1. CHAPTER 1 INTRODUCTION TO DIGITAL LOGIC
EKT 121 / 3 ELEKTRONIK DIGIT 1
CHAPTER 1
INTRODUCTION TO DIGITAL LOGIC

2. Number & codes (1) Digital vs. Analog Numbering systems Octal (Base 8)
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

3. 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

4. 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).

5. 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

6. Digital vs. Analog (cont.)1
Systems which process discrete (step by step) values
Systems which are capable of processing a continuous range of values varying with respect to time 2
Digital representation the quantities - digits (0/1)
Analog representation a quantity – I / V / meter movement 3 4
Example: Digital watch, PSP, iPod, Handphone, digital computers and calculators
Example: audio amplifiers, magnetic tape recording and playback equipment

7. 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

8. Digital Techniques Advantages: Limitations: 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

9. Analog Waveform

10. Digital Waveform

11. 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

12. Numbering Systems Decimal Binary Octal Hexadecimal 0 ~ 9 0 ~ 1 0 ~ 7
0 ~ 9, A ~ F

13. Numbering Systems (cont.)
A B C D E F
Binary
Octal
Hex
Dec N U M B E R S Y T

14. Significant Digits
Binary : Most Significant Bit Least Significant Bit (MSB) (LSB) Hexadecimal: 1 D 6 3 A 7 Most Significant Digit Least significant Digit (MSD) (LSD)

15. Decimal numbering system (base 10)
Base 10 system: (0,1,2,3,4,5,6,7,8,9)
Example : 39710
Weights for whole numbers are positive power of ten that increase from right to left , beginning with 100
3 X 102 +
9 X 101 +
7 X 100
=>
=>
39710

16. 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:
Weights in a binary number are based on power of two, that increase from right to right to left, beginning with 20
1X 22 +
0 X 21 +
1 X 20
=>
=>
510

17. Octal Number System (base 8)
Base 8 system: (0,1,………,7)
multiplication and division algorithms for conversion to and from base 10
example : convert to decimal
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
=>
49410
=>
Readily converts to binary
Groups of three (binary) digits can be used to represent each octal number
example : convert to binary

18. Hexadecimal Number System (base 16)
BINARY
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 A 10
1010 B 11
1011 C 12
1100 D 13
1101 E 14
1110 F 15
1111
Base 16 system
Uses digits 0 ~ 9 &
letters A,B,C,D,E,F
Groups of four bits represent each base 16 digit

19. Hexadecimal Number System (2)
Base 16 system
multiplication and division algorithms for conversion to and from base 10
example : A9F16 convert to decimal
A 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
=>
271910
=>
Readily converts to binary
Groups of four (binary) digits can be used to represent each hexadecimal number
example : A9F16 convert to binary
A F

20. Number Conversion
Any Radix (base) to Decimal Conversion

21. Number Conversion (BASE 2 –> 10)
Binary to Decimal Conversion

22. Binary to Decimal Conversion
Convert ( )2 to its decimal equivalent:
Binary
Positional Values x x x x x x x x 27 26 25 24 23 22 21 20
Products
= 17310

23. Octal to Decimal Conversion
Convert 6538 to its decimal equivalent:
Octal Digits x x x
Positional Values
Products
= 42710

24. Hexadecimal to Decimal Conversion
Convert 3B4F16 to its decimal equivalent:
Hex Digits
B F x x x x
Positional Values
Products
= 15,18310

25. Number Conversion INTEGER DIGIT:
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 :
to Binary

26. Decimal to Binary Conversion
Remainder
2 5 = 2
1 2 =
6 =
3 =
MSB LSB
1 = =

27. Decimal to Binary Conversion
MSB
LSB
Carry
x 2 =
x 2 =
x 2 =
0.5 x 2 =
Answer:

28. Decimal to Octal Conversion
Convert to its octal equivalent:
427 / 8 = 53 R3 Divided by 8; R is LSD
53 / 8 = 6 R Divide Q by 8; R is next digit
6 / 8 = 0 R6 Repeat until Q = 0
6538

29. Decimal to Hexadecimal Conversion
Convert to its hexadecimal equivalent: 830 / 16 = 51 R / 16 = 3 R3 3 / 16 = 0 R3
= E in Hex
33E16

30. Decimal to Octal Conversion
Binary to Octal Conversion (vice versa)
Grouping the binary position in groups of three starting at the least significant position.

31. Octal to Binary Conversion
Each octal number converts to 3 binary digits
To convert 6538 to binary, just substitute code:

32. Example : Number Conversion
Convert the following binary numbers to their octal equivalent (vice versa).
47.38
Answer:
11.748

33. Binary to Hexadecimal Conversion
Binary to Hexadecimal Conversion (vice versa)
Grouping the binary position in 4-bit groups, starting from the least significant position.

34. 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

35. Number Conversion
Example:
Convert the following binary numbers to their hexadecimal equivalent (vice versa).
1F.C16
Answer:
10.816

36. Substitution Code (1) = 56AE6A16 5 6 A E 6 A
Convert ( )2 to hex using the 4-bit substitution code :
A E A
= 56AE6A16

37. Substitution Code (2)
Substitution code can also be used to convert binary to octal by using 3-bit groupings: =

38. Binary Addition 0 + 0 = 0 Sum of 0 with a carry of 0
Example:
???

39. Simple Arithmetic Addition Example: 100011002 5816 + 1011102 + 2416
Substraction
101102
Example:
5816
7C16

40. Binary Subtraction 0 - 0 = 0 1 - 1 = 0 1 - 0 = 1
10 -1 = with a borrow of 1
Example:
???

41. Binary Multiplication
0 X 0 = 0
0 X 1 = 0 Example:
1 X 0 =
1 X 1 = X
100110
000000

42. Binary Division
Use the same procedure as decimal division

43. 1’s complements of binary numbers
Changing all the 1s to 0s and all the 0s to 1s
Example:
Binary number
’s complement
****** same as applying NOT gate ******

44. 2’s complements of binary numbers
Step 1: Find 1’s complement of the number
Binary #
1’s complement
Step 2: Add 1 to the 1’s complement

45. Signed Magnitude Numbers …
Sign bit
31 bits for magnitude
0 = positive
1 = negative
***** This is your basic
Integer format

46. Sign numbers Left most is the sign bit Sign-magnitude 1’s complement
0 is for positive, and 1 is for negative
Sign-magnitude
= +25
sign bit magnitude bits
1’s complement
The negative number is the 1’s complement of the corresponding positive number
Example:
+25 is is

47. 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

48. 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.
Convert 10 to binary and BCD.

49. 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.

50. Digital Codes (3) The Gray Code Only 1 bit changes
Decimal
Binary
Gray Code
0000 1
0001 2
0010
0011 3 4
0100
0110 5
0101
0111 6
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 ?????

51. 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

52. Digital Codes (6) **** Assingment#1: due date 10/01/11 ****
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 ****

53. 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 numeric bits (on the right)

54. 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: , ,
Real binary numbers: , ,
Generally, FPNs are expressed in exponential notation. Eg:
can be written as 3 x 104
can be written as x 102
can be written as x 103
mantissa exponent

55. End of Numbers system