As introduced in §
3.12, the
complex numbers

are called the
th roots of unity because each of them satisfies
In particular,

is called a
primitive
th root of unity.
6.2
The

th roots of unity are plotted in the
complex plane in
Fig.
6.1 for

. It is easy to find them graphically
by dividing the unit circle into

equal parts using

points, with
one point anchored at

, as indicated in Fig.
6.1. When

is even, there will be a point at

(corresponding to a
sinusoid
with frequency at exactly half the
sampling rate), while if

is
odd, there is no point at

.
Figure 6.1:
The
roots of unity for
.
![\includegraphics[width=\twidth]{eps/dftfreqs}](http://www.dsprelated.com/josimages_new/mdft/img1016.png) |
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