Roots of Unity
Since
for every integer
,
we
can write
These are the
th roots of unity. The special case

is
called a
primitive
th root of unity,
since integer powers
of it give all of the others:
The

th roots of unity are so frequently used that they are often
given a special notation in the
signal processing literature:
where

denotes a primitive

th root of
unity.
3.7 We may also call

a
generator of the
mathematical
group consisting of the

th roots of unity and
their products.
We will learn later that the
th roots of unity are used to generate
all the sinusoids used by the length-
DFT and its inverse.
The
th complex sinusoid used in a DFT of length
is given by
where

,

, and

is the
sampling interval in seconds.
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