Orthogonality of Sinusoids
A key property of sinusoids is that they are orthogonal at different frequencies. That is,

For length sampled sinusoidal signal segments, such as used
by the DFT, exact orthogonality holds only for the harmonics of
the sampling-rate-divided-by-
, i.e., only for the frequencies (in Hz)



The complex sinusoids corresponding to the frequencies are


Nth Roots of Unity
As introduced in §3.12, the complex numbers


![$\displaystyle \left[W_N^k\right]^N = \left[e^{j\omega_k T}\right]^N
= \left[e^{j k 2\pi/N}\right]^N = e^{j k 2\pi} = 1.
$](http://www.dsprelated.com/josimages_new/mdft/img1010.png)


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
.
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
.
Next Section:
Orthogonality of the DFT Sinusoids
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Geometric Series