ECE 3110: Introduction to Digital SystemsPrevious class SummaryDigital Design LevelsSlide 4Slide 5Slide 6Slide 7Binary RepresentationNumber SystemsPositional NotationSlide 11Slide 12Common PowersLeast Significant Digit Most Significant DigitHex (base 16) to Binary ConversionHex to Binary, Binary to HexNext…ECE 3110: Introduction to Digital SystemsNumber Systems2Previous class SummaryElectronics aspects of digital designIntegrated Circuits (wafer, die, SSI, MSI, LSI, VLSI)PLDs: PLAs, PALs, CPLD, FPGAASIC3Digital Design LevelsMany representations of digital logicDevice Physics and IC manufacturingMoore’s Law [1965, Gordon Moore]: Transistor level --->Logic design, functional building blocksThe number of transistors per square inch in an IC doubles every year [18months].4Digital Design LevelsTransistor-level circuit diagramsExample: Multiplexor5Truth tablesGate-level Logic diagrams6Prepackaged building blocks, e.g. multiplexerEquations: Z = S  A+ S B7Various hardware description languagesABELVHDL8Binary RepresentationThe basis of all digital data is binary representation.Binary - means ‘two’1, 0True, FalseHot, ColdOn, OffWe must be able to handle more than just values for real world problems1, 0, 56True, False, MaybeHot, Cold, Warm, CoolOn, Off, Leaky9Number SystemsTo talk about binary data, we must first talk about number systemsThe decimal number system (base 10) you should be familiar with!Positional number system10Positional NotationValue of number is determined by multiplying each digit by a weight and then summing. The weight of each digit is a POWER of the BASE and is determined by position.11The decimal number system (base 10) you should be familiar with!A digit in base 10 ranges from 0 to 9.A digit in base 2 ranges from 0 to 1 (binary number system). A digit in base 2 is also called a ‘bit’.A digit in base R can range from 0 to R-1A digit in Base 16 can range from 0 to 16-1 (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). Use letters A-F to represent values 10 to 15. Base 16 is also called Hexadecimal or just ‘Hex’.12953.7810 = 9 x 102 + 5 x 101 + 3 x 100 + 7 x 10-1 + 8 x 10-2 = 900 + 50 + 3 + .7 + .08 = 953.78 1011.112 = 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2 = 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.75A2F16 = 10x162 + 2x161 + 15x160 = 10 x 256 + 2 x 16 + 15 x 1 = 2560 + 32 + 15 = 2607Base 10, Base 2, Base 1613Common Powers2-3 = 0.1252-2 = 0.252-1 = 0.520 = 121 = 222 = 423 = 824 = 1625 =3226 = 6427 = 12828 = 25629 = 512210 = 1024211 = 2048212 = 4096160 = 1 = 20161 = 16 = 24162 = 256 = 28163 = 4096 = 212210 = 1024 = 1 K220 = 1048576 = 1 M (1 Megabits) = 1024 K = 210 x 210230 = 1073741824 = 1 G (1 Gigabits)14Least Significant DigitMost Significant Digit5310 = 1101012 Most Significant Digit (has weight of 25 or 32). For base 2, also called Most Significant Bit (MSB). Always LEFTMOST digit.Least Significant Digit (has weight of 20 or 1). For base 2, also called Least Significant Bit (LSB). Always RIGHTMOST digit.15Hex (base 16) to Binary ConversionEach Hex digit represents 4 bits. To convert a Hex number to Binary, simply convert each Hex digit to its four bit value.Hex Digits to binary (cont):916 = 10012A16 = 10102B16 = 10112C16 = 11002D16 = 11012E16 = 11102F16 = 11112Hex Digits to binary:016 = 00002116 = 00012216 = 00102316 = 00112416 = 01002516 = 01012616 = 01102716 = 01112816 = 1000216Hex to Binary, Binary to HexA2F16 = 1010 0010 11112 34516 = 0011 0100 01012Binary to Hex is just the opposite, create groups of 4 bits starting with least significant bits. If last group does not have 4 bits, then pad with zeros for unsigned numbers.10100012 = 0101 00012 = 5116 Padded with a zero17Next…More conversionsAddition/SubtractionHW