Symmetries for Real Signals

If a time-domain signal $x$ is real, then its Fourier transform $X$ is conjugate symmetric (Hermitian):

$$\zbox{x \in \mathbb{R}^N\Leftrightarrow X(-k) = \overline{X(k)}}$$

or

$$\zbox{\hbox{Real} \leftrightarrow \hbox{Hermitian}}$$

Hermitian symmetry implies

• Real part Symmetric (even):
     re$$\left\{X(-k)\right\} =$$   re$$\left\{X(k)\right\}$$

• Imaginary part Antisymmetric (skew-symmetric, odd):
     im$$\left\{X(-k)\right\} = -$$im$$\left\{X(k)\right\}$$

• Magnitude Symmetric (even):
  $$\vert X(-k)\vert = \vert X(k)\vert$$

• Phase Antisymmetric (odd):
  $$\angle X(-k) = - \angle X(k)$$