4750 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 7, JULY 2019

Achievable Degrees of Freedom of Relay-AidedMIMO Cellular Networks Using Opposite

Directional Interference AlignmentHo-Youn Kim and Jong-Seon No

Abstract— In this paper, we propose an interference align-ment (IA) scheme for the multiple-input multiple-output (MIMO)uplink cellular network with the help of a relay which operatesin half-duplex mode. The proposed scheme only requires globalchannel state information (CSI) knowledge at the relay, withno transmitter beamforming and time extension at the userequipment (UE), which differs from conventional IA schemesfor cellular networks. We derive the feasibility condition of theproposed scheme for the general cellular network configurationand analyze the degrees-of-freedom (DoF) performance of theproposed IA scheme while providing a closed-form beamformerdesign at the relay. Extensions of the proposed scheme todownlink and full-duplex cellular networks are also proposedin this paper. The DoF performance of the proposed schemeis compared to those of linear IA scheme and relay-aidedinterference management scheme for a cellular network withno time extension. It is also shown that advantages similar tothose in the uplink case can be obtained for the downlink casethrough the duality of a relay-aided interfering multiple-accesschannel (IMAC) and an interfering broadcast channel (IBC).Furthermore, the proposed scheme for a full-duplex cellularnetwork is shown to have advantages identical to those of anumber of proposed half-duplex cellular cases.

Index Terms— Degrees-of-freedom (DoF), interference align-ment (IA), interfering broadcast channel (IBC), interferingmultiple-access channel (IMAC), multiple-input multiple-output(MIMO), opposite directional interference alignment (ODIA),relay.

I. INTRODUCTION

IT IS known that high degrees-of-freedom (DoF) can beachieved in a multi-user interference environment by means

of interference alignment (IA). The optimal DoF of single-input single-output (SISO) time-varying interference channelhas been derived through an asymptotic IA scheme [1].Research of IA in multiple-input multiple-output (MIMO)channels also has been done and the optimal DoF region for amulti-user MIMO interference channel has been revealed [2],

Manuscript received July 17, 2018; revised December 1, 2018 andMarch 11, 2019; accepted March 13, 2019. Date of publication March 22,2019; date of current version July 13, 2019. The associate editor coordinatingthe review of this paper and approving it for publication was S. Bhashyam.(Corresponding author: Ho-Youn Kim.)

The authors are with the Department of Electrical and Computer Engineer-ing, INMC, Seoul National University, Seoul 08826, South Korea (e-mail:ferui@ccl.snu.ac.kr; jsno@snu.ac.kr).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2019.2906920

showing that IA can also increase the DoF of MIMO commu-nication systems.

Unfortunately, conventional IA schemes cannot be directlyapplied to cellular networks because the channel state infor-mation (CSI) feedback to transmitter and time extension arerequired. For example, a subspace IA scheme for a SISOuplink cellular network, finding that the optimal DoF of onecan be achieved as the number of cellular users increases wassuggested [3]. However, the subspace IA scheme requires moretime extensions as the number of users increases. Thus thetransmission/reception delay of the information increases asthe number of users in the network becomes large, which isnot suitable for a cellular network. In general, user equip-ment (UE) is not practically able to handle/process beam-forming by CSI feedback and communicating nodes requireshort transmission/reception delays. In order to reduce theCSI feedback and time extension requirements, relay-aidedIA schemes have been proposed for wireless communicationnetworks, where the relay handles the CSI instead of thetransmitter, adding the potential to reduce the time exten-sion requirement in the MIMO interference channel. An IAscheme with two time slots and without CSIT feedback for aMIMO interference channel was proposed [4], but the effectivenoise increases due to zero-forcing and non-local (partial)CSI knowledge at the receiver (CSIR) is required. To overcomethe abovementioned limitations on this scheme, an oppositedirectional interference alignment (ODIA) scheme with relaywas introduced [5]. It was found that the ODIA can alsoachieve the optimal DoF in a symmetric MIMO interferencechannel. Interference neutralization (IN) schemes were pro-posed for relay-aided network, where relay helps to neutralizeinterference at destination by relay beamforming [6]. AlignedIN was proposed for two-user interference channel, whererelay beamforming and transmitter beamforming are combinedto align interference at relay and further neutralize alignedinterference at destination [7]. But, relay should be operated ininstantaneous full-duplex mode for aligned IN. Further, globalCSI feedback for every node and time extension are requiredfor transmitter beamforming.

Recently, researchers studied the achievable DoF for acellular network where a base station (BS) is equipped withdirectional or omni-directional antennas when considering thecellular network topology [8], [9]. In [10], the optimal DoFand an achievable IA scheme were derived in a MIMO cellular

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KIM AND NO: ACHIEVABLE DoF OF RELAY-AIDED MIMO CELLULAR NETWORKS USING ODIA 4751

network using earlier result in [11]. However, this scheme alsorequires infinite time extensions to achieve promised DoF fora large network, which is not practical for a cellular network.Furthermore, a one-shot linear IA scheme was proposed basedon subspace IA [12], but UEs may not be able to store theCSI of the entire network which should be used to aligninterferences.

Several studies have focused on the potential role of relaysin a cellular network with IA. An IA scheme for a two-cell downlink cellular network based on different half-duplexrelaying schemes with the decode-and-forward (DF) protocolwas proposed [13]. However, it is known that the relay requireshigh hardware complexity when the DF protocol is usedin the relay [14]. It was shown that in a quasi-static flat-fading environment, a MIMO cellular network can achievehigher DoF by using a full-duplex relay with finite timeextension [15]. In practice, self-interference of relay due tothe full-duplex operation at the relay is not negligible [16].To enable full-duplex transmission at relay, the relay mustcontain self-interference suppression function, which leads todisadvantage of complex hardware design.

In this paper, our main contribution is suggesting an inter-ference management scheme which is suitable for variouscellular networks, including the uplink, dowlink, and full-duplex cellular networks. Further, we focuse on minimizingthe load of BSs and UEs in the design of cellular networkinterference management by using a relay and the details aregiven as follows.

In order to solve the problems of applying IA to MIMOcellular networks, we propose an interfering multiple-accesschannel (IMAC)-ODIA scheme for an uplink cellular networkwith a half-duplex amplify-and-forward (AF) relay, where onlytwo time slots are needed regardless of the network size,to achieve interference alignment. The reason why AF relayis adopted is that DF relay requires high hardware complexitydue to decoding and demodulation for lots of messages at therelay. It has the advantage of no time extension and the abilityto be operated with various channel settings. Further, CSIT atthe UEs is not required in the proposed scheme and BSs isrequired to acquire only their local CSI. We evaluate the DoFperformance of the proposed IMAC-ODIA scheme and showthat for certain antenna configuration, the proposed schemecan achieve the optimal DoF for a MIMO cellular network.

Furthermore, we propose a relay-aided IA scheme for adownlink cellular network to resolve the global CSIR require-ment in the conventional IA scheme which requires a heavyload on the UEs. Using the uplink-downlink duality from ear-lier work [17], we propose a modification of the IMAC-ODIAscheme for the downlink, which leads to interfering broadcastchannel (IBC)-ODIA with features similar to those of theIMAC-ODIA scheme. Here, neither decorrelator constructionnor global CSIR is required at the UEs. We also extend ouridea to full-duplex cellular network, referred to as full-duplexODIA (FD-ODIA). It is important to handle interference toobtain full-duplexing gain and accordingly, we provide anIA beamformer design at each node and discuss its DoFperformance.

This paper is organized as follows. In Section II, we presentpreliminary information concerning the properties of the tensorproduct and system model of the cellular network with arelay. In Section III, the IA condition and the IMAC-ODIAscheme for the uplink cellular network are proposed. AnIBC-ODIA scheme for a downlink cellular network and itsextension to a full-duplex cellular network are also pro-posed in Section IV and Section V, respectively. Section VIincludes the discussion, and finally the conclusion is given inSection VII.

II. PRELIMINARY AND SYSTEM MODEL

In this section, we introduce several properties of the tensorproduct of matrices [18] and describe the cellular networkmodel with an AF relay. First, we focus on an uplink sym-metric cellular network modeled as a symmetric IMAC.

Throughout the paper, scalars are denoted in lowercase,vectors are written in boldface lowercase, and matrices areindicated by boldface capital letters. Several definitions per-taining to matrix A and vector a are given below.

- aij : (i, j) component of matrix A- ai: i-th component of vector a- nullleft

A and nullrightA : left and right null vectors of

matrix A, respectively- {A}i: i-th column vector of matrix A- {A}i:j : submatrix of A, i.e., [{A}i, {A}i+1, . . . , {A}j ]- rank(A) or rA: rank of matrix A- (·)T , (·)H , (·)left, (·)right, and (·)†: transpose, complex

conjugated transpose, left inverse, right inverse,and Moore-Penrose pseudo inverse, respectively

- 0I×J : I × J zero matrix and subscript can be omittedwhen the size of the zero matrix is not important.

- II : I × I identity matrix

- II×J : I × J rectangular identity matrix =[

IJ

0(I−J)×J

],

I ≥ J- K: real or complex field, R or C

A. Tensor Product and Kruskal RankFirst, we briefly introduce the pre-defined tensor products

of matrices, that is, the Kronecker and Khatri-Rao products.The Kronecker product of A and B is defined below.

Definition 1: A ⊗ B is the Kronecker product of A and Bdefined as

A ⊗ B =

⎡⎢⎣

a11B a12B · · ·a21B a22B · · ·

......

. . .

⎤⎥⎦ .

Let A = [A1 ... AD] and B = [B1 ... BD] be two par-titioned matrices with an equal number of partitions. TheKhatri-Rao product of A and B is then defined as the partition-wise Kronecker product as given below.

Definition 2: A�B is the Khatri-Rao product of partitionedmatrices A and B defined as

A � B =[A1 ⊗ B1 · · · AD ⊗ BD

].

4752 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 7, JULY 2019

Further, Kruskal rank and generalized Kruskal ranks areintroduced as follows.

Definition 3: The Kruskal rank of matrix A denoted byrankk(A) or kA, is the maximal number r such that any set ofr columns of A is linearly independent.

Definition 4: Let A = [A1 · · · AD] be the partitionedmatrix where submatrices Ai have same size. Then the gen-eralized Kruskal rank of partitioned matrix A denoted byrankk′(A, D) or k′

A,D, is the maximal number r such that anyset of r submatrices of A yields a set of linearly independentcolumns. A is full column rank matrix if A has generalizedKruskal rank of D, which is same to the total number ofsubmatrices.

Example 1: Consider a partitioned matrix A = [A1 A2],where A1 = A2 = I2. Then, the generalized Kruskal rank ofA with D = 2 is 1. Because if we choose any one matrix fromA1 or A2, it yields linearly independent columns as

[1 0

]T

and[0 1

]T. But if we choose two matrices together, column

set of [A1 A2] is no longer linearly independent. Therefore,generalized Kruskal rank of A with D = 2 cannot be largerthan 1.

We introduce a number of propositions for the generalizedKruskal rank from [18], which will be used in the nextsections.

Proposition 1 (Generalized Kruskal Rank of an UniformlyPartitioned Matrix [18]): Let A = [A1 · · · AD] be a matrixwhose entries are drawn i.i.d. from a continuous distributionand partitioned in D submatrices with Ar ∈ K

I×L. Then,the generalized Kruskal rank of A is min{� I

L�, D}.Proposition 2 (Generalized Kruskal Rank of Khatri-Rao

Product of Uniformly Partitioned Matrices [18]): Considerthe uniformly partitioned matrices A = [A1 · · · AD] withAr ∈ K

I×L and B = [B1 · · · BD] with Br ∈ KJ×M ,

1 ≤ r ≤ D. Then, we have

1) If k′A,D = 0 or k′

B,D = 0, then k′A�B,D = 0.

2) If k′A,D ≥ 1 and k′

B,D ≥ 1, then k′A�B,D ≥ min{k′

A,D +k′

B,D − 1, D}.

B. System Model: Cellular Network With a Single RelayIn this paper, the information-theoretic quantity of interest

is the DoF, where the DoF is defined as the number ofsuccessfully decodable data streams in the desired receiverthat are transmitted by the corresponding user. The term“successfully decodable” means that the desired data streamsare received in an interference-free space.

In a cellular network, the DoF per cell is of interest in gen-eral. It is defined as the inter-cell interference-free dimensionat the base station, regardless of whether it is the receiver orthe transmitter. Once the DoF per cell is derived, it can beused to determine the number of users for data transmissionper cell in a cellular network.

First, we focus on uplink cellular network, after whichdownlink cellular network will be considered. We consider afully connected symmetric cellular network with C cells andK users in each cell, where ‘symmetric’ means that each cellhas an identical configuration and cell structure. A relay islocated at the center of cellular network, where it can serve

Fig. 1. System model: Fully connected cellular uplink network with a singlerelay.

C cells. Cell j is served by a single base station denotedas BSj , j ∈ {1, . . . , C}. The user (k, j) denoted as UE(k,j),k ∈ {1, . . . , K}, is served by BSj as shown in Fig. 1. It isassumed that each UE has M transmit antennas, each BShas N receive antennas, and the relay has NR antennas. Thechannel from UE(k,i) to BSj is denoted by the N × Mmatrix Hj,(k,i), the channel from UE(k,j) to the relay isdenoted by the NR ×M matrix HUR

(k,j), and the channel fromthe relay to BSj is denoted by the N × NR matrix HRB

j .In the proposed schemes, only the relay requires global CSI.In addition, each BS has its local CSI, but UEs do not requireany CSI knowledge.

Throughout the paper, C cells are supported by a singlerelay, but in practice, relay can only cover smaller number ofcells due to its power limit. Then, large cellular network canbe decomposed into many smaller cellular networks, wherethe smaller femto/pico cells can be supported by a singlerelay. We assume that relay is operated in AF protocol asin [4], because DF relay requires decoder, demodulator, andremodulator resulting in very high hardware complexity.

Since conventional IA schemes require global CSI ateach UE, we can calculate CSI cost of network. Assume thatthere is one UE in one cell. Note that, for channel point ofview, a number of UEs can be replaced to one UE with manyantennas. Then, for C cells, there are C2 channels for allUE-BS pair. For relay, there are C channels for UE-Relay andRelay-BS pair each. If each UE has global CSI, then the costof global CSI at every UE can be calculated as C × C2. Forsingle relay case, global CSI cost at relay can be calculated as1× (C2 + 2C). Then, the cost of global CSI of the proposedsingle relay case can be reduced by C(C2 − C − 2). Fromabove, we can say that advantage of global CSI at relayincreases when the cellular network becomes larger.

The channel is assumed to be Rayleigh fading and theentries of the channel matrices are independent identicallydistributed complex Gaussian random variables with zero-mean and unit-variance. Due to the average power constraintin cellular networks, the data vector x(k,j) transmitted fromUE(k,j) is normalized such that the average transmit power of

KIM AND NO: ACHIEVABLE DoF OF RELAY-AIDED MIMO CELLULAR NETWORKS USING ODIA 4753

each UE is limited by P , i.e., E[||x(k,j)||] ≤ P , where E[·] and|| · || denote the expectation and norm functions, respectively.

We consider a relay-aided half-duplexing communicationscenario. Let yj,1 and yR be the corresponding received signalvectors of BSj and the relay at the first time slot as

yj,1 =K∑

k=1

Hj,(k,j)x(k,j) +C∑

i�=j

K∑k=1

Hj,(k,i)x(k,i)

+nj,1, j ∈ {1, . . . , C}, (1)

yR =C∑

j=1

K∑k=1

HUR(k,j)x(k,j) + nR, (2)

where nj,1 ∈ KN×1 and nR ∈ K

NR×1 are zero-meanunit-variance circularly symmetric additive white complexGaussian noise vectors at BSj and the relay, respectively. Thereceived signal vector of BSj at the second time slot is given as

yj,2 = HRBj TyR + nj,2, j ∈ {1, . . . , C}, (3)

where nj,2 ∈ KN×1 is also a zero-mean unit-variance circu-

larly symmetric additive white complex Gaussian noise vectorat BSj and T ∈ K

NR×NR is the relay beamforming matrix.Relay beamformer is implemented based on the global CSI.

Plugging (2) into (3) leads to

yj,2 =K∑

k=1

HRBj THUR

(k,j)x(k,j) +C∑

i�=j

K∑k=1

HRBj THUR

(k,i)x(k,i)

+(HRBj TnR + nj,2), j ∈ {1, . . . , C}. (4)

III. IMAC-ODIA FOR UPLINK CELLULAR NETWORK

Before we describe IMAC-ODIA for uplink cellular net-work, following example covers key idea of this section.

Example 2: Consider the uplink cellular network with(C, K, M, N) = (3, 2, 1, 2), so that there are three BSs withtwo antennas and two UEs in each cell with single antenna.In this network, each BS want to receive 1-dimensional signalfrom two UEs which are served by the BS. In this scenario,a BS cannot manage interference without UE beamformingand time extension, because antenna resource of BS is alreadyfully reserved for its serving UEs, i.e., N = KM . To handleinter-cell interferences, suppose there is a half-duplex relaywith six antennas which supports the uplink cellular network.Then, total number of six (= CN ) 1-dimensional signal fromeach cell can be successfully beamformed and transmitted toeach BS without dimension loss. The signals with appropriaterelay beamforming can be further used for interference can-cellation at each BS, where each BS can recover its desiredsignal.

First, we extend the idea of above example to describethe key idea of the proposed IMAC-ODIA scheme for anuplink cellular network, after which we simplify (1) and (4)using augmented channel matrices, which will be used for theproposed scheme. The IA conditions to be satisfied for theproposed scheme will also be described. Finally, we presentthe details of relay beamformer which meet the IA conditionsand analyze the achievable DoF of an uplink cellular networkwith the proposed IMAC-ODIA scheme.

A. IMAC-ODIA for a Symmetric Uplink Cellular Network

The conventional IA is to align interferences in a subspacewhich is linearly independent of the desired signal spaceby designing a beamformer at the transmitter. Then, using adecorrelator on the receiver side, the aligned interferences canbe perfectly cancelled. However, designing a beamformer atthe transmitter requires CSIT and increases the computationalcomplexity. In a cellular system, UEs are not likely to becapable of beamforming.

Thus, we propose an IMAC-ODIA scheme for an uplinkcellular network, where the interference vector received froma direct channel in the first time slot and the interference vectorreceived from the relay in the second time slot sum to zero.This is achieved by designing the beamforming matrix onlyat the relay as in Fig. 2. This implies that neither CSIT norcomplex computing is required at the UEs.

Using augmentation of the matrices and vectors in (1)and (2), the abovementioned IA description can be simpli-fied for the proposed IMAC-ODIA scheme. The augmentedchannel matrices for BSj are denoted as follows.

- Hj : interfering channel matrix from UEs of the other cellsto BSj

- HURj : interfering channel matrix from UEs of the other

cells to the relay- Hj : desired channel matrix from the UEs served by BSj

to BSj

- HUR

j : desired channel matrix from the UEs served by BSj

to the relayThese are defined as follows.

Hj =[Hj,(1,1) · · ·Hj,(K,j−1)

Hj,(1,j+1) · · ·Hj,(K,C)

] ∈ KN×(C−1)KM , (5)

HURj =

[HUR

(1,1) · · · HUR(K,j−1)

HUR(1,j+1) · · ·HUR

(K,C)

]∈ K

NR×(C−1)KM , (6)

Hj =[Hj,(1,j) · · · Hj,(K,j)

] ∈ KN×KM , (7)

HUR

j =[HUR

(1,j) · · · HUR(K,j)

]∈ K

NR×KM . (8)

Here, let xj and xj be the interfering data stream and thedesired data stream received at BSj , respectively. They shouldalso be converted into an augmented form as

xj =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

x(1,1)

...x(K,j−1)

x(1,j+1)

...x(K,C)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦∈ K

(C−1)KM×1, xj =

⎡⎢⎣

x(1,j)

...x(K,j)

⎤⎥⎦∈ K

KM×1.

From the augmented matrices and vectors, (1) and (4) can berewritten as

yj,1 = Hj xj + Hj xj + nj,1, j ∈ {1, . . . , C}, (9)

yj,2 = HRBj T(H

UR

j xj + HURj xj)

+(HRBj TnR + nj,2), j ∈ {1, . . . , C}. (10)

4754 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 7, JULY 2019

Fig. 2. Interference alignment by a relay in an uplink cellular network.

Ignoring the noise term and adding (9) and (10), the totalreceived signal vector at BSj becomes

yj,1 + yj,2 = (Hj + HRBj TH

UR

j )xj

+(Hj + HRBj THUR

j )xj , j ∈ {1, . . . , C}. (11)

In the proposed IMAC-ODIA scheme, the second term on theright-hand side in (11) is the total received interference signal,which should be cancelled. Thus, the IMAC-ODIA conditioncan be described as

Hj + HRBj THUR

j = 0, j ∈ {1, . . . , C}. (12)

This indicates the necessity of only the local CSIR at BSj

rather than the global CSIT and CSIR, though global CSI isrequired at the relay. At this point, our goal is to find therelay beamformer T which meets the requirements in (12). Thedesign of the relay beamformer T and its existence conditionwill be given in the next subsection.

B. Existence of a Relay Beamformer and Its Design

First, we introduce the important properties of theKronecker product as in the following propositions for themain theorem in this subsection.

Proposition 3 (Representation of the Matrix by theKronecker Product): Consider matrices A, B, C, and X andthe matrix equation

AXB = C. (13)

Here, (13) can be transformed by vectorization into

vec(AXB) = (BT ⊗ A)vec(X) = vec(C),

where vec(X) denotes the vectorization of matrix X by stack-ing the columns of X into a single column vector.

Proposition 4 (Transpose of the Kronecker Product): Formatrices A and B, the following property holds, that is,

(A ⊗ B)T = AT ⊗ BT .

Then, the following theorem provides the lower bound ofthe required number of antennas at the relay to implementIMAC-ODIA for a symmetric uplink cellular network.

Theorem 1 (Required Number of Antennas at the Relay forBeamforming): The relay beamformer T satisfying condition(12) exists if NR ≥ max{(C − 1)KM, CN}.

Proof: From Proposition 3, (12) can be transformed into

{(HURj )T ⊗HRB

j }vec(T) = vec(−Hj), j ∈ {1, . . . , C}. (14)

The C equations in (14) can be rewritten as a single equation

⎡⎢⎢⎢⎢⎣

(HUR1 )T ⊗ HRB

1

(HUR2 )T ⊗ HRB

2...

(HURC )T ⊗ HRB

C

⎤⎥⎥⎥⎥⎦ vec(T) =

⎡⎢⎢⎢⎣

vec(−H1)vec(−H2)

...vec(−HC)

⎤⎥⎥⎥⎦ . (15)

KIM AND NO: ACHIEVABLE DoF OF RELAY-AIDED MIMO CELLULAR NETWORKS USING ODIA 4755

Fig. 3. Matrix structure of HURi and HUR

j for i �= j.

Let H =

⎡⎢⎢⎢⎢⎣

(HUR1 )T ⊗ HRB

1

(HUR2 )T ⊗ HRB

2...

(HURC )T ⊗ HRB

C

⎤⎥⎥⎥⎥⎦ and h =

⎡⎢⎢⎢⎣

vec(−H1)vec(−H2)

...vec(−HC)

⎤⎥⎥⎥⎦. Then,

(15) can be represented as

H × vec(T) = h. (16)

Our goal is to find vec(T) satisfying (16). First, we haveto show its existence. Solutions of (16) exist if the followingcondition is satisfied as

rank([H h

]) = rank(H). (17)

Since all entries of h are drawn i.i.d. from a continuousdistribution, (17) can be satisfied if and only if H has fullrow rank, which means that HT has full column rank. FromProposition 4, HT can be rewritten as

HT =[HUR

1 ⊗ (HRB1 )T HUR

2 ⊗ (HRB2 )T

· · · HURC ⊗ (HRB

C )T]. (18)

In fact, HT in (18) is the Khatri-Rao product of the twomatrices HUR

and HRB , where

HUR =[HUR

1 HUR2 · · · HUR

C

],

HRB =[(HRB

1 )T (HRB2 )T · · · (HRB

C )T].

Note that HUR ∈ KNR×C(C−1)KM and HRB ∈ K

NR×CN .From Fig. 3, there are precisely (C − 2)K identical

NR × M submatrices for any pair of matrices in {HUR1 , HUR

2 ,

. . . , HURC } with probability one. This implies that the general-

ized Kruskal rank of HURdenoted by k′

HUR,Chas a maximum

value of one. Because any choice of two or more submatricesof HUR

, such as[HUR

i , HURj

]cannot be full column rank

matrix. From Proposition 1, k′HUR,C

can have two differentvalues

k′HUR,C

= min {� NR

(C − 1)KM�, C, 1}

=

{1, if NR ≥ (C − 1)KM

0, if NR < (C − 1)KM.

Furthermore, because HRB has submatrices, whose entries aredrawn i.i.d. from a continuous distribution, k′

HRB ,Cis also

given as

k′HRB ,C = min {�NR

N�, C} =

{C, if NR ≥ CN

�NR

N �, if NR < CN.

From Proposition 2, we can find the lower bound on thegeneralized Kruskal rank of HT = HUR � HRB . As long ask′

HUR,Cand k′

HRB ,Care not zeros, lower bound of k′

HT ,C=

k′HUR�HRB ,C

is given as

k′HT ,C = k′

HUR�HRB ,C≥ min {1 + C − 1, C} = C,

if NR ≥ max {(C − 1)KM, CN}.Considering that HT is partitioned into C submatrices, clearly,k′

HT ,C≤ C. Thus, the value of k′

HT ,Cis obtained as C, which

means that HT has full column rank from Definition 4 andthus (16) has the solution for vec(T). Thus, we prove thetheorem.

Note that H has its right inverse Hright = HT {HHT }−1.Then, the relay beamformer T is given as

T = vec−1NR

(nullrightH + Hrighth), (19)

where vec−1NR

(·) is defined as a devectorization function whichtransforms the vector into a matrix with a column size of NR.For simplification, nullright

H can be set as a zero vector.

C. Achievable DoF of a Symmetric Uplink CellularNetwork With IMAC-ODIA

First, we present the result of this subsection, whichdescribes the achievable DoF per cell for an uplink cellularnetwork with IMAC-ODIA.

Theorem 2 (Achievable DoF per Cell for an Uplink Cellu-lar Network With IMAC-ODIA): The achievable DoF per cellof a (C, K, M, N) uplink cellular network with IMAC-ODIAis given as

DoFcell =min{N, KM}

2,

where (C, K, M, N) should satisfy max{(C − 1)KM,CN} ≤ NR.

Proof: With the relay beamformer T in (19), the inter-cell interferences can be aligned at the relay and cancelled ateach receiver node. Thus, the DoF of BSj for two transmissiontime slots can be simply calculated as the rank of the effectivechannel of the desired data vector xj and divided by two.From (11) and (12), the effective channel Heff,j of BSj for

4756 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 7, JULY 2019

the desired signal defined as Heff,j xj = yj,1 + yj,2 and itsDoF are given as

Heff,j = Hj + HRBj TH

UR

j , (20)

DoFj =r�Heff,j

2.

Since the left inverse of HURj exists as {(HUR

j )T HURj }−1

(HURj )T , (12) can be transformed into HRB

j T =−Hj(H

URj )left + nullleft

HURj

. Therefore, (20) can be rewritten as

Heff,j = Hj−Hj(HURj )leftH

UR

j +nullleftHUR

jH

UR

j ∈ KN×KM .

(21)

Considering that the entries of each submatrix in the aug-

mented matrices Hj , Hj , (HURj )left, and H

UR

j are drawn i.i.d.from a continuous distribution, the rank of (21) can be derivedwith probability one as

r�Heff,j

= min{N, KM}.Finally, the achievable DoF per cell for a symmetric uplinkcellular network with IMAC-ODIA is given as

DoFcell =min{N, KM}

2.

D. IMAC-ODIA for an Asymmetric Uplink Cellular NetworkThe IMAC-ODIA for an asymmetric uplink cellular net-

work is described in this subsection, where ‘asymmetric’means that each cell has a different configuration. Con-sider an uplink cellular network with C cells and Kj

users served by a base station BSj with Nj antennasfor j ∈ {1, . . . , C}. User (k, j) is denoted by UE(k,j),k ∈ {1, . . . , Kj}, which is served by BSj . UE(k,j) isassumed to have M(k,j) antennas. Similarly, a relay is assumedto have NR antennas, and all channel parameters and matricesare defined in the same manner as the symmetric case withaugmented matrices (5) through (8). The number of requiredrelay antennas and the DoF of each cell are presented in theform of the following corollary.

Corollary 1 (IMAC-ODIA for an Asymmetric Uplink Cel-lular Network): The relay beamformer of IMAC-ODIA foran asymmetric uplink cellular network exists if the followinginequality is satisfied

NR ≥ max{( maxj∈{1,...,C}

∑i�=j

Ki∑k=1

M(k,i)), (C∑

j=1

Nj)}

and the achievable DoF for BSj is given as

DoFj =12

min{Nj,

Kj∑k=1

M(k,j)}.

The proof is identical to that of the symmetric case, andthus it is omitted here.

It is clear that many features of IMAC-ODIA make itpossible for a cellular network to adopt interference alignment.

Further, we extend the proposed scheme to a downlink cellularnetwork with the help of uplink-downlink duality for relay-aided cellular networks. This is discussed in the next section.

IV. IBC-ODIA FOR DONWLINK CELLULAR NETWORK

In this section, we extend the idea to a downlink cellularnetwork, which can be modeled as an IBC. Here, we onlyconsider a symmetric downlink cellular network because theIBC-ODIA scheme for an asymmetric case can be describedin the same manner as the IMAC-ODIA scheme for anasymmetric cellular network. Before we describe IBC-ODIAfor downlink cellular network, following example covers keyidea of this section.

Example 3: Consider the downlink cellular network with(C, K, M, N) = (3, 2, 1, 2), so that there are three BSs withtwo antennas and two UEs in each cell with single antenna.Therefore, there are two 2-dimensional inter-cell interferencesignals. Again, there is a half-duplex relay with six antennas.Similar to uplink cellular network case, each inter-cell inter-ference signal can be cancelled at UEs through relay beam-forming. But, there is still intra-cell interference remaining,which should be managed by additional BS beamforming.

Channel reciprocity refers to the relationship between uplinkand downlink channels, that is, each downlink channel matrixfor a pair of nodes is a complex conjugate transpose of thecorresponding uplink channel matrix for the same node pair.

A. System Model and Inter-Cell Interference AlignmentWe initially apply the result pertaining to duality from

earlier work [17] to the uplink cellular network withIMAC-ODIA, which gives us the motivation for IBC-ODIAfor a downlink cellular network. For the uplink case, thedesired data stream xj for BSj can be obtained from (11)and (20) as

xj = H†eff,j(yj,1 + yj,2), (22)

where H†eff,j serves as a decorrelator which handles intra-cell

interference. In Fig. 4(a), the uplink transmission process withIMAC-ODIA is described. If we assume channel reciprocityby a complex conjugate transpose operation, Fig. 4(a) is trans-formed into Fig. 4(b) without considering the data stream. Thisprovides the intuition for IBC-ODIA, where transmitter (BS)beamforming may be required for the IBC-ODIA scheme fora downlink cellular network, but no decorrelator is neededat the receiver (UE). Therefore, IBC-ODIA does not requireany complex operation at the UE compared to BS, whichsupports the feasibility of the proposed IBC-ODIA schemefor a downlink cellular network.

Unlike from the previous section, we describe the symmetricdownlink cellular network, that is, C cells and K users ineach cell. Cell j is served by a single base station BSj

for j ∈ {1, . . . , C}. User (k, j) is denoted by UE(k,j),k ∈ {1, . . . , K}, which is served by BSj as in Fig. 5. EachUE, BS, and relay is assumed to have M , N , and NR antennas,respectively. The channel from BSj to UE(k,i) is denoted bythe M ×N matrix H(k,i),j , the channel from BSj to the relay

KIM AND NO: ACHIEVABLE DoF OF RELAY-AIDED MIMO CELLULAR NETWORKS USING ODIA 4757

Fig. 4. Block diagrams of the proposed IMAC-ODIA scheme.

Fig. 5. Downlink cellular network with IBC-ODIA.

is denoted by the NR ×N matrix HBRj , and the channel from

relay to UE(k,j) is denoted by the M × NR matrix HRU(k,j).

xj denotes the transmitted data vector from BSj . Similarlyto the IMAC-ODIA scheme, the augmented matrices for theIBC-ODIA scheme are defined as

H(k,j) =[H(k,j),1 · · ·H(k,j),j−1

H(k,j),j+1 · · ·H(k,j),C

] ∈ KM×(C−1)N ,

HBRj =

[HBR

1 · · ·HBRj−1

HBRj+1 · · ·HBR

C

] ∈ KNR×(C−1)N

and the inter-cell interference vector for UEs at cell j isdefined as

xj =[xT1 · · · xT

j−1 xTj+1 · · · xT

C

]T ∈ K(C−1)N×1.

Similarly to the IMAC-ODIA case, by ignoring the noiseterm, the total received vector at UE(k,j) for two time slots is

given as

y(k,j),1 + y(k,j),2 = (H(k,j),j + HRU(k,j)TDLHBR

j )xj

+(H(k,j) + HRU(k,j)TDLHBR

j )xj ,

where TDL represents the relay beamforming matrix for down-link transmission. Note that TDL only aligns the inter-cellinterferences. The remaining intra-cell interferences will bealigned by beamforming at the BS, which will be describedlater. The BS beamforming for intra-cell IA originates fromthe intuition for uplink-downlink duality as discussed earlier.

For k ∈ {1, . . . , K} and j ∈ {1, . . . , C}, the interferencealignment condition should be satisfied as

H(k,j) + HRU(k,j)TDLHBR

j = 0. (23)

Further, TDL satisfying the above condition is given as

TDL = vec−1NR

(nullright

HDL+ (HDL)righthDL

),

4758 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 7, JULY 2019

where HDL =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(HBR1 )T ⊗ HRU

(1,1)

(HBR1 )T ⊗ HRU

(2,1)

...

(HBR1 )T ⊗ HRU

(K,1)

(HBR2 )T ⊗ HRU

(1,2)

...

(HBRC )T ⊗ HRU

(K,C)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

and

hDL =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

vec(−H(1,1))vec(−H(2,1))

...vec(−H(K,1))vec(−H(1,2))

...vec(−H(K,C))

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

The following theorem gives the number of antennasrequired at the relay for IBC-ODIA, where the proof is similarto that of IMAC-ODIA.

Theorem 3 (Required Number of Relay Antennas for theDownlink Cellular Network With IBC-ODIA): The relay beam-former TDL for the IBC-ODIA satisfying (23) exists if NR ≥max{(C − 1)N, CKM}.

With the proper design of TDL, the inter-cell interferencesare fully cancelled at the UEs. As noted above, intra-cellinterferences will be aligned by designing the beamformerat the BS. The beamformer design is described in the nextsubsection.

B. Intra-Cell Interference Alignment by BS Beamforming

The beamformer at the BS for the intra-cell IA can bedesigned jointly with the relay beamformer. For simplicity,each UE desires to receive d independent data streams from itsserving BS. The transmitted signal from BSj can be given as

xj = V(1,j)s(1,j) + V(2,j)s(2,j) + · · · + V(K,j)s(K,j),

where s(k,j) ∈ Kd×1 denotes the data stream for UE(k,j) from

BSj and V(k,j) ∈ KN×d is the beamforming matrix designed

at BSj for UE(k,j).The intra-cell interference at each UE is said to be fully

aligned by the BS beamforming if the following condition issatisfied as⎡

⎢⎢⎣H(1,j),j + HRU

(1,j)TDLHBRj

...H(K,j),j + HRU

(K,j)TDLHBRj

⎤⎥⎥⎦

[V(1,j) · · · V(K,j)

]

= AjVj =

⎡⎢⎢⎣

IM×d 0 0

0. . . 0

0 0 IM×d

⎤⎥⎥⎦ .

Since each entry of H(k,j),j is drawn i.i.d. from a continuousdistribution, Aj is either a full row or a full column rank matrixwith probability one with a right or left inverse matrix.

Hence, Vj is expressed as

Vj = A†j

⎡⎢⎣

IM×d 0 0

0. . . 0

0 0 IM×d

⎤⎥⎦

=[{A†

j}1:d {A†j}M+1:M+d

· · · {A†j}(K−1)M+1:(K−1)M+d

]. (24)

From (24), we can design Vj with A†j , where

A†j =

{Aright

j = ATj (AjAT

j )−1, if N ≥ KM

Aleftj = (AT

j Aj)−1ATj , if N < KM.

Note that N ≥ Kd should be guaranteed for successful datatransmission. Thus, V(k,j) makes it possible for BSj to delivermessages to UE(k,j) successfully with DoF d, where d ≤ Mshould be satisfied.

After designing the relay beamformer and the BS beam-former, the inter-cell and intra-cell interferences can be com-pletely removed at each UE. Accordingly, we summarizethe achievable DoF for a downlink cellular network withIBC-ODIA via the following theorem.

Theorem 4 (Achievable DoF for a Downlink Cellular Net-work With IBC-ODIA): The achievable DoF per cell andper UE for a (C, K, M, N) downlink cellular network withIBC-ODIA is expressed as

DoFDL,cell =min{Kd, N}

2≤ min{KM, N}

2,

DoFDL,UE =DoFDL,cell

K,

where (C, K, M, N) should satisfy max{(C − 1)N,CKM} ≤ NR.

Proof of Theorem 4 is similar to that of Theorem 2 and isthus omitted.

V. FD-ODIA FOR FULL-DUPLEX CELLULAR NETWORK

Furthermore, we focus on a cellular network, where full-duplex operation is feasible for BSs and UEs, that is, everyBS and UE can transmit and receive data at the same time slot.Due to the complexity of full-duplexing, it has not been a mainconcern for researchers, but it is known that full-duplexing,when possible, increases the throughput. However, the need fora management scheme of the additional interference caused byfull-duplex operation arises, which motivates the developmentof the proposed FD-ODIA scheme for a full-duplex cellularnetwork.

A. FD-ODIA for Full-Duplex Cellular Network

Recently, several researchers have investigated interferencealignment on full-duplex network due to the higher throughputachievable by the full-duplex mode [19]–[21]. In [19], it wasproved that allowing the full-duplexing in each node exactlydoubles the DoF performance compared to that of a half-duplexing multi-user interference channel. The DoF derivationprocess in [19] was based on network decomposition, wherea similar methodology can be applied to a cellular network.

KIM AND NO: ACHIEVABLE DoF OF RELAY-AIDED MIMO CELLULAR NETWORKS USING ODIA 4759

Fig. 6. Full-duplex network decomposition of a single full-duplex cell.

Fig. 7. Full-duplex cellular network served by a single half-duplex relay,where only interference-related channels are depicted.

In [20], if each node operates in full-duplex mode, then asingle full-duplex cell can be decomposed into two half-duplexcells, one on the uplink transmission and the other on thedownlink transmission as shown in Fig. 6.

Considering full-duplex network decomposition as well asthe fact that the proposed scheme can be operated on eitherthe uplink or downlink, we can apply the IMAC-ODIA andthe IBC-ODIA schemes proposed in the previous sections toa full-duplex cellular network, that is, FD-ODIA. Althoughthe relay operates in the half-duplex mode, the total DoF ofthe proposed FD-ODIA scheme can be doubled comparedto the half-duplex case. We describe the symmetric full-duplex cellular network model below. Throughout the section,we assume that BSs can cancel self-interferences, that is, thereis no self-interference for each cell.

Let C and K be the numbers of cells and users in eachcell, respectively. Cell j is served by a single base station infull-duplex mode denoted by BSj for j ∈ {1, . . . , C}. User(k, j) also operates in full-duplex mode denoted by UE(k,j),k ∈ {1, . . . , K}, which is served by BSj , as in Fig. 7. Notethat the UEs only receive their desired signals from theircorresponding BSs, implying the absence of device-to-devicecommunication. Each UE, each BS, and the relay is assumed

to have M , N , and NR antennas, respectively. Note that therelay operates in half-duplex mode and the full-duplex relaywill be discussed later. Allowing for full-duplex operationat each source and destination node generates the followingadditional channel matrices.

- Channel from UE(k,i) to BSj : N × M matrix HUBj,(k,i)

- Channel from UE(k1,i) to UE(k2,j): M × M matrixHUU

(k2,j),(k1,i)

- Channel from UE(k,j) to relay: NR × M matrix HUR(k,j)

- Channel from relay to UE(k,j): M × NR matrix HRU(k,j)

- Channel from relay to BSj : N × NR matrix HRBj

- Channel from BSi to BSj : N × N matrix HBBj,i

- Channel from BSj to UE(k,i): M × N matrix HBU(k,i),j

- Channel from BSj to relay: NR × N matrix HBRj

B. Achievable DoF of Full-Duplex CellularNetwork With FD-ODIA

Here, we derive the achievable DoF and the number ofantennas required at the relay for a full-duplex cellular networkwith FD-ODIA as in the following theorem.

Theorem 5 (Achievable DoF for a Full-Duplex CellularNetwork With FD-ODIA): The achievable DoFs per cell,per BS, and per UE of the (C, K, M, N) full-duplex cellularnetwork are given as

DoFFD,cell = DoFFD,BS + K × DoFFD,UE = min{KM, N},DoFFD,BS =

min{KM, N}2

,

DoFFD,UE =min{M, N

K }2

,

where (C, K, M, N) should satisfy C(KM + N) ≤ NR.Proof: We prove this theorem using the IMAC-ODIA and

IBC-ODIA schemes. As noted earlier, the full-duplex cellularnetwork with a relay can be decomposed into disjoint uplinkand downlink networks sharing the relay as in Fig. 7.

Note that additional interferences occur due to full-duplexing which degrade the full-duplex network DoFgain [22]–[25], i.e., UE-to-UE channels in the intra-cells orinter-cells now become interfering channels. These UE-to-UEinterferences will be considered in the further development ofrelay beamformer design.

To describe the FD-ODIA scheme, we start with matrixaugmentation with the same approach used earlier, where therelay only aligns the inter-cell interferences as

HBSj =

[HBB

j,1 · · · HBBj,j−1 HBB

j,j+1 · · ·HBB

j,C HUBj,(1,1) · · · HUB

j,(K,j−1) HUBj,(1,j+1)

· · · HUBj,(K,C)

]∈ K

N×(C−1)(KM+N),

HUE(k,j) =

[HBU

(k,j),1 · · · HBU(k,j),j−1 HBU

(k,j),j+1 · · ·HBU

(k,j),C HUU(k,j),(1,1) · · · HUU

(k,j),(k−1,j)

HUU(k,j),(k+1,j) · · · HUU

(k,j),(K,C)

]

∈ KM×{(CK−1)M+(C−1)N},

4760 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 7, JULY 2019

HBRj =

[HBR

1 · · · HBRj−1 HBR

j+1 · · · HBRC

HUR(1,1) · · · HUR

(K,j−1) HUR(1,j+1) · · · HUR

(K,C)

]

∈ KNR×(C−1)(KM+N),

HUR(k,j) =

[HBR

1 · · · HBRj−1 HBR

j+1 · · · HBRC

HUR(1,1) · · · HUR

(k−1,j) HUR(k+1,j) · · · HUR

(K,C)

]

∈ KNR×{(CK−1)M+(C−1)N}.

In addition, the interference vectors should be redefined forthe additional interferences as

xBSj =

[(xBS

1 )T · · · (xBSj−1)

T (xBSj+1)

T · · ·(xBS

C )T (xUE(1,1))

T · · · (xUE(K,j−1))

T

(xUE(1,j+1))

T · · · (xUE(K,C))

T]T

,

xUE(k,j) =

[(xBS

1 )T · · · (xBSj−1)

T (xBSj+1)

T · · ·(xBS

C )T (xUE(1,1))

T · · · (xUE(k−1,j))

T

(xUE(k+1,j))

T · · · (xUE(K,C))

T]T

.

Note that xBSj ∈ K

(C−1)(KM+N)×1, xUE(k,j) ∈

K{(CK−1)M+(C−1)N}×1 and xBS

j , xUE(k,j) are the transmitted

vectors from BSj and UE(k,j), respectively. The definitionsof the desired signal channel set {H} and the desired signalvector set {x} are exactly the same as those of the half-duplexuplink and downlink cellular network cases.

As noted above, the augmented interference matrices onlycontain the inter-cell interferences. Thus, additional BS beam-forming will align the intra-cell interferences at the UEs,which will be described later.

The IA condition for FD-ODIA is similar to the IMAC/IBC-ODIA case except for the fact that it should be satisfiedboth on the BSs and the UEs as

HBSj + HRB

j TFDHBRj = 0, j ∈ {1, . . . , C}, (25)

HUE(k,i)+HRU

(k,i)TFDHUR(k,i) = 0, k ∈ {1, . . . , K}, i ∈{1, . . . , C}.

(26)

The existence of the relay beamformer TFD for the FD-ODIAscheme is also proved in a manner similar to that of theIMAC/IBC-ODIA case except for the number of relay anten-nas. In fact, in the full-duplex network case, the number ofrelay antennas NR should satisfy

NR ≥ max{(C − 1)(KM + N), (C − 1)N

+(CK − 1)M, C(KM + N)} = C(KM + N),

where C(KM + N) is the total number of antennas in thecellular network.

Hence, TFD satisfying (25) and (26) is derived as

TFD =

vec−1NR

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(HBR1 )T ⊗ HRB

1,1...

(HBRC )T ⊗ HRB

C

(HUR(1,1))T ⊗ HRU

(1,1)

...

(HUR(K,C))

T ⊗ HRU(K,C)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

right ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

vec(−HBS1 )

...

vec(−HBSC )

vec(−HUE(1,1))

...

vec(−HUE(K,C))

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Note that the right null vector term in the argument ofvec−1

NR(·) is omitted.

Furthermore, it is necessary to design the beamformer VFDj

at BSj in the same manner as IBC-ODIA, which should satisfythe following intra-cell IA condition⎡

⎢⎣HBU

(1,j),j + HRU(1,j)TFDHBR

j

...HBU

(K,j),j + HRU(K,j)TFDHBR

j

⎤⎥⎦ [

VFD(1,j) · · ·&VFD

(K,j)

]

= AFDj VFD

j =

⎡⎢⎣

IM×d 0 0

0. . . 0

0 0 IM×d

⎤⎥⎦ .

Similar to the IBC-ODIA case, AFDj is either a full row or a

full column rank matrix with probability one depending on thenumber of antennas. In this case, VFD

j can be derived as

VFDj =

[{(AFDj )†}1:d {(AFD

j )†}M+1:M+d

· · · {(AFDj )†}(K−1)M+1:(K−1)M+d

],

where (AFDj )† is either the right or left inverse matrix of AFD

j .Since the interferences are fully aligned and cancelled at

each BS, the DoF of BSj for the full-duplex cellular networkcan be achieved as

DoFFD,BSj =

min{KM, N}2

,

and the DoF of UE(k,j) can also be achieved as

DoFFD,UE(k,j) =

min{KM, N}2K

=min{M, N

K }2

.

Thus, we prove the theorem.Note that the total achievable DoF of the proposed

FD-ODIA is doubled compared to the half-duplex case. It isalso interesting that the proposed scheme requires only ahalf-duplexing AF relay to achieve full-duplexing gain ina full-duplex cellular network with FD-ODIA. The designmethodology of the proposed IMAC/IBC-ODIA schemes candirectly be applied to the FD-ODIA scheme.

In the previous work [19], it was shown that causal MIMOfull-duplex relay based on the DF protocol cannot increasethe DoF but that non-causal and instantaneous full-duplex DFrelaying can increase the DoF. The result in Theorem 5 canbe directly applied to the case of full-duplex instantaneous AFrelay as in the following corollary.

KIM AND NO: ACHIEVABLE DoF OF RELAY-AIDED MIMO CELLULAR NETWORKS USING ODIA 4761

Corollary 2 (Achievable DoF With Full-Duplex Instanta-neous AF Relay): The achievable DoFs per cell, BS, andUE of the same full-duplex cellular network with full-duplexinstantaneous AF relaying are given as

DoFinstFD,cell = DoFinst

FD,BS + K × DoFinstFD,UE = 2 min{KM, N},

DoFinstFD,BS = min{KM, N},

DoFinstFD,UE = min{M,

N

K},

where the relay antenna requirement is the same as that inTheorem 5.

Remark 1 (Half-Duplex UE With Full-Duplex BS): Due tothe hardware complexity limitation, an UE cannot be operatedin full-duplex mode, and thus half-duplex UEs served by full-duplex BSs can be considered. In this case, the UEs can bepartitioned into either an uplink group or a downlink group.Assume that K1 UEs are in the uplink group and K2 UEs arein the downlink group with M antennas at each UE for bothgroups. Note that the BSs operate in the full-duplex mode withN antennas. Then the DoF of the above system can be givenas the following corollary.

Corollary 3 (DoF of a Cellular Network With Half-DuplexUEs Served by a Full-Duplex BS): The achievable DoF percell with the half-duplex UEs served by the full-duplex BS isgiven as

DoFHD+FD,cell = DoFHD+FD,BS + K2 × DoFHD+FD,UE, (27)

where DoFHD+FD,BS = min{K1M,N}2 and DoFHD+FD,UE =

min{M, NK2

}2 .

The DoF in (27) can be achieved with the IA schemesimilar to the previous FD-ODIA scheme. It can be achievedby FD-ODIA with the exclusion of non-existing BS-to-UEchannels, UE-to-BS channels, and UE-to-UE channels in therelay beamformer design, which vanishes automatically due tothe half-duplexing of UEs.

Thus, it should be noted that the proposed schemes are veryflexible, that is, IA schemes for various cellular networks canbe implemented with the proposed schemes by modifying thechannel assignment protocol in the relay.

VI. DISCUSSION

A. DoF Improvement Without Time Extension

With the proposed schemes, the total achievable DoF of ahalf-duplex uplink cellular network DoFODIA,uplink is given asC min{N,KM}

2 . For the (C, K, M, N) uplink cellular networks,DoFODIA,uplink of the proposed scheme with a relay and withoutUE beamforming is much larger than DoFlinear,uplink of thelinear scheme without a relay and with UE beamforming,which can be analyzed using the outer bound of DoF of thecellular network without time extension.

An (C, K, M, N) IMAC can be transformed into an(C, KM, N) IC with transmitter cooperation, that is, all UEsat each BS transmit their data jointly as a single super-UEas shown in Fig. 8. Let DoFLinear,coop denote the total DoF ofthe (C, KM, N) IC. Then, the following inequality is easilygiven as

DoFlinear,uplink ≤ DoFlinear,coop.

Fig. 8. IMAC-to-IC transformation with UEc, 1 ≤ c ≤ C, which is asuper-UE with KM antennas.

Further, using the result in [26], DoFlinear,coop is bounded as

DoFlinear,uplink ≤ DoFlinear,coop ≤ KM + N. (28)

However, the total DoF DoFODIA,uplink of the proposed schemeincreases proportional to the number of cells, which cannot beachieved in (28).

Performance comparison with the previous literature for(C, K, M, N) = (3, 1, 1, 1) uplink cellular network is givenin Table I, where we can find that our proposed schemeachieves comparable DoF to the previous interference align-ment/neutralization schemes. Note that, compared to theschemes in [7], [27], [28], and [29], our proposed scheme canalso achieve full DoF 3 for the given network if the relay canoperate in instantaneous full-duplex mode for uplink cellularnetwork. From [32], it is easy to find that DoFODIA,uplink =C min{N,KM}

2 is indeed the maximally achievable DoF in anuplink cellular network with a half-duplex relay.

Note that the information-theoretic upper bound of the(C, K, M, N) IMAC (≈ KMN

KM+N , see [10]) is approximatelytwo times larger than the achievable DoF of the proposedscheme in an extreme case (i.e., N KM or N � KM ),whereas the upper bound can only be achieved by infitnitetime extension.

Additionally, in an antenna regime where BSs and UEs ina cell have similar numbers of antennas (i.e., N ≈ KM ),the proposed IMAC/IBC-ODIA schemes achieve the optimalDoF of MIMO IMAC/IBC, which is believed to be achievedonly through asymptotic IA or IA aided by a full-duplex relay.

B. Achievable DoF per Cell Versus the Number of Antennas

The simulation environment is (C, K) = (3, 4)-uplinkcellular network. In the simulations, the channel elementsare i.i.d. random variables drawn from CN (0, 1). Since weevaluate the performance of achievable DoF per cell, the noiseat receiver node is ignored in simulation. In Fig. 9(a),we can check that the achievable DoF per cell cannot exceedmin{N,KM}

2 = 4 when the number of UE antennas gets largerthan two, N = 8. Similarly, in Fig. 9(b), given achievableDoF per cell performance also matches our result, M = 2.

4762 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 7, JULY 2019

TABLE I

ACHIEVABLE DOF FOR (C, K,M, N) = (3, 1, 1, 1) UPLINK CELLULAR NETWORK

Fig. 9. Simulation results of achievable DoF per cell.

In fact, achievable DoF per cell through IMAC-ODIA cannotbe directly increased by a single factor.

In Fig. 10, we described the achievable DoF per cellperformance with the number of relay antenna. From Fig. 10,we can find that if the number of relay antennas is less thanmax{(C − 1)KM, CN}, the DoF per cell performance isdegraded. DoF performance degradation occurs because whenthe number of relay antennas is not enough to serve thenetwork, some nodes suffer from very high interference fromrelay so that these nodes cannot successfully communicate.Further, we can think of the case when the number of relayantennas is larger than max{(C − 1)KM, CN}. In Fig. 10,we can easily check that the DoF per cell performance

Fig. 10. Achievable DoF versus the number of relay antennas NR .

saturates. Then the question is, “Is there any possible gainfrom additional relay antennas?” We answer this question.

First, we can think of antenna selection at relay, where therelay turns off residual antennas to save power. The antennaselection should be designed to maximize effective SINR atBSs to increase throughput. Second, we can use additionalantennas to increase power of desired signal, which leads toincrease of SINR at BSs. Consider that an uplink cellularnetwork with the IA-applied data stream through decorrelatorat the BS is expressed as (22) in the previous section. Ourgoal is to design a relay beamformer satisfying the followingadditional statement as

Heff,j = Hj + HRBj T+H

UR

j = (1 + α)Hj

⇒ HRBj T+H

UR

j = αHj , (29)

where α ≥ 0 and T+ denotes the relay beamformer withadditional gain. If (29) is satisfied, it can be interpreted asboosting the received signal power level by α, while the DoFperformance remains the same.

KIM AND NO: ACHIEVABLE DoF OF RELAY-AIDED MIMO CELLULAR NETWORKS USING ODIA 4763

We skip the details because the proof is identical to those inTheorems 1 and 2. The existence of T+ satisfying (29) can beguaranteed when NR ≥ max{CKM, CN}. Hence, the designof T+ is given as

T+ = vec−1NR

⎛⎜⎜⎜⎜⎝

⎡⎢⎢⎢⎣(HUR

1 )T ⊗ HRB1

(HUR2 )T ⊗ HRB

2...

(HURC )T ⊗ HRB

C

⎤⎥⎥⎥⎦

right ⎡⎢⎢⎢⎣

vec(H1)vec(H2)

...vec(HC)

⎤⎥⎥⎥⎦

⎞⎟⎟⎟⎟⎠ ,

where for simplicity, the right null vector term in the argumentof vec−1

NR(·) is omitted and Hj ∈ K

N×CKM and HURj ∈

KNR×CKM are correspondingly defined as

HURj =

[HUR

1,1 HUR2,1 · · · HUR

K,C ,],

Hj =[−Hj,(1,1) · · · −Hj,(K,j−1) αHj,(1,j) · · ·αHj,(K,j) −H1,(1,j+1) · · · −Hj,(K,C)

].

Then, the relay beamformer T+ for the proposedIMAC-ODIA can align the interferences, while also boostingthe desired signal power level to KM additional antennas atthe relay. Thus, from the additional relay antenna, we canachieve better throughput.

VII. CONCLUSION

In this paper, we proposed IA schemes for the cellularnetworks which operate with a half-duplex relay without UEbeamforming or CSI handling, representing the fundamentallimit when applying IA to cellular network. We proposedan uplink cellular network with IMAC-ODIA and derived itsachievable DoF. Using linear beamforming at the relay anduplink-downlink duality, we extended the proposed schemewith the BS beamformer design to a downlink cellular net-work without UE beamforming or CSI handling. Further,the proposed schemes can also be extended to a full-duplexcellular network, achieving doubled DoF compared to a half-duplex cellular network. Although it has DoF gap from theinformation-theoretic upper bound, we proved that the pro-posed schemes have network gain proportional to the networksize, which cannot be achieved in a cellular network withlinear IA. Moreover, for some antenna regime, the proposedschemes are able to achieve the optimal DoF.

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Ho-Youn Kim received the B.S. and Ph.D. degreesin electrical and computer engineering from SeoulNational University, Seoul, South Korea, in 2012and 2018, respectively. His research interests includeinterference alignment, caching, cellular network,and error correcting codes.

Jong-Seon No received the B.S. and M.S.E.E.degrees in electronics engineering from SeoulNational University, Seoul, Korea, in 1981 and 1984,respectively, and the Ph.D. degree in electrical engi-neering from the University of Southern California,Los Angeles, in 1988. He was a Senior MTSwith Hughes Network Systems from 1988 to 1990.He was also an Associate Professor with the Depart-ment of Electronic Engineering, Konkuk University,Seoul, from 1990 to 1999. He joined the faculty ofthe Department of Electrical and Computer Engi-

neering, Seoul National University, in 1999, where he is currently a Professor.His research interests includes error-correcting codes, sequences, cryptog-raphy, LDPC codes, interference alignment, and wireless communicationsystems. He was a recipient of IEEE Information Theory Society Chapter ofthe Year Award in 2007. He was elevated to an IEEE Fellow in researchengineering and science through the IEEE Information Theory Society,in 2011. From 1996 to 2008, he served as a Founding Chair of the SeoulChapter, IEEE Information Theory Society. He was a General Chair forSequence and Their Applications 2004 (SETA 2004), Seoul, Korea. He alsoserved as a General Co-Chair for International Symposium on InformationTheory and Its Applications 2006 (ISITA 2006) and International Symposiumon Information Theory 2009 (ISIT 2009) in Seoul, Korea. He became theCo-Editor-in-Chief of the Journal of Communications and Networks in 2012.