Inverse Fourier Transforms Calculator

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Example

Created on 2024-06-20Asked by William Nguyen (Solvelet student)

Find the inverse Fourier transform of

F(\omega) = \frac{1}{j\omega + a}

, where

a

is a positive constant.

Solution

To find the inverse Fourier transform of

F(\omega) = \frac{1}{j\omega + a}

, we use the inverse Fourier transform formula:

f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t} \, d\omega

Substitute

F(\omega)

f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{e^{j\omega t}}{j\omega + a} \, d\omega

We recognize this as a standard form whose inverse Fourier transform is known:

\frac{1}{j\omega + a} \xrightarrow{\mathcal{F}^{-1}} e^{-at} u(t)

Thus, the inverse Fourier transform of

\frac{1}{j\omega + a}

is:

f(t) = e^{-at} u(t)

Where

u(t)

is the Heaviside step function. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Amelia Thomas on Solvelet

1. Compute the inverse Fourier transform of the function

F(\omega) = \frac{3}{2 + i\omega}

,2. Determine whether the function

g(x) = \sqrt{4 - x^2}

has an inverse, and if it does, find its formula.,

DefinitionFourier Expression and inverse fourier, A way to reconstruct the time-domain signal from the its frequency domain one. The basic operation that you need to perform in this context is the inverse Fourier transform, which converts the frequency components for the original signal. This is crucial in the fields of signal processing, telecommunication, or any domain where the frequencies of the signals are of immense importance. For instance, if F(ω) is the Fourier transform of some function f(t), then the inverse Fourier transform of F(ω) is f(t)=2π1∫−∞∞F(ω)eiωtdω.

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