View source for Student summary spectral analysis 2D signals - Rhea

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== 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:
*<math>X(f) = \int_{-\infty}^{\infty}{x(t)e^{-j2\pi ft} dt } </math>
*<math>x(t) = \int_{-\infty}^{\infty}{X(2\pi f)e^{j2\pi ft} df } </math>
Similarily, in2D, we have:
*Forward transform- <math>F(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy }</math>                     
*Inverse transform- <math>f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{F(u,v)e^{j2\pi(ux+vy)} dudv } </math>                        
Like 1D signals, properties  such as the linearity, the shifting property, and so on, remained the same in 2D signals.
*'''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>'''
*'''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>''' 
*'''Reciprocity:'''<math>\displaystyle F(x,y)-------------CSFT ------f(-u,-v)</math> 
*'''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>'''
*'''Initial value: <math>F(0,0)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)dxdy } </math>'''
*'''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.'''

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>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,  <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>
*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>
*<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> \displaystyle Convolution Theorem</math>
*<math>\displaystyle f_1(x,y)**f_2(x,y)---CSFT---F_1(u,v)F_2(u,v) </math>
<math> \displaystyle Product Theorem</math> 
*<math>\displaystyle f_1(x,y)f_2(x,y)---CSFT---F_1(u,v)**F_2(u,v) </math>

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