Fourier transform calculator
A free online Fourier and inverse Fourier transform visualizer. Choose a signal and see both its time-domain graph and its frequency-domain magnitude spectrum side by side, with the frequency axis in hertz (Hz) or angular frequency (rad/s).
Forward and inverse transforms
In forward mode you pick a time signal — a rectangular or Gaussian pulse, a decaying exponential, a cosine — and the tool shows its spectrum X(f) = ∫ x(t) e^(−j2πft) dt. In inverse mode you shape a spectrum, such as an ideal low-pass rectangle, and see the time signal x(t) that it reconstructs — for the rectangle, a sinc.
Time–frequency reciprocity
A signal that is narrow in time has a wide spectrum, and a narrow spectrum corresponds to a signal spread out in time. Drag the width slider and watch the two graphs trade places — the same reciprocity behind the uncertainty principle and the bandwidth of a pulse.
Frequently asked questions
- What is the Fourier transform?
- The Fourier transform decomposes a time-domain signal into the frequencies it contains. X(f) = ∫ x(t) e^(−j2πft) dt gives the amplitude and phase of every frequency component, so you can see a signal's spectrum.
- What is the inverse Fourier transform?
- The inverse Fourier transform rebuilds a time signal from its spectrum: x(t) = ∫ X(f) e^(+j2πft) df. This tool's inverse mode lets you shape a spectrum (for example an ideal low-pass rectangle) and watch the corresponding time signal — a sinc — appear.
- What is the difference between Hz and rad/s on the frequency axis?
- Ordinary frequency f is in hertz (cycles per second); angular frequency ω is in radians per second, with ω = 2πf. The magnitude spectrum has the same shape either way — only the axis scale and label change. You can switch between them here.
- Why does a narrow pulse have a wide spectrum?
- This is the time–frequency reciprocity (a form of the uncertainty principle): the shorter a signal is in time, the more spread out its Fourier transform is in frequency, and vice-versa. Drag the width slider to watch it happen.