Joint Uplink-Downlink Cell Associations for Interference Networks with Local Connectivity

0.9 MB·

by  Manik Singhal

Joint Uplink-Downlink Cell Associations for
Interference Networks with Local Connectivity
Manik Singhal and Aly El Gamal
ECE Department, Purdue University
Email: {msingha,elgamala}@purdue.edu
Abstract—We study information theoretic models of interfer- (CoMP transmission and reception). CoMP transmission and
ence networks that consist of K Base Station (BS) - Mobile reception in cellular networks are applicable in the downlink
Terminal (MT) pairs. Each BS is connected to the MT carrying
and uplink, respectively. The model in [8] assumed that each
thesameindexaswellasLfollowingMTs,wheretheconnectivity
message can be available at M transmitters and can be
parameterL≥1.WefixthevalueofLandstudylargenetworks t
asK goestoinfinity.WeassumethateachMTcanbeassociated decoded through Mr received signals. It was shown that full
with Nc BSs, and these associations are determined by a cloud- Degrees of Freedom (DoF) can be achieved if Mt +Mr ≥
based controller that has a global view of the network. An K +1, where K is the number of transmitter-receiver paris
MT has to be associated with a BS, in order for the BS to
7 (users) in the network.
transmititsmessageinthedownlink,ordecodeitsmessageinthe
1 Recently in [9], alternative frameworks for cooperation in
uplink.Inpreviouswork,thecellassociationsthatmaximizethe
0
average uplink-downlink per user degrees of freedom (puDoF) both downlink and uplink were introduced. The new frame-
2
were identified for the case when L = 1. Further, when only works are based on the concept of message passing between

n the downlink is considered, the problem was settled for all base stations. In the downlink, quantized versions of the
a values of L when we are restricted to use only zero-forcing
analog transmit signals are being shared between base station
J interference cancellation schemes. In this work, we first propose
transmitters.Thesupportingkeyideaisthatinformationabout
5 puDoF inner bounds for arbitrary values of L when only the
2 uplink is considered, and characterize the uplink puDoF value multiple messages can be shared from one transmitter to
whenN ≥L.Wethenintroducenewachievableaverageuplink- another with the cost of sharing only one whole message
c
] downlinkpuDoF,andconjecturethatthenewschemeisoptimal (of the order of logP, where P is the transmit power), if
T forallvaluesofL,whenwerestrictourattentiontozero-forcing
we only share information needed to cancel the interference
I schemes.
. caused by the messages at unintended receivers, through dirty
s
c paper coding (see [10]). In the uplink, decoded messages are
I. INTRODUCTION
[ shared from one base station receiver to another, where they

Thefifthgenerationofcellularnetworksisexpectedtobring are used to cancel interference. It was shown in [9] that there
1
v newparadigmstowirelesscommunications,thatexploitrecent is a duality in this framework between schemes that are used
2 technologicaladvancementslikecloudcomputingandcooper- in the downlink and those that are used for the uplink, with
2 ative communication (also known as Coordinated Multi-Point the clear advantage that the same backhaul infrastructure can
5
or CoMP). In particular, the rising interest in Cloud Radio be used to support both scenarios.
7
Access Networks (CRAN) (see e.g., [1]-[6]) holds a promise In this work, we study locally connected networks, where
0
. for such new paradigms. These paradigms require new infor- the downlink has a similar model to the one in [8] that
1
mation theoretic frameworks to identify fundamental limits allows each message to be available at a specified number
0
7 and suggest insights that are backed by rigorous analysis. of transmitters, and the uplink has a similar model to the one
1 The focus of this work is to identify associations between in[9]thatallowssharingofdigitaldecodedmessagesbetween
: cell edge mobile terminals and base stations, that maximize receivers. We assume that in both downlink and uplink,
v
i the average rate across both uplink and downlink sessions, the backhaul carries digital messages between a centralized
X
while allowing for associating one mobile terminal with more controller and base stations. The message passing between
r than one base station and using cooperative transmission and base station receivers in the uplink can be implemented by
a
reception schemes between base stations in the downlink and first communicating the decoded message from its destination
uplink sessions, respectively. With a cloud-based controller, to the centralized controller, and then communicating the
optimal decisions for these associations can take into account message from the centralized controller to the base station(s)
the whole network topology, with the goal of maximizing a that will use it to cancel interference. We impose a constraint
sum rate function. that each message can be available at a specified maximum
Cloud-based CoMP communication is a promising new number of base stations. Our constraint is used to capture the
technology that could significantly enhance the rates of cell backhaul rate, as well as the overhead needed to associate
edge users (see [7] for an overview of CoMP). In [8], an a mobile terminal with a base station. The justification for
information theoretic model was studied where cooperation our choice is that sharing of analog signals face practical
wasallowedbetweentransmitters,aswellasbetweenreceivers constraintsbecauseofquantizationerrors,andmaynotfitwell
inawirelessdigitalinfrastructurewheredigitalmessagesfrom the paper for brevity unless it is needed. The signals Y and
i
multiplesessionscouldbecombinedandsharedoverthesame X correspond to the receive and transmit signals at the ith
i
backhaul link. base station and mobile terminal in the uplink, respectively,
When considering this work, it is important to note that and the ith mobile terminal and base station in the downlink
the assumptions in a theoretical framework need not reflect respectively. For consistency of notation, we will always refer
directly a practical setting, but are rather used to define a to H as the channel coefficient between mobile terminal i
i,j
tractable problem whose solution can lead to constructive and base station j.
insights. For example, it was shown in [11] that imposing
A. Channel Model
a downlink backhaul constraint where each message can be
available at a specified maximum number of transmitters We consider the following locally connected interference
(maximum transmit set size constraint), can lead to solutions network. The mobile terminal with index i is connected to
that are also useful to solve the more difficult and more base stations {i,i−1,··· ,i−L}, except the first L mobile
relevant to practice problem, where an average transmit set terminals, which are connected only to all the base stations
size constraint is used instead of the maximum. Also, in [12], with a similar or lower index. More precisely,
it was shown that solutions obtained for the locally connected
network models, that are considered in this work, can be used
H =0 iff i∈/ {j,j+1,··· ,j+L},∀i,j ∈[K], (2)
to obtain solutions for the more practical cellular network i,j
models,byviewingthecellularnetworkasasetofinterfering andallnon-zerochannelcoefficientsaredrawnfromacontin-
locallyconnectedsubnetworksanddesigningafractionalreuse uousjointdistribution.Finally,weassumethatglobalchannel
scheme that avoids interference across subnetworks. state information is available at all mobile terminals and base
stations.
A. Prior Work
In [13], the considered problem was studied for Wyner’s B. Cell Association
linear interference networks (channel model was introduced For each i ∈ [K], let C ⊆ [K] be the set of base stations,
i
in [14]). The optimal message assignment and puDoF value with which mobile terminal i is associated, i.e., those base
were characterized. Linear networks form the special case stations that carry the terminal’s message in the downlink and
of our problem when L = 1. Also, in [11], the downlink willhaveitsdecodedmessagefortheuplink.Thetransmitters
part of our problem was considered, and the optimal message in C cooperatively transmit the message (word) W to mobile
i i
assignment (cell association) and puDoF value were charac- terminal i in the downlink. In the uplink, one of the base
terized for general values of the connectivity parameter L, station receivers in C will decode M and pass it to the
i i
when we restrict our attention to zero-forcing (or interference remaining receivers in the set. We consider a cell association
avoidance)scheme.Here,allourresultsareforgeneralvalues constraintthatboundsthecardinalityofthesetC byanumber
i
of the connectivity parameter L. N ; this constraint is one way to capture a limited backhaul
c
capacity constraint where not all messages can be exchanged
B. Document Organization
over the backhaul.
In Section II, we present the problem setup. In Section III,
wediscusspreviousworkonzero-forcingCoMPtransmission |Ci|≤Nc,∀i∈[K]. (3)
schemes for the downlink. We then present inner bounds for
We would like to stress on the fact that we only allow full
thepuDoFoftheuplinkinSectionIV,andprovetheconverse
messages to be shared over the backhaul. More specifically,
foraspecialcaseinSectionV.InSectionVI,wepresentnew
splitting messages into parts and sharing them as in [15], or
achievable puDoF values when the average of the uplink and
sharing of quantized signals as in [9] is not allowed.
downlink is considred. We finally present concluding remarks
in Section VII. C. Degrees of Freedom
II. SYSTEMMODELANDNOTATION Let P be the average transmit power constraint at each
transmitter, and let W denote the alphabet for message W .
For each of the downlink and uplink sessions, we use the i i
ThentheratesR (P)= log|Wi| areachievableifthedecoding
standard model for the K−user interference channel with i n
errorprobabilitiesofallmessagescanbesimultaneouslymade
single-antenna transmitters and receivers,
arbitrarilysmallforalargeenoughcodingblocklengthn,and
(cid:88)K this holds for almost all channel realizations. The degrees of
Yi(t)= Hi,j(t)Xj(t)+Zi(t), (1) freedom d ,i ∈ [K], are defined as d = lim Ri(P).
i i P→∞ logP
j=1 The DoF region D is the closure of the set of all achievable
where t is the time index, X (t) is the transmitted signal of DoF tuples. The total number of degrees of freedom (η) is
j
transmitterj,Y (t)isthereceivedsignalatreceiveri,Z (t)is the maximum value of the sum of the achievable degrees of
i i
(cid:80)
the zero mean unit variance Gaussian noise at receiver i, and freedom, η =max d .
D i∈[K] i
H (t)isthechannelcoefficientfromtransmitterj toreceiver ForaK-userlocallyconnectedwithconnectivityparameter
i,j
i over time slot t. We remove the time index in the rest of L, we define η(K,L,N ) as the best achievable η on average
c
taken over both downlink and uplink sessions over all choices set C = {i − L,i − L − 1,··· ,N + 1}. It was shown
i c
of transmit sets satisfying the backhaul load constraint in (3). in[11]thatthepuDoFvalueof(5)achievedbythisschemeis
In order to simplify our analysis, we define the asymptotic that best achievable value in the downlink using the imposed
per user DoF (puDoF) τ(L,N ) to measure how η(K,L,N ) cooperation constraint and zero-forcing schemes.
c c
scales with K while all other parameters are fixed,
IV. UPLINK-ONLYSCHEME
η(K,L,N )
τ(L,N )= lim c . (4) We discuss in this section backhaul designs that optimize
c K→∞ K
only the uplink rate, and consider only zero-forcing coding
We further define τD(L,Nc) and τU(L,Nc) as the puDoF schemes. We present the following inner bound on the puDoF
when we optimize only for the downlink and uplink session, value that is characterised by a piecewise function as follows:
respectively.

1 L+1≤N ,
D. Zero-forcing (Interference Avoidance) Schemes  c
schWemeecso,nswidheerreinethaicshwmorekssthageeclaissseoitfhienrtenrfoetretnracnesamvoititdeadncoer τUzf(L,Nc)≥N2LN2c+N+c2+c1L 1L2≤≤NNcc≤≤L2L,−1. (6)
allocated one degree of freedom. Accordingly, every receiver Thecellassociationthatachievedtheaboveisasfollowing.
iseitheractiveorinactive.Anactivereceiverdoesnotobserve When N ≥ L+1, the optimal association is similar to the
c
any interfering signals. one specified in [13, Section 4], where each mobile terminal
We add the superscript zf to the puDoF symbol when we is associated with the L + 1 base stations connected to it.
impose the constraint that the coding scheme that can be used The last base station, with index K, in the network decodes
has to be a zero-forcing scheme. For example, τUzf(L,Nc) the last message and then passes it on to the L other base
denotes the puDoF value when considering only the uplink stations connected to the Kth mobile terminal, eliminating all
and impose the restriction to zero-forcing schemes. interferencecausedbythatmobileterminal.Eachbasestation
then decodes its message and passes it on to the other base
III. PRIORWORK:DOWNLINK-ONLYSCHEME
stations, eliminating the interference caused by the message.
In[11],theconsideredsettingwasstudiedforonlydownlink Thus, one degree of freedom is achieved for each user.
transmission. When restricting our choice of coding scheme In the second range L ≤N ≤L, the cell association that
2 c
to zero-forcing schemes, the puDoF value was characterized achievesapuDoFvalueof Nc+1 isasfollows.Thenetworkis
L+2
as, splitintosubnetworks,eachwithconsecutiveL+2transmitter-
2N
τDzf(L,Nc)= 2N +c L, (5) receiver pairs. In each subnetwork, we decode the last Nc+1
c words.Foreachi∈{L+2,L+1,··· ,L+2−N +1},message
c
and the achieving cell association was found to be the fol- W is associated with {i,i−1,··· ,L+2−N +1} ⊆ C .
i c i
lowing. The network is split into subnetworks; each with Thus the last N words are decoded. The base stations with
c
consecutive 2N + L transmitter-receiver pairs. The last L indicesintheset{2,3,··· ,L+2−N }areinactiveasthereis
c c
transmitters in each subnetwork are inactive to avoid inter- interference from the last transmitter in the subnetwork which
subnetwork interference. The zero-forcing scheme aims to cannotbeeliminated.ThefirstbasestationdecodesW .
L+2−Nc
deliver2N messagesfreeofinterferenceineachsubnetwork, To eliminate the interference caused by the transmitters in the
c
so that the acheived puDoF value is as in (5). In order to do setS ={L+2−N +1,L+2−N +2,··· ,L+1}atthefirst
c c
that with a cooperation constraint that limits each message basestationofthesubnetwork,weaddthefirstbasestationto
to be available at N transmitters, we create two Multi- each C ,∀i∈S. Now for messages with indices in the set S,
c i
ple Input Single Output (MISO) Broadcast Channels (BC) we have used up α =2+i−(L+2−N +1) associations;
i c
within each subnetwork; each with N transmitter-receiver thefactoroftwocomesfromthebasestationresolvingW and
c i
pairs, and ensure that interference across these channels is the first base station of the subnetwork. But each transmitter
eliminated. We now discuss the cell association in the first with indices in the set S\{L + 1} also interferes with the
subnetwork, noting that the remaining subnetworks follow an subnetwork directly preceding this subnetwork. ∀i∈S\{L+
analogous pattern. The first MISO BC consists of the first 1}, the message W interferes with the bottom L+1−i base
i
N transmitter-receiver pairs. For each i ∈ {1,2,··· ,N }, stations of the preceding subnetwork, which is precisely the
c c
message W is associated with base stations with indices in numberofassociationsleftfortherespectivemessagei.e.N −
i c
the following set, C ={i,i+1,··· ,N }. The second MISO α = L + 1 − i, thus inter-subnetwork interference can be
i c i
BC consists of the N transmitters with indices in the set eliminated at those base stations.
c
{N +1,N +2,··· ,2N } and the N receivers with indices In the third range 1 ≤ N ≤ L −1, the cell association
c c c c c 2
in the set {Nc+L+1,Nc+L+2,··· ,2Nc+L}. Note that that achieves the lower bound of 2N2Nc+cL is similar to the one
the middle L receivers in each subnetwork are deactivated describedinSectionIIIforthedownlink.Thenetworkissplit
to eliminate interference between the two MISO BCs. For into disjoint subnetworks; each with consecutive 2N + L
c
each i ∈ {N +L+1,N +L+2,··· ,2N +L}, message transmitter-receiver pairs. For the uplink, we consider two
c c c
W is associated with transmitters that have indices in the sets of indices for transmitters A = {1,2,··· ,N } and
i T c
B = {N + L + 1,N + L + 2··· ,2N + L}, and where δ =(L+1) mod 2.
T c c c
corresponding sets of receivers A = {1,2,··· ,N } and The coding scheme that achieves the inner bound for the
R c
B = {N + 1,N + L + 2··· ,2N }. For each i ∈ A , second range of (8) is essentially the union of the scheme
R c c c T
themessageW isassociatedwiththereceiversreceivingitin describedinSectionIIIandtheschemethatachievesthethird
i
A . Receiver i decodes W and the other associations in C range of (6). The network is split into disjoint subnetworks;
R i i
exist for eliminating interference. Similarly For each j ∈B , each with consecutive 2N +L transmitter-receiver pairs. We
T c
the message W is associated with the receivers receiving it consider two sets of base stations A ={1,2,··· ,N } and
j BS c
in B , but now receiver j − L decodes W and the other B ={N +1,N +L+2··· ,2N },andtwosetsofmobile
R j BS c c c
associations in C are for eliminating interference. terminals A = {1,2,··· ,N } and B = {N +L+
j MT c MT c
Weobservethatifwewerenotrestrictedtothezero-forcing 1,N +L+2··· ,2N +L}. Now for each i ∈ A , C =
c c MT i
coding scheme then for the third range, we could achieve 1 A . Similarly for each j ∈ B , C = B . Thus, for the
2 BS MT j BS
puDoF using the asymptotic interference alignment scheme downlink, we can get the optimal puDoF described in Section
of [16]. III, and for the uplink, we can get the inner bound stated in
the third range of (6).
V. PRELIMINARYCONVERSEPROOFWHENNc =L For the case where N ≥ L + 1, the coding scheme
c
In this section, we provide a converse proof for a special
that achieves the expression described in (8) is as follows.
case of the second range of (6). When N = L, The optimal
c First, we associate each mobile terminal with the L + 1
zero-forcing puDoF for the uplink can be characterised as:
base stations connected to it. This achieves the puDoF value
L+1 of unity during the uplink in the same way as the scheme
τUzf(L,L)= L+2. (7) that achieves it in Section IV. Hence, we know so far that
C ⊇ {i,i−1,i−2,··· ,i−L}∩[K],∀i ∈ [K]. During the
We begin by dividing the network into subnetworks of i
downlink, we divide the network into disjoint subnetworks;
L+2 consecutive transmitters-receiver pairs. We observe that
eachconsistsofN consecutivetransmitter-receiverpairs.This
in any subnetwork, if we have N +1 = L+1 consecutive c
c allows us to create in each subnetwork a MISO broadcast
active receivers (base stations), then the transmitter connected
channel. Let χ be the number of transmitter-receiver pairs
to all these receivers must be inactive, because a message’s
with an inactive node between the last active base station
interference cannot be canceled at N or more receivers. Let
c of one sub-network and the first active mobile terminal in
Γ be the set of subnetworks where all N +2 receivers
BS c the following subnetwork. Then we observe that in order to
are active, and Φ be the set of subnetworks with at most
BS eliminate inter-subnetwork interference, it has to be the case
N active receivers. Similarly, let Γ and Φ be the
c MT MT that χ ≥ L. Because the achieved DoF in any subnetwork is
subnetworks with N + 2 active transmitters and at most
c bound by the minimum of the number of active transmitters
N active transmitters, with respect to order. To be able to
c and the number of active receivers in the subnetwork, we set
achieve a higher puDoF than (7), it must be true that both
the number of inactive mobile terminals to be the same as
conditions hold: |Γ | > |Φ | and |Γ | > |Φ |. Now
BS BS MT MT the number of inactive base stations. Let that aforementioned
note that for any subnetwork that belongs to Γ , at most N
BS c number be (cid:15), then 2(cid:15) = χ ≥ L. Since minimizing (cid:15) will
transmitters will be active, because the interference caused
maximize the achieved DoF, we set (cid:15) = (cid:100)L(cid:101). As we are
by any message cannot be canceled at N or more receivers. 2
c leavingthefirst(cid:15)mobileterminalsinactiveinthesubnetwork,
Hence Γ ⊆ Φ . Further, the same logic applies to
BS MT thefirstbasestation(callitBSp)willbetransmittingmessage
concludethatforanysubnetworkwithN +1activereceivers,
c W tothethemobileterminalMTp+(cid:15).Forthisbroadcast
the number of active transmitters is at most N + 1, and p+(cid:15)
c channel to work, each active base station must be associated
hence Γ ⊆ Φ . It follows that if |Γ | > |Φ |, then
MT BS BS BS with all active mobile terminals in the subnetwork, so that all
|Γ |<|Φ |, and hence the statement is proved.
MT MT interfering signals can be eliminated at each mobile terminal
We conjecture that a similar argument holds to extend (7)
receiver.
andprovethatτUzf(L,Nc)= NLc++21 forthewholesecondrange When δ = 0, BS p+(cid:15) will be delivering W , whose
of (6), i.e., when L ≤N ≤L. p+L+1
2 c mobile terminal was not be associated with BS p through
VI. AVERAGEUPLINK-DOWNLINKDEGREESOFFREEDOM the uplink assignment, so we will need to add p to Cp+L+1.
In [13], the puDoF value τ(L = 1,Nc) was characterized. Thus we can only have Nc − (L + 1) − 1 active base
Here, we present zero-forcing schemes, with the goal of stations, among the base stations in the subnetwork that have
optimizing the average rate across both uplink and downlink indices greater than p+(cid:15). This is because if BS j, where
forarbitraryvaluesofL≥2.ThecorrespondingpuDoFinner j ≥ p+(cid:15)+Nc −(L+1), is active then to ensure that the
bounds are given by, interference caused by it does not propagate, we have to have
p∈C ,butthen|C |>N ,i.e.,mobileterminalj+(cid:15)has
j+(cid:15) j+(cid:15) c
τzf(L,Nc)≥(cid:40)12(cid:16)1+(cid:16)(cid:100)L2(cid:101)+δ+NNcc−(L+1)(cid:17)(cid:17) L+1≤Nc, tcoonbsetraasisnotcaialltoewdswfitohr.mSooreinbaesaecshtastuiobnnsettwhaotrwk,hwatethweibllahckavheaual
2N2Nc+cL 1≤Nc ≤L(8,) tionttaelrfeorfen∆ce,=ou(cid:15)t+ofNacto−tal(Lof+∆1+)(cid:15)w=orNds tmraonbsimleittteerdmwiniathlsoiunt
c
VII. CONCLUDINGREMARKS
It is important to note that most of the presented decoding
schemes for the uplink require a propagation delay that scales
with the number of users in the network K, even if the
employed subnetworks are small. We plan to consider delay
constraints in future work. We also would like to highlight
that this theoretical work is a preliminary effort to understand
optimal cell association decisions in cellular networks, with
cloud-based controllers that access the base stations through
a limited backhaul. Both channel model and cooperation
constraint assumptions may not be practical. Nevertheless, we
believethatthecapturedinsightscanleadtorigoroussolutions
Fig. 1: Scheme for average uplink (blue shade) and downlink for more practical settings.
(green shade) communication when N ≤L
c REFERENCES
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1 and 2 serve as examples for the above scheme. Figure 1
freedom of the K-user interference channel,” IEEE Trans. Inf. Theory,
uses values of Nc = 3, and L = 3. Using this scheme we vol.54,no.8,pp.3425-3441,Aug.2008.
get a puDoF of 2, which is equivalent to 2Nc . Figure 2
3 2Nc+L
uses values of N = L+2 for L = 5 and L = 4 for parts
c
a and b, respectively. The achieved puDoF values are 4 or 4,
7 6
respectively.