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Flip Operator

We define the flip operator by

$\displaystyle \hbox{\sc Flip}_n(x) \isdef x(-n), \protect$ (7.1)

for all sample indices $ n\in\mathbb{Z}$ . By modulo indexing, $ x(-n)$ is the same as $ x(N-n)$ . The $ \hbox{\sc Flip}()$ operator reverses the order of samples $ 1$ through $ N-1$ of a sequence, leaving sample 0 alone, as shown in Fig.7.1a. Thanks to modulo indexing, it can also be viewed as ``flipping'' the sequence about the time 0, as shown in Fig.7.1b. The interpretation of Fig.7.1b is usually the one we want, and the $ \hbox{\sc Flip}$ operator is usually thought of as ``time reversal'' when applied to a signal $ x$ or ``frequency reversal'' when applied to a spectrum $ X$ .

Figure 7.1: Illustration of $ x$ and $ \protect\hbox{\sc Flip}(x)$ for $ N=5$ for the two main domain conventions: a) $ n\in [0,N-1]$ . b) $ n\in [-(N-1)/2, (N-1)/2]$ .

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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 © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University