For a complex exponential function where is a complex number given by , we call as the complex frequency.

Expansion with Euler’s Formula

We can expand the complex exponential function with the complex exponential:

Since the conjugate of is , then

and

The Complex Frequency Plane

  • The real part determines the rate of growth or decay of the signal’s amplitude.
    • If , the amplitude grows exponentially.
    • If , the amplitude decays exponentially.
    • If , the amplitude remains constant.
  • The imaginary part determines the frequency of oscillation

For signals whose complex frequencies lie on the real axis (), the frequency of oscillation is zero. And consequently, these signals are monotonically increasing or decreasing. Conversely, signals whose complex frequencies lie on the imaginary axis () are sinusoids with constant amplitudes.

Here are some examples:

  1. complex frequency examples.png

Reference

  • Linear Systems and Signals, 3rd Edition, Chapter 1.3