FT-properties

Reference

• https://en.wikipedia.org/wiki/Fourier_transform

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1) Linearity

Reference

• https://en.wikipedia.org/wiki/Fourier_transform#Linearity

Definition

For any complex numbers $a \in \mathbb{C}$ and $b \in \mathbb{C}$,

$h(x) = af(x) + bg(x)$ $\xleftrightarrow{\mathcal{F}~(\textrm{Fourier transform})}$ $\hat{h}(\xi) = a \cdot \hat{f}(\xi) + b \cdot \hat{g}(\xi)$

Execution

    $ python demo_fourier_properties_1_linearity.py

Results

2) Shift in Spatial domain

Reference

• https://en.wikipedia.org/wiki/Fourier_transform#Translation_/_time_shifting

Definition

For any real number $x_0 \in \mathbb{R}$,

$h(x) = f(x-x_0)$ $\xleftrightarrow{\mathcal{F}~(\textrm{Fourier transform})}$ $\hat{h}(\xi) = e^{-2 \pi i x_0 \xi}\hat{f}(\xi)$

Execution

    $ python demo_fourier_properties_2_shift_in_spatial_domain.py

Results

3) Shift in Fourier domain

Reference

• https://en.wikipedia.org/wiki/Fourier_transform#Modulation_/_frequency_shifting

Definition

For any real number $\xi_0 \in \mathbb{R}$,

$h(x) = e^{2 \pi i x \xi_0}f(x)$ $\xleftrightarrow{\mathcal{F}~(\textrm{Fourier transform})}$ $\hat{h}(\xi) = \hat{f}(\xi - \xi_0)$

Execution

    $ python demo_fourier_properties_3_shift_in_Fourier_domain.py

Results

4) Convolution theorem

Reference

• https://en.wikipedia.org/wiki/Fourier_transform#Convolution_theorem

Definition

$h(x) = (f*g)(x) = \int_{-\infty}^{\infty}{f(y)g(x-y)dy}$ $\xleftrightarrow{\mathcal{F}~(\textrm{Fourier transform})}$ $\hat{h}(\xi) = \hat{f}(\xi) \cdot \hat{g}(\xi)$

where, * is convolution operator and $\cdot$ is element-wise multiplication.

Execution for 1D example

    $ python demo_fourier_properties_4_1d_convolution_vs_multiplication.py

Execution for 2D example

    $ python demo_fourier_properties_5_2d_convolution_vs_multiplication.py

Results

(a) 1D example (b) 2D example