Editing Discrete Fourier transform (section)

===Shift theorem===
Multiplying <math>x_n</math> by a ''linear phase'' <math>e^{\frac{i 2\pi}{N} nm}</math> for some integer ''m'' corresponds to a ''circular shift'' of the output <math>X_k</math>: <math>X_k</math> is replaced by <math>X_{k-m}</math>, where the subscript is interpreted [[modular arithmetic|modulo]] ''N'' (i.e., periodically).  Similarly, a circular shift of the input <math>x_n</math> corresponds to multiplying the output <math>X_k</math> by a linear phase. Mathematically, if <math>\{x_n\}</math> represents the vector '''x''' then

:if <math>\mathcal{F}(\{x_n\})_k=X_k</math>
:then <math>\mathcal{F}\left(\left\{ x_n \cdot e^{\frac{i 2\pi}{N}n m} \right\}\right)_k=X_{k-m}</math>
:and <math>\mathcal{F}\left(\left\{x_{n-m}\right\}\right)_k=X_k \cdot e^{-\frac{i 2\pi}{N}k m}</math>