Burst transmission symbol synchronization in the presence of cycle slip arising from different clock frequencies
 Somayeh Bazin^{1} and
 Mahmoud Ferdosizade Naeiny^{2}Email author
https://doi.org/10.1186/s1363401705255
© The Author(s) 2018
Received: 24 July 2017
Accepted: 22 December 2017
Published: 8 January 2018
Abstract
In digital communication systems, different clock frequencies of transmitter and receiver usually are translated into cycle slips. Receivers and transmitters may experience different sampling frequencies due to manufacturing imperfection, Doppler effect introduced by channel or having error in estimation of symbol rate. Timing synchronization in presence of cycle slip for a burst sequence of received information leads to severe degradation in system’s performance. Therefore, the necessity of prior detection and elimination of cycle slip is obvious. Accordingly, the main idea introduced in this paper is to employ the Gardner detector (GaD) not only to recover a fixed timing offset, but also its output is processed such that timing drifts can be estimated and corrected. By deriving a twostep algorithm, first the cycle slips arising from symbol rate offset is eliminated, and then symbol’s timing offset is synchronized in an iterative manner. GaD structure is used in a feedforward structure with the additional benefit that convergence and stability problems, which are typical challenges of the systems with feedback, are avoided. The proposed algorithm is able to compensate considerable symbol rate offsets at the receiver side. Results in terms of BER confirm the algorithm’s proficiency.
Keywords
1 Introduction
Timing recovery as a process of sampling at the right time are critical in digital communication receivers. The problem is formulated through maximum likelihood (ML) [1]. In direct application of ML method, message symbols and timing offset are estimated jointly. However, this solution conveys the exhaustive search methods that imposes a lot of computations and it makes the solution impractical. To avoid the complexity due to the exhaustive search in ML problem, iterative solutions are introduced [2–4]. 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. MLbased timing recovery methods usually ignore the time varying timing offsets and proceed under the assumption of fixed synchronization parameter estimation [5–7]. 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 [8].
Different clock frequencies in transmitter and receiver lead to linear increasing/decreasing of timing offset from symbol to symbol. While timing offset changes linearly for successive symbols, synchronizers may fail to track this timevarying delay. Getting far away from true value makes the estimator fall 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 voltagecontrolled 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 CS, it has to be eliminated before applying timing synchronization.
Although, several studies have considered the analysis of cycle slipping in synchronizers [9–11], few authors have proposed the solution [12–14]. Error tracking synchronizers, which are based on closed feedback loop, are more popular in low SNRs. 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 feedforward structure based on extracting timing delay estimation from the statistics of received samples, and then adjusting the time by interpolation is more suitable. In this work, in order to utilize the bandwidth efficiency of nondataaided (NDA) estimators and effective flexibility of interpolation, Gardner TED [15] and Farrow filter [1] are used in a feedforward structure.
In accordance with the above statements, this work is motivated by the objective of deriving a novel algorithm which employs GaD in a nonconventional manner so that not only the fixed timing offset is recovered, but also GaD’s output is processed in a way such that a considerable symbol rate offset can be estimated and corrected, which is not addressed in the literatures. At the first step of the proposed algorithm, CS is estimated and eliminated and then the remaining fixed timing offset is estimated and compensated. Simulation results shows that the performance of the proposed algorithm is very close to the theoretical lower bound (which is derived with the assumption of the perfect synchronization) even when there is a considerable symbol rate offset.
The general structure of this paper is as follows: in the next section, the problem of timing offset and CS are formulated, also the proposed algorithm to eliminate CS is derived in this section. Then Section 3 illustrates an iterative scheme for fixed timing offset synchronization after CS elimination. In Section 4, the BER performance of the proposed algorithm is derived using MonteCarlo simulation and it is compared with alternative algorithms and theoretical lower bound. The final section is about the conclusion of the introduced algorithm.
2 Problem formulation
2.1 Signal model
Where a_{ n } denotes the zero mean unit variance, independently and identically distributed (i.i.d) symbols that might be taken from any linear modulation scheme. θ and Δf are phase and carrier frequency offset, respectively. h(t) is a pulseshaping filter, n(t) is a complex zeromean additive white Gaussian noise with twosided power spectral density of N_{0}/2. Moreover τ, T, and N are unknown timing delay, symbol duration, and the number of transmitted symbols, respectively.
At the receiver side, the signal in (1) should be matched filtered and the transmitted symbols should be regenerated by sampling r(t) at \(kT  \hat \tau \) time instants, where \(\hat \tau \) is timing delay estimation provided by synchronizer. In softwaredefined radio (SDR) scheme, matched filtering (MF) is also implemented in digital domain. In order to implement MF in digital mode, the received signal, r(t), is sampled at a very high rate than the symbol rate. It means that the samples per symbol parameter is very higher than one. Then, the output of MF is resampled to generate one or two samples per symbol. At the output of MF, one sample per symbol is enough for symbol detection, but two samples per symbol are necessary for the calculation of Gardner timing error as will be discussed in next paragraphs.
where \(g\left (t \right) =h\left (t \right)*{h^{*}}\left ({  t} \right)= \smallint \limits _{ \infty }^{\infty } h\left ({\lambda } \right){h^{*}}\left (\lambda  t \right)d\lambda \) is the convolution of the pulse shape, h(t), with the matched filter with impulse response of h^{∗}(−t), w_{ n } is an i.i.d. zero mean Gaussian distributed variable with variance σ^{2}, A_{ n } is the nth noisy symbol, and μ stands for \(\tau  \hat \tau \).
Obviously, timing delay takes different values for different symbols of a received burst caused by the variable timing delay part which is increased linearly by k. Traditional approaches assume this variation is slow in comparison to the burst interval, and they approximate average timing delay over a number of symbol periods which it means that synchronization parameter can be considered as quasiconstant [16], however, ignoring this variation would degrade the performance as it will be well illustrated in simulation results.
2.2 Cycle slip in synchronization
The obvious fact is that the phase offset does not play an influential role in Gardner’s timing delay estimation. Likewise, the impression of carrier frequency offset on timing delay estimation is negligible, considering an assumption that ΔfT≪1. Therefore, carrier and phase offset can be omitted as far as \(\frac {{\Delta f}}{{BW}} \ll 1\) and timing offset can be synchronized regardless of prior carrier and phase synchronization.
As long as μ+kε<T/2, TED is capable of tracking the timing delay. For any special value of k that μ+kε exceeds from this interval, the synchronizer starts to relate the timing delay to the adjacent symbol and CS happens. This nonuniform delay detection results in a quasiperiodic behavior of u(k), which is wrapped for particular coefficients of K that Kε=T. This fact, which leads to the periodic function of u(k), is proved in the following Lemma.
Lemma 1
u(k) is a periodic function with the period of T/ε.
Proof
Which is always positive irrespective to what A_{ k } and A_{k+1} are, and it confirms that when the timing of the samples is late and less than T/2, then the Gardner error is positive.
2.3 Cycle slip detection and correction
In (16), K is approximated by N^{′}point DFT where N^{′}>N and N is the length of u(k). It means that the vector u is zero padded by N^{′}−N zeros and DFT is applied to achieve U. N^{′} can be assumed to be a power of 2, such that DFT operation can be done with low complexity fast Fourier transform (FFT). Finally, the index in which the absolute value of DFT is maximized, yields an approximation of K.
Once K was determined, new samples must be generated based on the new updated symbol rate, \(\hat {R}_{s}^{{{new}}}\). This is done in SRC block by the fractional interpolation of the samples. It is noteworthy that in the above mentioned algorithm K is a positive parameter, but η can be either positive or negative. Although there are some methods to determine the sign of η based on the phase of the U(q), it is simpler and more practical to test the positive and negative values of η.
3 Iterative timing recovery
In [18], an iterative timing recovery derivation from the maximum likelihood principle is proposed, where the timing information, extracted from Gardner’s TED, has been used iteratively to adjust sampling time. Here, this method is used in cooperation with digital filtering by interpolation in order to improve the sampling instants.
In the other words, averaging the evaluated timing error by Gardner’s TED over all the symbols of a burst provides an estimation of the constant timing delay. This can be used for timing adjustment in interpolator.
4 Results and discussion
To verify the efficiency of the proposed algorithm, simulation results are presented in this section. Simulations are carried out with BPSK and QPSKmodulated signals which are shaped by a square root raised cosine filter with rolloff factor of 0.5 in transmitter, and they are passed through AWGN channel. In order to implement the matched filtering in digital domain, the received signal is sampled with the rate of 10 samples per symbol and a sample rate conversion (SRC) is used at the output of matched filter for conversion of the sample rate to 2 samples per symbol. The interpolation filter order in the timing recovery block is two. It means that a simple linear interpolation is used in SRC block. It is noteworthy that the higher interpolation filter order leads to more accurate results and compensates the lower sample per symbol ratio in ADC. However, in the following simulations, since sample per symbol ratio is set to 10, increasing the order of the filter does not make a considerable difference. Therefore, linear interpolation is used to avoid the complexity. After SRC block 2N samples are zero padded by 4096−2N zeros and a 4096 points DFT is used to estimate the period of CS based on (15).
5 Conclusions

The GaD is not only employed to recover a fixed timing offset, but also its output is processed in a way such that timing drifts can be estimated and corrected.

Normally, GaD is used symbol by symbol in a feedback loop, whereas in this contribution it is suggested to apply it to a feedforward structure with the additional benefit that convergence and stability problems are avoided, as they are typical for feedback schemes.
It is shown that linearly increased timing delay causes alternate cycle slips in synchronizer. A burst sequence of timing error provided by GaD is used to indicate and eliminate CS. After CS correction, iterative 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 performance of the proposed algorithm.
Declarations
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