The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
CAPACITY EVALUATION OF VARIOUS MULTIUSER MIMO SCHEMES IN
DOWNLINK CELLULAR ENVIRONMENTS
Jingon Joung, Eun Yong Kim, Sung Hoon Lim, Yong-Up Jang, Won-Yong Shin,
Sae-Young Chung, Joohwan Chun, and Yong H. Lee
∗
Department of Electrical Engineering and Computer Science
Korea Advanced Institute of Science and Technology (KAIST)
373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea
TEL: +82-42-869-3437, FAX: +82-42-869-4030
E-mail: yohlee@ee.kaist.ac.kr
A BSTRACT
Presented in this paper is a study of the capacity evaluation
of various multiuser MIMO schemes in cellular environments.
The throughputs per user of the generalized zero-forcing with
rank adaptation and vector perturbation schemes are compared
with the capacity bound of the Gaussian MIMO broadcast
channel, obtained by dirty paper coding under proportional
fairness scheduling. The average cell throughputs of these
schemes are also compared. From these comparisons, this
study provides vital information for applying multiuser MIMO
schemes in multicell environments.
I.
I NTRODUCTION
Recently, the capacity bounds of Gaussian multiple-input
multiple-output (MIMO) broadcast channels have been studied
and shown to be achieved by dirty paper coding (DPC) [1]–[7],
and several practical progresses using source-channel coding
in the dirty paper channel have been made in this area [6], [7].
However, there are still open problems to achieve rates closer to
capacity with practical transceivers, whereas many suboptimal
schemes have been proposed to simplify the transceiver. In [4],
the capacity bound obtained from the DPC and the throughput of linear processing, such as zero forcing (ZF)-based orthogonal space-division multiplexing (OSDM) [8]–[10] and
time-division multiple-access (TDMA), have been compared
for downlink cellular systems.
In this paper, the downlink throughputs of a more general
linear OSDM scheme with rank adaptation [11] and nonlinear vector perturbation [12], [13] are evaluated and compared
numerically with the capacity bound in the multicell environment. To apply a point-to-point MIMO technique in a point-tomultipoint MIMO system, the proportional fairness (PF) scheduler in [14] is considered. First, the cumulative density functions (CDFs) of the throughput per user are shown, and then the
average cell throughputs are compared. The throughput gaps
from the capacity bound for suboptimal techniques are shown
using computer simulations. All of these results provide useful insights into the design and application of multiuser MIMO
techniques in cellular environments.
The organization of this paper is as follows. Section II describes the multiuser MIMO system model in multicell envi∗ This work was supported in part by the University Information Technology
Research Center Program of the government of Korea.
c
1-4244-0330-8/06/$20.00°2006
IEEE
ronments. Various multiuser MIMO schemes are then presented in Section III. Section IV shows the computer simulation results to demonstrate the performance evaluation of the
optimal and suboptimal methods. Finally, Section V concludes
the paper.
II.
M ULTIUSER MIMO S YSTEM M ODEL
The multiuser downlink MIMO system in cellular environments, where a common base station with NT transmit antennas transmits different signals to multiple mobile users, is
shown in Fig. 1. Let K be the number of users in the cell and
NR,k be the number of receive antennas at the k-th user.
To ensure that the users receive their data without coordination, an appropriate preprocessing of the data should be carried
out at the transmitter. Let F denote the preprocessing function
and xk be the Lk -by-1 data vector for the k-th user, where Lk is
the number of spatial modes supported to the k-th user. Then,
the transmitted signal is represented by:
s = F (x1 , x2 , . . . , xK ) ,
where s is an NT -by-1 vector.
In multicell environments, the received signal at the k-th
user, an NR,k -by-1 vector, is represented by
yk = Hk s + nk ,
where Hk is an NR,k -by-NT matrix that denotes the channel
matrix between the base station and the k-th user, and nk denotes the additive noise due to both the thermal noise and the
interferences from neighboring cells. It is assumed that the entries of Hk are independent and identically distributed (i.i.d.)
random variables whose means are zero and variances are determined by the path loss and shadowing factor of the k-th user.
Each user decodes the data vector as follows:
x̂k = Gk (yk ) ∀k,
where Gk is the postprocessing function for the k-th user.
We assume that the channel matrices for different users are
independent and all the channel matrices are quasi-static, flat
fading, and perfectly known at the base station. For simplicity, it is also assumed that every user has the same number of
receive antennas, i.e. NR,k = NR ∀k.
Throughout this paper, A† , tr(A), and |A| denote the conjugate transpose, trace operation, and determinant of matrix A,
respectively.
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
BS
N1
1
H1
1
x1 LM
NR,1
n1
⊕
⊕
MS 1
G1
M
1
M
L1
x̂1
1
1
H2
2
x 2 LM
F
2
s M
M1
x K LM
HK
MS 2
⊕
NR,2
NT
K
n2
N1
G2
M
⊕
1
M
L2
x̂ 2
N
N1
NR,K
MS K
nK
⊕
⊕
1
M
GK
M
LK
x̂ K
Figure 1: System model of a multiuser MIMO system
III.
T ECHNIQUES FOR M ULTIUSER MIMO D OWNLINK
In this section, we first present the capacity bound of the multiuser MIMO downlink systems, i.e. the DPC achievable rate
regions. Next, we present two suboptimal schemes: the generalized zero-forcing (GZF) and the vector perturbation (VP)
schemes. Finally, we explain the PF scheduler with rank adaptation that is also under consideration.
A.
Capacity Bound (DPC)
With the DPC it is possible to precancel interferences that are
known noncausally at the transmitter, resulting the same capacity as if there is no interference. When the DPC is applied to
a multiuser MIMO downlink, it can be used to precancel other
users’ coded signals with an appropriate ordering [3]. The DPC
achievable rate region is given by:
CDP C = Co
[
π,Σi
Rπ ,
(1)
S
where Co (A) is the convex hull operation of the set A , is
the union operation, Rπ = (Rπ(1) , ..., Rπ(K) ), π is the user
permutation vector, and
Rπ(i)
=
¯
¯
³P
´
¯
¯
†
¯INr + Hπ(i)
j≥i Σπ(j) Hπ(i) ¯
¯,
³P
´
log ¯¯
¯
†
Σ
H
¯INr + Hπ(i)
π(j)
j>i
π(i) ¯
i = 1, ..., K.
Here, IN is the N -dimensional indentity matrix, Σk is the covariance matrix of the user k, the union operation is performed
over all possible permutations (π(1), π(2), ..., π(K)), and all
possible non-negative covariance matrices that are constrained
to tr(Σ1 + Σ2 + ...ΣK ) ≤ P .
In this paper, we evaluate the achievable rates using the PF
scheduler, which needs to find the optimal point on the boundary of the rate region (1) that maximizes the weighted sum rates
for a given weight vector. Finding the optimal operating point
with (1) is formidable since it is neither convex nor concave.
Thus, this problem is transformed into a dual multiple access
channel (MAC) optimization problem that is concave of the covariance matrices; then, the following result is obtained [2]:
[
CDP C (P, H) =
CM AC (P, H† ),
P:
PK
i=1
Pi =P
which shows that the capacity of the Gaussian broadcast channel is the same as the union of the MAC capacity regions over
all individual power constraints P = (P1 , ..., P2 ) that sum to P .
Here, the MAC capacity region for a given power constraint
and channel instance, CM AC (P, H† ), is given by the union of
the rate regions over all possible non-negative covariance matrices, as follows:
(
S
P
R : i∈S Ri
CM AC (P, H† ) =
Qi ≥0
tr(Qi )≤Pi ∀i
)
¯
¯
P
¯
¯
†
≤ log ¯INt + i∈S Hi Qi Hi ¯ ∀S ⊆ {1, ..., K} ,
where Qi is the covariance matrix for the i-th user and R =
(R1 , R2 , ..., RK ). We locate the operating point on the boundary of the optimal region for a given weight vector, µ, by maximizing the weighed sum rates using the algorithm in [4], which
is calculated using the results of the dual MAC and standard
convex optimization procedures shown in (2), where R̄ is a
vector of user rates of the set R and µ is a weight vector given
by PF. The numerical methods for the optimization procedure
in (2) are shown in detail in [4].
B.
Suboptimal Schemes
1) Generalized Zero-Forcing (GZF)
For the downlink of multiuser MIMO systems, one of the suboptimal schemes is the linear OSDM. The OSDM enables the
users to receive their own data with zero co-channel interference. Recently, [10] proposed an iterative algorithm that finds
the preprocessing and postprocessing linear operators for the
OSDM, which maximize the effective channel gains. In this
paper, however, to avoid the burden from iterations, we use the
conventional ZF block-diagonalization [9], in which the multiuser MIMO channels are decomposed into multiple, singleuser MIMO channels. For the case of NR,k > Lk , the dominant Lk left singular vectors are considered as the effective left
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
max
R∈RDP C
µ · R̄ =
max
Qi :
PK
K−1
X
i=1 tr(Qi )=P i=1
¯
¯
¯
¯
K
i
¯
¯
¯
¯
X
X
¯
¯
¯
¯
†
†
Hπ(l)
Qπ(l) Hπ(l) ¯
(µπ(i) − µπ(i+1) ) log ¯I +
Hπ(l)
Qπ(l) Hπ(l) ¯ + (µπ(K) ) log ¯I +
¯
¯
¯
¯
singular vectors of Hk for the k-th user in order to preserve
the maximum spatial diversity. Then, we apply the singular
value decomposition approach [8] to each single-user MIMO
channel to achieve the maximum single-user MIMO capacity.
This scheme is termed GZF. In the aspect of complexity, GZF
has a merit in that it needs only linear processing at both the
transmitter and receivers.
Let F denote the preprocessing matrix at the base station
and Gk denote the postprocessing matrix at the k-th user. We
note that F can be separated as [F1 F2 · · · FK ], where Fk is
the preprocessing matrix for the k-th user, and the transmitted
signal can be represented by:
s=
K
X
Fk xk .
k=1
To obtain the optimal Fk and Gk , let Wk be a matrix with
orthogonal columns such that:
Wk ∈ null
¡£ †
H1 U1 · · · H†k−1 Uk−1
H†k+1 Uk+1
and
h
i†
Heff = H†1U1 · · · H†K UK [W1 · · · WK ] ,
°2
°
K
°
°X
°
′ °
F
(x
+
τ
l
)
l = arg min
°
° ,
k
k
k
°
l′ °
k=1
where τ is a scalar of a positive real value, lk is the integer
vector, and k·k denotes the vector 2-norm.
Using the same pre- and postprocessing matrices with GZF
as in (3), the postprocessed data vector for VP is given by:
x̂k = Gk Hk Fk (xk + τ lk ) + Gk nk .
(5)
Taking the modulo-τ operations for (5), the estimate of x with
VP is obtained and is the same as (4).
where the effective channel of the k-th user is given by:
Heff,k = U†k Hk Wk .
Let the columns of Ueff,k (Veff,k ) denote the left (right) singular vectors of Heff,k , such as:
†
Heff,k = Ueff,k Deff,k Veff,k
,
where Deff,k denotes a diagonal matrix that consists of singular
values of Heff,k . To achieve the maximum capacity, Fk and Gk
are
Fk , Wk Veff,k Ek , and Gk , U†eff,k U†k ,
(3)
where the power-loading matrix Ek is an Lk -by-Lk diagonal
matrix, whose diagonal elements are determined by the wellknown waterfilling algorithm [8]. Then, the postprocessed data
vector is represented by:
= G k yk
= Gk Hk Fk xk + Gk nk .
2) Vector Perturbation (VP)
In our comparison, VP, a one-dimensional dirty paper approach, is considered as a simple nonlinear technique. By
perturbing the data vector at the transmitter and taking modulo operation at the receivers, VP achieves an excellent performance gain over the linear processing schemes, especially at
high signal-to-noise ratios (SNRs) [12]. Recently, VP has been
extended to systems with multiple receive antennas and multiple spatial modes, and an optimum VP for minimizing the
mean square error has been proposed [13].
Once the precoding and decoding matrices for a ZF block diagonalization are determined, VP chooses the best perturbation
vector to minimize the power in the transmitted signal when
added to the data vector. In this paper, the data vector is perturbed based on the GZF. The perturbation vector is constrained
to an integer vector so as not to affect the zero co-channel interference condition [12]. The transmitted signal can be represented by:
K
X
Fk (xk + τ lk ),
s=
k=1
¤† ¢
· · · H†K UK ,
where null(A) denotes the null space of A and Uk consists of the dominant Lk left singular vectors of Hk . If
[W1 · · · WK ] is multiplied to the transmitted signal as the
preprocessing matrix, the effective block-diagonal channel matrix becomes:
x̂k
(2)
l=1
l=1
(4)
3) Scheduling with Rank Adaptation
When the total number of receive antennas is more than NT ,
both GZF and VP need an extra scheduling algorithm, while the
DPC provides the optimal user selection implicitly. Moreover,
if the number of receive antennas for a user can be more than
one, the rank adaptation technique [11] should be applied to
improve the sum rates.
In our comparison, the PF scheduler is employed under the
consideration of fairness among the users. The set of spatial
modes for users, (L1 , L2 , . . . LK ), are determined by bruteforce searching.
IV.
C OMPARISON AND D ISCUSSION
The CDFs of throughput per user and the average cell throughputs for the DPC, GZF, VP, and TDMA are evaluated. MIMO
channels are obtained by generating independent Gaussian random variables with a zero mean, and the results shown below are the averages over 350 independent trials under the PF
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
Table 1: Simulation Parameters
Values
Parameter
{2, 1}, {4, 2}
Inter-cell
1, 2, 4, 8
Tx antenna pattern
PF
Path loss model
Synchronous
Path loss exponent
47 dBm
Shadowing STD
−94 dBm
BS correlation
1000 m
Min. separation (d0 )
Rayleigh
Max. achievable SINR
1
1
0.9
0.9
0.8
0.8
0.7
0.7
CDF of throughput
CDF of average SINR
Parameter
# of antennas {NT ,NR,k }
# of users (K)
Scheduling
Sync. protocol
Tx power
Rx noise level
Radius of cell
Channel model
0.6
0.5
0.4
0.6
0.5
0.3
0.2
0.2
0.1
0.1
−6
−4
−2
0
2
4
6
8
Average SINR (dB)
10
12
14
16
0
18
K=1,2,4,8
0.4
0.3
0
−10 −8
Values
18 (2nd-tier)
Omni-direction
3GPP2/TSG-C.R1002
3.5
8.9 dB
0.5
100 m
17.8 dB
DPC
VP
GZF
TDM
0
1
2
3
4
Throughput (bits/sec/Hz)
5
6
7
(a)
Figure 2: SINR CDF of multicell environments in Table 1.
1
0.9
0.8
0.7
CDF of throughput
scheduler for each average SINR point. Every average SINR
was assigned independently 8, 000 times to all users according to the distribution in Fig. 2, which was generated under
the multicell environment outlined in Table 1. The mobile station locations are constrained to be distanced from base station
farther than minimum distance (d0 ) 100 m and the maximum
achievable SINR in the receiver is limited to 20 dB , as shown
in Table 1. BS correlation is defined as the correlation factor
among the inter-cells’ BSs and its value is 0.5.
Fig. 3(a) and 3(b) show the CDFs of the throughput per user
for {NT , NR } = {2, 1} and {4, 2}, respectively. Here, when
NT = 4 and K = 4, 10% of the high end users for each scheme
in an inter-cell achieve 4.65, 4.19, 3.92, and 2.86 bit/sec/Hz
using the DPC, VP, GZF, and TDMA, respectively. For the
low average SINR users, the gains of the GZF, VP, and DPC
over TDMA seem to be negligible. This is because the DPC,
GZF, and VP all use the solution of allocating all of the power
to the proportionally highest rate user in this regime, which
is the same as TDMA with PF. In contrast, in the high SINR
regime, the DPC, GZF, and VP have a significant gain over
TDMA since the multiplexing gain of TDMA is bounded by
min{NT , NR }, whereas those of the multiuser MIMO schemes
are bounded by min{NT , KNR }. There is an additional gain
0.6
0.5
K=1,2,4,8
0.4
0.3
0.2
DPC
VP
GZF
TDM
0.1
0
0
2
4
6
8
10
Throughput (bits/sec/Hz)
12
14
(b)
Figure 3: CDF of throughput of various multiuser MIMO
schemes when (a) {NT , NR } = {2, 1}, (b) {NT , NR } =
{4, 2}.
for the DPC over GZF and VP due to the optimal cancelling of
inter-user interferences.
The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’06)
VP, which also have potential to be improved further, are good
candidates for next generation communications. However, all
of the previous schemes need perfect channel state information
at the transmitter for implementation. Examining the robustness of these schemes against channel uncertainty and the exact complexity comparison of each system remains as work to
be undertaken.
12
DPC
VP
GZF
TDM
Average cell throughput (bits/sec/Hz)
11
10
9
8
7
6
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5
T
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Figure 4: Average cell throughput versus the number of users
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and FDMA, the GZF and VP have a significant throughput improvement of approximately 32(45)% and 34(53)%, respectively, with some additional complexity. Despite the gains that
the GZF and VP provide over TDMA, there is still a noticeable
gap from the optimal bound.
V.
C ONCLUSION
The CDFs of the throughput per user and the average cell
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optimal and various suboptimal schemes are shown. Here, it
is surmised that the suboptimal schemes such as the GZF and
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