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Mathematics of the DFT
    Fourier Theorems for the DFT
       Signal Operators
          Flip Operator

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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{\bf 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 $ \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]$.
\begin{figure}\input fig/flip.pstex_t \end{figure}

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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