New Results on Multiple-Input Multiple-Output Broadcast Channels with Confidential Messages

0.28 MB·English

by  Ruoheng Liu

1
New Results on Multiple-Input Multiple-Output
Broadcast Channels with Confidential Messages
Ruoheng Liu, Tie Liu, H. Vincent Poor, and Shlomo Shamai (Shitz)
Abstract—This paper presents two new results on multiple- needing to be kept asymptotically perfectly secret from the
input multiple-output (MIMO) Gaussian broadcast channels other, was characterized in [10].
with confidential messages. First, the problem of the MIMO
This paper presents two new results on MIMO Gaussian
1 Gaussianwiretap channelisrevisited.A matrixcharacterization broadcast channels with confidential messages1:
1 ofthecapacity-equivocationregionisprovided,whichextendsthe
0 previous result on the secrecy capacity of the MIMO Gaussian 1) The problem of the MIMO Gaussian wiretap channel
2 wiretapchanneltothegeneral,possiblyimperfectsecrecysetting. is revisited. A matrix characterization of the capacity-
n Ntweoxtr,etcheeivperrsobalnemd tohfreMeIiMndOepGenaduesnsitamnebsrsoaagdesc:asat ccohmanmnoenlsmweitsh- equivocationregionisprovided,whichextendstheresult
a sage intended for both receivers, and two confidential messages of [6] on the secrecy capacity of the MIMO Gaussian
J each intended for one of the receivers but needing to be kept wiretapchanneltothegeneral,possiblyimperfectsecrecy

1 asymptotically perfectly secret from the other, is considered. setting.
1 A precise characterization of the capacity region is provided, 2) The problem of MIMO Gaussian broadcast channels
generalizing the previous results which considered only two out
with two receivers and three independent messages, a
] of three possible messages.
T common message intended for both receivers, and two
I Index Terms—Multiple-input multiple-output (MIMO) com- mutually confidential messages each intended for one
. munication, wiretap channel, capacity-equivocation region,
s of the receivers but needing to be kept asymptotically
c broadcast channel, confidential message
perfectly secret from the other, is considered. A precise
[
characterizationofthecapacityregionisprovided,gener-

1
alizing the results of [9] and [10] which considered only
v I. INTRODUCTION
two out of three possible messages.
7
Information-theoretic security has been a very active area
0 Notation. Vectors and matrices are written in bold letters.
0 of research recently. (See [1] and [2] for overviews of re- All vectors by default are column vectors. The identity ma-
2 cent progress in this field.) In particular, significant progress trices are denoted by I, where a subscript may be used to
. has been made in understanding the fundamental limits of
1 indicate the size of the matrix to avoid possible confusion.
0 multiple-inputmultiple-output(MIMO)secretcommunication. The transpose of a matrix A is denoted by A⊺, and the trace
1 Morespecifically,thesecrecycapacityoftheMIMOGaussian of a square matrix A is denoted by Tr(A). Finally, we write
1 wiretap channel was characterized in [3]–[7]. The works [8] A(cid:22)B(or,equivalently,B(cid:23)A)wheneverB−Aispositive
:
v and [9] consideredthe problemof MIMO Gaussian broadcast semidefinite.
i channels with two confidential messages, each intended for
X
one receiver but needing to be kept asymptotically perfectly
ar secret from the other, and provided a precise characterization II. THECAPACITY-EQUIVOCATION REGION OF THE
ofthecapacityregion.ThecapacityregionoftheMIMOGaus- MIMO GAUSSIAN WIRETAP CHANNEL
sianbroadcastchannelwithtworeceiversandtwoindependent A. Channel Model
messages, a commonmessageintendedforbothreceiversand
Consider a MIMO Gaussian broadcast channel with two
a confidential message intended for one of the receivers but
receivers, one of which is a legitimate receiver and the other
is an eavesdropper. The received signals at time index m are
This research was supported by the National Science Foundation under
given by
Grant CNS-09-05398, CCF-08-45848 and CCF-09-16867, by the Air Force
OfficeofScientificResearchunderGrantFA9550-08-1-0480,bytheEuropean Y[m] = HrX[m]+Wr[m] (1)
CommissionintheframeworkoftheFP7NetworkofExcellenceinWireless Z[m] = H X[m]+W [m]
e e
Communications NEWCOM++, and by the Israel Science Foundation. The
material in this paper was presented in part at the IEEE International whereH andH are(real)channelmatricesatthelegitimate
r e
SymposiumonInformation Theory(ISIT),Austin,TX,June2010. receiver and the eavesdropper respectively, and {W [m]}
RuohengLiuiswithAlcatel-Lucent, MurrayHill,NJ07974,USA(email: r m
[email protected]). and {We[m]}m are independent and identically distributed
Tie Liu is with the Department of Electrical and Computer Engineer- (i.i.d.) additive vector Gaussian noise processes with zero
ing, Texas A&M University, College Station, TX 77843, USA (e-mail:
means and identity covariance matrices.
[email protected]).
H.VincentPooriswiththeDepartmentofElectricalEngineering,Princeton
University, Princeton, NJ08544,USA(e-mail:[email protected]). 1Themainresults ofthispaperwereinitially postedonthearXivwebsite
Shlomo Shamai (Shitz) is with the Department of Electrical Engineering, in January 2010 [11] and were subsequently reported at the 2010 IEEE
Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel International Symposium on Information Theory [12], [13]. Similar results
(e-mail: [email protected]). wereindependently reportedbyEkremandUlukusin[14]and[15].
2
n
Y Legitimate ^
W
receiver
n Channel
X
W Transmitter
p(y, z| x)
n
Z Eavesdropper Re –ε ≤ 1n_ H(W|Zn)
(a) Rate-equivocation setting
Yn Legitimate ^ ^
(W,W)
receiver s p
Xn Channel
(Ws,Wp) Transmitter
p(y, z| x)
Zn Eavesdropper 1n_I(Ws;Zn)→0
(b) Simultaneous private-confidential communication
Fig.1. Wiretap channel.
The transmitter has a single message W, which is uni- the equivocation R is set to equal the communication rate
e
formly distributed over {1,...,2nR} where R is the rate of R. In this case, the secrecy constraint (2) can be equivalently
communication. The goal of communication is to deliver W written as
1
reliablytothelegitimatereceiverwhilekeepingitinformation- I(W;Zn)≤ǫ (4)
n
theoretically secure from the eavesdropper. Following the
classical work [16], [17], for every ǫ>0 it is required that i.e., message W needs to be asymptotically perfectly secure
from the eavesdropper. Under the asymptotic perfect secrecy
1
H(W|Zn)≥R −ǫ (2) constraint (2), the maximum rate of communication is called
e
n
thesecrecycapacity.FortheMIMOGaussianwiretapchannel
forsufficientlylargen,wherenistheblocklengthofcommu-
(1), a matrix characterization of the secrecy capacity was
nication,Zn :=(Z[1],...,Z[n]),andRe representstheprede- obtainedin[3]–[5]underanaveragetotalpowerconstraintand
termined level of security of message W at the eavesdropper
in [6] and [7] under a more general matrix power constraint.
known as equivocation. The capacity-equivocation region is
Similar matrix characterizations of the capacity-equivocation
thesetofrate-equivocationpairs(R,Re)thatcanbeachieved region, however, were unknown.
by any coding scheme. In the literature, this communication
scenario is usually known as the rate-equivocation setting of
the MIMO Gaussian wiretap channel; see Fig. 1(a) for an B. Main Results
illustration. The main result of this section is a matrix characterization
of the capacity-equivocation region of the MIMO Gaussian
Csisza´r and Ko¨rner [17] studied the rate-equivocation set-
wiretapchannel.Morespecifically,considerthe MIMOGaus-
ting of a general discrete memoryless wiretap channel. A
sian wiretap channel (1) under the matrix power constraint
single-letter expression for the capacity-equivocation region
was derived [17, Theorem 1], which can be written as the set n
1
of nonnegative rate-equivocation pairs (R,R ) satisfying (X[m]X⊺[m])(cid:22)S (5)
e n
m=1
R ≤ min{R,I(V;Y|U)−I(V;Z|U)} X
e (3) where S is a positive semidefinite matrix. Let
R ≤ I(V;Y)
1
for some p(u,v,x,y,z) = p(u)p(v|u)p(x|v)p(y,z|x). Here, C(S,Hr)= 2log|I+HrSH⊺r| (6)
p(y,z|x) is the transition probability of the discrete memory-
be the Shannon capacity of a MIMO Gaussian point-to-point
less wiretap channel, and U and V are two auxiliary random
channel with channel matrix H and under the matrix power
variables.Intheory,acomputableexpressionforthecapacity- r
constraint (5), and let
equivocation region can be obtained by evaluating the single-
letter expression (3) for the MIMO Gaussian wiretap channel 1 I+H BH⊺
C (S,H ,H )= max log r r (7)
(1). However, such an evaluation is generally difficult due to s r e 0(cid:22)B(cid:22)S2 (cid:12)I+HeBH⊺e(cid:12)
the presence of the auxiliary random variables U and V. (cid:12) (cid:12)
be the secrecy capacityof a MIMO Ga(cid:12)ussian wiretap(cid:12)channel
Severalrecentworks[3]–[7] studiedthe special case where withlegitimatereceiverandeavesdroppe(cid:12)rchannelmat(cid:12)ricesH
r
3
4
The capacity-equivocation region of the MIMO Gaussian
wiretap channel under an average total power constraint is
3.5
summarized in the following corollary. The result is a direct
3 consequence of Theorem 1 and [18, Lemma 1].
Corollary 1: The capacity-equivocation region of the
2.5
MIMO Gaussian wiretap channel (1) under the average total
power constraint
Re 2
n
1
1.5 (X[m]⊺X[m])≤P (9)
n
m=1
1 X
is given by the set of nonnegative rate-equivocation pairs
0.5 (R,R ) satisfying
e
00 1 2 R3 4 5 6 RRe ≤≤ Cm(inS{,RH,rC)s(S,Hr,He)} (10)
(a) Capacity-equivocation region for some S(cid:23)0, Tr(S)≤P.
4 C. Proof of the Main Results
Next, we prove Theorem 1. As mentioned previously,
3.5
directly evaluating the single-letter expression (3) for the
3 MIMO Gaussian wiretap channel (1) is difficult due to the
presence of the auxiliary random variables. We thus resort to
2.5
anindirectapproachthatconnectstherate-equivocationsetting
ofaMIMOGaussianwiretapchanneltotheproblemofsimul-
Rs 2
taneously communicating private and confidential messages.
1.5 The problem of simultaneously communicating private and
confidential messages over a discrete memoryless wiretap
1
channel is illustrated in Fig. 1(b). Here, the transmitter has
a private message W , which is uniformly distributed over
0.5 p
{1,...,2nRp}, and a confidential message Ws, which is
0 uniformly distributed over {1,...,2nRs}. The confidential
0 1 2 3 4 5 6
R messageW isintendedforthelegitimatereceiverbutneedsto
p s
be keptasymptoticallyperfectly secret fromthe eavesdropper.
(b) Private-confidential messagecapacity region
That is, for any ǫ>0 it is required that
Fig.2. MIMOGaussianwiretap channelundermatrixpowerconstraint.
1
I(W ;Zn)≤ǫ (11)
s
n
andHe respectivelyandunderthematrixpowerconstraint(5) for sufficiently large block length n. The private message Wp
[6], [7]. We then have the following result. is also intended for the legitimate receiver, but is not subject
to any secrecy constraint. The private-confidential message
Theorem 1: Thecapacity-equivocationregionoftheMIMO
capacity region is the set of private-confidential rate pairs
Gaussian wiretap channel (1) under the matrix power con-
(R ,R ) that can be achieved by any coding scheme.
straint(5) is givenbythe setof nonnegativerate-equivocation p s
The following lemma provides a single-letter characteriza-
pairs (R,R ) satisfying
e
tion of the private-confidentialmessage capacity regionof the
Re ≤ min{R,Cs(S,Hr,He)} (8) discrete memoryless wiretap channel.
R ≤ C(S,Hr) Lemma 1: The private-confidential message capacity re-
where C(S,H ) and C (S,H ,H ) are defined as in (6) and gion of the discrete memoryless wiretap channel p(y,z|x) is
r s r e
given by the set of nonnegative private-confidentialrate pairs
(7), respectively.
(R ,R ) satisfying
Fig. 2(a) illustrates the capacity-equivocation region of a p s
MIMO Gaussian wiretap channel with channel matrices R ≤ I(V;Y|U)−I(V;Z|U)
s (12)
R +R ≤ I(V;Y)
1.8 2.0 3.3 1.3 s p
H = and H =
r 1.0 3.0 e 2.0 −1.5 for some p(u,v,x,y,z) = p(u)p(v|u)p(x|v)p(y,z|x), where
(cid:18) (cid:19) (cid:18) (cid:19)
(which yields a nondegraded wiretap channel) and matrix U and V are auxiliary random variables.
The achievability part of the lemma can be proved by
power constraint
considering a coding scheme that combines superposition
5.0 1.25
S= . coding, random binning, and rate splitting. In particular, part
1.25 10.0
of the private message will be used in the binning scheme
(cid:18) (cid:19)
4
H Zn
1 1
^ ^
× + Y1n (W0,W1)
Receiver 1
1n_I(W2;Y1n )→0
Xn
(W0,W1,W2) Transmitter H2 Z2n
^ ^
× + Y2n (W0,W2)
Receiver 2
1n_I(W1;Y2n )→0
Fig.3. MIMOGaussianbroadcast channel withcommonandconfidential messages.
to protect the confidential message against the eavesdropper. oftheprivate-confidentialrateregion(13)followsfromthatof
The converse proof follows standard information-theoretic (12)bysettingV=X=U+G,whereUandGdenotetwo
argument.ThedetailsoftheproofaredeferredtoAppendixA. independentGaussianvectorswithzeromeansandcovariance
A simple inspection of the capacity-equivocation region matrices S−B∗ and B∗, respectively.
(3) and the private-confidential message capacity region (12) The fact that R ≤ C (S,H ,H ) for any achievable
s s r e
reveals the following interesting fact: confidential rate R follows from the secrecy capacity result
s
Fact 1: Anonnegativeratepair(R,Re)=(Rp+Rs,Rs)is of[6]and[7]ontheMIMOGaussianwiretapchannelundera
anachievablerate-equivocationpairforadiscretememoryless matrix power constraint,by ignoringthe private message W .
p
wiretap channel if and only if (Rp,Rs) is an achievable ThefactthatRs+Rp ≤C(S,Hr)foranyachievableprivate-
private-confidentialrate pair for the same channel. confidential rate pair (R ,R ) follows from the well-known
p s
The “if” part of the fact is easy to verify: Simply use capacity result on the MIMO Gaussian point-to-pointchannel
the same code for both communication scenarios and view under a matrix power constraint, by viewing (W ,W ) as a
p s
(Wp,Ws) as the single message W for the rate-equivocation single message and ignoring the asymptotic perfect secrecy
setting. Note that constraint (11) on the confidential message W .
s
1 1 Remark 1: It is particularly worth mentioning the corner
H(W|Zn) = H(W ,W |Zn)
n n p s point (Rp,Rs) of the private-confidential message capacity
1 region (13) as given by
≥ H(W |Zn)
s
n
≥ R −ǫ (Rp,Rs)=(C(S,Hr)−Cs(S,Hr,He),Cs(S,Hr,He)).
s
= R −ǫ. Here, under the matrix power constraint, both messages W
e s
and (W ,W ), viewed as a single private message, can
Thus, the same code satisfying the secrecy constraint(11) for p s
transmit simultaneously at their respective maximum rates. In
simultaneousprivate-confidentialcommunicationalsosatisfies
particular,transmittinganadditionalprivatemessageW does
thesecrecyconstraint(2)fortherate-equivocationsetting.The p
not incur any rate loss for communicating the confidential
“only if” part of the fact comes as a mild surprise, as in the
message W .
rate-equivocationsettingwhichpartofmessageissecuredoes s
Now, Theorem 1 follows immediately from Lemma 2 and
not need to be specified a priori and may even depend on
a Fourier-Motzkin elimination with R = R +R and R =
the realization of the channel noise. We note here that the p s e
R .Forcomparison,theprivate-confidentialmessagecapacity
above interesting fact was first mentioned in [19, pp. 411– s
region of the same MIMO Gaussian wiretap channel as used
412] without proof.
for Fig. 2(a) is illustrated in Fig. 2(b).
In light of Fact 1, next we first establish a matrix charac-
terization of the private-confidential message capacity region
using the existing matrix characterization [6], [7] on the
secrecy capacity of the MIMO Gaussian wiretap channel. III. MIMO GAUSSIAN BROADCAST CHANNELSWITH
The result will then be mapped to the rate-equivocation COMMON AND CONFIDENTIAL MESSAGES
setting using the aforementioned equivalence between these
A. Channel Model
two communication scenarios.
Lemma 2: Theprivate-confidentialmessagecapacityregion
Consideratwo-receiverMIMOGaussianbroadcastchannel.
of the MIMO Gaussian wiretap channel (1) under the matrix
The transmitter is equipped with t transmit antennas, and
powerconstraint(5)isgivenbythesetofnonnegativeprivate-
receiver k, k =1,2, is equipped with r receive antennas. A
k
confidential rate pairs (R ,R ) satisfying
p s discrete-time sample of the channel at time m can be written
R ≤ C (S,H ,H ) as
Rs+Rps ≤ Cs(S,Hrr). e (13) Yk[m]=HkX[m]+Zk[m], k =1,2 (14)
Proof: LetB∗ be an optimalsolution to the optimization where H are the (real) channel matrices of size r ×t, and
k k
problem on the right-handside of (7). Then, the achievability {Z [m]} are i.i.d. additive vector Gaussian noise processes
k m
5
with zero means and identity covariance matrices.2
As illustrated in Fig. 3, the transmitter has a commonmes-
sage W0 and two independent confidential messages W1 and 5
W . The commonmessage W is intendedfor both receivers.
2 0
The confidential message W is intended for receiver k but 4
k
needstobekeptasymptoticallyperfectlysecretfromtheother
3
receiver. Mathematically, for every ǫ>0 we must have
R0
1 1 2
I(W ;Yn)≤ǫ and I(W ;Yn)≤ǫ (15)
n 1 2 n 2 1
1
for sufficiently large block length n. Our goal here is
to characterize the entire capacity region C(H ,H ,S) = 0
1 2 0
{(R ,R ,R )} that can be achieved by any coding scheme,
0 1 2 0
whereR ,R andR arethecommunicationratescorrespond- 1
0 1 2 R 1
ingtothecommonmessageW0 andtheconfidentialmessages 2 2 2 R1
W and W , respectively.
1 2 3 3
With both confidential messages W and W but without
1 2 (a) Capacity regionC(H1,H2,S)
the common message W , the problem was studied in [8] for
0
the multiple-input single-output (MISO) case and in [9] for
2.5
general MIMO case. Rather surprisingly, it was shown in [9]
R=0
that, under a matrix power constraint both confidential mes- 0
sages can be simultaneously communicated at their respected
2
maximumrates. With the commonmessage W0 and only one R0=1.60
confidential message (W or W ), the capacity region of the
1 2
MIMO Gaussian wiretap channel was characterized in [10] 1.5
R=2.40
using a channel-enhancement approach [18] and an extremal 0
entropy inequality of Weingarten et al. [21]. R2
1
R=3.45
0
B. Main Results
0.5
The main result of this section is a precise characterization
ofthecapacityregionoftheMIMOGaussianbroadcastchan-
nelwithamorecompletemessagesetthatincludesacommon
0
0 0.5 1 1.5 2 2.5 3
message W and two independent confidential messages W
0 1 R
1
and W .
Theo2rem 2: The capacity region C(H ,H ,S) of the (b) (R1,R2)-crosssections
1 2
MIMOGaussian broadcastchannel(14) with a commonmes- Fig. 4. MIMO Gaussian broadcast channel with common and confidential
sageW andtwoconfidentialmessagesW andW underthe messages.
0 1 2
matrix power constraint(5) is givenby the set of nonnegative
rate triples (R ,R ,R ) such that
0 1 2
withrespectto H andH , eventhoughitisnotimmediately
R0 ≤ min 12log H1(HS1−SBH0⊺1)H+I⊺1r+1Ir1 , evident from the e1xpressio2ns themselves.
n 1(cid:12)log H2SH⊺2+Ir2(cid:12) Remark 3: By setting B = S−B we can recover the
RR1 ≤≤ 121lloogg|IIrr12++122H(cid:12)(cid:12)lHo2(gS1|B−(cid:12)(cid:12)(cid:12)IHrB122H0()+SH⊺1−H⊺2|B−20−B)H1⊺2H+(cid:12)(cid:12)I⊺2r|2(cid:12)(cid:12)(cid:12)o (16) crWeos0nufilatdneodnftti[ha1le0m,cTeoshnsefiaodgreeenmWtia12l].mth0eastsaingceluWde1s1bthuet wcoimthmouotnthmeesostahgeer
2 2 (cid:12)(cid:12)(cid:12) Ir212+loHg2BI1r1HI+r⊺21H+1H((cid:12)(cid:12)(cid:12)S1−BB1H0)⊺1H⊺1 chaFnign.e4l(ma)atirlliucestsraatnedstthheecmapaatrciixtyproewgeiorncoCn(Hstr1a,inHt2a,sSg)ivfoenrtbhye
forRseommaerkB20:(cid:23)By0,sBett1in(cid:23)g 0Ban=d 0B(cid:12)(cid:12)(cid:12)w0+e cBan1 r(cid:22)ecSo.ver th(cid:12)(cid:12)(cid:12)e result of H1 = 11..08 32..00 , H2 = 32..30 −11.3.5
0 (cid:18) (cid:19) (cid:18) (cid:19)
[9, Theorem 1] that includes both confidential messages W1 and S= 5.0 1.25 .
and W2 but without the common message W0. Similar to [9, 1.25 10.0
Theorem 1], for any given B the upper bounds on R and (cid:18) (cid:19)
0 1 (ThechannelparametersarethesameasthoseusedforFig.2.)
R can be simultaneously maximized by a same B . In fact,
2 1 In Fig. 4(b), we have also plotted the (R ,R )-cross section
the upper bounds on R and R in (16) are fully symmetric 1 2
1 2 of C(H ,H ,S) for several given values of R . Note that
1 2 0
2ThechannelmodelisthesameasthatinSectionII-A.However,different when R0 = 0, the (R1,R2)-cross section is rectangular, im-
notation isusedherefortheconvenience ofpresentation. plying that under a matrix power constraint, both confidential
6
messages W and W can be simultaneously transmitted at zeromeansandcovariancematricesB ,B andS−B −B
1 2 0 1 0 1
their respective maximum rates [9]. For R > 0, however, respectively, and
0
the (R ,R )-cross sections are generally non-rectangular as
1 2 F:=BH⊺(I +H BH⊺)−1H .
different boundary points on the same cross section may 1 r1 1 1 1
correspond to different choice of B .
0 To show that the rate region (19) over all possible B (cid:23)0,
Thecapacityregionunderanaveragetotalpowerconstraint 0
B (cid:23) 0 and B +B (cid:22) S is indeed the capacity region, we
issummarizedinthefollowingcorollary.Theresultisadirect 1 0 1
shall consider proof by contradiction and resort to a channel-
consequence of Theorem 2 and [18, Lemma 1].
Corollary 2: The capacity region C(H ,H ,P) of the enhancement argument akin to that in [10].
1 2
MIMO Gaussian broadcast channel (14) with a common Morespecifically,assumethat(R†,R†,R†)isanachievable
0 1 2
messageW andtwoconfidentialmessagesW andW under rate triple that lies outside the rate region (19) for any given
0 1 2
the average total power constraint (9) is given by B (cid:23) 0, B (cid:23) 0 and B +B (cid:22) S. Since (R†,R†,R†) is
0 1 0 1 0 1 2
achievable, we can bound R† by
C(H ,H ,P)= C(H ,H ,S). (17) 0
1 2 1 2
S(cid:23)0,T[r(S)≤P R† ≤min 1log S+N1 , 1log S+N2 =Rmax.
0 2 N 2 N 0
(cid:18) (cid:12) 1 (cid:12) (cid:12) 2 (cid:12)(cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
C. Proof of the Main Results Moreover, if R† = (cid:12)R† = 0,(cid:12)then Rm(cid:12)ax can be(cid:12) achieved by
1 (cid:12) 2 (cid:12) 0(cid:12) (cid:12)
Next, we prove Theorem 2. Following [18], we shall focus setting B =S and B =0 in (19). Thus, by the assumption
0 1
on the canonical case in which the channel matrices H and that(R†,R†,R†) is outside the rate region(19) for any given
1 0 1 2
H are square and invertible and the matrix power constraint B (cid:23) 0, B (cid:23) 0 and B +B (cid:22) S, we can always find
2 0 1 0 1
S is strictly positive definite. In this case, multiplying both λ ≥0 and λ ≥0 such that
1 2
sides of (14) by H−1, the MIMO Gaussian broadcastchannel
k λ R†+λ R† =λ R⋆+λ R⋆+ρ (21)
(14) can be equivalently written as 1 1 2 2 1 1 2 2
for some ρ>0, where λ R⋆+λ R⋆ is given by
Y [m]=X [m]+Z [m], k =1,2 (18) 1 1 2 2
k k k
where {Z [m]} are i.i.d. additive vector Gaussian noise max(B0,B1) λ1f1(B1)+λ2f2(B0,B1)
processes kwith zmero means and covariance matrices N = subject to f0(B0)≥R0†
H−1H−⊺. Similarly, the rate region (16) can be equivalekntly B0 (cid:23)0 (22)
k k B (cid:23)0
written as 1
B +B (cid:22)S.
0 1
R ≤ min 1log S+N1 ,1log S+N2
0 2 (S−B0)+N1 2 (S−B0)+N2 Here, the functions f0, f1 and f2 are defined as
RR1 ≤≤ 121llooggn(cid:12)B(S1−N+B1N(cid:12)(cid:12)(cid:12)01)+(cid:12)N−221−log1(cid:12)l(cid:12)(cid:12)(cid:12)Bog1N+2N(S2−(cid:12)(cid:12)(cid:12)(cid:12)B0)+N1 . (cid:12)(cid:12)(cid:12)o f0(B0):=min 21log (S−SB+N)+1 N ,
BC(0HN(cid:23)e21x,0tH,,w2B,e1S2s)(cid:23)hof0owra(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ntthhdaeBBt1ct+0haNn+e(cid:12)(cid:12)o2rnBaitc1e(cid:12)(cid:12)(cid:12)a(cid:22)lreMgS2(cid:12)(cid:12)iIogMnivO(cid:12)(cid:12)(cid:12)(e1s9GtB)h(cid:12)(cid:12)a1eou+vscNseair1panaacll(cid:12)(cid:12)(cid:12)ibtyrpoorasedsg(ci1ibao9lsne)t f1(B1):= 2112lloogg(cid:26)(cid:12)(cid:12)(cid:12)(cid:12)B(S1N−+S(cid:12)(cid:12)(cid:12)(cid:12)BN+01N)+2−N2102l(cid:12)(cid:12)(cid:12)(cid:12)o(cid:27)g B11(cid:12)(cid:12)(cid:12)(cid:12)N+N2
channel(18).Extensionstothegeneralmodel(14)followfrom (cid:12) 1 (cid:12) (cid:12) 2 (cid:12)
1 (cid:12)(S−B )(cid:12)+N (cid:12) (cid:12)
thewell-knownlimitingargument[6],[10],[18]andhenceare and f (B ,B ):= log(cid:12) 0 (cid:12) 2 (cid:12) (cid:12)
2 0 1 2 (cid:12) B +N(cid:12) (cid:12) (cid:12)
omitted from the paper. (cid:12) 1 2 (cid:12)
1 (cid:12) (S−B )+N(cid:12)
To prove the achievability of the rate region (19), recall − lo(cid:12)g 0 (cid:12)1 .
thattheproblemofatwo-receiverdiscretememorylessbroad- 2 (cid:12) (cid:12) B1+N1 (cid:12) (cid:12)
cast channel with a common message and two confidential (cid:12) (cid:12)
common messages was studied in [22]. There, a single-letter Let (B⋆0,B⋆1) be an optimal(cid:12)(cid:12)solution to the (cid:12)(cid:12)optimization
program (22). By assumption, the matrix power constraint
expressionforanachievablerateregionwasestablished,which
S is strictly positive definite in the canonical model. Thus,
is given by the set of rate triples (R ,R ,R ) such that
0 1 2 (B⋆,B⋆) must satisfy the following Karush-Kuhn-Tucker
R0 ≤ min[I(U;Y1),I(U,Y2)] (KK0T)1conditions:
R ≤ I(V ;Y |U)−I(V ;V ,Y |U) (20)
1 1 1 1 2 2
R2 ≤ I(V2;Y2|U)−I(V2;V1,Y1|U) (β1+λ2)[(S−B⋆0)+N1]−1+β2[(S−B⋆0)+N2]−1+M0
whereU,V andV areauxiliaryrandomvariablessatisfying =λ2[(S−B⋆0)+N2]−1+M2 (23)
1 2
the Markov relation (U,V1,V2) → X → (Y1,Y2). The (λ1+λ2)(B⋆1+N1)−1+M1
proposed coding scheme is a natural combination of double =(λ +λ )(B⋆+N )−1+M (24)
1 2 1 2 2
binning[23]andsuperpositioncoding.Thus,theachievability
M B⋆ =0, M B⋆ =0, and M (S−B⋆−B⋆)=0 (25)
of the rate region (19) follows from that of (20) by setting 0 0 1 1 2 0 1
V =U +FU , V =U , and X=U+U +U where whereM ,M andM arepositivesemidefinitematrices,and
1 1 2 2 2 1 2 0 1 2
U, U and U are three independent Gaussian vectors with β , k =1,2, are nonnegative real scalars such that β >0 if
1 2 k k
7
and only if Following (24), we may also obtain
1log S+Nk =R†. (λ1+λ2)(B⋆1+N)−1 =(λ1+λ2)(B⋆1+N2)−1+M2
2 (S−B⋆)+N 0 (31)
(cid:12) 0 k(cid:12)
It follows that (cid:12)(cid:12) (cid:12)(cid:12) which implies that Ne (cid:22)N .
(cid:12) (cid:12) 2
(β1+β2)R0†+λ1R1†+λ2R2† Consider the following enhanced aligned MIMO Gaussian
β S+N β S+N broadcast channel e
1 1 2 2
= log + log
2 (cid:12)(S−B⋆0)+N1(cid:12) 2 (cid:12)(S−B⋆0)+N2(cid:12) Y1a[m] = X[m]+Z1a[m]
+λ1(cid:18)(cid:12)(cid:12)(cid:12)21log(cid:12)B⋆1N+1N1(cid:12)(cid:12)(cid:12)(cid:12)− 21log(cid:12)B(cid:12)(cid:12)(cid:12)⋆1N+2N2(cid:12)(cid:19) (cid:12)(cid:12)(cid:12) YYe21ab[[mm]] == XX[[mm]]++ZZe12ba[[mm]] (32)
1 (cid:12)(S−B⋆)(cid:12)+N (cid:12) (cid:12) Y [m] = X[m]+Z [m]
+λ log(cid:12) 0 (cid:12) 2 (cid:12) (cid:12) 2b 2b
2 2 (cid:12) B⋆+N(cid:12) (cid:12) (cid:12) e e
(cid:18) (cid:12) 1 2 (cid:12) where{Z [m]},{Z [m]},{Z [m]}and{Z [m]}arei.i.d.
1 (S(cid:12)−B⋆)+N (cid:12) 1a 1b 2a 2b
− log (cid:12) 0 1 (cid:12)+ρ. (26) additivevectorGaussiannoise processeswith zeromeansand
2 (cid:12) (cid:12) B⋆1+N1 (cid:12)(cid:19)(cid:12) covarianece matrices N, N1, Neand N2, respectively.
(cid:12) (cid:12)
Next, we shall(cid:12)(cid:12) find a contradi(cid:12)(cid:12)ction to (26) through the The message set configuration is the same as that for
following three steps. channel (27). Since eN (cid:22) {Ne ,N }, we conclude that the
1 2
capacity region of channel (32) is at least as large as that of
1) Split each receiver into two virtual receivers: Consider
channel (27) under thee same matrix power constraint.
the following canonical MIMO Gaussian broadcast channel
with four receivers: Furthermore, from (31) we have
Y1a[m] = X[m]+Z1a[m] [(S−B⋆)+N](B⋆+N)−1
Y [m] = X[m]+Z [m] 0 1
Y1b[m] = X[m]+Z1b[m] (27) =[(S−B⋆0)+N2](B⋆1+N2)−1 (33)
2a 2a e e
Y2b[m] = X[m]+Z2b[m] and hence
where{Z1a[m]},{Z1b[m]},{Z2a[m]}and{Z2b[m]}arei.i.d. (S−B⋆0)+N = (S−B⋆0)+N2 . (34)
caodvdSaiutripivapenocvseeecmtthoaarttrGtihcaeeustsrsNaina1sn,mnNiott1ies,reNhpar2soacthnersdeseeNsin2wd,iertpehsezpneedrceotnivmtemelyae.nsssaagneds Combining(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(23)Ba⋆1n+d (Ne31)e, w(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)e m(cid:12)(cid:12)(cid:12)(cid:12)ay oBb⋆1ta+inN2 (cid:12)(cid:12)(cid:12)(cid:12)
W0, W1 and W2, where W0 is intended for both receivers 1b (λ1+λ2)[(S−B⋆0)+N]−1
and 2b, W1 is intended for receiver 1a but needs to be kept =(λ +β )[(S−B⋆)+N ]−1
asymptotically perfectly secret from receiver 2b, and W2 is +(2λ +1β )[(S−e0B⋆)+N1 ]−1+M . (35)
intended for receiver 2a but needs to be kept asymptotically 1 2 0 2 0
perfectly secret from receiver 1b. Mathematically, for every Substituting (30) and (34) into (26), we have
ǫ>0, we must have
(β +β )R†+λ R†+λ R†
1 1 1 2 0 1 1 2 2
nI(W1;Y2nb)≤ǫ and nI(W2;Y1nb)≤ǫ (28) = β1 log S+N1 + β2 log S+N2
2 (S−B⋆)+N 2 (S−B⋆)+N
for sufficiently large block length n. Note that receivers 1a (cid:12) 0 1(cid:12) (cid:12) 0 2(cid:12)
and 1b are statistically identical to receiver 1 in channel (18), +λ (cid:12)(cid:12)(cid:12)1log (S−B⋆0)+(cid:12)(cid:12)(cid:12) N − 1l(cid:12)(cid:12)(cid:12)og (S−B⋆0)+N(cid:12)(cid:12)(cid:12) 2
stshoaumaserecoarsnecctlheuiadvteerotsfh2actahtahannendecla2pb(a1tc8oi)tyruenrceedgieviroenrth2oefisncahmcahenannmenlae(tlr2i(7x1)8p)iso.wWtheeer +λ1 21log(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(S−BNe⋆0)+Ne(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)− 21log(cid:12)(cid:12)(cid:12)(cid:12)(S−BN⋆02)+N1(cid:12)(cid:12)(cid:12)(cid:12)!
con2s)trCaionnt.struct an enhanced channel: Let N be a real sym- +ρ.2 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ne e(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 2 (cid:12)(cid:12)(cid:12)(cid:12) N1 (3(cid:12)(cid:12)(cid:12)(cid:12)6!)
metric matrix satisfying
e 3) Outerboundtheenhancedchannel: Next,weconsidera
1 −1
N:= N−1+ M (29) discretememorylessbroadcastchannelwithfourreceiversand
(cid:18) 1 λ1+λ2 1(cid:19) three independent messages and provide a single-letter outer
bound on the capacity region.
whichimpliestehatN(cid:22)N .SinceM B⋆ =0,following[18,
1 1 1
Lemma 11] we have Lemma 3: Considera discrete memorylessbroadcastchan-
(λ +λ )(B⋆+eN)−1 =(λ +λ )(B⋆+N )−1+M nel p(y1a,y1b,y2a,y2b|x) with four receivers and three in-
1 2 1 1 2 1 1 1 dependent messages (W ,W ,W ): W is intended for both
0 1 2 0
and receiveers 1b ande2b, W1 is intended for receiver 1a but needs
e
to be kept asymptotically perfectly secret from receiver 2b,
|B⋆+N||N |=|B⋆+N ||N|. (30)
1 1 1 1 and W2 is intended for receiver 2a but needs to be kept
e e
8
asymptotically perfectly secret from receiver 1b. Assume that IV. CONCLUDING REMARKS
In this paper we have presented two new results on MIMO
X →Y →(Y ,Y ) and X →Y →(Y ,Y )
1a 1b 2b 2a 1b 2b
Gaussianbroadcastchannelswithconfidentialmessages,lead-
form two Markov chains. Then, any achievable rate triple ingtoamorecomprehensiveunderstandingofthefundamental
e e
(R0,R1,R2) must satisfy limits of MIMO secret communication.
First, a matrix characterizationof the capacity-equivocation
R ≤ min[I(U;Y ),I(U,Y )]
0 1b 2b
region of the MIMO Gaussian wiretap channel has been
R ≤ I(X;Y |U)−I(X;Y |U) (37)
1 1a 2b obtained,generalizingthe previousresults[3]–[7] whichdealt
R ≤ I(X;Y |U)−I(X;Y |U)
2 2a 1b only with the secrecy capacity of the channel. The result
e
for some p(u,x), where U is an auxiliary random variable. has been obtained via an interesting connection between the
e
The proof follows standard information-theoretic argument rate-equivocationsettingandsimultaneousprivate-confidential
and is deferred to Appendix B. communication over a discrete memoryless wiretap channel,
Now,wecancombineallpreviousthreestepsandobtainan which allows a matrix characterization of the entire capacity-
upperboundontheweightedsumrate (β +β )R†+λ R†+ equivocation region based on the existing characterization of
1 2 0 1 1
λ R†.Byassumption,(R†,R†,R†)isanachievableratetriple secrecy capacity for the MIMO Gaussian wiretap channel.
2 2 0 1 2
Next,theproblemofMIMOGaussianwiretapchannelswith
for channel (18). Then, following Lemma 3 we have
tworeceiversandthreeindependentmessages,acommonmes-
(β1+β2)R0†+λ1R1†+λ2R2† sageintendedforbothreceivers,andtwomutuallyconfidential
β β messageseachintendedforoneofthereceiversbutneedingto
≤ 1 log|2πe(S+N )|+ 2 log|2πe(S+N )|
2 1 2 2 bekeptasymptoticallyperfectsecurefromtheother,hasbeen
λ N λ N considered. A precise characterization of the capacity region
+ 1 log 2 + 2 log 1 +η(λ ,λ ) (38)
2 N 2 N 1 2 hasbeenobtainedviaachannel-enhancementargument,which
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) is a natural extension of the channel-enhancement arguments
where (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) e (cid:12) (cid:12) e (cid:12) of [9] and [24].
η(λ ,λ ):=λ h(X+Z |U)+λ h(X+Z |U)
1 2 1 1a 2 2a
−(λ +β )h(X+Z |U)−(λ +β )h(X+Z |U). APPENDIX A
2 1 e 1b 1 2e 2b PROOF OF LEMMA1
Note that 0≺N(cid:22){N1,N2}, 0≺B⋆0 (cid:22)S, and B⋆0M0 =0. We first prove the achievability part of the lemma by con-
By [21, Corollary 4] and (35), we have sideringa codingschemethatcombinessuperpositioncoding,
e randombinning,andratesplitting.Fixp(u)p(v|u)p(x|v).Split
η(λ ,λ )≤(λ +λ )log 2πe(S−B⋆)+N
1 2 1 2 0 theprivatemessageW intotwoindependentsubmessagesW′
p p
−(λ +β )lo(cid:12)g|2πe(S−B⋆)+N(cid:12) | and W′′.
2 1 (cid:12) 0 e(cid:12) 1 p
−(λ +β )lo(cid:12)g|2πe(S−B⋆)+N(cid:12) |. (39) Codebook generation. Fix δ > 0. Randomly and indepen-
1 2 0 2 dently generate 2n(R′p+δ) codewords of length n according
Combining (38) and (39), we have to pn. Label each of the codewords as un, where j is the
U j
codeword number. We will refer to the codeword collection
(β +β )R†+λ R†+λ R†
1 2 0 1 1 2 2 {un} as the U-codebook.
≤ β21 log(cid:12)(S−SB+⋆0N)+1 N1(cid:12)+ β22 log(cid:12)(S−SB+⋆0N)+2 N2(cid:12) indFjeoprjenedacehntlcyodgeewneorradteu2njn(iRns+thRe′p′U+T-c)ocdoedbeowoko,rdrsanodfolmenlygthannd
++λλ1 (cid:12)(cid:12)(cid:12)211lloogg(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)((SS−−BBNe⋆0⋆0))++(cid:12)(cid:12)(cid:12) NNe(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)−− 211ll(cid:12)(cid:12)(cid:12)oogg(cid:12)(cid:12)(cid:12)(cid:12)((SS−−BBN⋆0⋆02))++NN(cid:12)(cid:12)(cid:12) 21(cid:12)(cid:12)(cid:12)(cid:12)! atwchcooacdrtoedrwesdaoicinrnhdgtso.tsouF2bunQ-rRbtnihsi=ne1rbcpipnoVasn|rUttasi=toiinuosjtn[hi2a]e.ntaTRcehaacncbodhidonembwiinlnyotorcdpo2san.nrttRiaLti′pia′nobsnseul2tbhn-ee(baRiccn′ph′os+dTsoeo)f-
wthhaitcthheis2ra atec2roengtiroa(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ndi(c1ti9o)noNevtoer(3al6le)pa(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)osssρib2>leB0.(cid:12)(cid:12)(cid:12)(cid:12)W(cid:23)e0th,uBNs1c(cid:23)on0claund(cid:12)(cid:12)(cid:12)(cid:12)de! tlthhdeeecncoooddteeeswwtoohrreddssnuuabsm-bbvinejn,rkn,wlu,tmithwbienhreewraeicthhkinsdueebna-ocbhtiensb.iWtnh,eeanwbdiinlltnrdeuefmneorbteetros,
B +B (cid:22) S is indeed the capacity regio0n of the1canonical the codeword collection {vjn,k,l,t}k,l,t as the V-subcodebook
0 1 corresponding to un. Fig. 5 illustrates the overall codebook
MIMO Gaussian broadcast channel (18). This completes the j
structure.
proof of Theorem 2.
Encoding.Tosendamessagetriple(w ,w′,w′′),thetrans-
Remark 4: Note that in the enhanced channel (32), both s p p
mitter first chooses the codeword un from the U-codebook.
legitimatereceivers1aand2ahavethesamenoisecovariance wp′
Next, the transmitter looks into the V-subcodebook corre-
matrices. This fact greatly simplified the capacity analysis of
spondingto un andrandomly(accordingto a uniformdistri-
theenhancedchannelandiskeytothesuccessoftheproposed w′
p
bution)choosesacodewordvn fromthew′′thsub-bin
channel enhancement approach. We mention here that the of the w th bin. Once a vnwp′,ws,wp′′,its chosen, apn input se-
sv.asm.-ceotmecmhnoinqumeewssaasgaelscoapuasecdityinr[e2g4io]ntoodfetrhiveeMthIeMsOumG-paruisvsaitaen- quencexsn isgeneratedaccowrp′d,winsg,wtp′o′,t ni=1pX|V=vwp′,ws,wp′′,t[i]
broadcast channel. and is then sent through the channel.
Q
9
*’%&"+,-.",,/ secrecy condition (11) implies that for every ǫ>0,
0’+,-.",,/ %&"’"#$( H(Ws|Zn)≥H(Ws)−nǫ. (45)
%&"’"#$) On the other hand, Fano’s inequality [20, Ch. 2.11] implies
"#$1
+,-.1,2-( !!! that for every ǫ0 >0,
+,-.1,2-) %&"’"#$2nR′p′ H(Ws,Wp|Yn)≤ǫ0log 2n(Rs+Rp)−1 +h(ǫ0)
!!! :=nδ. h i (46)
!!!
Applying (45) and (46), we have
+,-.1,2-3
!!! nRs =H(Ws)
%&"’"#$(
≤ H(W |Zn)+nǫ + nδ−H(W ,W |Yn)
+,-.1,2-2nR′p %&"’"#$) "#$2nRs ≤H(W s,W |Zn)−H(W ,W |Yn)s+np(ǫ+δ). (47)
!!! (cid:2) s p (cid:3) (cid:2) s p (cid:3)
By the chain rule of the mutual information [20, Ch. 2.5],
%&"’"#$2nR′p′
n(R −ǫ−δ)≤I(W ,W ;Yn)−I(W ,W ;Zn)
s s p s p
n
Fig.5. Codebookstructure. = I(W ,W ;Y |Yi−1)
s p i
i=1
X(cid:2)
−I(W ,W ;Z |Zn )
s p i i+1
Decodingat receiver 1. Given yn, receiver 1 looksinto the n
1 = I(W ,W ;Y |Yi−1,Zn(cid:3) )
codebooks U and V and searches for a pair of codewords s p i i+1
(unj,vjn,k,l,t) that are jointly typicalwith y1n. In the case when Xi=1(cid:2)−I(W ,W ;Z |Yi−1,Zn ) (48)
s p i i+1
R′ < I(U;Y) (40)
p where the last equality follows from [17, Lemma 7]. (cid:3)Let
and R +R′′+T < I(V;Y|U) (41)
s p
U := Yi−1,Zn and V :=(W ,W ,U ) (49)
with high probability the transmitted codeword pair i i+1 i s p i
(un ,vn ) is the only one that is jointly typical and we hav(cid:0)e from (48)(cid:1)that
wiwthp′ ynw.p′,ws,wp′′,t n
1
Security at receivers 2 and 3. Fix ǫ>0. In the case when n(Rs−ǫ−δ)≤ [I(Vi;Yi|Ui)−I(Vi;Zi|Ui)]. (50)
i=1
R′′+T >I(V;Z|U) (42) X
p
we have [17, Theorem 1]
1
I(W ;Zn|W′)≤ǫ (43)
n s p
for sufficiently large n. Since W and W′ are independent,
s p
we have from (43) that
Next, we consider an upper bound on the sum private-
1 1 confidential message rate R +R . By (46),
I(W ;Zn) ≤ I(W ;Zn,W′) s p
n s n s p
n(R +R )=H(W ,W )
1 s p s p
= nI(Ws;Zn|Wp′) ≤I(Ws,Wp;Yn)−nδ. (51)
≤ ǫ
Applying the chain rule of the mutual information [20,
i.e., the message W is asymptotically perfectly secure at the Ch. 2.5], we have
s
eavesdropper. n
To summarize, for any given p(u)p(v|u)p(x|v) and any n(Rs+Rp−δ)≤ I(Ws,Wp;Yi|Yi−1)
T ≥ 0, any rate triple (R ,R′,R′′) that satisfies (40)–(42) i=1
s p p Xn
is achievable. Note that
≤ I(W ,W ,Yi−1,Zn ;Y )
s p i+1 i
Rp =Rp′ +Rp′′. (44) Xi=1
n
Eliminating T, R′ and R′′ from (40)–(42) and (44) using = I(V ;Y ). (52)
2 2 i i
Fourier-Motzkin elimination, we may conclude that any rate i=1
X
pair (Rs,Rp) satisfying (12) is achievable. Applying the standard single-letterization procedure (e.g., see
Toprovetheconversepartofthelemma,wefirstconsideran [20, Ch. 14.3])to (50) and(52), we have thedesiredconverse
upperboundon the confidentialmessage rate R . The perfect result for Lemma 1.
s
10
APPENDIX B Substituting (54b) into (59), we may obtain
PROOF OF LEMMA3
nR ≤H(W |W ,Yn)−H(W |W ,Yn)
1 1 0 2b 1 0 1a
The perfect secrecy condition (28) implies that for every +n(ǫ+δ +δ )
0 1
ǫ>0, =I(W ;Yn|W )−I(W ;Yn|We)
1 1a 0 1 2b 0
H(W1|Y2nb)≥H(W1)−nǫ (53a) +n(ǫ+δ0+δ1). (60)
and H(W |Yn)≥H(W )−nǫ. (53b) e
2 1b 2 Applying [17, Lemma 7], (60) can be rewritten as
Ontheotherhand,Fano’sinequality[20,Chapter2.11]implies n
that for every ǫ0 >0, nR1 ≤ I(W1;Y1a,i|W0,Y1ia−1,Y2nb,i+1)
i=1
max[H(W |Yn), H(W |Yn)] X(cid:2)
0 1b 0 2b −I(W ;Y e|W ,Yi−e1,Yn ) +n(ǫ+δ +δ )
≤ǫ log 2nR0 −1 +h(ǫ ):=nδ (54a) 1 2b,i 0 1a 2b,i+1 0 1
0 0 0 n
H(W1|Y1na)(cid:0) (cid:1) ≤ I(Xi;Y1a,i|W0e,Y1ia−1,Y2nb,i+1(cid:3))
and H(≤W2ǫ0|Yelo2nag)(cid:0)2nR1 −1(cid:1)+h(ǫ0):=nδ1 (54b) −Xi=I1((cid:2)Xi;Y2be,i|W0,Y1ia−e1,Y2nb,i+1) +n(ǫ+δ0+δ(16)1)
(cid:3)
≤ǫ log 2nR2 −1 +h(ǫ ):=nδ . (54c) e
0 0 2 where (61) follows from the Markov chain
e
Let (cid:0) (cid:1)
W →X →Y →Y .
1 i 1a,i 2b,i
U :=(W ,Yi−1,Yn ) (55)
i 0 1b 2b,i+1 Moreover, due to the Markov chain
e
which satisfies the Markov chain (W ,Y ,Yn )→Yi−1 →Yi−1 (62)
0 1a,i 2b,i+1 1a 1b
U →X →(Y ,Y ,Y ,Y ). (56)
i i 1a 2a 1b 2b we can further bound R as
e 1 e
We first bound R based on (54a) as follows: n
0 e e nR ≤ I(X ;Y |W ,Yi−1,Yi−1,Yn )
nR =H(W ) 1 i 1a,i 0 1a 1b 2b,i+1
0 0 i=1
≤I(W0;Y1nb)+nδ0 −XI((cid:2)Xi;Y2be,i|W0,Y1ia−e1,Y1ib−1,Y2nb,i+1)
n
= I(W ;Y |Yi−1)+nδ +n(ǫ+δ0+δ1) (cid:3)
0 1b,i 1b 0 n e
Xi=1 = I(X ;Y |U ,Yi−1)
n i 1a,i i 1a
≤ I(W ,Yi−1,Yn ;Y )+nδ i=1
0 1b 2b,i+1 1b,i 0 X(cid:2)
Xi=1 −I(Xi;Y2be,i|Ui,Y1ia−e1) +n(ǫ+δ0+δ1) (63)
n n
= I(Ui;Y1b,i)+nδ0. (57) = I(Xi;Y1a,i|Uei)−I(cid:3)(Xi;Y2b,i|Ui)
Xi=1 Xi=1(cid:2) (cid:3)
Similarly, we have − I(Yi−1;eY |U )−I(Yi−1;Y |U )
1a 1a,i i 1a 2b,i i
nR0 ≤In(W0;Y2nb)+nδ0 +n(cid:2)n(ǫe+δ0+eδ1) e (cid:3)
= I(W0;Y2b,i|Y2nb,i+1)+nδ0 ≤ I(Xi;Y1a,i|Ui)−I(Xi;Y2b,i|Ui)
Xi=1 Xi=1(cid:2) (cid:3)
n +n(ǫ+δ e+δ ) (64)
0 1
≤ I(W ,Yi−1,Yn ;Y )+nδ
0 1b 2b,i+1 2b,i 0 where (63) followsfrom the definition of U in (55), and (64)
i=1 i
Xn follows from the fact that Y2b,i is degraded with respect to
= I(Ui;Y2b,i)+nδ0. (58) Y1a,i so I(Y1ia−1;Y2b,i|Ui)≤I(Y1ia−1;Y1a,i|Ui).
i=1
X Following the same steps as those in (59)–(64), we may
Next, we bound R based on (53a) and (54b) as follows: e e e e
1 obtain
nR =H(W ) n
1 1
nR ≤ I(X ;Y |U )
≤ H(W |Yn)+nǫ + nδ −H(W |Yn) 2 i 2a,i i
1 2b 1 1 1a i=1
=(cid:2)H(W1|W0,Y2nb)+(cid:3)I(W(cid:2)1;W0|Y2nb)−H(W(cid:3)1|Y1na) −XI((cid:2)Xi;Y1be,i|Ui) +n(ǫ+δ0+δ2). (65)
e
+n(ǫ+δ )
1 (cid:3)
e Finally, applyingthe standardsingle-letterizationprocedure
≤H(W |W ,Yn)+H(W |Yn)−H(W |W ,Yn)
1 0 2b 0 2b 1 0 1a (e.g.,see[20,Chapter14.3])to(57),(58),(64)and(65)proves
+n(ǫ+δ ). (59)
1 the desired result (37) for Lemma 3.
e