International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 51-52   | 1 | 2 |

## Section 1.3.2.7.5. Properties of the discrete Fourier transform

G. Bricognea

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#### 1.3.2.7.5. Properties of the discrete Fourier transform

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The DFT inherits most of the properties of the Fourier transforms, but with certain numerical factors (Jacobians') due to the transition from continuous to discrete measure.

 (1) Linearity is obvious. (2) Shift property. If and  , where subtraction takes place by modular vector arithmetic in and , respectively, then the following identities hold: (3) Differentiation identities. Let vectors ψ and Ψ be constr­ucted from as in Section 1.3.2.7.3 , hence be related by the DFT. If designates the vector of sample values of at the points of , and the vector of values of at points of , then for all multi-indices  or equivalently (4) Convolution property. Let and (respectively ψ and Ψ) be related by the DFT, and define Then and Since addition on and is modular, this type of convolution is called cyclic convolution. (5) Parseval/Plancherel property. If ϕ, ψ, Φ, Ψ are as above, then (6) Period 4. When N is symmetric, so that the ranges of indices and can be identified, it makes sense to speak of powers of and . Then the standardized' matrices and are unitary matrices whose fourth power is the identity matrix (Section 1.3.2.4.3.4 ); their eigenvalues are therefore and .