2013 Joint Time-Domain Correlation and Code-Aided

Joint Time-Domain Correlation and Code-Aided Carrier Synchronization AlgorithmJinhua Sun, Xuemei Wang State Key Laboratory of Integrated Service NetworksXidian UniversityXi’an, Chinajhsun@Xiaojun WuSchool of Information Engineering Chang’ an UniversityXian, ChinaAbstract—Considering low estimation accuracy and small synchronization range of carrier synchronization algorithms in low signal-to-noise ratio (SNR) for short burst communication systems, a joint time-domain correlation and code-aided carrier synchronization algorithm is proposed. Firstly, a frame structure in which the pilot symbols are separated with data symbols is used. Then, a new frequency estimation method using the summation of the entire cross correlation is deduced based on the time-domain correlation frequency estimation criterion. Based on this, the summation of the cross correlation of the pilot symbols after removing modulation is utilized to perform coarse synchronization and of the pilot symbols and decoding soft information is utilized to perform fine synchronization. So a joint time-domain correlation and code-aided carrier synchronization algorithm is obtained. Simulations by the Turbo code verify that the proposed algorithm can obtain higher estimation accuracy with only 32 QPSK pilot symbols, and the SNR degradation of Turbo decoder is within 0.5dB when the normalized frequency offset is smaller than 1.5×10-3.Index Terms—Carrier synchronization; pilot symbol; Turbo code; soft-output; short burst communicationI.I NTRODUCTIONModern communication systems are required to provide services based on high data rates burst-mode packet-data transmission, capable of operation at low SNR, such as Turbo code and Low Density Parity code. However, their decoding process is very sensitive to carrier synchronization errors, requiring exact knowledge of the carrier frequency and phase, so even slight frequency and phase offsets can cause severe performance loss [1].Conventional data-aided carrier synchronization algorithm, such as ML algorithm which is presented in [2], has frequency estimation range of half the symbol rates, but the frequency estimation accuracy can be improved by increasing the pilot symbols, so it is unsuitable for short burst communication. Recently, according to the structure feature of Shannon limit code, the carrier parameters estimation performance can be improved obviously by joint decoding and carrier synchronization, so code-aided synchronization algorithms are proposed in [3-11]. ML-ISDD algorithm [3] and ML-ITDD [4] algorithm both perform ML estimation utilizing pilot symbols and soft-information from Turbo decoder to improve the carrier parameters estimation performance at low SNR. Simulation results show that the estimation performance of the two algorithms converge to Cramer-Rao low bound (CRLB in [12]) within phase offset standard deviation 10φσ< and small frequency offset (-4710sfTΔ≤×) with only 22 pilot bits. However, when the frequency offset-4710sfTΔ>×, the residual frequency offset is still large after initial coarse synchronization due to limited pilot symbols, and the reliability of decoding soft-information is very poor even after several iterations. Besides, the peak of the periodogram searching is utilized in both coarse estimation and fine estimation process, on the other hand, ML-ISDD algorithm needs non-linear transformation of decoding soft-information, and ML-ITDD algorithm needs to make hard decision, recoding, so they have a higher computation complexity for the system. In addition, non-data-aided carrier synchronization algorithms based on maximum-mean-square soft-output (M2S2O) proposed in [5] and expectation-maximization (EM) proposed in [6] have a higher computation complexity, making its implementation prohibitive in many cases. Especially for the M2S2O algorithm, although it can correct the large range of frequency offset and phase offset, and achieves optimal BER performance, yet it needs two-dimensional search of frequency offset and phase offset. A carrier synchronization scheme for DVB-RCS system is proposed in [7], the frequency synchronization scheme uses FFT、L&R algorithm and the summation of correlation method successively in coarse synchronization process, and a low-complexity frequency estimator based on linear regression is used in fine synchronization process. The scheme is complex in coarse synchronization process, and can converge to the performance of EM algorithm with sufficient data blocks. So for short burst communication at low SNR, how to utilize a shorter pilot symbols and joint decoding soft-information to estimate large frequency offset is a hot topic for current carrier synchronization.For short burst communication system, references [8, 9, 13, and 14] present the effect of pilot placement on carrier parameters estimation range and estimation accuracy. To achieve large estimation range and high estimation accuracy, we choose Preamble-Middle (PM) pilot structure. References [7-9, 14] have presented the idea of coarse carrier synchronization based on the time-domain correlation of pilot symbols. In this paper the estimation algorithm based on time-domain correlation is applied to the fine synchronization978-1-4799-1027-4/13/$31.00 ©2013 IEEEprocess for the first time. The data frame is removed modulation by pilot symbols and soft-information from iterative decoder and then used to estimate carrier parameters, which forms the joint time-domain correlation and Turbo iterative decoding carrier synchronization algorithm. Firstly, frequency estimation method using the summation of the cross correlation is deduced based on the time-domain correlation frequency estimation criterion. Then, the summation of the cross correlation of the pilot symbols after removing modulation is utilized to perform coarse synchronization and of the pilot symbols and decoding soft information is utilized to perform fine synchronization. Simulation results show that the algorithm can correct large frequency offset with short pilot symbols, and then achieve the optimal BER performance. Compared to references [3, 4], the proposed algorithm abandons the frequency searching process based on ML algorithm, and it estimates the frequency offset value utilizing time-domain method directly, so it reduces the computation complexity, on the other hand, the summation of the correlation is firstly introduced into Turbo fine synchronization, which is of low complexity compared to frequency-domain estimation method.The remainder of this paper is organized as follows: Section Ⅱ and Section Ⅲ present the system model and carrier synchronization algorithm based on the summation of cross correlation respectively. In section Ⅳ the joint time-domain correlation and code-aided carrier synchronization algorithm is presented in detail. Section Ⅴ presents the simulations results for the proposed method and section Ⅵ concludes the whole paper.II. S YSTEM M ODELThe system model considered in the paper is shown in Fig.1. The description below includes the parameters used for the simulations described in the subsequent sections.MUXDEMUX×Fig.1. System modelA short packet dis first encoded by a parallel concatenated convolution code (PCCC) turbo encoder with recursivesystematic convolutional encoders. A pilot sequence is inserted by the multiplexer (MUX) into the encoder output code word, to enable initial data-aided (DA) carrier synchronization at the receiver. The MUX output frame is modulated into complextransmitted symbols frames. The transmitted frame is passed through an AWGN channel, a random phase shift[,)φππ∈−(constant per packet) and a carrier frequency shiftΔf . Assuming perfect symbol timing recovery, perfect framesynchronization and normalized frequency offset 1s fT Δ<<, the received signal can be written as (in equivalent base-band notation):exp((2)),0,1,...,1(1)k k s k r s j fkT w k K πφ=Δ++=− where s T is the symbol period, K is the number of transmitted symbols and k w represents complex-valued additive white Gaussian noise with two-sided power spectral density 02N . In the receiver, the received signal is passed to the demodulator and then to the demultiplexer (DEMUX), which extracts the received pilot sequence and passes it on to the carrier frequency and phase estimation process for initial coarse carrier offsets estimation. Next, the received data sequence, after initial frequency and phase offsets compensation, is passed again to the demodulator and the demultiplexer, which in this stage extracts the received codeword and passes it on to the extended turbo decoder. The pilot sequence is reused with the soft-output information based on the extended Turbo decoder to perform fine carrier offsets estimation, and this step is performed during the iterative decoding process. Finally, after certain iterations, the decoderproduces an estimation ˆdof the transmitted N -bit data packet. III. C ARRIER S YNCHRONIZATION A LGORITHM B ASED ON THES UMMATION OF C ROSS C ORRELATIONReference [9] analyzes frequency estimation range andestimation accuracy of different frame structures in detail, such as Preamble-Postamble(PP) structure (half of pilot symbols at the beginning of the burst and the other half at the end)、Preamble-Middle(PM) structure (half of pilot symbols at the beginning of the burst and the other half in the middle of the burst)、Preamble-Middle-Preamble-Middle (PMPM) structure (the pilot symbols is divided into serveral blocks and inserted into the data blocks). To satisfy the requirements of large frequency estimation range and high estimation accuracy for the short burst communication system, we choose PM structure. Fig.2 shows PM structure, L is the number of pilot symbols, N is the number of data symbols,D is the distance of the two adjacent pilot blocks./2L /2L /2NFig.2. PM StructureFor PSK signals, assuming k α is the posterior probabilityof k s , if k α is enough reliable, it satisfies that *k k s E αα= (s E is the symbol energy, ∗is conjugate). Therefore,(2)*(2)s j fkT k k k kz r e n πφαΔ+==+where k k k n w α∗=⋅.For PM structure, defining removed-modulation signals of the two blocks as follows:(2)1*,0,1,2,...,/21(3)s j fkT k k k k z r e n k K πφαΔ+==+=−(2)2*,/2,/21,...,1(4)s j fkT k k k k z r e n k K K K πφαΔ+==+=+−Then, the cross correlations of the two removed-modulation signals are summed,/211/211(2)(2)1*2*0/20/2/2112()(2)(2)**0/2()()()(5)s s s s s K K K K j fkT j fmT k m k m k m K k m K K K j fT k m j fT k j fT m m k k m k m K R z z en e n ee n e n n n πφπφππφπφ−−−−Δ+Δ+====−−−Δ−−Δ+Δ+====++=+++∑∑∑∑∑∑Ignoring noise, so it can be obtained,/211/21/212()2220/2222221111sin(/2)sin()s ssss s ss ssK K K K j fT k m j fDT j fkT j fmT k m K k m j fKT j fKT j fDT j fT j fT j fDT s s R ee ee e e ee e fKT e fT ππππππππππ−−−−−Δ−Δ−ΔΔ====−ΔΔΔΔ==−−=−−⎛⎞Δ=⎜⎟Δ⎝⎠∑∑∑∑(6)Because 2sin(/2)sin()s s fKT fT ππ⎛⎞Δ⎜⎟Δ⎝⎠is always positive, we can get theexpression of frequency estimation:{}/2111*20/2/21/21*0011ˆarg arg 221arg (7)2K K k m k m K s s K K m D k k m s f R z z DT DT z z DT πππ−−==−−+==⎧⎫Δ==⎨⎬⎩⎭⎧⎫=⎨⎬⎩⎭∑∑∑∑Refer to the estimation variance derivation of D -spaced estimator in [14], we can prove that the variance of the normalized frequency estimation of equation (7) is as follows:{}22222200111ˆ()(8)/2(/)ss s E f f T E N D K E N π⎛⎞Δ−Δ=+⎜⎟⎝⎠where, K is the number of transmitted symbols, 0/s E N is theratio of the symbol energy to noise power spectral density. The phase estimation can be expressed as follows:/21/21ˆˆ22()00ˆarg (9)s s K K j k fT j k D fT k k D k k z ez e ππθ−−−Δ−+Δ+==⎧⎫=+⎨⎬⎩⎭∑∑If the received signal is known, such as pilot symbols,then k k s α=, the upper limit of equations (7) and (9) is the half number of pilot symbols; if the received signal is unknown, such as data symbols, and then k αis the posterior probability of the transmitted data symbols.IV. J OINT T IME -D OMAIN C ORRELATION AND C ODE -A IDEDC ARRIER S YNCHRONIZATION A LGORITHM A. Extraction of the posterior probability of data symbolsDuring the iterative decoding process, it needs to make decision for the log-likelihood ratio of the posterior probability k Λ from Turbo decoder after each iteration. There are two methods to obtain posterior information k α based on different decision rules.Method 1: Hard decision is made directly to the log-likelihood ratio of the poseterior probability, 1k α= if k Λ is equal or greater than 0; 1k α=− if k Λis less than 0. If we want to obtain the posterior information of QPSK signal, it only needs to modulate the hard decision sequence into QPSK symbols;Method 2: The posterior information symbol is obtained through non-linear transformation, i.e. tanh(/2)k k α=Λ. If we want to obtain the posterior information of QPSK signal, it only needs to transform the posterior probability from turbo decoder into two parallel information, and form complex signal, i.e. 212tanh(/2)tanh(/2)k k k j α−=Λ+Λ.When the frequency offset and phase offset is within the estimation range of the proposed algorithm, the residual frequency offset after the coarse estimation can make Turbo decoder converge, so the reliability of posterior probability from decoder is higher, i.e.1k Λ . There is tanh(/2)1k Λ≈when 1k Λ , tanh(/2)1k Λ≈− when 1k Λ− . In the case,the two methods are equivalent, but hard decision is simpler in implementation, so we use method 1 in the following sections.B. Joint time-domain correlation and code-aided carrier synchronization algorithmBefore the received signal is passed into the decoder, it needs coarse synchronization that guarantees the turbo decoding to converge with the residual frequency offset, and then the BER performance is improved further after fine synchronization. The diagram of the joint time-domain correlation and code-aided carrier synchronization algorithm is shown in Fig. 3.As shown in Fig.3, the received frame {},0,1,...,1k r k K =− is passed to an initial carrier parameters estimator and the initial frequency offset and phase offset estimation values are obtained. To improve the frequency estimation accuracy of coarse synchronization, according to equations (7) and (9), pilot symbols are utilized to estimate frequency and phase values coarsely. The whole received frame is compensated with the initial estimated carrier parameters, and then delivered to the demodulator, the demultiplexer, and the extended turbo decoder successively. If the code rate is 1/3, firstly, we can obtain three soft-outputs of information symbol d Λ(in the form of log-likelihood ratio), the first parity symbol 1p Λ and the second parity symbol 2p Λ and make decisions, then multiplex the hard decision symbols and local pilot symbols based on the PM structure and pass them to the modulator for obtainingmodulated signals ˆs. Finally, according to equations (7) and (9), removed-modulation signals of pilot and hard decision symbols are utilized for fine synchronization. The received signals are compensated and delivered to the next iteration. This process is repeated until certain iterative numbers satisfied.During the fine synchronization, the posterior probability of pilot symbols and data symbols are extracted as follows:ˆk k kk data symbols k pilot symbols α∈⎧=⎨∈⎩s sy1d Λ2d Λ2p Λ1e Λ2e Λ1p Λ××Fig.3 The diagram of the joint time-domain correlation and code-aided carrier synchronizationV. S IMULATION R ESULTS AND A NALYSISA. The analysis of BER performanceThe performance of the algorithm is tested based on the parameters as follows: the code is a rate-1/3 PCCC turbo code with recursive systematic convolutional encoders feed-forward and feedback generators 8(5,7)g = respectively, the packet size 256N = bits and the iteration number is 6. The receiver utilizes the extracted pilot symbols to compensate the received signal after coarse estimation, and then perform iterative decoding. The carrier parameters are updated in each iteration process.Fig. 4 shows the BER performance of turbo decoder with different normalized frequency offsets. When the normalized frequency offsets are 51.010−×, 41.010−×, 42.010−× and 43.010−×, the SNR losses are about 0.25dB 、0.5dB 、1dB 、2.3dB respectively compared to the condition with no frequency offset. Therefore, it can be seen that the turbo code is very sensitive to the residual frequency offsets, and the degradation in BER performance is small when the normalized frequency offset 42.010s fT −Δ<×. Fig. 5 presents the effect of pilot symbols’ number on the coarse estimation at 0/1b E N dB =. It can be seen that the frequency estimation accuracy of the coarse estimation of the proposed algorithm is superior to the ML algorithm. When the pilot sequence achieves 22 bits, the root mean square error (RMSE) of the normalized residual frequency offset of the proposed algorithm is a little more than 42.010−×, and the RMSE of ML algorithm is 32.010−×. According to fig.4, when the normalized residualfrequency offset is more than 42.010−×, the degradation of turbo decoding performance is significant, and the loss of SNR is more than 1dB. When the pilot sequence achieves 60 bits, the RMSE of the proposed algorithm is less than 41.210−× and approaches to 41.010−×, and the RMSE of the ML algorithm is near to 45.010−×. So PM structure with a length of 64 bits pilot sequence is utilized for the following simulation.To evaluate the effectiveness of coarse synchronization and fine synchronization, fig. 6 shows the frequency estimation error variance for the proposed algorithm in comparison with the CRLB and fig. 7 presents the BER performance of the proposed synchronization algorithm. The theoretical derivation of the estimation variance of the normalized frequency offset is approximate, so there is error bewteen the simulation curve and the theoretical value. It can be seen that for 0/2b E N dB ≥ and31.510s fT −Δ<×, although there is a certain distance between the estimation RMSE and the CRLB, the estimation error is within the convergent range of the decoder. When 31.810s fT −Δ=× and 0/2b E N dB ≥, the RMSE of the normalized frequency offset is approximate to 41.010−×, the residual frequency offset is so large that BER performance is degraded as fig.4 shows; however, when 32.010s fT −Δ=×, the RMSE reaches to 43.010−× at 0/ 3.5b E N dB =, the performance loss is severe as fig.4 shows. Therefore, it can be seen that the proposed algorithm can correct carrier offset effectively when the normalized frequency offset 31.510s fT −Δ≤×, and the SNR loss is within 0.5dB when BER is in the range of 2610~10−−. However, ML-ISDD/ML-ITDDalgorithm can only work when the frequency offset is small, such as -46.010s fT Δ=×. If the frequency offset is larger, such as -49.010s fT Δ=×, the SNR loss is about 1.5dB with in the range of 2610~10−−BER performance. Because of the larger residual frequency offsets after initial coarse synchronization with ML-ISDD/ML-ITDD algorithm, the reliability of the soft information which is utilized to the iterative decoding process is very poor, so it cannot track carrier frequency precisely andhas a significant degradation in BER performance.Fig.4.Turbo BER performance with frequency offsetFig.5.The effect of pilot sequence number on frequency estimationFig.6.Frequency estimation mean square root error for the proposed algorithmFig.7.Turbo BER performance of the proposed algorithmB. The algorithm complexitySimilar to ML-ISDD algorithm and ML-ITDD algorithm, the proposed algorithm needs coarse estimation and fine estimation which need ML frequency searching for ML-ISDD and ML-ITDD algorithm. Its essence is searching the peak of the periodogram. However, the proposed algorithm utilized cross correlation of removed-modulation signal to estimate carrier parameters, which is an estimation method in time-domain and abandons the process of frequency searching in frequency-domain. On the other hand,compared to ML-ITDD and ML-ISDD algorithm, the proposed algorithm has no recoding and non-linear transformation. Therefore, it has a lower computation complexity and is easy to implement.VI. C ONCLUSIONTo satisfy the requirement of using short pilot symbols to estimate large frequency offset for short burst communication system, a joint time-domain correlation and Turbo iterative decoding carrier synchronization scheme is proposed. To obtain the larger synchronization range, PM pilot structure is adopted. Firstly, frequency estimation based on time-domain correlation is deduced, and then the residual frequency offset is reduced after coarse synchronization utilizing pilot symbols, finally, the fine synchronization is performed with the aid of joint pilot symbols and decoding information in the Turbo iterative decoding process. Simulation results show that the higher frequency estimation accuracy in coarse process is obtained with only 32 QPSK symbols, meanwhile, Turbo code can decode correctly when 31.510s fT −Δ≤×, the SNR loss is within 0.5dB with in the range of 261010−−∼BER performance.A CKNOWLEDGMENTThis work was supported by the National Natural Science Foundation of China (60902039, 61271175), the Fundamental Research Funds for the Central Universities (K50511010014, K5051201043, and CHD2011JC088).R EFERENCES[1]Jordan M A, Nichols R A, The effects of channel characteristicson turbo code performance [C]//Military Communication conference, vol.1,1996：17-21[2]Umberto Mengali, Aldo N D Andrea. SynchronizationTechniques for Digital Receivers [M]. New York and London,Plenum Press, 1997:79-97.[3]Rahamim Y, Freedman A, Reichman A. ML Iterative Soft-Decision-Directed (ML-ISDD): A Carrier Synchronization System for Short Packet Turbo Coded Communication [J]. 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