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Roots of Unity
Since
, we can write
where
is any positive integer.
The
complex numbers
are called
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 under
ordinary complex multiplication.
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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