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+ | [[Category:Formulas]] | ||
+ | [[Category:Fourier transform]] | ||
+ | [[Category:ECE301]] | ||
+ | [[Category:ECE438]] | ||
− | + | <center><font size= 4> | |
+ | '''[[Collective_Table_of_Formulas|Collective Table of Formulas]]''' | ||
+ | </font size> | ||
+ | |||
+ | Table of Continuous-space (CS) Fourier Transform Pairs and Properties | ||
+ | |||
+ | click [[Collective_Table_of_Formulas|here]] for [[Collective_Table_of_Formulas|more formulas]] | ||
+ | |||
+ | </center> | ||
+ | |||
+ | ---- | ||
+ | |||
+ | |||
+ | {| | ||
+ | |- | ||
+ | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Continuous-Space (2D) Fourier Transform (CSFT): definition and inverse transform | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | [[continuous-space_Fourier_transform|Forward transform]] | ||
+ | | <math>\mathcal{F}(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy }</math> | ||
+ | |[[continuous-space_Fourier_transform|(info)]] | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Inverse transform | ||
+ | | <math>f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{F(u,v)e^{j2\pi(ux+vy)} dudv } </math> | ||
+ | | | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |- | ||
+ | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | 2D Continuous Space Fourier Transform(CSFT) Properties | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle f(x,y)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math> \displaystyle F(u,v) </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Linearity''' | ||
+ | | <math>\displaystyle af_1(x,y)+bf_2(x,y) </math> | ||
+ | | | ||
+ | | <math>\displaystyle aF_1(u,v)+bF_2(u,v)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Scaling''' | ||
+ | | <math>f(\frac{x}{a},\frac{y}{b}) </math> | ||
+ | | | ||
+ | | <math>\displaystyle|ab|F(au,bv)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Shifting''' | ||
+ | | <math>\displaystyle f(x-x_o,y-y_o) </math> | ||
+ | | | ||
+ | | <math>\displaystyle F(u,v)e^{-j2\pi(ux_o+vy_o)}</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Modulation''' | ||
+ | | <math>\displaystyle f(x,y)e^{j2\pi(xu_o+yv_o)} </math> | ||
+ | | | ||
+ | | <math>\displaystyle F(u-u_o,v-v_o)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Reciprocity''' | ||
+ | | <math>\displaystyle F(x,y)</math> | ||
+ | | | ||
+ | | <math>\displaystyle f(-u,-v)</math> | ||
+ | | | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |- | ||
+ | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other Properties | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Parseval’s relation''' | ||
+ | | <math>\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|f(x,y)|^2dxdy }=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|F(u,v)|^2dudv }</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Initial Value''' | ||
+ | | <math>F(0,0)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)dxdy }</math> | ||
+ | | | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |- | ||
+ | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | 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.''' | ||
+ | |- | ||
+ | | <math>\displaystyle F(u,v)=A(u,v)e^{j\theta(u,v)}</math> | ||
+ | |- | ||
+ | |<math>\displaystyle F(u,v)=F^{*}(-u,-v)</math> | ||
+ | |- | ||
+ | |<math> \displaystyle A(u,v)=A(-u,-v)</math> | ||
+ | |- | ||
+ | |<math>\displaystyle \theta(u,v)=-\theta(-u,-v) </math> | ||
+ | |- | ||
+ | |<math>f(x,y)=2 \int_0 ^{\infty}\int_{-\infty}^{\infty}{ A(u,v)cos[2 \pi(ux+vy)+ \theta(u,v)] dudv}</math> | ||
+ | | | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |- | ||
+ | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Separability | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle g(x)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math> \displaystyle G(u) </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle h(x)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math> \displaystyle H(v) </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle f(x,y)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math> \displaystyle F(u,v) </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle f(x,y)=g(x)h(y)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math> \displaystyle F(u,v)=G(u)H(v) </math> | ||
+ | |- | ||
+ | |For example, | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle rect(x,y)=rect(x)rect(y)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math> \displaystyle sinc(u)sinc(v)=sinc(u,v) </math> | ||
+ | | | ||
+ | |} | ||
+ | |||
+ | *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). | ||
+ | |||
+ | {| | ||
+ | |- | ||
+ | | align="right" style="padding-right: 6em;" | | ||
+ | | <math>\displaystyle circ(x,y)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math> \displaystyle jinc(u,v) </math> | ||
+ | |- | ||
+ | |} | ||
+ | *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. | ||
+ | |||
+ | {| | ||
+ | |- | ||
+ | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Important Continuous Space Fourier Transform Pairs | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle x(t)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math> \mathcal{X}(f) </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle \delta (x,y) </math> | ||
+ | | | ||
+ | | <math>\displaystyle 1</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle 1 </math> | ||
+ | | | ||
+ | | <math>\displaystyle \delta (u,v)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle rect(x) </math> | ||
+ | | | ||
+ | | <math>\displaystyle sinc(u) \delta (v)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle \delta (x) </math> | ||
+ | | | ||
+ | | <math>\displaystyle \delta (v)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>e^{j2\pi(u_ox+v_oy)} </math> | ||
+ | | | ||
+ | | <math>\displaystyle \delta (u-u_o,v-v_o)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math>\displaystyle cos[2\pi(u_ox+v_oy)]</math> | ||
+ | | | ||
+ | | <math>\frac{1}{2} [\delta (u-u_o,v-v_o)+\delta (u+u_o,v+v_o)]</math> | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |- | ||
+ | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | 2D Continuous Space Fourier Transform in Polar Form | ||
+ | |- | ||
+ | | align="right" style="padding-right: 6em;" | | ||
+ | | <math> \displaystyle x=r \cos (\theta)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math>\displaystyle u= \rho \cos (\phi)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 6em;" | | ||
+ | | <math> \displaystyle y=r \sin (\theta)</math> | ||
+ | | <math>\longrightarrow</math> | ||
+ | | <math>\displaystyle v= \rho \sin (\phi)</math> | ||
+ | | | ||
+ | |} | ||
+ | |||
+ | {| | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Forward transform | ||
+ | | <math> 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 }</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Inverse transform | ||
+ | | <math> 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 }</math> | ||
+ | | | ||
+ | |} | ||
+ | |||
+ | |||
+ | {| | ||
+ | |- | ||
+ | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | some properties related to the polar representations | ||
+ | |- | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Rotation | ||
+ | | <math>\displaystyle f(r,\theta +\alpha )</math> | ||
+ | | | ||
+ | |<math>\displaystyle F(\rho ,\phi + \alpha)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Circular Symmetry | ||
+ | | <math>\displaystyle f(r,\theta)=f_o(r)</math> | ||
+ | | | ||
+ | |<math>\displaystyle F(\rho , \phi)=F_o(\rho)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Circular Symmetry | ||
+ | | <math>\displaystyle f(r,\theta)=f_o(r)</math> | ||
+ | | | ||
+ | |<math>\displaystyle F(\rho , \phi)=F_o(\rho)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Convolution Theorem | ||
+ | | <math> \displaystyle f_1(x,y) \circledast f_2(x,y)</math> | ||
+ | | | ||
+ | |<math> \displaystyle F_1(u,v)F_2(u,v)</math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | '''Product Theorem | ||
+ | | <math> \displaystyle f_1(x,y)f_2(x,y)</math> | ||
+ | | | ||
+ | |<math> \displaystyle F_1(u,v) \circledast F_2(u,v)</math> | ||
+ | |} | ||
+ | |||
+ | |||
+ | ---- | ||
+ | [[ECE438|Go to Relevant Course Page: ECE 438]] | ||
+ | |||
+ | [[ECE637|Go to Relevant Course Page: ECE 637]] | ||
+ | |||
+ | [[Collective_Table_of_Formulas|Back to Collective Table of Formulas]] | ||
+ | |||
+ | [[Category:Formulas]] | ||
+ | [[Category:continuous-space Fourier transform]] |
Latest revision as of 14:32, 23 April 2013
Table of Continuous-space (CS) Fourier Transform Pairs and Properties
click here for more formulas
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