Flip Operator

We define the flip operator by

$$\hbox{\sc Flip}_n(x) \isdef x(-n), \protect$$ (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) $n\in [-(N-1)/2, (N-1)/2]$ .