Symmetry of the DTFT for Real Signals

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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 of every real signal satisfies

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

(3.16)

I.e.,

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

(3.17)

In other terms, if a signal

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.

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:

$\begin{eqnarray*} \hbox{\sc DTFT}_\omega(x) & \isdef & \sum_{n=-\infty}^{\infty}x(n) e^{-j\omega n}\\ & = & \sum_{n=-\infty}^{\infty}x(n) \left[\cos(\omega n) + j\sin(\omega n)\right]\\ & = & \sum_{n=-\infty}^{\infty}x(n) \cos(\omega n) + j\sum_{n=-\infty}^{\infty}x(n)\sin(\omega n)\\ & = & \sum_{n=-\infty}^{\infty}x(n) \cos(\omega n)\\ & = & \hbox{real and even} \end{eqnarray*}$

This is true since cosine is even, sine is odd, even times even is even, even times odd is odd, and the sum over all samples of an odd signal is zero. I.e.,

$\begin{eqnarray*} \sum_{n=-\infty}^{\infty}x(n)\cos(\omega n) &=& \sum_{n=-\infty}^{\infty}\hbox{(even in $n$)}\;\cdot\;\hbox{(doubly even)}\\ &=& \sum_{n=-\infty}^{\infty}\hbox{(doubly even)} = \hbox{(even in $\omega$)} \end{eqnarray*}$

and

$\begin{eqnarray*} \sum_{n=-\infty}^{\infty}x(n)\sin(\omega n) &=& \sum_{n=-\infty}^{\infty}\hbox{(even in $n$)}\;\cdot\;\hbox{(doubly odd)}\\ &=& \sum_{n=-\infty}^{\infty}\hbox{(doubly odd)} = 0. \end{eqnarray*}$

If is real and even, the following are true:

$\begin{eqnarray*} \hbox{\sc Flip}(x) & = & x \qquad \hbox{($x(-n)=x(n)$)}\\ \overline{x} & = & x\\ [5pt] \hbox{\sc Flip}(X) & = & X\\ \overline{X} & = & X\\ \angle X(\omega) & =& 0 \hbox{ or } \pi \end{eqnarray*}$

Similarly, if a signal is odd and real, then its DTFT is odd and purely imaginary. This follows from Hermitian symmetry for real signals, and the fact that the DTFT of any odd signal is imaginary.

$\begin{eqnarray*} \hbox{\sc DTFT}_\omega(x) & \isdef & \sum_{n=-\infty}^{\infty}x(n) e^{-j\omega n}\\ & = & \sum_{n=-\infty}^{\infty}x(n) \cos(\omega n) + j\sum_{n=-\infty}^{\infty}x(n)\sin(\omega n)\\ & = & j\sum_{n=-\infty}^{\infty}x(n) \sin(\omega n)\\ & = & \hbox{imaginary and odd} \end{eqnarray*}$

where we used the fact that

$\begin{eqnarray*} \sum_{n=-\infty}^{\infty}x(n)\cos(\omega n) &=& \sum_{n=-\infty}^{\infty}\hbox{(odd in $n$)}\;\cdot\;\hbox{(doubly even)}\\ &=& \sum_{n=-\infty}^{\infty}\hbox{(odd in $n$, even in $\omega$)} = 0 \end{eqnarray*}$

and

$\begin{eqnarray*} \sum_{n=-\infty}^{\infty}x(n)\sin(\omega n) &=& \sum_{n=-\infty}^{\infty}\hbox{(odd in $n$)}\;\cdot\;\hbox{(doubly odd)}\\ &=& \sum_{n=-\infty}^{\infty}\hbox{(even in $n$, odd in $\omega$)} = \hbox{(odd in $\omega$)}. \end{eqnarray*}$

Next Section:
Shift Theorem for the DTFT
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Time Reversal