IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997 1613
Robust Frequency and Timing
Synchronization for OFDM
Timothy M. Schmidl and Donald C. Cox, Fellow, IEEE
Abstract— A rapid synchronization method is presented for
an orthogonal frequency-division multiplexing (OFDM) system
using either a continuous transmission or a burst operation over
a frequency-selective channel. The presence of a signal can be
detected upon the receipt of just one training sequence of two
symbols. The start of the frame and the beginning of the symbol
can be found, and carrier frequency offsets of many subchannels
spacings can be corrected. The algorithms operate near the
Cram
´
er–Rao lower bound for the variance of the frequency
offset estimate, and the inherent averaging over many subcarriers
allows acquisition at very low signal-to-noise ratios (SNR’s).
Index Terms— Carrier frequency, orthogonal frequency-
division multiplexing, symbol timing estimation.
I. INTRODUCTION
I
N AN orthogonal frequency-division multiplexing (OFDM)
system, synchronization at the receiver is one important
step that must be performed. This paper describes a method
to acquire synchronization for either a continuous stream of
data as in a broadcast application or for bursty data as in
a wireless local area network (WLAN). In both cases the
receiver must continuously scan for incoming data, and rapid
acquisition is needed. The ratio of the number of overhead bits
for synchronization to the number of message bits must be kept
to a minimum, and low-complexity algorithms are needed.
Synchronization of an OFDM signal requires finding the
symbol timing and carrier frequency offset. Symbol timing
for an OFDM signal is significantly different than for a single
carrier signal since there is not an “eye opening” where a
best sampling time can be found. Rather there are hundreds or
thousands of samples per OFDM symbol since the number of
samples necessary is proportional to the number of subcarriers.
Finding the symbol timing for OFDM means finding an
estimate of where the symbol starts. There is usually some
tolerance for symbol timing errors when a cyclic prefix is
used to extend the symbol. Synchronization of the carrier
frequency at the receiver must be performed very accurately,
or there will be loss of orthogonality between the subsymbols.
OFDM systems are very sensitive to carrier frequency offsets
since they can only tolerate offsets which are a fraction of the
Paper approved by M. Luise, the Editor for Synchronization of the
IEEE Communications Society. Manuscript received April 16, 1996; revised
February 11, 1997. This work was supported in part by a National Science
Foundation Graduate Fellowship. This work was presented in part at the IEEE
International Conference on Communications (ICC), Dallas, TX, June 1996.
T. M. Schmidl is with DSP Research and Development Center at Texas
Instruments Incorporated, Dallas, TX 75243 USA (e-mail: schmidl@ti.com).
D. C. Cox is with the STAR Laboratory, Department of Electrical
Engineering, Stanford University, Stanford, CA 94305-4055 USA (e-mail:
dcox@nova.stanford.edu).
Publisher Item Identifier S 0090-6778(97)09083-1.
spacing between the subcarriers without a large degradation in
system performance [1].
There have been several papers on the subject of synchro-
nization for OFDM in recent years. Moose gives the maximum
likelihood estimator for the carrier frequency offset which is
calculated in the frequency domain after taking the FFT [2].
He assumes that the symbol timing is known, so he just has to
find the carrier frequency offset. The limit of the acquisition
range for the carrier frequency offset is
the subcarrier
spacing. He also describes how to increase this range by using
shorter training symbols to find the carrier frequency offset.
For example shortening the training symbols by a factor of
two would double the range of carrier frequency acquisition.
This approach will work to a point, but the estimates get
worse as the symbols get shorter because there are fewer
samples over which to average, and the training symbols need
to be kept longer than the guard interval so that the channel
impulse response does not cause distortion when estimating
the frequency offset.
Nogami and Nagashima [4] present algorithms to find the
carrier frequency offset and sampling rate offset. They use
a null symbol where nothing is transmitted for one symbol
period so that the drop in received power can be detected to
find the beginning of the frame. The carrier frequency offset
is found in the frequency domain after applying a Hanning
window and taking the FFT. The null symbol is also used in
[11]. This extra overhead of using a null symbol is avoided by
using the technique described in this paper. If instead of a con-
tinuous transmission mode, a burst mode is used, it would be
difficult to use a null symbol since there would be no difference
between the null symbol and the idle period between bursts.
Van de Beek [3] describes a method of using a correlation
with the cyclic prefix to find the symbol timing. If this method
were used to find the symbol timing, while using one of the
previous methods to find the carrier frequency offset, there
would still be a problem of finding the start of the frame to
know where the training symbols are located.
Classen introduces a method which jointly finds both the
symbol timing and carrier frequency offset [5]. However, it
is very computationally complex because it uses a trial and
error method where the carrier frequency is incremented in
small steps over the entire acquisition range until the correct
carrier frequency is found. It is impractical to do the exhaustive
search and go through a large amount of computation at each
possible carrier frequency offset.
This paper introduces some modifications of Classen’s
method which both greatly simplify the computation necessary
for synchronization and extend the range for the acquisition
0090–6778/97$10.00 1997 IEEE
Authorized licensed use limited to: Oxford University Libraries. Downloaded on April 20,2010 at 12:04:16 UTC from IEEE Xplore. Restrictions apply.
1614 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997
Fig. 1. Block diagram of OFDM transmitter.
Fig. 2. Block diagram of OFDM receiver.
of carrier frequency offset. The method in this paper avoids
the extra overhead of using a null symbol, while allowing
a large acquisition range for the carrier frequency offset. By
using one unique symbol which has a repetition within half a
symbol period, this method can be used for bursts of data to
find whether a burst is present and to find the start of the burst.
Acquisition is achieved in two separate steps through the
use of a two-symbol training sequence, which will usually
be placed at the start of the frame. First the symbol/frame
timing is found by searching for a symbol in which the
first half is identical to the second half in the time domain.
Then the carrier frequency offset is partially corrected, and
a correlation with a second symbol is performed to find
the carrier frequency offset.
II. OFDM P
RINCIPLES
The OFDM signal is generated at baseband by taking the
inverse fast Fourier transform (IFFT) of quadrature amplitude
modulated (QAM) or phase-shift keyed (PSK) subsymbols
(Fig. 1). In the figure, the block P/S represents
a parallel-to-serial converter. An OFDM symbol has a useful
period
and preceding each symbol is a cyclic prefix of length
, which is longer than the channel impulse response so
that there will be no intersymbol interference (ISI) [6]. The
frequencies of the complex exponentials are
, and
the useful part for
subcarriers is given by
(1)
The baseband signal is quadrature modulated, up-converted to
the radio frequency (RF) and transmitted through the channel.
At the receiver (Fig. 2), the signal is down-converted to an
intermediate frequency (IF), and quadrature demodulated. The
block S/P represents a serial-to-parallel converter. A carrier
frequency offset of
causes a phase rotation of .
If uncorrected this causes both a rotation of the constellation
and a spread of the constellation points similar to additive
white Gaussian noise (AWGN). A symbol-timing error will
have little effect as long as all the samples taken are within
the length of the cyclically-extended OFDM symbol.
III. E
STIMATION OF SYMBOL TIMING
A. Symbol Timing Estimation Algorithm
The symbol timing recovery relies on searching for a
training symbol with two identical halves in the time do-
main, which will remain identical after passing through the
channel, except that there will be a phase difference be-
tween them caused by the carrier frequency offset. The two
halves of the training symbol are made identical (in time
order) by transmitting a pseudonoise (PN) sequence on the
even frequencies, while zeros are used on the odd frequen-
cies. This means that at each even frequency one of the
points of a QPSK constellation is transmitted. In order to
maintain an approximately constant signal energy for each
symbol the frequency components of this training symbol are
multiplied by
at the transmitter, or the four points of
the QPSK constellation are selected from a larger constel-
lation, such as 64-QAM, so that points with higher energy
can be used. Transmitted data will not be mistaken as the
start of the frame since any actual data must contain odd
frequencies. Note that an equivalent method of generating
this training symbol is to use an IFFT of half the normal
size to generate the time domain samples. The repetition is
not generated using the IFFT, so instead of just using the
even frequencies, a PN sequence would be transmitted on
all of the subcarriers to generate the time domain samples
which are half a symbol in duration. These time-domain
samples are repeated (and properly scaled) to form the first
training symbol.
The second training symbol contains a PN sequence on the
odd frequencies to measure these subchannels, and another
PN sequence on the even frequencies to help determine
frequency offset. Table I illustrates the use of PN sequences
in the training sequence for an OFDM signal with nine
subcarriers with the points chosen from a subset of a 64-
QAM constellation. The selection of a particular PN sequence
should not have much effect on the performance of the
synchronization algorithms. Instead the PN sequence can be
chosen on the basis of being easy to implement or having a low
peak-to-average power ratio so that there is little distortion in
the transmitter amplifier.
Complex samples
are taken by mixing the received
signal down to the IF, splitting the signal into two branches,
multiplying by both the in-phase and quadrature local oscil-
lators, and low-pass filtering and sampling to get baseband
in-phase and quadrature components (Fig. 2). This can be
expressed mathematically by writing the IF local oscillator
for the in-phase branch as
(2)
Authorized licensed use limited to: Oxford University Libraries. Downloaded on April 20,2010 at 12:04:16 UTC from IEEE Xplore. Restrictions apply.
SCHMIDL AND COX: ROBUST FREQUENCY AND TIMING SYNCHRONIZATION FOR OFDM 1615
TABLE I
I
LLUSTRATION OF USE OF PN SEQUENCES FOR TRAINING SYMBOLS
and the IF local oscillator for the quadrature branch at the
receiver as
(3)
Let the output of the mixer after down-conversion be
. The
demodulated signal before the sampler can be expressed as
(4)
where
means to low-pass filter the terms in the
argument. The output of the in-phase branch is considered to
be real and the output of the quadrature branch is considered to
be imaginary. This is a mathematical convention to represent
the in-phase and quadrature components as a complex number.
After sampling, the complex samples are denoted as
.
Consider the first training symbol where the first half is
identical to the second half (in time order), except for a phase
shift caused by the carrier frequency offset. If the conjugate of
a sample from the first half is multiplied by the corresponding
sample from the second half (
seconds later), the effect of
the channel should cancel, and the result will have a phase
of approximately
. At the start of the frame,
the products of each of these pairs of samples will have
approximately the same phase, so the magnitude of the sum
will be a large value.
Let there be
complex samples in one-half of the first
training symbol (excluding the cyclic prefix), and let the sum
of the pairs of products be
(5)
which can be implemented with the iterative formula
(6)
Note that
is a time index corresponding to the first sample in
a window of
samples. This window slides along in time as
the receiver searches for the first training symbol. The received
energy for the second half-symbol is defined by
(7)
Fig. 3. Example of the timing metric for the AWGN channel (
SNR = 10
dB).
which can also be calculated iteratively. may be used
as part of an automatic gain control (AGC) loop. A timing
metric can be defined as
(8)
Fig. 3 shows an example of the timing metric as a window
slides past coincidence for the AWGN channel for an OFDM
signal with 1000 subcarriers, a carrier frequency offset of 12.4
subcarrier spacings, and an signal-to-noise ratio (SNR) of 10
dB, where the SNR is the total signal (all the subcarriers) to
noise power ratio. The timing metric reaches a plateau which
has a length equal to the length of the guard interval minus
the length of the channel impulse response since there is no
ISI within this plateau to distort the signal. For the AWGN
channel, there is a window with a length of the guard interval
where the metric reaches a maximum, and the start of the frame
can be taken to be anywhere within this window without a loss
in the received SNR. For the frequency selective channels, the
length of the impulse response of the channel is shorter than
the guard interval by design choice of the guard interval, so
the plateau in the maximum of the timing metric is shorter
than for the AWGN channel.
This plateau leads to some uncertainty as to the start of
the frame. For the simulations in this paper, OFDM symbols
are generated with 1000 frequencies,
500 to 499. They are
slightly oversampled at a rate of 1024 samples for the useful
part of each symbol. In an actual hardware implementation, the
ratio of the sampling rate to the number of frequencies would
be higher to ease filtering requirements. The guard interval is
set to about 10% of the useful part, which is 102 samples.
B. Performance of Symbol Timing Estimator
There are two issues to consider when evaluating the
performance of the symbol timing estimator. First, since the
timing metric is also used to determine whether the training
sequence has been received, there is a probability of either
missing a training sequence and not detecting the signal or
falsely detecting a training sequence when none is there. In
this paper the distribution of the timing metric at the correct
start of the frame is calculated. Using this distribution, the
number of samples that need to be processed during the
detection phase can be determined and a threshold can be
Authorized licensed use limited to: Oxford University Libraries. Downloaded on April 20,2010 at 12:04:16 UTC from IEEE Xplore. Restrictions apply.
1616 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 12, DECEMBER 1997
set for this detection. Second, there is some degradation in
performance if the symbol timing estimate deviates from the
correct region. Simulations are performed to find the effect of
extra interference that is introduced by poor symbol timing
estimates for two types of channels.
1) Distribution of Timing Metric: Let each complex sam-
ple
be made up of a signal and a noise
component. Let the variance of the real and imaginary com-
ponents be:
(9)
(10)
so that the SNR is
. To find the mean and variance of the
estimator at the best symbol timing, first look at
which
can be written as
(11)
At the correct symbol timing, this can be broken into parts that
are in-phase and quadrature to the
product
which has phase
. This is just another way of looking at
the problem with a new set of axes with one axis in the
direction and the other axis perpendicular to it. For usable
values of SNR, when the magnitude is taken the quadrature
part will be small compared to the in-phase part and can be
neglected, so
(12)
where
means the component in the direction.
The quadrature part is neglected because the Rician distri-
bution can be approximated by a Gaussian when taking the
envelope of a dominant signal with Gaussian noise [8]. The
dominant in-phase part is so much larger than the quadrature
part that the envelope can be approximated by the in-phase
part. Note that the first term of
is much larger
than the second term since the
products
all have phase
and add in-phase, while all the other terms
add with random phases. By the central limit theorem (CLT),
is Gaussian with mean since each of the
terms has an expected value of , and
all the other terms have an expected value of zero.
For
, the magnitude of each term could be taken by
adding the squares of the real and imaginary parts. Instead,
we can define a new set of orthogonal axes in which one axis
is in the direction of the term
and the other is
perpendicular to it.
(13)
Here,
means to take the component in
the direction of
. By the CLT, is also Gaussian
with mean
. Note that for usable values of
SNR, the mean is much greater than the standard deviation.
Thus, for the Gaussian approximation, the probability of
is insignificantly small making the approximation
of the nonnegative
by a Gaussian reasonable. Another
equivalent way of thinking about the distribution is that
is Rician with the mean much larger than the standard
deviation. In this case the Gaussian approximation may be
used [8].
Define the square root of
to be
(see (14) at the bottom of the page).
Since the standard deviations of both the numerator and
denominator are much smaller than the means, the approx-
imation
can be used.
Another way to explain this is that there is a Gaussian random
variable in the numerator and a Gaussian random variable in
the denominator, and since the standard deviation of both of
these Gaussian random variables is much smaller than their
mean values, the above approximation can be used to write
their ratio as a single Gaussian random variable. As long as
this approximation holds,
is Gaussian with
(15)
This can be justified because linear operations on a Gauss-
ian random variable will result in another Gaussian random
variable [9].
When calculating the variance, note that
(16)
and
(14)
Authorized licensed use limited to: Oxford University Libraries. Downloaded on April 20,2010 at 12:04:16 UTC from IEEE Xplore. Restrictions apply.
SCHMIDL AND COX: ROBUST FREQUENCY AND TIMING SYNCHRONIZATION FOR OFDM 1617
Since these terms are the same in both the numerator and
denominator, they do not contribute to the overall variance.
Then
(17)
is , which is the square of a Gaussian
random variable. Since the variance is much smaller than the
mean, this also can be approximated by a Gaussian random
variable since linear operations performed on a Gaussian ran-
dom variable will produce another Gaussian random variable:
(18)
where
is used to denote a Gaussian random variable
with a mean of
and a variance of .
The expected value is
(19)
and the variance is
(20)
At high SNR the mean is approximately 1 and the variance is
approximately
.
The value of
also can give an estimate of the SNR,
which is
(21)
This equation is derived from (19). This can be written as
. The denominator
is Gaussian, and its reciprocal will also be Gaussian since
the standard deviation is much smaller than the mean for low
values of SNR. Again, this is a linear operation on a Gaussian
random variable (after the approximation). Then
(22)
This estimator works well for the SNR below 20 dB. Above
this level,
is so close to 1 that an accurate estimate of
the SNR can not be determined, but only that the SNR is high.
For example, if
, then dB.
This can be used to set a threshold so that very weak signals
will not be decoded, or it can be used in a WLAN to feed back
to the transmitter to indicate what data rate will be supported
so that an appropriate constellation and code can be chosen. A
lookup table can be implemented based on
, so that
no square roots or divisions need to be performed.
Even if there is a frequency selective channel, all the signal
energy will go into the signal component term except when
the length of the channel impulse response becomes so large
that it is longer than the cyclic prefix. At this point, the energy
located at longer delays becomes interference and would be
added to the noise terms.
At a position outside the first training symbol, the terms in
the sum
add with random phases since there is not
a periodicity for samples spaced by
samples. For the purpose
of computing the mean and variance of the timing metric, the
samples can be considered to be composed of just noise terms
since noise terms will also add with random phases, so the
statistics will be independent of the SNR. This assumption
is verified by the simulation results shown in Fig. 6 which
will be explained later in this section. The real and imaginary
parts of
are Gaussian by the CLT. The sum of
the square of two zero-mean Gaussian random variables, each
with a variance of 1 is a chi-square random variable with
two degrees of freedom and is represented by the symbol
.
The mean of
is 2 and the variance is 4 [9]. To simplify
the computations, let the variance of the real and imaginary
components of
be:
(23)
Both
and are Gaussian-
distributed with zero mean and a variance of
. Incor-
porating this scaling factor, we have
(24)
The mean and variance of
are
(25)
(26)
The denominator
has a Gaussian distribution by
the CLT, and its square is also Gaussian because the standard
deviation is much smaller than the mean. Again, this is a
linear operation on a Gaussian random variable (using the
approximation), so the result is also Gaussian. Thus,
(27)
where the
operator means “is distributed.” The ratio of these
two random variables (after dividing both the numerator and
denominator by a constant) is
(28)
Here, the variance of the Gaussian random variable is propor-
tional to
and can be neglected. The mean and variance
Authorized licensed use limited to: Oxford University Libraries. Downloaded on April 20,2010 at 12:04:16 UTC from IEEE Xplore. Restrictions apply.
