MIMO Multiway Relaying with Clustered Full Data Exchange: Signal Space Alignment and Degrees of Freedom

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1 009/TWC , IEEE Transactions on Wireless Communications IO ultiway Relaying with Clustered Full Data Exchange: Signal Space Alignment and Degrees of Freedom Xiaojun Yuan, ember, IEEE Abstract Recently, much research interest has been attracted towards the design of efficient communication mechanisms for multiple-input multiple-output IO multiway relay channels mrcs In this paper, we investigate achievable degrees of freedom DoF of the IO mrc with clusters and K users per cluster, where each user is equipped with antennas and the relay with antennas Our analysis is focused on a new data exchange model, termed clustered full data exchange, ie, each user in a cluster wants to learn the messages of all the other users in the same cluster ovel signal alignment techniques are developed to jointly and systematically construct the beamforming matrices at the users and the relay for efficient implementation of physical-layer network coding Based on that, we derive an achievable DoF of the IO mrc with an arbitrary network configuration of and K, as well as with an arbitrary antenna configuration of and We show that our proposed scheme achieves the DoF capacity when K and K + The DoF results derived in this paper can K serve as fundamental benchmarks in evaluating the performance of practical communication systems over IO mrcs, and provide guidance and insights into the design of wireless relay networks I ITRODUCTIO Physical-layer network coding PC has been intensively investigated in the past several years The earliest model for PC is the two-way relay channel TWRC, in which two users exchange information via the help of a single relay [] The peer-to-peer based communication protocol for TWRC requires four phases to complete one round of information exchange PC reduces the number of the required phases to two by allowing users to transmit or receive signals simultaneously, which implies potentially doubled network throughput Since its advent, abundant progresses on PC design for TWRC have been reported in the literature; see []-[7] and the references therein Particularly, it was shown in [4] that, with nested lattice coding, the capacity of the TWRC can be achieved within 2 bit ater, the authors in []-[7] considered multiple-input multiple-output IO TWRCs, in which every node is equipped with multiple antennas It was revealed that the asymptotic capacity of the IO TWRC at high anuscript received December, 203; revised arch 8, 204; accepted ay 6, 204 This work was supported by grants from the University Grants Committee of the Hong Kong Special Administrative Region, China Project o 4872 and AoE/E-02/08 The associate editor coordinating the review of this paper and approving it for publication was D iyato X Yuan was with the Institute of etwork Coding, The Chinese University of Hong Kong He is now with the School of Information Science and Technology, ShanghaiTech University, Shanghai, China yuanxj@shanghaitecheducn signal-to-noise SR can be achieved within a finite gap per spatial dimension for arbitrary antenna configurations Tremendous success of PC over TWRC intrigues intensive research on PC for more general relay networks In this regard, a natural generalization of TWRC is called a multiway relay channel mrc, in which users are grouped into clusters and each user in a cluster wants to communicate with other users in the same cluster [8] This setup generally models a variety of communication scenarios For example, in a social network, groups of users want to share files via a relay station Each user in a group only has a distinct portion of a common file desired by all the other users in the same group; many such groups need to be served simultaneously by the relay station This setup is also relevant to ad hoc wireless networks in which multiple distributed nodes want to communicate with a central controller to share available local information Two special models of the mrc have been studied in the literature [8]-[2]: in the full data exchange model, each user wants to learn the messages from all the other users in the network; in the pairwise exchange model, the network consists of multiple pairs, and the two users in each pair want to exchange information with each other Various relaying protocols and design criteria have been studied for mrcs operating under these two exchange models In particular, it was shown in [8] that the sum-rate capacity of the network can be achieved within a finite-bit gap for any number of clusters Initial work on mrcs was limited to a single-antenna setup, ie, each node in the network is equipped with a single antenna Recently, much attention has been attracted to the IO mrc, in which each node in the network is equipped with multiple antennas to allow spatial multiplexing [3]-[20] For example, the IO technique has been introduced into the Y channel in [] a special mrc with one cluster and three users per cluster, and also into the multipair TWRC in [4] a special mrc with multiple clusters and two users in each cluster An important research avenue on IO mrcs is to analyze the degrees of freedom DoF of the network, which characterizes the high signal-to-noise ratio SR performance []-[20] It is known that interference alignment, in which interference signals are aligned to occupy a minimal number of temporal/spectral/spatial dimensions, is the key technique to achieve the DoF capacity of various wireless multi-terminal networks [2][22] As for relay networks, a similar notion, termed signal space alignment, was proposed in [] It was shown that, by aligning the signal streams of the users with information exchange to a common direction, c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

2 009/TWC , IEEE Transactions on Wireless Communications 2 the spatial dimensions at the relay can be efficiently utilized, so as to achieve the DoF capacity of the relay network This paper considers a general IO mrc with clusters and K users per cluster ote that a similar channel setup has been previously studied in [7], where the users in a cluster exchange private data in a pairwise manner In contrast, this paper is focused on a new data exchange model, termed clustered full data exchange, in which each user in a cluster wants to learn all the messages from other users in the same cluster This data exchange model arises in many practical scenarios, such as teleconferencing and data sharing in a social network The difficulty in the design of efficient communication mechanisms for clustered IO mrcs is largely attributed to the following feature of such networks: within each cluster, signal space alignment is necessary to exploit the potential advantage of PC; meanwhile, for each cluster, the signals from the other clusters are interference, implying that temporal and spatial interference alignment is necessary to minimize the signal dimensions occupied by the interference Therefore, efficient protocols over clustered IO mrcs involve both network coding and interference alignment, which poses a serious challenge for system design In this paper, we investigate the achievable DoF of the IO mrc with clustered full data exchange We assume a symmetric antenna setup, in which each user is equipped with antennas and the relay with antennas The main contribution of this paper is to develop a novel systematic signal-alignment technique for efficient PC design over the considered IO mrc Specifically, we call a bunch of K signal spatial streams as a unit if it consists of one spatial signal stream from every user The signal streams in a unit are aligned to form a certain spatial structure referred to as a pattern that allows signal separation at the user ends The number of spatial dimensions occupied by a pattern characterizes the efficiency of this pattern in utilizing the signal space of the relay Intuitively, a higher ratio of allows higher freedom at the user ends to align the signals, and therefore, a more efficient pattern can be constructed The signal alignment problem is then to pack as many units with the most efficient patterns as possible to occupy the overall signal space of the relay In this way, the DoF analysis reduces to counting the maximum number of units that can be packed Based on this technique, an achievable DoF can be derived for the considered IO mrc with an arbitrary network configuration of and K, as well as with an arbitrary antenna configuration of and By comparing the derived achievable DoF with the cut-set outer bound, we show that the proposed scheme achieves the DoF capacity when K and K + K We also show that the derived DoF is always piecewise linear and is bounded by either or, implying that either the users or the relay has redundant antennas This is similar to the DoF results of the IO interference channel obtained in [22] The remainder of this paper is organized as follows Section II describes the system model The achievable DoF of the considered IO mrc with = 2 clusters and K = 3 users per cluster is derived in Section III In Section IV, we generalize the results in Section III to an arbitrary configuration of and K Finally, we close the paper in Section V with some Fig User Relay User User K Uplink User K Downlink Cluster Cluster 2 The IO mrc with = 2 clusters and K users in each cluster concluding remarks highlighting our main results A otation II PREIIARIES The following notation is used throughout this paper Scalars are denoted by lowercase regular letters, vectors by lowercase bold letters, and matrices by uppercase bold letters For any matrix A, A T and A denote the transpose and the Hermitian transpose, respectively; spana denotes the column space; nulla denotes the right nullspace; tra denotes the trace of a square matrix A dims denotes the dimension of a space S; S U and S U denote the intersection and the direct sum of two spaces S and U, respectively; R n m and C n m denote the n-by-m dimensional real space and complex space, respectively; + denotes max{, 0}; C 0, σ 2 denotes the circularly symmetric complex Gaussian distribution with mean zero and variance σ 2 B System odel In this paper, we consider a discrete memoryless symmetric IO mrc in which multiple users, partitioned into clusters, are simultaneously served by a single relay, and each user in a cluster wants to multicast its message to all other users in the same cluster We assume that full-duplex communication is employed, ie, all the users and the relay can transmit and receive signal simultaneously We also assume that the direct links between the users are ignored due to channel impairments, such as shadowing and deep fading 2 et be the number of clusters and K be the number of users in each cluster The IO mrc with = 2 is illustrated in Fig Users in cluster j, j I {,, }, are denoted by T j,, T jk Each user is equipped with antennas, and the relay with antennas For any k I K {,, K}, the channel matrix from user k of cluster j to the relay is denoted by H C, and the channel matrix from the The DoF results in this paper directly hold for the case of half-duplex communication by including a multiplicative factor of 2 2 This assumption can be relaxed in half-duplex communication, for that the system model for the considered two-phase transmission protocol is always given by, even if the users are located close to each other such that the user-to-user links cannot be ignored c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

3 009/TWC , IEEE Transactions on Wireless Communications 3 relay to user k of cluster j is denoted by G T C We assume that the elements of H and G, j, k, are independently drawn from a continuous distribution Thus the channel matrices are of full column or row rank, whichever is smaller, with probability one We also assume that the channel state information CSI is perfectly known at all nodes 3 Each round of information exchange consists two transmission phases, namely, the uplink phase and the downlink phase Correspondingly, the Gaussian IO mrc is modeled as Y R = j= k= K H X + Z R a Y = G T X R + Z, j I, k I K, b where X C T and Y C T are the input and the output at user T, respectively; X R C T and Y R C T are the input and the output at the relay, respectively; T denotes the number of channel uses in an uplink or downlink transmission frame; Z R C T and Z C T, respectively, are the additive white Gaussian noise AWG matrices at the relay and at user T, with the elements independently drawn from C 0, σ 2 The power constraints of the transmitted signals at user T and at the relay are respectively given by T trx X P, j I, k I K 2a T trx RX R P R 2b where P and P R are the power budgets at user T and at the relay, respectively C inear Processing at Users and Relay We now present detailed operations at the users and the relay Throughout this paper, we assume clustered full data exchange That is, each user T, j I, k I K, sends a common message W to all the other users in cluster j, and wants to learn the messages W, k I K \{k}, from all the other users in cluster j First consider the uplink phase et X Cm T be a codeword matrix of user T one-to-one mapped to W, where m is the number of spatial streams with m Denote by U C m the corresponding precoding matrix Then, the channel input of user T is given by X = U X, j I, k I K 3 The relay operation is described as follows For convenience, we introduce the following notation: j = [H j U j,, H jk U jk ], j I 4a = [,, ] 4b j = [,, j, j+,, ], j I 4c 3 In practice, the relay node is usually computationally more powerful than the user nodes Thus, pilot-based channel estimation can be implemented at the relay to acquire the CSI of the whole network Based on the acquired CSI, the relay computes its own precoding matrix, as well as the precoding matrices of the users Then, the relay delivers the precoding information to the intended user nodes via the downlink transmission ote that detailed overhead analysis for CSI acquisition is out of the scope of this paper For each cluster j, the relay extracts the signal component of Y R orthogonal to the signals of the other clusters To this end, let P j C be the projection matrix that projects a vector into the subspace null j Then, for each cluster j, the relay obtains P j Y R = P j H j X j + P j Z R a where X j = [X j,, X jk ]T In the above, the signals from cluster j j disappear due to the projection We now consider the downlink phase Similarly to 4, we denote j = [G j V j,, G jk V jk ], and j = [,, j, j+,, ], where V C m is the receive processing matrix of user T et W j C be the projection matrix that projects a vector into the subspace null j The relay sends out the signal Wj T P jy R for each cluster j, ie, the relay s transmit signal is given by X R = α Wj T P j Y R 6 j= where α is a scaling factor to meet the relay s power constraint The relay-to-user signal model is then given by VY T = αvg T T Wj T P j Y R + VZ T 7a j = K = αvg T T Wj T P j H U X + Z R k = +V T Z 7b Upon receiving V T Y, each user T computes the message estimate of W, denoted by Ŵ k, for all k k From 7, the equivalent channel from user T to user T is given by V T GT WT j P jh U This equivalent channel is the product of two symmetric components, namely, the uplink equivalent channel P j H U and the downlink equivalent channel V T GT WT j This symmetry implies that any beamforming design in the uplink phase directly carries over to the downlink phase Therefore, we mostly focus on the uplink beamforming design in what follows D Degrees of Freedom This paper is focused on analyzing the DoF of the considered IO mrc Roughly speaking, the DoF of a network is the number of independent signal streams that can be supported by the network To make this notion rigorous, we introduce some definitions in what follows For notational convenience, we assume P = P R = P, j I, k I K and denote SR = P/σ 2, without compromising the generality of the DoF results derived in this paper et R be the information rate carried in W We say that user T achieves a sum rate of C = K k =,k k R, if Pr{Ŵ k W } tends to zero for k I K \{k} as T The achievable sum rate C is in general a function of SR, denoted as C SR, j I, k I K We define the total achievable DoF as d sum lim SR K j= k= C SR log SR c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

4 009/TWC , IEEE Transactions on Wireless Communications 4 where C SR is in bit, and log denotes the logarithm with base 2 Also, we define the corresponding achievable DoF per user as d user K d sum 9 and the achievable DoF per relay dimension as h 3 h 2 h h 2 h 22 h 23 Pattern dim-6 h 2 h 2 h 22 h 23 h Pattern 2 dim- h 2+h 22+h 23 h +h 2+h 3 h 3 h 3 h 2 h h 2 h 22 h 23 Pattern dim-2 d relay d sum 0 ater, we will see that d relay measures the efficiency of utilizing the signal space at the relay A cut-set outer bound on the DoF of the considered IO mrc is given as d sum mink, K, or equivalently d user min, spanh,h2 spanh2,h22 h 2 h 3 h 22 h 2 h h 23 Pattern 3 dim-4 spanh,h2 h h 22 h 22 h 23 h 2 h 2 h 3 spanh2,h22 Pattern 4 dim-3 h 23 The above outer bound can be intuitively explained as follows On one hand, each user has antennas and can decode at most independent data streams Thus, d user is upperbounded by On the other hand, the relay has antennas and simultaneously serves clusters Thus, the relay can deliver at most independent data streams to the users in each cluster This outer bound will be used as a benchmark in the following analysis Also, the DoF capacity is achieved when an achievable DoF meets the outer bound III THE CASE OF = 2 AD K = 3 In this section, we consider a IO mrc with = 2 clusters and K = 3 users in each cluster We will generalize our results to arbitrary values of and K in the next section A Preliminary Discussions We start with some intuitions on signal space alignment for a IO mrc We refer to the l-th row of the codeword matrix X, denoted by x l, as the l-th spatial stream of user T From and 3, we see that x l relay in the direction of h l H u l impinges upon the, where ul is the l-th column of U Similarly, let v l be the l-th column of V Then, the relay transmits the l-th spatial stream to user T in the direction of g l G v l streams of all users, ie, {x l We refer to the l-th spatial j, k}, as a unit Clearly, a unit in general contains K spatial streams, one from each user Further, we refer to the spatial structure formed by {h l j, k} and {g l j, k} as a pattern We design a pattern in such a way that each user achieves one DoF Suppose that each user T transmits only one spatial stream From 7, we see that each user T receives one linear equation with the equivalent channel coefficient for the link between T and T given by g T WT j P jh 4 ote that g, W j, P j, h are statistically independent of each other Thus, g T WT j P jh is nonzero with probability one, provided that both null j and null j are of at least dimension one or equivalently, rankp j and rankw j In this way, 4 Here the unit index in h and g is omitted, as there is only one unit considered in the design Fig 2 An illustration of Patterns to each user T obtains one linear combination of the signals from the other K users in cluster j in one channel use after cancelling the self signal x l known by user T Combining K channel uses, each user in cluster j has K independent combinations and is able to decode K messages from the other users in the same cluster, achieving a DoF of d user = per channel use The following five patterns satisfy the above design criteria for the case of = 2 and K = 3 Pattern : {h j {, 2}, k I K } span a subspace of dimension 6 dim-6, and so do {g j {, 2}, k I K } 2 Pattern 2: {h j {, 2}, k I K } span a subspace of dim-, and so do {g j {, 2}, k I K }; for any cluster j, {h k I K } span a dim-3 subspace, and so do {g k I K } 3 Pattern 3: {h j {, 2}, k I K } span a subspace of dim-4, and so do {g j {, 2}, k I K }; for any cluster j, {h k I K } span a dim-2 subspace, and so do {g k I K } 4 Pattern 4: {h j {, 2}, k I K } span a subspace of dim-3, and so do {g j {, 2}, k I K }; for any cluster j, {h k I K } span a dim-2 subspace, and so do {g k I K } Pattern : {h j {, 2}, k I K } span a subspace of dim-2, and so do {g j {, 2}, k I K }; for any cluster j, {h k I K } span a dim- subspace, and so do {g k I K } An illustration of the above five patterns is given in Fig 2, where only the uplink channel vectors are demonstrated by noting the uplink/downlin symmetry It can been seen that, for these patterns, the corresponding projection matrices {P j } and {W j } are of at least rank one As an example, for Pattern 2, the overall signal space occupied by {h j {, 2}, k I K } is of dim- Each cluster j occupies a signal subspace of dim- 3 Thus, null j is at least of dim-2, ie, P j is at least of rank 2 Due to the uplink/downlink symmetry, the rank of W j is also at least 2 The reasoning for the other patterns are similar Then, from 7 and the discussions therein, we c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

5 009/TWC , IEEE Transactions on Wireless Communications TABE I PATTERS FOR THE IO RC WITH = 2 AD K = 3 Pattern Dimension d sum d relay Requirement 6 6 A > 6 > > 2 3 conclude that each of the above five patterns can achieve the same total DoF of 6 evertheless, the dimension of the relay s signal space spanned by these channel vectors differs from pattern to pattern, which leads to a varying DoF per relay dimension ie, d relay ; see Table I Clearly, the greater d relay is, the more efficiently the relay s signal space is utilized Thus, the priority of a pattern is ranked by d relay, with Pattern of the highest priority Besides, we note that the last column of Table I gives the requirement on the antenna setup to construct these patterns in a IO mrc The details on how to obtain these requirements will be elaborated in Subsection III-C B ain Result We now consider the general case that each user transmits multiple spatial streams over a IO mrc with = 2 and K = 3 We will design the beamforming matrices {U } and {V } to align signals in such a way that the equivalent channel vectors {H u l j, k} and {G v l j, k} form one of the five patterns described in Subsection III-A Further, as aforementioned, the priorities of these patterns are ranked by d relay Our target is to determine the most efficient way of constructing units to occupy the relay signal space, with the result given below emma : For the -by- IO mrc with = 2 and K = 3 operating in the clustered full data exchange mode, an achievable DoF per user is given by min,, 3 3 min d user =,, 4 3 < < 4 9 3, min 3, 2, > The proof of emma can be found in the next subsection From emma, we observe the following interesting property Property : The achievable DoF is proportional to the relay s antenna number for any given antenna ratio, where f is a function of That is, d user = f This property generally holds for IO mrcs by noting the fact that doubling both and always doubles the DoF of the IO mrc This property will be used to prove an important lemma shortly The DoF results in emma is obtained by aligning signals in the -dimension signal space seen at the relay All the antennas at the relay are utilized in signaling Physically, it is always allowed to disable a portion of antennas at a node We next show that, for certain configurations of and, disabling a portion of relay antennas and aligning signals in a relay s signal space with a smaller dimension improves the achievable DoF To show this, we first present the antennadisablement lemma below emma 2: For the considered IO mrc, assume that a DoF of d user = d 0 is achievable at a certain antenna configuration of = 0, = 0 Then, every point of x = 0, y = d user on the line segment of y = d 0 for x [ 0 0, is achievable by disabling 0 user antennas; also, every point of x = 0, y = d user on the line segment of y = d00 0 x for x 0, 0 0 ] is achievable by disabling relay antennas Proof: The first half of the lemma ie, the statement on disabling user antennas trivially holds We focus on the second half By assumption, d user = d 0 is achievable at = 0, = 0 Then, from Property, d user = d0 0 is achievable at any, satisfying = 0 0 ow consider an antenna configuration, = 0 with < 0 To prove emma 2, it suffices to show that the DoF of d user = d 0 0 is achievable As 0 < 0 0, we can reduce the number of active relay antennas to = 0 0 by disabling relay antennas Then, = 0 0, implying that a DoF of d user = d 0 0 = d 0 0 = d x is achievable This completes the proof of emma 2 Combining emmas and 2, we obtain the following main result of this section Theorem 3: For the -by- IO mrc with = 2 and K = 3, an improved achievable DoF per user is given by min,, 3 d user = min 4,, 3 3 min,, < < 9 > 9 3 The function of the achievable DoF against is plotted in Fig 3 It is interesting to see how the achievable DoF given by emma is improved by using emma 2 As illustrated, the relay antenna disablement lemma increases the achievable DoF in both ranges of 4, 4 9 and 9, 6 Also, we see that the improved achievable DoF curve is piecewise linear and is bounded either by the user antenna number or by the relay antenna number, which is analogous to the DoF results for the IO interference channel [22] In Fig 3, we also include the cut-set outer bound for comparison We see that the achievable bound and the outer bound coincide when 6 and, implying that the DoF capacity is achieved in these ranges of values However, for < < 6, there is still a gap between the c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

6 009/TWC , IEEE Transactions on Wireless Communications 6 Achievable DoF per User /2 /3 /4 / 0 0 / Cut-set bound DoF by emma DoF by Theorem 3 4/ /3 /2 4/9 /2 /9 2/3 /6 / Fig 3 The achievable DoF per user and the DoF cut-set bound with respect to the antenna ratio for the IO mrc operating in the clustered full data exchange model The considered IO mrc consists of = 2 clusters, and K = 3 users per cluster achievable DoF and the outer bound arrowing this gap will be an interesting topic for future research C Proof of emma As aforementioned, we need to jointly design the transmit beamforming vectors {u l j, k, l}, the receive beamforming vectors {v l j, k, l}, and the relay s projection matrices, so as to achieve the DoF specified in emma We will mostly focus on the uplink beamforming design, as the downlink design is straightforward by noting the uplink/downlink symmetry Our goal is to design as many units as possible to pack the relay s signal space, under the constraint that the messages in each unit are decodable at their intended users Case of 6 : As 6, both span{h j, k} and span{g j, k} are of dim- 6 with probability one This implies that the relay has enough signal space to allow each user to transmit spatial streams, which amounts to units with Pattern Thus, an achievable DoF per user is 2 Case of 6 < 3 : In this case, is sufficiently large to construct Pattern 2 Denote H = [H, H 2, H 3, H 2, H 22, H 23 ] C 6 From channel randomness, nullh is of dimension 6 with high probability Thus, there exists a full column-rank matrix U C 6 6 satisfying HU = 0, ie j= k= K H U = 0, 4 [ where U is partitioned ] as U = U T, U T 2, U T 3, U T 2, U T 22, U T T 23 with U C 6, j, k From 4, the rank of C 66 defined in 4b is at most min{, 6 } From the randomness of H, U is of full rank for sure Then, using emma 0 in Appendix A, we further see that is of rank min{, 6 } with probability one First consider 6 <, or equivalently, < Denote by u l the l-th column of U Then K H u l = 0, for l =,, 6 j= k= Thus, span{h u l j, k} is of at most dim-, for l As rank = 6, span{h u l j, k} is of dim- with probability one, for l Therefore, {H u l j, k} forms a unit with Pattern 2, for l =,, 6 These 6 units occupy a dim-6 subspace with probability one The remaining 6 = 6 dimensions are used for constructing units with Pattern Specifically, each user has 6 = unused dimensions In other words, each user can transmit extra data streams at most Hence the total number of the spatial streams transmitted by all the users is 6, which is exactly the number of the unused dimensions at the relay Thus, we can construct units with Pattern Then, each user transmits spatial streams in total Therefore, the achievable DoF per user is We now consider the remaining case of In this case, the overall signal space of dim- are entirely occupied by the units with Pattern 2 This corresponds to a maximum DoF per user of We emphasize that is not necessarily an integer We can use the technique of symbol extension to achieve a fractional DoF For example, consider channel uses Then, the overall signal space is enlarged to be of dim- Then, units with Pattern 2 can be constructed provided that, which achieves a total DoF of 6, or equivalently, a DoF per user per channel use of Similar symbol extension techniques will be used to achieve a fractional DoF without further notice To summarize, the maximum DoF per user is given by min, with probability one when 6 < 3 3 Case of 3 < < 4 9 : In this case, is sufficiently large to construct units with Pattern 3 Denote H j = [H j, H j2, H j3 ] C 3, for j =, 2 From channel randomness, nullh j is of dimension 3 > 0 for sure Thus, there exists a full column-rank matrix U j C 3 3 satisfying H j U j = 0, for j =, 2, ie K k= H U = 0, for j =, 2, 6 where U j is partitioned as U j = [ U T j, UT j2, T j3] UT with U C 3, j, k From 6, the rank of C 66 defined in 4b is at most min{, 43 } From emma 0 in Appendix A and the fact that H and H 2 are mutually independent, we see that is of rank min{, 43 } with probability one Consider the case of > 43, or equivalently, < 2 Then, rank = 46 with probability one Denote by u l the l-th column of U Then, For > 6, there is enough freedom to construct units with Pattern to occupy the overall signal space Thus, units with Pattern can be constructed to occupy any signal subspace left by the units with pattern 2 Similar arguments will be implicitly used throughout this paper c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

7 009/TWC , IEEE Transactions on Wireless Communications 7 K k= H u l = 0, for j =, 2, and l =,, 3 Thus, span{h u l j, k} is of at most dim-4, for l =,, 3 Further, since rank = 46, we have dimspan{h u l j, k} = 4 with probability one, for l =,, 3 Therefore, {H u l j, k} forms a unit with Pattern 3, for l =,, 3 In total, these 3 units with Pattern 3 occupy a dim- 43 subspace with probability one The remaining 43 dimensions are for constructing 43 units with Pattern 2 Thus, the achievable DoF per user is d user = = 3 For the remaining case of 2 < 4 9, the overall space of dim- are all used for constructing units with Pattern 3 This corresponds to a maximum DoF per user of 4 Thus, we conclude that the maximum DoF per user is given by min 3, 4 with probability one when 3 < < Case of : In this case, we will construct units to achieve d relay = 2 Unlike the preceding cases in which signal alignment is done in a unit-by-unit manner, we will jointly align the signals of multiple units here, with the reason explained as follows The unit-by-unit signal-alignment technique implies that the signals from different units can be distinguished at the relay ie, the relay is able to decode combinations of the signal streams in one unit without seeing any interference from the other units However, this decodability is not necessarily required to accomplish clustered full data exchange As a matter of fact, the relay is allowed to decode combinations of the signal streams belonging to different units, provided that these streams are from a same cluster This means that joint signal alignment of multiple units can potentially outperform unit-by-unit based signal alignment, as seen in what follows We start with unit-by-unit signal alignment To construct Pattern 4, we need to design beamforming vectors satisfying H j u j + H j2 u j2 + H j3 u j3 = 0, j =, 2, 7a H u + H 2 u 2 + H 2 u 2 + H 22 u 22 = 0, or equivalently, in a matrix form as H H 2 H H 2 H 22 H 23 H H 2 0 H 2 H b u = Bu = 0 8 where u = [u T, u T 2, u T 3, u T 2, u T 22, u T 23] T As B C 3 6 is of full rank for sure, we require that > 2 to ensure that 8 has nontrivial solutions We next show that the above requirement on can be relaxed to 4 9 when joint signal alignment of multiple units is considered Our target is to construct full-rank beamforming matrices {U C t } satisfying H j U j + H j2 U j2 + H j3 U j3 = 0, j =, 2, 9a H U Ũ +H 2 U 2 Ũ 2 +H 2 U 2 Ũ 2 +H 22 U 22 Ũ 22 =09b where t represents the number of units involved, and Ũ C t t is a full-rank matrix We will determine t and Ũ shortly in this subsection In the above, the existence of {Ũ} implies that, when the relay decodes a combination from a certain cluster, signal streams from different units are allowed to interfere with each other, provided that these signal streams are from the same cluster From 9, span with defined in 4b is of dimension 3t; span j is of dimension 2t Then, null 3 j span is of dimension t, and so the projection matrix P j is of rank t This implies that the relay can decode t independent linear combinations of the messages from each cluster As the received signal at the relay occupies a subspace of dim-3t, the above scheme achieves d relay = 6t 3t = 2 which is equal to the d relay of Pattern 4 We now describe how to construct {U } satisfying 9 We show that, when = 4 9, this can be done with t = 3 From the uplink/downlink symmetry, it suffices to only consider 9 From emma in Appendix A, spanh j, H j2 spanh j3 is of dimension 3 = 3 with probability one Thus, there exist U C 3, for j =, 2 and k =, 2, 3, satisfying 9a Further, spanh j U j, H j2 U j2 is of dim- 2 3 with probability one oting that spanh U, H 2 U 2 and spanh 2 U 2, H 22 U 22 are independent of each other, we obtain from emma that dimspanh U, H 2 U 2 spanh 2 U 2, H 22 U 22 = 3 with probability one et H C 3 give a basis of the intersection of spanh U, H 2 U 2 spanh 2 U 2, H 22 U 22 Then, since span H spanh j U j, H j2 U j2, for j =, 2, there exist Ũ C 3 3, for j =, 2 and k =, 2, satisfying H = H U Ũ + H 2 U 2 Ũ 2 = H 2 U 2 Ũ 2 H 22 U 22 Ũ Then, 9b is also met Therefore, we conclude that a per-user DoF of 3 is achievable when Case of > 2 3 : In this case, is large enough to construct units with Pattern Specifically, the intersection of three subspaces spanh, k =, 2, 3, is of dim-3 2 with probability one That is, with high probability, there exist unitary matrices U C 3 2 for j, k satisfying H j U j = H j2 U j2 = H j3 U j3, j =, 2 2 From 2, C 63 2 is of rank min, 23 2 with probability one ow suppose > 23 2, or equivalently, < 6 Then, is of rank 23 2 with probability one Denote by u l the l-th column of U From 2, {H u l j =, 2; k =, 2, 3} form a unit with Pattern, and in total, there are 3 2 of such units, occupying a signal subspace of dim-23 2 The remaining signal space of dim is used to support Pattern 4 Thus, d user = = 3 We now consider the remaining case of 6 The overall signal space is wholly occupied by units with Pattern This corresponds to a maximum DoF per user of 2 Therefore, the maximum DoF per user is given by min 3, 2, which concludes the proof of emma IV GEERAIZATIO TO ARBITRARY AD K In this section, we generalize the DoF results to an arbitrary network configuration of and K We start with some notions arising from the special case studied in the preceding section c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

8 009/TWC , IEEE Transactions on Wireless Communications 8 A Definitions We define a corner point of a DoF curve as follows Definition : Given an achievable DoF curve of d user = f, with = 0, a point 0, d user is said to be achievable if d user f, 0 Definition 2: Corner Point Given an achievable DoF curve of d user = f, with = 0, an achievable point 0 0, d 0 is said to be a corner point if 0, d 0 0 is not achievable for any > 0, and 0, d 0 is not achievable for any < 0 A corner point is achieved when the overall relay s signal space is occupied by units following a single pattern For the case of = 2 and K = 3, five patterns are considered in deriving the achievable DoF in emma, and therefore, there are five candidate corner points, ie, 6, 6,,, 2, 4, 4 9, 3, and 6, 2, corresponding to Patterns to, respectively Some of them namely,,, 4 9, 3, and 6, 2 are indeed corner points of the DoF curve given in Theorem 3 see Fig 3; the others namely, 6, 6 and 2, 4 are not as they are obscured by the real corner points To obtain the achievable DoF in Theorem 3, it suffices for us to identify the corner points and then to apply the antennadisablement lemma to these corner points To be specific, define the g-function as { bx g a,b x = a, for x < a 22 b, for x a We consider the construction of Type-I candidate corner points for a general setup of and K The results are presented below, with the proofs given in IV-C and IV-D emma 4: For the considered IO mrc with clusters and K users per cluster and 2K, K ], the following candidate corner points, d user are achievable: = max l l :2 l l l l l + K 24a l lk l+ d user = lk l + l, for l = 2,, 24b emma : For the considered IO mrc with clusters and K users per cluster and 0, 2K ], the following candidate corner points, d user are achievable: = l + 2a tl + K tl + K l tl + d user = tl+k l, l =, t=2,, 2b t We now consider the construction of Type-II corner points, with the result given below The proof can be found in IV-E emma 6: For the considered IO mrc with clusters and K users per cluster > K, the following candidate corner points, d user are achievable: K k + = K kl + kl + d user = 26b where a and b are constant coefficients Then, we can represent the achievable DoF in Theorem 3 using corner points as d user = max g,, g 4 9, 3, for k =,, K, and l = 2,, With emmas 4-6, we present an achievable DoF of the g 6, 2 IO mrc with an arbitrary setup of, K,, as 23 follows Denote by I the set of all the candidate corner points We categorize all corner points into two types, namely, Type-I and Type-II, based on the value of the antenna ratio, d user specified in 24, 2, and 26 Then, we obtain the following main theorem of this paper Type-I corner points are those that fall into the range of K In this Theorem 7: For the considered -by- IO mrc with range, we have K Then the nullspace of H j = [H j, H j2,, H jk ] C K clusters and K users per cluster, the following DoF per user is achievable: contains no vector except the trivial zero vector, implying that, no matter how a pattern is constructed, the K channel d user = max g a,b a,b I, 27 vectors of a common cluster j always span a subspace of dim- K Therefore, to construct patterns corresponding to Type-I where the g-function is defined in 22 corner points, we only need to consider signal space alignment Remark : Theorem 3 is a special case of Theorem 7 between clusters For the case of = 2 and K = 3,, Specifically, for = 2 and K = 3, we obtain the following is a Type-I corner point, as seen from Fig 3 corner points given in emmas 4 and : Type-I corner point Type-II corner points are those that fall into the range of > K In this, d user =, given by 24 with l = 2; Type-II corner range, we have K > Thus, there are point 4 9, 3 given by 26 with l = 2 and k = 2; and Type-II nontrivial vectors in the null space of [H j, H j2,, H jk ], corner point 6, 2 given by 26 with l = 2 and k = implying that we need to consider signal space alignment not Then, Theorem 3 follows by evaluating 27 only between clusters, but also within each cluster For the etting = 3 and K = 3 in Theorem 7, we obtain the case of = 2 and K = 3, Type-II corner points include following result 4 9, 3 and 6, 2, as seen from Fig 3 Corollary 8: For the -by- IO mrc with = 3 To generalize the DoF results to the case of arbitrary and and K = 3, an achievable DoF per user is given by d user = K, we need to systematically identify possible corner points, min, 8, for 6 ; d user = min 3 4, 2, for 6 < as elaborated in what follows 2 9 ; d user = min 3, 7, for 2 9 < 2 7 ; d user = min 2 B ain Result, 2 9, for 2 7 < 4 27 ; d user = min 3 7, 3, for > a Proof: For = 3 and K = 3, the following corner points are identified: 8, 8 given by 2 with t = 2; 8 4, 2 given c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

9 009/TWC , IEEE Transactions on Wireless Communications 9 /3 [7], the total DoF of the IO mrc is bounded by 2 which does not scale with the number of users Achievable DoF per User 2/9 /6 /7 /8 Cut-set bound DoF by Corollary 8 0 /8/6 2/9 2/7/3 4/9 4/27 7/9 8/4 /2 / Fig 4 The achievable DoF per user and the DoF cut-set bound with respect to the antenna ratio for the IO mrc operating in the clustered full data exchange model The considered IO mrc consists of = 3 clusters, and K = 3 users per cluster by 24 with l = 2; 2, 7 given by 24 with l = 3; 4 9, 2 9 given by 26 with l = 2 and k = 2; and 7 9, 3 given by 26 with l = 3 and k = Then, the corollary follows by 27 Remark 2: The achievable DoF against given by Corollary 8 is plotted in Fig 4 For comparison, we also provide the cut-set outer bound in From Fig 4, the achievable bound matches the cut-set outer bound when 7 9 and 8, implying that the DoF capacity of the consider IO mrc is achievable in these ranges of This result can be generalized to the case of arbitrary and K, as detailed below Corollary 9: For the considered -by- IO mrc with clusters and K users in each cluster, the DoF capacity of d user = is achievable when ] 0, K ; and the [ DoF capacity of d user = is achievable when K + K, ] Proof: For 0, K, K, is a corner point given by 2 with l = and t =, [ which achieves the DoF outer bound in ; for K + K,, K + K, is a corner point given by 26 with l = and k =, which again achieves the DoF outer bound Remark 3: We see from Corollary 9 that d sum = Kd user is proportional to K ]ie, the number [ of users per cluster, when 0, K and K + K, It can be verified that, with the achievable DoF given in Theorem 7, the linearity of d sum in K holds for any, configurations Intuitively, this linearity is due to the fact that: in clustered full data exchange, each relay antenna may transmit combinations of K messages in a cluster one from each user in the cluster in K time slots; each user in the cluster receives K combinations, and is able to decode the K messages from the other users by utilizing the self-message, which results in K DoF per relay antenna, or K DoF in total ote that this linearity of DoF in K in general does not hold for other data exchange models For example, for pairwise data exchange in C Proof of emma 4 Without loss of generality, we assume that the pattern to be constructed involves l [2, ] active clusters, and index these clusters as j [], j [2],, j [l] in the ascending order Denote H j = [H j, H j2,, H jk ] Our target is to construct beamforming vectors {u j[],, u j[l] } satisfying the following equation: H j[] H j[2] u j[] H j[2] u j[2] H j[3] H j[l ] H j[l] u j[l] =B u=0 28 In the above, B contains l block-rows For each index i, the i-th block-row of B corresponds to a vector in null[h j[i], H j[i+] ] The existence of such vectors is guaranteed by the fact that, for 2K, ] K, null[hj, H j ], for j, j, is of dimension 2K 0 for sure The maximum number of block-rows of B is limited to, since it is required that every new block-row added into B must contain one and only one new H j 6 Also, each unit following the pattern in 28 contains lk spatial streams, with one from each of the users in the l active clusters; the subspace occupied by each unit is of dimension lk l +, as B in 28 contains l block-rows Furthermore, for each active cluster j [t], the interference consists of the l K spatial streams from the other l clusters, satisfying the following equation: H j[t 2] H j[t ] H j[t ] H j[t+] u j[t ] u j[t+] = 0 29 The above matrix has l 2 block-rows and is of full row rank Then, the interference of cluster j t spans a subspace of dimension l K l 2 = l K + Recall that the total signal space spanned by the pattern 28 is of dimension lk + Therefore, the relay sees an interference-free subspace of dimension K, and can obtain one combination of the K signal streams from cluster j t without interference for sure As the cluster index j t is arbitrarily chosen, we conclude that a unit following pattern 28 is decodable for sure at the relay as well as at the user ends due to the uplink/downlink symmetry From emma 2, the number of independent units can be constructed following pattern 28 with a given selection of 6 Otherwise, there must exist an H j repeated in B at different columns For example, suppose H j[3] is replaced by H j[] in 28 Then, subtracting the first block-row of B by the second block-row, we obtain H j[] U j[] H j[] U j[3] = 0, or equivalently, H j[] U j[] U j[3] = 0 ote that U j[] U j[3], as they correspond to different spatial streams in a pattern Therefore, H j[] U j[] U j[3] = 0 implies that the nullspace of H j[] C K is not trivial, ie, K > This contradicts with the condition of 2K, ] K in emma c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

10 009/TWC , IEEE Transactions on Wireless Communications 0 the index set {j [], j [2],, j [l] } is given by the dimension of the nullspace of B However, by considering all possible index selections, there are a family of l different patterns in the form of 28 We need to determine how many linearly independent units following this pattern family can be constructed, as detailed below First, from the channel randomness, B in 28 is of full row rank with probability one, and thus the nullspace of B is of dimension lk l for sure Then, lk l linearly independent units can be constructed for any given selection of the index set {j [], j [2],, j [l] }, as ensured by emma 0 Since there are l different ways to select the index set, the total number of units is limited by c l = l lk l 30 Second, each block row of B in 28 corresponds to an equation of the form H j u j + H j u j = 0 with j j For any given index pair j, j with j j, null[h j, H j ] has 2K linearly independent nullspace vectors of the form [u T j, ut j ]T for sure This means, the equation H j u j + H j u j = 0 cannot appear in all the constructed units for more than 2K times This gives another limitation on the number of units that can be constructed Specifically, there are in total l different ways to select l active clusters from clusters; there are 2 l 2 of these selections in which both clusters j and j are selected et c 2 be the total number of units constructed Then, the number of H j u j + H j u j = 0 used in constructing c 2 units is given by 2 l 2 c 2 l = c 2 2 l 3 The above number cannot exceed 2K ie, the number of independent vectors in null[h j, H j ] Therefore, the total number of units following pattern 28 is limited by 2 c 2 = 2 2K 2 l 32 Third, there are more constraints on constructing units with the pattern in 28 et j [], j [2],, j [l ] without loss of generality, in the ascending order be a set of l l distinct indexes selected from {,, } Consider the following subpattern similar to 28 as H j H [] j [2] H j [2] H j [3] H j [l ] H j [l ] u = B u = 0 33 The key observation here is that every block-row of B can be obtained by linearly combining the block-rows of B, provided that {j [], j [2],, j [l ] } {j [], j [2],, j [l] } To see this, it suffices to focus on the first block-row of B As {j [], j [2] } {j [], j [2],, j [l] }, we have j [] = j [p] and j [2] = j [p ] for certain integers p and p with p < p Consider the following combination of the p-th to p -th block-rows of B : row p row p+ +row p p p row p This yields H j[p] u j[p] + p p H j[p ]u j[p ] = 0, or equivalently, [H j[p] H j[p ] ][u T j [p] p p u T j [p ] ] T = 0, which corresponds to the first block-row of B With the above observation, we see that every vector u satisfying 28 corresponds to a unique vector u satisfying the sub-pattern in 33, provided that {j[], j [2],, j [l ] } {j [], j [2],, j [l] } This means, for all the linearly independent units following pattern 28, the sub-pattern in 33 cannot be used for more than l K l times as the dimension of nullb is l K l for sure The chance of a random selection of {j[], j [2],, j [l ] } satisfying {j[], j [2],, j [l ] } {j [], j [2],, j [l] } is given by l l l l l l = 34 l Therefore, the total number of linearly independent units following 28 cannot exceed l l K l c l = l, for l = 2, 3,, l 3 l ote that 30 and 32 are two special cases of 3 by letting l = l and l = 2, respectively Fourth, besides those in the form of 33, we shall consider more sub-patterns to guarantee that linearly independent units are constructed, as detailed below et B be a block matrix with each block-row given by an arbitrary combination of the block-rows of B in 28 We require that the nullspace of B contains enough linearly independent vectors to support the linearly independent units constructed Without loss of generality, we assume that the sub-pattern B involves l clusters, and index these clusters as {j[], j [2],, j [l ]} For {j[], j [2],, j [l ] }, we construct B in the form of 33 Further, we have the following facts: first, every block-row of B can be expressed as a linear combination of the blockrows of B, implying that the rank of B is no greater than the rank of B, or equivalently, from the rank-nullity theorem, the dimension of nullb is no less than the dimension of nullb ; second, sub-patterns B and B have the same times of occurrence in the pattern family of 28 by noting the fact that B and B involve the same set of active clusters Therefore, provided that the requirement for sub-patterns of the form B is met, the nullspace requirement for the subpatterns of the form B is automatically met Based on all the above discussions, we conclude that the maximum number of linearly independent units following pattern 28 is given by c min = min 2 l l c l To achieve a corner point, these c min units span the overall relay s signal space of dimension Thus, we have c min lk l + =, yielding in 24a Also, as each unit spans a subspace of dimension lk l +, the corresponding d user is given by d user =, which concludes the proof lk l+ l D Proof of emma et t be an integer in [2, ], and l = t For ], we construct a pattern matrix B 2 with tl + t+k, tk c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

11 009/TWC , IEEE Transactions on Wireless Communications active clusters indexed by j [], j [2],, j [tl+] as below: H j[] H j[t+] H j[t+] B 2 = H j[2t+] H j[2t+] H j[3t+] 36 In the above, B 2 has l block-rows; every block-row contains t + nonzero blocks; every new block-row added into B 2 contains t new H j s until the unused H j s are not enough to construct a new block-row, which implies l = t We first verify that 36 defines a valid pattern Clearly, each unit with 36 contains tl + K spatial streams; the subspace spanned by each unit is of dimension tl + K l For any index j, the interference of cluster j contains tlk spatial streams which span a subspace of dimension tlk l Therefore, for each cluster j, the relay sees an interferencefree subspace of dimension K, and can obtain one combination of the K signal streams from cluster j for sure Thus, a unit with pattern 28 is decodable at the relay as well as at the user ends from the uplink/downlink symmetry We next determine the number of linearly independent units that can be constructed with pattern 36 From emma 2 and the discussions therein, the maximum number of linearly independent units allowed is given by the nullspace dimension of the pattern matrix ote that nullb 2 is of dimension tl+ K l with probability one Therefore, we are able to construct c = tl+k l units following pattern 36 7 To achieve a corner point, these c units span the overall relay s signal space of dimension Thus, we have ctl + K l =, yielding given in 2a Also, as each unit is of dimension tl + K l, the corresponding d user is given by d user =, which concludes the proof tl+k l tl+ E Proof of emma 6 We generalize the construction of Pattern 4 in Section III as follows et l be the number of active clusters, m be the number of spatial streams transmitted by each active user, and k be an arbitrary integer [, K] For the uplink channel, we design the beamforming matrices {U C m } to satisfy H j U j + + H U + H jk+t U jk+t = 0, 37 for j =,, l, and t =,, K k, or equivalently H j H H jk+ 0 U j =B 3 U j =0 38 H j H 0 H jk U jk for j =,, l From 37, spanh jk+t U jk+t spanh j U j,, H U, for t =,, K k, or equivalently, spanh j U j,, H jk U jk = spanh j U j,, H U is of dimension mk Thus, 7 It is automatically ensured that the nullspace of any sub-pattern matrix B 2 with each block-row of B 2 being a certain combination of the blockrows of B 2 is able to support c units From linear algebra, we always have rankb 2 rankb 2 Thus, the dimension of nullb 2 is no less than that of nullb 2, implying that nullb 2 is not the bottleneck That is, the maximum number of units following pattern 36 is limited by c to ensure that m linearly independent units with pattern in 37 can be constructed, it is required that the nullspace of B 3 C K k K in 37 is of dimension m From the channel randomness, this can be met when K K k = m 39 On the other hand, to ensure that the relay can decode m linear equations for each cluster j, the dimension of the interference space spanned by the signals from the other l active clusters cannot exceed mk, by noting that the signal subspace of each cluster j j, ie, spanh j U j,, H j KU j K, is of dimension mk Thus mkl = m 40 Combining 39 and 40, we obtain m= kl + and = K k+ 4 K kl + oting that only l of clusters are active, we obtain d user = m l = kl + l 42 Considering l = 2,, and k =,, K, we conclude the proof of emma 6 V COCUSIO In this paper, we analyzed achievable DoF of the -by- IO mrc with clusters and K users per cluster, operating in the clustered full data exchange mode We developed a novel systematic signal alignment technique to jointly construct the beamforming matrices at the users and the relay for efficient implementation of PC Based on that, an achievable DoF was derived for the considered IO mrc with an arbitrary configuration of, K,, We also showed that our proposed scheme is DoF-optimal for the considered IO mrc, when K and The study of IO mrcs is still in an initial stage The fundamental performance limits of such channels are far from being well understood For example, the derived achievable DoF in this paper in general serves as a lower bound of the DoF capacity of the considered IO mrc To narrow the gap towards the DoF capacity will be an important future research topic oreover, DoF analysis only characterizes the system performance at high SR Optimal beamforming designs for the IO mrc at finite SR remains a challenging problem, and will be of interest for future research APPEDIX A SOE USEFU EAS K + K Consider a full-rank matrix A = [A,, A K ], where A i C i for i =,, K We make the following assumptions on the matrix size: i, i, and K i= i > Then, A is a wide matrix with full row rank; also, each A k is a tall matrix et U C K i= i K i i= be a nullspace matrix of A, ie, the columns of U give a basis of nulla Partition U as U = [U T,, U T K ]T with U i c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

12 009/TWC , IEEE Transactions on Wireless Communications 2 C i K C K K i= i Denote à = [A U,, A K U K ] i= i Then emma 0: Assume that U k is of full rank, ie, ranku k = min i, K i= i, for k =,, K Then, rankã is given by K i= min i, K i= i K i= i + Proof: et v = [v T,, vk T ]T be an arbitrary vector in nullã, where v i C K i= i Then, we obtain 0 = Ãv = Adiag{U,, U K }v Thus, diag{u,, U K }v belongs to nulla As U spans nulla, there exists x C K i= i such that diag{u,, U K }v = Ux, or equivalently, U i v i = U i x, for i =,, K We now determine the number of free variables in v which is equal to the dimension of nullã ote that U i is of rank min i, K i= i for i =,, K We consider the following two cases First, if i K i= i, the left inverse of U i exists Then, U i v i = U i x implies v i = x, or equivalently, v i is uniquely determined by x Second, if i < K j= j, from U i v i = U i x, each v i has K j= j i free variables for any fixed x Combining these two cases, we see that each v i has max K j= j i, 0 free variables for any given x Considering all {v i }, together with the fact that x C K i= i is arbitrary, we obtain that v has K K K i + max j i, 0 43 i= i= j= free variables, or equivalently, the dimension of nullã is given by 43 By using the rank-nullity theorem, we conclude that the rank of à is given by K i= min i, K i= i K i= i + From emma 0, we have the following result emma : Assume that A i C i for i =, 2 are random matrices with the coefficients independently drawn from an arbitrary continuous distribution Also assume i for i and + 2 > The intersection of spana and spana 2, denoted by spana spana 2, is a subspace of dimension with probability one Proof: From the randomness of A and A 2, null[a, A 2 ] is of dimension with probability one et U = [U T, U T 2 ] T C be a nullspace matrix of [A, A 2 ] Again from the randomness of A and A 2, U and U 2 are of full rank for sure From emma 0, we obtain that spana U, A 2 U 2 is of dimension min + 2 +, for sure Then, emma follows by noting spana spana 2 = spana U, A 2 U 2 We next show that, due to the channel randomness, the maximum number of independent units allowed for a pattern eg, in 28, 33, 36, and 38 is limited by the nullspace dimension of the corresponding pattern matrix We focus on the pattern B u = 0 in 28 The proofs for the other patterns are similar and omitted for brevity emma 2: et r be the dimension of nullb Then, the maximum number of linearly independent units following pattern 28 is given by minr, for sure lk l+ Proof: From the channel randomness, rankb = lk, l for sure From the rank-nullity theorem, nullb is of dimension r = lk l + et U = [U T j [], U T j [2],, U T j [l] ] T C lk r be a nullspace matrix of B Further partition each U j[i] C K r as U j[i] = [U T j [i],, UT j [i] K ]T, where U j[i] k C r Denote j[i] = [H j[i] U j[i],, H j[i] KU j[i] K], and = [ j[],, j[l] ] From the discussions below 28, each unit is of dimension lk l+ Thus, to prove emma 2, it suffices to show rank = minlk l + r, We first show that there exists such a nullspace matrix U that every component U is of full rank, ie, ranku = min, r for j, k To see this, we rewrite 28 as H j[] U j[] = H j[2] U j[2] = = l H j[l] U j[l] 44 or equivalently, for i =,, l, spanh j[i] U j[i] =spanh j[] spanh j[2] spanh j[l] 4 Recursively using emma, we see that the dimension of S = spanh j[] spanh j[2] spanh j[l] is mink, r for sure As H j[],, H j[l] are randomly generated, their intersection S is also random in C Further, for any index i {,, l}, S is a random subspace of dimension mink, r in spanh j[i] From linear algebra, U j[i] = [U T j [i],, UT j [i] K ]T is the coordinate matrix to describe a basis of S in spanh j[i] From the randomness of S, U j[i] k for k =,, K are of full rank for sure We now determine rank for the case of r lk l+ et v = [vj T [],, vj T [l] ] T be a vector in null, where v j[i] C Kr, i =,, l To show rank = lk l + r, it suffices to show that null is of dimension l r, or equivalently, v has l r free variables et v j [] =v j[] and v j [i] =v j[i] v j [i ], for i = 2,, l 46 From B U = 0, we obtain H j[i] U j[i] + H j[i+] U j[i+] = 0, for i =,, l Thus, for i =,, l, we obtain or equivalently H j[i] U j[i] v j [i] + H j[i+] U j[i+] v j [i] = 0, 47 v j [i] = 0, 48 where v j [i] = [0,, 0, v j T [i], v j T [i], 0,, 0] T C lkr is a block-vector with only the i-th and i + -th block-entries being nonzero As v null, we have v = 0; also, by definition, we have v = l i= v j [i] Subtracting the l equations in 48 from v = 0, we obtain H j[l] U j[l] v j [l] = 0, which implies v j [l] = 0 by noting that both H j[l] and U j[l] are full column rank 8 Therefore, only v j [i] for i =,, l may contain free variables 8 ote that r = lk l K implies K, which exceeds the range of considered in constructing pattern 28 Thus, we have r = lk l < K, which implies U j[i] C K r is a tall matrix Then, U j[i] is of full column rank for sure c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See

13 009/TWC , IEEE Transactions on Wireless Communications 3 Further denote v j = [v T j,, v T jk ]T with v Cr, and rewrite 47 as K K H j[i] ku j[i] kv j [i] k + H j[i+] ku j[i+] kv j [i] k = 0 49 k= k= for i =,, l We next show that each v j [i] contains exactly r free variables To see this, it suffice to show that v j [i] k, k = 2,, K, are all uniquely determined, provided that v j [i] is given We prove by contradiction Suppose that, for a given v j [i], there exist distinct v j [i] k and v j [i] k, k = 2,, K, satisfying 49 Then, we obtain K k=2 H j [i] ku j[i] k v j [i] k v j [i] k + K k=2 H j [i+] ku j[i+] k v j [i] k v j [i] k = 0, or equivalently C[ v j [i] 2 v j [i] 2 T,, v j [i] K v j [i] K T ] T = 0 0 where C = [H j[i] 2U j[i] 2,, H j[i] KU j[i] K, H j[i+] 2U j[i+] 2, K, H j[i+] KU j[i+] K] From 44, k= H j [i]ku j[i]k + K k= H j [i+]ku j[i+]k = 0, implying that spanh j[i] U j[i], H j[i+] U j[i+], C = spanh j[i] U j[i], C is of dimension min2k r, = 2K r for sure, where 2K r is implied by r lk l+ Thus, C C 2K 2r is of full column rank for sure, or equivalently, 0 does not have non-trivial solutions, which leads to absurdity Therefore, each v j [i] contains exactly r free variables Considering i =,, l, we obtain that v have l r free variables, ie, the dimension of null is given by l r That is, r linearly independent units can be constructed What remains is the case of r > lk l+ In this case, as each unit spans a subspace of dimension lk l +, the maximum number of linear independent units is limited by, which concludes the proof lk l+ [8] D Gündüz, A Yener, A Goldsmith, and H V Poor, The multiway relay channel, IEEE Trans Inform Theory, vol 9, no, pp -63, Jan 203 [9] T Cui, T Ho, and J Kliewer, Space-time communication protocols for n-way relay networks, in Proc IEEE Global Commun Conf GlobeCom, ew Orleans, T, ov 2008 [0] Ong, S J johnson, and C Kellett, The capacity region of multiway relay channels over finite fields with full data exchange, IEEE Trans Inform Theory, vol 7, no, pp , ay 20 [] A U T Amah and A Klein, Regenerative multi-group multi-way relaying, IEEE Trans Veh Technol, vol 60, no 7, pp , Sep 20 [2] A Chaaban, A Sezgin, and S Avestimehr, Approximate sum-capacity of the Y-channel, IEEE Trans Inform Theory, vol 9, no 9, pp , Sep 203 [3] F Wang, X Yuan, S iew, and D Guo, Wireless IO switching: Weighted sum mean square error and sum rate optimization, IEEE Trans Inf Theory, vol 9, no 9, pp297-32, Sep 203 [4] Z Fang, X Wang, and X Yuan, Beamforming design for multiuser two-way relaying: A unified approach via max-min SIR, IEEE Trans Signal Processing, vol 6, no 23, pp 84-82, Dec 203 [] ee, J-b im, and J Chun, Degrees of freedom of the IO Y channel: Signal space alignment for network coding, IEEE Trans Inf Theory, vol 6, no 7, pp , July 200 [6] K ee, ee, and I ee, Achievable degrees of freedom on K-user Y channels, IEEE Trans Wireless Commun, vol, no 3, pp 20-29, ar 202 [7] Y Tian and A Yener, Degrees of freedom for the IO multi-way relay channel, submitted to IEEE Trans Inform Theory, Aug 203 [8] K ee, ee, and I ee, Achievable degrees of freedom on IO two-way relay interference channels, IEEE Trans Wireless Commun, vol 2, no 4, pp , Apr 203 [9] R Tannious and A osratinia, The interference channel with IO relay: Degrees of freedom, in Proc IEEE Int Symp Inf Theory ISIT, Toronto, Canada, July 2008 [20] Z Xiang, Tao, J o, and X Wang, Degrees of Freedom for IO Two-Way X Relay Channel, IEEE Trans Signal Processing, vol 6, no 7, pp 7-720, April 203 [2] V R Cadambe and S A Jafar, Interference alignment and degrees of freedom of the K-user interference channel, IEEE Trans Inform Theory, vol 4, no 8, pp , Aug 2008 [22] C Wang, T Gou, and S A Jafar, Subspace alignment chains and the degrees of freedom of the three-user IO interference channel, submitted to IEEE Trans Inform Theory, Sep 20 ACKOWEDGET The author would like to thank Dr Rui Wang for insightful discussions The author would also like to thank the anonymous 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[7] X Yuan, T Yang, and I B Collings, ultiple-input multiple-output two-way relaying: A space-division approach, IEEE Trans Inform Theory, vol 9, no 0, pp , Oct 203 Xiaojun Yuan S received the BS degree in Electronic and Information Systems from Shanghai Jiaotong University, the S degree in Circuit and Systems from Fudan University, and the PhD degree in Electrical Engineering from the City University of Hong Kong in 2008 From 2009 to 20, he was a research fellow at the Department of Electronic Engineering, the City University of Hong Kong He was a visiting scholar at the Department of Electrical Engineering, the University of Hawaii at anoa in spring and summer 2009, as well as in the same period of 200 From 20 to 204, he was a research assistant professor at the Institute of etwork Coding, The Chinese University of Hong Kong He is now an assistant professor with the School of Information Science and Technology, ShanghaiTech University His research interests cover a broad range of wireless communications, signal processing, and information theory including multiple-input multiple-output IO communications, rateless coding design, non-linear equalization, physical-layer network coding, cooperative communications, etc He has published over 60 peer-reviewed research papers in the leading international journals and conferences, and has served on a number of technical programs for international conferences c 203 IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See