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 \(a,b \in \mathbb{C}\).

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 \(t_0 \in \mathbb{R}\).

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 \(\omega_0 \in \mathbb{R}\).

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 \(a \in \mathbb{R}\).

Proof:

Will be added later.

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

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

Proof:

Will be added later.

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

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

Proof:

Will be added later.

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

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

Proof:

Will be added later.

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

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

Proof:

Will be added later.