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: $ Insert formula here $
- X(f)=
- X(t)=
Similarily, in2D, we have:
- Forward transform- F(u,v)=
- Inverse transform- f(x,y)=
Like 1D signals, properties such as the linearity, the shifting property, and so on, remained the same in2D 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^(j2pixo)
- Modulation:f(x,y)e ------------CSFT---------F(u-uo,v-vo)
|Reciprocity: F(x,y)-----------------CSFT ------f(-u,-v) |Parseval’s relation: |Initial value: |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:
