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Properties of the Fourier Transform
Let us name
a function, and
its Fourier transform. We
use here the simple, non-unitary convention
- Definition:
- Linearity:
, then
- Translation:
, then
- Shifting:
, then
- Scaling:
, then
(the so-called time reversal property is obtained with
- Conjugation: if
, then
- Integration: With
in the definition
- Convolution:
then
.
The Fourier Transform of a product of two functions is the convolution of
the Fourier Transforms of the functions.
- Uncertainty principle: the more concentrated
is, the
more spread out its Fourier transform
must be. In particular,
the scaling property of the Fourier transform may be seen as saying: if
we squeeze a function in
, its Fourier transform stretches out in
.
It is not possible to arbitrarily concentrate both a function and its
Fourier transform.
The definition, illustrated here with scalars, also holds for
being 2-D vectors in Euclidean space, the product in the definition
being a standard dot product of these vectors.
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Gildas manager
2023-06-01