1Slide 1EE40 Fall 2006Prof. Chang-HasnainEE40Lecture 35Prof. Chang-Hasnain12/5/07Reading: Ch 7, SupplementaryReaderSlide 2EE40 Fall 2006Prof. Chang-HasnainWeek 15• OUTLINE– Need for Input Controlled Pull-Up– CMOS Inverter Analysis– CMOS Voltage Transfer Characteristic– Combinatorial logic circuits– Logic– Binary representations– Combinatorial logic circuits• Reading– Chap 7-7.5– Supplementary Notes Chapter 42Slide 3EE40 Fall 2006Prof. Chang-HasnainDigital Circuits – Introduction• Analog: signal amplitude is continuous with time.• Digital: signal amplitude is represented by a restrictedset of discrete numbers.– Binary: only two values are allowed to represent the signal: Highor low (i.e. logic 1 or 0).• Digital word:– Each binary digit is called a bit– A series of bits form a word• Byte is a word consisting of 8-bits• Advantages of digital signal– Digital signal is more resilient to noise◊ can more easilydifferentiate high (1) and low (0)• Transmission– Parallel transmission over a bus containing n wires.• Faster but short distance (internal to a computer or chip)– Serial transmission (transmit bits sequentially)• Longer distanceSlide 4EE40 Fall 2006Prof. Chang-Hasnain◊ Analog-to-digital (A/D) & digital-to-analog (D/A)conversion is essential (and nothing new)think of a piano keyboard• Most (but not all) observables are analogthink of analog vs. digital watchesbut the most convenient way to represent & transmitinformation electronically is to use digital signalsthink of telephonyAnalog vs. Digital Signals3Slide 5EE40 Fall 2006Prof. Chang-HasnainVoltage with normal piano key stroke Voltage with soft pedal applied50 microvolt 220 Hz signal-60-40-2002040600 1 2 3 4 5 6 7 8 9 10 11 12t in millisecondsV in microvolts50 microvolt 440 Hz signal-60-40-2002040600 1 2 3 4 5 6 7 8 9 10 11 12t in millisecondsV in microvolts25 microvolt 440 Hz signal-60-40-2002040600 1 2 3 4 5 6 7 8 9 10 11 12t in millisecondsV in microvoltsAnalog Signal Example: Microphone VoltageAnalog signal representing piano key A,below middle C (220 Hz)Slide 6EE40 Fall 2006Prof. Chang-HasnainDigital Signal RepresentationsBinary numbers can be used to represent any quantity.We generally have to agree on some sort of “code”, andthe dynamic range of the signal in order to know the formand the number of binary digits (“bits”) required.Example 1: Voltage signal with maximum value 2 Volts• Binary two (10) could represent a 2 Volt signal.• To encode the signal to an accuracy of 1 part in 64(1.5% precision), 6 binary digits (“bits”) are neededExample 2: Sine wave signal of known frequency andmaximum amplitude 50 µV; 1 µV “resolution” needed.4Slide 7EE40 Fall 2006Prof. Chang-HasnainDecimal Numbers: Base 10Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9Example:3271 = (3x103) + (2x102) + (7x101) + (1x100)This is a four-digit number. The left handmost number (3 in this example) is oftenreferred as the most significant numberand the right most the least significantnumber (1 in this example).Slide 8EE40 Fall 2006Prof. Chang-HasnainNumbers: positional notation• Number Base B ⇒ B symbols per digit:–Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9–Base 2 (Binary): 0, 1• Number representation:–d31d30 ... d1d0 is a 32 digit number–value = d31 × B31 + d30 × B30 + ... + d1 × B1 + d0 × B0• Binary: 0,1 (In binary digits called “bits”)11010 = 1×24 + 1×23 + 0×22 + 1×21 + 0×20= 16 + 8 + 2= 26–Here 5 digit binary # turns into a 2 digit decimal #5Slide 9EE40 Fall 2006Prof. Chang-HasnainHexadecimal Numbers: Base 16• Hexadecimal:0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F–Normal digits + 6 more from the alphabet• Conversion: Binary⇔Hex–1 hex digit represents 16 decimal values–4 binary digits represent 16 decimal values⇒1 hex digit replaces 4 binary digitsSlide 10EE40 Fall 2006Prof. Chang-HasnainDigital Signal RepresentationsBinary numbers can be used to represent any quantity.We generally have to agree on some sort of “code”, andthe dynamic range of the signal in order to know the formand the number of binary digits (“bits”) required.Example 1: Voltage signal with maximum value 2 V andminimum of 0 V.• Binary two (10) could represent a 2 Volt signal.• To encode the signal to an accuracy of 1 part in 64(1.5% precision), 6 binary digits (“bits”) are neededExample 2: Sine wave signal of known frequency andmaximum amplitude 50 µV; 1 µV “resolution” needed.6Slide 11EE40 Fall 2006Prof. Chang-HasnainResolution• The size of the smallest element that canbe separated from neighboring elements.The term is used to describe imagingsystems, the frequency separationachieved by spectrometers, and so on.Slide 12EE40 Fall 2006Prof. Chang-HasnainDecimal-Binary Conversion• Decimal to Binary– Repeated Division By 2• Consider the number 2671.– Subtraction – if you know your 2N values byheart.• Binary to Decimal conversion1100012 = 1x25 +1x24 +0x23 +0x22 + 0x21 + 1x20 = 3210 + 1610 + 110 = 4910 = 4x101 + 9x1007Slide 13EE40 Fall 2006Prof. Chang-HasnainPossible digital representation for the sine wave signal:Analog representation: Digital representation:Amplitude in µVBinary number1 0000012 0000103 0000114 0001005 0001018 00100016 01000032 10000050 11001063 111111Example 2 (continued)Slide 14EE40 Fall 2006Prof. Chang-HasnainBinary Representation• N bit can represent 2N values: typicallyfrom 0 to 2N-1– 3-bit word can represent 8 values: e.g. 0, 1, 2,3, 4, 5, 6, 7• Conversion– Integer to binary– Fraction to binary (13.510=1101.12 and0.39210=0.0110012)• Octal and hexadecimal8Slide 15EE40 Fall 2006Prof. Chang-Hasnain• Logic gates– Combine several logic variable inputs toproduce a logic variable output• Memory– Memoryless: output at a given instantdepends the input values of that instant.– Momory: output depends on previous andpresent input values.Slide 16EE40 Fall 2006Prof. Chang-HasnainBoolean algebras• Algebraic structures– "capture the essence" of the logical operations AND,OR and NOT– corresponding set for theoretic operationsintersection, union and complement– named after George Boole, an English mathematicianat University College Cork, who first defined them aspart of a system of logic in the mid 19th century.– Boolean algebra was an attempt to use algebraictechniques to deal with expressions in thepropositional calculus.– Today, Boolean algebras find many applications
View Full Document
Unlocking...