From Non-Causality to Causality: Degrees ofFreedom of the Interference Relay Channel

Anas Chaaban and Aydin SezginEmmy-Noether Research Group on Wireless Networks

Institute of Telecommunications and Applied Information TheoryUlm University, 89081, Ulm, Germany

Email: anas.chaaban@uni-ulm.de.aydin.sezgin@uni-ulm.de

Abstract-The achievable degrees of freedom (DOF) in aninterference relay channel are examined. We highlight somedifferences between two relaying strategies in two differentsetups. Namely, we study the DOF achieved using interferencealignment and interference cancellation in a cognitive relay setupand in a half-duplex relay setup. The difference between the twosetups is causality, which leads to a difference in the achievableDOF. We show that 3/2 DOF are achievable almost surely in a3-user complex Gaussian interference relay channel with quasi­static channel coefficients and with one antenna at each node.This result holds for both the cognitive as well as for the morepractical half-duplex non-cognitive relay. However, while this isachieved in the cognitive relay setup using one symbol duration, itneeds two symbol extensions in the half-duplex case. On the otherhand, we show that in the cognitive case, interference cancellationperforms better than interference alignment, since it achieves 2DOF almost surely. Moreover, interference cancellation achievesfull, i.e, K DOF for a family of K -user interference channelswith a cognitive relay. Interestingly, for the half-duplex setup theoutcome is swapped. Here, interference alignment outperformsinterference cancellation, since the latter achieves less than 3/2DOF almost surely in the 3 user case.

Index Terms-Degrees of freedom, asymmetric complex sig­naling, interference management.

I. INTRODUCTION

Although wireless multi-node networks are being studiedsince three decades [1], they have drawn increased researchattention recently. The analysis of such networks includescharacterizing its capacity or at least its first order approxima­tion, the degrees of freedom (DOF). This quantity, the DOF, isthe main figure that matters in characterizing the sum capacityof a network at high transmit power [2].

In a recent result [2], it was shown that the K user ICwith time varying or frequency selective channel coefficientshas K /2 DOF, achieved by interference alignment. The firstinstance of phase alignment appears also in that paper. Inter­estingly, it was shown in [3] that the use of a relay can bebeneficial in order to reduce the number of symbol extensionsspanning multiple channel realizations needed to performalignment in the interference network [2]. Indeed, relays caneven help to increase the degrees of freedom [4], which wasuntil then thought to be not the case. On the other hand, for

This work is supported by the German Research Foundation, DeutscheForschungsgemeinschaft (DFG), Germany, under grant SE 1697/3.

978-1-4244-7417-2/10/$26.00 ©2010 IEEE

some setups it was proved that relays do not increase theDOF [5]. The focus of this paper is to study the achievableDOF in an interference relay channel. We show that relayscan be utilized to increase the achievable DOF in the 3 userinterference channel with constant channel coefficients, i.e.with quasi-static fading and with one antenna at each node. Wealso examine the effect of causality on the achievable DOF inrelay setups. Achievable DOF and achievability schemes varyfrom the non-causal cognitive relay setup to the causal half­duplex one. We restrict ourselves to linear techniques, whichhave been shown to be DOF optimal for a couple of channelsincluding the MIMO IC [6] and the MIMO X channel [7].

For the IC (without a relay) with constant channel coef­ficients and one antenna at each node, it was shown in [8]that 6/5 DOF are achievable almost surely using asymmetriccomplex signaling and phase alignment. However, the authorsalso show that 3/2 degrees of freedom are achievable only fora set of channels of measure zero. We adapt the asymmetriccomplex signaling scheme proposed in [8] to the IC with acognitive relay (IC-CR). Then, we prove that if the users andthe relay jointly perform interference alignment, 3/2 degreesof freedom are achievable almost surely.

However, we also show that interference cancellation out­performs interference alignment in the IC-CR. An interferencecancellation scheme achieve 2 DOF almost surely in a K userIC-CR. Moreover, this scheme achieves full, i.e. K DOF fora family of K -user IC-CR.

Interestingly, when we replace the cognitive relay with ahalf-duplex relay, we loose this advantage of interference can­cellation over interference alignment. Interference alignmentcan still be performed in the interference channel with a half­duplex relay (IC-HDR) in a quasi-static environment (i.e. withconstant channel coefficients), and achieves 3/2 DOF almostsurely. A similar result has been shown for the time-varyingchannel in [3]. On the other hand, interference cancellationcan not achieve 3/2 DOF almost surely.

We notice that 2 symbol extensions are required to performinterference alignment in the half-duplex relay setup whileonly one is needed in the cognitive relay setup. Moreover,while interference cancellation can achieve 2 DOF in thecognitive relay setup, it can not achieve 3/2 DOF in the halfduplex case almost surely.

where

where the second term vanishes as P -... 00 .

3

Yj = 2..= Ihij! U (¢ij )X i + !hRjI U (¢Rj )X R + z; (4)i=l

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III. ACHI EVABLE OOF IN THE IC-CR

The question we ask ourselves in this section is: How manyOaF does the 3 user complex Gaussian IC-CR have? In otherwords, how many parallel interference-free channels can beachieved for this channel at high P? Keep in mind that allnodes have a single antenna, and that channel coefficients arenot varying over time.

In an attempt to answer these questions, we consider twoschemes : Interference alignment using asymmetric complexsignaling, and interference cancellation. We also comparethese schemes based on the OaF they can achieve.

A. Interference Alignment using Asymmetric Complex Signal­ing

The main idea of asymmetric complex signaling is thatthe complex valued received signals Yj can be treated as atwo dimensional real signal. Thus interference alignment canbe done over this two dimensional space. The input-outputequations in (1) can be written in terms of real quantities as[8]

The rest of the paper is organized as follows: In sectionII, we introduce the IC-CR. We investigate the asymmetriccomplex signaling interference alignment scheme in III-A, andthe interference cancellation scheme in III-B. An example isgiven in section IV. The interference channel with a half­duplex relay is introduced in section 4, and the achievableOOF are studied in VI. Finally we conclude with section VII.

II. THE 3 USER IC-CR MODEL

Consider a 3 user IC-CR with single antenna at each node,where transmitter i wants to communicate a message Ui toits respective receiver i , i E {I , 2, 3} as shown in Figure I.The cognitive relay has non-causal access to messages of all3 users, and tries to facilitate the communications betweentransmitters and receivers.

Each user encodes his message Ui to a signal Xi, and therelay encodes Ul , U2 and U3 to the signal XR. The input-outputrelations for this channel at time t are given by

3

Yj(t) = 2..= hijX i(t) + hRjX R(t) + Zj (t ), (1)i = l

i, j E {I , 2, 3}, hij, hRj, Xi (t) , XR(t ), Zj (t) E Co Furthermore,we assume that hi j and bn, are quasi-static, thus, do notallow the possibility of interference alignment using channelextensions over time varying channels. This channel is showngraphically in Figure 1. The time index t will be droppedwhere we consider only one time slot, and will be used onlywhen necessary.

UI - X l \. J.:,-------

<:h22~h112 tR

h 32 U l, U2 ,U3

U 3 - X 3

Fig. 1. The 3 user IC with cognitive relay.

The OOF are defined as the limit as P -... 00 of the ratioof the network sum capacity C(P) to 10g(P), where P is thetotal transmit power and the noise variance at each receiver isnormalized to one [9]. Or in other words, it states how doesthe sum capacity scale with respect to log(P).

. C(P)d = lim -I(P) · (2)

P s-v ix: og

Finding the OaF of a channel leads to a first order approxi­mation of the sum capacity of the form

C(P) ------7 dlog(P) + o(Iog(P)) , (3)P -+oo

and ¢ij is the phase of the channel from transmitter i toreceiver j . Here X i and Yj are two vectors in jR2, withcomponents corresponding to the real and imaginary part ofXi and Yj respectively.

Each transmitter needs to communicate message Ui toits corresponding receiver. The real valued representation ofthe complex signal Xi , i.e. X i , is constructed from Ui bymultiplying with a precoding matrix Vi E jR2,

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The relay also constructs the vector X R by using 3 precodingmatrices VRi E jR2 for each message Ui, i E {I , 2, 3}

X R = [VRl VR2 VR3][Ul U2 U3r , (7)

where the precoding matrices Vi and VRi are designed in away that facilitates interference alignment. Now we can write

3 3

Yj = 2..= !hijI U (¢ij )Viui + IhRjI U(¢Rj ) 2..= VRiUi + z,i=l i = l

3

= 2..= ( lhij !U(¢ij)Vi + IhRj !U (¢Rj )VRi ) Ui + z; (8)i=l

and we state the following theorem.

Theorem 1. The three user complex Gaussian IC-CR withquasi-static channel coefficients, and with one antenna at eachnode has 3/2 degrees offreedom almost surely.

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Proof: The degrees of freedom are achieved using jointinterference alignment by the users and the relay. In order toalign the interference, we should have1

span(l h31IU(¢31)V3 + IhRlIU(¢Rl)VR3) (9)

= span(l h21IU(¢21)V2+ IhRlIU(¢Rl)VR2)

Span(lhI21U(¢12)VI + IhR2IU(¢R2)VRl) (10)

= span(lh32IU(¢32)V3 + IhR2IU(¢R2)VR3)

span(lh23IU(¢23)V2 + IhR3IU(¢R3)VR2) (11)

= span(lhI3IU(¢13)VI + IhR3IU(¢R3)VRl)

Moreover, the interference must occupy only one dimension,leaving the other dimension free for the useful signal, so wecan rewrite the interference alignment conditions above as

Ih31IU(¢31)V3+ IhR1IU(¢Rl)VR3

= Al(l h21IU(¢21)V2+ IhRlIU(¢Rl)VR2) (12)

IhI2IU(¢12)V1+ IhR2IU(¢R2)VR1

= A2 (lh32IU(¢32)V3 + IhR2IU(¢R2)VR3) (13)

Ih23IU(¢23)V2+ IhR3IU(¢R3)VR2

= A3 (l hd U(¢13)VI + IhR3IU(¢R3)VRl) (14)

for some AI, A2'A3 E JR. This assures that the interferingsignals at each receiver occupy one dimension.

However, in order to decode the useful signal interferencefree, the vector along which it arrives at the receiver mustbe linearly independent of the one along which interferencearrives. Thus, the following matrices must have full rank (i.e.two)

(lh 11IU(¢II)V1+ IhRIIU(¢Rl)VRl)T r (15)(lh 21IU(¢21)V2+ IhR1IU(¢Rl)VR2)T

(lh22IU(¢22)V2+ IhR2IU(¢R2)VR2)Tr (16)(lh I2IU(¢12)V1+ IhR2IU(¢R2)VR1)T

(lh33IU(¢33)V3 + IhR3IU(¢R3)VR3)T r (17)(lh I3IU(¢13)V1+ IhR3IU(¢R3)VR1)T

The first column of each of these matrices represents thedesired signal, while the second represents the interference.Let us randomly choose precoding vectors VI, V2, and V3,and solve (12), (13), and (14) for VR1, VR2, and VR3 to get

TT _ ( A H 13- H 12)V I +(H2I-H23)qA2 V2+(H32-H3I)A2 v3(18)VRI - I-A

V; - (H13- H12)A3 VI +(AH2I-H23)V2+(H32-H3I)A2A3 V3(19)R2 - I-A

TT _ (H13-H12)AIA3 VI +(H2I- H23)AI V2+(AH32-H3I)V3(20)VR3 - I-A

with H i j = i~;:)1 U (¢ij - ¢Rj) a 2x 2matrix, and A =AIA2A3. Thus, if the users choose their precoding matrices by

1The span of a set of vectors represents the vector space that can begenerated as a linear combination of these vectors.

random, the relay can use the above expressions to calculateits precoding matrices, and the interference is aligned.

Now the columns of (15) must be linearly independent, so

QI (lhnIU(¢n)VI + IhRlIU(¢Rl)VRl)

+ Q2(lh21IU(¢21)V2 + IhRlIU(¢Rl)VR2) = 0, (21)

if and only if al == a2 == O. Let us assume that (21) is satisfiedfor some aI, a2 -=I- O. We can rewrite this as

al (h11VI + hRlVRl) + a2(h 21V2 + hn:VR2) == 0 (22)

where Vi == IViIej 8i and VRi == IvRiIej 8Ri are the complex

representations of Vi and VRi respectively (i.e. Vi == Vi,1 +jVi,2). Notice from (18) that all VRi are independent of hii.Equation (22) means that the two complex quantities have thesame phase modulo 1r, or in other words, the two phases musthave the same tangent, i.e.

Ih11Vl1 sin(¢11 + (}1) + IhR1VRlisin(¢Rl + (}Rl)Ih11Vl1 COS(¢11 + (}1) + IhR1VRliCOS(¢Rl + (}Rl)Ih21V21 sin(¢21 + (}2) + IhR1VR21 sin(¢Rl + (}R2)Ih21V21 COS(¢21 + (}2) + IhR1VR21 COS(¢Rl + (}R2)

Since the channel parameters are randomly chosen froma continuous distribution, h11 does not satisfy this conditionalmost surely. Therefore, we must have al == a2 == o.

Consequently, the vectors along which the useful signal andthe interference arrive at receiver 1 are linearly independent,i.e. (22) holds if and only if al == a2 == o. Similarly forreceiver 2 and 3.

As a result, each receiver can project its received signalinto the null space of the interference, achieving one degreeof freedom. Thus, three degrees of freedom are achieved overa two dimensional space, resulting in 3/2 DOF. •

Notice that for the 3 user IC-CR, channel knowledge isonly required at the relay, while for the 3 user IC, channelknowledge was required at all nodes. The advantage of thisachievable scheme is that the transmitters can choose theirprecoding matrices Vi by random, and the relay constructs thesuitable VRi according to its knowledge of the channels.

Another advantage of this scheme is that 3/2 DOF areachievable without any channel extension (i.e. in one timeslot). Ifwe replace the cognitive relay with a half-duplex relay,we need 2 time slots to achieve 3/2 DOF as shown in sectionVI (and in [3] for time varying channels).

As a corollary to this theorem, we have

Corollary 1. The K user complex Gaussian IC-CR withconstant channel coefficients and with one antenna at eachnode has at least 3/2 DOF

Proof: Simply by shutting off all transmitters but 3, andusing the relay as in a 3 user IC-CR, we achieve 3/2 DOF. •

B. Interference Cancellation

Another scheme that we can use in the IC-CR is interferencecancellation. An interference cancellation scheme for the 2

VR I =(28)

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302010P (dB )

o

lin: = 1,

E~ 10E:::l

(/J 5

N 20

~0.

s: 15.=

[- 2.05 ] v [ - 1.38 ] v: [0.71]- 0.02 ' R2 = - 1.93 ' R 3 = - 1.52

Notice that at high P, the sum rate achieved with in­terference cancellation increases with slope 2, while that ofinterference alignment increases with slope 3/2. This is an in­teresting result, showing that if the relay has knowledge about

Figure 2 shows the directions of the received signals at allthree receivers.

The solid lines denote the desired signals. Notice that thetwo interfering signals are aligned, while the desired signal islinearly independent from the interference. Thus each receiveris able to project its received signal on a direction orthogonalto the interference, and hence achieve one OOF over a twodimensional space, leading to 3/2 OOF in total.

Since this channel satisfies the conditions in Theorem 4, wecan use interference cancellation to achieve 2 OOF. The sumrate achievable by both schemes are plotted in Figure 3 as afunction of P.

IV. N UM ERI CA L EXAMPLE

Consider the 3 user IC with hij = 1 Vi, j E {1,2,3}.Clearly, we can not achieve 3/2 OOF in this channel usingasymmetric complex signaling [8]. Now assume we have acognitive relay, with channels to the receivers given e.g. by:

Corollary 2. The K user fC-CR with one antenna at eachnode satisfying

hi jh i i -I- hRi-h ' Vi -I- j, i ,j E {l , . . . , K }.

Rj

has at least 2 DOF

for example. Then, we can use asymmetric complex signalingto achieve 3/2 degrees of freedom.

We use

VI = [ ~~o~~9 ],V2 = [ ~O~;~5 ], V3 = [ 0~~6 ]

and

Fig. 3. Achievable sum rate using interference alignment and interferencecancellation as a function of P.

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user IC-CR appears in [10]. We focus on the 3 user IC-CR,and propose interference cancellation schemes for this case.

For a family of 3 user IC-CR, a simple interference can­cellation scheme is able to achieve 3 degrees of freedom. Westate the result in the following theorem.

Theorem 4. fn a 3 user fC-CR with single antenna nodes, 2DOF are achievable as long as

hi jn« -I- hRi-h ' Vi -I- j , i , j E {I , 2, :3} . (33)R j

Proof By shutting off one transmitter, the 3 user IC-CRreduces to the 2 user IC-CR, where 2 OOF are achievable[10]. •

As the channel coefficients are random, the conditions inTheorem 4 are satisfied almost surely. As a corollary for theK user IC-CR, we have the following.

Theorem 3. In a K user fC-CR with single antenna nodes,and with coefficients satisfy ing

h i j h ik- -- , Vj, k-l- i , i ,j,k E{l , ... , K },h R j lin»

K degrees offreedom are achievable as long as

h ··b« -I- h R i -, t

J, for some i -I- i . i , j E {I , . .. , K} . (32)

tRj

However, the family of IC-CR in Theorem 3 constitutes aset of channels of measure O. For the more general case, wehave

Theorem 2. fn a 3 user fC-CR, ifh 31 h 32 h 21 h 23

h Rl h R 2 ' h Rl h R 3 '

then 3 DOF are achievable as long as

h l 2 h 21 h 31h l1 -I- lins -h ' h 22 -I- h R2 -h ' h 33 -I- hR3-h . (25)

R 2 R I RI

Proof The achievable scheme uses interference cancel­lation. The relay sends the following signal

n livz n h 21 n h 31 nx R = - - - X l - --X2 - --x 3 ,

h R 2 h R I h R I

users 1, 2, and 3 send x r, x~ , and x~ respectively. Thus, thereceived signals in (1) reduce to

n (' h h 12 ) n nYI = tl1 - RI -h X l + ZlR2

n (' h h21 ) n nY2 = t22 - R 2-h X 2 + Z2RI

n (h h h 31) n nY 3 = 33 - R3 -h X3 + Z3 .RI

Therefore, each user achieves one DOF as long as

bvz h 21 h 31h l1 -I- h Rl-,- , h 22 -I- h R2-,-, h 33 -I- h R3-,- . (30)

tR2 tRI tRI

•We can also generalize this result to the K user IC-CR to

get the following:

Ul +U 3i U 2.. , I

II I

II i

I

~l U I

II, II, II

T I U 3I

U 3 II ~

I ..U2

, t Ul

Fig. 2. The directions of the received signals at receiver I (left), receiver 2 (middle), and receiver 3 (right).

the messages in advance, simple interference cancellation getsalready 2 OOF and thus outperforms interference alignment.Another interesting observation is that the source nodes do notneed any CSI at all for aligning interference, their transmitvectors are chosen randomly in this example.

V. THE 3 USER IC-HOR MODEL

Let us consider now the 3 user IC with a half-duplex relaywith constant channel coefficients. In this case, the relay hasonly causal knowledge of the signals of the sources. Thesesignals are received by the relay in a MAC channel as shownin Figure 4. In this case, the nodes communicate over 2 timeslots. In the first time slot, the source communicate with therelay and the destinations, thus

X l U<::---++---~..----~ YI

X2 Y 2

X 3 Y3

Fig. 4. The 3 user IC with a non-cognitive relay.

In the second time slot, the sources and the relay transmit tothe destinations, so

3

Yj(2) = L hi]Xi(2) + hR]XR(2) + zj(2) . (37)i=l

3

Yj(l) = L hij Xi(l) + Zj(l)i = l

3

Yr(1) = L hiRXi(l) + ZR(1) .i = l

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Proof Consider a 2 time slot protocol. The sources areactive in both time slots, and the relay is active only inthe second. The relaying scheme is an amplify and forwardscheme. So, in the second time slot, the sources send x i(2)and the relay sends (3YR(l ) where (3 E R Thus the receivedsignals in the second time slot can be written as

3

Yj (2) = L [hij Xi(2) + (3hRj hiRXi (1)]+ (3hRj zR(l ) + zj(2) .i = l

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The received signals in the two time slots form the followingvector channel

Thus, the IC-HOR reduces to an 3 user MIMO IC with 2antennas at each node. As long as the channel matrices H ijare invertible, i.e. hij -I- 0, which is true almost surely, weachieve 3/2 OOF as shown in [2]. This is achieved by choosing

where in the nth time slot n E {1,2}, xi (n ) and Yi(n) arethe transmit and received signals by the i th tansmitter andreceiver respectively. We use hiR to denote the channel fromtransmitter i to the relay, and ZR to denote the noise at therelay.

VI. ACHI EVA BLE OOF IN THE IC-HOR

In order to compare the IC-CR and the IC-HOR, we studythe achievable OOF of the IC-HOR. It was shown in [3] thatwith time varying channels, 3/2 OOF are achievable almostsurely. We consider here the quasi-static channel, and we showthat 3/2 OOF are achievable as well using amplify and forwardrelaying as in [3].

Theorem 5. Using interference alignment, 3/2 DOF areachievable in a 3 user IC-HDR with constant channel coeffi­cients almost surely.

where

and

n

Yj = L u.,«, + z..i= l

[h . .

H .. - tJtJ - {3 hiRhRj

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precoding matices Vi and postcoding matrices U, such that

Choose {3 such that h i j + {3h iRhR j == 0, Vi i- j, i.e. {3 ==

-h12/h 1RhR 2 due to the construction of the channel. In thiscase, the received signal becomes

Yj == (h j j + {3h j RhRj)Xj + (3h R j zR + Zj. (47)

Therefore, 3 nOF are achievable over 2 time slots, resultingin 3/2 nOF for this family of channels. •

The result above holds only for channels satisfying (45).However, for an arbitrary channel, only one interfering signalcan be canceled at only one user, leading to less nOF.

[1] T. Cover, A. E. Gamal, "Capacity Theorems for the Relay Channel",IEEE Trans. Inform. Theory, vol. 25, no. 5, pp. 572-584, 1979.

[2] V. R. Cadambe, S. A. Jafar, "Interference Alignment and Degrees ofFreedom of the K-User Interference Channel", IEEE Trans. Inform.Theory, vol. 54, no. 8, pp. 3425-3441, August 2008.

[3] K. Gomadam, V. R. Cadambe, S. A. Jafar, "Approaching the Capacity ofWireless Networks through Distributed Interference Alignment", arXiv:0803.3816, e-print.

[4] H. Ghozlan, H. EI Gamal, and G. Kramer, "The MIMO wireless switch:relaying can increase the multiplexing gain," 2009 IEEE Int. Symp. Inf.Theory, Seoul, Korea, pp. 1448-1452, June 28-July 3, 2009.

[5] V. R. Cadambe, S. A. Jafar, "Can Feedback, Cooperation, Relays andFull Duplex Operation Increase the Degrees of Freedom of WirelessNetworks? ", Proc. of IEEE International Symposium on InformationTheory (ISIT) 2008.

[6] S. A. Jafar, M. Fakhereddin, "Degrees of Freedom for the MIMOInterference Channel", IEEE Transactions on Information Theory, July2007, Vol. 53, No.7, Pages: 2637-2642.

[7] S. A. Jafar, S. Shamai, "Degrees of Freedom Region for the MIMO XChannel", IEEE Transactions on Information Theory, Vol. 54, No.1,Jan. 2008, Pages: 151-170.

[8] V. Cadambe, S. A. Jafar, C. Wang, "Interference alignment withasymmetric complex signaling, Settling the Host-Madsen NosratiniaConjecture", e-print, arxiv:0904.0274.

[9] V. R. Cadambe, S. A. Jafar, "Degrees ofFreedom of WirelessNetworks­What a Difference Delay Makes", Proceedings of Asilomar Conferenceon Signals, Systems and Computers 2007.

[10] S. Sridharan, S. Vishwanath, S. A. Jafar, S. Shamai, "On the Capacityof Cognitive Relay assisted Gaussian Interference Channel", in Proc.of IEEE International Symposium on Information Theory, ISIT 2008,Toronto, Ontario, Canada, July 2008.

supported by a single antenna cognitive relay. An interfer­ence alignment scheme with asymmetric complex signalingis proposed for the three user complex Gaussian interferencechannel with a cognitive relay. With this scheme, only onetime slot is required to achieve interference alignment, wherethe users and the relay jointly align interference, leaving onereal dimension free of interference at each receiver.

We have shown that interference cancellation performsbetter than interference alignment in the 3 user interferencechannel with a cognitive relay from nOF point of view.

However, if we replace the non-causal cognitive relay witha causal half-duplex relay, this advantage of interferencecancellation over interference alignment is lost. In this case,3/2 nOF are achievable using interference alignment in twotime slots almost surely, while interference cancellation cannot achieve 3/2 nOF almost surely.

As future work, we work on answering the followingquestions: Can we achieve K degrees of freedom for allK user interference channels with a cognitive relay? On theother hand, can we always achieve K /2 nOF in a K usercomplex Gaussian interference channel with a cognitive relayand constant channel coefficients using interference alignmentwith asymmetric complex signaling?

REFERENCES

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Vi i- j

i,jE{1,2,3}

Uf Hij Vi == 0,

UiH u«Vi i- 0,

This follows since the sources communicate with the relay ina MAC with single antenna nodes. Notice also that if we arewilling to cancel the interference, the relay should not alterits received signal in a way that changes Xi. This is becausereceiver i should receive axj from the relay in order to cancelthe effect of x j received from source j, where i i- j anda E JR. Thus, while using interference cancellation, we willstick to the amplify and forward relay to see if we can achievemore that 1 nOF.

Theorem 6. If the IC-HDR satisfies

hij/hiRhRj == hjk/hjRhRk, Vi i- j,j i- k, i,j, k E {I, 2, 3}(45)

then 3/2 DOF are achievable using interference cancellation.

Proof: Let us also consider communication over 2 timeslots. User i transmits Xi in both time slots. As a result, thereceived signal in the second time slot is given by

3

Yj == L hijXi + {3h iRhRj Xi + {3hRjZR + Zj. (46)i=l

•Interestingly, contrary to the IC-CR, some cooperation is

needed here between the sources and the relay in order toalign interference. Thus, both the sources and the relay musthave CSI in order to generate the desired beam directions forinterference alignment.

Notice that if we are hoping to achieve more than 1 nOF,we should not use decode and forward at the relay. This isbecause if the relay needs to decode the messages, then anOF restriction results, i.e.

3

Ldi :S 1.i=l

VII. CONCLUSION

While in the 3 user complex Gaussian interference channelwith constant channel coefficients and with single antenna ateach node, interference alignment with asymmetric complexsignaling achieves 3/2 nOF for a set of channels of measurezero, it achieves 3/2 nOF almost surely if the users are being