An Achievable Rate Region for Interference Channels with Conferencing
An Achievable Rate Region for Interference Channels
with Conferencing
Yi Cao and Biao Chen
Department of EECS, Syracuse University, Syracuse, NY 13244
Email: ycao01@syr.edu, bichen@ecs.syr.edu
Abstract— In this paper, we propose an achievable rate region
for discrete memoryless interference channels with conferencing
at the transmitter side. We employ superposition block Markov
encoding, combined with simultaneous superposition coding,
dirty paper coding, and random binning to obtain the achievable
rate region. We show that, under respective conditions, the
proposed achievable region reduces to Han and Kobayashi’s
achievable region for interference channels, the capacity region
for degraded relay channels, and the capacity region for the
Gaussian vector broadcast channel. Numerical examples for the
Gaussian case are given.
Index terms — interference channels, dirty paper coding,
superposition block Markov encoding, random binning.
I. INTRODUCTION
The capacity region of an interference channel (IC), where
the information sources at the two transmitters are statistically
independent, has been a long standing problem. Carleial was
the first to use the superposition code idea [1] to obtain an
inner bound for IC. This inner bound was later improved by
Han and Kobayashi [2] who gave an achievable rate region
that is the largest reported to this date. Recently, a simplified
description of the Han-Kobayashi (HK) rate region for the
general IC is derived by Chong-Motani-Garg in [3].
A related and less well investigated problem is when the
information sources at the two transmitters are correlated, i.e.,
interference channel with common information (ICCI) [4], [5].
In [4], an achievable rate region, an outer bound, and a limiting
expression for the capacity region were obtained. Later, the
capacity region of this channel under strong interference was
found in [5]. Recently, improved achievable regions for general
ICCI [6], [7] and three new outer bounds for the capacity
region of Gaussian ICCI [8] were proposed. However, all those
results are based on the assumption that the common message
is available noncausally.
In this work, we investigate the problem of user cooperation
in interference channels for the causal case. Here, each user
not only transmits his own message to the intended receiver,
but also serves as a relay to help transmit part of the other
user’s message. We apply the superposition block Markov
encoding, which was used previously for the relay channel
[9] and for user cooperation in multiple access channels [10].
Our proposed achievable rate region is a generalized form of
the HK region for IC [2], the capacity region of degraded relay
channels [9], and the capacity region of the Gaussian vector
broadcast channel (GVBC) [11].
This paper is organized as follows. In section II, we present
the channel model and review some existing results. In section
III, we propose an achievable region for general IC with
transmitter conferencing. In section IV, numerical examples
are used to compare the proposed region with the HK region
and the capacity region of GVBC. We conclude in section V.
II. PRELIMINARIES AND EXISTING RESULTS
A. Definitions
A memoryless discrete IC with conferencing (ICC) is de-
noted by (X1,X2, p,Y1,Y2, Ỹ1, Ỹ2), where X1,X2 are two
finite alphabet sets for the channel input, Y1,Y2 are two finite
alphabet sets for the channel output, Ỹ1, Ỹ2 are two finite
Fig. 1. Interference channel with conferencing at the transmitter side.
alphabet sets for the received signals at the transmitters (which
also serve as relays), and p is the channel transition prob-
ability p(y1, y2, ỹ1, ỹ2|x1, x2). Here we assume the channel
is memoryless and encoders 1 and 2 are allowed to depend
only on their own messages and the past values of ỹ2 and ỹ1.
Let M1 = {1, 2, · · ·,M1} and M2 = {1, 2, · · ·,M2} be the
message sets of sender 1 and sender 2, respectively. Thus, for
w1 ∈ M1 and w2 ∈ M2, the joint probability mass function
of M1 ×M2 ×Xn1 ×Xn2 ×Yn1 ×Yn2 × Ỹn1 × Ỹn2 is given by
p(w1, w2,x1,x2,y1,y2, ỹ1, ỹ2)
= p(w1)p(w2)
i=1 p(x1i|w1, ỹ21, · · ·, ỹ2i−1)
×p(x2i|w2, ỹ11, · · ·, ỹ1i−1)p(y1i, y2i, ỹ1i, ỹ2i|x1i, x2i)
Suppose w1 ∈ M1 and w2 ∈ M2 are sent by transmitters
1 and 2 respectively, g1 and g2 are the decoding functions at
receivers 1 and 2; the average probabilities of decoding error
of this channel are defined as
e,1 ≡
w1,w2
Pr(g1(Y1) 6= w1|w1, w2 sent) (2)
e,2 ≡
w1,w2
Pr(g2(Y2) 6= w2|w1, w2 sent) (3)
http://arxiv.org/abs/0704.1873v1
The capacity region of ICC is the closure of all rate pairs
(R1, R2) such that P
e,1 → 0, P
e,2 → 0 as codeword length
n→ ∞, where R1 = 1n logM1 and R2 =
logM2.
B. Existing Results
1) Chong-Motani-Garg recently derived a simplified de-
scription of the HK region for IC [3], as summarized below.
Proposition 1: Let P∗1 be the set of probability distributions
P ∗1 (·) that factor as
P ∗1 (q, u1, u2, x1, x2) = p(q)p(x1u1|q)p(x2u2|q). (4)
For a fixed P ∗1 ∈ P∗1 , let RcHK(P ∗1 ) be the set of (R1, R2) sat-
isfying (9)-(15) in Theorem 2 of [3]. Then
P∗1 ∈P
RcHK (P ∗1 )
is equivalent to the HK region.
2) The capacity of the degraded relay channel is given in
proposition 2 [9].
Proposition 2: A relay channel consists of an input x1, a
relay output y1, a channel output y, and a relay sender x2
(whose transmission is allowed to depend on the past symbols
of y1). If y is a degraded form of y1 [9], then
C = max
p(x1,x2)
min{I(X1, X2;Y ), I(X1;Y1|X2)}. (5)
3) The capacity region of GVBC is computed using a
covariance matrix constraint on the inputs X = (X1, X2)
of the form E[XXT ] ≤ S. In order to mimic the individual
power constraints P1 and P2 on the two users for the vector
case, the input covariance matrix S is of the form S =
, for some −
P1P2 ≤ c ≤
P1P2. Then, the
capacity region of GVBC is given below [11].
Proposition 3: For each such S and all positive semi-definite
matrices B and D, where B +D ≤ S, both rate pairs
R1 ≤ 12 log
|H1BH
1 +Q1|
, R2 ≤ 12 log
|H2(B+D)H
2 +Q2|
|H2BH
R1 ≤ 12 log
|H1(B+D)H
1 +Q1|
|H1DH
1 +Q1|
, R2 ≤ 12 log
|H2DH
2 +Q2|
are achievable, where H1 = (1, a21) and H2 = (a12, 1). The
convex hull of the union of these pairs over all possible S,B
and D matrices is the capacity region of GVBC.
III. MAIN RESULTS
We first give a brief outline of our encoding-decoding
strategy. We split each user’s message into two parts: M and
W , where M is to be sent directly to the intended receiver,
and W is the cooperative message to be sent to the receiver
via the cooperation of the other user (relay). Our cooperation
strategy is based on superposition block Markov encoding with
the assumption that W can be perfectly decoded by the relay.
The purpose of introducing M is to achieve a reasonable rate
region (no less than IC without conferencing) even when the
conferencing channel is poor. For the message M , we apply
simultaneous superposition coding [2] and further split it into
two parts: private message V and common message U .
For the cooperation in transmitting W , we jointly consider
B blocks, each of n symbols. Each user transmits a sequence
of B − 1 messages w1, · · ·, wB−1 in B blocks, with no
new message in the last block. Note that as B → ∞,
(B − 1)/B is arbitrarily close to 1, hence the penalty on rate
is negligible. Now we take user 1 as an example to show
the whole process. Suppose there are 2nR13 codewords of W1
for user 1 to transmit. We establish a random partition by
randomly throwing them into 2nR10 cells. This partition is
made known to both transmitters and receivers. Suppose user 1
sends message w1,b−1 at block b−1. At the end of block b−1,
following our assumption, user 2 can perfectly decode w1,b−1
and calculate the cell index s1b to which w1,b−1 belongs. At
block b, both user 1 and 2 spend some power transmitting
s1b. This provides the basis for cooperatively resolving the
remaining Y1 uncertainty about w1,b−1. After decoding s1b at
the end of block b, receiver Y1 intersects its ambiguity set
D(y1(b− 1)) (i.e., the set of all codewords w that are jointly
typical with y1,b−1 [9]) with cell s1b and gets the unique
correct codeword w1,b−1 with a probability close to 1.
Since both users 1 and 2 can perfectly decode each other’s
messages W1, W2 and then calculate the corresponding cell
indices S1, S2, we can employ dirty paper coding (DPC) to
transmit V1, U1 and S1 treating S2 as a known interference
at transmitter 1. Similarly at transmitter 2, we can transmit
V2, U2 and S2 treating S1 as a known interference. Thus,
introducing auxiliary random variables M1, N1, G1, H1 and
M2, N2, G2, H2 for DPC, we summarize the achievable region
for ICC in the theorem below.
Thoerem 1: Let Z1 = (Y1, Y2, Ỹ1, Ỹ2, X1, X2,M1, N1, G1,
H1, V2, U2,W2, S2, Q) and let P∗1 be the set of distribution on
Z1 that can be factored into the form
p(q)p(u2|q)p(w2|q)p(s2|q)p(v2|u2s2q)
×p(n1|s2q)p(g1|s2q)p(h1|s2q)p(m1|n1h1s2q)
×p(x1|m1g1s2q)p(x2|v2w2h1q)p(y1y2ỹ1ỹ2|x1x2)
Let S(Z1) be the set of (R1, R2) such that R1 = R11+R12+
R13 and R2 = R22 +R21 +R23 satisfying:
R11 ≤ L11 − I(M1;S2|N1H1Q) (9)
R12 ≤ L12 − I(N1;S2|Q) (10)
R13 ≤ L13 − I(G1;S2|Q) (11)
R10 ≤ L10 − I(H1;S2|Q) (12)
L11 ≤ I(Y1N1H1U2;M1|Q) (13)
L11 + L12 ≤ I(Y1H1U2;M1N1|Q) (14)
L11 + L10 ≤ I(Y1N1U2;M1H1|Q) (15)
L11 +R21 ≤ I(Y1N1H1;M1U2|Q) (16)
L11 + L12 + L10 ≤ I(Y1U2;M1N1H1|Q)) (17)
L11 + L12 +R21 ≤ I(Y1H1;M1N1U2|Q) (18)
L11 + L10 +R21 ≤ I(Y1N1;M1H1U2|Q) (19)
L11 + L12 + L10 +R21 ≤ I(Y1;M1N1H1U2|Q) (20)
L13 ≤ R10 + I(Y1M1N1H1U2;G1|Q) (21)
L13 ≤ I(Ỹ1H1S2;G1|Q) (22)
R22 ≤ I(Y2U2S2N1;V2|Q) (23)
R22 +R21 ≤ I(Y2S2N1;V2U2|Q) (24)
R22 +R20 ≤ I(Y2U2N1;V2S2|Q) (25)
R22 + L12 ≤ I(Y2U2S2;V2N1|Q) (26)
R22 +R21 +R20 ≤ I(Y2N1;V2U2S2|Q) (27)
R22 +R21 + L12 ≤ I(Y2S2;V2U2N1|Q) (28)
R22 +R20 + L12 ≤ I(Y2U2;V2S2N1|Q) (29)
R22 +R21 +R20 + L12 ≤ I(Y2;V2U2S2N1|Q) (30)
R23 ≤ R20 + I(Y2V2U2S2N1;W2|Q) (31)
R23 ≤ I(Ỹ2H1S2;W2|Q) (32)
Let R∗1 =
S(Z1). Swap index 1 and 2 in
all of the above statements and inequalities and we get
R∗2 =
S(Z2). Then the achievable region R∗ =
convhull(R∗1 ∪R∗2) and the cardinality ||Q|| ≤ 33.
Proof : We only need to prove the achievability of R∗1.
Codebook Generation: Let q = (q(1), · · ·, q(n)) be a
random sequence of Qn distributed according to
t=1 p(q
(t)).
Generate 2nR21 i.i.d (independent and identically dis-
tributed) codewords u2(j2) for common messages, 2
i.i.d codewords s2(l2) for cell indices, and 2
nR23 i.i.d
codewords w2(k2) for cooperative messages according to
t=1 p(u
2 |q(t)),
t=1 p(s
2 |q(t)) and
t=1 p(w
2 |q(t)),
respectively. For each pair of (u2(j2), s2(l2)), generate
2nR22 i.i.d codewords v2(i2, j2, l2) for private messages ac-
cording to
t=1 p(v
(t))1. Generate 2nL12 i.i.d
codewords n1(η1) for common messages according to
t=1 p(n
1 |q(t)) and randomly place them into 2nR12 bins2;
generate 2nL10 i.i.d codewords h1(ω1) for cell indices accord-
ing to
t=1 p(h
1 |q(t)) and randomly place them into 2nR10
bins; generate 2nL13 i.i.d codewords g1(ψ1) for cooperative
messages according to
t=1 p(g
1 |q(t)) and randomly place
them into 2nR13 bins. For each pair of (n1(η1),h1(ω1)),
generate 2nL11 i.i.d codewords m1(ξ1, η1, ω1) for private mes-
sages according to
t=1 p(m
(t)) and randomly
place them into 2nR11 bins.
To apply superposition block Markov encoding, we also
need two random partitions. Randomly place the above gen-
erated 2nR23 codewords w2(k2) into 2
nR20 cells, and those
2nL13 codewords g1(ψ1) into 2
nR10 cells.
Encoding: In block b, user 2 wants to send new indices
i2b, j2b and k2b. For cooperatively resolving the remaining Y2
uncertainty about w2(k2,b−1) in the previous block b − 1, it
also sends the cell index of w2(k2,b−1), denoted by l2b. At
the same time, user 1 wants to send new indices i1b, j1b, k1b
and the cell index of g1(ψ1,b−1), denoted by l1b. Since user
1 can also perfectly calculate l2b at the end of block b − 1,
it looks into bins j1b, k1b and l1b for codewords n1(η1b) and
g1(ψ1b) and h1(ω1b) that are jointly typical with s2(l2b),
1Note that the codebook generation for the direct transmission part follows
that of [3] instead of [2].
2Random binning is used both for the superposition block Markov encoding
(relay) part and for DPC. To distinguish, we use ”cell” when referring to
superposition block Markov encoding and ”bin” when referring to DPC.
respectively. For the previously found (n1(η1b),h1(ω1b)),
encoder 1 looks into bin i1b for codeword m1(ξ1b, η1b, ω1b)
such that (q, s2(l2b),n1(η1b),h1(ω1b),m1(ξ1b, η1b, ω1b))
are jointly typical. For the above bin searching, if there
is more than one such codeword, pick the one with the
smallest index; if there is no such codeword, declare
an error. Then, user 1 sends x1 generated according to
t=1 p(x
1 |m1(ξ1b, η1b, ω1b)(t)g1(ψ1b)(t)s2(l2b)(t)q(t))
and user 2 sends x2 generated according to
t=1 p(x
2 |v2(i2b, j2b, l2b)(t)w2(k2b)(t)h1(ω1b)(t)q(t)).
Decoding: User 2, as a relay to user 1, wants to correctly
recover the new index k1b sent in block b. Since it already
knows h1(ω1b) and s2(l2b) during encoding, it looks for all
the sequences g1(ψ1), such that
{q, s2(l2b),h1(ω1b),g1(ψ1), ỹ1b} ∈ A(n)ǫ (QS2H1G1Ỹ1)
If those g1(ψ1) have the same bin index k1b, we declare
k1b =
k1b. Otherwise, we declare an error. On the other hand, user
1 determines the unique w2(k2b), such that
{q, s2(l2b),h1(ω1b),w2(k2b), ỹ2b} ∈ A(n)ǫ (QS2H1W2Ỹ2).
At the receiver side, we assume Y1 knows i1,b−1, j1,b−1, l1,b−1
and j2,b−1, and it can construct m1(ξ1,b−1, η1,b−1, ω1,b−1),
n1(η1,b−1), h1(ω1,b−1) and u2(j2,b−1), which are jointly
typical with y1,b−1. Now it wants to first decode bin indices
i1b, j1b, l1b and the common message index j2b. It looks for
m1(ξ1, η1, ω1),n1(η1),h1(ω1),u2(j2), such that
{q,m1(ξ1, η1, ω1),n1(η1),h1(ω1),u2(j2),y1b}
∈ A(n)ǫ (QM1N1H1U2Y1). (35)
If those sequences satisfying (35) have the same bin indices
and message index respectively, we declare î1b = i1b, ĵ1b =
j1b, l̂1b = l1b and ĵ2b = j2b. Otherwise, declare an error.
Assuming cell index l1b is successfully decoded at Y1, then
we declare k̂1,b−1 = k1,b−1 if those sequences g1(ψ1) ∈
C(l1b) ∩ D(y1(b − 1)) have the same bin index k1,b−1. Here
C(l1b) denotes the set of g1(ψ1) in cell l1b, and D(y1(b− 1))
is the ambiguity set, i.e., sequences of g1(ψ1) such that
{q,m1(ξ1,b−1, η1,b−1, ω1,b−1),n1(η1,b−1),h1(ω1,b−1),
u2(j2,b−1),g1(ψ1),y1,b−1} ∈ A(n)ǫ (QM1N1H1U2G1Y1). (36)
For Y2, the decoding process is the same and we skip the
details.
Analysis of error probability: We first consider P
and we still use the story in block b. Let P0 denote
the probability that there is no m in bin i1b, such
that (q, s2(l2b),n1(η1b),h1(ω1b),m1(ξ1, η1b, ω1b)) are jointly
typical. Then,
P0 ≤ (1− 2−n(I(M1;S2|N1H1Q)+3ǫ))2
n(L11−R11)
≤ e−2
−n(I(M1;S2|N1H1Q)+3ǫ−L11+R11+1/n)
So, (9) guarantees P0 → 0 as n→ ∞. Similarly, bounds (10)-
(12) guarantee that encoder 1 can find codewords n1(η1b),
g1(ψ1b) and h1(ω1b), which are jointly typical with s2(l2b),
respectively.
Now we calculate the error probability for user 2 (as a
relay) to decode k1b. Denote the sent codeword g1(ψ1b) as
g1(k1b, k
∗) since it is picked from bin k1b. k
∗ denotes the
index of g1(ψ1b) in bin k1b. Let E1(k1, k) denote the event
(33) and let P1 denote the probability for user 2 to make a
decoding error. Then
P1 ≡ Pr{Ec1(k1b, k∗) or
k1 6=k1b
E1(k1, k)} (39)
≤ Pr{Ec1(k1b, k∗)}+
k1 6=k1b,k
Pr{E1(k1, k)}(40)
k1 6=k1b,k
Pr{E1(k1, k)} (41)
For k1 6= k1b, we know
Pr{E1(k1, k)}
(qs2h1g1ỹ1b)∈A
p(q)p(g1|q)p(s2h1ỹ1b|q)
≤ |A(n)ǫ |2−n(H(Q)−ǫ)2−n(H(G1|Q)−ǫ)2−n(H(S2H1Ỹ1|Q)−ǫ)
≤ 2−n(H(Q)+H(G1|Q)+H(S2H1Ỹ1|Q)−H(QG1S2H1Ỹ1)−4ǫ)
≤ 2−n(I(S2H1Ỹ1;G1|Q)−4ǫ)
Therefore, P1 ≤ ǫ + 2−n(I(S2H1Ỹ1;G1|Q)−L13−4ǫ). ǫ can be
arbitrarily small by letting n→ ∞. Thus, bound (22) assures
P1 → 0 as n→ ∞.
For the decoding of i1b, j1b, l1b and j2b by Y1, it is a
direct application of the simultaneous superposition coding
[2]. However, regarding our codebook generation scheme,
particularly the construction of m1, we will get a somewhat
simpler description, similar to that of Chong-Motani-Garg [3].
This leads to the bounds (13)-(20) and we skip the details here.
Let E2(k1, k) denote the event that g1(k1, k) ∈ C(l1b) ∩
D(y1(b − 1)) and E21(k1, k) denote the event (36). We also
define an indicator function I(k1, k). If g1(k1, k) satisfies
(36), I(k1, k) = 1; otherwise, I(k1, k) = 0. The number of
sequences in D(y1(b− 1)) with bin index k1 6= k1,b−1 is
||D(y1(b− 1)|| (42)
k1 6=k1,b−1,k
E(I(k1, k)) (43)
k1 6=k1,b−1,k
Pr{E21(k1, k)} (44)
k1 6=k1,b−1,k
2−n(I(Y1M1N1H1U2;G1|Q)−4ǫ) (45)
≤ 2−n(I(Y1M1N1H1U2;G1|Q)−L13−4ǫ) (46)
Now, let P2 denote the probability of error for Y1 to decode
k1,b−1. Denote the actually sent codeword g1(ψ1,b−1) in block
b− 1 as g1(k1,b−1, k∗). Then,
P2 ≡ Pr{Ec2(k1,b−1, k∗) or
k1 6=k1,b−1
E2(k1, k)}(47)
k1 6=k1,b−1,k
Pr{E2(k1, k)} (48)
≤ ǫ+ ||D(y1(b − 1)|| · 2−nR10 (49)
≤ ǫ+ 2−n(I(Y1M1N1H1U2;G1|Q)+R10−L13−4ǫ) (50)
Bound (21) guarantees P2 → 0 as n → ∞. Thus, with
bounds (9)-(22), it is guaranteed that Y1 will correctly decode
i1b, j1b, l1b, j2b and k1,b−1 at the end of block b with a
probability arbitrarily close to 1. Then, the information state
of receiver Y1 propagates forward, yielding the total decoding
error probability P
e,1 → 0 as n → ∞. The analysis of P
is similar to P
e,1 , which leads to bounds (23)-(32). Thus, R∗1
is achievable. Proof of R∗2 is identical, hence R∗ is achievable
for ICC via time sharing.
The cardinality bound on Q, i.e., |Q| ≤ 33 is obtained by
applying the Caratheodory Theorem to a set of inequalities
that bound the rate pair (R1, R2), obtained via Fourier-
Motzkin elimination to Eqs. (9)-(32). Q.E.D.
Remarks: From the encoding-decoding strategy of the above
theorem, the achievable region R∗ is actually a generalization
of the HK region for IC, the capacity region of degraded relay
channels, and the capacity region of GVBC. It reduces to those
extreme cases under the conditions elaborated below.
1)When the conferencing channel between the two users is
very poor, the bound in (22) and (32) can be very small. In
this case, allocating power for cooperation will actually reduce
the rates otherwise achievable via direct transmission. As a
result, the encoders will not allocate any power to transmit
W1 and W2, so W1 = W2 = S1 = S2 = 0 and G1 = G2 =
H1 = H2 = 0. Then, both R∗1 and R∗2 reduce to the region
in Proposition 1, which equals to the HK region.
2)When the conferencing channel between the two users is
good enough, it is not necessary to transmit messages directly
to the receiver, because cooperative transmission with the other
user will always yield a better rate. In this case, the encoders
will let V1 = V2 = U1 = U2 = 0 and M1 = M2 = N1 =
N2 = 0. Now, if user 2 refrains from transmitting its own
message and only serves as a relay to user 1 (i.e., W2 = S2 =
G2 = H2 = 0), both R∗1 and R∗2 reduce to the capacity region
of the degraded relay channel in Proposition 2. Similarly, if
user 1 serves only as a relay to user 2, it also reduces to the
capacity region of the degraded relay channel.
3)When the conferencing channel between the two users is
ideal (i.e., the conferencing channel capacity is infinite), the
bounds in (22) and (32) are no longer needed. So, W1 =
W2 = G1 = G2 ≈ 0. Combining the result in case 2 that
V1 = V2 = U1 = U2 = 0 and M1 =M2 = N1 = N2 = 0, we
can easily check that R∗1 reduces to
R1 ≤ I(Y1;H1|Q)− I(H1;S2|Q) (51)
R2 ≤ I(Y2;S2|Q) (52)
and R∗2 reduces to
R1 ≤ I(Y1;S1|Q) (53)
R2 ≤ I(Y2;H2|Q)− I(H2;S1|Q) (54)
For the Gaussian case, the rate region (51)-(52) becomes (6)
and (53)-(54) becomes (7). So, in this case, R∗ reduces to the
capacity region of GVBC.
4)During the review process of this paper, we became aware
of [12], which essentially tackles the same problem using
a different approach. In [12], user cooperation results in a
common information (in the sense of [4]) at the encoders
and the decoder uses backward decoding (similar to that
of [10]) instead of the random partitioning (i.e., binning)
we use in our approach. Except for some extreme cases, it
appears no subset relation can be established. The obtained
achievable region in [12] is simpler as it does not involve
a large number of auxiliary variables; however, the scheme
in [12] is strictly suboptimal for certain extreme cases (e.g.,
degraded relay channels, IC with degraded message sets with
weak interference, and MIMO BC) whereas our achievable
region can be easily shown to be optimal in each of these
cases.
IV. NUMERICAL EXAMPLES
The standard form of a Gaussian interference channel is:
Y1 = X1 + a21X2 + Z1 (55)
Y2 = a12X1 +X2 + Z2 (56)
where Z1 and Z2 are arbitrarily correlated zero mean, unit
variance Gaussian random variables. Suppose the power con-
straints of X1 and X2 are P1 and P2, respectively. For the
conferencing channel with perfect echo cancellation, we have
Ỹ1 = K1X1 + Z̃1, Ỹ2 = K2X2 + Z̃2 (57)
where Z̃1 and Z̃2 are both zero mean, unit variance Gaussian
variables. By reciprocity, we assume the channel coefficient
K1 = K2. Since the computation of R∗ is formidable, here
we constrain all the inputs to be Gaussian distributed and set
Q = φ in order to compare our region with G′ in (5.9) of [2]
and the capacity region of GVBC. We denote this modified
region as R. Consider, that for certain αt, βt, γt, θt, µt ∈ [0, 1],
with αt+βt+γt+ θt+µt = 1, where t = 1, 2, the following
hold:
Ut ∼ N(0, βtPt),Wt ∼ N(0, γtPt), St ∼ N(0, 1) (58)
Vt = V
t + Ut +
θtPtSt, where V
t ∼ N(0, αtPt) (59)
X1 = V1 +W1 +
µ1P1S2, X2 = V2 +W2 +
µ2P2S1 (60)
After applying Fourier-Motzkin Elimination on those bounds
(9)-(32), we find that for each set of (α1, β1, γ1, θ1, µ1) and
(α2, β2, γ2, θ2, µ2), both S(Z1) and S(Z2) are delimited by
straight lines of slope 0,− 1
,−1,−2,∞ as in the original
HK region. Exhausting all the parameters between [0, 1], and
taking the convex hull of all those S(Z1) and S(Z2), we get
the achievable region R for ICC in Fig.2.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
HK region
K1=K2=1
K1=K2=4
Gaussian vector BC
Fig. 2. Comparison of R with HK region and Gaussian vector broadcast
channel capacity. P1 = 6, P2 = 1.5, a12 = a21 = 0.74
Remarks:
1) When there is no conferencing between the two users,
the achievable region reduces to the HK region. When the
quality of the conferencing channel improves, it increases our
achievable region for ICC within the limit of the capacity of
GVBC.
2) When the channel coefficient is K1 = K2 = 4, the region
R is already very close to the upper bound; when K1 = K2 =
1, which is equal to the channel coefficient of the transmitter to
the receiver, cooperation achieves a slightly better rate region
than independent transmission.
3) For the channel coefficient K1 = K2 = 4, the corre-
sponding relay channels (i.e., one of the users only serves as
a relay) are degraded, thus the intercepts of the bound at both
axes are the capacities of respective relay channels.
REFERENCES
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Globecom’07, Washington DC, Nov. 2007.
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[11] H. Weingarten, Y. Steinberg, and S. Shamai, “The Capacity Region of
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(IFC-GF),” in ITA’07 Workshop, UCSD, Feb. 2007.
	Introduction
	Preliminaries and Existing Results
	Definitions
	Existing Results
	Main Results
	Numerical Examples
	References
