# Fourier Transform

The Fourier transform analyses the "frequency
contents" of a signal.

Its many properties make it suitable for studying
linear time invariant operators, such as differentiation.

It is a global representation of a signal.

## Fourier transform

The **Fourier
transform** of f in L^{2 }is

The inverse Fourier transform represents f as a
sum of sinusoids

## Properties

The Fourier transform has many algebraic properties. Note
that sinusoidal waves are eigenvectors of the differentiation
operator.

This makes it possible for the Fourier transform
to give indications on the regularity of a signal.

## Implementation

To reduce the number of operations, the
Fast Fourier
Transform separates odd and even
frequencies when computing a discrete Fourier transform.

## A global representation

The Fourier transform is a global representation
of the signal. It cannot analyze it local frequency contents or its
local regularity. The convergence condition on the Fourier transform
only gives the **worst** order of regularity. It ignores local regular
behaviours.

There exist,
however, a definition of the instantaneous
frequency of an analytic signal.

It is useless in practice because it fails to
detect the **summation** of two signals. It is nonetheless a convenient means of
defining synthetic signals for numerical experimentations.

Trying to discriminate each of the stacked
frequencies leads to a frequency analysis that is localized in
frequency as well as in time. This requires some understanding of the
**time-frequency localization of
signal**.

Time-Frequency
Localization