# Denoising Method Based on Sparse Representation for WFT Signal

Authors: Xu Chen; Guoyu Lin; Yuxin Zhang
Journal: Journal of Sensors
 (2014)

---

## Abstract

Affected by external noise and various nature disturbances, Wheel Force Transducer (WFT) signal may be completely submerged, and the sensitivity and the reliability of measurement can be strongly decreased. In this paper, a new wavelet packet denoising method based on sparse representation is proposed to remove the noises from WFT signal. In this method, the problem of recovering the noiseless signal is converted into an optimization problem of recovering the sparsity of their wavelet package coefficients, and the wavelet package coefficients of the noiseless signals can be obtained by the augmented Lagrange optimization method. Then the denoised WFT signal can be reconstructed by wavelet packet reconstruction. The experiments on simulation signal and WFT signal show that the proposed denoising method based on sparse representation is more effective for denoising WFT signal than the soft and hard threshold denoising methods.


---

## Body

## 1. Introduction

Wheel Force Transducer (WFT) can measure the forces and torques (longitudinal forceFx, lateral force Fy, vertical force Fz, heeling moment Mx, twist torque My, and aligning torque Mz, which are shown in Figure 1) applied in the wheel dynamically, so it is an important vehicle test device for the research of body vibration, suspension, and wheel dynamics as well as the performance matching of vehicle transmission and braking system [1, 2]. Figure 2 shows the overall structure of WFT which is composed of the elastic body, reforming rims, intermediate flange, sample module, and transfer module. The wheel rims are reformed to connect with the elastic body by bolt, and the intermediate flange is used to protect bridge circuit in the elastic body from water and dirt. The sample module acquires the datum of the wheel’s force/torques and rotation angle and then sends them to the transfer module by wireless. Finally the transfer module forwards the datum to the data acquisition devices by CAN bus.Figure 1
Diagram of forces and torques applied in the wheel.Figure 2
Overall structure of WFT.Because the elastic body rotates with the rolling wheel, the outputs ofFx bridge and Fz bridge are similar to sinusoidal signals and the phase difference between them is 90 degree. Figure 3 shows the typical outputs of the two bridges. But all the signals obtained by WFT mounted on wheel are affected by external noise and various nature disturbances when the vehicle is moving. According to the previous research [3, 4], the main causes of these noises include the following (1) Vehicle braking, road roughness and suspension vibration may result in the distortion of WFT signal. (2) As the sensing element, the strain gages are connected to form a Wheatstone bridge. The output of the bridge is only a few millivolts and should be amplified hundreds of times to match the request of data collection. Meanwhile, the electromagnetic noise of the circuit and Johnson noise of the resistance will also be amplified. So even in stationary state the outputs of WFT still fluctuate irregularly. Based on these, the WFT signal is nonstationary signal with random noise which greatly degrades the measurement accuracy of WFT.Figure 3
Typical outputs ofFx bridge and Fz bridge.In recent years, the wavelet transform has been proposed as an alternative technique that has been applied to the signal processing [5–9]. Wavelet transform involves multiresolution decomposition of the measured data into wavelet coefficients, each having unique time and frequency information. Compared with Fourier transform, wavelet transform suppresses noises more effectively than Fourier Transform. Due to its merits of low entropy, multiresolution, and decorrelation, wavelet transform (WT) can be used to recover original WFT signal from severe noises [10, 11]. There exist three major forms of wavelet denoising methods. The first method, provided by Mallat and Zhong [12–15], is based on the extrema in the wavelet coefficients which reflect the propagation properties of the signal and noise across wavelet decomposition scales. The second method is called the relativity method [16–21] in which the wavelet coefficients are kept or eliminated according to the magnitude values of their relativities across the neighboring scales. The third method relies on the threshold operation [22, 23] in which the wavelet coefficients whose amplitude values are larger than a given threshold are kept or shrunk while the remaining coefficients are eliminated. Commonly referred to as “hard” or “soft” threshold method, the last one which is the most widely used has the advantages of simplicity and less computer burden. But based on the different distribution of analyzed signal and noises under different resolution and by thresholding empirical wavelet coefficients individually, threshold denoising method ignores the structural properties in the wavelet coefficients [24].In this paper, based on the compressive sensing theory and sparse theory, a new wavelet packet denoising algorithm with sparse representation (WPDSR) is presented. The proposed algorithm is efficient in recovering the noiseless signal from noisy signal and well suited for WFT signal. The method presented here has two major advantages. First, it employs wavelet packet transform (WPT) for signal decomposition. Similarly to WT, WPT also has the framework of multiresolution analysis. WPT can simultaneously decompose detail and approximation parts of the signal, but WT only decomposes approximation. Therefore, the presented method is more suitable for analyzing signal because of its good resolution regardless of high and low frequencies [25]. Second, sparse representation technique is adopted, aiming at faithfully estimating the noiseless wavelet packet coefficients without losing significant information. It should be noticed that WPDSR is quite different from the denoising method based on sparse decomposition. The latter should construct the over-complete dictionary, based on which the minimum nonzero (most sparse) atoms are solved to reconstruct the noiseless signal. However, WPDSR just selects the sparse and effective wavelet packet coefficients from the original coefficients to reconstruct the noiseless signal without constructing any dictionary.The rest of the paper is organized as follows. Compared with the hard and soft threshold methods, our proposed method is depicted in Section2. Then the details of the algorithm are elaborated upon in Section 3 by introducing the sparse representation of the wavelet packet coefficients and showing the denoising framework. In Section 4, to illustrate the efficiency and effectiveness of the proposed denoising method, we review the selection of wavelet base and decomposition level briefly and then report the experimental results on the denoising effectiveness by comparing our proposed methods with hard and soft threshold method. Finally, we draw the conclusions and discuss the future work in Section 5.


## 2. Denoising Principle

Assume that the observed signal is contaminated with additive noise as shown in (1):
(1)f(t)=s(t)+n(t),
where f(t) and s(t) are the observed and noiseless signals, respectively, and n(t) is a random noise with standard normal distribution N(0,σ2). After wavelet transform, the transformation of (1) to the wavelet domain is
(2)fjk=wjk+njk,
where fjk, wjk, and njk are the wavelet coefficients of observed signal f(t), noiseless signal s(t), and noise signal n(t), respectively, j denotes the decomposition level, and k denotes the index of the coefficient at jth level. From (2), it is clear that the wavelet coefficients of the observed signal can themselves be considered as a noisy version of the wavelet coefficients of the noiseless signal.Based on the wavelet transform, Donoho proposed the threshold denoising method [21]. The basic of the method is that the coefficients smaller than threshold are considered to be generated by noise and should be set to zero, meanwhile the coefficients larger than threshold are considered to be generated by origin signal and should be remained. Generally, there are soft and hard threshold denoising methods based on the different threshold value quantization, which are defined as follows in (3) and (4), respectively [24]:
(3)wj,k*={sign⁡(wjk)·(|wjk|-λ),|wjk|≥λ,0,|wjk|<λ,(4)wj,k*={wjk,|wjk|≥λ,0,|wjk|<λ.In these methods, threshold is usually defined as:(5)λ=σ2log⁡(N)N,
where σ=median(|wj,k|)/0.675 is the noise standard deviation and N is the length of the signal.In hard threshold method,wj,k* is not continuous, so it brings oscillating to the reconstructed signal; while in soft threshold method, though wj,k* is continuous overall, it has constant deviation with wavelet coefficients and the reconstructed signal appears so smooth that the accuracy comes down to some extent [26]. Moreover, from (3) and (4), it can be seen that in the threshold denoising method, the thresholding process is executed point by point. So the structural property of wavelet coefficients is omitted.From the principle of the hard and soft threshold methods, the hidden information can be induced as follows.(1)
The wavelet coefficient of noiseless signal is larger than that of noise.(2)
The number of wavelet coefficient of noiseless signal is less than that of observed signal.From the hidden information and many experiments, we can get an inference that if the wavelet base is selected correctly, the wavelet coefficients of the noiseless signal are sparse. However, if the signal is contaminated by noises, the sparsity of its wavelet coefficients is destroyed [27]. Based on that, we think it is possible to denoise the signal by recovering the sparsity of its wavelet coefficients. And taking the overall structural property of wavelet coefficients into consideration, it will get better denoising performance than hard and soft threshold methods.


## 3. Wavelet Packet Denoising Algorithm with Sparse Representation

The threshold denoising method uses a coordinate-wise processing scheme, which ignores the structural property of the wavelet coefficients. Instead of thresholding wavelet coefficients individually, we manage a denoising method based on sparse representation according to the compressive sensing theory. The presented denoising algorithm called wavelet packet denoising algorithm with sparse representation (WPDSR) has the following steps (shown in Figure4).Figure 4
Structure of wavelet packet denoising algorithm with sparse representation.(
1) Wavelet Packet Decomposition. Choose a proper wavelet base function and ascertain the wavelet decomposition level N, and then decompose the signal at N levels by wavelet packet. But there is no unified theory about how to choose wavelet function in the process of denoising. The universal way is to choose and compare the different wavelet functions and find out the best wavelet and its base function.(
2) Coefficients Processing. The key step in WPDSR is the recovery of the noiseless coefficients from the noisy coefficients. We propose to apply sparse representation during this key step. Idealy the wavelet coefficients of noiseless signal should have a small set of large-amplitude values (i.e., sparsity). If the signal is heavily contaminated by noise, the sparsity of the wavelet coefficients decreases [28, 29]. So we approach this problem by employing the powerful sparse representation technique, aiming at faithfully recovering the sparsity of the wavelet coefficients.(
3) Wavelet Packet Reconstruction. The wavelet packet reconstruction is realized to get the denoised signal based on its resulting coefficients.Among these three steps, the most important one is how to process the coefficients based on sparse representation. It, to some extent, directly relates to the denoising effectiveness.


### 3.1. Wavelet Packet Algorithm

The following is the decomposition algorithm of wavelet packet:(6)dj2n[t]=∑k∈Zhl-2kdj+1[l],dj2n+1[t]=∑k∈Zgl-2kdj+1[l],
where {hk} is the low-pass filter coefficient and {gk} is the high-pass filter coefficient. After passing through a low-pass filter LF and a high-pass filter HF and then being sampled down by a factor of two, the signal can be decomposed into two components, one representing the rapidly time varying features of the signal (output of the HF) and the other representing the slowly time varying features (output of the LF) at a specific scale. These two components are called the detail and the approximation coefficients, respectively. Then the decomposition process is iterated on both the detail and approximate subsequence until the detail and approximate coefficients are suited to the scales for a particular application.Next is the reconstruction algorithm of wavelet packet:(7)dj+1n=∑k∈Zhk-2ldj2n[l]+∑k∈Zgk-2ldj2n+1[l].


### 3.2. Sparse Representation

According to the theory of harmonic analysis, a one-dimensional discrete time signalf whose length is N can be expressed as linear combinations of a set of orthonormal basis (ψi:i=1,…,m) as follows
(8)f=∑i=1Nxiψi,
where ψi is a column vector and can be, for example, a wavelet basis or a Fourier basis, which depends on the application. x=[x1,x2,…,xN] is a N×1 vector which is the weighted coefficients of signal f, and xi=〈f,ψi〉. If there are only K nonzero elements in x, then x can be considered as the K-sparse representation of f.


### 3.3. Compressive Sensing

Assume that there exists a measurement matrixΦ∈RM×N (with M less than N) and a measurement y which is defined as y=Φx, where x is an unknown signal. According to compressive sensing theory [30], if x is K-sparsity, then the unknown signal x can be recovered by solving the following equation:
(9)x^=argmin⁡∥x∥0s.t.y=Φx,
where ∥·∥0 is L0 norm which means the number of the nonzero elements in x.


### 3.4. WPDSR

#### 3.4.1. Denoising Mode

The theories of sparse representation and compressive sensing can be applied to the signal denoising. For wavelet basis is a kind of orthonormal basis, if the signal is noiseless, according to Section3.2 the wavelet coefficients of the signal is K-sparsity. However according to Section 2, if the signal is contaminated by noise, the sparsity of their wavelet coefficients will decrease. This means that the nonzero elements of wavelet coefficients will be beyond K. Suppose s and f represent the noiseless signal and noisy signal and their elements satisfy (1) or (2). Let w* denote the wavelet packet coefficients of s and w denote the wavelet packet coefficients of f. According to Section 2, if we can recover w* from w, then we can recover the noiseless signal s from the noisy signal f.According to Section3.3, because w* is sparse but unknown and w can be obtained by WPT, then if we can find a suitable measurement matrix Φ, then w* can be solved by the following equation:
(10)w^*=argmin⁡∥w*∥0s.t.w=Φw*.Considering the reconstruction error, (10) can be converted to (11) by replacing the equality constraints with inequality constraints:
(11)w^*=argmin⁡∥w*∥0s.t.∥w-Φw*∥≤ε,
where ε is the error tolerance.It should be noticed that one of the preconditions of (9) or (10) is that the measurement matrix Φ should satisfy the restricted isometry constants. Being independent of any orthonormal basis approximately [31, 32], Gaussian random matrix satisfies the restricted isometry constants and can be exploited as the measurement matrix Φ.So it is clear that the problem of recovering the noiseless signals is converted to an optimization problem of recovering the sparsity of their wavelet packet coefficients. If the optimal solution is acquired, the noise can be removed from the signal.


#### 3.4.2. Solution of the Denoising Model

The equation shown in (11) cannot be solved in general because it is an NP-hard problem in combinatorial mathematics. Here we apply augmented Lagrange multiplier method to handle the problem with inequality constraint in order to avoid the ill condition of the augmented Lagrange function and the dependency of the penalty parameters of the traditional penalty function. The augmented Lagrange function for this constrained optimization problem shown in (11) can be defined as follows [33]:
(12)Lα(w*,β,γ)=∥w∥0+γ[max⁡(∥w-ϕw*∥2-ε,-β2γ)]2+β[max⁡(∥w-ϕw*∥2-ε,-β2γ)],
where γ is a penalty parameter and the corresponding Lagrange multiplier β is associated with inequality constraint. To solve this constrained optimization problem, the iterative algorithm is adopted. In each iteration, the Lagrange multiplier β and the penalty parameter γ are updated in order to improve the convergence of the algorithm. The Lagrange multiplier β is typically updated as follows:
(13)βj+1=βj+2γ[max⁡(∥w-ϕw*∥2-ε,-β2γ)].And the penalty parameterγ is increased by a constant rate until it reaches the predetermined maximum value γmax⁡ as shown below:
(14)γj+1={c×γj,ifγj<γmax⁡,γmax⁡,otherwise,
where c is the positive constant increasing rate and γmax⁡ is the maximum penalty multiplier corresponded with inequality constraint.


#### 3.4.3. Denoising Steps

The wavelet packet coefficients of the noiseless signal are correlated across levels, so it is likely that the nonzero wavelet packet coefficients are located in the similar area at different levels. Therefore, the wavelet packet coefficients at thejth level can be predicted from the coefficients at the (j+1)th level, which helps to select the initial value iteratively at each level. Based on it, the detailed steps of the proposed denoising algorithm based on sparse representation are listed in Algorithm 1, and some priori knowledge and symbol are given below:(1)
J is the wavelet packet decomposition level;(2)
wj (j=1,…,J) is the wavelet packet coefficients of noisy signal at jth level;(3)
w*j (j=1,…,J) is the reconstructed wavelet packet coefficients at jth level;(4)
Φ is the measurement matrix; as shown in Section 3.4.1 Gaussian random matrix is used;(5)
βj (j=1,…,J) is the iterative initial value at jth level; according to the correlation of wavelet packet coefficients across levels shown above, βj is set as
(15)βj={wJ,ifj=J,w*j+1,ifj<J.Algorithm 1:Flowchart of the proposed denoising algorithm.
(1) choose the proper wavelet basic;
(2) calculatewj(j=1,…,J) using WPD;
(3) letj=J;
(4) whilej <> 1
(1) letw0j=0 and w1j=βj;
(2) while∥w1j-w0j∥>ε do
w0j=w1j;
computew1j by (12);
(3) letw*j=w1j;
(4)j=j-1;
(5) reconstruct the signal byw*j;


## 3.1. Wavelet Packet Algorithm

The following is the decomposition algorithm of wavelet packet:(6)dj2n[t]=∑k∈Zhl-2kdj+1[l],dj2n+1[t]=∑k∈Zgl-2kdj+1[l],
where {hk} is the low-pass filter coefficient and {gk} is the high-pass filter coefficient. After passing through a low-pass filter LF and a high-pass filter HF and then being sampled down by a factor of two, the signal can be decomposed into two components, one representing the rapidly time varying features of the signal (output of the HF) and the other representing the slowly time varying features (output of the LF) at a specific scale. These two components are called the detail and the approximation coefficients, respectively. Then the decomposition process is iterated on both the detail and approximate subsequence until the detail and approximate coefficients are suited to the scales for a particular application.Next is the reconstruction algorithm of wavelet packet:(7)dj+1n=∑k∈Zhk-2ldj2n[l]+∑k∈Zgk-2ldj2n+1[l].


## 3.2. Sparse Representation

According to the theory of harmonic analysis, a one-dimensional discrete time signalf whose length is N can be expressed as linear combinations of a set of orthonormal basis (ψi:i=1,…,m) as follows
(8)f=∑i=1Nxiψi,
where ψi is a column vector and can be, for example, a wavelet basis or a Fourier basis, which depends on the application. x=[x1,x2,…,xN] is a N×1 vector which is the weighted coefficients of signal f, and xi=〈f,ψi〉. If there are only K nonzero elements in x, then x can be considered as the K-sparse representation of f.


## 3.3. Compressive Sensing

Assume that there exists a measurement matrixΦ∈RM×N (with M less than N) and a measurement y which is defined as y=Φx, where x is an unknown signal. According to compressive sensing theory [30], if x is K-sparsity, then the unknown signal x can be recovered by solving the following equation:
(9)x^=argmin⁡∥x∥0s.t.y=Φx,
where ∥·∥0 is L0 norm which means the number of the nonzero elements in x.


## 3.4. WPDSR

### 3.4.1. Denoising Mode

The theories of sparse representation and compressive sensing can be applied to the signal denoising. For wavelet basis is a kind of orthonormal basis, if the signal is noiseless, according to Section3.2 the wavelet coefficients of the signal is K-sparsity. However according to Section 2, if the signal is contaminated by noise, the sparsity of their wavelet coefficients will decrease. This means that the nonzero elements of wavelet coefficients will be beyond K. Suppose s and f represent the noiseless signal and noisy signal and their elements satisfy (1) or (2). Let w* denote the wavelet packet coefficients of s and w denote the wavelet packet coefficients of f. According to Section 2, if we can recover w* from w, then we can recover the noiseless signal s from the noisy signal f.According to Section3.3, because w* is sparse but unknown and w can be obtained by WPT, then if we can find a suitable measurement matrix Φ, then w* can be solved by the following equation:
(10)w^*=argmin⁡∥w*∥0s.t.w=Φw*.Considering the reconstruction error, (10) can be converted to (11) by replacing the equality constraints with inequality constraints:
(11)w^*=argmin⁡∥w*∥0s.t.∥w-Φw*∥≤ε,
where ε is the error tolerance.It should be noticed that one of the preconditions of (9) or (10) is that the measurement matrix Φ should satisfy the restricted isometry constants. Being independent of any orthonormal basis approximately [31, 32], Gaussian random matrix satisfies the restricted isometry constants and can be exploited as the measurement matrix Φ.So it is clear that the problem of recovering the noiseless signals is converted to an optimization problem of recovering the sparsity of their wavelet packet coefficients. If the optimal solution is acquired, the noise can be removed from the signal.


### 3.4.2. Solution of the Denoising Model

The equation shown in (11) cannot be solved in general because it is an NP-hard problem in combinatorial mathematics. Here we apply augmented Lagrange multiplier method to handle the problem with inequality constraint in order to avoid the ill condition of the augmented Lagrange function and the dependency of the penalty parameters of the traditional penalty function. The augmented Lagrange function for this constrained optimization problem shown in (11) can be defined as follows [33]:
(12)Lα(w*,β,γ)=∥w∥0+γ[max⁡(∥w-ϕw*∥2-ε,-β2γ)]2+β[max⁡(∥w-ϕw*∥2-ε,-β2γ)],
where γ is a penalty parameter and the corresponding Lagrange multiplier β is associated with inequality constraint. To solve this constrained optimization problem, the iterative algorithm is adopted. In each iteration, the Lagrange multiplier β and the penalty parameter γ are updated in order to improve the convergence of the algorithm. The Lagrange multiplier β is typically updated as follows:
(13)βj+1=βj+2γ[max⁡(∥w-ϕw*∥2-ε,-β2γ)].And the penalty parameterγ is increased by a constant rate until it reaches the predetermined maximum value γmax⁡ as shown below:
(14)γj+1={c×γj,ifγj<γmax⁡,γmax⁡,otherwise,
where c is the positive constant increasing rate and γmax⁡ is the maximum penalty multiplier corresponded with inequality constraint.


### 3.4.3. Denoising Steps

The wavelet packet coefficients of the noiseless signal are correlated across levels, so it is likely that the nonzero wavelet packet coefficients are located in the similar area at different levels. Therefore, the wavelet packet coefficients at thejth level can be predicted from the coefficients at the (j+1)th level, which helps to select the initial value iteratively at each level. Based on it, the detailed steps of the proposed denoising algorithm based on sparse representation are listed in Algorithm 1, and some priori knowledge and symbol are given below:(1)
J is the wavelet packet decomposition level;(2)
wj (j=1,…,J) is the wavelet packet coefficients of noisy signal at jth level;(3)
w*j (j=1,…,J) is the reconstructed wavelet packet coefficients at jth level;(4)
Φ is the measurement matrix; as shown in Section 3.4.1 Gaussian random matrix is used;(5)
βj (j=1,…,J) is the iterative initial value at jth level; according to the correlation of wavelet packet coefficients across levels shown above, βj is set as
(15)βj={wJ,ifj=J,w*j+1,ifj<J.Algorithm 1:Flowchart of the proposed denoising algorithm.
(1) choose the proper wavelet basic;
(2) calculatewj(j=1,…,J) using WPD;
(3) letj=J;
(4) whilej <> 1
(1) letw0j=0 and w1j=βj;
(2) while∥w1j-w0j∥>ε do
w0j=w1j;
computew1j by (12);
(3) letw*j=w1j;
(4)j=j-1;
(5) reconstruct the signal byw*j;


## 3.4.1. Denoising Mode

The theories of sparse representation and compressive sensing can be applied to the signal denoising. For wavelet basis is a kind of orthonormal basis, if the signal is noiseless, according to Section3.2 the wavelet coefficients of the signal is K-sparsity. However according to Section 2, if the signal is contaminated by noise, the sparsity of their wavelet coefficients will decrease. This means that the nonzero elements of wavelet coefficients will be beyond K. Suppose s and f represent the noiseless signal and noisy signal and their elements satisfy (1) or (2). Let w* denote the wavelet packet coefficients of s and w denote the wavelet packet coefficients of f. According to Section 2, if we can recover w* from w, then we can recover the noiseless signal s from the noisy signal f.According to Section3.3, because w* is sparse but unknown and w can be obtained by WPT, then if we can find a suitable measurement matrix Φ, then w* can be solved by the following equation:
(10)w^*=argmin⁡∥w*∥0s.t.w=Φw*.Considering the reconstruction error, (10) can be converted to (11) by replacing the equality constraints with inequality constraints:
(11)w^*=argmin⁡∥w*∥0s.t.∥w-Φw*∥≤ε,
where ε is the error tolerance.It should be noticed that one of the preconditions of (9) or (10) is that the measurement matrix Φ should satisfy the restricted isometry constants. Being independent of any orthonormal basis approximately [31, 32], Gaussian random matrix satisfies the restricted isometry constants and can be exploited as the measurement matrix Φ.So it is clear that the problem of recovering the noiseless signals is converted to an optimization problem of recovering the sparsity of their wavelet packet coefficients. If the optimal solution is acquired, the noise can be removed from the signal.


## 3.4.2. Solution of the Denoising Model

The equation shown in (11) cannot be solved in general because it is an NP-hard problem in combinatorial mathematics. Here we apply augmented Lagrange multiplier method to handle the problem with inequality constraint in order to avoid the ill condition of the augmented Lagrange function and the dependency of the penalty parameters of the traditional penalty function. The augmented Lagrange function for this constrained optimization problem shown in (11) can be defined as follows [33]:
(12)Lα(w*,β,γ)=∥w∥0+γ[max⁡(∥w-ϕw*∥2-ε,-β2γ)]2+β[max⁡(∥w-ϕw*∥2-ε,-β2γ)],
where γ is a penalty parameter and the corresponding Lagrange multiplier β is associated with inequality constraint. To solve this constrained optimization problem, the iterative algorithm is adopted. In each iteration, the Lagrange multiplier β and the penalty parameter γ are updated in order to improve the convergence of the algorithm. The Lagrange multiplier β is typically updated as follows:
(13)βj+1=βj+2γ[max⁡(∥w-ϕw*∥2-ε,-β2γ)].And the penalty parameterγ is increased by a constant rate until it reaches the predetermined maximum value γmax⁡ as shown below:
(14)γj+1={c×γj,ifγj<γmax⁡,γmax⁡,otherwise,
where c is the positive constant increasing rate and γmax⁡ is the maximum penalty multiplier corresponded with inequality constraint.


## 3.4.3. Denoising Steps

The wavelet packet coefficients of the noiseless signal are correlated across levels, so it is likely that the nonzero wavelet packet coefficients are located in the similar area at different levels. Therefore, the wavelet packet coefficients at thejth level can be predicted from the coefficients at the (j+1)th level, which helps to select the initial value iteratively at each level. Based on it, the detailed steps of the proposed denoising algorithm based on sparse representation are listed in Algorithm 1, and some priori knowledge and symbol are given below:(1)
J is the wavelet packet decomposition level;(2)
wj (j=1,…,J) is the wavelet packet coefficients of noisy signal at jth level;(3)
w*j (j=1,…,J) is the reconstructed wavelet packet coefficients at jth level;(4)
Φ is the measurement matrix; as shown in Section 3.4.1 Gaussian random matrix is used;(5)
βj (j=1,…,J) is the iterative initial value at jth level; according to the correlation of wavelet packet coefficients across levels shown above, βj is set as
(15)βj={wJ,ifj=J,w*j+1,ifj<J.Algorithm 1:Flowchart of the proposed denoising algorithm.
(1) choose the proper wavelet basic;
(2) calculatewj(j=1,…,J) using WPD;
(3) letj=J;
(4) whilej <> 1
(1) letw0j=0 and w1j=βj;
(2) while∥w1j-w0j∥>ε do
w0j=w1j;
computew1j by (12);
(3) letw*j=w1j;
(4)j=j-1;
(5) reconstruct the signal byw*j;


## 4. Experiments and Results

To verify the effectiveness of the presented method, the simulation WFT signal and real WFT signal are exploited, and SNR (signal-to-noise ratio) and CC (correlation coefficient) are used to evaluate the denoising performance, which are formulated as follows [34]:
(16)SNR(db)=10×log10Energy(signal)Energy(noise),CC=∑i=1N(F(i)-F-)(R(i)-R-)∑i=1N(F(i)-F-)2∑i=1N(R(i)-R-)2,
where F and R are the denoised and noise-free signals, respectively, F- and R- are the mean value of F and R, respectively, and N is the length of the signal F.SNR measures the ratio between the energy of signal and that of noise, and the smaller the SNR is, the better the denoising effect is. CC measures the similarity between the denoised signal and the noise-free signal, so the larger the CC is, the more similar they are and the better the performance is.Because the outputs ofFx bridge and Fz bridge of WFT are similar to the sinusoidal signals with random noise, we select them as the simulation signal, and the frequency is 50 Hz, 55 Hz, 60 Hz, 65 Hz, and 70 Hz, respectively, with different SNR.


### 4.1. Selection of Wavelet and Decomposition Level

The selection of an optimal wavelet and the decomposition level is two important issues of the wavelet packet decomposition.The first task to be accomplished is to choose the optimal wavelet. In this paper, a method based on minimum prominent decomposition coefficients [35] is extended to choose the optimal wavelet from a set of wavelet bases candidates including Daubechies, Symlets, and Coiflets, considering the features such as regularity and vanishing moments. For each candidate wavelet, the method first decomposes the signal into the wavelet packet domain down to a predetermined level of 4. Secondly, the average of absolute values (denoted as η) of detail wavelet packet coefficients is calculated for each decomposition level and then summated across all decomposition levels, and it is formulated as η=∑j=1Jabs(dj2n)/Nj, where j (j=1,…,J) is the decomposition level and dj2n is the nth (n=1,…,Nj) detail coefficient at jth level. η indicates how closely the candidate wavelet is describing the selected signal, and the smaller η is, the better the performance is. Then this procedure is applied to all signals by all candidates wavelets. The wavelet giving the lowest η is chosen as the best wavelet. Table 1 shows the results of the simulation signal with 5 dB SNR, and the three smallest η are displayed in bold. From Table 1, it can be seen that wavelet “db10” could achieve the best performance because of the lower η. The similar results are obtained for other signals with different SNR, which are not listed here in detail for lack of space forbids. So “db10” is chosen as the wavelet for WFT signal decomposition.Table 1
η of test signals with 5 dB SNR by different wavelet.
Wavelet
Frequency (Hz)
50
55
60
65
70
db2
1.4931
1.5620
1.5549
1.5129
1.4843
db4
1.1897
1.2464
1.2392
1.1830
1.1897
db6
1.0958
1.1454
1.1058
1.0634
1.0950
db8
1.0840
1.1089
1.0548
1.0112
1.0716
db10
1.0375
1.0904
1.0633
0.9943
1.0374
sym2
1.4931
1.5620
1.5549
1.5129
1.4843
sym4
1.1938
1.2521
1.2455
1.1881
1.1904
sym6
1.0991
1.1534
1.1357
1.0785
1.1159
sym8
1.0558
1.0788
1.0553
1.0394
1.1072
sym10
1.0229
1.0896
1.0634
1.0364
1.1336
coif2
1.2084
1.2639
1.2353
1.1895
1.2048
coif3
1.1405
1.2096
1.1836
1.1142
1.1283
coif4
1.1243
1.1508
1.1498
1.1009
1.1614
coif5
1.1372
1.1499
1.0808
1.0635
1.1697Besides the optimal choice of the wavelet, how to select the best decomposition level is also important. In the above experiments the decomposition level is preset (J=4). To asses the performance of the wavelet at other decomposition levels, all the parameters above are kept unchanged except the decomposition level, then we can see how the decomposition level influences the denoising effect of the proposed method. Figures 5 and 6 show the influence of decomposition level on SNR and CC for the simulation signal with 5 dB SNR, from which it can be seen that SNR and CC could hardly increase when the decomposition level gets beyond 5. The similar results are obtained for signals with different SNR. As we know the less the wavelet decomposition level is, the less the computational operation of signal denoising will be. So according to the experimental results, we choose the optimal decomposition scale (J=5) during the following denoising experiments.Figure 5
Influence of decomposition level on SNR.Figure 6
Influence of decomposition level on CC.


### 4.2. Comparison of Different Denoising Methods for Simulation Signal

To evaluate the denoising performance of WPDSR, our proposed method is compared with the threshold method including soft and hard. In the experiments, the parameters of the proposed WPDSR are set empirically:(1)
the error toleranceε is set as 1;(2)
the maximum penalty multiplierγmax⁡ is set as 10;(3)
the positive constantc is set as 0.1.Table2 shows SNR and CC of the denoised signal by the different denoising methods. From Table 2, it can be seen that the presented denoising method leads to the largest SNR and CC after denoising. This means that WPDSR outperforms the other two methods for both noise reduction and signal restoration.Table 2
Comparison of SNR and CC by using the three different denoising methods.
SNR before denoising
Frequency
SNR after denoising (dB)
CC after denoising
Hard threshold
Soft threshold
The proposed  method
Hard threshold
Soft threshold
The proposed  method
10 dB
50
11.8975
12.5750
13.3241
0.9901
0.9908
0.9912
55
12.1520
12.8504
13.4851
0.9798
0.9813
0.9845
60
11.7591
12.4382
13.5607
0.9732
0.9735
0.9756
65
11.6483
12.0538
13.2084
0.9789
0.9800
0.9812
70
11.5608
12.4672
13.6375
0.9816
0.9833
0.9863
5 dB
50
8.0593
8.8428
9.0593
0.9843
0.9854
0.9893
55
8.3879
8.8534
9.7245
0.9620
0.9632
0.9689
60
8.3189
9.0547
9.8915
0.9609
0.9637
0.9664
65
8.0125
8.8504
9.7581
0.9636
0.9658
0.9692
70
7.5483
8.1592
9.2461
0.9709
0.9724
0.9759
0 dB
50
6.1360
6.8254
7.5268
0.9738
0.9755
0.9780
55
5.8967
6.5249
7.0423
0.9586
0.9614
0.9631
60
5.1586
5.4860
6.7808
0.9694
0.9721
0.9743
65
6.1404
6.5328
7.0524
0.9638
0.9645
0.9672
70
5.1453
5.8918
6.9245
0.9652
0.9674
0.9695
−5 dB
50
3.5524
3.9805
4.5246
0.9631
0.9647
0.9663
55
2.8593
3.4395
4.1250
0.9646
0.9651
0.9684
60
2.7695
3.0256
3.8925
0.9659
0.9683
0.9705
65
3.1369
3.5284
4.2386
0.9571
0.9588
0.9594
70
3.1245
3.8925
4.8425
0.9609
0.9613
0.9635
−10 dB
50
1.2534
1.8562
2.5851
0.9476
0.9489
0.9498
55
1.5962
2.1256
2.8408
0.9549
0.9553
0.9563
60
1.2564
1.7592
2.3614
0.9423
0.9428
0.9446
65
0.9547
1.5410
2.0596
0.9362
0.9365
0.9372
70
1.5814
1.9854
2.5468
0.9438
0.9443
0.9456As an example, the denoising results of the simulation noisy signal with SNR = 5 dB and frequency = 50 Hz using the proposed method as well as the other mentioned methods are shown in Figure7. It is obvious that the curve denoised by the proposed method is much smoother than the curve denoised by the other two methods. And the mean of squared error (MSE) from hard threshold, soft threshold, and our method are 0.5376, 0.4880, and 0.4184, respectively, so the absolute error from the proposed method is much smaller than that from the other two methods.Comparison of the denoising performance for simulation WFT signal with SNR = 5 dB.
(a)
Noise-free signal
(b)
Noisy signal with SNR = 5 dB
(c)
Result of hard threshold
(d)
Absolute error from hard threshold
(e)
Result of soft threshold
(f)
Absolute error from soft threshold
(g)
Result of the proposed method
(h)
Absolute error from proposed method


### 4.3. Comparison of Different Denoising Methods for WFT Signal

To verify the effectiveness of the proposed method in the real application, the real WFT signal is denoised by the above methods. Figure8 shows the denoising results of the Fx bridge output by the proposed method, soft threshold method, and hard threshold method. It can be seen that the WPDSR obtains superior visual quality.Figure 8
Comparison of the denoising performance for real WFT signal.To further show the performance of the proposed method acting on the WFT signal processing, we use the original and the denoised bridge outputs to obtain the longitudinal forceFx and twist torque My applied to the wheel. Figure 9 shows the comparison of the original and the denoised force/torque in an acceleration course. There exist two gear shifts in this experiment. During shifting T1-T2 the driver presses the clutch, and then the engine and the gearbox detach. Then the driver presses the accelerator pedal in T2-T3. After that shifting T3-T4 follows. In this acceleration process the output torque of the gearbox causes Fx and My to change correspondingly.Figure 9
F
x and My in an acceleration process.From Figures8 and 9, it is shown that the proposed denoising method can effectively remove the noise from the output of WFT and the force/torque applied to the wheel can be measured effectively by using the proposed method.


## 4.1. Selection of Wavelet and Decomposition Level

The selection of an optimal wavelet and the decomposition level is two important issues of the wavelet packet decomposition.The first task to be accomplished is to choose the optimal wavelet. In this paper, a method based on minimum prominent decomposition coefficients [35] is extended to choose the optimal wavelet from a set of wavelet bases candidates including Daubechies, Symlets, and Coiflets, considering the features such as regularity and vanishing moments. For each candidate wavelet, the method first decomposes the signal into the wavelet packet domain down to a predetermined level of 4. Secondly, the average of absolute values (denoted as η) of detail wavelet packet coefficients is calculated for each decomposition level and then summated across all decomposition levels, and it is formulated as η=∑j=1Jabs(dj2n)/Nj, where j (j=1,…,J) is the decomposition level and dj2n is the nth (n=1,…,Nj) detail coefficient at jth level. η indicates how closely the candidate wavelet is describing the selected signal, and the smaller η is, the better the performance is. Then this procedure is applied to all signals by all candidates wavelets. The wavelet giving the lowest η is chosen as the best wavelet. Table 1 shows the results of the simulation signal with 5 dB SNR, and the three smallest η are displayed in bold. From Table 1, it can be seen that wavelet “db10” could achieve the best performance because of the lower η. The similar results are obtained for other signals with different SNR, which are not listed here in detail for lack of space forbids. So “db10” is chosen as the wavelet for WFT signal decomposition.Table 1
η of test signals with 5 dB SNR by different wavelet.
Wavelet
Frequency (Hz)
50
55
60
65
70
db2
1.4931
1.5620
1.5549
1.5129
1.4843
db4
1.1897
1.2464
1.2392
1.1830
1.1897
db6
1.0958
1.1454
1.1058
1.0634
1.0950
db8
1.0840
1.1089
1.0548
1.0112
1.0716
db10
1.0375
1.0904
1.0633
0.9943
1.0374
sym2
1.4931
1.5620
1.5549
1.5129
1.4843
sym4
1.1938
1.2521
1.2455
1.1881
1.1904
sym6
1.0991
1.1534
1.1357
1.0785
1.1159
sym8
1.0558
1.0788
1.0553
1.0394
1.1072
sym10
1.0229
1.0896
1.0634
1.0364
1.1336
coif2
1.2084
1.2639
1.2353
1.1895
1.2048
coif3
1.1405
1.2096
1.1836
1.1142
1.1283
coif4
1.1243
1.1508
1.1498
1.1009
1.1614
coif5
1.1372
1.1499
1.0808
1.0635
1.1697Besides the optimal choice of the wavelet, how to select the best decomposition level is also important. In the above experiments the decomposition level is preset (J=4). To asses the performance of the wavelet at other decomposition levels, all the parameters above are kept unchanged except the decomposition level, then we can see how the decomposition level influences the denoising effect of the proposed method. Figures 5 and 6 show the influence of decomposition level on SNR and CC for the simulation signal with 5 dB SNR, from which it can be seen that SNR and CC could hardly increase when the decomposition level gets beyond 5. The similar results are obtained for signals with different SNR. As we know the less the wavelet decomposition level is, the less the computational operation of signal denoising will be. So according to the experimental results, we choose the optimal decomposition scale (J=5) during the following denoising experiments.Figure 5
Influence of decomposition level on SNR.Figure 6
Influence of decomposition level on CC.


## 4.2. Comparison of Different Denoising Methods for Simulation Signal

To evaluate the denoising performance of WPDSR, our proposed method is compared with the threshold method including soft and hard. In the experiments, the parameters of the proposed WPDSR are set empirically:(1)
the error toleranceε is set as 1;(2)
the maximum penalty multiplierγmax⁡ is set as 10;(3)
the positive constantc is set as 0.1.Table2 shows SNR and CC of the denoised signal by the different denoising methods. From Table 2, it can be seen that the presented denoising method leads to the largest SNR and CC after denoising. This means that WPDSR outperforms the other two methods for both noise reduction and signal restoration.Table 2
Comparison of SNR and CC by using the three different denoising methods.
SNR before denoising
Frequency
SNR after denoising (dB)
CC after denoising
Hard threshold
Soft threshold
The proposed  method
Hard threshold
Soft threshold
The proposed  method
10 dB
50
11.8975
12.5750
13.3241
0.9901
0.9908
0.9912
55
12.1520
12.8504
13.4851
0.9798
0.9813
0.9845
60
11.7591
12.4382
13.5607
0.9732
0.9735
0.9756
65
11.6483
12.0538
13.2084
0.9789
0.9800
0.9812
70
11.5608
12.4672
13.6375
0.9816
0.9833
0.9863
5 dB
50
8.0593
8.8428
9.0593
0.9843
0.9854
0.9893
55
8.3879
8.8534
9.7245
0.9620
0.9632
0.9689
60
8.3189
9.0547
9.8915
0.9609
0.9637
0.9664
65
8.0125
8.8504
9.7581
0.9636
0.9658
0.9692
70
7.5483
8.1592
9.2461
0.9709
0.9724
0.9759
0 dB
50
6.1360
6.8254
7.5268
0.9738
0.9755
0.9780
55
5.8967
6.5249
7.0423
0.9586
0.9614
0.9631
60
5.1586
5.4860
6.7808
0.9694
0.9721
0.9743
65
6.1404
6.5328
7.0524
0.9638
0.9645
0.9672
70
5.1453
5.8918
6.9245
0.9652
0.9674
0.9695
−5 dB
50
3.5524
3.9805
4.5246
0.9631
0.9647
0.9663
55
2.8593
3.4395
4.1250
0.9646
0.9651
0.9684
60
2.7695
3.0256
3.8925
0.9659
0.9683
0.9705
65
3.1369
3.5284
4.2386
0.9571
0.9588
0.9594
70
3.1245
3.8925
4.8425
0.9609
0.9613
0.9635
−10 dB
50
1.2534
1.8562
2.5851
0.9476
0.9489
0.9498
55
1.5962
2.1256
2.8408
0.9549
0.9553
0.9563
60
1.2564
1.7592
2.3614
0.9423
0.9428
0.9446
65
0.9547
1.5410
2.0596
0.9362
0.9365
0.9372
70
1.5814
1.9854
2.5468
0.9438
0.9443
0.9456As an example, the denoising results of the simulation noisy signal with SNR = 5 dB and frequency = 50 Hz using the proposed method as well as the other mentioned methods are shown in Figure7. It is obvious that the curve denoised by the proposed method is much smoother than the curve denoised by the other two methods. And the mean of squared error (MSE) from hard threshold, soft threshold, and our method are 0.5376, 0.4880, and 0.4184, respectively, so the absolute error from the proposed method is much smaller than that from the other two methods.Comparison of the denoising performance for simulation WFT signal with SNR = 5 dB.
(a)
Noise-free signal
(b)
Noisy signal with SNR = 5 dB
(c)
Result of hard threshold
(d)
Absolute error from hard threshold
(e)
Result of soft threshold
(f)
Absolute error from soft threshold
(g)
Result of the proposed method
(h)
Absolute error from proposed method


## 4.3. Comparison of Different Denoising Methods for WFT Signal

To verify the effectiveness of the proposed method in the real application, the real WFT signal is denoised by the above methods. Figure8 shows the denoising results of the Fx bridge output by the proposed method, soft threshold method, and hard threshold method. It can be seen that the WPDSR obtains superior visual quality.Figure 8
Comparison of the denoising performance for real WFT signal.To further show the performance of the proposed method acting on the WFT signal processing, we use the original and the denoised bridge outputs to obtain the longitudinal forceFx and twist torque My applied to the wheel. Figure 9 shows the comparison of the original and the denoised force/torque in an acceleration course. There exist two gear shifts in this experiment. During shifting T1-T2 the driver presses the clutch, and then the engine and the gearbox detach. Then the driver presses the accelerator pedal in T2-T3. After that shifting T3-T4 follows. In this acceleration process the output torque of the gearbox causes Fx and My to change correspondingly.Figure 9
F
x and My in an acceleration process.From Figures8 and 9, it is shown that the proposed denoising method can effectively remove the noise from the output of WFT and the force/torque applied to the wheel can be measured effectively by using the proposed method.


## 5. Conclusions

Although the standard wavelet threshold denoising method is simple and practical, it does not work well for signals with a low SNR; besides, it uses coordinate-wise processing without taking the structural information of the wavelet coefficients into account. In this paper, the new denoising method based on wavelet packet transform and sparse representation is used to overcome these problems. It can estimate the noise-free wavelet packet coefficients by solving an optimization problem and is suitable for signals with a low SNR. Experiments were conducted on different simulation signals, which were corrupted by various noise levels, to assess the performance of the proposed algorithm in comparison with soft and hard threshold denoising methods. The analysis of the results indicated that the proposed method outperforms the universal hard threshold and soft threshold methods and it could be competent for denoising WFT signal with heavy noise. Through the relevant graphs and data, the superiority of the presented method is illustrated clearly. Furthermore, the proposed method also can be used in other applications and should acquire satisfactory results.



---

*Source: 145870-2014-02-13.xml*