Linear successive user allocation in the Multi-Cell MIMO environment

Abstract

An interference management method for coordinating downlink transmission in a Multiple-Input Multiple-Output (MIMO) cellular network is proposed. The problem is to efficiently manage inter-cell interference in a multi-cell environment, by so-called transmitter cooperation, in order to reduce the diminishing effects of interference on the networks performance. To allow for application in deployable networks, an utmost concern of the presented algorithm is to provide a low-complexity solution avoiding costly combinatorial or non-convex optimization problems. The problem is solved by a network wide successive allocation of data streams and choosing linear transmit and receive filters for each data stream, such that interference is completely avoided. By embedding our new algorithm in a more general framework for interference coordination, we can show relevant gains for network performance where especially the cell-edge users profit.

Full text

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Linear Successive User Allocation in the Multi-Cell
MIMO Environment
Andreas Dotzler,∗Wolfgang Utschick,∗and Guido Dietl†
∗Associate Institute for Signal Processing, Technische Universit¨ at M¨ unchen, 80290 Munich
E-Mail: {dotzler,utschick}@tum.de, Tel.: +49-89-289-28510, Fax: +49-89-289-28504
†DOCOMO Communications Laboratories Europe GmbH, Landsberger Str. 312, 80687 Munich
E-Mail: dietl@docomolab-euro.com, Tel.: +49-89-56824-245 Fax: +49-89-56824-301
Abstract—An interference management method for coordi-
nating downlink transmission in a Multiple-Input Multiple-
Output (MIMO) cellular network is proposed. The problem
is to efficiently manage inter-cell interference in a multi-cell
environment, by so-called transmitter cooperation, in order to
reduce the diminishing effects of interference on the networks
performance. To allow for application in deployable networks, an
utmost concern of the presented algorithm is to provide a low-
complexity solution avoiding costly combinatorial or non-convex
optimization problems. The problem is solved by a network wide
successive allocation of data streams and choosing linear transmit
and receive filters for each data stream, such that interference
is completely avoided. By embedding our new algorithm in a
more general framework for interference coordination, we can
show relevant gains for network performance where especially
the cell-edge users profit.
I. INTRODUCTION
In the downlink of a cellular network inter-cell interference
(ICI) can be a severely limiting factor, especially users at the
cell edge are affected and might be excluded from network
service. A possible solution to completely eliminate ICI is
the joint encoding of information over multiple transmitters
[1], [2], so-called network MIMO. In case full channel state
information (CSI) and all data is available at a central con-
troller, joint encoding over geographically distributed antennas
renders the network into a super-cell, and network MIMO can
efficiently exploit all spatial degrees of freedom to eliminate
ICI. Network MIMO requires a huge amount of additional
complexity and signaling compared to single cell signal
processing and might be difficult to implement in practice.
Therefore, methods aiming at elimination of interference by
cooperation of the transmitters, while every user is served by
a single transmitter, are attractive for deployable networks. In
order to cancel interference, user signals are orthogonalized
in the signal space constituted by the available resources, for
example time, frequency, and space. A simple scheme that
completely removes ICI is to exclusively allocate frequency
bands to transmitters and apply any single cell algorithm
on the allocated carriers, which corresponds to the classical
frequency reuse planning, a very simple form of interfer-
ence management, missing the opportunity for cooperation in
the spatial domain. The availability of multiple antennas at
transmitter and receiver allows multiple users to be served
interference free at the same time on the same frequency by
spatial multiplexing. Interference coordination by adjusting
the transmission space of each user is well understood and
can be solved optimally for a single cell [3], [4]. In con-
ventional cellular network design, signal processing in the
spatial domain is only performed per cell, but interesting
research towards extending spatial multiplexing over multiple
transmitters is emerging. For networks where the transmitters
have multiple antennas and the receivers are equipped with
a single antenna (MISO), the optimal solution is known [5],
however the presented algorithm has prohibitive complexity
for larger networks. Methods based on pricing for the in-
terference caused [6], thresholds for a forbidden interference
direction [7], or so-called interference temperatures [8] have
potential for implementation with reasonable complexity.
Coordinated transmission strategies for full MIMO sys-
tems are mainly available for smaller scenarios, for example
two or multiple interfering point-to-point transmissions [9],
[10]. In [11] a gradient projection method on the covari-
ance matrices of the input signals to interfering point-to-
point MIMO links is used to compute a local maximum.
Recent results on the maximum available degrees of freedom
in interference networks [12], and interference alignment as
technique to achieve them [13], appear to be a promising
research direction. However existing algorithms to compute
transmit and receive filters such that interference is aligned,
for example the maxSINR algorithm [14], do not provide
a user selection method, which is an important feature of
our algorithm. In our method every user is served by a
single cell and transmitters cooperate by jointly adjusting the
transmission space of each stream in order to eliminate or
reduce interference. By choosing linear transmit and receive
filters for each data stream, interference can be completely
avoided. The data streams transmitted are determined by a
network wide successive allocation, inspired by the LInear
Successive Allocation (LISA) [15], [16]. Depending on the
network regarded, complete cancellation of interference by
zero-forcing is not desired as it drastically reduces the number
of active data streams. After showing encouraging results in a
small network, we embed our algorithm in a framework [17],

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that enables to apply the algorithm only to these users that
benefit enough from removing interference, leading to a sig-
nificant improvement of the network performance, especially
for the cell-edge users. We summarize the main features of
our algorithm:
• linear receive and transmit filter to avoid the high com-
putational complexity of Dirty Paper Coding
• successive user allocation to avoid costly combinatorial
optimization problems
• reduced data exchange compared to network MIMO by
coordination of single-cell transmission strategies
II. SYSTEM MODEL AND PROBLEM FORMULATION
The cellular system is given by a set of transmit arrays
S,S = |S|, and a set of users K,K = |K| distributed
throughout the covered area. User assignment to a transmitter
is done by a cell selection scheme formally described by a
mapping f : K → S. We usually assume the assignment to a
transmitter to be fixed for each user and therefore f partitions
the users such that
K = K1∪ K2∪ ... ∪ KS and Ki∩ Kj= ∅ if i ?= j.
MTX,sis the number of transmit antennas of a transmitter s and
MRX,kis the number of receive antennas of user k. Although
not a prerequisite for our algorithm, we assume the same
number of antennas for each transmitter and each user and
therefore
MTX,s= MTX, ∀ s ∈ S,
and
MRX,k= MRX, ∀ k ∈ K.
The channel matrices are
{Hks}k∈K,s∈S∈ CMRX×MTX,
where Hks is the channel matrix between transmitter s
and user k. The received signal of user k consists of the
desired signal, intra-cell, and inter-cell interference and can
be expressed as
yk= Hkf(k)xk+
?
i∈Kf(k)\k
?
Hkf(k)xi
???
intra-cell interference
+
?
i∈K\Kf(k)
?
Hkf(i)xi
???
inter-cell interference
+η,
where xi ∈ CMTXis the transmit signal for user i and
η ∼ CN(0,σ2IMRX) represents white Gaussian noise, where
IMRXdenotes the MRX× MRX identity matrix and σ2is
the power of the noise received by every antenna. Assuming
Gaussian modulation, the covariance matrix of the transmit
symbol xiis E{xixH
the covariance matrix of the noise plus interference for user k
is given by
i} = Qi. As we assume linear precoding,
Zk= σ2IMRX+
?
i∈K\k
Hkf(i)QiHH
kf(i),
and the information theoretic rate for user k is given by


In the work presented, communicationin the network is carried
out by transmitting data streams over scalar channels that are
established by linear transmit and receive filters. An operation
point of the network is therefore defined by the set of data
streams D,D = |D|,
rk= log2
???Hkf(k)QkHH
kf(k)+ Zk
???
|Zk|

.
D = {(π(1),u1,v1,p1),...,(π(D),uD,vD,pD)},
where each stream d is described by the assignment to a user
π(d) ∈ K, receive filter ud∈ CMRX, transmit filter vd∈ CMTX,
and power allocation pd∈ R+. The receive and transmit filters
are chosen as normalized vectors and the rate of a data stream
d can be calculated as


σ2+
??uH
Our contribution is a scheme that selects the stream set D,
aiming at the maximization of a utility U : D ?→ R+ that
measures the network performance, in our case a weighted
sum of the user rates
?
d∈D
where the user weights w ∈ RK
of the users, or act as an interface that can be used to establish
a fair rate allocation among the users. As the scheme intends
to completely avoid interference, the filters are selected under
a zero-forcing constraint, meaning that
rd= log2


1 +
??uH
i∈D\d
dHπ(d),f(π(d))vd
?
??2pd
dHπ(d),f(π(i))vi
??2pi




. (1)
U(D) =
wπ(d)rd,
+may represent the priorities
uH
dHπ(d),f(π(i))vi= 0 ∀ i ∈ D \ d,
for all d ∈ D. We define the gain of the scalar channel for
each stream as
1
σ2
The data rate of a stream is then given by
λd=
??uH
dHπ(d),f(π(d))vd
??2.
(2)
rd= log2(1 + λdpd).
Assuming that the transmit and receive filter for all data
streams are fixed, the optimal power allocation is calculated
per transmitter according to the water-filling rule. For a
transmitter s the power allocation is given by
pd= max?wπ(d)µs− λ−1
where
Ds= {d ∈ D : f(π(d)) = s}
d,0?, ∀ d ∈ Ds,
(3)
is the set of streams transmitted by transmitter s and µs
is chosen such that the complete power budget Ps of the
transmitter s is used, i. e.
?
d∈Ds
pd= Ps.

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To facilitate more convenient notation we define a function
WF(D), that updates the power allocation according to the
water-filling rule,
D ← WF(D).
(4)
It remains to describe how to decide for the set of data
streams. The coordination algorithm is a scheme to succes-
sively allocate the data streams, aiming at the maximization
of a weighted sum-rate, while the transmit and receive filters
are chosen to eliminate interference.
III. SUCCESSIVE ALLOCATION OF DATA STREAMS
In a nutshell, the LISA algorithm [15], [16] is a zero-forcing
beamforming scheme developed for an isolated MIMO broad-
cast system. In a first stage, the LISA algorithm performs a
heuristic data stream allocation by successively adding the user
that leads to the maximum performance increase, while being
orthogonal to the previously allocated streams. The receive
beamforming vectors found by the stream allocation are kept
fixed and the transmit beamforming vectors of the resulting
MISO system are chosen under a zero-forcing constraint. The
power allocation for the interference free scalar sub-channels
is found via water-filling. Based on this recipe we develop a
successive zero-forcing scheme for the multi-cell scenario. In
each iteration, a stream is selected by deciding for the user, the
transmit filter, and the receive filter. The first stream selected
is found by solving
{π(1),u1,v1} = argmax
k∈K,u,v
˜Uk
?uHHkf(k)v?
||u||2,||v||2= 1,
s.t.
where ˜Uk is a monotonic increasing function suitable to
determine which datastream promises the highest performance
improvement, see Section III-A. The solution is the user who’s
channel has the maximum principal singular value, and u1
and v1are chosen as the corresponding left and right singular
vectors. Besides finding the user to allocate the data stream, the
corresponding receive filter is determined as well and is kept
fixed. When continuing to allocate streams, projector matrices
P1,...,PSper transmitter are used to assure orthogonality
to the previously assigned streams. The projector matrices are
initialized by identity matrices and updated after each step of
the stream allocation. Assuming the m-th stream is allocated
to user k and the receive filter is um, the projection matrices
are updated as follows:
P(m+1)
s
= P(m)
s
−P(m)
s
HH
mHksP(m)
ksumuH
mHksP(m)
HH
ksum
s
uH
s
∀ s ∈ S.
Selection of the m-th stream is done such that stream dm
does not disturb the previously allocated data streams, which
is assured by the projection matrices P(m)
1
,...,P(m)
S
:
{π(m),um,vm} = argmax
k∈K,u,v
˜Uk
?
uHHkf(k)P(m)
f(k)v
?
s.t.
||u||2,||v||2= 1.
(5)
At this point of the stream allocation the power allocation
is meaningless so we can set pm= 0 and update the set of
streams as follows:
D′(m)= D(m−i)∪ (π(m),um,vm,0).
Assume D′(m)as the result of the stream allocation, it is clear
that a stream dn does not cause interference to the streams
d1,...,dn−1, due to the way it was selected. However, it
well interferes with the streams dn+1,...,dm and therefore
the transmit filters, {vn}n∈{1,...,m−1}, are updated according
to a zero forcing constraint
vn= argmax
v
uH
nHπ(n),f(π(n))v
s.t.
uH
eHπ(e),f(π(n))v = 0 ∀ e ∈ D′(m)\ dn,
||v||2= 1.
(6)
Thus the MIMO system is decomposed into a system of
effective interference free scalar subchannels where the gains
are given in Equation (2). Now we can calculate the optimal
power allocation and update the stream set
D(m)= WF(D′(m)).
It is known from the single cell LISA algorithm that
with each newly allocated user the number of zero-forcing
constraints in Equation (6) increases and all channel gains
diminish from one step to the next. It can therefore happen
that the losses in channel gain for the streams already allocated
leads to a stronger decrease in sum-rate than the gain through
the newly allocated stream. For this reason we check in each
iteration, whether the addition of another stream still leads to
an increase in sum-rate, i. e.
?
If not, the algorithm terminates and D(m−1)is the operating
point of the network chosen by our algorithm. The algorithm
terminates anyway, when m = Mtx, as for m > Mtxthere is no
solution to (6). In Algorithm 1 we summarize the coordination
algorithm.
U
D(m)?
> U
?
D(m−1)?
.
A. Estimation Function
The intuition behind the successive stream allocation is to
add the user promising the maximum increase in system per-
formance, while being orthogonal to the previously allocated
users. The performance increase is measured by a heuristic
estimation function˜Uk, which is evaluated for each user and
is a function of the expected channel gain. The channel gain
is related to the maximum principal singular value γ(m)
the channel Hkf(k) projected into the subspace given by
range
P(m)
f(k)
, which can be calculated, together with the
potential transmit and receive filters, as
k
of
?
?
γ(m)
k
=max
u,v
s.t.
uHHkf(k)P(m)
f(k)v
||u||2,||v||2= 1.
A requirement for the estimation function is to account for
the individual power constraints of each transmitter and the

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Input: S,K,f : K ?→ S,{Hks}k∈K,s∈S,{Ps}s∈S,σ2
Output: D
m = 1
P(m)
s
= I, ∀ s ∈ S
while m ≤ MTXdo
{π(m),um,vm} =argmax
k∈K,u,v
s.t.
D(m)= D(m−i)∪ (π(m),um,vm,0)
for d ∈ D(m)do
vd← argmax
v
s.t.
˜Uk
?
uHHkf(k)P(m)
f(k)v
?
||u||2,||v||2= 1
uH
dHπ(d),f(π(d))v
uH
eHπ(e),f(π(d))v = 0 ...
... ∀ e ∈ D(m)\ d,
||v||2= 1
end
D(m)← WF(D(m))
if U
break
else
for s ∈ S do
P(m+1)
s
?
D(m)?
< U
?
D(m−1)?
then
=
P(m)
s
−P(m)
s
HH
π(m)sumuH
mHπ(m)sP(m)
mHπ(m)sP(m)
HH
π(m)sum
s
uH
s
end
m ← m + 1
end
end
return D(m−1)
Algorithm 1: Coordination Algorithm
weights of the users. In order to correctly account for the
user performance, one has to calculate a waterfilling solution
for each user and use the obtained power allocation in the
estimation function. In order to keep complexity low, we
assume that power is distributed uniformly between all streams
sent by one transmitter, which is accurate in case of high
transmit power. Assuming the power allocation the newly
added stream is
|Df(k)|+1, we can estimate the weighted rate
of the new stream as
?
σ2
|Df(k)| + 1
Pf(k)
wklog2
1 +
1
Pf(k)
???γ(m)
k
???
2?
,
which defines our estimation function˜Uk.
IV. SIMULATION RESULTS
The performance of the coordination algorithm is evaluated
by Monte-Carlo-Simulations for a wrap-around configuration,
following the guidelines in [18]. We use channel matrices gen-
erated for a realistic channel model, implemented as described
in [19] and use the parameters in Table I. We regard sectorized
cells with three antenna arrays per site. An average of 10 users
per sector are uniformly distributed in the area covered by the
network and we average over 20 drops.
010203040
0
5
10
15
20
25
P in [dBm]
Avg. Sp. Efficiency per User [bps/Hz]
Uncoordinated Approach
Coordination Algorithm
Fig. 1.Simulation Results – One Site
The spectral noise density is -174 dBm/Hz and we gradually
increase the transmit power from 0 dBm to 40 dBm. As a
reference, we use uncoordinated transmission in each sector,
where the degradation due to the interference is accepted. We
chose a linear precoding method, the LISA algorithm [15],
[16], applied in each sector instead of the capacity achieving
Dirty Paper Coding approaches. Unfortunately the very in-
teresting coordination methods mentioned in the introduction,
[6], [7], [8], do not easily extend to a scenario multiple receive
antennas.
In a first step we regard an isolated site, with three sectors
and a total of 30 users. We set the weights of all users to
one and Figure 1 shows the average spectral efficiency per
user. We can see that the performance of the uncoordinated
approach is limited by interference, while the coordination
algorithm is interference free and shows a significant gain.
We will however see that these gains do not directly translate
to larger scenarios: Figure 2 shows the simulation results for
the full 57-sector scenario with a total of 570 users. Again,
as in the small network, the LISA approach is interference
limited while the performance of the coordination algorithm
grows with increasing transmit power. The intersection point,
where completely removing interference pays off for such
a network seams to be far away from the usual operating
point of a wireless network. Reducing interference comes at
scenario
inter-site dist.
center freq.
bandwidth
antenna conf.
sectors
users per sector
urban macro-cell
500 m
2 GHz
20 MHz
4x4 MIMO
19 · 3 = 57
10
user speed
min dist. to site
height site
height user
building height
street width
antenna spacing
30 km/h
25 m
25 m
1.5 m
20 m
20 m
0.5 λ
TABLE I
SIMULATION PARAMETERS

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010 203040
0
0.5
1
1.5
2
P in [dBm]
Avg. Sp. Efficiency per User [bps/Hz]
Uncoordinated Approach
Coordination Algorithm
Fig. 2.Simulation Results – 57 Sectors
the price of having less users active. Or in other words, the
demand of complete orthogonality of the transmission signals
is too strict for larger networks. Finding the optimal cost-
benefit ratio between interference avoidance and the number
of active streams is essential for efficient operation of wireless
networks. We therefore embed the coordination algorithm into
a larger resource allocation framework [17], which assures
that the coordination algorithm is only applied for those users
that benefit enough. The algorithm iteratively manipulates
the weights of the users, which can be interpreted as multi-
user scheduling, and assigns resources to the obtained rate
configurations, such that the average data rates maximize a
network wide utility. Here we chose proportional fairness,
which corresponds to maximize the sum of the logarithms
of the users rates, F(r) =
allows various physical layer modes to be combined optimally,
by jointly selecting the rate configuration within each mode
and the resources assigned to the modes. In Figure 3 we
illustrate such a combined approach. Some users in favorable
position, in the illustration users number 1, 3, and 6, obtain
sufficiently high rates although hit by interference. Serving
these users by the uncoordinatedsingle-cell approach increases
the number of active user. However, the users at the cell-edge,
user number 2, 4, and 5, would be excluded from network
service unless interference is reduced. Therefore, these users
are served on distinct resources by a coordination algorithm,
like the one introduced in this work. We combine the single
cell LISA algorithm and the novel coordination algorithm
such that the users located in the center of a cell are served
by the LISA algorithm, while the users at the cell-edge are
served interference-free by the coordination algorithm. The
classification as cell-edge user or cell-center user is inher-
ently performed by the algorithm, which combines the two
strategies by optimally assigning distinct resource blocks to
the two modes, for details the reader is referred to [17]. Other
?
k∈Klogrk. The framework
1
2
3
4
5
6
Frequency
Time
.........
...
...
...
...
Uncoordinated Approach
Coordination Algorithm
Fig. 3.Illustration Combined Approach
methods to introduce the necessary non-orthogonality might
be clustering the cooperation or introducing thresholds instead
of strict zero-forcing conditions, see our comments on future
work in Section V.
Figure 4 shows the performance measured by the propor-
tional fairness utility for the combined approach and shows
that this approach is no more interference limited. As the
absolute values of the utility F(r) are difficult to interpret we
include a commonly accepted performance measure: Figure 5
shows the average spectral efficiency of the 10% worst users
in the system, usually users at the cell-edge. We notice a
significant gain and can see that while in the uncoordinated
approach these users are more or less shut off, with the
combined approach the cell-edge users are able to meet the
targets of future wireless systems.