Flip Operator

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Flip Operator

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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

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Flip Operator



Definition: We define the flip operator by

\

which, by modulo indexing, is $x(N-n)$. The $\ operator reverses the order of samples $1$ through $N-1$ of a sequence, leaving sample $0$alone, as shown in Fig. 8.1a. Thanks to modulo indexing, it can also be viewed as “flipping” the sequence about the vertical axis, as shown in Fig. 8.1b. The interpretation of Fig. 8.1b is usually the one we want, and the $\ operator is usually thought of as “time reversal” when applied to a signal $x$ or “frequency reversal” when applied to a spectrum $X$.
Figure 8.1:Illustration of $x$ and $\for $N=5$ and two different domain interpretations: a) $n\. b) $n\.
\

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