# 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.