Fourier Transform Calculator

Compute continuous Fourier transform formulas and magnitude/phase at a selected angular frequency.

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What This Calculator Does

Compute continuous Fourier transform formulas and magnitude/phase at a selected angular frequency.

It combines Signal Type, Amplitude A, Pulse Width T, Gaussian sigma to estimate F(ω) Formula, F(ω) Complex Value, Magnitude |F(ω)|.

Formula & Method

Core equation:

F(\omega)=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}dt

using common analytic transform pairs.

Notation used in the formulas: $R$ = F(ω) Formula; $x_{1}$ = Signal Type; $x_{2}$ = Amplitude A; $x_{3}$ = Pulse Width T; $x_{4}$ = Gaussian sigma; $x_{5}$ = Sinc parameter b; $x_{6}$ = Decay a.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use this calculator for quick scenario analysis. Start with baseline values, change one driver at a time, and compare how sensitive the results are to each input shown above.

Reference Book

Signals and Systems

Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab · Pearson

A classic reference for convolution, Fourier analysis, Laplace transforms, z-transforms, sampling, and LTI systems.

View Book

Inputs Used

Signal Type: Used directly in the calculation.
Amplitude A: Used directly in the calculation.
Pulse Width T: Used directly in the calculation.
Gaussian sigma: Used directly in the calculation.
Sinc parameter b: Used directly in the calculation.
Decay a: Used directly in the calculation.
Angular Frequency ω: Used directly in the calculation.

Related Learning

Use these guided proofs, examples, and explanations to go deeper on the ideas behind this tool.

Test convergence of an improper integral
An improper integral is decided by a limit, not by the antiderivative alone. This example tests $\int_1^{\infty} \frac{1}{x^2}\,dx$ and compares it with the divergent harmonic-tail integral.
What the definite integral means geometrically
The definite integral $\int_a^b f(x)\,dx$ measures signed area between a curve and the $x$-axis. This page builds that idea from Riemann sums, connects it to antiderivatives, and shows how to read integral notation.

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