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Conjugation and Reversal



Theorem: For any $ x\in{\bf C}^N$ ,

$\displaystyle \zbox {\overline{x} \;\longleftrightarrow\;\hbox{\sc Flip}(\overline{X}).}
$



Proof:

\begin{eqnarray*}
\hbox{\sc DFT}_k(\overline{x})
&\isdef & \sum_{n=0}^{N-1}\overline{x(n)} e^{-j 2\pi nk/N}
= \sum_{n=0}^{N-1}\overline{x(n) e^{j 2\pi nk/N}} \\ [5pt]
&=& \overline{\sum_{n=0}^{N-1}x(n) e^{-j 2\pi n(-k)/N}}
\isdef \hbox{\sc Flip}_k(\overline{X})
\end{eqnarray*}



Theorem: For any $ x\in{\bf C}^N$ ,

$\displaystyle \zbox {\hbox{\sc Flip}(\overline{x}) \;\longleftrightarrow\;\overline{X}.}
$



Proof: Making the change of summation variable $ m\isdeftext N-n$ , we get

\begin{eqnarray*}
\hbox{\sc DFT}_k(\hbox{\sc Flip}(\overline{x}))
&\isdef & \sum_{n=0}^{N-1}\overline{x(N-n)} e^{-j 2\pi nk/N}
= \sum_{m=N}^{1} \overline{x(m)} e^{-j 2\pi (N-m)k/N} \\
&=& \sum_{m=0}^{N-1}\overline{x(m)} e^{j 2\pi m k / N}
= \overline{\sum_{m=0}^{N-1}x(m) e^{-j 2\pi m k/N}}
\isdef \overline{X(k)}.
\end{eqnarray*}



Theorem: For any $ x\in{\bf C}^N$ ,

$\displaystyle \zbox {\hbox{\sc Flip}(x) \;\longleftrightarrow\;\hbox{\sc Flip}(X).}
$



Proof:

\begin{eqnarray*}
\hbox{\sc DFT}_k[\hbox{\sc Flip}(x)] &\isdef & \sum_{n=0}^{N-1}x(N-n) e^{-j 2\pi nk/N}
= \sum_{m=0}^{N-1}x(m) e^{-j 2\pi (N-m)k/N} \\
&=& \sum_{m=0}^{N-1}x(m) e^{j 2\pi mk/N} \isdef X(-k) \isdef \hbox{\sc Flip}_k(X)
\end{eqnarray*}

Corollary: For any $ x\in{\bf R}^N$ ,

$\displaystyle \zbox {\hbox{\sc Flip}(x) \;\longleftrightarrow\;\overline{X}}$   $\displaystyle \mbox{($x$\ real).}$



Proof: Picking up the previous proof at the third formula, remembering that $ x$ is real,

$\displaystyle \sum_{m=0}^{N-1}x(m) e^{j 2\pi mk/N}
= \overline{\sum_{m=0}^{N-1}\overline{x(m)} e^{-j 2\pi mk/N}}
= \overline{\sum_{m=0}^{N-1}x(m) e^{-j 2\pi mk/N}}
\isdef \overline{X(k)}
$

when $ x(m)$ is real.

Thus, conjugation in the frequency domain corresponds to reversal in the time domain. Another way to say it is that negating spectral phase flips the signal around backwards in time.

Corollary: For any $ x\in{\bf R}^N$ ,

$\displaystyle \zbox {\hbox{\sc Flip}(X) = \overline{X}}$   $\displaystyle \mbox{($x$\ real).}$



Proof: This follows from the previous two cases.



Definition: The property $ X(-k)=\overline{X(k)}$ is called Hermitian symmetry or ``conjugate symmetry.'' If $ X(-k)=-\overline{X(k)}$ , it may be called skew-Hermitian.

Another way to state the preceding corollary is

$\displaystyle \zbox {x\in{\bf R}^N\;\longleftrightarrow\;X\;\mbox{is Hermitian}.}
$


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA