Training Sequences Design for Symbol Timing
Estimation in MIMO Correlated Fading Channels
Yik-Chung Wu and Erchin Serpedin
Department of Electrical Engineering, Texas A&M University, College Station, TX 77843-3128, USA.
Email: {ycwu, serpedin}@ee.tamu.edu
Abstract—In this paper, the problem of training sequences de-
sign for symbol timing estimation in MIMO channel is addressed.
In particular, we consider correlated fading between antennas.
The optimal training sequences are derived by minimizing
the modified Cramer-Rao bound (MCRB) with respect to the
training data. It is found that when the transmit pulse is a root
raised cosine pulse and there is no correlation among antennas,
the optimal training sequences resemble the Walsh sequences.
Furthermore, it is also found that the the impact of not knowing
the antenna correlations when designing training sequences is
very small.
I. I NTRODUCTION
Symbol timing synchronization is an important issue in
multiple-input multiple-output (MIMO) communication sys-
tems, as most of the research works in MIMO systems
assume perfect symbol timing information at the receiver
[1]-[8]. Symbol timing synchronization in MIMO uncorre-
lated flat fading channel was first studied by Naguib et al.
[12], where orthogonal training sequences are transmitted at
different transmit antennas to simplify the maximization of
the oversampled approximated log-likelihood function. This
algorithm was extended by the authors in [13] and [14] to
increase its estimation accuracy. Recently, the true maximum
likelihood (ML) symbol timing estimator in MIMO channel
based on training data was derived in [15].
Although the symbol timing estimation algorithms in
MIMO channels have been well established, the problem
of how to design the training sequences for symbol timing
estimation in MIMO channels is not fully addressed. In [13],
it was mentioned that the training sequences should be the
Walsh sequences with the largest number of sign changes.
Unfortunately, this choice of training sequences is based on an
intuitive argument, so there is no optimality associated with it.
In [14], the training sequences are designed to minimize the
contribution of the inter-symbol interference (ISI) term in the
approximated log-likelihood function, resulting the so-called
“perfect sequences” [16]. However, if the approximated log-
likelihood function is not used in the estimation (e.g., for the
true ML estimator [15]), the training sequences obtained may
not be optimal anymore.
In this paper, we try to derive optimal training sequences
that are independent of the estimation method. Toward this
end, the training sequences are found by minimizing the
modified Cramer-Rao bound (MCRB) with respect to the
training data. It is found that when the transmit pulse is a root
raised cosine pulse and there is no correlation among antennas,
the optimal orthogonal training sequences resemble the Walsh
sequences. Furthermore, it is also shown that the knowledge
of antenna correlations is not important for designing training
sequences.
II. SIGNAL MODEL
Consider a MIMO communication system with N transmit
and M receive antennas. At each receiving antenna, a super-
position of faded signals from all the transmit antennas plus
noise is received. The complex envelope of the received signal
at the j
th
receive antenna can be written as
r
j
(t)=
E
s
NT
N
i=1
h
ij
n
d
i
(n)g(t − nT − ε
o
T )+ η
j
(t),
j =1, 2, ..., M
(1)
where E
s
/N is the symbol energy; h
ij
’s are the complex
channel coefficients between the i
th
transmit antenna and the
j
th
receive antenna; d
i
(n) is the zero-mean complex valued
symbol transmitted from the i
th
transmit antenna; g(t) is the
transmit filter with unit energy; T is the symbol duration;
ε
o
∈ [0, 1) is the unknown timing offset and η
j
(t) is the
complex-valued circularly distributed Gaussian white noise at
the j
th
receive antenna, with power density N
o
. Throughout
this paper, it is assumed that the channel is frequency flat and
quasi-static.
After passing through the anti-aliasing filter, the received
signal is then sampled at rate f
s
=1/T
s
, where T
s
T/Q
(Q is the oversampling ratio). The received vector r
j
, which
consists of L
o
Q consecutive received samples (L
o
is the
observation length), can be expressed as (without loss of
generality, we consider the received sequence start at t =0)
r
j
= ξ A
εo
ZH
T
j,:
+ η
j
, (2)
where
1
ξ
E
s
/N T ,
r
j
[r
j
(0) r
j
(T
s
) ... r
j
((L
o
Q − 1)T
s
)]
T
, (3)
A
ε
[a
−Lg
(ε) a
−Lg+1
(ε) ... a
Lo+Lg−1
(ε)] , (4)
a
i
(ε) [g(−iT − εT ) g(T
s
− iT − εT )
... g((L
o
Q − 1)T
s
− iT − εT )]
T
, (5)
Z [d
1
d
2
··· d
N
], (6)
d
i
[d
i
(−L
g
) d
i
(−L
g
+ 1) ··· d
i
(L
o
+ L
g
− 1)]
T
, (7)
1
Notation x
T
denotes the transpose of x, and x
H
denotes the transpose
conjugate of x.
Globecom 2004 81 0-7803-8794-5/04/$20.00 © 2004 IEEE
IEEE Communications Society