## 11.1. Theoretical background

### 11.1.1. *Fourier transform shortcomings: interpretation difficulties*

The Fourier transform is involved in a wide range of signal processing applications. However, despite its rigorous mathematical definition it may lead to some physical interpretation difficulties. It is, for example, clear from its definition that the evaluation of a spectrum value *X(v)* requires the knowledge of all signal history, from – ∞ to + ∞. In the same way, the signal value at any time *t* is expressed by the inverse Fourier transform as a superposition of an infinite number of complex exponentials, i.e. eternal waves perfectly delocalized in the time domain. If this mathematical point of view is able to reveal interesting signal properties in many cases, it may also be sometimes rather inappropriate.

This is typically the case with transient signals, which are known to be bounded in time. The Fourier analysis can provide this image of the signal, but in a somehow artificial manner, i.e. as a sum of an infinite number of virtual sinusoids which cancel each other. Consequently, we obtain a “dynamic” zero on a time interval where the signal is vanishing, while from a physical point of view this should be a “static” zero, since the signal does not exist on this interval. Furthermore, the Fourier analysis expresses a finite energy signal as a linear superposition of infinite energy basic signals.

These physical interpretation difficulties suggest that the Fourier transform should ...