Back to ECE438 course page


Note: If you are planning to make major changes to this page, please create a duplicate one with a new title (copy and paste the source code) and work on that one. This way the initial work from 2009 will be preserved and you will still be able to make the modifications you wish to make. This site is not like wikipedia where a single page on each subject is created: we have several different pages on the same subjects and we think it is perfectly ok.

The modifications by user:chen558 are now on a new page: Summary_spectral_analysis_2D_signals_Chen_Fall_2010.

-pm


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_{-\infty}^{\infty}{x(t)e^{-j2\pi ft} dt } $
  • $ x(t) = \int_{-\infty}^{\infty}{X(2\pi f)e^{j2\pi ft} df } $

Similarily, in2D, we have:

  • Forward transform- $ F(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy } $
  • Inverse transform- $ f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{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_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|f(x,y)|^2dxdy }=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{|F(u,v)|^2dudv } $
  • Initial value: $ F(0,0)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)dxdy } $
  • 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.
    • $ \displaystyle F(u,v)=A(u,v)e^{j\theta(u,v)} $
    • $ \displaystyle F(u,v)=F^{*}(-u,-v) $
    • $ \displaystyle A(u,v)=A(-u,-v) $
    • $ \displaystyle \theta(u,v)=-\theta(-u,-v) $
    • $ f(x,y)=2 \int_0 ^{\infty}\int_{-\infty}^{\infty}{ A(u,v)cos[2 \pi(ux+vy)+ \theta(u,v)] dudv} $

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:

  • $ \displaystyle \delta (x,y)---CSFT---1 $
  • $ \displaystyle 1---CSFT--- \delta (u,v) $
  • $ \displaystyle rect(x)---CSFT---sinc(u) \delta (v) $
  • $ \displaystyle \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)] $

In 2D, we can also change the coordianting system from rectangular to polar form,

    • $ x=rcos(\theta) $----------------$ u=\rho cos(\phi) $
    • $ y=rsin(\theta) $----------------$ v=\rho sin(\phi) $
  • Forward transform-$ F(\rho,\phi)=\int_0 ^{2\pi}\int_0 ^{\infty}{f(r,\theta)e^{-j2\pi\rho r cos(\phi -\theta)}r dr d \theta } $
  • Inverse transform-$ f(r,\theta)=\int_0 ^{2\pi}\int_0 ^{\infty}{F(\rho,\phi)e^{j2\pi\rho r cos(\phi -\theta)}\rho d \rho d \phi } $

Some properties related to the polar representations

  • Rotation $ \displaystyle f(r,\theta +\alpha )-------CSFT-------F(\rho ,\phi + \alpha) $
  • Circular Symmetry $ \displaystyle f(r,\theta)=f_o(r)---------->F(\rho , \phi)=F_o(\rho) $

Convolution Theorem

  • $ \displaystyle f_1(x,y)**f_2(x,y)---CSFT---F_1(u,v)F_2(u,v) $

Product Theorem

  • $ \displaystyle f_1(x,y)f_2(x,y)---CSFT---F_1(u,v)**F_2(u,v) $

Back to ECE438 course page

Alumni Liaison

Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett