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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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*<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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*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>'''  
+
|-
−
*'''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>
+
|}
−
*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>
 +
| <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

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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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Basic linear algebra uncovers and clarifies very important geometry and algebra.

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