Digital Logic Design 1 Digital Logic Design Digital
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Digital Logic Design 1
Digital Logic Design ° Digital - Concerned with the interconnection among digital components and modules » Best Digital System example is General Purpose Computer ° Logic Design - Deals with the basic concepts and tools used to design digital hardware consisting of logic circuits » Circuits to perform arithmetic operations (+, -, x, ÷) 2
Digital Signals ° Decimal values are difficult to represent in electrical systems. It is easier to use two voltage values than ten. ° Digital Signals have two basic states: 1 (logic “high”, or H, or “on”) 0 (logic “low”, or L, or “off”) ° Digital values are in a binary format. Binary means 2 states. ° A good example of binary is a light (only on or off) on off Power switches have labels “ 1” for on and “ 0” for off. 3
Digital Logic Design ° Bits and Pieces of DLD History ° George Boole - Mathematical Analysis of Logic (1847) An Investigation of Laws of Thoughts; Mathematical Theories of Logic and Probabilities (1854) ° Claude Shannon - Rediscovered the Boole “ A Symbolic Analysis of Relay and Switching Circuits “ Boolean Logic and Boolean Algebra were Applied to Digital Circuitry ----- Beginning of the Digital Age and/or Computer Age World War II Computers as Calculating Machines Arlington (State Machines) “ Control “ 4
Motivation ° Microprocessors/Microelectronics have revolutionized our world • Cell phones, internet, rapid advances in medicine, etc. ° The semiconductor industry has grown tremendously 5
Objectives ° Number System, Their Uses, Conversions ° Basic Building Blocks of Digital System ° Minimization ° Combinational And Sequential Logic ° Digital System/Circuit Analysis and Design ° State Minimizations ° Integrated Circuits ° Simulations 6
Text Book ° Primary Text: “Digital Design” By M. Morris Mano and Michael D. Ciletti ° Complementary Material “Logic and Computer Design Fundamentals” By M. Morris Mano & Charles R Kime. 7
Digital Logic Design Lecture 1 Number Systems 8
Number Systems ° Decimal is the number system that we use ° Binary is a number system that computers use ° Octal is a number system that represents groups of binary numbers (binary shorthand). It is used in digital displays, and in modern times in conjunction with file permissions under Unix systems. ° Hexadecimal (Hex) is a number system that represents groups of binary numbers (binary shorthand). Hex is primarily used in computing as the most common form of expressing a humanreadable string representation of a byte (group of 8 bits). 9
Overview ° The design of computers • It all starts with numbers • Building circuits • Building computing machines ° Digital systems ° Understanding decimal numbers ° Binary and octal numbers • The basis of computers! ° Conversion between different number systems 10
Analog vs. Digital Consider a faucet Digital Water can be flowing or NOT flowing from the faucet Two States • On • Off Analog How much water is flowing from the faucet? Advantages of Digital Replication • Analog Try replicating the exact flow from a faucet • Digital Try replicating ON or OFF 11
Advantages of Digital o Error Correction/Detection • Small errors don’t propagate o Miniaturization of Circuits o Programmability • Digital computers are programmable ° Two discrete values are used in digital systems. ° How are discrete elements represented? • Signals are the physical quantities used to represent discrete elements of information in a digital system. ° Electric signals used: • Voltage • Current 12
Advantages of Digital/Representation of Binary Values ° Why are there voltage ranges instead of exact voltages? o Two possible values • Variations in circuit behavior & noise • • • 1, 0 On, Off True, False High, Low Heads, Tails Black, White 13
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Digital Computer Systems ° Digital systems consider discrete amounts of data. ° Examples • 26 letters in the alphabet • 10 decimal digits ° Larger quantities can be built from discrete values: • Words made of letters • Numbers made of decimal digits (e. g. 239875. 32) ° Computers operate on binary values (0 and 1) ° Easy to represent binary values electrically • Voltages and currents. • Can be implemented using circuits • Create the building blocks of modern computers 15
Understanding Decimal Numbers ° Decimal numbers are made of decimal digits: (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) ° But how many items does a decimal number represent? • ° What about fractions? • • ° 8653 = 8 x 103 + 6 x 102 + 5 x 101 + 3 x 100 97654. 35 = 9 x 104 + 7 x 103 + 6 x 102 + 5 x 101 + 4 x 100 + 3 x 10 -1 + 5 x 10 -2 In formal notation -> (97654. 35)10 Why do we use 10 digits, anyway? 16
Understanding Octal Numbers ° Octal numbers are made of octal digits: (0, 1, 2, 3, 4, 5, 6, 7) ° How many items does an octal number represent? • ° (4536)8 = 4 x 83 + 5 x 82 + 3 x 81 + 6 x 80 = (1362)10 What about fractions? • (465. 27)8 = 4 x 82 + 6 x 81 + 5 x 80 + 2 x 8 -1 + 7 x 8 -2 ° Octal numbers don’t use digits 8 or 9 ° Who would use octal number, anyway? 17
Understanding Binary Numbers ° Binary numbers are made of binary digits (bits): • ° How many items does an binary number represent? • ° (110. 10)2 = 1 x 22 + 1 x 21 + 0 x 20 + 1 x 2 -1 + 0 x 2 -2 Groups of eight bits are called a byte • ° (1011)2 = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 = (11)10 What about fractions? • ° 0 and 1 (11001001) 2 Groups of four bits are called a nibble. • (1101) 2 18
Why Use Binary Numbers? ° Easy to represent 0 and 1 using electrical values. ° Possible to tolerate noise. ° Easy to transmit data ° Easy to build binary circuits. AND Gate 1 0 0 19
Binary In Binary, there are only 0’s and 1’s. These numbers are called “Base-2” ( Example: 0102) Base 2 = Base 10 Binary to Decimal 000 = 0 001 = 1 010 = 2 011 = 3 100 = 4 101 = 5 110 = 6 111 = 7 We count in “Base-10” (0 to 9) ° Binary number has base 2 ° Each digit is one of two numbers: 0 and 1 ° Each digit is called a bit ° Eight binary bits make a byte ° All 256 possible values of a byte can be represented using 2 digits in hexadecimal notation. 20
Binary as a Voltage ° Voltages are used to represent logic values: ° A voltage present (called Vcc or Vdd) = 1 ° Zero Volts or ground (called gnd or Vss) = 0 A simple switch can provide a logic high or a logic low. 21
A Simple Switch ° Here is a simple switch used to provide a logic value: Vcc Vcc, or 1 Gnd, or 0 There are other ways to connect a switch. 22
Binary digits Bit: single binary digit Byte: 8 binary digits Bit 100101112 Radix Byte 23
Conversion Between Number Bases Octal(base 8) Decimal(base 10) Binary(base 2) Hexadecimal ° Learn to convert between bases. ° Already demonstrated how to convert from binary to decimal. ° Hexadecimal described in next lecture. (base 16) 24
Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexadecimal 16 0, 1, … 9, A, B, … F No No 25
Conversion Among Bases The possibilities: Decimal Octal Binary Hexadecimal 26
Convert an Integer from Decimal to Another Base For each digit position: 1. Divide decimal number by the base (e. g. 2) 2. The remainder is the lowest-order digit 3. Repeat first two steps until no divisor remains. Example for (13)10: Integer Remainder Quotient 13/2 = 6/2 = 3/2 = 1/2 = 6 3 1 0 + + ½ 0 ½ ½ Coefficient a 0 = 1 a 1 = 0 a 2 = 1 a 3 = 1 Answer (13)10 = (a 3 a 2 a 1 a 0)2 = (1101)2 27
Convert an Fraction from Decimal to Another Base For each digit position: 1. Multiply decimal number by the base (e. g. 2) 2. The integer is the highest-order digit 3. Repeat first two steps until fraction becomes zero. Example for (0. 625)10: Integer 0. 625 x 2 = 0. 250 x 2 = 0. 500 x 2 = 1 0 1 Fraction + + + 0. 25 0. 50 0 Coefficient a -1 = 1 a -2 = 0 a -3 = 1 Answer (0. 625)10 = (0. a-1 a-2 a-3 )2 = (0. 101)2 28
The Growth of Binary Numbers n 2 n 0 20=1 8 28=256 1 21=2 9 29=512 2 22=4 10 210=1024 3 23=8 11 211=2048 4 24=16 12 212=4096 5 25=32 20 220=1 M Mega 6 26=64 30 230=1 G Giga 7 27=128 40 240=1 T Tera 29
Binary Addition ° Binary addition is very simple. ° This is best shown in an example of adding two binary numbers… 1 1 1 1 0 1 + 1 0 1 1 1 ----------1 0 1 0 0 1 1 1 carries 30
Binary Subtraction ° We can also perform subtraction (with borrows in place of carries). ° Let’s subtract (10111)2 from (1001101)2… 1 0 10 10 1 10 0 0 10 borrows 0 0 1 1 1 ------------1 1 0 31
Binary Multiplication ° Binary multiplication is much the same as decimal multiplication, except that the multiplication operations are much simpler… 1 0 1 1 1 X 1 0 1 -----------0 0 0 0 0 1 0 1 1 1 -----------1 1 1 0 0 32
Convert an Integer from Decimal to Octal For each digit position: 1. Divide decimal number by the base (8) 2. The remainder is the lowest-order digit 3. Repeat first two steps until no divisor remains. Example for (175)10: Integer Remainder Quotient 175/8 = 21/8 = 21 2 0 + + + 7/8 5/8 2/8 Coefficient a 0 = 7 a 1 = 5 a 2 = 2 Answer (175)10 = (a 2 a 1 a 0)2 = (257)8 33
Convert an Fraction from Decimal to Octal For each digit position: 1. Multiply decimal number by the base (e. g. 8) 2. The integer is the highest-order digit 3. Repeat first two steps until fraction becomes zero. Example for (0. 3125)10: Integer 0. 3125 x 8 = 0. 5000 x 8 = 2 4 Fraction + + 5 0 Coefficient a -1 = 2 a -2 = 4 Answer (0. 3125)10 = (0. 24)8 34
Summary ° Binary numbers are made of binary digits (bits) ° Binary and octal number systems ° Conversion between number systems ° Addition, subtraction, and multiplication in binary 35
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