Unit Circle Visualization
The unit circle is essential in understanding the relationship between complex exponentials and sinusoids. This interactive visualization demonstrates how a point moving around the unit circle relates to sine and cosine waves.
Angle (radians):
0
\( e^{j\theta} = \cos(\theta) + j\sin(\theta) = 1 + j0 \)
Understanding the Unit Circle in DSP
In Digital Signal Processing, the unit circle in the complex plane has special significance:
Every point on the unit circle corresponds to a complex sinusoid with a specific frequency
The angle \(\theta\) determines the frequency of the sinusoid: \(\omega = \theta / T\), where \(T\) is the sampling period
Moving counterclockwise around the circle represents increasing frequency
A full rotation around the circle represents the sampling frequency \(f_s\)
Complex exponentials \(e^{j\omega n}\) are eigenfunctions of LTI systems
Common Angles and Their Values
Angle (radians)
Angle (degrees)
\(\cos(\theta)\)
\(\sin(\theta)\)
\(e^{j\theta}\)
0
0°
1
0
1 + j0
\(\pi/4\)
45°
\(\sqrt{2}/2\)
\(\sqrt{2}/2\)
\(\sqrt{2}/2 + j\sqrt{2}/2\)
\(\pi/2\)
90°
0
1
0 + j1
\(\pi\)
180°
-1
0
-1 + j0
\(3\pi/2\)
270°
0
-1
0 - j1
\(2\pi\)
360°
1
0
1 + j0