Chapter 1. Binary Systems 1-1. Outline. ! Introductions. ! Number Base Conversions. ! Binary Arithmetic. ! Binary Codes. ! Binary Elements 1-2

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1 Chapter 1 Binary Systems 1-1 Outline! Introductions! Number Base Conversions! Binary Arithmetic! Binary Codes! Binary Elements 1-2

3C Integration 傳輸與介面 IA Connecting 聲音與影像 Consumer Screen Phone Set Top Box Mobile Phone Handheld Game Game Console Net TV Auto PC Smart Mobile Phone DSC Handheld PC PDA Thin

!!!!!! 1947: Bardeen, Brattain & Shockley invented the transistor, foundation of the IC industry 1958: Kilby invented integrated circuits (ICs) 1968: Noyce and Moore founded

2 3C Integration 傳輸與介面 IA Connecting 聲音與影像 Consumer Screen Phone Set Top Box Mobile Phone Handheld Game Game Console Net TV Auto PC Smart Mobile Phone DSC Handheld PC PDA Thin Client Palm PC MP3 Computing 資料處理 1-3 Milestones for IC Industry!!!!!!! 1947: Bardeen, Brattain & Shockley invented the transistor, foundation of the IC industry 1958: Kilby invented integrated circuits (ICs) 1968: Noyce and Moore founded Intel 1971: Intel announced 4-bit 4004 microprocessors (2300 transistors) 1976/81: Apple / IBM PC 1985: Intel began focusing on microprocessors Today, Intel-P4 has > 10M transistors!! up to 1.13 GHz; 0.18 um Semiconductor/IC: #1 key field for advancing into Y2K (Business Week, Jan. 1995) 1-4 2

3 The First Transistor 1-5 Pioneers of the Electronic Age 1-6

4 Moore s Law! Logic capacity doubles per IC per year at regular intervals (1965)! Logic capacity doubles per IC every 18 months (1975) 1-7 The Dies of Intel CPUs Pentium Pro 1-8

5 Semiconductor Technology Roadmap Year Technology node (nm) On-chip local clock (GHz) Microprocessor chip size (mm 2 ) Microprocessor transistor/chip Microprocessor cost/transistor (x10-8 USD) DRAM bits per chip Wiring level Supply voltage (V) Power (W) M M M G M 580 4G M G M G B G B 22 1T Source: SIA Product Creation Product Idea Design Fabrication 1-10

6 Design on Different levels S Q (a) silicon (b) circuit R Q (c) gate F-F Mux ALU Reg Parallel bus Data available chip Serial in Serial out clock (d) Register (e) Chip 1-11 Design Styles 1-12

7 IC Fabrication 1-13 Silicon Wafers 1-14

8 Wafer and Dices Source: Sawing a Wafer into Chips 1-16

9 IC and Die 1-17 Chip Packing 1-18

Pentium-MMX with PGA Packaging 1-19 Analog v.s. Digital analog waveform digital waveform! Analog system:! Inputs and outputs are represented by continuous values! More close to real-world signals!

10 Pentium-MMX with PGA Packaging 1-19 Analog v.s. Digital analog waveform digital waveform! Analog system:! Inputs and outputs are represented by continuous values! More close to real-world signals! Often used as interface circuits! Digital system:! Inputs and outputs are represented by discrete values! Easier to handle and design! More tolerable to signal degradation and noise! Binary digital systems form the basis of all hardware design today 1-20

11 Outline! Introductions! Number Base Conversions! Binary Arithmetic! Binary Codes! Binary Elements 1-21 Binary Numbers! Decimal numbers 10 symbols (0, 1, 2,, 9) A10 = a n-1 a n-2 a 1 a 0. a -1 a -2 a -m = Ex: (7392) 10 = 7 x x x x10 0! Binary numbers 2 symbols (0, 1) A 2 = n 1 i= m i ai *10 0 ai n 1 i ai *2 ai i= m = 0,1 9 Ex: ( ) 2 = 1 x x x x x x x2-2 = (26.75)

12 Powers of Two n 2 n n 2 n n 2 n Other Number Bases! In general, a base r (radix r) number system :! Coefficients a n is multiplied by powers of r! Coefficients a n range in value from 0 to r - 1! Example : A r = n 1 i= m ai * r! (4021.2) 5 = 4 x x x x x5-1 = (511.4) 10! (127.4) 8 = 1 x x x x8-1 = (87.5) 10 i 0 a r 1! (B65F) 16 = 11 x x x x16 0 i = (46687)

13 Number Base Conversions (1/2)! Convert (41) 10 to binary 41/2 = 20 remainder = 1 20/2 = 10 = 0 10/2 = 5 = 0 5/2 = 2 = 1 2/2 = 1 = 0 1/2 = 0 = = answer (41) 10 =(101001) 2! Convert (0.6875) 10 to binary x 2 = x 2 = x 2 = x 2 = = answer (0.6875) 10 =(0.1011) Number Base Conversions (2/2)! Convert (153) 10 to octal 153/8 = 19 remainder = 1 19/8 = 2 = 3 2/8 = 0 = = answer (153) 10 =(231) 8! Convert (0.513) 10 to octal x 8 = x 8 = x 8 = x 8 = x 8 = x 8 = = answer (0.513) 10 =( )

14 Fast Conversions! The conversion from and to binary, octal, and hexadecimal is important but much easier! To octal : 3 digits per group ( ) 2 = ( ) ! To hexadecimal : 4 digits per group ( ) 2 = (2C6B.F06) 16 2 C 6 B F 0 6! To binary : inverse process ( ) 8 = ( ) (306.D) 16 = ( ) D 1-27 Octal & Hexadecimal Numbers Base 10 Base2 Base 8 Base16 Base10 Base2 Base 8 Base A B C D E F 1-28

15 Outline! Introductions! Number Base Conversions! Binary Arithmetic! Binary Codes! Binary Elements 1-29 Binary Addition! Binary Addition 0+0=0 sum of 0 with a carry of (45) 0+1=1 sum of 1 with a carry of (39) 1+0=1 sum of 1 with a carry of (84) 1+1=10 sum of 0 with a carry of 1! Binary Addition with carry 1+0+0=01 sum of 1 with a carry of =10 sum of 0 with a carry of =10 sum of 0 with a carry of =11 sum of 1 with a carry of 1 carry bit 1-30

16 Binary Subtraction! Binary Subtraction 0 0 = = = = 1 (borrow 1 from the higher bit) EX : (45) (39) (6) 1-31 Binary Multiplication! Binary Multiplication 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = (5) EX : x (5) (25) 1-32

17 Binary Division! Binary Division EX : = Complements! Unsigned binary arithmetic is quite similar to decimal operations! But how about the negative numbers???! Use the complements of positive numbers! Complements can simplify the subtraction operation and logical manipulation! Two types of complements :! Diminished radix complement! Radix complement 1-34

18 Diminished Radix Complement! Given a number N in base r having n digits, the (r-1) s complement of N is defined as (r n -1)-N! The 9 s complement of (546700) 10 is = ! The 9 s complement of (012398) 10 is = ! The 1 s complement of ( ) 2 is ! The 1 s complement of ( ) 2 is bit inverse only!! 1-35 Radix Complement! The r s complement of an n-digit number N in base r is defined as r n -N, for N 0 and 0 for N=0! Equal to its (r-1) s complement added by 1! The 10 s complement of is ! The 10 s complement of is ! The 2 s complement of is ! The 2 s complement of is bit inverse and added with

19 Subtraction with Complements! The subtraction of two n-digit unsigned numbers M N in base r can be done as follows: (1):Add the minuend, M, to the r s complement of the subtrahend, N. M+(r n N)=M N + r n (2):If M N, the sum will produce an end carry, r n, which can be discarded ; what is left is the result M N (3):if M < N, the sum does not produce an end carry and is equal to r n (N M), which is the r s complement of (N M). To obtain the answer in a familiar form, take the r s complement of the sum and placed a negative sign in front 1-37 Examples of Subtraction Given two binary numbers X = and Y = ! X Y (use 2 s complement) 2 s complement of Y = = X Y = X + Y carry is discarded X Y = ! Y X (use 2 s complement) 2 s complement of X = = Y X = Y + X no end carry generated Y X =

20 Signed Binary Numbers! Signed 2 s complement use 2 s comp. Decimal Signed 2 s complement Signed 1 s complement Signed magnitude ! Signed 1 s complement use 1 s comp ! Signed magnitude first digit: sign others: magnitude Arithmetic Addition/Subtraction! The negative numbers are in 2 s complement form! If the sum is negative, it s also in 2 s complement form ! For subtraction, take 2 s complement of the subtrahend (±A) (+B) = (±A) + (-B) (±A) (-B) = (±A) + (+B) 1-40

21 Outline! Introductions! Number Base Conversions! Binary Arithmetic! Binary Codes! Binary Elements 1-41 BCD Code! BCD = Binary Coded Decimal! 4 bits for one digit Decimal symbol 0 1 BCD Digit Ex: (185) 10 = ( ) BCD = ( ) 2! 1010 to 1111 are not used and have no meaning in BCD! Can perform arithmetic operations directly with decimal numbers in digital systems

22 BCD Addition! When the binary sum is equal to or less than 1001 (without a carry), the corresponding BCD digit is correct ! Otherwise, the addition of 6 = (0110) 2 can convert it to the correct digit and also produce the required carry " (0 1001) BCD 10 " (1 0000) BCD 10+6 = 16 = (10000) 2 The addition of 6 can convert to correct digit 1-43 Other Decimal Codes DECIMAL DIGIT Unused bit combinations BCD (8421) weight EXCESS

23 Gray Code Only change one bit when going from one number to the next!! Gray code Decimal equivalent Gray code Decimal equivalent ASCII Character Code 1-46

24 Error-Detecting Code! To detect error in data communication, an extra parity bit is often added! Just count the total number of 1 can decide the status of parity bit! Even parity = 1 " the number of 1 is even (include parity)! Odd parity = 1 " the number of 1 is odd (include parity)! If the received number of 1 does not match the parity bit, an error occurs with even parity with odd parity ASCII A = ASCII T = parity bit 1-47 Outline! Introductions! Number Base Conversions! Binary Arithmetic! Binary Codes! Binary Elements 1-48

25 Binary Elements! Binary cell! Possess two stable states and store one bit of information! Registers! A group of binary cells! A register with n cells can store any discrete quantity of n-bit information! Binary logic! Define the operations with variables that take two discrete values! The operations are implemented by logic gates! Register transfers (to where) and data operations (do what) form the digital systems 1-49 Register Transfer of Information MEMORY UNIT J O H N Memory Register PROCESSOR UNIT 8 cells 8 cells 8 cells 8 cells Processor Register INPUT UNIT KEYBOARD J O H N 8 cells CONTROL Input Register 1-50

Logical Operations! Truth tables of logical operations: AND X Y X Y 0 0 0 0 1 0 1 0 0 1 1 1 OR X Y X + Y 0 0 0 0 1 1 1 0 1 1 1 1 NOT X X 0 1 1 0!

26 Logical Operations! Truth tables of logical operations: AND X Y X Y OR X Y X + Y NOT X X ! Logic gates: electronic circuits that perform the corresponding logical operations 1-51 Properties of Logic Gates! Logic gates handle binary signals and also generate binary signals! AND and OR gates may have more than two inputs 1-52

Digital Systems Roberto Muscedere Images 2013 Pearson Education Inc. 1

Digital Systems Roberto Muscedere Images 2013 Pearson Education Inc. 1 Digital Systems Digital systems have such a prominent role in everyday life The digital age The technology around us is ubiquitous, that is we don t even notice it anymore Digital systems are used in: