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[[ECE438_(BoutinFall2009)|Back to ECE438 course page]]
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[[ECE438 (BoutinFall2009)|Back to ECE438 course page]]  
  
 
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== Spectral Analysis of 2D Signals (Nov.16) ==
 
  
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This recitation covers the material from Nov. 4 to Nov. 13.  So far, we have introduced the basic knowledge of 2D signals. As a review, let us start from the Continuous-Space Fourier Transform(CSFT) definitons and its inverse transform.
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== Continuous Space Fourier Transform of 2D Signals  ==
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In 1D, we have:
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−
*<math>X(f) = \int_{-\infty}^{\infty}{x(t)e^{-j2\pi ft} dt } </math>
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{|
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*<math>x(t) = \int_{-\infty}^{\infty}{X(2\pi f)e^{j2\pi ft} df } </math>
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|-
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Similarily, in2D, we have:
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! style="background: none repeat scroll 0% 0% rgb(228, 188, 126); font-size: 110%;" colspan="2" | Continuous Space Fourier Transform (2D Fourier Transform)
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*Forward transform- <math>F(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy }</math>                    
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|-
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*Inverse transform- <math>f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{F(u,v)e^{j2\pi(ux+vy)} dudv } </math>                      
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | 1D Continuous Space Fourier Transform(CSFT) definitions and its inverse transform
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Like 1D signals, properties such as the linearity, the shifting property, and so on, remained the same in 2D signals.
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|-
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*'''Linearity:'''<math>\displaystyle af_1(x,y)+bf_2(x,y)------CSFT------ aF_1(u,v)+bF_2(u,v) </math>
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| align="right" style="padding-right: 1em;" | Continous Space Fourier Transform
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*'''Scaling:<math>f(\frac{x}{a},\frac{y}{b})--------------CSFT--------|ab|F(au,bv)</math>'''
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| <math>\mathcal{X}(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft} dt </math>
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*'''Shifting: <math>f(x-x_o,y-y_o)------------CSFT-------F(u,v)e^{-j2\pi(ux_o+vy_o)} </math>'''  
+
|-
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*'''Modulation:<math>f(x,y)e^{j2\pi(xu_o+yv_o)}----------CSFT---------F(u-u_o,v-v_o)</math>'''
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| align="right" style="padding-right: 1em;" | Inverse Continous Space Fourier Transform
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*'''Reciprocity:'''<math>\displaystyle F(x,y)-------------CSFT ------f(-u,-v)</math>  
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| <math>x(t)=\int_{-\infty}^{\infty}\mathcal{X}(2\pi f)e^{j2\pi ft} df </math>  
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*'''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>'''
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|
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*'''Initial value: <math>F(0,0)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)dxdy } </math>'''
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|}
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*'''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>
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{|
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** <math>\displaystyle F(u,v)=F^{*}(-u,-v)</math>
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|-
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** <math> \displaystyle A(u,v)=A(-u,-v)</math>
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | 2D Continuous Space Fourier Transform(CSFT) definitions and its inverse transform
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**<math>\displaystyle \theta(u,v)=-\theta(-u,-v) </math>
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|-
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** <math>f(x,y)=2 \int_0 ^{\infty}\int_{-\infty}^{\infty}{ A(u,v)cos[2 \pi(ux+vy)+ \theta(u,v)] dudv}</math>
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| align="right" style="padding-right: 1em;" | Forward transform  
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Before we go to the important transform pairs, the separability is a very important property of 2D signals. It enables us to transform 2D signals to our familiar 1D signals. given,  
+
| <math>\mathcal{F}(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy }</math>
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**<math>g(x)-----CSFT-----------G(u)</math>  
+
|-
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**<math>h(y)---- CSFT-----------H(v)</math>  
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| align="right" style="padding-right: 1em;" | Inverse transform  
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**<math>f(x,y)-- CSFT------------F(u,v)</math>  
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| <math>f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{F(u,v)e^{j2\pi(ux+vy)} dudv } </math>  
−
If a function can be rewritten as <math>\displaystyle f(x,y)=g(x)h(y)</math>; then, its fourier transform is <math>\displaystyle   F(u,v)=G(u)H(v) </math>.
+
|
−
*For example, <math>\displaystyle rect(x,y)=rect(x)rect(y)----------CSFT------------sinc(u)sinc(v)=sinc(u,v)</math>
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|}
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*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).
+
   
−
Another special function is the circ function and the jinc function.
+
{|
−
*<math>\displaystyle circ(x,y)------------CSFT----------------jinc(u,v)</math>
+
|-
 +
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | 2D Continuous Space Fourier Transform(CSFT) Properties
 +
|-
 +
| 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;" | '''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>
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| <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: 1em;" |
 +
| <math>\displaystyle circ(x,y)</math>
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| <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.
 
*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.
−
Other important transform pairs:
+
 
−
*<math>\displaystyle \delta (x,y)---CSFT---1</math>
+
<br> Before we go to the important transform pairs, the separability is a very important property of 2D signals. It enables us to transform 2D signals to our familiar 1D signals. given,
−
*<math>\displaystyle  1---CSFT--- \delta (u,v)</math>
+
 
−
*<math>\displaystyle  rect(x)---CSFT---sinc(u) \delta (v)</math>
+
 
−
*<math>\displaystyle  \delta (x)--CSFT---\delta (v) </math>
+
Other important transform pairs:  
−
*<math> e^{j2\pi(u_ox+v_oy)}---CSFT--- \delta (u-u_o,v-v_o)</math>
+
 
 +
*<math>\displaystyle \delta (x,y)---CSFT---1</math>  
 +
*<math>\displaystyle  1---CSFT--- \delta (u,v)</math>  
 +
*<math>\displaystyle  rect(x)---CSFT---sinc(u) \delta (v)</math>  
 +
*<math>\displaystyle  \delta (x)--CSFT---\delta (v) </math>  
 +
*<math> e^{j2\pi(u_ox+v_oy)}---CSFT--- \delta (u-u_o,v-v_o)</math>  
 
*<math> cos[2\pi(u_ox+v_oy)]---CSFT---\frac{1}{2} [\delta (u-u_o,v-v_o)+\delta (u+u_o,v+v_o)] </math>
 
*<math> cos[2\pi(u_ox+v_oy)]---CSFT---\frac{1}{2} [\delta (u-u_o,v-v_o)+\delta (u+u_o,v+v_o)] </math>
−
In 2D, we can also change the coordianting system from rectangular to polar form,
+
 
−
**'''<math>x=rcos(\theta)</math>----------------<math>u=\rho cos(\phi)</math>'''
+
In 2D, we can also change the coordianting system from rectangular to polar form,  
−
**'''<math>y=rsin(\theta)</math>----------------<math>v=\rho sin(\phi)</math>'''
+
 
−
*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>
+
**'''<span class="texhtml">''x'' = ''r'''</span>'''''c''''o''''s''(θ)----------------<span class="texhtml">''u'' = ρ''c''''o''''s''(φ)</span>  
 +
**'''<span class="texhtml">''y'' = ''r'''</span>'''''s''''i''''n''(θ)----------------<span class="texhtml">''v'' = ρ''s''''i''''n''(φ)</span>  
 +
*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>  
 
*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>
 
*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>
−
Some properties related to the polar representations
+
 
−
*Rotation <math>\displaystyle f(r,\theta +\alpha )-------CSFT-------F(\rho ,\phi + \alpha) </math>
+
Some properties related to the polar representations  
−
*Circular Symmetry <math>\displaystyle f(r,\theta)=f_o(r)---------->F(\rho , \phi)=F_o(\rho)</math>
+
 
−
Convolution Theorem
+
*Rotation <math>\displaystyle f(r,\theta +\alpha )-------CSFT-------F(\rho ,\phi + \alpha) </math>  
 +
*Circular Symmetry <math>\displaystyle f(r,\theta)=f_o(r)---------->F(\rho , \phi)=F_o(\rho)</math>
 +
 
 +
Convolution Theorem  
 +
 
 
*<math>\displaystyle f_1(x,y)**f_2(x,y)---CSFT---F_1(u,v)F_2(u,v) </math>
 
*<math>\displaystyle f_1(x,y)**f_2(x,y)---CSFT---F_1(u,v)F_2(u,v) </math>
−
Product Theorem
+
 
 +
Product Theorem  
 +
 
 
*<math>\displaystyle f_1(x,y)f_2(x,y)---CSFT---F_1(u,v)**F_2(u,v)</math>
 
*<math>\displaystyle f_1(x,y)f_2(x,y)---CSFT---F_1(u,v)**F_2(u,v)</math>
  
−
[[ECE438_(BoutinFall2009)|Back to ECE438 course page]]
+
[[ECE438 (BoutinFall2009)|Back to ECE438 course page]]

Revision as of 13:58, 30 November 2010

Back to ECE438 course page

Continuous Space Fourier Transform of 2D Signals

Continuous Space Fourier Transform (2D Fourier Transform)
1D Continuous Space Fourier Transform(CSFT) definitions and its inverse transform
Continous Space Fourier Transform $ \mathcal{X}(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft} dt $
Inverse Continous Space Fourier Transform $ x(t)=\int_{-\infty}^{\infty}\mathcal{X}(2\pi f)e^{j2\pi ft} df $
2D Continuous Space Fourier Transform(CSFT) definitions and its inverse transform
Forward transform $ \mathcal{F}(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy } $
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 x(t) $ $ \longrightarrow $ $ \mathcal{X}(f) $
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.


Before we go to the important transform pairs, the separability is a very important property of 2D signals. It enables us to transform 2D signals to our familiar 1D signals. given,


Other important transform pairs:

  • $ \displaystyle \delta (x,y)---CSFT---1 $
  • $ \displaystyle 1---CSFT--- \delta (u,v) $
  • $ \displaystyle rect(x)---CSFT---sinc(u) \delta (v) $
  • $ \displaystyle \delta (x)--CSFT---\delta (v) $
  • $ e^{j2\pi(u_ox+v_oy)}---CSFT--- \delta (u-u_o,v-v_o) $
  • $ cos[2\pi(u_ox+v_oy)]---CSFT---\frac{1}{2} [\delta (u-u_o,v-v_o)+\delta (u+u_o,v+v_o)] $

In 2D, we can also change the coordianting system from rectangular to polar form,

    • 'x = rc'o's(θ)----------------u = ρc'o's(φ)
    • 'y = rs'i'n(θ)----------------v = ρs'i'n(φ)
  • 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 )-------CSFT-------F(\rho ,\phi + \alpha) $
  • Circular Symmetry $ \displaystyle f(r,\theta)=f_o(r)---------->F(\rho , \phi)=F_o(\rho) $

Convolution Theorem

  • $ \displaystyle f_1(x,y)**f_2(x,y)---CSFT---F_1(u,v)F_2(u,v) $

Product Theorem

  • $ \displaystyle f_1(x,y)f_2(x,y)---CSFT---F_1(u,v)**F_2(u,v) $

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