1. Logic and Computer Design Fundamentals
Chapter 1 – Digital Systems and Information
Charles Kime & Thomas Kaminski
© 2008 Pearson Education, Inc.
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2. Overview Digital Systems, Computers, and Beyond
Information Representation
Number Systems [binary, octal and hexadecimal]
Arithmetic Operations
Base Conversion
Decimal Codes [BCD (binary coded decimal)]
Alphanumeric Codes
Parity Bit
Gray Codes
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3. DIGITAL & COMPUTER SYSTEMS - Digital System
Takes a set of discrete information inputs and discrete internal information (system state) and generates a set of discrete information outputs.
Discrete Information Processing System
Discrete
Inputs
Discrete Outputs
System State
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4. example: ticket counter
In a simple scenario, any digital system can be considered either as being combinational or sequential
Combinational Circuit
always gives the same output for a given set of inputs
ex: adder always generates sum and carry, regardless of previous inputs
Sequential Circuit
stores information
output depends on stored information (state) plus input
so a given input might produce different outputs, depending on the stored information
example: ticket counter
advances when you push the button
output depends on previous state
useful for building “memory” elements and “state machines”
3-4

5. Combinational vs. Sequential
Two types of locks. One combinational and the other sequential
30
25
20 5
10 4 1 8
15
Sequential
Success depends on
the sequence of values
(e.g, 2R(13) , 1L(22) ,
1R ( 3).
Combinational
Success depends only on
the values, not the order in
which they are set.
3-5

6. Digital Computer Example
Memory
Control
unit
CPU
Datapath
Inputs:
Outputs: LCD
keyboard, mouse, microphone
screen, speakers
Input/Output
Synchronous or Asynchronous?
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7. And Beyond – Embedded Systems
Computers as integral parts of other
products
Examples of embedded computers
Microcomputers
Microcontrollers
Digital signal processors
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8. Embedded Systems Examples of Embedded Systems Applications Cell phones
Automobiles
Video games
Copiers
Dishwashers
Flat Panel TVs
Global Positioning Systems ( GPS)
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9. Digital systems operate with signals which have only two values, known as logical 0 (zero) and logical 1 (one).
Other equivalent ways of understanding and/or representing the two (i.e. binary) values of a digital system include:
FALSE
OFF
DOWN
LOW
OPEN 
TRUE
ON UP HIGH
CLOSED
Digital logic circuits operate with binary values of current or voltage, that is, with two values representing the logical 0 and logical 1 of a digital system.
2-9

10. Computer is a binary digital system.
Now we must define the different rages of logic 0 and logic 1 for different Semiconductor technologies. For example for MOS we may consider the following arrangement
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11. floating state or tri-state )
As can be seen
0V ≤ Logic 0 ≤0.5 V
2.4 V ≤ logic 1 ≤ 2.9V
Explanation:
Illegal state
( high impedance,
floating state or tri-state )
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12. What are other physical quantities
Binary Values: Other Physical Quantities
Voltage
Magnetic Field Direction
Surface Pits/Light
What are other physical quantities
represent 0 and 1?
• CPU
• Disk
• CD
• Dynamic RAM
Electrical Charge
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13. Lesson 1: number systems

14. For example the decimal number of 543.21 is represented as:
Number Systems
The decimal number system is employed in everyday arithmetic to represent number by strings of digits. For example depending on its position in the string, each digit has an associated value of an integer raised to the power of 10.
For example the decimal number of
is represented as:
 (5x122) +(4x101) +(3x100 ) +(2x10-1)
+(1x10 -2)
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15. The base is b=10 since we deal with the decimal number.
Number Systems ( Cont.)
The base is b=10 since we deal with the
decimal number.
The digits 5,4,3,2,1, satisfy the following
condition:
 0 ≤ 5,4,3,2,1 < 10
With the smallest possible digit being 0 while the largest possible digit being 9.
Problem 1.1. What is the presentation of
the following number 724.5?
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16. j      An - 1An - 2 … A1A0 . A- 1 A- 2 … A- m  1 A- m Ai r 
NUMBER SYSTEMS – Representation
In general our notation for an arbitrary base (radix) r for a string of digits such :
An - 1An - 2 … A1A0 . A- 1 A- 2 … A- m  1 A- m
in which 0  Ai < r and . is the radix point can
be presented as: j
  
i = n - 1
j = - 1  
(Number) =
Ai r i 
Aj r r
i = 0 j = - m
(Integer Portion) + (Fraction Portion)
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17. Now we are interested in 4 bases: Base 10 for which r =10
Number Systems ( Cont.)
Now we are interested in 4 bases:
Base 10 for which r =10
Base 2 for which r =2
Base 8 for which r = 8
Base 16 for which r = 16.
Then by replacing the value of r ( some books use b instead of r) in the general equation we can obtain the equivalent of that number in the required base.
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18. For each base we adopt the following conventions:
Number Systems ( Cont.)
For each base we adopt the following
conventions:
 Base 10 digits : 0,1,2,3,……9.
Base 2, digits 0,1
 Base 8 digits 0,1,2,3,4… 7
 Base 16 digits 0,1,2,3,4,…8,9,A,B,C,D,E,F.
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19. Are used for the above 16 bases.
Number Systems ( Cont.)
What is new is that :
A represents 10
B represents 11
C represents 12
D represents 13
E represents 14
F represents 15
Are used for the above 16 bases.
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20. We give “ Special names” to our bases: Base 10 is the decimal system
Number Systems ( Cont.)
We give “ Special names” to our bases:
Base 10 is the decimal system
Base 2 is the binary system
Base 8 is the octal system
Base 16 is the hexadecimal system.
From now on we must specify the number system of interest in order to specify a number.
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21. Lesson 2: number system conversion

22. Converting Decimal to Any base
Repeatedly divide the number by base r and save the remainders. The digits for the new radix are the remainders in reverse order of their computation.
Example: Convert to N2
Example: Convert to N16
Example: Convert to N8
625 – 512 = 113 => 9
113 – 64 = 49 => 6
49 – 32 = 17 => 5
17 – 16 = => 4
1 – = => 0
Placing 1’s in the result for the positions recorded and 0’s elsewhere,

23. Converting Any base to Decimal
To convert to decimal, use decimal arithmetic to form
S (digit × respective power of base r).
Example: Convert to N10:  
Example: Convert 7258 to N10:  
Example: Convert to N10:  
Example: Convert A2E16 to N10:
Powers of 2: 43210
=>
1 X = 16
+ 1 X = 8
+ 0 X = 0
+ 1 X 21 = 2
+ 0 X 20 = 0
2610

24. Convert from base X to base Y
First convert from base X to decimal
Then convert from decimal to base Y
Example: convert to hexadecimal
Example: convert to binary
Example: convert to octal
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25. The binary system The digit 0, 1 in the binary system have a special
name. They are called bits.
Bits represent the smallest memory element used
in every computer.
Problem 1.3
How many bits are needed to represent
A decimal digit?
A binary digit ?
An octal digit?
A hexadecimal digit?
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26. Number Systems ( Cont.) In computer literature we define memories as:
 2 10 = 1024 = 1 K ( Kilo) bytes
 2 20 = 2 10 x 2 10 = 1 K x 1 K = 1M ( Mega)
 2 30 = x 2 10 x = 1 K x 1 K x 1K = 1G (
giga)
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27. A) binary to octal, hexadecimal and B) octal, hexadecimal to binary.
Number Systems ( Cont.)
Working with binary, octal and hexadecimal numbers is very easy. We can easily convert
A) binary to octal, hexadecimal and
B) octal, hexadecimal to binary.
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28. Convert the following binary number into its octal equivalent.
Number Systems ( Cont.)
A For converting
I. Binary to octal
Group “ every three bits ( beginning from the right , ( LSB) and convert each group of 3 bits into octal.
Problem 1.4
Convert the following binary number into its octal equivalent.
(101111)2 = ( ?) 8
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29. Number Systems ( Cont.) II. Binary to hexadecimal
Group “ every four bits” ( beginning from the right
( LSB) and convert each group of 4 bits into hexadecimal.
Problem 1.5
 ( ) 2 = ( ?) 16
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30. Octal to binary “ ungroup every digital and convert it into 3 bits
(b) for converting
Octal to binary “ ungroup every digital and convert it into 3 bits
Problem 1.6
( 75) 8 = ( ?) 2
Hexadecimal to binary
“ Ungroup” every digit and convert it into 4 bits.
Problem 1.7
 ( FE) 16 = ( ?) 2
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31. Numbers in Different Bases Good idea to memorize!
Decimal
(Base 10)
Binary
(Base 2)
Octal
(Base 8)
Hexadecimal
(Base 16) 00
00000 01
00001 02
00010 03
00011 04
00100 05
00101 06
00110 07
00111 08
01000 10 09
01001 11
01010 12 0A
01011 13 0B
01100 14 0C
01101 15 0D
01110 16 0E
01111 17 0F
10000 20
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32. Algorithm for converting from decimal into binary form
Break every decimal into two:
integer part . Fractional part
For the integer part we follow the procedure that we outlined before:
See the next page for details:
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33. Number Systems ( Cont.) Quotient Remainder Digit Int ( N/2) = q0
N-2q0 = r 0 r0
Int ( q0/ 2) = q1
q0-2q1 = r 1
r 1
Int ( q1/2) = q2
q1-2q2 = r2 r2
Int ( q2/2) = q3
q2-2q3 = r3 r3 .
Until q n+1 becomes
zero
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34. The resulting binary from is
Number Systems ( Cont.)
The resulting binary from is
 rn rn-1 …..r3 r2 r1 r0
for the integer part.
Perhaps an easier way to remember the algorithm would be the following way
( see the explanation in the class ).
This can be generalized for converting the fractional part.
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35. Number Systems ( Cont.) Problem 1.10  ( 0.625) 10 = ( ?) 2
 ( 0.625) 10 = ( ?) 2
Problem 1.11
 ( 28.21) 10 = ( ?) 2
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36. What does (543.21)10 represent in a) hexadecimal? b) octal?
Number Systems ( Cont.)
Problem 1.2 :
What does (543.21)10 represent in
a) hexadecimal?
b) octal?
Problem 1.10
 ( 0.625) 10 = ( ?) 8
Problem 1.11
 ( 28.21) 10 = ( ?) 16
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37. Lesson 3: number system arithmetic

38. Single Bit Binary Addition with Carry

39. Multiple Bit Binary Addition
Extending this to two multiple bit examples:
Augend
Addend
Sum= ?
Carries
Augend
Addend
Sum

40. Number Systems ( Cont.) Problem 1.16  ( 01100) + ( 10001) =?????
 ( 01100) + ( 10001) =?????
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41. Single Bit Binary Subtraction with Borrow
Given two binary digits (X,Y), a borrow in (Z) we get the following difference (S) and borrow (B):
Borrow in (Z) of 0:
Borrow in (Z) of 1: Z X 1
- Y -0 -1 BS
0 0
1 1
0 1 Z 1 X
- Y -0 -1 BS 11
1 0
0 0
1 1

42. Multiple Bit Binary Subtraction
Extending this to two multiple bit examples:
Minuend
Subtrahend
Difference= ?
Borrows
Minuend
Subtrahend – –10011
Difference

43. Binary Multiplication

44. Arithmetic Operations
Problem 1.15
( 1011) x ( 101) see the solution in the class
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45. Arithemtic operations in other systems
Basically it is the same as decimal, except you need to use the tables of that system.
Problem 1.12
( 59F) 16 + ( E46) 16 = ????
Problem 1.13
( 3FE) 16 + ( E2)16 = ????
Octal Multiplication
Problem 1.14
Perform the multiplication ( 762 ) 8 x ( 45) 8
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