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PGT 104 DIGITAL ELECTRONIC. CHAPTER 1 INTRODUCTION TO DIGITAL LOGIC. Number & codes (1). Digital vs. Analog Numbering systems Decimal (Base 10) Binary (Base 2) Hexadecimal ( Base 16) Octal ( Base 8) Number conversion Binary arithmetic

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pgt 104 digital electronic

PGT 104DIGITAL ELECTRONIC

CHAPTER 1

INTRODUCTION TO DIGITAL LOGIC

number codes 1
Number & codes (1)
  • Digital vs. Analog
  • Numbering systems
    • Decimal (Base 10)
    • Binary (Base 2)
    • Hexadecimal (Base 16)
    • Octal (Base 8)
  • Number conversion
  • Binary arithmetic
  • 1’s and 2’s complements of binary numbers
slide3

Number & codes (2)

  • Signed/Unsigned numbers
  • Arithmetic operations with signed numbers
  • Coded
    • Binary-Coded-Decimal (BCD)/ 8421
    • ASCII
    • Gray
    • Excess-3)
  • Error Detecting and Correction Codes
  • Floating Point Numbers
digital vs analog
Digital vs. Analog
  • Two ways of representing the numerical values of quantities :

i) Analog (continuous)

ii) Digital (discrete)

  • Analog : a quantity represented by voltage, current or meter movement that is proportional to the value that quantity.
  • Digital : the quantities are represented not by proportional quantities but by symbols called digits (0/1).
digital vs analog cont
Digital vs. Analog (cont.)
  • Digital system:
    • combination of devices designed to manipulate logical information or physical quantities that are represented in digital forms
  • Analog system:
    • contains devices manipulate physical quantities that are represented in analog forms
digital vs analog cont2
Digital vs. Analog (cont.)
  • Why digital ?
  • Problem with all signals – noise
    • Noise isn\'t just something that you can hear - the fuzz that appears on old video recordings also qualifies as noise. In general, noise is any unwanted change to a signal that tends to corrupt it.
    • Digital and analogue signals with added noise:

Digital : easily be recognized even

among all that noise : either 0 or 1

Analog : never get back a perfect copy of the original signal

digital techniques
Digital Techniques
  • Advantages:
    • Easier to design
    • Information storage is easy
    • Accuracy and precision are greater
    • Operation can be programmed - simple
    • Digital circuits less affected by noise
    • More digital circuitry can be fabricated on IC chips
  • Limitations:
    • In real world there are analog in nature and these quantities are often I/O that are being monitored, operated on, and controlled by a system. Thus, conversion and re-conversion in needed
introduction to numbering systems
Introduction to Numbering Systems
  • We are familiar with decimal number systems

for daily used such as calculator, calendar,

phone or any common devices use this

numbering system :

Decimal = Base 10

  • Some other number systems:
    • Binary = Base 2
    • Octal = Base 8
    • Hexadecimal = Base 16
numbering systems
Numbering Systems
  • Decimal
  • Binary
  • Octal
  • Hexadecimal
  • 0 ~ 9
  • 0 ~ 1
  • 0 ~ 7
  • 0 ~ 9, A ~ F
numbering systems cont

Dec

Hex

Octal

Binary

000000010010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

0123456789ABCDEF

0123456789101112131415

00010203040506071011121314151617

Numbering Systems (cont.)

N

U

M

B

E

R

S

Y

S

T

E

M

S

significant digits
Significant Digits

Binary : 1 0 1 1 0 1

Most Significant Bit Least Significant Bit

(MSB) (LSB)

Hexadecimal: 1 D 6 3 A 7

Most Significant Digit Least significant Digit

(MSD) (LSD)

decimal numbering system base 10
Decimal numbering system (base 10)

Weights for whole numbers are positive power of ten that increase from right to left , beginning with 100

  • Base 10 system: (0,1,2,3,4,5,6,7,8,9)
  • Example : 39710

3 9 7

+

7 X 100

3 X 102

9 X 101

+

=>

300 + 90 + 7

39710

=>

binary number system base 2
Binary Number System (base 2)
  • Base 2 system: (0 , 1)
    • used to model the series of computer electrical signals represent the informations.
    • 0 represents the no voltage or an ‘off’ state
    • 1 represents the presence of voltage or an ‘on’ state
  • Example: 1012

1 0 1

Weights in a binary number are based on power of two, that increase from right to right to left, beginning with 20

+

0 X 21

+

1 X 20

1X 22

=>

4 + 0 + 1

=>

510

octal number system base 8
Octal Number System (base 8)
  • Base 8 system: (0,1,………,7)
    • multiplication and division algorithms for conversion to and from base 10
    • example: 7568 convert to decimal

7 5 6

Weights in a binary number are based on power of eight that increase from right to right to left, beginning with 80

+

7X 82

5 X 81

+

6 X 80

=>

448 + 40 + 6

49410

=>

  • Readily converts to binary
    • Groups of three (binary) digits can be used to represent each octal number
    • example : 7568 convert to binary
  • 7 5 6

1111011102

hexadecimal number system base 16
Hexadecimal Number System (base 16)
  • Base 16 system
    • Uses digits 0 ~ 9 &

letters A,B,C,D,E,F

    • Groups of four bits represent each base 16 digit
hexadecimal number system 2
Hexadecimal Number System (2)
  • Base 16 system
    • multiplication and division algorithms for conversion to and from base 10
    • example : A9F16 convert to decimal

A 9 F

Weights in a hexadecimal number are based on power of sixteen that increase from right to right to left,beginning with 160

10X 162

9 X 161

+

15 X 160

+

=>

2560 + 144 + 15

271910

=>

  • Readily converts to binary
    • Groups of four (binary) digits can be used to represent each hexadecimal number
    • example : A9F16 convert to binary
  • A 9 F

1010100111112

number conversion
Number Conversion
  • Any Radix (base) to Decimal Conversion
number conversion base 2 10
Number Conversion (BASE 2 –> 10)
  • Binary to Decimal Conversion
binary to decimal conversion
Binary to Decimal Conversion
  • Convert (10101101)2 to its decimal equivalent:

Binary 1 0 1 0 1 1 0 1

Positional Values

x

x

x

x

x

x

x

x

27

26

25

24

23

22

21

20

Products

128 + 0 + 32 + 0 + 8 + 4 + 0 + 1

= 17310

octal to decimal conversion
Octal to Decimal Conversion
  • Convert 6538 to its decimal equivalent:

Octal Digits

6 5 3

x

x

x

Positional Values

82 81 80

Products

384 + 40 + 3

= 42710

hexadecimal to decimal conversion
Hexadecimal to Decimal Conversion
  • Convert 3B4F16 to its decimal equivalent:

Hex Digits

3 B 4 F

x

x

x

x

Positional Values

163 162 161 160

12288 + 2816 +64 +15

Products

= 15,18310

number conversion1
Number Conversion
  • Decimal to Any Radix (Base) Conversion
      • INTEGER DIGIT:

Repeated division by the radix & record the remainder

      • FRACTIONAL DECIMAL:

Multiply the number by the radix until the answer is in integer

  • example:

25.3125 to Binary

decimal to binary conversion
Decimal to Binary Conversion

Remainder

2 5 = 12 + 1

2

1 2 = 6 + 0

2

6 = 3 + 0

2

3 = 1 + 1

2

MSB LSB

1 = 0 + 1

2 2510 = 1 1 0 0 1 2

decimal to binary conversion1
Decimal to Binary Conversion

MSB

LSB

Carry . 0 1 0 1

0.3125 x 2 = 0.625 0

0.625 x 2 = 1.25 1

0.25 x 2 = 0.50 0

0.5 x 2 = 1.00 1

Answer: 1 1 0 0 1.0 1 0 1

decimal to octal conversion
Decimal to Octal Conversion

Convert 42710 to its octal equivalent:

427 / 8 = 53 R3 Divided by 8; R is LSD

53 / 8 = 6 R5 Divide Q by 8; R is next digit

6 / 8 = 0 R6 Repeat until Q = 0

6538

decimal to hexadecimal conversion
Decimal to Hexadecimal Conversion

Convert 83010 to its hexadecimal equivalent:

830 / 16 = 51 R 14

51 / 16 = 3 R3

3 / 16 = 0 R3

= E in Hex

33E16

decimal to octal conversion1
Decimal to Octal Conversion
  • Binary to Octal Conversion (vice versa)
      • Grouping the binary position in groups of three starting at the least significant position.
octal to binary conversion
Octal to Binary Conversion
  • Each octal number converts to 3 binary digits

To convert 6538 to binary, just substitute code:

6 5 3

110 101 011

example number conversion
Example : Number Conversion
  • Convert the following binary numbers to their octal equivalent (vice versa).
    • 1001.11112
    • 47.38
    • 1010011.110112

Answer:

      • 11.748
      • 100111.0112
      • 123.668
binary to hexadecimal conversion
Binary to Hexadecimal Conversion
  • Binary to Hexadecimal Conversion (vice versa)
      • Grouping the binary position in 4-bit groups, starting from the least significant position.
binary to hexadecimal conversion1
Binary to Hexadecimal Conversion
  • The easiest method for converting binary to hexadecimal is using a substitution code
  • Each hex number converts to 4 binary digits
number conversion2
Number Conversion
  • Example:
    • Convert the following binary numbers to their hexadecimal equivalent (vice versa).
      • 10000.12
      • 1F.C16

Answer:

        • 10.816
        • 00011111.11002
substitution code 1
Substitution Code (1)

Convert (010101101010111001101010)2 to hex using the 4-bit substitution code :

0101 0110 1010 1110 0110 1010

5 6 A E 6 A

= 56AE6A16

substitution code 2
Substitution Code (2)

Substitution code can also be used to convert binary to octal by using 3-bit groupings:

010 101 101 010 111 001 101 010

2 5 5 2 7 1 5 2

= 255271528

binary addition
Binary Addition

0 + 0 = 0 Sum of 0 with a carry of 0

0 + 1 = 1 Sum of 1 with a carry of 0

1 + 0 = 1 Sum of 1 with a carry of 0

1 + 1 = 10 Sum of 0 with a carry of 1

Example:

11001 111

+ 1101 + 11

100110 ???

simple arithmetic
Simple Arithmetic

Example:

5816

+ 2416

7C16

  • Addition

Example:

100011002

+ 1011102

101 1 10102

  • Substraction

Example:

10001002

- 1011102

101102

binary subtraction
Binary Subtraction

0 - 0 = 0

1 - 1 = 0

1 - 0 = 1

10 -1 = 1 0 -1 with a borrow of 1

Example:

1011 101

- 111 - 11

100 ???

binary multiplication
Binary Multiplication

0 X 0 = 0

0 X 1 = 0 Example:

1 X 0 = 0 100110

1 X 1 = 1 X 101

100110

000000

+ 100110

10111110

binary division
Binary Division
  • Use the same procedure as decimal division
1 s complements of binary numbers
1’s complements of binary numbers
  • Changing all the 1s to 0s and all the 0s to 1s

Example:

1 1 0 1 0 0 1 0 1 Binary number

0 0 1 0 1 1 0 1 0 1’s complement

****** same as applying NOT gate ******

2 s complements of binary numbers
2’s complements of binary numbers
  • 2’s complement
    • Step 1: Find 1’s complement of the number

Binary # 11000110

1’s complement 00111001

    • Step 2: Add 1 to the 1’s complement

00111001

+ 1

00111010

signed magnitude numbers
Signed Magnitude Numbers

110010..

…00101110010101

Sign bit

31 bits for magnitude

0 = positive

1 = negative

***** This is your basic

Integer format

sign numbers
Sign numbers
  • Left most is the sign bit
    • 0 is for positive, and 1 is for negative
  • Sign-magnitude

0 0 0 1 1 0 0 1 = +25

sign bit magnitude bits

  • 1’s complement
    • The negative number is the 1’s complement of the corresponding positive number
    • Example:

+25 is 00011001 -25 is 11100110

sign numbers1
Sign numbers
  • 2’s complement
    • The positive number – same as sign magnitude and 1’s complement
    • The negative number is the 2’s complement of the corresponding positive number.

Example:

Express +19 and -19 in

i. sign magnitude

ii. 1’s complement

iii. 2’s complement

digital codes 1
Digital Codes (1)
  • BCD (Binary Coded Decimal) / 8421 Code
      • Represent each of the 10 decimal digits (0~9) as a 4-bit binary code.

Example:

    • Convert 15 to BCD.

1 5

0001 0101

    • Convert 10 to binary and BCD.
digital codes 2
Digital Codes (2)
  • ASCII (American Standard Code for Information Interchange) Code
    • Used to translate from the keyboard characters to computer language
    • A world standard alphanumeric code for microcomputers and computers
    • A 7-bit code representing 27 (128) diff. characters (26 upper case, 26 lower case, 10 numbers, 33 special characters/symbol, 33 ctrl characters
    • 8-bit version ASCII (USACC-II 8 or ASCII-8) represent max. of 256 characters.
digital codes 3
Digital Codes (3)
  • The Gray Code
    • Only 1 bit changes
    • Can’t be used in arithmetic circuits
  • Can convert from Binary to Gray Code and vice versa.
  • How to convert ?????
digital codes 4
Digital Codes (4)
  • Excess-3 Code
      • Used to express decimal numbers.
      • The code derives its name from the fact that each binary code is the corresponding 8421 code plus 3
digital codes 6
Digital Codes (6)
  • Error Detecting and Correction Code
      • Required for reliable transmission and storage of digital data.
        • Error Detecting Codes
            • Parity (Even and Odd)
            • Check sums
        • Error Correcting Codes
            • Hamming Code ????

**** Assingment#1: due date 10/01/11 ****

digital codes 7
Digital Codes (7)
  • EBCDIC (Extended Binary Coded Decimal Interchange) Code
      • Mainly used with large computer systems like mainframe.
      • An 8-bit code and accommodates up to 256 characters
      • Divided into 2 portions:

4 zone bits (on the left) and 4 numeric bits (on the right)

floating point numbers fpn
Floating Point Numbers (FPN)
  • A real number or FPN is a number which has both an integer and a fractional part.
  • Examples:
    • Real decimal numbers: 123.45, 0.1234, -0.12345
    • Real binary numbers: 1100.1100, 0.1001, -1.001
  • Generally, FPNs are expressed in exponential notation. Eg:
      • 30000.0 can be written as 3 x 104
      • 312.45 can be written as 3.1245 x 102
      • 1010.001 can be written as 1.010001 x 103

mantissa exponent

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