarychannel with input y
1
and output
y
2
=By
1 +
where N
C (0;
), independent of z
1
and with
=
z2
B
z1 B
H
, which by
assumption is a valid covariance matrix. Among all channels with assigned marginal
transitionpdfs, thereexists the \cascade" channel withjointtransitionpdfp(y
1
;y
2 jx)=
p 0
(y
2 jy
1 )p(y
1
jx), where
p(y
1
jx) = N
C (H
1 x;
z1 )
p 0
(y
2 jy
1
) = N
C (By
1
;
)
For the cascade channel,x!y
1
!y
2
is aMarkov chain and wehave
I(x;y
1
;y
2
)=I(x;y
1
)+I(x;y
2 jy
1
)=I(x;y
1 )
By maximizingI(x;y
1
) with respect to the input distribution (subject tothe inputcon-
straint),we get by denition max
x:trace(x)=A I(x;y
1 )=R
coop
1
.
Lemma 2 is actually \included" in Lemma 1, since R coop
1
is achieved by a particular
choice of
z
in the RHS of (24). However, it puts in evidence the interesting fact that,
under certainconditions onH, the RKI strategyis asymptoticallyoptimalfor high SNR
even if the channel has rankm <r. In particular, we havethe following
Corollary 2. Let H have rank m r and, after a suitable row permutation, assume
that
H=
H
1
BH
1
(29)
where H
1 2 C
mt
has rank m and kBk 2
1. Then, the RKI scheme is asymptotically
optimal forthe t1:r GBC inthe limitfor large SNR,i.e., lim
A!1
(R R rki max
)=0.
Proof. We apply Lemma 2 with the choice
z1
= I
m
and
z2
= I
r m
. If kBk
2 1,
then I
r m
BB H
is non-negative denite and, by Lemma 2, R R coop
1
, the coopera-
tive throughput of the tm subchannel dened by H
1
. Let R rki max
1
be the maximum
RKI throughput for this subchannel. Since H
1
is full row-rank, by Theorem 3 we have
that lim
A!1 (R
coop
1
R rki max
1
) = 0. Since R rki max
1
R
rki max
R , this implies that
lim
A!1
(R R rki max
)=0, asdesired.
The decomposition (29) (if it exists) is essentially unique, as stated by the following
result in linearalgebra (new up to the authors' knowledge):
Lemma 3. Let H2C rt
have rankm and assume that, aftera suitable rowpermuta-
tion, it can be decomposed into the submatrices H
1 2 C
mt
of rank m and H
2
= BH
1
as in(29), with kBk
2
1. Then, this decomposition isunique , inthe sense that forany
H 0
1 2C
mt
and H 0
2
obtained by exchanging some rows of H
1
with some rows of H
2 such
that rank (H 0
1
)=m, we havekH 0
2 (H
0
1 )
+
k
2 1.
Proof. See Appendix C.
Establishing if a given matrix H admits the decomposition (29) is a problem of
independent interest which can be formulated as follows: given a set of vectors S =
fv
1
;:::;v
r g in C
t
, spanning an m minfr;tg dimensional subspace, nd (if it exists)
a set S 0
of m linearly independent vectors such that all other vectors in S can be writ-
ten as linear combinationsof the vectors in S 0
and such that the matrix of combination
coeÆcients has 2-normnot larger than 1.
The following simple example shows that there exist matrices for which decomposi-
tion (29) is not possible. For these channels, we cannot claim that the RKI strategy is
asymptoticallyoptimal.
Example. LetH2C 3t
ofrank2,andassumethath 3
=
1 h
1
+
2 h
2
. Ifj
1 j
2
+j
2 j
2
>1
and j
2 j
2
<j
1 j
2
<1+j
2 j
2
orj
1 j
2
<j
2 j
2
<1+j
1 j
2
,itisnot possibletoexpressanyof
the rows h 1
, h 2
and h 3
as a linear combination of the other two with coeÆcients 0
and
00
suchthat j 0
j 2
+j 00
j 2
1. The set of coeÆcients
1
;
2
satisfyingthe abovecondition
is clearly non-empty,so, such matrices exist.
Modied RKIstrategy. In ordertotighten theRKI throughputlowerbound wecan
consider the following modied RKI strategy. Let H = GQ and construct the trans-
mitted signal as x = Q H
Ru where R is a mm upper triangular matrix satisfying
trace(RR H
)Aand whereE[u
i u
H
i ]=I
m
. Thisyieldsthe setofm interference channels
y
k;i
=w
k;k u
k;i +
X
j<k w
k;j u
j;i +
X
j>k w
k;j u
j;i +z
k;i
; k =1;:::;m (30)
where W denotes the upper mm block of GR, and where no information is sent to
users k=m+1;:::;r. Theencoderconsiders theinterference signal P
j<k w
k;j u
j;i
caused
by users j <k as known non-causally and the k-thdecoder treats the interference signal
P
j>k w
k;j u
j;i
caused by users j >k as additionalnoise. Byapplying a coding for known
interference strategy (e.g., the lattice precoding scheme of [17]) and by using minimum
Euclidean distance decoding at each k-th receiver, from the results of Appendix B and
the resultsof [36] it is not hard toshow that the followingthroughput is achievable
R mod rki
= m
X
k=1
log 1+
jw
k;k j
2
1+ P
j>k jw
k;j j
2
!
(31)
This can be further maximized over all matrices R satisfying the trace constraint and
overallordered user subsets Sofsize m. The maximum throughputofthe modied RKI
strategy provides alowerbound toR generallytighterthan the basicRKI strategy,since
the former reduces to the latter by constraining R tobe diagonal.
5 The optimal throughput of the t 1 : 2 GBC
The simplestnon-trivial(i.e., non-degraded)GBCwithmultipletransmitantennasisthe
two-user case. Forthis channel,we havethe following closed-formresult:
Theorem 5. The maximum achievable throughput of the t1:2GBC is given by
R = (
log(1+jh 1
j 2
A) A A
1
log
(Adet (HH H
)+trace(HH H
)) 2
4jh 2
(h 1
) H
j 2
4det (HH H
)
A>A
1
(32)
where withoutloss of generality we assumejh 1
j jh 2
jand where
A
1
= jh
1
j 2
jh 2
j 2
det(HH H
)
Proof. Assumejh 1
jjh 2
j. Thecaseofrank(H)=1istrivial,sinceinthiscasethetwo
rows of Harelinearlydependent, then,the t1:2GBCreduces toastandarddegraded
GBC with input x=h 1
xand outputs
y
1
=x+z
1
; y
2
=(jh 2
j=jh 1
j)x+z
2
The throughput is clearly maximized by transmitting to the best user only [27], i.e., to
1 2
since in this case A
1
=+1and onlythe rst linein(32) is relevant, therefore,Theorem
5 holds inthe rank1 case.
Next,weconsiderthecaseofHofrank2. WeshalluseLemma1tondanupperbound
andthemodiedRKIstrategytondalowerbound,andshowthattheseboundscoincide
with (32).
Cooperative throughput upperbound. A looser version of Lemma 1 yields the upper-
bound
R min
z 2U
max
x 2A
log
det H
x H
H
+
z
det
z
(33)
where A is the set of all tt non-negative denite covariance matrices with trace A,
and U is the set of all positive denite covariance matrices satisfying the unit-diagonal
constraint (diagonal elements strictly equal to 1). Consider rst the maximization of
mutualinformationwithrespect to
x
foragiven
z
. Byletting
z
=U
z U
H
,with U
unitary and
z
diagonal, weobtain the equivalent problem
max
x2A
logdet H
z
x H
H
z +I
where H
z
= 1=2
z U
H
H. More explicitly,since
z
=
1
1
we obtain
U= 1
p
2
1 1
;
z
=
1+r 0
0 1 r
where we letr =jj and =exp ( j\).
Let
1 and
2
denotethe eigenvalues ofH
z H
H
z
. Then,the maximizationwithrespect
to
x
2A yieldsthe function of r; and A
f(r;;A)= r
X
i=1
[log(
i )]
+
(34)
where solvesthe equation
2
X
i=1
1
i
+
=A (35)
Explicitly,we have
H
z H
H
z
= 1
2
"
h +
+2
1+r
h +j2!
p
1 r 2
h j2!
p
1 r 2
h +
2
1 r
#
where we dene h +
= jh 1
j 2
+jh 2
j 2
, h = jh 1
j 2
jh 2
j 2
and h 2
(h 1
) H
= +j!. The
eigenvalues
1;2
are given by
1;2
= 1
T p
T 2
4D
withT =trace(H
z H
H
z
)andD=det (H
z H
H
z
). ItisimmediatetocheckthatDisindepen-
dent of , while T is adecreasing function of . Weconclude that minimizingcapacity
isgiven by
?
=exp ( j\h 2
(h 1
) H
),sothat =jh 2
(h 1
) H
jismaximum. Withthischoice,
we have
T = h
+
2rjh 2
(h 1
) H
j
1 r 2
; D= jh
1
j 2
jh 2
j 2
jh 2
(h 1
) H
j 2
1 r 2
In order toobtain the eigenvalues ina convenient form,it isuseful torepresent the rows
h 1
and h 2
inanorthonormalbasis. Applying Gram-Schmidtorthogonalizationtoh 1
and
h 2
(in the order) we obtain h 1
=g
1;1 q
1
, h 2
=g
2;1 q
1
+g
2;2 q
2
, and where q 1
and q 2
(the
rows of Q) are orthonormal. The explicit expression of the eigenvalues is nowgiven by
1;2
=
1
2(1 r 2
)
jg
1;1 j
2
+jg
2;1 j
2
+jg
2;2 j
2
2r
(jg
1;1 j
2
jg
2;1 j
2
jg
2;2 j
2
) 2
4r(jg
1;1 j
2
+jg
2;1 j
2
+jg
2;2 j
2
)
+ 4r 2
jg
1;1 j
2
(jg
2;1 j
2
+jg
2;2 j
2
)+4 2
1=2 i
(36)
Next,wehavetominimizethemaximum mutualinformationdened by (34)and by(35)
with respect to the noise correlation parameter r 2[0;1) (recall that minimizationwith
respect to is already achieved). We partition the SNR range [0;1) into two intervals,
called infollowing thelow-SNR and the high-SNRregions, anddened by therange ofA
for which 1=
2
or >1=
2
, respectively (notice that
1
2
holds for any channel
matrixand valueofr). Then,weconsider separatelytheminimizationoftheupperbound
on the two regions.
Low SNR region. Letr =r
?
=jg
2;1
=g
1;1
j==jg
1;1 j
2
. Then, weobtain
1
=jg
1;1 j
2
;
2
= jg
1;1 g
2;2 j
2
jg
1;1 j
2
jg
2;1 j
2
For
0AA
1
= jg
1;1 j
2
jg
2;1 j
2
jg
2;2 j
2
jg
1;1 g
2;2 j
2
= jh
1
j 2
jh 2
j 2
det(HH H
)
(37)
the resulting mutual information is log(1+jg
1;1 j
2
A), which coincides with the rst line
of (32). This is achievable under the broadcast channel constraint (non-cooperative re-
ceivers) by MRC beamformingto user 1 only (the best user), and therefore it is clearly
a tight upper bound. For jh 1
j 2
= jh 2
j 2
we have A
1
= 0 and under this condition the
low-SNR case isirrelevant.
High SNR region. Inthis case, (34) and (35) become
f(r;
?
;A) = log(
1
)+log(
2 )
=
1
2
A+ 1
1 +
1
2
(38)
By substituting weobtain
f(r;
?
;A) = 2log
p
1
2
A+
1 +
2
1
2
2log2
= 2log
p
D(A+T=D)
2log2 (39)
Trace and determinant T and D are writtenin termsof the g
i;j 's as
T = jg
1;1 j
2
+jg
2;1 j
2
+jg
2;2 j
2
2r
1 r 2
; D= jg
1;1 g
2;2 j
2
1 r 2
Bydirectsubstitutionoftheaboveexpressionsinto(39)andbydierentiatingwithrespect
to r we obtaina stationary pointin
r
?
=
2jg
1;1 g
2;1 j
Ajg
1;1 g
2;2 j
2
+jg
1;1 j
2
+jg
2;1 j
2
+jg
2;2 j
2
=
2jh 2
(h 1
) H
j
Adet (HH H
)+trace(HH H
)
Noticethatr
?
isadecreasingfunctionofA,andlim
A!1 r
?
=0. Therefore,theworst-case
noise in the large-SNR case is white. Also, for A = A
1
we obtain r
?
= jg
2;1
=g
1;1 j 1.
Then, the solutionr
?
iscompatiblewith the positive denitenessconstrainton
z
forall
nite A. Finally, we observe that the worst-case noise correlation r
?
is continuous in A
for all A 0, since the limitsof r
?
for A !A
1
from the left and from the right are the
same. Eventually, by substituting r
?
found above into (38) we obtain the second lineof
(32).
ModiedRKIthroughputlowerbound. ThethroughputachievablebythemodiedRKI
strategy is given by
R mod rki
=log
1+ jg
1;1 r
1;1 j
2
a
1
1+jg
1;1 r
1;2 j
2
a
2
+log 1+jg
2;1 r
1;2 +g
2;2 r
2;2 j
2
a
2
(40)
where we let
R=
p
a
1 r
1;1 p
a
2 r
1;2
0
p
a
2 r
2;2
subject to the constraint trace(RR H
)=A, whichis writtenexplicitlyas
jr
1;1 j
2
a
1 +(jr
1;2 j
2
+jr
2;2 j
2
)a
2
=A
This must be maximized with respect to a
1
;a
2
and the coeÆcients r
1;1
;r
1;2
;r
2;2 . To
this purpose, we reparameterize the problem by letting b = jg
2;1
=g
2;2
j, z = jr
1;2
=r
2;2 j,
q =z 2
=(1+z 2
) and p = (bz+1) 2
=(1+z 2
), X
1
= jr
1;1 j
2
a
1
and X
2
=(jr
1;2 j
2
+jr
2;2 j
2
)a
2 .
Then, we obtain
R mod rki
=log
1+ jg
1;1 j
2
X
1
1+jg
1;1 j
2
qX
2
+log(1+jg
2;2 j
2
pX
2
) (41)
with the constraint X
1 +X
2
= A. For z =0, the modied RKI strategy reduces to the
standardRKIstrategyandforX
1
=AandX
2
=0itachieves(32)inthelow-SNRregion.
Therefore, weshall consider onlythe high-SNRregion AA
1 .
By dividing all elements of G by jg
1;1
j and by replacing the input constraint A by
a
=Ajg
1;1 j
2
weobtain the equivalent problem
R mod rki
=log
1+ x
1
1+qx
2
+log(1+px
2
) (42)
where x
1 +x
2
= a and where, by letting z = p
q=(1 q) and b = p
= with =
jg
2;1
=g
1;1 j
2
and =jg
2;2
=g
1;1 j
2
, we have the relation
p=
p
q+ p
(1 q)
2
The high-SNR condition AA
1
translates intothe condition a (1 )=.
For any xed q 2 [0;1], we let x
1
= a x and x
2
= x in (42) and maximize the
resultingexpression forx2[0;a]. Byletting
@
@x R
mod rki
=0and makingthe substitution
y =1+qx, we obtainthe solution
y= s
(1+aq)(p q)
p(1 q)
(43)
which isvalid if pq and y1. The rst condition yieldsthe inequality
p
q+ p
(1 q)
2
q
which implies
0qq
max
=
(1 p
) 2
+ 1
The second condition yields the inequality
p
q+ p
(1 q)
2
1+aq
1+a
By lettingq =z 2
=(1+z 2
)this isturned intothe secondorder inequality
( 1)z 2
+2 p
z+ 1=(1+a)0
which impliesz 2[z
1
;z
2
], where
z
1;2
= p
q
1
1+a
1
Fora(1 )=itiseasytocheck that (1 )=(1+a)0,thereforethe above
solution always exists in the high-SNR region. It is easy tocheck that
0q
1
= z
2
1
1+z 2
1 q
2
= z
2
2
1+z 2
2 q
max
Therefore, the solution(43)isvalidintheintervalq2[q
1
;q
2
]andthe maximumthrough-
put canbeobtained byrst substituting(43)into(42)and thenmaximizingwithrespect
to q2[q
1
;q
2
]. After substitution, the function tobe maximizedis
2log
"
1
q q
( p
q+ p
(1 q)) 2
(1+aq) r
( p
q+ p
(1 q)) 2
q
(1 q)
!#
(44)
Bysubstitutingagain q=z 2
=(1+z 2
)intothe aboveexpression andlettingthe derivative
with respecttoz equal tozerowe obtaina5-th orderequation,whoserootscan be given
(quite fortuitously!) inclosed form asz
1
;z
2
given above, z
3;4
= p
= and
z
5
=
2 p
(a+1)+1
It can bechecked that z
5 2[z
1
;z
2
], and therefore itis the sought maximum.
Finally, by substituting the resulting
q
?
= z
2
5
1+z 2
5
=
4
((a+1)+1 ) 2
+4
into(44), we obtainthe maximumthroughput as
R mod rki
=log
(a+1++) 2
4
4
(45)
which coincides with the second lineof (32). This concludes the proof.
Example: average throughput inRayleighfading. WeuseTheorem5tocompute
the maximum average throughput
R of the composite channel when H has i.i.d. entries
N
C
(0;1)(independent Rayleighfading). Subjecttothe short-termconstraint,wehave
simply
R (A) =E
H
[R (A)] where R (A) is given by (32) and where we put in evidence its
dependence on the input constraint A. The maximum average throughput subject to a
long-termconstraint isobtained by solving
max E
H [R (a)]
subject to E
H
[a]=A; a 0
(46)
ByusingthestandardLagrange-Kuhn-Tuckertechnique[32],aftersomealgebraweobtain
the optimaltransmitpowerallocationinthe form
a(H)= 8
<
:
[ 1=jh 1
j 2
]
+
for jh
1
j 4
jh 2
(h 1
) H
j 2
jh 1
j 2
det (HH H
)
+ q
2
+ 4jh
2
(h 1
) H
j 2
H 2
trace (HH H
)
H
otherwise
(47)
where satisesE
H
[a(H)]=A. The condition jh
1
j 4
jh 2
(h 1
) H
j 2
jh 1
j 2
det (HH H
)
isequivalenttoa(H)
A
1
, whereA
1
isgiven inTheorem5. Hence,by substituting(47) inthe placeofAin(32),
we obtainexplicitlythe optimalthroughputsubject tothe long-termpowerconstraintas
R (A) =E
H [f
(H)] where
f
(H)=
8
<
:
[log (jh 1
j 2
)]
+
for jh
1
j 4
jh 2
(h 1
) H
j 2
jh 1
j 2
det (HH H
)
log h
det (HH H
)
+ q
2
+ 4jh
2
(h 1
) H
j 2
det(HH H
) 2
i
log2 otherwise
(48)
Figs. 1 and 2 show
R in the case t = r =2 for the short and the long-termconstraints,
respectively, vs. E
b
=N
0
. Forthe sake of comparison, we show alsothe 22 cooperative
and ZFthroughputs.
In this work, E
b
=N
0
forthe t1:r GBC isdened as[37]
E
b
N
0
= tA
R
(49)
the factor t in the numerator of (49) is introduced totake intoaccount that, undermild
conditions onH, the averagereceived energy perchannel use increases linearly with t for
MRCbeamformingtoanygivenuser (e.g.,ifHhasi.i.d. elementswithunit secondorder
moment the individual user channels have average gain t). For the cooperative system,
since tr and rt channelsyield the same throughput [3], we adopt the denition
E
b
N
0
=
maxfr;tgA
R
(50)
For the short-term constraint, there exists aminimum (E
b
=N
0 )
min
>0 below which
R is
zero.
3
Thiscan be calculatedby lettingA #0 in(49)and in(50). Forthe 21:2 GBC
(basic RKI, ZF and modied RKIschemes) we obtain
E
b
N
0
min
=
2log2
E[jh 1
j 2
]
=
2log2
(11=4)
= 2:97dB
where weused the fact that jh 1
j 2
isdistributed asthe maximum of two independentand
identically distributed central Chi-squared random variables with 4 degrees of freedom.
For the 22cooperative system we have
E
b
N
0
min
=
2log2
E[c
1 ]
=
2log2
(7=2)
= 4:02dB
wherewe usedthe factthatc
1
isthe maximumeigenvalueof the22Wishart matrix[3]
HH H
.
From Figs.1 and 2 it isclearly visible that the basic RKI scheme is optimalboth for
E
b
=N
0
#(E
b
=N
0 )
min
andforE
b
=N
0
!1. ForsmallSNR,itis(asymptotically)equivalent
3
For the long-term constraint (E
b
=N
0 )
min
= 0 since jh 1
j 2
has a distribution with unbounded sup-
0 2 4 6 8 10
-8 -6 -4 -2 0 2 4 6 8 10
R (bit/s/Hz)
E b /N 0 (dB)
Independent Rayleigh fading, short-term constraint Mod-RKI (optimal GBC)
RKI Zero-Forcing Cooperative
Figure1: Throughputof the 21:2 GBCwith independent Rayleighfadingand short-
term input constraint.
toZFand both strategiesreducetosimpleMRCbeamformingtothebest user. Forlarge
SNR, it is (asymptotically) equivalent to the cooperative single-user multiple-antenna
capacity. This is a consequence of the fact that for independent Rayleigh fading the
channel matrix H has full-rank almost surely, therefore Theorem 3 applies to almost all
realizations of the channel.
6 Performance of the basic RKI strategy with Rayleigh
fading
In this section we focus on the basic RKI scheme when the channel matrix has i.i.d.
entries N
C
(0;1),and weconsiderthe throughputofthecompositechannelsubjecttoa
long-termpowerconstraint. Weconsider boththenitedimensionalandthelarge-system
limit,i.e., whenr;t !1 whilethe ratior=tusers/transmitantennaconvergesto agiven
constant (referred to in the following as antenna loading). Moreover, we assume that
no eort is made to optimize the user ordering. In other words, the receiver works with
the natural ordering of the rows of H and considers the coeÆcients fd
i
: i = 1;:::;mg
corresponding to the QRdecomposition H=GQ.
It interesting to notice that since the composite channel is symmetric with respect
to any user by time-sharing with uniform probability over all possible user subsets and
0 2 4 6 8 10
-8 -6 -4 -2 0 2 4 6 8 10
R (bit/s/Hz)
E b /N 0 (dB)
Independent Rayleigh fading, long-term constraint Mod-RKI (optimal GBC)
RKI Zero-Forcing Cooperative
Figure 2: Throughput of the 21: 2GBC with independent Rayleigh fadingand long-
term input constraint.
orderings, every user inthesystem achieves the sameaverage per-userrate
=
R =r with
no lossof optimality inthe total throughput.
We make use of the followingresults [39, 40, 41, 42]:
Lemma 4. LetH2C rt
have i.i.d. entries N
C
(0;1),and letg
i;i
bethei-th diagonal
elementof Ginthe QRdecompositionH=GQ. Then, the randomvariablesd
i
=jg
i;i j
2
arestatisticallyindependentandd
i
2
2(t i+1)
,where 2
2k
denotesthecentralChi-squared
distribution with 2k degrees of freedom, whosepdf isf(z)=z k 1
e z
=(k 1)!.
Lemma 5. Let = i=r 2 [0;1] denote the normalized user index. If H 2 C rt
(for
r=t = ) has i.i.d. circularly-symmetric elements with mean 0, variance 1 and bounded
fourth moment,then
lim
r!1 1
t d
i
=[1 ]
+
(51)
with probability 1.
Inthecaseofniter;t,Corollary1yieldssolutionof P
m
i=1 E
h
[ 1
di ]
+ i
=A,where,
from Lemma4, we have
m
X
i=1 E
1
d
i
+
= m
X
i=1 Z
1
1=
1
z
z t i
e z
(t i)!
dz
= m
X
i=1 1
(t i)!
[ (t i+1;1=) (t i;1=)] (52)
where (n;x)
= R
1
x z
n 1
e z
dz, and where (0;x) = E
i
(1;x) and, for integer n 1,
(n;x)=(n 1)!e x
P
n 1
j=0 x
j
=j!.
The resultingRKI average throughputis given by
R rki
= m
X
i=1 E
[log(d
i )]
+
= m
X
i=1 1
t i+1
Z
1
1
log(z) z
t i
e z=
(t i)!
dz
= m
X
i=1 J
t i+1
() (53)
where we let[43]
J
k (a)
= a k
Z
1
1
log(z) z
k 1
e z=a
(k 1)!
dz
= E
i
(1;1=a)+ k 1
X
`=1 1
` e
1=a
` 1
X
j=0 a
j
j!
(54)
WiththesameassumptionsonthestatisticsofH,theelementsb
i
intheZFthroughput
formula (14) are identically distributed 2
2(t k+1)
, for any subset of active users of
cardinality k m. Here, the cardinality k of the active user set can be chosen in order
to optimizefurther the ZF throughput. By replicating the calculation done above inthe
case ofk activeusers weobtaintheZF throughputsubjecttothe long-termconstraintas
R zf
= max
k=1;:::;m kJ
t k+1 (
k
) (55)
where
k
is the solution of the waterllingequation
k
(t k)!
[ (t k+1;1=) (t k;1=)]=A
analogous to (52).
The throughput of the cooperative system can be obtained from the pdf of a single
eigenvalue of the mm Wishart matrix
W=
HH H
for rt
H
(56)
