136 lines
No EOL
2.9 KiB
Markdown
136 lines
No EOL
2.9 KiB
Markdown
# Fourier transformations
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## Definition of the Fourier transform
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> *Definition*: let $f, F: \mathbb{R} \to \mathbb{C}$, the Fourier transform of $f$ is given by
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>
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> $$
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> F(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t}dt,
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> $$
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>
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> for all $\omega \in \mathbb{R}$. The inverse Fourier transform of $F$ is given by
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>
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> $$
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> f(t) = \frac{1}{2\pi} \int_{-\infty}^\infty F(\omega) e^{i \omega t} d\omega,
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> $$
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>
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> for all $t \in \mathbb{R}$. Therefore $f$ and $F$ form a Fourier transform pair denoted by
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>
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> $$
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> f \overset{\mathcal{F}}\longleftrightarrow F,
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> $$
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>
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> therefore we have
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>
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> $$
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> \begin{align*}
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> &f(t) = \mathcal{F}^{-1}[F(\omega)], \quad &\forall t \in \mathbb{R}, \\
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> &F(\omega) = \mathcal{F}[f(t)], \quad &\forall \omega \in \mathbb{R}.
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> \end{align*}
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> $$
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## Properties of the Fourier transform
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> *Proposition*: let $f, g, F, G: \mathbb{R} \to \mathbb{C}$, we have linearity given by
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>
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> $$
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> af(t) + bg(t) \overset{\mathcal{F}}\longleftrightarrow aF(\omega) + bG(\omega),
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> $$
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>
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> with $a,b \in \mathbb{C}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$, we have time shifting given by
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>
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> $$
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> f(t - t_0) \overset{\mathcal{F}}\longleftrightarrow F(\omega) e^{-i\omega t_0},
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> $$
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>
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> with $t_0 \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$, we have frequency shifting given by
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>
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> $$
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> e^{i \omega_0 t} f(t) \overset{\mathcal{F}}\longleftrightarrow F(\omega - \omega_0)
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> $$
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>
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> with $\omega_0 \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$, we have time or frequency scaling given by
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>
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> $$
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> f(t/a) \overset{\mathcal{F}}\longleftrightarrow |a| F(a\omega)
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> $$
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>
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> with $a \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f, g, F, G: \mathbb{R} \to \mathbb{C}$, we have time convolution given by
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>
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> $$
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> f(t) * g(t) \overset{\mathcal{F}}\longleftrightarrow F(\omega) G(\omega).
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f, g, F, G: \mathbb{R} \to \mathbb{C}$, we have frequency convolution given by
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>
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> $$
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> f(t) g(t) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2\pi} F(\omega) * G(\omega).
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$ be differentiable, we have time differentation given by
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>
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> $$
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> f'(t) \overset{\mathcal{F}}\longleftrightarrow i \omega F(\omega).
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$ be differentiable, we have time integration given by
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>
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> $$
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> \int_{-\infty}^t f(u)du \overset{\mathcal{F}}\longleftrightarrow \frac{1}{i\omega} F(\omega) + \pi F(0)\delta(\omega).
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br> |