Interference Alignment: From Degrees of Freedom to Constant-Gap Capacity Approximations

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 8, AUGUST 2013 4855

Interference Alignment: From Degrees of Freedom toConstant-Gap Capacity Approximations

Urs Niesen, Member, IEEE, and Mohammad Ali Maddah-Ali, Member, IEEE

Abstract—Interference alignment is a key technique for com-munication scenarios with multiple interfering links. In severalsuch scenarios, interference alignment was used to characterizethe degrees of freedom of the channel. However, these degree-of-freedom capacity approximations are often too weak to make ac-curate predictions about the behavior of channel capacity at finitesignal-to-noise ratios ( ). The aim of this paper is to signifi-cantly strengthen these results by showing that interference align-ment can be used to characterize capacity to within a constantgap. We focus on real, time-invariant, frequency-flat X-channels.The only known solutions achieving the degrees of freedom of thischannel are either based on real interference alignment or on layer-selection schemes. Neither of these solutions seems sufficient for aconstant-gap capacity approximation. In this paper, we proposea new communication scheme and show that it achieves the ca-pacity of the Gaussian X-channel to within a constant gap. To aidin this process, we develop a novel deterministic channel model.This deterministic model depends on the most-signif-icant bits of the channel coefficients rather than only the singlemost-significant bit used in conventional deterministic models. Theproposed deterministic model admits a wider range of achievableschemes that can be translated to the Gaussian channel. For thisdeterministic model, we find an approximately optimal commu-nication scheme. We then translate this scheme for the determin-istic channel to the original Gaussian X-channel and show that itachieves capacity to within a constant gap. This is the first con-stant-gap result for a general, fully-connected network requiringinterference alignment.

Index Terms—Capacity approximation, deterministic model, in-terference alignment, network information theory, X-channel.

I. INTRODUCTION

I NTERFERENCE alignment has been used to achieveoptimal degrees of freedom (capacity pre-log factor) in

several common wireless network configurations such asX-channels [1]–[4], interference channels [5], [6], interferingmultiple-access and broadcast channels [7], multiuser systemswith delayed feedback [8]–[10], and distributed computation[11], among others. The main idea of interference alignmentis to force all interfering signals at the receivers to be aligned,

Manuscript received December 20, 2011; revised March 12, 2013; acceptedApril 01, 2013. Date of publication May 16, 2013; date of current version July10, 2013. The material in this paper was presented in part at the InternationalSymposium on Information Theory, July 2012.The authors are with Bell Labs, Alcatel-Lucent, Holmdel, NJ 07974 USA

(e-mail: [email protected]; [email protected]).Communicated by S. A. Jafar, Associate Editor for Communications.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2013.2259140

Fig. 1. Different alignment approaches and their relation.

thereby maximizing the number of interference-free signalingdimensions.

A. Background

Alignment approaches can be divided into two broad cate-gories (see Fig. 1).1) Vector-space alignment ([1], [5] among others): In this ap-proach, conventional communication dimensions, such astime, frequency, and transmit/receive antennas, are used toalign interference. At the transmitters, precoding matricesare designed over multiple of these dimensions such thatthe interference at the receivers is aligned in a small sub-space. If the channel coefficients have enough variationacross the utilized time/frequency slots or antennas, thensuch precoding matrices can be found.

2) Signal-scale alignment ([6], [12] among others): If thetransmitters and receivers have only a single antenna andthe channel coefficients are time invariant and frequencyflat, the vector-space alignment method fails. Instead, onecan make use of another resource, namely the signal scale.Using lattice codes, the transmitted and received signalsare split into several superimposed layers. The transmittedsignals are chosen such that all interfering signals areobserved within the same layers at the receivers. Thus,alignment is now achieved in signal scale.

Signal-scale interference alignment can be further subdi-vided into two different, and seemingly completely unrelated,approaches: alignment schemes motivated by signal-strengthdeterministic models [12], [13] and real interference alignment[6].For the signal-strength deterministic approach, the channel is

first approximated by a deterministic noise-free channel. In thisdeterministic model, all channel inputs and outputs are binaryvectors, representing the binary expansion of the real-valuedsignals in the Gaussian case. The actions of the channel aremodeled by shifting these vectors up or down, depending on themost-significant bit of the channel gains, and by bitwise addi-tion of interfering vectors. The signal layers are represented by

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4856 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 8, AUGUST 2013

the different bits in the binary expansion of the signals. In thesecond step, the signaling schemes and the outer bounds devel-oped for this simpler deterministic model are used to guide thedesign of efficient signaling schemes for the original Gaussianproblem.This deterministic approach has proved instrumental in de-

riving constant-gap capacity approximations for several chal-lenging multiuser communication scenarios such as single-mul-ticast relay networks [14], two-user interference channels withfeedback [15] or with transmit/receive cooperation [16], [17],and lossy distributed source coding [18]. In all these commu-nication scenarios, interference alignment is not required. Forcommunication scenarios in which interference alignment is re-quired, the deterministic approach has been less helpful. In fact,it has only been successfully used to obtain constant-gap ca-pacity approximations for the fairly restrictive many-to-one in-terference channel, in which only one of the receivers experi-ences interference while all others are interference free [12].Even for the X-channel, one of the simplest Gaussian networksin which interference alignment is required, only weaker (gen-eralized) degree-of-freedom capacity approximations were de-rived using the deterministic approach [13]. The resulting com-munication scheme for the Gaussian X-channel is quite com-plicated and cannot be used to derive a constant-gap capacityapproximation.For the real interference-alignment approach, each trans-

mitter modulates its signal using a scaled integer lattice suchthat at each receiver, all interfering lattices coincide, while thedesired lattice is disjoint. Each receiver recovers the desiredsignal using a minimum-distance decoder. A number-theoreticresult concerning the approximability of real numbers by ratio-nals, called Groshev’s theorem, is used to analyze the minimumconstellation distance at the receivers. For almost all channelgains, this scheme is shown to achieve the full degrees offreedom of the Gaussian X-channel and the Gaussian interfer-ence channel [6]. While this scheme is asymptotically optimalfor almost all channel gains, there are infinitely many channelgains for which the scheme fails, for example, when the channelgains are rational. Moreover, this approach can again not beused to derive stronger constant-gap capacity approximations.At first glance, real interference alignment appears to rely

on the irrationally of the channel coefficients, preventing thedesired integer input signals from mixing with the undesiredinteger interference signals. This raises the concern that thescheme might be severely affected by the presence of mea-surement errors or quantization of the channel coefficients. Inaddition, arbitrarily close to any irrational channel realizationis a rational channel realization. How are we then to engineera communication device based on this scheme? Quoting fromSlepian’s 1974 Shannon Lecture [19]: “Most of us would treatwith great suspicion a model that predicts stable flight for anairplane if some parameter is irrational but predicts disaster ifthat parameter is a nearby rational number. Few of us wouldboard a plane designed from such a model.”Some of these concerns follow from the fact that real interfer-

ence alignment is somehow isolated from other known signalingschemes and only poorly understood. Unlike the vector-spaceand the deterministic approaches, no vector-space interpreta-

tion is known for real interference alignment, making it harderto obtain intuition. On the other hand, it is known that the de-grees of freedom of the interference channel are discontinuousat all rational channel coefficients [20]. It should therefore notbe surprising that the rates achieved by real interference align-ment share this characteristic. Rather, it appears that it is thedegree-of-freedom capacity approximation that is too weak toallow accurate predictions about the behavior of channel ca-pacity at finite signal-to-noise ratios ( ), and that the dis-continuity of the degrees of freedom in the channel coefficientsis mainly caused by taking a limit as approaches infinity.Thus, a stronger capacity approximation is needed.

B. Summary of Results

The main contributions of this paper are as follows.1) New Deterministic Channel Model: We develop a novel

deterministic channel model, in which each channel gain ismodeled by a lower triangular, binary Toeplitz matrix. Theentries in this matrix consist of the first bitsin the binary expansion of the channel gain in the corre-sponding Gaussian model. This contrasts with the traditionalsignal-strength deterministic model, which is based only onthe single most-significant nonzero bit. The proposed lowertriangular deterministic model is rich enough to explain thereal interference-alignment approach. Thus, it unites the sofar disparate deterministic and real interference-alignmentapproaches mentioned above (see Fig. 1). Moreover, as ourproposed deterministic model is based on a vector space, it en-ables an intuitive interpretation of real interference alignment.2) New Mathematical Tools: The solution for the proposed

lower triangular deterministic model can be translated to theGaussian setting. To analyze the resulting scheme for theGaussian setting, we develop new tools. In particular, to proveachievability for the Gaussian case, we extend Groshev’s the-orem to handle finite as well as channel gains of differentmagnitudes, and we prove a strengthening of Fano’s inequality.3) New Notion of Capacity Approximation: We introduce the

new notion of a constant-gap capacity approximation up to anoutage set. Specifically, the aim is to provide a constant-gapcapacity approximation uniform in the and the channelgains as long as these channel gains are outside a computableoutage set of arbitrarily small measure. This new notion of aconstant-gap approximation up to an outage set can lead to amore concise capacity characterization as we will see next.4) Constant-Gap Result for the Gaussian X-Channel: We

apply these ideas to the Gaussian X-channel by deriving a con-stant-gap capacity approximation up to outage for this channel.This is the first constant-gap result for a general, fully connectednetwork requiring interference alignment. To simplify the expo-sition, we focus in this paper on the most relevant situation, inwhich the direct links of the X-channel are stronger than thecross links—the tools and techniques developed here apply tothe other settings as well.To develop this result, we first consider the lower triangular

deterministic version of the X-channel and design a signalingscheme that achieves its capacity up to a constant gap, as longas the binary channel matrices satisfy certain rank conditions(see Theorems 1 and 5 in Section IV). We then show that the

NIESEN AND MADDAH-ALI: INTERFERENCE ALIGNMENT: FROM DEGREES OF FREEDOM TO CONSTANT-GAP CAPACITY APPROXIMATIONS 4857

translated version of the solution for the deterministic modelachieves the capacity of the Gaussian X-channel to within aconstant gap up to the aforementioned outage set (see Theo-rems 3 and 6 in Section IV). In addition, we show that, similarto the MIMO broadcast channel [21], capacity is not sensitiveto channel quantization and measurement errors smaller than

.One implication of these results is that the complicated

solution achieving the degrees of freedom of the GaussianX-channel in [13] is a result of oversimplification in thesignal-strength deterministic model rather than the propertiesof the original Gaussian channel itself. Moreover, the resultsin this paper imply that the discontinuity of the degrees offreedom of the Gaussian X-channel with respect to the channelcoefficients is due to the large limit and is not present atfinite .

C. Organization

The remainder of this paper is organized as follows. Section IIintroduces the new deterministic channel model. Section III for-malizes the Gaussian networkmodel and the problem statement.Section IV presents the main results of the paper—Sections Vand VI contain the corresponding proofs. Section VII containsthe mathematical foundations for the analysis of the decodingalgorithms. Section VIII concludes the paper.

II. DETERMINISTIC CHANNEL MODELS

Developing capacity-achieving communication schemesfor multiuser communication networks is often challenging.Indeed, even for the relatively simple two-user interferencechannel, finding capacity is a long-standing open problem. Forthe Gaussian network, the difficulty is due to the interactionbetween the various components of these networks, such asbroadcast, multiple access, and additive noise. For example,the two-user interference channel mentioned before has twobroadcast links, two multiple-access links, and two additivenoise components.The problem of characterizing capacity can be substantially

simplified if these noise components are eliminated so that theoutput at the receivers becomes a deterministic function of thechannel inputs at the transmitters [14], [22]. Such networks arecalled deterministic networks. This observation motivates theinvestigation of noisy networks by approximating themwith de-terministic networks [14], [23], [24].This approximation has two potential advantages. First, the

capacity of the deterministic network may directly approximatethe capacity of the original Gaussian network. Second and moreimportant, the deterministic model may reveal the essential in-gredients of an efficient signaling scheme for the noisy network.In other words, the capacity achieving signaling scheme for thedeterministic network may be used as a road map to designsignaling schemes for the Gaussian network. If the determin-istic approximation is well chosen, then the resulting signalingscheme for the Gaussian network is close to capacity achieving.The first critical step in this approach is thus to find an appro-

priate deterministic channel approximating the Gaussian one.This deterministic channel model should satisfy two criteria:simplicity and richness. These two requirements are conflicting.

Indeed, oversimplification of the Gaussian model can sacrificethe richness of the deterministic model. Conversely, keeping toomany of the features of the Gaussian model can result in a deter-ministic model that is rich but too difficult to analyze. Strikingthe right balance between these two requirements is the key todeveloping a useful deterministic network approximation.One of the approaches that achieves this goal is the

signal-strength deterministic model proposed by Avestimehr etal. [14]. We review this deterministic model in Section II-A.We introduce our new lower triangular deterministic modelin Section II-B. Section II-C compares the two deterministicmodels, explaining the shortcomings of the former and the needfor the latter.

A. Signal-Strength Deterministic Model [14]

We start with the real point-to-point Gaussian channel

(1)

with additive white Gaussian noise and unit av-erage power constraint at the transmitter. Here, is a nonnega-tive integer, and . Observe that all channel gains (andhence ) greater than or equal to one can be expressed inthe form for and satisfying these conditions. Since fora constant-gap approximation the other cases are not relevant,(1) is essentially the general case.1

To develop the deterministic model and for simplicity, weassume that and are positive and upper bounded by one.We can then write and in terms of their binary expansions

(2a)

(2b)

The Gaussian point-to-point channel (1) can then be approxi-mated as

or, more succinctly

see Fig. 2(a).The approximations in this derivation are to ignore the impact

of , the noise, as well as all the bitswith exponent less than zero. These bits with exponent lessthan zero are approximated as being completely corrupted by

1If the magnitude of the channel gains is less than one, then capacity is lessthan one bit per channel use and hence not relevant for capacity approximationup to a constant gap. Moreover, since capacity is only a function of the magni-tude of the channel gains, negative channel gains are not relevant either.


Fig. 2. Comparison of the signal-strength deterministic model [14], and thelower triangular deterministic model proposed in this paper. In the figure, solidlines depict noiseless binary links of capacity one bit per second. Dashed linesdepict noiseless links of either capacity one or zero bits per channel use (de-pending on whether the corresponding entry in the channel matrix is oneor zero). Links with the same color/shade have the same capacity. (a) Signal-strength deterministic model. (b) Lower triangular deterministic model.

noise, whereas the bits with higher exponent are approximatedas being received noise free. Therefore, we can approximate theGaussian channel with a deterministic channel consisting ofparallel error-free links from the transmitter to the receiver, eachcarrying one bit per channel use.Having reviewed the signal-strength model for the point-to-

point case, we now turn to the Gaussian multiple-access channel

(3)

where is additive white Gaussian noise.2 Asbefore, we impose a unit average transmit power constrainton and . Moreover, is a nonnegative integer,and , . The signal-strength deterministic modelcorresponding to the Gaussian channel (3) is

(4)

where denotes addition over , i.e., modulo two.We note that in this model, the contributions of and

are entirely ignored, real addition is replaced with bitwisemodulo-two addition, and noise is eliminated. As mentionedearlier, this simple model has been used to characterize thecapacity region of several challenging problems in networkinformation theory to within a constant gap. However, it fallsshort for some other settings. For example, for certain relay net-works with specific channel parameters, this model incorrectlypredicts capacity zero. Similarly, for interference channels withmore than two users and for X-channels, this model fails topredict the correct behavior for the Gaussian case.

B. Lower Triangular Deterministic Model

The signal-strength deterministic model recalled in the lastsection ignores the contribution of in the Gaussianpoint-to-point channel (1). Indeed, is approximated by 1. Inthis section, we introduce a new deterministic channel model,

2For ease of exposition, we consider here the symmetric case where both linkshave the same approximate strength .

termed lower triangular deterministic model, in which the ef-fect of is preserved. As we will see later, the new deterministicmodel admits a wider range of solutions—a fact that will be crit-ical for the approximation of Gaussian networks with multipleinterfering signals.Consider again the Gaussian point-to-point channel (1).Write

the channel parameter in terms of its binary expansion

Observe that , due to the assumption that .Then, from (1) and (2), we have

so that

The approximation here is to ignore the noise as well as all bitsin the convolution of and with expo-nent less than zero. These bits with exponent less than zero areapproximated as being completely corrupted by noise, whereasthe bits with higher exponent are approximated as being re-ceived noise free.This suggests to approximate the Gaussian point-to-point

channel (1) by a deterministic channel between the binary inputvector

and the binary output vector

connected through the channel operation

(5)

with

......

. . ....

...


Fig. 3. Lower triangular deterministic model for a point-to-point channel withand .

Fig. 4. Permissible signaling schemes for both deterministic models.(a) Signal-strength deterministic model. (b) Lower triangular deterministicmodel.

as depicted in Fig. 2(b). Here, we have normalized the receivedvector to contain the bits from 1 to . This is a determin-istic channel with finite input and output alphabets. Note thatall operations in (5) are over , i.e., modulo two. Similarly, theGaussian multiple-access channel (3) can be approximated bythe deterministic channel model

(6)

Example 1: For a concrete example, consider the Gaussianpoint-to-point channel (1) with channel gain 21, so thatand . The bits in the binary expansion of are

, , , , ,, and the corresponding lower triangular deterministic

model is depicted in Fig. 3. For channel input , the channeloutput is

Fig. 5. Illustration of a signaling scheme that succeeds for the lower triangularmodel (assuming the subspace condition (7) holds), but fails for the signal-strength model. (a) Signal-strength deterministic model. (b) Lower triangulardeterministic model.

C. Comparison of Deterministic Models

We now compare the signal-strength deterministic model re-viewed in Section II-A and the lower triangular deterministicmodel introduced in Section II-B. As an example, we considerthe Gaussian multiple-access channel (3) with signal strength

. The corresponding deterministic models are given by(4) and (6). Assume that transmitter one wants to send three bits, , and to the receiver. At the same time, transmitter two

wants to send one bit .Some signaling schemes work for both deterministic models

(4) and (6). For example, in both models transmitter one canuse the first three layers to send , , and , while trans-mitter two can use the last layer to send , as shown in Fig. 4.For the signal-strength model, the decoding scheme is trivial.For the lower triangular model, the receiver starts by decodingthe highest layer containing only . Having recovered , thereceiver cancels out its contribution in all lower layers. The de-coding process continues in the same manner with at thesecond-highest layer, until all bits are decoded.There are, however, some signaling schemes that are

only decodable in the lower triangular model, but not in thesignal-strength model. An example of such a signaling schemeis depicted in Fig. 5. In this scheme, transmitter one uses againthe first three layers to send , , and . Unlike before,transmitter two now also uses the first layer to send . FromFig. 5(a), we can see that, in the signal-strength model, receiver


Fig. 6. Modulation scheme for the Gaussian model suggested by the signaling scheme for the lower triangular deterministic model depicted in Fig. 5(b). At the de-coder, blue dots correspond to input tuples with , and red dots correspond to input tuples with .Here, and .

one observes and cannot recover and from thereceived signal. However, this scheme can be utilized success-fully in the lower triangular model as long as the subspacesspanned by the message bits at the receivers are linearly inde-pendent. In this case, the subspace spanned by the first threecolumns of and the subspace spanned by the first columnsof need to be linearly independent. This is the case if andonly if

(7)

The event (7) depends not only on , but also on the bits in thebinary expansion of and . Thus, this scheme is successfulfor all channel gains , where is the eventthat (7) does not hold. The set can be understood as an outageevent: if the channel gains are in , the achievable scheme failsto deliver the desired target rate of 4 bits per channel use.Noting that the scheme depicted in Fig. 4(b) always works

while the scheme depicted in Fig. 5(b) only works under someconditions, one might question the relevance of the second classof solutions. The answer is that this second class of solutionsmake use of the “diversity” provided by the lower order bits ofthe channel gains. It is precisely this diversity that is required forefficient communication over the X-channel to be investigatedin Section IV.As pointed out earlier, the second step in using the determin-

istic approach is to translate the solution for the deterministicmodel to a solution for the original Gaussian model. We nowshow how this can be done for the signaling scheme shown inFig. 5(b). The proposed scheme for the Gaussian multiple-ac-cess channel is depicted in Fig. 6. In this scheme, the input con-stellation at transmitter one is the set , and theinput constellation at transmitter two is the set . Sincethe additive Gaussian receiver noise has unit variance, we ex-pect the receiver to be able to recover the coded input signalsroughly when

(8)

for all , , , suchthat . In words, we require the minimumconstellation distance as seen at the receiver to be greater thantwo.We note that condition (8) for the Gaussian channel corre-

sponds to condition (7) for the deterministic model. As in the de-terministic case, this scheme fails to work whenever the channel

Fig. 7. Outage set (indicated in black) for the modulation scheme in Fig. 6with . The set consists of all channel gains such that (8) failsto hold for some channel inputs. The figure makes clear that, for finite SNR, the outage set is not determined by the rationality or irrationality of thechannel gains .

gains are in the set not satisfying (8), and one can bound theLebesgue measure of this outage event . It is worth empha-sizing that condition (8) has nothing to do with the rationalityor irrationality of the channel coefficients as can be seen fromFig. 7.Remark: In the special case in which each transmitter has

the same message size, the modulation scheme shown in Fig. 6is the same as the modulation scheme used in real interferencealignment [6], [20]. The objective in [6] is to achieve only thedegrees of freedom of the channel, and therefore, the schemethere is designed and calibrated for the high- regime. As aresult, the modulation scheme in [6] is not sufficient to provea constant-gap capacity approximation. Rather, as we will seein Section IV, asymmetric message sizes and judicious layerselection guided by the proposed lower triangular deterministicmodel together with a more careful and more general analysisof the receivers are required to move from a degrees of freedomto a constant-gap capacity approximation.

III. NETWORK MODEL

In the remainder of this paper, we focus on the X-channel,which is formally introduced in this section. We start with nota-tional conventions in Section III-A. We introduce the GaussianX-channel in Section III-B, and the corresponding lower trian-gular deterministic X-channel in Section III-C.


A. Notation

Throughout this paper, we use small and capital bold font todenote vectors and matrices, i.e., and . For a real number

, we use to denote . For a set ,denotes -dimensional Lebesgue measure. Fi-

nally, all logarithms are expressed to the base two and capacitiesare expressed in bits per channel use.

B. Gaussian X-Channel

The Gaussian X-channel consists of two transmitters and tworeceivers. The channel output at receiver andtime is

(9)

where is the channel input at transmitter , whereis the channel gain from transmitter to receiver ,

and where is additive white Gaussian receivernoise. The channel gains consist of two parts, and .We assume that and that for each , .Since varies over as varies over

, we see that any real channel gain greater than one can bewritten in this form. As discussed in Section II-A, this impliesthat (9) models essentially the general Gaussian X-channel.3

Writing the channel gains in the form decomposesthem into two parts capturing different aspects. The parameter

captures the magnitude or coarse structure of the channelgain. Indeed, the of the link from transmitter to receiveris approximately . On the other hand, the parametercaptures the fine structure of the channel gain. As we will

see soon, the impact of these two parameters on the behavior ofchannel capacity is quite different. We denote by

the collection of .Each transmitter has one message to communicate to each

receiver. So there are a total of four mutually independent mes-sages with , . We impose a unit average powerconstraint on each of the two encoders. Denote by the rateof message and by the sum capacity of the GaussianX-channel.An important special case of this setting is the symmetric

Gaussian X-channel, for which for all , so that

(10)

With slight abuse of notation, we denote the sum capacity of thesymmetric Gaussian X-channel by .In the following, we will be interested in a particular modula-

tion scheme for the Gaussian channel, which we describe next.Fix a time slot ; to simplify notation, we will drop the depen-dence of variables on whenever there is no risk of confusion.

3Indeed, channel gains with magnitude less than one are not relevant for aconstant-gap capacity approximation, and can hence be ignored. Similarly, neg-ative channel gains have no effect on the achievable schemes and outer boundspresented later, and can therefore be ignored as well.

Assume each message is modulated into the signal .Transmitter one forms the channel input

(11a)

Similarly, transmitter two forms the channel input

(11b)

The received signals are then given by

(12a)

(12b)

Receiver one is interested in the signals and . The othertwo signals and are interference. We see from (12a) thatthe interfering signals and are received with the samecoefficient . The situation is similar for receiver two.It will be convenient in the following to refer to the effective

channel gains including the modulation scheme as , i.e.,

(13a)

(13b)

(13c)

Here, for , corresponds to the desired signal, and for corresponds to the interference

terms. Since , we have . We can thenrewrite (12) as

(14a)

(14b)

C. Deterministic X-Channel

As in the discussion in Section II-B, it is insightful to considerthe lower triangular deterministic equivalent of the modulatedGaussian X-channel (14). To simplify the discussion, we as-sume for the derivation and analysis of the deterministic channelmodel that the channel gains defined in (13) are ininstead of —the Gaussian setting will be analyzed for thegeneral case.Let us first consider the symmetric X-channel (10), i.e.,

for all and . Let

......

. . ....

... (15)

be the deterministic channel matrix corresponding to the binaryexpansion of the channel gain with and


. Since by assumption so that ,the diagonal entries of are equal to one.The lower triangular deterministic equivalent of the modu-

lated Gaussian X-channel (14) is then given by

(16a)

(16b)

where the channel input and the channel output are allbinary vectors of length , and where all operations are over.4

Let us then consider the general X-channel (9). To simplifythe presentation, we focus in the following on the case wherethe direct links are stronger than the cross links,5 i.e.,

It will be convenient to split the channel input into “common”and “private” portions, i.e.,

where and for . The lowertriangular deterministic equivalent of the modulated GaussianX-channel (14) is then

(17a)

(17b)

where all operations are again over (see Figs. 8 and 9). Here,the lower triangular binary matrices are defined in analogyto (15). The matrix is of dimension andis of dimension for all . Comparingthe general deterministic model (17) to the symmetric one (16),we see that the difference in the values of results in theinputs observed over the cross links to be shifted down.As a consequence, the private portions of the channel inputsare visible at only the intended receiver, whereas the commonportions are visible at both receivers.As before, there are four independent messages . Each

transmitter consists of two6 encoders mapping one of the two

4This definition of the deterministic model corresponds to a power con-straint of 16 in the Gaussian model. This is mainly for convenience of no-tation. Since the additional factor 16 in power only increases capacity by aconstant number of bits per channel use, this does not significantly affect thequality of approximation.5This assumption is made for ease of exposition. Since the labeling of

the receivers is arbitrary, all results carry immediately over to the case. The models and tools developed in this

paper for these two cases can be applied to the other cases as well.6Observe that in the definition of capacity of the modulated deter-

ministic X-channel (17), we use two encoders at each transmitter (one for eachof the two messages). This differs from the definition of capacity of theGaussian X-channel (10), where we use a single encoder. Thus, in the deter-ministic case, we force the messages to be encoded separately, while we allowjoint encoding of the two messages in the Gaussian case. This restriction is in-troduced because the aim of the deterministic model is to better understand themodulated Gaussian X-channel (12), which already handles the joint encodingof the messages through the modulation process.

Fig. 8. Deterministic model at receiver one. The figure shows the signal ob-served at receiver one decomposed into its four components (see (17a)). For sim-plicity, the matrices are omitted. The interference terms and areobserved at receiver one multiplied by the same matrix . The desired terms

and are multiplied by different matrices and , respectively.

Fig. 9. Deterministic model at receiver two (see (17b)). The matricesare again omitted. The interference terms and are observed at receiverone multiplied by the same matrix . The desired terms and aremultiplied by different matrices and , respectively.

messages to a sequence of channel inputs . Denote bythe rate of message and by the sum capacity

of the (modulated) deterministic X-channel (17). For the specialcase of the symmetric deterministic X-channel (16), the sumcapacity is denoted by .

IV. MAIN RESULTS

The main result of this paper is a constant-gap approxima-tion for the capacity of the Gaussian X-channel. To simplify thepresentation of the relevant concepts and results, we start withthe analysis of the Gaussian X-channel with symmetricin Section IV-A. We then consider the Gaussian X-channel witharbitrary in Section IV-B.

A. X-Channel With Symmetric

We start with the analysis of the deterministic X-channel—aswe will see in the following, the insights obtained for this modelcarry over to the Gaussian X-channel. The capacity of thesymmetric deterministic X-channel is characterized by the nexttheorem.Theorem 1: For every and , there exists a

set of Lebesgue measure

such that for all channel gains thesum capacity of the (modulated) symmetric deterministicX-channel (16) satisfies

for some positive universal constant .


Fig. 10. Allocation of bits for the deterministic X-channel with symmetric as seen at receiver one. The white regions correspond to zero bits; the shadedregions carry information. Observe that the interference signals and are aligned.

Theorem 1 is a special case of Theorem 5 presented inSection IV-B. We hence omit its proof.Theorem 1 approximates the capacity of the modulated de-

terministic X-channel (16) up to a constant gap for all channelgains outside the set of arbitrarily small mea-sure. The event can be interpreted as an outageevent, as in this case the proposed achievable scheme fails todeliver the target rate of . Here, parametrizesthe tradeoff between the measure of the outage set and thetarget rate: decreasing decreases the measure of the outageevent , but at the same time also decreases the target rate

. We point out that can be chosen indepen-dently of the number of input bits ; hence, the approximationgap is uniform in .Theorem 1 can be used to derive the more familiar result on

the degrees of freedom of the deterministicX-channel. Setting results in the measures

to be summable over . Applying the Borel–Cantellilemma yields then the following corollary to Theorem 1.Corollary 2: For almost all channel gains ,

the (modulated) symmetric deterministic X-channel (16) hasdegrees of freedom, i.e.,

We emphasize that, while Corollary 2 is simpler to stateand perhaps more familiar in form, Theorem 1 is considerablystronger. Indeed, Theorem 1 provides the stronger constant-gapcapacity approximation for the sum capacity , whereasCorollary 2 provides the weaker degrees of freedom capacityapproximation. Moreover, Theorem 1 provides bounds for finiteon the measure of the outage event , whereas Corollary 2

provides only asymptotic information about its size.We now describe the communication scheme achieving the

lower bound in Theorem 1 (see Fig. 10). Use the first com-ponents of each vector to transmit information, and set thelast components to zero. The sum rate of this commu-nication scheme is hence . Receiver one is interested inand . These vectors are received in the subspace spanned bythe first columns of and , respectively. On the otherhand, the messages and that receiver one is not inter-ested in, and that can hence be regarded as interference, are bothreceived in the same subspace spanned by the first columnsof . Thus, the two interference vectors are aligned in a sub-space of dimension . The situation at receiver two is similar.Assume that the three subspaces spanned by the first

columns of , , and are linearly independent. Then,

receiver one can recover the two desired vectors by projectingthe received vector into the corresponding subspaces in orderto zero force the two interfering vectors. We show that for mostchannel gains, this linear independence of the three subspacesholds for . The outage event in Theorem 1 is thusprecisely the event that at either of the two receivers the threesubspaces spanned by the first columns of , , and

are not linearly independent.We now turn to the Gaussian X-channel. The results for the

deterministic X-channel suggest that the modulation scheme(11) should achieve a sum rate of

over the Gaussian channel as . Furthermore, it suggeststhat a -bit quantization of the channel gains available atboth transmitters and receivers should be sufficient to achievethis asymptotic rate. This intuition turns out to be correct, as thenext theorem shows.Theorem 3: For every and , there exists a

set of Lebesgue measure at most

such that for all channel gains , the sumcapacity of the symmetric Gaussian X-channel (10) satisfies

for some positive universal constant . Moreover, the lowerbound is achievable with a -bit quantization of the channelgains available at both transmitters and receivers.Theorem 3 is a special case of Theorem 6 presented in

Section IV-B. We hence omit its proof.Theorem 3 provides a constant-gap capacity approximation

for the symmetric Gaussian X-channel (10). The constant in theapproximation is uniform in the channel gainsoutside the set of arbitrarily small measure, and uniform in. The event can again be interpreted as an outageevent, and parametrizes the tradeoff between the measure ofthe outage set and the target rate of the achievable scheme.Since can be chosen independently of , the approximationgap is uniform in the , i.e., uniform in .Remark: It is worth pointing out that the outage set can

be explicitly computed: given channel gains , there is analgorithm that can determine in bounded time if these channelgains are in the outage set . More precisely, is the unionof “strips” similar to Fig. 7 in Section II-C. Member-ship of in the outage set is mostly determined by the


most-significant bits in the binary expansion of the channelgains . In particular, for any finite (and hence finite ),the question of rationality or irrationality of the channel gains

is largely irrelevant to determining membership in .The theorem shows furthermore that the proposed achiev-

able scheme for the Gaussian X-channel is not dependent onthe exact knowledge of the channel gains, and a quantized ver-sion, available at all transmitters and receivers, is sufficient. Infact, the scheme achieving the lower bound uses mismatchedencoders and decoders. The encoders perform modulation withrespect to the wrong channel model

(18)

where is a -bit (or, equivalently, -bit) quanti-zation of the true channel gain . In other words, the channelinputs are

The decoders perform maximum-likelihood decoding alsowith respect to the wrong channel model (18). Thus, both theencoders and the decoders treat the channel estimates as ifthey were the true channel gains. This shows that the proposedachievable scheme is actually quite robust with respect tochannel estimation and quantization errors.As before, we can use Theorem 3 to derive more familiar

results on the degrees of freedom of the Gaussian X-channel.Consider a sequence of indexed by , and set

. Then, the measures are summable over. Applying the Borel–Cantelli lemma as before yields

the following corollary to Theorem 3.Corollary 4: For almost all channel gains ,

the symmetric Gaussian X-channel (10) has degrees offreedom, i.e.,

Since the of the channel is approximately so that, the quantity in Corollary 4

is indeed the degree-of-freedom limit. Corollary 4 recovers theresult in [6]. We emphasize again that Theorem 3 is consider-ably stronger than Corollary 4. Indeed, Theorem 3 proves theconstant-gap capacity approximation

with pre-constant in the term uniform in the channelgains outside . This is considerably stronger thanthe degree-of-freedom capacity approximation in Corollary 4,which shows only that

with pre-constant in the term depending on . More-over, Theorem 3 provides bounds on the measure of the outageevent for finite , not just asymptotic guarantees as in Corol-lary 4.

B. X-Channel With Arbitrary

In the last section, we considered the Gaussian X-channelwith across each link of order . Thus, all links hadapproximately the same strength. We now turn to the GaussianX-channel with arbitrary . As before, we start with theanalysis of the deterministic X-channel. The next theoremprovides an approximate characterization of the sum capacity

of the general deterministic X-channel with bit levels .Theorem 5: For every and

with , there exists a setof Lebesgue measure

such that for all channel gains , thesum capacity of the (modulated) general deterministicX-channel (17) satisfies

for some positive universal constant , and where

and

The proof of Theorem 5 is presented in Section V. For thespecial case of symmetric channel , for all ,, Theorem 5 reduces to Theorem 1 in Section IV-A.We now provide a sketch of the communication scheme

achieving the lower bound in Theorem 5 (see Figs. 11 and 12).Observe from Figs. 8 and 9 in Section III-C that theleast-significant bits of are not visible at the secondreceiver. Therefore, we can use these bits to privately carry

bits from the first transmitter to the first receiverwithout affecting the second receiver. The rate of this privatemessage is denoted by . The remaining rate is denoted by

, i.e.,

where

Similarly, the least-significant bits of arenot visible at the first receiver. Therefore, we can use this partto privately carry bits from the second transmitter tothe second receiver without affecting the first receiver. The rateof this private message is denoted by . The remaining rateis denoted by , i.e.,


Fig. 11. Allocation of bits as seen at receiver one. Here, and are the de-sired bits and are received multiplied by the matrices and (not shownin the figure), respectively. The vectors and are interference and areboth received multiplied by the same matrix . Observe that the interferencesignals and are aligned.

Fig. 12. Allocation of bits as seen at receiver two. Here, and are the de-sired bits and are received multiplied by the matrices and (not shownin the figure), respectively. The vectors and are interference and areboth received multiplied by the same matrix .

where

It remains to choose the values of , , , and .Our proposed design rules are as follows.1) We dedicate the most-significant bits of to carryinformation from transmitter one to receiver one.

2) Similarly, we dedicate the most-significant bits ofto carry information from transmitter two to receiver two.

3) We always set the most-significant bits ofto zero. The next bits of carry information fromtransmitter two to receiver one. As shown in Fig. 12, thisguarantees the (partial) alignment of with at thesecond receiver.

4) We always set the most-significant bits ofto zero. The next bits of carry information fromtransmitter one to receiver two. As shown in Fig. 11, thisguarantees the (partial) alignment of with at thefirst receiver.

Optimizing the values of the rates subject to the condi-tion that both receivers can decode the desired messages yieldsthe lower bound in Theorem 5. The details of this analysis canbe found in Section V-A.Generalizing these ideas from the deterministic to the

Gaussian model, we obtain the following constant-gap capacityapproximation for the Gaussian X-channel with general asym-metric channel gains.

Theorem 6: For every andwith , there exists a set

of Lebesgue measure

such that for all channel gains the sumcapacity of the general Gaussian X-channel (9) satisfies

for some positive universal constant , and where is asdefined in Theorem 5. Moreover, the lower bound on isachievable with a -bit quantization of the channelgains available at both transmitters and receivers.The proof of Theorem 6 is presented in Section VI. For the

special case of symmetric channel , for all , ,Theorem 6 reduces to Theorem 3 in Section IV-A. ComparingTheorems 6 and 5, we see that, up to a constant gap, the GaussianX-channel and its lower triangular deterministic approximationhave the same capacity. Thus, the lower triangular deterministicmodel captures the relevant features of the Gaussian X-channel.The lower bound in Theorem 6 is achieved by encoders and

decoders that have access to only a -bit quantiza-tion of the channel gains . As before, the encodersand decoders are mismatched, in the sense that they are oper-ating under the assumption that is the correct channel gain.This shows again that the proposed communication scheme isquite robust with respect to channel estimation and quantizationerrors.

V. PROOF OF THEOREM 5 (DETERMINISTIC X-CHANNEL)

This section contains the proof of the capacity approximationfor the deterministic X-channel in Theorem 5. Achievability ofthe lower bound in the theorem is proved in Section V-A; theupper bound is proved in Section V-B.

A. Achievability for the Deterministic X-Channel

This section contains the proof of the lower bound in The-orem 5. Without loss of generality, we assume that .We use the achievable scheme outlined in Section IV-B (seeFigs. 11 and 12 there). We want to maximize the sum rate

where

is the total rate from transmitter to receiver . The constraintis that each receiver can solve for its own desired messages plusthe visible parts of the aligned interference bits.If the subspaces spanned by the columns of cor-

responding to information-bearing bits of are linearlyindependent, then there exists a unique channel input to thedeterministic X-channel that results in the observed channeloutput. The decoder declares that this unique channel inputwas sent. The next lemma provides a sufficient condition for


this linear independence to hold and hence for decoding to besuccessful.Lemma 7: Let and such that

. Assume , , ,, , satisfy

(19a)

(19b)

(19c)

and

(20a)

(20b)

(20c)

Then, the bit allocation in Section IV-B for the (modulated) de-terministic X-channel (17) allows successful decoding at bothreceivers for all channel gains except for aset of Lebesgue measure

If , then (19b) can be removed (i.e., doesnot need to be verified); and if , (19c) can be removed.Similarly, if , (20b) can be removed; and if

, (20c) can be removed.The proof of Lemma 7 is reported in Section VII-A.We now choose rates satisfying these decoding conditions.

For ease of notation, we will ignore the termsthroughout—the reduction in sum rate due to this additionalrequirement is at most . The optimal allocation ofbits at the transmitters depends on the value . Wetreat the cases

separately. Since by the assump-tion , this covers all possiblevalues of .Case I : We set

Fig. 13. Allocation of bits in case II. Here, , , ,and . The transmitters send private messages at rates and

. Transmitter two sends a common message to receiver two at rate.

In words, we solely communicate using the private channel in-puts and . Recall that, by our assumptions throughoutthis section, . Hence,and , so that this rate allocation is valid. The calcula-tion in Appendix A verifies that this rate allocation satisfies thedecoding conditions (19) and (20) in Lemma 7. Hence, both re-ceivers can recover the desired messages. The sum rate can beverified to be

(21)

Case II : We set

as shown in Fig. 13. Since , we have ,and hence, this rate allocation is valid.The calculation in Appendix A verifies that this rate alloca-

tion satisfies the decoding conditions (19) and (20) in Lemma7. Hence, both receivers can decode successfully. The sum ratecan be verified to be

(22)

Case III : We set


Fig. 14. Allocation of bits in case III. Here, , , ,and . The transmitters send private messages at rates and

. Transmitter one sends a common message to receiver one at rate

. Transmitter two sends a common message to receiver two at rate. The rates over the cross links are and . Observe

that the interference terms are partially aligned at each receiver.

Fig. 15. Allocation of bits in case IV. Here, , ,. The transmitters send private messages at rates and

. Transmitter one sends a common message to receiver one at rate ,and transmitter two sends a common message to receiver two at rate. The rates over the cross links are and . In case IV,

the interference terms are completely aligned at receiver two, but only partiallyaligned at receiver one.

as depicted in Fig. 14. Using and, it can be verified that and ,

and hence, this rate allocation is valid.The calculation in Appendix A verifies the decoding condi-

tions (19) and (20) in Lemma 7. The sum rate can be verified tobe

(23)

Case IV : We set

as shown in Fig. 15. Using that , it canbe verified that , , and are nonnegative so that thisrate allocation is valid.

Fig. 16. Allocation of bits in case V. Here, , ,, and . The private messages to receiver one and two have rates

and . The remaining messages to receiver one have rate, and are both entirely aligned at receiver two. The remaining

messages to receiver two have rate , and are both entirelyaligned at receiver one.

The calculation in Appendix A verifies the decoding condi-tions (19) and (20) in Lemma 7. The sum rate can be verified tobe at least

(24)

where the loss of three bits is due to the floor operation in thedefinition of , , and .Case V : We set

as shown in Fig. 16. From , it follows that ,, , and are nonnegative so that this rate allocation

is valid.The calculation in Appendix A verifies the decoding condi-

tions (19) and (20) in Lemma 7. The sum rate is at least

(25)

where the loss of four bits is due to floor operation in the defi-nition of , , , and .Combining (21)–(25) and accounting for the -bit

loss in Lemma 7 shows that assuming

with


If , we can simply relabel the two transmitters andreceivers, and the same argument holds. This relabeling of re-ceivers introduces the function instead of in thelower bound. Together, this concludes the proof of the lowerbound in Theorem 5.

B. Upper Bound for the Deterministic X-Channel

The section contains the proof of the upper bound in Theorem5. We start with a lemma upper bounding various linear combi-nations of achievable rates for the deterministic X-channel.Lemma 8: Any achievable rate tuple

for the (modulated) deterministic X-channel (17) satisfies thefollowing inequalities:

(26a)

(26b)

(26c)

(26d)

(26e)

(26f)

(26g)

(26h)

(26i)

(26j)

The proof of Lemma 8 is reported in Appendix B. Inequal-ities (26a)–(26f) are based on an argument from [13, Th. 4.4].Inequalities (26g)–(26j) are novel.The upper bounds in Lemma 8 can be understood intuitively

as multiple-access bounds for a channel where the receivers areforced to decode certain parts of the interference (see Figs. 8 and9 in Section IV-B). For example, inequality (26a) correspondsto the multiple-access bound

(27)

at receiver one, combined with the inequality

Similarly, inequality (26e) corresponds to the multiple-accessbound

at receiver one, combined with the multiple-access bound

(28)

at receiver two. Finally, inequality (26g) corresponds to themul-tiple-access bounds (27) and

at receiver one, combined with the multiple-access bound (28)at receiver two. The proof of Lemma 8 makes this intuitive rea-soning precise. A detailed discussion of this type of cut-set in-terpretation can be found in [23].We proceed with the proof of the upper bound in Theorem 5

for the deterministic X-channel. Under the assumption

(29)

the first four inequalities (26a)–(26d) in Lemma 8 yield the fol-lowing upper bound on sum capacity:

(30)

Again, using (29), inequality (26e) in Lemma 8 shows that

(31)

Inequalities (26d) and (26g) in Lemma 8 combined with (29)yield

(32)

Similarly, from (26c) and (26j) in Lemma 8

(33)

The sum capacity is hence at most the minimum of the upperbounds (30)–(33), i.e.,

concluding the proof.

VI. PROOF OF THEOREM 6 (GAUSSIAN X-CHANNEL)

This section contains the proof of the capacity approximationfor the Gaussian X-channel in Theorem 6. Achievability of the


lower bound in the theorem is proved in Section VI-A; the upperbound is proved in Section VI-B.

A. Achievability for The Gaussian X-Channel

Here, we prove the lower bound in Theorem 6 by trans-lating the achievable scheme for the deterministic model to theGaussian model. For ease of exposition, we assume in most ofthe analysis that all channels gains are exactly known at thetwo transmitters and receivers. The changes in the argumentsnecessary for the mismatched case, in which the transmittersand receivers have access only to a quantized version ofthe channel gain , are reported in Appendix C.Recall that each transmitter has access to two messages

and . The transmitter forms themodulated symbol fromthe message . From these modulated signals, the channelinputs

are constructed.We now describe the modulation process from to

in detail. Each is of the form

with . Since and ,the resulting channel input satisfies the unit average powerconstraint at the transmitters.In analogy to the achievable scheme for the deterministic

channel, we only use certain portions of the bits in thebinary expansion of ; the remaining bits are set to zero.The allocation of information bits depends on the channelstrength and is chosen as in the deterministic case describedin Sections IV-B and V-A, and as illustrated in Figs. 11 and12. In particular, the messages are again decomposed intocommon and private portions, i.e.,

We denote by the modulation rate of in bits persymbol in analogy to the deterministic case.To satisfy the power constraint (as discussed above), we im-

pose that the two most significant bits of each common mes-sage are zero. For reasons that will become clear in the nextparagraph, we also impose that the two most significant bits foreach private message are zero. This reduces the modulation rateby at most 12 bits per channel use compared to the deterministiccase.The channel output at receiver one is

with denoting the product of two channel gains as definedin (13) in Section III-B. The situation is similar at receiver two.The channel output is grouped into three parts. The first partcontains the two desired signals and . The second partcontains the interference signals and . Note that theseinterference terms are received with the same coefficientand are hence aligned. The third part contains noise and theprivate portion of the message . By construction

so that

We will treat this part of the interference as noise.Set

The goal of the demodulator at receiver one is to find estimatesof , from which estimates for the desired channel inputsand can be derived. The demodulator searches for ,, that minimize

We point out that the demodulator decodes only the sum ofthe two interfering symbols, but not the individual interferingsymbols themselves. The demodulator at receiver two works inanalogy.We now lower bound the minimum distance

(34)

between the noiseless received signal generated by the correctand by any other triple . The next

lemma provides a sufficient condition for this minimum dis-tance to be large at both receivers.Lemma 9: Let and such that

. Assume , , ,, , satisfy

(35a)

(35b)

(35c)


and

(36a)

(36b)

(36c)

where . Then, the bit allocation in Section IV-B ap-plied to the Gaussian X-channel (9) results in a minimum con-stellation distance at each receiver for all channel gains

except for a set of Lebesguemeasure

If , then (35b) can be removed (i.e., doesnot need to be verified); and if , (35c) can be removed.Similarly, if , (36b) can be removed; and if

, (36c) can be removed.The proof of Lemma 9 is reported in Section VII-B. Observe

that, up to the constants, Lemma 9 is exactly of the same form asLemma 7 in Section V-A for the lower triangular deterministicX-Channel, highlighting again the close connection between thetwomodels. In the following discussion, we will assume that thechannel gains are outside the outage set, i.e., .Recall that we have chosen the same allocation of informa-

tion bits in the binary expansion of as in the determin-istic case analyzed in Section V-A. Since the most-significantbit of each is zero, the binary expansion of is alsoof the form analyzed there. Moreover, since the conditions inLemma 9 used here are the same as the conditions in Lemma 7used in the deterministic case, we conclude that Lemma 9 canbe applied if we further reduce the rates to accommodate theconstant in Lemma 9. This can be achieved,for example, by reducing the modulation rate by a further

per symbol. Accounting forthe loss of 12 bits per channel use due to the power constraint,the sum rate of the modulation scheme is then

(37)

with as defined in Theorem 5 for the deterministicX-channel, and where the additional loss of 4 bits results fromrounding in the bit allocation for the deterministic scheme asdiscussed in Section V-A.Lemma 9 is sufficient to show that the probability of demodu-

lation error is small. To achieve a vanishing probability of error,we use an outer code over the modulated channel. The distribu-tion of is chosen to be uniform over the set allowed by themodulator constraints and independent of all other modulatorinputs. Let denote the rate of this outer code from trans-mitter to receiver . We now lower bound the rate as afunction of the modulation rate .

We have

We will argue below that

(38)

so that

(39)

On the other hand

Together with (39), this shows that

Since there is a one-to-one relationship between and ,this implies that the outer code can achieve a rate of

The same argument can be used for the other rates as well,showing that

for all , . Hence, the outer codes achieves a sumrate of at least

Using (37), this shows that except for a set of measure at most

with which is what needed to be shown.Remark: The rate of the outer code can be lower

bounded in terms of the modulation rate using Fano’sinequality. This is the approach taken, for example, in [6] and[20]. However, this approach results in a gap that depends on, and is hence not strong enough for a constant-gap approx-

imation of capacity. Instead, we use a stronger argument (seethe proof of (38) below) that yields a gap independent of .


Fig. 17. Illustration of the mapping . The parameter ranges over theintegers. The parameter ranges over all possible values of . Observe that foreach fixed value of , ranges over all possible values of as a function of. Similarly, for each fixed value of , ranges over a subset of the possiblevalues of as a function of . The distance between any two points is at least32.

This argument is a key step in the derivation of the lower boundon capacity.It remains to prove (38). It will be convenient to define

and similarly for with respect to . Observe that the channeloutput is then equal to plus signals treated as noise. Sincewe assume that the channel gains are outside the outage set ,Lemma 9 implies that there is a one-to-one relationship betweenand , and between and . Hence

(40)

Set

Wewill show that is small by arguing that isclose to one for and decays exponentially quickly for. More precisely, define a mapping , with a possiblevalue of and an integer, as follows. Set . If is anegative integer, set to be the th closest possible valueof to the left of . If is a positive integer, set to be theth closest possible value of to the right of . This mapping isillustrated in Fig. 17. We will show that decaysexponentially in .Rewrite as

Recall that, by Lemma 9, the distance between two possiblevalues of is at least . In order to decode to ifthe correct value of is , the noise terms needs to havemagnitude at least . From this observation, we can obtainan upper bound on .As mentioned previously, this analysis is based on the as-

sumption that both transmitters and receivers have access to. The analysis in Appendix C shows that the only differ-

ence under mismatched encoding and decoding, in which thetransmitters and receivers use -bit quantized channelgains instead of , is a decrease in the minimum con-stellation distance . In particular, (79) in Appendix C showsthat for

where in the last inequality, we have used the Chernoff boundon the Q-function. Hence

(41)showing that decays exponentially in .We next argue that this implies that is small. We

have

(42)

Applying [25, Th. 9.7.1]

Combined with (41) and (42), this implies

(43)

Now, since for every fixed value of , takes each pos-sible value of at most once as a function of (see Fig. 17), wehave

Moreover


Substituting this into (43) yields

Together with (40), this proves (38).

B. Upper Bound for the Gaussian X-Channel

This section proves the upper bound in Theorem 6. We startwith a lemma upper bounding various linear combinations ofachievable rates for the Gaussian X-channel.Lemma 10: Any achievable rate tuple

for the Gaussian X-channel (9) satisfies the inequalities in (44)shown at the bottom of the page.The proof of Lemma 10 is reported in Appendix D. Inequali-

ties (44a)–(44f) are from, [13, Lemma 5.2, Th. 5.3]. Inequalities(44g)–(44j) are novel.

We proceed with the proof of the upper bound in Theorem6 for the Gaussian X-channel. Note that for and

and

Hence, (44a) yields

(44a)

(44b)

(44c)

(44d)

(44e)

(44f)

(44g)

(44h)

(44i)

(44j)


In a similar manner, we can upper bound the right-hand sides ofall terms in Lemma 10 by quantities depending only on . Forexample, (44e) yields

and (44g) yields

Comparing this to the upper bounds in Lemma 8 inSection V-B for the lower triangular deterministic X-channel,we see that Lemma 10 for the Gaussian X-channel is identicalup to a constant gap. This highlights again the close connectionbetween the two models. Using the same derivation as for thedeterministic case, Lemma 10 can thus be used to show thatunder the assumption

the sum capacity of the Gaussian X-channel satisfies

This concludes the proof of the upper bound.

VII. MATHEMATICAL FOUNDATIONS FOR RECEIVER ANALYSIS

This section lays the mathematical groundwork for the anal-ysis of the decoders used in Sections V-A and VI-A. For the de-terministic channel model, decoding is successful if the variousmessage subspaces are linearly independent. Conditions for thislinear independence to hold are presented in Section VII-A. Forthe Gaussian case, decoding is successful if the minimum dis-tance between the different messages as seen at the receiversis large. As we will see, this problem can be reformulated as anumber-theoretic problem. Conditions for successful decodingin the Gaussian case are presented in Section VII-B.

A. Decoding Conditions for the Deterministic Channel

We start by analyzing a “generic” receiver (i.e., the bit allo-cation seen at either receiver one or two). To this end, we as-sume there are two desired vectors and and one inter-ference vector . The interference vector consists of twosignal vectors that are aligned and can therefore be treated as asingle vector. These three vectors are multiplied by the lowertriangular channel matrices , , and created via the bi-nary expansion of the channel gains , , as before.We assume that certain components of the vectors are set

to zero. To formally capture this, we need to introduce some

Fig. 18. Vector in the set . White regions represent bits set tozero.

Fig. 19. Generic receiver as analyzed in Lemma 11. White regions correspondto zero bits; shaded regions carry information. Bits are labeled from 1 to ,starting from the top.

notation. Let and be two nonnegative integers such that. Define

as illustrated in Fig. 18. We consider vectors inthe set

with , as illustrated in Fig. 19. Here, andare to be interpreted as the common and private portions of

the desired signal transmitted over the direct link; is to beinterpreted as the desired signal transmitted over the cross link;and is to be interpreted as the aligned interference.The next lemma states that the subspaces spanned by the cor-

responding columns of are linearly independent for mostchannel gains .Lemma 11: Let , , such that ,

and let , , , . Define the event

and set


For any satisfying

we have

Observe that is the set of channel gains , , such thatthe corresponding subspaces spanned by the selected columnsof , , are linearly dependent. In other words, isthe set of channel gains resulting in decoding error. Thus, thelemma states that if the rates satisfy certain conditions, thenthe subspaces under consideration are linearly independent withhigh probability, and hence, decoding is successful.The condition on the rates in Lemma 11 can be interpreted

as follows. Let be some natural number. Since the matricesare lower triangular, the subspaces spanned by the last

columns of are the same for all . Thus, a nec-essary condition for the linear independence of the three sub-spaces is that the total number of possible nonzero componentsof with and is at mostfor every . By the structure of the set , thiscondition can be verified by considering only three values of ,namely (see Fig. 19). Thus, a necessary con-dition for the linear independence of the subspaces is

After some algebra, these three conditions can be rewrittenequivalently as

Thus, Lemma 11 shows that, up to the constant andfor most channel gains , these necessary conditionsare also sufficient for the linear independence of the subspaces.Before we provide the proof of Lemma 11, we show how it

can be used to prove Lemma 7 in Section V-A.Proof of Lemma 7: We start by reformulating the conditions

in Lemma 11 for each receiver. Consider first receiver one inLemma 7. From Fig. 11 in Section IV-B, we see that the corre-sponding message rates in Lemma 11 are given by

The choice of the bit levels in Lemma 11 depends on thevalues of and . If , , we need to set

see again Fig. 11.The conditions in Lemma 11 (with replaced by to guar-

antee that the outage event at each receiver has measure at most) are then that

(45a)

(45b)

(45c)

If , then the second column in Fig. 19 is empty, andhence, the third condition in Lemma 11 does not need to be ver-ified. Formally, note that in this case, the value of is irrelevantto the decoding process. We may hence assume without loss ofgenerality that is equal to (thus still satisfying ).As a consequence, only conditions (45a) and (45b) need to bechecked. If , then the value of is irrelevant to the de-coding process, and we can assume it to be equal to (thus stillsatisfying ). As a consequence, only conditions (45a)and (45c) need to be checked.The decoding conditions for receiver two follow by

symmetry.Denote by the collection of triples

such that decoding fails at receiver one. Similarly, definewith respect to receiver two. Finally, let be the

union of and . If the two sets of decoding conditions aresatisfied, then Lemma 11 shows that

for , where here and in the following we use thenotation to emphasize that Lebesgue measure is computedin . Then

i.e., the collection of channel gains forwhich decoding fails is small. This concludes the proof ofLemma 7.It remains to prove Lemma 11.Proof of Lemma 11: We start with a few preliminary obser-

vations. Note that, by the assumptions on


which implies that

(46)

From Fig. 19, we see that this guarantees that if , then

(47)

for , where for a binary vector we use the notationto denote the smallest index such that with the

convention that . Moreover, we see from the samefigure that .We now remove the dependence of on the private signal .

Since is lower triangular with unit diagonal (so that bits areonly shifted downward), we have . Hence

can hold only if

where we have used that .Furthermore, we have for that

can hold only if .Defining the sets

and

we hence have

We can then upper bound using the union bound

(48)Observe that the right-hand side does not depend on the privatesignal .We continue by analyzing each term in the summationon the right-hand side of (48) separately.Since we are integrating with respect to Lebesgue measure

over , we can equivalently assume that, , are independent and uniformly distributed over .

The bits in the binary expansion of these numbersare then binary random variables with the following properties.

for , , and are i.i.d.Bernoulli(1/2) (see, e.g., [26, Exercise 1.4.20]). The lower trian-gular Toeplitz matrix is then constructed from these binary

random variables. Note that this implies that the three matrices, , are independent and identically distributed.Fix a binary vector and consider the product for some

, , and with addition again over . We nowdescribe the distribution of . Since is lower triangularwith unit diagonal, whenever , and

. Moreover, the components forare i.i.d. Bernoulli(1/2).

Assume first that

(49)

The summand in (48) can be written as

(50)

where the probabilities are computed with respect to the randommatrices . Using that , the three factorsinside the summation are nonzero only if

From this, we obtain that

where for the last equality we have used (49) and thatis larger than by (47). Since by (49),this shows that can hold only if

.If these conditions on the and are satisfied, then

Substituting this into (50) shows that

whenever , and

otherwise.Assume more generally that . Then,

a similar argument shows that

(51)


whenever there are two distinct indices , achieving the min-imum , and

(52)

otherwise. In particular, the set has measure zerowhenever at least two of the are equal to zero.Setting

we can then rewrite (48) as

By (52), the set has measure zero whenever there is only asingle minimizing . Together with the assumption

, this shows that we can restrict the lower boundariesof the sets in the various sums. For example

where we have changed to ,and similarly for the other three summations. Together with (51)this yields that

We consider each of the four terms in turn.

For the first term, we have

Using that

the right-hand side can be further upper bounded by

We can upper bound the remaining three terms in a similarfashion, yielding

This shows that if

and (in order to guarantee (46)) if

then

completing the proof of the lemma.

B. Decoding Conditions for the Gaussian Channel

In this section, we analyze a “generic” receiver for theGaussian case. To this end, we prove a variation of a well-knownresult from Diophantine approximation called Groshev’s the-orem (see, e.g., [27, Th. 1.12]).


Define

where we assume that the binary expansion of and is iden-tical. Set

is the set of real numbers such that their binary expansions,when viewed as vectors of length , are in the set as illus-trated in Fig. 19 in Section VII-A. Thus, is the direct transla-tion of the set of possible channel inputs for the deterministicsetting to the Gaussian setting. The next lemma states that if thechannel inputs are chosen from , then the resulting minimumconstellation distance as observed at the receivers is large formost channel gains .Lemma 12: Let , , such that ,

and let , , , . Define the event

and set

For any satisfying

we have

Lemma 12 is the equivalent for the Gaussian channel ofLemma 11 for the deterministic channel. Note that, except forthe constants, the conditions on the rates in the two lemmas areidentical.We now prove Lemma 9 in Section VI-A using Lemma 12.Proof of Lemma 9: We will use Lemma 12 with instead

of and the same rate allocations as in the deterministic case(see Figs. 11 and 12 in Section IV-B). Let bethe collection of triples such that decoding issuccessful at receiver . Define as the collection of channelgains such that the corresponding arein . Finally, let denote the union of and . Followingthe same arguments as in the proof of Lemma 7 from Lemma 11presented in Section VII-A, it can be shown that if the decodingconditions in Lemma 9 are satisfied, then Lemma 12 guaranteesthat

for .The next lemma allows us to transfer this statement about the

products of channel gains to the corresponding statementabout the original channel gains . For ease of notation, thestatement of the lemma uses as a shorthand for as definedin (13) for some fixed value of .Lemma 13: Let be a subset of channel gains

such that . Define

Then, .The proof of Lemma 13 is reported in Appendix E. Applying

Lemma 13 to the sets and corresponding to the outageevents defined above, this implies that

Hence

proving Lemma 9.We continue with the proof of Lemma 12. Instead of di-

rectly analyzing the set in the statement of Lemma 12, itwill be convenient to work with an equivalent set. Note that

can be written as

By the definition of (see also Fig. 19 in Section VII-A), wecan decompose

with

for and

for , where

We now remove the dependence of on . We can furtherrewrite using the triangle inequality as


where all sets are defined over , and wherewe have defined

Setting

we then have

The next lemma analyzes the set with .Lemma 14: Let , , , and , ,. Define the event

and set

Then

with

Remark: The special case of Lemma 14 with, , and corresponds to the(converse part of) Groshev’s theorem, see, e.g., [27, Th. 1.12].Hence, Lemma 14 extends Groshev’s theorem to asymmetricand nonasymptotic settings.Before we present the proof of Lemma 14, we show how to

prove Lemma 12 with the help of Lemma 14.Proof of Lemma 12: We consider the three cases

, , andseparately.Assume first that . Define

Note that , , and that if

(53)

as required by Lemma 14. The quantities and in Lemma14 can be upper bounded as

since implies that , and as

Applying Lemma 14 yields then

where we have used that and that implying

Substituting the definitions of , , and yields that

Together with (53), this shows that if

then

Since and , the third condition is redundantand can be removed, showing the result in Lemma 12. We pointout that the third condition in Lemma 12 is not active if

. This is consistent with it not appearing in thederivation here.Assume next that . Define


(54)


We can hence apply Lemma 14 by appropriately relabeling in-dices (i.e., by swapping indices 0 and 1). The quantities andcan be upper bounded as

and

Applying Lemma 14 yields then that

Substituting the definitions of and yields that


then

Since and , , the second condition is redun-dant and can be removed, showing the result in Lemma 12. Ascan be verified, the second and third conditions in Lemma 12are not active when , consistent with themnot appearing in the derivation here.Finally, assume that . Define


(55)

We can hence apply Lemma 14 by relabeling indices as before(this time by swapping indices 0 and 2). The quantities andcan be upper bounded as

and

since implies .Applying Lemma 14 yields then that

Substituting the definitions of , , and , yields that


then

showing the result in Lemma 12. It can be verified that, unlikein the other two cases, all three conditions in Lemma 12 can beactive when . This is again consistent withthe derivation here. This proves Lemma 12.It remains to prove Lemma 14. The proof builds on an argu-

ment in [28].Proof of Lemma 14: Define

for , and

For , set

Observe that is a subset of and that


Fig. 20. Illustration of the set with , ,, , and . In the figure, we assume that

. The set consists of strips of slope.

We treat the cases andseparately. Assume first and . If

then

where we have used that . Hence, .We can therefore assume without loss of generality that

for any value of . By a similar argument, we can assume that

for .The set consists of at most

strips of slope and width in thedirection, including several partial strips (see Fig. 20). The areaof this set is at most

(56)

We now consider the case and . Asbefore, we can assume without loss of generality that

for any value of , and that

for . By the same analysis as in the last paragraph, weobtain that

(57)

Finally, when and , then ,and hence

(58)

We can upper bound

The right-hand side can be split into the following five terms:

Combined with (56), (57), and (58), this yields

(59)


We now upper bound the four terms in the right-hand side of(59).For the first term in (59), observe that

so that

(60)

For the second term in (59), observe that

and hence

Moreover

with

Using these two facts, we can upper bound

(61)

Similarly, for the third term in (59)

(62)

and for the fourth term

(63)

with

Substituting (60)–(63) into (59) yields

completing the proof.

VIII. CONCLUSION

In this paper, we derived a constant-gap capacity approxima-tion for the Gaussian X-channel. This derivation was aided bya novel deterministic channel model used to approximate theGaussian channel. In the proposed deterministic channel model,the actions of the channel are described by a lower triangularToeplitz matrices with coefficients determined by the bits in thebinary expansion of the corresponding channel gains in the orig-inal Gaussian problem. This is in contrast to traditional deter-ministic models, in which the actions of the channel are onlydependent on the single most-significant bit of the channel gainsin the original Gaussian problem. Preserving this dependence onthe fine structure of the Gaussian channel gains turned out to becrucial to successfully approximate the Gaussian X-channel bya deterministic channel model.Throughout this paper, we were only interested in obtaining

a constant-gap capacity approximation. Less emphasis wasplaced on the actual value of that constant. For a meaningfulcapacity approximation at smaller values of SNR, this constantneeds to be optimized. More sophisticated lattice codes (asopposed to the ones over the simple integer lattice used in thispaper) could be employed for this purpose (see, e.g., [29]).Furthermore, all the results in this paper were derived for allchannel gains outside an arbitrarily small outage set. Analyzingthe behavior of capacity for channel gains that are inside this


outage set is hence of interest. An approach similar to the onein [30] could perhaps be utilized to this end.Finally, the analysis in this paper focused on the Gaussian

X-channel as an example of a fully connected communicationnetwork in which interference alignment seems necessary.The hope is that the tools developed in this paper can be usedto help with the analysis of more general networks requiringinterference alignment. Ultimately, the goal should be to movefrom degree-of-freedom capacity approximations to strongerconstant-gap capacity approximations.

APPENDIX AVERIFICATION OF DECODING CONDITIONS

This appendix verifies that the rate allocation in Section V-Afor the deterministic X-channel satisfies the decoding conditions(19) and (20) in Lemma 7.Case I : Recall

This choice of rates satisfies (19a) and (20a). Since these are theonly two relevant conditions in this case, this shows that bothreceivers can recover the desired messages.Case II : Recall

At receiver one, (19a) and (19b) are satisfied since

Condition (19c) does not need to be checked here. At receivertwo, (20a) is satisfied since

Conditions (20b) and (20c) do not need to be checked here.Hence, both receivers can decode successfully.Case III : Recall

To check the decoding conditions (19) and (20), we first arguethat

(64a)

(64b)

The first equality trivially holds if . Assuming then that, we have

If , then this is nonnegative. Assuming then ,we obtain

where we have used that and that. This proves (64a). Using a similar argument, it can

be shown that , proving (64b). To check thedecoding conditions (19) at receiver one, observe now that

satisfying (19b). Moreover

satisfying (19a). Finally, if , then

satisfying (19c); and if , then (19c) is irrelevant. Usinga similar argument, it can be shown that the decoding conditions(20) at receiver two hold.Case IV : Recall

To check the decoding conditions, note first that

since

by assumption. For receiver one, we then have

satisfying (19a). Moreover

where we have used . Hence, (19b) issatisfied. Finally


where we have again used . Hence, (19c)is satisfied. Together, this shows that decoding is successful atreceiver one. At receiver two, we have

satisfying (20a), and

satisfying (20b). Finally

where we have used . Hence, (20c)is satisfied. Together, this shows that decoding is successful atreceiver two.Case V : Recall

For decoding at receiver one, we need to verify the decodingconditions (19). We have

satisfying (19a). Moreover

where we have used This satisfies(19b). Finally

where we have used that . Hence, (19c) issatisfied. A similar argument shows that the decoding conditions(20) at receiver two hold. Hence, decoding is successful at bothreceivers.

APPENDIX BPROOF OF LEMMA 8 IN SECTION V-B

Throughout this proof, we make use of the fact that, for the(modulated) deterministic X-channel (17), the definition of ca-pacity imposes that

is only a function of .We start with (26a). Define as the contribution of the

second transmitter at the first receiver, i.e.,

Let denote the contribution of the second transmitter at thesecond receiver, i.e.,

Similarly, we define and as the contributions of the firsttransmitter at the first and second receivers, respectively. Withthis, we can rewrite the received vector at receiver as

For block length , we have

(65)

where the first step follows from Fano’s inequality. In addition,using again Fano’s inequality

(66)

Adding (65) and (66) yields

For the first term on the right-hand side, we have

For the second term, recall that is a function of only ,and hence


Since is lower triangular with nonzero diagonal, it is in-vertible, implying that

Together, this shows that

(67)

Therefore, as and , we have (26a). Similarly, wecan prove (26b)–(26d).We now establish the upper bound (26e). Starting with Fano’s

inequality

(68)

Similarly, we have

(69)

Adding (68) and (69), we obtain

where in the last line we have used that is only a function of. For the first term, we can use invertibility of the matricesto obtain (70) shown at the bottom of the next page. Since

the matrices are lower triangular, the right-hand side of(70) is upper bounded by

By an analogous argument

Together, this shows that

proving (26e) as and . Similarly, we can prove(26f).We now establish the bound (26g). By Fano’s inequality

(71)

Moreover, using again Fano’s inequality

(72)


Combined with (67) derived earlier, we obtain


Since is a deterministic function of , we have

(73)

From (73), we obtain

Letting and yields the upper bound (26g).Similarly, we can prove (26h)–(26j).

Remark: Equation (73) is a key step in the derivationof the outer bound (26g). If we had used the standard bound

, we would have obtained alooser bound than (26g).

APPENDIX CANALYSIS OF MISMATCHED ENCODERS AND DECODERS

The proof of Theorem 6 in Section VI-A assumes that theprecise channel gains are available at all encoders and de-coders. Here, we assume instead that these channel gains areonly known approximately at any node in the network. As wewill see, the only effect of this change in available channel stateinformation is to decrease the minimum constellation distanceseen at the receivers.Formally, assume both transmitters and receivers have only

access to estimates of satisfying

(74)

In other words, all transmitters and receivers have access to a-bit quantization of the channel gains. Since we

know a priori that , we can assume without lossof generality that as well.Each transmitter forms the modulated symbol from

the message . From these modulated signals, the channelinputs

are formed. In other words, the transmitters treat the estimatedchannel gains as if they were the correct ones; the encodersare thus mismatched. The modulation process from tois the same as in the matched case analyzed in Section VI-A.Since and , the resulting channel inputsatisfies the unit average power constraint at the transmitters.

The channel output at receiver one is

As in the matched case, the received signal consists of desiredsignals, interference signals, and signals treated as noise. For thethird term treated as noise, we have

(75)

since

The demodulator at receiver one searches forminimizing

Note that the entire demodulation process depends solely on theestimated channel gains and not on the actual channel gains

. Furthermore, the demodulator is the maximum-likelihooddetector only if the estimated channel gains coincide with theactual channel gains. Thus, the demodulator is mismatched.We now analyze the probability of error of this mismatched

demodulator. There are two contributions to this probability oferror. One is due to noise, the other one due to mismatcheddetection. Set

and define similarly, but with respect to . Weneed to upper bound

with as defined in in Section VI-A. Let be the min-imum distance between any two noiseless estimated receivedsignals (as assumed by the mismatched demodulator using

), i.e., between any two possible values of . Let bethe maximum distance between the noiseless received signal

and the estimated received signalwith the same channel inputs. Then

(76)

where we have used (75).

(70)


We start by upper bounding themismatch distance .We have

(77)

where we have used (74), that , and that .We continue by lower bounding the distance between the

estimated received signal (i.e., as assumed by the mismatcheddetector) generated by the correct and by anyother triple . By the triangle inequality

(78)

where denotes the minimum distance (34) in the matched caseas analyzed in Section VI-A. Here we have used that

by (74), and similarly for the other two terms.Combining (77) and (78) shows that

By (76), this implies

(79)

APPENDIX DPROOF OF LEMMA 10 IN SECTION VI-B

The inequalities (44a)–(44f) have been already proved in,[13, Lemma 5.2, Th. 5.3]. Here, we present the proof for in-equalities (44g)–(44j). First, we establish the bound (44g).Define as the contribution of transmitter at receivercorrupted by receiver noise , i.e.,

For block length , we have

(80)

where the first step follows from Fano’s inequality. Again fromFano’s inequality, we have

(81)


(82)

Using Fano’s inequality at receiver two, we have

(83)

Moreover, Fano’s inequality at receiver one yields

(84)


(88)


(85)

Adding (85) and (82) derived earlier, we obtain

(86)

Since

(87)

we obtain from (86) that

where the last inequality follows from the fact that i.i.d.Gaussian random variables maximize conditional differentialentropy. Letting and proves (44g). Inequalities(44h)–(44j) can be proved similarly.

Remark: We point out that, as in the deterministic case,(87) is a key step to the derivation of the outer bound for theGaussian X-channel.

APPENDIX EPROOF OF LEMMA 13 IN SECTION VII-B

By Fubini’s theorem, we have (88) shown at the top of thepage for . The situation is analogous for .

ACKNOWLEDGMENT

The authors would like to thank G. Kramer for pointing themto Slepian’s 1974 Shannon Lecture [19] and the reviewers fortheir careful reading of the manuscript and their thoughtfulcomments.

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UrsNiesen (S’03–M’09) received theM.S. degree from the School of Computerand Communication Sciences at the École Polytechnique Fédérale de Lausanne(EPFL) in 2005 and the Ph.D. degree from the department of Electrical En-gineering and Computer Science at the Massachusetts Institute of Technology(MIT) in 2009. Currently, he is amember of technical staff in theMathematics ofNetworks and Communications Research Department at Bell Labs, Alcatel-Lu-cent. His research interests are in the areas of communication and informationtheory.

Mohammad Ali Maddah-Ali (S’02–M’08) received the B.Sc. degree fromIsfahan University of Technology, Isfahan, Iran, in 1997 and the M.A.Sc.degree from the University of Tehran, Tehran, Iran, in 2000, both in electricalengineering with highest rank in classes. From 2002 to 2007, he was withthe Coding and Signal Transmission Laboratory (CST Lab), Department ofElectrical and Computer Engineering, University of Waterloo, Waterloo, ON,Canada, working toward the Ph.D. degree. From March 2007 to December2007, he was a Postdoctoral Fellow at the Wireless Technology Laboratories,Nortel Networks, Ottawa, ON, Canada, in a joint program between CSTLab and Nortel Networks. From January 2008 to August 2010, he was aPostdoctoral Fellow at the Wireless Foundations Center, the Department ofElectrical Engineering and Computer Sciences in the University of Californiaat Berkeley. Since September 2010, he has been at Bell Laboratories, Al-catel-Lucent, Holmdel, NJ, as a communication network research scientist. Hisresearch interests include wireless communications and multiuser informationtheory. Dr. Maddah-Ali received several awards including Natural Science andEngineering Research Council of Canada (NSERC) Postdoctoral Fellowship.

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Interference Alignment: From Degrees of Freedom to Constant-Gap Capacity Approximations

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