Continuous Space Fourier Transform (frequences in hertz) - Rhea

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Continuous-Space (2D) Fourier Transform (CSFT): definition and inverse transform
Forward transform	$\mathcal{F}(u,v) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)e^{-j2\pi(ux+vy)} dxdy }$	(info)
Inverse transform	$f(x,y) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{F(u,v)e^{j2\pi(ux+vy)} dudv }$

2D Continuous Space Fourier Transform(CSFT) Properties
	$\displaystyle f(x,y)$	$\longrightarrow$	$\displaystyle F(u,v)$
Linearity	$\displaystyle af_1(x,y)+bf_2(x,y)$		$\displaystyle aF_1(u,v)+bF_2(u,v)$
Scaling	$f(\frac{x}{a},\frac{y}{b})$		$\displaystyle\|ab\|F(au,bv)$
Shifting	$\displaystyle f(x-x_o,y-y_o)$		$\displaystyle F(u,v)e^{-j2\pi(ux_o+vy_o)}$
Modulation	$\displaystyle f(x,y)e^{j2\pi(xu_o+yv_o)}$		$\displaystyle F(u-u_o,v-v_o)$
Reciprocity	$\displaystyle F(x,y)$		$\displaystyle f(-u,-v)$

Other Properties
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	$F(0,0)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{f(x,y)dxdy }$

Symmetry Properties for Continuous Space Fourier Transform
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.
$\displaystyle F(u,v)=A(u,v)e^{j\theta(u,v)}$
$\displaystyle F(u,v)=F^{*}(-u,-v)$
$\displaystyle A(u,v)=A(-u,-v)$
$\displaystyle \theta(u,v)=-\theta(-u,-v)$
$f(x,y)=2 \int_0 ^{\infty}\int_{-\infty}^{\infty}{ A(u,v)cos[2 \pi(ux+vy)+ \theta(u,v)] dudv}$

Separability
	$\displaystyle g(x)$	$\longrightarrow$	$\displaystyle G(u)$
	$\displaystyle h(x)$	$\longrightarrow$	$\displaystyle H(v)$
	$\displaystyle f(x,y)$	$\longrightarrow$	$\displaystyle F(u,v)$
	$\displaystyle f(x,y)=g(x)h(y)$	$\longrightarrow$	$\displaystyle F(u,v)=G(u)H(v)$
For example,
	$\displaystyle rect(x,y)=rect(x)rect(y)$	$\longrightarrow$	$\displaystyle sinc(u)sinc(v)=sinc(u,v)$

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).

\displaystyle circ(x,y)

\longrightarrow

\displaystyle 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.


$\displaystyle x(t)$	$\longrightarrow$	$\mathcal{X}(f)$
$\displaystyle \delta (x,y)$		$\displaystyle 1$
$\displaystyle 1$		$\displaystyle \delta (u,v)$
$\displaystyle rect(x)$		$\displaystyle sinc(u) \delta (v)$
$\displaystyle \delta (x)$		$\displaystyle \delta (v)$
$e^{j2\pi(u_ox+v_oy)}$		$\displaystyle \delta (u-u_o,v-v_o)$
$\displaystyle cos[2\pi(u_ox+v_oy)]$		$\frac{1}{2} [\delta (u-u_o,v-v_o)+\delta (u+u_o,v+v_o)]$

2D Continuous Space Fourier Transform in Polar Form
	$\displaystyle x=r \cos (\theta)$	$\longrightarrow$	$\displaystyle u= \rho \cos (\phi)$
	$\displaystyle y=r \sin (\theta)$	$\longrightarrow$	$\displaystyle v= \rho \sin (\phi)$

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	$\displaystyle f(r,\theta +\alpha )$	$\displaystyle F(\rho ,\phi + \alpha)$
Circular Symmetry	$\displaystyle f(r,\theta)=f_o(r)$	$\displaystyle F(\rho , \phi)=F_o(\rho)$
Circular Symmetry	$\displaystyle f(r,\theta)=f_o(r)$	$\displaystyle F(\rho , \phi)=F_o(\rho)$
Convolution Theorem	$\displaystyle f_1(x,y) \circledast f_2(x,y)$	$\displaystyle F_1(u,v)F_2(u,v)$
Product Theorem	$\displaystyle f_1(x,y)f_2(x,y)$	$\displaystyle F_1(u,v) \circledast F_2(u,v)$

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