The Role of Transmitter Cooperation in Linear Interference Networks with Block Erasures

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by  Yasemin Karacora

The Role of Transmitter Cooperation in Linear
Interference Networks with Block Erasures
Yasemin Karacora, Tolunay Seyfi and Aly El Gamal
ECE Department, Purdue University
Email: {ykaracor,tseyfi,elgamala}@purdue.edu
Abstract—In this work, we explore the potential and optimal number of such transmitters, without any constraint on their
use of transmitter cooperation in large wireless networks with identity, what would be the optimal assignment of messages
deepfadingconditions.Weconsideralinearinterferencenetwork
to transmitters and corresponding transmission scheme that
withK transmitter-receiverpairs,whereeachtransmittercanbe
maximizes the average rate over all possible realizations of
connected to two neighboring receivers. Long-term fluctuations
(shadow fading) in the wireless channel can lead to any link the network?. To simplify analysis, we consider the linear
being erased with probability p. Each receiver is interested in interferencenetworkintroducedin[1],whereeachtransmitter
7
1 one unique message that can be available at two transmitters. canonlybeconnectedtothereceiverhavingthesameindexas
The considered rate criterion isthe peruser degrees of freedom
0 well as one following receiver. The channelcapacity criterion
(puDoF) as K goes to infinity. Prior to this work, the optimal
2 we consider is the pre-log factor of the sum capacity at high
assignmentofmessagestotransmitterswereidentifiedinthetwo

n limits p→ 0 and p→1. We identify new schemes that achieve Signal to Noise Ratio (SNR), also known as the degrees of
a average puDoF values that are higher than the state of the art freedom(DoF).Becauseourgoalistounderstandtheoptimal
J forasignificantpartoftherange0<p<1.Thekeyideatoour patternoftransmittercooperationthatscalesinlargenetworks,

6 results is to understand that the role of cooperation shifts from
weconsidertheDoFnormalizedbythenumberoftransmitter-
2 increasingtheprobabilityofdeliveringamessagetoitsintended
receiver pairs, and take the limit as that number goes to
destination at high values of p, to interference cancellation at
] low values of p. Our schemes are based on an algorithm that infinity;wecallthistheperuserdegreesoffreedom(puDoF).
T achieves the optimal puDoF value, when restricted to a given In [2], the considered setting was studied, where first the
I message assignment as well as the use of interference avoidance case where each message can only be available at a single
.
s zero-forcing schemes. transmitterwasanalyzed.Theoptimalassignmentofmessages
c
[ to transmitters, and the value of puDoF were identified as
I. INTRODUCTION a function of the erasure probability p. In this work, we
1
The anticipated developmentof 5G communication is rais- extend the work of [2] by studying the case where each
v
ing interest in new problems in communication theory. In message can be available at two transmitters, and transmitter
4
2 particular, we now have this general vision of smart coop- cooperation is allowed. The optimal message assignment in
5 erative cloud-based networks that can autonomously adjust the limits p → 1 and p → 0 were identified in [2] and [3].
7 to varying environmental conditions. Traditional results in As p → 1, each message is assigned to the two transmitters
0
network information theory hence need to be used as tools connected to its destination, to maximize the probability of
.
1 to build a theoretical framework that identifies useful insights successful delivery. As p → 0, the puDoF value goes to 4,
0 5
and fundamental limits for these new networks. Our focus and is achieved by splitting the network into subnetworks;
7
in this work is to analyze information theoretic models of each has five transmitter-receiver pairs. In order to avoid
1
: interference networks that capture the effect of deep fading interference between the subnetworks, the last transmitter in
v
conditions through introducing random link erasure events in each subnetwork is inactive. And hence, each of the first
i
X blocksofcommunicationtimeslots.Morespecifically,inorder and last message in each subnetwork is only assigned to
r toconsidertheeffectoflong-termfluctuations(deepfadingor one of the two transmitters connected to its destination, and
a shadowing), we assume that communication takes place over the other assignment is used at a transmitter not connected
blocks of time slots, and independentlink erasures take place to its destination, but connected to another receiver that is
withaprobabilitypineachblock.Further,short-termchannel prone to interference caused by this message. Further, the
fluctuations allow us to assume that in each time slot, all non middle message in each subnetwork is not transmitted. We
erased channel coefficients are drawn independently from a find, through simulations, in this work that assigning that
continuous distribution; this is known as the assumption that middlemessagetoonlyonetransmitterconnectedtoits desti-
the channel is generic. nation,andanothertransmitternotconnectedtoitsdestination,
We are interested in understanding the role of transmitter leads to better rates than assigning it to the two transmitters
cooperation (also known as Coordinated Multi-Point (CoMP) connected to its destination at low values of p. That implies
Transmission) in these dynamic interference networks. In that a fraction of 3 of the messages are assigned to only one
5
particular, if each message can be assigned to more than of the two transmitters connected to their destination, and the
one transmitter, with a restriction only on the maximum remaining 2 are assigned to the two transmitters connected
5
to their destination. We show in this work, that at any value simultaneouslymadearbitrarilysmallastheblocklengthgoes
of p from 0 to 1, the assignment achieving the highest toinfinity,andthisholdsforalmostallrealizationsofnon-zero
puDoF using our proposed scheme, has a fraction of f(p) channelcoefficients.ThesumcapacityC (P)isthemaximum
Σ
of messages that are assigned to only one of the transmitters value of the sum of the achievable rates. The total number of
connectedtotheirdestination,andanothertransmitterusedfor degrees of freedom (η) is defined as limsup CΣ(P). For
P→∞ logP
interference cancellation, and the remaining fraction 1−f(p) a K-userchannel,anda probabilityofblockerasurep, we let
of messages are assigned to the two transmitters connectedto η (K)betheaveragevalueofη overpossiblechoicesofnon-
p
their destination. The value of f(p) decreases monotonically zerochannelcoefficients.Wefurtherdefinetheasymptoticper
from 3 to 0 as p increasesfrom 0 to 1, which agreeswith the user DoF (puDoF) τ to measure how η (K) scale with K.
5 p p
intuition about the shifting role of cooperative transmission
η (K)
p
from canceling interference to increasing the probability of τ = lim (3)
p
K→∞ K
successful delivery as p increases from 0 to 1.
D. Zero-forcing (Interference Avoidance) Schemes
II. SYSTEMMODELAND NOTATION
Weconsiderinthisworktheclassofinterferenceavoidance
We use the standard model for the K−user interference schemes, where each message is either not transmitted or
channel with single-antenna transmitters and receivers, allocated one degree of freedom. Accordingly, every receiver
K iseitheractiveorinactive.Anactivereceiverdoesnotobserve
Y (t)= H (t)X (t)+Z (t) (1) any interfering signals.
i X i,j j i
j=1
III. OPTIMAL ZERO-FORCING SCHEME
where t is the time index, X (t) is the transmitted signal of
j WepresentAlgorithm1thattakesasinputagivenclusterof
transmitterj,Y (t)isthereceivedsignalatreceiveri,Z (t)is
i i users that are labeled as {1,2,··· ,N}, and has transmit sets
the zero mean unit variance Gaussian noise at receiver i, and
{T ,i ∈ [N]}. The output of the algorithm is the transmit
H (t)isthechannelcoefficientfromtransmitterj toreceiver i
i,j signals {X ,i ∈ [N]} that employs zero-forcing transmit
i overthe time slott. We removethe time indexin the restof i
beamforming to maximize the puDoF value for users within
the paper for brevity unless it is needed. Finally, we use [K]
the cluster, while guaranteeing that there is no interference
to denote the set {1,2,...,K}
caused by the last user of the cluster at the first receiver of
A. Channel Model the following cluster.
For each message W , we define four binary variables;
Eachtransmittercanonlybeconnectedtoitscorresponding i
namely b ,j ∈{i−2, i−2, i, i+1}. These are initialized
receiver as well as one following receiver, and the last trans- i,j
to zero. We look at every message starting from W to
mitter can only be connected to its corresponding receiver. 1
W and evaluate the conditions under which a message can
In order to consider effect of long-term fluctuations (shad- N
be sent and decoded at its desired receiver, such that no
owing), we assume that communication takes place over
interferenceoccurs.Ifa decisionismadeto sendmessageW
blocks of time slots, and let p be the probability of block i
from transmitter j, the corresponding variable b is set to
erasure. In each block, we assume that for each j, and each i,j
one. Since message W can be sent to its destination using
i ∈ {j,j+1}, H = 0 with probability p. Moreover, short- i
i,j
either transmitter i or i − 1, there are two cases that are
termchannelfluctuationsallowustoassumethatineachtime
consideredinthealgorithm.Inthefollowingwearediscussing
slot,allnon-zerochannelcoefficientsaredrawnindependently
and justifying both cases. Note, that user one and two are
froma continuousdistribution.Finally,we assume thatglobal
considered separately in (6-16), since they representa special
channel state information is available at all transmitters and
case due to their position at the beginning of the cluster.
receivers.
Case1:Inthefirstpartofthefor-loopstartingatline17,we
B. Message Assignment check if message W can be sent from transmitter i−1. This
i
For each i ∈ [K], let Wi be the message intended for is only possible if Hi,i−1 exists and message Wi is available
receiver i, and T ⊆[K] be the transmit set of receiver i, i.e., at transmitter i−1. Furthermore, we have to make sure that
i
thosetransmitterswiththeknowledgeofWi. Thetransmitters whilesendingWi,transmitteri−1doesnotcauseinterference
in T cooperativelytransmit the message W to the receiver i. at receiver i−1. First, assume that transmitter i−1 does not
i i
The messages {Wi} are assumed to be independent of each send Wi−1. There are three possibilities, for which message
other.Eachmessagecanonlybeavailableattwo transmitters. Wi canbedecodedwithoutinterference.Thetrivialoneisthat
i.e., the link between transmitter i−1 and receiver i−1 does not
|T |≤2,∀i∈[K] (2) exist. Another possible scenario is that receiver i−1 is not
i
able to decode its desired message anyway, i.e. W is not
i−1
C. Degrees of Freedom
sent from transmitter i−2. As long as these conditions are
The total power constraint across all the users is P. In satisfied, the variable b is set to 1. Otherwise, if W does
i,i−1 i
each block of time slots, the rates R (P) are achievable interferewith W atreceiveri−1,we mightstill beable to
i i−1
if the decoding error probabilities of all messages can be remove the interference by sending a signal from transmitter
i−2 such that it will cancelthe interferenceat receiveri−1.
Thisispossibleaslongasthefollowingconditionshold:First,
MessageW mustbeavailableattransmitteri−2aswell(i.e.
i
(i−2)∈T ).Furthermore,wehavetomakesurethatthesignal
i
Algorithm 1
sent for interference cancellation does not cause interference
at receiver i−2. This is guaranteed, if either H = 0 1: for i=1:N do
i−2,i−2
or receiver i−2 is not able to decode its desired message 2: Define bi,i−2 =bi,i−1 =bi,i =bi,i+1 =0
anyway. In this case, not only b but also b is set to 3: end for
i,i−1 i,i−2
1. 4: if H1,1 =1 ∧ 1∈T1 then
5: b1,1 =1
6: end if
45: for i=1:N do
4476:: GSeetnXerait=e X0i,j, j ∈ {i−1,i} from Wj using an optimal 897::: eifnHdb2i2f,,11==11 ∧ 1∈T2 ∧ b1,1 =0 then
AWGN channel point-to-point code (see e.g., [4])
544089::: eifnbdiX,iifi=←1Xthie+nXi,i 111102::: if Hif2,2Hb=22,,121==∧012∨∈bT12,1 ∧=0b2t,h1e=n0 then
5521:: if biX+1i,i←=X1it+heXni+1,i 111435::: eelnsdebii2ff,22=∈1T;1b1th,2e=n 1;
53: end if
16: end if
54: end for
17: for i=3:N do
55: for i = 2:N do
555876::: eifnbdiX−if1i,i←=X1it−heHni,i−1HXi,ii−1,i−1 211098::: if Hifi,ib−ii−1f1=H,i−i1−11=∧,i−01(i=t−he01n)∨∈bTii−t1h,ie−n2 =0 then
59: end for 21: bi,i−1 =1
60: for i = 1:N-1 do 22: else if (i−2) ∈ Ti ∧ [Hi−2,i−2 = 0 ∨
(b =0 ∧ b )=0] then
6621:: if biX+2i,i←=X1it−heHni+1,iH+i1+X1,ii+2,i+1 2243:: i−2,i−2 endbiif,i−i1−=2,i−13bi,i−2 =1
63: end if 25: end if
64: end for 26: end if
27:
Case 2: Now we consider the case of sending message Wi 28: if Hi,i =1 ∧ i∈Ti ∧ bi,(i−1) =0 then
from transmitter i (line 35-51). Here, the trivial conditions 29: if [bx,i−1 = 0, ∀ x ∈ {i−2,i−1,i,i+1}] ∨
to make this possible are that Hi,i exists, message Wi is Hi,i−1 =0 then
availableattransmitteri,andW isnotbeingdeliveredthrough 30: if i=N then
i
transmitteri−1.Thistime,wehavetomakesurethatreceiver 31: if HN+1,N =0 ∨ HN+1,N+1 =0 then
icandecodemessageWi withoutanyinterference.Thisholds 32: bi,i =1
ifeitherthelinkH doesnotexistoriftransmitteri−1is 33: end if
i,i−1
notactive.Thenbi,i issetto1.ForuserN,wemakesurethat 34: elsebi,i =1
there is no inter-cluster interference. Similar to the previous 35: end if
casewe canalsocanceltheinterferencefromtransmitteri−1 36: else if i ∈ Ti−1 ∧ bx,i−1 = 0, ∀ x ∈ {i−2,
aslongasmessageW isavailableattransmitteriandW i,i+1} then
i−1 i−1
is the only message that causes interference at receiver i. If 37: if i=N then
theseconditionsholdandiffori=N thereisnointer-cluster 38: if HN+1,N =0 ∨ HN+1,N+1 =0 then
interference, both bi,i and bi−1,i are set to 1. 39: bi,i =1, bi−1,i =1,
We now prove the following result to justify Algorithm 1. 40: end if
Theorem 1: Givenanyassignmentofmessagestotransmit- 41: end if
ters, such that each message can only be available at two 42: end if
transmitters, Algorithm 1 leads to the optimal zero-forcing 43: end if
transmission scheme if the whole network is selected as the 44: end for
input cluster.
Proof: First, we show that the general structure of our
Algorithm leads to the optimal transmission scheme. Then,
we justify the conditions in detail for the first three messages
and finally show how the argument generalizes. This simplifies the optimal Algorithm in two ways. On the
We consider the messages in ascending order from W to onehand,wecangothroughthelinksonebyoneandcheckif
1
W , and check which transmitter can deliver message W it is possible to send a message to its desired receiverwithout
N i
such thatit can be decodedat its desiredreceiverand without interfering with any of the previous active messages. If it is
interfering at any previous active receiver. If this is true, we possible, we will always decide to send the message. On the
will transmit the message. In the following, we prove by other hand, decisions that we already made do not have to
inductionthatthisprocedureleadstotheoptimaltransmission be changed later, because at each step we make sure to avoid
scheme.Inafirststep,weconsiderthebasecase,i.e.weprove conflicts with previously activated messages. This procedure
thatsendingW fromTransmitter1 is alwaysoptimalas long is applied in Algorithm 1, as we illustrate below.
1
as it is available 1∈T and H 6=0. It is sufficient for the proof to consider only clusters where
1 1,1
WedefineΩ tobethesubsetofalllinksH throughwhich alldiagonallinksexist,i.e.allH =1,j ∈{1, 2,..., K−
i,j j+1,j
a message W can be sent and decodedat its desired receiver, 1}. This is because otherwise, we can split the network and
i
and call it the feasible set. In other words, all links in Ω studybothclustersseparately.Thiswillbeexplainedingreater
satisfy the trivial conditions for transmission; namely j ∈ T detail in the journal version of the paper.
i
and H 6= 0. Let S ⊂ Ω\H be an arbitrary set of links In the following, we derive the decision conditions for the
i,j 1,1
that can be used simultaneously to deliver messages to their first three messages in a cluster.
desired receivers while eliminating interference. If H ∈ Ω, sending W is optimal, as shown in the base
1,1 1
Starting with anyset S, if H ∈Ω, we either addH to case of the proof by induction. Hence, set b =1.
1,1 1,1 1,1
S or replace the first link in S by H if there is a conflict. If H ∈ Ω, we have two possibilities. If b = 1, we
1,1 2,1 1,1
We claim thatthisreplacementcannotdecreasethe DoF. This cannot send W from transmitter 1 as well without causing
2
isbecauseononehand,thefirstactivereceiverinthenetwork interference at the first receiver. Otherwise, if b = 0, it is
1,1
doesneverobserveinterference.Also,ifwesendW fromthe optimal to send W from the first transmitter.
1 2
first transmitter,thiscan onlycause interferenceat the second If H ∈ Ω, there are three cases to consider. First, if
2,2
receiver, but as H , j ∈{1,2} is either not in S or it is the b = 1 and b = 0, we have interference from W at
2,j 1,1 2,1 1
first linkin S andhenceis replacedby H , the transmission the second receiver. However, this can be canceled as long as
1,1
of W does not prevent any other message corresponding to W is available at transmitter 2. If this is true, set b = 1
1 1 2,2
subsequentlinksinS frombeingdecodedattheirdestination. and b = 1. Otherwise, if b = 0 and b = 1, it is not
1,2 1,1 2,1
As a consequence, it is always optimal to transmit W from necessary to send W from transmitter 2. Finally, if b = 0
1 2 1,1
Transmitter 1 as long as 1∈T and H =1. and b = 0, we basically have the same situation as for
1 1,1 2,1
Next, we extend the proof to all users by induction. The user 1. Thus b is set to 1. Note that due to the previous
2,2
induction hypothesis is as follows. We consider an arbitrary conditions,thecasewhereb =1andb =1neveroccurs.
1,1 2,1
link H ∈ Ω. Let S ⊂ Ω be the set of links H , through Now, we consider sending message W .
i,j 1 k,l 3
which a subset of the messages {W ,k<i} can be delivered If H ∈ Ω, the following four scenarios are possible. If
k 3,2
simultaneously to their destinations, while eliminating inter- b = 0 and b = 0, which implies that b = 0, the case
2,1 2,2 1,2
ference. Assume that all links in S are chosen optimally, oftransmittingW becomessimilartothecasewe considered
1 3
i.e. the number of delivered messages cannot be increased by above for user 1. Hence, b = 1. If b = 0, b = 1
3,2 1,1 2,1
changing any of these links. and b = 0, which implies that b = 0, then we need to
2,2 1,2
Then, we do the induction step. Let S ⊂Ω be any set of considerH . As long asH =0, we can send W without
2 2,2 2,2 3
links H , throughwhich a subset of the messages {W ,k > causinginterferenceatthesecondreceiver.Thus,setb =1.
k,l k 3,2
i} can be transmitted simultaneously such that they can be Otherwise, if H = 1, W sent from the second transmitter
2,2 3
decoded at their destination. Also, the links in S are chosen interferes with W . We can cancel the interference only if
2 2
optimallytomaximizethenumberofdeliveredmessages.Ifit 1 ∈ T , so then set b = 1 and b = 1. If b = 1,
3 3,2 3,1 1,1
ispossibletosendW throughH withoutcausingaconflict b = 0 and b = 1, which implies that b = 1, message
i i,j 2,1 2,2 1,2
with any of the messages, that are sent through the links in W cannot be sent, because this would cause interference at
3
S , the samelogic appliesto H asto H in the base case. the second receiver, that cannot be canceled using the first
1 i,j 1,1
Moreprecisely,ifW doesnotinterfereatanypreviousactive transmitter without interfering with W . Finally, if b = 0,
i 1 1,1
receiver and it can be decoded at receiver i while eliminating b =0 and b =1, which implies that b =0, we could
2,1 2,2 1,2
interference, H can be either added to S or replace the basically send both W and W from the second transmitter
i,j 2 2 3
first link in S , in order to obtain an optimal set of links and cancel interference using the first and third transmitter.
2
for the transmission of the messages {W ,k ≥ i}. This is However, the sum DoF of both messages will remain one.
k
possible since again, W does not cause interference at any Therefore, in this case we set b =0.
i 3,2
active receiver with an index k > i, because H is either Next, we check the possibilities to send W from the third
i+1,j 3
not in S or it is the link that is replaced by H anyway. transmitter if H ∈ Ω. If b = 0 and b = 0, then also
2 i,j 3,3 3,2 2,2
Therefore,sendingamessageW throughalinkH isalways b =0 and the third receiver does not observe interference.
i i,j 1,2
optimal as long as it is possible to decode W at receiver i Thus, we set b =1. If b =0 and b =1, our decision
i 3,3 3,2 2,2
without causing interference at a previously active receiver. willdependonwhethermessageW issent.Ifb =0,which
1 1,1
implies that b = 0, the interference from W at the third f(p) = 4 = 1 we consider a cluster with N = 25 users
1,2 2 100 25
receiver can be canceled as long as 3∈T . If this is true, set whereonlythefirstmessageisassignedtothefirstandsecond
2
b = 1 and b = 1. Otherwise, if b = 1 and therefore transmitter, while all other messages W are known at the
3,3 3,2 1,1 i
b = 1, the interference caused by sending W cannot be transmitters i−1 and i, respectively.
1,2 1
canceled, because M =2 and hence, W cannot be assigned As a result, the maximum puDoF that is achievable with
1
to transmitter 3. Note that as long as b = 1, sending W the set of message assignments described above is shown in
3,2 3
from transmitter 3 does not increase the DoF. Figure 1. Compared to the schemes presented in [2], there
Since each message can only be available at two transmit- exist message assignments with a better performance. These
ters, it is not necessary to consider the users before receiver are presented in Table I. Note that in [2] it was shown that
i−2todecidewhetherW canbetransmitted.Amoredetailed assignmentwith f(p)= 2 is optimalforp→0.Interestingly,
i 5
explanation on this will be given in the journal version. As we find an assignment with f(p)= 3 (see the green curve in
5
a consequence, the conditions for sending message W , if Fig. 1) that achievesthe same puDoF for p=0, but performs
3
generalized to W , basically apply to all following messages slightly better on the interval (0,0.15). From our results in
i
as well. There are only two additionalaspects that have to be Table I, we observe that the optimal fraction f(p) decreases
considered.First, if we generalizethe case whereb =1 for monotonically from 3 to 0 as p goes from 0 to 1.
3,1 5
interference cancellation, we have to make sure that receiver
i − 2 is not active. That means for i − 2 > 1, not only
1
b = 0 but also b = 0 has to be true. Finally,
i−2,i−2 i−2,i−3 K = 5, f(p) = 3/5
for any message sent from the last transmitter in the cluster, 0.8 K , f(p) = 0
maximum DoF
inter-cluster interference must be avoided. Therefore, either
0.6
HN,N+1 or HN+1,N+1 has to be erased. puDoF0.4
IV. SIMULATION
0.2
Using Algorithm 1, we can determine the optimal trans-
0
mission scheme for a given network realization and message 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
p
assignment.Inthissection,weapplythisalgorithmtocompute
the DoF as a function of the erasure probability p for several Fig.1. TheplotshowsthepuDoFasafunctionoftheerasureprobabilityp
foundbyapplyingAlgorithm1to6000randomgeneratedchannelrealizations
cluster sizes and message assignments. In particular, we find
foreachvalueofp∈{0,0.01,0.02,···,1}.Thebluecurvecorrespondsto
schemes that outperform those presented in [2] for a wide themessageassignmentandtransmissionstrategypresentedin[2],whichwas
range of the open interval 0<p<1. showntobeoptimalasp→1.TheredcurveisthemaximumpuDoFthatis
achievable withthemessageassignments weconsidered foroursimulation.
To compute the average puDoF at a certain p for a given
messageassignment,wesimulateasufficientlylargenumbern
ofchannelrealizations,wherelinksareerasedwithprobability
Range of p Best performing message assignment
p,andapplyAlgorithm1toeachrealization.ThepuDoFvalue
0 to 0.15 N =5, f(p)= 3
is thencomputedasthe averagenumberof decodedmessages 5
0.16 to 0.29 N =100, f(p)= 1
divided by the cluster size N. 2
0.3 N =100, f(p)= 49
The simulation is done for a set of message assignments 100
0.31 to 0.32 N =100, f(p)= 12
with different fractions f(p) of messages that are assigned to 25
onetransmitterconnectedtotheirdesiredreceiverandanother 0.33 to 0.58 N =100, f(p)= 1010
transmitter that can be used to cancel interference, while the 0.59 to 1 N →∞, f(p)=0 (as in [2])
remaining fraction of 1 − f(p) of messages are assigned TABLEI
to both transmitters that are connected to their destination. MESSAGEASSIGNMENTSWITHTHEBESTPERFORMANCEOUTOFTHESET
Furthermore, we vary the cluster size N. More precisely, we OFASSIGNMENTSTHATWASSTUDIEDWITHINTHISPAPER.
use the following assignment strategy:
{1, 2} i=1
{N−2, N−1} i=N ∧ f(p)·N >1 REFERENCES
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100
to 99 , calculating the puDoF as a function of p for each of (DoF) of Locally Connected Interference Channels with Coordinated
100 Multi-Point (CoMP) Transmission,” in Proc. IEEE International Con-
these message assignments. Additionally, we vary the cluster
ferenceonCommunications (ICC),Ottawa, Jun.2012.
size and thus also the message assignment by reducing the [4] T.M.CoverandJ.A.Thomas,ElementsofInformationTheory,2ndEd.
fraction by its greatest common divisor. As an example, for Wiley,2006.