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Fourier Transform

Decomposes a function or signal into the frequencies that make it up — converting between the time (or space) domain and the frequency domain.

Formula

\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx

Variables

f(x)Original signal

f̂(ξ)Frequency-domain representation

ξFrequency

iImaginary unit

Example

A musical chord in time → sharp peaks at each note’s frequency

How it works

The Fourier transform rewrites a function as a weighted mixture of complex exponentials, where each frequency the size of the coefficient tells you how much of that frequency is present. It works because complex exponentials are eigenfunctions of shift-invariant operations and form an orthogonal basis for many spaces of signals, so projecting onto them separates oscillatory content cleanly. In practice, the integral measures correlation with a pure wave at frequency , and the factor e^{-2pi i x } is chosen so the transform and its inverse line up neatly with common engineering conventions.

When to use it

Use it when you want to analyze oscillations, filter noise, solve differential equations, or understand how a signal behaves in the frequency domain. It is especially useful in physics, signal processing, optics, and PDEs where derivatives or convolutions become simpler after transforming.

Worked example, step by step

Take f(x) = e^{-\pi x^2}, a standard Gaussian. We compute \hat{f}(\xi) = \int_{-\infty}^{\infty} e^{-\pi x^2} e^{-2\pi i x \xi} \, dx. Combine the exponents: -\pi x^2 - 2\pi i x\xi = -\pi(x^2 + 2ix\xi). Complete the square: x^2 + 2ix\xi = (x+i\xi)^2 + \xi^2. So the integrand becomes e^{-\pi[(x+i\xi)^2 + \xi^2]} = e^{-\pi \xi^2} e^{-\pi (x+i\xi)^2}. Shifting the contour does not change the Gaussian integral, so \int_{-\infty}^{\infty} e^{-\pi (x+i\xi)^2} dx = \int_{-\infty}^{\infty} e^{-\pi u^2} du = 1. Therefore \hat{f}(\xi) = e^{-\pi \xi^2}. Final answer: the Fourier transform of a Gaussian is another Gaussian, \hat{f}(\xi) = e^{-\pi \xi^2}.

Common mistakes to avoid

Forgetting the transform uses a complex exponential, so the result is generally complex-valued
Mixing up the sign or the 2\u03c0 convention in the exponential
Assuming the formula applies to every function without checking integrability or distributional interpretation

Did You Know?

The Fourier transform underpins JPEG images, MP3 audio, MRI scans and Wi-Fi — almost all modern signal processing.

fourier transform signal frequency analysis

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Reviewed & maintained by the FormulaWon editorial teamLast reviewed September 2026