# A Timing Estimation Method Based-on Skewness Analysis in Vehicular Wireless Networks

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A new cross-correlation, summing up and skewness analysis (denoted CSS) method is proposed which can be used to estimate the time offset in positioning sensors of vehicular wireless networks.
- (2)
- The CSS method provides better precision than some well-known time estimation techniques, especially in low signal to noise ratio (SNR), and multi-path environments, which is very important for vehicular wireless networks.
- (3)
- The CSS method can also be used in other wireless communications which use OFDM with repeated preamble symbols, such as IEEE 802.11a.

## 2. OFDM Preamble Structure

#### 2.1. Modulation Technique

_{FFT}) in IEEE 802.11p is 6.4 μs, which is twice that of IEEE 802.11a. The duration of GI is 1.6 μs.

#### 2.2. Preamble Symbols

_{FFT}/4 = 1.6 μs. Therefore, the duration of the ten short preamble symbols is 10 × 1.6 = 16 μs.

## 3. Related Work

#### 3.1. SC Method

^{2}, R

^{2}(d) and M(d) with L = 32 in an ideal channel with a time delay of 191 samples. For the IEEE802.11p protocol, there are 160 short preamble samples, so the first 160 − 2L = 96 samples are considered to be the GI. The timing metric M(d) forms a plateau which has a length equaling to the length of the GI (N

_{g}= 96 samples) minus the length of channel delay spread. In an AWGN channel, there will be no delay spread, so the length of the plateau is 96 samples.

#### 3.2. RMB Method

_{b}/N

_{0}(the energy per bit to noise power spectral density ratio) = 30 dB, time delay = 191 samples and L = 16.

#### 3.3. MathWorks Method

_{b}/N

_{0}= 0 dB and 30 dB are simulated in Figure 4 and the time delay is 191 samples. In these cases, the first peak should be located at ${L}_{first}=$191 + 16/2 + 1 = 200th sample. As shown in Figure 4, When E

_{b}/N

_{0}= 30 dB, the nine peaks are very sharp and the first peak appears at the 200th sample. When E

_{b}/N

_{0}= 30 dB, all nine peaks exceed the threshold, but when E

_{b}/N

_{0}= 0 dB, there are only two peaks exceeding the threshold, so the timing offset cannot be estimated accurately. In order to solve this problem, the following two steps are put forward in the following Section.

#### 3.4. Liu et al. Method

## 4. Proposed Timing Estimation Method

_{0}is the correlation duration, p(t) is the reference template, the short preamble symbols, and r(t) is the received signal.

#### 4.1. Cross-Correlation

_{b}/N

_{0}is low, the noise can affect the number of peaks that exceed a specific threshold, and it is difficult to determine the optimal threshold value, this is why the performance of the MathWorks method is relatively low.

#### 4.2. Summing-Up

_{b}/N

_{0}decreases, the timing metric M(d) of MathWorks method will become more and more hard to be identified, but the new timing metric G(t) is better. As shown in Figure 7, when E

_{b}/N

_{0}= −5 dB, the 9 peaks of M(d) are very hard to be identified, but the maximum of G(t) is very clear, and is located at 200th sample, and then can be estimated correctly.

_{b}/N

_{0}decreases more and more, the G(t) will become more difficult to identify by a fixed threshold. As shown in Figure 8, when E

_{b}/N

_{0}= −10 dB, the maximum of G(t) is very hard to locate in the 200th sample. That is to say, the maximum of G(t) is not the proper sample index. At the same time, it is very hard to determine the threshold $\xi $ to find the proper index. Therefore, in the following part of this Section, a dynamic threshold method is put forward.

#### 4.3. Skewness-Analysis and Threshold

_{b}/N

_{0}, a dynamic threshold using the skewness analysis of IEEE 802.11p short preamble is proposed. In this Section, the skewness and standard deviation (STD) of the G(t) are analyzed. Then, in order to determine the best threshold (ξ

_{best}), the relationship between estimation error and the skewness of G(t) was investigated. Finally the relationship between skewness and dynamic threshold is set up to estimate TOA.

#### 4.3.1 Statistical Characteristics of the Summing-Up Samples

- (1)
- Standard Deviation

_{b}is the number of G(t) samples and x

_{i}is the ith value of G(t).

- (2)
- Skewness

_{b}/N

_{0}increases, S will tend to increase. If there are no signals (or E

_{b}/N

_{0}is too low), S will be zero.

_{b}/N

_{0}from −10 dB to 20 dB, 2000 channel realizations were generated. The average results of STD and skewness are normalized and shown in Figure 9 which showa that the skewness increases as the E

_{b}/N

_{0}increases but the STD increases at first and then decreases as the E

_{b}/N

_{0}increases. Therefore, skewness is a monotonic function for a large range of E

_{b}/N

_{0}values, and it is more suitable for TOA estimation than STD, because it can better reflect changes in E

_{b}/N

_{0}.

#### 4.3.2. Relationship between Estimation Error, Skewness and Threshold

_{best}) based on skewness, the relationship between estimation error, skewness and threshold was investigated. 2000 ITU channel realizations with each E

_{b}/N

_{0}= {−10, 5,…, 20} dB were simulated. With each channel realizations, the thresholds of {0.05, 0.10, 0.15, 0.20,…,1.0} are compared with G(t) to find the first threshold crossing sample index, as shown in Equation (12).

_{best}.

#### 4.3.3. Relationship between Skewness and Threshold

_{best}for each value of S by using the method of least-squares where S is the x-coordinate and ξ

_{best}is the y-coordinate. The result of the fitting will generate the coefficients of the polynomial with the minimized summed square of residuals. The ith residual r

_{i}for the ith pair of (S, ξ

_{best}) is defined as:

_{i}is the best threshold and ŷ

_{i}is the fitted threshold value for the ith S, so the summed square of residuals S

_{S}is given by:

_{best}).

^{3}−0.088866S

^{2}+ 0.17697S + 0.82368

## 5. Simulation Setup and Result Analysis

#### 5.1. Simulation Setup

#### 5.1.1. System Model

_{c}= 5.9 GHz is the center frequency of the transmitted signal, and c is the speed of light.

_{g}should equal to 0, so L = 80 is used (denoted by SC-80). The timing metric of SC-80 is presented in Figure 12 for an ideal environment with a time delay of 191 samples, and this indicates that there is no plateau. The other parameters for the simulation are shown in Table 1.

Parameters | Value |
---|---|

Modulation mode | BPSK |

Number of subcarriers | 52 |

Symbol duration | 8 μs |

Guard time | 1.6 μs |

FFT period | 6.4 μs |

Preamble duration | 32 μs |

Subcarrier spacing | 0.15625 MHz |

Vehicle speed | 100 km/h |

Channel mode | ITU-A |

#### 5.1.2. Transmission Channel

Tap | Channel A | Channel B | Doppler | ||
---|---|---|---|---|---|

Relative Delay (ns) | Average Power (dB) | Relative Delay (ns) | Average Power (dB) | Spectrum | |

1 | 0 | 0.0 | 0 | –2.5 | Classic |

2 | 310 | –1.0 | 300 | 0 | Classic |

3 | 710 | –9.0 | 8900 | –12.8 | Classic |

4 | 1090 | –10.0 | 12,900 | –10.0 | Classic |

5 | 1730 | –15.0 | 17,100 | –25.2 | Classic |

6 | 2510 | –20.0 | 20,000 | –16.0 | Classic |

#### 5.1.3. Performance Metric

_{n}is the peak index corresponding to the nth actual time delay, ${\widehat{D}}_{n}$ is the peak index corresponding to the nth estimated time delay, and K is the number of simulation iterations.

#### 5.2. Performance Results and Analysis

_{b}/N

_{0}from −15 dB to 20 dB. In the remainder of this paper, six methods are considered, those are the CSS method with dynamic threshold (denoted by CSS-dynamic), the CSS method with fixed threshold = 1 (denoted by CSS-fixed), the MathWorks method, the SC-80 method, the RMB method, and the Liu et al. method (denoted by Liu).

#### 5.2.1. Estimation Error

_{b}/N

_{0}values.

_{b}/N

_{0}= −10 dB, the MAE of the CSS-dynamic method is 66 samples, while MAE are only 101 samples for the CSS-fixed method, 102 samples for the SC-80 method, 136 samples for the RMB method, 162 samples for the Liu method, and 192 samples for the MathWorks method. The MAE of the MathWorks method is the highest when E

_{b}/N

_{0}is lower than 9 dB. When E

_{b}/N

_{0}is less than −2 dB the CSS-fixed method is almost the same as the SC-80 method. As shown in Figure 14, because of the multipath, when E

_{b}/N

_{0}is greater than 13 dB, the MAEs of the CSS-dynamic, CSS-fixed, and MathWorks methods are close to 0. Nevertheless, the MAE of the SC-80 method is close to three samples and the MAE of RMB is close to four samples. For the Liu method when E

_{b}/N

_{0}is greater than 18 dB, so its MAEs drop down to two samples.

#### 5.2.2. Correct Percentage

_{b}/N

_{0}= 0 dB, the percentage of the CSS-dynamic and CSS-fixed methods having no error are nearly 59%, whereas for the same percentage with MathWorks, E

_{b}/N

_{0}should exceed 8 dB, but for the RMB, Liu and SC-80 methods the percentages are too low for all values of E

_{b}/N

_{0}. When E

_{b}/N

_{0}is greater than 14 dB, the percentages of the CSS-dynamic, CSS-fixed and MathWorks methods are close to 100%, but the others are very low which can also be explained using Figure 14. This is because when E

_{b}/N

_{0}is greater than 14 dB the MAE of the other three methods are not close to 0 samples. At the same time, for the methods of SC-80 and RMB, the percentage increases at first as E

_{b}/N

_{0}increases, and then decreases to three or four samples. The reason for the behavior can be explained using Figure 16 and Figure 17 which show the performance in a channel with no noise.

- (1)
- When E
_{b}/N_{0}is high, the multipath in the ITU channel causes the peaks to move to the right, making the timing estimation incorrect. Take the SC-80 method as an example, which is illustrated in Figure 16. The percentage will be close to 0 but will not reach 0. On the other hand, Figure 17 gives the corresponding results for the CSS method. This shows that the maximum peak with the CSS method does not move in the multipath channel, and only some smaller peaks are generated, so the percentage will be nearly equal to 100% for a high E_{b}/N_{0}. - (2)
- When E
_{b}/N_{0}is very low (close to −10 dB), the noise levels are too high, so it is too difficult to locate the peak, so the percentage is again close to 0. - (3)
- For E
_{b}/N_{0}around 0 dB, the signal energy is close to that of the noise, so it is easier to locate the peak. Now because of the randomization of noise, it’s likely that the estimated time happens to be the true time delayed.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Cui, X.; Li, J.; Wu, C.; Liu, J.-H.
A Timing Estimation Method Based-on Skewness Analysis in Vehicular Wireless Networks. *Sensors* **2015**, *15*, 28942-28959.
https://doi.org/10.3390/s151128942

**AMA Style**

Cui X, Li J, Wu C, Liu J-H.
A Timing Estimation Method Based-on Skewness Analysis in Vehicular Wireless Networks. *Sensors*. 2015; 15(11):28942-28959.
https://doi.org/10.3390/s151128942

**Chicago/Turabian Style**

Cui, Xuerong, Juan Li, Chunlei Wu, and Jian-Hang Liu.
2015. "A Timing Estimation Method Based-on Skewness Analysis in Vehicular Wireless Networks" *Sensors* 15, no. 11: 28942-28959.
https://doi.org/10.3390/s151128942