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Figure 3.12. Haar wavelet.

batch process data. A noisy process signal (CO2 evolution rate) was decomposed in four scales using Haar wavelet in Figure 3.13. The low frequency component (dominating nonlinear dynamics) of the original signal (uppermost figure) is found in the scaling coefficients at the last scale whereas the high frequency components that are mostly comprised of noise appear at wavelet coefficients at different scales.

Wavelets are widely used to remove the noise from signals by extracting the low frequency content and removing the high frequency content above a threshold value. The denoised signal is obtained by reconstructing the signal by applying inverse wavelet transform to the scaling and thresholded wavelet coefficients. Thresholding, a crucial step of wavelet denoising, can be applied either as soft or hard thresholding. Hard thresholding (Eq. 3.34) removes the wavelet coefficients smaller than the threshold and replaces them with zero:

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