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

This is true whether they are complex or real, and whatever amplitude
and phase they may have. All that matters is that the frequencies be
different. Note, however, that the durations must be infinity (in general).
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)
These are the only frequencies that have a
whole number
of periods in
samples (depicted in Fig.
6.2 for

).
6.1
The
complex sinusoids corresponding to the frequencies

are
These sinusoids are generated by the

th
roots of unity in the
complex plane.
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) |
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
![$ [W_N^k]^n$](http://www.dsprelated.com/josimages_new/mdft/img1018.png)
,

. 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
![$ [-N/2,N/2-1] = [-4,3]$](http://www.dsprelated.com/josimages_new/mdft/img1020.png)
instead of
![$ [0,N-1]=[0,7]$](http://www.dsprelated.com/josimages_new/mdft/img1021.png)
. 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 SinusoidsPrevious Section: Geometric Series