Industry 4.1. Группа авторов

at the second level of resolution with the length of coefficients being N/4. In other words, a recursive relationship exists between the approximation and detail coefficients at successive levels of resolution. Therefore, the general decomposition form of wavelet coefficients of length N at the j level can be expressed as follows:

      (2.2)

      (2.3)

where

       jscale parameters j = 1, 2, …, L;

       Lnumber of decomposition levels;

       N data length of a discrete signal;

       n translation parameter ;

        nth approximation coefficient at level j;

        nth detail coefficient at level j;

       g[k] DWT low‐pass filters; and

       h[k] DWT high‐pass filters.

Figure 2.16 illustrates an example of three‐level DWT decomposition. A signal with available bandwidth 2,000 Hz is decomposed into the sets and , where {A₁, D₁}, {A₂, D₂}, and {A₃, D₃} represent the approximate (gray (A_i) blocks) and detail (white (D_i) blocks) coefficients in levels 1–3, respectively.

Figure 2.16 Three‐level decomposition tree of the DWT.

       Thresholding

      The performance of wavelet de‐noising depends on the determination of two factors, the threshold value λ and the threshold function. In this section, the wavelet de‐noising algorithm adopts the soft thresholding method to adaptively filter the specific spectrums of the noisy signals to obtain modified wavelet coefficients . Let at level j.

      The soft thresholding method sets every wavelet coefficient c_j[n] to zero if |c_j[n]| is less than or equal to a chosen threshold λ; otherwise, the threshold is subtracted from any c_j[n]. Then, all modified coefficients = at level j can be defined as:

      (2.4)

      (2.5)

where

       cj[n] nth wavelet coefficients at level j;

        nth modified wavelet coefficients at level j;

       λ threshold of cj[n]; and

       MAD(cL − 1[n]) mean absolute deviation of cL − 1[n].

       Reconstruction

      The de‐noised sensor data can be reconstructed from all modified wavelet coefficients at the level of resolution L. The reconstruction process is in the opposite direction to the decomposition process; that is, the process proceeds with level j=L, L−1, …, till 1. As such, intermediate modified approximation coefficients at each level of resolution, j−1 can be recovered by up‐sampling modified approximation and detail coefficients at the level of resolution j. Finally, is obtained with j = 1 as follows:

      (2.6)

      2.3.3 Feature Extraction

      Feature extraction [6–8] is the process to generate a smaller linear or nonlinear combination set to represent the original high‐dimensional data set. Thus, the de‐noised sensing signals need to be transformed into meaningful signal features (SFs), which can adequately describe the physical meaning of the signal and maintain relevant information of the machining operations [6]. However, monitoring machining conditions based on a single SF is not enough [7]. To properly describe machining precisions, a set of multiple SFs is required to provide further insight into coordination [8].

      This section introduces feature extraction approaches that are commonly used in the time, frequency, and time–frequency domains. Feature selection is a process used to define a small and concise feature subset through the removal of redundant features from a feature set and it is introduced in Section 2.3.3.1(A.1). In addition, an Autoencoder (AEN) as a popular ANN‐based feature extraction method [11, 12] is introduced in Section 2.3.3.4.

      2.3.3.1 Time Domain

      SFs extracted from time‐domain signals are very intuitive to understand how the signals change in the past or at a specific time. Statistical SFs and correlation‐based SFs are two main methods used to describe these changes in practical manufacturing applications.

      In signal processing, SFs of the statistical description can essentially identify changes from the shape of the waveform; while SFs of cross‐correlation and autocorrelation, based on the Pearson product‐moment correlation coefficient [13], investigate similarity relationship between time‐varying signals. The feature extraction methods of statistical SFs, cross‐correlation SFs, and autocorrelation SFs are presented as follows.

       Statistical SFs

      If the machining parameters (such as feed rate, spindle speed, depth of cut, etc.) are fixed and the precision after machining is within specifications, then the machining operations are a kind of a quasi‐static condition [8]. Under this condition, Yang et al. [8] summarized that the nine most common SFs for various types of sensor signals are derived from all the elements of .

For the de‐noised time‐domain signals that contain N elements (data length = N), these nine SFs are represented and defined as in Table 2.3,