Compute continuous, inverse, DFT, IDFT and FFT with real math. Live trapezoidal integration, native DFT formula, Cooley-Tukey FFT, plus magnitude, phase and interactive charts.
A Fourier Transform Calculator is a computational tool that converts a time-domain signal into its frequency-domain representation. This mathematical operation, named after the French mathematician Jean-Baptiste Joseph Fourier, is fundamental in signal processing, physics, engineering, and data analysis. The Fourier transform decomposes a function or signal into its constituent frequencies, revealing the spectral content that is often hidden in the time domain.
The calculator on this page supports multiple variants: the continuous Fourier transform (CFT) for analog signals, the Discrete Fourier Transform (DFT) for sampled data, and the Fast Fourier Transform (FFT) which is an efficient algorithm for computing the DFT. It also includes the inverse Fourier transform (IFT), which reconstructs the original signal from its frequency-domain representation. This makes it an essential tool for anyone working with waveforms, audio signals, image processing, or any application involving spectral analysis.
Using the Fourier Transform Calculator is straightforward. Select the transform type from the dropdown menu: Continuous Fourier Transform, Inverse Fourier Transform, DFT, IDFT, or FFT. For continuous and inverse transforms, enter a mathematical expression in terms of t (e.g., exp(-t^2) or sin(2*Math.PI*t)). The calculator validates your expression in real-time, providing instant feedback on syntax errors.
For discrete transforms (DFT, IDFT, FFT), enter a sequence of numbers separated by commas (e.g., 0,1,2,3,4,3,2,1). You can adjust parameters like the integration range L, the number of frequency points, and the integration steps for the continuous transforms. Once you click "Calculate FT", the tool computes the transform using trapezoidal integration for continuous cases or the Cooley-Tukey FFT algorithm for discrete cases. Results are displayed as interactive charts showing the time-domain signal, magnitude spectrum, phase plot, and real/imaginary components.
A Fourier transform table is a reference that lists common functions alongside their Fourier transforms. This is invaluable for engineers and physicists who frequently work with transforms. Some classic pairs include:
This calculator uses these principles internally, providing you with accurate transforms that match known analytical results. You can test these pairs directly by inputting the corresponding time-domain expressions and comparing the computed spectra with theoretical expectations.
Examples of Fourier Transform are everywhere in modern technology. In audio engineering, the Fourier transform is used for equalization, noise reduction, and audio compression (MP3, AAC). In telecommunications, it's the foundation of OFDM (Orthogonal Frequency Division Multiplexing) used in Wi-Fi and 4G/5G networks. In medical imaging, MRI and CT scans rely on Fourier techniques to reconstruct images from raw data.
Here are three practical examples you can try with this calculator:
sin(2*Math.PI*5*t) + sin(2*Math.PI*20*t). The magnitude spectrum will clearly show two peaks at 5 Hz and 20 Hz, demonstrating the transform's ability to separate superimposed signals.1/(1+t^10) or a step function) and observe the sinc-like spectrum. This helps understand how sharp transitions in time create broad frequency content.The Discrete Fourier Transform (DFT) computes the frequency spectrum of a finite sequence of samples. Its direct implementation has O(n²) complexity, which becomes impractical for large sequences. The Fast Fourier Transform (FFT) is an algorithm that computes the DFT in O(n log n) time using the Cooley-Tukey method, which recursively divides the problem into smaller sub-problems. This calculator implements both, allowing you to compare the results. The FFT is available for sequences whose length is a power of two; for other lengths, the calculator automatically falls back to the DFT.
This Fourier Transform Calculator stands out because it combines accuracy, speed, and visual feedback in a single, free tool. It features real-time input validation, multiple transform modes, and high-resolution charts that update instantly. Whether you're a student learning signal processing, an engineer working on a project, or a researcher analyzing data, this tool provides the functionality you need without requiring expensive software licenses. The built-in export options let you download results as text or copy them for further analysis.
Additionally, the calculator is fully responsive, works offline once loaded, and respects your privacy with no data sent to servers. It's built with performance in mind, using canvas-based rendering for charts and efficient numerical algorithms.
Choose your transform type and enter a time-domain expression or discrete sequence. Real-time validation ensures your input is correct.
The calculator uses trapezoidal integration for continuous transforms or the Cooley-Tukey FFT algorithm for discrete sequences, delivering fast, accurate results.
Explore the magnitude, phase, real, and imaginary components through interactive charts. Download results for offline analysis or reporting.
exp(-t^2) or sin(2*Math.PI*t)), set the integration range and frequency parameters, then click "Calculate FT". The tool will display the magnitude and phase spectra, along with the real and imaginary components.From Z-Transforms to Laplace Transforms, Gamma Functions to Gradient Calculators — find everything you need for your academic and professional projects.