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• Digital Circuit Design

Digital Systems and Binary Numbers

Lan-Da Van (范倫達), Ph. D.

Department of Computer Science

National Chiao Tung University Taiwan, R.O.C.

Spring, 2017

ldvan@cs.nctu.edu.tw

http://www.cs.nctu.edu.tw/~ldvan/

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-2

Digital Systems

Binary Numbers

Number-Base Conversion

Signed Binary Numbers

Binary Codes

Binary Storage and Registers

Binary Logic

Outline

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-3

Digital age and information age

Digital computers

– general purposes

– many scientific, industrial and commercial applications

Digital systems

– smartphone

– digital camera

– electronic calculators, PDA's

– digital TV

Discrete information-processing systems

– manipulate discrete elements of information

Digital System

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-4

A Digital Computer Example

Synchronous or

Asynchronous?

Inputs: keyboard,

mouse, modem,

microphone,

posture

Outputs: LCD,

modem, speakers

Memory

Control unit

Datapath

Input/Output

CPU

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-5

Signal

An information variable represented by physical quantity

For digital systems, the variable takes on digital values

 Two level, or binary values are the most prevalent values

Binary values are represented abstractly by:

 digits 0 and 1

 words (symbols) False (F) and True (T)

 words (symbols) Low (L) and High (H)

 words On and Off.

Binary values are represented by values or ranges of

values of physical quantities

• Lecture 1

Digital Circuit Design

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Signal

Analog signal

 t->y: y=f(t), y:C, n:C

Discrete-time signal

 n->y: y=f(nT), y:C, n:Z

Digital signal

 n->y: y=D{f(nT)}, y:Z,n:Z

)3(

)1(

)2(

)1(

2)1110(

2)1000( 2)1011( 2)1000(

t n n

y y y

Analog Signal Discrete-Time Signal Digital Signal

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-7

Binary Numbers

Decimal number

5 4 3 2 1 0 1 2 3

5 4 3 2 1 0 1 2 310 10 10 10 10 10 10 10 10a a a a a a a a a   

           

… a5a4a3a2a1a0.a1a2a3…

Decimal point

3 2 1 07,329 7 10 3 10 2 10 9 10       

Example:

ja

Power

General form of base-r system 1 2 1 1 2

1 2 1 0 1 2

n n m

n n ma r a r a r a r a a r a r a r    

                  

Coefficient: aj = 0 to r  1

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-8

Example: Base-2 number

Binary Numbers

2 10

4 3 2 1 0 1 2

(11010.11) (26.75)

1 2 1 2 0 2 1 2 0 2 1 2 1 2 

             

Example: Base-5 number

5

3 2 1 0 1

10

(4021.2)

4 5 0 5 2 5 1 5 2 5 (511.5)          

Example: Base-8 number

Example: Base-16 number

3 2 1 0

16 10(B65F) 11 16 6 16 5 16 15 16 (46,687)        

4

10 1012

8

)5.87(84878281

)4.127(

 

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-9

Binary Numbers

Example: Base-2 number

2 10(110101) 32 16 4 1 (53)    

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"

 240 (1.0995116e+12) is Tera, denoted “T"

 250 (1.1258999e+15) is Peta, denoted “P"

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-10

Arithmetic operations with numbers in base r follow the same rules as decimal numbers.

Binary Numbers

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-11

Binary Arithmetic

Augend: 101101

Sum: 1010100

Subtraction

Minuend: 101101

Subtrahend: 100111

Difference: 000110

The binary multiplication table is simple:

0  0 = 0 | 1  0 = 0 | 0  1 = 0 | 1  1 = 1

Extending multiplication to multiple digits:

Multiplicand 1011

Multiplier  101

Partial Products 1011

0000 -

1011 - -

Product 110111

Multiplication

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-12

 Numbers with different bases: Table 1.2.

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-13

Example1.1

Convert decimal 41 to binary. The process is continued until the integer quotient

becomes 0.

Number-Base Conversions

10/2

5/2

2/2

1/2

5

2

1

0

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-14

 The arithmetic process can be manipulated more conveniently as follows:

Number-Base Conversions

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-15

Example 1.2

Convert decimal 153 to octal. The required base r is 8.

Number-Base Conversions

• Lecture 1

Digital Circuit Design

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Example1.3

 To convert a decimal fraction to a number expressed in base r, a similar

procedure is used. However, multiplication is by r instead of 2, and the

coefficients found from the integers may range in value from 0 to r  1

Convert (0.6875)10 to binary.

The process is continued until the fraction becomes 0 or until the number of digits has

sufficient accuracy.

Number-Base Conversions

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-17

Example1.4 Convert (0.513)10 to octal.

 From Examples 1.1 and 1.3: (41.6875)10 = (101001.1011)2

 From Examples 1.2 and 1.4: (153.513)10 = (231.406517)8

Number-Base Conversions

• Lecture 1

Digital Circuit Design

Lan-Da Van DCD-01-18

 Conversion from binary to octal can be done by positioning the binary number into

groups of three digits each, starting from the binary point and proceeding to the left

and to the right.

 Conversion from binary to hexadecimal i