Balder W. Holst

Hello there!

My name is Balder and this is my personal website. Here i write down what i learn to maybe help others some day. For now i will just act like people are reading my articles, and make them for fun.

This website has three main sections:

I write these articles for fun, and even though i try to make sure that they are correct, i am not a perfect human and there are sure to be mistakes. Below is a list of most recent articles in all categories.

Recent Posts

Delay through filter (gruppeløstid)
Delay through filter (gruppeløstid) The delay of different frequencies is different. We can try to mitigate this in our filter. The delay can be found as the derivative of phase as a function of frequency $\phi(\omega)$. $$T_{g} = -\frac{d\phi(\omega)}{d\omega} \s [\text{s}]$$ Because of this we want *linear phase*: If not, the Step Response will ocilate:
0001/01/01 · Notes
Den Inverse af en Matrix
Den Inverse af en Matrix $$A^{-1} \cdot A = I$$ $I$ : Identitetsmatrix Løs ligningssystem med den Inverse Matrix Udgangspunkt: ligningssystem $$A\vec{x} = \vec{b}$$ “Dividerer med $A$” ($A^{-1}$ eksisterer kun hviss $\det(A) \neq 0$) $$A^{-1} \cdot A\vec{x} = A^{-1} \cdot \vec{b}$$ $$I\vec{x} = A^{-1} \cdot \vec{b} \arrow \vec{x} = A^{-1}\cdot \vec{b}$$ At finde den Inverse $$ A^{-1} =\left( \begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \ a_{21} & a_{22} & a_{23} & a_{24} \ a_{31} & a_{32} & a_{33} & a_{34} \ a_{41} & a_{42} & a_{43} & a_{44} \ \end{array} \right)^{-1} $$ $$\Downarrow$$ $$ \left( \begin{array}{cccc|cccc} a_{11} & a_{12} & a_{13} & a_{14} & 1 & 0 & 0 & 0 \ a_{21} & a_{22} & a_{23} & a_{24} & 0 & 1 & 0 & 0 \ a_{31} & a_{32} & a_{33} & a_{34} & 0 & 0 & 1 & 0 \ a_{41} & a_{42} & a_{43} & a_{44} & 0 & 0 & 0 & 1 \ \end{array} \right) \s \sim \s \left( \begin{array}{cccc|cccc} 1 & 0 & 0 & 0 & b_{11} & b_{12} & b_{13} & b_{14} \ 0 & 1 & 0 & 0 & b_{21} & b_{22} & b_{23} & b_{24} \ 0 & 0 & 1 & 0 & b_{31} & b_{32} & b_{33} & b_{34} \ 0 & 0 & 0 & 1 & b_{41} & b_{42} & b_{43} & b_{44} \ \end{array} \right) $$ $$\Downarrow$$ $$A^{-1} = \left( \begin{array}{cccc} b_{11} & b_{12} & b_{13} & b_{14} \ b_{21} & b_{22} & b_{23} & b_{24} \ b_{31} & b_{32} & b_{33} & b_{34} \ b_{41} & b_{42} & b_{43} & b_{44} \ \end{array} \right)$$
0001/01/01 · Notes
Den retningsafledte
Den retningsafledte Hældningen i retning af > $\vec{v}$> . I et punkt $(x,y)$ og i en retning $\vec{v}$. $$D_{\frac{\vec{v}}{|\vec{v}|}} = \frac{\vec{v}}{|\vec{v}|} \bullet \nabla f(x,y)$$
0001/01/01 · Notes
Densitet
Densitet $$\rho = \frac{m}{R}$$ Enhed $$\frac{\text{kg}}{\text{m}^3}$$
0001/01/01 · Notes
Det Komplekse Plan
Det Komplekse Plan Calculus 9th.pdf>page=1084 $$\C = \set{x + yi : x,y \in \R}$$ Plottes i et Argand diagram, med det reelle komponent ($Re(w)$) på “$x$-aksen” og den imaginære del ($Im(w)$) på “$y$-aksen”.
0001/01/01 · Notes
Determinanen for Matricer
Determinanen for Matricer Fortæller om en ($n\times n$) matrix $A$ har fuld rang. The matrix must be square. $$\det(A) \neq 0 \arrow \text{Fuld rang!}$$ For $2\times 2$ matricer $$M=\left( {\begin{array}{cccc} a & b \ c & d \ \end{array} } \right)$$ $$det(M) = detM = |M| = ad-cb$$ For $3\times 3$ matricer $$M= \left( {\begin{array}{cccc} a & b & c \ d & e & f \ g & h & i \end{array} } \right)$$ $$det(M) = |M| = a \cdot \left|\left( {\begin{array}{cccc} e & f \ h & i \ \end{array} } \right)\right| - b \cdot \left|\left( {\begin{array}{cccc} d & f \ g & i \ \end{array} } \right)\right| + c \cdot \left|\left( {\begin{array}{cccc} d & e \ g & h \ \end{array} } \right)\right|$$ $$= a \cdot e \cdot i + b \cdot f \cdot g + d \cdot h \cdot c - c \cdot e \cdot g - b \cdot d \cdot i - f \cdot h \cdot a$$
0001/01/01 · Notes
DH-Parameters
DH-Parameters See the slides. A standard for placing frames on a robot. Every joint can be describes as a combination of prismatic and revolute joints. Prismatic joint: Pure translation in one axis Revolute joint: Pure rotation in one axis Variables These values are derived for every link. $i$ denotes the link index. $a_{i-1}$: Link length, shortest distance between two joint axis. $\alpha_{i-1}$: Link twist, the angle between the axis around $a_{i-1}$.
0001/01/01 · Notes
DHCP
DHCP See slides.
0001/01/01 · Notes
Differensligninger
Differensligninger Describe the output ($y$) as a function of an input ($x$) and previous values of $y$. $$ y(n) = \sum_{i=0}^{N}a_{i}x(n-i) - \sum_{i=1}^{N}b_{i}y(n-i) $$ Første Orden $N=1$ $$y(n) = a_{0}x(n) + a_{1}x(n-1) - b_{1}y(n-1)$$ Anden Orden $N=2$ $$y(n) = a_{0}x(n) + a_{1}x(n-1) + a_{2}x(n-2) - b_{1}y(n-1) - b_{2}y(n-2)$$ Overføringsfuntion Samme som i Laplace Transformation. $$H(z) = \frac{Y(z)}{X(z)}$$ Differensligning til overføringsfunction $$y(n) = 2y(n-1) + 3x(n) \Rightarrow Y(z) = 2Y(z) \cdot z^{-1} + 3X(n)$$ Example: transfer function from difference equation
0001/01/01 · Notes
Differential Equations
Differential Equations En equation with a function as a solution. En equation containing a derivative. A nice overview of ODE’s can be found here. Noter list from #differentialligninger sort file.name
0001/01/01 · Notes