Electrical Circuit Analysis

Summary of the Properties of the Fourier Transform

Electrical Circuit Analysis > Fourier Transform > Properties of the Fourier Transform

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Table. 1: Properties of the Fourier transform
Property	$ f(t) $	$ F(\omega) $
Linearity	$ a_{1} f_{1}(t)+a_{2} f_{2}(t) $	$ a_{1} F_{1}(\omega)+a_{2} F_{2}(\omega) $
Scaling	$ f(a t) $	$ \frac{1}{\|a\|} F\left(\frac{\omega}{a}\right) $
Time shift	$ f(t-a) u(t-a) $	$ e^{-j \omega a} F(\omega) $
Frequency shift	$ e^{j \omega_{0} t} f(t) $	$ F\left(\omega-\omega_{0}\right) $
Modulation	$ \cos \left(\omega_{0} t\right) f(t) $	$ \frac{1}{2}\left[F\left(\omega+\omega_{0}\right)+F\left(\omega-\omega_{0}\right)\right] $
Time differentiation	$ \frac{d f}{d t} $	$ j \omega F(\omega) $
	$ \frac{d^{n} f}{d t^{n}} $	$ (j \omega)^{n} F(\omega) $
Time integration	$ \int_{-\infty}^{t} f(t) d t $	$ \frac{F(\omega)}{j \omega}+\pi F(0) \delta(\omega) $
Frequency differentiation	$ t^{n} f(t) $	$ (j)^{n} \frac{d^{n}}{d \omega^{n}} F(\omega) $
Reversal	$ f(-t) $	$ F(-\omega) $ or $ F^{*}(\omega) $
Duality	$ F(t) $	$2 \pi f(-\omega) $
Convolution in $ t $	$ f_{1}(t) * f_{1}(t) $	$ F_{1}(\omega) F_{2}(\omega) $
Convolution in $ \omega $	$ f_{1}(t) f_{1}(t) $	$ \frac{1}{2 \pi} F_{1}(\omega) * F_{2}(\omega) $

Table. 2: Fourier transform pairs.
$f (t)$	$F (ω)$
$δ(t)$	$1$
$1$	$2πδ(ω)$
$u(t)$	$πδ(ω) + { 1 \over jω}$
$u(t + τ) − u(t − τ)$	$2 {sin ωτ \over ω}$
$\|t\|$	${ -2 \over ω^2}$
$sgn(t)$	${ 2 \over jω}$
$e^{−at}u(t)$	${ 1 \over a + jω }$
$e^{at}u(-t)$	${ 1 \over a - jω }$
$t^ne^{−at}u(t)$	${ n! \over (a + jω)^{n+1} }$
$e^{−a\|t\|}$	${ 2a \over a^2 + ω^2}$
$e^{jω_0t}$	$2πδ(ω − ω_0)$
$\sin ω_0t$	$jπ[δ(ω + ω_0) − δ(ω − ω_0)]$
$\cos ω_0t$	$π[δ(ω + ω_0) + δ(ω − ω_0)]$
$e^{−at}\sin ω_0t u(t)$	${ω_0 \over (a + jω)^2 + ω_0^2}$
$e^{−at}\cos ω_0t u(t)$	${a + jω \over (a + jω)^2 + ω_0^2}$

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