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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_{-\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) $

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