Figure 3.10. C02 concentration profile before and after MPCA based de-noising.
constitute an orthonormal basis. For this reason, although there are other choices for discretization, dyadic discretization is used frequently . The discretized wavelet function becomes j and k are the scale and translation parameters, respectively.
According to Mallat's multiresolution theory , any square integrable signal can be represented by successively projecting it on scaling and wavelet functions. The scaling function is shown as
The scaling coefficients a^k (Eq. 3.30) which are the low frequency content of the signal are obtained by the inner product of the signal with the scaling function The wavelet coefficients dhk (Eq.3.31) which are the high frequency content of the signal are obtained by the inner product of the original signal with the wavelet function "i.
where < •, • > indicates the inner product operation. Mallat  developed a fast pyramid algorithm for wavelet decomposition based on successive filtering and dyadic downsampling. Figure 3.11 represents this process for one scale. The input signal X is filtered by a low pass filter L(n) and a high pass filter H(n) in parallel obtaining the projection of the original signal onto wavelet function and scaling function. Dyadic downsampling is applied to the filtered signal by taking every other coefficient of the filtered output. The same procedure is repeated for the next scale to the downsampled output of L(n) shown as Al, since the low pass output includes most of the original signal content. By applying this algorithm successively, scaling coefficients a3 and wavelet coefficients dj at different scales j can be found as a3 = Laj_i, dj = Haj-i . (3.32)
Increasing the scale yields scaling coefficients that become increasingly smoother versions of the original signal. The original signal can be computed recursively by adding the wavelet coefficients at each scale and the scaling coefficients at the last scale.
Haar wavelet  is the simplest wavelet function that can be used as a baa is function to decompose the data into its scaling and wavelet coefficients. It is defined as
C 1 , 0 < t < 1/2 f(i) = -1 , 1/2 < t < 1 (3.33)
[ 0 , otherwise and its graphical representation is shown in Figure 3.12. The scaling and wavelet coefficients for the Haar wavelet are [1,1] and [1,-1], respectively. Haar wavelet transform gives better results if the process data contain jump discontinuities. Most of batch process data by nature contain such discontinuities which make Haar wavelet a suitable basis function for decomposing
Figure 3.11. Discrete wavelet decomposition.
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