Linear and quadratic time-frequency analysis

The purpose of signal processing is to study, design and build signal processing systems, a signal being a model representing a phenomenon evolving in time, space... and designed to carry information. Signal theory is concerned with the development of these models, as well as the study of the various analysis tools that can be applied to them. Many physical phenomena and devices produce signals: speech synthesizers, radar and sonar antennas, cameras, television transmitters, sensors at the interface between a physical medium and a measurement system, stock market quotations...

The main categories of signals are defined by characteristics related to the medium (range of variation of the variables), the set of values and the mode of signal generation. Throughout this article, we restrict ourselves to the study of deterministic, continuous-time (or permanent), real or complex scalar, univariable, temporal signals; discrete and sampled versions of the various transforms presented are provided for information purposes and with a view to possible implementation on a computer.

A classic method for analyzing a deterministic signal is to associate a particular transform with its initial representation. For quantities satisfying the superposition principle, and depending on the need or nature of the signal, one of the various forms of Fourier, Laplace or z-transform is usually used. The resulting representation is made up of the complex coefficients of a linear combination of basic quantities, reconstructing the original quantity.

While Fourier frequency analysis uses two conjugate, global representations, one temporal and the other frequency-based, nature is rich in signals for which the useful information is conveyed not only by the frequencies emitted, but also by the temporal structure of the signal itself; the example of music is typical of this (a musical score gives both the frequency of the notes and the order in which they are to be played). A representation of the signal as a function of time alone gives little indication of its frequency behavior, while its Fourier analysis masks the instant of emission and duration of each of the signal's composite elements. For the purposes of signal processing, it has therefore been sought to associate with a time or frequency signal representations that depend on both the time and frequency variables, and thus simultaneously possess both the time and frequency characteristics. These transforms are known as time-frequency representations (RTFs), and should not be confused with partial frequency representations of a bivariable space-time signal (which depend on the time and frequency associated with the spatial variable).

The transition from RTF to the time-frequency domain should...