Relay-Aided MIMO Cellular Networks Using Opposite Directional Interference Alignment
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Relay-Aided MIMO Cellular Networks UsingOpposite Directional Interference Alignment
Hoyoun Kim and Jong-Seon No
Abstract
In this paper, we propose an interference alignment (IA) scheme for the multiple-input multiple-output (MIMO) uplink cellular network with the help of a relay which operates in half-duplex mode. Theproposed scheme only requires global channel state information (CSI) knowledge at the relay, with notransmitter beamforming and time extension at the user equipment (UE), which differs from conventionalIA schemes for cellular networks. We derive the feasibility condition of the proposed scheme for thegeneral cellular network configuration and analyze the degrees-of-freedom (DoF) performance of theproposed IA scheme while providing a closed-form beamformer design at the relay. Extensions of theproposed scheme to downlink and full-duplex cellular networks are also proposed in this paper. TheDoF performance of the proposed schemes is compared to that of a linear IA scheme for a cellularnetwork with no time extension. It is also shown that advantages similar to those in the uplink casecan be obtained for the downlink case through the duality of a relay-aided interfering multiple-accesschannel (IMAC) and an interfering broadcast channel (IBC). Furthermore, the proposed scheme for afull-duplex cellular network is shown to have advantages identical to those of a number of proposedhalf-duplex cellular cases.
Index Terms
Degrees-of-freedom (DoF), interference alignment (IA), interfering broadcast channel (IBC), in-terfering multiple-access channel (IMAC), multiple-input multiple-output (MIMO), opposite directionalinterference alignment (ODIA), relay.
H. Kim and J.-S. No are with the Department of Electrical and Computer Engineering, INMC, Seoul National University,Seoul 08826, Korea. e-mail: [email protected], [email protected].
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I. I
NTRODUCTION
It is known that high degrees-of-freedom (DoF) can be achieved in a multi-user interferenceenvironment by means of interference alignment (IA). The optimal DoF of single-input single-output (SISO) time-varying interference channel has been derived through an asymptotic IAscheme [1]. Research of IA in multiple-input multiple-output (MIMO) channels also has beendone and the optimal DoF region for a multi-user MIMO interference channel has been revealed[2], showing that IA can also increase the DoF of MIMO communication systems.Unfortunately, conventional IA schemes cannot be directly applied to cellular networks becausethe channel state information (CSI) feedback to transmitter and time extension are required. Forexample, a subspace IA scheme for a SISO uplink cellular network, finding that the optimal DoFof one can be achieved as the number of cellular users increases was suggested [3]. However,the subspace IA scheme requires more time extensions as the number of users increases. Thusthe transmission/reception delay of the information increases as the number of users in thenetwork becomes large, which is not suitable for a cellular network. In general, user equipment(UE) is not practically able to handle/process beamforming by CSI feedback and communicatingnodes require short transmission/reception delays. In order to reduce the CSI feedback and timeextension requirements, relay-aided IA schemes have been proposed for wireless communicationnetworks, where the relay handles the CSI instead of the transmitter, adding the potential toreduce the time extension requirement in the MIMO interference channel. An IA scheme withtwo time slots and without CSIT feedback for a MIMO interference channel was proposed [4],but the effective noise increases due to zero-forcing and global CSI knowledge at the receiver(CSIR) is required. To overcome the abovementioned limitations on this scheme, an oppositedirectional interference alignment (ODIA) scheme with relay was introduced [5]. It was foundthat the ODIA can also achieve the optimal DoF in a symmetric MIMO interference channel.Recently, researchers studied the achievable DoF for a cellular network where a base station(BS) is equipped with directional or omni-directional antennas when considering the cellularnetwork topology [6] [7]. In [8], the optimal DoF and an achievable IA scheme were derived ina MIMO cellular network using earlier result in [9]. However, this scheme also requires infinitetime extensions to achieve promised DoF for a large network, which is not practical for a cellularnetwork. Furthermore, a one-shot linear IA scheme was proposed based on subspace IA [10], but
October 15, 2018 DRAFTUBMISSION FOR... 3 finding and providing feedback with regard to the optimized variables to align the interferencesmay place a heavy load on the UEs.Several studies have focused on the potential role of relays in a cellular network with IA.An IA scheme for a two-cell downlink cellular network based on different half-duplex relayingschemes with the decode-and-forward (DF) protocol was proposed [11]. However, it is knownthat the relay design becomes complex when the DF protocol is used in the relay [12]. It wasshown that in a quasi-static flat-fading environment, a MIMO cellular network can achieve higherDoF by using a full-duplex relay with finite time extension [13]. In practice, relay echo due to thefull-duplex operation at the relay can severely degrade the performance of the cellular network.In this paper, in order to solve the problems of applying IA to MIMO cellular networks,we propose an interfering multiple-access channel (IMAC)-ODIA scheme for an uplink cellularnetwork with a half-duplex amplify-and-forward (AF) relay, where only two time slots are neededregardless of the network size, to achieve interference alignment. It has the advantage of no timeextension and the ability to be operated with various channel settings. Further, CSIT at the UEsare not required in the proposed scheme. We evaluate the DoF performance of the proposedIMAC-ODIA scheme and show that for certain antenna configuration, the proposed scheme canachieve the optimal DoF for a MIMO cellular network.Furthermore, we propose a relay-aided IA scheme for a downlink cellular network to resolvethe global CSIR requirement in the conventional IA scheme which requires a heavy load on theUEs. Using the uplink-downlink duality from earlier work [14], we propose a modification of theIMAC-ODIA scheme for the downlink, which leads to interfering broadcast channel (IBC)-ODIAwith features similar to those of the IMAC-ODIA scheme. Here, neither decorrelator constructionnor global CSIR is required at the UEs. We also extend our idea to full-duplex cellular network,referred to as full-duplex ODIA (FD-ODIA). It is important to handle interference to obtainfull-duplexing gain and accordingly, we provide an IA beamformer design at each node anddiscuss its DoF performance.This paper is organized as follows. In Section II, we present preliminary information concern-ing the properties of the tensor product and system model of the cellular network with a relay.In Section III, the IA condition and the IMAC-ODIA scheme for the uplink cellular network areproposed. An IBC-ODIA scheme for a downlink cellular network and its extension to a full-duplex cellular network are also proposed in Section IV and Section V, respectively. Section VI
October 15, 2018 DRAFTUBMISSION FOR... 4 includes the discussion, and finally the conclusion is given in Section VII.II. P
RELIMINARIES AND S YSTEM M ODEL
In this section, we introduce several properties of the tensor product of matrices [15] anddescribe the cellular network model with an AF relay. First, we focus on an uplink symmetriccellular network modeled as a symmetric IMAC.Throughout the paper, scalars are denoted in lowercase, vectors are written in boldface low-ercase, and matrices are indicated by boldface capital letters. Several definitions pertaining tomatrix A and vector a are given below.- a ij : ( i, j ) component of matrix A - a i : i -th component of vector a - null left A and null right A : 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 r A : rank of matrix A - ( · ) T , ( · ) H , ( · ) left , ( · ) right , and ( · ) † : transpose, complex conjugated transpose, left inverse,right inverse, and Moore-Penrose pseudo inverse, respectively- I × J : I × J zero matrix and subscript can be omitted when the size of the zero matrix isnot important.- I I : I × I identity matrix- I I × J : I × J rectangular identity matrix = I J ( I − J ) × J , I ≥ J - K : real or complex field, R or C A. Tensor Product and Kruskal Rank
First, we briefly introduce the pre-defined tensor products of matrices, that is, the Kroneckerand Khatri-Rao products. The Kronecker product of A and B is defined below. Definition 1: A ⊗ B is the Kronecker product of A and B defined as October 15, 2018 DRAFTUBMISSION FOR... 5 A ⊗ B = a B a B · · · a B a B · · · ... ... . . . . Let A = [ A ... A D ] and B = [ B ... B D ] be two partitioned matrices with an equal numberof partitions. The Khatri-Rao product of A and B is then defined as the partition-wise Kroneckerproduct as given below. Definition 2: A (cid:12) B is the Khatri-Rao product of partitioned matrices A and B defined as A (cid:12) B = (cid:104) A ⊗ B ... A D ⊗ B D (cid:105) . Further, Kruskal rank and generalized Kruskal ranks are introduced as follows.
Definition 3:
The Kruskal rank of matrix A denoted by rank k ( A ) or k A , is the maximal number r such that any set of r columns of A is linearly independent. Definition 4:
The generalized Kruskal rank of partitioned matrix A = [ A ... A D ] denoted byrank k (cid:48) ( A ) or k (cid:48) A , is the maximal number r such that any set of r submatrices of A yields a setof linearly independent columns.We introduce a number of propositions for the generalized Kruskal rank from [15], whichwill be used in the next sections. Proposition 1 (Generalized Kruskal rank of an uniformly partitioned matrix [15]):
Let A =[ A ... A D ] be a matrix whose entries are drawn i.i.d. from a continuous distribution and parti-tioned in D submatrices with A r ∈ K I × L . Then, the generalized Kruskal rank of A is min {(cid:98) IL (cid:99) , D } . Proposition 2 (Lemma 3.2 [15]):
Consider the partitioned matrices A = [ A ... A D ] with A r ∈ K I × L r and B = [ B ... B D ] with B r ∈ K J × M r , ≤ r ≤ D . Then,1) If k (cid:48) A = 0 or k (cid:48) B = 0 , then k (cid:48) A (cid:12) B = 0 .2) If k (cid:48) A ≥ and k (cid:48) B ≥ , then k (cid:48) A (cid:12) B ≥ min { k (cid:48) A + k (cid:48) B − , D } . October 15, 2018 DRAFTUBMISSION FOR... 6 UE (cid:4666)(cid:2869),(cid:3036)(cid:4667) UE (cid:4666)(cid:3012),(cid:3036)(cid:4667) ⋯ BS (cid:3036) UE (cid:4666)(cid:2869),(cid:3037)(cid:4667) UE (cid:4666)(cid:3012),(cid:3037)(cid:4667) ⋯ BS (cid:3037) Cell (cid:1862)Cell (cid:1861) (cid:1861) ∈ (cid:4668)1, … , (cid:1862) (cid:3398) 1, (cid:1862) (cid:3397) 1, … , (cid:1829)(cid:4669)
Relay
Fig. 1: System model: Fully connected cellular uplink network with a single relay.
B. System Model: Cellular Network with a Single Relay
In this paper, the information-theoretic quantity of interest is the DoF, where the DoF isdefined as the number of successfully decodable data streams in the desired receiver that aretransmitted by the corresponding user. The term “successfully decodable” means that the desireddata streams are received in an interference-free space.In a cellular network, the DoF per cell is of interest in general. It is defined as the interference-free dimension at the base station, regardless of whether it is the receiver or the transmitter. Oncethe DoF per cell is derived, it can be used to determine the number of users for data transmissionper cell in a cellular network.First, we focus on uplink cellular network, after which downlink cellular network will beconsidered. We consider a fully connected symmetric cellular network with C cells and K usersin each cell, where ‘symmetric’ means that each cell has an identical configuration. Cell j isserved by a single base station denoted as BS j , j ∈ { , ..., C } . The user ( k, j ) denoted as UE ( k,j ) , k ∈ { , ..., K } , is served by BS j as shown in Fig. 1. It is assumed that each UE has M transmit October 15, 2018 DRAFTUBMISSION FOR... 7 antennas, each BS has N receive antennas, and the relay has N R antennas. The channel fromUE ( k,i ) to BS j is denoted by the N × M matrix H j, ( k,i ) , the channel from UE ( k,j ) to the relay isdenoted by the N R × M matrix H UR ( k,j ) , and the channel from the relay to BS j is denoted by the N × N R matrix H RBj . In the proposed schemes, only the relay requires global CSI. In addition,each BS has its local CSI, but UEs do not require any CSI knowledge.The channel is assumed to be Rayleigh fading and the entries of the channel matrices areindependent identically distributed complex Gaussian random variables with zero-mean andunit-variance. Due to the average power constraint in cellular networks, the data vector x ( k,j ) transmitted from UE ( k,j ) is normalized such that the average transmit power of each UE is limitedby 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 communication scenario. Let y j, and y R be thecorresponding received signal vectors of BS j and the relay at the first time slot as y j, = K (cid:88) k =1 H j, ( k,j ) x ( k,j ) + C (cid:88) i (cid:54) = j K (cid:88) k =1 H j, ( k,i ) x ( k,i ) + n j, , j ∈ { , ..., C } , (1) y R = C (cid:88) j =1 K (cid:88) k =1 H UR ( k,j ) x ( k,j ) + n R , (2)where n j, ∈ K N × and n R ∈ K N R × are zero-mean unit-variance circularly symmetric additivewhite complex Gaussian noise vectors at BS j and the relay, respectively. The received signalvector of BS j at the second time slot is given as y j, = H RBj Ty R + n j, , j ∈ { , ..., C } , (3)where n j, ∈ K N × is also a zero-mean unit-variance circularly symmetric additive whitecomplex Gaussian noise vector at BS j and T ∈ K N R × N R is the relay beamforming matrix.Relay beamformer is implemented based on the global CSI.Plugging (2) into (3) leads to y j, = K (cid:88) k =1 H RBj TH UR ( k,j ) x ( k,j ) + C (cid:88) i (cid:54) = j K (cid:88) k =1 H RBj TH UR ( k,i ) x ( k,i ) + ( H RBj Tn R + n j, ) , j ∈ { , ..., C } . (4) October 15, 2018 DRAFTUBMISSION FOR... 8
RelayUE (cid:4666)(cid:2869),(cid:3037)(cid:4667) UE (cid:4666)(cid:3012),(cid:3037)(cid:4667) ⋯ BS (cid:3037) UE (cid:4666)(cid:3038),(cid:3036)(cid:4667) (cid:1863) ∈ (cid:4668)1, … , (cid:1837)(cid:4669)(cid:1861) ∈ (cid:4668)1, … , (cid:1862) (cid:3398) 1, (cid:1862) (cid:3397) 1, … , (cid:1829)(cid:4669) Interference from other cells Interference term from relayUplink data from serving cell
Fig. 2: Interference alignment by a relay in an uplink cellular network.III. IMAC-ODIA
FOR U PLINK C ELLULAR N ETWORK
First, we describe the key idea of the proposed IMAC-ODIA scheme for an uplink cellularnetwork, after which we simplify (1) and (4) using augmented channel matrices, which willbe used for the proposed scheme. The IA conditions to be satisfied for the proposed schemewill also be described. Finally, we present the details of relay beamformer which meet the IAconditions and analyze the achievable DoF of an uplink cellular network with the proposedIMAC-ODIA scheme.
A. IMAC-ODIA for a Symmetric Uplink Cellular Network
The conventional IA is to align interferences in a subspace which is linearly independent ofthe desired signal space by designing a beamformer at the transmitter. Then, using a decorrelatoron the receiver side, the aligned interferences can be perfectly cancelled. However, designing abeamformer at the transmitter requires CSIT and increases the computational complexity. In a
October 15, 2018 DRAFTUBMISSION FOR... 9 cellular system, UEs are not likely to be capable of beamforming.Thus, we propose an IMAC-ODIA scheme for an uplink cellular network, where the inter-ference vector received from a 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 thebeamforming matrix only at the relay as in Fig. 2. This implies that neither CSIT nor complexcomputing is required at the UEs.Using augmentation of the matrices and vectors in (1) and (2), the abovementioned IAdescription can be simplified for the proposed IMAC-ODIA scheme. The augmented channelmatrices for BS j are denoted as follows.- ¯ H j : interfering channel matrix from UEs of the other cells to BS j - ¯ H URj : interfering channel matrix from UEs of the other cells to the relay- (cid:98) H j : desired channel matrix from the UEs served by BS j to BS j - (cid:98) H URj : desired channel matrix from the UEs served by BS j to the relayThese are defined as follows. ¯ H j = (cid:104) H j, (1 , ... H j, ( K,j − H j, (1 ,j +1) ... H j, ( K,C ) (cid:105) ∈ K N × ( C − KM , (5) ¯ H URj = (cid:104) H UR (1 , ... H UR ( K,j − H UR (1 ,j +1) ... H UR ( K,C ) (cid:105) ∈ K N R × ( C − KM , (6) (cid:98) H j = (cid:104) H j, (1 ,j ) ... H j, ( K,j ) (cid:105) ∈ K N × KM , (7) (cid:98) H URj = (cid:104) H UR (1 ,j ) ... H UR ( K,j ) (cid:105) ∈ K N R × KM . (8)Here, let ¯ x j and ˆ x j be the interfering data stream and the desired data stream received at BS j ,respectively. They should also be converted into an augmented form as October 15, 2018 DRAFTUBMISSION FOR... 10 ¯ x j = x (1 , ... x ( K,j − x (1 ,j +1) ... x ( K,C ) ∈ K ( C − KM × , ˆ x j = x (1 ,j ) ... x ( K,j ) ∈ K KM × . From the augmented matrices and vectors, (1) and (4) can be rewritten as y j, = (cid:98) H j ˆ x j + ¯ H j ¯ x j + n j, , j ∈ { , ..., C } , (9) y j, = H RBj T ( (cid:98) H URj ˆ x j + ¯ H URj ¯ x j ) + ( H RBj Tn R + n j, ) , j ∈ { , ..., C } . (10)Ignoring the noise term and adding (9) and (10), the total received signal vector at BS j becomes y j, + y j, = ( (cid:98) H j + H RBj T (cid:98) H URj )ˆ x j + ( ¯ H j + H RBj T ¯ H URj )¯ x j , j ∈ { , ..., C } . (11)In the proposed IMAC-ODIA scheme, the second term on the right-hand side in (11) is the totalreceived interference signal, which should be cancelled. Thus, the IMAC-ODIA condition canbe described as ¯ H j + H RBj T ¯ H URj = 0 , j ∈ { , ..., C } . (12)This indicates the necessity of only the local CSIR at BS j rather than the global CSIT and CSIR,though global CSI is required at the relay. At this point, our goal is to find the relay beamformer T which meets the requirements in (12). The design of the relay beamformer T and its existencecondition will be given in the next subsection. B. Existence of a Relay Beamformer and Its Design
First, we introduce the important properties of the Kronecker product as in the followingpropositions for the main theorem in this subsection.
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Proposition 3 (Representation of the matrix by the Kronecker product):
Consider matrices A , B , C , and X and the matrix equation AXB = C . (13)Here, (13) can be transformed by vectorization intovec ( AXB ) = ( B T ⊗ A ) vec ( X ) = vec ( C ) , where vec( X ) denotes the vectorization of matrix X by stacking the columns of X into a singlecolumn vector. Proposition 4 (Transpose of the Kronecker product):
For matrices A and B , the followingproperty holds, that is, ( A ⊗ B ) T = A T ⊗ B T . Then, the following theorem provides the lower bound of the required number of antennas atthe relay to implement IMAC-ODIA for a symmetric uplink cellular network.
Theorem 1 (Required number of antennas at the relay for beamforming):
The relay beamformer T satisfying condition (12) exists if N R ≥ max { ( C − KM, CN } . Proof:
From
Proposition 3 , (12) can be transformed into { ( ¯ H URj ) T ⊗ H RBj } vec ( T ) = vec ( − ¯ H j ) , j ∈ { , ..., C } . (14)The C equations in (14) can be rewritten as a single equation ( ¯ H UR ) T ⊗ H RB ( ¯ H UR ) T ⊗ H RB ... ( ¯ H URC ) T ⊗ H RBC vec ( T ) = vec ( − ¯ H ) vec ( − ¯ H ) ...vec ( − ¯ H C ) . (15)Let H = ( ¯ H UR ) T ⊗ H RB ( ¯ H UR ) T ⊗ H RB ... ( ¯ H URC ) T ⊗ H RBC and h = vec ( − ¯ H ) vec ( − ¯ H ) ...vec ( − ¯ H C ) . Then, (15) can be represented as October 15, 2018 DRAFTUBMISSION FOR... 12 (cid:1782)(cid:3365) (cid:3036)(cid:3022)(cid:3019) (cid:3404) (cid:1782) (cid:4666)(cid:2869),(cid:2869)(cid:4667)(cid:3022)(cid:3019) (cid:1782) (cid:4666)(cid:2870),(cid:2869)(cid:4667)(cid:3022)(cid:3019) … (cid:1782) (cid:4666)(cid:3012),(cid:3036)(cid:2879)(cid:2869)(cid:4667)(cid:3022)(cid:3019) (cid:1782) (cid:4666)(cid:2869),(cid:3036)(cid:2878)(cid:2869)(cid:4667)(cid:3022)(cid:3019) … (cid:1782) (cid:4666)(cid:2869),(cid:3037)(cid:4667)(cid:3022)(cid:3019) … (cid:1782) (cid:4666)(cid:3012),(cid:3037)(cid:4667)(cid:3022)(cid:3019) … (cid:1782) (cid:4666)(cid:3004),(cid:3012)(cid:4667)(cid:3022)(cid:3019) (cid:1782)(cid:3365) (cid:3037)(cid:3022)(cid:3019) (cid:3404) (cid:1782) (cid:4666)(cid:2869),(cid:2869)(cid:4667)(cid:3022)(cid:3019) (cid:1782) (cid:4666)(cid:2870),(cid:2869)(cid:4667)(cid:3022)(cid:3019) … (cid:1782) (cid:4666)(cid:2869),(cid:3036)(cid:4667)(cid:3022)(cid:3019) … (cid:1782) (cid:4666)(cid:3012),(cid:3036)(cid:4667)(cid:3022)(cid:3019) … (cid:1782) (cid:4666)(cid:3012),(cid:3037)(cid:2879)(cid:2869)(cid:4667)(cid:3022)(cid:3019) (cid:1782) (cid:4666)(cid:2869),(cid:3037)(cid:2878)(cid:2869)(cid:4667)(cid:3022)(cid:3019) … (cid:1782) (cid:4666)(cid:3004),(cid:3012)(cid:4667)(cid:3022)(cid:3019)
Only different submatrix parts
Fig. 3: Matrix structure of ¯ H URi and ¯ H URj for i (cid:54) = j . H × vec ( T ) = h . (16)Our goal is to find vec( T ) satisfying (16). First, we have to show its existence. Solutions of(16) exist if the following condition is satisfied asrank ( (cid:104) H h (cid:105) ) = rank ( H ) . (17)Since all entries of h are drawn i.i.d. from a continuous distribution, (17) can be satisfied if andonly if H has full row rank, which means that H T has full column rank. From Proposition 4 , H T can be rewritten as H T = (cid:104) ¯ H UR ⊗ ( H RB ) T ¯ H UR ⊗ ( H RB ) T ... ¯ H URC ⊗ ( H RBC ) T (cid:105) . (18)In fact, H T in (18) is the Khatri-Rao product of the two matrices ¯ H UR and H RB , where ¯ H UR = (cid:104) ¯ H UR ¯ H UR ... ¯ H URC (cid:105) , H RB = (cid:104) ( H RB ) T ( H RB ) T ... ( H RBC ) T (cid:105) . Note that ¯ H UR ∈ K N R × C ( C − KM and H RB ∈ K N R × CN .From Fig. 3, there are precisely ( C − K identical N R × M submatrices for any pair of matricesin { ¯ H UR , ¯ H UR , ..., ¯ H URC } with probability one. This implies that the generalized Kruskal rank of October 15, 2018 DRAFTUBMISSION FOR... 13 ¯ H UR denoted by k (cid:48) ¯ H UR has a maximum value of one. From Proposition 1 , k (cid:48) ¯ H UR can have twodifferent values k (cid:48) ¯ H UR = min {(cid:98) N R ( C − KM (cid:99) , C, } = , if N R ≥ ( C − KM , if N R < ( C − KM.
Furthermore, because H RB has submatrices, whose entries are drawn i.i.d. from a continuousdistribution, k (cid:48) H RB is also given as k (cid:48) H RB = min {(cid:98) N R N (cid:99) , C } = C, if N R ≥ CN (cid:98) N R N (cid:99) , if N R < CN. From
Proposition 2 , we can find the lower bound on the generalized Kruskal rank of H T =¯ H UR (cid:12) H RB . As long as k (cid:48) ¯ H UR and k (cid:48) H RB are not zeros, lower bound of k (cid:48) H T = k (cid:48) ¯ H UR (cid:12) H RB is givenas k (cid:48) H T = k (cid:48) ¯ H UR (cid:12) H RB ≥ min { C − , C } = C, if N R ≥ max { ( C − KM, CN } . Considering that H T is partitioned into C submatrices, clearly, k (cid:48) H T ≤ C . Thus, the value of k (cid:48) H T is obtained as C , which means that H T has full column rank and (16) has the solution forvec( T ). Thus, we prove the theorem.Note that H has its right inverse H right = H T { HH T } − . Then, the relay beamformer T is givenas T = vec − N R ( null right H + H right h ) , (19)where vec − N R ( · ) is defined as a devectorization function which transforms the vector into a matrixwith a column size of N R . For simplification, null right H can be set as a zero vector. C. Achievable DoF of a Symmetric Uplink Cellular Network with IMAC-ODIA
First, we present the result of this subsection, which describes the achievable DoF per cell foran uplink cellular network with IMAC-ODIA.
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Theorem 2 (Achievable DoF per cell for an uplink cellular network with IMAC-ODIA):
Theachievable DoF per cell of a ( C, K, M, N ) uplink cellular network with IMAC-ODIA is givenas DoF cell = min { N, KM } . Proof:
With the relay beamformer T in (19), the inter-cell interferences can be aligned atthe relay and cancelled at each receiver node. Thus, the DoF of BS j for two transmission timeslots can be simply calculated as the rank of the effective channel of the desired data vector ˆ x j and divided by two. From (11) and (12), the effective channel (cid:98) H eff ,j of BS j for the desiredsignal is defined as (cid:98) H eff ,j ˆ x j = y j, + y j, and its DoF are given as (cid:98) H eff ,j = (cid:98) H j + H RBj T (cid:98) H URj , (20)DoF j = r (cid:98) H eff ,j . Since the left inverse of ¯ H URj exists as { ( ¯ H URj ) T ¯ H URj } − ( ¯ H URj ) T , (12) can be transformed into H RBj T = − ¯ H j ( ¯ H URj ) left + null left ¯ H URj . Therefore, (20) can be rewritten as (cid:98) H eff ,j = (cid:98) H j − ¯ H j ( ¯ H URj ) left (cid:98) H URj + null left ¯ H URj (cid:98) H URj ∈ K N × KM . (21)Considering that the entries of each submatrix in the augmented matrices (cid:98) H j , ¯ H j , ( ¯ H URj ) left , and (cid:98) H URj are drawn i.i.d. from a continuous distribution, the rank of (21) can be derived with probabilityone as r (cid:98) H eff ,j = min { N, KM } . Finally, the achievable DoF per cell for a symmetric uplink cellular network with IMAC-ODIAis given as DoF cell = min { N, KM } . October 15, 2018 DRAFTUBMISSION FOR... 15
D. IMAC-ODIA for an Asymmetric Uplink Cellular Network
The IMAC-ODIA for an asymmetric uplink cellular network is described in this subsection,where ‘asymmetric’ means that each cell has a different configuration. Consider an uplink cellularnetwork with C cells and K j users served by a base station BS j with N j antennas for j ∈{ , ..., C } . User ( k, j ) is denoted by UE ( k,j ) , k ∈ { , ..., K j } , which is served by BS j . UE ( k,j ) is assumed to have M ( k,j ) antennas. Similarly, a relay is assumed to have N R antennas, and allchannel parameters and matrices are defined in the same manner as the symmetric case withaugmented matrices (5) through (8). The number of required relay antennas and the DoF of eachcell are presented in the form of the following theorem. Theorem 3 (IMAC-ODIA for an asymmetric uplink cellular network):
The relay beamformerof IMAC-ODIA for an asymmetric uplink cellular network exists if the following inequality issatisfied N R ≥ max { ( max j ∈{ ,...,C } (cid:88) i (cid:54) = j K i (cid:88) k =1 M ( k,i ) ) , ( C (cid:88) j =1 N j ) } and the achievable DoF for BS j is given asDoF j = 12 min { N j , K j (cid:88) k =1 M ( k,j ) } . The proof is identical to that of the symmetric case, and thus it is omitted here.It is clear that many features of IMAC-ODIA make it possible for a cellular network to adoptinterference alignment. Further, we extend the proposed scheme to a downlink cellular networkwith the help of uplink-downlink duality for relay-aided cellular networks. This is discussed inthe next section. IV. IBC-ODIA
FOR D ONWLINK C ELLULAR N ETWORK
In this section, we extend the idea to a downlink cellular network, which can be modeled asan IBC. Here, we only consider a symmetric downlink cellular network because the IBC-ODIAscheme for an asymmetric case can be described in the same manner as the IMAC-ODIA schemefor an asymmetric cellular network. Channel reciprocity refers to the relationship between uplinkand downlink channels, that is, each downlink channel matrix for a pair of nodes is a complexconjugate transpose of the corresponding uplink channel matrix for the same node pair.
October 15, 2018 DRAFTUBMISSION FOR... 16 (cid:1824)(cid:3548) (cid:3037) (cid:1824)(cid:3364) (cid:3037) (cid:1782)(cid:3553) (cid:3037) (cid:3397) (cid:1782) (cid:3037)(cid:3019)(cid:3003) (cid:1794)(cid:1782)(cid:3553) (cid:3037)(cid:3022)(cid:3019) (cid:1782)(cid:3365) (cid:3037) (cid:3397) (cid:1782) (cid:3037)(cid:3019)(cid:3003) (cid:1794)(cid:1782)(cid:3365) (cid:3037)(cid:3022)(cid:3019) (cid:1782)(cid:3553) (cid:2915)(cid:2916)(cid:2916), (cid:3037)(cid:2993) (cid:1824)(cid:3548) (cid:3037)
UEs BS (cid:1862) (a) Block diagram of uplink transmission with IMAC-ODIA (cid:1782)(cid:3553) (cid:3037) (cid:3397) (cid:1782) (cid:3037)(cid:3019)(cid:3003) (cid:1794)(cid:1782)(cid:3553) (cid:3037)(cid:3022)(cid:3019) (cid:1782)(cid:3365) (cid:3037) (cid:3397) (cid:1782) (cid:3037)(cid:3019)(cid:3003) (cid:1794)(cid:1782)(cid:3365) (cid:3037)(cid:3022)(cid:3019) (cid:3009) (cid:1782)(cid:3553) (cid:2915)(cid:2916)(cid:2916), (cid:3037)(cid:2993) (cid:3009)
UEs BS (cid:1862) (b) Complex conjugate transpose of the uplink transmission
Fig. 4: Block diagrams of the proposed IMAC-ODIA scheme.
A. System Model and Inter-Cell Interference Alignment
We initially apply the result pertaining to duality from earlier work [14] to the uplink cellularnetwork with IMAC-ODIA, which gives us the motivation for IBC-ODIA for a downlink cellularnetwork. For the uplink case, the desired data stream ˆ x j for BS j can be obtained from (11) and(20) as ˆ x j = (cid:98) H † eff ,j ( y j, + y j, ) , (22)where (cid:98) H † eff ,j serves as a decorrelator which handles intra-cell interference. In Fig. 4(a), theuplink transmission process with IMAC-ODIA is described. If we assume channel reciprocityby a complex conjugate transpose operation, Fig. 4(a) is transformed into Fig. 4(b) withoutconsidering the data stream. This provides the intuition for IBC-ODIA, where transmitter (BS)beamforming may be required for the IBC-ODIA scheme for a downlink cellular network, but nodecorrelator is needed at the receiver (UE). Therefore, IBC-ODIA does not require any complexoperation at the UE compared to BS, which supports the feasibility of the proposed IBC-ODIAscheme for a downlink cellular network.First, we describe the symmetric downlink cellular network, that is, C cells and K users ineach cell. Cell j is served by a single base station BS j for j ∈ { , ..., C } . User ( k, j ) is denotedby UE ( k,j ) , k ∈ { , ..., K } , which is served by BS j as in Fig. 5. Each UE, BS, and relay isassumed to have M , N , and N R antennas, respectively. The channel from BS j to UE ( k,i ) is October 15, 2018 DRAFTUBMISSION FOR... 17 UE (cid:4666)(cid:2869),(cid:3036)(cid:4667) UE (cid:4666)(cid:3012),(cid:3036)(cid:4667) ⋯ BS (cid:3036) (cid:1861) ∈ (cid:4668)1,…, (cid:1862) (cid:3398) 1, (cid:1862) (cid:3397) 1,…, (cid:1829)(cid:4669) Relay UE (cid:4666)(cid:2869),(cid:3037)(cid:4667) UE (cid:4666)(cid:3012),(cid:3037)(cid:4667) ⋯ BS (cid:3037) Inter-cell Interference Inter-cell interference term from relayDownlink data including intra-cell interference
Fig. 5: Downlink cellular network with IBC-ODIA.denoted by the M × N matrix H ( k,i ) ,j , the channel from BS j to the relay is denoted by the N R × N matrix H BRj , and the channel from relay to UE ( k,j ) is denoted by the M × N R matrix H RU ( k,j ) . x j denotes the transmitted data vector from BS j . Similarly to the IMAC-ODIA scheme,the augmented matrices for the IBC-ODIA scheme are defined as ¯ H ( k,j ) = (cid:104) H ( k,j ) , ... H ( k,j ) ,j − H ( k,j ) ,j +1 ... H ( k,j ) ,C (cid:105) ∈ K M × ( C − N , ¯ H BRj = (cid:104) H BR ... H BRj − H BRj +1 ... H BRC (cid:105) ∈ K N R × ( C − N and the inter-cell interference vector for UEs at cell j is defined as October 15, 2018 DRAFTUBMISSION FOR... 18 ¯ x j = x ... x j − x j +1 ... x C ∈ K ( C − N × . Similarly to the IMAC-ODIA case, by ignoring the noise term, the total received vector atUE ( k,j ) for two time slots is given as y ( k,j ) , + y ( k,j ) , = ( H ( k,j ) ,j + H RU ( k,j ) T DL H BRj ) x j + ( ¯ H ( k,j ) + H RU ( k,j ) T DL ¯ H BRj )¯ x j , where T DL represents the relay beamforming matrix for downlink transmission. Note that T DL only aligns the inter-cell interferences. The remaining intra-cell interferences will be aligned bybeamforming at the BS, which will be described later. The BS beamforming for intra-cell IAoriginates from the intuition for uplink-downlink duality as discussed earlier.For k ∈ { , ..., K } and i ∈ { , ..., C } , the interference alignment condition should be satisfiedas ¯ H ( k,j ) + H RU ( k,j ) T DL ¯ H BRj = 0 . (23)Further, T DL satisfying the above condition is given as T DL = vec − N R (cid:16) null right H DL + ( H DL ) right h DL (cid:17) , where H DL = ( ¯ H BR ) T ⊗ H RU , ( ¯ H BR ) T ⊗ H RU , ... ( ¯ H BR ) T ⊗ H RUK, ( ¯ H BR ) T ⊗ H RU , ... ( ¯ H BRC ) T ⊗ H RUK,C and h DL = vec ( − ¯ H , ) vec ( − ¯ H , ) ...vec ( − ¯ H K, ) vec ( − ¯ H , ) ...vec ( − ¯ H K,C ) . October 15, 2018 DRAFTUBMISSION FOR... 19
The following theorem gives the number of antennas required at the relay for IBC-ODIA,where the proof is similar to that of IMAC-ODIA.
Theorem 4 (Required number of relay antennas for the downlink cellular network with IBC-ODIA):
The relay beamformer T DL for the IBC-ODIA satisfying (23) exists if N R ≥ max { ( C − N, CKM } .With the proper design of T DL , the inter-cell interferences are fully cancelled at the UEs. Asnoted above, intra-cell interferences will be aligned by designing the beamformer at the BS. Thebeamformer design is described in the next subsection. B. Intra-Cell Interference Alignment by BS Beamforming
The beamformer at the BS for the intra-cell IA can be designed jointly with the relaybeamformer. For simplicity, each UE desires to receive d independent data streams from itsserving BS. The transmitted signal from BS j can be given as x j = V (1 ,j ) s (1 ,j ) + V (2 ,j ) s (2 ,j ) + ... + V ( K,j ) s ( K,j ) , where s ( k,j ) ∈ K d × denotes the data stream for UE ( k,j ) from BS j and V ( k,j ) ∈ K N × d is thebeamforming matrix designed at BS j for UE ( k,j ) .The intra-cell interference at each UE is said to be fully aligned by the BS beamforming ifthe following condition is satisfied as H (1 ,j ) ,j + H RU (1 ,j ) T DL H BRj ... H ( K,j ) ,j + H RU ( K,j ) T DL H BRj (cid:104) V (1 ,j ) ... V ( K,j ) (cid:105) = A j V j = I M × d . . .
00 0 I M × d . Since each entry of H ( k,j ) ,j is drawn i.i.d. from a continuous distribution, A j is either a full rowor a full column rank matrix with probability one with a right or left inverse matrix.Hence, V i is expressed as V j = A † j I M × d . . .
00 0 I M × d = (cid:104) { A † j } d { A † j } M +1: M + d · · · { A † j } ( K − M +1:( K − M + d (cid:105) . (24) October 15, 2018 DRAFTUBMISSION FOR... 20
From (24), we can design V j with A † j , where A † j = A right j = A Tj ( A j A Tj ) − , if N ≥ KM A left j = ( A Tj A j ) − A Tj , if N < KM.
Note that N ≥ Kd should be guaranteed for successful data transmission. Thus, V ( k,j ) makes itpossible for BS j to deliver messages to UE ( k,j ) successfully with DoF d , where d ≤ M shouldbe satisfied.After designing the relay beamformer and the BS beamformer, the inter-cell and intra-cellinterferences can be completely removed at each UE. Accordingly, we summarize the achievableDoF for a downlink cellular network with IBC-ODIA via the following theorem. Theorem 5 (Achievable DoF for a downlink cellular network with IBC-ODIA):
The achievableDoF per cell and per UE for a ( C, K, M, N ) downlink cellular network with IBC-ODIA isexpressed as DoF DL,cell = min { Kd, N } ≤ min { KM, N } , DoF
DL,UE = DoF
DL,cell
K .
Proof of
Theorem
Theorem
FOR F ULL -D UPLEX C ELLULAR N ETWORK
Furthermore, we focus on a cellular network, where full-duplex operation is feasible for BSsand UEs. Due to the complexity of full-duplexing, it has not been a main concern for researchers,but it is known that full-duplexing, when possible, increases the throughput. However, the needfor a management scheme given the additional interference caused by full-duplex operationarises, which motivates the development of the proposed FD-ODIA scheme for a full-duplexcellular network.
A. FD-ODIA for Full-Duplex Cellular Network
Recently, several researchers have investigated interference alignment on full-duplex networkdue to the higher throughput achievable by the full-duplex mode [16] [17] [18]. In [16], it
October 15, 2018 DRAFTUBMISSION FOR... 21 UE (cid:4666)(cid:3038),(cid:3036)(cid:4667) BS (cid:3036) Full-duplex network UE (cid:4666)(cid:3038),(cid:3036)(cid:4667) BS (cid:3036) Uplink network: UE (cid:4666)(cid:3038),(cid:3036)(cid:4667) to BS (cid:3036) UE (cid:4666)(cid:3038),(cid:3036)(cid:4667) BS (cid:3036) Downlink network: BS (cid:3036) to UE (cid:4666)(cid:3038),(cid:3036)(cid:4667) Full-duplex network decomposition into two half-duplex networks Self-interferences
Fig. 6: Full-duplex network decomposition of a single full-duplex cell.was proved that allowing for full-duplexing in each node exactly doubled the DoF performancecompared to that of a half-duplexing multi-user interference channel. The DoF derivation processin [16] was based on network decomposition, where a similar methodology can be applied to acellular network. In [17], if each node operates in full-duplex mode, then a single full-duplexcell can be decomposed into two half-duplex cells, one on the uplink transmission and the otheron the downlink transmission as shown in Fig. 6.Considering full-duplex network decomposition as well as the fact that the proposed schemecan be operated on either the uplink or downlink, we can apply the IMAC-ODIA and theIBC-ODIA schemes proposed in the previous sections to a full-duplex cellular network, that is,FD-ODIA. Although the relay operates in the half-duplex mode, the total DoF of the proposedFD-ODIA scheme can be doubled compared to the half-duplex case. We describe the symmetricfull-duplex cellular network model below.Let C and K be the numbers of cells and users in each cell, respectively. Cell j is served October 15, 2018 DRAFTUBMISSION FOR... 22 UE (cid:4666)(cid:3038),(cid:3036)(cid:4667) (cid:1863) ∈ (cid:4668)1, … , (cid:1837)(cid:4669)(cid:1861) ∈ (cid:4668)1, … , (cid:1862) (cid:3398) 1, (cid:1862) (cid:3397) 1, … , (cid:1829)(cid:4669) BS (cid:3036) UE (cid:4666)(cid:3038),(cid:3037)(cid:4667) (cid:1863) ∈ (cid:4668)1, … , (cid:1837)(cid:4669) BS (cid:3037) Full-duplex cell (cid:1861)
Half-duplex relay Full-duplex cell (cid:1862)(cid:1782) (cid:3022)(cid:3022) (cid:1782) (cid:3022)(cid:3019) , (cid:1782) (cid:3019)(cid:3022) (cid:1782) (cid:3003)(cid:3022) , (cid:1782) (cid:3022)(cid:3003) (cid:1782) (cid:3003)(cid:3003) (cid:1782) (cid:3003)(cid:3019) , (cid:1782) (cid:3019)(cid:3003)
Fig. 7: Full-duplex cellular network served by a single half-duplex relay, where onlyinterference-related channels are depicted.by a single base station in full-duplex mode denoted by BS j for j ∈ { , ..., C } . User ( k, j ) alsooperates in full-duplex mode denoted by UE ( k,j ) , k ∈ { , ..., K } , which is served by BS j , asin Fig. 7. Note that the UEs only receive their desired signals from their corresponding BSs,implying the absence of device-to-device communication. Each UE, each BS, and the relay isassumed to have M , N , and N R antennas, respectively. Note that the relay operates in half-duplexmode and the full-duplex relay will be discussed later. Allowing for full-duplex operation at eachsource and destination node generates the following additional channel matrices.- Channel from UE ( k,i ) to BS j : N × M matrix H UBj, ( k,i ) - Channel from UE ( k ,i ) to UE ( k ,j ) : M × M matrix H UU ( k ,j ) , ( k ,i ) - Channel from UE ( k,j ) to relay: N R × M matrix H UR ( k,j ) - Channel from relay to UE ( k,j ) : M × N R matrix H RU ( k,j ) October 15, 2018 DRAFTUBMISSION FOR... 23 - Channel from relay to BS j : N × N R matrix H RBj - Channel from BS i to BS j : N × N matrix H BBj,i - Channel from BS j to UE ( k,i ) : M × N matrix H BU ( k,i ) ,j - Channel from BS j to relay: N R × N matrix H BRj
B. Achievable DoF of Full-Duplex Cellular Network with FD-ODIA
Here, we derive the achievable DoF and the number of antennas required at the relay for afull-duplex cellular network with FD-ODIA as in the following theorem.
Theorem 6 (Achievable DoF for a full-duplex cellular network with FD-ODIA):
For N R ≥ C ( KM + N ) , the achievable DoFs per cell, per BS, and per UE of the ( C, K, M, N ) full-duplex cellularnetwork are given asDoF FD,cell = DoF
FD,BS + K × DoF
FD,UE = min { KM, N } , DoF
FD,BS = min { KM, N } , DoF
FD,UE = min { M, NK } . Proof:
We prove this theorem using the IMAC-ODIA and IBC-ODIA schemes. As notedearlier, the full-duplex cellular network with a relay can be decomposed into disjoint uplink anddownlink networks sharing the relay as in Fig. 7. Note that additional interference occurs due tofull-duplexing, that is, UE to UE channels in the intra-cells or inter-cells now become interferingchannels.IBC-ODIA requires additional beamforming at the BS, in contrast to IMAC-ODIA. Thus, it isnatural that the FD-ODIA scheme should include beamforming at the BS. To describe the FD-ODIA scheme, we start with matrix augmentation with the same approach used earlier, wherethe relay only aligns the inter-cell interferences as
October 15, 2018 DRAFTUBMISSION FOR... 24 ¯ H BSj = (cid:104) H BBj, ... H BBj,j − H BBj,j +1 ... H BBj,C H UBj, (1 , ... H UBj, ( K,j − H UBj, (1 ,j +1) ... H UBj, ( K,C ) (cid:105) ∈ K N × ( C − KM + N ) , ¯ H UE ( k,j ) = (cid:104) H BU ( k,j ) , ... H BU ( k,j ) ,j − H BU ( k,j ) ,j +1 ... H BU ( k,j ) ,C H UU ( k,j ) , (1 , ... H UU ( k,j ) , ( k − ,j ) H UU ( k,j ) , ( k +1 ,j ) ... H UU ( k,j ) , ( K,C ) (cid:105) ∈ K M ×{ ( CK − M +( C − N } , ¯ H BRj = (cid:104) H BR ... H BRj − H BRj +1 ... H BRC H UR (1 , ... H UR ( K,j − H UR (1 ,j +1) ... H UR ( K,C ) (cid:105) ∈ K N R × ( C − KM + N ) , ¯ H UR ( k,j ) = (cid:104) H BR ... H BRj − H BRj +1 ... H BRC H UR (1 , ... H UR ( k − ,j ) H UR ( k +1 ,j ) ... H UR ( K,C ) (cid:105) ∈ K N R ×{ ( CK − M +( C − N } . In addition, the interference vectors should be redefined for the additional interferences as ¯ x BSj = x BS ... x BSj − x BSj +1 ... x BSC x UE (1 , ... x UE ( K,j − x UE (1 ,j +1) ... x UE ( K,C ) ∈ K ( C − KM + N ) × , ¯ x UE ( k,j ) = x BS ... x BSj − x BSj +1 ... x BSC x UE (1 , ... x UE ( k − ,j ) x UE ( k +1 ,j ) ... x UE ( K,C ) ∈ K { ( CK − M +( C − N }× . October 15, 2018 DRAFTUBMISSION FOR... 25
Note that x BSj and x UE ( k,j ) are the transmitted vectors from BS j and UE ( k,j ) , respectively. Thedefinitions of the desired signal channel set { (cid:98) H } and the desired signal vector set { (cid:98) x } areexactly the same as those of the half-duplex uplink and downlink cellular network cases.As noted above, the augmented interference matrices only contain the inter-cell interferences.Thus, additional BS beamforming will align the intra-cell interferences at the UEs, which willbe described later.The IA condition for FD-ODIA is similar to the IMAC/IBC-ODIA case except for the factthat it should be satisfied both on the BSs and the UEs as ¯ H BSj + H RBj T FD ¯ H BRj = 0 , j ∈ { , ..., C } , (25) ¯ H UE ( k,i ) + H RU ( k,i ) T FD ¯ H UR ( k,i ) = 0 , k ∈ { , ..., K } , i ∈ { , ..., C } . (26)The existence of the relay beamformer T FD for the FD-ODIA scheme is also proved in a mannersimilar to that of the IMAC/IBC-ODIA case except for the number of relay antennas. In fact,in the full-duplex network case, the number of relay antennas N R should satisfy N R ≥ max { ( C − KM + N ) , ( C − N + ( CK − M, C ( KM + N ) } = C ( KM + N ) , where C ( KM + N ) is the total number of antennas in the cellular network.Hence, T FD satisfying (25) and (26) is derived as T FD = vec − N R ( ¯ H BR ) T ⊗ H RB , ... ( ¯ H BRC ) T ⊗ H RBC ( ¯ H UR (1 , ) T ⊗ H RU (1 , ... ( ¯ H UR ( K,C ) ) T ⊗ H RU ( K,C ) right vec ( − ¯ H BS ) ...vec ( − ¯ H BSC ) vec ( − ¯ H UE (1 , ) ...vec ( − ¯ H UE ( K,C ) ) . Note that the right null vector term in the argument of vec − N R ( · ) is omitted.Furthermore, it is necessary to design the beamformer V FD j at BS j in the same manner asIBC-ODIA, which should satisfy the following intra-cell IA condition October 15, 2018 DRAFTUBMISSION FOR... 26 H BU (1 ,j ) ,j + H RU (1 ,j ) T FD H BRj ... H BU ( K,j ) ,j + H RU ( K,j ) T FD H BRj (cid:104) V FD (1 ,j ) ... V FD ( K,j ) (cid:105) = A FD j V FD j = I M × d . . .
00 0 I M × d . Similar to the IBC-ODIA case, A FD j is either a full row or a full column rank matrix withprobability one depending on the number of antennas. In this case, V FD j can be derived as V FD j = (cid:104) { ( A FD j ) † } d { ( A FD j ) † } M +1: M + d · · · { ( A FD j ) † } ( K − M +1:( K − M + d (cid:105) , where ( A FD j ) † is either the right or left inverse matrix of A FD j .Since the interferences are fully aligned and cancelled at each BS, the DoF of BS j for thefull-duplex cellular network can be achieved asDoF FD,BS j = min { KM, N } , and the DoF of UE ( k,j ) can also be achieved asDoF FD,UE ( k,j ) = min { KM, N } K = min { M, NK } . Thus, we prove the theorem.Note that the total achievable DoF of the proposed FD-ODIA is doubled compared to thehalf-duplex case. It is also interesting that the proposed scheme requires only a half-duplexingAF relay to achieve full-duplexing gain in a full-duplex cellular network with FD-ODIA. Thedesign methodology of the proposed IMAC/IBC-ODIA schemes can directly be applied to theFD-ODIA scheme.In the previous work [16], it was shown that causal MIMO full-duplex relay based on the DFprotocol cannot increase the DoF but that non-causal and instantaneous full-duplex DF relayingcan increase the DoF. The result in
Theorem 6 can be directly applied to the case of full-duplexinstantaneous AF relay as in the following corollary.
October 15, 2018 DRAFTUBMISSION FOR... 27
Corollary 1 (Achievable DoF with full-duplex instantaneous AF relay):
The achievable DoFsper cell, BS, and UE of the same full-duplex cellular network with full-duplex instantaneous AFrelaying are given asDoF instFD,cell = DoF instFD,BS + K × DoF instFD,UE = 2 min { KM, N } , DoF instFD,BS = min { KM, N } , DoF instFD,UE = min { M, NK } , where the relay antenna requirement is the same as that in Theorem 6 . Remark 1 (Half-duplex UE with full-duplex BS):
Due to the hardware complexity limitation,an UE cannot be operated in full-duplex mode, and thus half-duplex UEs served by full-duplexBSs can be considered. In this case, the UEs can be partitioned into either an uplink groupor a downlink group. Assume that K UEs are in the uplink group and K UEs are in thedownlink group with M antennas at each UE for both groups. Note that the BSs operate inthe full-duplex mode with N antennas. Then the DoF of the above system can be given as thefollowing corollary. Corollary 2 (DoF of a cellular network with half-duplex UEs served by a full-duplex BS):
Theachievable DoF per cell with the half-duplex UEs served by the full-duplex BS is given asDoF
HD+FD,cell = DoF
HD+FD,BS + K × DoF
HD+FD,UE , (27)where DoF HD+FD,BS = min { K M,N } and DoF HD+FD,UE = min { M, NK } .The DoF in (27) can be achieved with the IA scheme similar to the previous FD-ODIAscheme. It can be achieved by FD-ODIA with the exclusion of non-existing BS-to-UE channels,UE-to-BS channels, and UE-to-UE channels in the relay beamformer design, which vanishesautomatically due to the half-duplexing of UEs.Thus, it should be noted that the proposed schemes are very flexible, that is, IA schemesfor various cellular networks can be implemented with the proposed schemes by modifying thechannel assignment protocol in the relay. October 15, 2018 DRAFTUBMISSION FOR... 28 UE (1,1) BS 𝐶, 𝐾, 𝑀, 𝑁 − IMACUE (𝐾,1) ⋮ UE (1,𝐶) BS 𝐶 UE (𝐾,𝐶) ⋮ ⋮ UE BS 𝐶, 𝐾𝑀, 𝑁 − ICUE 𝐶 BS 𝐶 ⋮ Network transformation from IMAC to IC through UE(transmitter) cooperation
Fig. 8: IMAC-to-IC transformation with UE c , ≤ c ≤ C , which is a super-UE with KM antennas.VI. D ISCUSSION
A. DoF Improvement without Time Extension
With the proposed schemes, the total achievable DoF of a half-duplex uplink cellular net-work DoF
ODIA,uplink is given as C min { N,KM } . For the ( C, K, M, N ) uplink cellular networks,DoF ODIA,uplink of the proposed scheme with a relay and without UE beamforming is much largerthan DoF linear,uplink of the linear scheme without a relay and with UE beamforming, which canbe analyzed using the outer bound of DoF of the cellular network without time extension.An ( C, K, M, N ) IMAC can be transformed into an ( C, KM, N ) IC with transmitter coop-eration, that is, all UEs at each BS transmit their data jointly as a single super-UE as shownin Fig. 8. Let DoF
Linear,coop denote the total DoF of the ( C, KM, N ) IC. Then, the followinginequality is easily given as DoF linear,uplink ≤ DoF linear,coop . October 15, 2018 DRAFTUBMISSION FOR... 29
Further, using the result in [19], DoF linear,coop is bounded asDoF linear,uplink ≤ DoF linear,coop ≤ KM + N. (28)However, the total DoF DoF ODIA,uplink of the proposed scheme increases proportional to the num-ber of cells, which cannot be achieved in (28). From [20], it is easy to find that DoF
ODIA,uplink = C min { N,KM } is indeed the maximally achievable DoF in an uplink cellular network with a half-duplex relay.Note that the information-theoretic upper bound of the ( C, K, M, N ) IMAC ( ≈ KMNKM + N , see[8]) is approximately two times larger than the achievable DoF of the proposed scheme in anextreme case (i.e., N (cid:28) KM or N (cid:29) KM ), whereas the upper bound cannot be achieved bythe linear IA scheme without time extension.Additionally, in an antenna regime where BSs and UEs in a cell have similar numbers ofantennas (i.e., N ≈ KM ), the proposed IMAC/IBC-ODIA schemes achieve the optimal DoF ofMIMO IMAC/IBC, which is believed to be achieved only through asymptotic IA or IA aidedby a full-duplex relay. B. Possibility of Gain from Additional Relay Antennas
Suppose that a relay antenna setup is sufficient to serve an entire network, that is, there aremore antennas than required. Then the question is, “Is there any extra gain from additional relayantennas?” We answer this question in this subsection.Consider that an uplink cellular network with the IA-applied data stream through decorrelatorat the BS is expressed as (22) in the previous section. Our goal is to design a relay beamformersatisfying the following additional statement as (cid:98) H eff ,j = (cid:98) H j + H RBj T + (cid:98) H URj = (1 + α ) (cid:98) H j ⇒ H RBj T + (cid:98) H URj = α (cid:98) H j , (29)where α ≥ and T + denotes the relay beamformer with additional gain. If (29) is satisfied, itcan be interpreted as boosting the received signal power level by α , while the DoF performanceremains the same. October 15, 2018 DRAFTUBMISSION FOR... 30
We skip the details because the proof is identical to those in Theorems 1 and 2. The existenceof T + satisfying (29) can be guaranteed when N R ≥ max { CKM, CN } . Hence, the design of T + is given as T + = vec − N R ( H UR ) T ⊗ H RB ( H UR ) T ⊗ H RB ... ( H URC ) T ⊗ H RBC right vec ( H ) vec ( H ) ...vec ( H C ) , where for simplicity, the right null vector term in the argument of vec − N R ( · ) is omitted and H j ∈ K N × CKM and H URj ∈ K N R × CKM are correspondingly defined as H URj = (cid:104) H UR , H UR , · · · H URK,C , (cid:105) , H j = (cid:104) − H j, (1 , · · · − H j, ( K,j − α H j, (1 ,j ) · · · α H j, ( K,j ) − H , (1 ,j +1) · · · − H j, ( K,C ) (cid:105) . Then, the relay beamformer T + for the proposed IMAC-ODIA can align the interferences, whilealso boosting the desired signal power level to KM additional antennas at the relay. Thus, fromthe additional relay antenna, we can achieve better throughput.VII. C ONCLUSIONS
In this paper, we proposed IA schemes for the cellular networks which operate with a half-duplex relay without UE beamforming or CSI handling, representing the fundamental limit whenapplying IA to cellular network. We proposed an uplink cellular network with IMAC-ODIA andderived its achievable DoF. Using linear beamforming at the relay and uplink-downlink duality,we extended the proposed scheme with the BS beamformer design to a downlink cellular networkwithout UE beamforming or CSI handling. Further, the proposed schemes can also be extendedto a full-duplex cellular network, achieving doubled DoF compared to a half-duplex cellularnetwork. Although it has DoF gap from the information-theoretic upper bound, we proved thatthe proposed schemes have network gain proportional to the network size, which cannot be
October 15, 2018 DRAFTUBMISSION FOR... 31 achieved in a cellular network with linear IA. Moreover, for some antenna regime, the proposedschemes are able to achieve the optimal DoF.A
CKNOWLEDGEMENT
This work was supported by R
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