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11 presents the resulting dominant eigenvalues for the two cases. Example One: Decomposition of Multi-Component Signals The first example involves a multi-component signal composed of two sinusoidal patterns and a linear pattern [69]. When standard Fourier-based techniques are used to analyze data, linear trends typically result in misleading information. As a result, linear patterns are typically removed prior to analyzing signal characteristics. However, the temporal or spatial occurrence of linear trends can provide valuable information about the status of the manufacturing process.

Nonstationarities typically obscure the true nature of data and result in misleading information about the data. In addition, the nature of the nonstationarities themselves are often difficult to determine; this can provide valuable information about potential failures in the manufacturing machine. In this example, we generate signals to simulate the occurrence of faulty patterns during the normal operation of a manufacturing process. A total of m ϭ 1 … M, M ϭ 60, snapshots are assumed to be collected from the manufacturing process, each with N ϭ 100 sampled points.

The first frequency component, identified with the first eigenvector, is the component corresponding to the feed marks during the milling process. 23 Original milling profile vs. KL-reconstructed milling profile. 24 KL-filtered milling profile. during milling. Recall that, since the eigenfunctions have normalized magnitudes, the magnitude of the eigenfunctions does not represent their significance in the milling profiles. Each of these frequency components can be monitored by means of the corresponding coefficient vectors to determine their severity and potential harm to the manufacturing equipment and product.

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