HUANG et al.: RATE-ADAPTIVE FEEDBACK WITH BCS IN MULTIUSER MIMO BEAMFORMING SYSTEMS 4841
ergodic downlink capacity, and use it to facilitate the
rate-adaptive feedback according to the transmission rate
requirement of downlink services.
The rest of this paper is organized as follows. In section II,
the feature of multiuser MIMO channel is described, and the
VAR model is used to describe the temporal correlation among
continuous CSIs. In section III, Bayesian CS model is built
to compress the dimension of the channel prediction error. In
section IV, the feedback rate-distortion function and ergodic
downlink capacity are derived, and the relationship between the
feedback rate and downlink capacity will be defined. Such a
rate-adaptive feedback can be used to support arbitrary trans-
mission rate. In section V, simulation results are given to show
the efficiency of t he proposed compressive feedback scheme in
terms of the feedback rate, distortion, ergodic downlink capac-
ity, and bit error rate (BER) performance, followed by the
conclusion of this paper in section VI.
II. VAR-B
ASED MIMO CHANNEL MODEL
In this section, we first describe the structure of MIMO
channel, which is a prerequisite knowledge for the proposed
BCS feedback scheme. Furthermore, the VAR model will be
introduced to describe the correlation between continuous CSIs
[15]. From this VAR model, we will introduce the proposed
spatiotemporal compression, in order to reduce the feedback
error and the dimension of channel vector (thus achieve a
low-dimensional codebook).
A. MIMO Channel
In this paper, we consider a general MIMO wireless commu-
nication system with M transmit antennas at the base-station as
well as K users (each with a single antenna). Since the users are
in different places, their channel vectors are uncorrelated with
each other. The channel vector of user i , h
i
, can be modeled
as [13]
h
T
i
= αh
T
iid
R
1
2
T
X
(1)
where α is a scalar that reflects the channel gain; h
iid
repre-
sents a M × 1 vector consisting of independent and identically
distributed (i.i.d.) complex Gaussian random components with
zero-mean and unit variance, and (·)
T
represents transpose
operation; R
T
X
is a M × M correlation matrix at the transmit-
ter side (i.e., the base-station). We also assume that uniformly
separated linear antenna arrays are installed at the base-station,
and the spatial correlations in h
i
are solely determined by the
Jakes model [8]. Every element R
ij
in R
T
X
represents the cor-
relation coefficient between the i
th
and the j
th
antennas at the
base-station, and can be formulated as
R
ij
= J
0
(
2πd
ij
λ
) (2)
where d
ij
is the distance between the i
th
and the j
th
antennas
at the base-station, λ is the wavelength of the central carrier,
and J
0
(·) represents the zer o
th
-order Bessel function of the
first kind. From Eqn. (1), one can see that each component
of h
i
is the combination of vector h
iid
with different weights.
Thus there are internal correlations in the channel vector h
i
.
We assume that a perfect channel vector h
i
can be estimated
at the user i. This paper assumes single antenna in each user.
However, it can be easily extended to multi-antenna users by
using methods such as [16].
The CDI feedback information is the index of the quantized
channel vector, which is a codevector in the codebook. As in
other related works (especially [7]), we assume that the forward
error correction (FEC) can be adopted to mitigate the transmis-
sion errors. Hence, the successful collection of CDI feedbacks
can be guaranteed. After successfully collecting and selecting
(i.e., multiuser selection) CSI feedbacks from multiple users
(i.e., H = [h
1
, h
2
,...,h
M
]
T
), the base-station performs pre-
coding (i.e., ZFDPC or ZFBF), and obtains spatial multiplexing
gain. The signal received by a user i can be represented as
r
i
= h
T
i
x + n
i
, i = 1, 2,...,M (3)
where x is the transmitted symbol-vector that contains infor-
mation symbols of the selected users with an average power
constraint E{
x
2
}=P, and r
i
is the signal received by user i;
and n
i
represents additive white Gaussian noise (AWGN) with
variance σ
2
. Please note that x (i.e., M × 1 vector) is a precoded
result over original information symbol s (i.e., M × 1 vector),
through ZFDPC or ZFBF with the knowledge of CSI feedback
information H.
B. VAR Model
From the above description, we can see that it is critical for
a base-station to get to know the downlink CSIs, in order to
obtain the multiplexing gain and multiuser gain. For a specific
user i , it only knows its own CSI h
i
, and needs to feedback
such a channel vector to the base-station. In the following dis-
cussions, we omit index i of h
i
, and denote the channel vector
at time instant t as h
t
. As we mentioned earlier, it is very useful
to compress h
t
in order to obtain a lower dimension representa-
tion (thus obtaining low-dimension codebook). As we know, for
a time-varying channel (especially slowly varying one), there
are correlations between continuous CSIs [17]. Recently, much
research work on channel tracking has been done [8]–[11], [18],
and LP (i.e., VAR model) has been proved as t he best model to
describe the correlations among the CSI time sequences [9].
In a fast fading channel, the VAR model may not track the
CSI very well, and the prediction error will have large variance
in statistics. However, the proposed feedback scheme can still
benefit from the BCS model, which can compress the dimen-
sionality of the feedback CSI and thus a lower dimensional
codebook can be used in VQC.
AR( p) model (with order p) is commonly used to describe
the temporally correlated fading channel in single input single
output system (SISO) [3], [18]. Here, we adopt its extension
version (i.e., VAR( p)) to describe the vector channel, that is
h
t
= φ
1
h
t−1
+ φ
2
h
t−2
+···+φ
p
h
t− p
+ μ
t
(4)
where μ
t
is a M × 1 vector with complex zero-mean, covari-
ance A, multivariate normal distribution, and μ
t
represents