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As demonstrated above, the transform amounts to solving a familiar eigenvector equation to determine the fundamental eigenvectors in the new transform domain. Detection of linear vectors is the first step in decomposing a general function g(x, t) into its fundamental functions. Linear trends often obscure data; for example, linear trends appear falsely as a low frequency component using Fourier transform techniques. Typically, pre-processing of the data is necessary to remove the linear trends by means of linear regression techniques, as shown in the fractal-wavelet technique.

This can be viewed as the value of each snapshot projected onto the eigenfunctions in the new transform domain. It is important to note that the Karhunen-Loève transform results in M ϭ 30 eigenfunctions. The dimensionality in the new domain is determined by the rank of the covariance matrix, which is determined by the number of input snapshots. 11, only a subset (M ϭ 5) of these eigenfunctions is significant, containing the majority of the total energy. As expected, the reconstruction of the original profiles using these five eigenfunctions is perfect, as shown in Fig.

Real-time monitoring of vibration using the wigner distribution. Sound and Vibration, p. 22, May 1990. 47. Resnikoff, H. L. Wavelets and adaptive signal processing. Optical Engineering, 31(6):1229, June 1992. 48. Rioul, O. and Vetterli, M. Wavelets and signal processing. IEEE Signal Processing Magazine, p. 14, October 1991. 49. Rohrbaugh, R. A. Application of time-frequency analysis to machinery condition assessment. In The twenty-seventh Asilomar Conference on Signals, Systems, & Computers, 2:1455, 1993.

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