Flip Operator
We define the flip operator by
![$\displaystyle \hbox{\sc Flip}_n(x) \isdef x(-n), \protect$](http://www.dsprelated.com/josimages_new/mdft/img1145.png) |
(7.1) |
for all sample indices
![$ n\in{\bf Z}$](http://www.dsprelated.com/josimages_new/mdft/img79.png)
.
By
modulo indexing,
![$ x(-n)$](http://www.dsprelated.com/josimages_new/mdft/img1146.png)
is the same as
![$ x(N-n)$](http://www.dsprelated.com/josimages_new/mdft/img1147.png)
. The
![$ \hbox{\sc Flip}()$](http://www.dsprelated.com/josimages_new/mdft/img1148.png)
operator
reverses the order of samples
![$ 1$](http://www.dsprelated.com/josimages_new/mdft/img111.png)
through
![$ N-1$](http://www.dsprelated.com/josimages_new/mdft/img745.png)
of a sequence, leaving
sample 0 alone, as shown in Fig.
7.1a. Thanks to modulo
indexing, it can also be viewed as ``flipping'' the sequence about the
time 0, as shown in
Fig.
7.1b. The interpretation of Fig.
7.1b is usually the one we
want, and the
![$ \hbox{\sc Flip}$](http://www.dsprelated.com/josimages_new/mdft/img1149.png)
operator is usually thought of as ``time reversal''
when applied to a
signal ![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
or ``frequency reversal'' when applied to a
spectrum ![$ X$](http://www.dsprelated.com/josimages_new/mdft/img55.png)
.
figure[htbp]
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