IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 5, MAY 2007
321
A New OFDM Synchronization Symbol for Carrier
Frequency Offset Estimation
Amine Laourine, Student Member, IEEE, Alex Stéphenne, Senior Member, IEEE, and
Sofiène Affes, Senior Member, IEEE
Abstract—This letter proposes a new data-aided carrier frequency offset (CFO) estimation scheme for orthogonal frequency
division multiplexing (OFDM) communications suitable for
frequency-selective channels. The proposed method is based on
the transmission of a specially designed synchronization symbol
that generates a particular signal structure between the received
observation samples at the receiver. This structure is exploited to
derive a closed-form expression of the CFO. The proposed solution offers a wide acquisition range with reduced computational
load. Simulations over frequency-selective channels confirm the
superiority of the proposed method compared to recent data-aided
synchronization algorithms.
Index Terms—Frequency-division multiplexing, synchronization.
I. INTRODUCTION
RTHOGONAL frequency division multiplexing (OFDM)
performance suffers from a pronounced sensitivity against
carrier frequency offsets (CFOs) [1]. The CFO should therefore
be estimated and compensated before any further processing in
the receiver. Different data-aided schemes of CFO estimation
in OFDM systems have been appraised. These methods hinge
on the periodic transmission of known data blocks that allows
for the estimation of the CFO from the estimation of the phase
rotation between these blocks at the receiver. The most known
of these methods is the M&M method [3], which constitutes an
improvement to Schmidl’s [2]. The M&M method was also selected by [4] to provide a joint robust estimation of the timing
and the frequency offset. More recently, a new synchronization symbol proposed in [5] allowed for performance gains over
Schmidl’s algorithm.
In this letter, we propose a new synchronization symbol structure and a new CFO estimator that offer a wide acquisition
range and a high accuracy at a very reduced computational cost.
Based on the original shape of the new synchronization symbol,
a closed-form CFO estimator is derived. Our method has an acthe subcarrier spacing,
quisition range that can reach
where is a user-selected parameter characterizing our synchronization symbol, and is the number of OFDM subcarriers. As
O
Manuscript received July 4, 2006; revised September 21, 2006. This work
was supported by a Canada Research Chair in High-Speed Wireless Communications. The associate editor coordinating the review of this manuscript and
approving it for publication was Dr. Gerald Matz.
A. Laourine and S. Affes are with the INRS-EMT, Montréal, QC H5A 1K6,
Canada (e-mail: laourine@emt.inrs.ca; affes@emt.inrs.ca).
A. Stéphenne is with Ericsson Canada, Montreal, QC H4P 2N2, Canada
(e-mail: stephenne@ieee.org).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LSP.2006.888353
supported by simulations, our algorithm provides higher accuracy compared to the well-known M&M method [3] as well as
the method in [5]. More specifically, our estimator proved to be
robust against low SNRs (below 0 dB). Such low SNR values
are of great practical interest since next-generation wireless systems attempt to operate in MIMO multi-access multi-cell environments, which are characterized by low operating SNRs. In
this regard, OFDM needs reliable synchronization techniques
that operate well in a low SNR regime.
This letter is organized as follows. In Section II, the OFDM
system model is described. The designed synchronization
symbol and the resulting structure in the received samples are
described in Section III. The CFO estimator and the different
derivations are presented in Section IV. Section V analyzes the
performance of the proposed estimator. Numerical examples are
illustrated in Section VI. Conclusions are given in Section VII.
II. SYSTEM MODEL
subcarWe consider a discrete-time OFDM system with
riers. At the transmitter, complex-valued symbols
, which
belong to a QAM or PSK constellation, modulate orthogonal
subcarriers using the inverse fast Fourier transform (IFFT). Before transmission, a cyclic prefix is appended at the beginning
of the signal, yielding
if
if
where
.
At the receiver, the cyclic prefix is discarded, leading to the
following received samples:
(1)
(2)
(3)
where
is the transfer function of the
channel at the frequency of the th subcarrier,
corresponds to
the channel length, is the relative carrier frequency offset (the
ratio of the actual frequency offset to the intercarrier spacing),
and
is an additive complex white Gaussian noise (with
variance ) independent of
.
III. SYNCHRONIZATION SYMBOL DESIGN
To estimate the CFO, we propose to transmit the following
synchronization symbol:
1070-9908/$25.00 © 2007 IEEE
(4)
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IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 5, MAY 2007
or equivalently, for
), we may neglect the
, and by using (13), we have that
For high SNR (
term
(5)
(18)
where
is a user-selected even integer such that
. The choice of
has to be taken under
certain considerations such as the desired acquisition range,
the desired accuracy, and the PAPR that can be tolerated by the
system.
Based on the special structure of this synchronization symbol,
we will prove that there is a useful relationship between
and
that will allow us to easily derive a closed-form
estimator of the CFO. First, let us consider the noiseless part of
in the following:
(6)
where
CFO estimator:
. We therefore propose the following
(19)
is a window of real weights
where
designed to minimize the estimation variance and such that
. One can notice that this estimator will
induce a very small computational load. Indeed, it involves a
unique phase angle calculation, and its very form indicates that
it can be implemented efficiently using an FFT, a component
that already exists in the OFDM receiver. Moreover, the acqui, which can be large if one
sition range of this estimator is
chooses a low value of . In the section that follows, we will
derive closed-form expressions of both the window
and
the theoretical variance bounds of our estimator.
(8)
V. PERFORMANCE ANALYSIS
Inserting (18) in (19), we obtain that
(10)
is alIn a system with many subcarriers, the inequality
ways fulfilled. In this case,
for
,
which is equivalent to saying that adjacent subcarriers experience approximately the same channel. Moreover, due to the very
form of
and since
, it can be easily shown that
. Therefore
or in other words, if
, we have that
(20)
where
(21)
(11)
The second term
oped [using (5)]
could be further devel-
.
, we may approximate
and
Since
by
(12)
Since is even, then
this last equation in (11), we obtain:
. Substituting
(13)
(22)
Therefore, we have
Based on this last equation, we can now state that
(23)
(14)
Since
IV. PROPOSED CFO ESTIMATOR
The proposed CFO estimator is based on the correlation between the received samples spaced by lags. Let us consider
the following correlation product:
where
it is easy to see that
(16)
is the Kronecker function
and that
(24)
LAOURINE et al.: NEW OFDM SYNCHRONIZATION SYMBOL FOR CFO ESTIMATION
323
2) For
where
, the inverse of
is such as
(32)
(25)
Similarly,
, where
The corresponding optimal window is given by
(26)
From the Central Limit Theorem (CLT), we have that
and
. For high SNR, we may consequently state
that [6]
for
for
for
.
The corresponding variance is such that
.
A. Averaging
(27)
Since the acquisition range of the proposed estimator is
, then it is desirable to select small values of in order
to have the widest range. Therefore, we propose here a method
to improve the accuracy for values of
. Indeed, from
(13), we have the following recursion formula:
Consequently, we have
(33)
(28)
is an integer such that
where
suggests forming the following estimator:
. This equation
Therefore
i.e., the estimator is unbiased for high SNR
The variance of the proposed CFO estimator can be rewritten as
(34)
(29)
where
where
, and is an
matrix with entries equal to
. Following [7], the window that minimizes the variance
is such that
(30)
where
stands for an
-dimensional row vector consisting of only ones. The corresponding estimation variance is
(31)
Explicit forms of the variance and the optimal window were discussed extensively in [7]. Here, we just briefly cite some important results.
1) For
, we have that
, and therefore
(35)
and
is an
matrix with entries equal
to
. Notice that here again
this estimator lends itself well to an efficient implementation
via the FFT. The acquisition range of the estimator given by
(34) is
the subcarrier spacing. Therefore, an initial acquisition of the CFO has to be performed by , and then, we
compensate this value by multiplying the received signal
by
. Finally, the CFO estimate will be given by averaging in an effective way over all the possible values of , i.e.,
(36)
where
and
(
returns the integer part of ),
SNR
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IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 5, MAY 2007
Fig. 1. Mean square error versus SNR over an AWGN channel. [Theoretical
variance is given by (29).]
VI. NUMERICAL EXAMPLES
We consider an OFDM system with 64 subcarriers fed by
16-QAM constellations and a cyclic prefix of 16 samples in
). These are the settings of a typical OFDM-based
length (
WLAN. The CFO was fixed to 0.3 (so that it falls within the
range of all the estimators, i.e., no acquisition is required), and
2000 independent trials were performed to obtain the meansquare error estimates. Fig. 1 shows the results obtained over
an AWGN channel. First, we notice that averaging increases accuracy by as much as 8 dB for an MSE of 8
, with this
improvement being much higher at low SNR (results are not
shown at very low SNR for the sake of clarity). Another important remark is that the proposed estimators achieve the CRB
even for a moderate SNR (the CRB formula can be found in [3]).
In particular, the averaged version of the estimator will achieve
the CRB at a small SNR (around 5 dB). This figure shows also
that if a smaller acquisition range is allowed, choosing
will give accurate results without the use of averaging. In order
to show the reliability of our estimation, we made a comparison
with the well-known M&M scheme and the recent method developed in [5] (the last method is tuned to its best performance,
i.e.,
). These simulations were conducted in a multipath
environment having four paths with path delays of 0, 8, 10, and
12 samples (the CP length was set to be superior to the channel
length in order to avoid intersymbol interference). The amplitude of the th path is a Rayleigh random variable and varies
independently of the others according to an exponential power
delay profile, i.e.,
, where is the delay
of the th path. The power of each channel realization is then
normalized, which corresponds to a perfect power control situation. The obtained results are depicted in Fig. 2 (the CRB in
the AWGN channel is also plotted as a benchmark of the performance). When the acquisition range of our method and the
M&M one is equal to 16 times the subcarrier spacing, i.e.,
for the proposed scheme and
for the M&M method,
our estimator clearly outperforms the M&M, especially in the
low SNR region—so crucial for multi-antenna transceivers in
Fig. 2. Mean square error versus SNR over a multipath channel.
multi-cellular and multi-access environments—where the gap
is remarkable. Indeed, for an MSE equal to 7
, the SNR
gain is about 6 dB. For an acquisition range of eight times the
and
, the gap is still huge
subcarrier spacing, i.e.,
for low SNR, up to 5 dB in some regions. The superiority of our
method compared to the scheme in [5] is also obvious, for inor
at an MSE equal to 4
, where
stance, for
the SNR gain is around 3 dB.
VII. CONCLUSION
In this letter, a new CFO estimation scheme for OFDM has
been presented. The estimation is based on the transmission of
a specially designed synchronization symbol. The particular
structure in the received samples of our synchronization symbol
allows us to derive a closed-form expression for the CFO. This
estimator can provide very high accuracy over a wide acquisition range while keeping a very low computational complexity.
Simulations have also proved that this new technique gives
higher accuracy than recent methods developed in the literature.
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