The reason for devoting an entire chapter to the basic mathematical tools of spectral analysis and leaving the applications to mechanics to the next and last chapter of this volume, is the extraordinary importance of the subject in many fields of physics, including mechanics, and in signal processing, including signals of mechanical origin. Historically, spectral analysis stemmed from Newton's work on the decomposition of white light through a prism. At the outset, it is appropriate to be more specific about the concept of signal. As defined for instance in [HAR 98], a signal is a physical effect which propagates from a material object and can be described mathematically. In particular, a signal is deterministic if it can be defined as a single function, or distribution. Spectral analysis of time signals consists basically of decomposing them into a series of harmonic oscillations. An harmonic component is defined by three scalar quantities: frequency, phase and amplitude. Therefore, as a fundamental result of the spectral analysis, description of the signal is shifted from the time domain to the frequency domain, the phase and amplitude being generally frequency dependent. The mathematical tools necessary to perform the transformation are the Fourier series and the Fourier integral.