Symmetry of the DTFT for Real Signals
Most (if not all) of the signals we deal with in practice are real signals. Here we note some spectral symmetries associated with real signals.DTFT of Real Signals
The previous section established that the spectrum

![]() |
(3.16) |
I.e.,
![]() |
(3.17) |
In other terms, if a signal

- The real part is even, while the imaginary part is odd:
- The magnitude is even, while the phase is odd:
Real Even (or Odd) Signals
If a signal is even in addition to being real, then its DTFT is also real and even. This follows immediately from the Hermitian symmetry of real signals, and the fact that the DTFT of any even signal is real:![\begin{eqnarray*}
\hbox{\sc DTFT}_\omega(x)
& \isdef & \sum_{n=-\infty}^{\infty}x(n) e^{-j\omega n}\\
& = & \sum_{n=-\infty}^{\infty}x(n) \left[\cos(\omega n) + j\sin(\omega n)\right]\\
& = & \sum_{n=-\infty}^{\infty}x(n) \cos(\omega n) + j\sum_{n=-\infty}^{\infty}x(n)\sin(\omega n)\\
& = & \sum_{n=-\infty}^{\infty}x(n) \cos(\omega n)\\
& = & \hbox{real and even}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img142.png)



![\begin{eqnarray*}
\hbox{\sc Flip}(x) & = & x \qquad \hbox{($x(-n)=x(n)$)}\\
\overline{x} & = & x\\ [5pt]
\hbox{\sc Flip}(X) & = & X\\
\overline{X} & = & X\\
\angle X(\omega) & =& 0 \hbox{ or } \pi
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/sasp2/img145.png)



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Shift Theorem for the DTFT
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Time Reversal