Method for the extraction of shock signal features based on the upper limit of density integral

2.1. Amplitude density for monotonic functions

Assume that a signal $f\left(t\right)$ is defined in the interval $\left[a,b\right]$ and satisfies $f\mathrm{\text{'}}\left(t\right)\ge$ 0, $t\in \left[a,b\right]$ where $A=\mathrm{m}\mathrm{i}\mathrm{n}\left\{f\left(t\right)|t\in \left[a,b\right]\right\}$, $B=\mathrm{m}\mathrm{a}\mathrm{x}\left\{f\left(t\right)|t\in \left[a,b\right]\right\}$. The functional image of $f\left(t\right)$ is presented in Fig. 1.

Fig. 1Functional image of ft

Then, Eq. (1) is established by considering points $\left[t,y\right]$ and $\left(t+\mathrm{\Delta }t,y+\mathrm{\Delta }y\right)$ in $f\left(t\right)$:

We define a function $r\left(y\right)$ that satisfies Eq. (2):

Both ends of Eq. (2) are divided by $\mathrm{\Delta }y$. The limit is taken as Eq. (3). Then, Eq. (3) becomes equal to Eq. (4):

$r\left(y\right)$ is defined as the amplitude density function of $f\left(t\right)$ in the interval $\left[a,b\right]$. $r\left(y\right)$ reflects the probability of $y$ that $f\left(t\right)$ takes in the interval $\left[a,b\right]$. $r\left(y\right)\ge$ 0 exists, and Eq. (5) is true. The opposite number of the right end in Eq. (4) when $f\mathrm{\text{'}}\left(t\right)\le$ 0 is taken. Eq. (6) is thus established:

2.2. Amplitude density for piecewise monotonic functions

Assume that a signal $f\left(t\right)$ that is defined in the interval $\left[a,b\right]$ can be divided into$n$amounts of monotonic intervals and the derived function $f\mathrm{\text{'}}\left(t\right)$, $t\in \left[a,b\right]$ exists where:

The functional image of $f\left(t\right)$ is illustrated in Fig. 2. The piecewise function in every monotonic interval is assumed to be expressed as $f_i\left(t\right)$, $i=$ 1, 2, ..., $n$.

Fig. 2Function ft that is defined in finite monotonic intervals

Considering the amplitude density of $f\left(t\right)=y$, the piecewise function $f_k\left(t\right)$, $k\in C$ and $C\subseteq \left\{\mathrm{1,2},...,n\right\}$ is assumed to take the value$y$. On the basis of our analysis in section 2.1, we show that the amplitude density $r\left(y\right)$, where $f\left(t\right)=y$, should be established as (7):

The interval of the definition of the piecewise inverse function $f_i^{-1}\left(y\right)$, $i=$ 1, 2, ..., $n$, is extended. The result of the extension is plotted in Fig. 3. For every extreme point or end point in $f\left(t\right)$, the minimum points are extended to $y=A$, and the maximum points are extended to $y=B$. The piecewise inverse function that has been extended is still expressed as $f_i^{-1}\left(y\right)$, $y\in \left[A,B\right]$, $i=$ 1, 2, …, $n$ to facilitate expression in this paper. Therefore, Eq. (7) is equal to Eq. (8). We can still consider $r\left(y\right)$ as the amplitude density function of $f\left(t\right)$:

Fig. 3Extended piecewise inverse functions

2.3. ULDI

Both ends of Eq. (8) are subjected twice to the definite integral operation in the interval $\left[A,y_2\right]$ and $\left[A,y\right]$, $y_2\in \left[A,B\right]$, of which the result is Eq. (9):

let:

then, Eq. (9) is equal to Eq. (10):

We can ignore the influence of inverse functions without differential coefficients at the extreme points or end points of $f\left(t\right)$ when we calculate Eq. (10). Therefore, the value of $f_i^{-1}\left(y\right)$, $y\in \left[A,B\right]$ and $i=$ 1, 2, ..., $n$ at the local area of these points is readjusted to ensure that ${\left(f_i^{-1}\right)}^{\mathrm{\text{'}}}\left(y\right)$ exists, whereas the value of $\mathrm{\Phi }\left(y\right)$ remains unchanged. We can handle this operation in this manner because the value of the integral is unchanged by standard integration and changing the integrand on a set of zero measure does not change the value of the integral (measure theory).

The area of the shaded regions in Fig. 4 is regarded as $S\left(y\right)$. Eq. (11) is true. Furthermore, we regard the curve in Fig. 5 as $P_y\left(t\right)$, which is formed by $f\left(t\right)$ and the constant$y$:

$P_y\left(t\right)$ is easily obtained when$y$is determined. On the other hand, $S\left(y\right)$ can be expressed as Eq. (12):

We can obtain Eq. (13) by combining Eqs. (10), (11) and (12). Then, Eqs. (14) and (15) are true:

$\mathrm{\Phi }\left(y\right)$, $\mathrm{\Phi }\mathrm{\text{'}}\left(y\right)$and $r\left(y\right)$ are uniquely determined by each other when $f\left(t\right)$ is determined. We designate $\mathrm{\Phi }\mathrm{\text{'}}\left(y\right)$ as the ULDI of a signal. The data obtained in engineering applications are discrete. The error of the first-order differential operation is low in the calculation of the differential operation of Eq. (14). When solving the second-order differential operation in Eq. (15), the local of the first-order differential section steps into a jump. The second-order differential is therefore not consistent with the actual value if the differential region is small. On the other hand, if the differential region is large, the accuracy of the calculation will decrease. Signal features will be extracted by utilising Eq. (14) to avoid this problem.

Fig. 4Area Sy of the shaded regions

Fig. 5Curve of Pyt

3.1. ULDI of shock signals

Only the actual effective time period of the original signal is intercepted for analysis to avoid the interference of invalid signals with effective signals. The shock signals in the time domain are plotted in Fig. 8. The ULDIs of each signal are shown in Fig. 9. The area [–1000, 0]×[0, 0.3] of Fig. 9 is presented in Fig. 10 to properly illustrate the ULDI of each signal. From Fig. 10, we can conclude that under the same density integral, the amplitudes that correspond to different working modes are different.

Fig. 8Signals in the time domain

a)

b)

c)

3.2. Signal features

For a continuous shock signal $f\left(t\right)$, the ULDI $\mathrm{\Phi }\mathrm{\text{'}}\left(y\right)$ is a strictly monotonically increasing function of $y$. Therefore, for a determined value of $\mathrm{\Phi }\mathrm{\text{'}}\left(y\right)$, we can obtain the amplitude $y$ that corresponds to $\mathrm{\Phi }\mathrm{\text{'}}\left(y\right)$ through bisection, which is a method that is commonly used to identify zero points. The amplitude$y$that corresponds to a determined value of ${\left({\mathrm{\Phi }}_{i,j}^m\right)}^{\mathrm{\text{'}}}\left(y\right)$ can be denoted as $y_{i,j}^m$, $m\in \left\{\mathrm{1,2},3\right\}$, $i\in \left\{\mathrm{1,2},3...\right\}$, $j\in \left\{\mathrm{1,2},3...\right\}$, where $m$ means the index of signals generated by different patterns, $i$ means the index of signals generated by same pattern and $j$ means the index of amplitudes that correspond a signal.

Fig. 9Density integrals of signals

Fig. 10Area [–1000,0]×[0,0.3] of Fig. 9

Three signals that correspond to the different firing pattern with respect to the different values of $\mathrm{\Phi }\mathrm{\text{'}}\left(y\right)$ are shown in Table 2, which means $m\in \left\{\mathrm{1,2},3\right\}$, $i\in \left\{1\right\}$, $j\in \left\{\mathrm{1,2},...,8\right\}$ and $\mathrm{\Phi }\mathrm{\text{'}}\left(y\right)\in \left\{\mathrm{0.1,0.2},...,0.8\right\}$. Therefore, we can consider $F_i^m={\left(y_{i,1}^m,y_{i,2}^m,...,y_{i,N}^m\right)}^T$, $N=$ 8, as the feature of a signal. From Table 2, we can infer that the difference between the different work patterns first decreases and then increases as the density integral increases. This behaviour reflects the similarities and differences between each firing pattern.

3.3. Applying extracted features to a SVM

We further verified the effect of our proposed method and selected the data from one sensor for verification (there are 8 sensors in the actual test of sampling signals). We sampled 14 effective groups of data that correspond to each work pattern for application into an SVM, and limited by the service life of the automatic gun mechanism. Only small samples can be sampled. The steps of feature extraction and verification are as follows.

Step 1: on the base of original signals, intercept the signal segment when the latch sheet is working.

Step 2: the instantaneous frequency function of IMF2 is obtained by EMD decomposition of the signal that is intercepted.

Step 3: on the base of instantaneous frequency function of IMF2, we extracted the features that are similar to the features presented in Section 3.2.

Above what we analysed, there are 42 samples (features) for classification. At the process of training and testing SVM, we chose to leave one out cross validation. The results of the training are presented in Table 3, and the results of testing are provided in Table 4 [23-25].

The dimensionality of the features $F_i^m={\left(y_{i,1}^m,y_{i,2}^m,...,y_{i,N}^m\right)}^T$ used for the SVM is 11, and $\mathrm{\Phi }\mathrm{\text{'}}\left(y\right)\in \left\{\mathrm{0.1,0.15,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.85,0.9}\right\}\text{,}$$i\in \left\{\mathrm{1,2},...,42\right\}\text{,}$$j\in \left\{\mathrm{1,2},...,11\right\}\text{,}$$N=$ 11. We selected the radial basis function (RBF) as the kernel of the SVM. The kernel is expressed as:

$\sigma >$ 0, of which $\sigma$ is the width and was selected as 5.

We validated the effectiveness of the proposed method by comparing its performance with that of a statistics-based method. We selected the same sample data and SVM parameters for classification. In general, we can represent the instantaneous frequency data that correspond to a shock signal as $\left\{x_1,x_2,...,x_N\right\}$. Table 5 shows the representation of each component of the feature extracted from signals on the basis of mathematical statistics. We can regard this feature as ${\left(\stackrel{-}{x},x_p,x_{rms},x_r,K,C,I,S\right)}^T$. The dimensionality of the features is 8. Training results are shown in Table 6, and testing results are shown in Table 7.

From Tables 3 and 4, we can infer that the SVM conflates features that correspond to normal patterns with those that correspond to fault I patterns. Although the classification results are acceptable, accuracy must be improved by combining other types of features. As inferred from Tables 4 and 7, the classification performance of the proposed method is superior to that of the traditional method that constructs features on the basis of mathematical statistics. Tables 3, 4, 6 and 7 show that the generalisation of the former method is better than that of the latter method.