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Chapter 1 (1.1 through 1.6) Digital Computers and Information Based on “Logic and Computer Design Fundamentals”, by Mano and Kime, Prentice Hall CSE260-1-1 Announcements Homework problem set 1 will be posted on the class website. Due: Wednesday 1/23/2013 CSE260-1-2 Overview: you will learn Information Representation Number Systems [binary, octal and hexadecimal] Base Conversion Decimal Codes [BCD (binary coded decimal)] Alphanumeric Codes Parity Bit Gray Codes CSE260-1-3 1-1 INFORMATION REPRESENTATION in Digital Systems- Signals Information variables represented by physical quantities. For digital systems, the variables take on discrete values. Two levels or binary values are the most prevalent values in digital systems. Binary values are represented abstractly by: • • • • digits 0 and 1 words (symbols) False (F) and True (T) words (symbols) Low (L) and High (H) and words Off and On. CSE260-1-4 Signal Examples Over Time Time Analog Digital Asynchronous Synchronous with the clock Continuous in value & time Discrete in value & continuous in time Discrete in value & time CSE260-1-5 Advantage of Digital Circuit Immunity to noise Illustration: Noise added Receiver Sender Signal Signal CSE260-1-6 Immunity to noise: Noise Margin VOH NMH Forbidden VOL NML VIH VIL Threshold Region We cannot violate the Noise Margins, otherwise our digital assumption is not valid: thus Vo must be > VOH or < VOL. CSE260-1-7 Binary Values: Other Physical Quantities What are other physical quantities represent 0 and 1? • Logic Gates, CPU: Voltage • Disk Magnetic Field Direction • CD Surface Pits/Light • Dynamic RAM Electrical Charge CSE260-1-8 1-2 Number Systems Positive radix, positional number systems Examples: • Decimal (radix r =10) Ex: 24.3 = 2x101 + 4x100+3x10-1 • Binary (radix r =2) Digits (0-9) Ex: 1101.01 = ( . )10 • Octal (radix r = ) • Hexadecimal (r = Bits (0-1) ) Digits: 1,2,…9, A, B, C, D, E, F CSE260-1-9 Exercise (146.A)16 = ( ? )10 (247.4)8 = ( ? )10 CSE260-1-10 Powers of 2: 2n It will be convenient to remember these powers 2n n 0 1 2 3 4 5 6 7 8 9 10 1 2 4 8 16 32 64 128 256 512 1024 2n n -1 -2 -3 0.5 0.25 0.125 CSE260-1-11 Special Powers of 2 210 (=1024) is Kilo, denoted "K" 220 (=1,048,576) is Mega, denoted "M" 230 (1,073, 741,824) is ? Giga, denoted "G" 240 (1,099,511,627,776 ) is Tera, denoted “T" •Note: 8 bits (b) are also called a byte (B) Exercise: what is (1111 1111)2 equal to in decimal? Also what is the maximum number (in decimal) a two bytes long word can represent? CSE260-1-12 Range of numbers Binary number: ex. a 3-bit number: n=3 • 000, 001 … ,111 or in decimal system: 0, 1 … 7 Total of 8 numbers (=23) Range: from 0 to 7 (0 to 23-1) • In general a n-bit number represents: 2n different numbers Min: 0 Max number: 2n-1 For fractions: m bits after the radix point: • Min: 0 • Max number: (2m -1)/2m CSE260-1-13 Exercise 16 Gbyte (= 16 GB) of memory • How many (address) bits are required to address each byte? CSE260-1-14 Exercise Digital camera has 2048x2048 pixels, and each pixel stores 8 bits of information: • a. How many Mega pixels? • b. How many bits are stored per frame? • c. How many different intensity levels can be represented by each pixel? CSE260-1-15 Use of HEX system Short hand notation of large binary numbers: • Each HEX digits can be represented by exactly 4 bits • (16=24) Thus (10011110.0101)2 9 E . 5 Conversion from binary to HEX and HEX to binary is very easy: (10011101)2 = ( )16 (1010110110.11)2 = ( B39.716 = ( )2 )16 CSE260-1-16 Octal system Radix r = 8 8 digits: • 0, 1, 2,…7 Ex: 2758 = 2x82 + 7x8 + 5x1 = 128 + 56 + 5 = 18910 Each octal digit can be represented by 3 bits CSE260-1-17 1-3 Conversion Between Bases To convert from one base to another: 1) Convert the Integer Part 2) Convert the Fraction Part 3) Join the two results with a radix point CSE260-1-18 Conversion from Decimal to new Radix To Convert the Integral Part: Repeatedly divide the number by the new radix and save the remainders. The digits for the new radix are the remainders in reverse order of their computation. To Convert the Fractional Part: Repeatedly multiply the fraction by the new radix and save the integer digits that result. The digits for the new radix are the integer digits in order of their computation. If the new radix is > 10, then convert all integers > 10 to digits A, B, … CSE260-1-19 Example: convert (325.65)10 to hex Integer part: 32510 = ( 325/16 = 20 and rem = 5 20/16 = 1 and rem = 4 1/16 = 0 and rem = 1 Fractional part: .65 . )16 Least significant digit Most significant Thus 32510 = 14516 0.65x16 = 10.4 thus int = 10= A Most significant 0.4x16 = 6.4 thus int = 6 0.4x16 = 6.4 thus int = 6 Least significant Etc. Thus .6510 = A6616 325.6510 = 145.A6616 CSE260-1-20 Example: Convert 54.687510 To Base 2 Convert 54 to Base 2 Convert 0.6875 to Base 2: Join the results together with the radix point: CSE260-1-21 Conversions from Binary to Octal and Hexadecimal Grouping in groups of 3 or 4 bits Convert 10001011 into octal and hexadecimal: Octal: 10001011 2 1 38 Hexadecimal 10001011 8 B16 CSE260-1-22 Octal to Hexadecimal via Binary Conversion from Octal to Hexadecimal and vice versa Convert 2138 into an hexadecimal number: 2138 10001011 8 B16 CSE260-1-23 Exercise: Octal Hexadecimal Exercise: Hexadecimal to Octal: 3A.5 16 = ( )8 Octal to Binary to Hexadecimal 6 3 5.1778 = ( )16 CSE260-1-24 Conversion - Summary Divisions (or x) by 16 Ai.16i Decimal SAi.2i Binary Ai.8i Group in bits of 3 Hexadecimal Group in bits of 4 Octal Hex: through the binary representation Octal CSE260-1-25 Single Bit Binary Addition with Carry Given two binary digits (X,Y), a carry in (Z) we get the following sum (S) and carry (C): Carry in (Z) of 0: Carry in (Z) of 1: Z X +Y 0 0 +0 0 0 +1 0 1 +0 0 1 +1 CS 00 01 01 10 Z X +Y 1 0 +0 1 0 +1 1 1 +0 1 1 +1 CS 01 10 10 11 CSE260-1-26 Multiple Bit Binary Addition Extending this to two multiple bit examples: Carries Augend Addend Sum 0 0 01100 10110 +10001 +10111 Note: The 0 is the default Carry-In to the least significant bit. CSE260-1-27 Multiple Bit Binary Subtraction Extending this to two multiple bit examples: 0 0 Minuend 10110 10110 Subtrahend - 10010 - 10011 Borrows Difference Notes: The 0 is a Borrow-In to the least significant bit. If the Subtrahend > the Minuend, interchange and append a – to the result. CSE260-1-28 1-4 Binary Codes A n-bit binary code is a n-bit word which can represent up to 2n different elements. Example: 3-bit code can represent up to 8 different elements” CSE260-1-29 Binary Codes Example: A binary code for the seven colors of the rainbow Code 100 is not used Color Red Orange Yellow Green Blue Indigo Violet Binary Number 000 001 010 011 101 110 111 Given n binary digits (called bits), a binary code is a mapping from a set of represented elements to a subset of the 2n binary numbers or elements. CSE260-1-30 Binary Coded Decimal (BCD) The BCD code is the 8,4,2,1 code. This code only encodes the first ten values from 0 to 9. Each decimal digit is coded separately by 4 bits Example: • (325)10 = (0011 0010 0101)BCD 3 2 Exercise: (856)10 = ( 5 )BCD CSE260-1-31 1-5 ALPHANUMERIC CODES - ASCII Character Codes American Standard Code for Information Interchange This code is a popular code used to represent information sent as characterbased data. It uses 7-bits to represent: • 94 Graphic printing characters. • 34 Non-printing characters Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return) CSE260-1-32 ASCII Code: B7B6B5 B4B3B2B1 H=(1001000) CSE260-1-33 PARITY BIT Error-Detection Codes Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors. A simple form of redundancy is parity, an extra bit appended onto the code word in the most significant position to make the number of 1’s odd or even. Parity can detect all single-bit errors and some multiple-bit errors. A code word has even parity if the number of 1’s in the code word is even. A code word has odd parity if the number of 1’s in the code word is odd. TX 7 Parity bit RX CSE260-1-34 ASCII Parity Code Example TX 7 Parity bit Transmit Ha in ascii: 1001000 1100001 Transmit with even parity: 01001000 11100001 added parity bit RX At the receiver side: If an even parity is detected, send an ACK control = 00000110 If error was detected send negative acknowledge NAK = 10010101 CSE260-1-35 1-6 GRAY CODE – Decimal Binary 000 001 2 bit changes 010 011 2 bit changes 100 101 2 bit changes 110 111 Gray 000 001 011 010 110 111 101 100 Only 1 bit changes CSE260-1-36