IEEE TRANSACTIONS
ON INFORMATION
THEORY,
VOL.
IT-27,
NO.
1, JANUARY
1%
49
1
A New Achievable Rate Region for
the Interference Channel
TE SUN HAN, MEMBER, IEEE,AND KING0
Abstruct-Anew achievable rate region for the general interference
channel which extends previous results is presented and evaluated. Tbe
technique used is a generalization of superposition coding to the multivariable case. A detailed computation for tbe Gaussian cbaunel case clarifies
to wbat extent the new region improves previous ones. The capacity of a
class of Gaussian interference channels is also established.
I.
INTRODUCTION
T
HE INTERFERENCE
channel
is a channel
with
several pairs of input-output
terminals, where each
input communicates
with its respective output through the
common channel. The study of this kind of channel was
initiated
by C. E. Shannon
[l], and furthered
by R.
Ahlswede [2] who gave simple but fundamental
inner and
outer bounds
to the capacity
region. Recently
A. B.
Carleial [3] established
a considerably
improved achievable rate region for the memoryless interference
channel by
applying
the superposition
coding technique
of T. M.
Cover [4] which had originally been devised to study the
capacity region of the broadcast
channel. On the other
hand, H. Sato [5] obtained various inner and outer bounds
by transforming
the problem
to one for the associated
multiple-access
or broadcast channel. However, the problem of specifying a computable
expression of the capacity
region for the general interference
channel is still open,
although it has been solved for some very special cases
(Carleial [6], Benzel [7]).
This paper presents a refined and improved treatment
for the general interference
channel, thereby establishing a
new achievable rate region %* that contains as its subregions those of Carleial and Sato. We prove this coding
theorem in Section III (Theorems 3.1 and 3.2). The technique used should be regarded as a natural generalization
of Cover’s superposition
coding (and also of Ahlswede’s
random
coding) to the many variable case, where the
superiority
of simultaneous superposition
to sequential superposition
is pointed out (Remark 3, 2)).
In Section IV we give a simple and explicit expression
for constituent
subregions g(Z)
of the ‘%*, which enables
us to evaluate the region %* more easily. This expression
is attained
on the basis of the polymatroidal
structure
KOBAYASHI,
MEMBER, IEEE
(Remark 1) underlying
a collection of inequalities
specifying the region 3 *. In Section V we study the Gaussian
interference
channel
by applying
the result derived in
Sections III and IV, and numerically
evaluate achievable
rate regions for several typical values of the channel
parameters.
Comparison
with the computed
result of
Carleial reveals that our region considerably
improves the
previous ones. Finally, the capacity region of a class of
Gaussian
interference
channels
with less strong interference is established that extends the theorem of Carleial
[6] in the strong interference
case.
II.
In this section we shall define the interference
channel
and state the problem. We denote random variables by
X, y, ~4,. . 1 in finite
set
x, Y, u, * * * with values
%, %, %, * . * respectively.
A. Interference
Channel C
A discrete
interference
channel
is a quintuple
(%,,%‘2,w,%,,%z),
where !X,,!X* are two finite input
alphabet sets; %,, ‘!$ are two finite output alphabet sets;
and o is a collection of conditional
channel probabilities
of (Y,~Y~)~%
X% given (x,,x2)F%X
‘X2. The marginal distributions
w,, w2 of the o are given
~Y,Y,~x,x,)
by
~1(Y,lX*-%)’
Y2
2
4Y,Y2l+%)~
E%
Since we confine ourselves to memoryless
channels,
the
conditional
probability
w”( yi yz Ixi x2) of yi y2 E ‘%y X 9;
given xlxz E%; X95,” is
o”( y1y21x,x2)=
fi W(yI(‘)ypIXp
t=1
xf’),
where
xc2--(xp,.*-,xp)E%~,
Manuscript
received December 12, 1979; revised April 24, 1980.
T. S. Han is with the Department
of Information
Science, Sagami
Institute of Technology,
Tsujido Nishikaigan
l-l-25, Fujisawa, Japan
251.
K. Kobayashi
is with the Department
of Biophysical
Engineering,
Faculty of Engineering
Science, Osaka University,
Toyonaka,
Osaka,
Japan 560.
PRELIMINARIES
y,=(y~1);..,y,(“))E~,“,
a= 1,2.
Similarly for o;,o;.
Let %,={1,2;..,M,},
%,={1,2;..,M,}
sage sets for senders
1 and 2, respectively.
(n, M,, M,, X) is a collection of M, codewords
OOl&9448/81/0100-0049$00.75
01981
IEEE
be mesA code
xii E%;,
IEEE
50
TRANSACTIONS
ON INFORMATION
decoders
encoders
Fig.
1.
IT-27,NO. 1,JANUARY 1981
Interference
channel
decoders
Fig. 2.
C.
i E%,;
M2 codewords xZj E%;, j E $X,2; Ml disjoint decoding sets %Jli r%y, iE%,;
and M, disjoint decoding
sets %!I~
j L ‘?Jf, j E%,
such that
jkm E C, x GJ, x 3,
Intereference
channel
Cm.
such that
P,9 =
MI MI
,
&f,‘M
,z
z
2 1=1/‘=3
W;(‘tiIx,ix2j)‘A>
(2*1)
P,“z=
L
N\
1
Pe23 &
z
2 r=*
where c indicates
VOL.
channel
channel
$
f4(cB;j(xlix2j)
<A,
(2 .2)
j=l
the complement
“~(@lxlix2j)=
set, and for a= 1,2,
z1 o~(Ylxlix2j)’
YE@
We shall call Pe,, Pe2 the error probabilities
for the code
(n, Ml, M2, X) (calculated under the assumption
that each
message of i E $X7,, and j E $X2 is produced
with equal
probability).
The maps +i: d-+x,, for iE%,
and +2:
j++x2j for jEGsrc2 are the encoding functions for senders 1
and 2, respectively, whereas the maps #i: y, Hi if yi E%J~~
and #2: y2 ++j if y2 ~~~~ are the decoding functions
for
receivers 1 and 2, respectively.
The interference
channel ( !X i, Gx, , o, 9,) q2) is denoted
by C and models two senders communicating
information
to two receivers via a common channel (Fig. 1).
A pair (R,, R,) of nonnegative
real values is called an
achievable rate for the interference
channel C if for any
q> 0, O<X< 1, and for any sufficiently
large n, there
exists a code (n, M,, M2, X) such that
;logM,>R,-v,
a=1,2.
N
1
,F
w;(a~jkmIXlikX2jm)
“9
(2*5)
2 2 r/km
where the sums are taken over all ijkm EC, X%, Xc, X
GJL,. The maps +,: ik6-+xlik, and cp,: jmHx2j,
are the
encoding functions, whereas the maps +I : y, Wkm if yi E
and q2: y2 ++jkm if y, E$?J~~~,,, are the decoding
%ikm
functions.
The interference
channel
(‘Xi, X2, w, %i, ‘?J2)
used in this way is denoted
by C,, and models two
senders communicating
both “private”
and “common”
information
to two receivers, where i E Cl, j E C, are private
messages and k E 9Z 1, m E GJt2 are common messages (Fig.
2).
A quadruple
(S,, T,, S,, T2) of nonnegative
real values
is called an achievable
rate for C, if for arbitrary n > 0,
O<h < 1, and for any large n, there exists a code
(n, L,, N,, L,, N2, X) such that
ilogL,
+og
>s,-7,
a= 1,2,
(2.6)
N, > T, -7,
a= 1,2.
(2.7)
Lemma 2.1: If there is a code (n, L,, N,, L,, N,, A) for
C,, then there is a code (n, L, N,, L, N2, A) for C.
Proof:
Suppose
that
{xlik, xZjm, auk,,,, %2jk,,,} is an
for C,.
Setting
alik=
we have
(2.3) (n, L,, N,, L,, N2, X) code
ti32jm =
U>=lalikm>
u~~l%2jkm?
The set of all achievable rates is called the capacity region
of C. To establish
an achievable
rate region for the
channel C, it is convenient
to introduce a modified interference channel C, as in the next section.
B. Interference
THEORY,
Channel C,
The modified interference
channel
C, differs from C
only in the way of using the quintuple (%,, %,, w, %,, ‘?J2).
Instead of two message sets a,,
a2 let us consider four
message
sets Ci={l;..,Lr},
G3-C1={1;-.,N1},
C,=
(1,. . . 3 L2}, 973 = { 1; . . , N2}. An (n, L,, N,, L2, N2, A)
code is a collection
of L,N, codewords xlik E%;,
ikE
xZjm E %i, jm E F?.,X s2,
r., x 92,; L2N2 codewords
L , N, N, disjoint
decoding
sets ??I,+,, C !!4;, ikm E cl X
3, x%,;
and L,N,N,
disjoint decoding sets $i32jkm C%l,
~W~(~~ikmIXlikX2jm)~
4(~3fikInlikX2jm)
W~(~~jmlxlikx2jm
> 6W;(%jkmIXlikX2jm).
From conditions
(2.4) and (2.5) it follows that conditions
(2.1) and (2.2) are satisfied with i, j, M,, M2, replaced by
ik, jm, L,N,, L,N,, and hence {Xlik, xZjm, auk, %2j,} is
an (n, L,N,, L2N2, A) code for C.
Q.E.D.
Corollary 2.1: If (S,, T,, S,, T,) is an achievable rate for
C,, then (S, + T,, S, + T2) is an achievable rate for C.
C. Jointly
Typical Sequences
We summarize
here
sequences
(see Berger
the properties
of jointly
typical
[8], Cover [9], and Han and
I-IAN AND KOBAYASHI:
INTBBFERENCE
51
CHANNEL
Kobayashi
[IO]). Let Z,, * * * , Z, be dependent
random
variables with values in the finite sets, %,; * * , $!$ respectively. For a subset A of a= { 1; * * , r}, set Z, = (Zi)iEA,
%A =IIiEAzi,
and let Zi be an independent
identically
distributed (i.i.d.) n sequence of Z,. Setp(z) = Pr {Z, =z},
z E %A. An n sequence zA = (z(,), . * * , zcn)) E~J is called
jointly dypical for Z, if for all z EzA:
Fig. 3.
INzlL4)-nP(z)l
Test channel.
. ..g$$
A
where N(z IzA) is the number of t such that z =z(‘). We
denote by <(Z,)
the set of all jointly e-typical sequences
for Z,, and by T,(ZAIze), zB E%:, the set of all zA E%i
such that tAzB E T,(Z,Z,).
1~.
Lemma 2.2: Let A and B be disjoint subsets of Q, and
zAzs E T,(Z,Z,).
For any 0< l< 1 and sufficiently
large
n, we have
9
ii)
Pr{z,” E~(zAlzB)lz~=zB}
(I-c)exp[n(H(Z,IZ,)-2e)]
iii)
exp[-n(H(Z,IZ,)+26)]
<Pr{Z,R=zAIZg=zB}<exp[-n(H(Z,IZ,)
- 2r)].
<
l~(zAlzB)l
III.
(S,, T,, S,, T2) of nonnegative
T, <Z(Y,;
IV-2]U,W,Q),
(3 -2)
(3 -3)
(3 -4)
S,+T,<W',;U,W,lw,Q),
(3.5)
S,+T2<Z(Y,;U,W2IW,Q), (3.6)
T,+T,<Z(Y,;W,W,lu,Qh (3 *7)
S,+T,+T,gZ(Y,;U,W,W,lQ); (3.8)
S,<W'i;u,lW,w,Q),
(3 -9)
T,<ICY,; W,iU,W,Q), (3.10)
772
<ICY,;
w,lu,W,Q), (3.11)
S2+T,<ICY,;
u,W,Iw,Q)> (3.12)
(3.13)
S,+T2GZ0’2;
WW+',Q),
> 1-c;
RATE
such that
S,<ICY,;
U,IW,w,Q),
T,<Z(Y,;W,lu,W,Q>,
~exp[n(H(ZA1z,)+26)l;
AN ACHIEVABLE
real numbers
REGION
We shall establish an achievable rate region for C, and
(3.14)
then derive the associated
achievable
rate region for C.
T,+T,<Z(Y,;W,W,lu,Q>,
Let us consider auxiliary random variables Q, U,, IV,, U,,
(3.15)
S,+T,+T,<Z(Y,;U,W,W,JQ).
and W,, defined on arbitrary finite sets 2, %,, “w;, (?!L2,
Furthermore,
let S be the closure of uZE&5(Z).
and wz, respectively,
(Q is the time-sharing
parameter),
and also X, and X, defined on the input alphabet sets
Theorem 3.1: Any element of S is achievable
for the
%,, %* and Y,, Y,, defined on the output alphabet sets 9,
interference
channel C,.
and%,.Let??*bethesetofallZ=QUWUWXXYY 112
2
1212
Remark I: It is easy to see that S is convex. Inequalisuch that
ties (3.2)-(3.15)
may be represented
in a more compact
form
as
follows.
Set
r,
=
S,,
r,
=
T,,
r3
=
S,, r4 = T2, V, =U,,
independent
U,, W,, U,, and W, are conditionally
9
V,=W,,
V,=U,,
V,=W,,
x,=(1,2,4},
x,=(2,3,4},
given Q;
and
define
for
a
=
1,2,
ii) Xl =flWWl IQ>,X2=MU,W,IQ>;
Y, =y,lX,
= x,,
x2 = x2} =
iii) Pr {Y, = y,;
v,IV,,-,Q),for all SGZ,,
dS)=Z(y,;
dY,Y2I-%x2).
For each qE5l,f,(*Iq):
%, Xw,+%,,
andf,(.Iq):
a2 X
%2-+%2 are arbitrary deterministic
,functions, and f,, f2,
2, %,, ‘U,, %,, and “1Js,, range over all possible choices.
Note that a triple (f,, f2, o) defines a test channel of C,
(Fig. 3) which has G2L,,%,, G2L2,and “u;, as input alphabet
sets; 9, and owl
2 as output alphabet
sets, and channel
probabilities
w,( *I *) in state q E 2 as defined by
(3.2)-(3.8) (correwhere V’ = ( &)iE:s. Then inequalities
sponding to receiver 1) and inequalities
(3.9)-(3.15)
(corresponding
to receiver 2) may be rewritten as
p,(S)
izsI;
G P,(S),
for all SC&,
(3.16)
izsri
<p2(S),
for all SCZ,.
(3.17)
has the following
properties:
(3.18)
fJ,(+)=0;
P,(S)
ThenZ=QUWUWXXYY
E??* implies thatX,,X,;
112
2
1212
Y,, Y, are random variables induced
on %,, x2; %,, q2
from U,, W,, U,, W,, Q via this test channel.
For any Z E$?*, let S(Z) be the set of all quadruples
c P,(T),
p,(SuT>+p,(SnT)~p,(S>+p,(T);
SCT;
(3.19)
(3.20)
which means that Pa = (p,( e), Z,) forms a polymutroid
(a = 1,2) in the terminology
of combinatorics
(cf. Welsh
52
IEEE
[ 1 l]), and that the set S(Z)is the intersection
of the
independence
polyhedra
associated
with these two polymatroids.
This property
is made full use of in proving
Theorem 4.1. As for polymatroidal
aspects found in multiterminal information
theory, see Han [12], [13], [ 191 and
Han and Kobayashi
[lo].
Proof of Theorem 3.1: It is sufficient
to show the
achievability
of elements in S(Z) for each ZE‘!?*. Fix a
Z = QU,W,U,W,X,X,Y,Y,
E y*
and
take
any
(S,, T,, S,, T2) satisfying
conditions
(3.2)-(3.15).
Given
any n>O, define L,, N, (a= 1,2) by
$ogL,=S,-11.
(3.21)
+logN,=T,-q.
(3.22)
For notational
simplicity we use the following shorthand
for the probability
distributions
of Q, U,, U,, W,, W,:
TRANSACTIONS
ON INFORMATION
E,‘(lll),or
<Pr{Ef(lll)}+
functions
+2: c, X
U
E,(ikm)
2
Pr{E,(ikm)~E,(111)}.
ikm#lll
Pr{Ef(lll)}
<e.
On the other hand, by the symmetry
random variables we have
x
(3.28)
among
the relevant
Pr{Edikm)lWll))
ikm#lll
=(L,-l)Pr{E,(211)~E,(l11)}
+(N,-1)Pr{E,(121)]E1(111)}
+(N,
- I)Pr{E,(112)(E,(111)}
+(L,-l)(N,-1)Pr{E,(221)~E1(111)}
- 1)(N2 - 1)Pr{E,(212)]E1(111)}
+(N,-l)(N,-1)Pr{E,(122)~E1(111)}
+(L,-l)(N,-l)(N,-1)Pr{E,(222)~E1(111)}.
(3.23)
(3.29)
(3.24)
Let us first evaluate
Pr{ E,(21 l)] E,(l 1 l)}. Applying
Lemma 2.2, ii) to the case Z, = U,, Z, = Q W,W, Y, and
Lemma 2.2, iii) to the case Z, = U,, Z, = Q yields
where
f~(u,jw,,~q)=(f,(~lf~~~fll~“~)~~~~~f,(~I1~~~~~Is~“~)),
P4-WWlE,W~)) Gew[ -n(H(U,lQ)-2E)]
f~(u~~w~~~~)~(f~(u~~wz(~~~~*~)~~~~~f~(u~~w2(~~~~“~))~
The value of q is told to senders 1 and 2.
2) Decoding Rule: Suppose that receiver 1 has received
y, E%J ;. If ikm is the unique element of cl X %, X %2
such that
q~,iw,kwzm Y, E r,(Qu,w,w,r,L
(3.26)
From the way the random sequences q, uli, uzj, wlk, w,,
are generated
and by Lemma
2.2, i), with Z, =
QU,W,W,Y,,
and Z, is constant, it follows that
-+(L,
Is2I=N2.
+,: cl X%,-+%x;,
1981
(3.27)
We shall generate a random code 6? in the following way.
Let q=(q’*‘; . . , q(“)) be a random sequence of 2” distributed according to the probability
II:, ,pp(qCf)), and let
{uli =(~~~);. . , u$)):
i= 17. * * 2 L,} be L, independent
random
sequences
of %; each of which is distributed
according to the conditional
probability
n:=,p,l(uj:)lq(‘))
given q. Similarly, let {uzj =(z#;
. . , u!$)): j= 1;. *, L,},
k = 1,. . . , N,},
{Wan =
{wlk = (wl’;), . - . , w{;)):
(w&. *,wiz): m= 1;. *, N2} be independent random
sequences of %;, “Ilr,“, ‘?I.&“,respectively,
each of which is
according
distributed
to
II := , pu,( u $y I qCf)),
II~=,p,,(~,(:!jq(~)),
II~=,pw2(w$~Iq(‘)),
given q, respectively.
I) Encoding Rule: Let C,, %, , c,, and ?X2, be four
message sets such that
Define the encoding
9L2+%;
by
1, JANUARY
ikm+=lll
=w21Q=q}-
If&I=-&,
NO.
otherwise #2( y2) is arbitrary.
Here the value of q is also
told to receivers 1 and 2.
3) Evaluation of Error Probabili&: Since each message
emitted to the channel yields the same error probability,
we may confine ourselves to the situation where 1111 E
l?, X 3, X l?, X %, was sent. First we consider the decoding error probability
Fey (averaged over the random code
C?) for receiver 1. Suppose that y, ~9:
was received by
receiver 1, and let E,(ikm),
ikmEe,
X%,
X %z, denote
the event (3.25). Then we have
pw,(wlIq)=Pr{Wl
=w,lQ=q},
I%,,l=N,,
IT-27,
~2 E T,(Qu,KKr,);
q+jW,am
Py(u,Iq)=Pr{U,=u,lQ=qj,
Pu,(u21q)=Pr{U,
=U2IQ=q),
IC,I=L,,
VOL.
define the decoding
function
#,: 3; --&, X 3, X GJt, by
1c/,(y,) = ikm; otherwise let #,( y,) be an arbitrary element
of C, x Gs, x CR,. Similarly,
the decoding
function
#2: 9; +c, X 3, X ??K2 by G2( y2) =jkm,
if jkm
is the
unique element of C, x CR2 x %, such that
pp(q)=Pr{Q=ql,
pw,(w21q)=Pr{W,
THEORY,
(3.25)
.exp[n(H(UllWlW,YlQ)+2e)]
=exp[
-n(Z(W,KY,;
=exp[
-n(Z(Y,;
Using similar upper-bounding
terms in (3.29) and substituting
U,lQ>-4~11
U,lW,W,Q)-4~)].
techniques
for the other
(3.21) and (3.22) into L,
HAN AND KOBAYASHI:
INTERFERENCE
53
CHANNEL
and N, in (3.29), we have
2
I%{E,(ikm)~E,(111)}
ikm#lll
<exp[-n(Z(Y,;U,lW,W,Q)-S,
+v-4e)]
+ew[
-n(Z(Y,;
WllulW2Q)-Tl
+q--4c)]
+exp[
-n(Z(Y,;
W,lu,W,Q)-Tz
+77--4e)]
+~~p[-~(~(~,;~,~,l~2Q)-(~l
+7’,)+?1-4e)]
+ew[
-n(Z(Y,;
+ew[
-~(Z(Y,;W,W21~,Q>-(Tl+T2)+~-4~)]
+exp[
-n(Z(Y,;
Since E >0
sufficiently
u,W,lW,Q)-(S,+T,)+~-4~)]
U,W,W,lQ)-(S,
+T, +T,)+s--4e)].
can be made arbitrarily
small by letting
large, conditions
(3.2)-(3.8) yield
x
Pr{E,(ikm)]E,(l11)}
<A,
n be
(3.30)
ikm#lll
for a prescribed 0 <h < 1. Consequently
by (3.27) (3.28),
and (3.30), it follows that FA <2A.
For a receiver 2, we consider the event E,( jkm) specified by (3.26) instead of the event E,(ikm).
In the same
manner, the decoding error probability
Fe\ for receiver 2
can be evaluated
as Fe\ <2X on the basis of conditions
Q.E.D.
(3.9)-(3.15).
Now we can state an achievable
rate region for the
interference
channel C. Denote by a(Z),
Z ET*, the set
of all (R,, R2) such that R, =S, + T,, R, =S, + T, for
some (S,, T,, S,, T2)~S(Z),
and define %*=the
closure
of UZEy*%(Z).
Theorem
interference
3.2: Any element
channel C.
of ‘%* is achievable
Proof: The proof is immediate
Corollary 2.1.
from Theorem
convex closure
of
U
q(Z).
We now relate Theorem 3.2 to the previous results. To
this end it is helpful to define a simple subset of %*. Let
a,,(Z),
ZE‘?*, be the set of all (R,, R,) such that
R,<q+W,;U,lw,w,Q),
&<~2++I(Y2;4lw,w,Q>,
R,+R,<qz+Z(Y,;
u,lw,w,Q)+Z(y,;
u,lw,w,Q),
(3.32)
where
a,=min{Z(r,;w,lw,Q),Z(r,;w,lw,Q)},
u2=min{Z(r,;w,I~Q),Z(Y2;W21~Q)},
u12
=min{Z(Y,;
W,W,lQ),
Z(Y,;
w,7%lQ),
ICY,;
W,lw,Q)+Z(y,;
w,lW,Q)>
Z(Y,;W,lW,Q>+Z(r,;w,lw,Q>>.
(3.33)
for the
3.1 and
Remark 2: 1) %* is convex. 2) From the viewpoint of
polymatroids
the proof of Theorem 3.1 (or 3.2) may be
regarded as a natural generalization
of the superposition
coding technique exemplified by Cover [4] and Ahlswede
121. We treat here the situation
where an “atomic”
achievable
rate region S(Z) is the intersection
of two
polymatroids
P,, P2 of three dimensions
(specified
by
U,W,W, and U,W,W,, respectively;
cf. Remark I), while
Cover has treated the situation where an atomic achievable rate region is the intersection
of two polymatroids
of
two dimensions.
3) We may define another achievable rate
region for C instead of ‘%*: denote by ‘?? the set of all
Z=QU,W,U2W2X,X2Y,Y2E??*
such that Q=+,
($ is a
constant), and let
a=
[15]), and interference
channels (cf. Carleial [3] and Sato
[5]), a formulation
using the convex-hull
operation
as in
(3.31) has often been adopted instead of using the timesharing parameter Q as in Theorem 3.1 (originally due to
Cover 141). Here we prefer to use Q because the inverse
inclusion %,~a*
is not likely to hold in general. It is our
conjecture
that $i%* strictly extends ‘$L for the general
interference
channel, as is suggested by a numerical example for the Gaussian interference
channel (see the end of
Section V). (For this reason, e.g., in Han [ 12, theorem 5.11,
pi(S) = I(&; q IUs) should
be replaced
by p,(S) =
W&; yi IU,Q>. ‘I-his remark may also be relevant for the
broadcast
channel.)
The only case where it was ascertained that the convex-hull
formulation
is tantamount
to
the time-sharing
formulation
is the “noncompound”
multiple-access channel (see Han [ 12, lemma 4.11). Notice that
this channel has a single output terminal and accordingly
there is no intersection
of polymatroids.
(3.31)
Define
%z =closure
of
%a =convex
closure
(3.34)
lJ
a,(Z),
ZET*
of
U
a,(Z).
Corolkuy 3.1: The inclusion
relations
a* 1 a$ >%a
(‘%i~>%a) hold and hence ‘$g and %a are achievable rate
regions for C.
Proof: We can express any (R,, R2)E%,,(Z)
R,=S,+T,,
R2=S2+T2
for some (S,,T,,S,,T,)
that:
as
such
S, GZ(Y,;
u,IW,w,Q)>
(3.36)
S, GZ(y,;
u,IW,W,Q);
(3.37)
T,<u,,
T,<u,,
ZET
It is easy to see that %tc’%* because %* is convex. Hence
% is also an achievable rate region for C. In the literature
on coding for multiple-access
channels (cf. Ahlswede [14]
and Han [12]), broadcast channels (cf. Hajek and Pursley
(3.35)
ZET
T,+T,<u12.
(3.38)
It is easily ascertained
that conditions
(3.36)-(3.38)
implies conditions
(3.2)-(3.15)
so that %(Z)2%,,(Z).
Q.E.D.
IEEE
54
Carleial [3] has considered,
ing condition:
instead
TRANSACXIONS
ON ,NFORMAT,ON
THEORY,
VOL.
IT-27,
NO.
1, JANUARY
1981
1
of (3.38), the follow-
T <Z(Y,;W,lQh
T2<Z(Y,;W,lw,Q>
cl
or
c2
T,<Z(Y,;W,Iw,Q>,
TPW',T~IQ>
5
and
T,<W'-2;W,lQ),
T2<Z(r,;w,lw,Q)
or
b2
T,<Z(Y,;W,lW,Q>,
T&I(Y2;W2lQ>.
(3.39)
Let %c(Z) be the set of all (R,, R2) such that R, = S, + T,,
R, = S, + T, for some (S,, T,, S,, T2) satisfying
(3.36),
(3.37), (3.39),
and
define
at, = convex
closure
of
w h‘,c h coincides with the region established
uzdJb(Z)~
by Carleial.
9,
Corollary 3.2: The inclusion %a > %c holds, and hence
is an achievable rate region for C.
Proof: It is easily seen that condition
condition (3.38), so that a,,(Z) 1 %o( Z).
(3.39) implies
Corollary 3.3 (S&o [5]): Let qL, be the convex
of all (R,, R,) such that for some ZE??
R, ~W’,;
X,lX,),
R, Gmin{Z(Y,;
closure
Xl>, Z(Y,; Xl>},
R, GI(Y,;
X21X,).
(3.40)
Then ‘%,=>%s,
region for C.
and
hence
3s
is an
Proof Setting U,W,U,W, =X,$+X,,
cpX,X,+ in (3.39) we have (3.40).
Remark
3: 1) It is straightforward
and q2, in (3.38) satisfy
0, G 012,
02 <a,,,
which enables us to calculate
treme points P,, P2 of %a(Z):
Fig.4. Region‘X,,(Z) (solid line) and region 3,-(Z) (broken
_ - ,-.
P,, Pz are the maxlmal
extreme
points
of X,(Z);
line).
a, =
ICY,; ~,IW,~,Q)+
ICY,; W,IwA?,, a,=Z(Y,; u,IW,w,Q)+
Z(Y,; r,lw,Q),
b, = ICY,; &IW,W,Q, + ICY,; w,lW,Q),
b,=
Z(Y,; &IW,w,Q)+W’,;
w,lJ+‘,Q)> c,=Z(Y,; u,IW,w,Q)+
ICY,; U,IWP’,Q) + W,; wF’,lQh
cz = ICY,; u,IW,w,Q) +
W’s u,l~,w,Q)+W-z;
~,%lQ).
X2), Z(Y,; X2>}
or
R, Gmin{Z(Y,;
c
achievable
rate
or U,W,U,W,
explicitly
(3.41)
the maximal
IV.
=
to check that u,, a,,
(712<a, +a22
tween sequential and simultaneous
disappears. In general,
however, this is not likely to be so. In fact, in Section V
we shall show that a subregion
(Figs. 9 and 10) of %
applied to the Gaussian
interference
channel is strictly
larger than the corresponding region of Carleial [3]. We
conjecture that in general ‘$* fail:, .
ex-
In this section we give a simple
at(Z)
which is easier to compute,
geometrical
shape of a(Z).
The regions %a( Z), atc( Z) are illustrated in Fig. 4. Notice
that even a detailed inspection of the argument of Carleial
would have established the region a,,(Z).
2) The regions
?R*, 3, %g, ‘%a, ait, have all been established
based on
the superposition
technique
as well. But in deriving at,,
Carleial
used only a restricted
version of the general
superposition
coding, named sequential coding; in contrast
with this, the technique used in deriving a*, 3, a:, and
?!%a, should be called a simultaneous superposition
coding.
(Note that the subregions ‘%z and %a, can also be derived
by using sequential coding alone (first on W,, W,, next on
U,, U,), but the whole regions a*, %, can be established
only when we use simultaneous
ones.) For simpler systems
such as degraded broadcast
channels,
the difference be-
explicit expression
thereby revealing
for
the
Theorem 4.1: For any Z E $‘*, the region s(Z)
is equal
to the polyhedron
(see Fig. 5) consisting
of all pairs
(R,, R2) of nonnegative
real numbers such that
P,=(o,+Z(Y,;U,lw,w,Q>,u,2-~,
+Z(Y,;
v,Iw,w,Q)h
P2=(q2-u,+Z(Y,;
U,IW,W2Q),u2
+I(Y,;
U,lW,W,Q>).
(3.42)
SIMPLE EXPRESSION FOR Ck(Z)
R, G b,,
2R,
+
R,
R,
c
G ~2,
p,o,
R,
R, +
+
2R2
R,
< ~129
G ~20,
(4.1)
where
~,=4+I(Y,;4lw,w,Q>,
(4.2)
~2 =@
+I(Y,;
u,Iw,w,Q>,
(4.3)
‘~12 =a,z+W,;
u,IWw,Q)+Z(Y,;U,Iw,w,Q>,
(4.4)
plo=2a:+2V,;
4lw,w,Q)+Z(Y,;U,lw,w,Q>
+:-W'Z,Iw,Q)]+
+min{Z(Y,;
W,lW,Q),Z(Y,;
KlQ>
+[W'i;W,Iw,Q)-a:]+,
Z(Y,;W,Iw,Q>,Z(r,;w,w,lQ>-a:},
([x]+=xifx>O,
[x]+=Oifx<O),
(4.5)
HAN AND KOBAYASHI:
INTEBFEBENCE
55
CHANNEL
R,(z)
Fig. 5.
Shape of A(Z): Q(Z)=ABCDEOG,
%&Z)=HP,PzKO,
‘Xt,(Z)=HC,C,KO.
w-2;w, ‘t;Q),
Z(y,; ~,w,lQh~~}; (4.6)
and
W,Iw,QL4y,;
w,lW%Q)},(4.7)
KIw,Q)9V,;w,lVCQ>},(4.8)
U ,2 =min{Z(Y,
w,w,lQ)~W’i;w,w,lQ)~
Z(Y,;
w,lw,Q)+W2;
w,lw,Qh
ICY,;
W,lw,Q)+Z(r,;
w,lKQ)>. (4.9)
u: =min{Z(Y,
02 * =min{Z(Y,
Furthermore,
the region CR,* remains invariant
if we impose the following constraints
on the cardinalities
of the
auxiliary sets:
Fig. 6.
Relation
between
Q* (sqlid line) and 9X: (broken
2) We can easily ascertain
max{R,
+R,I(R,,
line).
from (3.32) and (4.4) that
R2)E%*}
=max{R,
R2)E%t;(;},
+R,I(R,,
and similarly
Proof:
See Appendix.
max{R,I(R,,
Remark 4: 1) By examining
the relations between
values of pr, p2, p12, plo, and p20, it is seen that
extreme points A, B, C, D of C%(Z) are given by
A =(P,,
PIO -2~,),
B=(P,o
-P,2,2P12
c=
D=(P,o
@PI,
-P203
-2p2,
the
the
= y;Z(Y,;
max{R21(R,,
R2>Ea*}
=max{R,I(R,,
= p~;l(y,;
-P,o),
P20 -Pl2)3
P2).
R,)E~*}=max{R,I(R,,
R,)@G}
X,1&),
R2)@34}
X,1X,).
Therefore,
although
%,* may be larger than ‘%s, the
region CR* must lie within the area delimited by three lines
of slope 0, - 1, cc supporting
the region ‘%E (Fig. 6). This
demonstrates
the kind of extension of an achievable rate
56
IEEE TRANSACTIONS
region that may be possible when we adopt simultaneous
superposition
coding (a*, %) instead of sequential superposition coding (%~,%o,%o)
(cf. Figs. 9 and 10 in Section v>.
V.
GAUSSIAN INTERFERENCE CHANNEL
So far we have treated only the discrete interference
channel
C and have established
a new achievable
rate
region for C. The result is applicable with obvious modifications to the Gaussian interference
channel.
In this section, we numerically
compute the achievable
rate region for the Gaussian interference
channel to get a
deeper insight into the properties of the region %* (or 3).
A memoryless
Gaussian
interference
channel
G is a
quintuple (%,, Gx,, w,~,,~2)with~~=~2=~1=~2=IW
(the field of real numbers),
and a channel probability
w
specified by
’
y, =axl +bx,
+n:,
(5.1)
y2 =cx, +dx,
+n;,
(5.2)
for x1 E%,, x2 E%,, y, @4,, and y, E%,, where nr, n:
are independent
Gaussian additive noises with mean zero
and variance N, and f12, respectively.
This kind of system
has been studied by Carleial [3] and Sato [5].
From the viewpoint of achievable rates, the channel G
is equivalent to the following “standard”
one:
ON INFORMATION
THEORY,
(5.3)
y2 =Gx,
(5 -4)
+x2+n2,
where we have set al2 =b2N2/d2Nl,
a2, =c2Nl/a2N2,
and
n, and n2 are independent
Gaussian
additive noises with
mean zero and variance one. In the sequel we confine
ourselves to channels of this form.
As usual we impose power constraints
on codewords
Xii, xzj (iE91L1,jE91L2):
Proof: The same argument
as in the proof of Theorem 3.1 establishes the theorem, except that the decoding
functions
#i and I/J~ should be based on the maximumQ.E.D.
likelihood criterion.
A. Computation
We are now in a position to evaluate the regions 4* and
8 numerically.
However,
the computation
needed
to
evaluate
the whole 9* may be formidable.
Even the
computation
for its subregion 8 seems to be impractical.
Therefore we impose the following customary
restriction
on the input signals.
Let the subclass
9’(Pi, P2) of
??(P,,P,)
be defined
by Z=$LJWUWXXYY
E
112
2
1212
??‘(P,, P2), if and only if Z@‘(P,,
P2), U,, W,, U,, W, are
Gaussian, and Xl = U, + W,, X2 = U, + W,. Let
9’ = convex closure of
u
wz>,
Z~~‘(P,, 4)
gh = convex closure of
u
Z-YP,,
On the other hand,
for G is:
the region
computed
u
ZevP,,
a‘t(z>9
U
4 = convex closure of
~c(Z).
[3]
(5.11)
Pz)
(5.6)
ICY,; W~IW,)=Y(&P,/(~+V,
+a12h2P2>),
w,;
w,w,)=Y((&P,+
+a12A2P2)),
a12~2p2)N+Vl
+a12X2P,)),
Vi; KIwI)=~(h2P2/(l+h2P2
+a2JlPl)),
Z(Y,; ~lIJ%)=y(a21~lPl/(1
w-2;
w*w2)=Y((r;,P2
+A2P2
+ a21hP1)/(1+~24
ICY,; K)=Y(&P~/(~+~,P~
ICY,;
+a,,A,P,)),
W,)=Y(~~P~/(~+A~P~
+a12P2)),
+a2,Pl>),
P2)
Z(r,;w,lU1w,)=y(a12T;,P,/(l+a,,X,p,)),
U
a(Z).
Z@YP,, P*)
by Carleial
ICY,;U,IW,W,)=y(X,P,/(1+a12A2P,)),
ICY,; v,IW,W,)=y(A2P,/(l+a2,AlPl)),
(5 *7)
(5.10)
are specified in Section III. Clearly
Here %o(Z), q&Z>
8’ > g; > g&. We compute 9’ and g; numerically
by applying Theorem 4.1 with ZET’( P,, P2) and compare them
with 8;. (Notice that it is reasonable
to compare these
regions because they are all based on the same range
G?‘(P,, P2) of the auxiliary random variables.)
If we put
y(x) = (l/2) log (1 + x), the relevant quantities in Theorem
4.1 are given by
ICY,; W,IW,)=Y(a12~2P2/(1+~lpl
Z@‘(P,,
$0(Z).
(5.9)
P2)
(5.5)
where xii =(x$i); * . , xi;)), and x2 j =(x9,?, . * . , x$‘J.‘). To
incorporate
the power constraints
(5.5) and (5.6), consider
a subclass $J’*(P,, P2) of ??* ($l’* was specified in Section
III) defined
as follows:
2 = QU,WlU2W2X,X2Y,Y2
E
??*(Pi, P2) if and only if ZE’??* and u’(X,)<P,,
u2(X2)
< P2. We write ZET(P,,
P2) if and only if ZE??*(P,,
P2)
and QF$J.
Paralleling the definitions
of %,* and ‘?Rin Section III,
set
9* = closure of
IT-27,NO. 1,JANUARY 1981
Theorem 5.1: Both 9* and 9 are achievable rate regions
for the Gaussian interference
channel G with power constraints (5.5) and (5.6), and 9* > 9.
$ = convex closure of
y,=x,+~x,+n,,
VOL.
(5-g)
V2;
wlIV,W,)=y(a21~1Pl/(1
+a2AP,)),
+azlh,P,))~
HAN AND KOBAYASHI:
INTERFERENCE
57
CHANNEL
az,=0.25.
Fig.10. P,=6.0,P2 =0.5,a,*= 1.0,
Fig.7. P,=P2 =2.0,al2=azl= 1.5.
R
0.458
0
Fig.8. P,=2.0,Pz =0.5,q2 =uzt= 1.0.
0.203
Fig.11. P,=6.0,P2= 1.5,
~12=a~,=0.55.
G'
Fig.9. P,=2.0,Pz=0.5,a12= 1.0,~,,=0.25.
where u2(U,)=X,P,,
u’(W,)=X,P,,
A, +x, = 1, where
a= 1,2.
We illustrate the results in Figs. 7- 14, where 9’, G& 9;
are indicated by solid lines, broken lines, and chain lines,
respectively
(note that some of these lines overlap). In
Figs. 7, 8, 11, and 14, 4’, $, 8; coincide. Figs. 9 and 10
show that 9’ strictly extends g; (in these cases 86 coincides
with 8;). Notice that 8’ lies within the area delimited by
three lines of slope 0, - 1, and cc, supporting
‘Z& On the
basis of these facts we conjecture that in general %#%o,
and ‘%* # ‘?Kz. Fig. 12 gives an example where 86 ( = 8’)
extends !J& In Fig. 13 we depict 9’ for various values of
a=a12 =a2, with P, = P2 =6 fixed. It is seen that for
l/3 <a < 1, (Fig. 13(a)), the region 9’ monotonically
shrinks as a decreases, but for 0 < a < l/3, (Fig. 13(b)), the
region spreads nearly monotonically
as a decreases, so
that a = l/3 is regarded
as a critical value. Such an
inversion phenomenon
is observed also for other values of
P= P, = P2, where the critical value a seems to be speci-
0
I
-’ Rl
0.973
Fig.12. P,=P,=6.0,a,2=az,=0.55.
fied by a(1 +aP) = 1. Fig. 14 is the same one as
given by Carleial .to show that the time division multiplex/frequency
division
multiplex
(TDu/FDM)
curve
(dotted line) with P, = P2 = 6, and parameters
A, x (= lA):
is not contained
in $. Since the region 9* specified by
(5.7) always contains
the TDM/FDM
curve in view of
IEEE
58
(
0.973
Rl*
TRANSACTIONS
ON INFORMATION
THEORY,
VOL.
IT-27,
NO.
1, JANUARY
1981
0
(4
Fig.14. P,=Pz =6.0,aI2=azl= l/3.
R2
ference
result.
0.0
0.97
0.02
al2 =a2, =O. Here we present
an extension
of his
Theorem 5.2: Let G be a Gaussian inteference
channel
with power constraints
P,, P2, and al2 > 1, a21 > 1. Then
the capacity region of G is the set $&t(G) of all (R,, R,)
such that
0.1
0.2
0.3
l/3\\
’
R,<(1/2)log(l
+f’,)t
(5.12)
R,
(5.13)
~<(1/2)log(l
+P2),
Rl+R2~min{(l/2)log(l+Pl+a12P2),
(1/2)log(l
(
0.
3
Rl
(3)
Fig.13. (a)P,=P,=6.0,1/3gn=a,z=a,,.(b)P,=Pz=6.0,0<a=
a,*=(I~,
< l/3.
the time-sharing
parameter
Q, this figure is an example
which shows that ?J*#S’ (= $), suggesting that a* #9l,
for the general interference
channel.
Remark 5: The regions 8, g’, 86, g& can cover the naive
TDM/FDM
in the sense of Bergmans and Cover [16] but
not necessarily
the nonnaive TDM/i;DM.
On the other
hand, 8* always covers the non-naive
TDM/FDM
as well
as the naive TDM/FDM.
B. Capacity
Channels
Region
of a Class
of Gaussian
Interference
Carleial [6] studied the Gaussian
interference
channel
with strong interference
(al2 > 1 + P,, a,, > 1 + P2), and
showed that the capacity region coincides with that of the
channel
with the same power constraint
and no inter-
+P2 +a,,P,)}.
(5.14)
Proof: It is immediate
that a(G)
coincides
with
a(Z)
with W, =X1, W, =X2, U,=+,
U2=+, and so
a(G)
is an achievable
rate region. Conversely,
suppose
that (R,, R2) is achievable
with a code (n, Ml, M,, A):
{xii, ‘%li}i”frl, {xzj, ‘?i?~~~}jM-2,,
where %, = { 1; * *, M,}, and
R,=(l/n)logM,,
where a=1,2.
Let I&: 9Ra-+%z
and
$a~ ‘9i+9Rk
be the associated
encoding
and decoding
functions (a= 1,2). If a message ij~ Gx, x $
is sent, the
sequences y,, y, received by receivers 1 and 2 are given by
Y*
=x*i + G
Y2 = G
x2j
x*i+x2j
+n*9
+n2,
where n, is an independent
identically
distributed
(i.i.d.)
n-sequence
of noise imposed
on the channel
(a = 1,2).
Define
772(y2)=-&
dY*)=-
&
{Y2-~2(~2C/Y2))}+~~2(~2C/,(Y2)),
{ Y1-91M
Yl))}
+ CL
+1M
YA
HAN AND KOBAYASHI:
INTERFERENCE
59
CHANNEL
and set
APPENDIX
z2 =xli + lq2
x2j + -it
23
To prove the theorem it suffices to determine
G-
z, = G
1
+ -----n
Gi
Xii +xzj
(R, =S,
E of the
Point
case the
1’
the coordinates
+ T,, R2 =S, + T2) of the extreme points G, A, B, C, D,
a( 2) (see Fig. 5).
G: Clearly R, =O, and so we put Sz = T2 =O. In this
inequalities (3.2)-(3.15) reduce to
Then
SI <ICY,;
(5.15)
Pr{m,(yJ=zl}
> leA,ij?
(5.16)
where Aoij is the decoding error probability
for receiver
when ij~‘%, ~9lL, is sent (a= 1,2). Note that
1
MI M2
~,lWGQ),
W',lw,Q)>
<ICY,;
w,lU,w,Q).
Similarly,
y2)+nzo
for receiver
E%ii}
> l-(X,ij+A2ij).
one we have
where n10 is an i.i.d. n-sequence of Gaussian noise n ,0 with
mean zero and variance 1 - l/a,, which is independent
of
n,. Therefore by using (#a, rII, nao), receiver a can exactly
reproduce both i E Gsrc, and j E $& with error probability
not larger than hlii +h2ij. By (5.17) the error probability
averaged over Em, x %, does not exceed 2A. Thus the
channel
G is reduced
to a compound
multiple-access
channel, of which the capacity region is known to coinQ.E.D.,
cide with 9,(G) (see, e.g., Wyner [17]).
It should be noted that Theorem 5.2 improves the result
of Carleial largely because Gaussian interference
channels
are usually considered
with P, > 1, P2 > 1 (noise powers
are one). A discrete counterpart
of this result has been
derived by Benzel [7].
Comment:
While writing this manuscript,
we were informed by H. Sato that he had independently
established
the result given in Theorem 5.2 ([ 181).
VI.
CONCLUDING REMARKS
In this paper we have established a new achievable rate
region for the interference
channel. The argument makes
full use of the polymatroidul
property which relies heavily
on the assumed
independence
of the auxiliary
random
variables U, , U,, IV,, W,. Marton [20] has recently devised
a new coding technique
for the broadcast
channel that
does not assume the independence
of these auxiliary random variables. One of the referees believes that the region
found by us can be improved
by allowing the auxiliary
variables to be correlated as in [20].
(A.21
(A.3)
<ICY,;
G
64.4)
Note that (A.l)-(A.3)
specify a polymatroidal
polyhedron
on
the (S,, Tl) plane. Since S, is irrelevant in (A.4) the value of S,
may be set to the largest value SF =I( Y,; U, 1W,W,Q) possible
under constraints (A.l)-(A.3).
In view of a2, > 1 we can consider a fictitious Gaussian
noise n20 with mean zero and variance 1 - l/a,, which is
of-n,. Let n2a be an i.i.d. n sequence of n,,.
independent
Then z2 +n,, has the same conditional
probability
distribution (given xii, xzj) as y,, so that Pr {z2 +n,, ~93~~) =
Pr { yi ~9~~) > 1 -Xuj. Hence it follows from (5.15) that
(A.1)
U,lW,w,Q),
TI <ICY,;
S,+T
a
(5.17)
Pr{n,(
4.1
PROOFOFTHEOREM
1
TI ~z(Y,;W,lw,Q>t
Then (A.l)-(A.4)
TI
reduce to
~W'i;~Iu2w2Q),
from which it follows that the largest value Tf of T, is a: and
hence R, = Sf + Tp, where
~~=~n{Z(Y,;W,lW,Q),Z(Y,;WllUz~Q>}.
Point E: By a similar
argument
Z(Y,; U,lW,W,Q)+a~,
where
we
have
R, =O,
R, =
~z*=~n{Z(Y,;W21W~Q),Z(Y~;W,JU,W~Q)}.
Point A: Clearly R, =Z(Y,; U,l W,W,Q)+of.
It is sufficient to
find the largest value of R, given R,. By setting
S1 =
Z(Y,; U,j W,W,Q), Tl =uf, the inequalities (3.2)-(3.15) become
r,~Z(YI;
WlW,QL
T2 <ICY,;
W,w,lQ)-4,
(A4
r, <ICY,;
W,lU,W,Q),
(A.71
T,
w,w,lu,Q)-6,
(A-8)
U2lW+YzQ),
64.9)
~I(y2;
S2 GZ(Y,;
S2 +T,
<I(Y,;
S2
<I(&;
S, +T2 <Z(Y,;
(A.51
U,w,IW,Q)>
(A.lO)
v,W,IW,Q)--al*,
(A.ll)
U,W,W,lQ)-a:.
(A. 12)
Since S2 is irrelevant in (A.5)-(A.@,
the value of S2 may be set
to the largest value $ possible under constraints
(A.9)-(A.12).
It is easy to see that
S,O=Z(Y,;U~I~,~,Q)-[U,*-Z(Y~;W,IW~Q)]+.
Then the largest value of T: of T2 for S2 =Szc is given by
T~=min{Z(Y2;W,lW,Q),Z(Y,;W21Q)
Z(Y,;WzIw,Q>,Z<Y,;w,w,lQ>-~t}.
Consequently
R 2 = Sf + Tt.
Point B: We shall say that (R,, R,) dominates
(R;, R;) if
R, > Rb (a= 1,2), and (R,, R,) is maximal in 8%(Z) if R, =Rb
whenever (R;, R;)E%(Z)
dominates (R,, R,). Clearly the extreme points A, B, C, D are maximal. Let (R, =SI + T,, R, =
S2 + T,) be a maximal point. Since all S,, T,, S,, T2 appear in
(3.2)-(3.15) with coefficients one or zero, at the maximal point
decreasing one of S, , T,, S,, T, by a small r > 0 increases each of
the others by exactly r or zero. Therefore, by noting that R, =
St + T,, R 2 = S2 + T,, we conclude that ?IL(Z) must be delimited
IEEE
60
TRANSACTIONS
l/2, - 1, - 2, co. Thus
of slope - 1 and a line
A. The latter line is
is given by (4.5). The
where
PI
By inspecting the forms of inequalities (3.2)-(3.15) on the basis
of properties
(3.18)-(3.20),
we see that the largest value of
S, + T, + S2 + T2 is attained with S, =Sp, S2 = St, where SF =
Z(Y,; U,lW,W,Q),
S: =Z(Y,; U,lW,W,Q).
Then (3.2)-(3.15) are
reduced to
[51
from above by straight lines of slope 0, point B must be the intersection of a line
of slope -2 passing through the point
. .
specified by 2 R, + R2 =p ,o where plo
former line is specified by R, + R 2 =P,~,
[31
141
PI~=~~~{R,+R~I(R,,R~)E~.(Z)}.
TI GZ(YI;
W,lKQ)>
TI +T2 <ICYI; W,%lQh
T2 <ICY,;
W2lW,Q)7
T,
T, <Z(y,; W,lw,Q),
7’1+T2 <ICY,; W,w,IQ>,
R,=S;+T/;
Sp=Z(Y,;U,lW,w,Q>-[up--I(Y,;w,lw,Q)]+,
TP=min{Z(Y,;W,IW2Q),Z(Y,;W,lQ)
W2;
w2Iw,Q)-~z*I+>
W,lW,Q)>Z(Y,;
WIW~IQ+J~}~
Point C: Similarly for point B: R, =2p,,
-pzO, R2 =p20 -p12,
where p12, pzo are specified by (4.4), (4.6).
Finally, the assertion about the cardinalities
of %,, WC,, Q2,
W2, 2, follows by applying
Caratheodory’s
theorem
to the
expressions (4.1)-(4.9).
Q.E.D.
REFERENCES
[l]
[91
[lOI
THEORY,
VOL.
IT-27,NO. 1,JANUARY 1981
R. Ahlswede, “The capacity region of a channel with two senders
and two receivers,” Annals Probabil., vol. 2, no. 5, pp. 805-814,
1974.
A. B. Carleial, “Interference
channels,” IEEE Trans. Znform. Theory, vol. IT-24, pp. 60-70, Jan. 1978.
T. M. Cover, “An achievable
rate region for the broadcasting
channel,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 399-404,
July 1975.
H. Sato, “Two-user
communication
channels,”
IEEE Trans. Znform. Theory, vol. IT-23, pp. 295-304, May 1977.
A. B. Carleial, “A case where intereference
does not reduce capacity,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 569-570, Sept.
1975.
R. Benzel, “The capacity
region of a class of discrete additive
degraded interference channels,” IEEE Trans. Znform Theov, vol.
IT-25, pp. 228-231, Mar. 1979.
T. Berger, “Multiterminal
source coding,” in The Information Theory Approach to Communications, G. Longo, Ed. Vienna; Springer,
pp. 172-231, CISM Courses and Lectures, No. 229, 1978.
T. M. Cover, “A proof of the data compression
theorem of Slepian
and Wolf for ergodic sources,” IEEE Trans. Inform. Theory, vol.
IT-21, pp. 226-228. Mar. 1975.
T. S. Han and K. Kobayashi, “A unified achievable rate region for
a general class of multiterminal
source coding systems,” IEEE
Trans. Inform. Theorv, vol. IT-26, PP. 277-288. May 1980.
D. J. A. Welsh, Ma&did Theory. -London: Academic,
1976.
T. S. Han, “The capacity region of general multiple-access channel
U21W~WzQ)+o2*,
+[Z(Y,;
[71
<W,;W,lW,Q)>
from which it follows that the largest values Tf + T: of Tl + T2
1s ut2 in view of inequalities
(3.41), where ui2 is specified by
(3.33). Hence the point B is specified by R, =plo -p12, R2 =
2p,, -p,,,,
where p12 is specified by (4.4).
Point D: Similarly for point A:
R,=Z(Y,;
VI
ON INFORMATION
C. E. Shannon, “Two-way communication
channels,” in Proc. 4th
Berkeley Symp. on Mathematical Statistics and Probabilify, Vol. 1.
Berkeley, CA: Univ. California Press, 1961, pp. 61 l-644.
with certain correlated sources,” Inform. Contr., vol. 40, no. 1, pp.
37-60, 1979.
“Slepian-Wolf-Cover
theorem for networks of channels,”
1131 -,
presented at IEEE Int. Symp. Information Theory, Grignano,
Italy, June 25-29, 1979; Znform. Contr., to be published.
R. Ahlswede, “Multi-way
communication
channels,” in Proc. 2nd
International Symp. on Znformation Theory, Tsahkadsor,
Armenia,
USSR, Sept. 2-8, 1971, pp. 23-52, Hungarian
Academy of Sciences, 1973.
of an achievable rate
1’51 B. E. Hajek and M. B. Pursley, “Evaluation
region for the broadcasting
channel,” IEEE Trans. Inform. Theory,
vol. IT-25, pp. 36-46, Jan. 1979.
WI P. P. Bergmans and T. M. Cover, “Cooperative broadcasting,”
ZEEE Trans. Inform Theory, vol. IT-20, pp. 317-324, May 1974.
results in the Shannon
theory,” IEEE
1171 A. D. Wyner, “Recent
Trans. Inform. Theory, vol. IT-20, pp. 2-10, Jan. 1974.
V81 H. Sato, “A study of a Gaussian two-user channel,” submitted to
IEEE Trans. Inform. Theory.
at the encoders,”
1191 T. S. Han, “Source coding with cross observation
IEEE Trans. Znform. Theory, vol. IT-25, pp. 360-361, May 1979.
[I41
1201 K. Marton, “A coding theorem for the discrete memoryless broadcast channel,” IEEE Trans. Znform. Theov. vol. IT-25, pp. 306-311,
May 1979.