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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

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

Alumni Liaison

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