For a complex exponential function
Expansion with Euler’s Formula
We can expand the complex exponential function with the complex exponential:
Since the conjugate of
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.
- If
- The imaginary part
determines the frequency of oscillation
For signals whose complex frequencies lie on the real axis (
Here are some examples:
Related
- Laplace transform - transform a signal from time domain into the complex frequency domain
Reference
- Linear Systems and Signals, 3rd Edition, Chapter 1.3