### Discrete Fourier transform (DFT) and Fast Fourier transform (FFT)

The Discrete Fourier transform (DFT) is obtained by decomposing a sequence of values into components of different frequencies. It converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence.

A Fast Fourier transform (FFT) is an algorithm that computes the DFT of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa.

Toolbox: DFT/IDFT calculatorSignal representation | Time domain | Frequency domain |
---|---|---|

DFT | ||

IDFT | ||

DFT/IDFT calculator (modify {x} or {_{n}X}, the other one is automatically updated)_{k} | ||

Time domain: {x}_{n} | ||

Frequency domain: {X}_{k} |

### Frequency domain analysis

The Fourier transform is commonly used to convert a time domain signal to the frequency domain.

The time domain signal can be seen as a sum of sine waves. A FFT will separate the constructing sine waves and return information about the frequency of these sine waves.

Toolbox: Frequency domain analysis**Be careful:** if f1 and f2 differs a lot (e.g. by more than 20 times), it will take very long time to process.

Time | Frequency | |
---|---|---|

*f*[Hz] =_{1}
| ||

*f*[Hz] =_{2}*A*=
| ||

Frequency domain analysis of user provided signal | ||

sampling frequency [Hz] = |