Sooner or later, every man in Australia runs into problems with impotency buy viagra australia like other bodily functions, must be in order.

## Untitled

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
On the Maximum Achievable Sum-Rate With
Successive Decoding in Interference ChannelsYue Zhao

*, Member, IEEE*, Chee Wei Tan

*, Member, IEEE*, A. Salman Avestimehr

*, Member, IEEE*,
Suhas N. Diggavi

*, Member, IEEE*, and Gregory J. Pottie

*, Fellow, IEEE*
**Abstract—****In this paper, we investigate the maximum achievable**
**sum-rate of the two-user Gaussian interference channel with**

Gaussian superposition coding and successive decoding. We first

examine an approximate deterministic formulation of the problem,

and introduce the complementarity conditions that capture the use

of Gaussian coding and successive decoding. In the deterministic

channel problem, we find the constrained sum-capacity and its

achievable schemes with the minimum number of messages, first

in symmetric channels, and then in general asymmetric channels.

We show that the constrained sum-capacity *oscillates ***as a function**
Fig. 1. Two-user Gaussian interference channel.

**of the cross link gain parameters between the information theo-**

retic sum-capacity and the sum-capacity with interference treated

as noise. Furthermore, we show that if the number of messages

of either of the two users is fewer than the minimum number
**required to achieve the constrained sum-capacity, the maximum**

achievable sum-rate drops to that with interference treated as
user Gaussian interference channels (cf. under

**noise. We provide two algorithms to translate the optimal schemes**
the constraints of successive decoding. While the information

**in the deterministic channel model to the Gaussian channel model.**
theoretic capacity region of the Gaussian interference channel is

**We also derive two upper bounds on the maximum achievable**

sum-rate of the Gaussian Han-Kobayashi schemes, which auto-
still not known, it has been shown that a Han-Kobayashi scheme

**matically upper bound the maximum achievable sum-rate using**
with random Gaussian codewords can achieve within 1 bit/s/Hz

**successive decoding of Gaussian codewords. Numerical evalua-**
of the capacity region and hence within 2 bits/s/Hz of the

**tions show that, similar to the deterministic channel results, the**
sum-capacity. In this Gaussian Han-Kobayashi scheme, each

**maximum achievable sum-rate with successive decoding in the**
user first decodes both users' common messages jointly, and

**Gaussian channels oscillates between that with Han-Kobayashi**

schemes and that with single message schemes.
then decodes its own private message. In comparison, the sim-plest commonly studied decoding constraint is that each user

**Index Terms—****Deterministic channel model, Gaussian interfer-**
treats the interference from the other users as noise, i.e., without

**ence channel, successive decoding, sum-rate maximization.**
any decoding attempt. Using Gaussian codewords, the corre-sponding constrained sum-rate maximization problem can be
Manuscript received March 26, 2011; revised December 23, 2011; accepted
formulated as a nonconvex optimization of power allocation,
January 24, 2012. Date of publication March 06, 2012; date of current version
which has an analytical solution in the two-user case It
May 15, 2012. The work of C. W. Tan was supported by grants from the Re-search Grants Council of Hong Kong, Project No. RGC CityU 112909, and
has also been shown that within a certain range of channel pa-
Qualcomm Inc. The work of A. S. Avestimehr was supported in part by the NSF
rameters for

*weak *interference channels, treating interference as
CAREER Award 0953117, NSF CCF1144000 grant, and by the AFOSR Young
noise achieves the information theoretic sum-capacity
Investigator Program Award FA9550-11-1-0064. The work of S. N. Diggaviwas supported in part by AFOSR MURI: "Information Dynamics as Founda-
For general interference channels with more than two users,
tion for Network Management", AFOSR MURI prime award FA9550-09-064,
there is so far neither a near optimal solution information theo-
subcontract to UCLA from Princeton University and by the NSF-CPS program
retically, nor a polynomial time algorithm that finds a near op-
by Grant 1136174. This paper was presented in part at the 2011 IEEE Interna-tional Symposium on Information Theory.

timal solution with interference treated as noise
Y. Zhao was with the Department of Electrical Engineering, University of
In this paper, we consider a decoding constraint—successive
California, Los Angeles (UCLA), Los Angeles, CA 90095 USA. He is now with
decoding of Gaussian superposition codewords—that bridges
the Department of Electrical Engineering, Stanford University, Stanford, CA94305 USA. He is also with the Department of Electrical Engineering, Princeton
the complexity between joint decoding (e.g., in Han-Kobayashi
University, Princeton, NJ 08544 USA (e-mail:

[email protected]).

schemes) and treating interference as noise. We investigate
C. W. Tan is with the Department of Computer Science, City University of
the maximum achievable sum-rate and its achievable schemes.

Hong Kong, Kowloon, Hong Kong (e-mail:

[email protected]).

A. S. Avestimehr is with the School of Electrical and Computer Engineering,
Compared to treating interference as noise, allowing successive
Cornell University, Ithaca, NY 14853 USA (e-mail:

[email protected]rnell.

cancellation yields a much more complex problem structure.

To clarify and capture the key aspects of the problem, we resort
S. N. Diggavi and G. J. Pottie are with the Department of Electrical En-
gineering, University of California, Los Angeles (UCLA), Los Angeles, CA
to the deterministic channel model In the information
90095 USA (e-mail:

[email protected];

[email protected]).

theoretic capacity region for the two-user deterministic interfer-
Communicated by S. Jafar, Associate Editor for Communication Networks.

ence channel is derived as a special case of the El Gamal-Costa
Color versions of one or more of the figures in this paper are available online
deterministic model and is shown to be achievable using
Digital Object Identifier 10.1109/TIT.2012.2190040
0018-9448/$31.00 2012 IEEE
ZHAO

*et al.*: MAXIMUM ACHIEVABLE SUM-RATE
We transmit messages using a superposition of Gaussian code-
are constant complex channel gains,
books, and use successive decoding. To capture the use of succes-
is the transmitted signal of the encoded messages from the th
sive decoding of Gaussian codewords, in the deterministic for-
mulation, we introduce the

*complementarity conditions *on the
There is an average power constraint equal to
bit levels, which have also been characterized using a conflict
. In the following, we first formulate the problem
graph model in We develop transmission schemes on the
of finding the sum-rate optimal Gaussian superposition coding
bit-levels, which in the Gaussian model corresponds to message
and successive decoding scheme, and then provide an illustra-
splitting and power allocation of the messages. We then derive
tive example to show that successive decoding schemes do not
the constrained sum-capacity for the deterministic channel, and
necessarily achieve the same maximum achievable sum-rate as
show that it

*oscillates *(as a function of the cross link gain pa-
rameters) between the information theoretic sum-capacity andthe sum-capacity with interference treated as noise. Furthermore,

*A. Gaussian Superposition Coding and Successive Decoding:*
the minimum number of messages needed to achieve the con-

*A Power and Decoding Order Optimization*
strained sum-capacity is obtained. Interestingly, we show that
Suppose the th user uses a superposition of
if the number of messages is limited to even

*one less *than this
the information rate of mes-
minimum capacity achieving number, the maximum achievable
. For the th user, the transmit signal
sum-rate drops to that with interference treated as noise.

We then translate the optimal schemes in the determin-
has a block length
, and is chosen from a codebook of size
istic channel to the Gaussian channel, using a rate constraintequalization technique. To evaluate the optimality of the trans-
that encodes message
, generated using independent
lated achievable schemes, we derive and compute two upper
and identically distributed (i.i.d.) random variables of
bounds on the maximum achievable sum-rate of Gaussian
With the power constraints
Han-Kobayashi Since a scheme using superpositioncoding with Gaussian codebooks and successive decoding isa special case of Han-Kobayashi schemes, these bounds auto-matically apply to the maximum achievable sum-rate with suchsuccessive decoding schemes as well. We select two mutually
exclusive subsets of the inequality constraints that characterizethe Gaussian Han-Kobayashi capacity region. Maximizing thesum-rate with each of the two subsets of inequalities leads to one
is the power allocated to message
of the two upper bounds. The two bounds are shown to be tight
The th receiver attempts to decode all
in different ranges of parameters. Numerical evaluations show
using successive decoding as follows. It chooses a decoding
that the maximum achievable sum-rate with Gaussian superposi-
messages from both users. It starts
tion coding and successive decoding oscillates between that with
decoding from the first message in this order (by treating all
Han-Kobayashi schemes and that with single message schemes.

other messages that are not yet decoded as noise,) then peeling
The remainder of the paper is organized as follows.
it off and moving to the next one, until it decodes all the mes-
formulates the problem of sum-rate maximization with suc-
sages intended for itself—
cessive decoding of Gaussian superposition codewords in
Denote the message that has order
Gaussian interference channels, and compares it with Gaussian
th message of the
th user. Then, for the successive
Han-Kobayashi schemes. reformulates the problem
decoding procedure to have a vanishing error probability as the
with the deterministic channel model, and then solves for the
, we have the following constraints on the
constrained sum-capacity. translates the optimal
rates of the messages:
schemes in the deterministic channel back to the Gaussianchannel, and derives two upper bounds on the maximumachievable sum-rate. Numerical evaluations of the achievabilityagainst the upper bounds are provided. concludesthe paper with a short discussion on generalizations of the
coding-decoding assumptions and their implications.

Now, we can formulate the sum-rate maximization problem as
II. PROBLEM FORMULATION IN GAUSSIAN CHANNELS
We consider the two-user Gaussian interference channel
shown in The received signals of the two users are
Note that involves both a

*combinatorial optimization *of
the decoding orders
and a

*nonconvex optimization *of the
Throughout this paper, when we refer to the Han-Kobayashi scheme, we
mean the Gaussian Han-Kobayashi scheme, unless stated otherwise.

. As a result, it is a hard problem from an
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
Fig. 2. Our approach to solving problem
optimization point of view which has not been addressed in the
ference between the maximum achievable sum-rate using
Gaussian successive decoding schemes and that using Gaussian
Interestingly, we show that an "indirect" approach can ef-
Han-Kobayashi schemes. This difference appears

*despite*
fectively and fruitfully provide approximately optimal solutions
the fact that the sum-capacity of a Gaussian multiple access
to the above problem Instead of directly working with the
channel is achievable using successive decoding of Gaussian
Gaussian model, we approximate the problem using the recently
codewords. In the remainder of this section, we show an illus-
developed deterministic channel model The approximate
trative example that provides some intuition into this difference.

formulation successfully captures the key structure and intuition
Suppose the th user
uses

*two *messages: a common
of the original problem, for which we give a complete analyt-
and a private message
. We consider a power
ical solution that achieves the constrained sum-capacity in all
allocation to the encoded messages, and denote the power allo-
channel parameters. Next, we translate this optimal solution in
Denote the achiev-
the deterministic formulation back to the Gaussian formulation,
. In a Han-Kobayashi
and show that the resulting solution is indeed close to the op-
scheme, at each receiver, the common messages and the in-
timum. This indirect approach of solving is outlined in
tended private message are

*jointly *decoded, treating the unin-
Next, we provide an illustration of the following point:
tended private message as noise. This gives rise to the achiev-
Although the constraints for the achievable rate region with
able rate region with any given power allocation as follows:
Han-Kobayashi schemes share some similarities with thosefor the capacity region of multiple access channels, succes-
sive decoding in interference channels does

*not *always havethe same achievability as Han-Kobayashi schemes, (whereas
time-sharing of successive decoding schemes does achieve thecapacity region of multiple access channels.)

*B. Successive Decoding of Gaussian Codewords versusGaussian Han-Kobayashi Schemes With Joint Decoding*
We first note that Gaussian superposition coding—succes-
sive decoding is a special case of the Han-Kobayashi scheme,
using the following observations. For the first user, if its message
is

*decoded *at the second receiver according
to the decoding order
, we categorize it into the

*common *in-
formation of the first user. Otherwise,
is treated as noise at
the second receiver, i.e., it appears

*after *all the messages of the
, and we categorize it into the

*private *infor-
mation of the first user. The same categorization is performed
messages of the second user. Note that every mes-
sage of the two users is either categorized as private informa-tion or common information. Thus, every successive decodingscheme is a special case of the Han-Kobayashi scheme, and
hence the capacity region with successive decoding of Gaussiancodewords is included in that with Han-Kobayashi schemes.

However, the inclusion in the other direction is untrue,
since Han-Kobayashi schemes allow joint decoding. In
we will give a characterization of the dif-
ZHAO

*et al.*: MAXIMUM ACHIEVABLE SUM-RATE
whereas translate to
In a successive decoding scheme, depending on the different
decoding orders applied, the achievable rate regions have dif-ferent expressions. In the following, we provide and analyze theachievable rate region with the decoding orders at receiver 1 and2 being
As a result, the maximum achievable sum-rates with the
tively. The intuition obtained with these decoding orders holds
Han-Kobayashi scheme and that with the successive decoding
similarly for other decoding orders. With any given power allo-
scheme are 10.19 and 5.56 bits, respectively. Here, the key
intuition is as follows: for a common message, its individualrate constraints at the two receivers in a successive decodingscheme are tighter than those in a joint decodingscheme In we will see that
lead to a nonsmooth behavior of the maximumachievable sum-rate using successive decoding of Gaussiancodewords. Finally, we connect the results shown in tothe results shown later in of

*Remark 2: *In the optimal symmetric power allocation
for a Han-Kobayashi scheme and that for a successive decodingscheme are
and 14.5 dB, respectively, leading to
sum-rates of 11.2 and 10.2 bits. This result corresponds to the
performance evaluation at
It is immediate to check that
III. SUM-CAPACITY IN DETERMINISTIC INTERFERENCE
To observe the difference between the maximum achievable

*A. Channel Model and Problem Formulation*
sum-rate with and that we examine thefollowing symmetric channel,
In this section, we apply the deterministic channel model
as an approximation of the Gaussian model on the two-user in-
terference channel. We define
in which we apply symmetric power allocation schemes with
, and a power constraint of

*Remark 1: *Note that
. As indicated in
of under this parameter setting, simply using successivedecoding of Gaussian codewords can have an arbitrarily large
maximum achievable sum-rate loss compared to joint decodingschemes, as
are the channel gains normalized by the
We plot the sum-rates with the private message power
noise power. Without loss of generality (WLOG), we assume
sweeping from nearly zero
to the maximum (30
. We note that the logarithms used in this paper
dB) as in As observed, the difference between the two
are taken to base 2. Now,
counts the bit levels of the signal
schemes is evident when the private message power
sent from the th transmitter that are above the noise level at the
sufficiently smaller than the common message power
th receiver. Further, we define
.) The intuition of why successive decoding of
Gaussian codewords is not equivalent to the Han-Kobayashischemes is best reflected in the case of
parameter setting, with
which represent the cross channel gains relative to the directchannel gains, in terms of the number of bit-level shifts. Toformulate the optimization problem, we consider

*real *numbers. (As will be shown later in Remark 5, with

*integer*bit-level channel parameters, our derivations automatically giveinteger bit-level optimal solutions.)

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
Fig. 3. Illustrations of the difference between the achievable sum-rate with Han-Kobayashi schemes and that with successive decoding of Gaussian codewords.

Fig. 4. Two-user deterministic interference channel. Levels A and B interfere at the first receiver, and cannot be fully active simultaneously.

In the desired signal and the interference signal at both
receivers are depicted.

are the sets of received *infor-*
*mation levels *at receiver 1 that are above the noise level, fromusers 1 and 2, respectively.

are the sets of received
information levels at receiver 2. A more concise representationis provided in
• The sets of information levels of the *desired *signals at re-
ceivers 1 and 2 are represented by the continuous intervals
lines, where the leftmost points correspond to the most sig-
Fig. 5. Interval representation of the two-user deterministic interference
nificant (i.e., highest) information levels, and the points at
correspond to the positions of the noise levels at both
• The positions of the information levels of the *interfering*
Note that an information level (or simply termed "*level*") is
signals are indicated by the dashed lines crossing between
a real *point *on a line, and the measure of a set of levels (e.g.,
the two parallel lines.

the length of an interval) equals the amount of information that
ZHAO *et al.*: MAXIMUM ACHIEVABLE SUM-RATE
this set can carry. The design variables are whether or not each
Problem does not include upper bounds on the number of
level of a user's received desired signal carries information for
. Such upper bounds can be added based on
this user, characterized by the following definition.

Remark 3. We will analyze the cases without and with upperbounds on the number of messages. We first derive the con-
*Definition 1:*
is the indicator function on whether the
strained sum-capacity in *symmetric *interference channels in the
carry information for the th user.

remainder of this section. Results are then generalized using
similar approaches to *general (asymmetric) *interference chan-
information for the th
*B. Symmetric Interference Channels*
As a result, the rates of the two users are
In this section, we consider the case where
. WLOG, we normalize the
amount of information levels by
. Note that in symmetric channels,
For an information level
, we call it an *active*
level for the th user, and otherwise an *inactive *level.

The constraints from superposition of Gaussian codewords
with successive decoding translate to the following*Complementarity Conditions *in the deterministic formulation.

are defined in The interpretation of
and (26) are as follows: for any two levels each from one ofthe two users, if they interfere with each other at any of the tworeceivers, they cannot be simultaneously active. For example, ininformation levels
from the first user and
From Lemma 1, it is sufficient to only consider the case with
second user interfere at the first receiver, and hence cannot be
, and the case with
fully active simultaneously. These complementarity conditions
by symmetry as in Corollary 3 later.

have also been characterized using a conflict graph model in
We next derive the constrained sum-capacity using succes-
sive decoding for
, first without upper bounds on the
*Remark 3: *For any given function
number of messages, then with upper bounds. We will see that
joint segment within
on it corresponds to a
in symmetric channels, the constrained sum-capacity
distinct message. Adjacent segments that can be so combined
. Thus, we also use the maximum
as a super-segment having
on it, are viewed as *one*
achievable *symmetric *rate, denoted by
as a function of ,
segment, i.e., the combined super-segment. Thus, for two seg-
as an equivalent performance measure.

is thus one half of
the optimal value of
*1) Symmetric Capacity Without Constraint on the Number*
separated by the point
have to correspond to two distinct
*of Messages:*
*Theorem 1: *In symmetric weak interference channels
Finally, we note that
, the constrained symmetric capacity, i.e., the maximum
achievable symmetric rate using successive decoding [with and (29)],
, is characterized by
Thus, we have the following result:
• In every interval
*Lemma 1: *The parameter settings
decreasing linear function.

• In every interval
correspond to the same set of complementarity conditions.

increasing linear function.

We consider the problem of maximizing the sum-rate
of the two users employing successive
decoding, formulated as the following continuous support
*Remark 4: *We plot
in compared with the infor-
(infinite dimensional) optimization problem:
mation theoretic capacity
The key idea in deriving the constrained sum-capacity is to
*decouple *the effects of the complementarity conditions. Beforewe present the complete proof of Theorem 1, we first analyzethe following two examples that illustrate this decoupling idea.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
Fig. 6. The symmetric capacity with successive decoding in symmetric deterministic weak interference channels.

*Example 1,*
*: *As in we divide the
with equal lengths.

From the complementarity conditions
can be achieved by letting
*Example 2,*
*: *As in we divide
lengths. For the same reasons as in the last example,
Fig. 7. Two examples that illustrate the proof ideas of Theorem 1. (a) The ex-
. (b) The example of =
can be achieved by letting
*Proof of Theorem 1:*
. We divide the interval
ZHAO *et al.*: MAXIMUM ACHIEVABLE SUM-RATE
Fig. 8. Segmentation of the information levels,
, where the first
segments have length
can be optimized *independently *of each other.

, and the last segment has length
(cf. With these, the complementarity conditionsare equivalent to the following:
. Hence can be solved
by separately solving the following two subproblems:
[Equations correspond to the shaded stripsin Similarly
We now prove that the optimal value of is
• (Achievability:)
is achievable with
We partition the set of all segments into two groups:
• Equation are constraints on
• Equation are constraints on
Consequently, instead of viewing the (infinite number of)optimization variables as
more convenient to view them as
By symmetry, the solution of can be obtained sim-ilarly, and the optimal value is
because there is no constraint between
well. Therefore, the optimal value of is
the complementarity conditions. In other words,
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
Fig. 9. Segmentation of the information levels,
As the above maximum achievable scheme is symmetric,
• (Achievability:)
is achievable with
the symmetric capacity is
is an *increasing linear *function of
By symmetry, the solution of can be obtained simi-
. Similarly to i), we divide the interval
larly. Thus, the optimal value of is
, where the first
maximum achievable scheme is also characterized by
segments have length , and the last segment has
and the symmetric rate is
complementarity conditions are equivalentto the following:
is a *decreasing linear *function of
. It can be verified
iii) It is clear that
, which is achievable with
which is achievable by
Similarly to i), with
, can be solved by separately
solving the following two subproblems:
We summarize the optimal scheme that achieves the con-
strained symmetric capacity as follows:
*Corollary 1: *When
, the constrained symmetric
capacity is achievable with
In the special cases when
We now prove that the optimal value of is
, the constrained symmetric capacity drops to

ZHAO *et al.*: MAXIMUM ACHIEVABLE SUM-RATE
Fig. 10. The symmetric capacity with successive decoding in symmetric deterministic strong interference channels.

also achievable by time sharing
Clearly, the maximum achievable symmetric rate achieved willbe lower than
. We start with the following two lemmas,
whose proofs are relegated to
We observe that the numbers of messages used by the two
*Lemma 2: *If there exists a segment with an even index
—in the above optimal schemes are as follows.

does not end at 1, such that
*Corollary 2:*
defined as in then
*Lemma 3: *If there exists a segment with an odd index
*Remark 5: *In the original formulation of the deterministic
are considered to be *integers*, and the
achievable scheme must also have integer bit-levels. In this case,
is a *rational *number. As a result, the optimal scheme
will consist of active segments
that have rational bound-
Recall that the optimal scheme requires that, for both
aries with the same denominator
. This indeed corresponds
users, *all *segments in
are fully inactive, and *all *segments
to an integer bit-level solution.

are fully active. The above two lemmas show the cost
From Theorem 1 (cf. it is interesting to see that the
of violating if one of the segments in
constrained symmetric capacity oscillates as a function of
active for either user (cf. Lemma 2), or one of the segments in
tween the information theoretic capacity and the baseline of .

becomes fully inactive for either user (cf. Lemma 3), the
This phenomenon is a consequence of the complementarity con-
resulting sum-rate cannot be greater than 1. We now establish
ditions. In we further discuss the connections of this
the following theorem.

result to other coding-decoding constraints.

*Theorem 2: *Denote by
the number of messages
Finally, from Lemma 1, we have the following corollary on
used by the th user. When
the maximum achievable symmetric rate with successive de-
, the maximum achievable sum-rate is 1.

coding in *strong *interference channels.

*Proof: *WLOG, assume that there is a constraint of
*Corollary 3: *In symmetric strong interference channels
i) First, the sum-rate of 1 is always achievable with
in compared with the
information theoretic capacity
ii) If there exists
, such that *either*
*2) The Case With a Limited Number of Messages: *In this
, then from Lemma 2,
subsection, we find the maximum achievable sum/symmetric
the achieved sum-rate is no greater than 1.

rate using successive decoding when there are constraints on the
in the interior of
maximum number of messages for the two users, respectively.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
Fig. 11. The maximum achievable symmetric rate with a limited number of messages. (a) Maximum achievable symmetric rate with L 2. (b) Maximumachievable symmetric rate with L 3.

*separates *the two segments
Comparing Theorem 2 to Corollary 2, we conclude that if the
first user. From Remark 3,
have to be two dis-
number of messages used for *either *of the two users is fewer
tinct messages provided that both of them are (at least partly)
than the number used in the optimal scheme (as in Corollary
active for the first user. On the other hand, there are
2), the maximum achievable symmetric rate drops to
is illustrated in with
the number of messages of the first user is upper bounded by
Complete solutions (without and with constraints on the
. In other words, there must be a segment in
number of messages) in *asymmetric *channels follow similar
is fully inactive for the first user. By Lemma 3, in this case, the
ideas, albeit more tediously. Detailed discussions are relegated
achieved sum-rate is no greater than 1.

ZHAO *et al.*: MAXIMUM ACHIEVABLE SUM-RATE
IV. APPROXIMATE MAXIMUM ACHIEVABLE SUM-RATE
WITH SUCCESSIVE DECODING IN GAUSSIAN
INTERFERENCE CHANNELS
In this section, we turn our focus back to the two-user
Gaussian interference channel, and consider the sum-ratemaximization problem Based on the relation between thedeterministic channel model and the Gaussian channel model,we *translate *the optimal solution of the deterministic channelinto the Gaussian channel. We then derive upper bounds onthe optimal value of and evaluate the achievability of ourtranslation against these upper bounds.

*A. Achievable Sum-Rate Motivated by the Optimal Schemein the Deterministic Channel*
As the deterministic channel model can be viewed as an ap-
proximation to the Gaussian channel model, optimal schemesof the former suggest approximately optimal schemes of thelatter. In this subsection, we show the translation of the op-timal scheme of the deterministic channel to that of the Gaussianchannel. We show in detail *two forms *(simple and fine) of thetranslation for symmetric interference channels
The translation for asymmetric channels can be derived simi-
Fig. 12. The optimal schemes in the symmetric deterministic interferencechannel. (a) Weak interference channel. (b) Strong interference channel.

larly, albeit more tediously.

*1) A Simple Translation of Power Allocation for the Mes-*
*sages: *Recall the optimal scheme for symmetric deterministicinterference channels (Corollary 1,) as plotted in
represent the segments (or *messages *as translated
deterministic scheme, the key property that ensures optimality
to the Gaussian channel) that are active for the th user. Recall
is the following:
*Corollary 4: *A message
that is decoded at *both *re-
ceivers is subject to the *same *achievable rate constraint at both
For example, in the optimal deterministic schemes (cf.

to the right (i.e., lower information levels)
is subject to an achievable rate con-
in the deterministic channel approximately corresponds to a
at the first receiver, and that of
power scaling factor of
in the Gaussian channel. Accord-
second receiver, with
. In weak interference
ingly, a simple translation of the symmetric optimal schemes
(cf. into the Gaussian channel is given as follows.

messages that are decoded at both receivers, whereas
are decoded only at their intended receiver (and treated
Algorithm 1: A simple translation by direct power scaling
as noise at the other receiver.) In strong interference channels,all messages are decoded at both receivers.

Step 1: Determine the number of messages
According to Corollary 4, we show that a finer translation
each user as the same number used in the optimal deterministic
of the power allocation for the messages is achieved by equal-
channel scheme.

izing the two rate constraints for every common message. (How-
ever, rates of different common messages are not necessarily thesame.) In what follows, we present this translation for weak in-
, and normalize the
terference channel and strong interference channel, respectively.

*Weak Interference Channel,*
*: *As the first
step of determining the power allocations, we give the followinglemma on the power allocation of message
, and normalize the
*2) A Finer Translation of Power Allocation for the Messages:*
In this part, for notational simplicity, we assume WLOG that
noise at the second (first) receiver, with
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
case, there is only one message for each user (as its private
Numerical evaluations of the above simple and finer trans-
message.) Rate constraint equalization is *not *needed.

lations of the optimal schemes for the deterministic channel
into that for the Gaussian channel are provided later in
at both receivers. To *equalize *their rate constraints at both
receivers, we must have the power allocation as follows:
*B. Upper Bounds on the Maximum Achievable Sum-Rate With*
*Successive Decoding of Gaussian Codewords*
Next, we observe that after decoding
In this subsection, we provide two upper bounds on the
optimal solution of for general (asymmetric) weak inter-
ceivers, determining
can be transformed to
ference channels. More specifically, the bounds are derived
an equivalent first step problem with
for the maximum achievable sum-rate with Han-Kobayashi
of the transformed problem gives the correct equal-
schemes, which automatically upper bound that with successive
izing solution for
of the original problem. In general, we
decoding of Gaussian codewords (as shown in
have the following recursive algorithm in determining
We will observe that, for weak interference channels, the two
bounds have complementary efficiencies, i.e., each being tightin a different regime of parameters. For strong interference
Algorithm 2.1, A finer translation by adapting
channels, the information theoretic capacity is known
using rate constraint equalization; weak interference channel
which is achievable by jointly decoding of all the messagesfrom both users.

Similarly to we denote by
and terminate.

sage of the th user, and
the common message
to be the power allocated to each *private *mes-
, 2. Then, the power of the common message
. Go to Step 1.

. WLOG, we normalize the channel parame-
. Denote the rates of
*Strong Interference Channel,*
*: *As the first
. The maximum achievable sum-rate of Gaussian
step of determining the power allocations, we give the following
Han-Kobayashi schemes is thus the following:
lemma on the power allocation of
(with the proof found
are always decoded at both re-
ceivers. Moreover,
, and the power allocation of
To bound we select two mutually exclusive *subsets *of
. In this case, there is only one
. Then, with each subset of the
message for each user. Rate constraint equalization is *not*
constraints, a relaxed sum-rate maximization problem can be
solved, leading to an *upper bound *on the original maximum
. To *equalize *the rate constraints
achievable sum-rate
) at both receivers, we must have the
The first upper bound on the maximum achievable sum-rate
power allocation as follows:
is as follows [whose proof is immediate from
*Lemma 6: *The maximum achievable sum-rate using Han-
Kobayashi schemes is upper bounded by
Next, we observe that after decoding
ceivers, determining
can be transformed to
an equivalent first step problem with
of the transformed problem gives the correct equal-
izing solution for
of the original problem. In general, we
have the following recursive algorithm in determining
Algorithm 2.2, A finer translation by adapting
using rate constraint equalization; strong interference channel
*Computation of the Upper Bound *Note that
and terminate.

. Go to Step 1.

ZHAO *et al.*: MAXIMUM ACHIEVABLE SUM-RATE
*Lemma 7: *The maximum achievable sum-rate using Han-
Kobayashi schemes is upper bounded by
the minimum of is
*Computation of the Upper Bound *Note that
Now, consider the halfspace
linear constraint
where is a function only of
, and is a function only of
can each be solved by taking the first order derivatives,
and checking the stationary points and the boundary points.

We combine the two upper bounds and as the fol-
lowing theorem.

*Theorem 3: *The maximum achievable sum-rate using
. Thus, depending on the sign
Gaussian superposition coding-successive decoding is upper
, we have the following two cases.

. Then, gives an *upper *bound
*C. Performance Evaluation*
. Consequently, to maximize the optimal solution is
. Thus, maximizing is equivalent to
We numerically evaluate our results in symmetric Gaussian
interference channels. The
is set to be 30 dB. We first eval-
uate the performance of successive decoding in weak interfer-
ence channels and then in strong interference channels.

*1) Weak Interference Channel: *We sweep the parameter
in which the objective is *monotonic*, and the solution is
approximate optimal transmission scheme is simply treatinginterference as noise without successive decoding.

. Then, gives a *lower *bound on
In the simple translation by Algorithm 1 and the finer
translation by Algorithm 2.1 are evaluated, and the two upperbounds derived above are computed. The maximum
achievable sum-rate with a single message for each user
is also computed, and is used as a baseline scheme for
Consequently, to maximize the optimal solution is
, which is a linear
We make the following observations:
. Substituting this into we need to solve the
• The finer translation of the optimal deterministic scheme
following problem:
by Algorithm 2.1 is strictly better than the simple trans-lation by Algorithm 1, and is also strictly better than theoptimal single message scheme.

• The first upper bound is tighter for higher
in this example), while the second upper bound
are constants determined by
is tighter for lower
in this example).

. Now, can be solved by taking the first
• A phenomenon similar to that in the deterministic chan-
derivative w.r.t.

, and checking the two stationary points and
nels appears: the maximum achievable sum-rate with
the two boundary points.

successive decoding of Gaussian codewords oscillates
In the other halfspace
, the same procedure as above can
between that with Han-Kobayashi schemes and that with
be applied, and the maximizer of within
can be found.

single message schemes.

Comparing the two maximizers within
• The largest difference between the maximum achievable
we get the global maximizer of
sum-rate of successive decoding and that of single mes-
The second upper bound on the maximum achievable sum-
sage schemes appears at around
rate is as follows [whose proof is immediate from
about 1.8 bits.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
Fig. 13. Performance evaluation in symmetric weak interference channel: achievability versus upper bounds.

Fig. 14. Maximum achievable sum-rate differences: Han-Kobayashi versus successive decoding at = 0:75, and successive decoding versus the optimal singlemessage scheme at = 0:66.

• The largest difference between the maximum achievable
shown with the deterministic channel model (cf. indicate
sum-rate of successive decoding and that of joint decoding
that these differences can go to infinity as
(Han-Kobayashi schemes) appears at around
because a rate point
on the symmetric capacity curve
. This corresponds to the same parameter setting as
in the deterministic channel has the following interpretation
discussed in (cf. We see that with
of generalized degrees of freedom in the Gaussian channel
, this largest maximum achievable sum-rate dif-
ference is about 1.0 bits.

For this particular case with
maximum achievable sum-rate differences (1.8 bits and 1.0
bits) may not seem very large. However, the capacity curves

ZHAO *et al.*: MAXIMUM ACHIEVABLE SUM-RATE
Fig. 15. Performance evaluation in symmetric strong interference channel: successive decoding versus information theoretic capacity.

In the finer translation by Algorithm 2.2 is evaluated
is the symmetric capacity in the two-user symmetric Gaussian
and compared with the information theoretic sum-capacity
channel as a function of
Interestingly, an oscillation phenomenon similar to that in the
deterministic channel case (cf. is observed.

finite gap of the achievable rates in the deterministic channelindicates a rate gap that goes to infinity as
V. CONCLUDING REMARKS AND DISCUSSION
Gaussian channel. To illustrate this, we plot the following max-
In this paper, we studied the problem of sum-rate maxi-
imum achievable sum-rate differences in the Gaussian channel,
mization with Gaussian superposition coding and successive
growing from 10 to 90 dB:
decoding in two-user interference channels. This is a hard
• The maximum achievable sum-rate gap between Gaussian
problem that involves both a combinatorial optimization of
superposition coding-successive decoding schemes and
decoding orders and a nonconvex optimization of power allo-
single message schemes, with
cation. To approach this problem, we used the deterministic
channel model as an educated approximation of the Gaussian
channel model, and introduced the complementarity condi-
tions that capture the use of successive decoding of Gaussian
codewords. We solved the constrained sum-capacity of the
As observed, the maximum achievable sum-rate gaps in-
deterministic interference channel under the complementarity
crease asymptotically linearly with
conditions, and obtained the constrained capacity achieving
schemes with the minimum number of messages. We showed
*2) Strong Interference Channel: *We sweep the parameter
that the constrained sum-capacity oscillates as a function of
. As the information theoretic
the cross link gain parameters between the information the-
sum-capacity in strong interference channel can be achieved by
oretic sum-capacity and the sum-capacity with interference
having each receiver jointly decode all the messages from both
treated as noise. Furthermore, we showed that if the number
users we directly compare the achievable sum-rate using
of messages used by either of the two users is fewer than its
successive decoding with this joint decoding sum-capacity (in-
minimum capacity achieving number, the maximum achievable
stead of upper bounds on it). This joint decoding sum-capacity
sum-rate drops to that with interference treated as noise. Next,
can be computed as follows:
we translated the optimal schemes in the deterministic channelto the Gaussian channel using a rate constraint equalization
technique, and provided two upper bounds on the maximumachievable sum-rate with Gaussian superposition coding andsuccessive decoding. Numerical evaluations of the translationand the upper bounds showed that the maximum achievablesum-rate with successive decoding of Gaussian codewordsoscillates between that with Han-Kobayashi schemes and thatwith single message schemes.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
Next, we discuss some intuitions and generalizations of the
successive decoding and that using joint decoding can be
significant for typical
s in practice.

Recently, the role of feedback in further increasing the in-
*A. Complementarity Conditions and Gaussian Codewords*
formation theoretic capacity region has been studied In these work, the deterministic channel model was also em-
The complementarity conditions in the deter-
ployed as an approximation of the Gaussian channel model,
ministic channel model has played a central role that leads to the
leading to useful insights in the design of near-optimal trans-
discovered oscillating constrained sum-capacity (cf. Theorem
mission schemes with feedback. We note that, in deterministic
1). The intuition behind the complementarity conditions is as
channels, allowing feedback implicitly assumes that modulo-2
follows: At any receiver, if two active levels from different users
sums can be decoded. In Gaussian channels, it remains an inter-
interfere with each other, then *no *information can be recovered
esting open question to find the maximum achievable sum-rate
at this level. In other words, the sum of interfering codewords
using successive decoding of Lattice codewords with feedback.

provides nothing helpful.

This is exactly the case when random Gaussian codewords
*C. Symbol Extensions and Asymmetric Complex Signaling*
are used in Gaussian channels with successive decoding,because the sum of two codewords from random Gaussian
We have focused on two-user complex Gaussian interference
codebooks cannot be decoded as a valid codeword. This is the
channels with constant channel coefficients, and have assumed
reason why the usage of Gaussian codewords with successive
that symbol extensions are not used, and *circularly symmetric*
decoding is translated to complementarity conditions in the
complex Gaussian distribution is employed in codebook gen-
deterministic channels. (Note that the preceding discussions
eration. With symbol extensions and *asymmetric *complex sig-
do not apply to *joint *decoding of Gaussian codewords as in
naling the maximum achievable sum-rate using successive
Han-Kobayashi schemes.)
decoding can be potentially higher. It has been shown that, inthree or more user interference channels, higher sum-degrees of
*B. Modulo-2 Additions, Lattice Codes and Feedback*
freedom can be achieved by interference alignment if symbolextensions and asymmetric complex signaling are used
In the deterministic channel, a relaxation on the comple-
In *two-user *interference channels, however, interference align-
mentarity conditions is that the *sum *of two interfering active
ment is not applicable, and it remains an interesting open ques-
levels can be decoded as their *modulo-2 sum*. As a result, the
tion to find the maximum achievable sum-rate with successive
aggregate of two interfering codewords still provides something
decoding considering symbol extensions and asymmetric com-
valuable that can be exploited to achieve higher capacity. This
plex signaling.

assumption is part of the original formulation of the determin-istic channel model with which the information theoretic
capacity of the two-user interference channel (cf. for
PROOFS OF LEMMA 2 AND 3
the symmetric case) can be achieved with Han-Kobayashischemes
*Proof of Lemma 2: *By symmetry, it is sufficient to prove
In Gaussian channels, to achieve an effect similar to de-
that does not end
coding the modulo-2 sum with successive decoding, *Lattice*
codes are natural candidates of the coding schemes. This is
Now, consider the sum-rate achieved within
because Lattice codebooks have the group property such that
can be partitioned into three
the *sum *of two lattice codewords can still be decoded as a valid
codeword. Such intermediate information can be decoded first
and exploited later during a successive decoding procedure,
in order to increase the achievable rate. For this to succeed in
degenerate.) Note that
interference channels, alignment of the signal scales becomes
• From the achievable schemes in the proof of Theorem 1,
essential However, our preliminary results have shown
the maximum achievable sum-rate within
that the ability to decode the sum of the Lattice codewords does
*not *increase the maximum achievable sum-rate for low and
s. In the above setting of
is typically considered as a high
in practice) numerical
• By the assumed condition,
computations show that the maximum achievable sum-rate
using successive decoding of lattice codewords with alignment
Therefore, under the assumed condition, the maximum achiev-
of signal scales is *lower *than the previously shown achievable
able sum-rate within
is achievable with
sum-rate using successive decoding of Gaussian codewords (cf.

for the entire range of
Furthermore, from the proof of Theorem 1, we know that
reason is that the cost of alignment of the signal scales turns out
the maximum achievable sum-rate within
is achievable with
to be higher than the benefit from it, if
is not sufficiently
high. In summary, no matter using Gaussian codewords or
the maximum achievable schemes within
Lattice codewords, the gap between the achievable rate using
ZHAO *et al.*: MAXIMUM ACHIEVABLE SUM-RATE
Fig. 16. C partitioned into three parts for Lemma 2.

Fig. 17. C partitioned into three parts for Lemma 3.

a sum-rate of 1 is achieved, and this is the maximum achievable
Furthermore, from the proof of Theorem 1, we know that
sum-rate given the assumed condition.

the maximum achievable sum-rate within
is achievable with
*Proof of Lemma 3: *By symmetry, it is sufficient to prove
the maximum achievable schemes within
Now, consider the sum-rate achieved within
a sum-rate 1 is achieved, and this is the maximum achievable
can be partitioned into three parts:
sum-rate given the assumed condition.

can be degenerate.) Note that:
• From the achievable schemes in the proof of Theorem 1,
SUM-CAPACITY OF DETERMINISTIC ASYMMETRIC
the maximum achievable sum-rate within
INTERFERENCE CHANNELS
In this section, we consider the general two-user interference
• By the assumed condition,
channel where the parameters
Therefore, under the assumed condition, the maximum achiev-
trary. Still, WLOG, we make the assumptions that
able sum-rate within
is achievable with
. We will see that our approaches in the symmetric
channel can be similarly extended to solving the constrained
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
sum-capacity in asymmetric channels, without and with con-
straints on the number of messages.

From Lemma 1, it is sufficient to consider the following three
By the same argument as in the proof of Theorem 1, the op-
timal solution of is given by
*A. Sum-Capacity Without Constraint on the Number of*
Also, the optimal solution of is given by
We provide the optimal scheme that achieves the constrained
sum-capacity in each of the three cases in respectively.

*: *This is by definition equivalent
Consequently, we have the following theorem.

As depicted in interval
is partitioned into
*Theorem 4: *A constrained sum-capacity achieving scheme
; the last segment ending at 1 has
the length of the proper residual. Intervalis partitioned into segments
; the last segment ending
at 1 has the length of the proper residual.

Similarly to as in the previous analysis for the symmetric
and the maximum achievable sum-rate is readily computable
channels, we partition the optimization variables
As depicted in interval
is partitioned into
; the last segment ending at 1 has the length of
the proper residual. Interval
is partitioned into
As there is no constraint between
ment ending at 1 has the length of the proper residual. (The in-
plementarity conditions similarly to and
dexing is not consecutive as we consider
the sum-rate maximization is decoupled into two separate
as degenerating to empty sets.)
does not conflict with any levels of
. On all the other segments, the
sum-rate maximization problem is
ZHAO *et al.*: MAXIMUM ACHIEVABLE SUM-RATE
n , n < n , n n , and n n , scheme I (nonoptimal).

By the same argument as in the proof of Theorem 1, the optimal
does not conflict with any levels of
solution of is given by
. On all the other segments, the
sum-rate maximization problem is again and the optimal
solution is given by
Thus, a sum-capacity achieving scheme is simply
Thus, a sum-capacity achieving scheme is
This is by definition equivalent to
. Note that by Lemma 1, it is sufficient to only consider the
, (because in case
is partitioned into segments
. As depicted in interval
partitioned into segments
; the last segment ending at 1 has the length of the
; the last segment ending at 1 has the length of the proper
proper residual. Interval
is partitioned into
residual. Interval
is partitioned into segments
; the last segment ending at 1 has the
the last segment ending at 1 has the length of the proper residual.

length of the proper residual.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
n , n < n , n n , and n n , scheme II (optimal).

Compare with Case 1 of and note the simi-
Noting the similarities between and we see that
larities between and we apply the same partition of
the optimal solution of the two cases are the same:
the optimization variables and the sum-rate maximization
is decoupled in the same way into two separate problems
*: *This is by equivalent to
and However, while the optimal solution of is still
. Note that by Lemma 1, it is sufficient to only
given by the optimal solution of is no longer given by
consider the case where
, (because in case
, the optimal solution of is given by
is partitioned into seg-
; the last segment ending at 1 has the
Thus, a sum-capacity achieving scheme is given by
length of the proper residual. Interval
, depicted as in
tioned into segments
*: *Comparing with Case 2 of
the last segment ending at 1 has the length of the proper
(cf. with the same definition of
and the same partition of
, the segmentation is
does not conflict with any levels of
. On all the other segments, the
ZHAO *et al.*: MAXIMUM ACHIEVABLE SUM-RATE
sum-rate maximization problem is again As
for the second user. On the other hand, there are
optimal solution is given by
, whereas the number of messages is
. In other words, for the second
user, there must be a segment with an odd index that is *fully in-*
Thus, a sum-capacity achieving scheme is
*active*. By Lemma 9, in this case,
Similarly to the symmetric case, we conclude that if the
number of messages used for *either *user is fewer than the
Summarizing the discussions of the six parameter settings (cf.

number used in the optimal scheme the maximum achiev-
and in this subsection, we observe:
able sum-rate drops to 1.

*Remark 6: *Except for Case 1 of the op-
timal schemes for the other cases all have the property that only
one message is used for each user.

PROOF OF LEMMA 4 AND 5
*The Case With a Limited Number of Messages: *In this sub-
*Proofs of Lemma 4: *At the first receiver, the mes-
section, we extend the sum-capacity results in
is decoded by treating all other messages
to the asymmetric channels when there are upper bounds onthe number of messages
for the two users, respec-
as noise, and has an
tively. From Remark 6, we only need to discuss Case 1 of
(cf. with its corresponding notations.

At the second receiver,
is first decoded and peeled off.

Similarly to the symmetric channels, we generalize Lemma 2
is also decoded at the second receiver (by treating
and 3 to the following two lemmas for the general (asymmetric)
as noise) it has an
channels, whose proofs are exact parallels to those of Lemma 2
. To equalize the rate constraints for
at both receivers, we need
does not end at 1, such that
does not end at 1, such that
. It implies that we
should not decode
at the second receiver, i.e.,
is the only message
of the th user, which is treated as
noise at the other receiver.

*Proof of Lemma 5:*
At the second receiver, the
We then have the following generalization of Theorem 2 to
is decoded by treating all other messages
the general (asymmetric) channels.

as noise, and has an
*Theorem 5: *Denote by
the number of messages used by
the th user in any scheme, and denote by
the dictated number
At the first receiver,
is first decoded and peeled off. Next,
of messages used by the th user in the constrained sum-capacity
is decoded by treating
achieving scheme Then, if
noise, and has an
the rate constraints for
at both receivers, we need
*Proof: *Consider
can be proved similarly.)
i) The sum-rate of 1 is always achievable with
. It implies that,
ii) If there exists
does not end at 1,
even if the common message
is allocated with all the
, then from Lemma 8,
power , it still has a higher rate constraint at the second (first)
receiver than at the first (second) receiver.

iii) If for every
does not end at 1, there
in the interior of
does not end at 1,
*separates *the two segments
[1] Y. Zhao, C. W. Tan, A. S. Avestimehr, S. N. Diggavi, and G. J. Pottie,
"On the sum-capacity with successive decoding in interference chan-
user. From Remark 3,
have to be two distinct
nels," in *Proc. IEEE Int. Symp. Inf. Theory*, St. Petersburg, Russia, Jul.

messages provided that both of them are (at least partly) active
2011, pp. 1494–1498.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 6, JUNE 2012
[2] R. H. Etkin, D. Tse, and H. Wang, "Gaussian interference channel ca-
**Chee Wei Tan **(M'08) received the M.A. and Ph.D. degrees in electrical

pacity to within one bit," *IEEE Trans. Inf. Theory*, vol. 54, no. 12, pp.

engineering from Princeton University, Princeton, NJ, in 2006 and 2008,
5534–5562, 2008.

[3] M. Ebrahimi, M. Maddah-Ali, and A. Khandani, "Power allocation and
Previously, he was a Postdoctoral Scholar at the California Institute of Tech-
asymptotic achievable sum-rates in single-hop wireless networks," in
nology (Caltech), Pasadena. He is currently an Assistant Professor at City Uni-
*Proc. Conf. Inf. Sci. Syst.*, Mar. 2006, pp. 498–503.

versity Hong Kong. He was a Visiting Faculty at Qualcomm R&D, San Diego,
[4] V. Annapureddy and V. Veeravalli, "Gaussian interference networks:
CA, in 2011. His research interests are in wireless and broadband communica-
Sum capacity in the low-interference regime and new outer bounds
tions, signal processing and nonlinear optimization.

on the capacity region," *IEEE Trans. Inf. Theory*, vol. 55, no. 7, pp.

Dr. Tan was the recipient of the 2008 Princeton University Wu Prize for Ex-
3032–3050, 2009.

cellence and 2011 IEEE Communications Society AP Outstanding Young Re-searcher Award. He currently serves as an Editor for the IEEE T
[5] A. Motahari and A. Khandani, "Capacity bounds for the Gaussian inter-
ON COMMUNICATIONS.

ference channel," *IEEE Trans. Inf. Theory*, vol. 55, no. 2, pp. 620–643,Feb. 2009.

[6] X. Shang, G. Kramer, and B. Chen, "A new outer bound and the
noisy-interference sum-rate capacity for Gaussian interference chan-
**A. Salman Avestimehr **(S'04–M'09) received the B.S. degree in electrical en-

nels," *IEEE Trans. Inf. Theory*, vol. 55, no. 2, pp. 689–699, Feb. 2009.

gineering from Sharif University of Technology, Tehran, Iran, in 2003 and the
[7] Z. Luo and S. Zhang, "Dynamic spectrum management: Complexity
M.S. degree and Ph.D. degree in electrical engineering and computer science,
and duality," *IEEE J. Sel. Topics Signal Process.*, vol. 2, no. 1, pp.

both from the University of California, Berkeley, in 2005 and 2008, respectively.

57–73, 2008.

He is currently an Assistant Professor at the School of Electrical and Com-
[8] C. W. Tan, S. Friedland, and S. H. Low, "Spectrum management in
puter Engineering, Cornell University, Ithaca, NY. He was also a Postdoctoral
multiuser cognitive wireless networks: Optimality and algorithm,"
Scholar at the Center for the Mathematics of Information (CMI), California In-
*IEEE J. Sel. Areas Commun.*, vol. 29, no. 2, pp. 421–430, Feb. 2011.

stitute of Technology, Pasadena, in 2008. His research interests include infor-
[9] A. Avestimehr, S. Diggavi, and D. Tse, "Wireless network information
mation theory, communications, and networking.

flow: A deterministic approach," *IEEE Trans. Inf. Theory*, vol. 57, no.

Dr. Avestimehr has received a number of awards, including the Presidential
4, pp. 1872–1905, Apr. 2011.

Early Career Award for Scientists and Engineers (PECASE) in 2011, the YoungInvestigator Program (YIP) award from the U. S. Air Force Office of Scientific
[10] G. Bresler and D. Tse, "The two-user Gaussian interference channel:
Research in 2011, the National Science Foundation CAREER award in 2010,
A deterministic view," *Eur. Trans. Telecommun.*, vol. 19, pp. 333–354,
the David J. Sakrison Memorial Prize from the EECS Department, University
of California, Berkeley, in 2008, and the Vodafone U.S. Foundation Fellows
[11] A. Gamal and M. Costa, "The capacity region of a class of determin-
Initiative Research Merit Award in 2005. He has been a Guest Editor for the
istic interference channels," *IEEE Trans. Inf. Theory*, vol. 28, no. 2, pp.

IEEE TRANSACTIONS ON INFORMATION THEORY Special Issue on Interference
343–346, Mar. 1982.

[12] Z. Shao, M. Chen, A. Avestimehr, and S.-Y. Li, "Cross-layer optimiza-
tion for wireless networks with deterministic channel models," *IEEETrans. Inf. Theory*, vol. 57, no. 9, pp. 5840–5862, Sep. 2011.

[13] H. Sato, "The capacity of the Gaussian interference channel under
**Suhas N. Diggavi **(S'93–M'98) received the Ph.D. degree in electrical engi-

strong interference," *IEEE Trans. Inf. Theory*, vol. 27, no. 6, pp.

neering from Stanford University, Stanford, CA, in 1998.

786–788, Nov. 1981.

After completing the Ph.D. degree, he was a Principal Member Technical
[14] S. Mohajer, S. N. Diggavi, C. Fragouli, and D. N. C. Tse, "Approx-
Staff in the Information Sciences Center, AT&T Shannon Laboratories, Florham
imate capacity of a class of Gaussian interference-relay networks,"
Park, NJ, after which he was on the faculty of the School of Computer and
*IEEE Trans. Inf. Theory*, vol. 57, no. 5, pp. 2837–2864, May 2011.

Communication Sciences, EPFL, where he directed the Laboratory for Informa-tion and Communication Systems (LICOS). He is currently a Professor in the
[15] C. Suh and D. Tse, "Feedback capacity of the Gaussian interference
Department of Electrical Engineering, University of California, Los Angeles.

channel to within 2 bits," *IEEE Trans. Inf. Theory*, vol. 57, no. 5, pp.

His research interests include wireless communications networks, information
2667–2685, May 2011.

theory, network data compression, and network algorithms. He has eight issued
[16] A. Vahid, C. Suh, and A. S. Avestimehr, "Interference channels with
rate-limited feedback," *IEEE Trans. Inf. Theory*, 2012, to be published.

Dr. Diggavi is a recipient of the 2006 IEEE Donald Fink prize paper award,
[17] V. R. Cadambe, S. A. Jafar, and C. Wang, "Interference alignment with
2005 IEEE Vehicular Technology Conference Best Paper award, and the
asymmetric complex signaling—Settling the Host-Madsen—Nos-
Okawa Foundation Research award. He is currently an Associate Editor for
ratinia conjecture," *IEEE Trans. Inf. Theory*, vol. 56, no. 9, pp.

ACM/IEEE TRANSACTIONS ON NETWORKING and the IEEE TRANSACTIONS
4552–4565, Sep. 2010.

ON INFORMATION THEORY.

**Gregory J. Pottie **(S'84–M'89–SM'01–F'05) was born in Wilmington, DE,

and raised in Ottawa, Canada. He received the B.Sc.degree in engineering

physics from Queen's University, Kingston, Ontario, Canada, in 1984, and the

M.Eng. and Ph.D. degrees in electrical engineering from McMaster University,

Hamilton, Ontario, in 1985 and 1988, respectively.

From 1989 to 1991, he was with the Transmission Research Department of
**Yue Zhao **(S'06–M'11) received the B.E. degree in electronic engineering from

Motorola/Codex, Canton, MA. Since 1991, he has been a faculty member of
Tsinghua University, Beijing, China, in 2006, and the M.S. and Ph.D. degrees
the Electrical Engineering Department, University of California, Los Angeles
in electrical engineering, both from the University of California, Los Angeles
(UCLA), serving in Vice-Chair roles from 1999 to 2003. From 2003 to 2009,
(UCLA), Los Angeles, in 2007 and 2011, respectively.

he served as Associate Dean for Research and Physical Resources of the Henry
He is currently a Postdoctoral Scholar with the Department of Electrical En-
Samueli School of Engineering and Applied Science. His research interests in-
gineering, Stanford University, Stanford, CA, and a Postdoctoral Research As-
clude wireless communication systems and sensor networks.

sociate with the Department of Electrical Engineering, Princeton University,
Dr. Pottie was secretary to the Board of Governors from 1997 to 1999 for
Princeton, NJ. In summer 2010, he was a Senior Research Assistant with the
the IEEE Information Theory Society. In 1998, he received the Allied Signal
Department of Computer Science, City University of Hong Kong, Hong Kong.

Award for outstanding faculty research for UCLA engineering. In 2005 he be-
His research interests include information theory, optimization theory and algo-
came a Fellow of the IEEE for "contributions to the modeling and applications
rithms, communication networks, and smart grids.

of sensor networks." In 2009, he was a Fulbright Senior Fellow at the University
Dr. Zhao is a recipient of the UCLA Dissertation Year Fellowship
of Sydney. He is a member of the Bruin Master's Swim Club (butterfly) and the
St. Alban's Choir (second bass).

Source: http://www.cs.cityu.edu.hk/~cheewtan/ZTADP_IT.pdf

Integrative Therapy in Dogs with Nervous System & Other Disorders R.M. Clemmons, DVM, PhD Associate Professor of Neurology & Neurosurgery Department of Small Animal Clinical Sciences Maintaining health is becoming increasingly difficult. All animals are born with a tremendous capacity to heal. In fact, most (up to 80%) patients who experience a temporary illness will

DIE ZEHN GEBOTE RATSCHLÄGE FÜR PATIENTEN NACH HERZINFARKT Sie haben vor kurzer Zeit einen Herzinfarkt erlebt und sind nach der Klinik und vielleicht auch nach der Rehabilitation wieder nach Hause und in die gewohnte Umgebung zurückgekehrt. Für die meisten Myokardinfarktpatienten ist nach wenigen Wochen ein nahezu normales Leben, fast wie vor dem Infarkt, möglich. Einige Patienten werden sich aber für die Zukunft Einschränkungen auferlegen müssen, wollen sie sich eine normale Lebenserwartung zurückgewinnen. Dies wird natürlich ihr zukünftiges Leben verändern. Bedenken Sie, dass der frühere amerikanische Präsident Johnson erst nach seinem Herzinfarkt Präsident der Vereinigten Staaten von Amerika geworden ist und viele andere Menschen trotz eines vorangegangenen Infarktes ein für die Gesellschaft und für sie selbst wertvolles und erfülltes Leben gestalten. All diesen Menschen ging es nach dem Herzinfarkt nicht besser, als es Ihnen jetzt ergeht. Man kann und soll also voll Hoffnung sein.