Interference Alignment Schemes Using Latin Square for K 3 ...ccl.snu.ac.kr/papers/journal_int/journal2018_12_1.pdf · structure by the Latin square is proposed for K 3 multiple-input

Received November 14, 2018, accepted December 13, 2018, date of publication December 18, 2018,date of current version January 11, 2019.

Digital Object Identifier 10.1109/ACCESS.2018.2888501

Interference Alignment Schemes Using LatinSquare for K × 3 MIMO X ChannelYOUNG-SIK MOON 1, JONG-YOON YOON1, JONG-SEON NO 1, (Fellow, IEEE),AND SANG-HYO KIM 2, (Member, IEEE)1The Department of ECE, INMC, Seoul National University, Seoul 08826, South Korea2The Department of ICE, Sungkyunkwan University, Suwon 16419, South Korea

Corresponding author: Young-Sik Moon ([email protected])

This work was supported by the National Research Foundation of Korea (NRF) through the Korean Government (MSIP) under GrantNRF-2016R1A2B2012960.

ABSTRACT In this paper, we study an interference alignment (IA) scheme with finite time extensionand beamformer selection method with low computational complexity for X channel. An IA scheme witha chain structure by the Latin square is proposed for K × 3 multiple-input multiple-output X channel.Since the proposed scheme can have a larger set of possible beamformers than the conventional schemes,its performance is improved by efficient beamformer selection for a given channel realization. Also,we propose a condition number-based beamformer selection method with low computational complexityand its performance improvement is numerically verified.

INDEX TERMS Beamforming, degrees of freedom (DoF), interference alignment (IA), latin square,multiple-input multiple-output (MIMO), X channel.

I. INTRODUCTIONInterference alignment (IA) is an important technique toman-age interference in the wireless communication networks.To resolve the interference problem, an interference align-ment schemewas recently proposed and has become a subjectof special interest in the area of wireless communications.Cadambe and Jafar [1] showed that each user in a multi-user interference channel can utilize half of all the networkresources, which corresponds to achieving the maximumdegrees of freedom (DoF). The key idea of this result isthe IA, which maximizes the overlap of all interferencesignal spaces at each receiver so that the dimension of theinterference-free space for the desired signals is maximized.

The IA has been further studied for various communicationenvironments [2]–[11]. There are studies focusing on beam-forming and precoding for IA with multiple antennas. Linearnetwork coding for interference alignment was consideredin [2], referred to as the precoding based network alignmentscheme (PBNA). On the other hand, there are works relatedto the channel state information at the transmitter (CSIT)used for IA. A blind interference alignment (BIA) schemeusing the reconfigurable antenna was proposed in [3]. In [5],the generalized degrees of freedom (GDoF) characterizationof the symmetric K user interference channel was obtainedunder finite precision CSIT. In addition to the problems

related to CSIT for IA, high computational complexity hasbeen a challenge in IA implementation. Low complexity non-iterative interference alignment (IA) schemes were investi-gated in [4]. Also, IA is studied to mitigate the interferenceproblem that occurs D2D communication in the heteroge-neous network [7], [8]. In [9], an IA scheme for M × N Xchannel was proposed and it was proved that the maximumDoF of theM ×N X channel isMN/(M +N − 1). However,an infinite time extension is required to achieve this maxi-mum DoF. In [10], an IA scheme for K × K X channel wasproposed without channel extension, where the beamformingvectors were constructed only by a spatial signature over unittime. Many types of research for the IA focus on proposingIA conditions to align the interferences and obtaining DoF forthe communication networks and there are several beamform-ing vectors satisfying the IA condition of the IA scheme. TheIA scheme to select a good beamforming vector in terms ofsum rate by using linear precoder design was studied in [11].

Although most of the studies for IA focus on networkthroughput, that is, DoF, reliability is also an important per-formance measure of wireless communication systems. Forreliability, various selection schemes have been studied suchas transmit antenna selection or equivalent path selectionbased on singular value decomposition (SVD). Since signalto noise ratio (SNR) at the receiver is the most critical factor,

43482169-3536 2018 IEEE. Translations and content mining are permitted for academic research only.

Personal use is also permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

VOLUME 7, 2019

https://orcid.org/0000-0002-8760-4119

https://orcid.org/0000-0002-3946-0958

https://orcid.org/0000-0002-0660-5516

Y.-S. Moon et al.: IA Schemes Using Latin Square for K × 3 MIMO X Channel

FIGURE 1. K × 3 MIMO X channel.

most of the selection schemes for IA are based on SNR atreceiver. However, calculating accurate received SNR at thetransmitter for beamformer design is practically draws highcomputational complexity.

In this paper, we study an IA scheme with finite timeextension and beamformer selectionmethodwith low compu-tational complexity for X channel. An IA schemewith a chainstructure by the Latin square is proposed for K × 3 multiple-input multiple-output (MIMO) X channel. The proposed IAscheme uses Latin squares to determine how to pair inter-ference signals and assign a single dimension. By aligningthe interference signal pairs determined by the Latin square,the proposed IA scheme has the expanded beamformingvector sets much larger than the conventional IA schemes.Therefore, the proposed IA scheme can find a beamformingvector set that is better suited for a given channel realization.Also, we propose a condition number (CN) based beam-former selection method with low computational complexityand its performance improvement is numerically verified.

The rest of this paper is organized as follows: In Section II,the system model of K × 3 MIMO X channel is described.Then, an expanded beamformer set using Latin square for theK × 3 MIMO X channel is proposed in Section III. A coupleof efficient beamformer selection methods are proposed inSection IV and its performance is numerically analyzed inSection V. Finally, conclusion is given in Section VI.

II. SYSTEM MODELWe consider the K × 3 MIMO X channel as in Fig. 1, whereeach transmitter j, j = 1, . . . ,K , transmits an independentmessage to each receiver i, i = 1, 2, 3. All nodes are equippedwith M = 2K antennas. The 2K × 1 transmit signal vectorxj from the j-th transmitter can be represented as a linearcombination of three different beamforming vectors

xj =3∑i=1

vijsij, (1)

where sij denotes a transmitted message from thej-th transmitter to the i-th receiver and vij denotes a 2K × 1beamforming vector for the message sij. The transmit signalvector xj has an average power constraint of Tr(xHj xj) ≤ Pj,

where Pj is the total transmit power of the j-th transmitter,Tr(·) denotes the trace function, and (·)H indicates the Her-mitian transpose.

Then, the received signal vector at the i-th receiver is givenas

Yi =K∑j=1

Hijvijsij +K∑j=1

3∑k=1k 6=i

Hijvkjskj + ni, (2)

where Hij is the 2K × 2K channel matrix from the j-th trans-mitter to the i-th receiver generated by identically distributedflat Rayleigh fading over one symbol period from a complexGaussian random variable with zero mean and unit varianceand ni is the circularly symmetric white Gaussian noise withzero mean and unit variance at the receiver i.

The first term and the second term in (2) denote thedesired signal and the interference signal at the receiver i,respectively. Clearly, the desired signal is composed of Kreceived signals and the interference signals are composedof 2K received signals.We consider the zero-forcing (ZF) decoder to remove the

interference signal at the receiver i. The ZF based decoder canseparate K × 3 X channel into 3K point-to-point channels.Therefore, the sum-rate of the proposed IA scheme is givenas

Rsum =3∑i=1

K∑j=1

log2(1+ PjR†ijHijvijv

†ijH

†ijRij),

where R†ij is ZF decoder for the desired symbol sij and Pjdenotes the transmit power at the transmitter j. It is wellknown that PjR

†ijHijvijv

†ijH

†ijRij is considered as the SNR of

the desired signal sij at the receiver i.

III. EXPANDED BEAMFORMER SET USING LATIN SQUAREFOR K × 3 MIMO X CHANNELIn this section, we propose an interference alignment schemefor K × 3 MIMO X channel, which can achieve DoF 3K .At each receiver, DoF M

2 (= K ) can be achieved, whichmeansthat the interference signals are aligned in the K dimensionalsubspace.

A. REQUIREMENTS ON THE BEAMFORMING VECTORSWe propose three requirements on the beamforming vectorsin K × 3 MIMO X channel as follows.i) The 2K interference signals at each receiver should be

aligned in M2 (= K ) dimensional signal space as

span(Hijvkj) = span(Himvlm), (3)

where i, j, k, l, and m are all distinct, i = 1, 2, 3.ii) Any two received interference signals from the same

channel should not be aligned along the same dimensionalsignal space as

span(Hijvkj) 6= span(Hijvlj), (4)

where i, k, and l are all distinct, i = 1, 2, 3.

VOLUME 7, 2019 4349


iii) Each interference signal pair at each receiver should bealigned as a chain structure, that is,

span(H1α1vδ1 ) = span(H1α2vδ2 ) for receiver 1

span(H2β1vδ2 ) = span(H2β2vδ3 ) for receiver 2 (5)

span(H3γ1vδ3 ) = span(H3γ2vδ1 ) for receiver 3,

where the requirement iii) can be satisfied in the proposed IAscheme by the Latin square and there are K chain structures.In the proposed IA scheme, the design of beamforming

vectors are subject to three requirements. At receiver i, halfof the antenna dimension is allocated to the desired signal,i.e., M

2 (= K ). The proposed IA scheme is feasible if 2Kinterference signals are aligned in the space of the other halfof the received signal space. By the requirement i), the 2Kinterference signals received by each receiver are alignedin the K dimensional signal space, which is a half of thesignal space of each receiver. According to the requirement i),the proposed IA scheme should align two interference signalsin one dimension. Therefore, the constraint on the antennaconfiguration is M = 2K .The requirement ii) means that two beamformers experi-

encing the same channel are not aligned in the same direc-tion. If the requirement ii) is not satisfied, the correspondingbeamformer will be aligned in the same signal space as thedesired signal at the other receiver. When two interferencesignals satisfying the requirements i) and ii) are aligned in thesame signal space, the available set of beamformers for the IAscheme can be expanded by using the chain structure in (5).In the next section, we will discuss the design of beamformersto satisfy the above three requirements.

B. DESIGN THE BEAMFORMING VECTORS USINGLATIN SQUAREFirst, we introduce the Latin square for beamformer design.The Latin square is a K × K array filled with K differentsymbols, each occurring exactly once in each row and exactlyonce in each column. For example, if the first row is fixed as[ABC], two different 3× 3 Latin squares are given asA B C

B C AC A B

,A B CC A BB C A

.Let V be the K × 3 beamforming matrix given as

V =

v11 v21 v31v12 v22 v32...

......

v1K v2K v3K

,where vij is a 2K × 1 beamforming column vector for signalfrom the j-th transmitter to the i-th receiver. The i-th col-umn of V represents the beamforming vectors of the signalstransmitted to the i-th receiver. That is, in the i-th receiver,the remaining two columns except for the i-th column repre-sent the beamforming vectors of 2K interference signals. Bythe requirement i), 2K interference signals in each receiver

should be aligned in K dimensions so that two interferencesignals should be aligned in the same dimensional signalspace. By the requirement ii), the beamformers in the samerow should not be aligned in the same signal space becausethey experience the same channel to the i-th receiver. Also,the beamformers in the same column should not be alignedin the same signal space to have a chain structure of therequirement iii).

Considering these conditions, the method to make K pairsof the 2K interference signals, where each pair of interferencesignals is aligned into one signal space, is proposed by usingthree columns in the K × K Latin square, where one columncorresponds to the desired signal and the other two columnscorrespond to the interference signals. Since the order ofaligned interference signals is irrelevant, the order of symbolsin the first row of the K × K Latin square can be fixed.Thus, the total number of K × K Latin squares with thefirst row fixed is (K − 1)!L(K ,K ), where L(K ,K ) is thenumber of reduced Latin squares of size K referenced in theonline encyclopedia of integer sequences (OEIS) A000315,that is, L(3, 3) = 1, L(4, 4) = 4, L(5, 5) = 56, L(6, 6) =9408, etc. [22]. The interference alignment pairs of the beam-forming vectors are obtained from three columns in theK×KLatin square with the first row fixed and thus the total numberof IA schemes is lower bounded by (K − 1)!L(K ,K ).

For simplicity, we design the beamforming vectors for oneof the available IA schemes as follows. In each IA schemewith K chain structures, each chain structure in (5) is rewrit-ten as

span(H−13γ2H3γ1H

−12β2H2β1H

−11α2H1α1vδ1 ) = span(vδ1 )

vδ2 = H−11α2H1α1vδ1

vδ3 = H−13γ1H3γ2vδ1 , (6)

where vδ1 is obtained from the eigenvectors of H−13γ2H3γ1H

−12β2

H2β1H−11α2H1α1 and vδ2 and vδ3 can be obtained as in (6).

Since the size of H−13γ2H3γ1H

−12β2H2β1H

−11α2H1α1 is 2K × 2K ,

vδ1 can be selected among 2K eigenvectors. Since there areK chain structures, we have (2K )K eigenvector sets, whereeach eigenvector set is composed of K eigenvectors. Let Bbe the beamformer set for each IA scheme and let b be thebeamforming vector set, where each element consists of 3Kbeamforming vectors as

b = {v11, v21, v31, · · · , v1K , v2K , v3K } ∈ B.

In each IA scheme, there are (2K )K beamforming vector setsderived from (2K )K eigenvector sets, that is, |B| = (2K )K .Let LK be the expanded beamformer sets by the Latin squares,where each element is a beamformer set Bi. The proposedIA scheme for K × 3 MIMO X channel has at least (K −1)!L(K ,K ) available IA schemes and thus we have at least(K − 1)!L(K ,K ) beamformer sets as

LK = {Bi|i = 1, · · · , (K − 1)!L(K ,K )}. (7)

In fact, (K − 1)!L(K ,K ) is large enough for the num-ber of IA schemes and we will use LK as the set of

4350 VOLUME 7, 2019


candidate IA schemes. Since each beamformer set con-tains the (2K )K beamforming vector sets, there are at least(K − 1)!L(K ,K )(2K )K expanded beamforming vector sets.By aligning the interference signal pairs determined by theLatin square, the proposed IA scheme has the expandedbeamforming vector sets much larger than the conventionalIA schemes, where the expanded beamforming vector setsderived from the Latin square as a chain structure are possiblefor only K × 3 MIMO X channel.

In order to select the best beamforming vector set, we haveto compute performance measure such as symbol error rate(SER) or sum-rate for (K − 1)!L(K ,K )(2K )K beamform-ing vector sets. Clearly, it requires tremendous amount ofcomputation. Thus, we propose how to efficiently select abeamforming vector sets to improve performance of SER andsum-rate for the K × 3 MIMO X channel in Section IV.

C. EXAMPLE OF THE PROPOSED SCHEME: 3 × 3 MIMO XCHANNELThe proposed IA scheme is applied to the 3 × 3 MIMO Xchannel which satisfies three requirements in (3), (4), and (5).In the 3 × 3 MIMO X channel, the received signals at threereceivers are given as

Y1 = H11v11s11 + H12v12s12 + H13v13s13+H11v21s21 + H11v31s31 + H12v22s22+H12v32s32 + H13v23s23 + H13v33s33 + n1

Y2 = H21v21s21 + H22v22s22 + H23v23s23+H21v11s11 + H21v31s31 + H22v12s12+H22v32s32 + H23v13s13 + H23v33s33 + n2

Y3 = H31v31s31 + H32v32s32 + H33v33s33+H31v11s11 + H31v21s21 + H32v12s12+H32v22s22 + H33v13s13 + H33v23s23 + n3. (8)

Six interference signals are received at each receiver and eachpair of interference signals is aligned in the same dimensionalsignal space to satisfy the requirement i). The choice fortwo interference signals aligning into one dimensional signalspace can be determined by the Latin square, where there aretwo distinct 3× 3 Latin squares. In other words, interferencealignment pairs can be determined by the 3× 3 Latin squaresasv11 v21 v31v12 v22 v32v13 v23 v33

←−A B CB C AC A B

or

A B CC A BB C A

.(9)

Each transmitter beamforms to align two interference signalsinto one dimensional signal space at each receiver. For exam-ple, the signals s11, s12, and s13 are desired signals at thereceiver 1 and the other six signals are interference signalsin (8). Thus, the first column of the beamforming matrix in(9) is the beamforming vectors for the desired signals of thereceiver 1. Each pair of interference signals corresponding toeach pair of the same symbols in the second and the third

columns in the Latin squares in (9) are aligned in the onedimensional interference signal space, that is, for the firstLatin square, each pair of (v23, v32) from A, (v21, v33) fromB, and (v22, v31) from C is aligned in the one dimensionalinterference signal space. That is, the interference signal pairsof three receivers can be obtained fromv21 v31

v22 v32v23 v33

←−B CC AA B

for receiver 1

v11 v31v12 v32v13 v33

←−A CB AC B

for receiver 2

v11 v21v12 v22v13 v23

←−A BB CC A

for receiver 3

and thus the three pairs of interference signals at each receiverare given as

Receiver 1; (v23, v32), (v21, v33), (v22, v31)

Receiver 2; (v11, v32), (v12, v33), (v13, v31)

Receiver 3; (v11, v23), (v12, v21), (v13, v22).

By aligning the interference signals in the form of the firstLatin square in (9), the following IA condition is obtainedfrom (8) as

span(H31v11) = span(H33v23)








span(H33v13) = span(H32v22). (10)

Using (10), the beamforming vectors for the IA scheme areobtained from

span(H−121 H22H−112 H13H

−133 H31v11) = span(v11)

v23 = H−133 H31v11v32 = H−122 H21v11 (11)

span(H−111 H13H−123 H22H

−132 H31v21) = span(v21)

v12 = H−132 H31v21v33 = H−113 H11v21 (12)

span(H−111 H12H−132 H33H

−123 H21v31) = span(v31)

v13 = H−123 H21v31v22 = H−112 H11v31. (13)

Clearly, v11 is obtained from the eigenvectors ofH−121 H22H

−112 H13H

−133 H31(= E1) and v23 and v32 are also

obtained from (11). Also, v21 and v31 are obtained from

VOLUME 7, 2019 4351


the eigenvectors of H−111 H13H−123 H22H

−132 H31(= E2) and

H−111 H12H−132 H33H

−123 H21(= E3), respectively. Therefore,

the nine beamforming vectors can be obtained from (11),(12), and (13). Since the size of Ei is 6 × 6, the number ofeigenvectors that can be selected for each of v11, v21, andv31 is six. There are 63 beamforming vector sets derivedfrom 63 eigenvector sets. Thus, the selectable number ofbeamforming vector sets for the IA scheme is 63.

Also, by aligning the interference signals in the form ofthe second Latin square in (9), the following IA condition isobtained as three chain structures









span(H32v12) = span(H33v23) (14)

and we have

span(H−121 H23H−113 H12H

−132 H31v11) = span(v11)

v22 = H−132 H31v11v33 = H−123 H21v11 (15)

span(H−131 H33H−123 H22H

−112 H11v21) = span(v21)

v13 = H−133 H31v21v32 = H−112 H11v21 (16)

span(H−111 H13H−133 H32H

−122 H21v31) = span(v31)

v12 = H−122 H21v31v23 = H−113 H11v31. (17)

Similarly, the nine beamforming vectors can also be obtainedfrom (15), (16), and (17) and the selectable number of beam-forming vector sets for the IA scheme is also 63. In the 3× 3MIMOX channel, there are two available IA schemes, whereeach IA scheme has 63 beamforming vector sets. Thus, thereare 2 × 63 expanded beamforming vector sets by two Latinsquares.

D. EXAMPLE OF THE PROPOSED SCHEME: 4 × 3 MIMOX CHANNELFor 4× 3 MIMO X channel, one of the available IA schemesis considered. Eight interference signals are received at eachreceiver and each pair of interference signals is aligned inthe same dimensional signal space to satisfy the requirementi). The choice for two interference signals aligning into onedimensional signal space can be determined by the 4×4 Latinsquare, where there are at least 3! Latin squares. Interferencealignment pairs can be determined by any three columns of

the 4× 4 Latin squares given asv11 v21 v31v12 v22 v32v13 v23 v33v14 v24 v34

←−A B CB C DC D AD A B

, (18)

where one column corresponds to the desired signal and theother two columns correspond to the interference signals.Each transmitter beamforms to align two interference signalsinto one dimensional signal space at each receiver. Thus,the first column of the beamforming matrix is the beamform-ing vectors for the desired signals of the receiver 1. Eachpair of interference signals corresponding to each pair ofthe same symbols in the second and the third columns arealigned in the one dimensional interference signal space, thatis, for the Latin square in (18), each pair of (v24, v33) fromA, (v21, v34) from B, (v22, v31) from C , and (v23, v32) from Dis aligned in the one dimensional interference signal space atthe receiver 1.

By aligning the interference signals in the form of the Latinsquare in (18), the following IA condition is obtained as fourchain structures












span(H34v14) = span(H33v23). (19)

Using (19), the beamforming vectors for the IA scheme areobtained from

span(H−131 H34H−114 H13H

−123 H21v11) = span(v11)

v33 = H−123 H21v11v24 = H−134 H31v11 (20)

span(H−131 H32H−122 H24H

−114 H11v21) = span(v21)

v34 = H−114 H11v21v12 = H−132 H31v21 (21)

span(H−121 H23H−133 H32H

−112 H11v31) = span(v31)

v22 = H−112 H11v31v13 = H−123 H21v31 (22)

span(H−134 H33H−113 H12H

−122 H24v14) = span(v14)

v32 = H−122 H24v14v23 = H−133 H34v14. (23)

4352 VOLUME 7, 2019


Here, v11 is obtained from the eigenvectors ofH−131 H34H

−114 H13H

−123 H21(= E1) and v21, v31, and v14 are

obtained from the eigenvectors of H−131 H32H−122 H24H

−114 H11

(= E2), H−121 H23H

−133 H32H

−112 H11(= E3), and H

−134 H33H

−113

H12H−122 H24(= E4), respectively. Therefore, 12 beamforming

vectors can be obtained from (20), (21), (22), and (23). Sincethe size of Ei is 8× 8, the number of eigenvectors that can beselected for each of v11, v21, v31, and v14 is eight. There are84 beamforming vector sets derived from 84 eigenvector sets.Thus, the selectable number of beamforming vector sets forthe IA scheme is 84. In the 4 × 3 MIMO X channel, thereare at least 3! IA schemes, where each IA scheme has 84

beamforming vector sets. Therefore, there are 3!84 expandedbeamforming vector sets generated by Latin squares.

IV. EFFICIENT BEAMFORMER SELECTION METHODSWe can select the beamforming vector set b∗ ∈ B by usingthe following methods:

• MinMax SNR based beamformer selection method;

b∗ = argmaxb∈B

mini|(R†Hv)i|, i = 1, 2, · · · , 3K , (24)

where R†, H , and v denote the zero-forcing matirix,the channel matrix, and the beamforming vector for thedesired signal i, respectively.

• Sum-rate maximization based beamformer selectionmethod;

b∗ = argmaxb∈B

(Rsum), (25)

where Rsum denotes the sum-rate at the given system.• Condition number (CN) based beamformer selectionmethod;

b∗ = argminb∈B

(F(κ(b))), (26)

where κ(b) is the CN for the IA scheme with b and F(·)is a function mapping fromCN toMinMax SNR or sum-rate.

A. UTILIZATION OF CONDITION NUMBERInstead of measuring the SNR or sum-rate with high compu-tational complexity, we introduce other measurements for thebeamformer selection deriving suboptimal solutions with lesscomputational complexity. That is, we use the CN as a mea-surement to estimate SNR or sum-rate for the K × 3 MIMOX channel. The CN of the nonsingular matrix is defined as aratio of the maximum singular value to the minimum singularvalue, where the singular value is a weight that indicates howmuch the range space in the matrix is biased in the givendirection. That is, the CN is the ratio of the singular value inthe most biased direction to that in the least biased directionof the matrix. When the CN reaches the minimum value of 1,the corresponding matrix is becomes an orthogonal matrix.Therefore, the closer the CN is to 1 (the smaller), the closerto the orthogonality of the matrix. In general, it is known that

the orthogonalization of the received signals can achieve theimprovement of performance in the interference channel.

For each receiver, the signal space matrix is defined as

Ai = [D1,D2, · · · ,DK , I1, I2, · · · , IK ], i = 1, 2, 3,

(27)

where Di and Ii are 2K × 1 desired signal vectorsand aligned interference signal vectors, respectively. Eachreceiver decodes the desired signal by zero-forcing. However,the desired signal is attenuated by zero-forcing, which is mea-sured by the CN of the signal space matrix in (27). In orderto consider the orthogonality relation rather than the magni-tude of the signal space matrix, we use a normalized signalspace matrix by normalizing each signal for an orthogonalitymeasure as

Ai = [D1

|D1|,D2

|D2|, · · · ,

DK|DK |

,I1|I1|

,I2|I2|

, · · · ,IK|IK |

]. (28)

Then, the CN is obtained for each normalized signal spacematrix of each receiver and used as a measure of the perfor-mance evaluation, whose computational complexity can bereduced compared to the SNR or sum-rate computation.

B. CORRELATION BETWEEN CONDITION NUMBER ANDMINIMUM SNRFor the case of 3 × 3 MIMO X channel, there are |B| = 63

beamforming vector sets for an IA scheme by a Latin square.For numerical analyses in this section, we generate a Rayleighchannel realization Hij and assume that the number of trans-mitters is fixed to K = 3 so that each node has six antennas.Also, we assume that each transmitter has the same transmitpower constraintP and the unit noise variance. Then, the SNRfor the i-th desired signal can be computed as |(R†Hv)i|2.For each beamforming vector set, the minimum SNR and theestimated minimum SNR from CN at Eb/N0 = 15 dB arecomputed as follows:

• Find the minimum SNR.

– For each of the 63 beamforming vector sets, sortmini|(R†Hv)i|2, i = 1, 2, · · · , 9, in the descending

order.

• Estimate the minimum SNR from CN.

– For each bj ∈ B, compute CNby maxi∈(1,2,3)

(κi(bj)), j =

1, 2, · · · , 63 of the normalized signal space matrixobtained from 63 beamforming vector sets, whereκi(bj) is the CN of the i-th receiver with the beam-forming vector set bj in B.

– Let bj be the sorted beamforming vector in theascending order of max

i(κi(bj)).

– For each of bj, obtain mini|(R†Hv)i|2, i =

1, 2, · · · , 9.

• Compare the minimum SNR and the estimated mini-mum SNR from CN.

VOLUME 7, 2019 4353


FIGURE 2. Comparison of the directly obtained minimum SNR and theestimated minimum SNR from CN in the 3 × 3 X channel.

FIGURE 3. Comparison of the MinMax SNR distribution for various CNu inthe 3 × 3 X channel.

Some reasons for the fluctuation in Fig. 2 are as follows.First, CN can be considered as a measurement of orthogonal-ity among received signals. Second, we need only the orthog-onality between the desired signal and the interference signal.But CN also computes a measure of orthogonality among theinterference signals. By considering the beamforming vectorsets within a certain range of the minimum SNR estimatedfrom CN, it is possible to compromise the fluctuation inestimating the minimum SNR by CN as follows:• Choose a set of bj’s with the smallest u CN’s and com-pute the minimum SNR of each bj among them.

• Select the b∗ that gives the maximum SNR amongthem.

We generate an ensemble of 10000 Rayleigh channel real-izations Hij to compare the MinMax SNR distribution.Fig. 3 compares the MinMax SNR distribution for the setsof bj with the smallest CN’s, denoted by CNu with set sizeu, u = 1, 3, 10, 20. For CN10, the shape of the SNRdistribution is similar to that of the optimal case and it isnearly optimal for CN20. Thus, MinMax SNR by CN20 isalmost the same as the Optimal MinMax SNR.

FIGURE 4. Comparison of the directly obtained minimum SNR and theestimated minimum SNRs from CN and OCN in the 3 × 3 X channel.

C. INTERFERENCE ORTHOGONALIZEDCONDITION NUMBERWith a large number of users, the number of beamform-ing vector sets increases exponentially and thus the set sizeCNu should be large. Thus we propose a modified CN thatimproves the method to obtain CN from signal space. In orderto avoid computation of the orthogonality between the inter-ference signals, a new signal space matrix can be made,which considers only the orthogonality between the desiredand interference signals. Thus, the signal space matrix canbe constructed by pre-orthogonalizing the interference signalspace [ I1

|I1|, I2|I2|, · · · , IK

|IK |] in the signal space matrix in (28)

using Gram-Schmidt orthogonalization. Then, its conditionnumber is called orthogonalized condition number (OCN),which is useful for large K .

In Fig. 4, compared to the fluctuation of the estimatedminimum SNR from CN, the fluctuation of the estimatedminimum SNR from the OCN is reduced.

D. CORRELATION BETWEEN CONDITION NUMBERAND SUM-RATEAs in the previous case, the procedure to obtain the maximumsum-rate among all beamforming vector sets can be replacedby the CN based beamformer selection with less computa-tional complexity.

For the case of 3× 3 MIMO X channel, there are |B| = 63

beamforming vector sets for an IA scheme by a Latin square.For each beamforming vector set, the maximum sum-rate andthe estimated maximum sum-rate from CN are computed asfollows:• Find the sum-rate.

– For the 63 beamforming vector sets, sort the sum-rates for all beamforming vector sets in the descend-ing order.

• Estimate the sum-rate from CN.– For each bj ∈ B, compute CN by

∑3i=1(κi(bj)),

j = 1, 2, · · · , 63 of the normalized signal spacematrix obtained from 63 beamforming vector sets,

4354 VOLUME 7, 2019


FIGURE 5. Comparison of the directly obtained sum-rate with theestimated sum-rate from CN in the 3 × 3 X channel.

where κi(bj) is the CN of the i-th receiver with thebeamforming vector set bj in B.

– Let bj be the sorted beamforming vector in theascending order of

∑3i=1(κi(bj)).

– For each of bj, obtain the sum-rate.

• Compare the sum-rate and the estimated sum-rate fromCN.

In Fig. 5, the sum rates for each beamforming vector set arecompared with the estimated sum rate from CN at Eb/N0 =

15 dB. The red curve denotes the sorted sum rates in descend-ing order and the blue curve denotes the sorted sum rates byCNs. We first sort the beamforming vector sets based on theCN and plot them in Fig. 5.

Comparing the directly obtained sum-rate with the esti-mated sum-rate from CN, there are some fluctuations forestimating the maximum sum-rate from CN as in Fig. 5.By considering the beamforming vector sets within a certainrange of the sum-rate estimated from CN, it is possible tocompromise the fluctuation in estimating the maximum sum-rate by CN as follows:

• Choose a set of bj with the smallest CN’s and computethe sum-rate of each bj among them.

• Select the b∗ that gives the maximum sum-rate amongthem.

E. COMPARISON OF COMPUTATIONAL COMPLEXITYIn the IA schemes for the MIMO X channel, the processesother than beamformer selection are the same and the com-putational complexity for beamformer selection is dominant.TheMinMax SNR based beamformer selectionmethod or thesum-rate maximization based beamformer selection methodis used to evaluate the performance of the 3K desired signals.Both methods require the zero forcing vector to evaluatethe performance of each desired signal, which requires anSVD operation of 2K × 2K matrix. Therefore, the 3K SVDoperations of the 2K × 2K matrix are required. On the

other hand, the computational complexity of the proposedCN based beamformer selection method uses the CN definedin the received signal space instead of the SNR computationfor each desired signal. The minimum andmaximum singularvalues for 2K × 2K matrix are required to calculate the CNof the received signal space. In K × 3 X channel, there arethree received signal spaces and thus they require the threeSVD operations of the 2K×2K matrices. Thus, the proposedCN based beamformer selection method reduces its com-putational complexity compared to the MinMax SNR basedbeamformer selection method or the sum-rate maximizationbased beamformer selection method by factor of K when Kis large.

reduces by factor of K

V. PERFORMANCE ANALYSIS OF EXPANDEDBEAMFORMER SETS WITH CN BASEDBEAMFORMER SELECTIONIn this section, the performance in terms of SER and sum-rateof K × 3 MIMO X channels with the CN based beamformerselection method are analyzed. For simplicity, we considerthe case of K = 3 so that each node has six antennas. Fornumerical analyses in this section, we generate a Rayleighchannel realization Hij and assume that each transmitter hasthe same transmit power constraint P.

A. SER BY CN BASED BEAMFORMER SELECTIONFor the case of 3 × 3 MIMO X channel, the proposed CNbased beamformer selection method for SER has the follow-ing four steps.• For each bj ∈ B, compute CN by max

i∈(1,2,3)(κi(bj)), j =

1, 2, · · · , 63 of the normalized signal space matrixobtained from 63 beamforming vector sets, where κi(bj)is the CN of the i-th receiver with the beamformingvector set bj in B.

• Let bj be the sorted beamforming vector in the ascendingorder of max

i(κi(bj)).

• Choose a set of bj’s with the smallest u CN’s.• b∗ = argmax

bmini|R†Hv|i, i = 1, 2, · · · , 9.

In Fig. 6, Optimal MinMax SNR uses the method to obtainthe minimum SNR for each beamforming vector set andselect the maximum SNR among them for a Latin square.Randomu refers to the method to obtain the minimum SNRfor each of randomly selected u beamforming vector sets andselect the maximum SNR among them. For CN13, it hasalmost similar SER performance to Optimal MinMax SNR.Also, compared to Randomu, it is very effective to select thebeamforming vector sets by CN. Therefore, it can be seenthat good SER performance can be achieved by the CN basedbeamformer selection without calculating the SNR of all thebeamforming vector sets.

B. SUM-RATE BY CN BASED BEAMFORMER SELECTIONFor the case of 3 × 3 MIMO X channel, the proposedCN based beamformer selection method for the sum-rate is

VOLUME 7, 2019 4355


FIGURE 6. SER comparison between the CN based beamformer selectionmethod and the MinMax SNR based beamformer selection method for aLatin square in the case of 3 × 3 MIMO X channel.

FIGURE 7. Sum-rate comparison between the CN based beamformerselection method and the sum-rate maximization based beamformerselection method for a Latin square in the case of 3 × 3 MIMO X channel.

obtained as

b∗ = argminbj∈B

(3∑i=1

(κi(bj))), j = 1, 2, · · · , 63. (29)

In Fig. 7, Optimal sum-rate uses the method to obtainthe sum-rate for each beamforming vector sets and selectthe maximum sum-rate among them. Randomu refers tothe method to obtain the sum-rate of randomly selected ubeamforming vector sets and select the maximum sum-rateamong them. For CN1, it has almost similar sum-rate perfor-mance to Optimal sum-rate. Also, compared to Randomu, it isvery effective to select the beamforming vector sets by CN.Therefore, it can be seen that high sum-rate performance canbe achieved by the CN based beamformer selection withoutobtaining the sum-rate of all the beamforming vector sets.

C. LATIN SQUARE EXPANSION GAINThe proposed IA scheme for K × 3 MIMO X channel hasat least (K − 1)!L(K ,K ) IA schemes by the K × K Latin

FIGURE 8. SER comparison of the proposed IA scheme with the Latinsquare expansion and the conventional scheme for the case of 3 × 3MIMO X channel.

squares. Since there are (2K )K beamforming vector sets foreach IA scheme, the total number of the expanded beamform-ing vector sets is at least (K − 1)!L(K ,K )(2K )K and theseexpanded beamforming vector sets are called the Latin squareexpansion. If computational power is sufficient, the selectionrange for the expanded beamforming vector sets for the pro-posed IA scheme can be expanded by using multiple Latinsquares.

For the 3 × 3 MIMO X channel, the proposed IA schemegives two groups of valid IA schemes from two different 3×3Latin squares. For each group, there are 63 = 216 beam-forming vector sets. Fig. 8 shows the SER performance forthe beamforming vector sets by the Latin square expansion.Prop216 selects the best beamforming vector set in terms ofSER among the 216 beamforming vector sets generated byone of the two Latin squares. Prop432 selects the beamform-ing vector set that provides the best SER performance amongthe 432 beamforming vector sets generated by two Latinsquares. The Latin square expansion provides more beam-forming vector sets and thus SER performance is improvedcompared to the conventional one. As a further work, it canbe researched how to choose the best IA scheme among atleast (K − 1)!L(K ,K ) IA schemes.

VI. CONCLUSIONSIn this paper, we proposed an IA scheme using the Latinsquare and CN based beamformer selection method with lowcomputational complexity in the K × 3 MIMO X channel.It was shown that the proposed IA scheme can be expanded bythe K×K Latin squares. Since the proposed IA scheme has achain structure, the proposed scheme has a larger beamform-ing vector sets than the conventional IA scheme. To selecta good beamforming vector set among many beamformingvector sets, the CN based beamformer selection with lowcomputational complexity was proposed. SER and sum-rateperformances can be improved by the proposed IA schemeusing Latin squares with the CN based beamformer selection.

4356 VOLUME 7, 2019


REFERENCES[1] V. R. Cadambe and S. A. Jafar, ‘‘Interference alignment and degrees of

freedom of the K-user interference channel,’’ IEEE Trans. Inf. Theory,vol. 54, no. 8, pp. 3425–3441, Aug. 2008.

[2] C. Meng, A. K. Das, A. Ramakrishnan, S. A. Jafar, A. Markopoulou,and S. Vishwanath, ‘‘Precoding-based network alignment for three unicastsessions,’’ IEEE Trans. Inf. Theory, vol. 61, no. 1, pp. 426–451, Jan. 2015.

[3] M. Morales-CÃľspedes, J. Plata-Chaves, D. Toumpakaris, S. A. Jafar, andA. G. Armada, ‘‘Blind interference alignment for cellular networks,’’ IEEETrans. Signal Process., vol. 63, no. 1, pp. 41–56, Jan. 2015.

[4] C. Wang, C. Qin, Y. Yao, Y. Li, and W. Wang, ‘‘Low complexity inter-ference alignment for mmWave MIMO channels in three-cell mobilenetwork,’’ IEEE J. Sel. Areas Commun., vol. 35, no. 7, pp. 1513–1523,Jul. 2017.

[5] A. G. Davoodi and S. A. Jafar, ‘‘Generalized degrees of freedom of thesymmetric K user interference channel under finite precision CSIT,’’ IEEETrans. Inf. Theory, vol. 63, no. 10, pp. 6561–6572, Oct. 2017.

[6] B. Yuan, H. Sun, and S. A. Jafar, ‘‘Replication-based outer bounds: Onthe optimality of ‘half the cake’ for rank-deficient MIMO interferencenetworks,’’ IEEE Trans. Inf. Theory, vol. 63, no. 10, pp. 6607–6621,Oct. 2017.

[7] S. Zeng, C. Wang, C. Qin, and W. Wang, ‘‘Interference alignment assistedby D2D communication for the downlink of MIMO heterogeneous net-works,’’ IEEE Access, vol. 6, pp. 24757–24766, 2018.

[8] C. Qin, S. Zeng, C. Wang, D. Pan, W. Wang, and Y. Zhang, ‘‘A distributedinterference alignment approach based on grouping in heterogeneous net-work,’’ IEEE Access, vol. 6, pp. 2484–2495, 2018.

[9] V. R. Cadambe and S. A. Jafar, ‘‘Interference alignment and the degrees offreedom of wireless X networks,’’ IEEE Trans. Inf. Theory, vol. 55, no. 9,pp. 3893–3908, Sep. 2009.

[10] S.-H. Park and Y.-C. Ko, ‘‘K-user MIMO X network system with per-fect interference alignment,’’ in Proc. IEEE Int. Conf. Commun. (ICC),Jun. 2011, pp. 1–5.

[11] H. J. Sung, S.-H. Park, K.-J. Lee, and I. K. Lee, ‘‘Linear precoder designsfor K-user interference channels,’’ IEEE Trans. Wireless Commun., vol. 9,no. 1, pp. 291–301, Jan. 2010.

[12] B. Zhou, B. Bai, Y. Li, D. Gu, and Y. Luo, ‘‘Chordal distance-baseduser selection algorithm for the multiuser MIMO downlink with perfector partial CSIT,’’ in Proc. Int. Conf. Adv. Inf. Netw. Appl., Mar. 2011,pp. 77–82.

[13] A. Goldsmith, S. A. Jafar, N. Jindal, and S. Vishwanath, ‘‘Capacity lim-its of MIMO channels,’’ IEEE J. Sel. Areas Commun., vol. 21, no. 5,pp. 684–702, Jun. 2003.

[14] M. A. Maddah-Ali, A. S. Motahari, and A. K. Khandani, ‘‘Communi-cation over MIMO X channels: Interference alignment, decomposition,and performance analysis,’’ IEEE Trans. Inf. Theory, vol. 54, no. 8,pp. 3457–3470, Aug. 2008.

[15] S. A. Jafar and S. Shamai (Shitz), ‘‘Degrees of freedom region of theMIMO X channel,’’ IEEE Trans. Inf. Theory, vol. 54, no. 1, pp. 151–170,Jan. 2008.

[16] A. Soysal and S. Ulukus, ‘‘Optimality of beamforming in fading MIMOmultiple access channels,’’ IEEE Trans. Commun., vol. 57, no. 4,pp. 1171–1183, Apr. 2009.

[17] F. Li and H. Jafarkhani, ‘‘Space-time processing for X channels usingprecoders,’’ IEEE Trans. Signal Process., vol. 60, no. 4, pp. 1849–1861,Apr. 2012.

[18] R. Prasad and A. Chockalingam, ‘‘Capacity bounds for the Gaussian Xchannel,’’ in Proc. Inf. Theory Appl. Workshop (ITA), Feb. 2013, pp. 1–10.

[19] K. Ko and J. Lee, ‘‘Multiuser MIMO user selection based on chordaldistance,’’ IEEE Trans. Commun., vol. 60, no. 3, pp. 649–654, Mar. 2012.

[20] S. Jafar and A. Goldsmith, ‘‘Transmitter optimization and optimality ofbeamforming for multiple antenna systems with imperfect feedback,’’IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1165–1175, Jul. 2004.

[21] T. Yoo and A. Goldsmith, ‘‘On the optimality of multiantenna broad-cast scheduling using zero-forcing beamforming,’’ IEEE J. Sel. AreasCommun., vol. 24, no. 3, pp. 528–541, Mar. 2006.

[22] Number of Reduced Latin Squares of Order N; Online Encyclopedia ofInteger Sequences (OEIS) A000315. Accessed: Jul. 26, 2018. [Online].Available: https://oeis.org/A000315

YOUNG-SIK MOON received the B.Sc. andPh.D. degrees in electrical and computer engineer-ing from Seoul National University, Seoul, SouthKorea, in 2011 and 2018, respectively. His areas ofresearch interests include interference alignment,space time code, and error correcting codes.

JONG-YOON YOON received the B.S. degreefrom the Department of EE, Yonsei University,Seoul, South Korea, in 2013. He is currently pur-suing the Ph.D. degree in electrical and computerengineering with Seoul National University. Hisareas of research interests include interferencealignment, space time code, and topological inter-ference management.

JONG-SEON NO received the B.S. andM.S. degrees in electronics engineering fromSeoul National University, Seoul, South Korea,in 1981 and 1984, respectively, and the Ph.D.degree in electrical engineering from the Univer-sity of Southern California, Los Angeles, in 1988.He was a Senior MTS with Hughes Network Sys-tems, from 1988 to 1990. He was also an AssociateProfessor with the Department of Electronic Engi-neering, Konkuk University, Seoul, South Korea,

from 1990 to 1999. He joined the faculty of the Department of Electricaland Computer Engineering, Seoul National University, in 1999, where he iscurrently a Professor. His area of research interests include error-correctingcodes, sequences, cryptography, LDPC codes, interference alignment, andwireless communication systems. Hewas a recipient of the IEEE InformationTheory Society Chapter of the Year Award, in 2007. From 1996 to 2008, heserved as a Founding Chair of Seoul Chapter, the IEEE Information TheorySociety. He was a General Chair for Sequence and Their Applications 2004(SETA2004) in Seoul. He also served as a General Co-Chair for InternationalSymposium on Information Theory and Its Applications 2006 (ISITA 2006)and International Symposium on Information Theory 2009 (ISIT 2009)in Seoul. He was elevated to IEEE Fellow in Research Engineer/Scientistthrough the IEEE Information Theory Society, in 2011. He became the Co-Editor-in-Chief of the Journal of Communications and Networks, in 2012.

SANG-HYO KIM received the B.Sc., M.Sc.,and Ph.D. degrees in electrical engineering fromSeoul National University, Seoul, South Korea,in 1998, 2000, and 2004, respectively. From2004 to 2006, he was a Senior Engineer withSamsung Electronics. He visited the University ofSouthern California as a Visiting Scholar, from2006 to 2007. In 2007, he joined the Collegeof Information and Communication Engineering,Sungkyunkwan University, Suwon, South Korea,

where he is currently a Professor. In 2015, he had one-year visit to theUniversity of California, San Diego, as a Visiting Scholar. His researchinterests include coding theory, wireless communications, machine-learning-inspired communications. He has served as an Editor for the Transactions onEmerging Telecommunications Technologies and the Journal of Communica-tions and Networks, since 2013.

VOLUME 7, 2019 4357

Documents

Interference Alignment Schemes Using Latin Square for K 3 ...ccl.snu.ac.kr/papers/journal_int/journal2018_12_1.pdf · structure by the Latin square is proposed for K 3 multiple-input