Retrospective Interference Alignment for the MIMO Interference Broadcast Channel

0.87 MB·English

by  Marc Torrellas

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Retrospective Interference Alignment
for the MIMO Interference Broadcast Channel
Marc Torrellas, Adrian Agustin, and Josep Vidal
Signal Theory and Communications Department
Universitat Polite`cnica de Catalunya (UPC), Barcelona
marc.torrellas.socastro, adrian.agustin, josep.vidal @upc.edu
{ }
Abstract—The degrees of freedom (DoF) of the multiple-input Feedback link for delayed CSIT
mwdsiediutleahltyi(ep2ddleelc-caoehylualestndpnuaCentldS(sMItT2aIt)Me.uRsiOneerf)tosrIronmpstepearretficoectrnievelelniscaIeanrvetBaeririlonfaeavbrdeleescntaicasgettattCAhedhleiagtnwnranmhneeelsnnm(tIoiBthntCaelyrs) x1 , x2 BS 1 H1,1 H2,1 UE 1 xxˆˆ1
shownthebenefitsintermsofDoFofexploitingdelayedCSITfor UE 2 2
interference, broadcast and also for the IBC. However, previous
H
works studying the IBC with delayed CSIT do not exploit 1,2 UE xˆ3
the fact that the users of each cell are served by a common 3
transmitter. This work presents a four-phase precoding strategy x3 , x4 BS
taking this into consideration. Assuming that transmitters and 2 H UE xˆ4
receivers are equipped with M,N antennas, respectively, new 4,2 4
DoFinnerboundsareproposed,outperformingtheexistingones
for ρ= M >2.6413.
N Fig. 1. The IBC with delayed CSIT, 2 cells, 2 users per cell, M = 4
Index Terms—MIMO, Interference Alignment, Degrees of antennasattheBSsandsingle-antenna(N =1)UEs.Solidlinesdenotethe
freedom, Delayed CSIT, Interference Broadcast Channel. direct links, providing intended signals, while dashed lines denote the cross
links,providinginterferingornon-desiredsignals.
I. INTRODUCTION
but completely outdated, referred to as delayed CSIT. The
Channelcapacitycharacterizationremainsstillunknownfor publication of [3] inspired many works that extended these
many wireless interference network settings. In this regard, ideastoInterferencechannel(IC)andXchannel(IC)[4][5][6].
the study of channel degrees of freedom (DoF) has become Initially proposed for the MISO BC, the signals are aligned
a crowded research avenue during the last decade. The DoF over the space-time domain by developing a transmission
represents the scaling of capacity with respect to the signal- along multiple phases, and exploiting linear beamforming and
to-noise ratio (SNR) at the high SNR regime, and allows to user scheduling. During the first phase, the base-station (BS)
elucidate the impact of different antenna settings or feedback sends linear combinations (LCs) of all symbols of each user
assumptions on channel capacity. One of the key elements in in a TDMA fashion, i.e. first the symbols of user equipment
this context is the emergence of interference alignment (IA), (UE) one (UE ), then symbols of UE , and so on. In this
1 2
a new strategy for managing the signal dimensions (time, case,eachUEobserveslessLCsthanthenumberofsymbols,
frequency, space) in pursuit of DoF maximization [1]. The thus the symbols cannot be linearly decoded yet. However,
conceptconsistsindesigningthetransmittedsignalsinsucha due to the randomness of wireless channels, different and
way that they are overlapped (or aligned) at the non-intended independent LCs of each set of symbols are observed at each
receivers.Therefore,theinterferenceliesonareduceddimen- UE. Those LCs are known at one user UE, and desired at
sional subspace, thus releasing some dimensions to allocate another. Therefore, they may be retransmitted during next
desired signals. A very extensive survey of IA applications phases in such a way that they can be removed at the non-
can be found in [2]. intended UEs, and provide additional LC of desired symbols
Usually, IA schemes require current channel state informa- to the intended UE. Interestingly, this can be interpreted as an
tion at the transmitter side (CSIT), an assumption not always interference alignment strategy acting retrospectively. For this
validinwirelessnetworks,especiallywhenchannelcoherence reason, this type of IA is denoted as Retrospective IA.
time is low. Recently, Maddah-Ali and Tse (MAT) introduced This work investigates the DoF of the 2-user 2-cell (M,N)
in [3] for the broadcast channel (BC) a new framework where MIMO IBC with delayed CSIT, see Fig. 1, where BSs and
IA concepts can be exploited even when the CSIT is perfect UEs are equipped with (M,N) antennas, respectively. This
scenario can be modeled as a 4-user (M/2,N) IC where two
This work has been supported by projects TROPIC FP7 ICT-2011-8- pairs of BSs cooperate. The first results assuming delayed
318784(EuropeanCommission),MOSAICTEC2010-19171/TCM,DISNET
CSIT were provided by [7] for N = 1 and M = 2,3,4,
TEC2013-41315-R(MinisteriodeEconomiayCompetitividad,SpanishGov-
ernmentandERDFs),and2014SGR-60(CatalanAdministration). but their scheme does not exploit the benefits of having two
cooperating transmitters per cell. Consequently, their results phase 1 phase 2 phase 3 phase 4
basicallyconsistinapplyingthetechniquesfortheIC,reported
in [8]. This work proposes a four-phase transmission for the
specifictopologyoftheIBC,takingintoaccountthattheusers round 1 ... round r ... round R
2
of each cell are served by the same BS. The achieved DoF
S slots. Only symbols of users
outperform any previous work for ρ= MN >ρA ≈2.6413. 2 in A ( 2 ,r ) are transmitted
A. Notation
Fig.2. Generalstructureofthetransmissionframe.Therearefourphases.
Boldface and lowercase types denote column vectors (x), PhasepisdividedinRp rounds,inturndividedinSp timeslots.
while row vectors are underlined (x). Boldface and uppercase
signalsintendedtoUE ,designedsubjecttoamaximumtrans-
types are used for matrices (X). Sets and subspaces are i
mission power constraint, and with V(p,r) =0, i / (p,r).
denoted by uppercase calligraphic types ( ). i ∀ ∈A
X Note that since the focus of this paper is on DoF analysis, all
We define 0 and I as the all-zero and identity matrices,
noise terms are omitted.
respectively,withsuitabledimensionsaccordingtothecontext.
The channel gains for each slot and each link between
For vectors and matrices, ()T and ()H are the transpose and
· · transmitter and receiver are described by an N M matrix.
transposeandconjugateoperators,respectively.Moreover,the Then,thechannelmatrixH(p,r) CNSp×MSp in×(2)isformed
following two predefined operations are defined: j,i ∈
as the block diagonal composition of S of such matrices,
p
X X 0 thus contains the channel gains from antennas of BSi to UEj
stack(X,Y)= , bdiag(X,Y)= . during all time slots of the (p,r)th round. A flat block fading
(cid:34)Y(cid:35) 0 Y channelmodelisassumed,i.e.channelsarei.i.d.as (0,1),
CN
  and completely uncorrelated in time and space.
span(X) is usually used to define the column subspace, con-
After the whole communication, UE collects all the re-
taining all possible linear combinations (LCs) of the columns j
ceivedsignals.Sincelineartransmittersandreceiversareused,
of x. However, in this work we use the row subspace, defined
as =rspan(X)=span XT .Finally,theset contains the objective is to obtain at least b LCs of desired symbols
X X\Y free of interference at each receiver. In this regard, let write
the elements that belong to but not to .
(cid:0) X(cid:1) Y thechannelmodelin(2)inamorecompactformbygrouping
II. SYSTEMMODEL the magnitudes of the different phases and rounds, as follows:
We consider the 2-cell 2-user IBC, see Fig. 1. In this y(p) =G(p) xT,...,xT T =stack y(p,1),...,y(p,Rp) ,
scenario, two BSs equipped with M antennas transmit inde- j j 1 4 j j
pendentmessagestoeachofitstwoassociatedUEs,equipped G(cid:2)(p) = H(p)V(cid:3) (p), ..., H(cid:16)(p)V(p) , (cid:17)
withN antennaseach.UEsarelabeledfrom1to4,thusBS j j,1 1 j,2 4
c(i)
(cid:104) (cid:105)
sineravefsraUmEeic,ownistihsticn(gi)o=f P(cid:100)i/=2(cid:101)4. Cphoamsemsuonficdautiroatnioins Rcarrrioedunodust, H(jp,i) =bdiag H(jp,i,1), H(jp,i,2), ..., H(jp,i,Rp) ,
p
in turn divided in Sp time slots each, see Fig. 2. The total Vi(p) =stack(cid:16)Vi(p,1), Vi(p,2), ..., Vi(p,Rp)(cid:17),
number of slots used for data transmission is denoted by
(cid:16) (cid:17)
P Hj,i =bdiag H(j1,i), H(j2,i), H(j3,i), H(j4,i) ,
4
τ = τp, τp =RpSp, Rp = p . (1) Vi =stack (cid:16)Vi(1), Vi(2), Vi(3), Vi(4) (cid:17),
p=1 (cid:18) (cid:19)
(cid:88) (cid:16) (cid:17)
The (p,r)th round, i.e. round r of phase p, is dedicated to where H(jp,i)∈CNτp×Mτp, Vi(p)∈CMτp×b, Hj,i∈CNτ×Mτ,
a specific group of p users (served UEs), denoted by (p,r). and V CMτ×b are equivalent magnitudes for the global
Theusersnotin (p,r) aredenotedasthelisteninguserAs.The channeil∈andprecodingmatrices,andmatrixG(p) isthesignal
A j
particular users in each (p,r) may be found in the second space matrix for phase p [9].
A
row of Table I. Note that Rp corresponds to the number of We assume a delayed CSIT model, i.e. transmitters have
possible p-tuples that can be formed, and for simplicity, they onlyaccesstothepreviousphasesCSI.Giventheformulation,
are ordered by selecting first the users with the lowest index. this means that the following channels
Theoutputatthejthreceiverduringtheslotsofthe(p,r)th
H((cid:37)) p−1, i,j,
round is described by: { j,i}(cid:37)=1 ∀
4 are available at the beginning of the phase p at both BSs.
yj(p,r) =s(jp,r)+i(jp,r)=H(jp,c,(rj))Vj(p,r)xj + Hj(p,c,(ri))Vi(p,r)xi Finally, desired signals are separated from interference by
(cid:88)ii=(cid:54)=1j (2) applying a linear zero-forcing filter. Then, using standard
derivations [10] and assuming full-rank and generic channels,
where yj(p,r)∈ CNSp×1 is the received signal at UEj during the achieved normalized DoF are given by:
the(p,r)thround,x Cb×1 containsthebuncorrelatedunit-
powered complex-vailu∈ed data symbols intended to the ith re- d(in) = 1 b. (3)
j N τ
ceiver, V(p,r) CMSp×b is the precoding matrix carrying the
i ∈
TABLEI
SERVEDUES(A(p,r))ANDAVAILABLEOVERHEARDINTERFERENCE(OHI)ATUE1FOREACHROUND.
Phase1 Phase2 Phase3 Phase4
{1} {2} {3} {4} {}1,2 {}1,3 {}1,4 {}2,3 {}2,4 {}3,4 {}1,2,3 {}1,2,4 {}1,3,4 {}2,3,4 {}1,2,3,4
T1,2 T1,3 T1,4 (cid:104)u(31,2),u(21,3)(cid:105) (cid:104)u(41,2),u(21,4)(cid:105) (cid:104)u(41,3),u(31,4)(cid:105) u(31,2),u(21,3) u(41,2),u(21,4) (cid:104)w1,2,w1,3,w1,4(cid:105)
cellinterference.ThistransformstheIBCintotwoindependent
12
25 2-user BCs, where the MAT scheme [3] for 2 users can be
applied. On the other hand, [7] showed that the DoF values
3 and 2 can be achieved for ρ=3,41, respectively. The DoF
3 8 5
7 fortherestofsettingsarederivedfromthefactthatincreasing
2 the number of transmit antennas cannot reduce the DoF.
5
We also depict the minimum between two trivial upper
3
8 boundsasbenchmark.First,assumingtransmittercooperation,
d(in) weconverttheIBCintoa4-userMIMOBCwith2M antennas
j
1 atthetransmitter,whoseDoFwerederivedin[3].Thisbound
3 is shown for ρ > 4. The second outer bound corresponds to
3
Outerbound[3][11] assuming current CSIT [11], and applies for ρ< 4.
Proposedinnerbound 3
Previousinnerbound[3][7] IV. PROPOSEDSCHEME
WithoutCSIT
AchievabilityofinnerboundsinTheorem1isbasedonthe
1
4 1 1 4 2 ρA 3ρB ρC 4 transmission protocol described in this section. It is shown
2 3 the precoding matrix design for the different phases, and that
ρ=M/N
the required LCs of desired symbols are provided to each
Fig. 3. Normalized DoF per user inner and outer bounds for the 2-cell
2-user MIMO IBC with delayed CSIT for ρ > 1. The optimal DoF for UE for correct decodability. Using this strategy forces some
2
ρ< 1 areattainedwithouttheneedofCSIT,see[11].Theproposedscheme constraints over the different system parameters, deferred to
2
performanceisdepictedonlyfortheregionwherepreviousinnerboundsare Appendix A, or the full paper [12]. Therefore, the optimal
outperformed,i.e.ρ>ρA. parameters are obtained for each antenna setting by means of
the following DoF maximization problem:
III. MAINRESULT
1 b
The next theorem summarizes the proposed DoF inner maximize (5)
bound as a function of the antenna ratio ρ= M: {b,S1,S2,S3}∈Z+ N 4S1+6S2+4S3+S4
N
s.t. max MS ,4NS b
Theorem 1: For the 2-cell 2-user MIMO IBC with delayed 1 1 ≥
CSIT and antenna ratio ρ, MS2(cid:0) NS1 (cid:1)
≥
MS 2NS
ρ3 ρ ρ<ρ 3 ≥ 2
ρ3+3ρ2+3ρ+8 A ≤ B N S1+3S2+6max S2,S3 b
 3ρ2 ρ ρ<ρ ≥
d(jin) = 5ρ126+ρ9+ρ7ρ2+03 ρBC≤≤ρ<4C (4) wtiohne,rewSe4ad=drmesisn(cid:0)th6eS2sp,e(cid:0)Nbci−ficSc1a−se3(SM2(cid:1),.ND)u=(cid:0)e t(o4,s1p)a(cid:1),c(cid:1)ewhliemreita37-
DoFperusercanbeachievedbydeliveringb=12symbolsto
3 ρ 4
normalizedDoFperusercan7beachieved,w≥hereρA 2.6413, eaancdhSu4se=ra6l.ongτ =28timeslots,withS1 =3,S2 =S3 =1,
≈
ρB 3.2196, ρC 3.5414. A. Phase 1
≈ ≈
The first phase is divided in four rounds, each of them
Proof: See Section IV.
dedicated to transmit LCs of symbols of one user during 3
time slots. The precoding matrices used during this round
Fig.3summarizescurrentandpreviousDoFinnerandouter
are predefined before the communication. Consequently, after
bounds using delayed CSIT, as well as the optimal DoF when
the first phase we have 3 LC of desired symbols, thus 9
there is no CSIT for comparison purposes. Previous inner
additionalLCsarerequired.Interestingly,thoseremainingLCs
bounds are outperformed for ρ>ρ .
A
For ρ < 3, previous inner bound curves correspond to the
1The DoF value for ρ = 4 can also be achieved by applying the ideas
application of time division multiplexing to manage the inter- reportedin[8]forthe4-userMISOIC.
are distributed along the listening receivers, who observe 3 C. Phase 3
LCsofeachnon-desiredsetofsymbols.TheseLCswillbethe
Eachofthefoursingle-slotroundsofthisphaseisdedicated
basis to align the interference during the next phases, while
toatripletofusers.Precodingmatricesaredesignedbasedon
providing new LCs of desired symbols. They represent the
theOHIobservedduringthepreviousphasewhentheUEsact
overheard interference (OHI) for the first phase, denoted by
as listening users. In contrast, now the objective is twofold:
T =H(1,i) V(1,i) C3×12, (6) 1) totransmitLCsofdesiredsignals,asinthepreviousphases,
j,i j,c(i) i ∈ but also 2) to decouple the OHI from the second phase.
On the one hand, transmitted signals are designed with the
i.e. the signals intended to UE and observed at UE , see
i j
objectiveofbeingalignedwiththepreviousreceivedOHI,thus
TableIfortheOHIcollectedatUE .Moreover,noticethatall
1
providingadditionalfreeofinterferenceLCofdesiredsignals.
these 12 distributed LCs can be assumed linearly independent
However, the generated interference will be aligned only at
with probability one due to the channel randomness. Finally,
one of the receivers due to the effect of the two distributed
forbetterreadibilityweshowthesignalspacematrixforUE :
1
transmitters, and the rest of receivers will obtain new LCs of
G(1) =bdiag(T ,T ,T ,T ) (7) interference, as explained next in this section. Then, at those
1 1,1 1,2 1,3 1,4
receiverswherenotalltheinterferenceisaligned,thereceived
B. Phase 2 signalswillbeusefultodecoupletheiravailableOHIasshown
for UE in Table I.
1
Users are served by pairs during this phase, with six
single-slot rounds. The first phase OHI is exploited to deliver TABLEII
aLCofdesiredsymbolsalignedatthenon-intendedUE.Con- OHITRANSMITTEDDURINGEACHTHIRDPHASEROUND
sidertheroundrdedicatedtoUE andUE ,i.e. (2,r)= i,j .
i j
A { } r A(3,r) A(3,r)(1) A(3,r)(2) A(3,r)(3)
Then, transmitted signals are designed such that
1 {1,2,3} {u(3),u(2)} {u(3),u(1)} {u(2),u(1)}
2,1 3,1 1,2 3,2 1,3 2,3
rspan H(j2,c,(ri))Vi(2,r) ⊆Tj,i, rspan H(i,2c,(rj))Vj(2,r) ⊆Ti,j, 2 {1,2,4} {u(24,1),u(42,1)} {u(14,2),u(41,2)} {u(12,4),u(21,4)}
(cid:16) (cid:17) (cid:16) (cid:17) 3 {1,3,4} {u(4),u(3)} {u(4),u(1)} {u(3),u(1)}
i.e. the signals obtained at each receiver are aligned with 3,1 4,1 1,3 4,3 1,4 3,4
the previously received OHI. One simple solution using only 4 {2,3,4} {u(34,2),u(43,2)} {u(23,1),u(32,1)} {u(23,1),u(32,1)}
delayed CSIT is given by
In order to illustrate these objectives, let us consider the
Vi(2,r) =Θ(i2,r)Tj,i, Vj(2,r) =Θ(j2,r)Ti,j, (8) precoding matrix design for the third round2, dedicated to
users in (3,3) = 1,3,4 :
where Θ(2,r),Θ(2,r) C4×3 are some random full rank A { }
i j ∈
matrices ensuring the transmit power constraint. V(3,3) =θ(3,3)u(4)+θ(3,3)u(3),
1 1 3,1 2 4,1
The OHI obtained during this phase is generally written as
V(3,3) =θ(3,3)u(4)+θ(3,3)u(1), (11)
3 1 1,3 3 4,3
u(jk,i) =h(k2,c,r()i)Vi(2,r)∈C1×12, (9) V4(3,3) =θ2(3,3)u(13,4)+θ3(3,3)u(31,4),
with rspan u(jk,i) ⊂ Tj,i, and r such that A(2,r) = {i,j}. where θ(i3,r)∈C4×1is some vector with random entries, thus
In other wo(cid:16)rds, u(cid:17)(jk,i) is the LC of Tj,i observed at UEk. For the interference generated at UE1 is:
instance, u(1) =h(2,6)Θ(2,6)T is the LC of T observed
4,3 1,2 3 4,3 4,3 i(3,3)= h(3,3) V(3,3)x +V(3,3)x = (12)
at UE . It is worth pointing out that some of those terms are 1 1,2 3 3 4 4
1
available coupled, as shown for UE in Table I. (cid:16) (cid:17)
1 x
Foreaseofexposition,thesignalspacematrixforthisphase = h(3,3) θ(3,3)u(4)x +θ(3,3)a 3 +θ(3,3)u(3)x
1,2 1 1,3 3 3 2 1,4 4
at UE is next shown: (cid:34)x (cid:35)
1 (cid:16) 4 (cid:17)
with a = u(1),u(1) . By exploiting the phase 2 OHI, see
u(1) u(1) 0 0 4,3 3,4
2,1 1,2 TableI,the(cid:104)termsprop(cid:105)ortionaltoacanberemoved,whilethe
u(1) 0 u(1) 0  rest of terms are aligned with the first phase OHI. Then, all
G(12) = 3,1 . 1.,3 . (10) theinterferencetermsin(12)areremoved,andoneadditional
 .. ..  LC of desired symbols is retrieved. Unfortunately, the trick in
 
 0 0 u(41,3) u(31,4) (s1a2m)ectarnanbsemditotenre, io.ne.lyalilftahlel tihneteirnfeterrefnecreentceermcsomarees ofrbosmervthede
 
Finally, from the received signals represented by (7) and (10),
2Due to space limitation, we intentionally focus on this third round in
each user is able to retrieve 3 new LCs of desired symbols by
ordertomakeuseof(10).Thedesignfortherestofroundsiseasilyderived
combining the current received signals and previous OHI. accordingtoTableII.
TABLEIII
throughthesamechannel,asinaBC.Toseethis,considerthe
OHITRANSMITTEDDURINGTHELASTPHASE
interference generated at this UE during the (3,1)th round:
i(3,1)=h(3,1) θ(3,1)u(3)+θ(3,1)u(1) x + (13) i wIi(3),i wIi(2),i wIi(1),i
1 1,1 1 1,2 3 3,2 2 1 {u(3),u(2)} {u(4),u(2)} {u(4),u(3)}
+h(13,2,1)(cid:16)θ(23,1)u(12,3)+θ(33,1)u(21,3)(cid:17)x3 2 {u2(13,,12),u3(31,,12)} {u2(14,,12),u4(41,,12)} {u3(34,,12),u4(43,,12)}
Remark1:Itisimpo(cid:16)rtanttoseethatsinceu((cid:17)1) andu(1) are 3 {u(12,3),u(21,3)} {u(14,3),u(41,3)} {u(24,3),u(42,3)}
known coupled (see Table I)), they cannot be3s,2imultane2,o3usly 4 {u(12,4),u(21,4)} {u(13,4),u(31,4)} {u(23,4),u(32,4)}
canceledin(13).However,thisroundallowsUE todecouple
1
them, as expressed in Table I, to be exploited during the last
5-dimensional orthogonal subspace where the signals can be
phase.
projected. Then, all the interference is removed, and 5 LCs of
Finally, the OHI for this phase is defined as follows:
desired signals can be retrieved from this phase.
w =h(3,r)V(3,r) C1×12, (14) Overall, the total number of free of interference LCs of
j,i j,c(i) i ∈ desired symbols at the end of the communication is equal to
with r defined such that i / (3,r). In other words, 3+3+1+5=12=b,thusd(in) = 12 = 3 DoFareachieved.
∈ A j 28 7
w is the LC of symbols of UE observed at UE
j,i i j V. CONCLUSION
during the unique third round where UE is a listen-
j
ing user. Note that rspan w rspan stack u(k),u(l) , A four-phase protocol based on linear precoding and
where all indices take dijff,iere⊂nt values. Forl,iexamk,iple, decoding is proposed for the 2-cell 2-user MIMO IBC
w =h(3,1)V(3,1) =h(3(cid:0),1) θ(cid:1)(3,1)u(3)+(cid:0) θ(3,1(cid:0))u(1) , a(cid:1)n(cid:1)d with delayed CSIT. Assuming transmitters and receivers are
4,2 4,1 1 4,1 1 1,2 3 3,2 equipped with M and N antennas, respectively, the proposed
its span lies on rspan stack u(cid:16)(13,2),u(31,2) . (cid:17) scheme improves best-known DoF inner bounds for the case
D. Phase 4 (cid:16) (cid:0) (cid:1)(cid:17) ρ= MN >2.641. Further work is oriented to study other an-
tenna settings, and scenarios with more cells or users per cell.
The last phase consists of a single round of 6 slots, during Moreover, since the optimal parameters for our scheme are
which all users are simultaneously served. In this case the derived by means of a maximization problem, some practical
precoding matrices transmit a rank-3 LC of symbols, thus constraints may be introduced, e.g. maximum number of time
there is some redundancy, that will be used not only to slots, which is also an interesting line of future research.
cancel the transmitted interference, but also to retrieve the
desired LCs not properly decoded during the third phase. The REFERENCES
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k=1
(cid:88) Theory,Jul.2011.
where θ(4) C4×1, i = 1,...,4 i , i(k) is the kth [3] M. Maddah-Ali and D. Tse, “Completely Stale Transmitter Channel
element iof∈i. TableIIII sho{ws the d}if\f{ere}ntILCs transmitted StateInformationisStillVeryUseful,”IEEETrans.Inf.Theory,vol.58,
I pp.4418–4431,Jul.2012.
during this phase. We next show how all the interference is
[4] C.VazeandM.Varanasi,“TheDegreesofFreedomRegionandInter-
removed at UE1, detailed in rows 2, 3, 4. First, exploiting all ference Alignment for the MIMO Interference Channel With Delayed
the available OHI (Table I) most of the terms are removed. CSIT,”IEEETrans.Inf.Theory,vol.58,pp.4396–4417,Jul.2012.
[5] D. T. H. Kao and A. Avestimehr, “Linear Degrees of Freedom
A concise description of this procedure may be found in
of the MIMO X-Channel with Delayed CSIT,” ArXiv e-prints, vol.
Appendix B. After this, UE1 obtains arXiv:1405.1091v1[cs.IT],May2014.
[6] M.Abdoli,A.Ghasemi,andA.Khandani,“OntheDegreesofFreedom
of K-User SISO Interference and X Channels with Delayed CSIT,”
x
H¯(4)V¯(4)x +H(4)θ(4)w x +H(4)θ(4) w ,w 3 IEEETrans.Inf.Theory,vol.59,pp.6542–6561,Jun.2013.
1,1 1 1 1,1 1 1,2 2 1,2 1 1,3 1,4 [7] J.ParajuliandG.Abreu,“Aspace-timeTxschemefortwo-cellMISO-
(cid:34)x (cid:35)
4 BCwithdelayedCSIT,”IEEEISIT,pp.1912–1916,Jun.2014.
rank1 (cid:2) (cid:3)
[8] M.Torrellas,A.Agustin,andJ.Vidal,“OntheDegreesoffreedomof
(16)
(cid:124) (cid:123)(cid:122) (cid:125) theK-userMISOInterferenceChannelwithimperfectdelayedCSIT,”
where H¯(4)V¯(4) C6×12 is a full rank matrix containing the IEEEICASSP,pp.1155–1159,May2014.
1,1 1 ∈ [9] M. Torrellas, A. Agustin, J. Vidal, and O. Munoz-Medina, “The DoF
desired signals received during phase 4, and rounds 1 and ofthe3-User(p,p+1)MIMOInterferenceChannel,”IEEETrans.on
2 of the third phase, i.e. the 5 LCs of desired signals not Commun.,vol.62,pp.3842–3853,Nov.2014.
resolved up to this point due to interference. Now, using the [10] T.CoverandJ.Thomas,Elementsofinformationtheory. Wiley,2006.
[11] G. Sridharan and W. Yu, “Degrees of freedom of MIMO cellular
OHI from the last third phase round ( w1,2,w1,3,w1,4 ) the networks with two cells and two users per cell,” IEEE ISIT, pp.
receiverisabletoremovetheremainin(cid:104)ginter-cellinterfe(cid:105)rence 1774–1778,Jul.2013.
[12] M. Torrellas, A. Agustin, and J. Vidal, “Retrospective Interference
terms. However, there remain some non-aligned intra-cell AlignmentfortheMIMOInterferenceBroadcastChannel,”Submittedto
interference terms. Fortunately, H(4)θ(4)w C6×12 has a IEEEISIT,Arxive-prints,vol.arXiv:1501.04204v1[cs.IT],Jan.2015.
1,1 1 1,2∈
APPENDIXA Taking into account that no more than these two constraints
DERIVATIONOFTHEDOFMAXIMIZATIONPROBLEM involve S4, we pick its minimum feasible, given by:
The proposed strategy forces some constraints over the dif- b
S =min 6S , S 3S
ferent system parameters, as shown in the DoF maximization 4 2 1 2
N − −
problem in (5). Next, each of those constraints is derived for (cid:18) (cid:19)
APPENDIXB
a general antenna setting.
UE1 DETAILEDPERSPECTIVEOF
1) Transmit rank in phase 1: During the first phase, MS
1 THEPRECODINGSCHEME
linear combinations of the b symbols are transmitted using M
For a better understanding of the precoding scheme, this
antennas, and during S slots. Then, for linear decodability
1
appendixshowshowthedifferentsignalsareobservedatUE ,
of the desired symbols, no more symbols than the number of 1
the different terms of overheard interference in Table I are
transmit dimensions can be sent, thus we force MS b.
1
≥ obtained, and finally the interference is canceled. To this end,
2) Enough linear combinations in phase 1: After the first we write in the last page the signal space matrix for UE
1
phase, NS1 linear combinations of each user’s symbols are during the first three phases, and the received signals during
distributedalongeachUE.Notethatnomorefreshlinearcom- the last phase, together with Table I again, for the sake of
binationsofdesiredsymbolswillbeintroducedtothesystem, reader’sconvenience.Dashedlinesseparateblockrowsofeach
sincetherestofphasesconsistsontheretransmissionofthose phase.
linear combinations. Therefore, we force that 4NS1 b, i.e. FromeachUEperspective,theroundsofthefirsttwophases
≥
if each user have access to all 4NS1 linear combinations of can be classified in two classes: serving or listening round.
its symbols, it should be able to linearly decode them. Duringtheservingrounds,extralinearcombinationsofdesired
3)Transmitrankinphase2:SinceNS linearcombinations signals are obtained, while the interference (if any) is aligned,
1
of overheard interference are retransmitted in the S slots thus can be removed. During the listening rounds, only in-
2
of each phase 2 round, we force MS NS , in order to terference is observed, to be used as overheard interference
2 1
≥
ensure that the rank of transmitted overheard interference is during the next phases. Note that after the second phase some
preserved. of the overheard interference is coupled. For example, during
the fourth round at UE , the overheard interference obtained
4) Enough linear combinations in phase 2: After the sec- 1
is given by
ond phase, S +3S linear combinations of desired signals
1 2
are obtained at each user, and 6S2 of desired symbols are y1(3,2) =i(13,2) =u(31,2)x2+u(21,3)x3. (17)
distributed along the rest of receivers. Hence, we must have
Consequently, we denote the overheard interference known
N(S +9S ) b.
1 2
≥ from this phase as u(1),u(1) . The rest of overheard inter-
5) Transmit rank in phase 3: Similarly to the first and third 3,2 2,3
constraints, we force MS 2NS . ference terms acquir(cid:104)ed at UE1(cid:105)is specified in Table I.
3 2
≥ In contrast to the first two phases, three different types of
6) Enough linear combinations for the whole communica-
roundsmaybedefinedduringthethirdphase.IncaseofUE ,
1
tion: At most, N S +3S +6max S ,S linear combi-
1 2 2 3 the third and fourth rounds are serving and listening rounds,
nationsofdesiredsymbolsaredeliveredtoeachuser.Then,we
(cid:0) (cid:0) (cid:1)(cid:1) respectively,followingthedefinitionabove.However,thefirst
musthaveN S +3S +max S ,S b.Thisconstraint
1 2 2 3 ≥ and second rounds correspond to an intermediate situation,
generalizes the fourth constraint, which is then omitted.
(cid:0) (cid:0) (cid:1)(cid:1) denoted as mixed rounds. In those cases, desired signals are
The problem in (5) results from collecting all these con- received but combined with interference terms that cannot be
straints, and taking as objective function the number of sym- removed. This is because they are transmitted from different
bols b divided by the signal dimensions Nτ. BSs, i.e. observed through different channels, and because
Next, we show how the value of S can be obtained in the overheard interference is known coupled. For example,
4
closed form. First, we have that 6S linear combinations of consider the first round, where the received signal contains
2
overheard interference are transmitted per UE during the last the term
phase, thus we force:
h(3,1)θ(3,1)u(1)x +h(3,1)θ(3,1)u(1)x .
1,1 3 3,2 2 1,2 3 2,3 3
N 6S N S .
2 4
· ≤ · UE only knows (17), i.e. it does not know each of the terms
1
Moreover, after the third phase N(S +3S +S ) linear
1 2 3 individually, thus it will not be able to remove them together.
combinations of desired symbols are provided to each UE,
Then, what the receiver can do is to at least remove one of
while the last phase delivers N(S S ) additional linear
4 3 them in order to decouple these two overheard interference
−
combinationstoeachofthem.Thislastexpressioncanbeeas-
terms. For this reason, Table I shows that after this round
ily derived by taking into account the zero-forcing operation. the available overheard interference is u(1),u(1) instead of
Then, we must have { 3,2 2,3}
u(1),u(1) , i.e. the overheard interference is decoupled.
3,2 2,3
N(S +3S +S +(S S )) b
1 2 3 4− 3 ≥ (cid:104) And fina(cid:105)lly, the last phase consists of a single round. The
N(S +3S +S ) b receivedsignalsatUE duringits6slotsareexplicitlywritten
1 2 4 1
≥
in(18)inthelastpage.ForsomematrixA(differentforeach
case),thesubspacesoccupiedbyinterferenceduringthisphase
can be classified in three classes:
• Au(k), k = 1,i = 1: All those terms can be removed
1,i ∀ (cid:54) (cid:54)
using the first phase overheard interference, since by
definition rspan u(k) .
j,i ⊂Tj,i
• Au(1), j = 1,i(cid:16): All(cid:17)those terms can be removed using
j,i ∀ (cid:54)
the second phase overheard interference, which are now
decoupled thanks to the mixed rounds of the third phase.
• Other terms that cannot be individually removed.
Aftersuppressingthetwofirstclassesofinterferenceterms,
the remaining interference may be written as
x
H(4)Θ(4)w x +H(4)Θ(4) w ,w 3
1,1 4 1,2 2 1,2 4 1,3 1,4
(cid:34)x (cid:35)
4
(cid:2) (cid:3)
where the last term can be removed using the overheard
interference obtained from the last round of the third phase.
However,thereremainssomeintra-cellinterference,asshown
in (16). Those remaining terms, in this case containing
symbols intended to UE are removed through zero-forcing
2
concepts, as explained in Section IV-D, after equation (16).
H(1,1)V(1,1) 0 0 0
1,1 1
0 H(1,2)V(1,2) 0 0
1,1 2
T
1,2
0 0 H(1,3)V(1,3) 0
1,2 3
T
1,3
0 0 0 H(1,4)V(1,4)
1,2 4
H(2,1)θ(2,1)T H(2,1)θ(2,1)T 0 0
1,1 1 2,1 1,1 2 1,2
H(2,2)θ(2,2)T 0 H(2,2)θ(2,2)T 0
1,1 1 3,1 1,2 3 1,3
H(2,3)θ(2,3)T 0 0 H(2,2)θ(2,3)T
1,1 1 4,1 1,2 4 1,4
0 H(2,4)θ(2,4)T H(2,4)θ(2,4)T 0
1,1 2 3,2 1,2 3 2,3
G(11) 0 H(12,1,5)θ(22,5)T4,2 0 H(12,2,5)θ(42,5)T2,4
G(12)= 0 0 H(12,2,6)θ(32,6)T4,3 u(1) H1(2,2,6)θ4(2,6)T3,4 u(1)
G(3) 4,3 3,4
1
  u(3) u(3) u(2)
H(3,1) θ(3,1),θ(3,1) 2,1 H(3,1) θ(3,1),θ(3,1) 1,2 H(3,1) θ(3,1),θ(3,1) 1,3 0
1,1 1 2 u(2) 1,1 1 3 u(1) 1,2 2 3 u(1)
(cid:31) (cid:30) 3,1 (cid:31) (cid:30) 3,2 (cid:31) (cid:30) 2,3
     
u(4) u(4) u(2)
H(3,2) θ(3,2),θ(3,2) 2,1 H(3,2) θ(3,2),θ(3,2) 1,2 0 H(3,2) θ(3,2),θ(3,2) 1,4
1,1 1 2 u(2) 1,1 1 3 u(1) 1,2 2 3 u(1)
(cid:31) (cid:30) 4,1 (cid:31) (cid:30) 4,2 (cid:31) (cid:30) 2,4
     
u(4) u(4) u(3)
H(3,3) θ(3,3),θ(3,3) 3,1 0 H(3,3) θ(3,3),θ(3,3) 1,3 H(3,3) θ(3,3),θ(3,3) 1,4
1,1 1 2 u(3) 1,2 1 3 u(1) 1,2 2 3 u(1)
(cid:31) (cid:30) 4,1 (cid:31) (cid:30) 4,3 (cid:31) (cid:30) 3,4
     
u(4) u(4) u(3)
0 H(3,4) θ(3,4),θ(3,4) 3,2 H(3,4) θ(3,4),θ(3,4) 2,3 H(3,4) θ(3,4),θ(3,4) 2,4
1,1 1 2 u(3) 1,2 1 3 u(2) 1,2 2 3 u(2)
w (cid:31) (cid:30) 4,2 (cid:31) (cid:30) 4,3 (cid:31) (cid:30) 3,4
1,2      
Phase1 Phase2 Phase3 Phase4
{1} {2} {3} {4} {}1,2 {}1,3 {}1,4 {}2,3 {}2,4 {}3,4 {}1,2,3 {}1,2,4 {}1,3,4 {}2,3,4 {}1,2,3,4
T1,2 T1,3 T1,4 (cid:104)u(31,2),u(21,3)(cid:105) (cid:104)u(41,2),u(21,4)(cid:105) (cid:104)u(41,3),u(31,4)(cid:105) u(31,2),u(21,3) u(41,2),u(21,4) (cid:104)w1,2,w1,3,w1,4(cid:105)
y(4) = H(4) Θ(4)w +Θ(4)w +Θ(4)w x + Θ(4)w +Θ(4)w +Θ(4)w x + (18)
1 1,1 1 4,1 2 3,1 3 2,1 1 1 4,2 2 3,2 4 1,2 2
H(4)(cid:16)(cid:0)Θ(4)w +Θ(4)w +Θ(4)w (cid:1)x +(cid:0)Θ(4)w +Θ(4)w +Θ(4)w (cid:1)x (cid:17)
1,2 1 4,3 2 3,3 4 1,3 3 2 3,4 3 2,4 4 1,4 4
(cid:16)(cid:0) (cid:1) (cid:0) (cid:1) (cid:17)
u(3) u(4) u(4)
= H(4) Θ(4)H(3,1) θ(3,1),θ(3,1) 2,1 +Θ(4)H(3,2) θ(3,2),θ(3,2) 2,1 +Θ(4)H(3,3) θ(3,3),θ(3,3) 3,1 x +
1,1 1 4,1 1 2 u(2) 2 3,1 1 2 u(2) 3 2,1 1 2 u(3) 1
(cid:16) (cid:104) (cid:105) 3,1 (cid:104) (cid:105) 4,1 (cid:104) (cid:105) 4,1 (cid:17)
      
u(3) u(4) u(4)
Θ(4)H(3,1) θ(3,1),θ(3,1) 1,2 +Θ(4)H(3,2) θ(3,2),θ(3,2) 1,2 +Θ(4)H(3,4) θ(3,4),θ(3,4) 3,2 x +
1 4,1 1 3 u(1) 2 3,1 1 3 u(1) 4 1,1 1 2 u(3) 2
(cid:16) (cid:104) (cid:105) 3,2 (cid:104) (cid:105) 4,2 (cid:104) (cid:105) 4,2 (cid:17)
      
u(2) u(4) u(4)
H(4) Θ(4)H(3,1) θ(3,1),θ(3,1) 1,3 +Θ(4)H(3,3) θ(3,3),θ(3,3) 1,3 +Θ(4)H(3,4) θ(3,4),θ(3,4) 2,3 x +
1,2 1 4,2 2 3 u(1) 3 2,2 1 3 u(1) 4 1,1 1 3 u(2) 3
(cid:16) (cid:104) (cid:105) 2,3 (cid:104) (cid:105) 4,3 (cid:104) (cid:105) 4,3 (cid:17)
      
u(2) u(3) u(3)
Θ(4)H(3,2) θ(3,2),θ(3,2) 1,4 +Θ(4)H(3,3) θ(3,3),θ(3,3) 1,4 +Θ(4)H(3,4) θ(3,4),θ(3,4) 2,4 x
2 3,2 2 3 u(1) 3 2,2 2 3 u(1) 4 1,1 2 3 u(2) 4
(cid:16) (cid:104) (cid:105) 2,4 (cid:104) (cid:105) 3,4 (cid:104) (cid:105) 3,4 (cid:17)
      