Collective Table of Formulas

Table of Continuous-space (CS) Fourier Transform Pairs and Properties

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Continuous-Space (2D) Fourier Transform (CSFT): definition and inverse transform
Forward transform $ \mathcal{F}(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy } $ (info)
Inverse transform $ f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{F(u,v)e^{j2\pi(ux+vy)} dudv } $
2D Continuous Space Fourier Transform(CSFT) Properties
$ \displaystyle f(x,y) $ $ \longrightarrow $ $ \displaystyle F(u,v) $
Linearity $ \displaystyle af_1(x,y)+bf_2(x,y) $ $ \displaystyle aF_1(u,v)+bF_2(u,v) $
Scaling $ f(\frac{x}{a},\frac{y}{b}) $ $ \displaystyle|ab|F(au,bv) $
Shifting $ \displaystyle f(x-x_o,y-y_o) $ $ \displaystyle F(u,v)e^{-j2\pi(ux_o+vy_o)} $
Modulation $ \displaystyle f(x,y)e^{j2\pi(xu_o+yv_o)} $ $ \displaystyle F(u-u_o,v-v_o) $
Reciprocity $ \displaystyle F(x,y) $ $ \displaystyle f(-u,-v) $
Other Properties
Parseval’s relation $ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|f(x,y)|^2dxdy }=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|F(u,v)|^2dudv } $
Initial Value $ F(0,0)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)dxdy } $
Symmetry Properties for Continuous Space Fourier Transform
If f(x,y) is real, the magnitued of F(u,v) is an even function; the angle of F(u,v) is an odd function.
$ \displaystyle F(u,v)=A(u,v)e^{j\theta(u,v)} $
$ \displaystyle F(u,v)=F^{*}(-u,-v) $
$ \displaystyle A(u,v)=A(-u,-v) $
$ \displaystyle \theta(u,v)=-\theta(-u,-v) $
$ f(x,y)=2 \int_0 ^{\infty}\int_{-\infty}^{\infty}{ A(u,v)cos[2 \pi(ux+vy)+ \theta(u,v)] dudv} $
Separability
$ \displaystyle g(x) $ $ \longrightarrow $ $ \displaystyle G(u) $
$ \displaystyle h(x) $ $ \longrightarrow $ $ \displaystyle H(v) $
$ \displaystyle f(x,y) $ $ \longrightarrow $ $ \displaystyle F(u,v) $
$ \displaystyle f(x,y)=g(x)h(y) $ $ \longrightarrow $ $ \displaystyle F(u,v)=G(u)H(v) $
For example,
$ \displaystyle rect(x,y)=rect(x)rect(y) $ $ \longrightarrow $ $ \displaystyle sinc(u)sinc(v)=sinc(u,v) $
  • Notes: If we are trying to draw rect(x,y) from a top view, it will just look like a square. In the 3D plot, keep the top view as a base, making the height as 1. The plot is a cube. Similar as sinc(u,v).
$ \displaystyle circ(x,y) $ $ \longrightarrow $ $ \displaystyle jinc(u,v) $
  • Notes: if we are trying to draw circ(x,y) from a top view, it will look like a circle with a radius of ½. In the 3D plot, we keep the top view as a base, making the height as 1. The plot is a cylinder.
Important Continuous Space Fourier Transform Pairs
$ \displaystyle x(t) $ $ \longrightarrow $ $ \mathcal{X}(f) $
$ \displaystyle \delta (x,y) $ $ \displaystyle 1 $
$ \displaystyle 1 $ $ \displaystyle \delta (u,v) $
$ \displaystyle rect(x) $ $ \displaystyle sinc(u) \delta (v) $
$ \displaystyle \delta (x) $ $ \displaystyle \delta (v) $
$ e^{j2\pi(u_ox+v_oy)} $ $ \displaystyle \delta (u-u_o,v-v_o) $
$ \displaystyle cos[2\pi(u_ox+v_oy)] $ $ \frac{1}{2} [\delta (u-u_o,v-v_o)+\delta (u+u_o,v+v_o)] $
2D Continuous Space Fourier Transform in Polar Form
$ \displaystyle x=r \cos (\theta) $ $ \longrightarrow $ $ \displaystyle u= \rho \cos (\phi) $
$ \displaystyle y=r \sin (\theta) $ $ \longrightarrow $ $ \displaystyle v= \rho \sin (\phi) $
Forward transform $ F(\rho,\phi)=\int_0 ^{2\pi}\int_0 ^{\infty}{f(r,\theta)e^{-j2\pi\rho r cos(\phi -\theta)}r dr d \theta } $
Inverse transform $ f(r,\theta)=\int_0 ^{2\pi}\int_0 ^{\infty}{F(\rho,\phi)e^{j2\pi\rho r cos(\phi -\theta)}\rho d \rho d \phi } $


some properties related to the polar representations
Rotation $ \displaystyle f(r,\theta +\alpha ) $ $ \displaystyle F(\rho ,\phi + \alpha) $
Circular Symmetry $ \displaystyle f(r,\theta)=f_o(r) $ $ \displaystyle F(\rho , \phi)=F_o(\rho) $
Circular Symmetry $ \displaystyle f(r,\theta)=f_o(r) $ $ \displaystyle F(\rho , \phi)=F_o(\rho) $
Convolution Theorem $ \displaystyle f_1(x,y) \circledast f_2(x,y) $ $ \displaystyle F_1(u,v)F_2(u,v) $
Product Theorem $ \displaystyle f_1(x,y)f_2(x,y) $ $ \displaystyle F_1(u,v) \circledast F_2(u,v) $



Go to Relevant Course Page: ECE 438

Go to Relevant Course Page: ECE 637

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Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva