CEC 220 Digital Circuit DesignNumber Systems & Conversions

Friday, January 9 CEC 220 Digital Circuit Design Slide 1 of 16

Number Systems & Conversions

Friday, January 9 CEC 220 Digital Circuit Design

• Number Systems and Conversions Digital Systems and Switching Circuits Number Systems and Conversion Binary, Hexadecimal, and Octal representations Examples

Slide 2 of 16

Number Systems & ConversionsDigital Systems and Switching Circuits

Friday, January 9 CEC 220 Digital Circuit Design

• What is the difference between analog and digital systems/signals?

• Analog – Continuous Natural Phenomena (Pressure, Temperature, Speed…)

Difficult to realize processing using electronics

• Digital – Discrete Binary Digit Signal Processing as Bit unit

Easy to realize processing using electronics

High performance due to Integrated Circuit Technology

Slide 3 of 16

Number Systems & ConversionsDigital Systems and Switching Circuits

Friday, January 9 CEC 220 Digital Circuit Design

• What is a binary signal?

• Binary = Two values (0, 1)• Each digit is referred to as a “bit”

• Number representation with only two values (0, 1)

• Can be implemented with simple electronics devices

• For Example• Voltage: High = (1) and Low = (0)• Switch: On = (1) and Off = (0)

Slide 4 of 16

Number Systems & ConversionsDigital Systems and Switching Circuits

Friday, January 9 CEC 220 Digital Circuit Design

• What is a switching circuit?

• Combinational Circuit:• Outputs depend on only present inputs, not on past inputs

• Sequential Circuit:• Outputs depend on both present inputs and past inputs

• Sequential circuits have “memory” !!!!!

Slide 5 of 16

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

4 3 2 1 0 1 2 3( . )RN a a a a a a a a

2 1 0 1 29 10 5 10 3 10 7 10 8 10

21011.11

Decimal:(base 10)

Binary:(base 2)

Base “R”

10953.78

3 2 1 0 1 21 2 0 2 1 2 1 2 1 2 1 2 1 1

8 0 2 12 4

10

311 11.75

4

4 3 2 1 04 3 2 1 0

1 2 31 2 3

a R a R a R a R a R

a R a R a R

Slide 6 of 16

9 100 5 10 3 1 7 /10 8 /100 900 50 3 7 /10 8 /100

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

• Examples Base 8 to Decimal

Base 16 to Decimal

8147.3

10103.375

2 1 0 11 8 4 8 7 8 3 8 3

64 32 78

162A F 2 1 010 16 2 16 15 16

102607

2560 32 15

Slide 7 of 16

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

• Conversion of Decimal to base R

1 2 1 01 2 1 0 1 2 1 0

n nn n n nR

N a a a a a a R a R a R a R a R

1 2 11 2 1 1

n nn n

Na R a R a R a Q

R

2 3 111 3 2 2

n nn n

Qa R a R a R a Q

R

3 421 3 3

n nn n

Qa R a R a Q

R

.

.

.

0remainder a

1remainder a

2remainder a

Quotient

Slide 8 of 16

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

• Example: Convert Decimal 53 (i.e. base 10) to base 2 (i.e., binary)

532

262

132

62

32

12

rem. = 1 = a0

rem. = 0 = a1

rem. = 1 = a2

rem. = 0 = a3

rem. = 1 = a4

0 rem. = 1 = a5

210 11010153

Slide 9 of 16

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

• Conversion of a Decimal fraction to base R

1 2 30 m RF a a a a

1112

31

21 FaRaRaRaaFR mm

2221

321 FaRaRaaRF mm

333

32 FaRaaRF mm

.

.

.

1 2 31 2 3

mma R a R a R a R

Slide 10 of 16

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

• Example: Convert Decimal 0.625 to base 2 (i.e., binary)

)1(

250.1

2

625.

1

a

F

)0(

500.0

2

250.

2

1

a

F

)1(

000.1

2

500.

3

2

a

F

10 20.625 0.101

Slide 11 of 16

10 20.625 0.???

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

• Example: Convert 231.34 to base 7 First convert to decimal

Convert the integer portion (4510 = ???7)

Convert the decimal portion ( .7510 = .???7)

104 75.454

31431623.231

457

67

0 rem. = 6

rem. = 3

.75

7

(5).25

.75

7

(5).25

.25

7

(1).75

.25

7

(1).75

4 10 7231.3 45.75 63.5151

Slide 12 of 16

…

763

70.5151

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

Binary

000

001

010

011

100

101

110

111

QUESTION:How many binary “bits” do we need to represent a single Octal digit?

Slide 13 of 16

Octal

0

1

2

3

4

5

6

7

Binary Hex Decimal

0000 0 0

0001 1 1

0010 2 2

0011 3 3

0100 4 4

0101 5 5

0110 6 6

0111 7 7

1000 8 8

1001 9 9

1010 A 10

1011 B 11

1100 C 12

1101 D 13

1110 E 14

1111 F 15

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

Binary Hex Decimal

0000 0 0

0001 1 1

0010 2 2

0011 3 3

0100 4 4

0101 5 5

0110 6 6

0111 7 7

1000 8 8

1001 9 9

1010 A 10

1011 B 11

1100 C 12

1101 D 13

1110 E 14

1111 F 15

QUESTION:How many binary “bits” do we need to represent a single Hexadecimal digit?

Slide 14 of 16

C

Number Systems & ConversionsNumber Systems and Conversion

Friday, January 9 CEC 220 Digital Circuit Design

• Converting to/from Binary, Hex, and Octal An example of converting Binary to Octal

An example of converting Binary to Hexadecimal

Slide 15 of 16

511 2 7

1001101.0101112 = 115.278

D4 5

1001101.0101112 = 4D.5C16

Next Lesson

Friday, January 9 CEC 220 Digital Circuit Design

• Binary Arithmetic• Representation of Negative Numbers

Slide 16 of 16