Abstract:

Our goal in this paper is to present a new mathematics tool called Wavelet Transform or Time-Frequency Analysis. We try to show both the pros and cons of this new tool against the secular Fourier Transform.

At the end of this paper the reader will be able to understand the fundamental ideas of Wavelet Transform and how this idea has spread among academics.

To motivate the reader, the first sections talks about a model that we have assumed as being easy to be implemented. In our model we have imagined any musician or any music lover that inputs the signal (music) in the computer and the computer runs the pre-processing by WT and a Neural Network recognises the pattern and the output will be the score. This kind of work already exists but it works with Fourier Transform.

We wrote a section that talk about the history of WT, its mainly mentor and other famous mathematicians who have been working with this modern tool.

Finally, we will show the necessary calculus to understand WT, but in a simple way with no hard equations. Further on we show, in a general sense, the Uncertainty Principle, which governs the size of the window. In particular, it will be observed in this arficie that the time-frequency, window of any Short-Time Fourier Transform is rigid, and is not very effective for detecting signals with high frequencies against WT, which presents a variable window and is capable of detecting any kind of frequency even in non-stationary signals or better, in non-continuous function.