Interference Alignment in MIMO InterferenceRelay Channels†

Xiang Chen, S.H. Song, and Khaled Ben Letaief, Fellow, IEEEDept. of Electronic and Computer Engineering, HKUST, Kowloon, Hong Kong

Email: {benxchen, eeshsong, eekhaled}@ust.hk

Abstract—The degrees of freedom (DoF) has been recognizedas a powerful metric to characterize the capacity of interferencechannels in the high signal-to-noise (SNR) region. In this paper,by utilizing linear interference alignment, we investigate the DoFof multiple-input and multiple-output (MIMO) interference relaychannels without symbol extensions. An innovative algorithm ispresented to align the interference, where the filter matricesat the sources, relays and destinations are determined in aniterative manner. Based on the assumption that improperness ofthe alignment condition implies its unsolvability, an upper boundfor the achievable DoF tuple by linear interference alignment isderived, and then utilized to examine the performance of theproposed alignment algorithm. Simulation results show that theiterative algorithm can achieve the upper bound in medium tohigh DoF regions.

I. INTRODUCTION

The capacity characterization of a general interference chan-nel has remained an open problem for decades. Recently,the degrees-of-freedom (DoF) metric has been receiving sig-nificant attention as it provides a method to characterizethe capacity in the high signal-to-noise (SNR) region. Manyinvestigations have been conducted to determine the achievableDoF for different interference networks, as well as to achievethe DoF by interference alignment [1], [2], [3]. In [4], [5], itwas shown that interference alignment can also be utilized toattain partial diversity gains.

The K-user time-varying interference channel can almostsurely achieve K/2 DoF, by utilizing interference alignmentwith infinite symbol extensions [1]. When the nodes areequipped with multiple antennas and symbol extension isallowed, the achievable DoF is approximately multiplied bythe number of antennas per node [3]. This indicates that the ca-pacity grows linearly with the number of users and the numberof antennas in the high SNR region for the multiple-input andmultiple-output (MIMO) interference channels. When symbolextension is not implemented, interference can only be alignedin the spatial domain with MIMO transmissions. For suchMIMO interference channels, neither the exact achievable DoFnor an optimal alignment algorithm is available for the generalcase. An upper bound for the achievable DoF was obtained in[6], [7], [8], where the upper bound does not grow linearly withthe number of users. An iterative algorithms was proposed forthe interference alignment with a given DoF tuple in [9], where

†This work is supported by the Hong Kong Grant Council under Grant #610311.

the algorithm cannot guarantee the convergence to the globaloptimum.

In the single-hop interference channels, the best we can dois to allocate half of the signal space to the desired signalsand the other half to the interference when utilizing symbolextensions. Without symbol extensions, the interference canonly be aligned in the spatial domain, and is more difficult toalign due to the constraint of the limited antenna numbers. Infact, the sum achievable DoF becomes saturated when the pairof the users increases [6], [7], [8]. Such limitation in the single-hop channels can be overcome by employing parallel relays.For such interference relay channels, partial interference canbe canceled or aligned by the relays, and the dimensionof the remaining interference at the destinations is reduced.Therefore, the DoF saturation can be avoided when we haveenough relays, and a much higher DoF can be attained.

Although the relays provide additional freedoms to handleinterference, they also render the optimization more compli-cated and the DoF more difficult to characterize. For decode-and-forward (DF) relaying, a two-hop parallel relay networkcan be treated as two cascaded X channels, and the full DoFcan be achieved when the number of relays increases to infinity[2]. For amplify-and-forward (AF) relaying, [10] considereda simple single-antenna case with 2 sources, 2 relays and 2destinations, and the min-cut outer bound of DoF was shownto be 2 with symbol extensions. With more than two pairs ofusers, [11] investigated the DoF region of a class of multi-hopinterference relay channels, where the results relied on a veryspecific relation between the number of hops and the numberof relays in each hop. For a general MIMO interferenceparallel AF relay networks, with or without symbol extensions,the results on the DoF and interference alignment algorithmare still not available in the literature.

In this paper, we consider a general MIMO interferencerelay channels without symbol extensions. All the sources,relays and destinations are equipped with multiple antennas.Each source communicates with its corresponding destinationthrough relays. To investigate the DoF performance for theconcerned interference channels, an iterative algorithm is firstproposed to determine the filter matrices at all nodes to attaina given feasible DoF tuple. Then, based on the assumptionthat improperness of the alignment condition implies its un-solvability, an upper bound of the achievable DoF tuple isderived for the MIMO interference relay channels. It mustbe noted that the above assumption was proved valid for

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conventional MIMO interference channels by [7], [8], but isunclear for the concerned relay channels in general. So far,we did not witness any counter examples to the assumptionduring our extensive simulations. For the special case whereeach S-D pair has an equal number of antennas and achievesthe full DoF, it is shown that the upper bound is the exactachievable DoF. Another observation is that in the medium tohigh DoF regions, the achieved DoF by the proposed iterativealignment algorithm is the same as the derived upper bound,which indicates that the optimal DoF tuple is achieved by theproposed algorithm in these regions.

The rest of this paper is organized as follows. In SectionII, the system model is introduced. The iterative algorithmfor the interference alignment is presented in Section III. Theupper bound of the achievable DoF tuple is provided in SectionIV. Numerical results are shown in Section V, and the paperconcludes with Section VI.

II. SYSTEM MODEL

Consider a K1×K2×K1 MIMO interference relay channelas shown in Fig. 1, where K1 sources transmit informa-tion to their dedicated destinations, respectively, through K2

relay nodes. Source Sm, relay Rl, and destination Dn areequipped with Mm, Ll, Nn antennas, respectively, wherem, n = 1, 2, · · · ,K1 and l = 1, 2, · · · ,K2. Direct links fromthe sources to the destinations are not considered. The relaysoperate in a full-duplex AF mode, which makes an easier com-parison with single hop interference channels. However, theinterference between different hops is assumed to be handledby other techniques and is not considered [10], [11]. Half-duplexing will simply cost half of the achievable DoF. Channelextension in the time and frequency domains is not applicable.Assume that all the sources, relays, and destinations have thechannel knowledge of the whole system [1], [7], [11].

S1

S2

SK1

R1

R2

RK2

D1

D2

DK1

Figure 1. System model.

In the first hop, the received signal at the relay Rl is givenby

yl,1 =

K1∑m=1

FlmVmsm + nl,1, (1)

where sm ∈ Cdm×1 denotes dm independently encoded datastreams transmitted from Sm and is normalized such that

E{smsHm} = I, Vm ∈ CMm×dm denotes the linear precodingfilter at Sm, and nl,1 ∼ CN (0, I) represents the circularlysymmetric additive white Gaussian noise (AWGN) vector withzero mean and unit variance. The channel coefficient matrixfrom Sm to Rl is denoted by Flm. In the second hop,the relays amplify their received signals and forward themto the destinations. The relaying filter at Rl is denoted byWl ∈ CLl×Ll . It should be noted that we put no powerconstraints on Wl, since it will not affect the feasibility ofinterference alignment and the achievable DoF. The receivedsignal at the destination Dn is given by

yn,2 =

K2∑l=1

GnlWlyl,1 + nn,2, (2)

where Gnl represents the channel coefficient matrix from Rl

to Dn, and the noise vector nn,2 is defined similarly as nn,1.The zero-forcing filter at Dn is denoted by Un ∈ CNn×dn .

Define the equivalent channel matrix from Sm to Dn asHnm ,

∑K2

l=1 GnlWlFlm. All channel matrices Flm andGnl are assumed to be generic. Then, the linear interferencealignment conditions are described by [6]

UHn HnmVm = 0, ∀m 6= n and (3)

rank(UH

n HnnVn

)= dn, ∀n. (4)

Condition (3) guarantees that all the interfering signals atDn are aligned in a subspace of Nn − dn dimensions, andcan be zero-forced by Un. Condition (4) guarantees that Dn

is able to decode all dn intended data streams successfully.When both conditions (3) and (4) can be satisfied, linearinterference alignment is feasible for the given DoF tuple(d1, d2, · · · , dK1

). For single-hop systems, it has been provedthat condition (4) is automatically satisfied almost surelyif the channel matrices do not have any special structure[6], [9]. For our concerned relay systems, the equivalentchannel Hnn =

∑K2

l=1 GnlWlFln contains the matrix to beoptimized (Wl), which may jeopardize the generic structure.However, with the proposed iterative algorithm, the simulationresults show that condition (4) is satisfied for most of thesettings because the channel matrices Fln and Gnl are generic.Also, losing condition (4) will not affect the upper-boundingproperty of the bound obtained in Section IV. Therefore, inthe following derivations, we only consider the feasibility ofcondition (3).

III. ITERATIVE ALGORITHM FOR INTERFERENCEALIGNMENT

In this section, an iterative algorithm will be presented todetermine the filters of all nodes to achieve a given DoF tuple(d1, d2, · · · , dK1

), which also provides an alternative way toverify the feasibility of the interference alignment.

A. Interference Leakage

The total interference leakage at Dn due to all undesiredsources is given by [9]

In = Tr{UH

n QnUn

}, (5)

631

where

Qn =

K1∑m=1,m6=n

1

dmHnmVmVH

mHHnm (6)

is the interference covariance matrix at Dn. Notethat In in (5) is nonnegative, and so are its termsTr{UH

n HnmVmVHmHH

nmUn

}. Therefore, the condition

In = 0 is equivalent to the zero-forcing condition (3) at Dn.If a group of {Vm,Un,Wl} can be found such that the suminterference leakage at all destinations is zero, then conditionin (3) is satisfied and interference alignment is feasible.Therefore, we formulate the optimization problem as follows:

min{Vm,Un,Wl}

K1∑n=1

In

s.t. VHmVm = Idm , UH

n Un = Idn , ∀m,n,(7)wHw = 1,

where w =[

vec {W1}T · · · vec {WK2}T

]Tand

vec(·) denotes vectorization of a matrix. The constraints givenin (7) will naturally lead to the proposed iterative algorithm,and will not affect the feasibility of the interference alignmentand the achievability of DoF [9]. Individual constraints on eachWl are not necessary, because the interference is either zeroor nonzero for the concerned feasibility problem.

B. Iterative AlgorithmAn alternating minimization procedure has been proposed

in [12]. The idea is to minimize the objective function withrespect to one variable at a time while keeping the othersfixed. Therefore, we will obtain {Un}, {Vm} and {Wl} inthe following three steps, respectively.

1) Zero-forcing Filters {Un}: By fixing {Vm} and w, theoptimization problem in (7) can be written as

min{Un}

∑K1

n=1 In =∑K1

n=1 Tr{UH

n QnUn

}s.t. UH

n Un = Idn, ∀n.

(8)

It can be observed that In only contains Un for each n =1, · · · ,K1, and thus In can be minimized individually by thecorresponding Un. Therefore, the columns of the optimal Un

are formed by the dn least dominant eigenvectors of Qn, andthe minimum In is equal to the sum of the dn least dominanteigenvalues of Qn [13].

2) Precoding Filters {Vm}: The sum interference leakageat all destinations can be reshaped as

K1∑n=1

In =

K1∑n=1

Tr{UH

n QnUn

}=

K1∑m=1

Tr{VH

mQmVm

}=

K1∑m=1

Im, (9)

where

Qm =1

dm

K1∑n=1,n6=m

HHnmUnU

Hn Hnm. (10)

Similarly, by fixing {Un} and w, we can prove that for allm = 1, · · · ,K1, the columns of the optimal Vm are formed bythe dm least dominant eigenvectors of Qm, and the minimumIm is equal to the sum of the dm least dominant eigenvaluesof Qm.

3) Relaying Filters {Wl}: By fixing {Un} and {Vm}, theoptimization problem in (7) can be written as

minw

wH(∑K1

n=1

∑K1

m=1,m6=n Anm

)w

s.t. wHw = 1,(11)

where Anm is given in (12) on the top of next page, and ⊗denotes the Kronecker product of two matrices. The optimal wis the least dominant eigenvector of

∑K1

n=1

∑K1

m=1,m6=n Anm,and the minimum sum interference leakage is equal to theleast dominant eigenvalue of

∑K1

n=1

∑K1

m=1,m6=n Anm. Then,the optimal {Wl} can be determined by the optimal w. Wheninterference alignment is feasible, the objective function in(11) is minimized and equal to zero. Because each term ofwHAnmw is nonnegative, wHAnmw = 0 for all m 6= nwhen interference alignment is attained.

The iterative procedure is summarized in Algorithm 1 asfollows.

Algorithm 1 : Iterative Algorithm for Interference Alignment1) Randomly generate {Vm} and {Un}, such that

VHmVm = Idm

and UHn Un = Idn

, ∀m,n.

2) Calculate the optimal {Wl} and update all filters.If∑K1

n=1 In = 0, stop iteration.

3) Calculate the optimal {Vm}and update all filters.If∑K1

n=1 In = 0, stop iteration.

4) Calculate the optimal {Un}and update all filters.If∑K1

n=1 In = 0, stop iteration.

5) Go to step 2).

C. Convergence and Optimality

Because each step in Algorithm 1 will reduce the suminterference leakage, which is bounded above by zero, theconvergence is guaranteed. However, convergence to the globalminimum cannot be guaranteed. The simulation results inSection V suggest that Algorithm 1 may achieve the globalminimum for medium to high DoF regions.

The above iterative algorithm can be naturally employed todetermine the filters for a feasible DoF tuple. On the otherhand, it can also be used to check the feasibility of a DoFtuple under a given system setting. Therefore, we can obtaina lower bound of the achievable DoF tuple numerically.

IV. UPPER BOUNDING THE ACHIEVABLE DOF TUPLE

In order to evaluate the DoF performance of Algorithm 1,an upper bound of the achievable DoF tuple is derived in thissection.

632

Anm =1

dm

(F∗1mV∗mVT

mFT1m

)⊗(GH

n1UnUHn Gn1

)· · ·

(F∗1mV∗mVT

mFTK2m

)⊗(GH

n1UnUHn GnK2

)...

. . ....(

F∗K2mV∗mVT

mFT1m

)⊗(GH

nK2UnU

Hn Gn1

)· · ·

(F∗K2m

V∗mVTmFT

K2m

)⊗(GH

nK2UnU

Hn GnK2

) . (12)

A. Properness of Polynomial Systems

To analytically determine the solvability of the polynomialsystem in condition (3), which implies the feasibility of thelinear interference alignment, we first provide the definitionof properness and improperness of a polynomial system [6].Consider a polynomial system

f1 = 0, · · · , fP = 0, (13)

where f1, · · · , fP are polynomial functions in multiple vari-ables x1, · · · , xQ, and the constant terms of f1, · · · , fP areall zeros. If the number of equations is less than the numberof variables, i.e., P < Q, the polynomial system in (13) iscalled proper. Otherwise, the system is improper. Thus, thepolynomial system in (3) is proper if and only if∑m∈K

dm (Mm +Nm − 2dm) +

K2∑l=1

L2l >

∑m,n∈K,m6=n

dmdn

(14)for all subset K ⊆ {1, 2, · · ·K1}, where the left term of (14)is the number of variables, and the right term is the numberof equations.

By considering the information theoretic outer bounds,a feasible interference alignment also needs to satisfy thefollowing two constraints:

min {Mm, Nm} ≥ dm, ∀m, (15)

andK2∑l=1

Ll ≥K1∑m=1

dm. (16)

(15) is straightforward because of (4). (16) is due to the factthat the sum DoF of all users should not exceed the DoF ofthe channels.

B. Feasibility of Interference Alignment

If the assumption that, improperness of the polynomialsystem in (3) implies its unsolvability, is correct, then (14-16)provide a true upper bound for the achievable DoF tuple. Forthe conventional MIMO interference channels, this assumptionwas proposed by [6] and proved by [7], [8]. Unfortunately, forthe concerned interference relay channels, only the followingfull-DoF case in Proposition 1 can be proved. However, wewere not able to observe any feasible system to be improperin the simulations. It is therefore safe to state that the obtainedbound can well indicate a rough region of the DoF tuple, andthe insight will be useful.

If dm = Mm = Nm for all m = 1, · · · ,K1, where eachpair of users achieves their maximum DoF, then no signalspace is left to align the interference at the destinations.

Thus, all interference must be canceled by the relays. Inthis case, the polynomial system in (3) reduces to a groupof homogeneous linear equations, and (14-16) provide theexact value of the achievable DoF. Specifically, we have thefollowing proposition.

Proposition 1. When dm = Mm = Nm for all m, themaximum sum DoF of

∑K1

m=1 Mm is achievable if the relayconfiguration satisfies the following two conditions:∑K2

l=1 L2l >

K1∑n=1

K1∑m=1,m6=n

MmMn,

and∑K2

l=1 Ll ≥∑K1

m=1 Mm.

(17)

When Mm = Nn = Ll = J and dm = J for all m, n =1, 2, · · · ,K1 and l = 1, 2, · · · ,K2, (17) can be simplified as

K2 ≥ K1 (K1 − 1) + 1, (18)

agrees with the J = 1 case in [14]. The condition in (18)indicates that in order to achieve full DoF for each user, thesystem requires at least K1 (K1 − 1) + 1 number of relays.

V. NUMERICAL RESULTS

In this section, numerical results will be presented toillustrate the accuracy of the lower and upper bounds obtainedin Section III and IV, respectively. For simplicity, all nodesare assumed to be equipped with J antennas. For this equal-antenna settings, it is optimal to evenly allocate the DoFto each active S-D pair. An S-D pair allocated with zeroDoF will not transmit, and will not cause any interference.After exhaustively searching different system settings with theproposed iterative algorithm, lower bounds can be obtainednumerically. The upper bounds are determined by (14-16).

In Fig. 2, a system with 4 pairs of users is consideredfor different number of antennas. It can be observed that formedium to high DoF regions, the lower and upper boundsare the same, which indicates that the iterative algorithm inSection III achieves the optimal DoF. For the low DoF region,the lower bound is not far from the upper bound. Comparedto the conventional single-hop MIMO interference channel,in which the sum DoF is no more than 2JK1

K1+1 , the MIMOinterference relay channel can achieve a much higher DoF.

Fig. 3 compares the systems with the same number of relays.It can be observed that the sum DoF is limited by the relaysand becomes saturated when K1 increases. It also shows thatthe lower and upper bounds will bifurcate when K1 increases.This is because when K1 is large, the DoF of each userapproaches to one or zero, and the upper bound is not accuratefor low DoF regions. The non-decreasing curves also suggestthat shutting down partial S-D pairs will not increase the sumDOF under the simulated scenarios.

633

The algorithm in this paper can also be used to determinethe minimum required number of relays and antennas per relay.Fig. 4 illustrates the relation between the required number ofrelays and the target sum DoF. Assume that each S-D pairat least achieves one DoF. It can be observed that when Jincreases, there is no significant growth in the required numberof relays. In fact, when all S-D pairs achieve the full DoF, therequired K2 is always 13, which is equal to K1 (K1 − 1)+1.Note that this agrees with the Proposition 1 in Section IV.

0 2 4 6 8 10 12 142

4

6

8

10

12

14

16

18

20

K2

Sum

DoF

Upper Bound

Lower BoundJ = 5

J = 4

J = 3

J = 2

Figure 2. Sum DoF versus number of relays, K1 = 4, Mi = Ni = Li = J .

0 2 4 6 8 10 12 142

4

6

8

10

12

14

K1

Sum

DoF

Upper BoundLower Bound

J = 4

J = 3

J = 2

Figure 3. Sum DoF versus number of S-D pairs, K2 = 5, Mi = Ni =Li = J .

VI. CONCLUSIONS

In this paper, we investigated linear interference alignmentand the achievable DoF of MIMO interference relay channels.First, an iterative algorithm was designed to determine theprecoding filters, amplifying filters and zero-forcing filters forthe sources, relays and destinations, respectively, to achievea given DoF tuple. Then, based on the assumption that im-properness of the alignment condition implies its unsolvability,an upper bound of the achievable DoF tuple was derived. Theupper bound was utilized to examine the DoF performanceof the proposed algorithm, and can determine the minimumrequired number of relays and number of antennas per relayfor a given DoF tuple. Numerical results demonstrated that the

4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

Sum DoF

Req

uire

d K

2

Iterative AlgorithmEqs. (14 − 16)

J = 2

J = 3

J = 4

J = 5

Figure 4. Required number of relays versus target sum DoF, K1 = 4,Mi = Ni = Li = J .

proposed iterative alignment algorithm can achieve the optimalDoF in medium to high DoF regions. It was also shown that byproviding additional freedoms to handle interference, the con-sidered relaying channels are able to avoid the DoF saturationof conventional single-hop MIMO interference channels.

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