SAGE Algorithm for Channel Estimation and Data Detection
with Tracking the Channel Variation in MIMO System
Takao Someya
†
Tomoaki Ohtsuki
††
†
Graduate School of Science and Technology, Tokyo University of Science
††
Dept. of Electrical Engineering, Tokyo University of Science
2641 Yamazaki, Noda, Chiba 278-8510 Japan
E-mail:† j7304638@ed.noda.tus.ac.jp ††ohtsuki@ee.noda.tus.ac.jp
Abstract— In recent years, Multiple-Input Multiple-Output
(MIMO) systems with some transmit and receive antennas have
attracted much attention in radio environments. In MIMO
systems, the channel estimation is important to distinguish
transmit signals from multiple transmit antennas. The Space-
Alternating Generalized Expectation-maximization (SAGE) al-
gorithm is known to be good for the channel estimation and
the data detection. However, the SAGE algorithm has not been
applied to MIMO systems. In this paper, we propose a SAGE
algorithm for the channel estimation and data detection in
MIMO systems. In addition, we propose a simplified SAGE
algorithm for the channel estimation and the data detection
with tracking the channel variation in MIMO systems. In the
simplified SAGE algorithm, we divide a transmit frame into
some subblocks and apply the SAGE algorithm to each subblock,
and we use the channel estimates in the previous subblock as the
initial channel estimates in the current subblock. According to
the division of the transmit frame, the computational complexity
is decreased. In addition, the simplified SAGE algorithm can
track the channel variation by using the channel estimates
transferred between the subblocks.
I. INTRODUCTION
In recent years, Multiple-Input Multiple-Output (MIMO)
systems with some transmit and receive antennas have at-
tracted much attention as a promising technique for achiev-
ing high bit-rate and high capacity transmission in radio
environments. However, when the channel state information
(CSI) is not perfect, MIMO systems are severely limited by
signal interference from other transmit antennas. Therefore,
in MIMO systems, to detect the transmitted signal from each
transmit antenna, an accurate CSI is needed at the receiver.
The Expectation-Maximization (EM) algorithm [1] is
known to be good for the channel estimation and the data
detection in Orthogonal Frequency-Division Multiplexing
(OFDM) systems [2] and Space-Time Block coded (STBC)
MIMO systems [3]. The EM algorithm is an iterative method
to approximate the maximum likelihood (ML) estimation
when a direct computation is computationally limited. The
EM algorithm makes use of the log-likelihood function in
a two-step iterative procedure. At the first step of the EM
algorithm, referred to as the Expectation-step (E-step), the
expectation of the log-likelihood function is calculated. In
the second step referred to as the Maximization-step (M-
step) the parameters are updated by maximizing the function
derived from the E-step. However, the EM algorithm updates
all the parameters for the channel estimation and the data
detection simultaneously, which results in a disadvantage
of slow convergence. In addition, the EM algorithm can
not track the channel variation well. The Space-Alternating
Generalized Expectation-maximization (SAGE) algorithm [4]
has been proposed for accelerating the convergence of the
Channel
Input
Output
Transmitter Receiver
DEMUX
MUX
1
2
N
M
1
2
Fig. 1. A Multiple-Input Multiple-Output (MIMO) system with N transmit
antennas and M receive antennas
EM algorithm. The SAGE algorithm updates the parameters
sequentially by alternating between the subset of parame-
ters. The SAGE algorithm was applied to Direct-Sequence
Code-Division Multiple-Access (DS-CDMA) systems [5] and
Space-Time Coding (STC) systems [6]. However, the SAGE
algorithm has not been applied to MIMO systems.
In this paper, we propose a SAGE algorithm for the channel
estimation and the data detection in MIMO systems. In
addition, we propose a simplified SAGE algorithm for the
channel estimation and the data detection with tracking the
channel variation in MIMO systems. In the simplified SAGE
algorithm, we divide a transmitted frame into some subblocks
and apply the SAGE algorithm to each subblock, and we use
the channel estimates in the previous subblock as the initial
channel estimates in the current subblock. According to the
division of the transmitted frame, the computational complex-
ity is decreased. In addition, the simplified SAGE algorithm
can track the channel variation by using the channel estimates
transferred between the subblocks. We show that the proposed
SAGE algorithm can achieve the better bit error rate (BER)
than the ML detection with training symbols. We also show
that the proposed simplified SAGE algorithm can achieve
the better BER with less computational complexity than the
proposed SAGE algorithm. In particular, we show that the
proposed simplified SAGE algorithm improves BER more
significantly with less computational complexity in the fast
fading environments than in the slow fading environments.
II. S
YSTEM MODEL
We consider the MIMO system with N transmit antennas
and M receive antennas shown in Fig. 1. One transmitted
frame of L symbols X
n
=[X
1
n
, ··· ,X
L
n
]
T
(n =1, ··· ,N)
is transmitted from the n-th transmit antenna. X
L
n
is the
transmitted symbol from the n-th transmit antenna at the L-
th symbol. The notation [·]
T
denotes the transpose operation.
A training sequence of p symbols is inserted in the head of
each transmitted frame. The training sequence is orthogonal
between each transmit antenna. At the k-th symbol, the
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