Spectral and Time–Frequency Analyses and Signal Processing

Even if the Fourier analysis dates back before the Fourier’s work and even if the different Fourier analysis developments have been done after him, Fourier is an icon whose influence is fundamental still today. In 1822, Fourier1 in his work entitled “Analytical Theory of Heat”, explained the way in which the linear equations of partial derivatives could describe the propagation of Heat in a simple form. In brief, it stated that any periodic function can be expressed as a sum of sinusoids, i.e. sines and cosine of different frequencies: This is the Fourier series. Then, by extension it is said that any periodic curve, even if it is discontinuous, can be decomposed into a sum of smooth curves. Consequently, an irregular or jagged curve and the sum of sinusoids are representations of the same thing, but one of them has an empirical nature and the other is the result of an algebraic decomposition. The decomposition method uses the amplitude of sinusoids by assigning to them coefficients and uses the phases. It is important to underline that we can reconstruct the function from the Fourier series (or Fourier transform) without loss of information.