Discrete wavelet transform for signals with finite-length

2 Discrete wavelet transform for signals with finite-length 

Prior to this chapter, we have considered multi-resolution analysis for the space of square integrable functions, L (R). However, in most practical applications such as spectral analysis, the data are obtained as a sequence of 

An example finite-length NMR spectrum is shown in Fig. 4. The example chosen here consists of N = 512 discrete sample values, a subset of a raw soft lobophytum compactum coral spectrum containing isolobophytolide. 

The DWT will require the computation of non-existent values as these values reside outside the interval. The main difficulty is not with the computation of the coefficients away from the interval boundaries, but with those that are at, or close to, the boundaries. The problem that arises is that of what to do with the boundaries of the interval as these, if not taken into account, can lead to artifacts in the filtered data. Similar problems also arise in the context of the discrete Fourier transform. 

There are many different approaches to handling boundaries. One possibility is to extend the sequence to an infinite one in some suitable way and then apply the standard DWT to the extension. Example extension techniques include extension by constant-padding, by periodicity, by reflection and by 

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