Difference between revisions of "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:

• $ 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) $

• $ rect(x)---CSFT---sinc(u) \delta (v) $

• $ \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)] $

Other representations of 2D signals: I will come back later!