Burst Transmission Symbol Synchronization in the Presence of Cycle Slip Arising from Different Clock Frequencies

In digital communication systems different clock frequencies of transmitter and receiver usually is translated into cycle slips. Receivers might experience different sampling frequencies from transmitter due to manufacturing imperfection, Doppler Effect introduced by channel or wrong estimation of symbol rate. Timing synchronization in presence of cycle slip for a burst sequence of received information, leads to severe degradation in system performance that represents as shortening or prolonging of bit stream. Therefor the necessity of prior detection and elimination of cycle slip is unavoidable. Accordingly, the main idea introduced in this paper is to employ the Gardner Detector (GAD) not only to recover a fixed timing offset, its output is also processed in a way such that timing drifts can be estimated and corrected. Deriving a two steps algorithm, eliminates the cycle slips arising from wrong estimation of symbol rate firstly, and then iteratively synchronize symbol timing of a burst received signal by applying GAD to a feed forward structure with the additional benefits that convergence and stability problems are avoided, as they are typical for feedback schemes normally used by GAD. The proposed algorithm is able to compensate considerable symbol rate offsets at the receiver side. Considerable results in terms of BER confirm the algorithm proficiency.


INTRODUCTION
Timing recovery as a process of sampling at the right times is critical in digital communication receivers. In principle, the problem is formulated through Maximum Likelihood (ML). The direct computation of ML problem, performs the task of jointly detection and estimation of the message bits and the timing offset. However this solution conveys the exhaustive search methods that imposes a lot of computation and makes the solution impractical. To avoid the complexity of direct computation of ML problem, iterative solutions are introduced [1][2][3]. The general idea of iterative timing recovery scheme is to improve the timing estimation accuracy by multiple exploiting the timing information provided by a set of samples and application of this estimation to regenerate a new set of samples that iteratively approaches to the local maximum of the likelihood function. ML based timing recovery methods usually ignore the time varying timing offsets and proceed under the assumption of fixed synchronization parameter estimation [4][5][6]. However in practice, the timing offset may vary with time, due to the different clock frequencies in transmitter and receiver caused by fractional error in baud rate estimation, manufacturing imperfection and etc. [7]. Different clock frequencies in transmitter and receiver leads to linear increasing of timing offset from symbol to symbol. While, timing offset increases linearly for successive symbols, Synchronizers may fail to track this time varying delay. Since getting far away from true value makes the estimator falls into the adjacent stable operating point and synchronizer starts to keep tracking this new stable operating point. Consequently one symbol inserted into or erased from the sequence. This is called Cycle Slipping (CS). There is also another source of CS which is the large phase variance of voltage controlled oscillators (VCOs) caused by low signal to noise ratio (SNR) that is not subjected in this paper. As long as cycle slips occur, system's performance decreases dramatically due to the relative loss of synchronization caused by symbol insertion or omission in the sequence. In order to alleviate the adverse effect of cycle slip it has to be eliminated before applying timing synchronization.
Although, several studies have considered the problem of cycle slipping in synchronizers [8][9][10], few authors have proposed the solution [11][12][13]. Typically, solutions concern about error tracking synchronizers in low SNRs which are based on closed feedback loop. While the good tracking performance of feedback schemes is not deniable, they require, in counterpart relatively long acquisition time that makes them unsuitable for burst transmission schemes .In this sense, a feed forward structure based on extracting timing delay estimation from the statistics of received samples, and then adjusting the time by some sort of interpolation is more suitable. In this work, in order to utilize the band width efficiency of Non-Data-Aided (NDA) estimators and effective flexibility of interpolation, Gardner TED [14] and Farrow filter [15] are used in a feed forward structure as timing delay estimator and interpolation filter respectively.
In accordance with the above statements, this work is motivated by the objective of deriving a novel algorithm which employs GAD in a non-conventional manner so that not only the fixed timing offset is recovered, GAD's output is also processed in a way such that considerable timing drifts can be estimated and corrected, which is not addressed in the literatures. CS is eliminated at the first step and then Gardner timing delay estimation in cooperation with Farrow based interpolation filter Makes the algorithm suitable for a more agile and efficient iterative receiver.

Signal model
Assume a traditional communication system, where the transmitted signal is corrupted by passing through AWGN channel which also imposes a timing delay and carrier frequency and phase offset to the received signal as follows: Where denotes the zero mean unit variance, independently and identically distributed (i.i.d.) symbols that might be taken from any linear modulation scheme. and ∆ are phase and carrier frequency offsets. ℎ( ) is a square root raised cosine pulse and ( ) is a complex zero-mean additive white Gaussian noise with two sided power spectral density of /2. Moreover , and are unknown timing delay, pulse duration and the length of the transmitted signal respectively.
At the receiver side, the signal in (1) should be matched filtered and the transmitted symbols should be regenerated by sampling ( ) in − ̂ time instants, where ̂ is timing delay estimation provided by synchronizer. Even if receiver has exact information about symbol rate, there still may exist some fractional difference between transmitter and receiver clock frequencies due to the implementation imperfection. However, blind receivers have to estimate symbol rate that always conveys some estimation error. In this case the regenerated symbols are located in − ̂ time instants, where: and is the difference between transmitter and receiver symbol duration and can be either positive value or negative one. The discrete samples are: Where is i.i.d. normally distributed variable with variance , ( ) is convolution of ℎ( ) with channel impulse response and the analog pre-filter response.
represents + and stands for − .
Obviously, timing delay varies for different symbols of a received burst, due to the variable timing delay part which is increased linearly by . Traditional approaches assume this variation is slow in comparison with symbol interval and they approximate the timing delay over a number of symbol periods that synchronization parameter can be considered as quasi-constant [16], however ignoring this variation would degrade the performance as it will be well illustrated in simulation results.

Cycle-Slips in synchronizer
Typically, Gardner's Timing Error Detection (TED) provides timing estimation to synchronize the received symbols using the samples at twice rate of the symbol rate, according to the following equation [14]: e phase off wise the imp nsidering an n be omitte ior carrier an | TED is ca s from this and CS hap f ( ) whi of ( ) is pr y, transmitt n -1 and +1 is depicted ol's offset c  Considering bandwidth of ( ) and doing some manipulation in order to discard ineffective sentences in (5), a simplified extend is obtained, As long as | + | ≤ Let ( ) represents the timing offset estimation of th symbol. For any arbitrary positive value of that 0 < < and + = − , ( ) is generated as follows: Or: Evidently: Noting to the fact that and are even +1 or -1, term − − − − can be ignored in (7) and consequently: Which is always positive irrespective to what and are, and confirm the correctness of delay estimation.
Suppose that for all th symbols that + >| |, ( ) represents the relevant timing offset estimation, Similarly assume is a positive value, 0 < < , that make the delay exceeds from , so that + = + . Relatively ( ) can be achieved through following equation: Regenerating of ( ) for + = − + corresponds to:   can be interpreted through trade of between DFT resolution and TED's accuracy. More precisely augmentation leads to the faster CS happening and better DFT resolution due to the more periodic terms that is provided by larger in a fix burst length. While, in the other hand it reduces the accuracy of timing error detected by TED. Note that in CS inexistence ( = 0) system performance does not approach to the perfect synchronization, since system had to recover and estimate constant timing delay.

CONCOLUSION
A new algorithm for timing recovery by prior elimination of cycle slip is proposed. It is shown that linearly increased timing delay causes alternate cycle slips in synchronizer. A burst sequence of timing error provided by Gardner TED is used to indicate and eliminate CS. After CS correction iterative bandwidth efficient timing recovery is applied to the sequence of burst samples. The satisfactory simulation results, evaluated in terms of BER and compared with theoretical and other approaches confirm the system's appropriate performance.