TABLE OF CONTENTS Chapter written by Eric GRIVEL and Yannick BERTHOUMIEU.
This chapter discusses frequential representations of signals and filters. We will introduce the Fourier transform of continuous-time signals by first presenting the Fourier series decomposition of periodic signals. Properties and basic calculation methods will be demonstrated. We will then present the frequential analysis of discrete-time signals from the discrete Fourier transform using the standard and most rapid versions. These concepts will then be illustrated using the example of speech signals from a common time-frequency-energy representation the spectrogram.
Let x(t) be a finite energy signal. We consider the scalar product 
of two functions ? i (t) and ? k (t) of finite energy, represented as follows:
| (3.1) |  |
where ? k * (t) denotes the complex conjugate of ? k (t).
A family { ? k (t)} of finite energy functions is called orthonormal if it verifies the following relations:
| (3.2) |  |
A family { ? k (t)} is complete if any vector of the space can be approximated as closely as possible by a linear combination of { ? k (t)}. A family { ? k (t)} is termed maximal when the sole function x(t) of orthogonal finite energy throughout ? k (t) is the null function. We can then decompose the signal x(t) on...
