mathematics-physics-wiki/docs/en/physics/mathematical-physics/signal-analysis/fourier-transform.md

Fourier transformations

Definition of the Fourier transform

Definition: let f, F: \mathbb{R} \to \mathbb{C}, the Fourier transform of f is given by

F(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t}dt,

for all \omega \in \mathbb{R}. The inverse Fourier transform of F is given by

f(t) = \frac{1}{2\pi} \int_{-\infty}^\infty F(\omega) e^{i \omega t} d\omega,

for all t \in \mathbb{R}. Therefore f and F form a Fourier transform pair denoted by

f \overset{\mathcal{F}}\longleftrightarrow F,

therefore we have

\begin{align*} &f(t) = \mathcal{F}^{-1}[F(\omega)], \quad &\forall t \in \mathbb{R}, \ &F(\omega) = \mathcal{F}[f(t)], \quad &\forall \omega \in \mathbb{R}. \end{align*}

Properties of the Fourier transform

Proposition: let f, g, F, G: \mathbb{R} \to \mathbb{C}, we have linearity given by

af(t) + bg(t) \overset{\mathcal{F}}\longleftrightarrow aF(\omega) + bG(\omega),

with a,b \in \mathbb{C}.

??? note "Proof:"

Will be added later.

Proposition: let f,F: \mathbb{R} \to \mathbb{C}, we have time shifting given by

f(t - t_0) \overset{\mathcal{F}}\longleftrightarrow F(\omega) e^{-i\omega t_0},

with t_0 \in \mathbb{R}.

??? note "Proof:"

Will be added later.

Proposition: let f,F: \mathbb{R} \to \mathbb{C}, we have frequency shifting given by

e^{i \omega_0 t} f(t) \overset{\mathcal{F}}\longleftrightarrow F(\omega - \omega_0)

with \omega_0 \in \mathbb{R}.

??? note "Proof:"

Will be added later.

Proposition: let f,F: \mathbb{R} \to \mathbb{C}, we have time or frequency scaling given by

f(t/a) \overset{\mathcal{F}}\longleftrightarrow |a| F(a\omega)

with a \in \mathbb{R}.

??? note "Proof:"

Will be added later.

Proposition: let f, g, F, G: \mathbb{R} \to \mathbb{C}, we have time convolution given by

f(t) * g(t) \overset{\mathcal{F}}\longleftrightarrow F(\omega) G(\omega).

??? note "Proof:"

Will be added later.

Proposition: let f, g, F, G: \mathbb{R} \to \mathbb{C}, we have frequency convolution given by

f(t) g(t) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2\pi} F(\omega) * G(\omega).

??? note "Proof:"

Will be added later.

Proposition: let f,F: \mathbb{R} \to \mathbb{C} be differentiable, we have time differentation given by

f'(t) \overset{\mathcal{F}}\longleftrightarrow i \omega F(\omega).

??? note "Proof:"

Will be added later.

Proposition: let f,F: \mathbb{R} \to \mathbb{C} be differentiable, we have time integration given by

\int_{-\infty}^t f(u)du \overset{\mathcal{F}}\longleftrightarrow \frac{1}{i\omega} F(\omega) + \pi F(0)\delta(\omega).

??? note "Proof:"

Will be added later.