Steering wheel angle filtering based on wavelet transform

Steering wheel angle filtering based on wavelettransformChen WeiSchool of Instrument Science and EngineeringSoutheast UniversityNanjing, Chinahenuchenwei.20@Song Xiang，Yu Jiahe，Li Xu School of Instrument Science and EngineeringSoutheast UniversityNanjing, Chinalixu.mail@Abstract—Traditional filtering methods can't eliminate the effect of interfering signals on steering wheel angle and extract useful signal effectively.Regarding to this, this paper presents a new method based on wavelets transform.According to the Mallat algorithm,the method first decomposes the signal for seven layers by selecting sym8 wavelet.Aiming at every layer detail signal after wavelets decomposing, the soft-threshold filtering method is employed to eliminate noises, then the inverse wavelet transform is used to reconstruct the filtered signal.The experimental results show that this method can effectively eliminate all kinds of noises and achieve a good performance.Keywords-steering wheel angle; wavelet transform; filterI.I NTRODUCTIONElectronic stability program（ESP）is an important progress of Automotive Active Safety and also is another innovative security technology after the car seat belts[1].It not only contains all the features of ABS, but also can control the posture of a vehicle and stabilize it, which will greatly reduce the risk of skidding and traffic accidents when the vehicle loses its stability. Because the steering angle signal is an important input parameter of ESP, its accuracy is directly related to the driving safety.In the process of driving, steering wheel angle sensor is susceptible to vibration, strong electromagnetism and other external factors, leading to the steering wheel angle signal contains more interference. In the analysis of this signal, we first need to preprocess the signal and extract the useful signal. Signal filtering methods commonly used are Fourier transform of the signal in time - frequency domain, Kalman filtering and neural network filtering. But for the steering wheel angle signal, because of the strong randomness of interfering signals and their frequency components are complex, the method of Fourier transform of the signal in time - frequency domain can not effectively distinguish between signal and noises when their frequency bands overlap[2]. Kalman filtering requires not only accurate models of state and observation of a system, but also the exact system noise and observation noise. However, these are difficult to get accurately in practice, resulting in low accuracy of the Kalman filtering[3]. Neural network filtering requires more priori data and a longer training time [4], which restrict the application of neural network filter.Thus，there are a lot of inconvenience when the traditional methods above are applied in practice. Therefore, we developed a filtering method to process the steering wheel angle signal, which owns the advantages of convenience and high- precision, overcoming the limitation of the methods above and becoming an urgent requirement in the area of automotive testing. Based on such consideration and according to the characteristics of interfering signals, we use wavelet transform to realize the steering wheel angle signal filtering in this passage.II.WAVELET THEORYA.Basic Theory of Wavelet TransformPresume )(tf is a square-integral function, ）（tψ is a basic wavelet and the Fourier transform. If ()ψωsatisfies the admissibility condition below[5]:2()RC dψψωωω=<∞⎰ (1) Then, we can get the wavelet transform of )(tf:1*2,(,),()()a bt bWf a b f a f t dtaψψ+∞--∞-==⎰ (2) In which *()t baψ-is the complex conjugate of ()t baψ-,a is a transformation scale factor, and b is a position factor. As the inverse transform of wavelet transform, the rebuild function of )(tf is:20,)(),(1)(adadbt b a Wf C t f b a ⎰⎰∞+∞∞-=ψψ(3) Where )(1)(,a b t at b a -=ψψ, R b a ∈>,0.B. Multiresolution analysisMultiresolution analysis is one of the most important concepts of wavelet transform. It researches the multiresolutional representation of a function from the aspect of the function, decomposing a signal into low-frequency components and high-frequency components of different resolutions. According to the Mallat algorithm[6], we assume 0,0k C as the original signal, LO_D 、HI_D 、LO_R 、HI_R aslow-pass decomposition filter bank, high-pass decomposition filter bank, low-pass reconstruction filter bank and high-pass reconstruction filter bank, thus the decomposition of 0,0k C is:)_(,,11D LO C D C M M k M k M *=++ (4) )_(,,11D HI C D d M M k M k M *=++ (5)Signal reconstruction algorithm is:R HI Ud R LO UC C M M M k M k M k M _)(_)(11,1,1,*+*=++++(6)Where M k M C , represents the low-frequency coefficient ofthe Mth layer,M k M d , represents the high-frequencycoefficient of the Mth layer, ‘*’represents convolution, ‘—’represents reverse, …D ‟ represents the down sampling by 2, ‘U ’represents the up sampling by 2.With the help of the method above, we can achieve rapid wavelet decomposition and reconstruction of a signal, thus Mallat algorithm is similar to the position of fast Fourier transform (FFT) in the classical Fourier transform.III. F ILTERING METHOD BASED ON WAVELET TRANSFORM This paper selects wavelet shrinkage threshold method to filter the steering wheel angle signal, which is one of the basic methods of wavelet transform filtering. This method can not only eliminate the noise, but also retain the characteristic peak point of the original signal with a good filtering effect, so it has been widely used. Using the wavelet shrinkage threshold value method to filter the signal may generally be divided into the following three steps [7]:(1) Wavelet decomposition of a signal. It needs to choose a wavelet base and determine the number of decomposition layers number N, and use the Mallat algorithm to decompose the signal into N layers.(2) Nonlinear thresholding of high-frequency coefficients. It needs to choose a threshold to process all the high-frequencycoefficients after N layer decomposition with soft-thresholdmethod.(3) Inverse wavelet transformation and the reconstruction of filtered signal. In order to rebuild the filtered signal, this step needs to do the inverse wavelet transform with the Nth layer low-frequency coefficient and the high-frequency coefficients after thresholding from the first layer to the Nth layer . A. Wavelet decomposition of a signalAccording to the characteristics of Steering wheel angle, We choose wavelet sym8 as the most suitable wavelet base. To be able to separate the low-frequency signal and high-frequency noises effectively, and in order to ensure the import singularities of steering wheel angle signal are not lost, the decomposing layers number N is determined to 7 after experimental analysis.First, we consider 0,0k C as the original steering wheelangle signal. The first layer low-frequency coefficient 1,1k C and high-frequency coefficient 1,1k d are computed bytaking down sample by 2 of the convolution of 0,0k C withD LO _ and D HI _ respectively; Similarly, the second layer low-frequency coefficient 2,2k C and high-frequencycoefficient 2,2k d are computed by taking down sample by 2 ofthe convolution of1,1k C with D LO _ and D HI _respectively. Repeating the similar process 7 times, the signal 0,0k C can be decomposed into 7 layers. Figure 1 shows thespecific process, where M =0,1…,6, ‘2↓’represents the down sample by 2, LO_D and HI_D can be determined by the function wfilters in MATLAB.1+1,kM M d +1+1,kM M C +,kC MFig.1 Wavelet decomposing structure of signalB. Nonlinear thresholding of high-frequency coefficientsAfter the high-frequency coefficients in different decomposing layers of steering wheel angle are extracted, threshold processing method is performed, which is the most critical step of the wavelet shrinkage threshold filtering. In order to maintain the whole waveform of the filtered signal unchanged, we use the soft threshold method to process the high-frequency coefficients, leaving all the low-frequency coefficients untouched. The soft threshold method is defined as[8]:Supported by Specialized Research Fund for the Doctoral Program of Higher Education (New Teacher) (No.200802861061) and Nature Science Foundation of Jiangsu Province of China (No. BK2010239),,,,,,0M M MM M M M k M k M k M k M k M k d d d d d d λλλλλ-≥⎧⎪⎪=<⎨⎪+≤-⎪⎩ (7) Where 1,2,...,7,1,2,...,k k l m ==, ,MM k d is the high-frequency coefficients after thresholding. The threshold is chosen through the adaptive threshold selection method based on Stein unbiased likelihood estimation. In practice, We can easily get the threshold λ via the function thselect in MATLAB.C. Inverse wavelet transformation and the reconstruction of filtered signalWith the low-frequency coefficient layer 7th and the high-frequency coefficients after thresholding from the first layer to the 7th layer, We carry out the inverse wavelet transformation to rebuild the filtered signal according to Mallat algorithm.The process of rebuild is shown in Figure 2.+1,M M k C + 1,M M kd ++ ,MM kCFig.2 The figure of the inverse wavelet transformWhere M =6, 5,…, 0，77,k C =77,k C ,‘2↑’represents the up sample by 2, 00,k Cis the filtered signal of steeringwheel angle. Similarly, LO_R and HI_R can be determined bythe function wfilters in MATLAB.IV. E XPERIMENTAL RESULTSThe steering wheel angle signal measured in a experiment is shown in Figure 3and the sampling interval is 20 milliseconds. From the figure, we can see the original signal contains a lot of gross errors and noises.Fig.3 Original steering wheel signalBecause of the gross errors, We firstly choose the Medianfiltering method to eliminate them. Then, steering wheel angle signal is decomposed into 7 layers with wavelet base sym8 .All thecoefficients are shown in Figure 4, in which a1,a2,…, a7 are low-frequency coefficients in different layer and d1,d2,…, d7 are the high- frequency coefficients in different layer.Fig.4 Low and high frequency coefficients of signal by 7-layerdecomposingFiltered signal of steering wheel angle and the originalsignal are shown in Figure 5, and the red line shows the filtered signal, the blue line for the original signal. We can find that this method not only almost completely eliminates the noises, but also ensures the important feature information of the signal not lost, achieving a quite favorable effect.Fig.5 The comparison of original signal and filtered signal by wavelettransformTo further illustrate the effect of wavelet transform filtering, we compare the results of Butterworth filtering and wavelet transform filtering. Butterworth filtering is a filtering method which is based on Fourier transform of the signal in time – frequency domain. The results of comparing shown in Figure 6, in which red line shows the Butterworth filtered signal, the blue line shows the signal which was filtered by the method mentioned in this passage. We could find that Butterworth filtering cannot completely filter out the noise interference, and there is obvious shock at the initial stage, hence the effect of filtering is much worse than the effect of wavelet filtering.Fig.6 The comparison of filter results of Butterworth and wavelettransformA CKNOWLEDGMENTIn this paper, due to the characteristic of steering wheel angle signal and interference signals, we use the wavelet transform to filter the steering wheel angle signal. According to the Mallat algorithm, the method firstly decomposes the signal into seven layers by selecting wavelet base sym8, then process the decomposed high-frequency coefficients obeying the rules of soft thresholding, and finally get the filtered signal by reconstruction. In conclusion, this method can remove the noises of steering wheel angle signal effectively, and achieved good filtering performance, providing a good guarantee for ESP, which has a high application value.R EFERENCES[1]Masami,Akio okada, “Analysia of vehicle stability control (VSC)'seffectiveness from accident data”, 18th ESV Conference, 2003,Nagoya ,Japan, pp.541-547.[2]Fadi Al-Badour, L.Cheded, and M.Sunar,“Non-stationary vibrationsignal analysis of rotating machinery via time-frequency and wavelettechniques”, 2010 10th International Conference on InformationSciences Signal Processing and their Applications (ISSPA),，2010，pp:21-24.[3] Giaouris, D.,Finch, J.W.,“Electric drives estimation and denoisingschemes based on wavelet transforms”，IEEE Industrial Electronics,IECON 2006 - 32nd Annual Conference on，2006，pp:1223-1228[4]Jiang Zongli, Introduction to Artificial Neural Networks. 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