ENG241 ENG241 Digital DesignDigital Design

Week #1Digital Computers and Information

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ResourcesResources

Chapter #1, Mano Sections 1.1 Digital Computers 1.2 Number Systems 1.3 Arithmetic Operations 1.4 Decimal Codes 1.5 Alphanumeric Codes

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TopicsTopics

Computing Devices and VLSI Design Signals (Digital vs. Analog) Digital Systems and Computers Number systems [binary, octal, hex] Base Conversion Arithmetic Operations Decimal Codes [BCD] Alphanumeric Codes

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PDA

Body

Entertainment

Household

Communication

Home Networking

Car

Medicine

PC

Super Computer

Computing Devices Everywhere!Computing Devices Everywhere!

Game console

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The Transistor RevolutionThe Transistor Revolution

First transistorBell Labs, 1948

Bipolar logic1960’s

• Intel 4004 processor • Designed in 1971• Almost 3000 transistors• Speed:1 MHz operation

© K

LMH

Lien

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1.3 VLSI Design Styles

Power (Vdd)-Rail

Ground (GND)-Rail

Contact

Vdd

GND

OUT

IN2

IN1OUT

IN2

IN1

OUTIN1

Vdd

GND

IN2

Diffusion layer

p-typetransistor

n-typetransistor

Metal layer

Poly layer

n+n+S

GD

+

DEVICE

CIRCUIT

GATE

MODULE

SYSTEM Specification

Functional design

Circuit design

Physical design

Test/Fabrication

Logic design

The VLSI Design CycleThe VLSI Design Cycle

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Signals

An information variable represented by a physical quantity (speech, Temp, humidty, noise, …)

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Signals

Signals can be analog or digital:

1.Analog signals can have an infinite number of values in a range;

2.Digital signals can have only a limited number of values.

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Analog Signals

Analog

Time

Continuous in value &

time

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Digital Signals

For digital systems, the variable takes on discrete values (i.e., not continuous)

Time

DigitalDiscrete in

value

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Signal Examples Over Time

Asynchronous

Synchronous

Discrete in value & continuous

in time

Discrete in value &

time

Digital

Digital (Binary) values are represented by: digits 0 and 1 / False (F) and True (T) words (symbols) Low (L) and High (H) words On and Off.

Time

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What are other physical quantities represent 0 and 1?

Binary Values: Other Physical Quantities

Magnetic Field DirectionSurface Pits/Light

Electrical Charge

CPU VoltageHDiskCDDynamic RAM

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Digital System Example:

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

1 30 0 5 6 4Count Up

Reset

Inputs: Count Up, Reset Outputs: Visual Display State: "Value" of stored digits

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A Digital Computer Example

Inputs: Keyboard, mouse, modem, microphone

Outputs: CRT, LCD, modem, speakers

Memory

Controlunit Datapath

Input/Output

CPU

Data/Instructions/code

clock

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Number Systems – Representation

A number with 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.

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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Decimal Number SystemDecimal Number System

Base (also called radix) = 10

● 10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Digit Position

● Integer & fraction

Digit Weight

● Weight = (Base) Position

Magnitude

● Sum of “Digit x Weight”

Formal Notation

1 0 -12 -2

5 1 2 7 4

10 1 0.1100 0.01

500 10 2 0.7 0.04

d2*B2+d1*B

1+d0*B0+d-1*B

-1+d-2*B-2

(512.74)10

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Octal Number SystemOctal Number System

Base = 8

● 8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }

Weights

● Weight = (Base) Position

Magnitude

● Sum of “Digit x Weight”

Formal Notation

1 0 -12 -2

8 1 1/864 1/64

5 1 2 7 4

5 *8 2+1 *8 1+2 *8 0+7 *8- 1+4 *8 -2

=(330.9375)10

(512.74)8

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Octal Number System: Example

For Example, (27)8 can be expressed as: ( )10

(17.1)8 can be expressed as: ( )10

2020

Hexadecimal Number SystemHexadecimal Number System

Base = 16

● 16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F }

Weights

● Weight = (Base) Position

Magnitude

● Sum of “Digit x Weight”

Formal Notation

1 0 -12 -2

16 1 1/16256 1/256

1 E 5 7 A

1 *162+14 *161+5 *160+7 *16-1+10 *16-2

=(485.4765625)10

(1E5.7A)16

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Hex to DecimalHex to Decimal

Just multiply each hex digit by decimal value, and add the results.

163 162 161 160

4096 256 16 1

(2ac)(2ac)1616

2 2 • 256• 256 + 10 + 10 • 16• 16 + 12 + 12 • 1• 1 = (684)= (684)1010

Dec Hex

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

10 a

11 b

12 c

13 d

14 e

15 f

2222

Binary Number SystemBinary Number System

Base = 2

● 2 digits { 0, 1 }, called binary digits or “bits”

Weights

● Weight = (Base) Position

Magnitude

● Sum of “Bit x Weight”

Formal Notation

Groups of bits 4 bits = Nibble

8 bits = Byte

1 0 -12 -2

2 1 1/24 1/4

1 0 1 0 1

1 *22+0 *21+1 *20+0 *2-1+1 *2-2

=(5.25)10

(101.01)2

1 0 1 1

1 1 0 0 0 1 0 1

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Binary Binary Decimal: Example Decimal: Example

7 6 5 4 3 2 1 0

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1

1 0 0 1 1 1 0 0

128 + 0 + 0 + 16 + 8 + 4 + 0 + 0 = 156 128 + 0 + 0 + 16 + 8 + 4 + 0 + 0 = 156

What is 10011100 in decimal?What is 10011100 in decimal?

7 6 5 4 3 2 1 0 position

Binary #

position

value

Why Binary?Why Binary?

This is easier to implement in hardware than a unit that can take on 10 different values.

● For instance, it can be represented by a transistor being off (0) or on (1).

● Alternatively, it can be a magnetic stripe that is magnetized with North in one direction (0) or the opposite (1).

Binary also has a convenient and natural association with logical values of:

● False (0) and

● True (1).2424

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Binary NumbersBinary NumbersExamples:

(00)2 (0)10

(01)2 (1)10

(10)2 (2)10

(010)2 (2)10

(11)2 (3)10

(100)2 (4)10

(1001010101000)2

Strings of binary digits (“bits”)One bit can store a number from 0 to 1n bits can store numbers from 0 to 2n-1

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The Power of 2 The Power of 2

n 2n

0 20=1

1 21=2

2 22=4

3 23=8

4 24=16

5 25=32

6 26=64

7 27=128

n 2n

8 28=256

9 29=512

10 210=1024

11 211=2048

12 212=4096

20 220=1M

30 230=1G

40 240=1T

Mega

Giga

Tera

Kilo

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Number Base ConversionsNumber Base Conversions

Decimal(Base 10)

Octal(Base 8)

Binary(Base 2)

Hexadecimal(Base 16)

Evaluate Magnitude

Evaluate Magnitude

Evaluate Magnitude

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Conversion Between BasesConversion 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

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Decimal (Decimal (IntegerInteger) to Binary Conversion) to Binary Conversion

Divide the number by the ‘Base’ (=2)

Take the remainder (either 0 or 1) as a coefficient

Take the quotient and repeat the division

Example: (13)10

Quotient Remainder Coefficient

Answer: (13)10 = (a3 a2 a1 a0)2 = (1101)2

MSB LSBMSB LSB

13/ 2 = 6 1 a0 = 1 6 / 2 = 3 0 a1 = 0 3 / 2 = 1 1 a2 = 1 1 / 2 = 0 1 a3 = 1

Decimal to Binary ConversionDecimal to Binary Conversion

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Decimal (Decimal (FractionFraction) to Binary Conversion) to Binary Conversion

Multiply the number by the ‘Base’ (=2)

Take the integer (either 0 or 1) as a coefficient

Take the resultant fraction and repeat multiplication

Example: (0.625)10

Integer Fraction Coefficient

Answer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2

MSB LSBMSB LSB

0.625 * 2 = 1 . 250.25 * 2 = 0 . 5 a-2 = 00.5 * 2 = 1 . 0 a-3 = 1

a-1 = 1

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Decimal to Octal ConversionDecimal to Octal Conversion

Example: (175)10

Quotient Remainder Coefficient

Answer: (175)10 = (a2 a1 a0)8 = (257)8

175 / 8 = 21 7 a0 = 7 21 / 8 = 2 5 a1 = 5 2 / 8 = 0 2 a2 = 2

Example: (0.3125)10

Integer Fraction Coefficient

Answer: (0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)8

0.3125 * 8 = 2 . 50.5 * 8 = 4 . 0 a-2 = 4

a-1 = 2

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Decimal to HexDecimal to Hex

163 162 161 160

4096 256 16 1

Dec Hex

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

10 a

11 b

12 c

13 d

14 e

15 f

(684)(684)1010

684/16 = 42 rem 12=c684/16 = 42 rem 12=c c

42/16 = 2 rem 10=a42/16 = 2 rem 10=a ac

2/16 = 0 rem 22/16 = 0 rem 2 2ac

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Hexadecimal (Base 16)Hexadecimal (Base 16)

Dec Bin Hex

0 0000 0

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

Dec Bin Hex

8 1000 8

9 1001 9

10 1010 a

11 1011 b

12 1100 c

13 1101 d

14 1110 e

15 1111 f

•Power of 2Power of 2

•Size of byteSize of byte

Why useWhy usebase 16?base 16?

Strings of 0’s and 1’s too hard to write Use base-16 or hexadecimal – 4 bits

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Hex to BinaryHex to Binary

Convention – write 0x (prefix) before number Hex to Binary – just convert digits

Bin Hex

0000 0

0001 1

0010 2

0011 3

0100 4

0101 5

0110 6

0111 7

1000 8

1001 9

1010 a

1011 b

1100 c

1101 d

1110 e

1111 f

0x2ac0x2ac

00100010 10101010 11001100

0x2ac = (001010101100)0x2ac = (001010101100)22

No magic – remember hex digit = 4 bitsNo magic – remember hex digit = 4 bits

(2ac)(2ac)1616

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Binary Binary −− Hexadecimal Conversion Hexadecimal Conversion

16 = 24

Each group of 4 bits represents a hexadecimal digit

Hex Binary0 0 0 0 01 0 0 0 12 0 0 1 03 0 0 1 14 0 1 0 05 0 1 0 16 0 1 1 07 0 1 1 18 1 0 0 09 1 0 0 1A 1 0 1 0B 1 0 1 1C 1 1 0 0D 1 1 0 1E 1 1 1 0F 1 1 1 1

Example:

( 1 0 1 1 0 . 0 1 )2

( 1 6 . 4 )16

Assume Zeros

Works both ways (Binary to Hex & Hex to Binary)

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Binary to HexBinary to Hex

Just convert groups of 4 bits

Bin Hex

0000 0

0001 1

0010 2

0011 3

0100 4

0101 5

0110 6

0111 7

1000 8

1001 9

1010 a

1011 b

1100 c

1101 d

1110 e

1111 f

(101001101111011)(101001101111011)22

10111011

55 33 77 bb

101001101111011 = 0x537b = (537b)101001101111011 = 0x537b = (537b)1616

0101 0101 0111 0111 0011 0011

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Octal Octal −− Hexadecimal Conversion Hexadecimal Conversion

Convert to Binary as an intermediate step

Example:

( 0 1 0 1 1 0 . 0 1 0 )2

( 1 6 . 4 )16

Assume Zeros

Works both ways (Octal to Hex & Hex to Octal)

( 2 6 . 2 )8

Assume Zeros

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AdditionAddition

Decimal Addition

5 5

55+

011

= Ten ≥ Base

Subtract a Base

11 Carry

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Binary AdditionBinary Addition

Column Addition

1 0 1111

1111 0+

0000 1 11

≥ (2)10

111111

= 61

= 23

= 84

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Binary SubtractionBinary Subtraction

Borrow a “Base” when needed

0 0 1110

1111 0−

0101 1 10

= (10)2

2

2

2 2

1

000

1

= 77

= 23

= 54

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Binary MultiplicationBinary Multiplication

Bit by bit

01 1 1 1

01 1 0

00 0 0 0

01 1 1 1

01 1 1 1

0 0 000

0110111 0

x

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Binary Numbers and Binary CodingBinary Numbers and Binary Coding

Flexibility of representation Within constraints below, 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 CodesNon-numeric Binary Codes

Binary Number 000001010011101110111

ColorRedOrangeYellowGreenBlueIndigoViolet

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Given M elements to be represented by a binary code, the minimum number of bits, n, needed, satisfies the following relationships:

2n > M > 2(n – 1)

n = log2 M where x , is called the ceiling function, i.e the integer greater than or equal to x.

Number of Bits RequiredNumber of Bits Required

Given M elements to be represented by a binary code, the minimum number of bits, n, needed, satisfies the following relationships:

Example: How many bits are required to represent decimal digits with a binary code?

• M = 10, hence n = ceiling (log2 10) = ceiling (3.3219) = 4

• Checking:

Number of Bits RequiredNumber of Bits Required

)(loglog

,22

22

1

MceilingMn

whereM nn

8210162 34 46

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Binary CodesBinary Codes

Group of n bits

● Up to 2n combinations

● Each combination represents an element of information

Binary Coded Decimal (BCD)

● Each Decimal Digit is represented by 4 bits

● (0 – 9) Valid combinations

● (10 – 15) Invalid combinations

Decimal BCD0 0 0 0 01 0 0 0 12 0 0 1 03 0 0 1 14 0 1 0 05 0 1 0 16 0 1 1 07 0 1 1 18 1 0 0 09 1 0 0 1

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Gray CodeGray Code

One bit changes fromone code to the nextcode

Different than Binary

Decimal Gray00 000001 000102 001103 001004 011005 011106 010107 010008 110009 110110 111111 111012 101013 101114 100115 1000

Binary0000000100100011010001010110011110001001101010111100110111101111

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Binary RepresentationsBinary Representations

A bit A bit is the most basic unit of information in a computer. It is a state of “on” or “off” in a digital circuit. Sometimes these states are “high” or “low” voltage instead

of “on” or “off..”A group of four bits four bits is called a nibblenibble (or nybble).

Bytes, therefore, consist of two nibbles: a “high-order nibble,” and a “low-order” nibble.

A bytebyte is a group of eight bitseight bits. A byte is the smallest possible addressable (can be found

via its location) unit of computer storage.A wordword is a contiguous group of bytescontiguous group of bytes.

Words can be any number of bits (16, 32, 64 bits are common).

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Conversion or Coding?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. 

(13)10 = (1101)2 (This is conversion) 

(13)BCD (0001|0011)BCD (This is coding)

Advantages/Disadvantages?

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BCD: Advantages/DisadvantagesBCD: Advantages/Disadvantages

Disadvantage: It is obvious that a BCD number needs

more bits than its equivalent binary value

(26)10 = (11010)2

(26)10 = (0010 0110)BCD

Advantages: Computer input/output data are handled by

people who use the decimal system. So it is easier to convert back/forth to BCD.

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Character CodesCharacter Codes

From numbers to letters ASCII

Stands for American Standard Code for Information Interchange

Only 7 bits defined Unicode

ASCII CodeASCII Code

American Standard Code for Information Interchange

Info 7-bit CodeA 1000001B 1000010...

.

.

.Z 1011010

a 1100001b 1100010...

.

.

.z 1111010

@ 1000000? 0111111+ 0101011

5353

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ASCII table

Error Detecting CodesError Detecting Codes

Parity

One bit added to a group of bits to make the total number of ‘1’s (including the parity bit) even or odd

● Even

● Odd

Good for checking single-bit errors

1 0 1 1 1 1 0 0 0 0 0 10

0 0 1 1 1 1 0 0 0 0 0 11

4-bit Example 7-bit Example

5555

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ReadingReading

Read Chapter 1 Make sure you’re comfortable with

material Check the lecture notes from

the Web site. Solve the assignment.

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HomeworkHomework

See Assignment #1 On Web I expect you to know number

systems well and be able to do conversions and arithmetic Decimal – Binary Binary – Decimal Decimal – Hex Hex – Decimal

Will be on test!

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