Ergodic Fading Interference Channels: Sum-Capacity and Separability

Abstract

The sum-capacity for specific sub-classes of ergodic fading Gaussian two-user
interference channels (IFCs) is developed under the assumption of perfect
channel state information at all transmitters and receivers. For the
sub-classes of uniformly strong (every fading state is strong) and ergodic very
strong two-sided IFCs (a mix of strong and weak fading states satisfying
specific fading averaged conditions) the optimality of completely decoding the
interference, i.e., converting the IFC to a compound multiple access channel
(C-MAC), is proved. It is also shown that this capacity-achieving scheme
requires encoding and decoding jointly across all fading states. As an
achievable scheme and also as a topic of independent interest, the capacity
region and the corresponding optimal power policies for an ergodic fading C-MAC
are developed. For the sub-class of uniformly weak IFCs (every fading state is
weak), genie-aided outer bounds are developed. The bounds are shown to be
achieved by treating interference as noise and by separable coding for
one-sided fading IFCs. Finally, for the sub-class of one-sided hybrid IFCs (a
mix of weak and strong states that do not satisfy ergodic very strong
conditions), an achievable scheme involving rate splitting and joint coding
across all fading states is developed and is shown to perform at least as well
as a separable coding scheme.

Full text

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arXiv:0906.0744v1 [cs.IT] 3 Jun 2009
1
Ergodic Fading Interference Channels:
Sum-Capacity and Separability
Lalitha Sankar, Member, IEEE, Xiaohu Shang, Member, IEEE, Elza
Erkip, Senior Member, IEEE, and H. Vincent Poor, Fellow, IEEE
Abstract
The sum-capacity of ergodic fading Gaussian two-user interference channels (IFCs) is developed
under the assumption of perfect channel state information at all transmitters and receivers. For the sub-
classes of uniformly strong (every fading state is strong) and ergodic very strong two-sided IFCs (a mix of
strong and weak fading states satisfying specific fading averaged conditions) the optimality of completely
decoding the interference, i.e., converting the IFC to a compound multiple access channel (C-MAC), is
proved. It is also shown that this capacity-achieving scheme requires encoding and decoding jointly
across all fading states. As an achievable scheme and also as a topic of independent interest, the capacity
region and the corresponding optimal power policies for an ergodic fading C-MAC are developed. For the
sub-class of uniformly weak IFCs (every fading state is weak), genie-aided outer bounds are developed.
The bounds are shown to be achieved by ignoring interference and separable coding for one-sided fading
IFCs. Finally, for the sub-class of one-sided hybrid IFCs (a mix of weak and strong states that do not
satisfy ergodic very strong conditions), an achievable scheme involving rate splitting and joint coding
across all fading states is developed and is shown to perform at least as well as a separable coding
scheme.
L. Sankar, X. Shang, and H. V. Poor are with the Department of Electrical Engineering, Princeton University, Princeton,
NJ 08544, USA. email: {lalitha,xshang,poor@princeton.edu}. E. Erkip is with the Department of Electrical and Computer
Engineering, Polytechnic Institute of New York University, Brooklyn, NY 11201, USA. email: elza@poly.edu. This research
was conducted in part when E. Erkip was visiting Princeton University.
This research was supported in part by the National Science Foundation under Grant CNS-06-25637 and in part by a fellowship
from the Princeton University Council on Science and Technology. The material in this paper was presented in part at the IEEE
International Symposium on Information Theory, Toronto, Canada, Jul. 2008 and at the 46thAnnual Allerton Conference on
Communications, Control, and Computing, Monticello, IL, Sep. 2008.
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Index Terms
Interference channel, ergodic fading, strong and weak interference.
I. INTRODUCTION
The interference channel (IFC) models a wireless network where every transmitter (user) communicates
with its unique intended receiver while causing interference to the remaining receivers. For the two-user
IFC, the topic of study in this paper and henceforth simply referred to as an IFC, the capacity region
is not known in general even when the channel is time-invariant, i.e., non-fading. Capacity results are
known only for specific classes of non-fading two-user IFCs where the classes are identified by the
relative strength of the channel gains of the interfering cross-links and the intended direct links. Thus,
strong and weak IFCs refer to the cases where the channel gains of the cross-links are at least as large
as those of the direct links and vice-versa.
The capacity region for the class of strong Gaussian IFCs is developed independently in [1], [2], [3]
and can be achieved when both receivers decode both the intended and interfering messages. In contrast,
for the weak channels, the sum-capacity can be achieved by ignoring interference when the channel
gains of one of the cross-links is zero, i.e., for a one-sided IFC [4]. More recently, the sum-capacity of a
class of noisy or very weak Gaussian IFCs has been determined independently in [5], [6], and [7]. Outer
bounds for the IFC are developed in [8] and [9] while several achievable rate regions for the Gaussian
IFC are studied in [10].
The best known inner bound is due to Han and Kobayashi (HK) [3]. Recently, in [9] a simple HK type
scheme is shown to achieve every rate pair within 1 bit/s/Hz of the capacity region. In [11], the authors
reformulate the HK region as a sum of two sets to characterize the maximum sum-rate achieved by
Gaussian inputs and without time-sharing. More recently, the approximate capacity of two-user Gaussian
IFCs is characterized using a deterministic channel model in [12]. The sum-capacity of the class of
non-fading MIMO IFCs is studied in [13].
Relatively fewer results are known for parallel or fading IFCs. In [14], the authors develop an achievable
scheme of a class of two-user parallel Gaussian IFCs where each parallel channel is strong using
independent encoding and decoding in each parallel channel. In [15], Sung et al. present an achievable
scheme for a class of one-sided two-user parallel Gaussian IFCs. The achievable scheme involves encoding
and decoding signals over each parallel channel independently such that, depending on whether a parallel
channel is weak or strong (including very strong) one-sided IFC, the interference in that channel is either
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viewed as noise or completely decoded, respectively. In this paper, we show that independent coding
across sub-channels is in general not sum-capacity optimal.
Recently, for parallel Gaussian IFCs, [16] determines the conditions on the channel coefficients and
power constraints for which independent transmission across sub-channels and treating interference as
noise is optimal. Techniques for MIMO IFCs [13] are applied to study separability in parallel Gaussian
IFCs (PGICs) in [17]. It is worth noting that PGICs are a special case of ergodic fading IFCs in which
each sub-channel is assigned the same weight, i.e., occurs with the same probability; furthermore, they
can also be viewed as a special case of MIMO IFCs and thus results from MIMO IFCs can be directly
applied.
For fading interference networks with three or more users, in [18], the authors develop an interference
alignment coding scheme to show that the sum-capacity of a K-user IFC scales linearly with K in the
high signal-to-noise ratio (SNR) regime when all links in the network have similar channel statistics.
In this paper, we study ergodic fading two-user Gaussian IFCs and determine the sum-capacity and
the corresponding optimal power policies for specific sub-classes, where we define each sub-class by
the fading statistics. Noting that ergodic fading IFCs are a weighted collection of parallel IFCs (sub-
channels), we identify four sub-classes that jointly contain the set of all ergodic fading IFCs. We develop
the sum-capacity for two of them. For the third sub-class, we develop the sum-capacity when only one
of the two receivers is affected by interference, i.e., for a one-sided ergodic fading IFC. While the four
sub-classes are formally defined in the sequel, we refer the reader to Fig. 1 for a pictorial representation.
An overview of the capacity results is illustrated in the sequel in Fig. 7.
A natural question that arises in studying ergodic fading and parallel channels is the optimality of
separable coding, i.e., whether encoding and decoding independently on each sub-channel is optimal in
achieving one or more points on the boundary of the capacity region. For each sub-class of IFCs we
consider, we address the optimality of separable coding, often referred to as separability, and demonstrate
that in contrast to point-to-point, multiple-access, and broadcast channels without common messages [19],
[20], [21], separable coding is not necessarily sum-capacity optimal for ergodic fading IFCs.
The first of the four sub-classes is the set of ergodic very strong (EVS) IFCs in which each sub-channel
can be either weak or strong but averaged over all fading states (sub-channels) the interference at each
receiver is sufficiently strong that the two direct links from each transmitter to its intended receiver are the
bottle-necks limiting the sum-rate. For this sub-class, we show that requiring both receivers to decode the
signals from both transmitters is optimal, i.e., the ergodic very strong IFC modifies to a two-user ergodic
fading compound multiple-access channel (C-MAC) in which the transmitted signal from each user is
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intended for both receivers [22]. To this end, as an achievable rate region for IFCs and as a problem of
independent interest, we develop the capacity region and the optimal power policies that achieve them
for ergodic fading C-MACs (see also [22]).
For EVS IFCs we also show that achieving the sum-capacity (and the capacity region) requires
transmitting information (encoding and decoding) jointly across all sub-channels, i.e., separable coding
in each sub-channel is strictly sub-optimal. Intuitively, the reason for joint coding across channels lies in
the fact that, analogous to parallel broadcast channels with common messages [23], both transmitters in
the EVS IFCs transmit only common messages intended for both receivers for which independent coding
across sub-channels becomes strictly sub-optimal. To the best of our knowledge this is the first capacity
result for fading two-user IFCs with a mix of weak and strong sub-channels. For such mixed ergodic
IFCs, recently, a strategy of ergodic interference alignment is proposed in [24], and is shown to achieve
the sum-capacity in [25] for a class of K-user fading IFCs with uniformly distributed phase and at least
K/2 disjoint equal strength interference links.
The second sub-class is the set of uniformly strong (US) IFCs in which every sub-channel is strong,
i.e., the cross-links have larger fading gains than the direct links for each fading realization. For this
sub-class, we show that the capacity region is the same as that of an ergodic fading C-MAC with the
same fading statistics and that achieving this region requires joint coding across all sub-channels.
The third sub-class is the set of uniformly weak (UW) IFCs for which every sub-channel is weak. As
a first step, we study the one-sided uniformly weak IFC and develop genie-aided outer bounds. We show
that the bounds are tight when the interfering receiver ignores the weak interference in every sub-channel.
Furthermore, we show that separable coding is optimal for this sub-class. The sum-capacity results for
the one-sided channel are used to develop outer bounds for the two-sided case; however, sum-capacity
results for the two-sided case will require techniques such as those developed in [16] that also determine
the channel statistics and power policies for which ignoring interference and separable coding is optimal.
The final sub-class is the set of hybrid IFCs for which the sub-channels are a mix of strong and weak
such that there is at least one weak and one strong sub-channel but are not EVS IFCs (and by definition
also not US and UW IFCs). The capacity-achieving strategy for EVS and US IFCs suggest that a joint
coding strategy across the sub-channels can potentially take advantage of the strong states to partially
eliminate interference. To this end, for ergodic fading one-sided IFCs, we propose a general joint coding
strategy that uses rate-splitting and Gaussian codebooks without time-sharing for all sub-class of IFCs.
For two-sided IFCs, the coding strategy we present generalizes to a two-sided HK-based scheme with
Gaussian codebooks and no time-sharing that is presented and studied in [26].
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In the non-fading case, a one-sided non-fading IFC is either weak or strong and the sum-capacity is
known in both cases. In fact, for the weak case the sum-capacity is achieved by ignoring the interference
and for the strong case it is achieved by decoding the interference at the receiver subject to the interference.
However, for ergodic fading one-sided IFCs, in addition to the UW and US sub-classes, we also have
to contend with the hybrid and EVS sub-classes each of which has a unique mix of weak and strong
sub-channels. The HK-based achievable strategy we propose applies to all sub-classes of one-sided IFCs
and includes the capacity-achieving strategies for the EVS, US, and UW as special cases.
The sub-class of uniformly mixed (UM) IFCs obtained by overlapping two complementary one-sided
IFCs, one of which is uniformly strong and the other uniformly weak, belongs to the sub-class of hybrid
(two-sided) IFCs. For UM IFCs, we show that to achieve sum-capacity the transmitter that interferes
strongly transmits a common message across all sub-channels while the weakly interfering transmitter
transmits a private message across all sub-channels. The two different interfering links however require
joint encoding and decoding across all sub-channels to ensure optimal coding at the receiver with strong
interference.
Finally, a note on separability. In [27], Cadambe and Jafar demonstrate the inseparability of parallel
interference channels using an example of a three-user frequency selective fading IFC. The authors use
interference alignment schemes to show that separability is not optimal for fading IFCs with three or
more users while leaving open the question for the two-user fading IFC. We addressed this question in
[28] for the ergodic fading one-sided IFC and developed the conditions for the optimality of separability
for EVS and US one-sided IFCs. In this paper, we readdress this question for all sub-classes of fading
IFCs. Our results suggest that in general both one-sided and two-sided IFCs benefit from transmitting
the same information across all sub-channels, i.e., not independently encoding and decoding in each
sub-channel, thereby exploiting the fading diversity to mitigate interference.
The paper is organized as follows. In Section II, we present the channel models studied. In Section
III, we summarize our main results. The capacity region of an ergodic fading C-MAC is developed in
Section IV. The proofs are collected in Section V. We discuss our results with numerical examples in
Section VI and conclude in Section VII.
II. CHANNEL MODEL AND PRELIMINARIES
A two-sender two-receiver (also referred to as the two-user) ergodic fading Gaussian IFC consists of
two source nodes S1 and S2, and two destination nodes D1 and D2 as shown in Fig. 2. Source Sk,
k = 1,2, uses the channel n times to transmit its message Wk, which is distributed uniformly in the set
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Two-user Erg. Fading Two-sided IFCs
US IFC:
every
sub-ch.
is strong
Two-user Erg. Fading One-sided IFCs
EVS IFC:
mix of weak and
strong sub-channels
Hybrid IFC: non-EVS
mix of weak and strong
Mixed
IFCs:
every
sub-ch
mixed
EVS IFC:
mix of weak and
strong sub-channels
Hybrid IFC:
non-EVS mix of
weak and strong
UW:
weak sub-channels
US IFC:
every
sub-ch.
is strong
UW:
weak sub-channels
Fig. 1. A Venn diagram representation of the four sub-classes of ergodic fading one- and two-sided IFCs.
1,1
h
1,2
h
2,2
h
1:
S
2:
S
1
D
2
D
2,1
h
11
N
WX
→
1
12
ˆ
IC:
ˆˆ
C-MAC:(,)
W
W W
2
12
ˆ
IC:
ˆˆ
C-MAC:(,)
W
W W
22
N
WX
→
1,1
h
1,2
h
2,2
h
1S
2 S
1
D
2
D
(a) Two-sided IFC
(b) One-sided IFC
Fig. 2. The two-user Gaussian two-sided IFC and C-MAC and the two-user Gaussian one-sided IFC.
{1,2,...,2Bk} and is independent of the message from the other source, to its intended receiver, Dk, at
a rate Rk= Bk/n bits per channel use. In each use of the channel, Sktransmits the signal Xkwhile
the destination Dkreceives Yk, k = 1,2. For X = [X1X2]T, the channel output vector Y = [Y1Y2]T
is given by
Y = HX + Z
(1)
where Z = [Z1Z2]Tis a noise vector with entries that are zero-mean, unit variance, circularly symmetric
complex Gaussian noise variables and H is a random matrix of fading gains with entries Hm,k, for all
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m,k = 1,2, such that Hm,kdenotes the fading gain between receiver m and transmitter k. We use h to
denote a realization of H. We assume the fading process {H} is stationary and ergodic but not necessarily
Gaussian. Note that the channel gains Hm,k, for all m and k, are not assumed to be independent; however,
H is known instantaneously at all the transmitters and receivers.
Over n uses of the channel, the transmit sequences {Xk,i} are constrained in power according to
n
?
Since the transmitters know the fading states of the links on which they transmit, they can allocate their
i=1
|Xk,i|2≤ nPk, for all k = 1,2. (2)
transmitted signal power according to the channel state information. A power policy P(h) is a mapping
from the fading state space consisting of the set of all fading states (instantiations) h to the set of non-
negative real values in R2
P(h) denotes the map for a particular fading state, we write P(H) to explicitly describe the policy for
+. The entries of P(h) are Pk(h), the power policy at user k, k = 1,2. While
the entire set of random fading states. Thus, we use the notation P(H) when averaging over all fading
states or describing a collection of policies, one for every h. The entries of P(H) are Pk(H), for all k.
For an ergodic fading channel, (2) then simplifies to
E[Pk(H)] ≤ Pk for all k = 1,2,
(3)
where the expectation in (3) is over the distribution of H. We denote the set of all feasible policies
P (h), i.e., the power policies whose entries satisfy (3), by P. Finally, we write P to denote the vector
of average power constraints with entries Pk, k = 1,2.
For the special case where both receivers decode the messages from both transmitters, we obtain a
compound MAC (see Fig. 2(a)). A one-sided fading Gaussian IFC results when either H1,2 = 0 or
H2,1= 0 (see Fig. 2(b)). Without loss of generality, we develop sum-capacity results for a one-sided
IFC (Z-IFC) with H2,1= 0. The results extend naturally to the complementary one-sided model with
H1,2= 0. A two-sided IFC can be viewed as a collection of two complementary one-sided IFCs, one
with H1,2= 0 and the other with H2,1= 0.
We write CIFC
and C-MAC, respectively. Our definition of average error probabilities, capacity regions, and achievable
?P1,P2
?and CC-MAC
?P1,P2
?to denote the capacity region of an ergodic fading IFC
rate pairs (R1,R2) for both the IFC and C-MAC mirror the standard information-theoretic definitions
[29, Chap. 14].
Non-fading IFCs can be classified by the relative strengths of the interfering to intended signals at
each of the receivers. A (two-sided non-fading) strong IFC is one in which the cross-link channel gains
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are larger than the direct link channel gains to the intended receivers [1], i.e.,
|Hj,k| > |Hk,k|
for all j,k = 1,2, j ?= k.
(4)
A strong IFC is very strong if the cross-link channel gains dominate the transmit powers such that (see
for e.g., [1], [2])
2 ?
k=1
C
?
|Hk,k|2Pk(H)
?
< C
?
2 ?
k=1
|Hj,k|2Pj(H)
?
for all j = 1,2,
(5)
where for the non-fading IFC, Pk(H) = Pkin (2). One can verify that (5) implies (4), i.e., a very strong
IFC is also strong.
A non-fading IFC is weak when (4) is not satisfied for all j,k, i.e., neither of the two complementary
one-sided IFCs that a two-sided IFC can be decomposed into are strong. A non-fading IFC is mixed
when one of complementary one-sided IFCs is weak while the other is strong, i.e.,
|H1,2| > |H2,2|
and
|H2,1| < |H1,1|
(6)
or
|H1,2| > |H2,2|
and
|H2,1| < |H1,1|.
(7)
An ergodic fading IFC is a collection of parallel sub-channels (fading states), and thus, each sub-
channel can be either very strong, strong, or weak. Since a fading IFC can contain a mixture of different
types of sub-channels, we introduce the following definitions to classify the set of all ergodic fading
two-user Gaussian IFCs (see also Fig. 1). Unless otherwise stated, we henceforth simply write IFC to
denote a two-user ergodic fading Gaussian IFC.
Definition 1: A uniformly strong IFC is a collection of strong sub-channels, i.e., both cross-links in
each sub-channel satisfy (4).
Definition 2: An ergodic very strong IFC is a collection of weak and strong (including very strong) sub-
channels for which (5) is satisfied when averaged over all fading states and for Pk(H) = P(wf)
where P(wf)
k
(Hkk) is the optimal waterfilling policy that achieves the point-to-point capacity for user k
in the absence of interference.
k
(Hkk),
Definition 3: A uniformly weak IFC is a collection of weak sub-channels, i.e., in each sub-channel
both cross-links do not satisfy (4).
Definition 4: A uniformly mixed IFC is a pair of two complementary one-sided IFCs in which one of
them is uniformly weak and the other is uniformly strong.
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Definition 5: A hybrid IFC is a collection of weak and strong sub-channels with at least one weak
and one strong sub-channel that do not satisfy the conditions in (5) when averaged over all fading states
and for Pk(H) = P(wf)
k
(Hkk).
Since an ergodic fading channel is a collection of parallel sub-channels (fading states) with different
weights, throughout the sequel, we use the terms fading states and sub-channels interchangeably. In
contrast to the one-sided IFC, we simply write IFC to denote the two-sided model. Before proceeding,
we summarize the notation used in the sequel.
• Random variables (e.g. Hk,j) are denoted with uppercase letters and their realizations (e.g. hk,j)
with the corresponding lowercase letters.
• Bold font X denotes a random matrix while bold font x denotes an instantiation of X.
• I denotes the identity matrix.
• |X| and X−1denotes the determinant and inverse of the matrix X.
• CN (0,Σ) denotes a circularly symmetric complex Gaussian distribution with zero mean and co-
variance Σ.
• K = {1,2} denotes the set of transmitters.
• E(·) denotes expectation; C(x) denotes log(1 + x) where the logarithm is to the base 2, (x)+
denotes max(x,0), I(·;·) denotes mutual information, h(·) denotes differential entropy, and RS
denotes?
III. MAIN RESULTS
k∈SRkfor any S ⊆ K.
The following theorems summarize the main contributions of this paper. The proof for the capacity
region of the C-MAC is presented in Section IV as are the details of determining the capacity achieving
power policies. The proofs for the remaining theorems, related to IFCs, are collected in Section V.
Throughout the sequel we write waterfilling solution to denote the capacity achieving power policy for
ergodic fading point-to-point channels [19].
A. Ergodic fading C-MAC
An achievable rate region for ergodic fading IFCs results from allowing both receivers to decode
the messages from both transmitters, i.e., by converting an IFC to a C-MAC. The following theorem
summarizes the sum-capacity CC-MACof an ergodic fading C-MAC.
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Theorem 1: The capacity region, CC-MAC
with average power constraints Pkat transmitter k, k = 1,2, is
?P1,P2
?, of an ergodic fading two-user Gaussian C-MAC
CC-MAC
?P1,P2
?=
?
P∈P
{C1(P (H)) ∩ C2(P (H))}
(8)
where for j = 1,2, we have
Cj(P (H)) =
?
(R1,R2) : RS≤ E
?
C
??
k∈S
|Hj,k|2Pk(H)
??
,for all S ⊆ K
?
.
(9)
The optimal coding scheme requires encoding and decoding jointly across all sub-channels.
Remark 1: The capacity region CC-MACis convex. This follows from the convexity of the set P and
the concavity of the log function.
Remark 2: CC-MACis a function of?P1,P2
policies, i.e., over all P (H) whose entries satisfy (3).
?due to the fact that union in (8) is over all feasible power
Remark 3: In contrast to the ergodic fading point-to-point and multiple access channels, the ergodic
fading C-MAC is not merely a collection of independent parallel channels; in fact encoding and decoding
independently in each parallel channel is in general sub-optimal as demonstrated later in the sequel.
Corollary 1: The capacity region CIFCof an ergodic fading IFC is bounded as CC-MAC⊆ CIFC.
B. Ergodic Very Strong IFCs
Theorem 2: The capacity region of an ergodic very strong IFC is
CEV S
IFC
=
?
(R1,R2) : Rk≤ E
?
C
?
|Hk,k|2Pwf
k
(Hk,k)
??
,k = 1,2
?
.
(10)
The sum-capacity is
2
?
k=1
E
?
C
?
|Hk,k|2Pwf
k
(Hk,k)
??
(11)
where, for all k, Pwf
k
(Hj,k) is the optimal waterfilling solution for an (interference-free) ergodic fading
link between transmitter k and receiver k such that, Pwf(Hk,k) satisfies
2
?
k=1
E
?
C
?
|Hk,k|2Pwf
k
(Hk,k)
??
< min
j=1,2E
?
C
?
2
?
k=1
|Hj,k|2Pwf
k
(Hk,k)
??
.
(12)
The capacity achieving scheme requires encoding and decoding jointly across all sub-channels at the
transmitters and receivers respectively. The optimal strategy also requires both receivers to decode
messages from both transmitters.
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Remark 4: In the sequel we show that the condition in (12) is a result of the achievable strategy,
and therefore is a sufficient condition. For the special case of fixed (non-fading) channel gains H, and
P∗
k= P1, (12) reduces to the general conditions for a very strong IFC (see for e.g., [1]) given by
|H1,2|2> |H2,2|2?
|H2,1|2> |H1,1|2?
1 + |H1,1|2P1
1 + |H2,2|2P2
?
?
(13a)
.
(13b)
In contrast, the fading averaged conditions in (12) imply that not every sub-channel needs to satisfy (13)
and in fact, the ergodic very strong channel can be a mix of weak and strong channels provided P(wf)
satisfies (12). This in turn implies that not every parallel sub-channel needs to be a strong (non-fading)
Gaussian IFC.
Remark 5: The set of strong fading IFCs for which every sub-channel is strong and the optimal
waterfilling policies for the two interference-free links satisfy (12) is strictly a subset of the set of
ergodic very strong IFCs.
Remark 6: As stated in Theorem 2, the capacity achieving scheme for EVS IFCs requires coding
jointly across all sub-channels. Coding independent messages (separable coding) across the sub-channels
is optimal only when every sub-channel is very strong at the optimal policy P(wf).
C. Uniformly Strong IFC
In the following theorem, we present the capacity region and the sum-capacity of a uniformly strong
IFC.
Theorem 3: The capacity region of a uniformly strong fading IFC for which the entries of every fading
state h satisfy
|h1,1| ≤ |h2,1|
and
|h2,2| ≤ |h1,2|
(14)
is given by
CUS
IFC
?P1,P2
?= CC-MAC
?P1,P2
?
(15)
where CC-MAC
the IFC. The sum-capacity is
?P1,P2
?is the capacity of an ergodic fading C-MAC with the same channel statistics as
?
The capacity achieving scheme requires encoding and decoding jointly across all sub-channels at the
max
P(H)∈Pminmin
j=1,2
?
E
?
C
??2
k=1|Hj,k|2Pk(H)
???
,
2
?
k=1
E
?
C
?
|Hk,k|2Pk(H)
???
.
(16)
transmitters and receivers, respectively, and also requires both receivers to decode messages from both
transmitters.
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12
Remark 7: In contrast to the very strong case, every sub-channel in a uniformly strong fading IFC is
strong.
Remark 8: The uniformly strong condition may suggest that separability is optimal. However, the
capacity achieving C-MAC approach requires joint encoding and decoding across all sub-channels. A
strategy where each sub-channel is viewed as an independentIFC, as in [14], will in general be strictly sub-
optimal. This is seen directly from comparing (16) with the sum-rate achieved by coding independently
over the sub-channels which is given by
max
P(H)∈PE
The sub-optimality of independent encoding follows directly from the fact that for two random variables
?
min
?
min
j=1,2
?
C
??2
k=1|Hj,k|2Pk(H)
??
,
2
?
k=1
C
?
|Hk,k|2Pk(H)
???
.
(17)
A(H) and ? B (H), E[min(A(H),B (H))] ≤ min(E[A(H)],E[B (H)])] with equality if and only if
for every fading instantiation h, A(H) (resp. B (H)) dominates B (H) (resp. A(H)). Thus, independent
(separable) encoding across sub-channels is optimal only when, at P∗(H), the sum-rate in every sub-
channel in (17) is maximized by the same sum-rate function.
D. Uniformly Weak One-Sided IFC
The following theorem summarizes the sum-capacity of a one-sided uniformly weak IFC in which
every sub-channel is weak.
Theorem 4: The sum-capacity of a uniformly weak ergodic fading Gaussian one-sided IFC for which
the entries of every fading state h satisfy
|h2,2| > |h1,2|
(18)
is given by
max
P(H)∈P
?
S(w,1)(P (H))
?
(19)
where
S(w,1)(P (H)) = E
?
C
?
|H1,1|2P1(H)
1 + |H1,2|2P2(H)
?
+ C
?
|H2,2|2P2(H)
??
.
(20)
Remark 9: One could alternately consider the fading one-sided IFC in which |h1,1| > |h2,1| and
h1,2 = 0 for the sum-capacity is given by (19) with the superscript 1 replaced by 2. The expression
S(w,2)(P (H)) is given by (20) after swapping the indexes 1 and 2.
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E. Uniformly Mixed IFC
The following theorem summarizes the sum-capacity of a class of uniformly mixed two-sided IFC.
Theorem 5: For a class of uniformly mixed ergodic fading two-sided Gaussian IFCs for which the
entries of every fading state h satisfy
|h1,1| > |h2,1|
and
|h2,2| ≤ |h1,2|
(21)
the sum-capacity is
max
P(H)∈P
?
min
?
E
?
C
??2
k=1|H1,k|2Pk(H)
??
,S(w,2)(P (H))
??
(22)
where S(w,2)(P (H)) is given by (20) by swapping indexes 1 and 2.
Remark 10: One could alternately consider the fading IFC in which |h1,1| ≤ |h2,1| and |h2,2| > |h1,2|.
The sum-capacity is given by (22) after swapping the indexes 1 and 2.
Remark 11: For the special case of Hk,k=√SNRejφkkand Hj,k=√INRejφjk, j ?= k, where φj,k
for all j and k is independent and distributed uniformly in [−π,π], the sum-capacity in Theorems 3 and
5 can also be achieved by ergodic interference alignment as shown in [25].
F. Uniformly Weak IFC
The sum-capacity of a one-sided uniformly weak IFC in Theorem 4 is an upper bound for that of
a two-sided IFC for which at least one of two one-sided IFCs that result from eliminating a cross-link
is uniformly weak. Similarly, a bound can be obtained from the sum-capacity of the complementary
one-sided IFC. The following theorem summarizes this result.
Theorem 6: For a class of uniformly weak ergodic fading two-sided Gaussian IFCs for which the
entries of every fading state h satisfy
|h1,1| > |h2,1|
and
|h2,2| > |h1,2|
(23)
the sum-capacity is upper bounded as
R1+ R2≤ max
P(H)∈Pmin
?
S(w,1)(P (H)),S(w,2)(P (H))
?
.
(24)
Remark 12: For the non-fading case, the sum-rate bounds in (24) simplify to those obtained in [9,
Theorem 3].
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G. One-sided IFC: General Achievable Scheme
For EVS and US IFCs, Theorems 2 and 3 suggest that joint coding across all sub-channels is optimal.
Particularly for EVS, such joint coding allows one to exploit the strong states in decoding messages.
Relying on this observation, we present an achievable strategy based on joint coding all sub-classes of
one-sided IFCs with H2,1= 0. The encoding scheme involves rate-splitting at user 2, i.e., user 2 transmits
w2= (w2p,w2c) where w2pand w2care private and common messages, respectively and can be viewed
as a Han-Kobayashi scheme with Gaussian codebooks and without time-sharing.
Theorem 7: The sum-capacity of a one-sided IFC is lower bounded by
max
P(H)∈P,αH∈[0,1]min(S1(αH,P (H)),S2(αH,P (H)))
(25)
where
S1(αH,P (H)) = E
?
C
?
|H1,1|2P1(H)
1 + |H1,2|2αHP2(H)
??
+ E
?
|H1,1|2P1(H) + |H1,2|2αHP2(H)
1 + |H1,2|2αHP2(H)
C
?
|H2,2|2P2(H)
??
,
(26)
S2(αH,P (H)) = E
?
C
?
|H2,2|2αHP2(H)
??
+ E
?
C
?
??
, (27)
such that αHis the power allocated by user 2 in fading state H to transmitting w2pand αH= 1 − αH,
αH ∈ [0,1]. For EVS one-sided IFCs, the sum-capacity is achieved by choosing αH = 0 for all H
provided S1
?
(25) for αH= 0 for all H. For UW one-sided IFCs, the sum-capacity is achieved by choosing αH= 1
0,P(wf)(H)
?
< S2
?
0,P(wf)(H)
?
. For US one-sided IFCs, the sum-capacity is given by
and maximizing S2(1,P (H)) = S1(1,P (H)) over all feasible P (H). For a hybrid one-sided IFC, the
achievable sum-rate is maximized by
α∗
H=



α(H) ∈ (0,1]
0
sub-channel H is weak
sub-channel H is strong.
(28)
and is given by (25) for this choice of α∗
H.
Remark 13: The optimal α∗
Hin (28) implies that in general for the hybrid one-sided IFCs joint
coding the transmitted message across all sub-channels is optimal. Specifically, the common message
is transmitted jointly in all sub-channels while the private message is transmitted only in the weak
sub-channels.
Remark 14: The separation-based coding scheme of [30] is a special case of the above HK-based
coding scheme and is obtained by choosing αH = 1 and αH = 0 for the weak and strong states,
respectively. The resulting sum-rate is at most as large as the bound in (25) obtained for α∗
H∈ (0,1] and
α∗
H= 0 for the weak and strong states, respectively.
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Remark 15: In [26], a Han-Kobayashi based scheme using Gaussian codebooks and no time-sharing
is used to develop an inner bound on the capacity region of a two-sided IFC.
IV. COMPOUND MAC: CAPACITY REGION AND OPTIMAL POLICIES
As stated in Corollary 1, an inner bound on the sum-capacity of an IFC can be obtained by allowing
both receivers to decode both messages, i.e., by determining the sum-capacity of a C-MAC with the same
inter-node links. In this Section, we prove Theorem 1 which establishes the capacity region of ergodic
fading C-MACs and discuss the optimal power policies that achieve every point on the boundary of the
capacity region.
A. Capacity Region
The capacity region of a discrete memoryless compound MAC is developed in [31]. For each choice
of input distribution at the two independent sources, this capacity region is an intersection of the MAC
capacity regions achieved at the two receivers. The techniques in [31] can be easily extended to develop the
capacity region for a Gaussian C-MAC with fixed channel gains. For the Gaussian C-MAC, one can show
that Gaussian signaling achieves the capacity region using the fact that Gaussian signaling maximizes
the MAC region at each receiver. Thus, the Gaussian C-MAC capacity region is an intersection of the
Gaussian MAC capacity regions achieved at D1and D2. For a stationary and ergodic process {H}, the
channel in (1) can be modeled as a parallel Gaussian C-MACs consisting of a collection of independent
Gaussian C-MACs, one for each fading state h, with an average transmit power constraint over all parallel
channels.
We now prove Theorem 1 stated in Section III-A which gives the capacity region of ergodic fading
C-MACs.
Proof of Theorem 1
We first present an achievable scheme. Consider a policy P (H) ∈ P. The achievable scheme involves
requiring each transmitter to encode the same message across all sub-channels and each receiver to jointly
decode over all sub-channels. Independent codebooks are used for every sub-channel. An error occurs at
receiver j if one or both messages decoded jointly across all sub-channels is different from the transmitted
message. Given this encoding and decoding, the analysis at each receiver mirrors that for a MAC receiver
[29, 14.3] and one can easily verify that for reliable reception of the transmitted message at receiver j,
the rate pair (R1,R2) needs to satisfy the rate constraints in (9) where in decoding wS= {wk: k ∈ S}
the information collected in each sub-channel is given by C
??
k∈S|Hj,k|2Pk(H)
?
, for all S ⊆ K.
June 3, 2009DRAFT