Performance Study of a Reduced Complexity Time Synchronization Approach for OFDM Systems Le¨ıla Nasraoui, Le¨ıla Najjar Atallah, Mohamed Siala Higher School of Communications University of Carthage, Tunisia Emails:
[email protected],
[email protected],
[email protected]
Abstract—This paper presents a performance analysis of a recently proposed preamble based reduced complexity twostage synchronization technique. The preamble, composed of two identical sub-sequences, is first used to determine an uncertainty interval based on Cox and Schmidl algorithm. Then, a differential correlation is carried using a new sub-sequence obtained by element wise multiplication of the preamble sub-sequence and a shifted version of it. This second step is exploited to fine tune the coarse estimate by carrying the differential correlation over the uncertainty interval. We here study the effect of the training sequence choice on the synchronization performance in the general case of multipath channels. We also discuss some complexity issues compared to previously proposed algorithms. We show that the frame start detection is greatly sensitive to the training sequence class and choice. Computational load evaluation ensure that the reduced complexity approach, which was found to provide almost equal performance to those obtained by higher complexity algorithms in [10, 11], has much lower complexity load comparable to that of simple sliding correlation based approaches. To further reduce the computational load, an optimal choice of the uncertainty interval, used in the fine stage, can also be adapted to the operating SNR. Index Terms—OFDM, synchronization, preamble, msequences, Gold sequences, sliding correlation, differential correlation
I. I NTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) technique is used to transmit data in parallel over a large number of orthogonal subcarriers. OFDM systems have emerged as one of the most practical techniques for both wired and wireless high data rate communications, over selective channels [1]. In addition, its implementation is simple and is based on the Fast Fourier Transform (FFT) algorithm. A time misalignment between the transmitter and the receiver may result in Inter Symbol Interference (ISI) and Inter Channel Interference (ICI) which severely deteriorates the reception performance. Many preamble based synchronization approaches have been proposed to estimate the Time Offset (TO) and the Frequency Offset (FO), either jointly or individually. The studies in [2-8] exploit preambles of specific structures with different polarity patterns along with a defined sequence type. The calculated metrics use either sliding correlation, characterized by its low computational load, or differential correlation of higher complexity with more accurate detection capacities. Other works [9-11], extending the cited above ones, aim not only to enhance the estimation accuracy but also to reduce the
complexity. In [9], Chou et al. proposed two modified preamble patterns obtained by zero padding for TO estimation of the preamble of Cox and Schmidl [2] and Minn et al. [3]. Those preambles have the potentials to reduce the computational complexity without degrading the system performance. In [10, 11], a Reduced Complexity (RC) two-stage approach was proposed. In the first stage, it exploits the repetitive structure and the sliding correlation to provide a coarse time estimate, respecting the algorithm of [2]. In the second stage, it carries a differential correlation around the coarse estimate using a new sequence deduced from the preamble sub-sequence. Even if the proposed method in [10, 11] is not restrictive to the type of the preamble sub-sequence, an arbitrary m-sequence was adopted. The obtained results show that the combined use of sliding and differential correlations leads to a better tradeoff between performance and complexity, compared to former approaches, including those based exclusively on either sliding or differential correlation. The contribution of this paper is first to analytically study the computational complexity in both the continuous and packet transmission modes. Then, an experimental study of the impact of sub-sequence choice on detection performance is realized. This gives an important insight on how the training symbol should be chosen in order to achieve better detection performance with the same amount of training overhead. We note that the duration of the synchronization process depends on the transmission mode, which in the continuous mode is regularly processed, depending on the preamble periodicity. In the bursty packet mode, the synchronization should be continuously carried, until preamble detection. The organization of this paper is as follows. Section II presents the system and channel model. Section III recalls the RC approach. Section IV studies the sequence type. In section V, complexity issues are discussed. Section VI is devoted to the simulation results and comparisons. The conclusion is provided in section VII. II. S YSTEM AND CHANNEL MODEL The OFDM symbol consists of a sum of Nu = 2m subcarriers that are modulated by using a linear modulation method, generally PSK or QAM. OFDM symbols are extended by a Cyclic Prefix (CP) of length Ng . The total symbol length is N = Nu + Ng . The k th transmitted baseband signal,
978-1-4673-1008-6/12/$31.00 2012 IEEE.
assuming ideal Nyquist pulse shaping, can be expressed as sk =
NX u −1
cn e
j2πkn/Nu
,
(1)
n=0
where cn , n = 0, ..., Nu − 1 stand for the transmitted data. We will denote by Num and Ngm respectively the useful part and the CP lengths relative to the preamble, which should verify N = Num + Ngm to keep the same total symbol length. Ngm and Ng are assumed to be longer than the channel impulse response. We consider a multipath channel of P discrete paths with length L and impulse response coefficients denoted by hl , l = 0, 1, ..., L − 1. We also assume a possible misalignment between the local oscillators of the transmitter and the receiver through an FO, whose normalized version with respect to subcarriers spacing is denoted by ν. The sampled received signal is, in this case, given by
B. Fine stage To get a finer estimation of the frame start, this stage carries a differential correlation. As this latter is computationally involved, we carry it over a reduced interval ∆τ around τˆc . The longer is ∆τ , the more complex is the processing and the more accurate is the time estimation. For the fine timing metric calculation, the sequence α is computed based on the sub-sequence initially used in the preamble as αk = p∗k pk+Lum +q = p∗k pk+1 , k ∈ [0, Lum − 1],
(3)
where pk stands for the k th preamble sample. The fine metric is given by 2 |Pf (d)| (4) Mf (d) = 2 , Rf (d) where Rf (d) =
Lum X−1
2
|Yd+m | ,
(5)
∗ αm Yd+m ,
(6)
m=0
rk = ej2πνk/Nu
L−1 X
hl sk−l + ωk ,
(2)
Pf (d) =
l=0
Lum X−1 m=0
where ωk is the k th sample of a zero-mean complex AWGN with variance σω2 . III. R EDUCED C OMPLEXITY APPROACH In this section, we summarize the RC two stage approach proposed in [10, 11]. This approach exploits a time domain preamble of two identical sub-sequences of length Lum = Num /2 each. In the receiver, a Brute Force (BF) approach consists in carrying a differential correlation using two new sequences. The first sequence, denoted by Y , is obtained as an element wise product of two delayed versions of the received signal. The second sequence, denoted by α, is obtained as an element wise product of two shifted versions of the preamble sub-sequence using the same shift q as for Y generation. In the Brute Force (BF) approach, where this treatment is carried during the whole synchronization process, the resulting computational load is very important. The aim of the RC approach is to reduce this complexity by splitting it into two stages: A. Coarse stage Based on the same idea of [2], this stage exploits the repetitive structure of the preamble to provide a coarse time estimate. The same metric, based on a sliding correlation, is here used. As shown in figure 1, this metric exhibits a plateau of length equal to the CP length minus the channel impulse response length. For the coarse time estimate determination, the portion of the metric plateau whose magnitude is above 90% of the maximum amplitude of the metric is first located. Then, a time interval of length Ngm /2 is added to its middle point to obtain the coarse TO estimate τˆc .
and Yk = rk∗ rk+Lum +q , k ∈ [ˆ τc − ∆τ, τˆc + ∆τ + Lum ].
(7)
It is worth noting that the coarse stage uses a shift of q = 0, where in the fine stage q must be chosen within [1, Lum −1]. The length of Y greatly influences the complexity and the detection accuracy. In the case of the BF approach, it is calculated continuously, in the case of bursty packet transmission mode, it is calculated over a chosen number of symbols around the expected preamble arrival, which results in an important complexity. Whereas, in the case of RC approach and in both transmission modes, it is calculated over a short interval of length Lum , each time the fine metric has to be computed. The timing metrics of the herein compared methods are presented in figure 1. The considered methods are : Cox and Schmidl (CS) [2], Minn’s sliding window method (MinnA) [3], Minn’s training symbol method (Minn-B) [3], Chou [9], Nasir [6], Precoding Method (PM) [7, 8], BF and RC [10, 11]. A system of N = 1126 is considered with Nu = 210 = 1024 useful samples, Ng = 102 CP length for the sliding correlation based approaches. Concerning the differential correlation based approaches, we here consider an m-sequence for the preamble generation with parameter Num = 2Lum = 2(29 − 1) = 1022 useful samples and Ngm = 104 CP length. In the figure, the frame start coincides with the 564th sample, as random data of length 460 samples are sent before the preamble. IV. T RAINING SEQUENCE TYPE The RC approach uses a preamble made of two identical sub-sequences of length Lum each. As will be confirmed through simulations, the choice of the preamble sub-sequence
978-1-4673-1008-6/12/$31.00 2012 IEEE.
of the first path to the last path is set to 12 dB. The primitive polynomials used to generate m-sequence 1 and m-sequence 2 are respectively P1 = 1 + x3 + x4 + x5 + x7 + x8 + x9 and P2 = 1 + x + x3 + x4 + x5 + x6 + x7 + x8 + x9 . As shown in the figure, in the multipath channel, some side-peaks appear at positions that do not correspond to any path for certain m-sequence choice. This problem is avoided by using one among the Gold sequences generated from the considered m-sequence. Indeed, as mentioned above, the weakness of the m-sequences is that the generated sequence α (3) used for the differential correlation coincides with a shifted version of the original sub-sequence pk , k = 0, ..., Lum − 1, thus resulting in the appearance of ghost paths in the metric. This behavior is avoided by the Gold sequences. Fig. 1. Timing metrics in the case of monopath channel under noiseless conditions
has a great impact on performance. In this paper, we concentrate on the study of the preamble sub-sequence choice. We envisage the class of random sequences and pseudo-noise sequences including m-sequences with the specific case of Gold sequences. As their name indicates, the m-sequences (maximal length sequences) are the sequences of maximum possible period of 2m − 1 obtained from an m-stage binary shift register with linear feedback. The principal property of m-sequences is their two-valued auto-correlation function (1, 1/Lum ), that is most exploited in the applications using m-sequences. The main feature of the auto-correlation is the ratio of the peak value to the modulus of the highest sidepeaks. If the auto-correlation presents high side-peaks, the robustness to noise and side-peaks is unsatisfactory and the synchronization may miss the first path and in some cases may detect an erroneous path. In this respect, m-sequences are adequate because of the two-valued auto-correlation which results in the absence of side-peaks. In the case of m-sequences, α given by (3) coincides with a shifted version of the original m-sequence. Thus, it is possible that some ghost side-peaks appear picking out false paths. It is worth to note that, for a given multipath channel, certain m-sequences are more adapted than other ones by avoiding side-peaks effect, thus allowing to greatly enhance the performance. Other sequence type derived from specific m-sequences and recognized as preferred pairs are known to have a minimal value of the out-of-phase auto-correlation. They are used to construct Gold sequences [13]. We here compare the performance of the frame start detection using different sequence types: random sequences, m-sequences and Gold sequences. Figure 2 shows the timing metric of the RC approach using random sequence of ±1, two m-sequences, each generated using a different primitive polynomial and a Gold sequence deduced from the second m-sequence. We consider a channel of 7 paths, spread over 42 samples and having an exponential profile. Paths are regularly separated by 6 samples and the ratio
Fig. 2. RC fine timing metric, using different sequence class/type, in the case of an exponential 7-tap multipath channel under noiseless conditions
V. C OMPUTATIONAL COMPLEXITY The main key of the detection accuracy of the method proposed in [11, 12] is the differential correlation carried with a shift different from the preamble sub-sequence length
978-1-4673-1008-6/12/$31.00 2012 IEEE.
Lum . As known, the differential correlation is computationally costly since it requires Lum − 1 complex multiplications per correlation step. Yet, the complexity of the synchronization is of crucial importance. To evaluate the complexity a comparison of the Number of Complex Multiplications (NCM) carried to detect the first preamble, for all the algorithms of interest, is made. The complexities of the estimation methods considered are presented in table I assuming a successful synchronization. The parameter Q denotes the time interval between the processing beginning and the first preamble arrival, M denotes the size, in samples, of the repetitive part in the preamble of Nasir’s approach and Nz is the size, in samples, of the zero padded part in Chou’s approach. A numerical example for the complexities is also given for the same system parameters in section III.B. An interval Q of 3 OFDM symbols is chosen, Nz = Nu /8 which provides the steepest roll-off of the metric and M = Nu /8. Table I shows that the method proposed by Chou et al. has the lowest computational load. The PM and the BF methods present the highest complexity. In between, the methods of Cox and Schmidl, Minn, Nasir and the RC approach have a relatively low complexity with an important accuracy enhancement realized by the RC approach [10, 11], as will be later shown through simulations. Once the first preamble is TABLE I N UMBER OF C OMPLEX M ULTIPLICATIONS Approach Cox and Schmidl Minn-A Minn-B Nasir et al. Chou et al. Precoding Method Brute Force Reduced Complexity
NCM Nu (Q + N ) 3 N (Q + N ) 2 u Nu (Q + N ) (2Nu − M )(Q + N ) (Nu − Nz )(Q + N ) Num (1 + Num )(Q + N ) 2 2 Num Num (1 + )(Q + N ) 2 2 Num (
Ngm 2
Numerical example 4.6 106 6.9 106 4.6 106 8.6 106 4 106 11.78 108 11.78 108 5.1 106
+ Q + N)
are random sequences of ±1, the two previously presented msequences generated using the primitive polynomials P1 and P2 , where the difference in the performance is prominent. To overcome this difference, a random choice of an m-sequence, realized for each iteration, is also considered. Finally, a Gold sequence, deduced from the randomly chosen m-sequence, is also considered. The methodology adopted to generate the Gold sequences consists on first finding a preferred pair of primitive polynomials of order m = 9. Second, the sequences corresponding to the polynomials are implemented by using a shift register architecture. Then, the Nu different phases of one of the generated sequences is used to find each of the Num + 2 Gold sequences. The measure used to evaluate the performance is the achieved Correct Detection Rate (CDR) of the frame start, where the estimated frame start exactly (no error tolerance) coincides with the real one. The time estimate variance is also evaluated. For each SNR value, 104 realizations are run. The figures clearly show the sub-sequence type influence on the performance. Figure 3 illustrates the CDR of the RC estimator with different sequence types used to generate the preamble. Despite the use of the same sequence class (m-sequences), the detection performance is greatly enhanced through an adequately chosen m-sequence. For a target CDR of 80% a gain of about 2 dB is achieved using m-sequence 1, with respect to m-sequence 2. As could be observed, the random choice of a sequence from the 48 m-sequences provides a detection rate between the two separately simulated m-sequences and almost equal to the detection rate provided by a random sequence of ±1. The best detection rate is provided by the Gold sequences, especially at very low SNR, which achieve a gain of about 2 dB compared to the other sequences. This enhancement is due to the disappearance of the ghost sub-peaks inherent to the m-sequences and problematic especially in the multipath channels.
detected, for the continuous transmission mode (e.g. DVB-T), the synchronization processing will be performed periodically resulting in an important complexity reduction for all the algorithms. In the case of the bursty packet transmission, where the synchronization processing must continuously be performed (e.g. WiFi), the PM and the BF approaches are penalized by the extremely high complexity. The RC approach provides the best compromise between complexity and performance, as it has a computational load almost equal to that required by the sliding correlation approaches and an accuracy almost equal to that provided by the other differential correlation based approaches [10, 11]. VI. S IMULATION R ESULTS In this section, we present simulation results to evaluate the performance of the time synchronization of the RC approach, using different sequence types, under multipath propagation channel. The simulation parameters adopted here are the same cited in the previous sections. The simulated sequences here
Fig. 3. Correct Detection Rate versus SNR using different sequences in the case of an exponential 7-tap multipath channel
978-1-4673-1008-6/12/$31.00 2012 IEEE.
Figure 4 shows the variance of the time estimate using the sequences cited above. In concordance to the previous results, the Gold sequences provide the lowest estimation variance, which vanishes at an SNR of 1 dB. The other sequences have almost the same estimation variances with a slight gain realized by m-sequence 2. All the curves vanish at an SNR of about 4 dB.
Fig. 4. Time estimation variance versus SNR using different sequences in the case of an exponential 7-tap multipath channel
[2] T.M. Schmidl, D. Cox, “Robust frequency and timing synchronization in OFDM,” IEEE Trans. on Comm., vol. 45, pp. 1613-1621, Dec. 1997. [3] H. Minn. P. Tarasak and V. K. Bhargava, “On timing offset estimation for OFDM systems,” IEEE Comm. Letters, vol. 4, no. 7, pp. 242-244, July 2000. [4] H. Minn, V. K. Bhargava, and K. B. Letaief, “A novel timing estimation method for OFDM systems,” IEEE Trans. Comm., vol. 2, no. 4, pp. 822839, July 2003. [5] L. N. Atallah, M. Siala, “A New scheme for preamble detection and frequency acquisition in OFDM systems,” in Proc. ICECS, pp.1008 1011, Dec. 2009. [6] A. A. Nasir, S. Durrani, R. A. Kennedy, “Performance of Coarse and Fine Timing Synchronization in OFDM Receivers”, ICFCC, vol. 2, pp. 412-416, May 2010. [7] L. Nasraoui, L. N. Atallah, and M. Siala, “A very efficient time and frequency synchronization method for OFDM systems operating in AWGN channels,” ComNet, pp. 1-5, Nov. 2010. [8] L. Nasraoui, L. N. Atallah, and M. Siala, “An efficient synchronization method for OFDM systems in multipath channels,” in Proc. ICECS, pp. 1152 - 1155, Dec. 2010. [9] C-P. Chou, W-J. Lin, and J-S.Lin, “Timing synchronization with insertion of zero-padding preambles for OFDM systems,” ICICS, pp. 1-5, Dec. 2009. [10] L. Nasraoui, L. N. Atallah, and M. Siala, “An Efficient ReducedComplexity Two-Stage Differential Sliding Correlation Approach for OFDM Synchronization in the AWGN Channel,” VTC Fall, pp. 1-5, Sept. 2011. [11] L. Nasraoui, L. N. Atallah, and M. Siala, “An Efficient ReducedComplexity Two-Stage Differential Sliding Correlation Approach for OFDM Synchronization in the Multipath Channel,” Accepted in WCNC. [12] A. Mitra, “On the Construction of m-Sequence via Primitive polynomials with a Fast Identification Method,” International Journal of Computer and Electrical Engineering, pp. 158-163, 2008. [13] S. Y. Hwang, G. Y. Park, H. J. Park and K. S. Jhang; “An improved implementation method of the Gold sequence generator,” ISCE, pp. 1-4, April 2008.
VII. C ONCLUSION A performance study of a preamble based two-stage reduced-complexity time estimation approach was investigated. The studied approach combines a coarse stage based on a sliding correlation, and a fine stage, based on a differential correlation. This approach which was proved to achieve considerable performance improvement [10, 11], compared to the exclusively sliding correlation based approaches, allows an important complexity reduction, compared to the exclusively differential correlation based approaches. In this paper, two aspects were analyzed, the computational load and the impact of sub-sequence choice on performance. More precisely, we evaluate the number of complex multiplications of the different benchmarks including the brute force approach, which carries a differential correlation all over the synchronization process. We have shown that the additional complexity, which is the cost for the frame start detection improvement of this method, is avoided in the RC approach. To study the effect of the preamble sub-sequence choice, the start frame correct detection rate and the estimation variance were evaluated, using different sequences. Namely, the case of random sequences, m-sequences and Gold sequences were simulated. It was shown that in the multipath channels, thanks to their good auto-correlation properties, the Gold sequences outperform the other sequence types. R EFERENCES [1] R. Prasad, OFDM for wireless communications systems, Artech House, 2004.
978-1-4673-1008-6/12/$31.00 2012 IEEE.
