Practical and Useful Tips on Discrete Wavelet Transforms [sp Tips & Tricks]

The discrete wavelet transform (DWT) [1] is one of the most powerful tools for time-frequency signal analysis. Its applicability is extremely relevant in various areas of science, as exemplified in [2], with digital signal processing (DSP) as the most notable one. After teaching this topic for many years, I have noted that neither young DSP students nor experienced researchers have perceived several interesting aspects of the DWT from a practical point of view. Thus, the objective of this article is to construe such relevant aspects, providing useful tips to calculate the transform in one (1-D) and two (2-D) dimensions. The entire discussion is also valid to the discrete wavelet-packet transform (DWPT), which extends the decomposition carried out by the DWT so that a finer time-frequency analysis takes place, and also to the discrete shapelet transform (DST) [3], which extends the properties of the DWT and DWPT so that a joint time-frequency-shape signal analysis becomes possible.