Birthday Calendar

Since recently I did not yet had a birthday calendar in my apartment. This was because I namely dislike the very ordinary “toilet kind” of birthday calendars, see below on the left. More than a year ago I stumbled upon a very original birthday calendar on Pinterest. Immediately I became spontaneous, it has a very nice look, it is original with its clothespins and easy to maintain as well as built.

A while ago I finally gave myself the time to replicate the idea. Using the clothespins I determined what the minimum size of the wooden plank had to be, 125 [cm] in length and 25 [cm] in width. At first I tried to find an old plank of a pallet. Those often have that old look which I wanted to go for. Unfortunately none of them did met the requirements w.r.t. the size. So in the end I just bought a plank from a hardware store and sawn it to the correct dimensions.

To do the writing of the header “birthday”, the months and the names of the people I bought a white and black acrylic marker called “DECO painter matt” of the brand Marabu. To make sure that the letters on the wood of the word “birthday” would be consistent, I made a small mold out of paper. The months I just drawn out of hand, same for the names and days on the clothespins.

DECO painter

In the example on Pinterest the clothespins are colored. This can be achieved by coloring the clothespins with paint or by buying colored clothespins. You can find the latter by searching on children toys. I did not color them and searched for old and worn clothespins. If you do not have those you can buy new ones and put them in the watery garden and wait a week or two.

The result

“Down” time games

When having a training, introduction or travel with a large group there might be moments in which everyone is a bit bored or might be waiting for something or someone. For these moments you can use so called “down” time games (in Dutch dodemoment spelletjes). These are small simple games which you can do with your group. In this blog I explain a couple of down time games for you.

Floating

You will need four chairs arranged in a square (close together). Four people sit on the chairs. Proceeding clockwise, they recline onto their neighbors’ laps (keeping their feet on the ground). The chairs are pushed out from under the sitting/reclining people, leaving a floating group. How long can they keep it up? When there are no chairs available, you can use four people to play the part of the chair by standing on their hands and feet. In the video below you see it in action.

The Human Knot

One person will move out of the group and shuts his eyes. The group will stand in a circle next to each other and holds hands. After this points hands should stay connected at all times. Following everyone will move crisscross, above or underneath each other to form a human knot. The person who was placed out of the group then has the task to untie the “knot”.

The Land of No Idea

Far, far away lays the Land of No Idea. They do have people there, but no children; they have membranes, but no skin; they have no vehicles, though they do have cars and scooters. They have computers, but no modems, monitors, or keyboards. Some words do exist while others do not. The answer lies in the name of the game; The Land of No Idea. The catch is in pronunciation of the word “idea”. The correct sentence would be; The Land of no “id”. So all words with either an “i” or “d” do not exist.

Monkey, monkey, elephant

A leader tells the children that he/she can do something the others can’t, and then shows them: He/she says the sentence “monkey monkey monkey monkey elephant monkey elephant monkey monkey monkey monkey” and explains the movements that go with it: Each time you say “monkey” you touch one of the fingers on your left hand with your right index finger. The sequence is: little finger, ring finger, middle finger, index finger and thumb, and then back again from the thumb to the little finger. When you say “elephant”, you slide your index finger from the top of your index finger to the top of your thumb, or the other way round. When the leader has explained it, he/she shows them how to do it once. He/she then makes an additional movement that is hardly noticeable. Everyone has to repeat the game with their fingers. It is only when the children figure out they also need to make the additional movement that they are told they have got it completely right. A suggestion: The ‘hardly noticeable’ movement could be that, after finishing the sentence, you fold your arms, and then say to one of the children: “Now it’s your turn.”

Black magic

All the players sit in a circle, and there are once again two game leaders (A and B). B leaves the room or space briefly. A asks someone from the circle to choose a nearby object, which B will have to guess. B is now called back. A begins to point out different nearby objects. For example: “Is it these trousers?” -“No”- “Is it that watch?” – “No”- and so on. B will continue to answer “No” to A’s questions, until A mentions the correct object – the one that the players previously agreed on. B will now answer “Yes”. How is that possible? How does it work? The clue is in the title – black magic. A will mention all kinds of objects that are not black. After mentioning several objects, A will mention one that is completely black. The next object that A mentions is the one that has been agreed on with the group. In other words, the game leaders know which object it will be beforehand: it is always the object that comes after the black object.

Who am I

Everyone takes a small piece of paper and writes down a name of a person that everyone knows, taking care not to show it to the person next to them. It could be someone famous or a personal acquaintance, or a fictional character, for example. You then attach the note onto the forehead of the person next to you using sticky tape. When everyone has a name on their forehead the game can begin. Taking turns, everyone asks yes/no questions about themselves, for example, “am I a woman?”, which the group then answers with “yes” or “no”. If the answer to the question is yes, the same person can ask another question; if it is no, the turn passes to the next person. The winner is the first person to work out who he or she is – the rest of the group can continue playing if they wish.

Black Stories

The game leader tells how a person died. The group then has to guess how the person has died and may only ask questions which can be answered with yes or no. Online you can find numerous amount of riddles or apps.

  • A guy is lying dead in a phone booth; the handset is off the hook, and there is glass lying on the ground outside the booth.

Solution: He called his wife to tell her how big the fish was that he had caught. The wife asks, “How big was it?” The man stretches out both arms to indicate the length (“THIS big!”), ramming both arms through the glass. In doing so, he cut his wrists and died.

  • A man is lying dead in the forest, wearing his swimming trunks.

Solution: The man had been swimming in a lake/ocean. A firefighting aircraft scooped him up and dumped him in a burning forest. The man died from the fall.

  • A man has hanged himself from the roof of a warehouse four meters high (he is hanging two meters above the ground). The warehouse is in the desert, and there is no ladder or similar object to be found. How did he do it?

Solution: He was standing on a block of ice two meters high. The ice melted and evaporated.

  • A man went out on the street, saw a hotel, and knew that he was bankrupt.

Solution: He was playing Monopoly.

  • A sailor goes into a restaurant, eats an albatross sandwich, and then falls over dead.

Solution: He had been shipwrecked a while ago. There had been nothing to eat on the uninhabited island. Several of his companions had died of hunger. He and his buddy were also ravenous. His buddy made two “albatross sandwiches,” which they quickly devoured. It now appears that the delicacy had not been albatross at all, but one of their former shipmates. This was too much for the man.

  • A man lives on the twelfth floor of an apartment building. In the morning, he takes the elevator to the ground floor to go to work. When he returns in the evening, he takes the elevator to the sixth floor and takes the stairs to the twelfth. When it’s raining, however, he takes the elevator to the twelfth floor. Why?

Solution: The man is a midget and can reach the button for the sixth floor, but no higher … unless he has an umbrella with him.

  • A man is sitting in the non-smoking section of the Swiss railways. He grabs his gun and shoots himself through the head. If he had been sitting in a smoking compartment, this would not have happened. Why not?

Solution: The man had been blind for years and was just returning from having surgery on his eyes. The surgeon had said that he could not yet remove the bandage from his eyes. Once in the train, however, he couldn’t wait, and he removed his bandage. At that precise moment, the train entered a tunnel and the compartment was shrouded in darkness. The man thinks that the operation had failed and that he will never be able to see, and so he shoots himself in the head. If there had been a smoker in the compartment, he would have seen the red tip of the burning cigarette.

  • The smallest man in the circus discovers sawdust under doormat: he now knows that he is growing!

Solution: As a midget, the smallest man in the circus earns his money with his stature. He does not notice that he is growing. To disguise this, his girlfriend regularly saws thin chips off the chair legs. She sweeps the sawdust under the doormat.

  • Romeo and Juliet are lying dead on the bed, and the window is closed.

Solution: Romeo is a cat, and Juliet is a fish. Romeo choked to death while eating Juliet.

Vikings

The group is in a circle. At the beginning one Viking is appointed. The two people sitting on the left and right of the Viking are the rowers. The rest is the sea. The Viking starts holding his hands like two horns to his head (like a viking helmet). Following he makes a growling Viking sound; aarrrrghh, raaaaaarrrrrrrw, et cetera. At the same time the rowers make a rowing movement (to the correct side) and say; row, row, row, row, et cetera. The people appointed as sea imitate waves with their hands and constantly say; woesh, woesh, woesh, et cetera. At a random point in time the Viking appoints a new Viking who adopts the new role and as such there will also be new rowers and a new sea.

Implementation of a filters

The discrete-time transfer function H(z) is often obtained from its counterpart, the continuous-time transfer function H(s) via discretization. A discrete-time transfer function has the following form:

(1)   \begin{equation*}H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{i = 0}^{N} b_i z^i}{\sum_{j = 0}^{M} a_j z^j}\end{equation*}

Herein, X(z) is the input and Y(z) is the output of the system, N and M are the degree of the numerator and denominator, respectively. Where N \leq M, meaning we are dealing with a proper transfer function. While (1) is valid for any order it is not recommended to directly use transfer functions of high order. These can namely introduce numerical problems very quickly. Rather factorize the numerator and denominator of (1) into a cascade of first and second order polynomials.

    \begin{equation*}H(z) = K \frac{\displaystyle \prod_{i = 0}^V \left( z + b_{0i} \right)}{\displaystyle \prod_{i = 0} ^W\left( z + a_{0i} \right)} \frac{\displaystyle \prod _{i = 0}^N \left( z^2 + b_{1i} z + b_{2i} \right)}{\displaystyle\prod_{i = 0}^M \left( z^2 + a_{1i} z + a_{2i} \right)}\end{equation*}

Now let us look at the simple discrete-time transfer function of order two:

    \begin{equation*}H(z) = \frac{b_0 z^2 + b_1 z + b_2}{z^2 + a_1 z + a_2}\end{equation*}

This function is non-causal, because it depends on future inputs. Therefor, both the numerator and denominator are multiplied by reciprocal of the highest order of z occuring in the denominator, in this case z^{-2}, to make the system causal. Hence, we obtain:

    \begin{equation*}H(z) = \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{1 + a_1 z^{-1} + a_2 z^{-2}}\end{equation*}

Followingly, using the linearity and time-shifting properties of the \mathcal{Z}-transform, i.e., \mathcal{Z}(a_1 x_1[n] + a_2 x_2[n]) = a_1 X_1(z) + a_2 X_2(z) and \mathcal{Z}(q^{-k}x[n]) = z^{-k}X(z), we obtain the difference equation. Remark that the shift operator q is defined as q^kx[n] = x[n+k], the forward shift operation and q^{-k}x[n] = x[n-k], the backward shift (delay) operator. As a result we obtain,

(2)   \begin{equation*}y[n] = \frac{b_0 + b_1 q^{-1} + b_2 q^{-2}}{1 + a_1 q^{-1} + a_2 q^{-2}} x[n]\end{equation*}

Rewriting (2) gives us:

    \begin{gather*}\left( 1 + a_1 q^{-1} + a_2 q^{-2} \right) y[n] = \left( b_0 + b_1 q^{-1} + b_2 q^{-2} \right) x[n] \\\Leftrightarrow y[n] + a_1 y[n-1] + a_2 y[n-2] = b_0 x[n] + b_1 x[n-1] + b_2 x[n-2] \\\Leftrightarrow y[n] = b_0 x[n] + b_1 x[n-1] + b_2 x[n-2] - a_1 y[n-1] - a_2 y[n-2]\end{gather*}

This last equation is the difference equation which we can easily implement on our digital platform. Numerous methods exists on how to implement a filter. Four of these methods are closely related to each other. These are:

  • Direct form I
  • Direct form II
  • Transposed direct form I
  • Transposed direct form II

Direct form I

The direct form I is an FIR filter followed by an IIR filter. That is to say, it implements Y(z) followed by \frac{1}{X(z)}

(3)   \begin{equation*}y[n] = b_0 x[n] + b_1 x[n-1] + b_2 x[n - 2] - a_1 y[n-1] - a_2 y[n-2]\end{equation*}

In an algorithm you can implement it as:

yk = b0 * xn + b1 * x1 + b2 * x2 - a1 * y1 - a2 * y2
x2 = x1
x1 = xn
y2 = y1
y1 = yn

Where, x1, y1, x2 and y2 are the four state variables.

Direct form II

The direct form I is an IIR filter followed by an FIR filter. Which implements \frac{1}{X(z)} followed by Y(z).

Signal flow diagram of direct form II. Where s[n] a state variable

(4)   \begin{align*}s[n] &= x[n] - a_1 s[n-1] - a_2 s[n-2] \\y[n] &= b_0 s[n] + b_1 s[n-1] + b_2 s[n-2]\end{align*}

In an algorithm you can implement it as:

s0 =      xn - a1 * s1 - a2 * s2
yn = b0 * s0 + b1 * s1 + b2 * s2
s2 = s1
s1 = s0

Where, s0, s1, s2 are the three state variables.

Direct form I transposed

Both direct forms can be converted to an equivalent transposed form via:

  • Reverse direction of each interconnection
  • Reverse direction of each multiplier
  • Change junctions to adders and vice-versa
  • Interchange the input and output signals
Signal flow diagram of direct form I transposed.

(5)   \begin{align*}y[n] &= b_0 v[n] + b_1 v[n-1] + b_2 v[n-2] \\v[n] &= x[n] - a_1 v[n-1] - a_2 v[n-2] \\\end{align*}



In an algorithm you can implement it as:

vn = xn + s2
yn = s4 + b0 * vn
s4 = s3 + b1 * vn
s3 = b2 * vn
s2 = s1 - a1 * vn
s1 = - a2 * vn

Note that this form is very inefficient. You can easily shift the delays to the center, obtaining the direct form II.

Direct form II transposed

Signal flow diagram of direct form II transposed.

(6)   \begin{align*}s_1[n] &= b_2 x[n-1] - a_2 y[n-1] \\s_2[n] &= b_1 x[n-1]  - a_1 y[n-1] + s_1[n-1] \\y[n] &= b_0 x[n] + s_2[n]\end{align*}

In an algorithm you can implement it as:

yn = s2 + b0 * xn
s2 = s1 + b1 * xn - a1 * yn
s1 = b2 * xn - a2 * yn

Symbol to represent Emergency Response Team (ERT) Bedrijfshulpverlening (BHV)

Emergency response officers (ERT), or in Dutch Bedrijfshulpverlening (BHV), within companies are trained in providing both first aid and limiting and combating (small) fires. All to limit the consequences of accidents. Various icons are used to resemble these organisations. One symbol which I have seen quiet often is the combination of first aid and a fire symbol.

However, since there was not yet a vector format available, I created one. Using the established first aid and fire sign from ISO 7010.

Emergency Response Team (ERT), or in Dutch Bedrijfshulpverlening (BHV), icon.

On the left the fire symbol with the red color. On the right half of the first aid cross with the green color. The red and green color form a gradient in the middle.

You can download the vector file here.

Filters in Control Systems

In this blog I want to elaborate on the various types of filters that are being used in Control Systems. These filters are often used in combination with a PI, PD or PID controller to obtain a robust controller. The filters which we will discuss are:

  • Low-Pass filter
  • High-Pass filter
  • Lead-Lag filter
  • Notch filter

Using these four filters we can create other filter types, such as a Band-Stop or Band-Pass filter. The behavior of each filter can be captured by a transfer function in the continuous-time using the Laplace domain s of either first and/or second order. We use the transfer functions to describes the filter H(s) = Y(s) / U(s) and as such the relation between the input U(s) and the output Y(s). Throughout this blog we will write the equations in the form of their angular frequency \omega in [rad/s], also known as radian frequency. However, we will specify filters using their frequency f in [Hz]. Using the radial frequency notation results in a more visual compact formula. Additionally, it is also possible to specify the filters in form of their time constant \tau in [s]. The following relation holds between the angular frequency, frequency and time constant.

    \begin{equation*} \omega \triangleq 2\pi f \triangleq \frac{1}{\tau} \end{equation*}

Many forms are used within literature, one book will use angular frequencies, the other will use time-constants. Finally, for second order filters, the only filter with possibly complex poles or zeros, can be written in various ways. We will specify second order filters in terms of the damping \beta of the corresponding frequency. It is also possible to describe these formulas using the Quality factor Q. Whereas \beta describes how oscillations decay in a system after a disturbance, Q describes how underdamped the system is. The following relation holds between \beta and Q

    \begin{equation*} \beta \triangleq \frac{1}{2Q} \end{equation*}

Low-pass filter

A low-pass filter is used to pass signals with a frequency lower than a certain cut-off frequency f_\text{lp}. Below the formulas for both the first- and second-order low-pass filter is given.

    \begin{equation*} H(s) = K\cdot\frac{\omega_{\text{lp}}}{s + \omega_{\text{lp}}} \quad\quad H(s) = K\cdot\frac{\omega_{\text{lp}}^2}{s^2 + 2\beta\omega_{\text{lp}}s + \omega_{\text{lp}}^2} \end{equation*}

Herein, K denotes the gain, f_\text{lp} denotes the low-pass cut-off frequency and \beta_\text{lp} denotes the damping. Whereas the first-order supresses with 20 [dB/dec], the second-order supresses with 40 [dB/dec]. Low frequent the filter gain is K.

High-pass filter

The complement of a low-pass filter is a high-pass filter. This filter is used to pass signals with a frequency higher than a certain cut-off frequency f_\text{hp}. Below the formulas for both the first- and second-order high-pass filter is given.

    \begin{equation*} H(s) = K\cdot\frac{s}{s + \omega_{\text{hp}}} \quad\quad H(s) = K\cdot\frac{s^2}{s^2 + 2\beta\omega_{\text{hp}}s + \omega_{\text{hp}}^2} \end{equation*}

Herein, K denotes the gain, f_\text{hp} denotes the high-pass cut-off frequency and \beta_\text{hp} denotes the damping. Likewise as the low-pass filter the first-order supresses frequencies with 20 [dB/dec] and the second-order with 40 [dB/dec].

Lead-lag filter

A lead-lag filter, also known as a lead-lag compensator, is often mainly used for phase compensation rather then magnitude. Below the formula for a lead or lag filter is shown.

    \begin{equation*} H(s) = K\cdot\frac{\omega_\text{p}}{\omega_\text{z}}\cdot\frac{s + \omega_\text{z}}{s + \omega_\text{p}} \end{equation*}

Herein, f_\text{p} and f_\text{z} denote the frequency of the pole and zero, respectively. The filter functions as a lead filter if f_\text{p} > f_\text{z}  and otherwise as a lag filter. The filter has its maximum or minimum phase at \sqrt{f_\text{p}f_\text{z}}. Finally, at f = \infty the filter has a gain of Kf_\text{p} / f_\text{z} or Kf_\text{z} / f_\text{p} in case of a lead or lag filter, respectively. Naturally, the filter can be cascaded with itself by which a the filter can be a lead and lag filter simultaneously.

Notch filter

A notch filter is often used to filter undesired resonance peaks. Below the formula for a notch filter is shown.

    \begin{equation*} H(s) = K\cdot\frac{\omega_\text{p}}{\omega_\text{z}}\cdot\frac{s^2 + 2\beta_\text{z}\omega_\text{z}s + \omega_\text{z}^2}{s^2 + 2\beta_\text{p}\omega_\text{p}s + \omega_\text{p}^2} \end{equation*}

Herein, f_\text{p} and f_\text{z} denote the frequency of the pole and zero, respectively. Likewise, \beta_\text{p} and \beta_\text{z} denote the damping of the pole and zero and K is as usual the gain. When f_\text{z} = f_\text{z} the notch will target one specific frequency. The gain at that frequency is given by \beta_\text{z} / \beta_\text{p}. When f_1 \neq f_2 the notch filter is also referred as a skewed notch and the difference between gain at low and high frequencies is given by (f_\text{p} / f_\text{z})^2.

Getting things done

A while ago I read the book Getting Things Done (GTD) written by David Allan. In my own personal workflow I was already applying GTD. However, after reading the book I further refined my personal workflow. My fellow colleagues were interested and also in need of GTD. Therefore, I made a short presentation about my own personal workflow, with tips & tricks. Feel free to use the presentation for your own benefits.

Your own Happy Salmon game

Recently we had a game night with some friends of mine. At the start of the night we played the game “Happy Salmon“. It is a very simple game but really energetic. Every player gets a set of cards. Each card resembles an action. The goal is to get rid of your cards as fast as possible.  You will have to find another player with the same action and perform it. To find another player you simply shout your action out loud in the open. The actions are either:

  1. High five, players high five each other
  2. Pound it, players fist bumb each other
  3. Switcheroo, players switch places
  4. Happy Salmon, players flip there hands against each other very fast, like two fish splashing.

The following video describes the chaos in my living room when we played the game.

Chaos!

As you can see the game is very energetic. However since we simply threw our cards in the air these would most likely do not hold that long. Therefor, I created my own “Happy Salmon” such that I could simply print out, cut it in pieces, and would not have to worry about cards being destroyed. In the process of creating these cards I also added some more actions. These new actions will make the game more challenging when playing with a large crowd. 

The game cards, shout out to Freepik for his icons: https://www.flaticon.com/authors/freepik

The cards background are transparent, meaning that if you simply use colored paper you will nice colored cards. Most pictures are self explaining but nevertheless, the following new actions are defined:

  1. Bro Bump, more or less the same as pound it but only with the knuckles and on shoulder height as in the picture. 
  2. Bro Shake, handshake in the air
  3. Pinky Swear, pinky swear with one another
  4. Hug, bit more intimate but should be fun nonetheless
  5. Handshake, seems obvious
  6. Teamspirit, for this action you need three people instead of two. You put each other hands on each other and cheer afterwards, something like, https://www.youtube.com/watch?v=jGmp3zdRLyY
  7. Teamwork, for this action you will need four people instead of the regular two. Simply hold each others arms as shown on the picture.

The cards are simply created in office. Below you can either download the source file or download the original card set or the card set with the new actions. Both files are for one player.

Have fun and good luck!

Which tripod to buy?

In August I went on holiday to Sri Lanka. During this trip I wanted to do some landscape and long exposure photographing. A tripod is really necessary for the latter. Furthermore, it was also a good excuse to myself to justify the reason to buy a tripod. However, as with any product, there are various on the market and sometimes it is difficult to find the best buy. Each one of them have different features and specifications, such as, expensive versus cheap, light versus heavy, small versus big, et cetera. As with most of the products which I buy I tried to minimize the price but maximize the features. The two most important requirements which I wanted to optimize, besides the cost, were

  1. Minimize the weight of the tripod but ensuring the stability and endurance of the tripod
  2. Maximize the unfolded height of the tripod but minimize the folded height

Based on those criteria you are quickly directed to the tripods which are branded in the market as travel tripods. Below a table with the tripods which I took into consideration.

Manfrotto
BeFree
MeFoto Roadtrip
travel
Sirui
T-025X
Oben
CT-3535
Minimal height [cm] 34 0 10.5 21.8
Maximal height [cm] 144 162.5 147.5 131
Folded height [cm] 40 39 31 30.5
Weight [kg] 1.4 1.6 0.9 1.1
Maximum load [kg] 4 8 6 4
Material Aluminium Aluminium Carbon Carbon
Head mount? Y Y Y Y
Quick release? Y Y Y Y
Monopod? N Y N N

After balancing the advantages and disadvantages I decided to go for the MeFoto Roadtrip. For several reasons:

  1. For one it had good reviews on the internet, one particular which I liked was the review by Jason in the YouTube video below, which I also recommend too you to watch.
  2. The tripod was not too expensive compared to other tripods.
  3. It had the highest unfolded height
  4. Of all the tripods in the table above it is the heaviest, however, I considered 1.6 [kg] still as lightweight.
  5. Because I wasn’t planning to use the tripod on a daily bases I decided that an tripod of aluminium, instead of a carbon fiber one, would be durable enough
  6. The bar to which the camera is attached has a hook on the bottom to which extra weight can be added to increase the stability of the tripod. Along with the feature that the bar could be mounted upside down, which added to possibility to take photo’s directly at ground level. Were two features which I particularly liked.

Apart from the reasons above the MeFoto Roadtrip had also most of the features which other tripods featured, therefor the decision was made.

If you are ever looking for a reason to buy a tripod consider the following reasons:

  1. a tripod helps when using long exposures, for instance, you will need this when doing astrophotography.
  2. taking time lapses, panoramas, panning and videos become a lot easier since you have great support for your camera.
  3. when using a tripod you actually think a lot more about the framing of your shot, this is because everything takes a bit longer, setting up the tripod, adjusting, et cetera.
  4. self portraits are of a higher quality then when using a selfie stick.

Digital Control Systems 1.02

In the previous blog we introduced the various types of signals present in a digital control system and highlighted the pros and cons. In this blog we take a a closer look at the digital control system.

A digital control system, can only cope with digital signals. That is because it is inherently discrete in time and space. This is caused by the processors clock rate and due to the finite word length of a digital system. For instance, a 32 [bit] computer which runs at 4 [GHz] has a time resolution of 0.25 [ns] and the value of an unsigned integer can only range from 0 to 232 − 1.

Figure 1. Digital control system in its most elementary form. Continuous-time signals are in solid and discrete-time signals are dashed.

Figure 1 shows the digital control system in its most elementary form. Herein \mathcal{S} is the sampler. The sampler converts the analog output signal of the plant y(t) at time t \in \mathbb{R} to a digital measurement signal y[k] at discrete-time k \in \mathbb{N}. As mentioned in the previous blog, the digital signal is obtained by sampling and quantization of the analog signal. In final, \mathcal{H} denotes the recronstructor, also known as a digital-analog convertor (DAC), it converts the digital control signal u[k] provided by the controller to an analog control signal u(t).

Sampler

Figure 2 shows that the sampler exists out of two devices namely; i) a sample and hold (SH) device and ii) a analog-digtal convertor (ADC).

Figure 2. The sampler and its sub systems.

The SH device knows two different states. The first state is called the track state at which the analog input signal is tracked by the SH device. The track state is also refereed to as sample state. However because of the ambiguous meaning of the word sample we will use the track, which is also commonly used in literature. The second state is called the hold state at which the analog input signal is being kept constant for a short period of time. The hold state is activated when the hold command is given. During the hold state the ADC is able to process and digitize the signal. The hold command can be triggered by any logic device. While there are many more aspects and subtleties to be considered with a SH device this gives a summarized overview of its functionality.

Figure 3. The track and hold of a SH. Dashed red shows the original signal, solid blue shows the SH signal.

Various different strategies exists at what time instance t_k the hold command is triggered. These are called sampling strategies.

  1. Periodic sampling; the sampling instances t_k are equally spaced, that is to say t_k = kh with k \in \mathbb{N} and h, given in [s], being the sampling period. See also Figure 2(a). The sampling period h is often also denoted as the sampling rate or sampling frequency f_s, given in [Hz]. It is related to the sampling period by h = \frac{1}{f_s}.
  2. Multi-order sampling; a pattern of sampling instances \{t_k\} are repeated periodically, as a result t_{k+r} = t_k. See Figure 2(b).
  3. Event-based sampling; the sampling instances t_k are generated based on a event in the system. For instance, when a certain measurement threshold has been crossed. See Figure 2(c) in which the signal is sampled at every \pm0.4x \pm 0.2 value.
  4. Random sampling; the sampling time instances t_k are chosen at random, see Figure 2(d).

Figure 2. Visualization of different sampling techniques. Red line resembles the original analog signal, blue the sampled signal. Matlab code.

Periodic sampling is the most common in industry. That is because of several reason; i) most deterministic behavior, ii) extensively been researched, iii) easiest to model and iv) easiest to obtain key performance indicators, for instance, in time- and frequency domain. Finally, proofing stability for the other sampling strategies is much harder then for periodic sampling.

All State of Trance episodes on Spotify

I am a staunch fan of A State of Trance of Armin van Buuren. Therefor, I made a list of all the State of Trance episodes available on spotify.

  1. https://open.spotify.com/album/65rxkH0MLVAUfvGJU2HPk4
  2. https://open.spotify.com/album/5jtGUCgpoqvSNZH8WcQf8F
  3. https://open.spotify.com/album/2RLKYKweji0JBAi2OtxOCt
  4. https://open.spotify.com/album/51MG0dQ7aAON9YSeVQuESN
  5. https://open.spotify.com/album/3VlEbY9dHahYMVDur5W5JP
  6. https://open.spotify.com/album/2jh0ZxkDItwwlXlR28IymD
  7. https://open.spotify.com/album/46QhHtpLmgtlu9OiloGtac
  8. https://open.spotify.com/album/4WEIgZligoc0lqNQZH1BHU
  9. https://open.spotify.com/album/6KEof7LPZLLHVzkrqLdMAT
  10. https://open.spotify.com/album/2tMcgPF3PAliylcaKdmchi
  11. https://open.spotify.com/album/6QRawIT0hMoRIbBoVTE7aF
  12. https://open.spotify.com/album/7fG3Ojc9gcaWlDiEK0Khn3
  13. https://open.spotify.com/album/0rlYWWv1cqZ0dVTrIMwsdl
  14. https://open.spotify.com/album/78Jg3VsL3s6MQbvKCJO4Kk
  15. https://open.spotify.com/album/3LHyRc4hqhN1Ux6Ank35oH
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  17. https://open.spotify.com/album/4LdnyKph8K1Tfndu2gm4Ww
  18. https://open.spotify.com/album/0v7140z14LzlPUkBkZajsO
  19. https://open.spotify.com/album/2hjnBKdWMaxqzcNZsjvLo2
  20. https://open.spotify.com/album/0lLQFUfcGewYHzb7eNk2sU
  21. https://open.spotify.com/album/4DVm0heIq4WhnZCrvRgUcQ
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  23. https://open.spotify.com/album/4ErwCMdhHH8Is6XcUnrgGZ
  24. https://open.spotify.com/album/55IIabrevVvjs1eMhYd48G
  25. https://open.spotify.com/album/5Ro8x51oqQ68yIj4Bk3MGC
  26. https://open.spotify.com/album/0vef8t0hukoNUgLB7hfbCh
  27. https://open.spotify.com/album/3Q0L1bOSN4l62aZb7jEb1q
  28. https://open.spotify.com/album/6IgdDmdw6gzLgI29zkFV5B
  29. https://open.spotify.com/album/68t45JtpzzFCF4a20fMr2e
  30. https://open.spotify.com/album/7drFmJQdsZJMlljBTYuB3J
  31. https://open.spotify.com/album/403W3zS9MhVD4hs3kg9Ylu
  32. https://open.spotify.com/album/4kMis3hXbMYU36seBGd8ds
  33. https://open.spotify.com/album/3u81K6jJ2BfgfJzu24RLWd
  34. https://open.spotify.com/album/2wtvQZtq6NBrOUoteDRbCH
  35. https://open.spotify.com/album/5o2pRlTv4dN7lLaBbGfikP
  36. https://open.spotify.com/album/4CeGI6njkAhaNf1NJO4Wbv
  37. https://open.spotify.com/album/5Up7R0jZYZUPUebZiQPzNt
  38. https://open.spotify.com/album/2lAD34Mk4U6NTbQaKk1uJi
  39. https://open.spotify.com/album/1fPPuhpzMEESzVXd40g7iJ
  40. https://open.spotify.com/album/3xIPtxGDbKlkHRjV6UgJVA
  41. https://open.spotify.com/album/3FxDAo5EDK125dNr8x6pah
  42. https://open.spotify.com/album/2fpveoRx9KpQ7FY1AzQkS8
  43. https://open.spotify.com/album/5eRQRAh2yCc8b8g7DbcSqf
  44. https://open.spotify.com/album/0I6fTQtxHssnaHhqyS1Z4n
  45. https://open.spotify.com/album/1EbNowqGV0ipSyIaHVpMqi
  46. https://open.spotify.com/album/6J36U4sw4wvM9PFld9wzQd
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  48. https://open.spotify.com/album/515DdJsne3J4PsGzErOEVK
  49. https://open.spotify.com/album/0c1mtJ3Oxt8K5eakvEUJWk
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  53. https://open.spotify.com/album/2KH2xVEbQlqCqNkme5c13W
  54. https://open.spotify.com/album/6rkZGXPlsm8O8H3AqLug4T
  55. https://open.spotify.com/album/3PAzlLPZ5yTCdhsMMqH0Kn
  56. https://open.spotify.com/album/4TsmmpCSwWpOwI6y8pK07E
  57. https://open.spotify.com/album/56qM5Y21wbvCW9l5GiAiaV
  58. https://open.spotify.com/album/2uiOpUMUYXU5n2BVgZ9Szf
  59. https://open.spotify.com/album/24GG2tFabhxy4r14tTTpbI
  60. https://open.spotify.com/album/4dikoEqBQ1roY7A6GpRkRc
  61. https://open.spotify.com/album/3GJsL1YDH1Z6t2xGaqWeHj
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  90. https://open.spotify.com/album/4sowBJxlo7V9wYNi3hadNf
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  108. https://open.spotify.com/album/1jsImIi3MIBmMh36cguMoJ
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  110. https://open.spotify.com/album/5coGuXVsHbVjZ4yUG49rmk
  111. https://open.spotify.com/album/7zchhV4YivxoTTTWgz1PpV
  112. https://open.spotify.com/album/7EXl4dzp9cG7a9gYU3JXJJ
  113. https://open.spotify.com/album/7rzDHsRcoHWZejvhonZzfD
  114. https://open.spotify.com/album/6ZKDhLih8cO2fnYz3V1oRy
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  116. https://open.spotify.com/album/6VKCEw0CBqmA8gbq6lPVbJ
  117. https://open.spotify.com/album/7uIIedI8pHtalzpj5TAKjH
  118. https://open.spotify.com/album/15kakQ3Psf5TjQAYRdkm08
  119. https://open.spotify.com/album/61iVMAFH25HzJ5CSKUcRGo
  120. https://open.spotify.com/album/6VEEBclHX7tfOgj6hZ4IZH
  121. https://open.spotify.com/album/5pt9XNu5xVeLdsIAoW4Tzh
  122. https://open.spotify.com/album/5kWc6YxKeRvjWfV2cfQ3AZ
  123. https://open.spotify.com/album/6nlwIC0KVQdD4MW6mhp0aw
  124. https://open.spotify.com/album/4dGibs037LXTFhGFsr80JL
  125. https://open.spotify.com/album/1ixayckKODoYT1r3Sdq0tp
  126. https://open.spotify.com/album/5iLFct6fhawyQEmIecEoDC
  127. https://open.spotify.com/album/0pLc2hbMrI9WHr9StUOBPl
  128. https://open.spotify.com/album/7s9TL1n2T0Q10xDpu5OsPr
  129. https://open.spotify.com/album/1ZedL2Ys7rN2thimKnish7
  130. https://open.spotify.com/album/6Fvyy7F0kQysVvrU08a9rp
  131. https://open.spotify.com/album/3BeGpIoZSZxLWUX0IkwUsq
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  736. https://open.spotify.com/album/72KQ0UvGyOg1drmEoETwsY
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  738. https://open.spotify.com/album/1JsvK0bBbLQ3a3RUmqvhh5
  739. https://open.spotify.com/album/4YYekPlejBy5X5wotcDTLm
  740. https://open.spotify.com/album/67VfXNm7j2hVZAuNfzusND
  741. https://open.spotify.com/album/7gx0LHSSKb1zvhDsR1qlXj
  742. https://open.spotify.com/album/7wsjDzKqwlwxDJMNCr0GBc
  743. https://open.spotify.com/album/0Msj6QLuO1z7nEyzvN6Irb
  744. https://open.spotify.com/album/7JJm8HwliCfxne8IMG31l9
  745. https://open.spotify.com/album/1Z1GtIgbbzZdP8YRtBSMFQ
  746. Not available
  747. https://open.spotify.com/album/0HO5wjGn159W9tymooKuwS
  748. https://open.spotify.com/album/3vjKvq5sTB7M2o9t6lQWFW
  749. https://open.spotify.com/album/1Fe70vQ7EXYjynE7lGe5a8
  750. https://open.spotify.com/album/0mzW1Ok0eKZA1712LmEjwD
  751. https://open.spotify.com/album/2FOHuwfARdoSXytzbAuzky
  752. https://open.spotify.com/album/0O9bl7Kf2pnKZIE2UggXBU
  753. https://open.spotify.com/album/6fChr0wDnGVlbZo86pwBpL
  754. https://open.spotify.com/album/6rVtiDj4a6NTzA6cBhqzHr
  755. https://open.spotify.com/album/39e7F6yj5AkQBcMBgqx0xn
  756. https://open.spotify.com/album/4zPuW0rjBqWdW8WAOQea2C
  757. https://open.spotify.com/album/53LKezwD1gPUkbrjctewfn
  758. https://open.spotify.com/album/0jCSFZ33BJ4tIpZFSunGUS
  759. https://open.spotify.com/album/5WQawhjdyFldRPPfXzZlXB
  760. https://open.spotify.com/album/44FmS1arWXSF8msTTNLs1F
  761. https://open.spotify.com/album/6nbjor30rBuJkMDxVphr8t
  762. https://open.spotify.com/album/1j1cGp0SbDBXhDD4z9DNDE
  763. https://open.spotify.com/album/7nC8BaG7nZlVTA1oMD3d8i
  764. https://open.spotify.com/album/7It5yDcYuAwuharaGBulbp
  765. https://open.spotify.com/album/2Ab3m1wIJ7Og24AQcXqpDr
  766. https://open.spotify.com/album/1bx0cWK0nT7sszYa9y6o0X
  767. https://open.spotify.com/album/2G6fE5GqRvZAqKE7wJfuui
  768. https://open.spotify.com/album/1n8icO3BwuLa30R491S6PN
  769. https://open.spotify.com/album/3NMUXjLgqWGO6UP8L9GrbX
  770. https://open.spotify.com/album/1cSfHWplybRbrz3fQwJrlu
  771. https://open.spotify.com/album/2q478NAobUtiFplZk3irAf
  772. https://open.spotify.com/album/4GEZUQe63QHMUbVRm2qVIQ
  773. https://open.spotify.com/album/3BPKRLsFLCLsTsPNUc2HRV
  774. https://open.spotify.com/album/1s9YLn6PR4KuUOp8Vs4Xkf
  775. https://open.spotify.com/album/335jfcqrdWbZVwDfkqT0lU
  776. https://open.spotify.com/album/7ixugpsgmcFCjKfeUDL8ro
  777. https://open.spotify.com/album/1vWciNuhHcR1YuH4xXe4l4
  778. https://open.spotify.com/album/2D1GpZF8RqGlKSklBbgj6w
  779. https://open.spotify.com/album/483mk4BHBg2ma9iFzUz3M5
  780. https://open.spotify.com/album/4Th7mnq2a7uf5k2S9z7EQ1
  781. https://open.spotify.com/album/7lL2bQQPCDse2q8i3dMUjx
  782. https://open.spotify.com/album/7HhEcdNmiStc70DT0qzVxY
  783. https://open.spotify.com/album/4EjvvPim1wcnlNhNZjaP9k
  784. https://open.spotify.com/album/5aSra0kDmy7LlHe59nLcJh
  785. https://open.spotify.com/album/4vptJxkxzrfP1tXlc8m121
  786. https://open.spotify.com/album/61jYmljQv0lmucbuKEiHzC
  787. https://open.spotify.com/album/4RTghptkwIbIq160pe3jZM
  788. https://open.spotify.com/album/76TidMxZbTVogBOoM0ulA8
  789. https://open.spotify.com/album/2ac6XqWYgPrL0jYjtJHUb8
  790. https://open.spotify.com/album/4YA82is6EknJnoJSklUx4v
  791. https://open.spotify.com/album/2QweMWdHvDYUCozExzBHmM
  792. https://open.spotify.com/album/3OJP8vSzNpEw7HVzi2rTOi
  793. https://open.spotify.com/album/0aIAiBK8UkUpA6eYb1mUdW
  794. https://open.spotify.com/album/2WbGX8vWla3GiQ5PThmJZF
  795. https://open.spotify.com/album/3cw0ijH8cLdIKrCoAh1Lul
  796. Not available
  797. https://open.spotify.com/album/57bfCFzrv7MgiUhpe9oVo8
  798. https://open.spotify.com/album/1g2NpgmdW9ZJEtnYo4AAEa
  799. https://open.spotify.com/album/1WmnnvWIyZ1ESt5BQpGZ6T
  800. https://open.spotify.com/album/5bcr898rqwVdJTZta6SFo0
  801. https://open.spotify.com/album/2XsBduMp5RdZOTsWMqKJtv
  802. https://open.spotify.com/album/3ci4Yh7hNhAecGf4UqfEl8
  803. https://open.spotify.com/album/2QalqDoSKRt8Xo1RW8Xj3V
  804. https://open.spotify.com/album/1ZNocPzhsKlUjBwgWWZWuq
  805. https://open.spotify.com/album/4NR9t5I0TVUi1Yc3pLYHsx
  806. https://open.spotify.com/album/6oYSflckfXRUmL1MHnkOAQ
  807. https://open.spotify.com/album/655Ly71N60JXaE4A3bkXQt
  808. https://open.spotify.com/album/0FDRIgmEv6n2sERggnDQzy
  809. https://open.spotify.com/album/6179RIIF3vXkAaMqB1N3YV
  810. https://open.spotify.com/album/5PUYJ30eHIvzquoEaYhl8V
  811. https://open.spotify.com/album/4fOnrjQFQ2asr2asX2940u
  812. https://open.spotify.com/album/0qgsuBeCvrhEfJRboNRPxE
  813. https://open.spotify.com/album/4J0g0v4rAINnkvem7e5Fcu
  814. https://open.spotify.com/album/73rRnHjmwZbj9vFoD0tcGZ
  815. https://open.spotify.com/album/1fjWSIa9zzBMmO8OlPSW8m