CS220 : Digital Design

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CS220 Digital Design
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Basic Information

• Title Digital Design
• Code CS220
• Lecture 3
• Tutorial 1
• Pre-Requisite Computer Introduction (CS201)

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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.

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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 9 3
- Combinational Logic 12 4
- Synchronous Sequential Logic 9 3
Total 45 15
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Assessment schedule
Assessment Methods Week Weighting of Assessments
First Midterm Exam 6 20
Second Midterm Exam 13 20
Home work Every Chapter 10
project 12 10
Final Exam After week15 40
Total Total 100
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List of References

• Essential 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. 1998.
• 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.

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1. Digital Systems and Binary Numbers

• 1.1 Digital Systems
• 1.2 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

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1.1 Digital Systems
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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.
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1.3 Number-Base Conversions (Integer Part)
Example
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1.3 Number-Base Conversions (Fraction Part)
Example
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Binary-to-Decimal Conversion
Example
Example
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1.4 Octal and Hexadecimal Numbers
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Decimal-to-Octal Conversion
Example
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Decimal-to-Hexadecimal Conversion
Example
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Octal-to-Decimal Conversion
Example
Example
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Hexadecimal-to-Decimal Conversion
Example
Example
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BinaryOctal and OctalBinary Conversions
Example
Example
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HexBinary and BinaryHex Conversions
Example
Example
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HexOctal and OctalHex Conversions

• For HexadecimalOctal 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 OctalHexadecimal 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.

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Example
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Arithmetic Operation

• augend 101101
• Added 100111
• ----------Sum
  1010100

• Addition

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Subtraction
minuend 101101 subtrahend
- 100111
------------- difference 000110

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Multiplication
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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 9s complement of 546700 is 999999
  46700453299
• the 1s complement of 1011000 is 0100111
• Note
• The (r-1)s complement of octal or hexadecimal
  numbers is obtained by subtracting each digit
  from 7 or F (decimal 15), respectively

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Radix Complement

• Given a number N in base r having n digit, the
  rs complement of Nis defined as (rn -N) for N ?0
  and as 0 for N 0 .
• The 10s complement of 012398 is 987602
• The 10s complement of 246700 is 753300
• The 2s complement of 1011000 is 0101000

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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 rs
  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 lt N, the sum does not produce an end carry
  and is (N - M). Take the rx complement of the
  sum and place a negative sign in front.

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Example

• Using 10s complement subtract 72532 3250
• M 72532
• 10s complement of N 96750
• sum 169282
• Discarded end carry 105 -100000
• answer 69282

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Example

• Using 10s complement subtract 3250 - 72532
• M 03250
• 10s complement of N 27468
• sum 30718
• Discarded end carry 105 -100000
• answer -(100000 - 30718) -69282
• The answer is (10s complement of 30718) -69282

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Example

• Using 2s complement subtract (a) 1010100
  1000011
• M 1010100
• N 1000011, 2s complement of N 0111101
• 1010100
• ? 0111101
• sum 10010001
  Discarded end carry 27-10000000
• answer 0010001

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Example

• (b) 1000011 1010100
• M 1000011
• N 1010100, 2s complement of N 0101100
• 1000011
• ? 0101100
• sum 1101111
• answer - (10000000 - 1101111)
  -0010001
• The answer is (2s complement of 1101111) -
  0010001

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Example
Using 1s complement, subtract 1010100 - 1000011
M 1010100
N 1000011, 1s complement of N 0111100
1010100
? 0111100
10010000
end-around carry 1
answer 0010001
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Example
Using 1s complement, subtract 1000011 - 1010100
M 1000011
N 1010100, 1s complement of N 0101011
1000011
? 0101011
1101110
Answer -0010001
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1.6 Signed Binary Numbers
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1.6 Signed Binary Numbers
1 - Sign and Magnitude representation 2 - 1s
Complement Representation 3 - 2s 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
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Representing negative numbers in the previous
three systems
1s Complement of a negative number can be
obtained by flipping all bits of the positive
binary number 2s Complement of a negative number
can be obtained by adding 1 to the 1s Complement
or by flipping bits of the positive binary number
after the first one from the right
Example represent -76
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Arithmetic Addition with Comparison
Arithmetic Addition

• 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) -(38 - 25) -13

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Arithmetic Addition without Comparison

• The addition of two signed binary number with
  negative numbers represented in signed 2s
  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).

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Arithmetic Addition without Comparison
06 11111010
06 00000110
?
?
?
?
13 00001101
13 00001101
?
?
?
?
07 00000111
19 00010011
?
?
?
?
06 11111010
06 00000110
?
?
?
?
?
?
13 11110011
13 11110011
?
?
?
19 11101101
07 11111001
?
?
?
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Arithmetic Subtraction

• (/-) A (B) (/-) A (-B)
• (/-) A (-B) (/-) A (B)
• Example
• (-6) (-13) 7
• In binary (1111010 11110011) (1111010
  00001101)
• 
  100000111
• after removing the carry out the result will be
  00000111

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1.7 Binary Codes
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Binary Coded Decimal (BCD)
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Binary Coded Decimal (BCD)
in this system each digit is represented in 4 bits
For example to represent in BCD
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BCD Addition
Example Evaluate the following operations in
BCD System 1 3 4 2 4 8 3 - 148 576
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BCD Addition
Example Evaluate the following operations in
BCD System 1 3 4 2 4 8 3 - 148 576
Error
We must add 6 (0110) to the result
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BCD Addition
Example Evaluate the following operations in
BCD System 1 3 4 2 4 8 3 - 184 576
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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) ?
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Decimal Arithmetic

• Addition for signed numbers
• Example (375) (- 240) 135 in BCD
• Apply 10s complement to the negative number
  only.
• Addition is done by summing all digits,including
  the sign digit,and discarding the end carry
  
• 
  0 375
• 
  9 760
• 
  ------------
• 
  0 135

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Decimal Arithmetic

• Subtraction for signed and unsigned numbers
• Apply 10s complement to the subtrahend and apply
  addition (same as binary case)

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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
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Gray Code
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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 1000001 A
66 1000010 B

90 1011010 Z

97 1100001 a
98 1100010 b

122 1111001 z
Upper case Letters are represented by ASCII (65
90) Lower case Letters are represented by ASCII
(97 122)
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Error Detecting Code

• with even
  parity with odd parity
• ASCII A 1000001 01000001
  11000001
• ASCII T 1010100 11010100
  01010100

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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