Threshold selection method for UWB TOA estimation based on wavelet decomposition and kurtosis analysis
- Juan Li^{1},
- Xuerong Cui^{1}Email authorView ORCID ID profile,
- Houbing Song^{2},
- Zhongwei Li^{1} and
- Jianhang Liu^{1}
https://doi.org/10.1186/s13638-017-0990-4
© The Author(s). 2017
Received: 27 August 2017
Accepted: 15 November 2017
Published: 28 November 2017
Abstract
In wireless sensor networks, ranging or positioning via ultra-wideband (UWB) has caused widespread research interests where the non-coherent energy detection (ED) method with low sampling rate and low complexity is widely studied. However, the traditional energy detection methods only analyze the signal energy in the time domain, so their error is relatively large. In this paper, the simulation results show that most of the signal energy concentrates in the low-frequency band, so a novel threshold selection method for time of arrival (TOA) estimation is proposed that analyzes the signals in both time domain and frequency domain. In this method, the received signal is decomposed by “db6” wavelet and the kurtosis of energy blocks of the low-frequency wavelet coefficients (K _{c}) is analyzed. At last, the mapping relationship between K _{c} and the normalized threshold for TOA estimation is created using polynomial fitting with degree 3. The simulation results show that the TOA estimation error of the proposed method is significantly less than the method without wavelet decomposition.
Keywords
1 Introduction
In recent years, with the development of wireless communication technologies [1, 2], the applications of wireless sensor networks are more and more widely used. The important premise of these applications is to obtain the precise position of the targets [3–5]. Therefore, the precise positioning of targets becomes the key problem to be solved urgently.
Ultra-wideband (UWB) is a new wireless communication technology [6–9], which is widely used in many fields, such as indoor short-distance communication, high-speed wireless local area networks (WLAN) [10], security monitoring, ranging, positioning, and so on. UWB is the most promising technology for indoor positioning and tracking [3]. Compared with other short-range communication technology, UWB has many advantages for short-range communication: first, UWB can provide up to GHz bandwidth; second, UWB can provide data rates of hundreds of megabits per second or even gigabits per second, so it is an ideal technology for wireless communication in wireless sensor networks [11, 12]; third, continuous transmission carrier is not needed in UWB communication, and the intermittent pulse is used to transmit data, which make short pulse duration, low power consumption, and high multipath resolution.
Wireless positioning methods can be divided into fingerprint positioning algorithm based on received signal strength indicator (RSSI) [13], geometric or range positioning algorithm based on the range, time of arrival (TOA) [14, 15], time difference of arrival (TDOA) [16], or angle of arrival (AOA) [17], and some fusion positioning methods together with inertial measurement units (IMUs) [5]. The signal fingerprint positioning algorithm [4, 18] is based on the mapping relationship between some parameters obtained from the received signal and the position information of the target node. The range-based positioning algorithm with round-trip-time (RTT) measurements [19] is often used to meet the requirement of high-precision positioning because of its high time delay resolution. However, obtaining the accurate ranging estimation is a very challenging problem due to the effects of thermal noise, multi-path fading, non-line of sight, and other factors in wireless transmission channel. For example, in non-line of sight (NLOS) environment, the range estimation based on TOA will typically be positively biased [3].
In recent years, ranging algorithms for UWB systems have been extensively studied. There are three main approaches. The first approach is matched filter (MF) based on coherent algorithm with high sampling rate [20]. The second is machine learning method based on some selected channel parameters. In [3], a ranging method based on kernel principal component is proposed, where the channel parameters are projected onto a high-dimensional nonlinear orthogonal space, and then the subset from these projections is used for ranging. The third is energy detection (ED) algorithm based on non-coherent receiver with low sampling rate and low complexity [8, 9, 21]. The matched filter approach is not applicable in many practical situations due to the high complexity and high hardware requirement. As opposed to the complex matched filter method, the energy detection is a non-coherent method for TOA estimation which consists of a square-law device, an integrator, a sampler, and a decision mechanism. The TOA value is estimated by the first signal sample exceeding a specific threshold which is deemed as the start of the received signal. Thus, the energy detection method is applicable in many cases because it is a method with low complexity and low sampling rate. In this method, how to select an appropriate threshold is a key issue. In literature [9], a threshold selection method based on kurtosis analysis of energy blocks was proposed, and in literature [21], a threshold selection method based on skewness analysis of energy blocks was also put forward. However, the TOA estimation accuracy of these methods is not very high because these parameters such as kurtosis of the received signals can only reflect statistical characteristics in time domain and ignore all the characteristics in frequency domain. At the same time, the received signals will be affected by the random noise, so the large randomness will result in the poor precision of kurtosis in time domain.
In this paper, the simulations of UWB signal spectrum under different signal to noise ratio (SNR) find out that the UWB signal energy is mainly distributed in the low-frequency band, while the energy of the white Gauss noise is evenly distributed over the entire frequency band. The wavelet transform is equivalent to two channel filter banks with low-pass and high-pass characteristics, so using the wavelet transform, most of the signal energy concentrates in the low-frequency coefficients, while the energy of white Gauss noise distributes in the coefficients of all frequency bands. Thus, in this paper, after the wavelet transform used in the received signal, the high-frequency coefficients are discarded, and only the low-frequency coefficients are used as the received signal energy to improve the accuracy of ranging. In this way, the white Gauss noise interference in the signal can be reduced effectively.
The remainder of this paper is organized as follows. In Section 2, the UWB ranging system model is presented. Section 3 discusses some ranging estimation algorithms based on traditional energy detection method in the time domain. Section 4 introduces the proposed threshold selection method for TOA estimation based on wavelet decomposition, energy detection, and kurtosis analysis. In Section 5, the simulation results and the performance discussion are presented, and Section 6 concludes the paper.
2 UWB ranging system models
2.1 Pulse waveform
In UWB ranging system, short pulses with sharp rising and falling edges are usually used as the transmitting signal to get shorter pulse duration (nanosecond level) or higher time delay resolution.
2.2 Modulation method
2.3 IEEE 802.15.4a channel model
Classification of channel models specified in IEEE 802.15.4a standard
Channel models | Channel description |
---|---|
CM1 | Residential environment with LOS communication (7–20 m) |
CM2 | Residential environment with NLOS communication (7–20 m) |
CM3 | Office environment with LOS communication (3–28 m) |
CM4 | Office environment with NLOS communication (3–28 m) |
CM5 | Outdoor environment with LOS communication (5–17 m) |
CM6 | Outdoor environment with NLOS communication (5–17 m) |
CM7 | Industrial environment with LOS communication (2–8 m) |
CM8 | Industrial environment with NLOS communication (2–8 m) |
CM9 | Open outdoor environment with NLOS communication (e.g., farm, snow-covered area) |
Channel parameters of CM1 and CM2
Channel parameters | CM1 (LOS) | CM2 (NLOS) |
---|---|---|
Frequency dependency of the channel | 1.12 | 1.53 |
Standard deviation of the log-normal shadowing of entire impulse response | 2.22 | 3.51 |
Mean number of clusters | 3 | 3.5 |
Cluster arrival rate | 0.047 | 0.12 |
Two ray arrival rates (rays per nanosecond) for mixture of Poisson processes | 1.54, 0.15 | 1.77, 0.15 |
Cluster decay factor (time constant, ns) | 22.61 | 26.27 |
2.4 TOA estimation error
3 Energy detection method
4 The proposed threshold selection method
4.1 Low-frequency wavelet coefficient
Wavelet transform has the characteristics of multi-resolution analysis, and the received signal is divided into low-frequency part and high-frequency part using the wavelet decomposition. In the next layer process of wavelet decomposition, the low-frequency part is further divided into low-frequency part and high-frequency part, but the high-frequency part is no longer decomposed.
- (1)
Either in time domain or in frequency domain, when SNR decreases, the amplitude of the noise increases obviously comparing with the amplitude of the signal.
- (2)
The energy of signal is mainly distributed in the low frequency band, while the energy of white Gauss noise is evenly distributed over the entire frequency band.
Because the wavelet decomposition is equivalent to two channel filter banks with low-pass and high-pass characteristics, after the wavelet decomposition, most energy of the received UWB signal concentrates in the coefficients of low-frequency bands, while the energy of the white Gauss noise distributes in the coefficients of different frequency bands. Therefore, in the energy detection receiver of this paper, after the wavelet transform, the high-frequency wavelet coefficients are discarded, and only the low-frequency wavelet coefficient is regarded as the received energy to improve the accuracy of ranging. In this way, the white Gauss noise interference in the signal can be reduced effectively.
4.2 Kurtosis of the energy blocks of low-frequency coefficients
Figures 7 and 8 illustrate that the characteristics of K _{s} and K _{c} with respect to different values of E _{ b }/N _{0} is almost the same for the two different channels. Furthermore, Figs. 7 and 8 show that K _{s} and K _{c} increase as E _{ b }/N _{0} increases, but K _{c} changes more rapidly. Therefore, K _{c} can better reflect the changes with different values of E _{ b }/N _{0} and it is more suitable for threshold selection.
4.3 Relationship between K_{c} and threshold
- (1)
Generate a large amount of receive signals for each channel model under different E _{b}/N _{0} values of {5 dB, 6 dB, …, 25 dB}.
- (2)
Calculate the average MAE values with respect to different normalized thresholds ξ _{norm} of {0.1, 0.2, …, 1.0} for each K _{c} value. With each channel realization, the thresholds are compared with Z _{ c }[n] to find the first sample index crossing the normalized threshold, as shown in Eq. (10). In the simulation, because of the random noise signal, there are different MAE values with respect to one specific normalized threshold and one specific E _{b}/N _{0} value, so the average MAE is calculated with respect to one specific normalized threshold. Moreover, because K _{c} is a real value, K _{c} is rounded to the nearest integer value.
- (3)
Select the normalized threshold with the lowest MAE as the best threshold ξ _{best} with respect to specific K _{c} for each channel model.
- (4)
A polynomial with degree 3 is fitted to the best threshold ξ _{best} for each value of K _{c} by using the method of least-squares where K _{c} is the x-coordinate and ξ _{best} is the y-coordinate. To obtain the coefficient estimates, the least-squares method minimizes the summed square of residuals. The ith residual r _{ i } for the ith pair of (K _{c}, ξ _{best}) is defined as
where ξ _{c1} and ξ _{c1} are the optimal thresholds for CM1 and CM2.
5 Performance results and discussion
5.1 TOA estimation error
- (1)
In CM1 and CM2, the MAE of the K _{c}-based and K _{s}-based threshold selection methods decrease as the E _{ b }/N _{0} increase. But when E _{b}/N _{0} is between 9 and 20 dB, the proposed method is better than that of the K _{s}-based method proposed in [9]. It can be found that in CM1 channel, when E _{b}/N _{0} is 14 dB, the MAE of K _{c}-based method is about 5 ns better than that of the K _{s}-based method. In CM2 channel, when E _{b}/N _{0} is 15 dB, MAE of K _{c}-based method is about 5.87 ns better than that of the K _{s}-based method.
- (2)
When E _{b}/N _{0} is less than 8 dB, the MAE of the two algorithms are very near. This is because now the energy of the noise is very high, which will affect the decomposed low-frequency coefficients seriously, so the advantage of the proposed method is little compared with the K _{s}-based method.
- (3)
When E _{b}/N _{0} is higher than 21 dB, the MAE of the two methods is almost the same. This is because now the energy of the noise is very low compared with the signal energy, which will not affect the two methods.
5.2 Computational complexity
Running time of different methods
Operation (2100 iterations) | Energy block | Kurtosis analysis | TOA estimation | Total | ||||
---|---|---|---|---|---|---|---|---|
Methods | K _{c} | K _{s} | K _{c} | K _{s} | K _{c} | K _{s} | K _{c} | K _{s} |
Time (s) | 22.581 | 18.219 | 0.451 | 0.390 | 0.241 | 0.248 | 23.582 | 19.188 |
6 Conclusions
In the UWB ranging system, the energy detection method based on non-coherent receiver is widely used. However, because of the interference, such as multi-path fading, thermal noise, inter-symbol interference, and reflection interference, the precision of ranging is not very high. The simulation results show that in the UWB signal decomposed by wavelet transform, most of the signal energy is concentrated in the low-frequency band, while the energy of noise is evenly distributed over the entire frequency band. Therefore, this paper proposes a new threshold selection method based on wavelet decomposition and kurtosis analysis, that is, the UWB signal is decomposed by wavelet transform, the threshold is obtained based on the kurtosis of energy blocks of low-frequency wavelet coefficient, and then the first energy block exceeding the threshold is treated as the TOA of signal. The simulation results show that the new method can obviously improve the precision of TOA estimation.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant No. 61671482, the Nature Science Foundation of Shandong Province No. ZR2014FL014, and the Fundamental Research Funds for the Central Universities Nos. 16CX02046A, 17CX02042A, and 14CX02212A.
Authors’ contributions
The authors have contributed jointly to all parts on the preparation of this manuscript, and all authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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