Flip Operator

We define the flip operator by

$$_n(x) \isdef x(-n) \protect$$ (7.1)

which, by modulo indexing, is also $x(N-n)$. The 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 vertical axis, as shown in Fig. 7.1b. The interpretation of Fig. 7.1b is usually the one we want, and the 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: Illustration of $x$ and $\protect$FLIP$(x)$ for $N=5$ and two domain interpretations: a) $n\in [0,N-1]$. b) $n\in [-(N-1)/2, (N-1)/2]$.