Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision		 Previous revision
introduction_to_digital_systems:boolean_algebra [2022/10/20 04:58] – [Bearbeiten - Panel] tfischerintroduction_to_digital_systems:boolean_algebra [2023/09/19 23:44] (current) – mexleadmin
Line 1: Line 1:
-====== 1. Boolean Algebra ======+====== 1 Boolean Algebra ======
  
 ===== 1.1 Motivation: Digital Electronics in Daily Life ===== ===== 1.1 Motivation: Digital Electronics in Daily Life =====
Line 14: Line 14:
  
 </callout> </callout>
-==== 1.1.1 Everything is one, exept the zero... ====+==== 1.1.1 Everything is one, except the zero... ====
  
 <WRAP center> <WRAP center>
Line 22: Line 22:
 </WRAP> </WRAP>
  
-Before we start to dive deeper into the digital systems, it is a good idea to approach the topic with a first practical example. For this, we have a look onto an USB cable and mentally cut the cable. +Before we start to dive deeper into digital systems, it is a good idea to approach the topic with a first practical example. For this, we look at a USB cable and mentally cut the cable. 
-We will see a total of four wires waiting for us. Two of them are called D+ and D-. These are the wires that are used for digital data transmission. With special tools - like an oscilloscope we can measure the time course of the voltage between D+ and D-. This voltage is shows two different levels and rises and falls in between. What we see are the **logic states** $0$ or $1$! ((In detail it is a bit more complicated, since USB is not using the voltage levels but the edges to encode the bits. Details [[https://www.youtube.com/watch?v=wdgULBpRoXk|here]])) +We will see a total of four wires waiting for us. Two of them are called D+ and D-. These are the wires that are used for digital data transmission. With special tools - like an oscilloscope, we can measure the time course of the voltage between D+ and D-. This voltage shows two different levels and rises and falls in between. What we see are the **logic states** $0$ or $1$! ((In detail it is a bit more complicated since USB is not using the voltage levels but the edges to encode the bits. Details [[https://www.youtube.com/watch?v=wdgULBpRoXk|here]])) 
  
 Beyond data transmission, we can also encode other things with binary numbers: for example, we could look at  Beyond data transmission, we can also encode other things with binary numbers: for example, we could look at 
Line 29: Line 29:
   * or a switch that is open or closed.   * or a switch that is open or closed.
   * or a light that is on or off.    * or a light that is on or off. 
-  * even whether it is raining or not raining, whether a dog is barking or not barking or other things - we can encode all of that in binary units.+  * even whether it is raining or not raining, whether a dog is barking or not barking, or other things - we can encode all of that in binary units.
  
-In reality, the measured voltage on the USB cable would also show noise and only roughly depict the two states. When we ignore the noise and only assume that an upper level ($HIGH$ or $H$) encode one state and the lower one ($LOW$ or $L$) the second state the diagram ist also called the **level diagram**. In contrast to analog systems, a signal can only take one of the two valid states in digital systems. Similarly, boolean algebra only uses the states $TRUE$ and $FALSE$. +In reality, the measured voltage on the USB cable would also show noise and only roughly depict the two states. When we ignore the noise and only assume that an upper level ($HIGH$ or $H$) encodes one state and the lower one ($LOW$ or $L$) the second state the diagram is also called the **level diagram**. In contrast to analog systems, a signal can only take one of the two valid states in digital systems. Similarly, boolean algebra only uses the states $TRUE$ and $FALSE$. 
  
 <WRAP column 100%> <panel type="danger" title="Note!"> <WRAP group><WRAP column 7%>{{fa>exclamation?32}}</WRAP><WRAP column 80%> <WRAP column 100%> <panel type="danger" title="Note!"> <WRAP group><WRAP column 7%>{{fa>exclamation?32}}</WRAP><WRAP column 80%>
Line 40: Line 40:
     * $HIGH$, $LOW$     * $HIGH$, $LOW$
   * In binary logic, every expression can only be **true or false**. If something is not true, it has to be false - and vice versa.   * In binary logic, every expression can only be **true or false**. If something is not true, it has to be false - and vice versa.
-  * Therefore, binary logic has some **limitations for real world applications**: It may only rain softly, a light might be dimmed and also the voltage (which represents the bit) might be in an in-between state. This situation has to be coped with beyond the binary logic. Beyond binary logic, there is often an **invalid state** in reality, which separates the logic states.+  * Therefore, binary logic has some **limitations for real-world applications**: It may only rain softly, a light might be dimmed, and also the voltage (which represents the bit) might be in an in-between state. This situation has to be coped with beyond the binary logic. Beyond binary logic, there is often an **invalid state** in reality, which separates the logic states.
 </WRAP></WRAP></panel> </WRAP> </WRAP></WRAP></panel> </WRAP>
  
  
-We will find out more of the details behind this states in the chapter [[Number Systems]]. In this chapter we will start to think about, how this binary signals in connection with some algebra can generate a base for fundamental logic building blocks. This building blocks will be stacked together to larger boxes in the next chapters and are the basis for microcontroller, microprocessors and virtually all digital electronics from watches over automotive controllers to super computers.+We will find out more of the details behind these states in the chapter [[Number Systems]]. In this chapter, we will start to think about, how these binary signals in connection with some algebra can generate a base for fundamental logic building blocks. These building blocks will be stacked together into larger boxes in the next chapters and are the basis for microcontrollers, microprocessors, and virtually all digital electronics from watches over automotive controllers to supercomputers.
  
 <imgcaption BildNr1 | Inner 'Life' of an Integrated Cicuit > <imgcaption BildNr1 | Inner 'Life' of an Integrated Cicuit >
Line 56: Line 56:
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
-The core of a computer is the processor in which the instructions are executed. This central processing unit (CPU) is also used in microcontrollers, which can be found around us in almost every device: Mobile phones, cars, bank cards, washing machines... Often there are even several microcontrollers installed in the devices.+The core of a computer is the processor in which the instructions are executed. This central processing unit (CPU) is also used in microcontrollers, which can be found around us in almost every device: Mobile phones, cars, bank cards, and washing machines... Often there are even several microcontrollers installed in the devices.
  
-In the microcontroller, in addition to the command-executing microprocessor (more precisely, the {{wp>arithmetic-logic unit}}), other peripherals such as memory, clock generation, analog-to-digital converter and much more are built in. This makes it a compact tool for many applications. If you look at the microcontroller under an optical microscope, you will see the following picture (<imgref pic1>). +In the microcontroller, in addition to the command-executing microprocessor (more precisely, the {{wp>arithmetic-logic unit}}), other peripherals such as memory, clock generation, analog-to-digital converter, and much more are built in. This makes it a compact tool for many applications. If you look at the microcontroller under an optical microscope, you will see the following picture (<imgref pic1>). 
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
Line 70: Line 70:
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
-But now let's take a look at the structure of the processor. The processor shown in <imgref pic2> uad <imgref pic3> was developed by [[https://spectrum.ieee.org/tech-history/silicon-revolution/chip-hall-of-fame-atmel-atmega8|two students]] in 1990 and consists of several tens of thousands of transistors. This chip paved the way to cheap, fast and yet easily programmable controllers, from the fax machine to the hobby basement, and can be found on the Arduino boards, among others. You will get to know and to program the ATmega328 - a distant successor with several hundred thousand transistors - in higher semesters. +But now let's take a look at the structure of the processor. The processor shown in <imgref pic2> and <imgref pic3> were developed by [[https://spectrum.ieee.org/tech-history/silicon-revolution/chip-hall-of-fame-atmel-atmega8|two students]] in 1990 and consists of several tens of thousands of transistors. This chip paved the way to cheap, fast, and yet easily programmable controllers, from the fax machine to the hobby basement, and can be found on Arduino boards, among others. You will get to know and program the ATmega328 - a distant successor with several hundred thousand transistors - in higher semesters. 
-The images depict various peripheral components: The processor, FLASH, EEPROM and fuses will be shown in the following chapters. The voltage supply (German 'Spannungsversorung'), is needed to generate additional higher voltages based on the supplied external voltage. +The images depict various peripheral components: The processor, FLASH, EEPROM, and fuses will be shown in the following chapters. The voltage supply (German 'Spannungsversorung'), is needed to generate additional higher voltages based on the supplied external voltage. 
  
 The following clip shows a zoom into the smallest parts of the controller. The following clip shows a zoom into the smallest parts of the controller.
Line 81: Line 81:
  
  
-One question is still unclear: how can we connect the zeros and ones in such a way, that the processor can calculate something like $23 + 42$? For this we need to have a look onto binary logic.+One question is still unclear: how can we connect the zeros and ones in such a way, that the processor can calculate something like $23 + 42$? For this, we need to have a look at binary logic.
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
Line 94: Line 94:
   - apply the Boolean rules of arithmetic.   - apply the Boolean rules of arithmetic.
   - simplify Boolean expressions.   - simplify Boolean expressions.
-  - understand the following terms: bit, the different (logic) gates, timing diagram, truth table.+  - understand the following terms: bit, the different (logic) gates, timing diagram, and truth table.
   - understand the purpose of the Tri-State gate and the "Z" state.   - understand the purpose of the Tri-State gate and the "Z" state.
   - understand the use of the "Don't care" state.   - understand the use of the "Don't care" state.
Line 102: Line 102:
 ==== 1.2.1 First Steps into Logic ==== ==== 1.2.1 First Steps into Logic ====
  
-We have already learned about the 'bit', and it's two-valued value. This can be connected to the ancient idea of a binary logic.  +We have already learned about the 'bit', and its two-valued value. This can be connected to the ancient idea of binary logic.  
-The Greek Philosopher Aristotle started to build up a systematic in order to conclude from statements like 'at night, it is dark outside' and 'it is night' to 'it has to be dark outside'. +The Greek Philosopher Aristotle started to build up a system to conclude from statements like "at night, it is dark outside" and "it is night" to "it has to be dark outside". 
-It might seem a bit unrelated to comtrollers and computers, at the first sight. But in this scientific interpretation of logic all logic statements are either true or false.+It might seem a bit unrelated to controllers and computers, at the first sight. But in this scientific interpretation of logic, all logic statements are either true or false.
  
-{{wp>George Boole}} developed a more mathematical way into handline logic. Based on his work the fundamental logic was solidified into axioms. +{{wp>George Boole}} developed a more mathematical way of handline logic. Based on his work the fundamental logic was solidified into axioms. 
-One axiom we have already seen: If something is true, it cannot be false ans vice versa (the 'theorem of contradiction'). +One axiom we have already seen: If something is true, it cannot be false and vice versa (the 'theorem of contradiction'). 
  
 Other axioms help to combine statements. This is called 'reasoning' or 'deduction'.  Other axioms help to combine statements. This is called 'reasoning' or 'deduction'. 
-  * This was already used some sentences before: If 'at night, it is dark outside' and 'it is night' is true, one can make the **implication** 'it has to be dark outside'. \\ But be aware, that is is not always reversible: It could also be a solar eclipse.+  * This was already used in some sentences before: If "at night, it is dark outside" and "it is night" is true, one can make the **implication** 'it has to be dark outside'. \\ But be aware, that this is not always reversible: It could also be a solar eclipse.
   * Another deduction is the **equivalence**: when '$A$' implies '$B$', and '$B$' implies '$A$' both statements are equivalent.    * Another deduction is the **equivalence**: when '$A$' implies '$B$', and '$B$' implies '$A$' both statements are equivalent. 
  
-In the following, we will have a look onto the basic logic combinations, which are needed for digital systems. These basic logic combinations of logic statements (or bits) are called **logic operators** and can be used similarly to the algebraic "plus" or "minus". The mathematical construct, which describes the systematic of these combinations is called **boolean algebra**.  +In the following, we will have a look at the basic logic combinations, which are needed for digital systems. These basic logic combinations of logic statements (or bits) are called **logic operators** and can be used similarly to the algebraic "plus" or "minus". The mathematical construct, which describes the system of these combinations is called **boolean algebra**.  
-In digital systems the logic operators can be represented as a circuit of transistors or switches. This can be blackboxed into **logic gates**. +In digital systems, the logic operators can be represented as a circuit of transistors or switches. This can be black-boxed into **logic gates**. 
  
-This tools will get more familiar in the next subcapter. +These tools will get more familiar in the next subchapter. 
  
 <WRAP> <WRAP>
-A good introduction in binary logic+A good introduction to binary logic
 {{youtube>7bVnsXHO6Uw?start=123}} {{youtube>7bVnsXHO6Uw?start=123}}
 </WRAP>  </WRAP> 
Line 133: Line 133:
 The first very simple circuit negates the input value. It is also called an inverter or negation ([[https://en.wikipedia.org/wiki/Negation|logic NOT]], [[https://en.wikipedia.org/wiki/Inverter_(logic_gate)|NOT gate]]). This logic operator always generates the output value $Y(0)=1$ from the digital input value $X=0$ and for $X=1$ correspondingly $Y(1)=0$. <imgref pic20> shows an example: The light is only on ($X=1$), when the input is off ($Y=0$) - this functionality is "hard-wired".  The first very simple circuit negates the input value. It is also called an inverter or negation ([[https://en.wikipedia.org/wiki/Negation|logic NOT]], [[https://en.wikipedia.org/wiki/Inverter_(logic_gate)|NOT gate]]). This logic operator always generates the output value $Y(0)=1$ from the digital input value $X=0$ and for $X=1$ correspondingly $Y(1)=0$. <imgref pic20> shows an example: The light is only on ($X=1$), when the input is off ($Y=0$) - this functionality is "hard-wired". 
  
-When you think about the inputs and outputs in the shown picture above, you realise, that the input is a voltage, but the output is a current. This is sometimes beneficial, but inside an digital systems commonly only voltages are used. +When you think about the inputs and outputs in the shown picture above, you realize, that the input is a voltage, but the output is current. This is sometimes beneficial, but within digital systems commonly only voltages are used. 
  
 <WRAP><well> <WRAP><well>
Line 142: Line 142:
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
-In order to control a (output) voltage with a (input) voltage two complement types of switches are combined similar to a voltage divider or a half bridge (see <imgref pic21>). One is normally open and the other one  normally closed. This can also set up with complement types of transistors. Thus, only one transistor (TRANSfer ResISTOR) becomes conductive at a time, the other one correspondingly high impedance.+To control an (output) voltage with an (input) voltage two complement types of switches are combined similar to a voltage divider or a half-bridge (see <imgref pic21>). One is normally open and the other one is normally closed. This can also be set up with complementary types of transistors. Thus, only one transistor (TRANSfer ResISTOR) becomes conductive at a time, the other one correspondingly high impedance.
  
-With this setup the logic voltages ($0V$, $5V$) are just switched complementary via the switches. This technique is also called **CMOS** technique: __C__omplementary __MOS__FET. In today's electronics, this technology is used throughout and has completely replaced older variants (e.g. TTL).+With this setup, the logic voltages ($0~\rm V$, $5~\rm V$) are just switched complimentary via the switches. This technique is also called **CMOS** technique: __C__omplementary __MOS__FET. In today's electronics, this technology is used throughout and has completely replaced older variants (e.g. TTL).
  
-Comparing the circuits in <imgref pic20> and <imgref pic21>, one could simply see, that double the amount of switches have to be used and the lower one is a bit trickier to understand. With this in mind, for the following operators only the first type of circuit will be shown.+Comparing the circuits in <imgref pic20> and <imgref pic21>, one could simply see, that double the number of switches have to be used and the lower one is a bit trickier to understand. With this in mind, for the following operators only the first type of circuit will be shown.
  
 <WRAP center>  <WRAP center> 
Line 154: Line 154:
 </WRAP> </WRAP>
  
-The circuit symbols of the NOT-gate are shown in <imgref BildNr06>. Depending on the (software) tools and schematics one of the type is printed.+The circuit symbols of the NOT-gate are shown in <imgref BildNr06>. Depending on the (software) tools and schematics one of the types is printed.
  
 <WRAP center> <WRAP center>
Line 163: Line 163:
  
 Besides the symbol other representation are also common (see also <imgref BildNr07>): Besides the symbol other representation are also common (see also <imgref BildNr07>):
-  * The **truth table** shows the input(s) $X$ on the left and the output(s) $Y$ on the right. There is a row for each distinct combination of inputs. This representation will get handy in the next chapters, in order to analyse more complex logic. Often instead they are also called **look-up table** or **LUT**. +  * The **truth table** shows the input(s) $X$ on the left and the output(s) $Y$ on the right. There is a row for each distinct combination of inputs. This representation will get handy in the next chapters, to analyze more complex logic. Often instead they are also called **look-up table** or **LUT**. 
-  * The **timing diagram** shows the sequential behavior. For this diagram, the input variables are stimulated with all possible state combinations. Also this will become handy especialy in the chapter [[Sequential Logic]].+  * The **timing diagram** shows the sequential behavior. For this diagram, the input variables are stimulated with all possible state combinations. Also, this will become handy, especially in the chapter [[Sequential Logic]].
   * In **math** the inversion has also multiple representations e.g.   * In **math** the inversion has also multiple representations e.g.
     * $Y = /X$ (used e.g. for input with a keyboard)     * $Y = /X$ (used e.g. for input with a keyboard)
Line 174: Line 174:
   * Inputs are always denoted with $X$   * Inputs are always denoted with $X$
   * Outputs are always denoted with $Y$    * Outputs are always denoted with $Y$ 
-  * There are multiple ways for representing the logic. The most common ones are: \\  electric ciruit with switches (or transistors), logic gate, truth table, timing diagram, mathematical representation.+  * There are multiple ways for representing logic. The most common ones are: \\  electric circuits with switches (or transistors), logic gates, truth tables, timing diagrams, and mathematical representation.
   * When you use a representation: use it uniformly.   * When you use a representation: use it uniformly.
  
Line 189: Line 189:
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
-The next circuit will generate an positive output only, if __all__ inputs are true. This is called a logical **__con__junction** ([[https://en.wikipedia.org/wiki/logical_conjunction|logic AND]], [[https://en.wikipedia.org/wiki/AND_gate|AND gate]]). When one ore more inputs are false the output is also false. <imgref pic22> shows an example: The light is only on ($Y=1$), when all inputs are on ($X0=1$, $X1=1$, ...). This is commonly used for safety circuits, e.g. when the workspace of a robot has multiple doors and all have to be closed in order to start.+The next circuit will generate a positive output only, if __all__ inputs are true. This is called a logical **__con__junction** ([[https://en.wikipedia.org/wiki/logical_conjunction|logic AND]], [[https://en.wikipedia.org/wiki/AND_gate|AND gate]]). When one or more inputs are false the output is also false. <imgref pic22> shows an example: The light is only on ($Y=1$), when all inputs are on ($X0=1$, $X1=1$, ...). This is commonly used for safety circuits, e.g. when the workspace of a robot has multiple doors and all have to be closed to start.
  
 <WRAP center>  <WRAP center> 
Line 203: Line 203:
 </WRAP> </WRAP>
  
-Again, the other representation are shown:+Again, the other representations are shown:
   * The truth table and timing diagram are depicted in <imgref pic24>.   * The truth table and timing diagram are depicted in <imgref pic24>.
   * In **math** the AND operation has again multiple representations e.g.   * In **math** the AND operation has again multiple representations e.g.
Line 211: Line 211:
     * $Y = X_0 \land X_1$ (used in logic)     * $Y = X_0 \land X_1$ (used in logic)
  
-For upcoming, more complex terms the algebraic notation ($Y = X_0 \cdot X_1$) usually lead to a better understanding.+For upcoming, more complex terms the algebraic notation ($Y = X_0 \cdot X_1$) usually leads to a better understanding.
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
Line 223: Line 223:
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
-The NOT gate is often used in front of or after other gates. When used after AND gates, this creates an 'NOT AND' or in short 'NAND' ([[https://en.wikipedia.org/wiki/Sheffer_stroke|logic NAND]], [[https://en.wikipedia.org/wiki/NAND_gate|NAND gate]]). This circuit will only generate an negative output only, if __all__ inputs are true. When one ore more inputs are false the output is true. <imgref pic25> shows an example: The light is only off ($Y=0$), when all inputs are high ($X0=1$, $X1=1$, ...). In the simulation one has to look in detail: the used switches are normally closed (closed when the input is low). Therefore the switches are only open, when the input is high.+The NOT gate is often used in front of or after other gates. When used after AND gates, this creates a 'NOT AND' or in short 'NAND' ([[https://en.wikipedia.org/wiki/Sheffer_stroke|logic NAND]], [[https://en.wikipedia.org/wiki/NAND_gate|NAND gate]]). This circuit will only generate a negative output, if __all__ inputs are true. When one or more inputs are false the output is true. <imgref pic25> shows an example: The light is only off ($Y=0$), when all inputs are high ($X0=1$, $X1=1$, ...). In the simulation one has to look in detail: the used switches are normally closed (closed when the input is low). Therefore the switches are only open when the input is high.
  
 <WRAP center>  <WRAP center> 
Line 231: Line 231:
 </WRAP> </WRAP>
  
-The circuit symbols are shown in <imgref pic26>. In order to shorten the ciruit, the NOT is often 'shrinked' only to a small circle after the gate.+The circuit symbols are shown in <imgref pic26>. To shorten the circuit, the NOT is often 'shrank' only to a small circle after the gate.
  
 <WRAP center> <WRAP center>
Line 239: Line 239:
 </WRAP> </WRAP>
  
-Again, the other representation are shown:+Again, the other representations are shown:
   * The truth table and timing diagram are depicted in <imgref pic27>.   * The truth table and timing diagram are depicted in <imgref pic27>.
   * In **math** the NAND operation has again multiple representations e.g.   * In **math** the NAND operation has again multiple representations e.g.
Line 258: Line 258:
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
-We already had a look onto the AND gate. So what about OR? Of course there is also this kind of operation. This is called a logical **__dis__junction** ([[https://en.wikipedia.org/wiki/Logical_disjunction|logic OR]], [[https://en.wikipedia.org/wiki/OR_gate|OR gate]]). When there is one __or more__ input true the output is also true.  <imgref pic28> shows an example: The light is only off ($Y=0$), when all inputs are off ($X0=0$, $X1=0$, ...). Doesn't it - at the first glimpe - seem similar to the NAND circuit? The main difference is that normally open switches are used. Only, when both switches are open the light is off.+We already had a look at the AND gate. So what about OR? Of course, there is also this kind of operation. This is called a logical **__dis__junction** ([[https://en.wikipedia.org/wiki/Logical_disjunction|logic OR]], [[https://en.wikipedia.org/wiki/OR_gate|OR gate]]). When there is one __or more__ inputs true the output is also true.  <imgref pic28> shows an example: The light is only off ($Y=0$), when all inputs are off ($X0=0$, $X1=0$, ...). Doesn't it - at first glimpse - seem similar to the NAND circuit? The main difference is that normally open switches are used. Only, when both switches are open the light is off.
  
 <WRAP center>  <WRAP center> 
Line 266: Line 266:
 </WRAP> </WRAP>
  
-The circuit symbols are shown in <imgref pic29>. The DIN symbol is derived from the fact, that one ore more inputs have to be true to get an true output.+The circuit symbols are shown in <imgref pic29>. The DIN symbol is derived from the fact, that one or more inputs have to be true to get a true output.
  
 <WRAP center> <WRAP center>
Line 274: Line 274:
 </WRAP> </WRAP>
  
-Again, the other representation are shown:+Again, the other representations are shown:
   * The truth table and timing diagram are depicted in <imgref pic30>.   * The truth table and timing diagram are depicted in <imgref pic30>.
   * In **math** the OR operation has again multiple representations e.g.   * In **math** the OR operation has again multiple representations e.g.
     * $Y = X0 + X1$ (used e.g. for input with a keyboard or in handwriting)     * $Y = X0 + X1$ (used e.g. for input with a keyboard or in handwriting)
-    * ''Y = X0 | X1'' (used in c language biswise)+    * ''Y = X0 | X1'' (used in c language bitwise)
     * $Y = X_0 \lor X_1$ (used in logic, the $\lor$ stands for the Latin //vel//, which means or)     * $Y = X_0 \lor X_1$ (used in logic, the $\lor$ stands for the Latin //vel//, which means or)
  
-For upcoming, more complex terms the algebraic notation ($Y = X_0 + X_1$) usually lead to a better understanding. Also here: we will see the connection to math in the next chapters.+For upcoming, more complex terms the algebraic notation ($Y = X_0 + X_1$) usually leads to a better understanding. Also here: we will see the connection to math in the next chapters.
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
Line 293: Line 293:
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
-Beside the OR there is also an "either ... or ..., __but__ not both". This is called exclusive or, in short XOR ([[https://en.wikipedia.org/wiki/Exclusive_or|logic XOR]], [[https://en.wikipedia.org/wiki/XOR_gate|XOR gate]]). Only when one input is true the output is true. <imgref pic28> shows an example: When none or when all inputs are on, the light is only off. The circuit looks a bit more complicated with two series branches in parallel and the use of both normally closed and normally open switches. On the other hand one can already think about, that one of the branches look similar to the setup for the AND gate, and the parallel setup similar to the NAND gate. Could it be possible to convert the gates into each other? We will see that next...+Beside the OR there is also an "either ... or ..., __but__, not both". This is called exclusive or, in short XOR ([[https://en.wikipedia.org/wiki/Exclusive_or|logic XOR]], [[https://en.wikipedia.org/wiki/XOR_gate|XOR gate]]). Only when one input is true the output is true. <imgref pic28> shows an example: When none or when all inputs are on, the light is only off. The circuit looks a bit more complicated with two series branches in parallel and the use of both normally closed and normally open switches. On the other hand, one can already think, that one of the branches looks similar to the setup for the AND gate, and the parallel setup is similar to the NAND gate. Could it be possible to convert the gates into each other? We will see that next...
  
 <WRAP center>  <WRAP center> 
Line 309: Line 309:
 </WRAP> </WRAP>
  
-Again, the other representation are shown:+Again, the other representations are shown:
   * The truth table and timing diagram are depicted in <imgref pic33>.   * The truth table and timing diagram are depicted in <imgref pic33>.
   * In **math** the XOR operation has again multiple representations e.g.   * In **math** the XOR operation has again multiple representations e.g.
Line 328: Line 328:
 ==== 1.2.7 The Tri-State Gate ==== ==== 1.2.7 The Tri-State Gate ====
  
-The [[https://en.wikipedia.org/wiki/Three-state_logic|tri-state gate]] is not a boolean gate, however it is still often used in logic circuits such as microcontrollers. +The [[https://en.wikipedia.org/wiki/Three-state_logic|tri-state gate]] is not a boolean gate, however, it is still often used in logic circuits such as microcontrollers. 
 The essence of the tri-state gate is - in short - to be able to output 'nothing'. Nothing means: neither high nor low. The essence of the tri-state gate is - in short - to be able to output 'nothing'. Nothing means: neither high nor low.
  
-One possible output of the tri-state gate - besides high and low - is 'high ohmic', which is often reffered as $Z$. In this case, the gate output is not controlled by the tri-state gate, but floats. The output can instead be controlled by an other external source. The main use of this gate is to disconnect one logic circuit from another one.+One possible output of the tri-state gate - besides high and low - is 'high ohmic', which is often referred to as $Z$. In this case, the gate output is not controlled by the tri-state gate but floats. The output can instead be controlled by another external source. The main use of this gate is to disconnect one logic circuit from another one.
  
 <WRAP><well> <WRAP><well>
Line 340: Line 340:
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
-<imgref pic28> shows the function of a tri-state gate: When the $EN$able input is set to high this gate becomes transparent and the output $Y$ equals the input $Y$. When $EN$ is set to low, the output is not pinned to the input anymore. In the simulation above with $EN = 0$ the output voltage is clamped by the voltage behind the resistor (here $3V$). Since this voltage could be any value this output cannot called low or high, but is called the (undefined) state $Z$.+<imgref pic28> shows the function of a tri-state gate: When the $EN$able input is set to high this gate becomes transparent and the output $Y$ equals the input $Y$. When $EN$ is set to low, the output is not pinned to the input anymore. In the simulation above with $EN = 0$ the output voltage is clamped by the voltage behind the resistor (here $3V$). Since this voltage could be any value this output cannot be called low or high, but is called the (undefined) state $Z$.
  
 <WRAP center>  <WRAP center> 
Line 348: Line 348:
 </WRAP> </WRAP>
  
-The circuit symbol is shown in <imgref pic901>. Usually the triangle sybmol ist used, the DIN / EN symbol is much less common and therefore here ignored. +The circuit symbol is shown in <imgref pic901>. Usually, the triangle symbol is used, the DIN / EN symbol is much less common and therefore here ignored. 
-Since the Tri-State gate is not an logical gate is does not have any mathematical representation.+Since the Tri-State gate is not a logical gate it does not have any mathematical representation.
  
 <WRAP center> <WRAP center>
Line 357: Line 357:
 </WRAP> </WRAP>
  
-In the <imgref pic920> the truth table is shown. In the case $EN=0$ the input $X$ does not matter.  +In the <imgref pic920> the truth table is shown. In this case, $EN=0$ the input $X$ does not matter.  
-The situation 'the input x does not matter' is usually simplified by the use of a **{{wp>don't care term}}** . \\+The situation 'the input x does not matter' is usually simplified by the use of a **{{wp>don't care term}}**. \\
 The 'don't care' is written down as a ''x'' or a ''-''. The 'don't care' is written down as a ''x'' or a ''-''.
  
 <callout title="Example of an Application for a Tri-State Gate"> <callout title="Example of an Application for a Tri-State Gate">
  
-Let's imagine, we want to connect two microcontroller $A$ and $B$ in order to enable a communication, i.e. transmitting and receiving data on both sides. \\ +Let's imagine, we want to connect two microcontrollers $A$ and $B$ to enable communication, i.e. transmitting and receiving data on both sides. \\ 
-One posibility would be to use two wires:+One possibility would be to use two wires:
   * One wire for data sent from $A$ to $B$   * One wire for data sent from $A$ to $B$
   * One wire for data sent from $B$ to $A$   * One wire for data sent from $B$ to $A$
-This is shown in <imgref pic910>. You can change the inputs $X0$ and $X1$ by clicking on the ''L'' and ''H'' nearby this labels. \\ +This is shown in <imgref pic910>. You can change the inputs $X0$ and $X1$ by clicking on the ''L'' and ''H'' nearby these labels. \\ 
-The big advantage of this configuration is, that both connected microcontroller can send data whenever they want. The biggest disadvantage is, that one need two wires.+The big advantage of this configuration is, that both connected microcontrollers can send data whenever they want. The biggest disadvantage is, that one needs two wires.
  
 <WRAP><well> <WRAP><well>
Line 377: Line 377:
  
 Once we think of only using one wire, it becomes more complicated: a single wire can only be driven by one digital input - only one can transmit data at any given time.  Once we think of only using one wire, it becomes more complicated: a single wire can only be driven by one digital input - only one can transmit data at any given time. 
-Therefore, we have to switch on both sides from reveive to transmit, e.g. by a Single Pole Double Throw switch. This can be seen in <imgref pic911> for the two situations "$A$ sends data to $B$" and "$B$ sends data to $A$". +Therefore, we have to switch on both sides from receive to transmit, e.g. by a Single Pole Double Throw switch. This can be seen in <imgref pic911> for the two situations "$A$ sends data to $B$" and "$B$ sends data to $A$". 
-Again, you can change the inputs $X0$ and $X1$ by clicking on the ''L'' and ''H'' nearby this labels. +Again, you can change the inputs $X0$ and $X1$ by clicking on the ''L'' and ''H'' nearby these labels. 
  
-The problem is, that in one time the output becomes an input, but boolean gates an algebra result everytime in boolean outputs.+The problem is, that at one time the output becomes an input, but boolean gates an algebra result every time in boolean outputs.
  
 <WRAP><well> <WRAP><well>
Line 388: Line 388:
 </well></WRAP> </well></WRAP>
  
-We still could try solve it with gates, as seen in <imgref pic912>: Each microcontroller has an enable signal (''EN0'', ''EN1''). Both outputs from the microcontrollers have to be combined with another gate in such a way, that the result shows the enabled signal. +We still could try to solve it with gates, as seen in <imgref pic912>: Each microcontroller has an enable signal (''EN0'', ''EN1''). Both outputs from the microcontrollers have to be combined with another gate in such a way, that the result shows the enabled signal. 
  
-The problem here: We are back to a two wire system. So, we need to "split up" the OR gate somehow..+The problem here: We are back to a two-wire system. So, we need to "split up" the OR gate somehow..
  
 <WRAP><well> <WRAP><well>
Line 398: Line 398:
 </well></WRAP> </well></WRAP>
  
-For this we can use the Tri-State gate: This enables to switch an output to high ohmic. This means this output does not provide any current anymore, so this output is not driven anymore.+For this we can use the Tri-State gate: This enables us to switch output to high ohmic. This means this output does not provide any current anymore, so this output is not driven anymore.
  
 <WRAP><well> <WRAP><well>
Line 422: Line 422:
 </callout> </callout>
  
-Since the timing diagram is of importance not only for the upcoming courses like Electronics (in the 3rd semester) I recommend to watch the following video sequence.+Since the timing diagram is of importance not only for the upcoming courses like Electronics (in the 3rd semester) I recommend watching the following video sequence.
  
 <WRAP> <WRAP>
Line 441: Line 441:
 </callout> </callout>
  
-Some of the circuits in the provious chapter looked suspiciously similar. We will now have a more deeper look onto this and try to convert some gates into each other.  +Some of the circuits in the previous chapter looked suspiciously similar. We will now have a deeper look into this and try to convert some gates into each other.  
-In this subchapter we will focus on combining NAND gates in order to build other gates. As we will see, based on the NAND and NOR gates any other gate and any other logic can be created.+In this subchapter, we will focus on combining NAND gates to build other gates. As we will see, based on the NAND and NOR gates any other gate and any other logic can be created.
  
 ==== 1.4.1 NAND in NOT ==== ==== 1.4.1 NAND in NOT ====
Line 452: Line 452:
 </well></WRAP> </well></WRAP>
  
-The conversion from NAND to NOT is relatively simple: When both inputs to NAND are the same (either '1' or '0') the output will be the negation of the input. This can also be seen in the truth table of the NAND gate: when the inputs as the same only the first and last row, have to be considered and lead to a inverting behaviour.+The conversion from NAND to NOT is relatively simple: When both inputs to NAND are the same (either '1' or '0') the output will be the negation of the input. This can also be seen in the truth table of the NAND gate: when the inputs as the same only the first and last rows, have to be considered and leading to an inverting behavior.
  
-A different approach to get an NOT is to set the second input to '1'. Here, the NOT can be 'deactivated' of with the second input. This can be tested in the <imgref pic34> by clicking on the input 'H' of the NAND gate on the right below.+A different approach to get a NOT is to set the second input to '1'. Here, the NOT can be 'deactivated' of with the second input. This can be tested in the <imgref pic34> by clicking on the input 'H' of the NAND gate on the right below.
  
 ==== 1.4.2 NAND in AND ==== ==== 1.4.2 NAND in AND ====
Line 464: Line 464:
 </well></WRAP> </well></WRAP>
  
-With the knwoledge from 'NAND to NOT' the NAND can be converted to AND: a negated NAND leads to an AND. It is roughly similar to 'not a no-go' is logically a 'go'.  +With the knowledge from 'NAND to NOT' the NAND can be converted to AND: a negated NAND leads to an AND. It is roughly similar to 'not a no-go' and is logically a 'go'.  
-Therefore, the NOT hat to be set behind the NAND.+Therefore, the NOT had to be set behind the NAND.
  
 ==== 1.4.3 NAND in OR ==== ==== 1.4.3 NAND in OR ====
Line 475: Line 475:
 </well></WRAP> </well></WRAP>
  
-When each input of a NAND gate is inverted the result acts like an OR gate. In order to understand this, one can again look onto the truth table of the NAND and the OR gate and try to invetigate what happens when the inputs of the NAND are negated. +When each input of a NAND gate is inverted the result acts like an OR gate. To understand this, one can again look at the truth table of the NAND and the OR gate and try to investigate what happens when the inputs of the NAND are negated. 
  
  
Line 493: Line 493:
 </callout> </callout>
  
-We have seen, that (at least) some of the gates can be represented by means of others. In order to approach this more systematically, we will now have a look onto the arithmetic rules of boolean algebra. These rules can be used to either build a logic cicruit out of the basis gates shown in chapter 1.2. On the other hand we are also able to simplify the logic circuits by ths rules.+We have seen, that (at least) some of the gates can be represented using others. To approach this more systematically, we will now have a look at the arithmetic rules of boolean algebra. These rules can be used to either build a logic circuit out of the basis gates shown in chapter 1.2. On the other hand, we are also able to simplify the logic circuits by these rules.
  
 ==== 1.5.1 The Set of Rules ==== ==== 1.5.1 The Set of Rules ====
Line 500: Line 500:
 It is possible to click on "math representation" and "algebraic representation" to switch both. It is possible to click on "math representation" and "algebraic representation" to switch both.
  
-Have a look for the algebraic representation. These are probably much simpler to remember!+Have a look at the algebraic representation. These are probably much simpler to remember!
  
 Be also aware, that the logical expressions sometimes are written as $X0$, $X1$, ... and sometimes with other letters, like $a$, $b$, ...  Be also aware, that the logical expressions sometimes are written as $X0$, $X1$, ... and sometimes with other letters, like $a$, $b$, ... 
Line 512: Line 512:
 |:::|$B \lor  B \rightarrow B                          $|:::    |  |:::|$B \lor  B \rightarrow B                          $|:::    | 
 |2  |**Duality**                                        | If $A$ is a statement of boolean algebra, so is $A^*$. \\ $A^*$ is obtained by exchanging $\land$ with $\lor$ and vice versa. |  |2  |**Duality**                                        | If $A$ is a statement of boolean algebra, so is $A^*$. \\ $A^*$ is obtained by exchanging $\land$ with $\lor$ and vice versa. | 
-|3  |**Neutral Element**                                | There exist a neutral element to the operators $\land$ and $\lor$. \\ Applying the oparator to a and the neutral element results in $a$. |+|3  |**Neutral Element**                                | There exist a neutral element to the operators $\land$ and $\lor$. \\ Applying the operator to a and the neutral element results in $a$. |
 |:::|$a \land 1 = a                                    $|:::    | |:::|$a \land 1 = a                                    $|:::    |
 |:::|$a \lor  0 = a                                    $|:::    |  |:::|$a \lor  0 = a                                    $|:::    | 
Line 524: Line 524:
 |:::|$a \land b = b \land a                            $|:::    |  |:::|$a \land b = b \land a                            $|:::    | 
 |:::|$a \lor  b = b \lor  a                            $|:::    |  |:::|$a \lor  b = b \lor  a                            $|:::    | 
-|7  |**Associative Law**                                | For the same operator bracketing can be moved. \\ associative means "to unite" or "to  connect" |+|7  |**Associative Law**                                | For the same operator bracketing can be moved. \\ associative means "to unite" or "to connect" |
 |:::|$a \land (b \land c) = (a \land b) \land c        $|:::    |  |:::|$a \land (b \land c) = (a \land b) \land c        $|:::    | 
 |:::|$a \lor  (b \lor  c) = (a \lor  b) \lor  c        $|:::    |  |:::|$a \lor  (b \lor  c) = (a \lor  b) \lor  c        $|:::    | 
Line 530: Line 530:
 |:::|$a \land (b \lor  c) = (a \land b) \lor  (a \land c)$|:::    |  |:::|$a \land (b \lor  c) = (a \land b) \lor  (a \land c)$|:::    | 
 |:::|$a \lor  (b \land c) = (a \lor  b) \land (a \lor  c)$|:::    |  |:::|$a \lor  (b \land c) = (a \lor  b) \land (a \lor  c)$|:::    | 
-|9  |**Law of Absorbtion**                                          | In bracketed formulas similar expressions can "absorb" each other. \\ This law can be drived from the laws (8), (5), (7)  |+|9  |**Law of Absorption**                                          | In bracketed formulas similar expressions can "absorb" each other. \\ This law can be derived from the laws (8), (5), (7)  |
 |:::|$a \land (a \lor  b) = a$ \\ $a \land (\bar{a} \lor  b) = a \land b $|:::    |  |:::|$a \land (a \lor  b) = a$ \\ $a \land (\bar{a} \lor  b) = a \land b $|:::    | 
 |:::|$a \lor  (a \land b) = a$ \\ $a \lor  (\bar{a} \land b) = a \lor  b $|:::    |  |:::|$a \lor  (a \land b) = a$ \\ $a \lor  (\bar{a} \land b) = a \lor  b $|:::    | 
Line 545: Line 545:
 |:::|$B +  B \rightarrow B                          $|:::    |  |:::|$B +  B \rightarrow B                          $|:::    | 
 |2  |**Duality**                                        | If $A$ is a statement of boolean algebra, so is $A^*$. \\ $A^*$ is obtained by exchanging $\cdot$ with $+$ and vice versa. |  |2  |**Duality**                                        | If $A$ is a statement of boolean algebra, so is $A^*$. \\ $A^*$ is obtained by exchanging $\cdot$ with $+$ and vice versa. | 
-|3  |**Neutral Element**                                | There exist a neutral element to the operators $\cdot$ and $+$. \\ Applying the oparator to a and the neutral element results in $a$. |+|3  |**Neutral Element**                                | There exist a neutral element to the operators $\cdot$ and $+$. \\ Applying the operator to a and the neutral element results in $a$. |
 |:::|$a \cdot 1 = a                                    $|:::    | |:::|$a \cdot 1 = a                                    $|:::    |
 |:::|$a +  0 = a                                    $|:::    |  |:::|$a +  0 = a                                    $|:::    | 
Line 557: Line 557:
 |:::|$a \cdot b = b \cdot a                            $|:::    |  |:::|$a \cdot b = b \cdot a                            $|:::    | 
 |:::|$a +  b = b +  a                            $|:::    |  |:::|$a +  b = b +  a                            $|:::    | 
-|7  |**Associative Law**                                | For the same operator bracketing can be moved. \\ associative means "to unite" or "to  connect" |+|7  |**Associative Law**                                | For the same operator bracketing can be moved. \\ associative means "to unite" or "to connect" |
 |:::|$a \cdot (b \cdot c) = (a \cdot b) \cdot c        $|:::    |  |:::|$a \cdot (b \cdot c) = (a \cdot b) \cdot c        $|:::    | 
 |:::|$a +  (b +  c) = (a +  b) +  c        $|:::    |  |:::|$a +  (b +  c) = (a +  b) +  c        $|:::    | 
Line 563: Line 563:
 |:::|$a \cdot (b +  c) = (a \cdot b) +  (a \cdot c)$|:::    |  |:::|$a \cdot (b +  c) = (a \cdot b) +  (a \cdot c)$|:::    | 
 |:::|$a +  (b \cdot c) = (a +  b) \cdot (a +  c)$|:::    |  |:::|$a +  (b \cdot c) = (a +  b) \cdot (a +  c)$|:::    | 
-|9  |**Law of Absorbtion**                                          | In bracketed formulas similar expressions can "absorb" each other. \\ This law can be drived from the laws (8), (5), (7)  |+|9  |**Law of Absorption**                                          | In bracketed formulas similar expressions can "absorb" each other. \\ This law can be derived from the laws (8), (5), (7)  |
 |:::|$a \cdot (a +  b) = a$ \\ $a \cdot (\bar{a} +  b) = a \cdot b $|:::    |  |:::|$a \cdot (a +  b) = a$ \\ $a \cdot (\bar{a} +  b) = a \cdot b $|:::    | 
 |:::|$a +  (a \cdot b) = a$ \\ $a +  (\bar{a} \cdot b) = a +  b $|:::    |  |:::|$a +  (a \cdot b) = a$ \\ $a +  (\bar{a} \cdot b) = a +  b $|:::    | 
Line 573: Line 573:
 </accordion> </accordion>
  
-The last 3 laws are probable a kind of unintuitive. Therefore, these are shown in other representations. in the following+The last 3 laws are probably kind of unintuitive. Therefore, these are shown in other representations. in the following
  
 ==== 1.5.2 Dive into Distributive Law  ==== ==== 1.5.2 Dive into Distributive Law  ====
  
-The distributive law can be shown on a simple example of daily life. \\+The distributive law can be shown as a simple example of daily life. \\
 When one says When one says
    "I'm happy with fries AND (a water OR a coke)"     "I'm happy with fries AND (a water OR a coke)" 
 he will be happy with he will be happy with
-   (fries AND a water) OR (fries AND a coke)+   (fries AND water) OR (fries AND a coke)
  
-Be aware, that the is no exclusiveness here. The person would also be happy with fries AND a water AND a coke!+Be aware, that the is no exclusiveness here. The person would also be happy with fries AND water AND a coke!
  
-The gate represenation is shown in <imgref pic37>. At the first glimpse, the output $Y$ of the upper circuit and the lower circuit look similar (they are indeed the same). +The gate representation is shown in <imgref pic37>. At first glimpse, the output $Y$ of the upper circuit and the lower circuit look similar (they are indeed the same). 
-Also the truth tables for the first gate ($Y'$, $Y''$, $Y'''$) and a larger truth table for all results is given. \\ +Also, the truth tables for the first gate ($Y'$, $Y''$, $Y'''$) and a larger truth table for all results are given. \\ 
-But the conversion of the upper one to the lower one is not intuitive here. That is why the multiple represenations and remembering the rules for boolean algebra is very important. Translating from one represenation into another one helps to use other tools in order to simplify systems!+But the conversion of the upper one to the lower one is not intuitive here. That is why the multiple representations and remembering the rules for boolean algebra are very important. Translating from one representation into another one helps to use other tools to simplify systems!
  
 <WRAP><well> <WRAP><well>
Line 596: Line 596:
  
  
-==== 1.5.3 Dive into Law of Absorbtion ====+==== 1.5.3 Dive into the Law of Absorption ====
  
-We will also try to transfer the law of absorbtion to a example of daily life. \\+We will also try to transfer the law of absorption to an example of daily life. \\
 When one says When one says
    "I'm happy with fries OR (fries AND a coke)"     "I'm happy with fries OR (fries AND a coke)" 
 he will be happy with he will be happy with
-   fries.. +   fries... 
 No matter whether there is coke with it.  No matter whether there is coke with it. 
  
-The absortion of the inverse ($a +  (\bar{a} \cdot b) = a +  b $) is also possible:\\+The absorption of the inverse ($a +  (\bar{a} \cdot b) = a +  b $) is also possible:\\
 When one says When one says
    "I'm happy with fries OR (no fries AND a coke)"     "I'm happy with fries OR (no fries AND a coke)" 
 he will be happy with he will be happy with
    fries OR a coke    fries OR a coke
-When he only gets fries the first part is true. When he only gets coke, the secont part is true. When he gets both, he of cource gets fries and therefore the firt part is true..+When he only gets fries the first part is true. When he only gets coke, the second part is true. When he gets both, he of course gets fries and therefore the first part is true...
  
-The gate represenation is shown in <imgref pic38>. When ignoring the blinking High and Low of the lines, also here the similarity of the upper and lower circuits may be not really intuitive.+The gate representation is shown in <imgref pic38>. When ignoring the blinking High and Low of the lines, also here the similarity of the upper and lower circuits may be not intuitive.
  
 <WRAP><well> <WRAP><well>
Line 623: Line 623:
 ==== 1.5.4 Dive into  DeMorgan's Rule ==== ==== 1.5.4 Dive into  DeMorgan's Rule ====
  
-At last, let's have a look onto DeMorgan's rule. The following instance shows it in daily life. \\+At last, let's have a look at DeMorgan's rule. The following instance shows it in daily life. \\
 When one says When one says
-   "I don't like fries AND i don't like coke" +   "I don't like fries AND I don't like coke" 
 he will be unhappy when he gets either fries, or coke, or (fries and coke). he will be unhappy when he gets either fries, or coke, or (fries and coke).
  
-The gate represenation is shown in <imgref pic39>. All four circuits show the same behaviour. \\ +The gate representation is shown in <imgref pic39>. All four circuits show the same behavior. \\ 
-On the left the NOT gate is explicitely shown. On the right the NOT gates are shrinked down to the circles either on the input or on the output. +On the left, the NOT gate is explicitly shown. On the right, the NOT gates are shrunk down to the circles either on the input or on the output. 
  
 <WRAP><well> <WRAP><well>
Line 640: Line 640:
   * any AND is substituted with OR plus   * any AND is substituted with OR plus
   * any OR is substituted with AND   * any OR is substituted with AND
-Important is, that also the most upper expression also have to be inverted.+Important is, that also most upper expressions have to be inverted.
  
 ==== 1.5.5 Example of a logic Simplification ==== ==== 1.5.5 Example of a logic Simplification ====
Line 653: Line 653:
  
 Based on the associative law, when purely AND gates or purely OR gates are stacked the inputs are interchangeable. \\ Based on the associative law, when purely AND gates or purely OR gates are stacked the inputs are interchangeable. \\
-Therefore, this stacking can be substituted with a gate symbol with multiple input. The two AND-gates shown in <imgref pic40> can be substituted with a single one. +Therefore, this stacking can be substituted with a gate symbol with multiple inputs. The two AND gates shown in <imgref pic40> can be substituted with a single one. 
  
-The essence auf the AND gate is: An AND gate - no matter how much inputs it has - will only output high, when all inputs are high. +The essence of the AND gate is: An AND gate - no matter how many inputs it has - will only output high when all inputs are high. 
  
 <WRAP><well> <WRAP><well>
Line 672: Line 672:
 ==== 1.6.2 Switchable Inverter ==== ==== 1.6.2 Switchable Inverter ====
  
-Sometimes it is important switch between inverting and not inverting an output. This can be done with the XOR gate. When the $EN$ in <imgref pic41> is active, the input $X$ will be inversed.+Sometimes it is important to switch between inverting and not inverting an output. This can be done with the XOR gate. When the $EN$ in <imgref pic41> is active, the input $X$ will be inversed.
  
 <WRAP><well> <WRAP><well>
Line 679: Line 679:
 </well></WRAP> </well></WRAP>
  
-One application for a switchable inverter would be an monochrome display, where every pixel can be set by one bit. When the display (or a small part like the cursor symbol) has to be inverted it would be great to do so with a simple gate. This can be seen in  +One application for a switchable inverter would be a monochrome display, where every pixel can be set by one bit. When the display (or a small part like the cursor symbol) has to be inverted it would be great to do so with a simple gate. This can be seen in  
 [[https://www.falstad.com/circuit/circuitjs.html?ctz=CQAgjCAMB0lwTAFkWRkBs8CsIDMMxd0AOI44ksATnXRHmJBy0iYFMBaMMAKDRy4MQNEB0SMRrRsR5hiVUWHStEueIrDrV6qfVxUeAJUXKQidIhMrx4erfjQcDnKxjEsfQqKqaQF0RSW-qy4IIyufnToAOy2EDGRUIqi8DwATsJ0uFjqImgRYJDp4ELZ6pqM+UmFxf5lfpZVrDUZang5eFpw1UUAMt6+-hw0rMG2AGYAhgA2AM5sTFA8-RzESuB0w6brEVNzCy7FW+3qx03gRRkcdR3Xjd3Nl0n1Kg8XR231HG3nNXKWq3WIg40WIlkkeFk7hS5EyJjhIVkQzIZmR8BUY1CrBcID28yWAHcUtEtENqKTLEUiRx4EgzMD4CT6XQqcT1DlGDSsHQOYTRNEOryQR1cGC+SCSEwhML2UJWSD0cz+ZLEJIeNSSSExQqVGrqSRKkMDXgxfLoiqjbDzJSkQDCpBGNhNiN6NznotdjN8UVjKt-NaAnl-BB1P5nPRHEkYB4lADAqi4+sAyEkji8Qt5SiA98VQ91aIs90-fdXPni2Yi-Hk2XjVhzYpIHS6yyy4Um-WuI3LE7xW3u27Ow7XS39VbLYxm0swFgAVVgfhwSy7Niy1VuKd0G71+LN3RgbvwL4+qIqm0OFhfG1PfsPa2qFpgeSE7378+uK-RTbqbTLJ+UnTVRHFI3T-LkeTlMs1jodZASiL9FFfQCrCVeVT1OKpiFLakFzhGlYTVNBQjEdZUABBhHSHFRogAfSUajIGogAPNZIDorB6OgMB6Oo+BaOgbBuMQGjcA4xt2IYgARABBAAVKSeAAIwbWFvg3C8-GnJYlLEM82jAUENNiIpGIbIcxAEdwIDELR6EsABZABLABjNIAHsnNcgA7AAXNzpmmNg0h4EzvlwKzEA3BRvgUbtLAkhzZgAB2mSYAE8AB1ZgJBzvIAC0yiS0gcgA3QLMt6VyAHNnOC7wqAUWkBCoML6CEWKQAAYVcgBXHzAtq1ZEBwVRNmiMazH0MxbM6nq+qCkLslCVRYjw4a0Cmukut67z+pCqhIEaKy1kqRpprstgAFtXLSVKBrGyxoiIkkFHrdq7O6xiBqQHBolYLh0Bepbpoky6PoGrBmFwWJogUGcVGmgBlXKHPGbzMrSNhqtmHb5o0VhoggVBGGiU66V6ABRCTMoAEwS5K0sygB6TLEocxi2GmTLJjSNI0oGzAiKEk9shAWJGkYABJAA5AA1cnDBkzL0vSzzMoACXl8meCAA| this more complex example]], where the display smiley can be inverted via the XOR gate during data transmission. The other logic component in this example will be explained in the following (sub)chapters. [[https://www.falstad.com/circuit/circuitjs.html?ctz=CQAgjCAMB0lwTAFkWRkBs8CsIDMMxd0AOI44ksATnXRHmJBy0iYFMBaMMAKDRy4MQNEB0SMRrRsR5hiVUWHStEueIrDrV6qfVxUeAJUXKQidIhMrx4erfjQcDnKxjEsfQqKqaQF0RSW-qy4IIyufnToAOy2EDGRUIqi8DwATsJ0uFjqImgRYJDp4ELZ6pqM+UmFxf5lfpZVrDUZang5eFpw1UUAMt6+-hw0rMG2AGYAhgA2AM5sTFA8-RzESuB0w6brEVNzCy7FW+3qx03gRRkcdR3Xjd3Nl0n1Kg8XR231HG3nNXKWq3WIg40WIlkkeFk7hS5EyJjhIVkQzIZmR8BUY1CrBcID28yWAHcUtEtENqKTLEUiRx4EgzMD4CT6XQqcT1DlGDSsHQOYTRNEOryQR1cGC+SCSEwhML2UJWSD0cz+ZLEJIeNSSSExQqVGrqSRKkMDXgxfLoiqjbDzJSkQDCpBGNhNiN6NznotdjN8UVjKt-NaAnl-BB1P5nPRHEkYB4lADAqi4+sAyEkji8Qt5SiA98VQ91aIs90-fdXPni2Yi-Hk2XjVhzYpIHS6yyy4Um-WuI3LE7xW3u27Ow7XS39VbLYxm0swFgAVVgfhwSy7Niy1VuKd0G71+LN3RgbvwL4+qIqm0OFhfG1PfsPa2qFpgeSE7378+uK-RTbqbTLJ+UnTVRHFI3T-LkeTlMs1jodZASiL9FFfQCrCVeVT1OKpiFLakFzhGlYTVNBQjEdZUABBhHSHFRogAfSUajIGogAPNZIDorB6OgMB6Oo+BaOgbBuMQGjcA4xt2IYgARABBAAVKSeAAIwbWFvg3C8-GnJYlLEM82jAUENNiIpGIbIcxAEdwIDELR6EsABZABLABjNIAHsnNcgA7AAXNzpmmNg0h4EzvlwKzEA3BRvgUbtLAkhzZgAB2mSYAE8AB1ZgJBzvIAC0yiS0gcgA3QLMt6VyAHNnOC7wqAUWkBCoML6CEWKQAAYVcgBXHzAtq1ZEBwVRNmiMazH0MxbM6nq+qCkLslCVRYjw4a0Cmukut67z+pCqhIEaKy1kqRpprstgAFtXLSVKBrGyxoiIkkFHrdq7O6xiBqQHBolYLh0Bepbpoky6PoGrBmFwWJogUGcVGmgBlXKHPGbzMrSNhqtmHb5o0VhoggVBGGiU66V6ABRCTMoAEwS5K0sygB6TLEocxi2GmTLJjSNI0oGzAiKEk9shAWJGkYABJAA5AA1cnDBkzL0vSzzMoACXl8meCAA| this more complex example]], where the display smiley can be inverted via the XOR gate during data transmission. The other logic component in this example will be explained in the following (sub)chapters.
  
-Another usage is the inversion of binary numbers in the arithmetic logical unit of the processor.+Another usage is the inversion of binary numbers in the arithmetic logic unit of the processor.
  
 ==== 1.6.3 Data Valve ==== ==== 1.6.3 Data Valve ====
  
-To deactivate a data flow a simple AND gate can be used. This is also useful for channelize data in multiple directions. +To deactivate a data flow a simple AND gate can be used. This is also useful for channelizing data in multiple directions. 
  
-In the upper part of <imgref pic42> a single data valve is shown. Only when $EN=1$, then the value of the data input $D$ will set as the output $Y$.+In the upper part of <imgref pic42> a single data-valve is shown. Only when $EN=1$, then the value of the data input $X$ will set as the output $Y$.
  
-The lower part of <imgref pic42> shows the combination of two data valves. In this circuit depending on $EN$ the output of the the data input $DX$ will either be $Y0$ or $Y1$. This is also visible in the truth table. The truth table van be shortened which is also shown.+The lower part of <imgref pic42> shows the combination of two data valves. In this circuit depending on $EN$ the output of the data input $X$ will either be $Y0$ or $Y1$. This is also visible in the truth table. The truth table can be shortened which is also shown.
  
 <WRAP><well> <WRAP><well>
Line 699: Line 699:
 ==== 1.6.4 Multiplexer and Demultiplexer ==== ==== 1.6.4 Multiplexer and Demultiplexer ====
  
-In the linked 'display example' in 1.6.2 there were a Multiplexer (MUX) and a Demultiplexer (DEMUX) visible. +In the linked 'display example' in 1.6.2 there was a Multiplexer (MUX) and a Demultiplexer (DEMUX) visible. 
  
-A **multiplexer** is a electronic switcher, which has several data inputs $D0$, $D1$ ... and one data output $Y$. Additionally to the data input there are also state inputs $S0$, $S1$, ... . The state inputs address which of the data input is routed to the data output. An example is given in <imgref pic43>. When $S0 = 0$ and $S1 = 0$ the zeroth data input $D0$ is the output, for $S0 = 1$ and $S0 = 0$ the first data input $D1$, for $S0 = 0$ and $S0 = 1$ the second. The 'inner live' of the multiplexer will be shown in the chapter 3.+A **multiplexer** is an electronic switcher, which has several data inputs $D0$, $D1$ ... and one data output $Y$. Additionally to the data input, there are also state inputs $S0$, $S1$, ... . The state inputs address which of the data input is routed to the data output. An example is given in <imgref pic43>. When $S0 = 0$ and $S1 = 0$ the zeroth data input $D0$ is the output, for $S0 = 1$ and $S0 = 0$ the first data input $D1$, for $S0 = 0$ and $S0 = 1$ the second. The 'inner life' of the multiplexer will be shown in chapter 3.
  
-A **demultiplexer** is the counterpart to the multiplexer: It has one data input $D$ and several data outputs $Y0$, $Y1$, ... . Again there are also state inputs $S0$, $S1$ ... . Here, the state inputs address to which data __output__ the single data input is routed. also this is shown in <imgref pic43>. +A **demultiplexer** is the counterpart to the multiplexer: It has one data input $D$ and several data outputs $Y0$, $Y1$, ... . Again there are also state inputs $S0$, $S1$ ... . Here, the state inputs address to which data __output__ the single data input is routed. Also, this is shown in <imgref pic43>. 
  
 <WRAP><well> <WRAP><well>
Line 710: Line 710:
 </well></WRAP> </well></WRAP>
  
-With the background given in subcapter 1.6.3 the demultiplexer can be derived. The example in the subchapter 1.6.3 an input were forwarded to 2 outputs. We will now have a look onto a demultiplexer with 4 outputs. For this instead of an AND gate with two inputs an AND gate with three inputs are used. The top-most input is the data input $D$ for all gates. The other two inputs address which of the AND gate will route the data input. When one has a detailed look onto the addressing inputs, one can see in any time that only one AND gate has both lower inputs high. +With the background given in subchapter 1.6.3, the demultiplexer can be derived. In the example, in subchapter 1.6.3 an input was forwarded to 2 outputs. We will now have a look at a demultiplexer with 4 outputs. For this instead of an AND gate with two inputs an AND gate with three inputs is used. The topmost input is the data input $D$ for all gates. The other two inputs address which of the AND gate will route the data input. When one has a detailed look at the addressing inputs, one can see at any time that only one AND gate has both lower inputs high. 
  
 <WRAP><well> <WRAP><well>
Line 725: Line 725:
  
   * A nice overview of core ideas for [[https://informatik.mygymer.ch/ef2019/rechnen-mit-strom|calculating with electricity]] has been compiled by Gymnasium Kirchenfeld (CH, in German)   * A nice overview of core ideas for [[https://informatik.mygymer.ch/ef2019/rechnen-mit-strom|calculating with electricity]] has been compiled by Gymnasium Kirchenfeld (CH, in German)
-  * Explanation of [[https://ca.wikipedia.org/wiki/CMOS|CMOS]] in the english Wikipedia+  * Explanation of [[https://ca.wikipedia.org/wiki/CMOS|CMOS]] in the English Wikipedia
   * Wiki page on [[https://en.wikipedia.org/wiki/Integrated_circuit|integrated circuits]]   * Wiki page on [[https://en.wikipedia.org/wiki/Integrated_circuit|integrated circuits]]
  
Line 781: Line 781:
 <panel type="info" title="Exercise 1.6.4. NAND-based gates"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 1.6.4. NAND-based gates"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-Realize the logic functions of NOT, AND, OR, NOR, XOR and XNOR exclusively with NAND gates.+Realize the logic functions of NOT, AND, OR, NOR, XOR, and XNOR exclusively with NAND gates. 
 + 
 + 
 +<button size="xs" type="link" collapse="Solution_1_6_4_1_Solution_path">{{icon>eye}} Solution </button><collapse id="Solution_1_6_4_1_Solution_path" collapsed="true"> 
 + 
 +**__NOT__** 
 + 
 +An inverter can be realized with a NAND gate when both inputs are connected to the same input. \\ 
 +When the input is $1$ the output will be $0$ and vice versa. 
 + 
 +<WRAP> 
 +<imgcaption Ex_1_6_4_1 | NOT based on NAND> 
 +</imgcaption> \\ 
 +{{url>https://www.falstad.com/circuit/circuitjs.html?hideSidebar=true&ctz=CQAgjCAMB0l3BWKsBMBOAHANjQFixmihtgOwa4gKRUi4DMVApgLRhgBQA7iCqSuFJZe-cESjcRAsOL4CUkDBIAyUsdPEyBNGgDMAhgBsAzk1qQOAWRD0K6uriVaovaAg5gEEOfduVnNAJI7tb0BDYKDkr0kYFuHKrE0ZHOMTouBiZmSBYAkrwkETRhyekwwRwAHuCE4NR0GNIIjJQolAByAPIAKgA6xgDm+gAuTP1cAJbDABb97QCC7QAi-UOjxlU1aCJK6IzEEK0dPasjTBxAA noborder}} 
 +</WRAP> 
 + 
 +**__AND__** 
 + 
 +With the realization of the NOT gate an AND gate is also simple: just put the NOT after a NAND. 
 + 
 +<WRAP> 
 +<imgcaption Ex_1_6_4_2 | AND based on NAND> 
 +</imgcaption> \\ 
 +{{url>https://www.falstad.com/circuit/circuitjs.html?hideSidebar=true&ctz=CQAgjCAMB0l3BWKsBMBmFA2dmEHYBOMPFAlPEBSSkAFjUoFMBaMMAKABkQUFMfIADnBkBw6hABmAQwA2AZ0Y1I7ALJ0w-MKNq1h2lFB7QE7MAgi8totGmuHqhpKfV3ht-rvd2jKE1x5BdxQJGxCjahkFJSQVcwgPEUNaTSTfI1MAdxBEgxyfEPF2bNy8flzRFW4rcDK04n5qSLlFZTMqQO9yzC706lNqoJyEQzy0EaMpFpiodgAPcEFRtAI6PCQwWmSeWhAAQQA5ABEAHXkAc2kAF0YzzIBLK4ALM4PD04vrxnl5xYZSQwoQQUUhIXYoXbvM6XG7sIA noborder}} 
 +</WRAP> 
 + 
 +**__OR__** 
 + 
 +With the realization of the NOT gate an AND gate is also simple: just put the NOT after a NAND. 
 + 
 +<WRAP> 
 +<imgcaption Ex_1_6_4_3 | OR based on NAND> 
 +</imgcaption> \\ 
 +{{url>https://www.falstad.com/circuit/circuitjs.html?hideSidebar=true&ctz=CQAgjCAMB0l3BWKsDMCBMAOAbNymwAWBTTQzAThAUmpEJWoFMBaMMAKABkR0Fte+cBXSDMUcCABmAQwA2AZyZ1IHALL0wAsAHYBhcuD0T00BBzAIIKFNuOEtRgbVFJzG2+Jv7D3k2e5eUhAUdFowERCwiVpZRWUkVUsIPgEwr1sxEwlzAHdefizU3nRCKA584ojRYvTyniqDYVEiMtpY+SUVQKwvDGaQ-vDpToTygA9wHBAKcUJsMrBICDLSkAB5ACUAHQUAcxkAFyZd3IBLQ4ALXYA5AEEbgBFdg+OFDkmwTEZ0SKsqX5IVZlLYvI4nBQACgUZwAtgAHORnKRnJgAE1OF2uCnuTwAlBUCtosETwE1VJVCq1SVp2hYrDSmn5qS46HkQplqcy8OV8szIn46kkGb0OQJPGL6Lw2fSasF5mUJQrsm4PlNGIRougdKJNRreCCdvtwT15ZhwpFyO1JHEuolTXNDNV6IZrbaxqoPNg5gsXT62rwAkA noborder}} 
 +</WRAP> 
 + 
 +**__NOR__** 
 + 
 +Again the NOR gate can be derived from the NOT gate and OR gate. 
 + 
 +<WRAP> 
 +<imgcaption Ex_1_6_4_4 | NOR based on NAND> 
 +</imgcaption> \\ 
 +{{url>https://www.falstad.com/circuit/circuitjs.html?hideSidebar=true&ctz=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-UOf1CkO+8OeRmWX5m90eN0VTn5JnOHX+BhO7MZEzmIA noborder}} 
 +</WRAP> 
 + 
 +**__XOR__** 
 + 
 +The XOR gate is a bit more complex. One path to a solution is the following:  
 +  - The XOR-gate only outputs a $0$ in the two cases $(X0=0, X1=0)$, $(X0=1, X1=1)$.  
 +  - Therefore, as a first idea, it would be great to have gates detecting $X0=0\; and\; X1=0$ such as $X0=1\; and\; X1=1$.  
 +    - The ladder one is $X0\cdot X1$, which is only $1$ for both inputs as $1$. \\ 
 +    - For detecting $X0=0\; and\; X1=0$ one can use $\overline{X0+X1}$, which is only $1$ for both inputs are $0$.  
 +  - These two gates have to be combined in such a way that, only for $(X0=0, X1=0)$, $(X0=1, X1=1)$ the final output is $1$. \\ In other words: only when the output of the AND- and the NOR-gates is $0$, the final output has to be $1$. \\For this, a NOR gate can be used. 
 +  - All of the mentioned gates can be built based on NAND gates. This is shown in <imgref Ex_1_6_4_5 >. 
 + 
 +<WRAP> 
 +<imgcaption Ex_1_6_4_5 | XOR based on NAND> 
 +</imgcaption> \\ 
 +{{url>https://www.falstad.com/circuit/circuitjs.html?hideSidebar=true&ctz=CQAgzCAMB0l3BWKsBMB2ALADgIwrFpAggJzYkBsFICkNIGECApgLQ44BQAMiPijRR0cJAQiFRJAMwCGAGwDOzepB58E1bFhAiBWyThCzFypKoAeOrJswNCDNGgZ8MIABoB5AEoAdBQHMZABdmPwB3AEsggAs-ADkAQTiAET9AkIVOSxFDHAoSPkodND0Xd2804OY1FA1wcT4UVzAGujpjJRUakvAJFCbetp0jeU6zThwECBQsbQw4cAo5hboBOgROML5Z8CxXdAECV1Utmbm89U0Lk8uGC9rqFrXNvh6j18OJG4PB7e0wL4vM71AQPEFQCa0P7gMCPJYw6irSQbLaw5Z0NG7Y4vTFPRb-WEQgCyDGw+LsGPhq2gG0mEFx8IwZMxSPWkOakH2kH+ez43MkaxoajAnJ0oluugMIxMXV4Iv2AzB-WO0lGpmJWL552oQm01I22SwBRQJCYRsKmjKnl8ASqfgAFJEYn4kqkFDIAHYAE3iFVtGQAlJDDGCEGhHqKw9QQ8jg7cKPz5SAE9oY2zJoYk6RDg1s8NBbSEJnc+L5utxWmhac6hR7jWVkC6gheaGqXGkxginjO9G+LGM+CewwFkPKyi7tQpldJ4CthgLhptTRWnGi2ImTQEzQN2O4xgKK4EBu19vXLuB-PJ1v94et+eixPBHMD0-87HUZHw+BI7ybkmExioophCqINAB4J5n+JY5mIv44rmordkUUF6F2DT6DcN7LnoL4aG0LxYVODC4TMIE0Hgp7kWIhI3CeR6HhRxDPHOC6Mgsi5kWWm7ojQ4qYSORRceI+FzuxEiXoIIl8ACknSQItaIkCMkKfG1yNle2hghh6l2P8oraR+h59DJwlkVmEhZqKqgknhlF5vRAo0pwQA noborder}} 
 +</WRAP> 
 + 
 +The shown circuit can even be simplified:  
 +  * Two Not-gates in series can be skipped since $\overline{\overline{X}}=X$. \\ With this knowledge, the XOR-gate needs 6 NAND gates. 
 +  * An alternative setup for the XOR-gate can be built by one NAND-, one OR-, and one AND-gate. This circuitry needs also 6 NAND gates. \\The circuitry is not shown. 
 +  * The circuity with the least NAND gates is shown in <imgref Ex_1_6_4_6 > 
 + 
 +<WRAP> 
 +<imgcaption Ex_1_6_4_6 | XOR based on NAND, alternatives> 
 +</imgcaption> \\ 
 +{{url>https://www.falstad.com/circuit/circuitjs.html?hideSidebar=true&ctz=CQAgzCAMB0l3BWKsBMB2ALADgIwrFpAggJzYkBsFICkNIGECApgLQ44BQAHiDlhhBYsICmggkcDECkEANAPIAlADoBnAOYBDAC7N1ACgByAQSMARddr1r1AdwCWOgBYB7AK471AEw8AjABt9NSMFABU1AEoePiwScBx4jHRwFBRpWRBFVU1dYIMtAL0AJwA7XQcAN2ZogFkGSnAKEVJ0sGaoGWgEThwEKQxsJpEMRvaRKXSkHr6pcfABBhxqAkFJ+hn+4fAEdIxlnfT1uh6AGRlhcEG+EjbruikAM0K1ZnpITjtDvgo6MF2+Cg6B8vv8jmgVgCUAcQQwKGsId8cIjYRh4T8-gDlsDOOd8EdrokCYJgSBngFXu9cTIENQOEdbnw8J0ni83kgPrMadQgX9IIJeXwZJ1NpNaTIwG1+RKjsLpr0tmBpdDIekVUKpjRPtyZBg6ChxWkSdqDTzICJTTd0pzFdK8JijkCNRsFVJ+kcOrSHh11vKuWi1p7mcsJsKTtqlQK4OBleaoBHlZlIzL4193T8WsHfqmaFnM+DqLD8XxrsXZDivmW9TryznLUSdfS6+Km8mm7C26XJSXjaC7dc2ygRB96l6MzRGSHOihuq6LiJ5mB2ttfVq8ZcwJPGZvrZ1yZSOdr9qqGEMwTnjwtBIMF4sO8vFx15h2n9nH8OI6+F0+USaN6WN2zIt-0ERd7j-BdESHBdGWAmC2ifWCjzPAEbyaCtT0g6g0NWeMgA noborder}} 
 +</WRAP> 
 + 
 + 
 + 
 +</collapse> 
 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
  
 <panel type="info" title="Exercise 1.6.5. NOR-based gates"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 1.6.5. NOR-based gates"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-Realize the logic functions of NOT, OR, AND, NAND, XOR and XNOR exclusively with NOR gates.+Realize the logic functions of NOT, OR, AND, NAND, XOR, and XNOR exclusively with NOR gates.
 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
  
Line 802: Line 878:
 <button size="xs" type="link" collapse="Solution_1_6_6_1_Solution_path">{{icon>eye}} Solution for 1. </button><collapse id="Solution_1_6_6_1_Solution_path" collapsed="true"> <button size="xs" type="link" collapse="Solution_1_6_6_1_Solution_path">{{icon>eye}} Solution for 1. </button><collapse id="Solution_1_6_6_1_Solution_path" collapsed="true">
  
-Often the formula can be easier analysed with in the more compact terminology. \\ +Often the formula can be easier analyzed with more compact terminology. \\ 
-Additionally the brackets can be ignored in the case of products - similar to the convention in math for $(a \cdot b) + (c \cdot d) = ab + cd$+Additionally, the brackets can be ignored in the case of products - similar to the convention in math for $(a \cdot b) + (c \cdot d) = ab + cd$
  
 $Y = \color{blue}{\overline{X_0}}\;\color{green}{\overline{X_1}}\;\color{red}{\overline{X_2}}$$ +  \color{cornflowerblue}{X_0 }\;\color{yellowgreen}{   {X_1}}\;\color{red}{\overline{X_2}}$$ +  \color{blue}{\overline{X_0}}\;\color{green}{\overline{X_1}}\;\color{salmon}{      {X_2}}$$ +  \color{cornflowerblue}{X_0 }\;\color{yellowgreen}{   {X_1}}\;\color{salmon}{      {X_2}}$ $Y = \color{blue}{\overline{X_0}}\;\color{green}{\overline{X_1}}\;\color{red}{\overline{X_2}}$$ +  \color{cornflowerblue}{X_0 }\;\color{yellowgreen}{   {X_1}}\;\color{red}{\overline{X_2}}$$ +  \color{blue}{\overline{X_0}}\;\color{green}{\overline{X_1}}\;\color{salmon}{      {X_2}}$$ +  \color{cornflowerblue}{X_0 }\;\color{yellowgreen}{   {X_1}}\;\color{salmon}{      {X_2}}$
Line 831: Line 907:
  
 $Y = \overline{\color{cornflowerblue}{X_0} \text{#} \color{yellowgreen}{X_1} }$ $Y = \overline{\color{cornflowerblue}{X_0} \text{#} \color{yellowgreen}{X_1} }$
-\end{align*}+
 </collapse> </collapse>
 <button size="xs" type="link" collapse="Solution_1_6_6_1_final_result">{{icon>eye}} Final result for 1. </button><collapse id="Solution_1_6_6_1_final_result" collapsed="true"> <button size="xs" type="link" collapse="Solution_1_6_6_1_final_result">{{icon>eye}} Final result for 1. </button><collapse id="Solution_1_6_6_1_final_result" collapsed="true">
Line 843: Line 919:
 <button size="xs" type="link" collapse="Solution_1_6_6_3_Solution_path">{{icon>eye}} Solution for 3. </button><collapse id="Solution_1_6_6_3_Solution_path" collapsed="true"> <button size="xs" type="link" collapse="Solution_1_6_6_3_Solution_path">{{icon>eye}} Solution for 3. </button><collapse id="Solution_1_6_6_3_Solution_path" collapsed="true">
  
-Again, the formula can be easier analysed with in the more compact terminology. \\ +Again, the formula can be easier analyzed with more compact terminology. \\ 
-Additionally the brackets can be ignored in the case of products - similar to the convention in math for $(a \cdot b) + (c \cdot d) = ab + cd$+Additionally, the brackets can be ignored in the case of products - similar to the convention in math for $(a \cdot b) + (c \cdot d) = ab + cd$
 \begin{align*} \begin{align*}
 Y &= \color{cornflowerblue}{X_0 }\;\color{yellowgreen}{   {X_1}}\;\color{salmon}{      {X_2}}\;\color{brown}{\overline{X_3}}  Y &= \color{cornflowerblue}{X_0 }\;\color{yellowgreen}{   {X_1}}\;\color{salmon}{      {X_2}}\;\color{brown}{\overline{X_3}} 
Line 909: Line 985:
 <panel type="info" title="Exercise 1.6.7. only using NAND"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 1.6.7. only using NAND"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-Rewrite the following expressions using only NAND, NOT and AND. Check the result for equality using truth tables.+Rewrite the following expressions using only NAND. Check the result for equality using truth tables.
  
   - $Y = X0 + X1*/X2$    - $Y = X0 + X1*/X2$
Line 917: Line 993:
   - $Y = /(/(X0+X1) + X0*X1)$   - $Y = /(/(X0+X1) + X0*X1)$
   - $Y = /(/X0+X1+/X2) + /(X0+/X1+X2)$   - $Y = /(/X0+X1+/X2) + /(X0+/X1+X2)$
 +
 +<button size="xs" type="link" collapse="Solution_1_6_7_5_Solution_path">{{icon>eye}} Solution for 5. </button><collapse id="Solution_1_6_7_5_Solution_path" collapsed="true">
 +
 +\begin{align*}
 +Y = \overline{\overline{X0+X1} + X0\cdot X1}
 +\end{align*}
 +
 +Generally, the first step would be to see which of the parts show already a NAND configuration. In the following, these are marked in $\color{magenta}{magenta}$.\\
 +In this case, there are no terms with NAND available.
 +
 +One of the next steps is to do substitutions with the DeMorgans rule: $\overline{a+b}=\overline{a}\cdot\overline{b}$. \\
 +This can be used on the outer negation:
 +
 +\begin{align*}
 +Y = \overline{\overline{X0+X1}} \cdot  \color{magenta}{\overline{X0 \cdot X1}}
 +\end{align*}
 +
 +This can be done again on $\overline{X0+X1}$:
 +
 +\begin{align*}
 +Y = \color{magenta}{\overline{\color{black}{\overline{X0}}\cdot \color{black}{\overline{X1}}}} \cdot  \color{magenta}{\overline{X0 \cdot X1}}
 +\end{align*}
 +
 +The last step is the substitution of the inverted parts: $\overline{X}=\color{magenta}{\overline{X\cdot X}}$
 +
 +\begin{align*}
 +Y = \color{magenta}{\overline{\overline{X0 \cdot X0}\cdot \overline{X1 \cdot X1}}} \cdot  \color{magenta}{\overline{X0 \cdot X1}}
 +\end{align*}
 +
 +
 +<WRAP center>
 +<imgcaption Solution_1_6_7_5 | Truth Table>
 +</imgcaption>
 +{{drawio>Solution1675tt.svg}}
 +</WRAP>
 +
 +</collapse>
  
 </WRAP></WRAP></panel> </WRAP></WRAP></panel>
  
-<panel type="info" title="Exercise 1.6.8. step by step example for logic simplification using boolean rules"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>+<panel type="info" title="Exercise 1.6.8. step-by-step example for logic simplification using boolean rules"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
 Hints for the clip:  Hints for the clip: 
   * The ladder logic is not necessary for our course.   * The ladder logic is not necessary for our course.
-  * Example 1 ist quiet exhausting including a recap of the boolean rules $\rightarrow$ you can skip this example+  * Example 1 is quite exhausting including a recap of the boolean rules $\rightarrow$ you can skip this example
   * Example 2 starts at 17:17   * Example 2 starts at 17:17
   * Example 3 starts at 21:00   * Example 3 starts at 21:00
   * Example 4 starts at 28:27   * Example 4 starts at 28:27
-  * Example 5-8 start at 32:33+  * Examples 5-8 start at 32:33
  
 {{youtube>59BbncMjL8I}} {{youtube>59BbncMjL8I}}
Line 938: Line 1051:
 <panel type="info" title="Exercise 1.6.9. XOR in Cryptography"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 1.6.9. XOR in Cryptography"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
-In the following EXCEL file an example of symmetric encryption can be found: {{introduction_to_digital_systems:xor_in_cryptography.xlsx}}+In the following EXCEL file, an example of symmetric encryption can be found: {{introduction_to_digital_systems:xor_in_cryptography.xlsx}}
  
   * Try to understand {{wp>XOR_cipher}} with this example.   * Try to understand {{wp>XOR_cipher}} with this example.