From the Fourier transform to the wavelet transform: the success of time-frequency analysis

Wavelets, like Fourier decomposition, are stars in signal analysis. This discipline tries to study and understand signals using mathematics. A signal is a broad word. It can be a function dependent on a variable, such as an ECG which is a time-dependent electrical intensity. Or, a signal can be an image which is a color function depending on two variables indicating the position of the pixel. In the history of mathematics, success first smiled on Fourier decomposition, wavelets being in some ways its descendants. Indeed, since Antiquity we have used the trigonometric functions “cosine” and “sine” to solve geometric problems. In the ۱۸th century, we began to understand their usefulness outside of geometry. And, more precisely, around ۱۸۱۰, Joseph Fourier stated that any function can be decomposed as an infinite sum of these two functions by means of simple calculations. Over the years, mathematicians have appropriated this result and its applications have multiplied. Its success is relentless. Wavelets were born much later, in the ۱۹۸۰s, from reflections on Fourier decomposition and thanks to the advent of digital technology. Yves Meyer, one of the founding fathers of wavelets, received the prestigious Abel Prize in ۲۰۱۷. In this paper, we briefly study the wavelet transform, from the Fourier transform to what we are faced as wavelet transform.