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The main contents of the chapter consist of the following: Binary numbers are made of binary digits (bits); binary and octal number systems; conversion between number systems; addition, subtraction, and multiplication in binary.
Digital Logic Design EEE241 Sajjad Ali Mushtaq sajjad@ciit.edu.pk sajjad.mushtaq@ieee.org 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, ÷) Digi tal ° Sig Decimal values are difficult to represent in electrical systems It is easier to use two voltage values than nals ten ° Digital Signals have two basic states: (logic “high”, or H, or “on”) (logic “low”, or L, or “off”) ° Digital values are in a binary format Binary means 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 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 “ Motivation ° Microprocessors/Microelectronics have revolutionized our world • Cell phones, internet, rapid advances in medicine, etc ° The semiconductor industry has grown tremendously 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 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 Digital Logic Design Lecture Number Systems 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 bits) Overvie w ° 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 Binary as a Voltage ° Voltages are used to represent logic values: ° A voltage present (called Vcc or Vdd) = ° Zero Volts or ground (called gnd or Vss) = 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 Vcc, or 1 Gnd, or 0 There are other ways to connect a switch 22 Binary digits Bit: single binary digit Byte: 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 (base16) 24 Number Systems System Decimal Base Symbols 10 0, 1, … 9 Used by Used in humans? computers? Yes No Binary 0, 1 No Yes Octal 0, 1, … 7 No No Hexa decimal 16 0, 1, … 9, A, B, … F No No 25 Con ver sio The possibilities: n Am ong Bas es Decimal Binary Octal Hexadecimal 26 Convert an Integer from Decimal to Another Base For each digit position: Divide decimal number by the base (e.g 2) The remainder is the lowest-order digit Repeat first two steps until no divisor remains Example for (13)10: Integer Remainder Quotient 13/2 = 6/2 = 3/2 = 1/2 = + + + + ½ ½ ½ Coefficient a0 = a1 = a2 = a3 = Answer (13)10 = (a3 a2 a1 a0)2 = (1101)2 27 Convert an Fraction from Decimal to Another Base For each digit position: Multiply decimal number by the base (e.g 2) The integer is the highest-order digit Repeat first two steps until fraction becomes zero Example for (0.625)10: Integer 0.625 x = 0.250 x = 0.500 x = 1 Fraction + + + 0.25 0.50 Coefficient a -1 = a -2 = a -3 = Answer (0.625)10 = (0.a-1 a-2 a-3 )2 = (0.101)2 28 The Growth of Binary Numbers n 2n n 2n 20=1 28=256 21=2 29=512 22=4 10 210=1024 23=8 11 211=2048 24=16 12 212=4096 25=32 20 220=1M Mega 26=64 30 230=1G Giga 27=128 40 240=1T 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 + 1 1 1 0 1 carries 30 Binary Subtraction ° We can also perform subtraction (with borrows in place of carries) ° Let’s subtract (10111)2 from (1001101)2… 10 10 10 0 10 borrows 0 1 1 1 -1 1 31 Binary Multiplication ° Binary multiplication is much the same as decimal multiplication, except that the multiplication operations are much simpler… 1 1 X 1 0 0 1 1 0 0 1 1 1 0 1 0 32 Convert an Integer from Decimal to Octal For each digit position: Divide decimal number by the base (8) The remainder is the lowest-order digit Repeat first two steps until no divisor remains Example for (175)10: Integer Remainder Quotient 175/8 = 21/8 = 2/8 = 21 + + + 7/8 5/8 2/8 Coefficient a0 = a1 = a2 = Answer (175)10 = (a2 a1 a0)2 = (257)8 33 Convert an Fraction from Decimal to Octal For each digit position: Multiply decimal number by the base (e.g 8) The integer is the highest-order digit Repeat first two steps until fraction becomes zero Example for (0.3125)10: Integer 0.3125 x = 0.5000 x = Fraction + + Coefficient a -1 = a -2 = 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 .. .Digital Logic Design ° Digital - Concerned with the interconnection among digital components and modules » Best Digital System example is General Purpose Computer ° Logic Design - Deals... 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 Digital Logic Design Lecture Number. .. for off 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