Complex Numbers and Oscillations

Imaginary Numbers¶

Imaginary numbers offer a soloution to the following problem:

$$\sqrt{-1} = ?$$

In mathematics, the $i$ has been introduced to represent the imaginary unit. Since $i$ is also used for current, it is often replaces with $j$ in electrical engineering. The imaginary unit solves the following equation:

$$ \displaystyle i^{2}=-1 $$

Complex Numbers¶

A complex number is comprised of a real part $a$ and an imaginary part $b$:

$$ z = a + bi $$

The real part of $z$ is denoted as:

$$ \mathcal{Re}(z) = a$$

The imaginary part of $z$ is denoted as:

$$ \mathcal{Im}(z) = b$$

The Gaussian Plane¶

The Gaussian plane (or complex plane) is used to visualize complex numbers in a so-called Argend diagram. These diagrams help understand the basic functionality of complex numbers:

Within the complex plane, the complex number is treated as a vector.

Magnitude¶

The magnitude - the lenght of the vector that represents the complex number - is calculated via Pythagoras' theorem: $$|z| = \sqrt{a^2 + b^2}$$

Phase (Argument)¶

The phase is the counter-clockwise angle between the real axis and the vector:

$$ \varphi = \tan^{-1}\left({\frac{a}{b}} \right)$$

Euler¶

Euler's formula is crucial when applying complex numbers in digital signal processing, like in the Discrete Fourier Transform.

Real and imaginary part

$$ e^{j\varphi} = \cos(\varphi) + j \sin(\varphi)$$