*2.1. Sparse Representation and Filter Banks*

Generally, the fault vibration signal (observation signal) *y* of rotating machinery such as rolling bearing, which contains three parts below: the fault transient impulses *x*, the systematic natural signal *f* and the additive noise *w*, i.e.,

$$y = \mathbf{x} + f + w \tag{1}$$

The core work of the fault diagnosis is to extract the fault transient impulses *x* from the noisy observation *y*. Assume that fault transient impulses and the systematic natural signal are estimated, we have,

$$
\stackrel{\wedge}{\mathbf{x}} \approx \mathbf{x}, \stackrel{\wedge}{f} \approx f \tag{2}
$$

Given an estimation ∧*x* of *x*, we can estimate *f* as follows,

$$\stackrel{\wedge}{f} = LPF(y - \stackrel{\wedge}{\mathbf{x}}) \tag{3}$$

where *LPF* is a specified low-pass filter. Substituting ∧ *f* = *LPF*(*y* − ∧*x*) into ∧*f* ≈ *f* , we have,

$$LPF(y-\overset{\wedge}{\dot{x}}) \approx f\tag{4}$$

Substituting *y* = *x* + *f* + *w* into *LPF*(*y* − ∧*x*) ≈ *f* , we have,

$$LPF(y-\overset{\wedge}{\mathbf{x}}) \approx y-\mathbf{x}-w \tag{5}$$

Substituting ∧*x* ≈ *x* into *LPF*(*y* − ∧*x*) ≈ *y* − *x* − *w*, we have,

$$LPF(y - \stackrel{\wedge}{\mathbf{x}}) \approx y - \stackrel{\wedge}{\mathbf{x}} - w \Leftrightarrow (I - LPF)(y - \stackrel{\wedge}{\mathbf{x}}) \approx w \tag{6}$$

Defining *HPF* = *I* − *LPF* = *H*, which is a high-pass filter, thus we have,

$$HPF(y - \overset{\wedge}{\dot{\mathbf{x}}}) \approx w \tag{7}$$

On the other hand, it should be noted that Equation (1) is a highly underdetermined equation, i.e., ill-posed or *N* − *P* hard problem [32,33], and has an infinite set of solutions because the number of unknowns is greater than the number of equations. Usually, convex optimization techniques are commonly used to estimate transient components from the observation signal, based on the aforementioned work [24–31], the estimation of *x* can be formulated as the optimization problem, i.e.,

$$\overset{\triangle}{\mathbf{x}} = \operatorname\*{argmin}\_{\mathbf{x}} \left\{ \frac{1}{2} \|H(\mathbf{y} - \mathbf{x})\|\_{2}^{2} + \lambda \|\mathbf{D}\mathbf{x}\|\_{1} \right\} \tag{8}$$

where *H* is a specified high-pass filter, i.e., the *HPF* in Equation (7), *λ* represents regularization term parameter, *D* is a matrix defined as *D* = ⎡ ⎢⎢⎢⎣ −1, 1 ... ... −1, 1 ⎤ ⎥⎥⎥⎦, which determines the

sparsity degree of the approximating value of *x*. Commonly, if *x* is a sparse component, i.e., most of the values in *x* tend to zero, the correspondingly optimization problem in Equation (8) can be estimated by L1-norm fused lasso optimization (LFLO) model [24–31], i.e.,

$$\hat{\mathbf{x}} = \operatorname\*{argmin}\_{\mathbf{x}} \left\{ \frac{1}{2} \|H(\mathbf{y} - \mathbf{x})\|\_{2}^{2} + \lambda\_{0} \|\mathbf{x}\|\_{1} + \lambda\_{1} \|\mathbf{D}\mathbf{x}\|\_{1} \right\} \tag{9}$$

where *λ*0 and *λ*1 are regularization parameters. The solution of LFLO model can be obtained by the soft-threshold method [34] and total variation de-noising (TVD) algorithm [35–37], we have,

$$\mathbf{x} = \text{soft}(\text{tvd}(y, \lambda\_2), \lambda\_1) \tag{10}$$

where function tvd(·, ·) is the TVD algorithm and the soft-threshold model is given as follows,

$$\text{soft}(\mathbf{x}, \lambda) = \begin{cases} |\mathbf{x} - \lambda \frac{\mathbf{x}}{|\mathbf{x}|}|\mathbf{x}| > \lambda \\ 0, & |\mathbf{x}| \le \lambda \end{cases} \tag{11}$$

In addition, the high-pass filter *H* described above could be formulated as follows [36–38],

$$H = \mathbf{A}^{-1}\mathbf{B} \tag{12}$$

where *A* and *B* are Toeplitz matrices.
