Fourier Transform Calculator

Step-by-Step Forward & Inverse Symbolic CAS Free · No Signup

Compute forward and inverse Fourier transforms with detailed step-by-step solutions. Features a complete Fourier pairs reference table, frequency domain analysis, 2D graphs, and a built-in Python compiler. Essential for signal processing, communications, and image analysis.

Fourier Transform
Enter a function of t
Type an expression above…
f(t)F(ω)Notes
Impulse
DC signal
a > 0
a > 0
Gaussian
Rect↔Sinc
Sinc↔Rect
Cosine
a > 0
t^2    w^3    (w+1)^2
sin(2*pi*t)    cos(pi*t)
e^(-3*t)    exp(-t^2)
Abs(t)    Abs(w)
Heaviside(t-2)    DiracDelta(t)
1/(1+w^2)    exp(-pi*w^2)
Multiplication: Use * explicitly: t*exp(-t) not texp(-t)
Powers: t^2 or (w+1)^2
Constants: pi, e, I (imaginary unit)

Result

ℱ

Enter an expression and click Compute

Compute forward or inverse Fourier transforms with step-by-step solutions.

f(t) vs t

Compute a transform to see the time-domain plot.

Python Compiler

What is the Fourier Transform?

The Fourier transform decomposes a time-domain signal f(t) into its constituent frequencies, producing a frequency-domain representation F(ω). It is defined as F{f(t)} = ∫−∞∞ f(t) e−2πiωt dt, where ω is the frequency variable.

The Fourier transform is fundamental in signal processing, communications, image analysis, quantum mechanics, and many areas of applied mathematics and engineering.

Key Properties

Linearity

F{af(t) + bg(t)} = aF(ω) + bG(ω). The transform distributes over addition and scalar multiplication.

Time-Shifting

F{f(t−t0)} = e−2πiωt0F(ω). A time delay corresponds to a phase shift in frequency.

Frequency-Shifting

F{e2πif0tf(t)} = F(ω−f0). Modulation in time shifts the spectrum in frequency.

Parseval's Theorem

∫|f(t)|²dt = ∫|F(ω)|²dω. Total energy is preserved between time and frequency domains.

Applications

📡

Signal Processing

Analyze frequency content of signals, design filters, and perform spectral analysis for communications systems.

🖼

Image Processing

2D Fourier transforms enable image compression (JPEG), edge detection, and spatial frequency filtering.

🎵

Audio Analysis

Spectrograms, pitch detection, audio equalization, and noise reduction all rely on Fourier analysis.

📱

Communications

OFDM, modulation schemes, bandwidth analysis, and channel characterization use Fourier techniques.

Frequently Asked Questions

The Fourier transform decomposes a time-domain signal f(t) into its constituent frequencies, producing a frequency-domain representation F(ω). It is defined as F{f(t)} = integral from -infinity to infinity of f(t)*e^(-2*pi*i*omega*t) dt. It is fundamental in signal processing, physics, and engineering.
The forward Fourier transform converts a time-domain function f(t) to the frequency domain F(ω). The inverse Fourier transform does the reverse, converting F(ω) back to f(t). Together they allow analysis and synthesis of signals in both domains.
The frequency domain is a representation of a signal in terms of its frequency components rather than time. The Fourier transform maps signals from the time domain to the frequency domain, revealing the amplitude and phase of each frequency present in the signal.
The Laplace transform uses a complex variable s = σ + jω and integrates from 0 to infinity (one-sided). The Fourier transform uses purely imaginary frequency ω and integrates from -infinity to infinity. The Fourier transform can be seen as a special case of the Laplace transform evaluated on the imaginary axis.
The convolution theorem states that the Fourier transform of a convolution of two functions equals the product of their individual Fourier transforms: F{f*g} = F(ω)·G(ω). This property is fundamental in signal processing and filter design.
Yes, this calculator is completely free with no signup required. You get symbolic computation via SymPy, step-by-step solutions, 2D graphs, LaTeX export, and a built-in Python compiler.

🔥 Explore More Math

ℒ

Laplace Transform Calculator

Forward & inverse Laplace transforms with steps and ROC
*

Convolution Calculator

Continuous & discrete convolution with steps and graphs
Z

Z-Transform Calculator

Forward & inverse Z-transforms for discrete-time signals

Support This Free Tool

Every coffee helps keep the servers running. Every book sale funds the next tool I'm dreaming up. You're not just supporting a site — you're helping me build what developers actually need.

500K+ users
200+ tools
100% private
☕ One-time support Buy me a coffee 📚 Learn & support 9-Book Bundle - $9 𝕏 Stay updated Follow @anish2good
Privacy Guarantee: Private keys you enter or generate are never stored on our servers. All tools are served over HTTPS.