All of these properties of the discrete Fourier transform (DFT) are applicable for discrete-time signals that have a DFT. Meaning these properties of DFT apply to any generic signal x(n) for which an X(k) exists. (x(n) X(k)) where .

Property |
Mathematical Representation |

Linearity | a_{1}x_{1}(n)+a_{2}x_{2}(n) a_{1}X_{1}(k) + a_{2}X_{2}(k) |

Periodicity | if x(n+N) = x(n) for all n
then x(k+N) = X(k) for all k |

Time reversal | x(N-n) X(N-k) |

Duality | x(n) Nx[((-k))_{N}] |

Circular convolution | |

Circular correlation | For x(n) and y(n), circular correlation r_{xy}(l) is
r |

Circular frequency shift | x(n)e^{2πjln/N} X(k+l)
x(n)e |

Circular time shift | x((n-l))N X(k)e^{-2πjlk/N}
or X(k)W |

Multiplication | |

Complex conjugate | x*(n) X*(N-k) |

Symmetry | For even sequences,
X(k) = x(n)Cos(2πnk/N) For odd sequences, X(k) = x(n)Sin(2πnk/N) |

Parseval’s theorem | x(n).y*(n) = X(k).Y*(k) |

**Proofs of the properties of the discrete Fourier transform**