ieee transactions on information theory, vol. 49 a new...

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-27, NO. 1, JANUARY 1% 1 49

A New Achievable Rate Region for the Interference Channel

TE SUN HAN, MEMBER, IEEE, AND KING0 KOBAYASHI, MEMBER, IEEE

Abstruct-A new 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 multivari- able 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

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.

(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. PRELIMINARIES

In this section we shall define the interference channel and state the problem. We denote random variables by x, Y, u, * * * with values X, y, ~4,. . 1 in finite set %, %, %, * . * 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

~Y,Y,~x,x,) of (Y,~Y~)~% X% given (x,,x2)F%X ‘X2. The marginal distributions w,, w2 of the o are given

by

~1(Y,lX*-%)’ 2 4Y,Y2l+%)~ Y2 E%

Since we confine ourselves to memoryless channels, the conditional probability w”( yi yz I xi x2) of yi y2 E ‘%y X 9; given xlxz E%; X95,” is

o”( y1y21x,x2)= fi W(yI(‘)ypIXp xf’), t=1

where

xc2 - -(xp,.*-,xp)E%~, y,=(y~1);..,y,(“))E~,“,

a= 1,2.

Similarly for o;,o;. Let %,={1,2;..,M,}, %,={1,2;..,M,} be mes-

sage sets for senders 1 and 2, respectively. A code (n, M,, M,, X) is a collection of M, codewords xii E%;,

OOl&9448/81/0100-0049$00.75 01981 IEEE

50 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-27, NO. 1, JANUARY 1981

channel channel

encoders decoders

Fig. 1. Interference channel C.

decoders

Fig. 2. Intereference channel Cm.

i E%,; M2 codewords xZj E%;, j E $X,2; Ml disjoint de- jkm E C, x GJ, x 3, such that coding sets %Jli r%y, iE%,; and M, disjoint decoding sets %!I~ j L ‘?Jf, j E%, such that P,9 =

, MI MI &f,‘M ,z z W;(‘tiIx,ix2j)‘A> (2*1)

2 1=1/‘=3 P,“z = L N\ N ,F w;(a~jkmIXlikX2jm) “9 (2*5) 1 1 2 2 r/km

Pe23 & z $ f4(cB;j(xlix2j) <A, (2 2) . where the sums are taken over all ijkm EC, X%, Xc, X 2 r=* j=l GJL,. The maps +,: ik6-+xlik, and cp,: jmHx2j, are the

where c indicates the complement set, and for a= 1,2, encoding functions, whereas the maps +I : y, Wkm if yi E

%ikm and q2: y2 ++jkm if y, E$?J~~~,,, are the decoding

“~(@lxlix2j)= z1 o~(Ylxlix2j)’ functions. The interference channel (‘Xi, X2, w, %i, ‘?J2) YE@ used in this way is denoted by C,, and models two

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. (2.3)

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 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 r., x 92,; L2N2 codewords xZjm E %i, jm E F?., X s2, 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,

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, >s,-7, a= 1,2, (2.6)

+og 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 (n, L,, N,, L,, N2, X) code for C,. Setting alik= U>=lalikm> ti32jm = u~~l%2jkm? we have

4(~3fikInlikX2jm) ~W~(~~ikmIXlikX2jm)~

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 the properties of jointly typical sequences (see Berger [8], Cover [9], and Han and

I-IAN AND KOBAYASHI: INTBBFERENCE 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:

INzlL4)-nP(z)l . ..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 Pr{z,” E~(zAlzB)lz~=zB} > 1-c; ii) (I-c)exp[n(H(Z,IZ,)-2e)]

< l~(zAlzB)l ~exp[n(H(ZA1z,)+26)l;

iii) exp[-n(H(Z,IZ,)+26)] <Pr{Z,R=zAIZg=zB}<exp[-n(H(Z,IZ,) - 2r)].

III. AN ACHIEVABLE RATE REGION

We shall establish an achievable rate region for C, and then derive the associated achievable rate region for C. Let us consider auxiliary random variables Q, U,, IV,, U,, and W,, defined on arbitrary finite sets 2, %,, “w;, (?!L2, and wz, respectively, (Q is the time-sharing parameter), and also X, and X, defined on the input alphabet sets %,, %* and Y,, Y,, defined on the output alphabet sets 9, and%,.Let??*bethesetofallZ=QUWUWXXYY 112 2 1212

such that

9 U,, W,, U,, and W, are conditionally independent given Q;

ii) Xl =flWWl IQ>, X2 =MU,W, IQ>; iii) Pr {Y, = y,; Y, =y,lX, = x,, x2 = x2} =

dY,Y2 I-%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

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

51

Fig. 3. Test channel.

(S,, T,, S,, T2) of nonnegative real numbers such that

S, <ICY,; U,IW,w,Q), T, <Z(Y,;W,lu,W,Q>, T, <Z(Y,; IV-2]U,W,Q),

S, +T, <W',; U,W,lw,Q), S,+T2<Z(Y,;U,W2IW,Q), T,+T,<Z(Y,;W,W,lu,Qh

S,+T,+T,gZ(Y,;U,W,W,lQ); S, <W'i; u,lW,w,Q), T, <ICY,; W,iU,W,Q), 772 <ICY,; w,lu,W,Q),

S2 +T, <ICY,; u,W,Iw,Q)> S, +T2 GZ0’2; WW+',Q), T,+T,<Z(Y,;W,W,lu,Q>,

S,+T,+T,<Z(Y,;U,W,W,JQ).

(3 -2)

(3 -3)

(3 -4)

(3.5)

(3.6)

(3 *7)

(3.8)

(3 -9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

Furthermore, let S be the closure of uZE&5(Z). Theorem 3.1: Any element of S is achievable for the

interference channel C,.

Remark I: It is easy to see that S is convex. Inequali- 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,, V,=W,, V,=U,, V,=W,, x,=(1,2,4}, x,=(2,3,4}, and define for a = 1,2,

dS)=Z(y,; v,IV,,-,Q), for all SGZ,,

where V’ = ( &)iE:s. Then inequalities (3.2)-(3.8) (corresponding to receiver 1) and inequalities (3.9)-(3.15) (corresponding to receiver 2) may be rewritten as

izsI; G P,(S), for all SC&, (3.16)

izsri <p2(S), for all SCZ,. (3.17)

p,(S) has the following properties:

fJ,(+)=0; (3.18)

P,(S) c P,(T), SCT; (3.19)

p,(SuT>+p,(SnT)~p,(S>+p,(T); (3.20)

which means that Pa = (p,( e), Z,) forms a polymutroid (a = 1,2) in the terminology of combinatorics (cf. Welsh


[ 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,:

pp(q)=Pr{Q=ql, Py(u,Iq)=Pr{U, =u,lQ=qj,

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

q+jW,am ~2 E T,(Qu,KKr,); (3.26)

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

E,‘(lll),or U E,(ikm) ikm+=lll

Pu,(u21q)=Pr{U, =U2IQ=q), <Pr{Ef(lll)}+ 2 Pr{E,(ikm)~E,(111)}.

pw,(wlIq)=Pr{Wl =w,lQ=q}, ikm#lll

pw,(w21q)=Pr{W, =w21Q=q}- (3.27)

We shall generate a random code 6? in the following way. From the way the random sequences q, uli, uzj, wlk, w,,

Let q=(q’*‘; . . , q(“)) be a random sequence of 2” dis- are generated and by Lemma 2.2, i), with Z, =

tributed according to the probability II:, ,pp(qCf)), and let QU,W,W,Y,, and Z, is constant, it follows that

{uli =(~~~);. . , u$)): i= 1 . * * L,} be L, independent 7 2 Pr{Ef(lll)} <e. (3.28)

random sequences of %; each of which is distributed according to the conditional probability n:=,p,l(uj:)lq(‘))

On the other hand, by the symmetry among the relevant

given q. Similarly, let {uzj =(z#; . . , u!$)): j= 1;. *, L,}, random variables we have

{wlk = (wl’;), . - . , w{;)): k = 1,. . . , N,}, {Wan = x Pr{Edikm)lWll)) (w&. * ,wiz): m= 1;. *, N2} be independent random ikm#lll

sequences of %;, “Ilr,“, ‘?I.&“, respectively, each of which is distributed

=(L,-l)Pr{E,(211)~E,(l11)} according to II := , pu,( u $y I qCf)),

II~=,p,,(~,(:!jq(~)), II~=,pw2(w$~Iq(‘)), given q, respec- +(N,-1)Pr{E,(121)]E1(111)}

tively. +(N, - I)Pr{E,(112)(E,(111)} I) Encoding Rule: Let C,, %, , c,, and ?X2, be four

message sets such that +(L,-l)(N,-1)Pr{E,(221)~E1(111)}

IC,I=L,, I%,,l=N,, If&I=-&, Is2I=N2. -+(L, - 1)(N2 - 1)Pr{E,(212)]E1(111)}

Define the encoding functions +,: cl X%,-+%x;, +2: c, X +(N,-l)(N,-1)Pr{E,(122)~E1(111)}

9L2+%; by +(L,-l)(N,-l)(N,-1)Pr{E,(222)~E1(111)}.

(3.23) (3.29)

where

(3.24)

f~(u,jw,,~q)=(f,(~lf~~~fll~“~)~~~~~f,(~I1~~~~~Is~“~)),

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 (3.25)

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

P4-WWlE,W~)) Gew[ -n(H(U,lQ)-2E)] .exp[n(H(UllWlW,YlQ)+2e)]

=exp[ -n(Z(W,KY,; U,lQ>-4~11

=exp[ -n(Z(Y,; U,lW,W,Q)-4~)].

Using similar upper-bounding techniques for the other terms in (3.29) and substituting (3.21) and (3.22) into L,

HAN AND KOBAYASHI: INTERFERENCE CHANNEL 53

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,; u,W,lW,Q)-(S,+T,)+~-4~)]

+ew[ -~(Z(Y,;W,W21~,Q>-(Tl+T2)+~-4~)]

+exp[ -n(Z(Y,; U,W,W,lQ)-(S, +T, +T,)+s--4e)].

Since E >0 can be made arbitrarily small by letting n be sufficiently large, conditions (3.2)-(3.8) yield

x Pr{E,(ikm)]E,(l11)} <A, ikm#lll

(3.30)

[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 time- sharing 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 I Us) should be replaced by p,(S) = W&; yi I U,Q>. ‘I-h is 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.

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 (3.9)-(3.15). Q.E.D.

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).

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

Theorem 3.2: Any element of ‘%* is achievable for the interference channel C.

Proof: The proof is immediate from Theorem 3.1 and Corollary 2.1.

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,=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)

Define

%z =closure of lJ a,(Z), ZET*

(3.34)

%a =convex closure of U a,(Z). ZET

(3.35)

Corolkuy 3.1: The inclusion relations a* 1 a$ >%a (‘%i~>%a) hold and hence ‘$g and %a are achievable rate regions for C.

a= convex closure of U q(Z). ZET

(3.31)

Proof: We can express any (R,, R2)E%,,(Z) as R,=S,+T,, R2=S2+T2 for some (S,,T,,S,,T,) such that:

S, GZ(Y,; u,IW,w,Q)> (3.36)

S, GZ(y,; u,IW,W,Q); (3.37)

T,<u,, T,<u,, 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).

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 Q.E.D.

54 IEEE TRANSACXIONS ON ,NFORMAT,ON THEORY, VOL. IT-27, NO. 1, JANUARY 1981

Carleial [3] has considered, instead of (3.38), the follow- 1

ing condition:

or

and

T <Z(Y,;W,lQh T2 <Z(Y,;W,lw,Q> cl c2

T,<Z(Y,;W,Iw,Q>, TPW',T~IQ> 5

T, <W'-2;W,lQ), T2 <Z(r,;w,lw,Q) or

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

uzdJb(Z)~ h‘ h w ,c coincides with the region established by Carleial.

b2

Corollary 3.2: The inclusion %a > %c holds, and hence c 9, is an achievable rate region for C.

Fig. 4. Region ‘X,,(Z) (solid line) and region 3,-(Z) (broken line). _ - ,-. Proof: It is easily seen that condition (3.39) implies P,, Pz are the maxlmal extreme points of X,(Z); a, =

condition (3.38), so that a,,(Z) 1 %o( Z). 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,=

Corollary 3.3 (S&o [5]): Let qL, be the convex closure Z(Y,; &IW,w,Q)+W’,; w,lJ+‘,Q)> c,=Z(Y,; u,IW,w,Q)+

of all (R,, R,) such that for some ZE?? ICY,; U,IWP’,Q) + W,; wF’,lQh cz = ICY,; u,IW,w,Q) + W’s u,l~,w,Q)+W-z; ~,%lQ).

R, ~W’,; X,lX,), R, Gmin{Z(Y,; X2), Z(Y,; X2>}

or tween sequential and simultaneous disappears. In general,

R, Gmin{Z(Y,; Xl>, Z(Y,; Xl>}, R, GI(Y,; X21X,). however, this is not likely to be so. In fact, in Section V we shall show that a subregion (Figs. 9 and 10) of %

(3.40) applied to the Gaussian interference channel is strictly

Then ‘%,= >%s, and hence 3s is an achievable rate larger than the corresponding region of Carleial [3]. We

region for C. conjecture that in general ‘$* fail:, .

Proof Setting U,W,U,W, =X,$+X,, or U,W,U,W, = cpX,X,+ in (3.39) we have (3.40).

Remark 3: 1) It is straightforward to check that u,, a,, and q2, in (3.38) satisfy

0, G 012, 02 <a,,, (712 <a, +a22 (3.41)

which enables us to calculate explicitly the maximal extreme points P,, P2 of %a(Z):

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)

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-

IV. SIMPLE EXPRESSION FOR Ck( Z)

In this section we give a simple explicit expression for at(Z) which is easier to compute, thereby revealing the geometrical shape of a(Z).

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

R, G b,, R, G ~2, R, + R, < ~129

2R, + R, c p,o, R, + 2R2 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 CHANNEL 55

R,(z)

Fig. 5. Shape of A(Z): Q(Z)=ABCDEOG, %&Z)=HP,PzKO, ‘Xt,(Z)=HC,C,KO.

w-2; w, and

u: =min{Z(Y,

02 * =min{Z(Y,

U ,2 =min{Z(Y,

‘t;Q), Z(y,; ~,w,lQh~~}; (4.6)

W,Iw,QL4y,; w,lW%Q)}, (4.7)

KIw,Q)9V,; w,lVCQ>}, (4.8)

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)

Furthermore, the region CR,* remains invariant if we impose the following constraints on the cardinalities of the auxiliary sets:

Proof: See Appendix.

Remark 4: 1) By examining the relations between the values of pr, p2, p12, plo, and p20, it is seen that the extreme points A, B, C, D of C%(Z) are given by

A =(P,, PIO -2~,),

B=(P,o -P,2,2P12 -P,o),

c= @PI, -P203 P20 -Pl2)3

D=(P,o -2p2, P2).

Fig. 6. Relation between Q* (sqlid line) and 9X: (broken line).

2) We can easily ascertain from (3.32) and (4.4) that

max{R, +R,I(R,, R2)E%*}

=max{R, +R,I(R,, R2)E%t;(;},

and similarly

max{R,I(R,, R,)E~*}=max{R,I(R,, R,)@G}

= y;Z(Y,; X,1&),

max{R21(R,, R2>Ea*} =max{R,I(R,, R2)@34}

= p~;l(y,; 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


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 Sec- tion 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 modifi- cations 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:

y,=x,+~x,+n,, (5.3)

y2 =Gx, +x2+n2, (5 -4)

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):

(5.5)

(5.6)

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 ‘?R in Section III, set

9* = closure of U a‘t(z>9 (5 *7) Z@‘(P,, P2)

4 = convex closure of U a(Z). (5-g) Z@YP,, P*)

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.

Proof: The same argument as in the proof of Theo- rem 3.1 establishes the theorem, except that the decoding functions #i and I/J~ should be based on the maximum- likelihood criterion. Q.E.D.

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>, (5.9) Z~~‘(P,, 4)

gh = convex closure of u $0(Z). (5.10) Z-YP,, P2)

On the other hand, the region computed by Carleial [3] for G is:

$ = convex closure of u ~c(Z). (5.11) ZevP,, Pz)

Here %o(Z), q&Z> are specified in Section III. Clearly 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,; U,IW,W,)=y(X,P,/(1+a12A2P,)), ICY,; v,IW,W,)=y(A2P,/(l+a2,AlPl)), ICY,; W~IW,)=Y(&P,/(~+V, +a12h2P2>),

ICY,; W,IW,)=Y(a12~2P2/(1+~lpl +a12A2P2)),

w,; w,w,)=Y((&P,+ a12~2p2)N+Vl +a12X2P,)),

Vi; KIwI)=~(h2P2/(l+h2P2 +a2JlPl)), Z(Y,; ~lIJ%)=y(a21~lPl/(1 +A2P2 +a,,A,P,)),

w-2; w*w2)=Y((r;,P2 + a21hP1)/(1+~24 +azlh,P,))~

ICY,; K)=Y(&P~/(~+~,P~ +a12P2)),

ICY,; W,)=Y(~~P~/(~+A~P~ +a2,Pl>),

Z(r,;w,lU1w,)=y(a12T;,P,/(l+a,,X,p,)),

V2; wlIV,W,)=y(a21~1Pl/(1 +a2AP,)),


Fig. 7. P, =P2 =2.0, al2 =azl = 1.5.

R

0

Fig. 8. P,=2.0, Pz =0.5, q2 =uzt = 1.0.

0.203 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-

Fig. 10. P, =6.0, P2 =0.5, a,* = 1.0, az,=0.25.

0.458

Fig. 11. P, =6.0, P2 = 1.5, ~12 =a~, =0.55.

I 0 0.973

-’ Rl

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 (= l- A):

is not contained in $. Since the region 9* specified by (5.7) always contains the TDM/FDM curve in view of


(

R2

0.97

(

(4

0.973 Rl*

0.02

0.1

0.2

0.3

l/3\\ ’

0.0

0.

(3)

3 Rl

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 Region of a Class of Gaussian Interference Channels

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-

0

Fig. 14. P, =Pz =6.0, aI2 =azl = l/3.

ference al2 =a2, =O. Here we present an extension of his result.

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

R, <(1/2)log(l +f’,)t (5.12)

R, ~<(1/2)log(l +P2), (5.13)

Rl+R2~min{(l/2)log(l+Pl+a12P2),

(1/2)log(l +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 x2j +n*9

Y2 = G x*i +x2j +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)=- & {Y2-~2(~2C/Y2))}+~~2(~2C/,(Y2)),

dY*)=- & { Y1-91M Yl))} + CL +1M YA


and set 1

z2 =xli + lq2 x2j + -it

G- 23

1 z, = G Xii +xzj + -----n

Gi 1’

Then

(5.15)

Pr{m,(yJ=zl} > leA,ij? (5.16)

where Aoij is the decoding error probability for receiver a when ij~‘%, ~9lL, is sent (a= 1,2). Note that

1 MI M2 (5.17)

In view of a2, > 1 we can consider a fictitious Gaussian noise n20 with mean zero and variance 1 - l/a,, which is independent of-n,. Let n2a be an i.i.d. n sequence of n,,. Then z2 +n,, has the same conditional probability distri- bution (given xii, xzj) as y,, so that Pr {z2 +n,, ~93~~) = Pr { yi ~9~~) > 1 -Xuj. Hence it follows from (5.15) that

Pr{n,( y2)+nzo E%ii} > l-(X,ij+A2ij).

Similarly, for receiver 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 coincide with 9,(G) (see, e.g., Wyner [17]). Q.E.D.,

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 in- formed 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].

APPENDIX PROOFOFTHEOREM 4.1

To prove the theorem it suffices to determine the coordinates (R, =S, + T,, R2 =S, + T2) of the extreme points G, A, B, C, D, E of the a( 2) (see Fig. 5).

Point G: Clearly R, =O, and so we put Sz = T2 =O. In this case the inequalities (3.2)-(3.15) reduce to

SI <ICY,; U,lW,w,Q), (A.1)

TI <ICY,; ~,lWGQ), (A.21

S,+T <ICY,; W',lw,Q)> (A.3)

G <ICY,; w,lU,w,Q). 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, 1 W,W,Q) possible under constraints (A.l)-(A.3). Then (A.l)-(A.4) reduce to

TI ~z(Y,;W,lw,Q>t TI ~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 we have R, =O, R, =

Z(Y,; U,lW,W,Q)+a~, where

~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 (A.51

T2 <ICY,; W,w,lQ)-4, (A4

r, <ICY,; W,lU,W,Q), (A.71

T, ~I(y2; w,w,lu,Q)-6, (A-8)

S2 GZ(Y,; U2lW+YzQ), 64.9)

S2 +T, <I(Y,; U,w,IW,Q)> (A.lO)

S2 <I(&; v,W,IW,Q)--al*, (A.ll)

S, +T2 <Z(Y,; 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


from above by straight lines of slope 0, - l/2, - 1, - 2, co. Thus point B must be the intersection of a line of slope - 1 and a line of slope -2 passing through the point A. The latter line is

. . specified by 2 R, + R2 =p ,o where plo is given by (4.5). The former line is specified by R, + R 2 =P,~, where

PI~=~~~{R,+R~I(R,,R~)E~.(Z)}.

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

TI GZ(YI; W,lKQ)> T, <W,;W,lW,Q)>

TI +T2 <ICYI; W,%lQh T, <Z(y,; W,lw,Q),

T2 <ICY,; W2lW,Q)7 7’1 +T2 <ICY,; W,w,IQ>, 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,; U21W~WzQ)+o2*,

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)

+[Z(Y,; w2Iw,Q)-~z*I+>

W2; 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] 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.

PI

[31

141

[51

VI

[71

[91

[lOI

1131

[I41

1’51

WI

1171

V81

1191

1201

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