On the Degrees-of-Freedom of the MIMO Three-Way Channel with Intermittent Connectivity

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by  Anas Chaaban

On the Degrees-of-Freedom of the MIMO
Three-Way Channel with Intermittent Connectivity
Anas Chaaban, Aydin Sezgin, and Mohamed-Slim Alouini
Abstract—The degrees-of-freedom (DoF) of the multi-antenna Node 3
three-way channel (3WC) with an intermittent node is studied. H31 Intermittent
Special attention is given to the impact of adaptation. A non- H13
17 adaptive transmission scheme based on interference alignment, H32H23 H21 Node 1
0 czoerrore-fsoprocnindgin,ganDdoFeraresugiroen-chisadnenreilvetrde.aTtmheenn,tiitsipsrsohpoowsendt,haantdthitiss Node 2 H12 sℓ
2
n sacdhdeimtioenatcohitehveesDtohFe sruegmio-DnooFf othfethneoninintteerrmmitittteennttcohnaen.nTelh,uisn, Fmiagt.ri1x:AandMsIM∈O{30W,1C}wisitthheaninintetremrmitittetnencyt-nsotadtee:aHt tiijmiestihnestacnhtanℓn.el
a adaptation is not necessary from those perspectives. To the ℓ
J contrary,itisshownthatadaptationisnecessaryforachievingthe
DoF region of the intermittent case. This is shown by deriving
7
an outer bound for the intermittent channel with nonadaptive availability of the intermittent node, which can be obtained
2
encoding, and giving a counterexample of an adaptive scheme by estimating its activity from the received signals. For this
] which achieves DoF tuples outside this bound. This highlights model,westudythedegrees-of-freedom(DoF),i.e.,thecapac-
T the importance of cooperation in this intermittent network.
ity scaling versus signal-to-noise ratio (SNR) in a dB scale.
I
. Wepayparticularattentiontothenecessityorthelackthereof,
s
c I. INTRODUCTION of adaptive encoding where the transmit signal of each node
[ is allowed to depend on its previously received signals. This
Multi-way, full-duplex, and device-to-device (D2D) com-
issue has been studied for various channels earlier [15]–[17].
1 municationsareimportanttechniquesthatareexpectedtogain
v more prominence in future communication systems. Multi- First,wedeviseanonadaptiveschemebasedoninterference
5 alignment and zero-forcing, where intermittency is treated as
waycommunicationreferstocommunicationbetweenmultiple
0 an erasure channel, and we derive its achievable DoF region.
nodes each acting as a source, a destination, and possibly a
0
Then,wederiveDoFupperboundsthatprovethatthisscheme
8 relay. Full-duplex operation is defined as when these three
achievesthesum-DoFof the channel.Itfollowsthatasfaras
0 functionalities take place over the same time/frequency re-
the sum-DoF is concerned, adaptation is not necessary. This
1. sources, and D2D communication refers to direct commu-
scheme also achieves the DoF region of the channel without
0 nication between users without or with limited base-station
intermittency, and hence, for the nonintermittent channel,
7 intervention.Thosetechniquesattractedandcontinuetoattract
1 adaptation is not necessary for achieving the DoF region.
research attention [1]–[5].
: Aftershowingtheunnecessityofadaptationinthosetwocases,
v Consider a setup where three nodes (D2D users e.g.) com-
we prove that adaptation is necessary to achieve the DoF
i municate with each other in a multi-way fashion. This setup
X region of the intermittent channel. To show this, we derive a
canbemodeledasathree-waychannel(3WC),anextensionof
r DoF outer bound that holds under nonadaptive encoding, and
a Shannon’stwo-waychannel[6]whichhasbeenstudiedin[7]–
provide an adaptive scheme that achieves rates that violate
[9].Therein,itisassumedthatthenodesarealwaysconnected.
thisouterbound.ThisprovesthatadaptationenlargestheDoF
This assumption is not always valid in practice since a node
region in the intermittent 3WC.
might have intermittent connectivity, e.g. due to shadowing,
Throughout the paper, we use xn for some i to denote
or if a D2D nodecauses strong interferenceto a cellular user, i
(x ,...,x ). The N ×N identity matrix is denoted I .
in which case it is not permitted to use its band [10]. i,1 i,n N
We write X ∼ CN(0,Q) to indicate that X is a complex
The impact of intermittency on the performance of various
Gaussian random variable with zero mean and covariance
networks was studied in [11]–[14] for instance. In this paper,
matrixQ.We write x+ to denotemax{0,x}forsomex∈R,
we study the impact of intermittency on the multiple-input
kxk todenotetheℓ -normofx,andH†,HH,andspan(H)
multiple-output (MIMO) 3WC. We consider a full-duplex i i
todenotethepseudo-inverse,theHermitiantranspose,andthe
MIMO 3WC with full message-exchange, where each node
the subspace spanned by the columns of H.
has an independent message to each of the other two nodes.
The permanent nodes have only causal knowledge of the
II. SYSTEMMODEL
A.ChaabanandM.-S.AlouiniarewiththeDivisionofComputer,Electri-
cal,andMathematical SciencesandEngineering(CEMSE)atKingAbdullah Consider a system where three MIMO full-duplex nodes
University of Science and Technology (KAUST), Thuwal, Saudi Arabia. communicatein a multi-way mannerusing the same medium,
Email:{anas.chaaban,slim.alouini}@kaust.edu.sa.
with one of the nodes being intermittently available (Fig. 1).
A. Sezgin is with the Institute of Digital Communication Systems at the
Ruhr-Universita¨t Bochum,Bochum,Germany.Email:[email protected]. For some transmission duration n ∈ N (in symbols), let sn
x1,ℓ xi,ℓ,i∈{2,3} Theorem 1: The DoF region of the 3WC satisfies D[in] ⊆
RAedsatrpictitveed E1,ℓE(1w,ℓ1(2w,1w21,3w,1s3n,,synℓ1)−1) Ei,ℓ(wEiji,,ℓw(wiki,js,ℓw−i1k,)yiℓ−1) Dsert,τof⊆dD∈aR,τ6, wsahtiesrfeyitnhgetahcehfioelvlaobwleinginfnoerrib,oju∈nd{2D,3r[i,}nτ],iri,s6=τthje:
TABLEI:Theencoding possibilities. Herew isthemessage tobe +
ij
sent from node i to node j. d +d +τd ≤τM , d +τd ≤τM , (4)
1i 1j ij 1 31 32 3
d +d +τd ≤τM , d +τd ≤τM , (5)
i1 j1 ij 1 13 23 3
denote the intermittency of node 1, where for ℓ∈{1,...,n}, di1+d1j +τdij ≤τM2. (6)
sℓ =1 means that node 1 is available, and sℓ =0 otherwise. Proof: The inclusion of D in D is obvious. The
r,τ a,τ
The state sn is a sequenceof independentand identically dis- achievability of D[in] is proved in the rest of this section.
r,τ
tributed (i.i.d) Bernoulli random variables S with probability
ℓ Note that the factor τ in the inequalities above imposes a
P(s =1)=τ and P(s =0)=1−τ ,τ¯. This sequence is
ℓ ℓ larger penalty on the streams going through the intermittent
known at node 1. However,knowledgeof s is only available
ℓ links. If we interpret τM in d +τd ≤τM as available
3 31 32 3
causally at node i ∈ {2,3}, i.e., node i does not know s at
ℓ resources,thenincreasingd by1‘eats’τ unitsofresources,
32
the beginning of the ℓ-th transmission, and can only obtain it
while increasing d by 1 ‘eats’ 1 unit of resources. Thus,
31
afterreceivingtheℓ-threceivedsignalfromwhichtheactivity
transmission between nodes 2 and 3 is ‘cheaper’ by a factor
of node 1 can be detected with certainty.
of τ, no matter how large M is as we shall see later. Next,
1
Nodei∈{1,2,3}isequippedwithMi transmitandreceive we prove the achievability of D[in].
r,τ
antennas. Its transmit signal at time index ℓ is represented by 1) Encoding: Eachnodesplitsitsmessagew intow[1]and
sxait,iℓs;fieasraeapliozwateironcoonfstraairnatn1domn veEc[tkoXr Xik,ℓ2]∈≤nCPM.i×C1leathrlayt wi[2j] to be sent using zero-forcingand interfereinjce alignijment,
x1,ℓ =0 if sℓ =0. The recePiveℓd=1signalsia,ℓre2 resSpienccteivenlyo.dEenc1odisingavpariolacbeeledsfaosr faollforwacst.ion of time, say
y1,ℓ =H21x2,ℓ+H31x3,ℓ+z1,ℓ, if sℓ =1 (1) m = ksnk0 out of the n transmissions, it encodes w1[q2]
y =H x +H x +z , (2) and w[q], q ∈ {1,2}, into codewords u[q]m and u[q]m with
2,ℓ 12 1,ℓ 32 3,ℓ 2,ℓ 13 12 13
i.i.d. CN(0,p I ) and CN(0,p I ) symbols, respec-
y3,ℓ =H13x1,ℓ+H23x2,ℓ+z3,ℓ, (3) 1 a[1q2] 1 a[1q3]
tively. Here a[q] is the vector length, and p is the power of
aCnMdky×1M,ℓi=rep0reisfensℓt t=he0c,hwanhneerel mHajtriic∈esCfMrojm×MniodaensdjHankdi ∈k each componeinjt of u[1q2]m and u[1q3]m. Then,1those codewords
to node i, respectively, and zi,ℓ is a realization of Zi,ℓ ∼ are extended to length n codewords x[1q2]n and x[1q3]n by
CN(0,σ2I ), i.i.d. with respect to ℓ. inserting zeros where sℓ = 0 (note that sn is known at node
We denoMtei P/σ2 by ρ and call it SNR throughout the 1).
Now,nodes2and3areavailableallthetime,buttheydonot
paper. We assume without loss of generality that M ≥ M .
2 3
have apriori knowledge of s . Thus, these nodes use standard
We also assume that M ≥ M . The channel matrices ℓ
1 2
random Gaussian codebooks to encode their messages, and
are generated randomly from a continuous distribution, held
treatthe channelto node1 as an erasure channelwith erasure
constantthroughoutthetransmission,andare knownglobally.
probability τ¯. Node 2 encodes w[q] and w[q], q ∈ {1,2},
The message sets, encoding, and decoding, and achievability 21 23
are defined in the standard Shannon sense [18]. The encoder into codewords x[2q1]n and x[2q3]n with i.i.d. CN(0,p2Ia[q])
at node i, Ei,ℓ, can be either adaptive where dependence of and CN(0,p2Ia[q]) symbols, respectively. Similarly, node213
xi,ℓ on yℓi−1 is allowed, or restricted (nonadaptive) where it encodes w[q] and23w[q], q ∈ {1,2}, into codewords x[q]n and
is not. These possibilities are shown in Table I. x[q]n with3i1.i.d. CN3(20,p I ) and CN(0,p I ) s3y1mbols,
The DoF region is the set of achievable DoF tuples d = re3s2pectively. 3 a[3q1] 3 a3[q2]
(d ,d ,d ,d ,d ,d )∈R6 definedasin[19].Roughly
12 13 21 23 31 32 + To satisfy the power constraint, the powers are chosen as
speaking, if a rate tuple (function of ρ)
p =(a[1]+a[2]+a[1]+a[2])−1m−1nP, (7)
R(ρ)=(R (ρ),R (ρ),R (ρ),R (ρ),R (ρ),R (ρ)) 1 12 12 13 13
12 13 21 23 31 32 p =(a[1]+a[2]+a[1]+a[2])−1P, (8)
i i1 i1 ij ij
where R (ρ) is the rate of the message from node i to
ij
i,j ∈ {2,3}, i 6= j. This encoding is restricted since it uses
node j, is achievable, then the DoF tuple d with d =
limsup Rij(ρ) is achievable. We denote the DoF reijgion neither sℓ−1 at nodes 2 and 3, nor yℓi−1 at nodes 1, 2, and 3.
ρ→∞ log(ρ) 2) Transmission: At time ℓ, node i sends
under restricted encoding and adaptive encoding for a given
2
τ by D and D , respectively,and we define the sum-DoF
r,τ a,τ x = V[q]x[q] +V[q]x[q] , (9)
as dr,τ =maxd∈Dr,τ kdk1 and da,τ =maxd∈Da,τ kdk1. i,ℓ Xq=1h ij ij,ℓ ik ik,ℓi
Next,we describea restrictedtransmissionscheme,andwe
derive its achievable DoF region. where j,k ∈ {1,2,3}\{i}, j 6= k, and V[q] ∈ CMi×a[iqj] is
ij
a beamforming matrix. Zero-forcing is achieved by choosing
III. RESTRICTED ENCODING TRANSMISSION SCHEME the matrices V[1] so that H V[1] = 0 for distinct i,j,k ∈
ij ik ij
In this section, we prove the following theorem. {1,2,3}. These matrices exist if
(M −M )+ ≥a[1], (10)
1Anypowerdiscrepancy isabsorbedintothechannel gains. i k ij
ensuringthatnodei has enoughantennasto send a[1] streams 4) Achievable DoF Region: The constraints can be com-
ij
tonodejwithoutinterferingwithnodek.Toavoidanyoverlap bined as follows
ofthetransmitsignalsin thetransmitsignalspace,werequire
(M −M )+ ≥a[1], (19)
i k ij
2
(a[iqj]+a[iqk])≤Mi. (11) 2 (a[q]+a[q])≤M , (20)
Xq=1 Xq=1 ij ik i
3) Decoding: Node 1 receives y1,ℓ =0 if sℓ =0 and min{a[2],a[2],(M +M −M )+}≥a¯[2], (21)
ij ji i j k ij
y1,ℓ = 3 2 Hj1V[jq1]x[jq1],ℓ+G23(cid:20)xx[[2223]]ℓ (cid:21)+z1,ℓ (12) 2 (a[jqi]+a[kqi])+a[j2k]+a[kqj]−a¯[j2k] ≤Mi. (22)
Xj=2Xq=1 32,ℓ Xq=1
otherwise, where G23 = [H21V[223], H31V[322]]. This signal for distinct i,j,k ∈ {1,2,3}, where a¯[2] and a¯[2] are
12 13
consists of four desired signals plus interference. To decode the dimensions of span(H V[2]) ∩ span(H V[2]) and
a desired signals, say x[211]n, node 1 zero-forcesthe remaining span(H V[2])∩span(H V13[2])1,2respectively.B2y3ad2d1ingthe
ssaigtinsafylsinbgyTm[21u1]lTti[2p11l]yHin=g Iyan1[211]byanadpost-coder T[211] ∈ Ca[211]×M1 τacah[22i1e])v.aSb1ule2bsDti1ot3uFtipnegrdstirjeainm3(,21w9e)3–1o(2b2ta)i,nudsiinjg(eM.g1. d≥21M=2τ≥a[211M]+3
and Fourier Motzkin’s elimination leads to the DoF region in
T[1][H V[2], H V[1], H V[2], G ]=0, (13)
21 21 21 31 31 31 31 23 Theorem 1. Details are omitted due to space limitations.
rank(T[1]H V[1])=a[1]. (14) Next, we study the optimality of this scheme.
21 21 21 21
After post-coding, node 1 is left with the signal y =
21,ℓ IV. OPTIMALITYDISCUSSION
T[1]H V[1]x[1] + T[1]z if s = 1 and y = 0
21 21 21 21,ℓ 21 1,ℓ ℓ 21,ℓ A. Sum-DoF
otherwise. This is an erasure channel over which the rate
I(x[1] ;y ,s ) = I(x[1] ;y |s ) = τI(x[1] ;y |s = We first consider the sum-DoF of the channel, and start by
21,ℓ 21,ℓ ℓ 21,ℓ 21,ℓ ℓ 21,ℓ 21,ℓ ℓ
1) is achievablefromnode2 to node1 for n large.2 Since we presenting the following DoF upper bounds.
used Gaussian i.i.d. codes, this rate is Lemma1:ThefollowingmustbesatisfiedbyanyDoFtuple
τlog I + p2T[1]H V[1]V[1]HHHT[1]H , (15) d∈Da,τ (and hence also d∈Dr,τ):
(cid:12) a[211] σ2 21 21 21 21 21 21 (cid:12) d +d +d ≤τM +τ¯M , (23)
(cid:12) (cid:12) 13 23 21 2 3
leading to(cid:12)a DoF of τa[1] as long as (14) is satisfied(cid:12) (cf. (8)). d +d +d ≤τM +τ¯M . (24)
21 31 32 12 2 3
A similar procedure can be applied for decoding x[2]n,
21 Proof:Forbrevity,wedenote(W ,W )byW ,anduse
x[1]n, and x[2]n, to achieve DoF of τa[2], τa[1], τa[2]. The ij ik i
31 31 21 31 31 ǫ , ǫ , and ǫ to denote quantities that vanish as n→∞.
existence of the post-coders T[iq1] which allow this procedure L1ent H˜2n23 be a3nn(M2 − M3) × M2 matrix so that Hˆ23 ,
is guaranteed as long as the columns of [HT , H˜T ]T has full rank M . Such a matrix exists almost
23 23 2
[H21V[211], H21V[221], H31V[311], H31V[321], G23] (16) surely. Also, let Y˜3,ℓ be defined as H˜23X2,ℓ+Z˜3,ℓ if Sℓ =
1 and 0 otherwise, where Z˜ ∼ CN(0,σ2I ), and
are linearly independent. Let a¯[223] be the dimension of define Yˆ =[YT , Y˜T ]T.3N,ℓow, considera3nyMc2o−dMe3for the
span(H V[2]) ∩ span(H V[2]). Then, span(G ) has 3,ℓ 3,ℓ 3,ℓ
21 23 31 32 23 3WC, and let us establish a bound on R +R +R .3 We
a[223]+a[322]−a¯[223] dimensions,andtheabovelinearindependence give (Y˜n3,W12) and (Yˆn3,W3,W23) as13side i2n3forma21tion to
is possible if we choose
nodes 3 and 1, respectively. By Fano’s inequality, we have
2 (a[q]+a[q])+a[2]+a[2]−a¯[2] ≤M . (17) n(R13+R23−ǫ1n)≤I(W13,W23;Yˆn3,Sn,W3,W12),
Xq=1 21 31 23 32 23 1 n(R21−ǫ2n)≤I(W21;Yn1,Yˆn3,Sn,W1,W3,W23).
Tominimizetheimpactofinterference,wechooseV[q] sothat Recallthateachnodecanestimate Sn withcertaintyfromthe
ij
a¯[2] is maximized. This can not be chosen arbitrarily large, received signals as assumed in the system model. Using the
23
as it has to be smaller than each of a[2] and a[2], and also chain rule, the independence of the messages of each other
smallerthanthedimensionofspan(H 2)3∩span(H32 ),which and of Sn, and combining the two bounds yields
21 31
is (M2+M3−M1)+ almost surely. Thus, n(R +R +R −ǫ )≤I(W ,W ;Yˆn|Sn,W ,W )
13 23 21 3n 2 13 3 3 12
min{a[223],a[322],(M2+M3−M1)+}≥a¯[223]. (18) +I(W21;Yn1|Yˆn3,Sn,W1,W3,W23).
The same arguments can be applied at nodes 2 and 3, for The second term in this bound is equal to
decodingtheir desiredsignals. Thisachievesτa[1], τa[2], a[1], n I(W ;Y |Yℓ−1,Yˆn,Sn,W ,W ,W ), which
12 12 32 ℓ=1 21 1,ℓ 1 3 1 3 23
and a[322] DoF at node 2, and τa[113], τa[123], a[213], and a[223] DoF at iPs no(log(ρ)),4 since given W1, W3, Yℓ1−1, Yn3, and
node 3, leading to similar constraints as (17) and (18).
3WewriteRij(ρ)simplyasRij forbrevity.
2Recallthatsn isknownatthedecoding stage. 4limρ→∞ o(lloogg((ρρ))) =0.
Sn, and using the adaptive encoder, we can construct a Lemma2:Underrestrictedencoding,wehaveD ⊂D[out]
r,τ r,τ
noisy version of Y for all ℓ with S = 1 given by defined as the set of d∈R6 satisfying d +τd ≤τM .
1,ℓ ℓ + 31 32 3
H21Hˆ−231 Y3,ℓ−Y˜H13X1,ℓ + H31X3,ℓ. On the other hand, Proof: Let Y˘2,ℓ =SℓY2,ℓ and let us give (Y˘n2,W2) to
using standhard step3s,ℓ i node 1 as side information. From Fano’s inequality, we have
I(W2,W13;Yˆn3|Sn,W3,W12) n(R31−ǫ1n)≤I(W31;Yn1,Y˘n2|Sn,W1,W2) (27)
n =I(W ;Y˘n|Sn,W ,W )
≤ I(X1,ℓ,X2,ℓ;Yˆ3,ℓ|Sℓ) (25) +I(W31 ;Y2 n|Sn,W1 ,W2 ,Y˘n). (28)
Xℓ=1 31 1 1 2 2
= n τI(X ,X ;Yˆ |S =1)+τ¯I(X ;Y |S =0) Given Y˘n2, W1, W2, and Sn, we can construct a noisy
1,ℓ 2,ℓ 3,ℓ ℓ 2,ℓ 3,ℓ ℓ version of Yn for S = 1 given by H H† (Y˘ −
Xℓ=1 H X )+H1 X ,ℓwhereH† existsalmo3s1tsu3r2ely.T2,hℓus,
≤n(τM2+τ¯M3)log(ρ)+no(log(ρ)), (26) I(W12 ;1Y,ℓn|Sn,2W1 2,,Wℓ ,Y˘n)=32nτo(log(ρ)), and hence
31 1 1 2 2
since the first and second terms represent (M1+M2)×M2 n(R −ǫ −τo(log(ρ)))≤I(W ;Y˘n|Sn,W ,W )
and M ×M MIMO channelswith M and M DoF almost 31 1n 31 2 1 2
2 3 2 3
n
surely (M3 ≤ M2), respectively. Combining terms, dividing =τ I(W ;Y |Sn,W ,W ,Y˘ℓ−1,S =1). (29)
by n and letting n→∞, this yields the bound 31 2,ℓ 1 2 2 ℓ
Xℓ=1
R13+R23+R21 ≤(τM2+τ¯M3)log(ρ)+o(log(ρ)), On the other hand, giving (W1,W31) to node 3 as side
information and using Fano’s inequality, we have
which consequently leads to the first DoF bound. The second
is obtained similarly by giving (Yn2,W2,W32) and W13 as n(τR32−ǫ2n)≤τI(W32;Yn2|Sn,W1,W2,W31) (30)
side information to nodes 1 and 2, respectively.
n
Based on Lemma 1, we can state the following theorem. =τ [h(Y |Sn,W ,W ,W ,Yℓ−1)
2,ℓ 1 2 31 2
Theorem 2: The sum-DoFof the intermittent3WC is given Xℓ=1
by dr,τ =da,τ =2τM2+2τ¯M3. −h(Y2,ℓ|Sn,W1,W2,W3,Y2ℓ−1)].
Proof: Achievability follows from Theorem 1 by using
Since conditioning does not increase entropy, the first term is
the simplexmethod[20] to maximizethe sum-DoFsubject to
aath[[211e1]]D==odFMc2o=−nsdMtra3in,=tas.[2023I]nin=ptahare[3t22i]sccu=hlaeMrm,e3it,dfaeonsldlcorwiab[1se22]db=yinsaSe[2t2e1t]cin.=gIIaaI.[2[11132T]]h==e oaunvpdepre,Xrsbin3nocfurenodmreesdWtrbicy1teahdn(dYenW2c,oℓ|3dS,innthg,eWcsaen1c,oWbned2ue,snWetdr3o1tp,oyY˘gte2ℓe−rnm1e)r.aistMeeqoXurean1-l
co32nverse13follow31s by adding the DoF bounds in Lemma 1. to h(Z2,ℓ)=h(Y2,ℓ|Sn,W1,W2,W3,Y˘ℓ2−1). Thus,
This agrees with intuition. To maximize the sum-DoF, one n
should capitalize on the stable links between nodes 2 and 3, n(τR −ǫ )≤τ I(W ;Y |Sn,W ,W ,W ,Y˘ℓ−1)
32 2n 32 2,ℓ 1 2 31 2
and use any remaining resources for communicating with the Xℓ=1
intermittentnode1.Thistheoremprovesthatadaptationisnot n
=τ I(W ;Y |Sn,W ,W ,W ,Y˘ℓ−1,S =1),
necessaryforachievingthesum-DoFoftheintermittent3WC. 32 2,ℓ 1 2 31 2 ℓ
The same does not hold true from a DoF region perspective Xℓ=1
as we shall see next. since for a given ℓ, this mutual information is equal to
I(W ;H X +Z |Sn,W ,W ,W ,Y˘ℓ−1) indepen-
32 32 3,ℓ 2,ℓ 1 2 31 2
dent of the state S . Combining the two bounds yields
B. DoF Region ℓ
In this section, we show that adaptation is necessary for n(R +τR −ǫ −τo(log(ρ)))
31 32 3n
achievingtheDoFregionofthe intermittent3WC. Thisresult n
is particularly interesting in light of the following statement. ≤τ I(W3;Y2,ℓ|Sn,W1,W2,Y˘ℓ2−1,Sℓ =1) (31)
Theorem3:TheschemeinSec.IIIachievestheDoFregion Xℓ=1
of the nonintermittent 3WC (τ =1) given by Da,1 =Dr[i,n1]. ≤nτM3log(ρ)+nτo(log(ρ)) (32)
Proof: The proof is based on upper bounds in [9], [21],
which follows using similar steps as in the proof of Lemma
and is omitted for lack of space.
1. This leads to the desired result.
Therefore, from a DoF-region point-of-view, adaptation is
Despite its simplicity, Lemma 2 suffices for our purpose.
not necessary in the nonintermittent case.5 Interestingly, the
Based on this lemma, the following theorem proves the
sameisnottrueintheintermittentcase.Toprovethis,firstwe
necessity of adaptation in the intermittent case.
need a DoF outer bound for the restricted intermittent 3WC,
Theorem 4: For an intermittent 3WC with M > M ,
1 3
andsecond,we need anadaptiveschemewhich achievesDoF D 6⊂D[out], and hence adaptation is necessary.
a,τ r,τ
tuplesoutsidethis outerbound.The first step is tackled in the
Proof: It suffices to show that ∃d ∈/ D[out] which is
following lemma. r,τ
achievable using an adaptive scheme. To is end, suppose that
only node 3 has a message to node 1, while node 2 acts
5Adaptationisstillnecessaryfromanachievableratepoint-of-view[7],but
thegainofadaptation doesnotscalewithρ. as a relay to support node 3 which employs block-Markov
encoding.Consider B transmission blocks, each consisting of (nonadaptive) encoding. Since the sum-DoF might be unfair
n channel uses, and let a ,a ∈ [0,1], be chosen so that in terms of per-user or per-stream DoF, the DoF region is of
2 3
a M ,a M ∈ N. In block 1, node 3 encodes a message highimportance.Inthiscase,wehaveshownthatadaptationis
2 2 3 3
w31(1)toacodewordxn31 withx31,ℓ ∈Ca3M3,andsendsitto in fact necessary, and that collaboration between nodes using
node1usinga M antennas.Node1receivesonlymsymbols adaptive encoding enlarges the DoF region beyond what can
3 3
corresponding to s = 1 where sn is the state sequence in be achieved using adaptive encoding.
1,ℓ 1
this block, with m ≤ n and m ≈ τ as n grows. However,
n
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