Chapter 1 Binary Systems

Transcription

1 EEA051 - Digital Logic 數位邏輯 Chapter 1 Binary Systems 吳俊興高雄大學資訊工程學系 September 2005

2 Chapter 1. Binary Systems 1-1 Digital Systems 1-2 Binary Numbers 1-3 Number Base Conversions 1-4 Octal and Hexadecimal Numbers 1-5 Complements 1-6 Signed Binary Numbers 1-7 Binary Codes 1-8 Binary Storage and Registers 1-9 Binary Logic 2

3 Chapter 1. Binary Systems Presents the various binary systems suitable for representing information in digital systems The binary number system is explained and binary codes are illustrated Examples are given for addition and subtraction of signed binary numbers and decimal numbers in BCD 3

4 1-1 Digital Systems Analog vs. digital analog continuous digital discrete the real world is mainly analog Why digital? digital systems are easier to design information storage is easy accuracy and precision is better operation can be programmed digital circuits are less affected by noise Example: digital camera 4

5 Digital Systems Digital age Digital systems telephone switching exchanges digital camera electronic calculators, PDA's digital TV, digital broadcast Digital computers many scientific, industrial and commercial applications Generality Discrete information-processing systems 5

6 Typical Control System (ADC) (DAC) 6

7 Representing Binary Quantities 7

8 Digital Signals and Timing Diagrams Signals: physical quantities, e.g. voltages and currents, to represent discrete elements of information in a digital system predominately implemented by transistors most use just two discrete values, said to be binary A binary digit, called a bit, has two values: 0 and 1 Binary codes: groups of bits Why binary? reliability: a transistor circuit is either on or off (two stable states) 8

9 1-2 Binary Numbers Numbers system: a n a 3 a 2 a 1 a 0.a -1 a -2 a -3 a -m Decimal number (base or radix = 10) (10 digits) 7,392 = 7* * * *10 0 Binary number (base = 2) ( ) 2 = (26.75) 10 9

10 Base-r System Base-r system (coefficients multiplied by powers of r) (4021.2) 5, (127.4) 8, (B65F) 16 Base-r Decimal (4021.2) 5 = (511.4) 10 Octal (127.4) 8 = (87.5) 10 Hexadecimal (B65F) 16 =(46,687) 10 Binary (110101) 2 =(53) 10 10

11 Binary Numbers Powers of Two K(kilo)=2 10, M(mega)=2 20, G(giga)=2 30, T(tera)= , 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 7, 2 8, ,

12 Binary Arithmetic Operations Arithmetic operations with numbers in base r follow the same rules as for decimal numbers (discussed later) 12

13 1-3 Number Base Conversions Decimal Base-r: converting a decimal number to a number in base r (four examples) 1. Convert decimal 41 to binary: (101001) 2 2. Convert decimal 153 to octal: (231) 8 3. Convert (0.6875) 10 to binary: (0.1011) 2 4. Convert (0.513) 10 to octal: ( ) 8 Combining: ( ) 10 = ( ) 2 ( ) 10 = ( ) 8 13

14 1-4 Octal and Hexadecimal Numbers Binary to octal: 2 3 =8 Binary to hexadecimal: 2 4 =16 Octal to binary Hexadecimal to binary Octal or hexadecimal representation is more desirable 14

15 Binary Octal Hexadecimal 15

16 1-5 Complements Used for simplifying the subtraction operation and for logical manipulation Two types of complement diminished radix complement: (r-1) s complement (r n -1)-N radix complement: r s complement r n -N Decimal number 10 s complement and 9 s complement Binary number 2 s complement and 1 s complement 16

17 Examples Diminished Radix Complement The 9 s complement of is = The 9 s complement of is = The 1 s complement of is The 1 s complement of is Radix Complement The 10 s complement of is The 10 s complement of is The 2 s complement of is The 2 s complement of is

18 Subtraction with r s Complements Subtraction of two n-digit unsigned numbers M-N in base r: 1. Add M to the r s complement of the subtrahend, N: M + (r n N)= sum 2. If M N, the sum will produce an end carry, r n, which can be discarded M + (r n N) = sum = (M N) + r n, so M-N = sum - r n 3. If M < N, the sum is the r s complement of (N-M) M + (r n N) = sum = r n (N-M), so M-N = -(r n -sum) 18

19 Subtraction with (r-1) s Complements Subtraction of two n-digit unsigned numbers M-N in base r: Add M to the (r-1) s complement of subtrahend N: M + ((r n -1) N) = sum If M N, the sum will produce an end carry, r n, which can be discarded M + ((r n -1) N) = sum = (M N) + (r n -1), so M-N = sum r n + 1 (end-around carry) If M < N, the sum is the r s complement of (N-M) M + ((r n -1) N) = sum = (r n -1) (N-M), so M-N = -((r n -1) - sum) 19

20 Examples 20

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22 1-6 Signed Binary Numbers Table

23 Arithmetic Addition Arithmetic Subtraction (±A) (+B) = (±A) + ( B) (±A) ( B) = (±A) + (+B) 23

24 n-bit binary code 1.7 Binary Codes 2 n distinct combinations BCD Binary Coded Decimal (4-bit) (185) 10 = ( ) BCD = ( ) 2 (396) 10 = ( ) BCD BCD addition Get the binary sum If the sum > 9, add 6 to the sum Obtain the correct BCD digit sum and a carry 24

25 Number Systems and BCD Code Decimal Binary Octal Hexadecimal A B C D E F BCD

26 BCD Addition =

27 Other Decimal Codes 27

28 Gray Code only one bit change between two consecutive numbers useful in Analog-to-Digital Converter 28

29 ASCII Character Code 29

30 Error-Detecting Code Parity bit: an extra bit included with a message to make the total number of 1 s either even or odd 30

31 1.8 Binary Storage and Registers A binary cell two stable state store one bit of information examples: flip-flop circuits, ferrite cores, capacitor A register a group of binary cells e.g. AX in x86 CPU Register Transfer a transfer of the information stored in one register to another one of the major operations in digital system an example 31

32 Transfer of information 32

33 The other major component of a digital system circuit elements to manipulate individual bits of information 33

34 1.9 Binary Logic Binary Logic Boolean algebra consists of binary variables and logical operations Binary variables two discrete values (true/false; yes/no; 1/0) Logical operations Three basic operations: AND, OR, NOT 34

35 Logic Gates Binary signals Electrical signals: voltages or currents two separate voltage levels: logic-1 and logic-0 the intermediate region is crossed only during state transition Logic circuits circuits = logical manipulation paths Computation and control combinations of logic circuits Logic gates electronic circuits that operate on one or more input signals to produce an output signal 35

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38 Summary Chapter 1. Binary Systems 1-1 Digital Systems 1-2 Binary Numbers 1-3 Number Base Conversions 1-4 Octal and Hexadecimal Numbers 1-5 Complements 1-6 Signed Binary Numbers 1-7 Binary Codes 1-8 Binary Storage and Registers 1-9 Binary Logic 38