# Robust Frame Synchronization for Low Signal-to-Noise Ratio Channels Using Energy-Corrected Differential Correlation

- Dong-Uk Lee
^{1}, - Pansoo Kim
^{2}and - Wonjin Sung
^{1}Email author

**2009**:345989

https://doi.org/10.1155/2009/345989

© Dong-Uk Lee et al. 2009

**Received: **1 July 2008

**Accepted: **1 February 2009

**Published: **19 February 2009

## Abstract

Recent standards for wireless transmission require reliable synchronization for channels with low signal-to-noise ratio (SNR) as well as with a large amount of frequency offset, which necessitates a robust correlator structure for the initial frame synchronization process. In this paper, a new correlation strategy especially targeted for low SNR regions is proposed and its performance is analyzed. By utilizing a modified energy correction term, the proposed method effectively reduces the variance of the decision variable to enhance the detection performance. Most importantly, the method is demonstrated to outperform all previously reported schemes by a significant margin, for SNRs below 5 dB regardless of the existence of the frequency offsets. A variation of the proposed method is also presented for further enhancement over the channels with small frequency errors. The particular application considered for the performance verification is the second generation digital video broadcasting system for satellites (DVB-S2).

## 1. Introduction

Reliable synchronization is one of the key factors determining the transmission performance in communication channels, and various schemes for time, frequency, and phase estimation for imperfect communication links have been proposed and implemented. Although time and frequency estimation can be jointly performed at an increased complexity, frequency synchronization is usually preceded by the symbol and frame synchronization. A classic result on coherent detection for the frame synchronization has been presented by Massey [1], which suggests the usage of a data correction term for the optimal maximum likelihood (ML) statistics. The result has subsequently been verified, extended to specific modulation schemes, and approximated to suboptimal solutions [2–5]. In many practical receivers, the suboptimal metrics become preferred choices due to their simplicity and reasonable performance. The approximation of ML statistics using the low SNR assumption leads to a simplified computation of the correction term in the form of received signal energy [1, 2, 5].

While these coherent detection schemes provide optimal or near-optimal performance in the static additive white Gaussian noise (AWGN) channel, a performance loss is experienced when the frequency error exists in the channel. Under such circumstances, differential correlation metrics provide robustness to frequency and phase errors. In particular, detection methods in [6] are derived from the approximated ML metrics, and give a lower detection error probability than other known schemes. Reduced-complexity schemes are also extensively studied [7–10]. Differential postdetection integration (DPDI) techniques [8] are shown to provide a good performance-complexity trade-off, and generalized DPDI including average postdetection integration (APDI) schemes has also been proposed and analyzed [9, 10].

On the other hand, the requirements for the initial synchronization performance are becoming more stringent. As advanced transmission schemes including efficient modulation and powerful error-correction coding are developed, target operating SNRs tend to decrease to allow data transmission even in hostile channel environment and to maximize the bandwidth usage. The recent DVB-S2 standard [11, 12] adopted the low-density parity-check (LDPC) coding for adaptive coding and modulation (ACM), which includes high-density amplitude phase shift keying (APSK) as well as conventional phase shift keying (PSK). These techniques lowered the minimum operating SNR down to dB for the lowest ACM level. Since initial synchronization should reliably be performed for all ACM levels, it is important to verify the detector performance at this low SNR range.

In this paper, we propose detection strategies for robust frame synchronization under the effects of severe noise and frequency offsets and verify the corresponding performance. The proposed detector is constructed via appropriate adjustments of the correction term, and variational methods for further performance enhancement are also suggested. It is demonstrated that the methods provide a substantial gain over existing schemes for the SNR range of interest. The organization of the paper is as follows. The signal model, frame structure, and channel conditions used for performance evaluation are introduced in Section 2. In Section 3, a brief review and comparison of existing decision metrics for the frame synchronization are given, and the proposed method is presented. In Section 4, we discuss properties and parameter optimization issues of the proposed method. The detection performance is evaluated for different channel conditions to quantify the gain in Section 5, and the conclusions are given in Section 6.

## 2. Signal Model

where is the modulated symbol with normalized power and is the AWGN sample with variance . Then the received SNR is determined as . Parameters and , respectively, denote the frequency offset and the phase offset, and is the symbol duration. We assume that takes one of the values from where represents the maximum amount of frequency offset, and takes one of the values from . The frame detection includes the correlation of consecutive received samples with the synchronization sequence symbols .

For the performance evaluation, the SoF of length is used as the synchronization sequence. A particular attention is given to mid to low SNR values, including the minimum required operational SNR of dB for DVB-S2. The maximum frequency offset is 20% of the transmission bandwidth; that is, the normalized maximum frequency offset is given by . Unless otherwise stated, the frequency offset is uniformly generated from the range .

## 3. Correlation Methods for Frame Synchronization

*correction term*. For small values of SNR , it has been shown that the correction term takes the form of

which accounts for the energy correction of the received sequence. It has also been confirmed by Gansman et al. [5] that the correction term of the ML detector reduces to a function of the received signal energy when the low SNR approximation is applied.

Such correlation has been utilized in the decision variable
suggested in [9], where parameter
(<*L*) determines the performance versus complexity trade-off. Related discussion can also be found in [10], which presents the variable
for enhanced detection performance under a severe effect of frequency offset.

by dropping the squares, which results in performance enhancement over the original for many cases of practical interest. The frame detectors using the decision variables in (6) and (7) will, respectively, be called C1 and C2 detectors. At an increased complexity caused by additional computation of the correction terms, C1 and C2 detectors outperform all the other aforementioned detection strategies in the presence of frequency offsets.

for the frame detection and call the corresponding detector L1. Multiple appearances of identical received samples in the decision variable result in highly correlated statistical characteristics, and an exact analytic evaluation of the statistics for each decision variable seems difficult. Nevertheless, the advantage of the proposed variable is immediately observable from the distributions experimentally obtained, as discussed in the following.

## 4. Properties and Variations of Proposed Correlation

### 4.1. Distribution of the Decision Variables

### 4.2. Utilization of the Vector Sum

To maximize the magnitude of the vector sum in (13), a smaller value of is desired as the frequency offset increases, since otherwise the sum of vectors with widespread angles results in a reduced magnitude. It can be observed that the magnitude of the vector sum begins to diminish as the terms with angular phase exceeding radians are added; thus parameter needs to be chosen to satisfy . As the frequency offset increases, a smaller number of -span differential correlation values contribute to the decision variable, and the reliability of the statistics decreases. Thus L2 exhibits improved performance over L1 when the frequency offset is small but is eventually outperformed by L1 as increases.

### 4.3. Weighted Energy Correction

using respective weight factors and . Since the performance of L3 and L4 detectors varies based on the choice of these parameters, proper parameter values need to be determined for the optimized performance at target operating points.

## 5. Performance Evaluation

where is the PDF for the synchronous decision variable. For the evaluation of FAR, random QPSK data symbols are generated and noise samples are added. The noise-corrupted data symbols are correlated with the SoF symbols; then the decision variable is computed and threshold tested. In the case of MDP, a similar procedure is performed, by generating noise corrupted SoF symbols instead of random data symbols for correlation.

The detector proposed by Nielsen [2] performs fully coherent correlation which is targeted for channels without frequency offsets. As indicated in Figure 10 (Nielsen), it outperforms all other schemes at zero frequency offset. Nevertheless, very rapid performance degradation is experienced as the offset increases, and more than 50% MDP occurs when the normalized frequency offset exceeds 0.05. Gansman's detector [5] (labeled Gansman) performs multi-span correlation with the energy correction term included. Because of the existence of the coherent correlation term in addition to differential terms, performance tends to become degraded as the frequency offset increases. Another mixed-type detector is APDI [8, 9] which efficiently controls the detection complexity and its MDP is also shown in the figure.

## 6. Conclusion

We have presented the correlation schemes for improved detection performance over various channel conditions. The proposed L3 detector is shown to provide a substantial gain over all existing detection methods, regardless of the existence of frequency offsets. Further performance enhancement is achievable by using the L4 detector when the amount of frequency offset is relatively small. Presented results confirm that an appropriate energy correction in correlating detectors has a significant impact on the detection performance.

## Declarations

### Acknowledgments

This work is supported in part by the IT R&D program of KCC/IITA 2007-S008-03, *Development of 21 GHz Band Satellite Broadcasting Transmission Technology,* and in part by the Special Research Grant of Sogang University.

## Authors’ Affiliations

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