1. CS220 : Digital Design

2. Basic Information Title: Digital Design Code: CS220 Lecture: 3
Tutorial: 1
Pre-Requisite: Computer Introduction (CS201)

3. Overall Aims of Course
By the end of the course the students will be able to:
Grasp basic principles of combinational and sequential logic design.
Determine the behavior of a digital logic circuit (analysis) and translate description of logical problems to efficient digital logic circuits (synthesis).
Understanding of how to design a general-purpose computer, starting with simple logic gates.

4. Contents Topics Contact Hours No. of Weeks
-Introduction to the course content, text book(s), reference(s) and course plane.
- Digital Systems and Binary numbers 9 3
- Boolean Algebra and Logic Gates 6 2
- Gate Level Minimization
- Combinational Logic 12 4
- Synchronous Sequential Logic
Total 45 15

5. Proportion of Final Assessment
Assessment schedule
Assessment
Assessment Task
Week Due
Proportion of Final Assessment 1
Major Exam 1
Week 6
15% 2
Major Exam II
Week13 3
Home works
For every chapter
10% 4
Quizzes
Week4,week11 5
project
Week 13 6
Final Examination
End of Semester
40%

6. List of References Essential Books Recommended Books
“DIGITAL DESIGN”, by Mano M. Morris, 4th edition, Prentice- Hall.
Recommended Books
“FUNDAMENTALS OF LOGIC DESIGN”, by Charles H. Roth, Brooks/Cole Thomson Learning.
“INTRODUCTION TO DIGITAL SYSTEMS”, by M.D. ERCEGOVAC, T. Lang, and J.H. Moreno, Wiley and Sons
“DIGITAL DESIGN, PRINCIPLES AND PRACTICES”, by John F.Wakely, Latest Edition, Prentice Hall, Eaglewood Cliffs, NJ.
“FUNDMENTALS OF DIGITAL LOGIC WITH VHDL DESIGN”, by Stephen Brown and Zvonko Vranesic, McGraw Hill.
“INTRODUCTION TO DIGITAL LOGIC DESIGN”, by John Hayes, Addison Wesley, Reading, MA.

7. 1. Digital Systems and Binary Numbers
1.3 Number-Base Conversions
1.4 Octal and Hexadecimal Numbers
1.5 Complements
1.6 Signed Binary Numbers
1.7 Binary Codes

8. 1.1 Digital Systems

9. 1.2 Binary Numbers
In general, a number expressed in a base-r system has coefficients multiplied by powers of r:
r is called base or radix.

11. 1.3 Number-Base Conversions (Integer Part)
Example:

12. 1.3 Number-Base Conversions (Fraction Part)
Example:

13. Binary-to-Decimal Conversion
Example:
Example:

14. 1.4 Octal and Hexadecimal Numbers

15. Decimal-to-Octal Conversion
Example:

16. Decimal-to-Hexadecimal Conversion
Example:

17. Octal-to-Decimal Conversion
Example:
Example:

18. Hexadecimal-to-Decimal Conversion
Example:
Example:

19. Binary–Octal and Octal–Binary Conversions
Example:
Example:

20. Hex–Binary and Binary–Hex Conversions
Example:
Example:

21. Hex–Octal and Octal–Hex Conversions
For Hexadecimal–Octal conversion, the given hex number is firstly converted into its binary equivalent which is further converted into its octal equivalent.
An alternative approach is firstly to convert the given hexadecimal number into its decimal equivalent and then convert the decimal number into an equivalent octal number. The former method is definitely more convenient and straightforward.
For Octal–Hexadecimal conversion, the octal number may first be converted into an equivalent binary number and then the binary number transformed into its hex equivalent.
The other option is firstly to convert the given octal number into its decimal equivalent and then convert the decimal number into its hex equivalent. The former approach is definitely the preferred one.

22. Example

23. Arithmetic Operation Addition augend 101101 Added: + 100111
Sum:

24. Subtraction minuend: 101101 subtrahend: - 100111 -------------
difference:

25. Multiplication

26. 1.5 Complements Diminished Radix Complement ((r-1)‘s complement)
Given a number N in base r having n digits, the (r - 1)’s
Complement of N is defined as (rn- 1) -N.
the 9’s complement of is – 46700=453299
the 1’s complement of is
Note:
The (r-1)’s complement of octal or hexadecimal numbers is obtained by subtracting each digit from 7 or F (decimal 15), respectively

27. Radix Complement
Given a number N in base r having n digit, the r’s complement of Nis defined as (rn -N) for N ≠0 and as 0 for N =0 .
The 10’s complement of is
The 10’s complement of is
The 2’s complement of is

28. Subtraction with Complement
The subtraction of two n-digit unsigned numbers M – N in base r can be done as follows:
M + (rn - N), note that (rn - N) is r’s complement of N.
If M  N, the sum will produce an end carry x, which can be discarded; what is left is the result M- N.
If M < N, the sum does not produce an end carry and is (N - M). Take the r’x complement of the sum and place a negative sign in front.

29. Example: Using 10’s complement subtract 72532 – 3250 M = 72532
10’s complement of N = 96750
sum =
Discarded end carry 105 =
answer: 69282

30. Example: Using 10’s complement subtract 3250 - 72532 M = 03250
10’s complement of N =
sum =
Discarded end carry 105 =
answer: -( ) =
The answer is –(10’s complement of 30718) =

31. Example Using 2’s complement subtract (a) 1010100 – 1000011
N = , 2’s complement of N = 
sum = Discarded end carry 27=
answer:

32. Example
(b) –
M =
N = , 2’s complement of N = 
sum =
answer: - ( ) =
The answer is –(2’s complement of ) =

33. Example
Using 1’s complement, subtract
M =
N = , 1’s complement of N = 
end-around carry =
answer: 33

34. Example
Using 1’s complement, subtract
M =
N = , 1’s complement of N = 
Answer: 34

35. 1.6 Signed Binary Numbers

36. 1.6 Signed Binary Numbers Example : represent +76 Notes
1 - Sign and Magnitude representation
2 - 1’s Complement Representation
3 - 2’s Complement Representation
Notes
1 - The previous representation are the same for positive numbers and different for negative numbers
2 - For a signed binary number the most significant bit is used for representing the sign of the number
We use 0 for positive numbers and 1 for negative numbers
Example : represent +76

37. Representing negative numbers in the previous three systems
1’s Complement of a negative number can be obtained by flipping all bits of the positive binary number
2’s Complement of a negative number can be obtained by adding 1 to the 1’s Complement or by flipping bits of the positive binary number after the first one from the right
Example : represent -76

38. Arithmetic Addition with Comparison
The addition of two numbers in the signed mgnitude system follow the rules of ordinary arithmetic.
If the signed are the same, we add the two magnitudes and give the sum the common sign.
If the signed are different, we subtract the smaller magnitude
from the larger and give the difference the sign of the larger
magnitude. EX. (+25) + (-38) = -( ) = -13

39. Arithmetic Addition without Comparison
The addition of two signed binary number with negative numbers represented in signed 2’s complement form is obtained from the addition of the two numbers, including their signed bits. A carry out of the signed bit position is discarded (note that the 4th case).

40. Arithmetic Addition without Comparison                        40

41. Arithmetic Subtraction
(+/-) A – (+B)= (+/-) A + (-B)
(+/-) A – (-B)= (+/-) A + (+B)
Example
(-6) – (-13)= +7
In binary: ( – )= ( ) =
after removing the carry out the result will be :

42. 1.7 Binary Codes

43. Binary Coded Decimal (BCD)

44. Binary Coded Decimal (BCD)
in this system each digit is represented in 4 bits
For example : to represent in BCD

45. BCD Addition Example : Evaluate the following operations in BCD System
1 – 3 + 4
2 – 4 + 8

46. We must add 6 (0110) to the result
BCD Addition
Example : Evaluate the following operations in BCD System
1 – 3 + 4
2 – 4 + 8
Error
We must add 6 (0110) to the result

47. BCD Addition Example : Evaluate the following operations in BCD System
1 – 3 + 4
2 – 4 + 8

48. In previous Example we added 0110 when the result was
Notes
1 – In BCD Addition , we add (0110)=(6) if the result value was greater than (1001)=(9) or if the result was more than 4 digits
In previous Example we added 0110 when the result was
1 - greater than 9 (1001)
2 - more than 4 digits (10000)
Note : result more than 4 digit is greater than 9(1001) 

49. Decimal Arithmetic Addition for signed numbers
Example: (+375) + (- 240) = in BCD
Apply 10‘s complement to the negative number only.
Addition is done by summing all digits,including the sign digit,and discarding the end carry

50. Decimal Arithmetic Subtraction for signed and unsigned numbers
Apply 10‘s complement to the subtrahend and apply addition (same as binary case)

51. For example : to represent in ex-3
Excess-3 (ex-3)
Excess-three (ex-3)is another system to represent a number
(ex-3) is like (BCD) in the way of representing number
i.e. each digit is represented in 4 bits
Except that : each digit is firstly incremented by three
For example : to represent in ex-3

53. Gray Code

54. ASCII character code
ASCII : American Standard Code for Information Interchange
ASCII code is used to represent characters , Symbols , …
ASCII code consists of 7-bits (to represent 128 character) #
ASCII Ch 65 A 66 B 90 Z 97 a 98 b
122 z
Upper case Letters are represented by ASCII (65 : 90)
Lower case Letters are represented by ASCII (97 : 122)

56. Error Detecting Code with even parity with odd parity
ASCII A
ASCII T

58. For more information about Number Systems and Conversations between them
Check these
1 – Our Logic Book
2 - Computer Organization's Lectures
3 – Any other References