The sampled
sinusoids generated by integer powers of the
roots of
unity are plotted in Fig.
6.2. These are the sampled sinusoids

used by the
DFT. Note that taking successively higher integer powers of the
point

on the unit circle
generates samples of the

th
DFT sinusoid, giving
![$ [W_N^k]^n$](http://www.dsprelated.com/josimages_new/mdft/img1018.png)
,

. The

th sinusoid generator

is in turn
the

th

th root of unity (

th power of the primitive

th root
of unity

).

Note that in Fig.
6.2 the range of

is taken to be
![$ [-N/2,N/2-1] = [-4,3]$](http://www.dsprelated.com/josimages_new/mdft/img1020.png)
instead of
![$ [0,N-1]=[0,7]$](http://www.dsprelated.com/josimages_new/mdft/img1021.png)
. This is the most
``physical'' choice since it corresponds with our notion of ``
negative
frequencies.'' However, we may add any integer multiple of

to

without changing the sinusoid indexed by

. In other words,

refers to the same sinusoid

for all integers

.
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