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Blind synchronization scheme using the conjugate characters of the OFDM BPSK-modulated symbol
Digital Signal Processing 21 (2011) 710–717
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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
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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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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):
φˆ =
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
(τˆ j − τ j )2 ,
(23)
j =1
where τ j is the actual symbol timing for jth simulation, τˆ 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 )
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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. [13] A.J. Coulson, Maximum likelihood synchronization for OFDM using a pilot symbol: algorithms, IEEE J. Selected Areas Commun. 19 (12) (2001) 2486–2494. [14] A.J. Coulson, Maximum likelihood synchronization for OFDM using a pilot symbol: analysis, IEEE J. Selected Areas Commun. 19 (12) (2001) 2495–2503. [15] F. Tufvesson, O. Edfors, M. Faulkner, Time and frequency synchronization for OFDM using PN-sequence preambles, in: Proceedings of IEEE Vehicular Technology Conference, vol. 4, 1999, pp. 2203–2207. [16] G. Ren, Y. Chang, H. Zhang, H. Zhang, Synchronization method based on a new constant envelop preamble for OFDM systems, IEEE Trans. Broadcast. 51 (1) (2005) 139–143. [17] J.J. van de Beek, M. Sandell, M. Isaksson, P.O. Börjesson, Low-complex frame synchronization in OFDM systems, in: Proceedings of IEEE International Conference on Universal Personal Communications, 1995, pp. 982–986. [18] W.L. Chin, S.G. Chen, A blind synchronizer for OFDM systems based on SINR maximization in multipath fading channels, IEEE Trans. Vehicular Technol. 58 (2) (2009) 625–635. [19] N. Lashkarian, S. Kiaei, Minimum variance unbiased estimation of frequency offset in OFDM systems, a blind synchronization approach, in: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 5, 2000, pp. 2945–2948. [20] A.J. Al-Dweik, A novel non-data-aided symbol timing recovery technique for 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.
[11] B. Yang, K.B. Letaief, R.S. Cheng, Z. Cao, Timing recovery for OFDM transmission, IEEE J. Selected Areas Commun. 18 (11) (2000) 2278–2291.
[21] C.R.N. Athaudage, A.D.S. Jayalath, Blind estimation of residual timing error in OFDM receivers: a non-data-aided maximum-likelihood approach, in: Proceedings of IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, vol. 4, 2005, pp. 2476–2480. [22] Z. Zhang, K. Long, M. Zhao, Y. Liu, Joint frame synchronization and frequency offset estimation in OFDM systems, IEEE Trans. Broadcast. 51 (3) (2005) 389– 394. [23] C.R.N. Athaudage, V. Krishnamurthy, A low complexity timing and frequency synchronization algorithm for OFDM systems, in: Proceedings of IEEE Global Telecommunications Conference, vol. 1, 2002, pp. 244–248. [24] Supplement to IEEE standard for information technology – Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications: high-speed physical layer in the 5 GHz band, IEEE Std. 802.11a, 1999. [25] J. Terry, J. Heiskala, OFDM Wireless LANs: A Theoretical and Practical Guide, SAMS Publishing, 2001.
(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. References [1] R. van Nee, Ramjee Prasad, OFDM Wireless Multimedia Communication, Artech House, 2000. [2] M. Engels, Wireless OFDM Systems: How to Make Them Work?, 1st edition, Kluwer Academic Publishers, 2002. [3] K. Fazel, S. Kaiser, Multi-Carrier and Spread Spectrum Systems, John Wiley & Sons Inc., 2003. [4] W.Y. Zou, Y. Wu, COFDM: An overview, IEEE Trans. Broadcast. 41 (1) (1995) 1–8. [5] G.R. Lee, J.H. Wen, The performance of subcarrier allocation scheme combined with error control coding in OFDM systems, IEEE Trans. Consumer Electronics 53 (3) (2007) 852–856. [6] Y.H. Peng, Y.C. Kuo, G.R. Lee, J.H. Wen, Performance analysis of a new ICI-selfcancellation-scheme in OFDM systems, IEEE Trans. Consumer Electronics 53 (4) (2007) 1333–1338. [7] J.H. Wen, G.R. Lee, J.W. Liu, T.L. Kung, Frame synchronization, channel estimation scheme and signal compensation using regression method in OFDM systems, Computer Commun. 31 (10) (2008) 2124–2130. [8] J.J. Van de Beek, M. Sandell, P.O. Borjesson, ML estimation of time and frequency offset in OFDM systems, IEEE Trans. Signal Process. 45 (7) (1997) 1800– 1805. [9] B. Park, H. Cheon, E. Ko, C. Kang, D. Hong, A blind OFDM synchronization algorithm based on cyclic correlation, IEEE Signal Process. Lett. 11 (2) (2004) 83–85. [10] C.Y. Yang, G.R. Lee, W.H. Kuan, J.H. Wen, Symbol synchronization at the BPSKOFDM receiver, in: Proceedings of IEEE Mobile WiMax, Symposium, 2007, 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.
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):
φˆ =
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
(τˆ j − τ j )2 ,
(23)
j =1
where τ j is the actual symbol timing for jth simulation, τˆ 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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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,
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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 )
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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. [13] A.J. Coulson, Maximum likelihood synchronization for OFDM using a pilot symbol: algorithms, IEEE J. Selected Areas Commun. 19 (12) (2001) 2486–2494. [14] A.J. Coulson, Maximum likelihood synchronization for OFDM using a pilot symbol: analysis, IEEE J. Selected Areas Commun. 19 (12) (2001) 2495–2503. [15] F. Tufvesson, O. Edfors, M. Faulkner, Time and frequency synchronization for OFDM using PN-sequence preambles, in: Proceedings of IEEE Vehicular Technology Conference, vol. 4, 1999, pp. 2203–2207. [16] G. Ren, Y. Chang, H. Zhang, H. Zhang, Synchronization method based on a new constant envelop preamble for OFDM systems, IEEE Trans. Broadcast. 51 (1) (2005) 139–143. [17] J.J. van de Beek, M. Sandell, M. Isaksson, P.O. Börjesson, Low-complex frame synchronization in OFDM systems, in: Proceedings of IEEE International Conference on Universal Personal Communications, 1995, pp. 982–986. [18] W.L. Chin, S.G. Chen, A blind synchronizer for OFDM systems based on SINR maximization in multipath fading channels, IEEE Trans. Vehicular Technol. 58 (2) (2009) 625–635. [19] N. Lashkarian, S. Kiaei, Minimum variance unbiased estimation of frequency offset in OFDM systems, a blind synchronization approach, in: Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 5, 2000, pp. 2945–2948. [20] A.J. Al-Dweik, A novel non-data-aided symbol timing recovery technique for 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.
[11] B. Yang, K.B. Letaief, R.S. Cheng, Z. Cao, Timing recovery for OFDM transmission, IEEE J. Selected Areas Commun. 18 (11) (2000) 2278–2291.
[21] C.R.N. Athaudage, A.D.S. Jayalath, Blind estimation of residual timing error in OFDM receivers: a non-data-aided maximum-likelihood approach, in: Proceedings of IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, vol. 4, 2005, pp. 2476–2480. [22] Z. Zhang, K. Long, M. Zhao, Y. Liu, Joint frame synchronization and frequency offset estimation in OFDM systems, IEEE Trans. Broadcast. 51 (3) (2005) 389– 394. [23] C.R.N. Athaudage, V. Krishnamurthy, A low complexity timing and frequency synchronization algorithm for OFDM systems, in: Proceedings of IEEE Global Telecommunications Conference, vol. 1, 2002, pp. 244–248. [24] Supplement to IEEE standard for information technology – Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications: high-speed physical layer in the 5 GHz band, IEEE Std. 802.11a, 1999. [25] J. Terry, J. Heiskala, OFDM Wireless LANs: A Theoretical and Practical Guide, SAMS Publishing, 2001.
(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. References [1] R. van Nee, Ramjee Prasad, OFDM Wireless Multimedia Communication, Artech House, 2000. [2] M. Engels, Wireless OFDM Systems: How to Make Them Work?, 1st edition, Kluwer Academic Publishers, 2002. [3] K. Fazel, S. Kaiser, Multi-Carrier and Spread Spectrum Systems, John Wiley & Sons Inc., 2003. [4] W.Y. Zou, Y. Wu, COFDM: An overview, IEEE Trans. Broadcast. 41 (1) (1995) 1–8. [5] G.R. Lee, J.H. Wen, The performance of subcarrier allocation scheme combined with error control coding in OFDM systems, IEEE Trans. Consumer Electronics 53 (3) (2007) 852–856. [6] Y.H. Peng, Y.C. Kuo, G.R. Lee, J.H. Wen, Performance analysis of a new ICI-selfcancellation-scheme in OFDM systems, IEEE Trans. Consumer Electronics 53 (4) (2007) 1333–1338. [7] J.H. Wen, G.R. Lee, J.W. Liu, T.L. Kung, Frame synchronization, channel estimation scheme and signal compensation using regression method in OFDM systems, Computer Commun. 31 (10) (2008) 2124–2130. [8] J.J. Van de Beek, M. Sandell, P.O. Borjesson, ML estimation of time and frequency offset in OFDM systems, IEEE Trans. Signal Process. 45 (7) (1997) 1800– 1805. [9] B. Park, H. Cheon, E. Ko, C. Kang, D. Hong, A blind OFDM synchronization algorithm based on cyclic correlation, IEEE Signal Process. Lett. 11 (2) (2004) 83–85. [10] C.Y. Yang, G.R. Lee, W.H. Kuan, J.H. Wen, Symbol synchronization at the BPSKOFDM receiver, in: Proceedings of IEEE Mobile WiMax, Symposium, 2007, 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.