Explaining the 4D Complex Chirp Visualization
Posted on August 27, 2025
The visualization you see is an attempt to represent a four-dimensional concept in a 3D space. The signal is a complex exponential:
\[s(t) = e^{j\theta(t)} = \cos(\theta(t)) + j\sin(\theta(t))\]Here, the phase θ(t)
is not linear. Its derivative, the instantaneous frequency, increases over time. This is called a “chirp” signal.
The Axes
- Real Axis (X): Represents the real part,
cos(θ(t))
. - Imaginary Axis (Y): Represents the imaginary part,
sin(θ(t))
. - Frequency Axis (Z): This axis is used to represent the passage of time, during which the frequency increases, causing the spiral to tighten.
This creates the distinctive “chirping” spiral you see.