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📖 Reference Material

Fourier Transform Reference

Formulas, tables, and properties at your fingertips. Everything you need to look up — fast.

↔ 20+ pairs

Fourier Transform Pairs

Complete table of 20+ common Fourier transform pairs — impulse, rectangular, Gaussian, sinc, exponential, and more. Time and frequency domain side by side.

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⚙ 15 properties

Transform Properties

Linearity, shifting, scaling, convolution theorem, Parseval's theorem, differentiation, modulation, and 10+ more properties with formulas.

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📋 Quick ref

Cheat Sheet

One-page quick reference with key formulas, essential pairs, critical properties, and practical tips. Designed to be bookmarked or printed.

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Quick Summary

Continuous Fourier Transform (CFT)

The continuous Fourier transform decomposes an integrable function into its constituent frequencies.

Forward Transform

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

Inverse Transform

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

Discrete Fourier Transform (DFT)

The DFT maps a finite sequence of equally-spaced samples into a same-length sequence of complex sinusoidal coefficients.

Forward DFT

X[k] = \sum_{n=0}^{N-1} x[n]\, e^{-j 2\pi k n / N}, \quad k = 0, 1, \ldots, N-1

Inverse DFT

x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k]\, e^{j 2\pi k n / N}, \quad n = 0, 1, \ldots, N-1

Fourier Series

A periodic function with period T can be expressed as a sum of harmonically related sinusoids.

Complex Exponential Form

f(t) = \sum_{n=-\infty}^{\infty} c_n\, e^{j n \omega_0 t}, \quad \omega_0 = \frac{2\pi}{T}

Coefficients

c_n = \frac{1}{T} \int_{0}^{T} f(t)\, e^{-j n \omega_0 t}\, dt

Try It Interactively

See these formulas in action with our interactive tools.

FFT Calculator Fourier Visualizer Signal Generator