ENG2410 Digital Design “Introduction to Digital...

ENG2410Digital Design

“Introduction to Digital Systems”

Fall 2017S. Areibi

School of EngineeringUniversity of Guelph

2

Resources

� Chapter #1, Mano Sections� 1.1 Digital Computers

� 1.2 Number Systems

� 1.3 Arithmetic Operations

� 1.4 Decimal Codes

� 1.5 Alphanumeric Codes

3

Topics

� 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

5

PDA

Body

Entertainment

Household

Communication

Home Networking

Car

Medicine

PC

Super Computer

Computing Devices Everywhere!“Embedded Systems”

Game console

What is the main enabler to such digital systems?

6

The Transistor Revolution

First transistorBell Labs, 1948

Bipolar logic1960’s

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

n+n+S

GD

+

DEVICE

CIRCUIT

GATE

MODULE

SYSTEMSpecification

Functional design

Circuit design

Physical design

Test/Fabrication

Logic design

The VLSI Design Cycle

© K

LMH

Lien

ig8

1.3 VLSI Design Styles Contact

Vdd

GND

OUT

IN2

IN1OUT

IN2

IN1

OUTIN1

Vdd

GND

IN2

Power (Vdd)-Rail

Ground (GND)-Rail

Diffusion layer

p-typetransistor

n-typetransistor

Metal layer

Poly layer

10

Signals

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

11

Signals

Signals can be analogor 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.

12

Analog Signals

Analog

Time

Continuous in

value & time

13

Digital Signals

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

Time

DigitalDiscrete in

value

14

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

15

What are other physical quantities represent 0 and 1?

Binary Values: Other Physical Quantities

CPU: VoltageHard Drive: Magnetic Field Direction

CD: Surface Pits/Light

Dynamic Ram: Electric Charge

16

A Digital Computer Example

Inputs: Keyboard, mouse, modem, microphone

Outputs: CRT, LCD, modem, speakers

Memory

Controlunit Datapath

Input/Output

CPU

Data/Instructions/codeAll in 01010010010

clock

18

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

19

Decimal Number System

�Base (also called radix) = 10

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

�Digit Position

● Integer & fraction

�Formal Notation

�Digit Weight

● Weight = (Base) Position

�Magnitude

● Sum of “Digit x Weight”

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

20

Octal Number System

�Base = 8

● 8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }

�Weights


�Formal Notation

�Magnitude


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

22

Hexadecimal Number System

�Base = 16

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

�Weights


�Formal Notation

�Magnitude


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

23

Hex to Decimal

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

��

163 162 161 160

4096 256 16 1

(2ac)16

2 • 256 + 10 • 16 + 12 • 1= (684)10

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

25

Binary Number System

�Base = 2

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

�Weights


�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

26

Binary → 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)10

What is 10011100 in decimal?

7 6 5 4 3 2 1 0 position

Binary #

position

value

27

Binary Numbers� Examples:

� (00)2 � (0)10

� (01)2 � (1)10

� (0000001)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 1

� n bits can store numbers from 0 to 2n-1

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Binary Fractions

bi bi-1 . . . b2 b1 b0 b-1 b-2 b-3 … b-j

1

2

4

2i-1

2i

2-j

1/8

1/4

1/2

Integer Values

Fractional Values

decimal number =

29

Example 1

1 1 0 . 1 12

1 x 22 + 1 x 21 + 0x20 + 1x2-1 + 1x2-2

6 and 3/4

30

Example 2

0 . 1 1 1 1 1 12

63/64

0 x 20 + 1 x 2-1 + 1x2-2 + 1x2-3 + 1x2-4 + 1x2-5 + 1x2-6

Note: (1) Numbers of the form 0.11111…2 are just below 1.0(2) Short form notation for such numbers is 1.0 - ε

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).31

32

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

34

Number Base Conversions

Decimal(Base 10)

Octal(Base 8)

Binary(Base 2)

Hexadecimal(Base 16)

Evaluate Magnitude

Evaluate Magnitude

Evaluate Magnitude

35

Conversion 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

Decimal to Binary Conversion

37

Decimal (Integer) 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 LSB

13/ 2 = 6 1 a0 = 16 / 2 = 3 0 a1 = 03 / 2 = 1 1 a2 = 11 / 2 = 0 1 a3 = 1

38

Decimal (Fraction) 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 LSB

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

a-1 = 1

39

Decimal to Octal Conversion

Example: (175)10

Quotient Remainder Coefficient

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

175 / 8 = 21 7 a0 = 721 / 8 = 2 5 a1 = 52 / 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

40

Decimal 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)10

684/16 = 42 rem 12=c c

42/16 = 2 rem 10=a ac

2/16 = 0 rem 2 2ac

41

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 2

•Size of byte

Why usebase 16?

� Strings of 0’s and 1’s too hard to write

� Use base-16 or hexadecimal – 4 bits

42

Hex 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

0x2ac

0010 1010 1100

0x2ac = (001010101100)2

No magic – remember hex digit = 4 bits

(2ac)16

43

Binary − 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

Pad with Zeros

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

44

Binary 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)2

1011

5 3 7 b

(101001101111011)2 = 0x537b = (537b)16

0101 0111 0011

45

Octal − 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

47

Addition

�Decimal Addition

5 5

55+

011

= Ten ≥ Base

� Subtract a Base

11 Carry

48

Binary Addition�Adding bits:

♦ 0 + 0 = 0

♦ 0 + 1 = 1

♦ 1 + 0 = 1

♦ 1 + 1 = (1) 0

♦ 1 + 1 + 1 = (1) 1

�Adding integers:

carry

0 0 0 . . . . . . 0 1 1 (1)2 = (7)10

+ 0 0 0 . . . . . . 0 1 1 (0)2 = (6)10

= = (13)10

carry

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

0

0 0 0 . . . . . 1

11

49

Binary Addition

�Column Addition

1 0 1111

1111 0+

0000 1 11

≥ (2)10

111111

= (61)10

= (23)10

= (84)10

51

Binary 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

52

� 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

53

� 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 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 Required

)(loglog

,22

22

1

MceilingMn

whereM nn

==>≥ −

8210162 34 =>≥=54

55

Binary Codes

�Group of n bits

● Up to 2n combinations

● Each combinationrepresents an elementof 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

56

Gray 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

57

Binary Representations

� 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 is called a nibble (or nybble).

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

� A byte is a group of eight bits.

� A byte is the smallest possible addressable (can be found via its location) unit of computer storage.

� A word is a contiguous group of bytes.

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

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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?

59

BCD: 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.

61

Character Codes

� From numbers to letters

� ASCII

� Stands for American Standard Code for Information Interchange

� Only 7 bits defined

� Unicode

ASCII Code

American Standard Code for Information Interchange

Info 7-bit CodeA 1000001B 1000010...

.

.

.Z 1011010

a 1100001b 1100010...

.

.

.z 1111010

@ 1000000? 0111111+ 0101011

62

63

ASCII table

64

Reading

� Read Chapter 1

� Make sure you’re comfortable with material

� Check the lecture notes from the

Web site.

� Solve the assignment.

65

Homework

� 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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