Fourier Transforms

Fourier transform equations

(4)\[ G(f_x, f_y) = F\{g\} = \int \int g(x, y) \exp \left[ -j 2 \pi (f_x x + f_y y) \right] dx dy \]

(5)\[ g(x, y) = \mathfrak{F}^{-1} \{G\} = \int \int G(f_x, f_y) \exp \left[ j 2 \pi (f_x x + f_y y) \right] dx dy \]

Fourier transform relations

	Fourier Domain
\(f(x)\)	\(F(f_x)\)
\(K f(x)\)	\(K F(f_x)\)
\(f(ax)\)	\(\frac{1}{a} F \left( \frac{f_x}{a} \right)\)
\(f(x-a)\)	\(\exp (-j 2 \pi f_x a) F(f_x)\)
\(\exp(j 2 \pi f_{xo}x) f(x)\)	\(F(f_x - f_{xo})\)

Useful functions

Table 1 Useful Functions

Function	Definition
rectangle	\(\mathrm{rect}(x) = \left\{ \begin{matrix} 1 & \lvert x \rvert \lt \frac{1}{2} \\ \frac{1}{2} & \lvert x \rvert = \frac{1}{2} \\ 0 & \mathrm{otherwise} \end{matrix} \right.\)
sinc	\(\mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}\)
signum	\(\mathrm{sgn}(x) = \left\{ \begin{matrix} 1 & x \gt 0 \\ 0 & x = 0 \\ -1 & x \lt 0 \end{matrix} \right.\)
triangle	\(\Lambda(x) = \left\{ \begin{matrix} 1 - \lvert x \rvert & \lvert x \rvert \leq 1 \\ 0 & \mathrm{otherwise} \end{matrix} \right.\)
comb	\(\mathrm{comb}(x) = \sum_{n=-\infty}^\infty \delta(x-n)\)
circle	\(\mathrm{circ}(\sqrt{x^2 + y^2}) = \left\{ \begin{matrix} 1 & \sqrt{x^2 + y^2} \lt 1 \\ \frac{1}{2} & \sqrt{x^2 + y^2} = \frac{1}{2} \\ 0 & \mathrm{otherwise} \end{matrix} \right.\)

Useful equivalencies

(6)\[ \int_{-\infty}^\infty \mathrm{sinc} (x) dx = \int_{-\infty}^{\infty} \mathrm{sinc}^2(x) dx = 1 \]

(7)\[ \int_{0}^{\infty} \lvert \frac{J_1(x)}{x} \rvert ^2 dx = \frac{4}{3\pi} \]

(8)\[ \int_0^\infty x | \frac{J_1(x)}{x} | ^2 dx = \frac{1}{2} \]

(9)\[ \lim_{x\to\infty} \left( J_1(x) \right) = \frac{x}{2} \]

Fourier transforms

Table 2 Fourier Transforms

Function	Definition	Transform
Impulse	\(\delta (x)\)	\(1\)
Uniform	\(A\)	\(2 \pi A \delta (f_x)\)
Cosine	\(\cos (\omega_{xo} x)\)	\(\pi \left[ \delta (\omega_x + \omega_{xo}) + \delta (\omega_x - \omega_{xo}) \right]\)
Sine	\(\sin (\omega_{xo})\)	\(j\pi \left[ \delta (\omega_x + \omega_{xo}) - \delta (\omega_x - \omega_{xo}) \right]\)
Rectangular	\(\mathrm{rect}(x)\)	\(\mathrm{sinc}(f_x)\)
Triangle	\(\Lambda (x)\)	\(\mathrm{sinc}^2(f_x)\)
Circle	\(\mathrm{circ} \left( \frac{r}{R} \right)\)	\(\pi R^2 \left[ 2 \frac{J_1 (2 \pi R f_x)}{2 \pi R f_x} \right]\)
Exponential	\(\exp \left( -\lvert x \rvert \right)\)	\(\frac{2}{1 + (2 \pi f_x)^2}\)
Gaussian	\(\exp \left( - \pi x^2 \right)\)	\(\exp \left( - \pi f_x^2 \right)\)
Chirp	\(\exp (j \pi x^2)\)	\(\exp \left( j \frac{\pi}{4} \right) \exp (-j \pi f_x^2)\)
Finite sum of pulses	\(\sum_{n=-s}^s \delta (x-n)\)	\(\frac{\sin (M \pi f_x)}{\sin (\pi f_x)}, M=2S+1\)
Infinite sum of pulses	\(\sum_{n=-\infty}^\infty \delta (x-n)\)	\(\sum_{n=-\infty}^\infty \delta (f_x - n)\)