DFT Sinusoids

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
,
. 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 instead of . 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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University