# Blind Coarse Timing Offset Estimation for CP-OFDM and ZP-OFDM Transmission over Frequency Selective Channels

- Vincent Le Nir
^{1, 2}Email author, - Toon van Waterschoot
^{1}, - Jonathan Duplicy
^{3}and - Marc Moonen
^{1}

**2009**:262813

https://doi.org/10.1155/2009/262813

© Vincent Le Nir et al. 2009

**Received: **9 May 2009

**Accepted: **15 December 2009

**Published: **19 January 2010

## Abstract

We present a blind coarse timing offset estimation technique for CP-OFDM and ZP-OFDM transmission over frequency selective channels. The technique exploits the cyclic prefix or zero-padding structure to estimate the starting position of the OFDM symbols without requiring any additional pilots. Simulation results are performed on various channel models with timing and frequency offsets. The presented technique is compared with the autocorrelation-based technique and various techniques in frequency selective channels. Our algorithm shows better performance results than those of the autocorrelation-based technique in NLOS channels, where the most predominant channel path is usually not the first arrival path.

## Keywords

## 1. Introduction

In this paper, we describe a technique to blindly estimate the timing offset in digital communications systems employing orthogonal frequency division multiplexing (OFDM). The starting position of the OFDM symbols is estimated using the cyclic prefix (CP) or zero-padding (ZP) structure of the transmitted signal without requiring any additional pilots. The aim of the paper is to provide an alternative coarse timing offset estimation technique to the autocorrelation-based technique [1–3] which does not perform well in non-line-of-sight (NLOS) frequency selective channels as well as for ZP-OFDM transmission.

The literature on timing offset estimation and carrier frequency offset (CFO) estimation can be divided in two categories: data-aided and non-data-aided techniques. Data-aided techniques use additional pilot symbols known at the receive side to estimate the timing and frequency offsets based on autocorrelation and other features [4–6]. Non-data-aided techniques do not require additional pilot symbols and can exploit the cyclic prefix structure of the transmitted signal in an autocorrelation metric [1, 3] (which is a simplified version of the maximum likelihood (ML) algorithm requiring the knowledge of the received signal-to-noise ratio (SNR) [2]). However, these techniques fail when the channel exhibits strong multipath components. Other non-data-aided techniques require the knowledge of the pulse shaping filter and exploit the cyclostationarity of the OFDM signal by a cyclic autocorrelation metric [7, 8], use bell-patterns to detect the symbol energy variations of the first subcarrier [9], require the knowledge of the maximum delay spread [10], or perform symbol timing and frequency offset estimation jointly [11].

In this paper, we propose a new non-data-aided approach for timing offset estimation which does not require the knowledge of the SNR [2], the pulse shaping filter [7, 8] or the maximum channel delay spread [10] and works well for CP-OFDM and ZP-OFDM transmission when the channel exhibits strong multipath components. For the mathematical derivations, we assume that the cyclic prefix duration or zero-padding duration is larger than the channel impulse response ; however we will show that the algorithm still exhibits good performance when (or ). Moreover, we assume the knowledge of the symbol duration , the cyclic prefix duration (or the zero-padding duration ) and the number of subcarriers which can indeed be estimated blindly with the algorithms described in [12]. The timing offset estimation technique exploits the cyclic prefix or zero-padding structure of the OFDM signal and tracks time domain symbol energy variations based on a transition metric. Contrary to the autocorrelation metric [1–3], the transition metric-based technique is able to estimate the timing offset in frequency selective channels with strong multipath components.

The paper is organized as follows. In Section 2, we present the OFDM signal model, we review the technique for timing offset estimation based on the autocorrelation metric, and then the techniques based on the transition metric are presented for CP-OFDM and ZP-OFDM transmission. In Section 3, simulation results are presented with realistic channels models. Finally, conclusions are drawn in Section 4.

## 2. Description of the Algorithm

The non-data-aided technique presented in this paper has been developed in the context of radio surveillance and cognitive radio systems for multicarrier modulations. The wideband received signal may contain multiple OFDM signals of interest. Therefore, the received signal is sampled in a large bandwidth to include existing and future OFDM standards, such as Wifi (2.4 GHz or 5 GHz), WiMAX (3.5 GHz), Long-Term Evolution (LTE), or WiMedia (ECMA-368) signals. The carrier frequencies, bandwidths, and average powers of the detected signals are estimated. After downconversion to baseband and low-pass filtering, each signal of interest is processed through a feature detection block to determine whether or not it is an OFDM signal and to estimate blindly its symbol duration , its cyclic prefix duration (or its zero-padding duration ) and its number of subcarriers [12]. Each signal of interest can be modeled as a received sequence of length such that

where is the transmitted signal vector oversampled by the ratio between the cut-off frequency of the low-pass filter and the transmitter maximum frequency, the 's are the oversampled multipath channel coefficients with the number of channel taps, is the vector of Additive White Gaussian Noise (AWGN), the receiver phase offset, the receiver frequency offset, and the receiver timing offset.

### 2.1. Timing Offset Estimation Techniques for CP-OFDM Transmission

However, maximizing the autocorrelation metric over a window of length will rather provide a timing offset estimate of the most predominant channel path than a timing offset estimate of the starting position for the OFDM symbols. Indeed, only a duration of is fully correlated between the received sequence and the conjugated received sequence shifted by the symbol duration . Therefore, the autocorrelation metric is expected to work well in line-of-sight (LOS) scenarios where the most predominant channel path is the first arrival path, but it will fail in NLOS scenarios where the most predominant channel path is usually not the first arrival path.

Instead of using an autocorrelation metric, we propose to use a metric-based on the difference between the received sequence and the same sequence shifted by the symbol duration (see also Figure 2). The correlated duration of the cyclic prefix is cancelled out in this operation. In this way, the cyclic prefix structure of the OFDM signal is exploited by tracking time domain symbol energy variations based on a transition metric between the end of the fully correlated duration and the beginning of a new OFDM symbol. Contrary to the autocorrelation metric [2, 3], the transition metric-based technique is able to estimate the timing offset in frequency selective channels even with strong multipath components. The received sequence of length is also divided into blocks of size where the transition metric is performed on the available blocks except the last block. The proposed transition metric is given by

One can see that the modulus operation is used outside the difference operation. This formula is especially suited for small CFOs. However, one can apply the modulus operation to each component of the formula to make the algorithm insensitive to large CFOs. An algorithm that minimizes the difference metric between the received sequence and the same sequence shifted by the symbol duration over a window duration has been presented in [4]. The authors use prolonged guard intervals assuming a quite large ISI free period and known pilot symbols to provide fine-timing synchronization. In this paper, the ratio between two consecutive averaged difference metrics is calculated which require only one ISI free symbol to estimate the true timing offset (or even no ISI free symbol when as long as the largest ratio index corresponds to the true timing offset). Moreover, it can be shown that averaging over several symbols in the same block has a negative impact on the timing offset estimate because this time smoothing will create an uncertainty on the starting position of the OFDM symbol [4].

The transition metric can be evalued for each index according to Figure 2 and ranging from to . As the transition metric is the ratio between two consecutive averaged metrics, we define the difference metric . When applying the signal model (1) to the difference metric (with ), it can be shown that

The transition metric can be rewritten as

The detection of the transition corresponding to (4) is given at delay :

while for other delays, the ratio can be considered very small especially at high SNR (ratio between two successive elements of (4)). In the case , the detection of the transition is also given at delay by

This proves that the proposed transition metric-based technique can exhibit good performance when as long as the channel has smaller power components for delays larger than the cyclic prefix duration than the first tap (which can be considered valid for an exponentially decreasing power delay profile). For low SNR, the performance of the algorithm can be improved by considering multiple ratios of the difference metric

The detection of the transition is also given at delay . As increases, the performance of the algorithm improves as long as the denominator falls in the correlated duration (ISI free part) .

### 2.2. Timing Offset Estimation Techniques for ZP-OFDM Transmission

A similar algorithm which tracks time domain symbol energy variations can be used for ZP-OFDM signals, where the autocorrelation metric also fails. We assume that the symbol duration and the zero-padding duration have been estimated blindly according to the algorithm described in [12]. We also assume that the zero-padding duration is larger than the channel impulse response . In this case, the zero-padding structure of the received OFDM signal is exploited by tracking time domain symbol energy variations (transition metric) between the end of the duration (noise only) and the beginning of a new OFDM symbol (the time domain symbol energy variations are not averaged over several symbols in the same block because this time smoothing has a negative impact on the timing offset estimate). The transition metric is performed on the available blocks except the last block and is given by

For ZP-OFDM signal, we define the difference metric . When applying the signal model (1) to the difference metric, it can also be shown that

The transition metric and the detection value at delay are also given by (5), (6). For the case , the detection value at delay is given by (7). Multiple ratios of the difference metric can also be considered to improve the performance at low SNR. The performance of the algorithm improves as long as the denominator falls in the ISI free part . In the next section, simulation results compare the autocorrelation metric and the transition metric for CP-OFDM signals. For ZP-OFDM signals, the transition metric can also be compared to a correlation metric-based on a power mask in the time domain. Indeed, knowing the symbol duration and the zero-padding duration , the power correlation metric finds the power mask of length that maximize the correlation with the received power in the time domain. Assuming that the received power has been normalized to unity, the power correlation metric is given by

with

## 3. Simulation Results

SUI channel models.

The coefficient variation shows the dispersion of the timing offset parameter from its true value normalized to the mean of the observed value at a particular SNR threshold. One can see that the autocorrelation metric-based technique [1, 3] and the MMSE metric-based technique [4] give better results than the transition metric-based technique for SUI-1 channels, with 3 dB difference at a lock-in probability of 0.8. As demonstrated in Section 3, for LOS channels the most predominant channel path is the first arrival path and the transition metric is averaged over 10 blocks while the autocorrelation metric is averaged over 10 blocks times the cyclic prefix duration. Although we have considered multiple ratios for the transition metric , the autocorrelation metric-based technique performs better than the transition metric-based technique owing to the exploitation of a larger number of symbols in the same OFDM block. Considering the knowledge of the channel delay spread , the performance of the MC1 and MMSE1 metric-based techniques [17] improves the SNR of the autocorrelation and MMSE metric-based techniques by 1 dB at a lock-in probability of 0.8. The derivative metric-based technique achieves the lowest SNR ( dB) at a lock-in probability of 0.8. However, the right figure shows that the derivative metric-based technique [18] gives the highest CV(RMSD) compared to the MC1 and MMSE1 metric-based techniques. The autocorrelation and MMSE metric-based techniques have lower CV(RMSD) than the derivative metric-based technique. The minimum CV(RMSD) is given by the transition based technique for SNR larger than 8 dB. For SUI-4 channels, the autocorrelation and MMSE metric-based techniques have similar performance compared to the transition metric-based technique for low SNRs. However, the transition metric-based technique gives significantly better results than the autocorrelation and MMSE metric-based techniques when the SNR is larger than 3 dB (we have considered multiple ratios for the transition metric ). In fact, the autocorrelation and MMSE metric-based techniques cannot exceed a lock-in probability of 0.5 even at high SNR because the most predominant channel path is usually not the first arrival path. The MC1, MMSE1 (considering the knowledge of the channel delay spread ), and derivative metric-based techniques achieve very good performance compared to the transition metric-based technique for SNRs lower than 13 dB. However, for higher SNRs, the transition metric-based technique outperforms the MC1, MMSE1, and derivative metric-based techniques which cannot exceed a lock-in probability of 0.8. The right figures show that the transition based technique gives closer estimates to the true value of the timing offset than the autocorrelation, MMSE, MC1, MMSE1, and derivative metric-based techniques at almost all SNR ranges for SUI-4 channels.

## 4. Conclusion

In this paper, we have described a technique to blindly estimate the timing offset in digital communications systems employing orthogonal frequency division multiplexing (OFDM). The starting position of the OFDM symbols has been estimated without any additional pilots using the cyclic prefix or zero-padding structure of the transmitted signal. This paper has provided an alternative coarse timing offset estimation technique to the autocorrelation-based technique which does not perform well in non-line-of-sight (NLOS) frequency selective channels as well as for ZP-OFDM transmission. The results confirm that the transition metric-based technique is able to estimate the timing offset in frequency selective channels with strong multipath components for CP-OFDM and ZP-OFDM transmission.

## Declarations

### Acknowledgment

This research work was carried out in the frame of the European FP7 UCELLS project. The scientific responsibility is assumed by its authors.

## Authors’ Affiliations

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