# Time-Frequency Localization

There is no finite energy function which is
compactly supported both in the time and frequency domains.

The time-frequency localization is measured in the
mean squares sense and is represented as a Heisenberg box.

The Fourier transform can be viewed as a
representation of a function as a sum of sinusoidal waves. These
sinusoids are very well localized in the frequency, but not in time,
since their support has an infinite length. This is a consequence of
periodicity.

To represent the frequency behavior of a signal
locally in time, the signal should be analyzed by functions which are
localized both in time and frequency, for instance, signals that are
compactly supported in the time *and
*Fourier domains.

This time-frequency
localization is limited by the following two results:

__The uncertainty theorem of Heisenberg.__

If *f *is in L^{2}, then its time root deviation and its Fourier root deviation are defined.
Then

A balance has to be reached between the time and
frequency resolution. In the limit case of a sinusoid,is zero and is
infinite.

The previous inequality is an equality if and only
if *f* is a
Gabor chirp .

__Compact supports__

If *f* is non zero with a compact support, then its Fourier
transform cannot be zero on a whole interval. Similarly, if its
Fourier transform is compactly supported, then it cannot be zero on a
time interval.

Hence, even if the Heisenberg constraints are
verified, *it is impossible to have an
function in L*^{2} which is compactly supported both in the time and Fourier
domains.

In particular, this means that **there is no instantaneous frequency analysis for finite
energy signals.**

*
*

*
*
**Time-frequency localization is thus achievable only in
the mean squares sense.**

This localization is represented as a Heisenberg
box.

For a family of vectors to be a basis of
L^{2}, it is
reasonable to expect that their Heisenberg boxes pave the time
frequency plane.

Two time frequency localization strategies are
presented in parallel; the first one leads to the windowed Fourier
transform, while the other one leads to the wavelet transform.

Windowed Fourier
transforms and wavelet transforms