NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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. The
operator reverses the order of samples
through
of a sequence, leaving sample
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
or ''frequency reversal'' when applied to a spectrum
.
for 
and two different domain interpretations: a) 
. b)
.