Conjugation and Reversal

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Conjugation and Reversal

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page  TOC  INDEX  Next page >>

Conjugation and Reversal



Theorem: For any $x\,

\

Proof:

\





Theorem: For any $x\,

\

Proof: Making the change of summation variable $m\, we get

\





Theorem: For any $x\,

\

Proof:

\



Corollary: For any $x\,

\

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

\

when $x(n)$ 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\,

\

Proof: This follows from the previous two cases.



Definition: The property $X(-k)=\ is called Hermitian symmetryor ''conjugate symmetry.'' If $X(-k)=-\, it may be calledskew-Hermitian.

Another way to state the preceding corollary is

\

<< Previous page  TOC  INDEX  Next page >>