On the Achievable Rate Regions for Interference Channels with Degraded
  Message Sets
On the Achievable Rate Regions for Interference
Channels with Degraded Message Sets†
Jinhua Jiang and Yan Xin
Department of Electrical and Computer Engineering
National University of Singapore, Singapore 117576
Email: { jinhua.jiang, elexy }@nus.edu.sg
Abstract
The interference channel with degraded message sets (IC-DMS) refers to a communication model
in which two senders attempt to communicate with their respective receivers simultaneously through a
common medium, and one of the senders has complete and a priori (non-causal) knowledge about the
message being transmitted by the other. A coding scheme that collectively has advantages of cooperative
coding, collaborative coding, and dirty paper coding, is developed for such a channel. With resorting to
this coding scheme, achievable rate regions of the IC-DMS in both discrete memoryless and Gaussian
cases are derived, which, in general, include several previously known rate regions. Numerical examples
for the Gaussian case demonstrate that in the high-interference-gain regime, the derived achievable rate
regions offer considerable improvements over these existing results.
Index Terms
Cognitive radio, cooperative communication, degrade message sets, dirty paper coding, Gel’fand-
Pinsker coding, interference channels, superposition coding.
† The work is supported by the National University of Singapore (NUS) under start-up grants R-263-000-314-101 and R-
263-000-314-112 and by a NUS Research Scholarship. The correspondence author of the paper is Dr. Yan Xin (tel. no. +65
6516-5513 and fax no. +65 6779-1103).
November 4, 2018 DRAFT
http://arxiv.org/abs/0704.0540v2
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 1
I. INTRODUCTION
The interference channel with degraded message sets (IC-DMS) refers a communication model
in which two senders attempt to communicate with their respective receivers simultaneously
through a common medium, and one of the senders has complete and a priori (non-causal)
knowledge about the message being transmitted by the other. Such a model generically
characterizes some realistic communication scenarios taking place in cognitive radio channels
[1], [2] or in wireless sensor networks over a correlated field [3], [4], which we illustrate in
Figs. 1(a) and 1(b).
From an information-theoretic perspective, the IC-DMS have been investigated in [1]–[4].
Specifically, several achievable rate results have been obtained in [1]–[4], and the capacity regions
for two special cases have been characterized in [2]–[4]. The main achievable rate region in
[1] was obtained by incorporating the Gel’fand-Pinsker coding [5] into the well-known coding
scheme applied to the interference channel (IC) [6], [7]. In this coding scheme, each of the two
senders splits its message into two sub-messages, and allows its non-pairing receiver to decode
one of the sub-messages. Knowing the two sub-messages and the corresponding codewords which
sender 1 wishes to transmit, sender 2 applies the Gel’fand-Pinsker coding to encode its own sub-
messages by treating the codewords of sender 1 as known interferences. It has been also shown
in [1, Corollary 2] that, an improved achievable rate region can be attained by time-sharing
between the early derived rate region and a so called fully-cooperative rate point achieved by
letting sender 2 use all its power to transmit sender 1’ messages. A different coding scheme was
adopted in [2] and [3], in which neither of the senders splits its message into two sub-messages,
and receiver 2 does not decode any transmitted information from sender 1. Since sender 2 knows
what sender 1 wishes to transmit, sender 2 is allowed to: 1) apply the Gel’fand-Pinsker coding to
encode its own message; and 2) partially cooperate with sender 1 using superposition coding. It
has been proven in [2], [3] that, this is the capacity-achieving scheme for the Gaussian IC-DMS
in the low-interference-gain regime, in which the normalized link gain between sender 2 and
receiver 1 is less than or equal to 1.
However, in practice, due to the mobility of the users or random distributions of the sensors,
Genie
(a) (b)
Receiver 1
Receiver 2
Receiver 1Primary user
Cognitive user Receiver 2
Sender 2
Sender 1
PSfrag replacements
Fig. 1. (a) A genie-aided cognitive radio channel [1], in which the Genie informs the cognitive user of what the primary
user will transmit; (b) A four-node wireless sensor network [3], in which sender 2 senses a larger area such that it knows what
information sender 1 obtains.
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 2
Sender 2
Receiver 2
Receiver 1
Sender 1
PSfrag replacements
Fig. 2. An interference channel with degraded message sets in which sender 2 is close to receiver 1.
sender 2 may be geographically located near to receiver 1, as illustrated in Fig. 2. It is likely, in
such a situation, that the Gaussian IC-DMS is in the high-interference-gain regime, in which the
normalized link gain between sender 2 and receiver 1 is greater than 1. In fact, the findings in
this paper reveal that the achievable rate region, which was proven to be the capacity region in
the low-interference-gain regime in [2] and [3], is strictly non-optimal for the Gaussian IC-DMS
in the high-interference-gain regime.
In this paper, we develop a new coding scheme for the IC-DMS to improve existing achievable
rate regions. Our coding scheme differs from one proposed in [2], [3] in the way that, sender
2 splits its message into two sub-messages, and encodes both sub-messages using Gel’fand-
Pinsker coding. Moreover, receiver 1 is required to jointly decode the message from sender 1
and one sub-message from sender 2. With this coding scheme, we derive our main achievable
rate region for the discrete memoryless case. For comparison purpose, we compromise either
the coding flexibility (fixing an auxiliary random variable as a constant), or the advantage of
simultaneous decoding [7], to obtain two subregions of the main achievable rate region. The
obtained subregions are shown to either include or be the same as the existing ones. We further
extend the obtained regions from the discrete memoryless case to the Gaussian case, and show
by numerical examples that our Gaussian achievable rate results strictly improve the existing
ones in the high-interference-gain regime.
The rest of the paper is organized as follows. In Section II, we introduce the channel model
of the IC-DMS, and the related terminologies. In Section III, we present the main achievable
result for the discrete memoryless case with a detailed proof. In Section IV, we derive two
subregions of the main achievable rate region, and we show that the derived subregions include
several existing results as special cases. Lastly, in Section V, we extend our rate regions from
the discrete memoryless case to the Gaussian case, and compare them with the existing results.
Notations: Random variables and their realizations are denoted by upper case letters and lower
case letters respectively, e.g., X and x. Bold lower (upper) case letters are used to denote vectors
(matrices), e.g., x and Σ. Calligraphic fonts are used to denote sets, e.g., X and R.
II. THE CHANNEL MODEL
Consider the IC-DMS (also termed as the genie-aided cognitive radio channel in [1])
depicted in Fig. 3, in which sender 1 wishes to transmit a message (or message index),
w1 ∈ M1 := {1, ...,M1}, to receiver 1 and sender 2 wishes to transmit its message,
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 3
Sender 1
Channel
Receiver 1
Receiver 2Sender 2
PSfrag replacements
p(y1, y2|x1, x2)
f1(·)
f2(·)
g1(·)
g2(·)
Fig. 3. An interference channel with degraded message sets.
w2 ∈ M2 := {1, ...,M2}, to receiver 2. Typically, this discrete memoryless IC-DMS is described
by a tuple (X1,X2,Y1,Y2, p(y1, y2|x1, x2)), where X1 and X2 are the channel input alphabets, Y1
and Y2 are the channel output alphabets, and p(y1, y2|x1, x2) denotes the conditional probability
of (y1, y2) ∈ Y1×Y2 given (x1, x2) ∈ X1×X2. The channel is discrete memoryless in the sense
p(y1,t, y2,t|x1,t, x2,t, x1,t−1, x2,t−1, ...) = p(y1,t, y2,t|x1,t, x2,t), (1)
for every discrete time instant t in a synchronous transmission. In terms of the channel input-
output relationship, the IC-DMS is the same as the IC. However, in the IC-DMS, sender 2 is
able to noncausally obtain the knowledge of the message w1, which will be transmitted from
sender 1. This is the key difference between the IC-DMS and IC in terms of the information
flow. We next present the following standard definitions with regard to the existence of codes
and achievable rates for the discrete memoryless IC-DMS channel.
Definition 1: An (M1,M2, n, Pe) code exists for the discrete memoryless IC-DMS, if and
only if there exist two encoding functions
f1 : M1 → X n1 , f2 : M1 ×M2 → X n2 ,
and two decoding functions
g1 : Yn1 → M1, g2 : Yn2 → M2,
such that max{P (n)e,1 , P
e,2 } ≤ Pe, where P
e,1 and P
e,2 denote the respective average probabilities
of error at decoders 1 and 2, and are computed as
e,1 =
p(ŵ1 6= w1|(w1, w2) were sent),
e,2 =
p(ŵ2 6= w2|(w1, w2) were sent).
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 4
Definition 2: A non-negative rate pair (R1, R2) is achievable for the IC-DMS, if for any given
0 < Pe < 1 and any sufficiently large n, there exists a (2
nR1 , 2nR2, n, Pe) code for the channel.
The capacity region of the IC-DMS is the set of all the achievable rate pairs for the channel,
and an achievable rate region is a subset of the capacity region.
It should be noted that from an information-theoretic standpoint, the IC can not be simply
treated as a special case of the IC-DMS in the sense that the capacity region of the IC-DMS, if
any, does not imply a capacity region of the IC.
III. AN ACHIEVABLE RATE REGION FOR THE DISCRETE MEMORYLESS IC-DMS
In this section, we present the main achievable rate region for the discrete memoryless IC-
DMS, which is the primary result in this paper.
Consider auxiliary random variables W , U , Ũ , V , Ṽ and a time-sharing random variable Q,
defined on arbitrary finite sets W , U , Ũ , V , Ṽ and Q respectively. Let P denote the set of all
joint probability distributions p(·) that factor in the form of
p(q, w, x1, u, ũ, v, ṽ, x2, y1, y2) =p(q)p(w, x1|q)p(u, ũ|w, q)p(v, ṽ|w, q)
· p(x2|ũ, ṽ, w, q)p(y1, y2|x1, x2), (2)
where w, u, ũ, v, ṽ, and q are realizations of random variables W , U , Ũ , V , Ṽ and Q.
Let R(p) denote the set of all non-negative rate pairs (R1, R2) such that the following
inequalities hold simultaneously
R1 ≤ I(W ; Y1U |Q), (3)
R2 ≤ I(UV ; Y2|Q)− I(U ;W |Q)− I(V ;W |Q), (4)
R1 + R2 ≤ I(UW ; Y1|Q) + I(V ; Y2U |Q)− I(U ;W |Q)− I(V ;W |Q); (5)
0 ≤ I(UW ; Y1|Q)− I(U ;W |Q), (6)
0 ≤ I(U ; Y2V |Q)− I(U ;W |Q), (7)
0 ≤ I(V ; Y2U |Q)− I(V ;W |Q), (8)
0 ≤ I(UV ; Y2|Q)− I(U ;W |Q)− I(V ;W |Q), (9)
for a given joint distribution p(·) ∈ P .
Let C denote the capacity region of the discrete memoryless IC-DMS, and let
p(·)∈P
R(p).
Theorem 1: The region R is an achievable rate region for the discrete memoryless IC-DMS,
i.e., R ⊆ C.
Proof: Before presenting the proof of Theorem 1, we state the following lemma as it will
be frequently used in the proof.
Lemma 1 ( [8, Theorem 14.2.3]): Let A
ǫ denote the typical set for the probability mass
distribution p(s1, s2, s3), and let P (S
1 = s1,S
2 = s2,S
3 = s3) =
p(s1i|s3i)p(s2i|s3i)p(s3i),
then P{(S′1,S′2,S′3) ∈ A
ǫ } .= 2−n(I(S′1;S′2|S′3)±6ǫ).
To prove this theorem we apply the notion of the asymptotic equipartition property (APE)
[8]. Our coding scheme is mainly based on the arguments of superposition coding [9] and
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 5
Gel’fand-Pinsker coding [5]. Specifically, sender 1 independently encodes its message w1 as a
whole; while sender 2 needs split its message into two parts, i.e., w2 = (w21, w22), and encode
them separately. Both w21 and w22 are encoded using the Gel’fand-Pinsker approach, but they are
processed differently at the receivers. The message w22 will be decoded by receiver 2 only, while
w21 will be decoded by both receivers. Moreover, knowing the message and codeword which
sender 1 is going to transmit, sender 2 not only can apply Gel’fand-Pinsker coding to deal with
the known interference, but also can cooperate with sender 1 to transmit w1 using superposition
coding. Let R21 and R22 denote the rates of w21 and w22 respectively, i.e., w21 ∈ {1, . . . , 2nR21}
and w22 ∈ {1, . . . , 2nR22}. If receiver 1 can decode w1 and receiver 2 can decode both w21 and
w22 with vanishing probabilities of error, then (R1, R21 +R22) is an achievable rate pair for the
IC-DMS.
To prove that the entire region R is achievable for the channel, it is sufficient to prove that
R(p) is achievable for a fixed joint probability distribution p(·) ∈ P .
A. Random Codebook Generation
Consider a fixed joint distribution p(·) ∈ P , and a random time-sharing codeword q of length
n, which is given to both senders and receivers. The codeword q is assumed to be generated
according to
i=1 p(qi).
Generate 2nR1 independent codewords w(j), j ∈ {1, . . . , 2nR1}, according to
i=1 p(wi|qi);
and for each w(j) generate one x1(j), according to
i=1 p(x1i|wiqi). Similarly, generate 2nR̃21
independent codewords u(l1), l1 ∈ {1, . . . , 2nR̃21}, according to
p(ui|qi), and generate 2nR̃22
independent codewords v(l2), l2 ∈ {1, . . . , 2nR̃22}, according to
i=1 p(vi|qi).
For each codeword pair (u(l1),w(j)), generate one codeword ũ(l1, j) according
i=1 p(ũi|ui(l1)wi(j)qi), and similarly for each codeword pair (v(l2),w(j)), gen-
erate one codeword ṽ(l2, j) according to
i=1 p(ṽi|vi(l2)wi(j)qi). Lastly, for each
codeword triple (u(l1),v(l2),w(j)), generate one codeword x2(l1, l2, j) according to
i=1 p(x2i|ũi(l1)ṽi(l2)wi(j)qi).
Now uniformly distribute 2nR̃21 codewords u(l1) into 2
nR21 bins indexed by k1 ∈
{1, . . . , 2nR21} such that each bin contains 2n(R̃21−R21) codewords; uniformly distribute 2nR̃22
codewords v(l2) into 2
nR22 bins indexed by k2 ∈ {1, . . . , 2nR22} such that each bin contains
2n(R̃22−R22) codewords.
The entire codebook is revealed to both senders and receivers.
B. Encoding and Transmission
We assume that the senders want to transmit a message vector (w1, w21, w22) = (j, k1, k2).
Sender 1 simply encodes the message as codeword x1(j) and sends the codeword with n channel
uses. Sender 2 will first need to look for a codeword u(l̂1) in bin k1 such that (u(l̂1),w(j),q) ∈
ǫ , and a codeword v(l̂2) in bin k2 such that (v(l̂2),w(j),q) ∈ A(n)ǫ . If sender 2 fails to
do so, it will randomly pick a codeword u(l̂1) from bin k1 or a codeword v(l̂2) from bin k2.
Sender 2 then transmits codeword x2(l̂1, l̂2, j) through n channel uses. We further assume that
the transmissions are perfectly synchronized.
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 6
C. Decoding
Receiver 1 first looks for all the index pairs (ĵ,
l1) such that (w(ĵ),u(
l1),y1,q) ∈ A(n)ǫ . If ĵ
in all the index pairs found are the same, receiver 1 determines w1 = ĵ, otherwise declares an
error.
Receiver 2 will first look for all index pairs (
l2) such that (u(
l1),v(
l2),y2,q) ∈ A(n)ǫ . If ¯̂l1
in all the index pairs found are indices of codewords u(
l1) from the same bin with index k̂1,
l2 in all the index pairs found are indices of codewords v(
l2) from the same bin with index
k̂2, then receiver 2 will decode that (w21, w22) = (k̂1, k̂2) were transmitted; otherwise, an error
is declared.
D. Evaluation of Probability of Error
We now derive upper bounds for the probabilities of the respective error events, which may
happen during the encoding and decoding process. Due to the symmetry of the codebook
generation and encoding processing, the probability of error is not codeword dependent. Without
loss of generality, we assume that (w1, w21, w22) = (1, 1, 1) were encoded and transmitted. We
next define the following three types of events:
Ea,b = (u(a),w(b),q) ∈ A(n)ǫ ,
Ėa,b = (w(a),u(b),y1,q) ∈ A(n)ǫ ,
Ëa,b = (u(a),v(b),y2,q) ∈ A(n)ǫ .
Let Pe(enc2), Pe(dec1), and Pe(dec2) denote the probabilities of error at the encoder of sender
2, the decoder of receiver 1, and the decoder of receiver 2, respectively.
[Evaluation of Pe(enc2).] An error is made if 1) the encoder at sender 2 can not find u(l̂1)
in bin 1 such that (u(l̂1),w(1),q) ∈ A(n)ǫ , and/or 2) it can not find v(l̂2) in bin 1 such that
(v(l̂2),w(1),q) ∈ A(n)ǫ . Then the probability of error at the encoder of sender 2 is bounded as
Pe(enc2) ≤ Pr
u(l̂1)∈bin 1
(u(l̂1),w(1),q) /∈ A(n)ǫ
 + Pr
v(l̂2)∈bin 1
(v(l̂2),w(1),q) /∈ A(n)ǫ
u(l̂1)∈bin 1
Pr(Ec
l̂1,1
v(l̂2)∈bin 1
Pr(Ec
l̂2,1
≤ (1− Pr(E
l̂1,1
n(R̃21−R21)
+ (1− Pr(E
l̂2,1
n(R̃22−R22)
≤ (1− 2−n(I(U ;W |Q)+6ǫ))2n(R̃21−R21) + (1− 2−n(I(V ;W |Q)+6ǫ))2n(R̃22−R22),
where (a) follows from the fact that we can obtain Pr(E
l̂1,1
) ≥ 2−n(I(U ;W |Q)+6ǫ) and Pr(E
l̂2,1
2−n(I(V ;W |Q)+6ǫ) by setting S′1 = U, S
2 = W , and S
3 = Q, and S
1 = V, S
2 = W , and S
3 = Q
in Lemma 1, respectively. Following the same argument in the proof of Lemma 2.1.3 of [10],
we conclude that Pe(enc2) → 0 as n → +∞, if
R̃21 ≥ R21 + I(U ;W |Q), (10)
R̃22 ≥ R22 + I(V ;W |Q), (11)
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 7
are satisfied. We further choose
R̃21 = R21 + I(U ;W |Q), (12)
R̃22 = R22 + I(V ;W |Q). (13)
Note that such a choice still ensures that Pe(enc2) → 0 as n → +∞.
[Evaluation of Pe(dec1)] An error is made if 1) Ėc1,l̂1
happens, and/or 2) there exists some ĵ 6= 1
such that Ė
happens. Note that
l1 is not required to be equal to l̂1, since it is unnecessary for
receiver 1 to decode l̂1 correctly. The probability of error at receiver 1 can be upper bounded as
Pe(dec1) ≤ Pr(Ėc1,l̂1
∪ĵ 6=1Ėĵ,ˆ̂l1)
≤ Pr(Ėc
1,l̂1
ĵ 6=1
Pr(Ė
= Pr(Ėc
1,l̂1
ĵ 6=1
Pr(Ė
ĵ,l̂1
ĵ 6=1,
l1 6=l̂1
P (Ė
≤ Pr(Ėc
1,l̂1
) + 2nR1Pr(Ė2,l̂1) + 2
n(R1+R̃21)Pr(Ė
l1 6=l̂1
). (14)
Choosing S′1 = W, S
2 = (Y1,U), and S
3 = Q in Lemma 1, we have Pr(Ė2,l̂1)
2−n(I(W ;Y1U |Q)±6ǫ). Likewise, we have Pr(Ė
l1 6=l̂1
= 2−n(I(WU ;Y1|Q)±6ǫ). In addition, it follows
from AEP that Pr(Ėc
1,l̂1
) → 0 as n → +∞. Thus, we infer from (14) that Pe(dec1) → 0 as
n → +∞, if
R1 ≤ I(W ; Y1U |Q), (15)
R1 + R̃21 ≤ I(WU ; Y1|Q), (16)
are satisfied.
[Evaluation of Pe(dec2)] An error is made if 1) Ëc
l̂1,l̂2
happens, and/or 2) there exists some
l2) in which either
l1 or
l2 is not an index of any codeword from the respective bin 1. The
probability of the second case is upper bounded by the probability of the event, Ë¯̂
for some
l2) 6= (l̂1, l̂2). Thus, the probability of error at receiver 2 is bounded as
Pe(dec2) ≤ Pr(Ëcl̂1,l̂2
l2)6=(l̂1,l̂2)
≤ Pr(Ëc
l̂1,l̂2
l2)6=(l̂1,l̂2)
P (Ë¯̂
= Pr(Ëc
l̂1,l̂2
l1 6=l̂1
Pr(Ë¯̂
l1,l̂2
l2 6=l̂2
Pr(Ë
l1 6=l̂1,
l2 6=l̂2)
Pr(Ë¯̂
≤ Pr(Ëc
l̂1,l̂2
) + 2nR̃21Pr(Ë¯̂
l1 6=l̂1,l̂2
) + 2nR̃22Pr(Ë
l2 6=l̂2
) + 2n(R̃21+R̃22)Pr(Ë¯̂
l1 6=l̂1,
l2 6=l̂2
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 8
Applying Lemma 1 to evaluate Pr(Ë¯̂
l1 6=l̂1,l̂2
), Pr(Ë
l2 6=l̂2
) and Pr(Ë¯̂
l1 6=l̂1,
l2 6=l̂2
) in (17), we
conclude that Pe(dec2) → 0 as n → +∞ if the following inequalities,
R̃21 ≤ I(U ; Y2V |Q), (18)
R̃22 ≤ I(V ; Y2U |Q), (19)
R̃21 + R̃22 ≤ I(UV ; Y2|Q), (20)
are satisfied.
According to (12), (13) and the fact that R2 = R21+R22, we first substitute R̃21 and R̃22 with
R21+ I(U ;W |Q) and R22+ I(V ;W |Q) in (15), (16) and (18)–(20), and subsequently substitute
R21 with R2 −R22 in the resulting inequalities. After these two substitution steps, we have
R1 ≤ I(W ; Y1U |Q), (21)
R1 +R2 −R22 ≤ I(WU ; Y1|Q)− I(U ;W |Q), (22)
R2 −R22 ≤ I(U ; Y2V |Q)− I(U ;W |Q), (23)
R22 ≤ I(V ; Y2U |Q)− I(V ;W |Q), (24)
R2 ≤ I(UV ; Y2|Q)− (I(U ;W |Q) + I(V ;W |Q)). (25)
Furthermore, applying Fourier-Motzkin elimination [11] to remove R22 from (21)–(25), we have
R1 ≤ I(W ; Y1U |Q), (26)
R2 ≤ I(UV ; Y2|Q)− (I(U ;W |Q) + I(V ;W |Q)), (27)
R2 ≤ I(U ; Y2V |Q)− I(U ;W |Q) + I(V ; Y2U |Q)− I(V ;W |Q), (28)
R1 + R2 ≤ I(WU ; Y1|Q)− I(U ;W |Q) + I(V ; Y2U |Q)− I(V ;W |Q). (29)
Since I(U ; Y2V |Q)+I(V ; Y2U |Q)−I(UV ; Y2|Q) = I(U ;V |Q)+I(U ;V |Y2Q) ≥ 0, (27) implies
(28) and thus (28) is redundant. To ensure that R1, R21 and R22 are non-negative, we enforce
four additional constraints (6)–(9). Therefore, the rate region R(p) is achievable for a fixed joint
probability distribution p(·) ∈ P , and Theorem 1 follows.
Remark 1: The proposed coding scheme exploits three coding methods to achieve any rate
pair in the rate region, R. The first method is cooperation that is realized by the superposition
relationship between w and x2 through p(x2|ũ2, ṽ2, w, q). The second is collaboration, by which
we mean that sender 2 separates its own message into two parts, i.e., w2 = (w21, w22), and
encodes w21 at a possibly low rate such that receiver 1 can decode it. By doing so, the effective
interference caused by the signals carrying the sender 2’s information may be reduced. The third
is Gel’fand-Pinsker coding, which we apply to encode both messages, w21 and w22, from sender
2 by treating the codeword w as known interference. This perhaps allows receiver 2 to be able
to decode the messages from sender 2 at the same rate as if the interference caused by sender
1 was not present [12].
IV. RELATING WITH EXISTING RATE REGIONS
In this section, we will show that Theorem 1 includes the achievable rate regions in [2], [3].
To demonstrate it, we compromise the advantages of the coding scheme developed in Section
III to obtain the following subregions of R.
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 9
A. A Subregion of R
Let P∗ denote the set of all joint probability distributions p(·) that factors in the form of
p(q, w, x1, u, v, ṽ, x2, y1, y2) =p(q)p(x1, w|q)p(u|q)p(v, ṽ|w, q)
· p(x2|u, ṽ, w, q)p(y1, y2|x1, x2). (30)
Note that the joint distribution (30) differs from (2) in the way that conditioned on Q, U is now
independent of any other auxiliary random variables, and Ũ is not present.
Let Rsim(p) denote the set of all non-negative rate pairs (R1, R2) such that
R1 ≤ I(W ; Y1|UQ), (31)
R2 ≤ I(UV ; Y2|Q)− I(V ;W |Q), (32)
R1 +R2 ≤ I(WU ; Y1|Q) + I(V ; Y2|UQ)− I(V ;W |Q); (33)
0 ≤ I(V ; Y1|UQ)− I(V ;W |Q), (34)
for a joint probability distribution p(·) ∈ P∗. Furthermore, let
Rsim =
p(·)∈P∗
Rsim(p).
Theorem 2: The rate region Rsim is achievable for the discrete memoryless IC-DMS, i.e.,
Rsim ⊆ R ⊆ C.
Proof: The proof can be devised from the proof of Theorem 1 by customizing the original
coding scheme for the new joint distribution (30). We change the encoding and decoding method
for the message w21 (corresponding to U), i.e., the Gel’fand-Pinsker coding used in the proof
of Theorem 1 was replaced by the conventional random coding. Specifically, we generate 2nR21
independent codewords u(k1), k1 ∈ {1, . . . , 2nR21}, according to
i=1 p(ui|qi). The encoding
and decoding are then adapted to the new codebook accordingly. Evaluating the probability of
error in the same way as was done in the proof of Theorem 1, we obtain
R̃22 −R22 ≥ I(V ;W |Q); (35)
R1 ≤ I(W ; Y1|UQ), (36)
R1 +R21 ≤ I(WU ; Y1|Q); (37)
R21 ≤ I(U ; Y2|V Q), (38)
R̃22 ≤ I(V ; Y2|UQ), (39)
R21 + R̃22 ≤ I(UV ; Y2|Q). (40)
Again, we choose R̃22 − R22 = I(V ;W |Q) in (35), and then substitute R̃22 with R22 +
I(V ; Y2|UQ) as well as R21 with R2 − R22 in the group of (36)–(40). By applying Fourier-
Motzkin elimination on the resulting inequalities to remove R22, and adding the constraints that
ensure the respective rates R1, R21 and R22 to be non-negative, we obtain (31)–(34). Therefore,
the region Rsim(p) is achievable for a given p(·) ∈ P∗, and the theorem follows.
Note that simultaneous decoding (simultaneous joint typicality) is applied at both decoders.
The advantage of simultaneous decoding over successive decoding is well demonstrated on the
IC by Han and Kobayashi in [7]. We next modify the coding scheme by applying successive
decoding instead of simultaneous decoding at both decoders to derive a subregion of Rsim.
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 10
B. A Subregion of Rsim
Let Rsuc(p) denote the set of all achievable rate pairs (R1, R2) such that
R1 ≤ I(W ; Y1|UQ), (41)
R2 ≤ min{I(U ; Y1|Q), I(U ; Y2|Q)}+ I(V ; Y2|UQ)− I(V ;W |Q); (42)
0 ≤ I(V ; Y1|UQ)− I(V ;W |Q), (43)
for a fixed joint probability distribution p(·) ∈ P∗. Define
Rsuc =
p(·)∈P∗
Rsuc(p).
Theorem 3: The rate region Rsuc is achievable for the discrete memoryless IC-DMS, i.e.,
Rsuc ⊆ Rsim ⊆ R ⊆ C.
Proof: The codebook generation, encoding and transmission remain the same as those used
to prove Theorem 2, whereas the decoding processes at both decoders are altered. Both decoders
decode w21 first, and then decoder 1 decodes w1 and decoder 2 decodes w22 respectively. Then
the following can easily be obtained
R̃22 −R22 ≥ I(V ;W |Q), (44)
R21 ≤ I(U ; Y1|Q), (45)
R1 ≤ I(W ; Y1|UQ), (46)
R21 ≤ I(U ; Y2|Q), (47)
R̃22 ≤ I(V ; Y2|UQ). (48)
From (44)–(48), it is straightforward to obtain (41)–(43). Therefore, the region Rsuc(p) is
achievable, and the theorem follows immediately.
Remark 2: Note that (45) is only necessary when the successive decoding is applied. This is
because every decoding step in a successive decoding scheme is expected to have a vanishing
probability of error.
In what follows, we further specialize the subregion Rsuc to obtain two more achievable rate
regions Rsp1 and Rsp2. Let P∗1 denote the set of all joint probability density distributions p(·)
that factor in the form of
p(q, w, x1, v, ṽ, x2, y1, y2) = p(q)p(x1, w|q)p(v, ṽ|w, q)p(x2|ṽ, w, q)p(y1, y2|x1, x2). (49)
Let Rsp1(p) denote the set of all non-negative rate pairs (R1, R2) such that
R1 ≤ I(W ; Y1|Q), (50)
R2 ≤ I(V ; Y2|Q)− I(V ;W |Q), (51)
for a fixed joint distribution p(·) ∈ P∗1 . Define
Rsp1 =
p(·)∈P∗1
Rsp1(p).
Corollary 1: The region Rsp1 is an achievable rate region for the discrete memoryless IC-
DMS, i.e., Rsp1 ⊆ Rsuc ⊆ Rsim ⊆ R ⊆ C.
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 11
Proof: Fixing the auxiliary random variable U as a constant, we reduce (41) and (42) to
(50) and (51), and the corollary follows immediately.
Remark 3: We note that the region Rsp1 is similar to the region Rin reported in [3, Theorem
3.1]. It seems that the region Rin is more general than the region Rsp1 in the sense that fixing
the auxiliary random variable U in Rin as a constant, one can obtain a region which is the
same as Rsp1. Nevertheless, after examining the coding scheme used in [3, Theorem 3.1], one
can find that there exists a one-one correspondence between codewords u(w2) and x2(w2), and
both codewords are jointly generated and decoded, i.e., p(u, x2) is used to generate two-letter
codewords. Thus, one can introduce one auxiliary random variable W such that there exists a
one-one mapping between W and U×X2, i.e., f : U×X2 ↔ W , and thus W has the probability
mass distribution p(w) = p(f−1(w)) = p(u, x2). Replacing all (X
2 (w2), U
n(w2)) by W
n(w2)
in the proof of [3, Theorem 3.1] will yield the same rate region. Equivalently speaking, for
any input distribution achieving a rate region characterized by [3, Theorem 3.1], one can find
a corresponding joint distribution in the form of (49) such that Corollary 1 yields exactly the
same rate region. Therefore, two rate regions Rsp1 and Rin are identical.
Let P∗2 denote the set of all joint probability distributions p(·) that factor in the form of
p(q, w, x1, u, x2, y1, y2) = p(q)p(x1, w|q)p(u|q)p(x2|u, w, q)p(y1, y2|x1, x2). (52)
Let Rsp2(p) denote the set of all non-negative rate pairs (R1, R2) such that
R1 ≤ I(W ; Y1|UQ), (53)
R2 ≤ min{I(U ; Y1|Q), I(U ; Y2|Q)}, (54)
for a fixed joint distribution p(·) ∈ P∗2 . Define
Rsp2 =
p(·)∈P∗2
Rsp2(p).
Corollary 2: The region Rsp2 is an achievable rate region for the discrete memoryless IC-
DMS, i.e., Rsp2 ⊆ Rsuc ⊆ Rsim ⊆ R ⊆ C.
Proof: The proof can be devised from the proof of Theorem 3 easily by fixing V as a
constant.
V. THE GAUSSIAN IC-DMS
In the preceding sections, we have derived several achievable rate regions for the discrete
memoryless IC-DMS. We now extend these results to obtain corresponding achievable rate
regions for the Gaussian IC-DMS (GIC-DMS).
A. The Channel Model of the GIC-DMS
In general, with no loss of information-theoretic optimality, the GIC-DMS can be converted
to the GIC-DMS in the standard form through invertible transformations [2], [6], [11]. We thus
only consider the GIC-DMS in the standard form, which is represented as follows
Y1 = X1 +
c21X2 + Z1,
Y2 = X2 +
c12X1 + Z2,
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 12
PSfrag replacements
x1(w1)
x2(w2, w1)
Fig. 4. A Gaussian interference channel with degraded message sets.
where Zi, i = 1, 2, is the additive white Gaussian noise with zero mean and unit variance, and√
c21 and
c12 are the normalized link gains in the GIC-DMS depicted in Fig. 4. Moreover,
the transmitted codeword xi = (xi1, . . . , xin), i = 1, 2, is subject to an average power constraint
given by
‖xit‖2 ≤ Pi.
Since it has been shown in the maximum-entropy theorem in [8] that Gaussian input signals are
optimal for Gaussian channels, we will consider Gaussian codewords Xni , i = 1, 2.
B. Achievable Rate Regions for the GIC-DMS
1) Gaussian Extension of R: We first extend R to its Gaussian counterpart denoted by G.
To obtain the rate region G, we map the random variables involved in the joint distribution (2)
to the corresponding Gaussian random variables with the following customary constraints:
P1) W , distributed according to N (0, 1),
P2) X1 =
P1W ,
P3) Ũ , distributed according to N (0, αβP2),
P4) Ṽ , distributed according to N (0, αβ̄P2),
P5) U = Ũ + λ1W ,
P6) V = Ṽ + λ2W ,
P7) X2 = Ũ + Ṽ +
ᾱP2W ,
where α, β ∈ [0, 1], α + ᾱ = 1, β + β̄ = 1, λ1, λ2 ∈ [0,+∞), and W , Ũ and Ṽ are mutually
independent. The input-output relationship of the GIC-DMS can be described by
c21ᾱP2
c21Ũ +
c21Ṽ + Z1, (55)
Y2 = Ũ + Ṽ +
ᾱP2 +
c12P1
W + Z2. (56)
To simplify the derivations, we fix the time-sharing random variable Q as a constant. The issue
of how this time-sharing random variable affects the achievable rate region is well addressed
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 13
in [13]. In the Gaussian case, the respective mutual information terms in (3) – (9) need be
evaluated with respect to the mappings defined by P1–P7. Since the computation procedure to
obtain G and the resulting description of G are fairly lengthy, we relegate them (Theorem 5) to
the Appendix.
2) Gaussian Extension of Rsuc: For illustration and comparison purpose, we next show how
to obtain the Gaussian counterpart of Rsuc in details. Following the first step in the previous
derivation, we also map the random variables involved in (30) to the Gaussian ones with the
following constraints:
M1) W , distributed according to N (0, 1),
M2) X1 =
P1W ,
M3) U , distributed according to N (0, αβP2),
M4) Ṽ , distributed according to N (0, αβ̄P2),
M5) V = Ṽ + λW ,
M6) X2 = U + Ṽ +
ᾱP2W ,
where α, β ∈ [0, 1], α + ᾱ = 1, β + β̄ = 1, λ ∈ [0,+∞), and W , U and Ṽ are mutually
independent. Using the mappings defined by M1–M6, we express the input-output relationship
for the GIC-DMS as:
c21ᾱP2
c21U +
c21Ṽ + Z1, (57)
Y2 = U + Ṽ +
ᾱP2 +
c12P1
W + Z2. (58)
Let Gsuc(α, β) denote the set of all the non-negative rate pairs (R1, R2) such that
c21ᾱP2
c21αβ̄P2 + 1
, (59)
log2(1 + αβ̄P2) + min
c21αβP2
c21ᾱP2
+ c21αβ̄P2 + 1
αβ̄P2 +
ᾱP2 +
c12P1
. (60)
Define
Gsuc =
α,β∈[0,1]
Gsuc(α, β).
Theorem 4: The region Gsuc is an achievable rate region for the GIC-DMS in the standard
form.
Proof: It suffices to prove that Gsuc(α, β) is achievable for any given α, β ∈ [0, 1]. Since
Gsuc is extended from Rsuc, we need compute the mutual information terms in (41) and (42).
The righthand side of (59) can be readily obtained through a straightforward computation of
I(W ; Y1|UQ) in (41). Recall that Q is a constant. It is also fairly straightforward to obtain the
two terms within the minimum operator in (60) through computing I(U ; Y1|Q) and I(U ; Y2|Q)
in (42). We next evaluate the only remaining term I(V ; Y2|UQ)− I(V ;W |Q) for a constant Q.
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 14
Defining Ỹ2 = Ṽ +
ᾱP2 +
c12P1
W + Z2, we have
I(V ; Y2|U)− I(V ;W ) = h(Y2|U)− h(Y2|UV )− I(V ;W )
= h(Ỹ2)− h(Ỹ2|V )− I(V ;W )
= h(Ỹ2) + h(V )− h(Ỹ2V )− I(V ;W ). (61)
With V = Ṽ + λW , we evaluate (61) as
I(V ; Y2|U)− I(V ;W )
αβ̄P2 +
ᾱP2 +
c12P1
log2(2πe(αβ̄P2 + λ
(2πe)2
αβ̄P2 +
ᾱP2 +
c12P1
(αβ̄P2 + λ
αβ̄P2 + λ
ᾱP2 +
c12P1
αβ̄P2
. (62)
It is easy to find that when
αβ̄P2
ᾱP2 +
c12P1
αβ̄P2 + 1
, (63)
the term I(V ; Y |U)− I(V ;W ) is maximized, and the maximum value is
max[I(V ; Y2|U)− I(V ;W )] =
log2(1 + αβ̄P2). (64)
This is in parallel with the result in [12].
Therefore, the rate region Gsuc(α, β) is achievable for any pair α, β ∈ [0, 1], and the theorem
follows.
In the following, we obtain two corollaries by setting β = 0 and β = 1 in Theorem 4,
respectively.
Corollary 3: The rate region Gsp1 is an achievable rate region for the GIC-DMS in the standard
form with Gsp1 :=
α∈[0,1] Gsuc(α, 0), i.e., Gsp1 is the union of the sets of non-negative rate pairs
(R1, R2) satisfying
c21ᾱP2
c21αP2 + 1
log2(1 + αP2),
over all α ∈ [0, 1].
Corollary 4: The rate region Gsp2 is an achievable rate region for the GIC-DMS in the standard
form with Gsp2 :=
α∈[0,1] Gsuc(α, 1), i.e., Gsp2 is the union of the sets of non-negative rate pairs
(R1, R2) satisfying
c21ᾱP2
R2 ≤min
c21αP2
c21ᾱP2
ᾱP2 +
c12P1
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 15
0 0.5 1 1.5 2
 (bits)
(iii)
Fig. 5. P1 = P2 = 6, c21 = 0.3, c12 = 0. (i) gives the rate region in Theorem 1 of [1]; (ii) gives the rate region in Corollary
2 of [1]; (iii) gives the rate region in Corollary 3 (equivalently, Theorem 4.1 of [2] and Theorem 3.5 of [3]).
over all α ∈ [0, 1].
Remark 4: Corollaries 3 and 4 correspond the Gaussian extensions of Corollaries 1 and 2
respectively. Particularly, the rate region depicted by Corollary 3 is the same as the rate regions
given in [2, Theorem 4.1] and [3, Theorem 3.5]. It has been proven in both [2] and [3] that
the rate region Gsp1 is indeed the capacity region for the GIC-DMS in the low-interference-gain
regime, i.e., c21 ≤ 1.
In addition, the set of achievable rate pairs given in [2, Lemma 4.2] is contained in the region
Gsp2 as a subset.
C. Numerical Examples
We next provide several numerical examples to illustrate improvements of our achievable rate
regions over the previously known results in [1]–[3]. Denote the achievable rate regions obtained
in [1, Theorem 1] and [1, Corollary2] by Gdmt1 and Gdmt2, respectively.
1) Comparing with Rate Regions in [1]: Fig. 5 compares the rate regions Gdmt1, Gdmt2, and
Gsp1 for an extreme case in which receiver 2 does not experience any interference from sender
1, i.e., c12 = 0. As can be seen from Fig. 5, the rate region Gsp1 strictly includes Gdmt1, as well
as Gdmt2 obtained through time-sharing between Gdmt1 and a fully-cooperative rate point. The
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 16
coding scheme used to establish Gdmt1 incurs certain rate loss due to the fact that sender 2 does
not use its power to help the sender 1’s transmissions even though it has complete and non-
causal knowledge about the message being transmitted by sender 1. In contrast, our proposed
coding scheme allows sender 2 to use superposition coding to help sender 1, and thus yields an
improved rate region.
In Fig. 6, we consider another case in which the transmit power of sender 1 is set to zero and
c21 ≤ 1. From the figure, we observe that the rate region Gdmt2 is strictly smaller than Gsp1. Note
that in this case, the GIC-DMS becomes a Gaussian degraded broadcast channel. According to
[8], the optimal coding scheme for this case is: sender 2 uses a portion of its power to transmit
the codeword conveying w1, and uses the remaining power to transmit the codeword conveying
w2, which is encoded by using the dirty-paper coding [14]. It is easy to verify that this scheme
is a special case of the coding scheme developed in Theorem 1.
0 0.2 0.4 0.6 0.8 1
 (bits)
Fig. 6. P1 = 0, P2 = 6, c21 = 0.5. (i) gives the rate region in Corollary 2 of [1]; (ii) gives the rate region in Corollary 3
(equivalently, Theorem 4.1 of [2] and Theorem 3.5 of [3]).
2) Comparing with Rate Regions in [2], [3]: As mentioned earlier, the rate region Gsp1, a
subregion of G, is the same as the one given in [2, Theorem 4.1] and the one given in [3,
Theorem 3.5], which is indeed the capacity region for GIC-DMS in the low-interference-gain
regime. In Figs. 7 and 8, we compare G with Gsp1 and Gsp2 in the high-interference-gain regime,
i.e., c21 > 1. As can be seen from the figures, the rate region G strictly includes both Gsp1 and
Gsp2 in this case. Comparing Fig. 7 with Fig. 8, we observe that the improvement of the rate
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 17
0 0.5 1 1.5 2 2.5 3
 (bits)
(iii)
Fig. 7. P1 = P2 = 6, c21 = 2, c12 = 0.3. (i) gives the rate region in Corollary 3 (equivalently, Theorem 4.1 of [2] and
Theorem 3.5 of [3]); (ii) gives the achievable rate region in Corollary 4; (iii) gives the achievable rate region in Theorem 5.
region G over Gsp1 becomes more pronounced as the link gain c21 increases. The improvement is
mainly because our coding scheme allows receiver 1 to decode partial information from sender
2, and thus reduces the effective interference experienced by receiver 1. In addition, it can be
seen from the figures that in the high-interference-gain regime, Gsp1 is not convex and thus is
only suboptimal.
VI. CONCLUSIONS
In this paper, we have investigated the IC-DMS from an information theoretic perspective.
We have developed a coding scheme that combines the advantages of cooperative coding,
collaborative coding and Gel’fand-Pinsker coding. With the coding scheme, we have derived
a new achievable rate region for such a channel, which not only includes existing results as
special cases, but also exceeds them in the high-interference-gain regime. However, we are not
able to establish a converse for the derived achievable rate region, because the achievable result
is closely related to the achievable results for the interference channel and the broadcast channel,
for which there is no converse available in general.
APPENDIX
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 18
0 0.5 1 1.5 2 2.5 3 3.5
 (bits)
(iii)
Fig. 8. P1 = P2 = 6, c21 = 6, c12 = 0.3. (i) gives the rate region in Corollary 3 (equivalently, Theorem 4.1 of [2] and
Theorem 3.5 of [3]); (ii) gives the achievable rate region in Corollary 4; (iii) gives the achievable rate region in Theorem 5.
AN ACHIEVABLE RATE REGION FOR THE GIC-DMS
In this appendix, we show how to extend R, the achievable rate region for the discrete
memoryless IC-DMS, to its Gaussian counterpart, G. Note that the mappings M1–M6 of the
auxiliary random variables are described in Section V. We first compute the following two
covariance matrices:
ΣWUY1 =
µ11 µ12 µ13
µ21 µ22 µ23
µ31 µ32 µ33
E{W 2} E{WU} E{WY1}
E{WU} E{U2} E{UY1}
E{WY1} E{UY1} E{Y 21 }
P1 λ1P1 η1
λ1P1 αβP2 + λ
1P1 λ1η1
c21αβP2
P1 λ1η1
c21αβP2 η
1 + c21αP2 + 1
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 19
ΣUV Y2 =
ν11 ν12 ν13
ν21 ν22 ν23
ν31 ν32 ν33
E{U2} E{UV } E{UY2}
E{UV } E{V 2} E{V Y2}
E{UY2} E{V Y2} E{Y 22 }
αβP2 + λ
1P1 λ1λ2P1 αβP2 + λ1η2
λ1λ2P1 αβ̄P2 + λ
2P1 αβ̄P2 + λ2η2
αβP2 + λ1η2
P1 αβ̄P2 + λ2η2
P1 αP2 + η
2 + 1
where
c21ᾱP2,
ᾱP2 +
c12P1,
and E{·} denotes the expectation of a random variable.
Define Γ(x) = log2(x)/2, and ξ = log2(2πe)/2. We express the respective differential entropy
terms as:
ha = h(W ) = ξ + Γ(µ11),
hb = h(UY1) = 2ξ + Γ
µ22 µ23
µ32 µ33
hc = h(WUY1) = 3ξ + Γ
µ11 µ12 µ13
µ21 µ22 µ23
µ31 µ32 µ33
hd = h(UV ) = 2ξ + Γ
ν11 ν12
ν21 ν22
he = h(Y2) = ξ + Γ(ν33),
hf = h(UV Y2) = 3ξ + Γ
ν11 ν12 ν13
ν21 ν22 ν23
ν31 ν32 ν33
hg = h(WU) = 2ξ + Γ
µ11 µ12
µ21 µ22
hh = h(Y1) = ξ + Γ(µ33),
hi = h(V ) = ξ + Γ(ν22),
hj = h(UY2) = 2ξ + Γ
µ11 µ13
µ31 µ33
hk = h(U) = ξ + Γ(ν11),
hl = h(V Y2) = 2ξ + Γ
µ22 µ23
µ32 µ33
where | · | denotes the determinant of a matrix.
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 20
The mutual information terms in (3)–(9) are then computed as:
I1 = ha + hb − hc,
I2 = hd + he − hf ,
I3 = Γ(1 +
λ21P1
I4 = Γ(1 +
λ22P1
αβ̄P2
I5 = hg + hh − hc,
I6 = hi + hj − hf ,
I7 = hk + hl − hf .
Let G(α, β, λ1, λ2) denote the set of all rate pairs (R1, R2) such that the following inequalities
are satisfied:
R1 ≤ I1, (65)
R2 ≤ I2 − I3 − I4, (66)
R1 +R2 ≤ I5 + I6 − I3 − I4; (67)
0 ≤ I5 − I3, (68)
0 ≤ I7 − I3, (69)
0 ≤ I6 − I4, (70)
0 ≤ I2 − I3 − I4. (71)
for given α, β ∈ [0, 1] and λ1, λ2 ∈ [0,+∞). Note that (65)–(71) are directly extended from
(3)–(9).
Theorem 5: The rate region G is achievable for the GIC-DMS in the standard form with
α,β∈[0,1];λ1,λ2∈[0,+∞)
G(α, β, λ1, λ2).
November 4, 2018 DRAFT
IEEE TRANSACTIONS ON INFORMATION THEORY (SUBMITTED) 21
REFERENCES
[1] N. Devroye, P. Mitran, and V. Tarokh, “Achievable rates in cognitive channels,” IEEE Trans. Inform. Theory, vol. 52, no. 5,
pp. 1813–1827, May 2006.
[2] A. Jovičić and P. Viswanath, “Cognitive radio: An information-theoretic perspective,” 2006. [Online]. Available:
http://arxiv.org/abs/cs/0604107
[3] W. Wu, S. Vishwanath, and A. Arapostathis, “On the capacity of interference channels with degraded message sets,”
2006. [Online]. Available: http://arxiv.org/abs/cs/0605071
[4] I. Maric, R. Yates, and G. Kramer, “The strong interference channel with unidirectional cooperation,” in Proc. UCSD
Workshop on Information Theory and its Applications, San Diego, CA, USA, Feb. 2006.
[5] S. I. Gel’fand and M. S. Pinsker, “Coding for channel with random parameters,” Problems of Control and Information
Theory, vol. 9, no. 1, pp. 19–31, 1980.
[6] A. B. Carleial, “Interference channels,” IEEE Trans. Inform. Theory, vol. IT-24, no. 1, pp. 60–70, Jan. 1978.
[7] T. S. Han and K. Kobayashi, “A new achievable rate region for the interference channel,” IEEE Trans. Inform. Theory,
vol. IT-27, no. 1, pp. 49–60, Jan. 1981.
[8] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: John Wiley & Sons, 1994.
[9] T. M. Cover, “An achievable rate region for the broadcast channel,” IEEE Trans. Inform. Theory, vol. IT-21, pp. 399–404,
July 1975.
[10] T. Berger, “Multiterminal source coding,” in The Information Theory Approach to Communications, ser. CISM Courses
and Lectures,G. Longo, Ed. Springer-Verlag, vol. 229, 1978, pp. 171–231.
[11] G. Kramer, “Review of rate regions for interference channels,” in Proc. IZS Workshop, Zurich, Feb. 2006.
[12] M. H. M. Costa, “Writing on dirty paper,” IEEE Trans. Inform. Theory, vol. IT-29, no. 3, pp. 439–441, May 1983.
[13] I. Sason, “On achievable rate regions for the Gaussian interference channel,” IEEE Trans. Inform. Theory, vol. 50, no. 6,
pp. 1345–1356, June 2004.
[14] W. Yu and J. Cioffi, “Trellis precoding for the broadcast channel,” in Proc. IEEE Global Telecommunications Conference
(Globecom), San Antonio, TX, Nov. 2001.
November 4, 2018 DRAFT
http://arxiv.org/abs/cs/0604107
http://arxiv.org/abs/cs/0605071
	Introduction
	The Channel Model
	An Achievable Rate Region for the Discrete Memoryless IC-DMS
	Random Codebook Generation
	Encoding and Transmission
	Decoding
	Evaluation of Probability of Error
	Relating with Existing Rate Regions
	A Subregion of R
	A Subregion of Rsim
	The Gaussian IC-DMS
	The Channel Model of the GIC-DMS
	Achievable Rate Regions for the GIC-DMS
	Gaussian Extension of R
	Gaussian Extension of Rsuc
	Numerical Examples
	Comparing with Rate Regions in Tarokh06:icdmscog
	Comparing with Rate Regions in jovicic06:cogICDMS,wuwei06icdms
	Conclusions
	References
