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Unformatted text preview: An All—Digital Frequency Synchronization Scheme
for OFDM Systems Li Zhenhong, Aame Mammelal V I I Electronics, PO. Box 1100, FIN—90571, Oulu, Finland
Tel.: +358 8 551 2111, fax: +358 8 551 2320
Email: [email protected], [email protected]ﬁ ABSTRACT The main purpose of this paper is to study the frequency
synchronization problem in OFDM systems. A new fre
quency synchronization method is achieved by means of an
all—digital synchronization loop including acquisition and
tracking modes. The frequency offset is estimated only once
per frame. The acquisition algorithm is based on periodi
cally inserted reference symbols, and the tracking algorithm
on the cyclic preﬁx of the OFDM symbols. The algorithm
performance is assessed in an AWGN channel with a low
SNR value and in the presence of a timing inaccuracy. The
simulations show that the algorithm is accurate and the sug
gested frequency synchronization scheme works well. I. INTRODUCTION The research and development work on the orthogonal fre
quency division multiplexing (OFDM) technique for high
speed digital data transmission has received considerable
attention and has made a great deal of progress [1]. As is
well known, OFDM systems are very sensitive to frequency
offsets [2], as an offset of a small fraction of the subcarrier
distance may lead to intolerable degradation. Frequency
synchronization therefore is one of the most important tasks
in an OFDM receiver. A frequency synchronization algorithm conventionally in
cludes acquisition and tracking algorithms, also called
coarse and ﬁne frequency synchronization algorithms, re
spectively, in some papers. Several methods focusing on
frequency tracking algorithms for OFDM systems have been
presented. P. H. Moose [3] introduced such an algorithm
using repeated data symbols, while the approaches proposed
in [45] exploit cyclic preﬁx to estimate the ﬁne frequency
offset. In [4] the algorithm was derived from the maximum
likelihood function and in [5] the available information
provided by the estimator is the imaginary part, which is an
approximation of the phase. The above algorithms are lim—
ited to tracking frequency offsets of less than half of the
subcarrier distance, however, and few proposals have con
cerned a large frequency offset. One of these is based on
iterative maximization of the likelihood function through the
stochastic gradient method [6]. In addition, techniques that
include dataaided acquisition and decisiondirected track 0780338715/97/$10.00 © 1997 IEEE 327 ing, are suggested in [78], in which known symbols are
utilized to cope with a large frequency offset. A brief gen
eral description of the tracking loop is provided in [8]. This paper presents a new alldigital frequency synchroni
zation scheme in which the synchronization principles de
rived for single carrier systems by F. M. Gardner [9] are
applied to OFDM systems after suitable modiﬁcations. The
frequency estimators in our scheme were suggested by P.
Mandarini and A. Falaschi, who analyzed the performance
of the ﬁne frequency estimator in [10]. Moreover, the syn
chronization task is divided into an acquisition and a track
ing mode. The acquisition mode, based on reference sym—
bols, provides an initial estimate for the tracking mode. The
tracking mode then exploits the cyclic preﬁx to estimate the
rotated phase, which contains the frequency offset informa—
tion. In particular, an alldigital synchronization loop struc
ture suitable for TDMA/OFDM systems is introduced in
detail in this paper. Since frequency offset is estimated once
per frame and the frequency is corrected in each received
sample, a hold circuit is used as an interpolator in the syn
chronization loop. This paper is organized as follows: A description of the
system model and the proposed frequency synchronization
scheme is provided in Section H, the frequency estimation
algorithm is presented in Section III, and the performance of
the synchronization scheme is then evaluated in detail with
simulations in section IV. Finally, Section V contains our
conclusions. II. SYSTEM MODEL The transmission system considered here is based on the
OFDM technique [11] with N" used subcarriers. After the
addition of N—Nu null data corresponding to the virtual carri
ers at the edges of the spectrum, the resulting N samples are
processed by an Inverse Fast Fourier Transform (IFFT) of
size N. A cyclic preﬁx made up of the last NS samples is
added at the beginning of each OFDM symbol. The N+Ng
samples generated during the lth OFDM symbol can be rep—
resented by 1 Nl xk =Fm§0Xm(l)e j27t(kNg )m/N OSk<N+Ng (1) where the data XMU) represent complex Mary PSK symbols
in the lth OFDM symbol. In order to estimate the coarse
frequency offset, a known reference symbol is transmitted
between every Fl OFDM data symbols. Thus one frame
consists altogether of F symbols. In the presence of a noise and frequency offset 5f, the re
ceived signal can be written by yk=xw“”f“”+nk a) where "n, are assumed to be independent zero mean complex
Gaussian random variables, T is the time interval between
the transmitted signal samples and (p is a random phase with
a uniform density in the range [75, 1c). Fig. 1 Highlevel block diagram of the frequency synchronization scheme A frequency synchronization loop generally consists of a
frequency corrector, a frequency estimator, a loop ﬁlter and
a number controlled oscillator (NCO). The frequency offset
is corrected with information provided by the frequency
estimator, ﬁltered by the loop ﬁlter and integrated by the
NCO as described in {9]. The functional block diagram of
the proposed frequency synchronization scheme in Fig. 1
contains suitable modifications based on [9]. The coarse
frequency estimator provides a coarse estimate as soon as
possible at the beginning of transmission, and the frequency
offset can have large values. Since we are not operating in a
burst transmission mode, the acquisition mode is activated
only once at the beginning of transmission. In contrast to the
acquisition mode, the tracking performance should show
excellent behaviour and only small frequency ﬂuctuations
have to be dealt with. The control signal, which is indicated
by the dotted lines in Fig. 1, is a binary signal dependent on
the estimated coarse frequency offset. Both the tracking mode and the acquisition mode work in a
closed loop, as the tracking algorithm needs the initial
coarse frequency estimate due to its limitations. Coarse fre
quency estimation is ﬁrst carried out in the acquisition
mode. In order to bring the frequency offset within the ex
pected range rapidly, the first frame is always used for
coarse frequency estimation. The control signal in the ﬁrst
frame is therefore fixed at zero and that in the next frames is /\
dependent on the estimated coarse frequency offset 61?an . Once the estimated coarse frequency offset is less than half 328 of the subcarrier distance, the control signal is ﬁxed at one,
the received signal goes into the tracking mode by a denial
tiplexer and ﬁne frequency estimation starts. The estimated A
ﬁne frequency offset & crack then goes into the synchroni zation loop based on the initial coarse frequency estimate by
a multiplexer. The frequency estimate is processed only
once per frame. The frequency synchronization loop equivalent digital
model is illustrated in Fig. 1. The digital loop ﬁlter transfer
function is given by 1 l m H(z):az_ l — z _
where (X is the loop ﬁlter gain and could be onacq or am,
which is dependent on the control signal. Due to the re
quirements of the acquisition mode, the gain oram should be
selected carefully. Thus an initial coarse frequency estimate
can be provided only after a few frames, and an am close to
1.0/F could be a good choice. In the tracking mode 0cmk is quite small in order to obtain a ﬁne estimate and small vari—
ance. Two zero—order holds are included in the proposed loop
structure, in order to upsample back to the sampling rate of
the received signal. The first one repeats the received signal
F times and the hold time for the second one is N+Ng. One
hold circuit placed after the loop ﬁlter with a hold time
equal to F  (N+Ng) would be an alternative choice. In order
to achieve similar performance to that of the loop shown in
Fig. l, the loop ﬁlter gain could be multiplied by F. The NCO is implemented according to the difference equa—
tion 90¢) = [0(k — 1) + e(k)]mod— 27: (4) where e(k) is the estimated and scaled frequency offset A
27m? including the coarse and ﬁne frequency estimates. When the loop is stable, 6(k) is approximately equal to
(27réfkT) mod— 2n . To ﬁt the causality requirements of the system, at least one symbol interval delay is unavoidable
before the frequency correction. The frequency offset can be
corrected by n=nrwm to
in which the delay is ignored. Once the estimated coarse frequency offset is larger than half
of the subcarrier distance, the acquisition mode will restart
and the corresponding steps will be repeated. III. FREQUENCY ESTIMATION ALGORITHM Since the frequency tracking will be usually out of control
when the frequency offset exceeds half of the subcarrier
distance, a strategy for initial acquisition must be developed to bring the offset within the limitations of the tracking al
gorithm. This section concentrates on the algorithms of the
frequency estimators. A maximum timing error less than
half of the preﬁx is assumed. A. Acquisition Algorithm The acquisition algorithm will provide a coarse frequency
estimate by analysis of the reference symbols after the FFI‘.
It is not so sensitive to the timing error, by virtue of the
protection given by the preﬁx. The reference symbol, which
is the same for each frame, consists of virtual carriers and
M, pilot tones spaced by D virtual carriers (M, and D are an
even and an odd number, respectively). The method of
modulation for the pilot tones is the same as for the data
symbols. In order to keep the transmitted power the same in
the reference and data symbols, the amplitude of the pilot
tones is multiplied by a factor
7 = N u / M , . (6)
We assume that Y[l], ..., Y[N] indicate the samples of the
reference symbol at the receiver after the FFT, the vector
V'] of D components is evaluated from Y[1] to Y[N]:
. _ . . 2 V[J]= 221;; Mm. +M. 0] (7)
with —(D— 1) / 2 S j S (D— 1) / 2 , where j: represents the
index of the first pilot tone in the reference symbol.
index The value of VD] and its j= jm =argmax{V[j]} will be searched through the
J maximum vector. Once the corresponding j1m is found, the values
v_,=v[jmx—1], V0=V[j ] and V,=V[jm+1] are
considered. In order to achieve the estimated coarse frequency offset,
the following processing is done: p1 = m, (8) 101 = WM—Vo, (9) d =(p1p_,)/2.O, (10)
$3,, = Aroma, +sgn{d},ﬂe7l) (11) where Af represents the subcarrier distance. According to
the above equations, the acquisition range is clearly depend
ent on the parameter D, which controls the possible maxi
mum index of j. We assume that the actual frequency offset
is equal to éf=AfU+§l (12) 329 where J is an integer and < 05. Thus jmax is an estimate of J and sgn{d is an estimate of g. The same reference symbol can be used for the coarse frame
synchronization, as explained in [12]. B. Tracking Algorithm The ﬁne frequency estimator, the frequency tracking algo
rithm provides a precise estimate of the frequency offset
based on the cyclic prefix of the OFDM symbols. We as
sume that y[0], ..., y[N+Ng—1] represent the samples of the
OFDM data symbol at the receiver before the FFT. By vir
tue of the periodic nature of the OFDM signal, the presence
of the cyclic preﬁx results in the equality x, = x,”v with
N S k S N+ N8 —1. In the absence of noise, i.e., n, = 0, we obviously obtain “k = Yr ‘Yk—N
= xk .x*(kN),ej27’&trackNT [2 eﬂﬂé’uack/Af
k s (13) =5), Thus a simple implementation of a low variance estimator A
for t? track is given by A warm = Af(tan‘1(22'§0“uk)) (14)
in the presence of noise and a frequency offset lower than
half of the subcarrier distance Af . The phase rotation due
to the frequency offset is estimated and scaled in (14) to
achieve the frequency estimation. The quality of the esti
mate depends on the signaltonoise ratio per bit (SNR). IV. SIMULATION RESULTS Many simulations have been performed, each with different
parameters. This section presents the key results, to support
the conclusions set out in Section V. All the simulations are performed in a COSSAP environ
ment, the COSSAP models developed are of the ﬂoating
point type. The subcarrier modulation method is QPSK. The
main parameters in the simulation models are: F = 64, N =
512, N" = 424, NR = 64, M, = 28, D = 15 and j: = 54. The
number of classes between the minimum and maximum
class values in the histograms is 50, including the negative
minimum value. A total of 20,000 events are taken into ac
count. The performance of the coarse frequency estimator is
evaluated with a simulation model which contains an
OFDM transmitter, an AWGN channel and a coarse fre
quency estimator. The corresponding results are shown with
histograms (number of events vs. coarse frequency estima
tion error normalized by the subcarrier distance) in Fig. 2, with SNR = 10.0 dB. Typically, we choose Q to be 0.00.5
with step 0.1. The histograms show two peaks for g, of 0.0 and 0.5, lo—
cated approximately symmetrically with respect to zero.
Though there may be a small bias, most of the values are
located in the range [—0.10, 0.10). r_____._____________._____.____ coarse frequency estimation error
with J = 5.0 .E 0.0
L 0.1 number of event: : , , . . . . , . . . . , . . l .
—o.1o o.05 0.00 0.05
come frequency munum crew nowd by th mbcartler dim .___________________J Flg. 2 Histograms of the normalized coarse frequency estimation error The numerical results give an inaccuracy of half of the sub
carrier distance with a probability of approximately 1.0,
even when the SNR is 5.0 dB and the PET processor win
dow starts somewhere within the preﬁx. The timing error in
the simulations was generated by delaying the received sig
nal by ‘6 samples, where 't is an integer. These results are
accurate enough for ﬁne frequency estimation. Furthermore,
this guarantee is conﬁrmed with one of the worst cases (é =
0.0 and 0.5 in Fig. 2) by the standard deviation (std) of the
normalized coarse frequency estimation error as shown in
Fig. 3. coarse frequency estimation error
withJ =5.0and§= 0.0 std of lhe normalized coaxso frequency animation error . , . . I . . .
5.0 10.0 15.0
signal—to—neise ratio per bit 0.0 I
[‘13] Fig. 3 Std of the normalized coarse frequency estimation error The behaviour of the tracking loop, including a fine fre
quency estimator and the proposed alldigital loop structure
with two hold circuits, is measured by the histograms and
the std of the tracking loop error. Random OFDM data sym
bols are used to estimate the frequency offset in the simula—
tion model with 8f :04. The simulated results are illustrated
in Fig. 4 and 5 when the loop is in the steady state. 330 When the preﬁx is sampled correctly, i.e., 1:: 0 even at low
SNR (SNR = 5.0 dB), the normalized frequency tracking
loop error is almost always less than 0.01 and its std is less
than 0.003. Even when the preﬁx is sampled with t: 32 and
SNR = 5.0 dB, almost all of the values are still in the range
[0.05, 0.05). The std of the normalized tracking loop error
is less than 0.01. frequency tracking loop error ":‘mao.oaédro
+—+:suns.odsuiavo
o—e:SKR runaway:2
sunseams 132 8000 — r:: 6000 numbar of event. 2000 0  l  t
0.050 . K . . .
0.025
mmwmmhymnwm .‘...x....l...
0.050 0.025 0.000 Fig. 4 Histograms of the normalized frequency tracking loop error with
the filter gain am = 0.003 T_'———_'——"‘__—_—_‘~‘1 frequency tracking loop error “~‘—: 7  O
x—x—z 7  32 . , .
15.0 lld of tho normalized frequency lnoklng loop error . I .
10.0
stands—min "do per hit. [dB] l___________._______ Fig. 5 Std of the normalized frequency tracking loop error with the ﬁlter
gain am = 0.003 20.0 The behaviour of the tracking loop is also affected to a cer
tain extent by the gain am. The larger am is, the shorter
the convergence time and the larger the variance in the
tracking loop error. Thus it is important to obtain a good
compromise between estimation accuracy and short conver
gence time. V. CONCLUSIONS This paper suggests an alldigital frequency synchronization
scheme suitable for OFDM systems and shows that the pro
posed closed synchronization loop structure provides fast
acquisition and accurate tracking in an AWGN channel. The
frequency synchronization scheme is practicable and is able
to deal with a large frequency offset. A similar structure can
be applicable to the future W'LAN systems, in which ex tremer large bit rates will be necessary and OFDM signals
should be used. Although the present simulations were limited to an AWGN
channel, the scheme can be easily applied to a fading multi
path channel. The corresponding performance of the algo
rithms will be analyzed in the future. ACKNOWLEDGMENT The authors gratefully acknowledge the ﬁnancial support
given by the MEDIAN project and they also wish to thank
Prof. P. Mandarini and A. Falaschi for the frequency esti
mation algorithms and to express their gratitude to the DSP
group members at VTI‘ Electronics for their comments. REFERENCES [1] W. Y. Zou, “COFDM: An Overview,” IEEE Trans
actions on Broadcasting, Vol.41, No. 1, March 1995,
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