Department of EECS Spring 2007 EE100 42 43 Rev 1 Introduction to Digital Systems 0 Acknowledgments Many thanks to Prof Bernhard Boser and National Instruments for funding this project in the Summer of 2007 Ferenc Kovac has been and will continue to be an excellent mentor Winthrop Williams designed the strain gauge lab a paradigm of the K I S S Keep it Simple Stupid philosophy We are using this as a sensor for interfacing to our digital system a PIC microcontroller Tho Nguyen and Karen Tran were the brave souls who first tried out the PIC microcontroller Kevin Mullaly s staff installed the microcontroller development tools Last but not the least a shout out to the authors on the internet who made this document possible Tony R Kuphaldt s free description of digital electronics http www ibiblio org obp electricCircuits Digital index html forms the crux of this document Combined with Richard Bowles excellent description of feedback in digital systems http richardbowles tripod com dig elec chapter1 chapter1 htm it is hoped that this document will serve as a self contained introduction to digital systems Go Bears 1 Introduction Logic circuits are the basis for modern digital computer systems To appreciate how computer systems operate you will need to understand digital logic and Boolean algebra This chapter provides only a basic introduction to Boolean algebra describing it in its entirety would take up an entire textbook I chose to concentrate on the basics of Boolean algebra rather than on optimizing concepts like Karnaugh Maps First we start out with the concept of digital vs analog 2 Digital vs Analog 2 The term digital refers to the fact that the signal is limited to only a few possible values In general digits signals are represented by only two possible voltages on a wire 0 volts which we called binary 0 or just 0 and 5 volts which we call binary 1 or just 1 We sometimes call these values low and high or false and true More complicated signals can be constructed from 1s and 0s by stringing them end to end like a necklace If we put three binary digits end to end we have eight possible combinations 000 001 010 011 100 101 110 and 111 In principle there is no limit to how many binary digits we can use in a signal so signals can be as complicated as you like The figure below shows a typical digital signal firstly represented as a series of voltage levels that change as time goes on and then as a series of 1s and 0s Department of EECS Spring 2007 EE100 42 43 Rev 1 Figure 1 A digital signal Analog electronics uses voltages that can be any value within limits of course it s difficult to imagine a radio with voltages of a million volts The voltages often change smoothly from one value to the next like gradually turning a light dimmer switch up or down The figure below shows an analog signal that changes with time Figure 2 An analog signal 3 Number Systems 1 a The Binary Number System The binary number system is a natural choice for representing the behavior of circuits that operate in one of two states on or off 1 or 0 For instance we studied a diode logic gate refer to the Diodes and Transistors handout online when we discussed diode circuits But before we study logic gates you need to be intimately familiar with the binary number system the system used by computers for counting Lets count from zero to twenty using the decimal number system and the binary number system Decimal 0 1 2 3 4 5 6 7 Binary 0 1 10 11 100 101 110 111 Department of EECS Spring 2007 EE100 42 43 Rev 1 8 9 10 11 12 13 14 15 16 17 18 19 20 1000 1001 1010 1011 1100 1101 1110 1111 10000 10001 10010 10011 10100 Notice though how much shorter decimal notation is over binary notation for the same number of quantities What takes five bits in binary notation only takes two digits in decimal notation An interesting footnote for this topic is the one of the first electronic digital computers the ENIAC The designers of the ENIAC chose to represent numbers in decimal form digitally using a series of circuits called ring counters instead of just going with the binary numeration system in an effort to minimize the number of circuits required to represent and calculate very large numbers This approach turned out to be counter productive and virtually all digital computers since then have been purely binary in design This is intuitively due to that fact that a binary number directly maps to the on and off state in digital systems Notice that the binary number system and digital logic are actually two different concepts A binary number is a number in base 2 it is independent of the concept of digital logic However the computer revolution is attributed to the very simple fact that mathematics in digital electronics can be represented by binary numbers This is the number system that we will primarily study along with the hexadecimal base 16 system for convenience of representing large digits To convert a number in binary numeration to its equivalent in decimal form all you have to do is calculate the sum of all the products of bits with their respective place weight constants To illustrate Convert 110011012 bits 1 weight 1 in decimal 2 notation 8 to decimal 1 0 0 1 6 3 1 8 4 2 6 form 1 0 1 4 2 1 The bit on the far right side is called the Least Significant Bit LSB because it stands in the place of the lowest weight the one s place The bit on the far left side is called the Most Significant Bit MSB because it stands in the place of the highest weight the one hundred twenty eight s place Remember a bit value of 1 means that the respective place weight gets added to the total value and a bit value of 0 means that the respective place weight does not get added to the total value With the above example we have 12810 6410 810 410 110 20510 If we encounter a binary number with a dot called a binary point instead of a decimal point we follow the same procedure realizing that each place weight to the right of the point is Department of EECS Spring 2007 EE100 42 43 Rev 1 one half the value of the one to the left of it just as each place weight to the right of a decimal point is one tenth the weight of the one to the left of it For example Convert 101 0112 to decimal form bits 1 0 1 0 1 1 weight 4 2 1 1 1 1 in decimal notation 2 4 8 410 110 0 2510 0 12510 5 37510 b The Hexadecimal Number System Because binary numeration requires so many bits to represent relatively small numbers compared to the economy of the decimal system analyzing the
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