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Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Fourier Theorems for the DTFT
          Symmetry of the DTFT for Real Signals
             Real Even (or Odd) Signals

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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... ...nfty}^{\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=-\inf... ...\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=-\inf... ...y odd)}\\ &=& \sum_{n=-\infty}^{\infty}\hbox{(doubly odd)} = 0. \end{eqnarray*}

If $ x$ is real and even, the following are true:

\begin{eqnarray*} \hbox{\sc Flip}(x) & = & x \qquad \hbox{($x(-n)=x(n)$)}\\ \ov... ... \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... ...}^{\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=-\inf... ...um_{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=-\inf... ...hbox{(even in $n$, odd in $\omega$)} = \hbox{(odd in $\omega$)}. \end{eqnarray*}

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Previous: Symmetry of the DTFT for Real Signals
Next: Shift Theorem for the DTFT
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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