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^{-j2pift} dt } $
- $ x(t) = \int{X(2pif)e^{j2pift} df } $
Similarily, in2D, we have:
- Forward transform- $ F(u,v) = \int{f(x,y)e^{-j2pi(ux+vy)} dxdy } $
- Inverse transform- $ f(x,y) = \int{F(u,v)e^{j2pi(ux+vy)} dudv } $
Like 1D signals, properties such as the linearity, the shifting property, and so on, remained the same in 2D signals.
- Linearity:$ af1(x,y)+bf2(x,y)------CSFT------ aF1(u,v)+bF2(u,v) $
- Scaling:$ f(x/a,y/b)---------------CSFT--------|ab|F(au,bv) $
- Shifting: $ f(x-xo,y-yo)------------CSFT-------F(u,v)e^{-j2pi(uxo+vyo)} $
- Modulation:$ f(x,y)e^{j2pi(xuo+yvo)}------------CSFT---------F(u-uo,v-vo) $
- Reciprocity:$ 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)-----1-D CSFT-----------G(u) $
$ h(y)----1-D CSFT-----------H(v) $ $ f(x,y)---2-D CSFT------------F(u,v) $
If a function can be rewritten as $ f(x,y)=g(x)h(y) $; then, its fourier transform is $ F(u,v)=G(u)H(v) $ .
For example, 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, we 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. 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:
