Symmetry of the DTFT for Real Signals

Most (if not all) of the signals we deal with in practice are real signals. The Fourier transform of real signals exhibits conjugate symmetry. That is,

$$\zbox {x(n)\in{\bf R}\Leftrightarrow X(-\omega) = \overline{X(\omega)}}$$

In other terms, if a signal $x(n)$ is real, its spectrum is Hermitian, or ``conjugate symmetric.''

Hermitian spectra have the following equivalent characterizations:

• The real part is even, while the imaginary part is odd:
  $$\begin{eqnarray*} \mbox{re}\left\{X(-\omega)\right\} &=& \mbox{re}\left\{X(\omeg... ...\left\{X(-\omega)\right\} &=& -\mbox{im}\left\{X(\omega)\right\} \end{eqnarray*}$$
  

• The magnitude is even, while the phase is odd:
  $$\begin{eqnarray*} \left\vert X(-\omega)\right\vert &=& \left\vert X(\omega)\right\vert\\ \angle{X(-\omega)} &=& -\angle{X(\omega)} \end{eqnarray*}$$
  

Note that an even function is symmetric about argument zero while an odd function is antisymmetric about argument zero.