Opportunistic Downlink Interference Alignment for Multi-Cell MIMO Networks

0.36 MB·

by  Hyun Jong Yang

IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 1
Opportunistic Downlink Interference
Alignment for Multi-Cell MIMO Networks
Hyun Jong Yang, Member, IEEE, Won-Yong Shin, Senior Member, IEEE,
Bang Chul Jung, Senior Member, IEEE, Changho Suh, Member, IEEE,
and Arogyaswami Paulraj, Fellow, IEEE
Abstract
Inthispaper,weproposeanopportunisticdownlinkinterferencealignment(ODIA)forinterference-
7
1 limitedcellulardownlink,whichintelligentlycombinesuserschedulinganddownlinkIAtechniques.
0 The proposed ODIA not only efficiently reduces the effect of inter-cell interference from other-cell
2 base stations (BSs) but also eliminates intra-cell interference among spatial streams in the same

n cell. We show that the minimum number of users required to achieve a target degrees-of-freedom
a (DoF) can be fundamentally reduced, i.e., the fundamental user scaling law can be improved by
J
using the ODIA, compared with the existing downlink IA schemes. In addition, we adopt a limited

6
feedback strategy in the ODIA framework, and then analyze the number of feedback bits required

forthesystemwithlimitedfeedbacktoachievethesameuserscalinglawoftheODIAasthesystem
]
T with perfect CSI. We also modify the original ODIA in order to further improve sum-rate, which
I achievesthe optimalmultiuserdiversity gain,i.e., loglogN, per spatial stream evenin the presence
.
s ofdownlinkinter-cellinterference,whereN denotesthenumberofusersinacell.Simulationresults
c
showthattheODIAsignificantlyoutperformsexistinginterferencemanagementtechniquesinterms
[
ofsum-rateinrealisticcellularenvironments.NotethattheODIAoperatesinanon-collaborativeand

1
decoupledmanner,i.e.,itrequiresnoinformationexchangeamongBSsandnoiterativebeamformer
v
optimization between BSs and users, thus leading to an easier implementation.
5
2
6 Index Terms
1
0 Inter-cell interference, interference alignment, degrees-of-freedom (DoF), transmit & receive
. beamforming, limited feedback, multiuser diversity, user scheduling.
1
0
7
1
:
v
i
X
r
a
H.J.YangiswiththeSchoolofElectricalandComputerEngineering,UNIST,Ulsan689-798,RepublicofKorea(E-mail:
[email protected]).
W.-Y.ShiniswiththeDepartmentofComputerScienceandEngineering,DankookUniversity,Yongin448-701,Republic
of Korea (E-mail: [email protected]).
B. C. Jung (corresponding author) is with the Department of Electronics Engineering, Chungnam National University,
Daejeon 305-764, Republic of Korea (E-mail: [email protected]).
C. Suh is with the Department of Electrical Engineering, KAIST, Daejeon 305-701, Republic of Korea (E-mail:
[email protected]).
A. Paulraj is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305 (email:
[email protected]).
IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 2
I. INTRODUCTION
Interference management has been taken into account as one of the most challenging
issues to increase the throughput of cellular networks serving multiple users. In multiuser
cellular environments, each receiver may suffer from intra-cell and inter-cell interference.
Interference alignment (IA) was proposed by fundamentally solving the interference problem
when there are multiple communication pairs [1]. It was shown that the IA scheme can
achieve the optimal degrees-of-freedom (DoF)1 in the multiuser interference channel with
time-varying channel coefficients. Subsequent studies have shown that the IA is also useful
and indeed achieves the optimal DoF in various wireless multiuser network setups: multiple-
input multiple-output (MIMO) interference channels [2], [3] and cellular networks [4], [5].
In particular, IA techniques [4], [5] for cellular uplink and downlink networks, also known
as the interfering multiple-access channel (IMAC) or interfering broadcast channel (IBC),
respectively, have received much attention. The existing IA framework for cellular networks,
however, still has several practical challenges: the scheme proposed in [5] requires arbitrarily
large frequency/time-domaindimensionextension,and the schemeproposed in [4] is based on
iterative optimization of processing matrices and cannot be optimally extended to an arbitrary
downlink cellular network in terms of achievable DoF.
In the literature, there are some results on the usefulness of fading in single-cell downlink
broadcast channels, where one can obtain multiuser diversity gain along with user schedul-
ing as the number of users is sufficiently large: opportunistic scheduling [6], opportunistic
beamforming [7], and random beamforming [8]. Scenarios exploiting multiuser diversity gain
have been studied also in ad hoc networks [9], cognitive radio networks [10], and cellular
networks [11].
Recently, the concept of opportunistic IA (OIA) was introduced in [12]–[14] for the K-cell
uplink network (i,e., IMAC model), where there are one M-antenna base station (BS) and
N users in each cell. The OIA scheme incorporates user scheduling into the classical IA
framework by opportunistically selecting S (S ≤ M) users amongst the N users in each cell
in the sense that inter-cell interference is aligned at a pre-defined interference space. It was
shown in [13], [14] that one can asymptotically achieve the optimal DoF if the number of
users in a cell scales as a certain function of the signal-to-noise-ratio (SNR). For the K-cell
downlink network (i.e., IBC model) assuming one M-antenna base station (BS) and N per-
cell users, studies on the OIA have been conducted in [15]–[20]. More specifically, the user
scaling condition for obtaining the optimal DoF was characterized for the K-cell multiple-
input single-output (MISO) IBC [15], and then such an analysis of the DoF achievability was
extended to the K-cell MIMO IBC with L receive antennas at each user [16]–[20]—full DoF
can be achieved asymptotically, provided that N scales faster than SNRKM−L, for the K-cell
MIMO IBC using OIA [19], [20].
In this paper, we propose an opportunistic downlink IA (ODIA) framework as a promising
interference management technique for K-cell downlink networks, where each cell consists of
one BS with M antennas and N users having L antennas each. The proposed ODIA jointly
takes into account user scheduling and downlink IA issues. In particular, inspired by the
precoderdesignin[4],weusetwocascadedbeamformingmatricestoconstructourprecoderat
eachBS.Todesignthefirsttransmitbeamformingmatrix,weuseauser-specificbeamforming,
which conducts a linear zero-forcing (ZF) filtering and thus eliminates intra-cell interference
among spatial streams in the same cell. To design the second transmit beamforming matrix,
we use a predetermined reference beamforming matrix, which plays the same role of random
1It is referred that ‘optimal’ DoF is achievable if the outer-bound on DoF for given network configuration is achievable.
IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 3
beamforming for cellular downlink [15], [19], [20] and thus efficiently reduces the effect
of inter-cell interference from other-cell BSs. On the other hand, the receive beamforming
vector is designed at each user in the sense of minimizing the total amount of received
inter-cell interference using local channel state information (CSI) in a decentralized manner.
Each user feeds back both the effective channel vector and the quantity of received inter-cell
interference to its home-cell BS. The user selection and transmit beamforming at the BSs and
the design of receive beamforming at the users are completely decoupled. Hence, the ODIA
operates in a non-collaborative manner while requiring no information exchange among BSs
and no iterative optimization between transmitters and receivers, thereby resulting in an easier
implementation.
The main contribution of this paper is four-fold as follows.
• We first show that the minimum number of users required to achieve S DoF (S ≤ M)
can be fundamentally reduced to SNR(K−1)S−L+1 by using the ODIA at the expense
of acquiring perfect CSI at the BSs from users, compared to the existing downlink IA
schemes requiring the user scaling law N = ω(SNRKS−L) [19], [20],2 where S denotes
the number of spatial streams per cell. The interference decaying rate with respect to N
for given SNR is also characterized in regards to the derived user scaling law.
We introduce a limited feedback strategy in the ODIA framework, and then analyze the
•
required number of feedback bits leading to the same DoF performance as that of the
ODIA assuming perfect feedback, which is given by ω(log SNR).
2
We present a user schedulingmethod for the ODIA to achieveoptimalmultiuserdiversity
•
gain, i.e., loglogN per stream even in the presence of downlink inter-cell interference.
To verify the ODIA schemes, we perform numerical evaluationvia computersimulations.
•
Simulation results show that the proposed ODIA significantly outperforms existing in-
terference management and user scheduling techniques in terms of sum-rate in realistic
cellular environments.
The remainder of this paper is organized as follows. Section II describes the system and
channel models. Section III presents the overall procedure of the proposed ODIA. In Section
IV, the DoF achievablility result is shown. Section V presents the ODIA scheme with limited
feedback. In Section VI, the achievability of the spectrally efficient ODIA leading to a better
sum-rate performance is characterized. Numerical results are shown in Section VII. Section
VIII summarizes the paper with some concluding remarks.
II. SYSTEM AND CHANNEL MODELS
We consider a K-cell MIMO IBC where each cell consists of a BS with M antennas and N
users with L antennas each. The number of selected users in each cell is denoted by S(≤ M).
It is assumed that each selected user receives a single spatial stream. To consider nontrivial
cases, we assume that L < (K−1)S+1, because all inter-cell interference can be completely
canceled at the receivers (i.e., users) otherwise. Moreover, the number of antennas at the users
is in general limited due to the size of the form factor, and hence it is more safe to assume
that L is relatively small compared to (K−1)S+1. The channel matrix from the k-th BS to
the j-th user in the i-th cell is denoted by H[i,j] ∈ CL×M, where i,k ∈ K , {1,...,K} and
k
j ∈ N , {1,...,N}. Each element of H[i,j] is assumed to be independent and identically
k
distributed (i.i.d.) according to CN(0,1). In addition, quasi-static frequency-flat fading is
assumed, i.e., channel coefficients are constant during one transmission block and change
2f(x)=ω(g(x))implies that limx→∞ fg((xx)) =0.
IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 4
u[1,1]HH[1,1]P ,η[1,1]
1 1
u[1,1]
User[1,1] H[1,1]PHv[2[12,,12]]P2v[2,1] H[31,1]P3v[3,2]
2 2 H[1,1]Pv[3,1]
BS 1 3 3
x[1,1]
x[1,2] [v[1V,1]1 v=[1,2]] (m3Pa-btry1ix-2) u[1,2] H[31,2]P3v[3,2]
User[1,2]
H[1,2]Pv[2,1]
2 2
H[1,2]Pv[3,1]
3 3
u[1,2]HH1[1,2]P1,η[1,2] H[21,2]P2v[2,2]
User[2,1]
BS 2
x[2,1]
V2= P2
x[2,2] [v[2,1] v[2,2]] (3-by-2
matrix)
User[2,2]
User[3,1]
BS 3
x[3,1]
x[3,2] V3= (3P-by3-2
[v[3,1] v[3,2]]
matrix)
User[3,2]
Fig. 1. The MIMO IBC model, where K =3, M =3, S =2, L=3, and N =2.
to new independent values for every transmission block. The j-th user in the i-th cell can
estimate the channels H[i,j], k = 1,...,K, using pilot signals sent from all the BSs.
k
The received signal vector at the j-th user in the i-th cell is expressed as:
K
y[i,j] = H[i,j]s +z[i,j], (1)
k k
k=1
X
where s ∈ CM×1 is the transmit signal vector at the k-th BS with unit average power, i.e.,
k
Eks k2 = 1, and z[i,j] ∈ CL×1 denotes additive noise, each element of which is independent
k
and identically distributed complex Gaussian with zero mean and the variance of N . The
0
2
average SNR is given by SNR = E H[i,j]s /E z[i,j] 2 = 1/N . Thus, in what follows
i i 0
we shall use the notation N = SN(cid:20)R(cid:13)−1 for n(cid:13)ota(cid:21)tionha(cid:13)l simp(cid:13)liicity.
0 (cid:13) (cid:13) (cid:13) (cid:13)
Figure 1 shows an example of th(cid:13)e MIMO(cid:13) IBC model, where K = 3, M = 3, S = 2,
L = 3, and N = 2. The details in the figure will be described in the subsequent section.
III. PROPOSED ODIA
We first describe the overall procedure of our proposed ODIA scheme for the MIMO IBC,
and then define its achievable sum-rate and DoF.
IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 5
A. Overall Procedure
The ODIA scheme is described according to the following four steps.
1) Initialization (Broadcast of Reference Beamforming Matrices): The reference beam-
formingmatrixat theBS in thek-th cell is givenby P = [p ,...,p ], wherep ∈ CM×1
k 1,k S,k s,k
is an orthonormal vector for k ∈ K and s = 1,...,S. That is, P is an orthonormal basis
k
for an S-dimensional subspace of CM×M. Each BS randomly generates P independently of
k
the other BSs. If the reference beamforming matrix is generated in a pseudo-random fashion,
i.e., it changes based on a certain pattern as if it changes randomly and the pattern is known
by the BSs as well as the users, BSs do not need to broadcast them to users. Then, the j-th
user in the i-th cell obtains H[i,j] and P , k = 1,...,K.
k k
2) Receive Beamforming & Scheduling Metric Feedback: In the second step, we explain
how to decide a user scheduling metric at each user along with given receive beamforming,
where the design of receive beamforming will be explained in Section IV. Let u[i,j] ∈ CL×1
denote the unit-norm weight vector at the j-th user in the i-th cell, i.e., u[i,j] 2 = 1. Note
that the user-specific beamforming V will be utilized only to cancel intra-cell interference
k
(cid:13) (cid:13)
out, and the inter-cell interference will be suppressed from user schedul(cid:13)ing, w(cid:13)hich will be
specified later. Thus, from the notion of P and H[i,j], the j-th user in the i-th cell can
k k
compute the following quantity while using its receive beamforming vector u[i,j], which is
given by
H 2
η˜[i,j] = u[i,j] H[i,j]P , (2)
k k k
(cid:13) (cid:13)
where i ∈ K, j ∈ N, and k ∈ K\i = {1(cid:13),...,i−1,i+(cid:13)1,...,K}. Using (2), the scheduling
(cid:13) (cid:13)
metric at the j-th user in the i-th cell, denoted by η[i,j], is defined as the sum of η˜[i,j]. That
k
is,
K
η[i,j] = η˜[i,j]. (3)
k
k=1,k6=i
X
As illustrated in Fig. 1, each user feeds the metric in (3) back to its home-cell BS. In
addition to the scheduling metric in (3), for each BS to design the user-specific beamforming
V , each user needs to feed back the information of the following vector
k
H H
f[i,j] , u[i,j] H[i,j]P . (4)
i i i
(cid:16) (cid:17)
3) User Scheduling: Upon receiving N users’ scheduling metrics in the serving cell, each
BS selects S users having the metrics up to the S-th smallest one. Without loss of generality,
the indices of selected users in every cell are assumed to be (1,...,S). Although η˜[i,j] is not
k
exactly the amount of the generating interference from the k-th BS to the j-th user in the i-th
cell due to the absence of V , it decouples the design of the user-specific precoding matrix
k
V from the user scheduling metric calculation, i.e., η[i,j] includes no information of V .
k k k
In addition, we shall show in the sequel that the inter-cell interference can be successfully
suppressed by using the metric η[i,j] even with V excluded and that the optimal DoF can be
k k
achieved.
At this point, it is worthwhile to note that the role of P is two-fold. First, it determines the
k
dimension of the effective received channel according to given parameter S. By multiplying
P to the channel matrix, the dimension of the effective channel is reduced to S rather than
k
M, whichresultsin reduced numberofinter-cell interference termsas wellas reduced average
IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 6
interference level for each interference term. We shall show in the sequel that P plays a role
k
in the end of rendering the user scaling law dependent on the parameter S.
Second, P separates the user schedulingprocedure from theuser-specific precoding matrix
k
design of V and also from the receiver beamforming vector design of u . By employing the
k k
cascaded precoding matrix design, the scheduling metric in (1) becomes independent of V
k
or u , and u can be obtained as a function of only H[i,j] and P as shown in (18).
k k k k
The reason why P is designed to change in a pseudo-random fashion is to increase the
k
fairness of the users scheduling by randomizing the scheduling metric of each user, but can
also be fixed if the fairness is not a matter or the channel changes fast enough. In addition,
if one wants to further improve the achievable rate, P may be channel-specifically designed
k
combined with the user scheduling, which however results in a collaborative and iterative
user scheduling and precoding matrix design.
In this and subsequent sections, we focus on how to simply design a user scheduling
method to guarantee the optimal DoF. An enhanced scheduling algorithm jointly taking into
account the vector to be fed back in (4) and the scheduling metric in (3) may provide a better
performance in terms of sum-rate, which shall be discussed in Section VI.
4) Transmit Beamforming & Downlink Data Transmission: As illustrated in Fig. 1, the
precoding matrix at each BS is composed of the product of the predetermined reference
beamforming matrix P and the user-specific precoding matrix V = v[i,1],...,v[i,S] , where
k i
v[i,s] ∈ CS×1, i ∈ K. Let us denote the transmit symbol at the i-th BS transmitted to the j-th
user by x[i,j], where E x[i,s] 2 = 1/S for s = 1,...,S. Denoting the(cid:2)transmit symbo(cid:3)l vector
by x = x[i,1],...,x[i,S] T, the transmit signal vector at the i-th BS is given by s = P V x ,
i (cid:12) (cid:12) i i i i
and the received signal(cid:12)vecto(cid:12)r at the j-th user in the i-th cell is written as
(cid:2) (cid:3)
K
y[i,j] = H[i,j]P V x + H[i,j]P V x +z[i,j]
i i i i k k k k
k=1,k6=i
X
S
= H[i,j]P v[i,j]x[i,j]+ H[i,j]P v[i,s]x[i,s]
i i i i
s=1,s6=j
desiredsignal X
intra-cellinterference
| {z }
K
+ H[i,j]P V x|+z[i,j]. {z } (5)
k k k k
k=1,k6=i
X
inter-cellinterference
H
The received signal vector a|fter receiv{ez beamfo}rming, denoted by y˜[i,j] = u[i,j] y[i,j], can be
rewritten as:
S
H H
y˜[i,j] = f[i,j] v[i,j]x[i,j] +f[i,j] v[i,s]x[i,s]
i i
s=1,s6=j
X
K
H H
+ f[i,j] V x +u[i,j] z[i,j], (6)
k k k
k=1,k6=i
X
where f[i,j]H = u[i,j]HH[i,j]P . By selecting users with small η[i,j] in (3), H[i,j]P tends to
k k k k k
be orthogonal to the receive beamforming vector u[i,j]; thus, inter-cell interference channel
matrices H[i,j]P V in (6) also tend to be orthogonal to u[i,j] as illustrated in Fig. 1.
k k k
IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 7
To cancel out intra-cell interference, the user-specific beamforming matrix V ∈ CS×Sis
i
given by
V = [v[i,1],v[i,2],...,v[i,S]]
i
u[i,1]HH[i,1]P −1 γ[i,1] 0 ··· 0
i i
= u[i,2]HH... [ii,2]Pi ·p0... pγ...[i,2] ·.·..· 0... , (7)
 
u[i,S]HH[i,S]P   0 0 ··· γ[i,S]
 i i  
 
 
p
where γ[i,j] denotes a normalization factor for satisfying the unit-transmit power constraint
for each spatial stream, i.e., γ[i,j] = 1/ P v[i,j] . In consequence, the received signal can be
p i
simplified to
(cid:13) (cid:13)
(cid:13) (cid:13)
K
H H
y˜[i,j] = γ[i,j]x[i,j]+ f[i,j] V x +u[i,j] z[i,j], (8)
k k k
k=1,k6=i
p X
inter-cellinterference
which thus does not contain the intra-cell interference term.
| {z }
As in [13], [21]–[25], we assume no loss in exchanging signaling messages such as
information of effective channels, scheduling metrics, and receive beamforming vectors.
B. Achievable Sum-Rate and DoF
From (8), the achievable rate of the j-th user in the i-th cell is given by
R[i,j] = log 1+SINR[i,j]
2
(cid:16) (cid:17)
γ[i,j] ·|x[i,j]|2
= log 1+
2 u[i,j]Hz[i,j] 2 +I˜[i,j]
 
 (cid:12) (cid:12) 
(cid:12) (cid:12)
(cid:12) (cid:12) γ[i,j]
= log 1+ , (9)
2 H 2
S + K S f[i,j] v[k,s]
SNR k=1,k6=i s=1 k
 
 (cid:12) (cid:12) 
H 2 P P (cid:12) (cid:12)
where I˜[i,j] , K f[i,j] V x . (cid:12) (cid:12)
k=1,k6=i k k k
Using (9), the achievable total DoF can be defined as [26]
(cid:12) (cid:12)
P (cid:12) (cid:12)
(cid:12) (cid:12) K S R[i,j]
DoF = lim i=1 j=1 . (10)
SNR→∞ P loPgSNR
IV. DOF ACHIEVABILITY
In this section, we characterize the DoF achievability in terms of the user scaling law with
theoptimalreceivebeamformingtechnique.Tothisend,westartwiththereceivebeamforming
design that maximizes the achievableDoF. For givenchannel instance, from (9), each user can
H 2
attainthemaximumDoF of1 ifand onlyiftheinterference K S f[i,j] v[k,s] ·SNR
k=1,k6=i s=1 k
(cid:12) (cid:12)
P P (cid:12) (cid:12)
(cid:12) (cid:12)
IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 8
remains constant for increasing SNR. Note that R[i,j] can be bounded as
γ[i,j]
R[i,j]≥log 1+ (11)
2 2 
S + K S f[i,j] kv[k,s]k2
SNR k=1,k6=i s=1 k
 
 (cid:13) (cid:13) 
P P (cid:13) (cid:13)
γ[i,j] (cid:13) (cid:13)
≥ log 1+ (12)
2 2 2
S + K S f[i,j] v(max)
SNR k6=i s=1 k i
 
 P P (cid:13)(cid:13) γ(cid:13)(cid:13)[i,j](cid:13)(cid:13) (cid:13)(cid:13) 
1 (cid:13) (cid:13)(cid:13)v(m(cid:13)ax)(cid:13)(cid:13)(cid:13)2 (cid:13)
= log (SNR)+log  + (cid:13) i (cid:13) , (13)
2 2 SNR S +I[i,j]
(cid:13)(cid:13)v(max)(cid:13)(cid:13)2
 (cid:13) i (cid:13) 
 
 
where v(max) in (12) is defined by
i
v(max) = argmax v[i′,j′] 2 : i′ ∈ K\i,j′ ∈ S , (14)
i
(cid:26) (cid:27)
(cid:13) (cid:13)
(cid:13) (cid:13)
S , {1,...,S}, and I[i,j] in (13) is defin(cid:13)ed by (cid:13)
K S
2
I[i,j] , f[i,j] ·SNR. (15)
k
k=X1,k6=iXs=1 (cid:13) (cid:13)
(cid:13) (cid:13)
Here, v(max) is fixed for given channel instance(cid:13), beca(cid:13)use v[i,j] is determined by H[i,j], j =
i i
1,...,S. Recalling that the indices of the selected users are (1,...,S) for all cells, we can
expect the DoF of 1 for each user if and only if for some 0 ≤ ǫ < ∞,
I[i,j] < ǫ, ∀j ∈ S,i ∈ K. (16)
To maximizetheachievableDoF, weaimtominimizethesum-interference K S I[i,j]
i=1 j=1
through receive beamforming at the users. Since I[i,j] = S η[i,j]SNR, we have
s=1 P P
K S K S S PK S
I[i,j] = η[i,j]SNR = S η[i,j]SNR. (17)
i=1 j=1 i=1 j=1 s=1 i=1 j=1
XX XXX XX
The equation (17) implies that the collection of distributed effort to minimize η[i,j] at the users
canreducethesumofreceivedinterference. Therefore,eachuserfindsthebeamformingvector
that minimizes η[i,j] from
K
2
u[i,j] = argminη[i,j] = argmin uHH[i,j]P . (18)
u u k k
k=X1,k6=i(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 9
Let us denote the augmented interference matrix by
G[i,j] , H[i,j]P ,..., H[i,j]P , H[i,j]P ,
1 1 i−1 i−1 i+1 i+1
"
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
H
..., H[i,j]P ∈ C(K−1)S×L, (19)
K K
#
(cid:16) (cid:17)
and the singular value decomposition of G[i,j] by
H
G[i,j] = Ω[i,j]Σ[i,j]Q[i,j] , (20)
where Ω[i,j] ∈ C(K−1)S×L and Q[i,j] ∈ CL×L consist of L orthonormal columns, and Σ[i,j] =
diag σ[i,j],...,σ[i,j] , where σ[i,j] ≥ ··· ≥ σ[i,j]. Then, the optimal u[i,j] is determined as
1 L 1 L
(cid:16) (cid:17)
u[i,j] = q[i,j], (21)
L
where q[i,j] is the L-th column of Q[i,j]. With this choice the scheduling metric is simplified
L
to
η[i,j] = σ[i,j]2. (22)
L
Since each column of P is isotropically and independently distributed, each element of the
k
effective interference channel matrix G[i,j] is i.i.d. complex Gaussian with zero mean and unit
variance.
Remark 1: In general, the conventional scheduling metric such as SNR or SINR in the IBC
is dependent on the precoding matrices at the transmitters, which makes the joint optimization
of the precoder design and user scheduling difficult to be separated from each other and
implemented with feasible signaling overhead and low complexity. The previous schemes
[2], [27] for the IBC only consider the design of the precoding matrices and receive filters
without any consideration of user scheduling.
With the cascaded precoding matrix design, however, the proposed scheme decouples the
user scheduling metric calculation and the user-specific precoding matrix V , as shown in
i
(2). In addition, the receive beamforming vector design can also be decoupled from V as
i
shownin (18). A similarcascaded precoding matrix design was used in [4] for someparticular
cases of the antenna configuration without the consideration of user scheduling. However, the
proposed scheme applies to an arbitrary antenna and channel configuration, where the inter-
cell interference is suppressed with the aid of opportunistic user scheduling. In addition, we
shall show in the sequel that the optimal DoF can be achievable under a certain user scaling
condition for an arbitrary antenna configuration without any iterative optimization procedure
between the users and BSs.
Remark 2: Note that although it is assumed in the proposed scheme that each user feeds
back the (1 × S)-dimensional vector f[i,j] to its home cell, the amount of CSI feedback is
i
equivalent to that in the conventional single-cell MU-MIMO scheme such as ZF or minimum
mean-squared error (MMSE) precoding. On the other hand, the previous iterative transceiver
design schemes [2], [27] based on local CSI for the IBC require all the selected users to
feed back the information of the receive beamformer to all the BSs in the network, which
results in K times more feedback compared to the single-cell MU-MIMO scheme even for
one iteration where the users feed back their receive beamformers and the BSs update their
transmit precoders once. Furthermore, the information of weight coefficients also needs to be
IEEETRANSACTIONSONWIRELESSCOMMUNICATIONS 10
fed back to all the BSs in [27]. We shall show via numerical simulations in the sequel that
even with K times less feedback the proposed scheme exhibits superior sum-rate compared
to the iterative scheme [27].
We start with the following lemma to derive the achievable DoF.
Lemma 1 (Lemma 1 [14]): The CDF of η[i,j], denoted by F (x), can be written as
η
F (x) = c x(K−1)S−L+1 +o x(K−1)S−L+1 , (23)
η 0
for 0 ≤ x < 1, where f(x) = o(g(x)) means lim (cid:0)f(x) = 0, and(cid:1)c˜ is a constant determined
x→∞ g(x) 0
by K, S, and L.
We further present the followinglemmafor theprobabilisticinterference levelof theODIA.
Lemma 2: The sum-interference remains constant with high probability for increasing
SNR, that is,
K S
P , lim Pr I[i,j] ≤ ǫ = 1 (24)
SNR→∞ ( )
i=1 j=1
XX
for any 0 < ǫ < ∞, if
N = ω SNR(K−1)S−L+1 . (25)
(cid:16) (cid:17)
Proof: See appendix A.
Now, the following theorem establishes the DoF achievability of the proposed ODIA.
Theorem 1 (User scaling law): The proposed ODIA scheme with the scheduling metric
(22) achieves the optimal KS DoF for given S with high probability if
N = ω SNR(K−1)S−L+1 . (26)
(cid:16) (cid:17)
Proof: If the sum-interference remains constant for increasing SNR with probability P,
the achievable rate in (13) can be further bounded by
R[i,j]
2
γ[i,j]/ S v(max)
1 i
≥ P·log (SNR)+log + (cid:18) (cid:13) (cid:13) (cid:19), (27)
2 2 SNR 1/ v(m(cid:13)(cid:13)ax) 2 +(cid:13)(cid:13)ǫ
  i 
  
(cid:13) (cid:13)
  
for any 0 ≤ ǫ < ∞. Thus, the achievable DoF in (10) c(cid:13)an be b(cid:13)ounded by
(cid:13) (cid:13)
DoF ≥ KS ·P. (28)
From Lemma 2, it is immediate to show that P tends to 1, and hence KS DoF is achievable
if N = ω SNR(K−1)S−L+1 , which proves the theorem.
From Theorem 1, itis shownthatthere existafundamentaltrade-offbetween theachievable
(cid:16) (cid:17)
DoF KS and required user scaling of N = ω SNR(K−1)S−L+1 . This trade-off can also be
observed in terms of the sum-rate even under a practical system setup, as we shall show in
(cid:16) (cid:17)
Section VII. Therefore, a higher S value can be chosen to achieve higher DoF or sum-rate if
there exist more users in the network.
The following remark discusses the uplink and downlink duality on the DoF achievability
within the OIA framework.