DTFT of Real Signals

The previous section established that the spectrum $ X$ of every real signal $ x$ satisfies

$\displaystyle \hbox{\sc Flip}(X)\eqsp \overline{X}.$ (3.16)


$\displaystyle \zbox {x(n)\in{\bf R}\;\longleftrightarrow\;X(-\omega) = \overline{X(\omega)}.}$ (3.17)

In other terms, if a signal $ x(n)$ is real, then its spectrum is Hermitian (``conjugate symmetric''). Hermitian spectra have the following equivalent characterizations:
  • The real part is even, while the imaginary part is odd:

\mbox{re}\left\{X(-\omega)\right\} &=& \mbox{re}\left\{X(\omega)\right\}\\
\mbox{im}\left\{X(-\omega)\right\} &=& -\mbox{im}\left\{X(\omega)\right\}

  • The magnitude is even, while the phase is odd:

\left\vert X(-\omega)\right\vert &=& \left\vert X(\omega)\right\vert\\
\angle{X(-\omega)} &=& -\angle{X(\omega)}

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

Next Section:
Real Even (or Odd) Signals