Multi-Way Information Exchange Over Completely-Connected Interference Networks with a Multi-Antenna Relay
aa r X i v : . [ c s . I T ] F e b Multi-Way Information Exchange OverCompletely-Connected Interference Networkswith a Multi-Antenna Relay
Namyoon Lee and Robert W. Heath Jr.Wireless Networking and Communications GroupDepartment of Electrical and Computer EngineeringThe University of Texas at Austin, Austin, TX 78712 USAE-mail : [email protected] and [email protected]
Abstract
This paper considers a fully-connected interference network with a relay in which multiple usersequipped with a single antenna want to exchange multiple unicast messages with other users in thenetwork by sharing the relay equipped with multiple antennas. For such a network, the degrees offreedom (DoF) are derived by considering various message exchange scenarios: a multi-user fully-connected Y channel, a two-pair two-way interference channel with the relay, and a two-pair two-wayX channel with the relay. Further, considering distributed relays employing a single antenna in thetwo-way interference channel and the three-user fully-connected Y channel, achievable sum-DoF arealso derived in the two-way interference channel and the three-user fully-connected Y channel. A majorimplication of the derived DoF results is that a relay with multiple antennas or multiple relays employinga single antenna increases the capacity scaling law of the multi-user interference network when multipledirectional information flows are considered, even if the networks are fully-connected and all nodesoperate in half-duplex . These results reveal that the relay is useful in the multi-way interference networkwith practical considerations.
September 28, 2018 DRAFT
I. I
NTRODUCTION
Multi-way communication using an intermediate relay is a promising wireless networkarchitecture with applications including cellular networks, sensor networks, and device-to-devicecommunications. The simplest multi-way relay network model is the two-way relay channel [1]-[7] in which a pair of two users wish to exchange messages by sharing a single relay. Althoughthe capacity of this simple channel is still unknown in general [2], physical layer network coding[3]-[5] and analog network coding [6]-[7] increase the achievable sum-rates of the two-way relaychannels because it allows users to exploit their transmit signal as side-information. Recently,the two-way relay channel has been generalized in a number of ways: multi-pair two-way relaychannels [8]-[12] and multi-user multi-way relay channels [11]-[16]. For the multi-pair two-wayrelay channel where multiple user pairs exchange messages with their partners by sharing acommon relay, the authors [12] characterized the capacity of multi-pair two-way relay networkfor a deterministic and Gaussian channel model. For the multi-user multi-way relay channelwith unicast messages exchange setup, the multiple-input multiple-output (MIMO) Y channelwas introduced where three users exchange independent unicast messages with each other viaan intermediate relay and characterized the degrees of freedom (DoF) of the channel by the ideaof signal space alignment for network coding [13]. This result was extended into the case of ageneral number of users as a K -user Y channel [17].In spite of extensive studies on different multi-way relay channels, relatively little work hasbeen addressed on the understanding of the capacity of multi-way relay channel, especially whenthe nodes are fully-connected in the network due to difficulty in managing interference. Notethat when the direct links are considered in the multi-way relay channel, it can be equivalentlyviewed as an interference network with a relay. In general, if the networks are fully-connected,i.e., they have a non-layered structure, then a node receives signals arriving along different paths,which causes more inter-user interference than that of the layered network. When the interferencenetworks have a layered structure, it has been shown that the relay can offer gain in the numberof DoF [18]-[20]. On the other hand, for the fully-connected interference network with relays[21]-[23], it was shown that relays cannot not improve the DoF of such a network regardless of September 28, 2018 DRAFT how many antennas the relay has with a few exceptions where the cognitive relay [26] or theinstantaneous relay [27] is considered. If the non-layered multi-hop interference network supportsuni-directional information flows [24]-[25], the non-layered structure incurs a DoF loss. Evenfor multi-way information flows, relays with infinitely many antennas do not increase the DoFof the fully-connected X network whose source nodes are disjoint from destination nodes [28] .In this paper, we provide counter examples of the claim that the relay cannot increase the DoFof the fully-connected interference network. We show that the relay is useful in improving theDoF of the multi-user interference networks when multi-way information exchange is allowedbetween users.In this paper, we consider a fully-connected interference network with a relay where K userswith a single antenna exchange unicast messages with each other via a relay with N multipleantennas. In particular, we assume that all nodes have half-duplex constraint due to hardwarelimitations, implying that transmission and reception occurs in different orthogonal time slots.We consider three different multi-way information exchange setups over the fully-connectedinterference network. • Fully-connected Y channel with a multi-antenna relay : First, we consider K -user fully-connected interference network with a relay equipped with N antennas. In such a channel,each user wishes to send K − unicast messages to all other users and also wishes todecode all other users’ messages. Since the same message exchange setup is considered inthe previous work on Y channel [13] and [17] without direct links between users, we referto it as “ a fully-connected Y channel .” • Two-pair two-way interference channel a multi-antenna relay : As a special case of thefour-user fully-connected Y channel, we consider a four-user fully-connected interferencenetwork with a relay where the four users form two-pairs. The two pairs exchange messageswith their partners via a multi-antenna relay. In particular, when the relay node is ignored,this channel model is equivalent with the two-way interference channels studied in [29] and[30]. • Two-way X channel with a multi-antenna relay : We also consider another four-user
September 28, 2018 DRAFT completely-connected interference network with a two-antenna relay. In this channel, eachuser wants to exchange two unicast messages with two different users in the network. Sincethis channel model can be viewed as a bi-directional X channel, we refer to it as “two-wayX channel with a relay.”
The main contribution of this paper is to derive sum-DoF bounds for certain networkedchannels. Specifically, for the general multi-way information exchange case, i.e., a fully-connected Y channel, it is demonstrated that the sum-DoF of K ( K − K − is the optimal if the relayhas N ≥ K − antennas by showing both converse and achievability. Further, it is shown that thesum-DoF of and are achievable for the two-pair two-way interference and X channel witha relay when the relay has N = 2 . These result are interesting because it has been shown thatthe sum-DoF for the fully-connected interference network cannot be improved by the use of therelays even if the relay has infinitely many antennas [22] and [28]. Our result, however, revealsthe fact that a relay with a finite number of antennas can increase the DoF of the network whenmulti-directional communication is considered between pairs. As an extension, by consideringmultiple distributed relays which of each has a single antenna, we derive the sum-DoF boundsfor both the two-pair two-way interference and three-user fully-connected Y channel. One majorimplication of the results is that the available DoF of fully-connected interference network thatsupports multi-directional information exchange can be improved substantially by allowing arelay with multiple antennas or the multiple distributed relays even if all nodes operate in half-duplex . To see how the relay is useful in terms of DoF for the multi-way communications, itis instructive to compare our result with the case when no relay is used by using the followingexamples: • Example 1 (Four-user fully-connected Y channel without a relay): Let us consider a four-user, fully-connected, and half-duplex Y channel where each user wants to exchange threeunicast messages with the other users in the network. If we assume that there is no relayin the network, the optimal DoF of such a channel equals as shown in [28], which canbe achieved by interference alignment. Meanwhile, our result shows that the sum-DoF of is achievable by involving a relay employing N = 3 antennas, which is a DoF
September 28, 2018 DRAFT improvement. • Example 2 (Two-pairs two-way interference channel without a relay) : When the relay isnot considered in the two-pairs two-way interference channel, it is known that the optimalDoF is one [31]. By involving the relay with two antennas in the network, however, it ispossible to achieve the sum-DoF of , which is DoF increase.Why does the relay provide DoF gain for multi-way communication ? The DoF gaincomes from two mechanisms. One is the side-information inherently given by multi-waycommunication, i.e., caching gain . The second is due to the fact that the relay can make sureeach user does not see the undesired interference signal or see the same interference shape usingjoint space-time precoding techniques. We refer to it as interference shaping gain . To acquiretwo different gains, the multiple antenna relay or multiple relays with a single antenna controlsthe information flow of the multi-way communication so that each user exploits side-informationefficiently, which leads to increase the DoF by the use of a relay. To show our results, multi-phase transmission schemes are proposed, which are inspired by wireless index coding [32] and[34].The rest of the paper is organized as follows. In Section II, a general system model ofmulti-way communications with a MIMO relay is described. To provide the intuition behindthe proposed transmission schemes, a motivating example is provided in Section III. In SectionIV, the optimal DoF for the fully-connected Y channel is addressed. Section V provides thesum-DoF inner bounds for the two-pair two-way interference and X channel with a relay byconsidering different message exchange scenarios in the four-user fully-connected interferencenetwork. To see the effect of relay antenna cooperation, achievable sum-DoF bounds are derivedfor the two-pair two-way interference channel and 3-user fully-connected Y channel when threedistributed relays are considered in Section VI. The paper is concluded in Section VII.Throughout this paper, transpose, conjugate transpose, inverse, Frobenius norm and trace of amatrix X are represented by X T , X ∗ , X − , k X k F and Tr ( X ) , respectively. In addition, C and R indicates a complex and real value. CN (0 , represents a complex Gaussian random variablewith zero mean and unit variance. September 28, 2018 DRAFT
User 1 User K User 2 User 4 User 3
Relay (AP) (cid:273) (cid:273)
Fig. 1. A K -user fully-connected interference network with one relay. User ℓ wants to send K − messages W k,ℓ and receive ˆ W ℓ,k for k ∈ U / { ℓ } in this network. II. S
YSTEM M ODEL
Let us consider an interference network comprised of K users with a single antenna eachand a relay with N antennas. All the users and the relay are completely-connected as illustratedin Fig. 1. User k , k ∈ U , { , , . . . , K } , wants to send K − unicast messages W ℓ,k for ℓ ∈ U / { k } , U c k to user ℓ and intends to decode K − messages W k,ℓ for ℓ ∈ U c k sent by allother users. In this channel, it is assumed that the relay and all nodes operate in half-duplexmode, implying that transmission and reception span orthogonal time slots.Let x ℓ [ n ] = f ( W k,ℓ ) for k ∈ U cℓ be the transmitted signal by user ℓ at time slot n where f ( · ) represents an encoding function. Also, let S n and D n denote the set of source and destinationnodes at time slot n . Due to the fully-connected property and the half-duplex constraint, whenthe users belonging the source set S n send their signals at the n -th time slot simultaneously, September 28, 2018 DRAFT user k ∈ D n and the relay receives the signals y k [ n ] = X ℓ ∈S n h k,ℓ [ n ] x ℓ [ n ] + z k [ n ] , k ∈ D n , (1) y R [ n ] = X ℓ ∈S n h R ,ℓ [ n ] x ℓ [ n ] + z R [ n ] , (2)where y k [ n ] and y R [ n ] ∈ C N × represent the received signal at user k and the relay; z k [ n ] and z R [ n ] denote the additive noise signal at user k and at the relay at time slot n whose elementsare Gaussian random variable with zero mean and unit variance, i.e., CN (0 , : and h k,ℓ [ n ] and h R ,ℓ [ n ] = [ h R ,ℓ [ n ] , . . . , h N R ,ℓ [ n ]] represent the channel coefficients from user ℓ to user k and thechannel vector from user ℓ to the relay, respectively.When the relay and user ℓ ∈ S n cooperatively transmit at the n -th time slot, at the same time,user k ∈ D n receives the signal as y k [ n ] = X ℓ ∈S n h k,ℓ [ n ] x ℓ [ n ] + h ∗ k, R [ n ] x R [ n ] + z k [ n ] , k ∈ D n , (3)where h ∗ j, R [ n ] = [ h j, R [ n ] , . . . , h Nj, R [ n ]] ∗ denotes the (downlink) channel vector from the relay touser k and x R [ n ] represents the transmit signal vector at the relay when the n -th channel is used.The transmit power at each user and the relay is assumed to be P , i.e., E [ | x j [ n ] | ] ≤ P and E [ k x R [ n ] k ] ≤ P . Further, it is assumed that all the entries of all channel values in h ℓ,k [ n ] , h R ,ℓ [ n ] , and h ∗ k, R [ n ] are drawn from a continuous distribution and the absolute value of all thechannel coefficients is bounded between a nonzero minimum value and a finite maximum value.The channel state information (CSI) is assumed to be perfectly known at the users in receivingmode and the relay has global channel knowledge for all links.User k sends an independent message W ℓ,k for one intended user ℓ with rate R ℓ,k ( P ) = log | W ℓ,k | n for ℓ, k ∈ U and ℓ = k , a rate tuple R = ( R , , R , , . . . , R K,K − ) ∈ R K ( K − is achievable if every receiver can decode the desired message with an error probabilitythat is arbitrarily small with sufficient channel uses n . Then, the sum-DoF characterizing the September 28, 2018 DRAFT approximate sum-rates in the high SNR regime is defined as d sum = K X k =1 ,k = ℓ K X ℓ =1 d k,ℓ = lim P →∞ P Kk =1 ,k = ℓ P Kℓ =1 R k,ℓ ( P )log ( P ) . (4)In this work, the sum-DoF is a key metric to compare the network performance of differentmessage setups. III. M OTIVATING E XAMPLE
Before deriving the main results, in this section, we provide an example that yield intuitionabout why the relay is useful for increasing the DoF in a three-user fully-connected Y channel.
A. Proposed Multi-Phase Transmissions
Consider a 3-user fully-connected Y channel with a N = 2 antennas relay. As illustrated inFig.2, in this channel each user sends two independent messages, one to each other user. Sincethere are direct links between users, this network model differs from previous work on the Ychannel [13] and [14] where the direct links between users are ignored. We will show that sum-DoF is achievable.
1) Phase One (Round-Robin Multiple-Access Channel (MAC) ):
This phase comprises ofthree time slots. In each channel use, two users send one message to one intended user so thatthe intended user has one equation that contains two desired data symbols. Specifically, at timeslot 1, user 2 and 3 send information symbols s , and s , for user 1. While user 1 and the relaylisten the signals, user 2 and 3 do not receive any signals in this time slot due to half-duplexconstraint, i.e. S = { , } and D = { } . When noise is ignored, user 1 and the relay have D [1] = h , [1] s , + h , [1] s , , (5) D R [1] = h R , [1] s , + h R , [1] s , . (6)Since the relay has two antennas, it resolves the transmitted data symbols s , and s , by usinga zero-forcing (ZF) decoder. September 28, 2018 DRAFT
User 1 Relay User 3
User 2
Fig. 2. Three-user fully-connected Y channel.
In the second time slot, user 1 and user 3 transmit data symbols s , and s , to user 2. Thereceived equations at user 1 and the relay are D [2] = h , [2] s , + h , [2] s , , (7) D R [2] = h R , [2] s , + h R , [2] s , . (8)By taking an advantage of multiple antennas, the relay decodes s , and s , .Finally, at time slot 3, user 1 and user 2 deliver data symbols s , and s , to user 3. Hence,the received equation at user 3 and the relay are given by D [3] = h , [3] s , + h , [3] s , , (9) D R [3] = h R , [3] s , + h R , [3] s , . (10)Similarly, the relay obtains s , and s , by using a ZF decoder. As a result, during the phaseone, each user obtains one equation consisted of two desired symbols and the relay acquires allsix independent data symbols in the network. September 28, 2018 DRAFT
2) Phase Two (Relay Broadcast):
The second phase spans one time slot. In this time slot, therelay sends a superposition of six data symbols obtained during the phase one. The transmittedsignal at the relay is given by x R [4] = X i =1 3 X j =1 ,j = i v i,j [4] s i,j , (11)where v i,j [4] ∈ C × denotes the beamforming vector used for carrying symbol s i,j at time slot . The main design principle of v i,j [4] is to control the interference propagation on the networkso that each user receives an equation that consists of desired data symbols or self interferencedata symbols which can be eliminated by using side-information at each user. For instance, user1 wants to receive an additional equation consisted of s , and s , and can cancel the self-interference signals caused by s , and s , by exploiting side-information. Thus, the relay picksthe beamforming vectors v , [4] and v , [4] carrying s , and s , so that user 1 does not receivethem. To accomplish this, v , [4] and v , [4] are selected as v , [4] ∈ null ( h ∗ , R [4]) and v , [4] ∈ null ( h ∗ , R [4]) . (12)Applying the same principle, the other relay beamforming vectors are designed as v , [4] ∈ null ( h ∗ , R [4]) , v , [4] ∈ null ( h ∗ , R [4]) , (13) v , [4] ∈ null ( h ∗ , R [4]) , and v , [4] ∈ null ( h ∗ , R [4]) . (14)To give some intuition on the proposed precoding solution, we rewrite the transmit signal at therelay as x R [4] = v c [4]( s , + s , ) + v c [4]( s , + s , ) + v c [4]( s , + s , ) , (15)where v ci [4] ∈ null ( h ∗ i, R [4]) . Thus, we can interpret the transmitted signal at the relay at thesecond phase as a class of superposition coding .
3) Decoding:
We explain a decoding method used by user 1. Recall that user 1 received anequation consisting of two desired symbols s , and s , at time slot 1 in the form of D [1] = h , [1] s , + h , [1] s , . In time slot 4, user 1 obtained an equation containing both two desired September 28, 2018 DRAFT0 and two self-interference data symbols given by y [4] = h ∗ , R [4] x R [4] , = h ∗ , R [4] { v c [4]( s , + s , )+ v c [4]( s , + s , ) } . (16)Assuming that user 1 preserves the transmitted information symbols s , and s , by cachingmemory as side-information and it has the effective channel h ∗ , R [4] v ci [4] for i ∈ { , , } , user1 can generate an interference equation M [4] = h ∗ , R [4] v c [4] s , + h ∗ , R [4] v c [4] s , . Thus, user 1extracts one equation that contains two desired symbols by eliminating the self-interference as y [4] − M [4] = h ∗ , R [4] v c [4] s , + h ∗ , R [4] v c [4] s , . (17)From self-interference cancellation, we acquired a new equation for the two desired symbols.Therefore, the two desired symbols are obtained by solving the following matrix equation, y [1] y [4] − M [4] = h , [1] h , [1] h ∗ , R [4] v c [4] h ∗ , R [4] v c [4] | {z } ˜H s , s , . (18)Note that beamforming vectors were selected independently of the direct channel between users,i.e. h , [1] and h , [1] . Therefore, the rank of the effective channel becomes 2 with probabilityone, which allows user 1 to decode two desired symbols s , and s , . For user 2 and user 3, thesame method applies. Consequently, it is possible to exchange a total of six data symbols within4 channel uses by using the relay employing multiple antennas in the 3-user fully-connected Ychannel. B. Interpretation of the Proposed Methods
Now we reinterpret our results from the perspective of index coding. The basic indexcoding problem is a follows. Suppose a transmitter has a set of information messages W = { W , W , . . . , W K } for multiple receivers and each receiver wishes to receive a subsetof W while knowing some other subset of W as side information. The underlying goal of indexcoding is to design the best encoding strategy at the transmitter using the side-information at the September 28, 2018 DRAFT1 receivers to minimize the number of transmissions, while allowing all receivers to obtain theirdesired messages.The proposed transmission schemes mimic the index coding algorithms developed in [32]-[34]. Specifically, until the relay has global knowledge of messages in the network, N userspropagate information into the network at each time slot. Since the relay has N antennas, itobtains N information symbols per one time slot and the remaining K − N other users inreceiving mode acquires one equation that has both desired and interfering symbols. When therelay obtains all the messages, it starts to control information flow by sending a useful signalto all users so that each user decodes the desired information symbols efficiently based on theirprevious knowledge: their transmitted symbols and the received equations. For instance, thereare six messages { W , , W , , W , , W , , W , , W , } in the three-user Y channel. During timeslot 1, 2 and 3, the relay acquires global message set { W , , W , , W , , W , , W , , W , } andeach user acquires following side information • User 1 knows { W , , W , } and L ( W , , W , ) , • User 2 knows { W , , W , } and L ( W , , W , ) , • User 3 knows { W , , W , } and L ( W , , W , ) ,where L k ( A, B ) denotes a linear function of the two messages A and B obtained at user k . Intime slot 4, the relay broadcasts the six mixed independent data symbols so that each user maydecode their desired data symbols by using the message they sent (caching) and the overheardinterference signals in the previous phases. This relay transmission allows for the users to obtaingains for caching and interference shaping gains in the network. Thus, we can interpret thisrelay precoding solution as a linear vector index coding in a complex field for a special class ofindex coding problems.IV. D O F OF K -U SER F ULLY -C ONNECTED
Y C
HANNEL
In this section, we consider a general setup where each user wants to exchange K − independent messages with all other users in the network as described in Section II. For such achannel, the following theorem is the main result of this section. September 28, 2018 DRAFT2
User 1 User K
User 2
User 4 User 3 Relay (AP) (cid:273) (cid:273)
Cooperation group
Fig. 3. A user cooperation scenario when all users except for user 1 cooperate by sharing the messages and antennas. Thus, K -user fully-connected Y channel is equivalently viewed as a non-separate two-way relay channel where one node has K − antennas but the other node has a single antenna. In this cooperation case, the messages between cooperating users set to benull, i.e., W i,j = φ for i, j ∈ U c . For this setup, cut-set bounds are applied. Theorem 1:
For the fully-connected Y channel where K users have a single antenna and arelay has N ≥ K − antennas, the maximum sum-DoF equals K . A. Converse
We provide the converse of Theorem 1 by using the cut-set theorem. Using the fact that usercooperation does not deteriorate the DoF of the channel, let us first consider a special case whereall users except user k fully cooperate. This cooperation allows us to view the fully-connectedY channel as a two-way relay channel equivalently where the user group has K − antennasbut user 1 has a single antenna as illustrated in Fig 3. Further, let us set the messages to benull between cooperating users, i.e., W i,j = φ for i, j ∈ U c , by the using the fact that thenull-messages cannot degrade the performance of the non-null messages. In this two-way relaychannel, the user group wants to send the message W ,k for k ∈ U c and user 1 wants to send September 28, 2018 DRAFT3 the messages W k, for k ∈ U c . The converse follows from the following lemma which serves anouter bound of the equivalent two-way relay channel. Lemma 1:
Let d k,ℓ be the DoF for message W k,ℓ for k, ℓ ∈ U . Then, the following inequalityholds: K X ℓ =1 ,ℓ = k d k,ℓ + K X k =1 ,k = ℓ d k,ℓ ≤ , for k, ℓ ∈ U . (19) Proof:
The detailed proof is provided in Appendix A. Note that for a single antenna at all nodes,an outer bound of the non-separate two-way relay channel is derived in [36]. We extend thisouter bound result to the case where the relay and a user have multiple antennas.To attain the converse result of Theorem 1, we add K inequalities from Lemma 1, whichgives us K X ℓ = k K X k =1 d ℓ,k ! ≤ K (20) ⇒ K X ℓ = k K X k =1 d ℓ,k ≤ K , (21)which completes the proof. B. Achievability
Communication takes place in two phases: 1) the multiple access (MAC) phase where K − users send an independent message to one intended user and the relay overhears the signal; 2)a relay broadcast phase where the relay transmits the signal obtained during the previous phaseto all the users.
1) Phase One (MAC phase):
The first phase comprises of K time slots, i.e., T = { , , . . . , K } . For k ∈ T time slot, user k and the relay listen the transmitted signals byall other users, i.e., S k = U c k and D k = { k } . Specifically, at time slot k all the users in U c k sendsinformation symbol { s k,ℓ } for k = ℓ to user k simultaneously. Thus, the received signals at user September 28, 2018 DRAFT4 k and the relay are given by y k [ k ] = K X ℓ =1 ,ℓ = k h k,ℓ [ k ] s k,ℓ + z k [ k ] , (22) y R [ k ] = K X ℓ =1 ,ℓ = k h R ,ℓ [ k ] s k,ℓ + z R [ k ] , (23)Note that user k acquires a linear equation consisting of the desired symbols. Meanwhile, therelay can decode K − information symbols from the users by using a ZF decoder since it has N ≥ K − antennas. As a result, during phase 1, each user obtains one desired equation andthe relay acquires global knowledge of the K ( K − messages on the network.
2) Phase Two (Relay Broadcast):
The second phase spans K − time slots, T = { K +1 , . . . , K − } . In each time slot, the relay sends a superposition of K ( K − data symbolsobtained during the phase one. The transmitted signal at the relay is given by x R [ n ] = K X ℓ = k K X k =1 v k,ℓ [ n ] s ℓ,k , n ∈ T , (24)where v ℓ,k [ n ] ∈ C N × denotes the precoding vector used for carrying symbol s ℓ,k at timeslot n . The principle for desinging beamforming vectors v ℓ,k [ n ] is to control the informationflow so that each user receives an equation consisting of desired data symbols and knowninterference. For instance, user 1 wants to receive an additional equation that consists of symbols { s , , s , , . . . , s ,K } and has the capability to remove the interference terms cased by itstransmitted symbols { s , , s , , . . . , s K, } by exploiting caching memory. Using this fact, therelay picks the precoding vector v ℓ,k [ n ] carrying s ℓ,k so that user j for j ∈ U / { k, ℓ } does notreceive it. To accomplish this, v ℓ,k [ n ] is selected as v ℓ,k [ n ] ∈ null h ∗ i , R [ n ] h ∗ i , R [ n ] ... h ∗ i K − , R [ n ] | {z } ( K − × N , (25) September 28, 2018 DRAFT5 where { i , i , . . . , i K − } = U / { ℓ, k } denotes an index set with K − elements. Note that therelay has N ≥ K − antennas and the elements of channel vectors are drawn from i.i.d. randomvariables, the precoding solution of v ℓ,k [ n ] exists with probability one. When the relay transmitsduring the second phase, the received signal at user j is given by y j [ n ] = h ∗ j, R [ n ] x R [ n ]= h ∗ j, R [ n ] K X ℓ = k K X k =1 v k,ℓ [ n ] s ℓ,k = h ∗ j, R [ n ] K X k =1 ,k = j v j,k [ n ] s j,k | {z } D j [ n ] + h ∗ j, R [ n ] K X i =1 ,i = j v i,j [ n ] s i,j | {z } M j [ n ] + z j [ n ] . (26)In (26), D j [ n ] contains the K − desired symbols seen by user j at time slot n ∈ T and M j [ n ] consists of known symbols transmitted by user j .
3) Decoding:
Let us consider a decoding for user j . Recall that user j received an equationconsisting of K − desired symbols { s j, , . . . s j,j − , s j,j +1 , . . . , s j,K } during time slot j ∈ T ,i.e., D j [ j ] = P Kk =1 ,k = j h j,k [ j ] s j,k . Further, for time slots that belong to the second phase n ∈ T ,it acquired K − additional linear equations each of which contains both K − desired and K − known data symbols y j [ n ] = D j [ n ] + M j [ n ] + z j [ n ] . (27)First, the receiver subtracts the known interference term by exploiting knowledge of side-information { s ,j , . . . s j − ,j , s j +1 ,j , . . . , s K,j } . Assuming that user j knows the effective downlinkchannel h ∗ j, R [ n ] v k,j [ n ] for k ∈ U / { j } , the receiver generates the same interference shape M j [ n ] .After interference cancellation, the remaining equation contains the desired K − desired symbols ˜ y j [ n ] = y j [ n ] − M j [ n ] = D j [ n ] + z j [ n ] , n ∈ T (28) September 28, 2018 DRAFT6
Finally, to decode K − intended information symbols, the equations in (22) and (28) areaggregated into a matrix form, y j [ j ]˜ y j [ K +1] ... ˜ y j [2 K − = h j, [ j ] · · · h j,j − [ j ] h j,j +1 [ j ] · · · h j,K [ j ]˜ h j, [ K +1] · · · ˜ h j,j − [ K +1] ˜ h j,j +1 [ K +1] · · · ˜ h j,K [ K +1] ... . . . ... ... . . . ... ˜ h j, [2 K − · · · ˜ h j,j − [2 K −
2] ˜ h j,j +1 [2 K − · · · ˜ h j,K [2 K − | {z } ˜H j s j, ... s j,j − s j,j +1 ... s j,K , (29)where ˜ h j,k [ n ] denotes the effective channel coefficient from the relay to user j carryinginformation symbol s j,k at time slot n ∈ T , i.e., ˜ h j,k [ n ] = h ∗ j, R [ n ] v j,k [ n ] . It is important tonote that the beamforming vectors were selected statistically independent with respect to thedirect channel between users, i.e. h j,k [ j ] for k ∈ U c j at time slot j ∈ T . Therefore, the effectivechannel matrix ˜H j has full rank, i.e., rank ( ˜H j ) = K − with probability one, which allows user j to decode K − desired symbols s j,k for k ∈ U c j at user j . By symmetry, user k for k ∈ U c j can apply the same decoding method to obtain K − desired data symbols. As a result, it ispossible to exchange a total of K ( K − data symbols within K + K − channel uses by usingthe relay in the fully-connected Y channel, which leads to achieve K ( K − K − = K sum-DoF. Thiscompletes the proof.Now, we make several remarks on the implication of our results. Remark 1 (No CSIT at users):
To achieve the optimal DoF, while CSIT at users is notneeded, the users require to know the effective (downlink) channel value from the relay touser j for j ∈ U , i.e., ˜ h j,k [ n ] = h ∗ j, R [ n ] v j,k [ n ] for performing self-interference cancellation. Thischannel knowledge can be obtained using demodulation reference signals that currently used incommercial wideband systems. Alternatively, CSIT at the relay plays in important role to attainthe DoF gains. Remark 2 (Decoding delay):
Note that each user cannot decode the desired informationsymbols until the relay transmissions are finished, which results in decoding delay. To reducethe delay, the proposed strategy may be implemented in a multi-carrier system that offers K − September 28, 2018 DRAFT7 independent parallel sub-channels. From this system, the relay can send all information symbolsrequired at all users for decoding within one time slot but over K − independent sub-channels. Remark 3 (Full-duplex operation):
In this work, we assumed that all nodes are in half-duplex.One interesting observation is that the DoF increase by the use of a relay shown this workcan also be translated to the gain overcoming half-duplex loss. For example, consider a K -userfully-connected X network, which is the same as the fully-connected Y channel without therelay. In this network, if the half-duplex is assumed, the optimal DoF is d X half = ( K/ K/ − K − asshown in [28]. Meanwhile, for the full-duplex assumption, the lower and upper bounds of theoptimal DoF were K ≤ d X full ≤ K ( K − K − [28]. Since our result assuming the half-duplex operationmeets the tight inner bound DoF for the full-duplex X network, i.e., K , the relay allows usersto overcome the loss due to half-duplex signaling in the network.V. F OUR -U SER F ULLY -C ONNECTED I NTERFERENCE N ETWORK WITH D IFFERENT M ESSAGES
In this section we consider two examples for the four-user fully-connected interferencenetwork. Specifically, we derive DoF inner bounds for two different channel models: two-pairtwo-way interference channel with a relay and two-pair two-way X channel with a relay.
A. Example 1: Two-Pair Two-Way Interference Channel with a Relay
In this example, we assume that there are four users in the network, i.e., K = 4 and therelay has two antennas N = 2 as depicted in Fig 4. Among the four users, two user pairs,user 1-user 3 and user 2-user 4, want to exchange the messages. Since each user exchangesthe message with its partner in a bi-directional way, we refer to it as the two-pair two-wayinterference channel with a relay. Note that this channel is equivalent to the four-user fully-connected interference network with a relay when the eight messages are set to be null, i.e., W , = W , = φ , W , = W , = φ , W , = W , = φ , and W , = W , = φ . Throughout thisexample, we show that four independent symbols s , = f ( W , ) , s , = f ( W , ) , s , = f ( W , ) ,and s , = f ( W , ) can be exchanged over three time slots using a new multi-phase transmissionmethod, which allows users to exploit side-information efficiently. September 28, 2018 DRAFT8
User 1 User 3 User 2 User 4 Relay (AP)
Fig. 4. The two-pair two-way interference channel with two antennas relay. Each user wants to exchange the messages withits partner by using a shared relay.
1) Phase 1 (Forward Interference Channel (IC) Transmission):
Phase one consists of onetime slot. In this phase, user 1 and user 2 transmit signals x [1] = s , and x [1] = s , overthe forward IC, i.e., S = { , } and D = { , } . Note that user 1 and 2 cannot receive eachother’s signals during time slot 1 due to the half-duplex constraint. When noise is neglected, thereceived signals at user 3, user 4, and the relay are given by y [1] = h , [1] s , + h , [1] s , , (30) y [1] = h , [1] s , + h , [1] s , , (31) y R [1] = h R , [1] s , + h R , [1] s , , (32)Since the relay has two antennas, it resolves the transmitted data symbols s , and s , by usinga zero-forcing (ZF) decoder .
2) Phase 2 (Backward IC Transmission):
Phase 2 uses one time slot. In time slot 2,information flow occurs over the backward channel, i.e., S = { , } and D = { , } . One may use another spatial decoders that can resolve two independent symbols at the relay with two antennas such asminimum mean square error (MMSE) or Vertical-Bell Laboratories Layered Space-Time (V-BLAST). For simplicity, ZF decoderwill be used in this paper.
September 28, 2018 DRAFT9
Specifically, user and user send signals x [2] = s , and x [2] = s , at time slot 2 throughthe backward IC. The signal received at the receivers of the backward channel are given by y [2] = h , [2] s , + h , [2] s , y [2] = h , [2] s , + h , [2] s , y R [2] = h R , [2] s , + h R , [2] s , . (33)From the backward transmission, user 1 and user 2 obtain a linear equation, while the relaydecodes s , and s , using a ZF decoder.
3) Phase 3 (The Relay Transmission) :
Phase 3 also uses one time slot. The real noveltyoccurs in this time slot using a technique that inspired by a wireless index coding. In this timeslot, the signal transmitted by the relay is x R [3] = v , [3] s , + v , [3] s , + v , [3] s , + v , [3] s , , (34)where v i,j [3] denotes the precoding vector for information symbol s i,j for i, j ∈ { , , , } . Therelay transmission aims to multicast the signal x R [3] so that all users can decode the desiredinformation symbol based on the side-information each user acquired from the previous timeslots. For example, user 1 wants to decode data symbol s , . Two different forms of side-information are acquired: the transmitted symbol s , at time slot 1 and the received signalfrom the backward transmission at time slot 2, i.e., y [2] = h , [2] s , + h , [2] s , . To exploitthis side-information when user 1 decodes the desired symbols efficiently, the relay should notpropagate the interference symbol s , to user 1. Hence, we select the relay precoding vector v , [3] carrying information symbol s , in a such that it does not reach to user 1 by selecting v , [3] ∈ null (cid:0) h ∗ , R [3] (cid:1) . (35)To accomplish the same objective for the other users, we choose the relay precoding vectors tosatisfy v , [3] ∈ null (cid:0) h ∗ , R [3] (cid:1) , (36) v , [3] ∈ null (cid:0) h ∗ , R [3] (cid:1) , (37) v , [3] ∈ null (cid:0) h ∗ , R [3] (cid:1) . (38) September 28, 2018 DRAFT0
Since the size of the channels h ∗ k, R [3] for k ∈ U is × , the beamforming solutions satisfyingthe equations in (35), (36), (37), and (38) exist almost surely. Thus, the received signals at usersat time slot 3 are given by y [3] = h ∗ , R [3] x R [3]= h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , | {z } Self-interference , (39) y [3] = h ∗ , R [3] x R [3]= h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , | {z } Self-interference , (40) y [3] = h ∗ , R [3] x R [3] , = h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , | {z } Self-interference , (41) y [3] = h ∗ , R [3] x R [3] , = h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , | {z } Self-interference . (42)
4) Decoding:
Successive interference cancellation is used to eliminate the back propagatingself-interference from the received signal at time slot 3. The remaining inter-user interference isremoved by a ZF decoder. For instance, user eliminates the self-interference h ∗ , R [3] v , [3] s , from y [3] as y [3] − h ∗ , R [3] v , [3] s , = h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , . (43)After canceling the self-interference, the received signals at time slot 2 and time slot 3 can berewritten in matrix form as y [2] y [3] − h ∗ , R [3] v , [3] s , = h , [2] h , [2] h ∗ , R [3] v , [3] h ∗ , R [3] v , [3] | {z } ˜H s , s , . (44)Since the beamforming vectors v , [3] and v , [3] were designed independently of h ∗ , R [3] andthe channel coefficients h , [2] and h , [2] were drawn from a continuous random distribution, September 28, 2018 DRAFT1 the effective channel matrix ˜H has full rank almost surely. This implies that it is possible todecode the desired symbol s , by applying a ZF decoder that eliminates the effect of inter-userinterference s , . Consequently, user 1 obtains the desired data symbol s , . By symmetry, theother users are able to decode the desired symbols by using the same decoding procedure. As aresult, a total of independent data symbols are delivered over three orthogonal channel uses,which leads to achieve the of sum-DoF, i.e., d sum = . Remark 4 (CSI knowledge and feedback) : To cancel interference, it is assumed thateach user has knowledge of the effective channel from the relay to the users, i.e., { h ∗ k, R [3] v , [3] , h ∗ k, R [3] v , [3] , h ∗ k, R [3] v , [3] , h ∗ k, R [3] v , [3] } for k ∈ U . This effective channel,however, can be estimated using demodulation reference signals formed in commercial widebandsystems. With the setting, the users do not need to know CSIT, implying that no CSI feedback isrequired between users. In contrast, the relay needs to know CSIT between the relay to the usersto generate precoding vectors. While this CSIT can by given by a feedback link if frequencydivision duplexing system is considered, it can be acquired without feedback when time divisionduplex system is applied due to channel reciprocity. Remark 5 (Another Transmission Method) : It is possible to achieve the of sum-DoF byapplying a different transmission scheme. During time slot 1, user 1 and user 3 send signals andthe other users and the relay listen, i.e., S = { , } and D = { , } . At time slot 2, while user2 and user 4 send the signals, other nodes overhear the transmitted signals, i.e., S = { , } and D = { , } . In the third time slot, the relay can perform space-time interference alignment [34],which creates the same interference shape between the currently observed and the previouslyacquired interference signal. Thus, each user cancels the received interference at time slot 3from the acquired interference equations during time slot 1 and 2. The details of the scheme isincluded in Appendix B. Remark 6 (Amplify-and-Forward (AF) vs Decode-and-Forward (DF)):
In this example, weassumed that the relay uses DF relaying. The same DoF ca be achieved by AF relaying. Let U [ n ] denotes the spatial decoder used at the relay for n ∈ { , } . When AF relaying is considered, September 28, 2018 DRAFT2
User 1 User 3 User 2 User 4 Relay (AP)
Fig. 5. The two-pair two-way X channel with two antennas relay. Each user wants to exchange two independent messages withthe other user group by using a shared relay. the transmitted signal at time slot 3 in (34) is modified as x R [3] = [ v , [3] , v , [3]] U [1] y R [1] + [ v , [3] , v , [3]] U [2] y R [2] . (45) B. Example 2: Two-Pair Two-Way X Channel with a Relay
In this example, we assume that K = 4 , and N = 2 . Although the physical channel model isthe same as Example 1, we consider a more complex information exchange scenario where user1 and 2 want to exchange two independent messages with both user 3 and 4. We refer to thisscenario as a two-pair two-way X channel with a multiple antenna relay. Note that this channelcan be interpreted as a 4-user multi-way interference network in which W , = W , = φ and W , = W , = φ . In this example, we will show that each user exchanges two independentsymbols with two different users over five time slots; a total sum-DoF are achievable. Thisachievability is shown by a generalization of space-time interference alignment [34].
1) Phase One (Forward IC transmission):
This phase consists of two time slots. In the firsttime slot, user 1 and user 2 send an independent symbol intended for user , i.e., x [1] = s , and x [1] = s , . In the second time slot, user 1 and user 2 transmit independent symbols intended September 28, 2018 DRAFT3 for user , i.e., x [2] = s , and x [2] = s , , i.e., S n = { , } and D n = { , } for n ∈ { , } .Let denote D j [ n ] and L j [ n ] denote the received equations at user j in the n -th time slot, whichcontain the desired and interference symbols, respectively. Neglecting noise at the receivers, user and user obtain two linear equations during two time slots, which are D [1] = h , [1] s , + h , [1] s , , (46) L [1] = h , [1] s , + h , [1] s , , (47) L [2] = h , [2] s , + h , [2] s , , (48) D [2] = h , [2] s , + h , [2] s , . (49)Due to the broadcast nature of the wireless medium, the relay is also able to listen thetransmissions by the users. Since it has two antennas, it is possible to decode two informationsymbols in each part of phase one, giving s , , s , , s , , and s , , by using a ZF decoder duringthe phase one.
2) Phase Two (Backward IC Transmission):
In the second phase, the role of transmitters andreceivers is reversed, i.e., S n = { , } and D n = { , } for n ∈ { , } . In time slot 3, user and user send an independent symbol intended for user 1, x [3] = s , and x [3] = s , . Fortime slot 4, user and user deliver information symbols intended for user 2, x [4] = s , and x [4] = s , . Therefore, user 1 and user 2 obtain two equations during the phase two, which aregiven by D [3] = h , [3] s , + h , [3] s , , (50) L [3] = h , [3] s , + h , [3] s , , (51) L [4] = h , [4] s , + h , [4] s , , (52) D [4] = h , [4] s , + h , [4] s , , (53)As with phase two, the relay decodes four data symbols s , , s , , s , , and s , by using a ZFdecoder. September 28, 2018 DRAFT4
3) Phase Three (Relay Broadcast):
The third phase uses only one time slot. In this phase,in contrast to previous examples, the relay exploits knowledge of the current downlink CSIfrom the relay to the users, i.e., h k, R [5] for k ∈ { , , , } and outdated CSI betweenusers i.e., { h , [1] , h , [1] , h , [2] , h , [2] } as well as the outdated CSI between the users, i.e., { h , [3] , h , [3] , h , [3] , h , [4] } . Using this information, in time slot 5, the relay transmit theeight data symbols { s , , s , , s , , s , , s , , s , , s , , s , } acquired during the phase oneand two as x R [5] = X j =3 2 X i =1 v j,i [5] s j,i + X j =1 4 X i =3 v j,i [5] s j,i , (54)where v j,i [5] ∈ C × denotes the beamforming vector to carry symbol s i,j during time slot 5.The main idea of the relay beamforming design is to provide the same interference signal shapeto all users observed through phase one and phase two so that each user can use the receivedinterference signal during phase two as side information.To illustrate, we explain the design principle of v , [5] carrying s , from a index codingperspective. Note that data symbol s , is only desired by user and it is interference to allthe other users except for user 1. This is because user 1 has already s , , so it can use it asside-information for self-interference cancellation. User 4 observed s , at time slot 1 in the formof L [1] = h , [1] s , + h , [1] s , . Therefore, user can cancel s , from the relay transmissionif it receives the same interference shape h , [1] s , . Unlike user 4, user 2 does not have anyknowledge of s , . Thus, the relay must design the beamforming vector carrying s , so that itdoes not reach to user 2. To satisfy both user 2 and 4, the relay designs v , [5] as h ∗ , R [5] v , [5] = 0 , h ∗ , R [5] v , [5] = h , [1] . (55)By applying the same design principle, we pick the other precoding vectors so that the followingconditions are satisfied as h ∗ , R [5] h ∗ , R [5] v , [5] = h , [2] , h ∗ , R [5] h ∗ , R [5] v , [5] = h , [1] , (56) September 28, 2018 DRAFT5 h ∗ , R [5] h ∗ , R [5] v , [5] = h , [2] , h ∗ , R [5] h ∗ , R [5] v , [5] = h , [3] , (57) h ∗ , R [5] h ∗ , R [5] v , [5] = h , [4] , h ∗ , R [5] h ∗ , R [5] v , [5] = h , [3] , (58) h ∗ , R [5] h ∗ , R [5] v , [5] = h , [4] . (59)Since we assume that the channel coefficients are drawn from a continuous distribution, wealways surely inverse. Therefore, we construct the relay transmit beamforming vectors as v , [5] = h ∗ , R [5] h ∗ , R [5] − h , [1] , v , [5] = h ∗ , R [5] h ∗ , R [5] − h , [2] , (60) v , [5] = h ∗ , R [5] h ∗ , R [5] − h , [1] , v , [5] = h ∗ , R [5] h ∗ , R [5] − h , [2] , (61) v , [5] = h ∗ , R [5] h ∗ , R [5] − h , [3] , v , [5] = h ∗ , R [5] h ∗ , R [5] − h , [4] , (62) v , [5] = h ∗ , R [5] h ∗ , R [5] − h , [3] , v , [5] = h ∗ , R [5] h ∗ , R [5] − h , [4] . (63)From the relay transmission, the received signals at the users are given by y [5] = h ∗ , R [5] X j =3 2 X i =1 v j,i [5] s j,i + X j =1 4 X i =3 v j,i [5] s j,i ! + z [5] , = ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , | {z } = h , [4] s , + h , [4] s , + z [5] , (64) y [5] = h ∗ , R [5] X j =3 2 X i =1 v j,i [5] s j,i + X j =1 4 X i =3 v j,i [5] s j,i ! + z [5] , = ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , | {z } = h , [3] s , + h , [3] s , , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + z [5] , (65) September 28, 2018 DRAFT6 y [5] = h ∗ , R [5] X j =3 2 X i =1 v j,i [5] s j,i + X j =1 4 X i =3 v j,i [5] s j,i ! + z [5] , = ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , | {z } = h , [2] s , + h , [2] s , + z [5] , (66) y [5] = h ∗ , R [5] X j =3 2 X i =1 v j,i [5] s j,i + X j =1 4 X i =3 v j,i [5] s j,i ! + z [5] , = ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , | {z } = h , [1] s , + h , [1] s , , + ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , + z [5] . (67)As shown in (64), (65), (66), and (67), each user acquires a equation that can be decomposedinto three sub-equations, each of which corresponds to desired, self-interference, and alignedinterference parts. For instance, for user 1, ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , denotesthe desired part as it contains desired information symbols s , and s , . The sub-equation ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , can be interpreted as back propagating self-interferencesignal from the relay because s , and s , were transmitted previously by user 1. Last, the sub-equation ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , represents interference signals because s , and s , are intended for user 2. By the proposed precoding, this interference sub-equation has thesame shape that was observed at time slot 4 by user 1 in the form of h , [4] s , + h , [4] s , .
4) Decoding:
Let us explain the decoding procedure for user 1. First, user 1eliminate the back propagating self-interference signals M [5] = ( h ∗ , R [5] v , [5]) s , +( h ∗ , R [5] v , [5]) s , from y [5] by using knowledge of the effective channel h ∗ , R [5] v , [5] and h ∗ , R [5] v , [5] and the transmitted data symbols s , and s , . Second, user 1 re-moves the effect of interference ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , by using the fact that ( h ∗ , R [5] v , [5]) s , + ( h ∗ , R [5] v , [5]) s , = L [4] . After canceling the known interference, theconcatenated input-output relationship seen by user 1 is y [3] y [5] − L [4] − M [5] = h , [3] h , [3] h ∗ , R [5] v , [5] h ∗ , R [5] v , [5] | {z } ˜H s , s , + z [3] z [5] − z [4] . (68) September 28, 2018 DRAFT7
User 1 User 3 User 2 User 4 Relay 1 Relay 3 Relay 2
Fig. 6. The two-pair two-way interference channel with three disturbed relays employing a single antenna.
Since beamforming vectors, v , [5] and v , [5] , were constructed independently from the directchannel h , [3] and h , [3] , then, rank (cid:16) ˜H (cid:17) = 2 . As a result, user 1 decodes two desired symbols s , and s , based on five channel uses. Similarly, the other users decode two desired informationsymbols by using the same method. Consequently, a total eight data symbols have been deliveredin five channel uses in the network, implying that a total d sum = is achieved.VI. E XTENSION TO M ULTIPLE R ELAYS WITH A S INGLE A NTENNA
So far, the scenario considered in this paper used a single relay with N antennas. This relaycan be viewed as N relays with a single antenna where the relays fully cooperate by sharingboth data and CSI. In this section, we consider multiple distributed relays which of each hasa single antenna. We assume that the relays have full CSI but do not share data. By providingtwo examples, we show the multiple relays with a single antenna can increase the DoF gain formulti-way interference networks. September 28, 2018 DRAFT8
A. Example 3: Two-Pair Two-Way Intererference Channel with Three Distributed Relays
In this example, we consider a two-pair two-way interference channel with three distributedrelays as illustrated in Fig. 6. Note that this network is similar to Example 1 in Section III,except that there are single-antenna relays instead of one two-antenna relay. In this example, weprove the following theorem.
Theorem 2:
For the two-pair two-way interference channel with three distributed relaysemploying a single antenna, a total of DoF is achievable. Proof : The achievability is shown by the proposed space-time interference neutralization.
1) Phase 1 (Forward IC Transmission):
In the first time slot, user 1 and user 2 send signals x [1] = s , and x [1] = s , . Ignoring noise, the received signals at user 3, user 4, and the relaysare given by y [1] = h , [1] s , + h , [1] s , , (69) y [1] = h , [1] s , + h , [1] s , , (70) y n R [1] = h n R , [1] s , + h n R , [1] s , , n ∈ R = { , , } (71)where y n R [1] and h n R ,i [1] denote the received signal at relay n ∈ R and the channel value fromuser i to relay n at time slot 1. In contrast to Example 1, the relays do not decode the transmitteddata symbols s , and s , because they have only a single antenna.
2) Phase 2 (Backward IC Transmission):
In the second time slot, user and user sendsignals x [2] = s , and x [2] = s , over the backward interference channel. The receivedsignals at user 1, user 2, and the three relays are given by y [2] = h , [2] s , + h , [2] s , y [2] = h , [2] s , + h , [2] s , y n R [2] = h n R , [2] s , + h n R , [2] s , . (72) September 28, 2018 DRAFT9
3) Phase 3 (The Relay Broadcast) :
In the third time slot, the three relays cooperatively sendthe received signals for the previous time slots. The signal transmitted by relay n ∈ R is x nR [3] = v n [1] y nR [1] + v n [2] y nR [2] (73) = v n [1]( h n R , [1] s , + h n R , [1] s , ) + v n [2]( h n R , [2] s , + h n R , [2] s , ) , (74)where v n [ t ] denotes a relay precoding coefficient used for the received signal y nR [ t ] for t ∈ { , } .Let g ∗ ℓ, R ,i = [ h ℓ, R [3] h R ,i [1] , h ℓ, R [3] h R ,i [1] , h ℓ, R [3] h R ,i [1]] for i ∈ { , } and g ∗ ℓ, R ,j =[ h ℓ, R [3] h R ,j [2] , h ℓ, R [3] h R ,j [2] , h ℓ, R [3] h R ,j [2]] for j ∈ { , } denote effective channel vector fromuser i ( j ) to user ℓ ∈ { , , , } via the three relays and v [ t ] = [ v [ t ] , v [ t ] , v [ t ]] ∗ for t ∈ { , } represents precoding vector used by the three relays in time slot t . The received signal at user ℓ ∈ { , , , } in time slot 3 is given by y ℓ [3] = X n =1 h nℓ, R [3] x n R [3] (75) = X n =1 h nℓ, R [3] (cid:8) v n [1]( h n R , [1] s , + h n R , [1] s , ) + v n [2]( h n R , [2] s , + h n R , [2] s , ) (cid:9) (76) = g ∗ ℓ, R , v [1] s , + g ∗ ℓ, R , v [1] s , + g ∗ ℓ, R , v [2] s , + g ∗ ℓ, R , v [2] s , , (77)The role of the three relays in this example is to control the information flow so that eachuser does not receive unknown interference signals. For instance, user 1 and user 2 do nothave knowledge of interference symbol s , and s , . Similarly, user 3 and user 4 do not knowinterference symbol s , and s , , respectively. Thus, the following interference neutralizationconditions are required g ∗ , R , g ∗ , R , v [1] = × and g ∗ , R , g ∗ , R , v [2] = × . (78)Note that the precoding solutions for v [1] and v [2] exist always in this case from the existenceof null space of the matrices g ∗ , R , g ∗ , R , ∈ C × and g ∗ , R , g ∗ , R , ∈ C × .
4) Decoding:
To explain the decoding, we consider user 1. From the interference neutraliza-tion conditions in (78), the received signal is y [3] = g ∗ , R , v [1] s , + g ∗ , R , v [2] s , + g ∗ , R , v [2] s , . (79) September 28, 2018 DRAFT0
We first subtract off contribution from the known signals to form y [3] − g ∗ , R , v [1] s , . Then,the concatenated input-output relationship is given by y [2] y [3] − g ∗ , R , v [1] s , = h , [2] h , [2] g ∗ , R , v [2] g ∗ , R , v [2] | {z } ˜G s , s , . (80)Since the effective channel matrix ˜G has full rank almost surely, user 1 decodes the desiredsymbol s , by applying a ZF decoder. By symmetry, the other users operate in a similar fashion,which implies that a total of DoF is achievable. Remark 7 (Comparision with results in [7])
Interference neutralization method was introducedin [7] for the K × L × K layered two-hop interference network. To realize interference neutral-ization for bi-directional information exchange between K -paris (2 K users), L ≥ K ( K −
1) + 1 relays with a single antenna are needed. For example, when K = 2 , the minimum required relaysare L = 5 to achieve DoF for the layered two-hop half-duplex interference network. When thefeasibility condition of interference neutralization does not hold, i.e.,
L < , there are no resultsso far that the relays increase the achievable DoF beyond one. Our result, however, shows thatwith fewer than five relays, it is possible to increase the achievable DoF beyond one by usingwhat we call space-time interference neutralization . B. Example 4: Three-User Fully-Connected Y Channel with Three Single Antenna Relays
Now, we consider a fully-connected 3-user Y channel with three distributed single-antennarelays as depicted in Fig. 7. By using space-time interference neutralization explained in Example3, we show the following theorem.
Theorem 3:
For the 3-user fully-connected Y channel with three distributed single-antennarelays, the optimal of DoF is achievable. Proof : Since the converse argument is direct from Theorem 1, achievability is shown by two-phase communication protocol. The first phase comprises three time slots, i.e., T = { , , } .For k ∈ T time slot, all users excepting user k transmit signals intended for user k . Thus, the September 28, 2018 DRAFT1
User 1 Relay 1 User 3 User 2 Relay 2 Relay 3
Fig. 7. 3-user fully-connected Y channel with three distributed relays employing a single antenna. received signals at user k and relay n ∈ { , , } are given by y k [ k ] = X ℓ =1 ,ℓ = k h k,ℓ [ k ] s k,ℓ + z k [ k ] , (81) y n R [ k ] = X ℓ =1 ,ℓ = k h n R ,ℓ [ k ] s k,ℓ + z n R [ k ] . (82)During phase one, each user k has one desired equation and each relay has three equations. Inthe second phase, one time slot is used for the relay transmission, T = { } . During time slot4, the three relays cooperatively send signals to the users based on what they obtained duringthe first phase. The transmitted signal at the n -th relay is given by x nR [4] = X k =1 v n [ k ] y nR [ k ] , (83)where v n [ k ] denotes the precoding coefficient used at the n -th relay for the k -th time slotobservation y nR [ k ] . When the three relays send at time slot 4, the received signal at user j is September 28, 2018 DRAFT2 given by y j [4] = X n =1 h nj, R [4] x nR [4]= X n =1 h nj, R [4] X k =1 v n [ k ] y nR [ k ]= X n =1 h nj, R [4] X k =1 v n [ k ] X ℓ =1 ,ℓ = k h n R ,ℓ [ k ] s k,ℓ ! (84) = h ∗ j, R [4] V R h H R , U c [1] H R , U c [2] H R , U c [3] i s , U c s , U c s , U c , (85)where h ∗ j, R [4] = [ h j, R [4] , h j, R [4] , h j, R [4]] ∈ C × denotes the channel vector from the three relaysto user j ∈ { , , } at time slot ; V R ∈ C × denotes a space-time relay network codingmatrix at time slot whose ( k, n ) element is defined as V R ( n, k ) = [ v n [ k ]] ; H R , U c k [ k ] ∈ C × denotes the effective channel matrix from user group U c k to the three relays at time slot k ∈ T ; s j, U c j represents the desired symbol vector at user j , which comes from user group U c j . Note thatuser j receives a linear combination of six symbols from the relays. Let us decompose these sixsymbols into three terms: two desired symbols s j, U c j , two self-interference symbols s U c j ,j , and twointer-user interference symbols s I c j = [ s k,ℓ ] where k = j or ℓ = j . Then, let define a permutationmatrix P j ∈ Z × that changes the order of transmitted symbols such that s , U c s , U c s K, U c = P j s j, U c j s U c j ,j s I c j . (86)Using the permutation matrix, we can rewrite the equation in (85) as y j [4] = h ∗ j, R [4] V R h H R , U c [1] H R , U c [2] H R , U c [3] i P j s j, U c j s U c j ,j s I c j (87) = h ∗ j, R [4] V R A j s j, U c j + h ∗ j, R [ t ] V R B j s U c j ,j + h ∗ j, R [ t ] V R C j s I c j , (88) September 28, 2018 DRAFT3 where the effective channel matrices A j ∈ C × , B j ∈ C × , and C j ∈ C × are defined as h A j B j C j i = h H R , U c [1] H R , U c [2] H R , U c [3] i P j . (89)To eliminate interference signal s I c j for user j , the relays cooperatively design space-time relaynetwork coding matrix V R so that the following interference neutralization condition is satisfied, h ∗ j, R [4] V R C j = × , for j ∈ U . (90)To solve the matrix equations in (90) for all K users, we convert them into vector forms byexploiting Kronecker product operation property, vec ( AXB ) = ( B T ⊗ A ) vec ( X ) . The combinedvector form of interference neutralization in (90) is given by C T ⊗ h ∗ , R [4] C T ⊗ h ∗ , R [4] C T ⊗ h ∗ , R [4] | {z } ¯F ∈ C × ¯v R |{z} × = × . (91)where ¯v R = vec ( V R ) is the vector representation of relay beamforming matrix V R by stackingthe column vectors of it. Because the elements of the channels are drawn from a continuousrandom variable and the size of the unified system matrix ¯F is × , ¯F has a null subspacealmost surely. Therefore, the relay beamforming vector eliminating all interference signals onthe network are obtained as ¯v R ∈ null ( ¯F ) . (92)By reshaping the vector solution ¯v R into a matrix, we obtain the network-wise space-time relayprecoding matrix V R .Last, let us consider the decoding procedure at user . Recall that user received an equationconsisting of two desired symbols s , U c = [ s , s , ] T in time slot 1. In addition, in time slot 4, user1 received a signal from the relay in the form of y [4] = h ∗ , R [4] V R A s , U c + h ∗ , R [4] V R B s U c , .Since user 1 has knowledge of s U c , , it cancels known interference symbols from y [4] . Thus, September 28, 2018 DRAFT4 the input-out relationship is given by y [1] y [4] − h ∗ , R [4] V R B s U c , = h , U c [1] h ∗ , R [4] V R A | {z } ˜H s , U c , (93)where h , U c [1] = [ h , [1] , h , [1]] . Since the effective channel matrix ˜H has full rank two, user1 decodes two desired symbols. By the symmetry, user 2 and 3 obtain their desired informationsymbols as well. As a result, it is possible to achieve DoF. This completes the proof.VII. C
ONCLUSION
In this paper, we studied a fully-connected interference network with relay nodes. Byconsidering the multi-way information flows, we characterized the sum-DoF for the fully-connected Y channel by yielding converse based on the cut-set theorem and achievability basedon the proposed multi-phase transmission scheme. Further, we derived DoF inner bounds fordifferent message setups and multiple relays with a single antenna case in the four-user fully-connected interference network. From the derived DoF results, we verified an intuition thatrelays are useful in increasing the DoF of multi-user interference network when the multi-wayinformation flows are considered, even if the relays and users operate in half-duplex. This DoFincrease is due to two gains: the caching gain that inherently given by multi-way communicationsand the interference shaping gain by the space-time relay transmission technique that controlsinformation flow in the network. A
PPENDIX AP ROOF OF L EMMA k ∈ U c where U c = U / { } intends to send messages W ,k k ∈ U c to user 1. User 1 also intends to send the messages W k, k ∈ U c to the usergroup. This user cooperation changes the K -user fully-connected Y channel into the two-way September 28, 2018 DRAFT5 relay channel where user group U c has K − antennas but user 1 has a single antenna. In thishalf-duplex non-separated two-way relay channel scenario, we need to show that P Kk =2 d k, + P Kℓ =2 d ,ℓ ≤ .In general, for the half-duplex non-separated two-way relay channel as illustrated Fig. ?? , sixdifferent network states exist. Let λ i , i ∈ { , , . . . , } denote the fraction of transmission timeused for network state i . For each state, the six- phase transmissions can occur. In the first phase n ∈ λ , the user group transmits, while both the relay and user 1 listen. Alternatively, in thesecond phase n ∈ λ , user 1 sends the signal, while the relay and the user group listen. In thethird phase n ∈ λ , the user group and user 1 transmit the signal at the same time to the relay.For the fourth phase n ∈ λ , the relay helps the transmission of the user group. In the fifth phase n ∈ λ , it also assists the transmission of user 1. Lastly, the relay broadcasts the signal to boththe user group and user 1 for the sixth phase n ∈ λ . For the half-duplex relay network with sixstates, we apply the cut-set theorem [35] to obtain the upper bounds of information flow fromthe user group to user 1 R , U c and from user 1 to the user group R U c , . Let us first consider theupper bound of the rate R , U c , which is R , U c ≤ min { R S , U c , R D , U c } , (94)where R SB,A and R DB,A denote information transfer rates from A to B across the cut around thesource and destination nodes, which are defined as R S , U c = λ I ( x U c ; y R , y , i = 1) + λ I ( x U c ; y R | x , i = 3) + λ I ( x U c ; y | x R , i = 5) , (95) R D , U c = λ I ( x U c ; y , i = 1) + λ I ( x R ; y , i = 4) + λ I ( x U c , x R ; y , i = 5) , (96)where x U c = { x [ n ] , . . . , x K [ n ] } , x = { x [ n ] } , x R = { y [ n ] } , y = { y [ n ] } , and y R = { y R [ n ] } for n ∈ λ i . Also, I ( a ; b ) and I ( a ; b | c ) denote mutual information between two random vectors a and b and mutual information conditioned on random vector c . Similarly, we have the upperbound of the rate R U c , as R U c , ≤ min { R S U c , , R D U c , } , (97) September 28, 2018 DRAFT6 where R S U c , = λ I ( x ; y R , y U c , i = 2) + λ I ( x ; y R | x U c , i = 3) + λ I ( x ; y U c | x R , i = 6) , (98) R D U c , = λ I ( x ; y U c , i = 2) + λ I ( x R ; y U c , i = 4) + λ I ( x , x R ; y U c , i = 6) . (99)From each mutual information expression term in (95), (96), (98), and (99), we have the DoFupper bounds of the rates R S , U c , R D , U c , R S U c , , and R D U c , as lim P →∞ R S , U c log P ≤ λ min { K − , N + 1 } + λ min { K − , N } + λ min { K − , } (100) = λ ( K −
1) + λ ( K −
1) + λ , (101) lim P →∞ R D , U c log P ≤ λ min { K − , } + λ min { N, } + λ min { K − N, } (102) = λ + λ + λ , (103) lim P →∞ R S U c , log P ≤ λ min { , K − N } + λ min { , N } + λ min { , K − } (104) = λ + λ + λ , (105) lim P →∞ R D U c , log P ≤ λ min { , K − } + λ min { N, K − } + λ min { N + 1 , K − } (106) = λ + λ ( K −
1) + λ ( K − , (107)where the equalities follows from N < K . Using above results, the maximum DoF forinformation transfer from user group U c to user 1 is K X k =2 d ,k = min ( lim P →∞ R S , U c log P , lim P →∞ R D , U c log P ) , (108) ≤ min { ( K − λ + λ ) + λ , λ + λ + λ } . (109)Similarly, the maximum DoF for information transfer from user 1 to user group U c is K X k =2 d k, = min ( lim P →∞ R S U c , log P , lim P →∞ R D U c , log P ) , (110) ≤ min { λ + λ + λ , λ + ( K − λ + λ ) } . (111) September 28, 2018 DRAFT7
Using (109) and (111), an upper bound of the sum-DoF is obtained by solving the followinglinear program, max λ ,...,λ min { ( K − λ + λ )+ λ , λ + λ + λ } + min { λ + λ + λ , λ +( K − λ + λ ) } subject to X i =1 λ i = 1 , λ i ≥ . (112)Although the optimal value of this linear programing problem can be obtained by usingoptimization techniques accurately, we can simply find the upper bound of the optimal value byusing the fact that min { α, β } + min { δ, γ } ≤ min { α + δ, α + γ, β + δ, β + γ } , which leads to theupper bound of the objective function in (112) as P i =1 λ i . Finally, we have an upper bound ofthe sum-DoF as K X k =2 d ,k + K X k =2 d k, ≤ max λ ,...,λ X i =1 λ i = 1 , (113)where the last equality follows from the fact P i =1 λ i = 1 .By symmetry, the sum-DoF bounds are the same for the other user cooperation scenarios.This completes the proof. A PPENDIX BA NOTHER T RANSMISSION S CHEME OF E XAMPLE DoFfor the two-pair two-way relay channel when N = 2 .In the first time slot, user 1 and user 3 send signals x [1] = s , and x [1] = s , . Due tohalf-duplex constraint, the received signals at user 2, user 4, and the relay are given by y [1] = h , [1] s , + h , [1] s , ,y [1] = h , [1] s , + h , [1] s , , y R [1] = h R , [1] s , + h R , [1] s , . (114)Using two antennas at the relay, it decodes the transmitted data symbols s , and s , . September 28, 2018 DRAFT8
In time slot 2, user and user send signals x [2] = s , and x [2] = s , and user 1, user 3,and the relay receive y [2] = h , [2] s , + h , [2] s , y [2] = h , [2] s , + h , [2] s , y R [2] = h R , [2] s , + h R , [2] s , . (115)From the second time slot transmission, user 1 and user 3 obtain a linear equation consisted ofinterference symbols, while the relay decodes s , and s , by using a ZF decoder.In the third time slot, the relay transmits signal, x R [3] = v , [3] s , + v , [3] s , + v , [3] s , + v , [3] s , . (116)To eliminate interference signals at the users based on the observed interference signals duringtime slot 1 and 2, the relay precoding vectors are designed so that the following space-timeinterference alignment conditions are satisfied, h h ∗ , R [3] h ∗ , R [3] i v , [3] = h , [1] h , [1] , (117) h h ∗ , R [3] h ∗ , R [3] i v , [3] = h , [1] h , [1] , (118) h h ∗ , R [3] h ∗ , R [3] i v , [3] = h , [2] h , [2] , (119) h h ∗ , R [3] h ∗ , R [3] i v , [3] = h , [2] h , [2] . (120)Since all channel elements are i.i.d., four precoding vectors can be obtained with high probability.Thus, the received signals at user 1 for time slot 3 is given by y [3] = h ∗ , R [3] x R [3]= h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , | {z } Self-interference + h ∗ , R [3] v , [3] s , + h ∗ , R [3] v , [3] s , | {z } y [2]= h , [2] s , + h , [2] s , . (121) September 28, 2018 DRAFT9
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