Digital Signal Processing 21 (2011) 710–717
Contents lists available at ScienceDirect
Digital Signal Processing
www.elsevier.com/locate/dsp
Blind synchronization scheme using the conjugate characters of the OFDM
BPSK-modulated symbol
Cheng-Ying Yang a , Gwo-Ruey Lee a,b,∗ , Wen-Hui Kuan b , Jyh-Horng Wen c
a
b
c
Department of Computer Science, Taipei Municipal University of Education, No. 1, Ai-Guo W. Rd., Taipei, Taiwan, ROC
The Institute of Electrical Engineering, National Chung Cheng University, No. 168, University Rd., Min-Hsiung, Chia-Yi, Taiwan, ROC
Department of Electrical Engineering, Tunghai University, No. 181 Section 3, Taichung Harbor Rd., Taichung, Taiwan, ROC
a r t i c l e
i n f o
a b s t r a c t
Article history:
Available online 28 January 2011
Previously, Beek’s scheme for timing and frequency offset estimation in the OFDM system employs cyclic
prefix (CP) has been proposed under the assumption of independent identified distributed (i.i.d.) OFDM
symbols. Actually, the real data in the OFDM modulated symbol, transferred by the inverse fast Fourier
transform (IFFT), has the characters of complex symmetry. With these characters, more information
in the whole OFDM symbol could be used for the timing and frequency offset estimation. In this
paper, two conjugate symmetry characters of the OFDM BPSK-modulated symbol are used to achieve
blind timing estimation algorithm in the OFDM systems. One is symbol-based symmetry and the other
is CP-based symmetry. With these two conjugate characters applied to the proposed algorithm, the
timing of the OFDM BPSK-modulated symbol could be derived. Under an AWGN channel, based on the
performance of lose symbol timing rate and estimator mean square error, the proposed algorithm is
with a tremendous improvement compared with Beek’s estimation method. Under a multipath fading
channel, the results show that performance including lose symbol timing rate and estimator MSE with
the proposed algorithm is better than those algorithms with Beek’s estimation method. In practical OFDM
applied system, the OFDM BPSK-modulated symbol could be used to replace the preamble or training
sequences in the standard to obtain an accurate timing and frequency offset estimation and to avoid the
data rate decreasing with the proposed algorithm.
Crown Copyright © 2011 Published by Elsevier Inc. All rights reserved.
Keywords:
OFDM systems
Symbol synchronization
Timing estimation
Conjugate characters
BPSK modulation
1. Introduction
Orthogonal frequency division multiplexing (OFDM) technique
in wireless communication becomes more important because of
the high data rate transmission [1–4]. OFDM technique has been
applied into many digital transmission systems such as digital audio broadcasting (DAB) system, digital video broadcasting terrestrial TV (DVB-T) system, asymmetric digital subscriber line (ADSL),
wireless local area network (WLAN), broadband wireless access
(BWA) network, worldwide interoperability for microwave access
(WiMax), ETSI/BRAN high performance metropolitan area network
(HIPERMAN) and ultra-wideband systems [2–4]. Orthogonal frequency division multiplexing technology is to split a high-rate data
stream into a number of lower rate streams that are transmitted
simultaneously over a number of subcarriers. Because the symbol
duration increases for the lower rate parallel subcarrier, the relative amount of dispersion in time causes by multipath delay spread
*
Corresponding author at: Department of Computer Science, Taipei Municipal
University of Education, No. 1, Ai-Guo W. Rd., Taipei, Taiwan, ROC. Fax: +886 2
23817242.
E-mail address: [email protected] (G.-R. Lee).
1051-2004/$ – see front matter Crown Copyright
doi:10.1016/j.dsp.2010.12.004
©
is decreased [1–4]. Besides, since the entire channel bandwidth is
divided into many narrow subbands, the spectrum of each individual data element normally occupies only a small part of available
bandwidth. When the number of subbands is sufficiently large, the
frequency response over each individual subcarrier is relatively flat.
So, the effect of frequency selective fading in OFDM systems can be
reduced [1,5]. Moreover, in OFDM systems, the spectrum of individual subcarrier is overlapped with minimum frequency spacing,
which is carefully designed so that each subcarrier is orthogonal to
the other subcarriers [6]. The bandwidth efficiency of OFDM is another advantage for the band limited communication system [1–4].
Therefore, OFDM with above advantage becomes more significant
in wireless communication nowadays.
However, the knowledge of symbol timing is required to demodulate the received OFDM modulated signal [1–5,7–10]. In the
receiver, the symbol boundaries and the optimal timing instants
are required to minimize the effects of inter-symbol interference
(ISI) [1,11]. Both data-aided and non-data-aided synchronization
algorithms have been proposed to achieve the symbol synchronization [12–17]. In the data-aided schemes, the symbol synchronization could be implemented with the aid of the dedicated
training symbols [13–16]. The pilot symbol-based synchronization
algorithms used to estimate timing and phase offset have been
2011 Published by Elsevier Inc. All rights reserved.
C.-Y. Yang et al. / Digital Signal Processing 21 (2011) 710–717
711
Fig. 1. The model of baseband OFDM system.
presented [13,14]. Based on the pseudo-random (PN) sequence
preambles, OFDM synchronization gives a better detection in terms
of the low false error and low missing error [15]. Based on the constant envelope preamble, the synchronization algorithm exploits
the correlation property of the PN sequence and the two identical parts of the preamble to estimate the timing offset. It improves
the accuracy of the timing offset estimation [16]. Although the
data-aided algorithms could provide a better estimation on symbol synchronizations, it suffers the bandwidth efficiency.
To avoid data rate decreasing in synchronization processing,
non-data-aided algorithms have been proposed [8–12]. Within the
non-data-aided algorithms, the cyclic property of the guard interval could be employed for the symbol synchronization without
any training symbol in the OFDM systems. Among those non-dataaided algorithms, the correlation between the cyclic prefix (CP)
and the OFDM data symbol is used to find the symbol timing
offset [17]. The estimator exploits the second-order cyclostationarity of the received signals and, then, it obtains the information
of symbol-timing offset by the cyclic correlation [9]. Based on
the maximum signal-to-interference-and-noise-ratio (SINR), a blind
synchronization for the symbol time offset estimation is proposed [18]. The inherent cyclic property of the received signal is
exploited to detect the synchronization parameter in the likelihood function of the observation vector [19]. A highly efficient
non-data-aided symbol timing recovery technique for OFDM systems is proposed [20]. The second-order statistics of interference is
used to blindly estimate the symbol timing. Besides, a novel nondata-aided maximum likelihood (ML) approach is proposed for the
estimation of the residual timing error in OFDM receivers [21]. The
novel approach effectively utilizes the finite alphabet property of
the received symbol constellation to perform a near perfect residual timing error estimation. Moreover, the maximum likelihood
estimator proposed by Jan-Jaap van de Beek uses the correlation
between the cyclic prefix and the OFDM symbol to find the symbol
timing under an AWGN channel. It uses the redundant information
contained within the cyclic prefix. The results show that Beek’s estimator could have a lower error variance when the number of
cyclic prefix samples is larger. The OFDM symbols contain sufficient information to perform synchronization [8].
However, Beek’s scheme for timing and frequency offset estimation in the OFDM system employs cyclic prefix (CP) has been
proposed under the assumption of independent identified distributed (i.i.d.) OFDM symbols. Actually, a complete OFDM symbol
is formed with the modulated symbol by an inverse fast Fourier
transform (IFFT) and a cyclic prefix extension [1–5]. It is called as
the OFDM BPSK-modulated symbol when the binary phase shift
keying (BPSK) mapping is selected in the system [10]. Those real
data in the OFDM modulated symbol, transferred by the inverse
fast Fourier transform (IFFT), has a character of complex symmetry. With the character, it could be used for the timing and
frequency offset estimation [10]. However, two conjugate symmetries could be obtained with the characters of the OFDM BPSKmodulated symbol modulated by fast Fourier transform. One is
the symbol-based symmetry and the other is CP-based symmetry
(will be described later). With these two symmetry characters, one
could find the symbol timing and the initial phase of the OFDM
BPSK-modulated symbol. In this paper, a two-stage non-data-aided
algorithm is proposed with the symbol-based symmetry and the
CP-based symmetry to find the estimated timing for accuracy. The
organization of this paper is as follows. The model of OFDM systems is described in Section 2. In Section 3, the proposed synchronization algorithm using the conjugate characters of OFDM BPSKmodulated symbol is presented to determine the symbol timing for
the OFDM systems. The simulation results are shown in Section 4.
Finally, a conclusion is given in Section 5.
2. The OFDM system description
An OFDM system could be treated as one of frequency division multiplex (FDM) techniques that are achieved by subdividing
the available bandwidth into multiple subchannels [1,4]. Parallel
data transmission is employed in the OFDM system. Then, each
parallel data transmission is modulated by the different subcarrier
frequencies using phase shift keying (PSK) or quadrature amplitude
modulation (QAM), i.e., an OFDM signal contains a sum of subcarriers with PSK or QAM modulation scheme. In general, OFDM
system contains the function of parallel transmission, signal mapping and IFFT/FFT [1–5]. In the OFDM systems, the BPSK scheme
is one candidate of the modulation schemes. In this paper, the signal mapping is only selected as BPSK scheme. Fig. 1 illustrates the
block diagram of the baseband, discrete-time FFT-based OFDM systems model.
Each parallel data is mapped with BPSK scheme and, then,
those data are modulated by an IFFT on N-parallel subcarriers.
With a cyclic prefix [1–4], the complete OFDM symbol is transmitted over a discrete-time channel. At the receiver, the data are
retrieved by a fast Fourier transform (FFT) and, then, demapped
with corresponding scheme to obtain the estimated data.
Without timing and frequency offset, the baseband transmitted
signal si (k) is as [22]
N −1
2π
1 si (k) = √
xi ,n e j N nk ,
N n =0
0 k N − 1,
(1)
where N denotes the IFFT window size, si (k) represents the kth
sample of the ith OFDM symbol, and xi ,n represents the data of
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C.-Y. Yang et al. / Digital Signal Processing 21 (2011) 710–717
si
N
2
N
+ τ + k = s∗i
+τ −k ,
2
where 1 k N
2
− 1.
(4)
Fig. 2. The structure of OFDM BPSK-modulated symbol with the conjugate character.
the nth subcarrier in the ith symbol interval. In Eq. (1), xi ,n is a
real value when BPSK mapping is used. Thus, in one OFDM BPSKmodulated symbol si (k), k = 0, 1, . . . , N − 1, the real and image
parts of transmitted signal have the following characters:
N
−1
2π
1
real si (k) = real √
xi ,n e j N nk
N n =0
N −1
1 =√
N n =0
xi ,n cos
2π
N
(2-1)
nk ,
N −1
1 j 2Nπ nk
xi ,n e
imag si (k) = imag √
N n =0
N −1
1 =√
N n =0
xi ,n sin
2π
N
(2-2)
nk ,
N −1
1 j 2Nπ n( N −k)
xi ,n e
real si ( N − k) = real √
N n =0
N −1
1 =√
N n =0
xi ,n cos
2π
N
nk ,
(2-3)
N −1
1 j 2Nπ n( N −k)
xi ,n e
imag si ( N − k) = imag √
N n =0
N n =0
N
(2-4)
where real(x) and imag(x) denote the real part and imaginary part
of complex number x, respectively. Also, Eq. (2) can be represented
as
si ( N − k) = s∗i (k),
si (τ + k) = s∗i (τ − k),
(3)
where s∗i (k) denotes the complex conjugate of si (k). In Eq. (3),
si ( N − k) and s∗i (k) are equal, i.e. si ( N − k) and si (k) are complex
conjugated. They are with the character of conjugate symmetry. As
mentioned previously, both si (k) and si ( N − k) belong to the ith
OFDM symbol. Hence, it is called as the conjugate character within
an OFDM BPSK-modulated symbol. With this character, the structure of OFDM BPSK-modulated symbol with a cyclic prefix could
be depicted in Fig. 2.
From this relationship between si ( N − k) and si (k), two symmetry properties could be developed. One is called a symbol-based
symmetry and the other is called a CP-based symmetry relationship. It is described as
where − CP k CP.
3. The proposed synchronization algorithm
Before demodulating the received OFDM signal, the receiver has
to make the symbol and frequency synchronization. Thus, the receiver should remove the cyclic prefix. However, the synchronization should be done to remove the prefix. Once, timing information
provided by the synchronization algorithm, one could exactly remove the prefix and, then, use FFT to extract the transmitted data.
Hence, synchronization is the most important work in the OFDM
system. Actually, the carrier frequency synchronization algorithm
in [23], for instance, could be used to compensate the effect of frequency offset. In this study, the carrier frequency synchronization
is not considered.
Consider the system model is in Fig. 1, the transmitted signal
si (k) could be presented as
k = 0, 1 , . . . , N − 1,
(6)
where A i ,k is the kth sampling amplitude for the ith OFDM symbol and θi ,k denotes as the phase of si (k). Based on the conjugate
character given in Eq. (3), si ( N − k) is complex conjugated with
si (k) and, then, it could be re-written as
si ( N − k) = s∗i (k) = A i ,k e − j θi,k .
(7)
With the consideration of timing offset, at the OFDM receiver, the
received signal of the ith OFDM symbol can be expressed as [8]
r i (k) = v i (k − τ ) + w i (k),
(8)
where τ is the timing offset caused by propagation delay, w i (k)
is Gaussian noise and v (k) is the desired signal with the effect of
frequency offset. In the OFDM system, v i (k) could be represented
as
v i (k) = si (k)e j (
2π
N
εk+φ) ,
(9)
where ε is the carrier frequency offset and φ is the initial phase
of the carrier with uniformly distributed within [0, 2π ]. Without considering w i (k) in Eq. (8), under a noise-free environment,
r i ( N /2 + τ − k) and r i ( N /2 + τ + k) could be expressed as
ri
N
2
2π
N
N
+ τ − k = si
− k e j ( N ε( 2 −k)+φ)
2
2π
(a) Symbol-based symmetry relationship
Based on the conjugate character of si ( N − k) = s∗i (k) in an
OFDM symbol, two opposite samples si ( N /2 + τ − k) and
si ( N /2 + τ + k) with the central sample si ( N /2 + τ ) have the
conjugate character [10], i.e.
(5)
Since the phase of the product, si (t ) · s∗i (t ), is zero, the timing
estimation could be predict with this character. In the next section, a synchronization algorithm with the conjugate characters to
predict the symbol timing and the initial phase of OFDM BPSKmodulated symbol is proposed.
si (k) = A i ,k e j θi,k ,
N −1
−1 2π
=√
xi ,n sin
nk ,
In the OFDM symbol, those samples with this conjugate character are within the data symbol.
(b) CP-based symmetry relationship
According the property of cyclic prefix, the cyclic prefix is conjugated with the first CP data within the data symbol. Two
opposite samples si (τ − k) and si (τ + k) with the central sample si (τ ) have the conjugate character, i.e.
= A i , N −k e j (π ε− N εk+φ+θi,N /2−k ) ,
2
2π
N
N
N
ri
+ τ + k = si
+ k e j ( N ε( 2 +k)+φ)
2
(10-1)
2
= A i , N −k e j (π ε+
2
2π
N
εk+φ−θi ,N /2−k ) .
(10-2)
C.-Y. Yang et al. / Digital Signal Processing 21 (2011) 710–717
The product of the two opposite samples r i ( N /2 + τ − k) and
r i ( N /2 + τ + k) with the central sample r i ( N /2 + τ ) is
ri
N
2
N
+ τ + k · ri
+ τ − k = A 2i , N /2−k e j (2φ+2π ε) ,
N
− 1.
2
(11)
In Eq. (11), the phase of the product, r i ( N /2 + τ − k) · r i ( N /2 + τ +
k), is a twice phase of sample r i ( N /2 + τ ), where 1 k N /2 − 1.
With the symbol-based symmetry relationship in Eq. (4), the proposed algorithm uses the slide windows with length N /2 − 1 to
find the symbol timing. First, multiply the corresponding symmetric samples within the slide window to find the angle of the
product as
B k,1 = angle r i (k + 1) · r i (k − 1) ,
B k,2 = angle r i (k + 2) · r i (k − 2) ,
..
.
B k, N /2−1 = angle r i (k + N /2 − 1) · r i (k − N /2 + 1) ,
(12)
where angle(x) represents the angle of the complex number x and
Bk = { B k,1 , B k,2 , . . . , B k, N /2−1 } is the set of these angles. The mean
of Bk is
E [Bk ] =
L
1
L
B k,m =
m =1
L
1
L
angle r i (k + m) · r i (k − m) ,
m =1
1 L N /2 − 1 ,
(13)
where E [ Z (k)] is the expectation value of the function Z (k), L is
the length of slide windows. The function f 1 (k) is defined as
f 1 (k) =
L
1 L
B k,m − E [Bk ], 1 L N /2 − 1.
(14)
m =1
In the slide windows, the central position is located at the sample
with the index k. With the symbol-based symmetry, the mean of
Bk in the case k = N /2 + τ can be obtained as (see Appendix A)
E [Bk |k= N /2+τ ] = 2φ + 2πε .
(15)
With the knowledge of frequency offset ε , the initial phase φ could
be obtained in Eq. (16):
φ̂ =
1
2
E [Bk |k= N /2+τ ] − 2πε .
(16)
On the other hand, as the similar concepts from Eq. (11) to
Eq. (14), the product of the two opposite samples r i (τ + k) and
r i (τ − k) is written as
r i (τ + k) · r i (τ − k) = A 2i ,k e j2φ ,
−CP < k < CP.
(17)
In Eq. (17), the phase of the product, r i (τ + k) · r i (τ − k), is a twice
phase of sample r i (τ ), where 1 k CP. The multiplication of the
corresponding symmetric samples within the slide window is used
to find the angle of the products as
C k,m = angle r i (k + m) · r i (k − m) ,
m = 1, 2, . . . , CP .
(18)
Ck = {C k,1 , C k,2 , . . . , C k,CP } is the vector of angles. And the mean
of Ck is
E [Ck ] =
CP
1 CP
m =1
C k,m =
CP
1 CP
angle r i (k + m) · r i (k − m) , (19)
m =1
and the function f 2 (k) is defined as
CP
C k,m − E [Ck ].
(20)
m =1
Thus, using two symmetry relationships, the cost function of the
proposed algorithm could be defined as
2
1k
f 2 (k) =
713
f (k) = f 1 (k) + f 2 k −
N
2
.
(21)
The cost function f (x), the superposition of function f 1 (k) and
f 2 (k) in the case k = N /2 + τ could be obtained as (see Appendix B)
f (k|k= N /2+τ )
= E angle r i ( N /2 + τ + m)ri ( N /2 + τ − m) − E [ B N /2+τ ]
+ E angle r i (τ + m)r i (τ − m) − E [C τ ] = 0.
(22)
As the above-mentioned, the proposed algorithm uses two different slide windows to obtain the cost function f (k). When k =
N /2 + τ in the corresponding slide windows, the cost function
f ( N /2 + τ ) is zero. However, the cost function f (k) could not
be zero under a noisy environment. Based on this property, the
proposed algorithm applies two slide windows to obtain the cost
function f (k). In the symbol duration, the proposed algorithm is
to choose a minimum cost function f (k) and, then, the timing offset τ could be predicted with the corresponding index k and the
initial phase φ . In practical application of OFDM system such as
IEEE 802.11a/g standard [24,25], the OFDM BPSK-modulated symbol could be used to replace the preamble or training sequences
in the Wireless LAN standard to obtain an accurate timing and frequency offset estimation and to avoid the data rate decreasing with
the proposed algorithm. Besides, the adaptive modulation schemes
used in the application could improve the transmission efficiency.
When the timing is determined, the high-level index modulation
could be replaced to provide higher data transmission services.
Based on the above algorithm, simulations are given in the following section.
4. Simulation results
Simulations for the proposed algorithm are performed over the
AWGN channel and the multipath fading channel. According to the
standard of IEEE 802.11a/g [24,25], the number of subcarriers and
the length of the cyclic prefix were N = 64 and CP = 16, respectively. In each simulation with 105 running times, it is assumed
timing delay τ to be 73 and, then, the actual symbol timing is located in the 73th sample. The multipath fading channel is referred
to the OFDM multipath channel model in IEEE 802.11a/g standard with the path number of three. The performance is evaluated
based on the lose symbol timing rate and, then, the comparison
with the one based on Beek’s estimation is made [8]. The lose
symbol timing rate indicates the probability of the missing symbol timing error. The estimator mean square error (MSE) of timing
estimation is defined as a performance measure of the estimator:
MSE =
Nt
1 Nt
(τ̂ j − τ j )2 ,
(23)
j =1
where τ j is the actual symbol timing for jth simulation, τ̂ j is the
estimated symbol timing for jth simulation and N t is the number
of simulation times.
In Fig. 3, under an AWGN channel, the histograms of estimated symbol timing for the proposed algorithm are accumulated via computer simulation at the signal-to-noise power ratio
SNR = 8 dB. The maximum likelihood algorithm uses the cyclic
prefix and the tail of OFDM symbol to estimate symbol timing.
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C.-Y. Yang et al. / Digital Signal Processing 21 (2011) 710–717
Sample index
Counts
Sample index
2
3–67
68
69
70
71
72
73
74
20
0
2
59
143
2410
8625
77 114
10 928
75
76
77
78
79
80
81–141
142
143
Counts
492
167
21
3
2
1
0
1
12
(a)
Sample index
Counts
20–24
25
26–72
73
74–120
121
122–130
0
4
0
99 991
0
5
0
(b)
Fig. 3. The histogram diagram of estimated symbol timing for (a) Beek’s algorithm and (b) the proposed algorithm.
Fig. 4. (a) Lose symbol timing rate and (b) estimator mean-squared error of Beek’s estimation and the proposed algorithm under an AWGN channel.
Thus, when the slide window has little deviation away from the
correct position, the correlation is still large. When noise is taken
into consideration, it’s easy to erroneous judge the sample around
the correct position as the symbol timing. For the proposed algorithm using two symmetry relationships, the estimated symbol
timing via computer simulation is almost located on the actual
symbol timing. The others are located on the 25th and 121th sample respectively. The estimated symbol timing located on the 25th
sample represents that the central point for two opposite samples
is v i −1 ( N /2 + τ ), and the estimated symbol timing located on the
121th sample represents that the central point for two opposition
samples is v i +1 (τ ). The miss probability of symbol timing with
the proposed algorithm using two symmetries is less than 10−4 at
SNR = 8 dB.
In Fig. 4, the proposed algorithm performs better than Beek’s
method over an AWGN channel. Beek’s algorithm [8] uses a cyclic
prefix and the tail of an OFDM symbol to estimate the symbol timing. When the slide window shifts to the correct position,
C.-Y. Yang et al. / Digital Signal Processing 21 (2011) 710–717
715
Fig. 5. (a) Lose symbol timing rate and (b) estimator mean-squared error of Beek’s estimation and the proposed algorithm under the multipath fading channel.
the value of the correlation varies slowly. However, when noise
is taken into consideration, it is easy erroneously to determine
that the sample around the correct position as the estimated symbol timing. The proposed algorithm uses the conjugate characters
of the OFDM BPSK-modulated symbol to estimate symbol timing.
Thus, when the central point of the slide windows shifts to the
correct sample, the value of the autocorrelation will seriously decay. The correct position can be obtained easily. As shown in Fig. 4,
the lose symbol timing rate and estimator MSE with the proposed
algorithm are better than that with Beek’s estimation algorithm
under an AWGN channel.
Simulation results under the multipath channel are shown in
Fig. 5. The performance including lose symbol timing rate and estimator MSE with the proposed algorithm are also excellent under
the multipath channel. The results show the conjugate character is
efficient to achieve the symbol synchronization. Also, the proposed
algorithm using two symmetry relationships could provide better
performances in the symbol synchronization.
5. Conclusion
In this paper, a blind timing estimation algorithm with applying
the conjugate characters is presented to determine the symbol timing in the OFDM systems. Two conjugate symmetries characters of
the OFDM BPSK-modulated symbol are obtained by the fast Fourier
transform. One is the symbol-based symmetry and the other is
CP-based symmetry. The proposed algorithm performs better than
that with Beek’s estimation algorithm does under the AWGN channel and the multipath channel. In one OFDM symbol, the samples
used in the proposed algorithm and the samples used in Beek’s
estimation algorithm are N + CP − 2 and CP, respectively, where
N = 64 and CP = 16 in the standard of IEEE 802.11a model. The
simulation results show that the proposed algorithm could provide
better performances including lose symbol timing rate and estimator MSE than those with Beek’s estimation algorithm. The other
advantage of proposed algorithm is that the initial phase could be
found at the same time when the symbol timing is found. Moreover, the proposed estimation algorithm is non-data-aided. It is
more desirable to avoid the data rate decreasing for the symbol
synchronization.
Appendix A
With the symbol-based symmetry in Eq. (4), the mean of Bk =
{ B k,1 , B k,2 , . . . , B k, N /2−1 } in the case k = N /2 + τ can be derived
as
E [Bk |k= N /2+τ ]
= E angle r i ( N /2 + τ + m)ri ( N /2 + τ − m)
2π
= E angle si ( N /2 + m)e j (π ε+ N εm+φ) + w i ( N /2 + τ + m)
2π
× si ( N /2 − m)e j (π ε− N εm+φ) + w i ( N /2 + τ − m)
2π
= E angle si ( N /2 + m)e j (π ε+ N εm+φ) + w i ( N /2 + τ + m)
2π
× s∗i ( N /2 + m)e j (π ε− N εm+φ) + w i ( N /2 + τ − m)
= E angle si ( N /2 + m)s∗i ( N /2 + m)e j (2π ε+2φ)
+ si ( N /2 + m)e j (π ε+
2π
N
εm+φ) w ( N /2 + τ − m)
i
2π
+ w i ( N /2 + τ + m)s∗i ( N /2 + m)e j (π ε− N εm+φ)
+ w i ( N /2 + τ + m ) w i ( N /2 + τ − m )
= E angle si ( N /2 + m)s∗i ( N /2 + m)e j (2π ε+2φ)
2π
+ angle si ( N /2 + m)e j (π ε+ N εm+φ) w i ( N /2 + τ − m)
2π
+ w i ( N /2 + τ + m)s∗i ( N /2 + m)e j (π ε− N εm+φ)
+ w i ( N /2 + τ + m ) w i ( N /2 + τ − m )
= E angle si ( N /2 + m)s∗i ( N /2 + m)e j (2π ε+2φ)
=
L
1 L
angle si ( N /2 + m)s∗i ( N /2 + m)e j (2π ε+2φ)
m =1
= 2φ + 2πε .
(I-1)
The result could be provided to obtain the initial phase φ in
Eq. (16).
Appendix B
Using two symmetry relationships, the cost function in the case
k = N /2 + τ could be derived as
f (k|k= N /2+τ )
= E angle r i ( N /2 + τ + m)ri ( N /2 + τ − m) − E [ B N /2+τ ]
+ E angle r i (τ + m)r i (τ − m) − E [C τ ]
2π
= E angle si ( N /2 + m)e j (π ε+ N εm+φ) + w i ( N /2 + τ + m)
2π
× si ( N /2 − m)e j (π ε− N εm+φ) + w i ( N /2 + τ − m)
− (2φ + 2πε )
716
C.-Y. Yang et al. / Digital Signal Processing 21 (2011) 710–717
2π
+ E angle si (m)e j ( N εm+φ) + w i (τ + m)
2π
× si (−m)e j (− N εm+φ) + w i (τ − m) − 2φ 2π
= E angle si ( N /2 + m)e j (π ε+ N εm+φ) + w i ( N /2 + τ + m)
2π
× s∗i ( N /2 + m)e j (π ε− N εm) + w i ( N /2 + τ − m)
− (2φ + 2πε )
2π
+ E angle si (m)e j ( N εm+φ) + w i (τ + m)
2π
× s∗i (m)e j (− N εm+φ) + w i (τ − m) − (2φ)
= E angle si ( N /2 + m)s∗i ( N /2 + m)e j (2π ε+2φ)
2π
+ angle si ( N /2 + m)e j (π ε+ N εm+φ) w i ( N /2 + τ − m)
2π
+ w i ( N /2 + τ + m)s∗i ( N /2 + m)e j (π ε− N εm+φ)
+ w i ( N /2 + τ + m) w i ( N /2 + τ − m) − (2φ + 2πε )
+ E angle si (m)s∗i (m)e j (2φ)
2π
+ angle si (m)e j ( N εm+φ) w i (τ − m)
+ w i (τ + m)s∗i (m)e j (−
2π
N
[12] M. Sandell, J.J. van de Beek, P.O. Borjesson, Timing and frequency synchronization in OFDM systems using the cyclic prefix, in: Proceedings of IEEE Int.
Symposium on Synchronization, 1995, pp. 6–9.
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OFDM using PN-sequence preambles, in: Proceedings of IEEE Vehicular Technology Conference, vol. 4, 1999, pp. 2203–2207.
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constant envelop preamble for OFDM systems, IEEE Trans. Broadcast. 51 (1)
(2005) 139–143.
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SINR maximization in multipath fading channels, IEEE Trans. Vehicular Technol. 58 (2) (2009) 625–635.
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OFDM systems, IEEE Trans. Commun. 54 (1) (2006) 37–40.
εm+φ)
+ w i (τ + m) w i (τ − m) − (2φ)
= E angle si ( N /2 + m)s∗i ( N /2 + m)e j (2π ε+2φ)
− (2φ + 2πε )
+ E angle si (m)s∗i (m)e j (2φ) − (2φ)
= E angle si ( N /2 + m)s∗i ( N /2 + m)e j (2π ε+2φ)
− (2φ + 2πε )
+ E angle si (m)s∗i (m)e j (2φ) − (2φ)
= (2φ + 2πε ) − (2φ + 2πε ) + (2φ) − (2φ)
= 0.
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IEEE J. Selected Areas Commun. 18 (11) (2000) 2278–2291.
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(II-1)
The result shows that the cost function f ( N /2 + τ ) is zero,
when k = N /2 + τ in the corresponding slide windows.
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pp. 12–16.
Cheng-Ying Yang was born in Taipei on October
13, 1964. He received the M.S. degree in Electronic
Engineering from Monmouth University, New Jersey,
in 1991, and Ph.D. degree from the University of
Toledo, Ohio, in 1999. He is a member of IEEE Satellite
& Space Communication Society. Currently, he is employed as an Associate Professor at Taipei Municipal
University of Education, Taiwan. His research interests are performance analysis of digital communication systems, error control coding, signal processing and computer security.
Gwo-Ruey Lee received the B.S. degree in Department of Electronic Engineering from the Fu-Jen
Catholic University, Taipei, Taiwan, in 2000, the M.S.
degree from Department of Communications Engineering, National Chung Cheng University, Chia-Yi,
Taiwan, in 2002 and the Ph.D. degree in Department of Electrical Engineering, National Chung Cheng
University, Chia-Yi, Taiwan, in 2008. From September
2009 to July 2010, he is a post-doctoral assistant at
Taipei Municipal University of Education. His current research interests
include OFDM systems, multi-carrier systems, personal communications,
spread-spectrum techniques, wireless broadband systems, radar systems
and its applications.
C.-Y. Yang et al. / Digital Signal Processing 21 (2011) 710–717
Wen-Hui Kuan received the M.S. degree from Department of Electrical Engineering, National Chung
Cheng University, Chia-Yi, Taiwan, in 2005. Her research interests include OFDM system and wireless
broadband systems.
Jyh-Horng Wen received the B.S. degree in electronic engineering from the National Chiao Tung University, Hsing-Chu, Taiwan, in 1979 and the Ph.D. degree in electrical engineering from National Taiwan
University, Taipei, in 1990. From 1981 to 1983, he
was a Research Assistant with the Telecommunication
Laboratory, Ministry of Transportation and Communications, Chung-Li, Taiwan. From 1983 to 1991, he was
a Research Assistant with the Institute of Nuclear En-
717
ergy Research, Taoyun, Taiwan. From February 1991 to July 2007, he was
with the Institute of Electrical Engineering, National Chung Cheng University, Chia-Yi, Taiwan, first as an Associate Professor and, since 2000,
as a Professor. He was also the Managing Director of the Center for
Telecommunication Research, National Chung Cheng University, from August 2001 to July 2004 and the Dean of General Affairs, National Chi
Nan University, from August 2004 to July 2006. Since August 2007, he
has been the Department Head of Electrical Engineering, Tunghai University, Taichung, Taiwan. He is an Associate Editor of the Journal of the
Chinese Grey System Association. His current research interests include
computer communication networks, cellular mobile communications, personal communications, spread-spectrum techniques, wireless broadband
systems, and gray theory. Prof. Wen is a member of the IEEE Communication Society, the IEEE Vehicular Technology Society, the IEEE Information Society, the IEEE Circuits and Systems Society, the Institute
of Electronics, Information and Communication Engineers, the International Association of Science and Technology for Development, the Chinese
Grey System Association, and the Chinese Institute of Electrical Engineering.