# User:Zhennan/FTcomparison

The table shows members of Fourier transform, with the characteristics of input signal and output signal ^{[1]}.

Technique | Time domain | Frequency domain |
---|---|---|

Continuous Fourier Transform | Continuous, non-periodic | Continuous, non-periodic |

Fourier Series | Continuous, periodic | Discrete, non-periodic |

Discrete-time Fourier transform | Discrete, non-periodic | Continuous, periodic |

Discrete Fourier transform | Discrete, periodic | Discrete, periodic |

It shows that,

- Discrete signal in one domain means periodic signal in the other domain.

- Continuous signal in one domain means non-periodic signal in the other domain.

Briefly speaking, **Fourier series** is used to represent a periodic function by a discrete sum of complex exponentials, while **Fourier transform** is then used to represent a general, nonperiodic function by a continuous superposition or integral of complex exponentials. Fourier transform can be viewed as the limit of the Fourier series of a function with the period approaches to infinity, so the limits of integration change from one period to (−∞,∞). **Discrete-time Fourier transform** is sampling of Continuous Fourier Transform in time domain. And **Discrete Fourier transform** is even sampling of Discrete-time Fourier transform in frequency domain.

**Reference**