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A Spatial Compatible User Evaluation Algorithm for
SDM Broadcast Channels
Zhisheng Niu and Lei Li
Department of Electronic Engineering
Tsinghua University, 100084, Beijing, P. R. China
niuzhs@tsinghua.edu.cn, lei-li02@mails.tsinghua.edu.cn
Abstract: CCI (cochannel interference) introduced by frequency reuse under SDMA (space
division multiple access) can be reduced by controlling the frequency sharing according to
the users’ spatial separability. Highly correlated users in spatial are avoided to be placed into
the same spatially multiplexing group. In this paper we propose a user spatial compatibility
evaluation criterion based on channel matrix structure analysis. We evaluate the interferences
by use of distances between subspaces spanned by users’ channel matrixes. Two kinds of
subspaces are considered in our analysis, row space and column space. Simulations results
showed the row space distance criterion outperformed other ones greatly because it fully
utilized the information from users’ channel matrixes. When dimensions of users’ channel
matrixes increase, the matrix structure analysis based criterions showed more efficiency.
Key words: Multiple input multiple output (MIMO), multiuser channel, radio resource
management
1 Introduction
MIMO (multiple input multiple output) has been attracting much attention due to the po-
tential to increase spectral efficiency of wireless resource greatly [1, 2, 3]. The bandwidth
reuse by multiple spatially compatible users in MIMO systems is through SDM (space divi-
1
sion multiplexing). However, with SDM, unlike with orthogonal multiplexing such as TDM
(time division multiplexing) and FDM (frequency division multiplexing) etc., users’ channels
are not completely orthogonal. CCI (cochannel interference) is a big challenge for multiuser
MIMO systems.
In many studies, transmit and receive weighted vectors are designed to eliminate the
cochannel (frequency and time slot) interference by complicated signal processing [4, 5, 6,
7]. However these often result in solving computationally intensive non-linear optimization
problems [8]. Another implementation is to divide the users into groups. The group division
measures are designed to avoid placing two highly correlated users into the same group. Users
in the same group can be multiplexed spatially, while groups must be multiplexed through
orthogonal multiplexing.
An estimate of the total correlation degrees between two users’ channels is proposed in
[9]. It is simple but not optimal. The SDMA methods, such as zero forcing, block diagonal-
ization [4], usually look into the users’ channel matrix structures to find an optimal transmit
and receive solution. Accordingly, in the spatial compatibility approximating algorithms the
users’ channel matrix structures should also be considered. In [10], several spatial compati-
bility checking approaches are compared and the results show that a subspace (spanned by
the spatial covariance matrix of each user)-oriented approximation is outperformed other ap-
proaches. Their study is in MISO (multiple input single output) system with beamforming
transmission. However if it is used in MIMO systems, the truncated subspaces will drop much
information. A better compatibility approximation should make full use of users’ channel
matrixes.
In this paper, we propose a spatial compatible user selection algorithm in SDM MIMO
systems based on users’ channel matrixes analysis. We avoid information discarding in the
algorithm in [10] by use of row spaces spanned by users’ channel matrixes. The interferences
2
are determined by the distance between the row spaces. Simulation results show that the
row space distance criterion achieves much better performance compared with column space
distance criterion and correlation based criterion in [9].
2 System Model
We consider a single cell downlink system. Suppose there are M active users. The base station
is equipped with NT antennas and each mobile station is equipped with NR antennas. For
spatial multiplexing, NT > NR is required. We consider linear coding SDM solutions with
NT
the maximum number of simultaneous users is N = NR
. Assume a rich scattered Rayleigh
channel. The channel is block fading, that is, it is constant over one time slot but varies from
time slot to time slot. The user selection algorithm is carried out at the very beginning of
each time slot.
Linear coding SDM systems separate users’ streams by different linear precoding and
decoding matrixes computed based on the users’ channel matrix. Let H i ∈ CNR ×NT denote
the channel matrix of user i and si ∈ CNi ×1 be transmitted signal vector of user i. The
received vector xi ∈ CNR ×1 can be represented as
N
xi = H i W i si + H i W j s j + ni , (1)
j=1,j=i
where W i ∈ CNT ×Ni is the precoding matrix computed for user i. The first item in (1) is
useful information and the second is undesirable item denoting the CCI introduced by SDM.
In order to recover the transmitted symbols, a decoding matrix Gi ∈ CNi ×NR is applied
at the receiver. The estimated signal vector is therefore given by
ˆ
si = Gi H i W i si
N
+ Gi H i W j s j + ni . (2)
j=1,j=i
3
Gi is also computed based on H i , as W i .
To reduce CCI, it is suggested to divide users into groups. Highly correlated users must
be placed into different groups. A multiple multiplexing strategy is applied, where SDM
within each group and orthogonal multiplexing, such as TDM, OFDM (orthogonal frequency
division multiplexing), etc. between the groups.
CCI varies as a function of the scattering environment, the distance between the base
station and users, the antenna geometrics, and the Doppler spread an so on. If some of two
users’ multipath channels are similar, for example they have a common scatterer, the two
users will highly correlate. Users’ channel matrixes are important because they contain much
of the channel state information.
3 Compatibility Approximation Criterions
From equation (2), we can get the spatial influence from user j on user i is Gi H i W j sj . The
interference between two users is determined by two factors, one is users’ channel matrixes
H i , the other one is the precoding and decoding solutions W j and Gi . However, because the
precoding solution is carried out after user grouping modular, W j and Gi are unavailable
currently. Hence it is unlikely to decide the exact spatial correlation. A suboptimal solution
is to exploit the subspace structures spanned by the channel matrixes, since both W j and
Gi are calculated based on H j and H i . We will propose two approximations to check the
users’ spatial compatibility then exclude the highly correlated users into a same group.
We rewrite the signal item and interference item in equation (2) as Gi H i W i si +Gi H i W j sj .
In order to pick up the useful signal si successfully, we expect the difference between Gi H i W i si
and Gi H i W j sj is large enough.
4
3.1 Column space distance criterion
We first explain this by means of the distance between column spaces spanned by H i and
H j . The column space of an NR × NT matrix H is defined as
Col H = {y ∈ CNR : y = Hs, s ∈ CNT }. (3)
Col H i and Col H j are two subspaces of the vector space CNR . Suppose in equation (3), s
is any signal vector the base station transmits, then Col H i stands for the range of signal
vector while Col H j represents the range of interference vector. The more distance between
these two subspaces the less interference the signal suffers.
The distance between two subspaces can be measured by orthogonal projection. The
orthogonal projection is correlated to SVD (singular value decomposition). For
˜ ˜
H = U ΛV H = [U r , Ur ]Λ[V r , V r ]H , (4)
where (·)H denotes conjugate transpose, r = rank H. U r U H is the orthogonal projection on
r
Col H [11]. When rank H i = rank H j = r, the distance between Col H i and Col H j is
defined as
dist(Col H i , Col H j ) = U r,i U H − U r,j U H
r,i r,j F, (5)
where · F denotes Frobinus norm.
However rank H i = rank H j does not always hold, because the rank of user’s channel
matrix is determined by the diffusion environment. In theory, it is a positive integer scaling in
the range of [1, NR ]. The more important problem is that, when rank H i = rank H j = NR ,
the subspaces will span to the whole vector space, and the distance will be 0.
It is suggested in [10] to truncate the subspaces into fix rank in such scenarios. We arrange
the singular values in descending order, and the left singular vectors in U accordingly. Select
the first min(rank H i , rank H j , NR − 1) number of left singular vectors into U min,i and
5
U min,j . Replace U r,i and U r,j in equation (5) by U min,i and U min,j separately. The reason to
reduce the dimension to be less than NR − 1 is to avoid 0 distance. Therefore the distance is
determined by
dist(Col H i , Col H j ) = U min,i U H − U min,j U H
min,i min,j F, i = j. (6)
However some useful information is discard when we truncate the subspaces. The simu-
lation results also showed poor performance.
3.2 Row space distance criterion
Since NT > NR is required for SDM, we consider making use of row spaces in stead of column
spaces. The definition of row space for an NR × NT matrix H is
RowH = {y ∈ CNT : y = H H s, s ∈ CNR }. (7)
Row H i and Row H j are two subspaces of the vector space CNT . The orthogonal projection
on Row H is V r V H . The distance between Row H i and Row H j is
r
dist(Row H i , Row H j ) = V r,i V H − V r,j V H
r,i r,j F, (8)
when rank H i = rank H j = r [11].
Because r ≤ NR < NT , the row spaces can not span to the NT dimension whole vector
space. Therefore, We need not to truncate the subspaces for fear of 0 distance any more. The
only consideration is rank defect channels. In fact, it does not usually happen. The distance
we use is
dist(Row H i , Row H j ) = V min,i V H − V min,j V H
min,i min,j F, i = j. (9)
where V min,i and V min,j are matrixes composed of the first min(rank H i , rank H j ) number
of right singular vectors.
6
3.2
3.0
2.8
2.6
Capacity (b/s/Hz)
2.4
2.2
2.0
Row space distance
1.8 Column space distance
Correlation division
1.6 Random
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Channel utilization
Figure 1: Comparison of users average capacity in as a function of channel utilization
1.0
Probability cumulative density function
0.8
NT=8 NT=12
NR=2 NR=3
0.6
0.4
0.2 Row space distance
Column space distance
Correlation division
Random
0.0
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
Capacity (bps/Hz)
Figure 2: Comparison of probability cumulative density functions
7
4 Numerical Examples
We evaluated the performance under a zero forcing transmission system [12]. The average
transmit SNR was set to be 10dB. For comparison, we also simulated a correlation based
HiHH 2
criterion, where the correlation between user i and j is calculated by j F
N Ri N Rj
[9].
In the simulation, we set a threshold to divide the users into groups according to their
compatibility evaluation results.
We first evaluate the average capacity as a function of channel utilization. In this case
we considered 8 transmit antennas at the base station and 2 receive antennas at each user.
Thus the number of maximum simultaneous users in SDM was 4. Suppose there were 24
active users in the system. After spatial compatible user selection algorithms, we put the
users from the divided groups into SDM slots for transmission. For example, if group I
held user {1, 3, 4, 6, 9}, it would occupy two SDM slots with the first slot holding user
{1, 3, 4, 6} and the second slot holding user {9}. We define the channel utilization as
total number of users . In fact, the total number of SDM slots occupied
·total number of SDM slots occupied
NT
NR
is determined by the threshold. The stricter the threshold set, the less interference users
suffer. The stricter the threshold set, the more groups users are divided into, and the less
the channel utilization is. We changed the thresholds to divide the users into from 1 to 24
groups in the spatial compatible user selection algorithms. The total number of SDM slots
occupied changed from 6 to 24. Thus the channel utilization varied from 100% to 25%.
Fig. 1 compares users average capacity in bps/Hz as functions of channel utilization.
At first, all the curves increase with the increasing of channel utilization. After the channel
utilization reaches 50% to 60%, they begin to decrease, because high channel utilization intro-
duces high cochannel interference. It is shown in Fig. 1 that the row space distance criterion
is much better than other ones. Its peak value is achieved when the channel utilization is
8
around 60%. The peak value of column space distance criterion is a bit higher than that of
the correlation based criterion and the corresponding channel utilization value is also higher.
However it is not very stable as it decreases quickly in the channel utilization range of 60%
to 70%.
Fig. 2 plots probability cumulative density functions. In this case we also compared the
scenario using 8 transmit antennas at the base station and 2 receive antennas at each user
and the scenario using 12 transmit antennas and 3 receive antennas. We set the thresholds of
all the criterions to make the channel utilization be 60%. It is shown in Fig. 2 that when the
number of antennas increases, the performance of column space distance criterion exceeds
the correlation criterion. This means that, matrix structure analysis is a more efficient way,
especially when dimensions of users’ channel matrixes increase. The increased capacities
overcome the cost of the complexity in SVD.
5 Conclusion
In this letter, we proposed a spatial compatible user selection algorithm in SDM MIMO
systems to reduce CCI introduced by frequency reuse under SDM. Two spatial compatibility
criterions are introduced, one is column space distance and the other is row space distance.
Both theory analysis and simulation results show that the late one is better than the first
one, for it makes full use of users’ channel matrixes. In the simulation we also compared our
criterions with the correlation based metrics. Results show the row space distance criterion
outperforms greatly. This means the criterion based on channel matrix structure analysis is
a better candidate for spatial compatibility approximation algorithms.
9
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