Frequency Domain

Frequency is usually seen as the rate of rotation of a complex sinusoid with time in the $IQ$ plane.

• From the magnitude perspective, this rate of rotation can be changed from very slow (close to $0$) to as fast as possible (close to $+\infty$).

• From the direction perspective, a clockwise rotation implies a negative frequency.

In view of the above observations, the complete range of frequencies of a complex sinusoid is from $-\infty$ to $+\infty$.

What does the frequency domain represent?

When a signal or function is drawn in the frequency domain, the graph depicts the complex magnitudes of those complex sinusoids with frequencies present in that signal.

We begin with the simplest case of a complex sinusoid, then move towards a real sinusoid, and then, handle the most general case of an arbitrary signal later.

Complex sinusoid

Since one complex sinusoid has a single frequency, it is drawn as a single narrow impulse on the $ IQ$ plane in the frequency domain in the figure below. The impulse is on $+F$ axis if the direction of rotation is counterclockwise and on the $-F$ axis when the direction of rotation is clockwise (i.e., the impulse is at $-F$ in the figure due to the clockwise rotation in the time domain).