Fourier Transform Calculator

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𝓕 { }

Time var: Freq var: Use step(t) for u(t), delta(t) for δ(t)

Direction: Forward 𝓕{f(t)} Inverse 𝓕⁻¹{F(ω)} Convention: ω (rad/s) f (Hz)

Steps: None Summary Detailed Full working

Try these examples

Causal exp: $exp(-2*t)*step(t)$ Gaussian: $exp(-t^2)$ Cosine: $cos(3*t)$ Sine: $sin(2*t)$ Two-sided exp: $exp(-abs(t))$ Impulse δ(t): $delta(t)$ Shifted δ(t-2): $delta(t-2)$ Rect pulse: $rect(t/4)$ Sinc: $sinc(t)$ Unit step u(t): $step(t)$ Damped cosine: $exp(-t)*cos(2*t)*step(t)$ t²·e⁻³ᵗ: $t^2*exp(-3*t)*step(t)$

$δ(t)$ $1$

$1$ $2πδ(ω)$

$e^{−at}u(t)$ $\frac{1}{a+i\omega}$

$e^{−a|t|}$ $\frac{2a}{a^2+\omega^2}$

$e^{−at^2}$ $\sqrt{\frac{\pi}{a}}e^{-\frac{\omega^2}{4a}}$

$\cos(\omega_0t)$ $\pi[\delta(\omega{-}\omega_0){+}\delta(\omega{+}\omega_0)]$

$\sin(\omega_0t)$ $-i\pi[\delta(\omega{-}\omega_0){-}\delta(\omega{+}\omega_0)]$

$\mathrm{rect}(t/\tau)$ $\tau\,\mathrm{sinc}(\omega\tau/2)$

$u(t)$ $\pi\delta(\omega)+\frac{1}{i\omega}$

$t^ne^{-at}u(t)$ $\frac{n!}{(a+i\omega)^{n+1}}$

Linearity $af+bg → aF+bG$

Time Shift $f(t-t₀) → e^{-iωt₀}F(ω)$

Freq Shift $e^{iω₀t}f(t) → F(ω-ω₀)$

Scaling $f(at) → \frac{1}{|a|}F(ω/a)$

Differentiation $f'(t) → iωF(ω)$

Integration $∫f → F(ω)/(iω)$

Convolution $f*g → F·G$

Multiplication $f·g → F*G/(2π)$

Parseval $∫|f|² = \frac{1}{2π}∫|F|²$

Duality $F(t) → 2πf(-ω)$

𝓕Fourier Transform Calculator

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