MIMO Multiway Relaying with Pairwise Data Exchange: A Degrees of Freedom Perspective

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by  Rui Wang

1
MIMO Multiway Relaying with Pairwise Data
Exchange: A Degrees of Freedom Perspective
Rui Wang, Member, IEEE and Xiaojun Yuan, Member, IEEE
Abstract—In this paper, we study achievable degrees of free- capacity of the TWRC can be achieved within 1 bit. Later,
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dom(DoF)ofamultiple-inputmultiple-output(MIMO)multiway the authors in [5]–[7] introduced the multiple-input multiple-
relay channel (mRC) where K users, each equipped with M
output (MIMO) technique into TWRCs. It was revealed that
antennas,exchangemessagesinapairwisemannerviaacommon
the space-division based network coding scheme proposed in
N-antenna relay node. A novel and systematic way of joint
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beamforming design at the users and at the relay is proposed [7] achieves the asymptotic capacity of the MIMO TRWC at
1
to align signals for efficient implementation of physical-layer high signal-to-noise ratio (SNR) within 1log(5) bit per relay
0 2 4
2 network coding (PNC). It is shown that, when the user number spatial dimension for an arbitrary antenna configuration.
K = 3, the proposed beamforming design can achieve the
A natural generalization of the TWRC is the multiway
g DoF capacity of the considered mRC for any (M,N) setups.
relaychannel(mRC),wheremultipleusersexchangemessages
u For the scenarios with K > 3, we show that the proposed
A signalingschemecanbeimprovedbydisablingaportionofrelay with the help of a single relay. Several mRC models have
antennas so as to align signals more efficiently. Our analysis beenstudiedintheliteraturerecently.Specifically,theauthors
7 reveals that the obtained achievable DoF is always piecewise in [10] studied a cellular two-way relaying model where a
2 linear, and is bounded either by the number of user antennas
base station exchanges private messages with multiple mobile
M or by the number of relay antennas N. Further, we show
T] that the DoF capacity can be achieved for MN ∈ (cid:16)0,KK(K−−12)(cid:105) umsRerCssviinawahricehlaythenoudsee;rsthaereagurtohuoprsedinint[o8]p,a[i9rs] ainnvdesthtiegattwedo
(cid:104) (cid:17)
I and M ∈ 1 + 1,∞ , which provides a broader range of usersineachpairexchangeinformationwitheachother;more
. N K(K−1) 2
s the DoF capacity than the existing results. Asymptotic DoF as generally,theauthorsin[11]studiedclusteredmRCs,inwhich
c
[ K →∞isalsoderivedbasedontheproposedsignalingscheme. the users in the network are grouped into clusters and each

userinaclusterwantstoexchangeinformationwiththeother
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Index Terms—Multiple-input multiple-output (MIMO), users in the same cluster. Approximate capacities of these
v
physical-layer network coding (PNC), multiway relay channel
9 mRC models were studied in [11] and [12], while the exact
(mRC), signal space alignment.
2 capacitycharacterizationsstillremainopen.Also,theseinitial
2 works on mRC are limited to the single-antenna setup, i.e.,
7 I. INTRODUCTION each node in the network is equipped with one antenna.
.
1 The MIMO technique has been introduced into mRCs to
Recently,anexponentialincreaseinthedemandsofwireless
0 allow spatial multiplexing. In a MIMO mRC, as each user in
service has imposed a significant challenge on the design of
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general transmits multiple spatial streams, a new challenge to
1 wireless networks. Advanced techniques, such as physical-
be addressed is to mitigate the inter-stream interference at the
: layer network coding (PNC), has been developed to achieve
v relay and at the user ends. Degrees of freedom (DoF) is a
high spectrum efficiency [1], [2]. The simplest model for
i
X PNC is the two-way relay channel (TWRC) where two users critical metric in characterizing the fundamental capacity of a
wireless network [20], [21]. The DoF of the MIMO mRC has
r exchange messages with the help of a relay node. With the
a beenpreviouslystudiedin[13]–[19].Forexample,theauthors
well-known two-phase PNC protocol, the relay node receives
in[14]investigatedtheDoFcapacityoftheMIMOYchannel
acombinationofthesignalstransmittedfromthetwousersin
(a special case of the MIMO mRC with three users) and
the first phase, and then broadcasts a network-coded message
showed that the DoF capacity can be achieved when M ≥ 2,
in the second phase. The desired message is then extracted at N 3
where M denotes the number of antennas at each user, and
eachuserendbyexploitingtheknowledgeoftheself-message.
N denotes the number of antennas at the relay. The authors
As compared to conventional one-way relaying where four
in [16] generalized the results in [14] by considering a three-
phases are required in one round of information exchange,
user asymmetric MIMO Y channel with different numbers of
PNC potentially achieves 100% improvement in spectrum
antennas at the users, and proved that the DoF capacity can
efficiency over TWRCs.
be achieved for arbitrary antenna setups. Recently, as parallel
AbundantprogresseshavebeenmadeonthePNCdesignfor
to the work in this paper, the work in [17] established the
TWRCs; see [3]–[7] and the references therein. In particular,
DoF capacity of the four-user symmetric MIMO Y channel
it was shown in [4] that, with nested lattice coding, the
for arbitrary antenna setups. Further, the authors in [18], [19]
R.WangiswiththeInstituteofNetworkCoding,TheChineseUniversity studiedmoregeneraldataexchangemodelsinwhichtheusers
ofHongKong,HongKong,Email:[email protected]. in the network are grouped into clusters, and each user in a
X. Yuan is with School of Information Science and Technology,
cluster exchanges information only with the other users in the
ShanghaiTech University, Shanghai, 200031, China, Email:
[email protected]. samecluster.Inparticular,theauthorsin[18]derivedsufficient
2
conditions on the antenna configuration to achieve the DoF is similar to the DoF capacity obtained for the MIMO
capacity of a clustered mRC with pairwise data exchange, interference channel in [21].
in which each user in a cluster sends a different message to • Finally,wederiveanasymptoticachievableDoFwhenK
each of the other users in the same cluster. Note that the data tendstoinfinity.Weshowthatthederivedachievabletotal
(cid:16) (cid:17)
exchange models considered in [14] and [15] can be regarded DoF is upper bounded by min N2 ,2N for arbitrary
N−M
astheone-clustercaseofthemodelstudiedin[18].Moreover, values of M and N.
the author in [19] derived an achievable DoF for a clustered
The rest of the paper is organized as follows. In Section
MIMOmRCwithfulldataexchange,i.e.,eachuserinacluster
II, we present the system model. In Section III, a DoF upper
delivers a common message to all the other users in the same
bound is introduced as the benchmark of the system design.
cluster.
The DoF capacity of the considered MIMO mRC with three
In this work, we study a symmetric MIMO mRC with
users is presented in Section IV. In Section V, we generalize
pairwise data exchange, and derive an achievable DoF for an
the results to the case of an arbitrary number of users. In
arbitrary setup of the antenna numbers (M,N) and the user
Section VI, an improved DoF result is presented by disabling
number K. Roughly speaking, the DoF of a network is the
a portion of relay antennas. Finally, we conclude the paper in
number of independent spatial streams that can be supported
Section VII.
bythenetwork.IntheMIMOmRCofconcern,multipleusers
Notation: Scalars, vectors, and matrices are denoted by
are simultaneously served by a common relay. To ensure that
lowercaseregularletters,lowercaseboldletters,anduppercase
multiple spatial streams are still separable at every user end, boldletters,respectively.ForamatrixA,AT andAH denote
the number of relay antennas is usually the bottleneck of the
the transpose and the Hermitian transpose of A, respectively;
network to achieve a higher DoF. Therefore, the challenge is tr(A) and A−1 stand for the trace and the inverse of A,
how to align the user and relay signals to efficiently utilize
respectively;diag(A ,A ,··· ,A )denotesablock-diagonal
1 2 n
the relay’s signal space. To this end, we propose a novel
matrixwiththei-thdiagonalblockspecifiedbyA wherenis
i
andsystematicbeamformingdesigntoachieveefficientsignal
aninteger;span(A)andnull(A)denotethecolumnspaceand
alignment. Specifically, we refer to a bunch of spatial streams
the nullspace of A, respectively; I denotes an n×n identity
n
as a unit, in which each pair of users who want to exchange
matrix;dim(S)denotesthedimensionofaspaceS;S∩U and
informationcontributetwospatialstreams,onefromeachuser;
S⊕U denotetheintersectionandthedirectsumoftwospaces
each spatial stream impinges upon (or is emitted from) the S and U, respectively; Rn×m and Cn×m denote the n×m
relay’santennaarrayatacertaindirection,andthesedirections
dimensionalrealspaceandcomplexspace,respectively;log(·)
formaspatialstructure,referredtoasapattern.Thedimension denotes the logarithm with base 2; [·]+ denotes max{·,0};
of the space spanned by the spatial streams in a pattern gives CN(µ,σ2) denotes the distribution of a circularly symmetric
a metric to evaluate the efficiency of this pattern. Then, the
complex Gaussian random variable with mean µ and variance
signal alignment problem is to construct units with the most σ2;(n)= n! denotesthebinomialcoefficientindexed
efficient patterns to occupy the overall relay’s signal space. m m!(n−m)!
by n and m.
An achievable DoF can be obtained by counting the number
of units that can be constructed for any given antenna setup
II. SYSTEMMODEL
of (M,N).
A. Channel Model
The main contributions of this work are summarized as
follows: Consider a discrete memoryless symmetric MIMO mRC
(M,N,K), where K users, each equipped with M antennas,
• We show that, for the considered MIMO mRC with
exchange messages in a pairwise manner with the help of
K = 3, the proposed signal alignment scheme achieves
a common N-antenna relay node, as illustrated in Fig. 1.
the DoF capacity for an arbitrary (M,N) setup, which
Full-duplex communication is assumed, i.e., all the nodes
coincides with the DoF result of [16], and improves
transmit and receive signal simultaneously.1 The direct links
the existing DoF capacity result in [14] by including
M ∈(cid:0)0,2(cid:1). between users are ignored due to physical impairments such
N 3 as shadowing and path loss of wireless fading channels.
• For the case of K > 3, we derive the DoF capacity
(cid:16) (cid:105) Eachroundofinformationexchangeisimplementedintwo
of the MIMO mRC for M ∈ 0, K−1 and M ∈
N K(K−2) N phases with equal time duration T. In the first phase (termed
(cid:104) (cid:17)
1 + 1,∞ . This result is stronger than the pre- theuplinkphase),alltheuserssimultaneouslytransmitsignals
K(K−1) 2
to the relay node. The received signal at the relay node can
vious result obtained in [18], where the achievability of
the DoF capacity is limited in the ranges of M ∈(cid:0)0, 1(cid:3) be written as
(cid:104) (cid:17) N K
and M ∈ 1 + 1,∞ . (cid:88)K
N K(K−1) 2 Y = H X +Z , k ∈I (cid:44){1,2,··· ,K} (1)
• For K > 3, we also derive an achievable DoF for an R k k R K
k=1
arbitrary setup of antenna numbers (M,N) satisfying
M ∈(cid:16) K−1 , 1 + 1(cid:17). Our analysis reveals that where Hk ∈ CN×M denotes the channel matrix from user
N K(K−2) K(K−1) 2 k to the relay; X ∈ CM×T is the transmit signal from
the achievable DoF is piecewise linear and is bounded k
either by the number of antennas at each user or by the
1AlltheDoFresultsobtainedinthispaperdirectlyholdforthehalf-duplex
number of antennas at the relay. This piecewise linearity casebyincludingamultiplicativefactorof 1.
2
3
User 1 B. Linear Signaling Scheme
In the considered mRC, each user k, k ∈ I , intends
K
to send a private message W(k,k(cid:48)) to user k(cid:48), ∀k(cid:48) (cid:54)= k.
The message W(k,k(cid:48)) is then encoded as f(W(k,k(cid:48))) =
User 2 User K
{s(k,k(cid:48)),s(k,k(cid:48)),··· ,s(k,k(cid:48))}, where f(·) is an encoding func-
1 2 L
tion; s(k,k(cid:48)) ∈C1×T denotes the spatial stream transmitted in
l
unitl;Listhenumberoftheunitswhichcanbesupportedby
Relay thenetwork.Thegoalofthisworkistoanalyzetheachievable
DoF of the considered MIMO mRC. Linear processing is
assumed to be applied at the transmitter, relay, and receiver
sides. The transmit signal at user k is denoted as
User 3 User k
L
(cid:88)
X = U S ,
k k,l k,l
l=1
where k denotes the user index; l denotes the unit
Uplink
index; U = [u(k,1),u(k,2),··· ,u(k,k−1),u(k,k+1),··· ,
Downlink k,l l l l l
u(k,K)] ∈ CM×(K−1) denotes the beamforming ma-
l
Fig.1. AnillustrationoftheMIMOmRCwithKusersoperatinginpairwise trix applied at user k for the l-th unit; Sk,l =
exchange. [s(k,1)T,s(k,2)T,··· ,s(k,k−1)T,s(k,k+1)T,··· ,s(k,K)T]T ∈
l l l l l
C(K−1)×T denotesthetransmitspatialstreamsoverT channel
user k; similarly, YR ∈ CN×T denotes the received signal uses; u(k,k(cid:48)) corresponds to the beamformer of spatial stream
at the relay node; ZR ∈CN×T is the additive white Gaussian s(k,k(cid:48)).lNote that the maximum number of spatial streams
noise (AWGN) matrix at the relay node and the elements are l
in a unit is K(K −1). But this number can be reduced to
independently drawn from the distribution of CN(0,σ2). The
R K(cid:48)(K(cid:48)−1), where K(cid:48)(≤K) is the number of active users in
transmit signal X at user k satisfies the power constraint of
k the unit.
1tr(X XH)≤P , k ∈I During the uplink, the equivalent channel matrix from user
T k k k K k to the relay can be expressed by
whereP isthemaximumtransmissionpowerallowedatuser
k (cid:104) (cid:105)
k. H U = h(k,1),h(k,2),··· ,h(k,k−1),h(k,k+1),··· ,h(k,K) .
k k,l l l l l l
In the second phase (termed the downlink phase), the relay (3)
sends the processed signals to all user ends. The received Note that the equivalent channel vectors of unit l, i.e.,
signal at user k is denoted by {h(k,k(cid:48))|∀k,k(cid:48),k (cid:54)= k(cid:48)}, form a spatial structure, referred to
l
as a pattern.
Y =G X +Z , k ∈I (2)
k k R k K
The transmit signal at the relay node can be written as
whereG ∈CM×N denotesthechannelmatrixfromtherelay
k
touserk;X ∈CN×T isthetransmitsignalattherelaynode; X =FY , (4)
R R R
Z ∈ CN×T is the AWGN noise matrix at user k with the
k
elements independently drawn from CN(0,σ2). The transmit where F denotes the linear beamforming
k
signal X satisfies the power constraint of matrix used at the relay. Similar to the up-
R
link, by using linear receive matrix V =
1 k,l
Ttr(XRXHR)≤PR, [vl(k,1),vl(k,2),··· ,vl(k,k−1),vl(k,k+1),··· ,vl(k,K)]T ∈
C(K−1)×M, the equivalent channel matrix in the downlink is
where P is the maximum transmission power allowed at the
R given by
relay.
andWGe aks,s∀ukm,eartheadtrtahwe ferloemmeantcsoonftinthueoucshadnisnterilbmutaiotrnic,ewshHichk Vk,lGk =(cid:104)gl(k,1),gl(k,2),··· ,gl(k,k−1),gl(k,k+1),··· ,gl(k,K)(cid:105)T .
implies that these channel matrices are of full column or row (5)
rank, whichever is smaller, with probability one. The channel Later, we will show that due to symmetry between the uplink
state information is assumed to be perfectly known at all and the downlink, the uplink design straightforwardly carries
nodes,followingtheconventionin[13]–[19].Itisworthnoting over to the downlink. Thus, we mostly focus on the uplink
that the considered MIMO mRC reduces to the MIMO two- design in this paper.
way relay channel (TWRC) when K = 2, and to the MIMO In what follows, we will see that by dividing the spatial
Y channel when K = 3. As the DoF capacity of the MIMO streams into a number of units, the signal alignment can be
TWRC is well understood, we henceforth focus on the case realized in a unit-by-unit fashion, which facilitates the system
of K ≥3. design.
4
C. Degrees of Freedom wherez(k,k(cid:48))T =v(k,k(cid:48))TZ .TheprincipleofPNCisapplied
k k
Let R(k,k(cid:48)) be the information rate carried in W(k,k(cid:48)), and in relay decoding. Specifically, for each user pair (k,k(cid:48)),
Wˆ(k,k(cid:48)) be the estimate of W(k,k(cid:48)) at user k(cid:48). We say that the relay decodes a linear mixture of s(k,k(cid:48)) and s(k(cid:48),k) as
user k achieves a sum rate of C = (cid:80)K R(k,k(cid:48)), if follows. Denote by H(k,k(cid:48)) ∈CN×4 the matrix formed by all
Pr{Wˆ(k,k(cid:48)) (cid:54)=W(k,k(cid:48))} tends to zkero as Tk(cid:48)=→1,k∞(cid:48)(cid:54)=.k the uplink channel vectors except h(k,k(cid:48)) and h(k(cid:48),k). Then,
define the projection matrix of pair (k,k(cid:48)) as P(k,k(cid:48)) =
We assume a symmetric mRC with P = P = ··· =
1 2 I −H(k,k(cid:48))(H(k,k(cid:48))HH(k,k(cid:48)))−1H(k,k(cid:48))H ∈CN×N.Foreach
P =P =P andσ2 =σ2 =···=σ2 =σ2 =σ2.Denote N
K R 1 2 K R pair (k,k(cid:48)), the relay projects the received signal vector Y
SNR = P . Let C (SNR), k ∈ I , be an achievable rate of R
σ2 k K onto the nullspace of span(H(k,k(cid:48))), yielding
user k. The total DoF of the mRC is defined as
dsum (cid:44)SNlRim→∞(cid:80)Kk=lo1gCSkN(SRNR). P(k,k(cid:48))YR =P(k,k(cid:48))(cid:18)(cid:88)K (cid:88)K h(k,k(cid:48))s(k,k(cid:48))T +ZR(cid:19)
k=1k(cid:48)=1,k(cid:48)(cid:54)=k
Also, we define the DoF per user and the DoF per relay =P(k,k(cid:48))(cid:0)h(k,k(cid:48))s(k,k(cid:48))T +h(k(cid:48),k)s(k(cid:48),k)T(cid:1)+P(k,k(cid:48))Z .
R
dimension respectively as (9)
1 1
d (cid:44) d and d (cid:44) d . (6) We now move to the relay-to-user phase modeled in
user K sum relay N sum (8b). Similarly to H(k,k(cid:48)), we denote G(k,k(cid:48)) ∈ CN×4
as the matrix formed by all the downlink channel vectors
III. ADOFOUTERBOUND except g(k,k(cid:48)) and g(k(cid:48),k). The projection matrix of pair
An outer bound on the total DoF of the MIMO mRC is (k,k(cid:48)) in the downlink is then defined as W(k,k(cid:48)) = I −
N
given in [18] as G(k,k(cid:48))(G(k,k(cid:48))HG(k,k(cid:48)))−1G(k,k(cid:48))H ∈ CN×N. The relay
sends out FY with F defined in (4) given by
d ≤min(KM,2N), (7a) R
sum
K K
or equivalently F=α(cid:88) (cid:88) W(k,k(cid:48))P(k,k(cid:48)), (10)
(cid:18) 2N(cid:19) k=1k(cid:48)=k+1
d ≤min M, . (7b)
user K whereαisascalingfactortomeettherelay’spowerconstraint.
In (10), the index k(cid:48) starts from k + 1 since a project
Theaboveouterboundcanbeintuitivelyexplainedasfollows. matrix P(k,k(cid:48)) is used to extract the signals s(k,k(cid:48)) and s(k(cid:48),k)
On one hand, the total number of independent spatial data
simultaneously. The received signal at user k is given by
streams supported by the MIMO mRC cannot exceed 2N, as
the relay’s signal space has N dimensions and thus the relay K K
y(k,k(cid:48))T =g(k,k(cid:48))T (cid:88) (cid:88) W(ln)P(ln)Y +z(k,k(cid:48))T
can only decode and forward N network-coded messages. On k R k
theotherhand,thenumberofindependentspatialdatastreams l=1n=l+1
(cid:16)
transmittedorreceivedbyeachusercannotexceedM,aseach =g(k,k(cid:48))TW(k,k(cid:48))P(k,k(cid:48)) h(k,k(cid:48))s(k,k(cid:48))T+
uassearobnenlychhmasaMrk iannttehnenfaosl.loTwheinogusteyrstbeomunddesinig(n7.)willbeused h(k(cid:48),k)s(k(cid:48),k)T +ZR(cid:17)+z(kk,k(cid:48))T.
(11)
IV. MIMOMRCWITHK =3 We note that g(k,k(cid:48)), W(k,k(cid:48)), P(k,k(cid:48)), and h(k(cid:48),k) are inde-
In this section, we focus the DoF of the MIMO mRC with pendent of each other. Therefore, the equivalent user-to-user
K =3. We propose a signal alignment scheme to achieve the channel coefficient g(k,k(cid:48))TW(k,k(cid:48))P(k,k(cid:48))h(k(cid:48),k) is non-zero
DoFcapacityoftheMIMOmRCwithK =3foranarbitrary withprobabilityone,providedthatW(k,k(cid:48)) andP(k,k(cid:48)) areof
antenna setup of (M,N). at least rank one. Then, each user k receives one linear com-
bination of the two signals in pair (k,k(cid:48)). By subtracting the
A. Preliminaries self-interference, each user can decode the desired messages
from the other two users, which achieves a per-user DoF of
We give some intuitions of the signal alignment by consid-
d = 2, or equivalently, a total DoF of d = 6 can be
ering only one unit. For brevity, we omit the unit index l in user sum
achieved.From(11),weseethatthesymmetryexistsbetween
this subsection. Recall that s(k,k(cid:48)) and s(k(cid:48),k) are exchanged
thedesignoftheuplinkandthedesignofthedownlink.Given
in a pairwise manner for any k (cid:54)= k(cid:48). For convenience, we thedesignofthebeamformeru(k,k(cid:48)) andtheprojectionmatrix
refer to s(k,k(cid:48)) and s(k(cid:48),k) as the signal pair (k,k(cid:48)). Denote by P(k,k(cid:48)), the receive vector v(k,k(cid:48)) and the projection matrix
h(k,k(cid:48)) and g(k,k(cid:48)) the equivalent channels in the uplink and W(k,k(cid:48)) in the downlink can be designed similarly, since
the downlink, respectively. The system model in (1) and (2) g(k,k(cid:48))TW(k,k(cid:48)) can be simply regarded as the transpose of
reduces to P(k,k(cid:48))h(k(cid:48),k). Therefore, it suffices to focus on design of the
Y = (cid:88)K (cid:88)K h(k,k(cid:48))s(k,k(cid:48))T +Z (8a) uplink in what follows.
R R We now describe four patterns involved (with a different
k=1k(cid:48)=1,k(cid:48)(cid:54)=k d )inachievingtheDoFcapacityoftheMIMOmRCwith
relay
y(k,k(cid:48))T = g(k,k(cid:48))TX +z(k,k(cid:48))T, k ∈I . (8b) K = 3. Denote U (cid:44) {h(k,k(cid:48))|k ∈ I ,k(cid:48) ∈ I ,k(cid:48) (cid:54)= k}. Let
k R k K K K
5
TABLEI
h(2,3)h(1,3)h(3,2)
PATTERNSFORTHEMIMOMRCWITHK=3
h(2,3)
h(1,3) h(2,1) h(3,2) h(1,3) h(1,2)h(2,1)h(3,1) Pattern Dimension dsum drelay Requirement
h(2,1) h(2,3) h(3,2) M
h(1,2) h(3,1) 1.1 6 6 1 >0
N
h(1,2) h(3,1) 6 M 1
1.2 5 6 >
5 N 3
3 M 1
(a) Pattern 1.1 with dim-6 (b) Pattern 1.2 with dim-5 1.3 4 6 >
2 N 3
M 1
h(3,2) 1.4 3 6 2 >
N 2
h(2,1) h(1,3) h(3,2)
span(h(1,2),h(1,3)) h(1,2) h(2,3) h(1,3) 2/3
h(3,1)
h(1,2) span(h(2,1)h,h((22,,33))) h(2,1) by N) 1/2
h(3,1) ed
z
span(h(3,2),h(3,1)) mali
or
(c) Pattern 1.3 with dim-4 (d) Pattern 1.4 with dim-3 F (n 1/3
o
D
er
s
Fig.2. AgeometricillustrationofPatterns1.1to1.4. U
er-
P
U\{h(k,k(cid:48)),h(k(cid:48),k)} be the vector set obtained by excluding Optimal Per-User DoF
h(k,k(cid:48)) and h(k(cid:48),k) from U.
0
0 1/3 1/2 2/3 1
1) Pattern1.1:U spansasubspacewithdimension6(dim- M/N
6) in CN.
2) Pattern 1.2: U spans a subspace with dim-5 in CN; for
Fig. 3. The DoF capacity for the MIMO mRC with K = 3 against the
any pair (k,k(cid:48)), U\{h(k,k(cid:48)),h(k(cid:48),k)} spans a subspace antennaratio M.
N
of dim-4.
3) Pattern 1.3: U spans a subspace with dim-4 in CN; the
B. Main Result
intersection of span(h(1,2),h(2,1)), span(h(2,3),h(3,2)),
We now consider the general case that each user transmits
and span(h(1,3),h(3,1)) is of dim-1, i.e., these three
multiplespatialdatastreamsoveraMIMOmRC,i.e.,multiple
planes go through a common line, so that U spans a
units co-exist in the relay’s signal space with each unit
subspace of dim-4.
consistingofK(K−1)spatialstreams.Ourgoalistoconstruct
4) Pattern 1.4: U spans a subspace with dim-3 in CN; for
any pair (k,k(cid:48)), h(k,k(cid:48)) and h(k(cid:48),k) span a subspace of units with the most efficient patterns to occupy the relay’s
signal space. The main result is summarized in the following
dim-1.
theorem.
The above four patterns are geometrically illustrated in
THEOREM 1. For the MIMO mRC (M,N,K) with K = 3,
Fig. 2. It can be verified that the projection matrices P(k,k(cid:48)),
the DoF capacity per user is given by
∀k,k(cid:48),k(cid:48) (cid:54)=k, for Patterns 1.1-1.4 are of at least rank one for
sure. For example, P(k,k(cid:48)) for Pattern 1.1 is of at least rank M, M < 2
N 3
twoforsure.Hencetheproposedsignalingschemeachievesa d = (12)
user 2N M 2
total DoF of 6. However, a different pattern spans a subspace  , ≥ .
3 N 3
withadifferentnumberofdimensions,whichyieldsadifferent
d as shown in Table I. In general, a pattern with a higher The per-user DoF capacity with respect to M is shown in
relay N
d ismoreefficientinutilizingtherelay’ssignalspace,and Fig.3.Weseethattheper-userDoFofd =M isachieved
relay user
hence is more desirable in the signal alignment design. The for M < 2, which means that the DoF is bounded by the
N 3
requirement on M to realize each specific pattern is given in numberofantennasattheuserends.Ontheotherhand,when
N
thelastcolumnofTableI.Notethattheserequirementswillbe M ≥ 2,theDoFisboundedbythenumberofrelayantennas.
N 3
discussed in detail in Subsection IV-C. It is also worth noting NotethattheDoFcapacityofthethree-userMIMOYchannel
that Pattern 1.2 and Pattern 1.3 have the same requirement on has been previously derived in [16]. However, we emphasize
M, but Pattern 1.3 achieves a higher d than Pattern 1.2. that the proposed signal alignment technique in our proof
N relay
Thus, Pattern 1.2 is ruled out by Pattern 1.3 in the proposed (cf., equations (14)-(16) and the discussions therein) is very
signal alignment scheme. different from the one in [16]. Also, our proposed technique
6
can be extended for the case of an arbitrary K, which is the d thanthelatter),wefocusontheconstructionofunitsfol-
relay
major contribution of this paper. lowing Pattern 1.3. Denote the intersection of span(H ,H )
1 2
and span(H ) by S(1.3). The dimension of S(1.3) is 3M −
3
C. Proof of Theorem 1 N > 0. We choose two vectors u3,l and u(l3,1) such that
H u and H u(3,1) are two linearly independent vectors in
We first note that d in (12) coincides with the DoF 3 3,l 3 l
outer bound in (7) withusKer = 3. Thus, to prove Theorem 1, S(1.3). By definition, both H3u3,l and H3ul(3,1) belong to
span(H ,H )=span(H )⊕span(H ).Thus,thereuniquely
it suffices to show the achievability of (12). We start with a 1 2 1 2
brief description of the overall transceiver design. We need to exist {ul(1,3),ul(2,3)} and {u1,l,u(l2,1)} satisfying
jointly design the transmit beamformers {u(k,k(cid:48))}, the receive
vectors {v(k,k(cid:48))}, and the relay projection ml atrices {P(k,k(cid:48))} H1u(l1,3)+H2ul(2,3)+H3u3,l =0 (14a)
and{W(k,lk(cid:48))}toefficientlyutilizetherelay’ssignalspalce.As H1u1,l+H2u(l2,1)+H3ul(3,1) =0. (14b)
l
adnifdfeWren(tkf,kro(cid:48))mnu(1ll0t)h,eheinretetrhfeerreenlcaey’nsoptroonjelyctiforonmmtahtreicoetshePrl(pka,kir(cid:48)s) wLeitthu((l134,2)),=weuo3,blt−ainul(3,1) and u(l1,2) =u1,l−ul(1,3). Together
l
in unit l but also from the other units. Taking P(k,k(cid:48)) as an
l H u(1,3)+H u(2,3)+H (u(3,2)+u(3,1))=0 (15a)
example,weseethatitprojectsavectorintothenullspaceof 1 l 2 l 3 l l
span(cid:0){Hk¯u¯(lk¯,k¯(cid:48))|∀¯l,k¯,k¯(cid:48);k¯(cid:48) (cid:54)= k¯}\{Hku(lk,k(cid:48)),Hkul(k(cid:48),k)}(cid:1). H1(ul(1,2)+ul(1,3))+H2ul(2,1)+H3u(l3,1) =0. (15b)
Hence the relay beamforming matrix F given in (4) is ex-
Subtracting (15b) by (15a), we further obtain
pressed as
L K K H u(1,2)+H (u(2,1)−u(2,3))−H u(3,2) =0. (15c)
F=α(cid:88)(cid:88) (cid:88) W(k,k(cid:48))P(k,k(cid:48)), (13) 1 l 2 l l 3 l
l l
We now show that three signal direction pairs
l=1k=1k(cid:48)=k+1
{H u(1,2),H u(2,1)}, {H u(2,3),H u(3,2)},
where L denotes the number of the units. Based on that, in 1 l 2 l 2 l 3 l
{H u(3,1),H u(1,3)} form a unit with Pattern 1.3 as
each unit, each user achieves a DoF of two, provided that the 3 l 1 l
projection matrices P(k,k(cid:48)) and W(k,k(cid:48)) are at least of rank shown in Fig. 2(c). From (15a), we see that two signal
one. Note that with thle transmit belamformer u(k,k(cid:48)) and the pairs (1,3) and (2,3) span a subspace of dim-3, which
l implies that the plane spanned by signal pair (1,3) and the
receive vector vl(k,k(cid:48)), the equivalent channel regarding the plane spanned by signal pair (2,3) go through a common
spatial stream s(k,k(cid:48)) in the uplink is denoted as h(k,k(cid:48)) = line. Further, from (15b) and (15c), we see that the plane
l l
H u(k,k(cid:48)), and the equivalent channel vector regarding the spanned by signal pair of (1,2) also goes through this
spaktiall stream s(lk(cid:48),k) in the downlink is denoted by gl(k,k(cid:48)) = common line, which implies that {Hku(lk,k(cid:48))|∀k,k(cid:48) (cid:54)= k}
GTv(k,k(cid:48)). Next we derive the DoF given in Theorem 1 by span a subspace of dim-4. Base on that, the dimension
divkidilng the overall range of M into multiple intervals. of null(cid:0){H1ul(1,3),H3ul(3,1),H2ul(2,3),H3u(l3,2)}(cid:1) ∩
1) Case of M ≤ 1: In tNhis case, the number of relay span(cid:0){H u(k,k(cid:48))|∀k,k(cid:48) (cid:54)= k}(cid:1) is of dim-1. Similarly,
N 3 k l
antennas N is no less than the number of antennas of all from (15b) and (15c), the intersection of nullspace of any
theusers,i.e.,span({H |∀k})isofdim-3M withprobability two of the three pairs in unit l and the subspace spanned by
k
one. This implies that the relay’s signal space has enough signals in unit l is of dim-1. Therefore, a linear combination
dimensions to support full multiplexing at the user end, i.e., of the signals for each signal pair can be decoded at the relay
each user transmits M spatial streams. M units with Pattern without interference.
1.1 (as shown in Fig. 2(a)) can be constructed. The geometric We now describe how to construct multiple linearly
structure in Fig. 2(a) indicates that each spatial stream in a independent units following Pattern 1.3. Let the columns of
unit occupies an independent direction in the relay’s signal U ∈ C3M×(3M−N) be a basis of null([H ,H ,H ]).
H 1 2 3
space. Then, in total the 6 spatial streams in a unit span a Partition U as U = (cid:2)UT ,UT ,UT (cid:3)T with
H H H1 H2 H3
subspace of dim-6. As the directions of signals with Pattern U ∈ CM×(3M−N). Then, u(1,3), u(2,3), and u
1.1arerandomlydrawnfromtherelay’ssignalspace,theinde- Hi l l 3,l
in (14a), are respectively chosen as the (2l − 1)-th
pendenceofdifferentunitscanbeguaranteedwithprobability column of U , U , and U . Further, u , u(2,1),
one. Clearly, the projection matrix P(lk,k(cid:48)) is of at least rank and u(3,1) in H(114b) aHre2 respectivHel3y chosen as 1th,le (2ll)-th
one. Thus, each unit achieves a DoF of 6. Considering all the l
column of U , U , and U . From (14), we see that
M units, we obtain that the achievable per-user DoF is M.2 span(cid:0)H u(1,3H),1H uH(22,3),H u H,3H u ,H u(2,1),H u(3,1)(cid:1)
2) Case of 1 < M ≤ 1: From Table I, this case 1 l 1 l 3 3,l 1 1,l 2 l 3 l
3 N 2
corresponds to Patterns 1.2 and 1.3. As Pattern 1.3 is more
orequivalently
efficient than Pattern 1.23 (i.e., the former achieves a higher
(cid:16)h(1,2)+h(2,1)+h(3,1)(cid:17)+(cid:16)h(2,3)+h(1,3)+h(3,2)(cid:17)=0,
2Asimilarproofofduser=M for MN ≤ 31 canbefoundin[18]. l l l l l l
3Pattern 1.2 can be constructed by designing the transmit beamforming which implies that the direction of h(1,2)+h(2,1)+h(3,1) is parallel to
vectorsinaunitas l l l
thedirectionofh(2,3)+h(1,3)+h(3,2) followingPattern1.2inFig.2(b).
H1(u(l1,2)+u(l1,3))+H2(u(l2,1)+u(l2,3))+H3(u(l3,1)+u(l3,2))=0, Thus,the6spatiallstreamslinaunitlspanasubspaceofdim-5.
7
is of dim-4, and Therefore, the maximum achievable per-user DoF is given by
span(cid:0)H u(1,3),H u(2,3),H u ,H u , min(M,23N).5
1 l 1 l 3 3,l 1 1,l The above obtained DoF coincides with the DoF upper
H2u(l2,1),H3u(l3,1)(cid:1) (16) bound in (7b), and therefore, this achievable DoF is exactly
= span(cid:0)H u(1,3),H u(1,2),H u(2,1),H u(2,3), the DoF capacity of the channel, which concludes the proof
1 l 1 l 2 l 2 l of Theorem 1.
H u(3,1),H u(3,2)(cid:1),
3 l 3 l
by noting u(3,2) = u − u(3,1) and u(1,2) = u − V. MIMOMRCWITHK >3
l 3,l l l 1,l
u(1,3). Thus, each unit l spans a subspace of dim-4. From In this section, we generalize Theorem 1 to the case of an
l
Lemma 3 in Appendix A, we see that the dimension of arbitrary number of users. We start with the case of K =4.
span(H U ,H U ,H U ) is min(2(3M −N),N).
Therefor1e, wHe1 can2 coHn2stru3ct mH3in(cid:0)3M−N,N(cid:1) linearly inde- A. Preliminaries
2 4
pendent units following Pattern 1.3.4
Again,westartwithsomeintuitionsofthesignalalignment
Suppose 2(3M −N) ≤ N, or equivalently, M ≤ 1. All
N 2 by assuming that each user transmits one independent spatial
3M−N units span a subspace of 3M−N ×4 = 2(3M −N)
2 2 stream to each of the other users in a unit. The relay’s
dimensions. The remaining N −2(3M −N) dimensions are
beamforming matrix is still given by (10).
used to construct N−2(36M−N) units with Pattern 1.1. Thus, The following patterns are involved in deriving the achiev-
the achievable per-user DoF is given by able DoF to be presented later. Also we omit the unit index
2(N −2(3M −N)) l for brevity in this subsection. Denote U (cid:44) {h(k,k(cid:48))|k ∈
duser =3M −N + 6 =M. (17) IK,k(cid:48) ∈ IK;k (cid:54)= k(cid:48)} with IK = {1,2,3,4}, and
U\{h(k,k(cid:48)),h(k(cid:48),k)} is the vector set obtained by excluding
Recall that the directions of signals with Pattern 1.1 are ran-
domly drawn from the relay’s signal space, the independence h(k,k(cid:48)) and h(k(cid:48),k) from U. Let U¯i (cid:44) {h(k,k(cid:48))|k ∈ IK,k(cid:48) ∈
oftheunitswithPattern1.3andtheunitswithPattern1.1can IK;k (cid:54)= k(cid:48);k (cid:54)= i,k(cid:48) (cid:54)= i} and U¯i\{h(k,k(cid:48)),h(k(cid:48),k)} be the
beguaranteedwithprobabilityone.Iftheoverallrelay’ssignal vector set obtained by excluding h(k,k(cid:48)) and h(k(cid:48),k) from U¯i.
space is occupied by units with Pattern 1.3, the maximum 1) Pattern 2.1: U spans a subspace with dim-12 in CN.
achievable per-user DoF is N × 2 = N. Therefore, the 2) Pattern 2.2: U spans a subspace with dim-9 in CN; for
achievable per-user DoF is giv4en by min(M2,N)=M. any pair (k,k(cid:48)), U\{h(k,k(cid:48)),h(k(cid:48),k)} spans a subspace
2
3) Case of M > 1: In this case, M is large enough to with dim-8.
N 2 N
constructunitswithPattern1.4.Theintersectionofspan(Hk) 3) Pattern 2.3: For each i, U¯i spans a subspace of dim-4
and span(H ) is of 2M −N > 0. Let h(k,k(cid:48)) be a vector in CN following Pattern 1.3.
in the intersekc(cid:48)tion of span(H ) and span(Hl ). There exist 4) Pattern 2.4: U spans a subspace with dim-6 in CN; for
{u(k,k(cid:48)),u(k(cid:48),k)} satisfying k k(cid:48) anypair(k,k(cid:48)),h(k,k(cid:48)) andh(k(cid:48),k) spanasubspacewith
l l dim-1.
Hku(lk,k(cid:48)) =Hk(cid:48)u(lk(cid:48),k) =h(lk,k(cid:48)), ∀k,k (cid:54)=k(cid:48), It can be readily shown that the projection matrix P(k,k(cid:48))
corresponding to Patterns 2.1 to 2.4 are of at least rank one
which implies that the two spatial streams of pair (k,k(cid:48)) in a
with probability one. Thus, Patterns 2.1, 2.2, and 2.4 achieve
unitspanasubspaceofdim-1,i.e.,twospatialstreamsofapair
a total DoF of 12, while Pattern 2.3 achieves a total DoF
arealignedinonedirectionasillustratedinFig.2(d).Then,in
of 24. Corresponding antenna requirement for each pattern
totalthe6spatialstreamsinaunitoccupyasubspaceofdim-
is given in Table II, which will be discussed in details in
3.Inthisway,wecanconstruct2M−N unitswithPattern1.4,
the Subsection V-C. It is worth mentioning that for a same
in total spanning a subspace of dim-3(2M −N). According
to Lemma 4 in Appendix A, we obtain that {h(k,k(cid:48))|∀l} are requirement on MN, some other patterns may possibly be
l constructed. However, they are ruled out due to a relatively
linearly independent with probability one. Further, due to the
randomnessofH ,theindependenceof{h(k,k(cid:48))|∀l,k,k(cid:48),k (cid:54)= low drelay, i.e., less efficiency in utilizing the relay’s signal
k l space. Again, the downlink patterns are omitted due to the
k(cid:48)} are guaranteed with probability one. Again, the remaining
uplink/downlink symmetry.
N − 3(2M − N) dimensions can be used for constructing
N−3(2M−N) units with Pattern 1.3. Thus an achievable per-
4 B. Main Result
user DoF is given by
PROPOSITION1. FortheMIMOmRC(M,N,K)withK =4,
2(N −3(2M −N))
d =2(2M −N)+ =M. the per-user DoF capacity of d = M is achieved when
user 4 user
M ≤ 3, and the per-user DoF capacity of d = N is
When the overall relay’s signal space is occupied by the units aNchieved8 when M ≥ 7 . For M ∈ (3, 7 ], aunsearchiev2able
with Pattern 1.4, a per-user DoF of N ×2= 2N is achieved. N 12 N 8 12
3 3 per-user DoF is given by
4Here we assume that 3M−N is an integer. Otherwise, we use symbol 3N 3 M 1
extension [20] to ensure tha2t the dimension of the above intersection is  8 , 8 < N ≤ 2
d =
dividablebytwo;seeAppendixBfordetails.Notethatthesymbolextension user 3M 3N 1 M 7
is used to achieved a fractional DoF throughout of the rest of this paper  − , < ≤ .
withoutfurtherexplicitnotification. 2 8 2 N 12
8
TABLEII columns of U , denoted by [u1T,u(2,1)T,u(3,1)T,u(4,1)T]T,
PATTERNSFORTHEMIMOMRCWITHK=4 H l l l l
[u(1,2)T,u2T,u(3,2)T,u(4,2)T]T, and
l l l l
Pattern Dimension dsum drelay Requirement [u(1,3)T,u(2,3)T,u3T,u(4,3)T]T, we have
l l l l
M
2.1 12 12 1 >0 H u1+H u(2,1)+H u(3,1)+H u(4,1) =0 (18a)
N 1 l 2 l 3 l 4 l
4 M 1 H u(1,2)+H u2+H u(3,2)+H u(4,2) =0 (18b)
2.2 9 12 > 1 l 2 l 3 l 4 l
3 N 4 H u(1,3)+H u(2,3)+H u3+H u(4,3) =0. (18c)
3 M 1 1 l 2 l 3 l 4 l
2.3 4 6 >
2 N 3 Letu(1,4) =u1−u(1,2)−u(1,3),u(2,4) =u2−u(2,1)+u(2,3),
l l l l l l l l
M 1 and u(3,4) =u3−u(3,1)+u(3,2). Subtracting (18a) by (18b)
2.4 6 12 2 > l l l l
N 2 and (18c), we have
H u(1,4)−H u(2,4)−H u(3,4)
1 l 2 l 3 l (19)
+H (u(4,1)−u(4,2)−u(4,3))=0.
by N) 1/2 We now show th4at leach unlit spansla subspace of dim-9
d
ze corresponding to Pattern 2.2. Eqs. (18) and (19) imply that
ali
m onedimensionissavedforthesubspacespannedbythesignals
or 3/8
oF (n r(e1l9a)teids stoimoinlaerutosetrh.eTohneesiugsneadlbayligPnamtteernnt s1h.3owshnoiwnn(1in8)(1a5n)d.
D
er Then, {H u(k,k(cid:48))|∀k,k(cid:48),k(cid:48) (cid:54)= k} span a subspace of dim-9,
Us k l
Per- while Ul\{h(lk,k(cid:48)),hl(k(cid:48),k)} span a subspace of dim-8. Hence,
ble we always have a nullspace of dim-1 to obtian the linear
a
ev DoF Upper Bound combinationofthesignalsineachpairattherelay.Inthisway,
chi Achievable DoF we can construct 4M−N units with Pattern 2.2, occupying a
A 3
subspace of dim-3(4M −N) in the relay’s signal space. The
0 remainingN−3(4M−N)dimensionsofrelay’ssignalspace
0 3/8 1/2 7/12 1
M/N areusedtoconstruct N−3(4M−N) unitswithPattern2.1.Thus,
12
an achievable per-user DoF is given by
4M −N 12 3(N −3(4M −N))
Fig.4. Anachievableper-userDoFfortheMIMOmRCwithK=4against × + =M.
theantennaratio M. 3 4 12
N
If the overall relay’s signal space is occupied by the units
The achievable per-user DoF for MIMO mRC with K =4 with Pattern 2.3, the maximum achievable per-user DoF of
is illustrated in Fig. 4. We observe that, different from the N ×3 = N is achieved. Therefore, the achievable per-user
9 3
case of K =3, the DoF bound given in Section III can only DoF is given by min(M,N)=M for 1 < M ≤ 1.
3 4 N 3
be achieved in the ranges of M ∈ (0,3] and M ∈ [ 7 ,∞); 3) Case of 1 < M ≤ 1: In this case, M is large enough
N 8 N 12 3 N 2 N
for M ∈ (3, 7 ), there is a certain DoF gap between the to construct the units with Pattern 2.3 shown in Table II.
N 8 12
achievable DoF and the capacity outer bound. Weformthefollowingfourthree-usergroups:{H1,H2,H3},
{H ,H ,H }, {H ,H ,H }, and {H ,H ,H }, and align
1 2 4 2 3 4 1 3 4
the signals within each three-user group. For each group, the
C. Proof of Proposition 1
signal alignment is conducted as Patten 1.3 for K =3, where
To prove Proposition 1, we consider four cases detailed
6 spatial streams occupy a relay’s signal subspace of dim-
below. 4. Similarly to Pattern 1.3, 3M−N units with Pattern 2.3 are
1) Case of MN ≤ 14: In this case, since N ≥ 4M, the constructed and span a subspac2e of 4×3M−N =2(3M−N)
relay’s signal space has enough dimensions to support full 2
dimensions. Considering the units from four groups, we have
multiplexing at the users, which implies that each user can
2(3M−N) units which span a subspace of dim-8(3M−N).
transmit M independent spatial streams, or equivalently, M
The independence of units can be proven by using the
units with Pattern 2.1 can be constructed. Therefore, a per-
result given in Lemma 5 in Appendix A. The remaining
user DoF of M can be achieved.
N −8(3M −N) dimensions of the relay’s signal space are
2) Case of 1 < M ≤ 1: As shown in Table II, this case
4 N 3 used to construct the units with Pattern 2.2, the achievable
corresponds to Pattern 2.2. As 4M−N >0, the nullspace of
per-user DoF can be expressed as
(cid:0) (cid:1)
span H ,H ,H ,H isofdim-(4M−N).Letthecolumns
1 2 3 4
of UH ∈CN×(4M−N) be a basis of null(cid:0)[H1,H2,H3,H4](cid:1). 3M −N ×3×2+ 3(N −8(3M −N)) =M.
Partition U as U = [UT ,UT ,UT ,UT ]T. From 2 9
Lemma 3,H span(cid:0)HH U ,HHU1 H,H2 UH3 ,HH4U (cid:1) is If the overall relay’s signal space is occupied by the units
of dim-3(4M − N)1foHr1sur2e. HA2rbitr3arilHy3 cho4osHe4three with Pattern 2.3, we achieve the maximum per-user DoF of
N × 3 × 2 = 3N. Hence, the achievable per-user DoF is
4 4 8
5Asimilarproofofduser= 23N for MN ≥ 32 canbefoundin[14],[18]. denoted by min(M,38N).
9
4) Case of M > 1: In this case, as 2M −N > 0, the signal alignment is to split all the users into different t-
N 2
signals in each pair can be aligned in one direction to occupy user groups and perform signal alignment in each group. In
a subspace of dim-1. In total, 2M −N units with Pattern 2.4 this way, we have (K) number of different t-user groups.
canbeconstructed,whichspanasubspaceofdim-6(2M−N). Further, each user is itncluded in (cid:0)K−1(cid:1) number of different
t−1
Similarly, the remaining N −6(2M −N) dimensions of the t-user groups. Denote the channel matrices for an arbitrarily
relay’ssignalspaceareusedtoconstructtheunitswithPattern chosen t-user group as {H ,H ,··· ,H }, the nullspace of
i1 i2 it
2.3. The achievable per-user DoF can be expressed as span(H ,H ,··· ,H ) is of dim-(tM − N) with proba-
i1 i2 it
bility one. Similarly to Pattern 2.3, an efficient way to align
3(N −6(2M −N)) 3M 3N
3(2M −N)+ = − . signals in one unit is let the spatial streams related to one
8 2 8
user be aligned together (without loss of generality, we term
If the whole relay’s signal space is occupied by the units this pattern as Pattern 3.t). The beamformers in unit l can be
with Pattern 2.4, we have the maximum per-user DoF of
designed to satisfy the conditions given in (22a)-(22c) shown
N × 3 = N. The achievable per-user DoF is given by
6 2 at the top of next page, where
min(3M − 3N,N), which is equivalent to
2 8 2 (cid:40)
1, i=1 or j =1
3M 3N 1 M 7 a(i,j) =
 − , < ≤ l −1, otherwise.
2 8 2 N 12
d =
user N M 7 Subtracting (22a) by equations from (22b) to (22c), we obtain
 , > .
2 N 12 the equation given in (22d) shown at the top of next page.
This completes the proof of Proposition 1. Based on (22), we see that total t − 1 dimensions can be
saved for the subspace spanned by the signals in unit l.
In total, we can construct tM−N units with Pattern 3.t in
D. Achievable DoF for A General K t−1
each group and the achievable per-user DoF in one unit is
WenowgeneralizetheachievableDoFresulttoanarbitrary t−1. Thus, the achievable per-user DoF is αt(tM−N). Note
t−1
K. Denote that the linear independence of the subspaces spanned by
α =(cid:0)K−1(cid:1)(t−1) and β =(K)(t−1)2. (20) the units with Pattern 3.t can be proven using the result in
t t−1 t t Lemma 5 in Appendix A. Moreover, for each t-user group,
PROPOSITION 2. For the MIMO mRC (M,N,K), the per- one unit spans a subspace of t(t−1)−(t−1) = (t−1)2
user DoF capacity of d = M is achieved when M ∈ dimensions. Considering all (K) number of groups, all the
(cid:16)0,KK(K−−12)(cid:105) and the peur(cid:104)-suerser DoF capac(cid:17)ity of duser =N2KN uNni−tsβspt(atnMa−sNu)bsdpimaceenosfioβnts(ttoM−f1−rteNla)yd’simseignnsaiolnssp.aTcheearreemuasiendintgo
is achieved when M ∈ 1 + 1,∞ . Further, for the t−1
N K(K−1) 2 constructtheunitswithPattern3.(t+1).Then,theachievable
remaining range of M, an achievable per-user DoF is given per-user DoF can be expressed as
N
by (cid:16) (cid:17)
α N − βt(tM−N)
d =min(cid:18)αt(tM −N) + αt+1(cid:16)N − βt(ttM−1−N)(cid:17),αtN(cid:19), duser = αt(ttM−−1 N) + t+1 βt+1 t−1 . (23)
user t−1 β β
t+1 t Whentherelay’ssignalspaceiswhollyoccupiedbyunitswith
(cid:18) (cid:21)
M 1 1 Pattern 3.t, we achieve a maximum achievable per-user DoF
∈ , ,
N t t−1 of αtN.Thus,anachievableper-userDoFisobtainedas(21).
(21) Wβhten t = K, we have only one group, which leads to
α =K−1 and β =(K−1)2. Note that for M < 1, we
where t=2,3,··· ,K−1.6 K K N K
cannot do any signal alignment, α and β defined in
K+1 K+1
Proof: For the case of M < 1, it is easy to obtain that (23) should be equal to K −1 and K(K −1), respectively.
N K
d = M, which is also the DoF capacity. On the other Substitute them into (21), we have d = min(M, N ) =
user user K−1
hand, for M ∈(1,∞), the achievable DoF is equal to the one M, which is the DoF capacity.
N
achieved at MN =1 as the number of relay antennas N is the When t = K −1, we have αK−1 = (K −1)(K −2) and
bottleneck. Thus, we only focus on the range of MN ∈(K1,1] βK−1 = K(K −2)2. Substitute(cid:16)αK, βK, αK(cid:17)−1, and βK−1
in the following proof. Moreover, we partition the range of into (21), we obtain d =min M,(K−1)N =M, which
user K(K−2)
(1,1] into intervals of (1, 1 ] with t = 2,3,··· ,K, and implies that the per-user DoF capacity of d =M can also
diKscuss the signal alignmentt dt−e1sign for MN ∈(1t,t−11] with an be achieved for M ∈(cid:16) 1 , K−1 (cid:105). user
arbitrary t. N K−1 K(K−2)
Similarly,wecanalsoverifythattheper-userDoFcapacity
Note that, M ∈ (1, 1 ] imples N < tM, which in- (cid:104) (cid:105)
N t t−1 of 2N can be achieved in M ∈ 1 + 1,1 by letting
dicates that only the spatial streams from t users can be K N K(K−1) 2
t=2. Then, we complete the proof Proposition 2.
aligned together. Based on that, the most efficient way of
When the number of the users K tends to infinity, the fol-
6The result given in (21) is applicable for a larger range than lowing asymptotic DoF can be obtained from Proposition 2.7
(cid:16) (cid:17)
K−1 , 1 + 1 since it includes a union of K−2 intervals as
(cid:16)KK1(−K1−,2K)1−K2(cid:105)(K∪−(cid:16)1K)1−22,K1−3(cid:105)∪···∪(cid:0)21,1(cid:3). Do7FWdhseunmKhe→re.∞,duser tendsto0.Thus,weconsiderthetotalachievable
10
t
H (cid:88)a(1,i)u(1,i)+H u(2,1)+···+H u(t−1,1)+H u(t,1) =0, (22a)
i1 l l i2 l it−1 l it l
i=2
t
H u(1,2)+H (cid:88) a(2,i)u(2,i)+···+H u(t−1,2)+H u(t,2) =0, (22b)
i1 l i2 l l it−1 l it l
i=1,i(cid:54)=2
.
.
.
t
H u(1,t−1)+H u(2,t−1)+···+H (cid:88) a(t−1,i)u(t−1,i)+H u(t,t−1) =0, (22c)
i1 l i2 l it−1 l l it l
i=1,i(cid:54)=t−1
(cid:32) t−1 (cid:33)
H u(1,t)+H u(2,t)+···+H u(t−1,t)+H u(t,1)−(cid:88)u(t,i) =0. (22d)
i1 l i2 l it−1 l it l l
i=2
DoF.
2
N) 1/(1-M/N) VI. IMPROVEDACHIEVABLEDOFUSINGRELAY
y ANTENNADEACTIVATION
b
d
e In the previous sections, we have shown that the proposed
z
mali beamformingdesignachievestheDoFcapacityfortheMIMO
F (nor 3/2 mRC with K (cid:16)= 3. However, a cert(cid:17)ain gap occurs in the
Do range of M ∈ K−1 , 1 + 1 when K >3. In this
otal 4/3 section, wNe showKt(hKa−t 2th)eKo(bKta−in1e)d ac2hievable DoF in Propo-
T
e 5/4 sition 2 can be enhanced by the technique of relay antenna
bl 6/5 1 + M/N
a
v deactivation, i.e., to leave a portion of relay antennas disabled
e
hi
c in the uplink and downlink transmissions. We emphasize that
A 1
the antenna deactivation technique in general cannot improve
the DoF of the considered relay channel. The improvement
0 1/51/4 1/3 1/2 1 presented below comes from the non-optimality of the signal
M/N alignment technique utilized in Section V.
To proceed, we first give the following property which can
be directly obtained from Proposition 2.
Fig.5. Atotal achievableDoF oftheMIMO mRCwith K →∞ against
theantennaratio M.
N PROPERTY1. For a MIMO mRC with an antenna configura-
tionof(M,N),theobtainedachievableDoFinProposition2
COROLLARY 1. For the MIMO mRC (M,N,K) with K → can always be represented by ψ(M)N where ψ(M) is a
N N
∞, the total DoF of d = 2N is achieved when M > 1 coefficient determined solely by M.
sum N 2 N
and d = N is achieved as M → 0. For M ∈ (0,1], the
sum N N 2 Property 1 implies that when the antenna configuration of
achievable total DoF contains discontinuities at M = 1,t =
N t a MIMO mRC varies from (M,N) to (σM,σN) where σ >
2,3,4,···. Specifically, when M ∈ (1, 1 ], a total DoF of
N t t−1 0 is an arbitrary coefficient, the obtained achievable DoF by
tt−N1 is achieved. Proposition 2 changes from ψ(M)N to ψ(M)σN. Then, we
N N
have the following lemma.
Remark 1: When K → ∞, the number of spatial streams
K(K − 1) tends infinity. The total achievable DoF is then LEMMA 1. For the MIMO mRC (M,N,K), assume that a
bounded by the number of relay antennas N, which further certain DoF of d = ϕN is achievable at (M,N) =
user 0
implies that only a portion of users can realize data exchange. (M ,N )whereM andN arecertainconstantintegers,and
0 0 0 0
The overall achievable total DoF with respect to M is il- ϕ∈(0,1]isacoefficient.Thend =ϕN M isachievable
lustrated in Fig. 5. From Corollary 1, we see that Nfor each for any M ≤ M by disabling ausferraction 10M−0 M of all the
antenna setup M = 1, the total achievable DoF jumps from relay antennas. 0 M0
N t
t+1 to t . It is interesting to verify that the discontinuous
t t−1 Proof: Consider an antenna setup of (M,N ) with M <
points (M,dsum) = (1,t+1) are went through by the line 0
(yth1te=,at−c1th1+i)eNvaMarbe/leeNNntv,oetwalolhpDieleodFttbhiyestntdhiiecsecclouynrbvtioenuuynod=uesd1p−boyMi1nt/thNses.(eMINntw,thodisNsucmucra)vse=es, aaMnllt0et.nhAneasrseNMltao0y<Nan(cid:48)MNte=00n,nwaMsMe.N80c0Tanhbeyrne,ddiuwscaeeblhtihnaevgenauNMmf(cid:48)rba=ecrtiooMNnf00a1.cBt−ivaesMMerd0eloaonyf
as shown in Fig. 5. Therefore, without loss of generality, we
refer to N2 as an upper bound of the total achievable DoF 8Here, we assume that MMN00 is an integer. Otherwise, the technique of
N−M symbolextensionshouldbeusedtoensurethatthenumberofdisabledrelay
and refer to N +M as a lower bound of the total achievable antennasisaninteger.