| Line 17: | Line 17: | ||
Like 1D signals, properties such as the linearity, the shifting property, and so on, remained the same in 2D signals. | Like 1D signals, properties such as the linearity, the shifting property, and so on, remained the same in 2D signals. | ||
| − | *'''Linearity:<math>af_1(x,y)+bf_2(x,y)------CSFT------ aF_1(u,v)+bF_2(u,v) </math> | + | *'''Linearity:'''<math>\displaystyle af_1(x,y)+bf_2(x,y)------CSFT------ aF_1(u,v)+bF_2(u,v) </math> |
| − | + | *'''Scaling:<math>f(\frac{x}{a},\frac{y}{b})--------------CSFT--------|ab|F(au,bv)</math>''' | |
| − | *'''Scaling:<math>f(\frac{x}{a},\frac{y}{b}) | + | |
| − | + | ||
*'''Shifting: <math>f(x-x_o,y-y_o)------------CSFT-------F(u,v)e^{-j2\pi(ux_o+vy_o)} </math>''' | *'''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>''' | |
| − | *'''Modulation:<math>f(x,y)e^{j2\pi(xu_o+yv_o)} | + | *'''Reciprocity:'''<math>\displaystyle F(x,y)-------------CSFT ------f(-u,-v)</math> |
| − | + | ||
| − | *'''Reciprocity:<math>F(x,y) | + | |
| − | + | ||
*'''Parseval’s relation:<math>\int{|f(x,y)|^2dxdy }=\int{|F(u,v)|^2dudv } </math>''' | *'''Parseval’s relation:<math>\int{|f(x,y)|^2dxdy }=\int{|F(u,v)|^2dudv } </math>''' | ||
| − | |||
*'''Initial value: <math>F(0,0)=\int{f(x,y)dxdy } </math>''' | *'''Initial value: <math>F(0,0)=\int{f(x,y)dxdy } </math>''' | ||
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Given, | Given, | ||
| − | + | **<math>g(x)-----CSFT-----------G(u)</math> | |
| − | + | **<math>h(y)---- CSFT-----------H(v)</math> | |
| − | + | **<math>f(x,y)-- CSFT------------F(u,v)</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, | + | *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). | *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> |
*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> \delta (x,y)---CSFT---1</math> | ||
| + | |||
| + | *<math> 1---CSFT--- \delta (u,v)</math> | ||
| + | |||
| + | *<math> e^{j2\pi(u_ox+v_oy)}---CSFT--- \delta (u-u_o,v-v_o)</math> | ||
| + | |||
*Other representations of 2D signals: | *Other representations of 2D signals: | ||
I will come back later! | I will come back later! | ||
Revision as of 16:15, 16 November 2009
Spectral Analysis of 2D Signals (Nov.16)
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.
In 1D, we have:
- $ X(f) = \int{x(t)e^{-j2\pi ft} dt } $
- $ x(t) = \int{X(2\pi f)e^{j2\pi ft} df } $
Similarily, in2D, we have:
- Forward transform- $ F(u,v) = \int{f(x,y)e^{-j2\pi(ux+vy)} dxdy } $
- Inverse transform- $ f(x,y) = \int{F(u,v)e^{j2\pi(ux+vy)} dudv } $
Like 1D signals, properties such as the linearity, the shifting property, and so on, remained the same in 2D signals.
- Linearity:$ \displaystyle af_1(x,y)+bf_2(x,y)------CSFT------ aF_1(u,v)+bF_2(u,v) $
- Scaling:$ f(\frac{x}{a},\frac{y}{b})--------------CSFT--------|ab|F(au,bv) $
- Shifting: $ f(x-x_o,y-y_o)------------CSFT-------F(u,v)e^{-j2\pi(ux_o+vy_o)} $
- Modulation:$ f(x,y)e^{j2\pi(xu_o+yv_o)}----------CSFT---------F(u-u_o,v-v_o) $
- Reciprocity:$ \displaystyle F(x,y)-------------CSFT ------f(-u,-v) $
- Parseval’s relation:$ \int{|f(x,y)|^2dxdy }=\int{|F(u,v)|^2dudv } $
- Initial value: $ F(0,0)=\int{f(x,y)dxdy } $
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,
- $ g(x)-----CSFT-----------G(u) $
- $ h(y)---- CSFT-----------H(v) $
- $ f(x,y)-- CSFT------------F(u,v) $
If a function can be rewritten as $ \displaystyle f(x,y)=g(x)h(y) $; then, its fourier transform is $ \displaystyle F(u,v)=G(u)H(v) $.
- For example, $ \displaystyle rect(x,y)=rect(x)rect(y)----------CSFT------------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).
Another special function is the circ function and the jinc function.
- $ \displaystyle circ(x,y)------------CSFT----------------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.
Other important transform pairs:
- $ \delta (x,y)---CSFT---1 $
- $ 1---CSFT--- \delta (u,v) $
- $ e^{j2\pi(u_ox+v_oy)}---CSFT--- \delta (u-u_o,v-v_o) $
- Other representations of 2D signals:
I will come back later!
