DTFT of Real Signals

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

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

I.e.,

$$\zbox {x(n)\in\mathbb{R}\;\longleftrightarrow\;X(-\omega) = \overline{X(\omega)}.}$$ (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:
  $$\begin{eqnarray*} \mbox{re}\left\{X(-\omega)\right\} &=& \mbox{re}\left\{X(\omega)\right\}\\ \mbox{im}\left\{X(-\omega)\right\} &=& -\mbox{im}\left\{X(\omega)\right\} \end{eqnarray*}$$
  

• 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.