Editing Discrete Fourier transform (section)

=== DFT of real and purely imaginary signals ===
* If <math>x_0, \ldots, x_{N-1}</math> are [[real number]]s, as they often are in practical applications, then the DFT <math>X_0, \ldots, X_{N-1}</math> is [[Even and odd functions|even symmetric]]:
:<math>x_n \in \mathbb{R} \quad \forall n \in \{0,\ldots,N-1 \} \implies X_k = X_{-k \mod N}^* \quad \forall k \in \{0,\ldots,N-1 \}</math>, where <math>X^*\,</math> denotes [[Complex conjugate|complex conjugation]].

It follows that for even <math>N</math> <math>X_0</math> and <math>X_{N/2}</math> are real-valued, and the remainder of the DFT is completely specified by just <math>N/2-1</math> complex numbers.
* If <math>x_0, \ldots, x_{N-1}</math> are purely imaginary numbers, then the DFT <math>X_0, \ldots, X_{N-1}</math> is [[Even and odd functions|odd symmetric]]:
:<math>x_n \in i \mathbb{R} \quad \forall n \in \{0,\ldots,N-1 \} \implies X_k = -X_{-k \mod N}^* \quad \forall k \in \{0,\ldots,N-1 \}</math>, where <math>X^*\,</math> denotes [[Complex conjugate|complex conjugation]].