# Entropy-Based TOA Estimation and SVM-Based Ranging Error Mitigation in UWB Ranging Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

#### 2.1. UWB Ranging System Models

Number of Channel Model | Channel Model Description |
---|---|

CM1 | LOS of indoor residential (7~20 m) |

CM2 | NLOS of indoor residential (7~20 m) |

CM3 | LOS of indoor office (3~28 m) |

CM4 | NLOS of indoor office (3~28 m) |

CM5 | LOS of outdoor (5~17 m) |

CM6 | NLOS of outdoor (5~17 m) |

CM7 | LOS of industrial (2~8 m) |

CM8 | NLOS of industrial (2~8 m) |

_{l}denotes the time delay of the lth cluster, ${\tau}_{k,l}$ denotes the time delay of the kth multipath component in the lth cluster, phase ${\varphi}_{k\text{,}l}$ is distributed uniformly in $[0,2\pi )$, L denotes the number of clusters.

#### 2.2. Ranging Error Analysis

## 3. Entropy-Based TOA Estimation in UWB Ranging

#### 3.1. The Theory of Entropy

**x**be a random variable with a probability mass function $P({x}_{i})$. The entropy is defined as:

#### 3.2. Proposed Entropy Based Method

**Figure 3.**RMSE as a function of the threshold factor in IEEE802.15.4a. (

**a**) CM1; (

**b**) CM2; (

**c**) CM3; (

**d**) CM4.

**E**can be obtained. As is known in information theory, a bigger entropy will be achieved when all non-repetitive elements occur with the same frequency, which means that the set is randomly distributed, like the noise regions in received signals. As to the signal region, the corresponding entropies are relatively low, because these samples can exceed the threshold in most cases.

_{max}= n

_{toa}, and the arrival time of FP can be determined by:

**E**can be obtained as shown in Figure 4c, and then the arrival time of FP can be precisely estimated. A great entropy decrease ocurrs when the signal arrives. Thus the arrival time of the FP can be determined by the decision of the sample which is followed by the great entropy decrease. In consideration of the simplicity and easiness to implement, the turning point of entropy series, as the red point shown in Figure 4c, is determined as the arrival time of the FP. In the following part, the performance of the proposed method will be evaluated with Monte Carlo simulations.

**Figure 4.**Illustration of the procedure in one realization (CM1 channel, SNR = 10 dB). (

**a**) Received signal; (

**b**) threshold crossing of nth sample; (

**c**) entropy of received signal.

#### 3.3. Ranging Performance and Discussion

**Figure 5.**Entropy of received signal of CM1 to CM4 (SNR = 10 dB). (

**a**) CM1; (

**b**) CM2; (

**c**) CM3; (

**d**) CM4.

**Figure 6.**Comparison of proposed method and conventional methods in CM1 to CM4, −10–20 dB. (

**a**) CM1; (

**b**) CM2; (

**c**) CM3; (

**d**) CM4.

## 4. SVM Regression-Based Ranging Error Mitigation

#### 4.1. Regression with Support Vector Machines

**X**is non-linear in the original space, it should be transformed into a N-dimensional feature vector spaces through a choice of a N-dimensional vector function $\varphi :{R}^{n}\to {R}^{N}$. In consideration of the decision function in Equation (26) contains the inner-product $\left(x\cdot {x}_{i}\right)$, the inner-product in N-dimensional space can be expressed as $\left(\varphi \left({x}_{i}\right)\cdot \varphi \left({x}_{j}\right)\right)$, and $\left(\varphi \left({x}_{i}\right)\cdot \varphi \left({x}_{j}\right)\right)$ ought to be calculated instead of being calculated respectively as $\varphi \left({x}_{i}\right)$ and $\varphi \left({x}_{j}\right)$. It is assumed that when there is a function $K\left({x}_{i}\cdot {x}_{j}\right)=\left(\varphi \left({x}_{i}\right)\cdot \varphi \left({x}_{j}\right)\right)$, which can transform the inner-product in N-dimensional space into the original space, the problem will be solved. Fortunately this kind of function does exist, known as the kernel function which used to calculate the inner-product in high dimension spaces, and the decision function can be written as:

**x**, the estimation of y can now be obtained. If some features can be extracted as the input vector

**x**, the error can be estimated by calculating y.

#### 4.2. Feature Selection and Mitigation Procedure

Name | Expression |
---|---|

Maximum Amplitude | ${R}_{\mathrm{max}}=\mathrm{max}\left|r(n)\right|$ |

Energy | ${E}_{r}={\displaystyle \sum _{n}{\left|r(n)\right|}^{2}}$ |

Mean Excess Delay | ${\tau}_{m}=\frac{{\displaystyle \sum _{l}{a}_{l}^{2}{\tau}_{l}}}{{\displaystyle \sum _{l}{a}_{l}^{2}}}$ |

RMS Delay Spread | ${\tau}_{RMS}=\sqrt{\frac{{\displaystyle \sum _{k}{a}_{k}^{2}{({\tau}_{k}-{\tau}_{m})}^{2}}}{{\displaystyle \sum _{k}{a}_{k}^{2}}}}$ |

Kurtosis | $K=\frac{E(r{(n)}^{4})}{{E}^{2}(r{(n)}^{2})}$$K=\frac{E(r{(n)}^{4})}{{E}^{2}(r{(n)}^{2})}$ |

Number of Significant Paths 1 (paths within X dB from the peak) | $\begin{array}{l}{N}_{1}={\displaystyle \sum _{n}\mathrm{sgn}\left(\left|r(n)\right|>threshold\right)}\text{}\\ (threshold={10}^{\frac{XdB}{20}}\mathrm{max}\left(\left|r(n)\right|\right))\end{array}$ |

Number of Significant Paths 2 (captures x% of energy in channel) | $\begin{array}{l}{N}_{2}=Index\left(\mathrm{min}\left({E}_{ce}(n)>x\%\times {E}_{r}\right)\right)\\ {E}_{ce}(n)\text{}is\text{\hspace{0.17em}}cumulative\text{}energy\text{}of\text{}r(n)\end{array}$ |

Estimated Distance | $\widehat{d}$ |

**S**consists of training samples. Every training sample is a vector consisting of eight elements (the features), as described in Table 2, along with the corresponding ranging error. Input the database

**S**into the SVM, and then a trained SVM regressor will be obtained. Another database

**S1**consisting of testing samples will be inputted into the trained SVM regressor. After that, the ranging error estimation $\widehat{\mathrm{\Delta}}$ is obtained, which can be used in mitigation. As mentioned above, the ranging estimate is positively biased, so the mitigation procedure can be simply performed by $\widehat{d}-\widehat{\mathrm{\Delta}}$.

#### 4.3. Mitigation Performance and Discussion

**S**consists of 12,400 samples (SNRs range from −10 dB to 20 dB with a step of 1 dB, 100 realizations under every channel with every SNR, 12,400 = 100 × 31 × 4), and each sample is a vector consisting of eight features along with the corresponding ranging error. The testing dataset

**S1**consists of another 12,400 samples and the predicting ranging error $\widehat{\mathrm{\Delta}}$ will be obtained from the trained SVM regressor. ε in loss-function is set as 0.003. Radial Basis Function (RBF) $K({x}_{i},{x}_{j})=\mathrm{exp}\left(-\gamma {\Vert {x}_{i}-{x}_{j}\Vert}_{2}^{2}\right)$ is used as the kernel function and γ is set as 2. For numerical reasons, the inputs are converted to the logarithmic domain prior to training. The output of the mitigation procedure is the predicting ranging error $\widehat{\mathrm{\Delta}}$. The mitigation results and the CDF of the residual ranging error are shown as Figure 9 and Figure 10.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Appendix

## Further Discussion of Threshold Selection in Entropy-Based TOA Estimation

**Figure A2.**Shape of the entropy series curve with various threshold factors (CM1, SNR = 10 dB) Threshold factor (

**a**) 0.1; (

**b**) 0.2; (

**c**) 0.3; (

**d**) 1; (

**e**)1.5; (

**f**) 2; (

**g**) 2.5; (

**h**) 3; (

**i**) 3.5; (

**j**) 10.

**Figure A3.**Shape of entropy series curve with various threshold factors (CM2, SNR = 10 dB) Threshold factor (

**a**) 0.1; (

**b**) 0.2; (

**c**)0.3; (

**d**) 1; (

**e**)1.5; (

**f**)2; (

**g**) 2.5; (

**h**) 3; (

**i**) 3.5; (

**j**) 10.

## Conflicts of Interest

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**MDPI and ACS Style**

Yin, Z.; Cui, K.; Wu, Z.; Yin, L.
Entropy-Based TOA Estimation and SVM-Based Ranging Error Mitigation in UWB Ranging Systems. *Sensors* **2015**, *15*, 11701-11724.
https://doi.org/10.3390/s150511701

**AMA Style**

Yin Z, Cui K, Wu Z, Yin L.
Entropy-Based TOA Estimation and SVM-Based Ranging Error Mitigation in UWB Ranging Systems. *Sensors*. 2015; 15(5):11701-11724.
https://doi.org/10.3390/s150511701

**Chicago/Turabian Style**

Yin, Zhendong, Kai Cui, Zhilu Wu, and Liang Yin.
2015. "Entropy-Based TOA Estimation and SVM-Based Ranging Error Mitigation in UWB Ranging Systems" *Sensors* 15, no. 5: 11701-11724.
https://doi.org/10.3390/s150511701