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Administrative Matters

Course: Digital Circuit Design Time/Location: 1B3EF-EC115 Instructor: E-mail: ylli@cs.nctu.edu.tw URL: http://people.cs.nctu.edu.tw/~ylli/ Office: 441 Office Hours: 4E or make an appointment by @ Teaching Assistants:

: eedc5tgb@gmail.com : gimball1220@gmail.com : wengshihyao@gmail.com : t790704re@gmail.com

Prerequisites: None

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Administrative Matters

Book: M. Morris Mano and Michael D. Ciletti, Digital

Design with an Introduction to the VERILOG HDL, 5th edition, 2013, PEARSON.

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Chap 1

Course Contents

Course Goals the fundamental to a digital system, how a digital system work, how to design a digital component, how to optimize a digital component design.

Course Contents Chap 1. Digital Systems and Binary Numbers Chap 2. Boolean Algebra and Logic Gates Chap 3. Gate-Level Minimization Chap 4. Combinational Logic Chap 5. Synchronous Sequential Logic Chap 6. Registers and Counters Chap 7. Memory and Programmable Logic Chap 8. Design at the Register Transfer Level (Optional)

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Grading Policy

Grading Examinations: 2 exams, 60% (25% and 35%) Handwriting and programming assignments: 20%

One person per team Quizzes: 20% Class participation: bonus

Course Web Site: eCampus Academic Honesty: Avoiding cheating at all cost.

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Overview

Integrated Circuits Overview Digital Systems and Computer Systems Information Representation Number Systems [binary, octal and hexadecimal] Base Conversion Decimal Codes [BCD (binary coded decimal),

parity] Alphanumeric Codes

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Integrated Circuits Overview

System hierarchy

Random (control)

logic

ALU

Digital circuits

ADconverter

RFcircuit

Analog circuits

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Integrated Circuits

Integrated Circuits (IC) imply to pack many components in a circuit. Small Scale Integration (SSI) A couple of

gates or a flip flop. Large Scale Integration (LSI) Functional unit

and memory with low computation power and capability.

Very Large Scale Integration (VLSI) millions of transistors (electrical switching device).

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Moores Law

Gordon Moore: co-founder of Intel. Predicted that the number of transistors per chip would grow

exponentially (double every 18 months).

Source: Intel

31000 2um134000 HMOS

275000 0.8um1260000 ?

3100000 0.8umBiCMOS7500000 0.35um CMOS

>9500000 0.25,0.18umCMOS

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IC History

Pentium 4

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VLSI Design Trends

Increasing demands from system level design

System On Chip (SOC)

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VLSI Design Trends

Digital circuits design trends Apply computer-aided design tools electrical design

automation (EDA) Design automation synthesis tools. Simulation tools. Verification tools.

Use Hardware Description Language (HDL) to design and verify circuits at early design stage Verilog 1983 Developed by Gateway Automation, 1989 Cadence

purchased Gateway Automation, 1995 IEEE Standard 1364 adopted.

VHDL 1983 IEEE Standard 1076. Why HDL ?

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VLSI Design Flow

(Hardware Description Language)

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VLSI Design Flow

Layout Die IC

cell levelchip level

SiO2, Si,Polisilicon,Aluminum,Copper,Photoresistant,Plasma,

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Digital System

Takes a set of discrete information inputs and discrete internal information (system state) and generates a set of discrete information outputs.

System State

DiscreteInformationProcessingSystem

DiscreteInputs Discrete

Outputs

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Types of Digital Systems

No state present Combinational Logic System Output = Function(Input)

State present State updated at discrete times (observe at fixed time)

=> Synchronous Sequential System State updated at any time (observe all the time)

=>Asynchronous Sequential System State = Function (State, Input) Output = Function (State)

or Function (State, Input)System State

DiscreteInformationProcessingSystem

DiscreteInputs Discrete

Outputs

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Digital System Example:

A Digital Counter (e. g., odometer):

1 30 0 5 6 4Count Up

Reset

Inputs: Count Up, ResetOutputs: Visual DisplayState: "Value" of stored digits

Synchronous or Asynchronous? Asynchronous

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A Digital Computer Example

Synchronous or Asynchronous?

Inputs: Keyboard, mouse, modem, microphone

Outputs: CRT, LCD, modem, speakers

Memory

Controlunit Datapath

Input/Output

CPU(Arithmetic & logic op.)

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Signal

Discrete elements of information (variables) are represented by physical quantities - signal.

For digital systems, the variable takes on discrete values. Two level, 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 On and Off.

Binary values are represented by values or ranges of values of physical quantities

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Signal Examples Over Time

Analog

Asynchronous

Synchronous

TimeContinuous in value &

timeDiscrete in

value & continuous

in timeDiscrete in

value & time

Digital

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5.0

4.0

3.0

2.0

1.0

0.0Volts

HIGH

LOW

HIGH

LOW

OUTPUT INPUT

Signal Example Physical Quantity: Voltage

Threshold Region

?

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Why Binary Rather Than Decimal?

5 V each of length 0.5V Need to recognize more levels more complex

and costly circuits are required More sensitive to noise - each of length 0.25V

for a given digit representation while considering noise on the voltages.

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What are other physical quantities represent 0 and 1? CPU Voltage Disk CD Dynamic RAM

Binary Values: Other Physical Quantities

Magnetic Field DirectionSurface Pits/Light

Electrical Charge

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Number Systems Representation

Positive radix, positional number systems A number with base or radix r is represented by

a string of digits:An - 1An - 2 A1A0 . A- 1 A- 2 A- m 1 A- m

in which 0 Ai < r and . is the radix point. An-1/A-m: most/least significant digit (msd/lsd).

The string of digits represents the power series:

(Number)r = j = - m

jj

i

i = 0i rArA

(Integer Portion) + (Fraction Portion)

i = n - 1 j = - 1

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Number Systems Examples

General Decimal BinaryRadix (Base) r 10 2Digits 0 => r - 1 0 => 9 0 => 1

0123

Powers of 4 Radix 5

-1-2-3-4-5

r0

r1

r2

r3

r4

r5

r -1

r -2

r -3

r -4

r -5

1101001000

10,000100,000

0.10.010.0010.00010.00001

124816320.50.250.1250.06250.03125

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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"

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Useful for Base ConversionExponent Value Exponent Value

0 1 11 2,0481 2 12 4,0962 4 13 8,1923 8 14 16,3844 16 15 32,7685 32 16 65,5366 64 17 131,0727 128 18 262,144

19 524,28820 1,048,57621 2,097,152

8 2569 51210 1024

Positive Powers of 2

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To convert to decimal, use decimal arithmetic to form (digit respective power of 2).

Example:Convert 110102 to N10:

Converting Binary to Decimal

110102 = 1 24 + 1 23 + 1 21 = 2610

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Method 1 Subtract the largest power of 2 (see slide 26) that gives a

positive remainder and record the power. Repeat, subtracting from the prior remainder and

recording the power, until the remainder is zero. Place 1s in the positions in the binary result

corresponding to the powers recorded; in all other positions place 0s.

Example: Convert 62510 to N2

Converting Decimal to Binary

625 512 = 113 = N1113 64 = 49 = N249 32 = 17 = N317 16 = 1 = N4

1 1 = 0 = N5

512 = 29

64 = 26

32 = 25

16 = 24

1 = 20(625)10 = 29 + 26 + 25 + 24 + 20 = (1001110001)2

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Commonly Occurring Bases

Name Radix DigitsBinary 2 0,1Octal 8 0,1,2,3,4,5,6,7Decimal 10 0,1,2,3,4,5,6,7,8,9Hexadecimal 16 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

The six letters (in addition to the 10integers) in hexadecimal represent:

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Decimal (Base 10)

Binary (Base 2)

Octal (Base 8)

Hexadecimal (Base 16)

00 00000 00 0001 00001 01 0102 00010 02 0203 00011 03 0304 00100 04 0405 00101 05 0506 00110 06 0607 00111 07 0708 01000 10 0809 01001 11 0910 01010 12 0A11 01011 13 0B12 01100 14 0C13 01101 15 0D14 01110 16 0E15 01111 17 0F16 10000 20 10

Numbers in Different Bases

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Conversion Between Bases

Method 2 To convert from one base to another:

1) Convert the Integer Part2) Convert the Fraction Part3) Join the two results with a radix point

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Conversion Details

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. If the new radix is > 10, then convert all remainders > 10 to digits A, B,

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,

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Example: Convert 46.687510 To Base 2

Convert 46 to Base 2

Convert 0.6875 to Base 2:

Join the results together with the radix point:

46/2 = 23 Remainder = 023/2 = 11+1/2 = 111/2 = 5 + 1/2 = 1

5/2 = 2 + 1/2 Remainder = 12/2 = 1 = 01/2 = 0 + 1/2 = 1

(46)10 = (101110)2

0.6875 2 = 1.3750 integer = 10.3750 2 = 0.7500 = 00.7500 2 = 1.5000 = 10.5000 2 = 1.0000 = 1

msd

lsd

msd

lsd(0.6875)10 = (0.1011)2

(46.6875)10 = (101110.1011)2

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Additional Issue - Fractional Part

Note that in this conversion, the fractional part became 0 as a result of the repeated multiplications.

In general, it may take many bits to get this to happen or it may never happen.

Example: Convert 0.6510 to N2 0.65 = 0.1010011001001 The fractional part begins repeating every 4 steps

yielding repeating 1001 forever! Solution: Specify number of bits to right of radix

point and round or truncate to this number.

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Checking the Conversion

To convert back, sum the digits times their respective powers of r. From the prior conversion of 46.687510

1011102 = 132 + 016 +18 +14 + 12 +01= 32 + 8 + 4 + 2= 46

0.10112 = 1/2 + 1/8 + 1/16= 0.5000 + 0.1250 + 0.0625= 0.6875

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Why Do Repeated Division and Multiplication Work?

46 = 125 + 024 + 123 + 122 + 121 + 02046/2 = 124 + 023 + 122 + 121 + 120 + 0/2

/2 = 123 + 022 + 121 + 120 + 1/2/2 = 122 + 021 + 120 + 1/2/2 = 121 + 020 + 1/2/2 = 120 + 0/2

01

11

1 0msd

lsd

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Why Do Repeated Division and Multiplication Work?

0.6875 = 12-1 + 02-2 + 12-3 + 12-40.68752 = 120+ 02-1 + 12-2 + 12-32 = 020 + 12-1 + 12-22 = 120 + 12-12 = 120

1011

msd

lsd

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Why Do Repeated Division and Multiplication Work? (continue)

Convert 46 to Base 2

Convert 0.6875 to Base 2:

46/2 = 23 + 0/2/2 = 11 + 1/2 + 0/22

/2 = 5 + 1/2 + 1/22 + 0/23

/2 = 2 + 1/2 + 1/22 + 1/23 + 0/24

/2 = 1 + 0/2 + 1/22 + 1/23 + 1/24 + 0/25

(46)10 = (101110)2

0.6875 2 = 1.3750 2 = 10.7500 2 = 101.5000 2 = 1011.0000

(0.6875)10 = (0.1011)2

(46.6875)10 = (101110.1011)2

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Octal (Hexadecimal) to Binary and Back

Octal (Hexadecimal) to Binary: Restate the octal (hexadecimal) as three (four)

binary digits starting at the radix point and going both ways.

Binary to Octal (Hexadecimal): Group the binary digits into three (four) bit

groups starting at the radix point and going both ways, padding with zeros as needed in the fractional part.

Convert each group of three (four) bits to an octal (hexadecimal) digit.

(010 110 001 101 011.111 100 000 110)2 = (26153.7406)8

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Octal to Hexadecimal via Binary

Convert octal to binary. Use groups of four bits and convert as

above to hexadecimal digits. Example: Octal to Binary to Hexadecimal

6 3 5 . 1 7 7 8

Why do these conversions work?

(110 011 101. 001 111 111)2(000110011101.001111111000)2( 1 9 D . 3 F 8 )16

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A Final Conversion Note

You can use arithmetic in other bases if you are careful:

Example: Convert 1011102 to Base 10 using binary arithmetic:Step 1 101110 / 1010 = 100 r 0110Step 2 100 / 1010 = 0 r 0100Converted Digits are 01002 | 01102

or 4 6 10

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Binary Numbers and Binary Coding

Flexibility of representation Within constraints below, we can assign any binary

combination (called a code word) to any data as long as data is uniquely encoded.

Information Types Numeric

Must represent range of data needed Very desirable to represent data such that simple,

straightforward computation for common arithmetic operations permitted

Tight relation to binary numbers Non-numeric

Greater flexibility since arithmetic operations not applied. Not tied to binary numbers

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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.

Example: Abinary codefor the sevencolors of therainbow

Code 100 is not used

Non-numeric Binary Codes

Binary Number000001010011101110111

ColorRedOrangeYellowGreenBlueIndigoViolet

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Complements of Numbers

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Complements of Numbers

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Complements of Numbers

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Complements of Numbers

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Complements of Numbers

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Complements of Numbers

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Signed Binary Numbers

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Signed Binary Numbers

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Signed Binary Numbers

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Signed Binary Numbers

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Signed Binary Numbers

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Binary Codes

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Binary Codes

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Binary Codes

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Binary Codes

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What interesting property is common to these two codes?

Excess 3 Code and 8, 4, 2, 1 Code

Decimal Excess 3 8, 4, 2, 10 0011 00001 0100 01112 0101 01103 0110 01014 0111 01005 1000 10116 1001 10107 1010 10018 1011 10009 1100 1111

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Warning: Conversion or Coding?

Do NOT mix up conversion of a decimal number to a binary number with coding a decimal number with a BINARY CODE.

1310 = 11012 (This is conversion) 13 0001|0011 (This is coding)

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ASCII Character Codes

American Standard Code for Information Interchange

This code is a popular code used to represent information such as character-based 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)

Other non-printing characters are used for record marking and flow control (e.g. STX and ETX start and end text areas).

(Refer to Table 1 -4 in the text)

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ASCII Properties

ASCII has some interesting properties: Digits 0 to 9 span Hexadecimal values 3016 to 3916 . Upper case A-Z span 4116 to 5A16 . Lower case a -z span 6116 to 7A16 .

Lower to upper case translation (and vice versa) occurs by flipping bit 6.

Delete (DEL) is all bits set, a carryover from when punched paper tape was used to store messages.

Punching all holes in a row erased a mistake!

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Binary Codes

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Binary Codes

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Binary Codes

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UNICODE

UNICODE extends ASCII to 65,536 universal characters codes For encoding characters in world languages Available in many modern applications 2 byte (16-bit) code words See Reading Supplement Unicode on the

Companion Website http://www.prenhall.com/mano

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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 to make the number of 1s 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 1s in the code word is even.

A code word has odd parity if the number of 1s in the code word is odd.

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4-Bit Parity Code Example

Fill in the even and odd parity bits:

The codeword "1111" has even parity and the codeword "1110" has odd parity. Both can be used to represent 3-bit data.

Even Parity Odd ParityMessage - Parity Message - Parity

000 - 0 000 -001 - 1 001 -010 - 1 010 -011 - 0 011 -100 - 1 100 -101 - 0 101 -110 - 0 110 -111 - 1 111 -

1 0 010110

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Terms of Use

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Pearson Education Legal Notice. Permission is given to incorporate these materials into classroom

presentations and handouts only to instructors adopting Logic and Computer Design Fundamentals as the course text.

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