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What do we mean by data? n Many definitions are possible depending on context Lecture 2: Introduction to Data u We will say that: Representation n data is a physical representation of information u Data can be stored n e.g.: computer disk, cash till Digital Electronics I u Data can be transmitted n e.g.: fax u Data can be processed n e.g.: cash till PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.1 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.3 Points Addressed in this Lecture Electronic Representation of Data u What do we mean by data? u Information can be very complicated u How can data be represented electronically? n e.g.: • Numbers Sounds u What number systems are often used and why? • Pictures Codes u How do number systems of different bases work? n We need a simple electronic representation 5 Volts u How do you convert a number between binary and u What can we do with electronics? Set up voltages and currents R decimal? n n Change the voltages and currents u A useful device is a switch Switch V n Switch Closed: V = 0 Volts n Switch Open: V = 5 Volts PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.2 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.4 u Information can be represented by a voltage level u The simplest information is TRUE/FALSE More Complex Symbols n This can be represented by two voltage levels: • 5 Volts for TRUE u We have seen a binary representation of information • 0 Volts for FALSE n YES / NO n 1/0 u A voltage signal which has only two possibilities is a BIT n ON / OFF n Bit stands for Binary Digit u What about more complicated information? u Binary means: only 2 possible values n COLOUR: black, red, orange, yellow, green, blue, indigo, violet. FALSE TRUE n ORDINARY NUMBERS: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. (0) (1) n CHARACTERS: a, b, c, d, e, f, g, h, ..., x, y, z. u Combine bits to make words u Advantages of using binary representation n A word is a binary number containing more than one bit n simple to implement in electronic hardware (switch) n Four bit word is called "nibble", eg: 1101 n good tolerance to noise n Eight bit word is called "byte", eg: 10110001 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.5 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.7 Binary, Octal, Decimal and Binary Coding of Information Hexadecimal n [Just a quick look for now - more on these number systems later] u If we have n possibilities u A binary digit has only 2 possibilities n for the COLOURS we had 8 possibilities so n = 8 0 1 n for the lower case alphabet we have 26 possibilities u An octal digit has 8 possibilities 0 1 2 3 4 5 6 7 u then we need log2 n bits n for the 8 COLOURS we need 3 bits u A decimal digit has 10 possibilities n for the lower case alphabet we need 5 bits (always round up) 0 1 2 3 4 5 6 7 8 9 u A hexadecimal (hex) digital has 16 possibilities 0 1 2 3 4 5 6 7 8 9 A B C D E F PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.6 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.8 Coding Example 1 - Colours Decimal Number System COLOUR Binary Code 5 Volts u Example: 259 in decimal is Hundreds Tens Units Black 000 2 1 0 2 x 10 + 5 x 10 + 9 x 10 2 5 9 Red 001 R R R Orange 010 u The decimal number system is the base 10 number system Yellow 011 1 0 1 n each column represents increasing powers of 10 Green 100 n a subscript can be used to indicate the base of a number, eg: (259)10 Blue 101 u In general, any number system can be used Indigo 110 n base 10, base 2, base 16 and base 8 are common Violet 111 Setting the switches u The number of different symbols used is the base as shown generates n base 10 has 10 symbols the code which n base 8 has 8 symbols represent blue. u Base 10 is most familiar to us PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.9 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.11 Coding Example 2 - Decimal Integers Binary Number System 5 Volts Decimal Binary Code u Uses 2 symbols by our previous rule 0 0000 R R R R n 0 and 1 1 0001 u Example: 10011 in binary is 24 23 22 21 20 2 0010 1 0 0 0 4 1 0 1 0 0 1 1 3 0011 1 x 2 + 1 x 2 + 1 x 2 =19 4 0100 u Binary is the base 2 number system 5 0101 6 0110 u Most common in digital electronics Setting the switches 7 0111 as shown generates 8 1000 the code which 9 1001 represent 8. etc PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.10 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.12 Integer and Fractional Parts Example u Binary numbers can contain fractional parts as well as u Convert (13)10 into binary integer parts 4 3 2 1 -3 2 2 2 2 20 2-1 2-2 2 13 ÷ 2 = 6 remainder 1 LSB Answer 3 2 1 0 0 1 1 0 1 1 6 ÷ 2 = 3 remainder 0 . 2 2 21 20 (19.375) 10 3 ÷ 2 = 1 remainder 1 . 1 1 0 1 Binary Point 1 ÷ 2 = 0 remainder 1 MSB u This 8-bit number is in Q3 format n 3 bits after the binary point u This algorithm can be extended to conversion to any base u How could 19.376 best be represented using an 8-bit binary number? u Other algorithms can be used as alternatives if you prefer n Quantization error PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.13 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.15 Conversion: decimal to binary Conversion: binary to decimal u Write a binary number N as u The simplest way 2is to represent the binary number as n 2 1 0 n 1 0 n an x 2 + ... + a2 x 2 + a1 x 2 + a0 x 2 n an x 2 + ... + a2 x 2 + a1 x 2 + a0 x 2 u The conversion can be done by substituting the a's with the u The problem is then to find the a's such that the above expression equals N. given bits then multiplying and adding: n This can be done by repeated division of N by 2. n eg: Convert (1101)2 into decimal 3 2 1 0 n 1 x 2 + 1 x 2 + 0 x 2 + 1 x 2 = (13) 10 u The remainders are the binary digits starting from the Least u Other algorithms can be used as alternatives if you prefer Significant Bit (LSB) PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.14 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.16 Binary Addition ASCII – Representing text 1 1 1 u First recall decimal addition u How to represent non-numeric information? A 1 2 3 4 +B 9 8 7 u Use coding: American Standard Code for Information Sum 2 2 2 1 Interchange (ASCII) – pronounced as “askee” u In binary addition we follow the same pattern but u Use 7-bits to represent alphanumeric characters n 0 + 0 = 0 carry-out 0 u Store in computers as 8-bit byte – 1 character/byte n 0 + 1 = 1 carry-out 0 u A string is a collection of characters: e.g. “hello world” is a n 1 + 0 = 1 carry-out 0 n 1 + 1 = 0 carry-out 1 string with 11 characters (must include “space”) 1 n 1 + 1 + carry-in = 1 carry-out 1 A 0 1 1 1 H E L L O W O R L D + B 0 1 1 0 48 45 4C 4C 4F 20 57 4F 52 4C 44 Sum 1 1 0 1 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.17 PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.19 u Note that we need to consider 3 inputs per bit of binary number n A, B and carry-in u Each bit of binary addition generates 2 outputs n sum and carry-out PAN/PYKC 8/10/02 Dept of EEE, Imperial College Digital Electronics I. Slide 1.18

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posted: | 5/25/2010 |

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