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. This is imortant in DSP when working with the Fourier transform.
Magnitude¶
The magnitude - the lenght of the vector that represents the complex number - is calculated via Pythagoras' theorem:
$$ r = |z| = \sqrt{a^2 + b^2}$$
Phase (Argument)¶
The phase is the counter-clockwise angle between the real axis and the vector. This follows the standard definition of angles in mathmatics - $0°$ and $360°$ point right. $\varphi$ is calculated as the inverse tangent, or $\mathrm{arctan}$ of the following argument:
$$ \varphi = \tan^{-1}\left({\frac{a}{b}} \right)$$
Exponential Representation¶
With phase and magintude, we can use the exponential representation for a complex number:
$$ z = r e^{j\varphi}$$