Circular Motion

Since the modulus of the complex sinusoid is constant, it must lie on a circle in the complex plane. For example,

$$x(t) = e^{j\omega t}$$

traces out counter-clockwise circular motion along the unit circle in the complex plane, while

$$\overline{x(t)} = e^{-j\omega t}$$

is clockwise circular motion.

We call a complex sinusoid of the form $e^{j\omega t}$, where $\omega>0$, a positive-frequency sinusoid. Similarly, we define a complex sinusoid of the form $e^{-j\omega t}$, with $\omega>0$, to be a negative-frequency sinusoid. Note that a positive- or negative-frequency sinusoid is necessarily complex.