Fast Fourier transform (FFT) illustrated

The FFT is an efficient algorithm to compute the discrete Fourier transform (DFT).

The FFT takes a time signal defined by discrete time points, e.g. N = 8 (left) and computes the spectrum (right).

The FFT transforms a time signal x to a frequency signal X. And vice versa, the IFFT transforms a frequency signal X to a time signal x.

FFT	IFFT
$X_{k} = \sum_{n = 0}^{N - 1} x_{n} e^{- j 2 π k \frac{n}{N}}$	$x_{k} = \frac{1}{N} \sum_{n = 0}^{N - 1} X_{n} e^{j 2 π k \frac{n}{N}}$

FFT / DFT and IFFT / DFT definitions

The spectrum analysis (FFT) and signal synthesis (IFFT) equations look quite similar apart from the term $\frac{1}{N}$ . This asymmetry leads to a power variation of the time and frequency signal.

See also

Create your individual signal spectra by using the FFT calculator. Click on the start button above!

Calculate the FFT of real and complex time domain signals. Plot time and frequency signals.

Enter the time domain samples. Press Submit to calculate the frequency domain result.

The data size must be a power of 2: 2, 4, 8, 16, 32...
The format is compatible with Excel: the sequence of complex numbers can be copied from and to spreadsheets.

Enter the time domain data – example formats. Or copy-paste values, e.g. from Excel.

Enter the time domain data - example formats. Or copy-paste values, e.g. from Excel.

Plot time and frequency signals

Some examples illustrate the Fast Fourier transform. Time signals and corresponding frequency signals are shown.

Time domain	Frequency domain
DC component	Spectral line at frequency f=0
Time signal - Dirac delta impulse	Constant spectrum
Time signal - cosine with frequency $\frac{f_{\max}}{2}$	Spectral lines at frequencies $\pm \frac{f_{\max}}{2}$
Time signal - cosine with frequency $f_{\max}$	Spectral line at frequency $f_{\max}$