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 as $t$ increases, while

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

is clockwise circular motion.

We may call a complex sinusoid $e^{j\omega t}$ a positive-frequency sinusoid when $\omega>0$. Similarly, we may 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.