Figure 1.8 shows Euler's relation graphically as it applies to
sinusoids. A point traveling with uniform velocity around a circle
with radius 1 may be represented by
in
the complex plane, where is time and is the number of
revolutions per second. The projection of this motion onto the
horizontal (real) axis is
, and the projection onto
the vertical (imaginary) axis is
. For
*discrete-time* circular motion, replace by to get
which may be interpreted as a
point which jumps an arc length
radians along the circle
each sampling instant.

For circular motion to ensue, the sinusoidal motions must be at the same frequency, one-quarter cycle out of phase, and perpendicular (orthogonal) to each other. (With phase differences other than one-quarter cycle, the motion is generally elliptical.)

The converse of this is also illuminating. Take the usual circular motion which spins counterclockwise along the unit circle as increases, and add to it a similar but clockwise circular motion . This is shown in Fig.1.9. Next apply Euler's identity to get

Thus,

This statement is a graphical or geometric interpretation of Eq. (1.11). A similar derivation (subtracting instead of adding) gives the

We call
a *positive-frequency sinusoidal
component* when
, and
is the
corresponding *negative-frequency component*. Note that both
*sine* and *cosine* signals have equal-amplitude positive- and
negative-frequency components (see also [83,53]). This
happens to be true of every *real* signal (*i.e.*, non-complex). To
see this, recall that every signal can be represented as a sum of
complex sinusoids at various frequencies (its *Fourier
expansion*). For the signal to be real, every positive-frequency
complex sinusoid must be summed with a negative-frequency sinusoid of
equal amplitude. In other words, any counterclockwise circular motion
must be matched by an equal and opposite clockwise circular motion in
order that the imaginary parts always cancel to yield a real signal
(see Fig.1.9). Thus, a real signal always has a magnitude
spectrum which is symmetric about 0 Hz. Fourier symmetries such as
this are developed more completely in [83].

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